Band parameters for III–V compound semiconductors and their alloys

JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 11 1 JUNE 2001

APPLIED PHYSICS REVIEW

Band parameters for III–V compound semiconductors and their alloysI. Vurgaftmana) and J. R. MeyerCode 5613, Naval Research Laboratory, Washington, DC 20375

L. R. Ram-MohanWorcester Polytechnic Institute, Worcester, Massachusetts 01609

~Received 4 October 2000; accepted for publication 14 February 2001!

We present a comprehensive, up-to-date compilation of band parameters for the technologicallyimportant III–V zinc blende and wurtzite compound semiconductors: GaAs, GaSb, GaP, GaN,AlAs, AlSb, AlP, AlN, InAs, InSb, InP, and InN, along with their ternary and quaternary alloys.Based on a review of the existing literature, complete and consistent parameter sets are given for allmaterials. Emphasizing the quantities required for band structure calculations, we tabulate the directand indirect energy gaps, spin-orbit, and crystal-field splittings, alloy bowing parameters, effectivemasses for electrons, heavy, light, and split-off holes, Luttinger parameters, interband momentummatrix elements, and deformation potentials, including temperature and alloy-compositiondependences where available. Heterostructure band offsets are also given, on an absolute scale thatallows any material to be aligned relative to any other. ©2001 American Institute of Physics.@DOI: 10.1063/1.1368156#

TABLE OF CONTENTS

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5816II. Relation to band structure theory and general

considerations . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .5817A. Multiband k"P method . . . . . . . . . . . . . . . . . . . .5817

1. Zinc blende materials . . . . . . . . . . . . . . . . . .58182. Nitrides with wurtzite structure . . .. . . . . . . 58193. Strain in heterostructures . . . . . . . . . . . . . . . 58214. Piezoelectric effect in III–V

semiconductors . . . . . . . . . . . . . . . . . . . . . . .58215. Band structure in layered

heterostructures . . . . . . . . . . . . . . . . . . . . . . .5822B. Relation ofk"P to other band structure

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58221. Empirical tight binding model . . . . . . . . . . . 58222. Effective bond orbital model . . . . . . . . . . . . 58233. Empirical pseudopotential model . . . . . . . . . 5823

III. Binary compounds . . . . . . . . . . . . . . . . . . . . . . . . .5823A. GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5823B. AlAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5824C. InAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5826D. GaP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5827E. AlP . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5828F. InP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5828G. GaSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5829H. AlSb . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5830I. InSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5831J. GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5832

1. Wurtzite GaN . . . . . . . . . . . . . . . . . . . . . . . .58322. Zinc blende GaN . . . . . . . . . . . . . . . . . . . . . . 5834

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K. AlN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5835L. InN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5836

IV. Ternary alloys . . .. . . . . . . . . . . . . . . . . . . . . . . . . .5837A. Arsenides . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .5838

1. AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58382. GaInAs . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .58393. AlInAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5840

B. Phosphides . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .58401. GaInP . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .58402. AlInP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58413. AlGaP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5841

C. Antimonides . . .. . . . . . . . . . . . . . . . . . . . . . . . . .58421. GaInSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58422. AlInSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58423. AlGaSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5843

D. Arsenides antimonides . . . . . . . . . . . . . . . . . . . .58431. GaAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58432. InAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58443. AlAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5844

E. Arsenides phosphides . . . . . . . . . . . . . . . . . . . . .58441. GaAsP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58442. InAsP . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .58453. AlAsP . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .5846

F. Phosphides antimonides . . . . . . . . . . . . . . . . . . .58461. GaPSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58462. InPSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58463. AlPSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5846

G. Nitrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58461. GaInN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58462. AlGaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58473. AlInN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5847

5 © 2001 American Institute of Physics

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5816 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 Appl. Phys. Rev.: Vurgaftman, Meyer, and Ram-Mohan

4. GaAsN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58485. GaPN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58496. InPN . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .58497. InAsN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5849

V. Quaternary alloys . . .. . . . . . . . . . . . . . . . . . . . . . . .5849A. Lattice matched to GaAs . . . . . . . . . . . . . . . . . .5850

1. AlGaInP . . .. . . . . . . . . . . . . . . . . . . . . . . . . .58502. GaInAsP . . .. . . . . . . . . . . . . . . . . . . . . . . . . .58503. AlGaInAs . . .. . . . . . . . . . . . . . . . . . . . . . . . .58514. GaInAsN . . . . . . . . . . . . . . . . . . . . . . . . . . . .5851

B. Lattice matched to InP . . . . . . . . . . . . . . . . . . . .58511. GaInAsP . . .. . . . . . . . . . . . . . . . . . . . . . . . . .58512. AlGaInAs . . .. . . . . . . . . . . . . . . . . . . . . . . . .58523. GaInAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . . .5852

C. Lattice matched to InAs . . .. . . . . . . . . . . . . . . . 58521. GaInAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . . .58522. AlGaAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . .58533. InAsSbP . . . .. . . . . . . . . . . . . . . . . . . . . . . . .5853

D. Lattice matched to GaSb . . . . . . . . . . . . . . . . . . . 58531. GaInAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . . .58532. AlGaAsSb . . . . . . . . . . . . . . . . . . . . . . . . . . .5854

E. Other substrates . . .. . . . . . . . . . . . . . . . . . . . . . .58541. AlGaAsP . . . . . . . . . . . . . . . . . . . . . . . . . . . .5854

F. Nitride quaternaries . . .. . . . . . . . . . . . . . . . . . . . 5854VI. Heterostructure band offsets . . .. . . . . . . . . . . . . . . 5854

A. GaAs/AlAs . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .5855B. GaInAs/AlInAs/InP . . .. . . . . . . . . . . . . . . . . . . . 5856C. Strained InAs/GaAs/InP and related

ternaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5856D. GaInP/AlInP/GaAs . . . . . . . . . . . . . . . . . . . . . . .5857E. GaP and AlP . . . . . . . . . . . . . . . . . . . . . . . . . . . .5858F. GaSb/InAs/AlSb . . . . . . . . . . . . . . . . . . . . . . . . .5858G. GaSb/InSb and InAs/InP . . .. . . . . . . . . . . . . . . . 5860H. Quaternaries . . .. . . . . . . . . . . . . . . . . . . . . . . . . .5861I. GaN, InN, and AlN . . . . . . . . . . . . . . . . . . . . . . . 5861

VII. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5862

I. INTRODUCTION

At present, III–V compound semiconductors provide tmaterials basis for a number of well-established commertechnologies, as well as new cutting-edge classes of etronic and optoelectronic devices. Just a few examplesclude high-electron-mobility and heterostructure bipotransistors, diode lasers, light-emitting diodes, photodetors, electro-optic modulators, and frequency-mixing comnents. The operating characteristics of these devices decritically on the physical properties of the constituent marials, which are often combined in quantum heterostructucontaining carriers confined to dimensions on the order onanometer. Because ternary and quaternary alloys maincluded in addition to the binary compounds, and the marials may be layered in an almost endless variety of confirations, a seemingly limitless flexibility is now availablethe quantum heterostructure device designer.

a!Author to whom correspondence should be addressed; [email protected]


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To fully exploit this flexibility, one clearly needs a reliable and up-to-date band parameter database for input toelectronic structure calculations and device simulatioHowever, after many years Volume 17 of the LandolBornstein series1 remains the most frequently quoted sourof III–V band parameters. Although that work containmuch of the required data for a broad range of materials,nearly 20 yr old, lacks detailed descriptions of many of timportant III–V alloys, and contains no information at all othe crucial band offset alignments for heterostructurespopular compilation by Casey and Panish2 covers the bandgaps for all III–V non-nitride binary materials and 12 ternaalloys, but the rapid progress in growth and characterizaof many of those materials taking place since its publicatin 1978 has decreased its usefulness. While a numberecent books and reviews on individual material systemsavailable,3–10 they are not necessarily complete or mutuaconsistent. A useful recent compilation of band structurerameters by Levinshteinet al.11 does not contain in-depthinformation on aluminum-containing elemental semicondtors, is often based on a limited number of original sourcand considers only six ternary and two quaternary alloys

The objective of the present work is to fill the gap in thexisting literature by providing a comprehensive and mually consistent source of the latest band parameters for athe common III–V zinc blende and wurtzite semiconducto~GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaAlN, and InN! and their ternary and quaternary alloys. Treviewed parameters are the most critical for band struccalculations, the most commonly measured and calculaand often the most controversial. They include:~1! direct andindirect energy gaps and their temperature dependences~2!spin-orbit splitting;~3! crystal-field splitting for nitrides;~4!electron effective mass;~5! Luttinger parameters and splitoff hole mass;~6! interband matrix elementEP and the as-sociatedF parameter which accounts for remote-band effein eight-bandk"P theory; ~7! conduction and valence bandeformation potentials that account for strain effectspseudomorphic thin layers; and~8! band offsets on an absolute scale which allows the band alignments betweencombination of materials to be determined. All paramesets are fully consistent with each other and are intendereproduce the most reliable data from the literature. For copleteness, lattice constants and elastic moduli for each mrial will also be listed in the tables, but in most cases will nreceive separate discussion in the text because they areerally well known and noncontroversial.

As a complement to the band parameter compilatiowhich are the main focus of this work, we also provideoverview of band structure computations. Thek"P method isoutlined, followed by brief summaries of the tight-bindinand pseudopotential approaches. The theoretical discussprovide a context for defining the various band parametand also illustrate their significance within each computional method. The tables provide all of the input parametthat are normally required for an eight-bandk"P calculation.While we have not attempted to cover every parametermay potentially be useful, most of those excluded~e.g., thekandq parameters necessary for structures in a magneticil:


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5817J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 Appl. Phys. Rev.: Vurgaftman, Meyer, and Ram-Mohan

and the inversion asymmetry parameter! are either poorlycharacterized or have well-known best values that havechanged much over time and are easily obtained from osources such as Landolt–Bornstein.

To the extent possible, we have fully treated the 12 mjor III–V binaries and their alloys. Other nominal III–V materials that are not covered include BN and other borcontaining compounds~which are commonly considered tbe insulators rather than semiconductors!, as well as thenarrow-gap InSbBi, InTlSb, InTlAs, and InTlP alloys, whicup to now have not achieved technological importance. Nof these materials has been integrated appreciably into anthe mainline systems, and in most cases a paucity of bstructure information precludes the recommendation of dnite parameter values. On the other hand, we attempt tovide a complete and up-to-date description of the nitrfamily of materials, including those with a wurtzite crystlattice, in light of its increasing prominence and the numous intense investigations currently being conducted.

It is naturally impossible for us to universally cite evearticle that has ever provided information or given valuesthe relevant band parameters. The reference list for sureview would number in the 10’s of thousands, which woube impractical even for a book-length treatment. We hatherefore judiciously selected those results that are mosttral to the purpose of the compilation. In some cases, wagreement on the value of a given parameter already exand/or previous works have critically and comprehensivreviewed the available information. Under those circustances, we have limited the discussion to a summary offinal conclusions, along with references to the earlier reviewhere additional information may be found. In other caswe discuss more recent data that have modified or alteredearlier findings. The most difficult topics are those for whithere is substantial disagreement in the literature. In thinstances, we summarize the divergent views, but noneless choose a particular result that is either judged to bemost reliable, or represents a composite combining a varof experimental and/or theoretical findings. Although suselections are inevitably subjective, since they require ansessment of the relative merits and reliabilities, in each cwe inform the reader of the basis for our judgment.

A guiding principle has been the maintenance of finternal consistency, both in terms of temperature depdences of the parameters for a given material and with regto variations with alloy composition. For examplcomposition-dependent parameter values forAl xGa12xAs alloy employ the best information at those itermediate compositions for which data are available,also invariably agree with the results for GaAs and Alwhen evaluated atx50 andx51. In a few cases, this harequired the introduction of additional bowing parameterscontexts where they are not usually applied. We have abeen as complete as possible in gathering all available inmation on the less common ternary and quaternary alloytems.

The review is organized as follows. Discussions of tk"P, tight-binding, and pseudopotential band structure calations are presented in Sec. II, along with some gen


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considerations regarding the major band parameters.main results for individual binary compounds, ternary alloand quaternary alloys are reviewed in Secs. III, IV, andrespectively. A table summarizes the parameters recmended for each material, while the text provides justifiction for the choices. Section VI then reviews the band offsthat are needed to calculate energy bands in quantum hestructure. Those results are presented in a format that reences all offsets to the valence band maximum of Inwhich allows a determination of the relative band alignmefor any possible combination of the materials covered in twork.

II. RELATION TO BAND STRUCTURE THEORY ANDGENERAL CONSIDERATIONS

A number of excellent books and review articles hasummarized the methods for calculating bulk12–14 andheterostructure15–20band structures. Since the present reviis intended to focus on band-parameter compilation ratthan theoretical underpinnings, this survey is aimed primrily at providing a context for the definitions of the variouparameters. The section is limited to a discussion of the bessentials for treating heterostructures and a brief descripof the practical approaches. A more detailed descriptionthe theory will be presented elsewhere.

A. Multiband k "P method

The most economical description of the energy bandssemiconductors is the effective mass approximation, whis also known as the envelope function approximationmultibandk"P method. It uses a minimal set of parametethat are determined empirically from experiments. By meaof a perturbative approach, it provides a continuation inwave vectork of the energy bands in the vicinity of somspecial point in the Brillouin zone~BZ!.

The electronic wave functions that satisfy the Sch¨-dinger equation with a periodic lattice potential in a bucrystal are given by Bloch’s theorem:

c~r !5eikrunk~r !. ~2.1!

The cell-periodic Bloch functionsunk(r ) depend on the bandindex n and the envelope function wave vectork. The wavefunctionsc(r ) form a complete set of states as do the wafunctions based on Bloch functions at any other wave vecincluding the wave vectors at special points in the BZ.21 Intreating the optical and electronic properties of direct gsemiconductors, it is natural to consider the zone-cenG-point Bloch functionsun0(r ) for our wave function expan-sions, and we drop the reference to thek50 index for thesefunctions.

For our purposes here, the general form of the wafunctions may be considered to be a linear combination ofinite number of band wave functions of the form

c~r !5 f n~z!eikxxeikyyun~r ![Fn~r !un~r !. ~2.2!

The envelope functionsFn(r ) are typically considered to beslowly varying, whereas the cell periodic and more osciltory Bloch functions satisfy Schro¨dinger’s equation withband-edge energies. We distinguish envelope functions


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responding to theG1 conductionc, G15 valencen, and theenergetically higher remote bands with an indexr: Fn

5$Fc ,Fn,Fr%. The remote bands are assumed to have baedge energiesuEr2Ecnu@uEc2Enu5Eg , where Eg is thedirect energy gap.

Next, Lowdin’s perturbation theory22 is applied in orderto eliminate the functionsFr in favor of perturbation terms inthe equations for the conduction and valence bands. Inbulk k"P theory, these terms correspond to the Kamodel23,24 for the band structure, with quadratic@O(k2)#terms within the conduction and the valence bands24,25 aswell as the appropriate interband~conduction-valence! terms.

1. Zinc blende materials

The zinc blende crystal consists of two interpenetratface-centered-cubic lattices, one having a group-III elematom ~e.g., Ga! and the other a group-V element atom~e.g.,As!. A zinc blende crystal is characterized by a single lattconstantalc.

The matrix elements of the momentum operator betwthe conduction and valence bands can be expressed in tof a single parameterP, originally defined by Kane:

P[2 i\

m0^SupxuX&, ~2.3!

where ^SupxuX& is the momentum matrix element betwethe s-like conduction bands andp-like valence bands. Itsvalue in a given material is usually reported in energy un~eV! as:

EP52m0

\2 P2. ~2.4!

The EP matrix element is one of the band parameters thaextensively reviewed in the subsequent section.

Through second-order perturbation theory, the highband contributions to the conduction band are parameterby the Kane parameterF, whose values are reviewed in thsubsequent sections:

F51

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r

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where ^Supxuur& is the momentum matrix element betwethe s-like conduction bands and remote bandsr, andm0 isthe free electron mass.

The second-order valence-band terms include thosepossible intermediater states that belong to energeticalhigher bands with the symmetry ofs(G1), p(G12), andd(G15) atomic orbitals.26 We define the following quantitiesin terms of matrix elements betweenp-like valence bands~whose symmetry here and in the following is indicated asX,Y, Z! and remote bands:

s52S 1

3m0D(

r

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If the conduction-band states were not included into oanalysis explicitly, the additional contribution of the nearbycbands would enter the sum over intermediate states insecond-order perturbation theory for thes parameter. Wedenote this revised value assL:

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Now, thesL, d, andp parameter set is simply related to thstandard definitions of the Luttinger parametersg1 , g2 , andg3 :

g152112sL14p14d,

g25sL2p12d, ~2.10!

g35sL1p2d.

All of the terms appearing in the valence-band Hamiltonmay be cast in terms of the three quantitiess, d, andp, orequivalently in terms of the Luttinger parameters.

The inclusion of electron spin and spin-orbit interactieffects are straightforward,23 and we obtain a 636 valence-band Hamiltonian. The matrix elements of the spin-orbitteraction, withs being the Pauli spin matrices here

Hso5\

4m02c2 DV3p"s, ~2.11!

are parameterized by the quantity called the spin-orbit spting

Dso53\ i

4m02c2 ^Xu

]V

]xpy2

]V

]ypxuY&. ~2.12!

The atomic potentialV appears in the above expressionThe spin-orbit interaction splits the sixfold degeneracy atzone center into fourfold degenerate heavy-hole~hh! pluslight-hole ~lh! bands ofG8 symmetry with total angular momentumJ53/2, and a doubly degenerate split-off~so! bandof G7 symmetry withJ51/2. In practice, the values of thDso parameter are determined experimentally.

For a bulk system, the eight-bandk"P model gives riseto eight coupled differential equations, which define tSchrodinger eigenvalue problem for the energy bands nthe center of the BZ. With the envelope functions repsented bye6 ikr , we have the usual 838 Hamiltonian withterms linear and quadratic ink. In a bulk semiconductorboth direct and indirect energy gaps in semiconductor mrials are temperature-dependent quantities, with the futional form often fitted to the empirical Varshni form27

Eg~T!5Eg~T50!2aT2

T1b, ~2.13!

where a and b are adjustable~Varshni! parameters. Al-though other, more physically justified and possibly quantatively accurate, functional forms have been proposed,28,29


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The effective massm* at the conduction and valencband edges can be obtained from the bulk energy disperV(k) as30

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m*. ~2.14!

Using this procedure, the conduction band effective masgiven in terms of the band parameters

m0

me*5~112F !1

EP~Eg12Dso/3!

Eg~Eg1Dso!. ~2.15!

Both EP @defined in Eq.~2.4!# andF @Eq. ~2.5!# appearing inEq. ~2.15! are usually taken to be independent of tempeture, which means that the temperature variation of thefective mass arises only through the temperature depdences of the energy gaps as in Eq.~2.13!. Unfortunately,despite their importanceEP andF are inherently difficult todetermine accurately, since the remote-band effects cacalculated but not directly measured. One alternative expmental technique is to rely on measuring the effectiveg fac-tor, which is not as influenced by remote bands as the eftive mass. In this article, we derive consistent sets oFparameters from the best available experimental reportelectron effective masses, energy gaps, spin-orbit splittinandmomentum matrix elements. There has been no prevattempt to compile reliable, self-consistentF parameters forsuch a broad range of materials.

In polar semiconductors such as the III–V compoundsis the nonresonant polaron31 mass that is actually measureThat quantity exceeds the bare electron mass by 1%–depending on the strength of the electron–phonon intetion. However, since the band structure is governed bybareelectron mass, we attempt to present the latter valuenote the approximate magnitude of the polaronic correcwhenever the information is available.

At the valence-band edge, the hh effective masses indifferent crystallographic directions are given by the retions

S m0

mhh*D z

5g122g2 ;

S m0

mhh*D @110#

51

2~2g12g223g3!; ~2.16!

S m0

mhh*D @111#

5g122g3.

These expressions show the relationship of the Luttingerrameters, which are not as physically meaningful but mconvenient to work with theoretically, to the hh effectivmasses that can typically be measured in a more direct mner. The lh and so hole effective masses are given by


le,a-

on

is

-f-n-

beri-

c-

ofs,us

it

%,c-e

ndn

he-

a-e

n-

S m0

mlh*D z

5g112g2 ,

S m0

mlh*D @110#

51

2~2g11g213g3!, ~2.17!

S m0

mlh*D @111#

5g112g3

and

m0

mso*5g12

EPDso

3Eg~Eg1Dso!. ~2.18!

Equation~2.18!, which relates the split-off hole mass to thLuttinger parameters, should in principle contain an adtional parameter to account for the effects of remote ba~analogous toF!. However, those remote bands are not nessarily the same ones that produce the largest correctiothe electron mass. At present, not enough data exist forof the III–V materials to fix the effect of the interaction witremote bands on the split-off hole mass.

In any theoretical model, being able to reproducecorrect effective masses at the center of the BZ ensuresthe curvature of the energy bands is properly reproducedthe context of heterostructures, using incorrect effectmasses could lead to severe deviations from experiment,in the bound state spectrum of a quantum well. For thin-laquantum structures, it is also important to have a good mofor the nonparabolic dispersion away from the center ofBZ. The eight-bandk"P model compares well with morerigorous calculations up to about a quarter of the way toBZ boundary, and extra bands may be included in ordeimprove the agreement.32,33 As an illustration, we show inFig. 1~a! the full-zone band structure in the vicinity of thenergy gap obtained for GaAs using the pseudopotenmethod~see Sec. II B 3!. Note the anticrossing of the conduction band with a higher band along theG –X direction,which sets one limit on the accuracy of perturbative aproaches. A more detailed plot of the band structure nearBZ center in GaAs is given in Fig. 1~b!.

The filled points in Fig. 2 showG-valley energy gaps asa function of lattice constant for zinc blende forms of thebinary III–V semiconductors reviewed in this work. Thconnecting curves represent band gaps for the randomnary alloys, although in a few cases~e.g., GaAsN and InPN!the extrapolated dependences extend well beyond the regthat are reliably characterized. We also emphasize thatG-valley direct gap is not necessarily the smallest, since seral of the materials are indirect-gap semiconductorswhich theX andL conduction-band valleys lie lower than thG minimum. For the non-nitride III–Vs, Fig. 3 shows a simlar plot of the lowest forbidden gap in each material, witX-valley andL-valley indirect gaps indicated by the dashand dotted lines, respectively.

2. Nitrides with wurtzite structure

The wurtzite crystal consists of two interpenetrating heagonal close-packed lattices, one having a group-III elematom ~e.g., Ga! and the other a group-V element atom~e.g.,


n-ethio

d

is

qe

ns

attter

ec-o.ianfortLut-efi-

iteand

o

he

-andbeenf thewith

on-


N!. A wurtzite crystal is characterized by two lattice costantsalc and clc. A major difference between zinc blendand wurtzite structures is that the in-plane behavior ofbands in a wurtzite crystal is different from the behavalong the@0001# axis ~thec axis!. TheG1c conduction bandsare s like at the center of the BZ, while the valence banbelong to the$G6v :$X,Y%1G1v:$Z%% representations. Thenearest higher-order conduction bands belong to theG6c

states of$X,Y% symmetry and theG3c states transforming likethe $Z% representation.

Due to the anisotropy of the crystal, there are two dtinct interband matrix elements arising from theG6v :$X,Y%and G1v :$Z% representations, defined by analogy with E~2.3!. These are in practice derived from the anisotropicfective mass using expressions similar to Eq.~2.15! ~assum-

FIG. 1. Diagram of the band structure in the vicinity of the energy gapGaAs: ~a! throughout the first Brillouin zone~reproduced with permissionfrom Ref. 81!, ~b! a magnified view near the zone center.


er

s

-

.f-

ing negligible crystal-field and spin-orbit splittings!. Al-though the different conduction-band energy contributiofrom the higher G6c5$X,Y% and G3c5$Z% intermediatestates lead to two distinctF parameters, no experiments thwould enable us to establish independent values for the lahave been reported. The compilations in the following stions take theF parameters in the wurtzite nitrides to be zer

The second-order valence-band terms in the Hamiltonare evaluated in a manner similar to the earlier discussionzinc blende structures. The procedure leads to six distincAparameters, which are to a large extent analogous to thetinger parameters in zinc blende materials. The detailed dnitions have appeared in the literature.34–38

In contrast to the zinc blende materials, the wurtzstructure does not give a triply degenerate valence b

f

FIG. 2. DirectG-valley energy gap as a function of lattice constant for tzinc blende form of 12 III–V binary compound semiconductors~points! andsome of their random ternary alloys~curves! at zero temperature. The energy gaps for certain ternaries such as AlAsP, InAsN, GaAsN, InPN,GaPN are extended into regions where no experimental data havereported. For GaAsN and InPN, the arrows indicate the boundaries oregions where the gap dependence on composition may be predictedany accuracy.

FIG. 3. Lowest forbidden gap as a function of lattice constant for nnitride III–V compound semiconductors~points! and their random ternaryalloys ~lines! at zero temperature. The materials withG-, X-, andL-valleygaps are indicated by solid, dotted, and dashed lines, respectively.


ne

-(

thd

s-a

etaIn

re

ca

ly

e

oft foringac-

or-gygeen

theeadro-al

heeis–Vrain.the

n-va-ingre-

tiondic-

ceo-nd

splitdo-

sw.

otby

rys-pyls.

nd

g

c-or-

ld

ne


edge. The crystal-field splitting leads to the band-edge egies:

^XuHcruX&5^YuHcruY&5Ev1D1 , ~2.19!

and

^ZuHcruZ&5Ev . ~2.20!

The spin-orbit splitting is parameterized by the relations

^XuH ~so!zuY&52 iD2 , ~2.21!

and

^YuH ~so!xuZ&5^ZuH ~so!yuX&52 iD3 . ~2.22!

Although in principle, two different spin-orbit splitting parameters arise, they are commonly assumed equalD2

5D3). However, the crystal-field splitting (D1) is in generalnot related to the spin-orbit splitting. TheDcr5D1 and Dso

53D2 parameters are tabulated for wurtzite materials infollowing sections. The definition of the spin-orbit ancrystal-field splitting in the wurtzite materials is further illutrated in Fig. 4~a!. A typical valence band structure forwurtzite material is shown in Fig. 4~b!.

3. Strain in heterostructures

To model the strain in a pseudomorphically grown herostructure such as a quantum well, quantum wire, or qutum dot, the elastic continuum theory is usually invoked.elastic continuum theory, the atomic displacements areresented by a local vector field,r 82r5u(r ). Ignoring thequadratic term for small deformations, we define the lostrain tensor« i j via ui(r )5« i j r j .39,40 The stress tensors i j

that generates the above strain is given by the relations i j

5l i jkl «kl . In crystals with cubic symmetry, there are onthree linearly independent constants:lxxxx5C11, lxxyy

5C12, lxyxy5C44. For the wurtzite structure, there are fivlinearly independent elastic constantslxxxx5C11, lzzzz

FIG. 4. ~a! Schematic illustration of the spin-orbit splitting and crystal-fiesplitting in wurtzite materials as compared to zinc blende materials.~b!Schematic diagram of the valence band structure of a wurtzite materialthe zone center with the heavy-hole~HH!, light-hole ~LH!, and crystal-hole~CH! valence bands explicitly identified.


r-

e

-n-

p-

l

5C33, lxxyy5C12, lxxzz5C13, lxzxz5C44. The values forthe elastic constants will be included in our compilationthe band parameters for completeness. Note that, at leasthe non-nitride materials, there is little controversy regardtheir values, and the datasets given below are widelycepted.

The physical deformation of the crystal leads to a disttion of the atomic locations, which in turn affects the enerlevels of the band electrons.14 The procedure for generatinadditional terms due to the presence of the strain has bdescribed in detail by Bir and Pikus14 and Bahder.41 In thecase of hydrostatic compression, the change inconduction-band-edge energyDEc due to the relative changin volumeDV/V5(«xx1«yy1«zz) can be parametrized bylinear relation between the change in energy and the hystatic strain. The constant of proportionality is the empiricdeformation potential constantac . Unfortunately, this pa-rameter is difficult to isolate experimentally. Instead it is tdeformation potential constanta, associated with the changin band gapEg due to a hydrostatic deformation, thatmeasured. Due to the nature of the atomic bonding in IIImaterials, the band gap increases for a compressive stUnder positive hydrostatic pressure, i.e., negative strain,change in energyDEg5a(«xx1«yy1«zz) must be positive.This implies a negative value fora[ac1av . Note that oursign convention forav is different from many other worksfound in the literature. It is generally believed that the coduction band edge moves upward in energy while thelence band moves downward, with most of the change bein the conduction band edge, although Wei and Zungercently argued that this is not always the case.42 The distribu-tion of the hydrostatic pressure shift between the conducand valence bands is generally based from theoretical pretions in this review.

From the Bir–Pikus strain interaction for the valenbands,14 it may be observed that the single deformation ptentialav , which parameterizes the shift of the valence-baedgeDEv5av(«xx1«xx1«xx), is insufficient to describe thefull effect of strain. Two additional potentialsb and d arenecessary to describe the shear deformations terms thatthe heavy/light-hole degeneracy. For the growth of pseumorphic layers along the@001# direction, only the value ofthe potentialb is relevant. All of the deformation potentialare tabulated for each material in the sections that folloFor the 636 valence-band strain Hamiltonian that is nreproduced here, the reader is referred to the textbooksBir and Pikus14 and by Chuang.43

The preceding discussion applies to the zinc blende ctal structure. In a wurtzite material, the crystal anisotroleads to two distinct conduction-band deformation potentiaFurthermore, six deformation potential constantsDi arisefrom a full treatment of the effect of strain on the six-bavalence-band structure as shown by Bir and Pikus.14 TheseDi are tabulated for each nitride material in the followinsections.

4. Piezoelectric effect in III –V semiconductors

Under an externally applied stress, III–V semicondutors develop an electric moment whose magnitude is prop

ar


ceg

i

in

nzoi

nc

tr

causulon

es

itheen

ous

ndi-entero-yeral,

dge

nan-lec-. Onntialherndndferi-lly

-nce

yelf.ove.theee

the

s-apsr atheires

to

itald asand

igh-t theredthe


tional to the stress.44,45 The strain-induced polarizationPs

can be related to the strain tensor« i j using piezoelectriccoefficientsei jk of the form

Pis5ei jk« jk . ~2.23!

The symmetry of the strain under interchange of its indiallows us to writeei jk in a more compact form. Convertinfrom the tensor notation to the matrix notation we write

$«11,«22,«33,~«23,«32%,~«31,«13%,~«12,«21%%

[$«1 ,«2 ,«3 ,«4 ,«5 ,«6%, ~2.24!

and

ei jk5H eim , ~ i 51,2,3;m51,2,3!;

12 eim , ~ i 51,2,3;m54,5,6!,

~2.25!

where it is standard practice to introduce the factor of 1/2front of theeim for m54,5,6 in order to obtain the followingform without factors of 1/2:

Pis5ei1«11ei2«21ei3«31ei4«41ei5«51ei6«6 . ~2.26!

In the zinc blende materials, only off-diagonal termsthe strain give rise to the electric polarization componentsPi

s

and

Pis5e14« jk , j Þk, ~2.27!

where e14 is the one independent piezoelectric coefficiethat survives due to the zinc blende symmetry. The pieelectric effect is negligible unless the epitaxial structuregrown along a less common direction such as@111#.

On the other hand, in a wurtzite crystal, the three distipiezoelectric coefficients aree315e32, e33, ande155e24 canbe derived from symmetry considerations. The piezoelecpolarization is given in component form by

Pxs5e15«13,

Pys5e15«12, ~2.28!

Pzs5e31«111e31«221e33«33.

where we have reverted to the tensor form for the strains« i j .Thus both diagonal and off-diagonal strain componentsgenerate strong built-in fields in wurtzite materials that mbe taken into account in any realistic band structure calction. The piezoelectric contribution to the total polarizatimay be specified alternatively in terms of matrixd related tomatrix e via the elastic constants

Pi5(j

ei j « j ,

~2.29!

Pi5(k

diksk5(j

(k

dikck j« j ,

where« j andsk are the components of the strain and strtensors in simplified notation@see Eq.~2.24!#, respectively.

The nitrides also exhibit spontaneous polarization, wpolarity specified by the terminating anion or cation at tlayer surface. Further details are available from recpublications46,47 and reviews.48,49 In the sections that follow,


s

n

t-

s

t

ic

nt

a-

s

h

t

we tabulate both piezoelectric coefficients and spontanepolarizations for the wurtzite nitride materials.

5. Band structure in layered heterostructures

So far we have considered the band parameters of ividual materials in the presence of strain. The misalignmof energy band gaps in adjacent layers of a quantum hetstructure is taken into account by specifying a reference laand defining a band offset function relative to it. In generthis potential energy is given as a functionVP(z) of thecoordinate in the growth direction and defines the band eprofile for the top of the valence band. WithVP(z)50 in thereference layer, the conduction band edge profile isVS(z)5Eg(z)2Eg(ref)1VP(z). Valence band offsets have beedetermined experimentally by optical spectroscopy of qutum well structures, x-ray photoelectron spectroscopy, etrical capacitance measurements, and other techniquesthe theoretical side, this is supplemented by pseudopotesupercell calculations that include atomic layers on eitside of a heterointerface. A critical review of valence baoffset determinations was given by Yu, McCaldin, aMcGill.50 In Sec. VII we will provide an updated review othe valence band offsets in zinc blende and wurtzite matals, putting the main emphasis on the compilation of a fuconsistent set of band offsets.

The coupled Schro¨dinger differential equations for a layered heterostructure may be solved using the finite-differemethod,51,52 the transfer matrix method,53–56 or the finite el-ement method,57–59which is a variational approach that mabe considered a discretization of the action integral itsFurther details are available from the references cited ab

The k"P model has also been extended to includeintrinsic inversion asymmetry of the zinc blendstructure,60–63 and, more phenomenologically, to include theffects ofG and Xz valley mixing in order to include sucheffects in modeling resonant double-barrier tunneling.64–69

These parameterizations are beyond the framework ofconsiderations presented here.

B. Relation of k "P to other band structure models

The k"P model is the most economical in terms of asuring agreement with the observed bulk energy band gand effective masses at the center of the BZ. However, fomore complete picture of the energy bands throughoutBZ, it is necessary to adopt another approach that requadditional information as input.

1. Empirical tight binding model

In the empirical tight-binding model~ETBM! introducedby Slater and Koster,70 the electronic states are consideredbe linear combinations of atomic (s,p,d,...) orbitals. TheHamiltonian’s matrix elements between the atomic orbstates are not evaluated directly, but are instead introducefree parameters to be determined by fitting the band gapsband curvatures~effective masses! at critical points in theBZ. Depending on the number of orbitals and nearest nebors used to represent the states, the ETBM requires thaoverlap integrals be determined in terms of the measudirect and indirect band gaps and/or effective masses in


-g

eria

dglinwintic

ohrs

te

zip

iMmrm

ep,hiM

ththlaa

er

retio

cffeel’artuthepic

ecta

l

ighlsu-ns.nicughs of

ost

pre-isof

lip-

of,pro-ure-

fol-

entsap

,ce

onics

eis-At

ce


bulk material.71,72 For example, thesp3s* basis with thesecond-nearest-neighbor scheme72 turns out to have 27 parameters for the zinc blende lattice structure, and the enerand effective masses obtained from the diagonalizationthe Hamiltonian and the resulting energy bands are nonlinfunctions of these parameters, which can be fitted by tand error or using, e.g., genetic algorithms.73 The lack of atransparent relationship between the input parameters anexperimentally determined quantities is probably the singreatest disadvantage of the tight-binding method in makcomplicated band structure calculations. In this review,make no attempt to give a standardized set of tight-bindparameters. The ETBM can also be applied to superlatband structure calculations.17,74,75

2. Effective bond orbital model

While the inclusion of additional bands and overlapshigher orbitals is a possible approach to improving the tigbinding modeling of energy bands in bulk semiconductothe effective bond orbital model76–78 ~EBOM! uses spin-doubled s,px ,py ,pz orbitals to generate an 838 Hamil-tonian. A crucial difference between the EBOM and relatight-binding formulations is that thes and p orbitals arecentered on the face-centered cubic lattice sites of theblende crystal rather than on both of the two real atomslattice site. The resulting somewhatad hoc formulation of-fers considerable computational savings in comparison wthe ETBM. However, the main significance of the EBOapproach derives from the fact that the resulting seculartrix has a small-k expansion that exactly reproduces the foof the eight-bandk"P Hamiltonian. This allows the EBOMinput parameters to be readily expressed in terms of theperimentally measured parameters, such as the band gasplit-off gap, and the zone-center mass of each band, whas not been accomplished using the more involved ETBIn fact, the EBOM can be thought of as an extension ofk"P method to provide an approximate representation ofenergy bands over the full BZ. Since short-period supertice bands sample wavevectors throughout the BZ, we mexpect the EBOM to be more accurate than thek"P modelfor thin-layer structures. However, the EBOM is considably less efficient computationally thank"P, especially forthicker superlattices. Each lattice position must be repsented in the supercell technique, i.e., no envelope funcapproximation is made.

3. Empirical pseudopotential model

The influence of core electrons in keeping the valenelectrons outside of the core may be represented by an etive repulsive potential in the core region. When this is addto the attractive ionic potential, the net ‘‘pseudopotentianearly cancels79,80 at short distances. The valence statesorthogonal to the core states, and the resulting band structheory corresponds to the nearly free-electron model. Inempirical pseudopotential model, the crystal potential is rresented by a linear superposition of atomic potentials, whare modified to obtain good fits to the experimental dirand indirect band gaps and effective masses. Further de


iesofarl

theegege

ft-,

d

ncer

th

a-

x-thech.

eet-y

-

-n

ec-d’eree-htils

are presented by Cohen and Chelikowsky81 and in the re-views by Heine and Cohen.82,83Ab initio approaches employcalculated band parameters~e.g., from the density-functionatheory! in lieu of experimental data. Combinations ofab ini-tio and empirical methods have been developed to a hlevel of sophistication.84 Extension of the pseudopotentiamethod to heterostructures entails the construction of apercell to assure the proper periodic boundary conditioWith atomic potentials as the essential input, the electroproperties of the heterostructure can be determined, althothe required computational effort far exceeds the demandthe k"P method. The relative merits of thek"P and pseudo-potential approaches have been assessed.85,86

III. BINARY COMPOUNDS

A. GaAs

GaAs is the most technologically important and the mstudied compound semiconductor material.3 Many bandstructure parameters for GaAs are known with a greatercision than for any other compound semiconductor. Thisespecially true of the fundamental energy gap with a value1.519 eV at 0 K.87 The analysis by Thurmond88 indicateda50.5405 meV/K andb5204 K @in Eq. ~2.13!#. A morerecent examination of a large number of samples by elsometry produced a very similar parameter set ofEg(T50)51.517 eV,a50.55 meV/K, andb5225 K.89 The two re-sults are well within the quoted experimental uncertaintyeach other and several other experimental determinations3,90

although somewhat different parameter sets have beenposed recently on the basis of photoluminescence measments: Eg(T50)51.519 eV, a50.895– 1.06 meV/K, andb5538– 671 K.91,92

The original controversy87 about the ordering of theLand X-valley minima was resolved by Aspnes,93 who pro-posed on the basis of numerous earlier experiments thelowing sets:Eg

L(T50)51.815 eV, aL50.605 meV/K, andEg

X(T50)51.981 eV, aX50.460 meV/K withb5204 K inboth cases. Schottky-barrier electroreflectance measuremyielded the widely accepted value for the split-off energy gin GaAs:Dso50.341 eV.94

Electron effective masses ofme* 50.0665m0 and0.0636m0 were observed atT560 and 290 K, respectivelyby Stradling and Wood95 using magnetophonon resonanexperiments.96–98 A low-temperature value of 0.065m0 wasdetermined for the bare electron mass once the polarcorrection was subtracted.99 While a somewhat larger masof '0.07m0 was derived theoretically in severalab initioand semiempirical band structure studies,100–102recent cyclo-tron resonance measurements103 indicate a low-temperatureresult of 0.067m0 at the band edge. This is the value wadopt following the recommendation of Nakwaski in hcomprehensive review,104 which is based on reports employing a wide variety of different experimental techniques.room temperature, the currently accepted value is 0.0635m0 ,as confirmed, for example, by photoluminescenmeasurements.105 We employ the effective masses for theLand X valleys given by Adachi3 and Levinshteinet al.,11

which were compiled from a variety of measurements.


bl

lly

o-

tic

toina

e-a

a

t

tb

f

t

et

picaa--ntlit

an

ha

iait

ifum

iontm-

se-asesP,signce-ionnd-of

ndsial-ose

of

m-

edlec-

syc-

itstersthe

hede-hatisof

me-zire-

theAt


Many different sets of Luttinger parameters are availain the literature.99 The most popular set:g156.85, g2

52.10, andg352.90 ~also given in Landolt–Bornstein1!,which was derived by Lawaetz106 on the basis of five-leve~14-band! k"P calculations, is in good agreement with earcyclotron resonance,107 magnetoreflection,108 andmagnetoexciton109 experiments. While two-photon magnetabsorption measurements110 indicated very little warping inthe valence band, those results are contradicted by opspectroscopy111 and Raman scattering112 studies of GaAsquantum wells as well as by fits to the shallow accepspectra.113,114Here we prefer a composite data set, whichwithin the experimental error of all accurate determinatioand reproduces well the measured hole masseswarping:99,104g156.98,g252.06, andg352.93.

While its value is usually less critical to the device dsign and band structure computations, the split-off hole min GaAs has also been measured and calculated usingriety of approaches.106,111A composite valuemso* 50.165m0

was derived by Adachi,3 which is in excellent agreemenwith the recent calculation by Pfeffer and Zawadzki.33 How-ever, we will use a slightly different value,mso* 50.172m0 , inorder to provide self-consistency with thek"P expression inEq. ~2.18!.

The first electron-spin-resonance measurements ofinterband matrix element in GaAs, which were reportedChadi et al.115 and Hermann and Weisbuch,116 yielded EP

528.8– 29.0 eV. However, Shantharamaet al.96,117 sug-gested that those analyses overestimated the influence omote bands outside of the 14-bandk"P model and gaveEP

525.060.5 eV. Theoretical studies99,118 have derived inter-mediate values. Since the analysis by Shantharamaet al. ap-pears to have internal consistency problems, we adoptHermann and Weisbuch value~implying F521.94!, whichhas been used with some success in the literature to dmine the optical gain in GaAs quantum-well lasers.119

The greatest uncertainty in the GaAs band structurerameters is associated with the deformation potentials, whare needed to calculate strain effects in pseudomorphicgrown layers. In this review, we will consider only deformtion potentials for theG valley. The total hydrostatic deformation potentiala is proportional to the pressure coefficieof the direct band gap, where the constant of proportionais approximately the bulk modulus. Some trends in the bgap pressure coefficients were noted by Weiet al.42,120 Theexperimental hydrostatic pressure dependence ofEg forGaAs implies a total deformation potentiala5ac1av'28.5 eV,121 where the minus sign represents the fact tthe band gap expands when the crystal is compressed~notethat our sign convention forav is different from a large num-ber of articles!. The conduction-band deformation potentac corresponds to the shift of the conduction band edge wapplied strain. Pseudopotential122 and linear-muffin-tin-orbital123 calculations yieldedac as large as218.3 eV,whereas various analyses of mobility data3,124using standarddeformation-potential scattering models are consistent wac falling in the range26.3 to213.5 eV. A recent study oacoustic–phonon scattering in GaAs/AlGaAs quantwells125 produced an estimate ofac5211.560.5 eV. On the


e

al

rssnd

ssva-

hey

re-

he

er-

a-h

lly

yd

t

lh

th

other hand, studies of the valence-band deformatpotential3,126,127suggest a smallav . In order to be consistenwith the experimental hydrostatic pressure shift, we recomend using the valuesac527.17 eV andav521.16 eV,which were derived from the ‘‘model-solid’’ formalism byVan de Walle.129 However, the first-principles calculationof Wei and Zunger42 show that the energy of the valencband maximum increases as the unit cell volume decrefor a number of III–V semiconductors including GaAs, GaGaSb, InP, InAs, and InSb, whereas it has the oppositein other materials. Whatever the direction of the valenband maximum, it is generally agreed that the conductband moves much faster with pressure. Heterolayer bastructure calculations are relatively insensitive to the valueav when it is close to zero.

The shear deformation potentialsb andd have been de-termined both experimentally and theoretically.3,121,126,128–131

Moreover, a ratio of the deformation potentialsd/b52.460.1 was recently derived from studies of acceptor-bouexcitons in biaxially and uniaxially strained GaAepilayers.132 The various values for the deformation potentb varied between21.66 and23.9 eV, although recent results tend toward the lower end of that range. We propthe following composite values:b522.0 eV and d524.8 eV, which are consistent with the vast majoritymeasurements and several calculations.

All of the recommended parameters for GaAs are copiled in Table I.

B. AlAs

Because of its frequent incorporation into GaAs-basheterostructures, AlAs is also one of the most important etronic and optoelectronic materials.3,6,8Unlike GaAs, AlAs isan indirect-gap semiconductor with theX–L –G ordering ofthe conduction valley minima. TheX-valley minimum is lo-cated at a wave vectork5(0.903,0,0). The exciton energiecorresponding to theG-valley energy gap were measured bMonemar133 to be 3.13 and 3.03 eV at 4 and 300 K, respetively. A small ~'10 meV! correction for the exciton bindingenergy is presumably necessary.134 Similar values were ob-tained by Onton,135 Garriga et al.,136 and Dumkeet al.137

Since direct measurements on AlAs are difficult owing torapid oxidation upon exposure to air, we choose paramethat are consistent with the more readily available data onAlGaAs alloy discussed in detail below as well as with tforegoing measurements on bulk AlAs. The temperaturependence of the direct energy gap, which is similar to tfor GaAs, was given in an extrapolated form by Logothetidet al.,138 although better agreement with the resultsMonemar133 can be obtained by increasingb to 530 K.

The low-temperatureX-G indirect gap in AlAs was mea-sured to be'2.23–2.25 eV.133,137,139We suggest the follow-ing temperature-dependence parameters, which are sowhat different from the empirical suggestion of Guzet al.,139 but are more consistent with the experimentalsults of Monemar:133 a50.70 meV/K andb5530 K. Notmany data are available on the temperature variation ofL –G gap, although it should be similar to that in GaAs.



Downloaded 05 D

TABLE I. Band structure parameters for GaAs.

Parameters Recommended values Range

alc ~Å! 5.6532513.8831025(T2300)

EgG ~eV! 1.519 1.420–1.435~300 K!

a~G! ~meV/K! 0.5405 0.51–1.06b~G! ~K! 204 190–671Eg

X ~eV! 1.981 ¯

a(X) ~meV/K! 0.460 ¯

b(X) ~K! 204 ¯

EgL ~eV! 1.815 ¯

a(L) ~meV/K! 0.605 ¯

b(L) ~K! 204 ¯

Dso ~eV! 0.341 0.32–0.36me* ~G! 0.067 0.065–0.07~0 K!, 0.0635–0.067~300 K!ml* (L) 1.9 ¯

mt* (L) 0.0754 ¯

mDOS* (L) 0.56 ¯

ml* (X) 1.3 ¯

mt* (X) 0.23 ¯

mDOS* (X) 0.85 ¯

g1 6.98 6.79–7.20g2 2.06 1.9–2.88g3 2.93 2.681–3.05mso* 0.172 0.133–0.388EP ~eV! 28.8 25.5–29.0F 21.94 0.76–~22!VBO ~eV! 20.80ac ~eV! 27.17 26.3–~218.3!av ~eV! 21.16 20.2–~22.1!b ~eV! 22.0 21.66–~23.9!d ~eV! 24.8 22.7–~26.0!c11 ~GPa! 1221 ¯

c12 ~GPa! 566 ¯

c44 ~GPa! 600 ¯

d

lt

olav

verth

n

ty

inerl ban

atssesss

wn

b-toadr,olegerandda-

terst

and

or-al

ues-

room temperature, a gap of'2.35 eV is generallyadopted.2,3,140,141A split-off gap of 0.275 eV was measureby Onton,135 although the extrapolations by Aubelet al.142

and Wrobelet al.143 indicated higher values.The G-valley electron effective mass in AlAs is difficu

to determine for the same reasons that the band gapsuncertain, and also because, in contrast to GaAs, it is impsible to maintain aG-valley electron population in thermaequilibrium. Various calculations and measurements hbeen compiled by Adachi3 and Nakwaski.104 As pointed outby Adachi, the indirect determinations144,145employing reso-nant tunneling diodes with AlAs barriers give an effectimass comparable to that in GaAs, but are less trustwothan the extrapolations from AlGaAs146,147 and theoreticalcalculations100,148 which indicateme* 50.15m0 . A slightlylower value of 0.124m0 was inferred from a fit to absorptiodata.137

Electron effective masses for theX valley with an ellip-soidal constant-energy surface were calculated usingpseudopotential method,148 and measured by Faradarotation149 and cyclotron resonance.104,150The former experi-ment in fact measured only the ratio between the longitudand transverse masses~of 5.7!, and there is some evidencthat the assumed value forml was inconsistent with othedata.104,151For reasons that are discussed in some detaiNakwaski,104 it appears that the recent results by Goir

ec 2002 to 140.105.8.31. Redistribution subject to AI

ares-

e

y

he

al

y

et al.152 represent the most reliable values availablepresent. For the longitudinal and transverse effective maof the L valley, we employ the calculated results of Heet al.148

The band edge hole masses in AlAs are also not knowith a great degree of precision. Both theoretical106,148 andexperimental111 sets of Luttinger parameters have been pulished. Unfortunately, the latter was obtained from a fitmeasurements on GaAs/AlGaAs quantum wells, which honly limited sensitivity to the AlAs parameters. Howevethe agreement between different calculations of the hmasses is rather good. We propose a composite Luttinparameter set based on an averaging of the heavy-holelight-hole masses from various sources and recommentions by Nakwaski:104 g153.76, g250.82, andg351.42.These values are quite similar to the composite paramesuggested by Adachi.3 For the split-off hole mass, we adopa value of 0.28m0 for consistency with theEP value of21.1 eV(F520.48) given by Lawaetz.106 This mass fallsmidway between the recommendations by PavesiGuzzi134 and Adachi.3

Very few determinations of the electron and hole defmation potentials in AlAs exist. Most of the experimentvalues are in fact extrapolations from AlGaAs~see below!.As for the case of GaAs, we recommend using the valac525.64 eV andav522.47 eV derived from the model


-

tori

m

eab

ur

e

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sew

mh

re

arena-rbit

rees

nedtron

hiskenermi

h

iesas

domandred

ereal-an

r-


solid formalism by Van de Walle.129 For the shear deformation potentials, the following values are suggested:b522.3 eV153 and d523.4 eV.3 The former is supported byellipsometry measurements of the heavy-light hole excisplittings in AlGaAs epitaxial layers, whereas further expements are necessary to confirm the latter.

All of the recommended parameters for AlAs are copiled in Table II.

C. InAs

InAs has assumed increasing importance in recent yas the electron quantum well material for InAs/GaSb/AlSbased electronic154 and long-wavelength optoelectronic155

devices. The vast majority of experimental low-temperatenergy gaps fall in the 0.41–0.42 eV range,156–159althoughsomewhat higher values have also been reported.160,161 Weadopt the value 0.417 eV that was obtained from recent msurements on a high-purity InAs sample,162 in which it waspossible to separate shallow impurity, exciton, and bandband transitions. The temperature dependence of the bgap has also been reported by several authors.2,27,163–165Al-though there is considerable variation in the propoVarshni parameters, most of the data agree reasonablywith the values given by Fanget al.:164 a50.276 meV/K andb593 K. Energies for theL andX conduction-band minimain InAs have not been studied experimentally. Our recomended values are based on the suggestions by Adac166

and Levinshteinet al.11 extrapolated from room temperatu

TABLE II. Band structure parameters for AlAs.


alc ~Å! 5.661112.9031025(T2300) ¯

EgG ~eV! 3.099 2.9–3.14

a~G! ~meV/K! 0.885 ¯

b~G! ~K! 530 ¯

EgX ~eV! 2.24 2.23–2.25

a(X) ~meV/K! 0.70 ¯

b(X) ~K! 530 ¯

EgL ~eV! 2.46 2.35–2.53

a(L) ~meV/K! 0.605 ¯

b(L) ~K! 204 ¯

Dso ~eV! 0.28 0.275–0.31me* (G) 0.15 0.06–0.15ml* (L) 1.32 ¯

mt* (L) 0.15 ¯

ml* (X) 0.97 ¯

mt* (X) 0.22 ¯

g1 3.76 3.42–4.04g2 0.82 0.67–1.23g3 1.42 1.17–1.57mso* 0.28 0.24–0.68EP ~eV! 21.1 ¯

F 20.48 ¯

VBO ~eV! 21.33 ¯

ac ~eV! 25.64 0.7–~25.64!av ~eV! 22.47 21.2–~22.6!b ~eV! 22.3 21.4–~23.9!d ~eV! 23.4 22.7–~26.0!c11 ~GPa! 1250 ¯

c12 ~GPa! 534 ¯

c44 ~GPa! 542 ¯


n-

-

rs-

e

a-

-nd

dell

-i

to 0 K. The temperature dependences of the indirect gapstaken to be identical to the direct gap, since no determitions appear to be available. The experimental spin-osplittings given in Landolt–Bornstein1 fall in the 0.37–0.41eV range. We take an average of 0.39 eV, which also agwell with the more recent experiments of Zverevet al.167

The electron effective mass in InAs has been determiby magnetophonon resonance, magnetoabsorption, cycloresonance, and band structure calculations.95,104,162,168–179

Owing to the strong conduction-band nonparabolicity in tnarrow-gap semiconductor, considerable care must be tato measure the mass at the band edge rather than at the Flevel.180 The majority of results at both low and higtemperatures fall between 0.0215m0

174 and 0.026m0 .171

Although a few theoretical and experimental studhave obtained low-temperature masses as high0.03m0 ,101,102,104,181these values were most likely influenceby the strong nonparabolicity. While a value near the bottof the reported range is usually recommended since the bedge mass represents a lower limit on the measuquantity,104 many of the results supporting such a mass win fact performed at room temperature. Since there aremost no credible reports of an effective mass lower th0.0215m0 at any temperature,104 our recommended low-temperature value is 0.026m0 which, accounting for the shiftof the energy gap, implies 0.022m0 at 300 K. The small~1%!polaronic correction is well within the experimental unce

TABLE III. Band structure parameters for InAs.


alc ~Å! 6.058312.7431025(T2300) ¯

EgG ~eV! 0.417 0.410–0.450

a~G! ~meV/K! 0.276 ¯

b~G! ~K! 93 ¯

EgX ~eV! 1.433 ¯

a(X) ~meV/K! 0.276 ¯

b(X) ~K! 93 ¯

EgL ~eV! 1.133 1.13–1.175

a(L) ~meV/K! 0.276 ¯

b(L) ~K! 93 ¯

Dso ~eV! 0.39 0.37–0.41me* (G) 0.026 0.023–0.03ml* (L) 0.64 ¯

mt* (L) 0.05 ¯

mDOS* (L) 0.29 ¯

ml* (X) 1.13 ¯

mt* (X) 0.16 ¯

mDOS* (X) 0.64 ¯

g1 20.0 6.79–7.20g2 8.5 1.9–2.88g3 9.2 2.681–3.05mso* 0.14 0.09–0.15EP ~eV! 21.5 21.5–22.2F 22.90 0–~22.90!VBO ~eV! 20.59 ¯

ac ~eV! 25.08 25.08–~211.7!av ~eV! 21.00 21.00–~25.2!b ~eV! 21.8 28–~22.57!d ~eV! 23.6 ¯

c11 ~GPa! 832.9 ¯

c12 ~GPa! 452.6 ¯

c44 ~GPa! 395.9 ¯


f

r-

vafu

inatisc

ng

vetu

r

erd,thte

bx-ote

t

-m

0-niat

te

ni

Au-

asdi-

om

ndn.

ein

la-

taley theasstedaP,

sst

tednts

am-s

con-

d

-

ma-

ly a

is

hat

n-

de

there-

i-


tainty in this case. The density-of-states effective massesthe X and L valleys are taken from Levinshteinet al.11 Byemploying typical experimentalml /mt ratios for relatedIII–V materials ~such as GaAs and GaSb! correspondinglongitudinal and transverse masses have been estimatedlisted in Table III.

Although Luttinger parameters for InAs were detemined experimentally by Kanskayaet al.,159 those param-eters appear to disagree with the heavy-hole masses forous directions given in the same reference. The values og1

and g3 given in that reference are very close to a previodetermination of Pidgeonet al.157 A number of theoreticalworks104,106,182predict a much higher degree of anisotropythe valence band. Surveying the data available to dNakwaski104 concludes that further experimental workneeded to resolve the matter. Noting the considerable untainty involved in this estimate, we suggest the followicomposite set:g1520, g258.5, andg359.2. Experimentalstudies183 have suggested a split-off mass of 0.14m0 in InAs.

The interband matrix element in InAs appears to habeen determined relatively accurately owing to the larggfactor in this narrow-gap semiconductor. Whereas early sies suggested a value of 22.2 eV,106,116 more recentlyEP

521.5 eV has gained acceptance.159,184We recommend thisvalue, which leads toF522.90. This is somewhat largethan the effect predicted by other workers,159 which also as-sumed a smaller low-temperature electron mass (0.024m0).

The hydrostatic deformation potential in InAs was detmined to bea526.0 eV.1 We take the conduction anvalence-band deformation potentials of Van de Walle129

who estimates that most of the energy shift occurs inconduction band. Somewhat different values were calculaby Blachaet al.122 and Wei and Zunger.42 The shear defor-mation potentials adopted from Landolt–Bornstein1 are ingood agreement with the calculations of Blachaet al.122 An-other set of deformation potentials has been calculatedWang et al.86 Unfortunately, at present there exist little eperimental data from which to judge the relative meritsthe above calculations. All of the recommended paramefor InAs are compiled in Table III.

D. GaP

Nitrogen-doped GaP has for a long time been used asactive material for visible light-emitting diodes~LEDs!.185

GaP is the only indirect-gap~with X–L –G ordering of theconduction-band minima! binary semiconductor we will consider that does not contain Al. The band structure is sowhat similar to that of AlAs, with theX-valley minimum atk5(0.95,0,0).5 Indirect X–G energy gaps of 2.338–2.35eV have been reported.186,187The main uncertainty in determining the various energy gaps for GaP is that excitorather than band-to-band absorption lines typically dominso that a calculated exciton binding energy~which presum-ably has a weak temperature dependence! must be added tothe experimental results. The situation is further complicaby the camel’s back structure of theX-valley conduction-band minimum.187 The most commonly employed Varshparameters, reported in Casey and Panish,2 are in excellent


or

and

ri-

s

e,

er-

e

d-

-

ed

y

frs

he

e-

ce,

d

agreement with piezomodulation spectroscopy results byvergneet al.188 The L-valley minimum is located'0.37 eVabove theX valley,4 although its temperature dependence hnot been determined. The low-temperature value for therect excitonic band gap was found to be 2.86–2.87 eV frabsorption measurements.133,137,189An exciton binding en-ergy of 20 meV is added to obtain the energy for interbatransitions, although the precise value is not well knowGaP has a small spin-orbit splitting of 0.08 eV.1,189

The electron effective mass in theG valley is estimatedfrom theory to be 0.09m0 ,11,96 although higher values havalso been reported.101,106The diamagnetic shifts measuredmagnetoluminescence experiments on GaAs12xPx alloyswith x,0.45190 suggest an effective mass of'0.13m0 forGaP, which is in good agreement with tight-binding calcutions by Shen and Fan.191

There have been a number of experimendeterminations192 of theX-valley longitudinal and transversmasses, although the measurements are complicated bcamel’s back structure that makes the longitudinal mhighly nonparabolic. This nonparabolicity must be accounfor in any treatment of the density of electron states in Gand is in fact more crucial than the precise value ofml farabove the camel’s back. Effective masses of 5 – 7m0 havebeen reported for the bottom of the camel’s back,193 whereasml* '2m0 high above.194 Estimates for the transverse marange from 0.19m0

182 to 0.275m0 ,195 although the presenconsensus puts the value at 0.25m0 .193 We have taken lon-gitudinal and transverse effective masses for theL-valleyminimum from Levinshteinet al.11

The Luttinger parameters for GaP were first calculaby Lawaetz.106 Subsequent cyclotron resonance experimerefined the masses along the@111#196–198and @100#198 direc-tions. Street and Senske also determined Luttinger pareters from acceptor binding energies.199 Those parameters acorrected in Landolt–Bornstein1 are in reasonably goodagreement with cyclotron resonance results, and may besidered the most reliable:g154.05, g250.49, and g3

51.25. A split-off hole mass of 0.23– 0.24m0 was calculatedby Lawaetz106 and by Krijn.101 We use a slightly highervalue of 0.25m0 in order to be consistent with the interbanmatrix element.

Lawaetz obtainedEP522.2 eV using a rather high theoretical estimate for theG-valley effective mass in GaP.106

Recently, more accurate determinations of the interbandtrix element in GaAsP have been published.190,191 By anal-ogy with the case of GaAs discussed above, these impmuch higherEP of 31.4 eV, which leads toF522.04. Al-though such an extrapolation from data on As-rich alloyssomewhat questionable, there is noa priori reason that aninterband matrix element in GaP so much different from tin GaAs is unreasonable.

The hydrostatic deformation potential for the direct eergy gap in GaP was measured by Mathieuet al.200 to bea529.9 eV. Van de Walle129 estimated that the valence-banshift is small compared with the conduction-band shift. Wrecommend his valence-band deformation potential, withconduction-band contribution corrected to reproduce thesult of Mathieuet al.200 A number of theoretical and exper


ee

m

dasbn

eai-

tio

olt

rb07ve

ec

lP/

andt fit-rmm-ari-f

teddInP/

cha

ma-

l-

calst

onsc-out.


mental values have been reported130,187,200for the shear de-formation potentials, which are generally in good agreemwith each other. We suggest the following composite valub521.6 eV andd524.6 eV.

All of the recommended parameters for GaP are copiled in Table IV.

E. AlP

AlP, with the largest direct gap of the III–V compounsemiconductors, is undoubtedly the most ‘‘exotic’’ and lestudied. Nevertheless, the essential characteristics haveknown for some time. It is unclear whether the conductioband minima follow theX–G –L201 or X–L –G182 ordering,since no actual measurements of theL-valley position appearto have been performed. The indirect energy gap of 2.5and its temperature dependence are given in CaseyPanish2 with appropriate corrections to the original determnations. A similar value has been obtained by extrapolafrom AlGaP alloys by Alferovet al.202 The direct gap of AlPwas measured by Monemar133 to be 3.63 eV at 4 K and 3.62eV at 77 K, while Bouret al.203 obtained an extrapolation t300 K of 3.56 eV. The required correction of these resudue to the exciton binding energy is unclear. The spin-osplitting in AlP should be small, on the order of 0.06–0.eV,101,204 although the actual value has apparently nebeen measured.

Almost no experimental data are available on the efftive masses in AlP. AG-valley mass of 0.22m0 was calcu-lated using the augmented spherical wave approach.101 An-other ab initio calculation182 yielded ml* 53.67m0 and mt*

TABLE IV. Band structure parameters for GaP.


alc ~Å! 5.450512.9231025(T2300) ¯

EgG ~eV! 2.88610.1081@1-coth(164/T)# 2.86–2.895

EgX ~eV! 2.35 2.338–2.350

a(X) ~meV/K! 0.5771 ¯

b(X) ~K! 372 ¯

EgL ~eV! 2.72 ¯

a(L) ~meV/K! 0.5771 ¯

b(L) ~K! 372 ¯

Dso ~eV! 0.08 0.08–0.13me* (G) 0.13 0.09–0.17ml* (X) 2.0 ~camel back! 2–7mt* (X) 0.253~camel back! 0.19–0.275ml* (L) 1.2 ¯

mt* (L) 0.15 ¯

g1 4.05 4.04–4.20g2 0.49 ¯

g3 2.93 ¯

mso* 0.25 0.23–0.25EP ~eV! 31.4 22.2–31.4F 22.04 0–~22.04!VBO ~eV! 21.27 ¯

ac ~eV! 28.2 26.3–~218.3!av ~eV! 21.7 20.2–~22.1!b ~eV! 21.6 21.66–~23.9!d ~eV! 24.6 22.7–~26.0!c11 ~GPa! 1405 ¯

c12 ~GPa! 620.3 ¯

c44 ~GPa! 703.3 ¯


nts:

-

teen-

Vnd

n

sit

r

-

50.212m0 for theX valley, although Issikiet al.205 obtainedbetter agreement with photoluminescence results for AGaP heterostructures using a somewhat smallerX-valleymass. We have adjusted the theoretical longitudinaltransverse masses by the same factor to conform to thating, although further studies are clearly needed to confiour projections. Composite values for the Luttinger paraeters and the split-off hole mass have been taken from vous calculations.101,106,182 An interband matrix element o17.7 eV (F520.65) is given by Lawaetz.106

The hydrostatic deformation potentials were calculaby Van de Walle,129 although a slight correction was founto be necessary when energy level alignments in a GaAlGaInP laser structure were fit.206 We selectb521.5 eV inaccordance with the calculations of O’Reilly130 and Krijn,101

although higher values have been computed by Blaet al.122 No values for the shear deformation potentiald ap-pear to have been reported. In the absence of other infortion, we recommend the value ofd524.6 eV derived forGaP~see previous subsection!.

All of the band structure parameters for AlP are colected in Table V.

F. InP

InP is a direct-gap semiconductor of great technologisignificance,4–7,10 since it serves as the substrate for mooptoelectronic devices operating at the communicatiwavelength of 1.55mm. Numerous studies of the band struture parameters for InP and its alloys have been carried

TABLE V. Band structure parameters for AlP.


alc ~Å! 5.467212.9231025(T2300) ¯

EgG ~eV! 3.63 3.62~77 K!, 3.56 ~300 K!

a~G! ~meV/K! 0.5771 ¯

b~G! ~K! 372 ¯

EgX ~eV! 2.52 2.49–2.53

a(X) ~meV/K! 0.318 ¯

b(X) ~K! 588 ¯

EgL ~eV! 3.57 ¯

a(L) ~meV/K! 0.318 ¯

b(L) ~K! 588 ¯

Dso ~eV! 0.07 0.06–0.07me* (G) 0.22 ¯

ml* (X) 2.68 2.68–3.67mt* (X) 0.155 0.155–0.212g1 3.35 ¯

g2 0.71 ¯

g3 1.23 ¯

mso* 0.30 0.29–0.34EP ~eV! 17.7 ¯

F 20.65 ¯

VBO ~eV! 21.74 ¯

ac ~eV! 25.7 25.54–~25.7!av ~eV! 23.0 23.0–~23.15!b ~eV! 21.5 21.4–~24.1!d ~eV! 24.6 ¯

c11 ~GPa! 1330 ¯

c12 ~GPa! 630 ¯

c44 ~GPa! 615 ¯


dt

n-urt

-

sio

drea

ew

thehene

d--6

ennong

ne

-

fe

sin

t

e-rte

es

ro-

losedotdif-

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Rochon and Fortin207 found the low-temperature direct bangap to be 1.423 eV, with most other reports agreeingwithin a few meV.1 Since exciton rather than interband trasitions are usually observed in such absorption measments, the binding energy of 5 meV has been added tospectral position of the resonance.1,207 The temperature dependence of the direct band gap has been reportedVarshni,27 Casey and Panish,2 Lautenschlageret al.,208 Hanget al.,209 and Pavesiet al.210 WhenT@b, the gap decreaselinearly with temperature and use of the Varshni expressis not necessary~i.e., b50!. Lautenschlageret al. andPavesiet al. did not use the Varshni functional form, anHang et al. focused primarily on fitting higher-temperatuband gaps up to 870 K. We recommend using the CaseyPanish temperature dependence~with a corrected value ofEg

at T50! if one’s interest is primarily in the temperaturrange from 0 K to somewhat above room temperature. Hoever, the Lautenschlageret al.211 and Hanget al.212 formsare more appropriate at temperatures well above 300 K.

There are greater uncertainties in the positions ofX-valley andL-valley conduction-band minima in InP. ThL-valley minimum is believed to lie 0.4–0.7 eV above tG-valley minimum, with the most reliable values favoring aapproximately 0.6 eV separation. The temperature depdence is unclear.1,4,11,201TheG –X separation has been stuied more closely,1 and the different works have been compared in detail.213 The inferred low-temperature value is 0.9eV, with dEG –X /dT520.37 meV/K. The spin-orbit split-ting was determined asDso50.108 eV by wavelength-modulated reflection214 and photovoltaic effect207 measure-ments.

The G-valley electron effective mass in InP has beinvestigated in great detail by cyclotron resonance, magtophonon resonance, and magnetospectroscopy of dtransitions. While values have spanned the ran0.068– 0.084m0 .97,98,108,173,207,215–225if early and ambiguousdeterminations are excluded and the polaron correctioestimated to retrieve the bare mass,99,218the reasonable rangfor me* (T50) narrows to 0.077– 0.081m0 . Averaging theresults from different groups, we obtainme* (T50)50.0795m0 , which is slightly smaller than the typically recommended values of 0.080– 0.081m0 .1,10 The effectivemasses for theL and X valleys are taken from Pitt,213 al-though there is some controversy on this point and a difent set of parameters was given by Levinshteinet al.11

Luttinger parameters for InP have been measured ucyclotron resonance,198 magnetoreflectance,109,226 the piezo-modulated photovoltaic effect,207 and magnetoabsorption.227

Furthermore, the valence-band warping was accurately demined by Alekseev et al. using hot-carrierphotoluminescence,228,229 and theoretical calculations havalso been performed.101,106 In arriving at our composite parameters, we follow the procedure of averaging the repovalues for heavy-hole and light-hole masses along the@100#and @111# directions. The resulting set ofg155.08, g2

51.60, andg352.10 is in good agreement with all of thmost reliable experimental results. The split-off hole mawas measured by Rochon and Fortin207 to be 0.21m0 .


o

e-he

by

n

nd

-

e

n-

e-ore

is

r-

g

er-

d

s

Lawaetz106 and Gorczycaet al.118 calculated the inter-band matrix elementEP in InP to be close to 20 eV. Thisresult is in good agreement with the value of 20.7 eV pposed by Hermann and Weisbuch.116 However, a number ofother experimental determinations have favored a value cto 16.5 eV.96,117,207,219,223This discrepancy is easily resolveonce it is realized that the majority of references did nconsider remote-band effects on the electron mass. Theference between the results of Hermann and Weisbuch116 andShantharamaet al.96 has already been discussed in conntion with the interband matrix element for GaAs. We adothe former value ofEP520.7 for InP, which impliesF521.31 ~the latter result would implyF'0!, although amore detailed examination of this issue is called for in vieof the large divergence between the two results.

There is a great deal of variation in the experimental atheoretical deformation potentials for InP as compiledAdachi.4 The most reliable values for the conduction-badeformation potential are probably those of Nolteet al.127

~27 eV! and Van de Walle129 ~25.04 eV!. In combinationwith the reported direct-gap deformation potential of26.6eV,1 these values imply a rather small shift in the valenband. The shear deformation potentialsb and d have beendetermined by Camasselet al.214 and are in good agreemenwith exciton reflectance measurements.230

All of the band structure parameters for InP are collecin Table VI.

G. GaSb

GaSb is often referred to an intermediate-gap semicductor, i.e., its gap of'0.8 eV is neither as wide as in GaA

TABLE VI. Band structure parameters for InP.


alc ~Å! 5.869712.7931025(T2300) ¯

EgG ~eV! 1.4236 1.420–1.432

a~G! ~meV/K! 0.363 0.51–1.06b~G! ~K! 162 190–671Eg

X ~eV! 2.384– 3.731024T 1.48–2.39

EgL ~eV! 2.014 1.82–2.12

a(L) ~meV/K! 0.363 ¯

b(L) ~K! 162 ¯

Dso ~eV! 0.108 0.108–0.13me* (G) 0.0795 0.068–0.084mDOS* (L) 0.47 0.25–0.47mDOS* (X) 0.88 0.32–0.88g1 5.08 4.61–6.28g2 1.60 0.94–2.08g3 2.10 1.62–2.76mso* 0.21 0.17–0.21EP ~eV! 20.7 16.6–20.7F 21.31 0.–~21.31!VBO ~eV! 20.94 ¯

ac ~eV! 26.0 23.4–~221!av ~eV! 20.6 20.4–~27.1!b ~eV! 22.0 21.0–~22.0!d ~eV! 25.0 24.2–~25.0!c11 ~GPa! 1011 ¯

c12 ~GPa! 561 ¯

c44 ~GPa! 456 ¯


Seds-ae

yesi

wV-

an

wd

ob

ra

e

ee

eurheelcarece

ve

d

pe

m

teica-

ntallys

sti-

plit-nn

of

n-ss

-usSb,

ol-

n-s it


and InP nor as narrow as in InAs and InSb. Since Gaforms an increasingly important component of mid-infraroptoelectronic devices,155 its various alloys have been invetigated in some detail. Duttaet al. have recently publishedcomprehensive review of the material and structural propties of GaSb.231

Photoluminescence measurements on high-purity lagrown by liquid-phase epitaxy yielded a free exciton trantion energy of 0.810 eV at 16 K.232 Correcting for the exci-ton binding energy and extrapolating the temperature,obtainEg(T50)50.812 eV, which agrees to within 1 mewith earlier determinations1 and also with the recent transmission measurements of Ghezziet al.233 Varshni param-eters for the direct gap have been given by CaseyPanish,2 Wu and Chen (0,T,215 K),232 Ghezziet al.,233

Bellani et al.,234 and Joullieet al.235 ~numerical values usedin that work are given in Ref. 231!. The four results that fitthe data to room temperature are in excellent agreementeach other, and recommended values have been obtainesimply averaging the Varshni parametersa andb.

At low temperatures, theL valley in GaSb is only0.063–0.100 meV higher than theG valley.1,11,231,235,236,237

The values near the bottom of that range, which weretained by electroreflectance235 and modulationspectroscopy,236 appear to be the most reliable. The tempeture dependence of theL-valley minimum is known to bestronger than that of theG valley.11,236The position and tem-perature dependence of theX-valley minimum were deter-mined by Lee and Woolley,238 although there was somspread in the earlier reports.1 The spin-orbit splitting wasmeasured by a number of techniques,1 with a value of 0.76eV236 being commonly accepted.

TheG-valley band-edge electron mass in GaSb has bstudied by a variety of techniques,175,233,239–245which pro-duced low-temperature values in the rather narrow rang0.039– 0.042m0 . The smallness of the spread is rather sprising in consideration of the indirect nature of many of tdeterminations, which are complicated by the relativstrong conduction-band nonparabolicity in GaSb. Onagain, care must be taken to separate results for the poland bare effective masses, although as is often the caspolaron correction is no greater than the experimental untainty. Averaging produces a bare effective mass ofme*50.039m0 at 0 K. Somewhat higher values for the effectimass have been calculated theoretically.100–106

Owing to the smallG –L energy separation in GaSb anthe much lower density of states at theG minimum, at roomtemperature a significant fraction of the electrons occuL-valley states. Effective masses for those states were msured by cyclotron resonance,11,244,245Faraday rotation,1 andpiezoresistance.1 On the basis of these results, we form coposite values ofmt* 50.10m0 andml* 51.3m0 . A large non-parabolicity at theL point has also been reported.245 Effec-tive masses for theX valley are taken from Levinshteinet al.11

The Luttinger parameters for GaSb have been demined using a number of experimental and theoretapproaches.106,239,240,242,243,246,247On the basis of these results, we suggest the composite valuesg1513.4, g254.7,


b

r-

rs-

e

d

ithby

-

-

n

of-

yeonther-

ya-

-

r-l

and g356.0, which are weighted toward the more receexperiments, but are also quite close to the ones typicused in the literature.248 The split-off hole mass in GaSb wameasured by Reineet al.239

The interband matrix element for GaSb has been emated by several workers.116,239–243Here the band structureis more sensitive to the adopted value ofEP than in mostmaterials, because the energy gap is nearly equal to the soff gap. Therefore, instead of using the result of Hermaand Weisbuch,116 we determine a composite valueEP

527.0 eV, which is within the experimental uncertaintynearly all the reports. This composite impliesF521.63,which is quite close to the results of Reineet al.239 and Rothand Fortin.243

An average value for the direct-gap deformation potetial a528.3 eV was determined for GaSb by uniaxial streand transmission experiments.1 We adopt the Van deWalle129 suggestion for the valence-band potential ofav520.8 eV, and adjust hisac slightly to produce the consistent value ofa. There is good agreement between variomeasurements of the shear deformation potentials in Gaas summarized in Landolt–Bornstein.1

All of the band structure parameters for GaSb are clected in Table VII.

H. AlSb

AlSb is an indirect-gap semiconductor with a lattice costant only slightly larger than that of GaSb. In recent year

TABLE VII. Band structure parameters for GaSb.


alc ~Å! 6.095914.7231025(T2300) ¯

EgG ~eV! 0.812 0.811–0.813

a~G! ~meV/K! 0.417 0.108–0.453b~G! ~K! 140 210–186Eg

X ~eV! 1.141 1.12–1.242a(X) ~meV/K! 0.475 ¯

b(X) ~K! 94 ¯

EgL ~eV! 0.875 0.871–0.92

a(L) ~meV/K! 0.597 ¯

b(L) ~K! 140 ¯

Dso ~eV! 0.76 0.749–0.82me* (G) 0.039 0.039–0.042ml* (L) 1.3 1.1–1.4mt* (L) 0.10 0.085–0.14ml* (X) 1.51 ¯

mt* (X) 0.22 ¯

g1 13.4 11–14.5g2 4.7 3–5.3g3 6.0 4.4–6.6mso* 0.12 0.12–0.14EP ~eV! 27.0 ¯

F 21.63 ¯

VBO ~eV! 20.03 ¯

ac ~eV! 27.5 ¯

av ~eV! 20.8 ¯

b ~eV! 22.0 ¯

d ~eV! 24.7 ¯

c11 ~GPa! 884.2 ¯

c12 ~GPa! 402.6 ¯

c44 ~GPa! 432.2 ¯


ig

d

..3r

i

e

ise

i-

g

egyeficarron

m

7

tiee

ol

sta

thSb

dla-f

o

.82

nd-ge

-a-ail-e

ractady

h-ther

esti-ofrt:

odedolec-

-

.2y


has found considerable use as the barrier material in hmobility electronic154 and long-wavelength optoelectronic155

devices. The direct gap in AlSb was measured using molation spectroscopy by Alibertet al.,236 spectroscopic ellip-sometry by Zollneret al.,249 and also using other methods1

Bulk AlSb samples were found to exhibit a gap of 2.35–2eV at liquid-helium temperature, and aT dependence similato that in GaSb. The conduction-band minima orderingbelieved to be the same as in GaP and AlAs:X–L –G, withthe L valley only 60–90 meV above theG valley236 ~alsoquite similar to GaSb!. The indirect gap associated with thlowestX valley was investigated by Sirota and Lukomskii,250

Mathieuet al.,251 and Alibertet al.236 It has been suggested1

that early estimates of the band gap needed to be revsince the exciton binding energy is 19 meV252 instead of theassumed value of 10 meV.250 We employ the resulting low-temperature value given in Landolt–Bornstein,1 along withcomposite Varshni parameters. The spin-orbit splittingtaken from Alibertet al.,236 which falls near the data compiled in Landolt–Bornstein.1

The electron mass in theG valley was measured usinhot-electron luminescence.253 While the L-valley andX-valley effective masses have been calculated,1,254 there ap-pear to be no definitive measurements. It is likely that thXvalley in AlSb exhibits a camel’s back structure by analowith AlAs and GaP. Little information is available on thhole masses in AlSb.253,255We form our composite values othe Luttinger parameters on the basis of theoretstudies101,106,182,254in conjunction with literature values fothe hole masses. The interband matrix element is taken fLawaetz,106 and the split-off mass is taken to be consistewith Eq. ~2.18!, although it could be argued that both paraeters have a high degree of uncertainty.

The deformation potentials in AlSb were measured atK using a wavelength modulation technique.256 Once again,we take the valence-band hydrostatic deformation potencalculated by Van de Walle,129 and make the rest of thresults consistent with what appears to be the sole expmental determination.

All of the band structure parameters for AlSb are clected in Table VIII.

I. InSb

InSb is the III–V binary semiconductor with the smalleband gap. For many years it has been a touchstone for bstructure computational methods,257 partly because of thestrong band mixing and nonparabolicity that result fromsmall gap. The primary technological importance of Inarises from mid-infrared optoelectronics applications.258

Numerous studies of the fundamental energy gap antemperature dependence have been conducted over thedecades.1,2,11,160,161,164,243,259–264While there is a broad consensus thatEg(T50)50.235 eV, several different sets oVarshni parameters have been proposed.2,164,259,264,265Aver-aging parameters from the most reliable references, wetain the composite set:a50.32 meV/K andb5170 K. TheL- andX-valley energies are taken from Adachi.166 The spin-


h-

u-

9

s

d,

s

l

mt-

7

al

ri-

-

nd

e

itsst 3

b-

orbit splitting energy was measured to be 0.81–0eV.183,260

Experimental and theoretical studies have found baedge electron masses for InSb in the ran0.012– 0.015m0 .95,106,168,171,260,262,265–275A bare effectivemass of 0.0135m0 at 0 K is in good agreement with a majority of the investigations. On the other hand, little informtion on the effective masses in the indirect valleys is avable. Levinshteinet al. quotes a density-of-states effectivmass of 0.25m0 for theL valley.11 We have found no explicittheoretical or experimental results for theX-valley effectivemasses, although in principle it should be possible to extthem from pseudopotential calculations that have alrebeen performed for bulk InSb.

A wide variety of experimental and theoretical tecniques such as magnetophonon resonance and omagneto-optical approaches have been employed to invgate the valence-band structureInSb.106,168,243,265,266,269,276–282By averaging the values fog1 , mhh* (001), andmhh* (111), we deduce the composite seg1534.8,g2515.5,g3516.5. These parameters are in goagreement with the majority of the values found in the citreferences. Owing to the narrow energy gap, the light-heffective mass in InSb is only slightly larger than the eletron mass.168,260,265,267,282,283The split-off hole mass is estimated to be 0.10– 0.11m0 .106,260

Our composite interband matrix element for InSb is 23eV.106,118,263,269,284A slightly higher value was deduced b

TABLE VIII. Band structure parameters for AlSb.


alc ~Å! 6.135512.6031025(T2300) ¯

EgG ~eV! 2.386 2.35–2.39

a~G! ~meV/K! 0.42 ¯

b~G! ~K! 140 ¯

EgX ~eV! 1.696 1.68–1.70

a(X) ~meV/K! 0.39 ¯

b(X) ~K! 140 ¯

EgL ~eV! 2.329 2.327–2.329

a(L) ~meV/K! 0.58 ¯

b(L) ~K! 140 ¯

Dso ~eV! 0.676 ¯

me* (G) 0.14 0.09–0.18ml* (L) 1.64 ¯

mt* (L) 0.23 ¯

ml* (X) 1.357 ¯

mt* (X) 0.123 ¯

g1 5.18 4.15–5.89g2 1.19 1.01–1.29g3 1.97 1.75–2.25mso* 0.22 ¯

EP ~eV! 18.7 ¯

F 20.56 ¯

VBO ~eV! 20.41 ¯

ac ~eV! 24.5 ¯

av ~eV! 21.4 ¯

b ~eV! 21.35 ¯

d ~eV! 24.3 ¯

c11 ~GPa! 876.9 ¯

c12 ~GPa! 434.1 ¯

c44 ~GPa! 407.6 ¯


te

b

luofisrerm

fo

taraN

de

hainhei

o

anr

-hichnd

he

s isyo-ced

cedra-

her-

s,-are

irectlyby

fortud-

oduent

n-b-

g

giestate

o-

was

ialstud-orp-wn

-

l-

anicnce


Hermann and Weisbuch,116 possibly due to an overestimaof the effectiveg factor. Somewhat lower values forEP arealso encountered in the literature.243,260Our correspondingFparameter~20.23! is close to other determinations.243,284

The deformation potentials in InSb have been studiedvarious optical and electrical techniques.1,285,286For the totalhydrostatic deformation potential, we take an average vaof 27.3 eV. According to the model-solid calculationsVan de Walle,129 the valence-band deformation potentialrather small by analogy with the other III–V materials. Theappears to be a consensus on values for the shear defotion potentials in InSb:b522.0 andd524.7.1,285–287

All of the recommended band structure parametersInSb are given in Table IX.

J. GaN

GaN is a wide-gap semiconductor that usually cryslizes in the wurtzite lattice~also known as hexagonal oa-GaN!. However, under certain conditions zinc blende G~sometimes referred to as cubic orb-GaN! can also be grownon zinc blende substrates under certain conditions. Unvery high pressure, GaN and other nitrides experiencphase transition to the rocksalt lattice structure.288 If the crys-tal structure of a nitride semiconductor is not stated in wfollows, the wurtzite phase is implied, whereas the zblende phase is always explicitly specified. A review of tphysical properties of GaN and other group-III nitride semconductors up to 1994 was edited by Edgar.289 The status ofGaN work in the 1970’s was summarized in twreviews290,291 as well as in Landolt-Bornstein.1 For a com-prehensive recent review of the growth, characterization,various properties of nitride materials, we refer the readethe article by Jainet al.292

TABLE IX. Band structures parameters for InSb.


alc ~Å! 6.479413.4831025(T2300) ¯

EgG ~eV! 0.235 ¯

a~G! ~meV/K! 0.32 0.299–0.6b~G! ~K! 170 106–500Eg

X ~eV! 0.63 ¯

EgL ~eV! 0.93 ¯

Dso ~eV! 0.81 0.8–0.9me* (G) 0.0135 0.012–0.015mDOS* (L) 0.25 ¯

g1 34.8 32.4–38.5g2 15.5 13.4–18.1g3 16.5 15.15–18mso* 0.11 ¯

EP ~eV! 23.3 ¯

F 20.23 ¯

VBO ~eV! 0 ¯

ac ~eV! 26.94 ¯

av ~eV! 20.36 ¯

b ~eV! 22.0 ¯

d ~eV! 24.7 ¯

c11 ~GPa! 684.7 ¯

c12 ~GPa! 373.5 ¯

c44 ~GPa! 311.1 ¯


y

e

a-

r

l-

era

tc

-

dto

1. Wurtzite GaN

Unlike any of the non-nitride wide-gap III–V semiconductors discussed above, GaN is a direct-gap material, whas led to its successful application in blue lasers aLEDs.293 It has been known since the early 1970’s that tenergy gap in wurtzite GaN is about 3.5 eV.294,295However,a precise determination from luminescence experimentnot straightforward, since what is usually measured at crgenic temperatures are the energies for various pronounexciton transitions. The identification of these closely sparesonances is nontrivial. For example, at very low tempetures, the lowest-orderA, B, andC exciton types related tothe three valence bands can be resolved, as well as higorderA(2s,2p) exciton transitions.296 The situation is furthercomplicated by the excitons bound to various impuritiesuch as neutral donors.290 All of these considerations contribute to the rather large experimental uncertainty in the bdirect energy gap at low temperatures.

Early measurements of the temperature-dependent dgap in wurtzite GaN, which are still the most frequentquoted in the nitride device literature, were performedMonemar.297,298Those experiments yielded a free-A-excitontransition energy of 3.475 eV and an estimate of 28 meVthe binding energy. Numerous other PL and absorption sies were published in the 1990s,296,299–308which broadenedthe range of reportedA-exciton transition energies at 0 K t3.474–3.507 eV. It has been suggested that this spread isto variations of the strain conditions present in the differeexperiments.309 If we average all of the available experimetal values, an exciton transition energy of 3.484 eV is otained. Experimental binding energies for theA excitonrange from 18 to 28 meV.296,298,302,303,306,310–312An accuratetheoretical determination313 is out of reach at present owinto the large uncertainty in the reduced mass~primarily asso-ciated with the poorly known hole effective mass!. We there-fore average the most reliable experimental binding enerdeduced from the difference between the ground-sand first-excited-state energies of the Aexciton.296,302,303,306,308,310,311This gives 23 meV for the ex-citon binding energy, which implies 3.507 eV for the zertemperature energy gap.

The temperature dependence of the GaN energy gapfirst reported by Monemar,298 who obtained a520.508 meV/K andb52996 K. While the signs of hisVarshni coefficients are opposite to all of the other materconsidered in this review, a large number of subsequent sies have derived less anomalous results. From optical abstion measurements on bulk and epitaxial layers groon sapphire, Teisseyre et al.314 obtained a50.939– 1.08 meV/K andb5745– 772 K. For the temperature variation of theA exciton resonance, Shanet al. reporteda50.832 meV/K andb5836 K.299 Petalaset al.315 fixed b5700 K and founda50.858 meV/K using spectroscopic elipsometry. Salvadoret al.316 obtaineda50.732 meV/K andb5700 K based on PL results. Manasreh304 reported a50.566– 1.156 meV/K andb5738– 1187 K from absorptionmeasurements on samples grown by MBE and metalorgchemical vapor deposition. The contactless electroreflectastudy of Li et al.307 led to a51.28 meV/K andb51190 K


cilcameerurelo

e

ga

t o

aeb

ncfi-th

lyriantio

il

-en

itl-

dles

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rob

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ted

so

ll-opy

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o-c-ethe

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for theA exciton transition energy. Finally, Zubrilovet al.317

suggesteda50.74 meV/K andb5600 K based on excitonluminescence spectra. It is not obvious how to reconthese diverse parameter sets, especially since in severalconsiderably different values are inferred even in the sastudy. Much of the difficulty stems from the fact that thresonances dominating the exciton spectra at low temptures are not readily distinguishable at ambient temperatOur recommended Varshni parameters represent a simplerage of the various reported values except the anomaresults of Monemar.298 This yieldsa50.909 meV/K andb5830 K, which are in good agreement with the parametsuggested by absorption measurements on AlGaN~a negli-gible composition dependence was reported!.318 It is fortu-nate that owing to the small relative change in the bandenergy ~only 72 meV between 0 and 300 K!, the precisechoice of Varshni parameters has only a modest impacthe device characteristics.

The indirect energy gaps in GaN are much larger ththe direct gap. Few studies have attempted to resolve thalthough various estimates of the critical points are availafrom theory and experiment.1,290,319–323Considering the hugeuncertainties in the indirect gaps and their lack of importato device applications, we do not recommended valuesthe wurtzite forms of the nitride materials. Zinc blende indrect gaps are specified in the tables, however, becauseare smaller and somewhat better known.

In contrast to most zinc blende materials, for which onthe spin-orbit splitting must be specified, in wurtzite mateals the crystal-field splitting is at least as important and cnot be ignored if one wishes to recover a realistic descripof the valence-band states~see Sec. II for details!.324 In thefollowing, we takeD25D35Dso/3 andD15Dcr . An earlystudy of Dingle et al. found Dcr522 meV and Dso

511 meV.294 A more recent and detailed analysis by Get al. yielded the valuesDcr510 meV andDso518 meV.301

Reynolds et al. obtained Dcr525 meV and Dso517 meVfrom a fit of their exciton data.311 Using a more precise description of the strain variation of the valence band-edgeergies, Chuang and Chang reanalyzed the Gilet al. data andderived Dcr516 meV andDso512 meV.325 From a similarapproach, Shikanaiet al. obtained Dcr522 meV and Dso

515 meV from their own data.Ab initio theoreticalcalculations326 support a rather small value for the spin-orbsplitting ~<10 meV!, but tend to overestimate the crystafield splitting. To obtain our recommended set ofDcr

519 meV andDso514 meV, we averaged all of the reportevalues with the exception of those from the first-principtheory.

The bottom of the conduction band in GaN is well aproximated by a parabolic dispersion relation, althoughslight anisotropy~resulting from the reduced lattice symmtry! is not ruled out.327 In early studies, Barker andIllegems328 obtained an effective mass of 0.20m0 fromplasma reflection measurements, RheinlanderNeumann329 inferred 0.27m0 from the Faraday rotation, anSidorovet al.330 derived values of 0.1m0– 0.28m0 , depend-ing on what primary scattering channel was assumed, ffits to the thermoelectric power. Other early values may


esese

a-e.av-us

rs

p

n

nm,le

eor

ey

--n

-

a

d

me

found in the reviews from the 1970s.291,290However, a con-siderable body of recent work has led to more precise evations of the effective mass. Meyeret al.331 and Witowskiet al.332 used measurements of the shallow-donor transitenergies to obtain masses of 0.236m0 and 0.222m0 , respec-tively. The latter result has the smallest error bounds quoin the literature~0.2%!. Drechsleret al. pointed out the im-portance of the polaron correction in GaN since it isstrongly polar~10%!, and derived a bare mass of 0.20m0

from cyclotron resonance measurements.333 Perlin et al.334

obtained similar results using infrared-reflectivity and Haeffect measurements, and additionally found the anisotrto be less than 1%. A slightly higher dressed mass of 0.23m0

was recently obtained by Wanget al.335 and Knapet al. Theformer may require a slight downward revision becauseelectron gas was confined in a quantum well, whereaslatter report apparently corrected for that effect.336 No appre-ciable correction appears to be necessary for the measment by infrared ellipsometry on bulkn-doped GaN reportedby Kasicet al.,337 in which a marginally anisotropic electroeffective mass with the values of 0.23760.006m0 and0.22860.008m0 along the two axes was obtained. FinallElhamri et al.,338 Saxleret al.,339 Wong et al.,340 and Wanget al.341 determined masses ranging from 0.18m0 to 0.23m0

from Shubnikov–de Haas oscillations in the twdimensional~2D! electron gas at a GaN/AlGaN heterojuntion. It was suggested338 that strain effects may have to somextent compromised the masses obtained by some ofother studies. Our recommendation of 0.20m0 for the baremass is close to the consensus from the investigationbulk materials, and to the average from the studies ofelectrons.

The experimental information presently available is sficient only to suggest an approximate band-edge effecmass for the holes. Factors contributing to this uncertaiinclude strong nonparabolicity near the valence-band eand the close proximity of heavy-, light-, and crystal-hobands@see the schematic diagram in Fig. 4~b!#.325 In order toderive the various parameters needed to characterize thelence band of a wurtzite material, one must resort to theoical projections.327,342 In the cubic approximation, these parameters may also be recast in terms of the Luttinparameters familiar from the case of the zinc blenmaterials.324 While early work suggested a GaN hole effetive mass of 0.8m0 .290,343,344consideration of the acceptobinding energies led Orton345 to suggest a much smallevalue of 0.4m0 . An even smaller mass of 0.3m0 was ob-tained by Salvadoret al. from a fit to the PL results.316 Onthe other hand, the excess-carrier lifetime measurementIm et al. indicated a very heavy hole mass of 2.2m0 .305 Merzet al. obtained an isotropically averaged heavy-hole mass0.54m0 from luminescence data.303 Fits of the exciton bind-ing energies yielded hole masses in the ran0.9– 1.2m0 .308,346Finally, an infrared ellipsometric study bKasic et al. yielded a hole mass of 1.4m0 for p-dopedGaN.337 It should be pointed out that most of these expemental values are somewhat lower than the theoretmasses derived by Suzukiet al.,327 which are commonlyused in band structure calculations, and are much lower t


c

lt

ud

oth

lereta

ahaa

thuranth

ic

dorcts

pn

rod

-

isithlt

aNf

b

t tore-

nts

them.

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n-ki

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d

al

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the pseudopotential results of Yeoet al.323 On the otherhand, first-principles calculations by Chenet al. produced asmaller density-of-states effective mass of 0.6m0 .302 An al-ternative set of effective-mass parameters was recentlyculated from a pseudopotential model by Dugdaleet al.347

Their electron mass is lower than the experimental resuwhich may indicate lower accuracy of theA parameters. Yetanother parameter set was extracted from empirical psepotential calculations by Renet al.38 That work succeeded intheoretically extracting the inversion parameterA7

593.7 meV/Å from a comparison with empirical pseudoptential calculations. Our recommendation is to useeffective-mass parameters of Suzukiet al.,327 which implyan average hole mass approximately equal to the free etron value (m0). It is hoped that a theoretical band structupicture that is fully consistent with the recent experimenresults will be developed in the near future.

In principle, there are two independent momentum mtrix elements in wurtzite GaN. However, the assumption tthe electron mass anisotropy is small implies that thesenearly equal. Insofar as no information is available oneffects of remote bands on the wurtzite GaN band structthat interaction is neglected. If one then derives an interbmatrix element directly from the electron effective mass,result isEP514 eV(F50). While there is one report305 ofEP57.7 eV in wurtzite GaN, that would imply a unrealistpositiveF parameter.

Six distinct valence-band deformation potentials, in adition to the strain tensor and the overall hydrostatic defmation potential, are necessary to describe the band struof GaN under strain. Using the cubic approximation, thecan be re-expressed in terms of the more familiarav , b, andd potentials.324 Christensen and Gorczyca319 reported a hy-drostatic deformation potentiala527.8 eV, which is ingood agreement with fits to the data of Gilet al. ~28.16eV!.301 A somewhat lowera526.9 eV was derived from anab initio calculation by Kimet al.348 Shanet al.349 noted thatthe hydrostatic deformation potential should be anisotrodue to the reduced symmetry of the wurtzite crystal, agave the values:a1526.5 eV anda25211.8 eV for the twocomponents. These are our recommended values. Numesets of valence-band deformation potentials have beenrived from both first-principles calculations325,342,350and fitsto experimental data.301,306,349 There are considerable discrepancies between the reported data, with variationsnearly a factor of 6 in some cases. Obviously, further workneeded to resolve this controversy. We form our composet of deformation potentials by selecting those valuesseem to be most representative of the majority of resuD1523.0 eV,350 D253.6 eV,350 D358.82 eV,306 D4

524.41 eV,306 D5524.0 eV,350 and D6525.1 eV ~de-rived by adopting the cubic approximation325!.

The determination of elastic constants for wurtzite Ghas been reviewed by Wright,351 who compared the results oa number of experiments352–355with two calculations.348,351

Overall, theory agrees best with the data of Polianet al.,353

who obtained the recommended values:C115390 GPa,C12

5145 GPa, C135106 GPa, C335398 GPa, and C44

5105 GPa. However, there are significant disagreements


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tween the various experimental results, so that in contrasthe zinc blende materials the elastic constants for GaNmain somewhat controversial.

Very few measurements of the piezoelectric coefficiein GaN have been reported. Guyet al.356,357 performed acareful study, which pointed out the differences betweencoefficients in a bulk material versus a strained thin filCoefficients ofd3353.7 pm/V andd13521.9 pm/V were de-duced for the bulk GaN from the single-crystal thin-filvalued3352.8 pm/V and the relationd1352d33/2. Anothermeasurement by Luenget al. yielded a thin-film valued33

52.13 pm/V.358 The latter measurement had the inherent ucertainty of an AlN buffer layer being present. Bykhovset al. attempted to derive thee31 and e33 coefficients fromthe e14 coefficient in zinc blende GaN, obtaining valuese31520.22 C/m2 and e3350.43 C/m2.359 Bernardini et al.employed a first-principles calculation to derivee31

520.49 C/m2 and e3350.73 C/m2.360 A calculation of Shi-mada et al. yielded values ofe31520.32 C/m2 and e33

50.63 C/m2.361 We recommend using thed coefficients fromthe experimental study of Guyet al. and use the assumeelastic constants to obtain thee coefficients: e31

520.35 C/m2 and e3351.27 C/m2, which turn out to besomewhat different from those calculated in the originreport.357

Only two first-principles calculations of the spontaneopolarization in GaN are available.360,362Very different valuesof Psp520.029 C/m2 andPsp520.074 C/m2 were reported.In one of the papers,362 it was noted that the computed spotaneous polarization is highly sensitive to the values ofinternal structure parameters such as the lengths ofatomic bonds. This consideration may prevent an accutheoretical evaluation of the spontaneous polarization foralistic nitride structures. Since onlydifferencesin the spon-taneous polarization are important in heterostructure bcalculations, we defer a full discussion of the experimenprobes of the spontaneous polarization in GaN/AlGaN qutum wells until the AlN section. The band structure paraeters for wurtzite GaN are compiled in Table X.

2. Zinc blende GaN

A number of theoretical and experimental studies ofenergy gap for the zinc blende phase of GaN have breported.315,363–371Some works rely on an explicit comparson with the better understood case of wurtzite GaN, wherthe most accurate appear to come from low-temperatureminescence measurements372–374 of the free-exciton peakwhich is estimated to be 26.5 meV below the energy gExperimentally, the low-temperature energy gaps range fr3.2 to 3.5 eV, although the most reliable values fall appromately midway, between 3.29 and 3.35 eV.365,367,368We rec-ommend a value of 3.299 eV obtained from averagingresults of the luminescence measurements. The temperdependence of the energy gap was studied in detailRamirez-Floreset al.368 and Petalaset al.315 Although thetwo studies obtain the sameb5600 K ~using the more reli-able model 1 in Ref. 315!, thea parameters are different, anwe recommend using an average value of 0.593 meV/K.though the indirect-gap energies have not been measure


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recent calculation of Fanet al.puts theX-valley andL-valleyminima at 1.19 and 2.26 eV above theG valley,respectively.369 Ramirez-Floreset al.368 have measured thspin-orbit splitting in zinc blende GaN to be 17 meV.

Electron spin resonance measurements indicated antron effective mass of 0.15m0 in zinc blende GaN.375 Sincethis appears to be the only experimental result, we adoptour recommendation. SimilarG-valley effective masses werderived from first-principles calculations by Chowet al.376

and Fanet al.369 Effective masses ofml* 50.5m0 and mt*50.3m0 were recently calculated for theX valley in GaN,370

which are similar to values obtained by Fanet al.369 Theconvergence of results from two different studies allowsto adopt these as our recommended values.

Although the hole effective masses in zinc blende Ghave not been measured, a number of theoretical sets oftinger parameters are available.369–378In order to derive ourrecommended values, we average the heavy-hole and lhole masses along@001# as well as the degree of anisotropg3–g2 . This results in the following parameter set:g1

52.67, g250.75, and g351.10. When all of thereported369–371,379split-off masses are averaged, we obtamso* 50.29m0 .

TABLE X. Recommended band structure parameters for wurtzite nitbinaries.

Parameters GaN AlN InN

alc ~Å! at T5300 K 3.189 3.112 3.545clc ~Å! at T5300 K 5.185 4.982 5.703Eg ~eV! 3.507 6.23 1.994a ~meV/K! 0.909 1.799 0.245b ~K! 830 1462 624Dcr ~eV! 0.019 20.164 0.041Dso ~eV! 0.014 0.019 0.001me

i 0.20 0.28 0.12me

' 0.20 0.32 0.12A1 26.56 23.95 28.21A2 20.91 20.27 20.68A3 5.65 3.68 7.57A4 22.83 21.84 25.23A5 23.13 21.95 25.11A6 24.86 22.91 25.96EP ~eV! 14.0 14.5 14.6F 0 0 0VBO ~eV! 22.64 23.44 21.59a1 ~eV! 26.5 29.0 23.5a2 ~eV! 211.8 29.0 23.5D1 ~eV! 23.0 23.0 23.0D2 ~eV! 3.6 3.6 3.6D3 ~eV! 8.82 9.6 8.82D4 ~eV! 24.41 24.8 24.41D5 ~eV! 24.0 24.0 24.0D6 ~eV! 25.1 25.1 25.1c11 ~GPa! 390 396 223c12 ~GPa! 145 137 115c13 ~GPa! 106 108 92c33 ~GPa! 398 373 224c44 ~GPa! 105 116 48e13 ~C/m2! 20.35 20.50 20.57e33 ~C/m2! 1.27 1.79 0.97Psp ~C/m2! 20.029 20.081 20.032


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Two theoretical values forEP in zinc blende GaN havebeen reported in the literature.371,379An average of the twoyields EP525.0 eV, which in turn impliesF520.92. Anote of caution is that these values have not been veriexperimentally.

Various calculations put the hydrostatic deformation ptential for zinc blende GaN in the range between26.4 and28.5 eV.42,319,369,370,376,380We choose an average valuea527.4 eV. The same procedure is followed in obtainithe recommended values ofav525.2 eV ~20.69 to213.6eV range! and b522.2 eV ~21.6 to 23.6 eV range!. Thevalue of d523.4 eV is an average between the only pulished values from Ohtoshiet al.380 and Van de Walle andNeugebauer.381 No experimental confirmations of any othese deformation potentials for zinc blende GaN appeaexist. Elastic constants ofC115293 GPa,C125159 GPa, andC445155 GPa are taken from the theoretical analysisWright.351 Very similar sets were calculated by Kimet al.382

and Bechstedtet al.362 The band structure parameters fzinc blende GaN are compiled in Table XI.

K. AlN

Although binary AlN is rarely used in practical deviceit represents the end point for the technologically importAlGaN alloy. As in the case of GaN, both wurtzite and ziblende forms of AlN can in principle be grown, although thgrowth of zinc blende AlN has not been reported. WurtzAlN has the distinction of being the only Al-containinIII–V semiconductor compound with a direct energy ga

eTABLE XI. Recommended band structure parameters for zinc blendetride binaries.

Parameters GaN AlN InN

alc ~Å! at T5300 K 4.50 4.38 4.98Eg

G ~eV! 3.299 4.9 1.94a~G! ~meV/K! 0.593 0.593 0.245b~G! ~K! 600 600 624Eg

X ~eV! 4.52 6.0 2.51a(X) ~K! 0.593 0.593 0.245b(X) ~meV/K! 600 600 624Eg

L ~eV! 5.59 9.3 5.82a(L) ~K! 0.593 0.593 0.245b(L) ~meV/K! 600 600 624Dso ~eV! 0.017 0.019 0.006me* (G) 0.15 0.25 0.12ml* (X) 0.5 0.53 0.48mt* (X) 0.3 0.31 0.27g1 2.67 1.92 3.72g2 0.75 0.47 1.26g3 1.10 0.85 1.63mso* 0.29 0.47 0.3EP ~eV! 25.0 27.1 25.0F 20.92 0.76 20.92VBO ~eV! 22.64 23.44 22.38ac ~eV! 22.2 26.0 21.85av ~eV! 25.2 23.4 21.5b ~eV! 22.2 21.9 21.2d ~eV! 23.4 210 29.3c11 ~GPa! 293 304 187c12 ~GPa! 159 160 125c44 ~GPa! 155 193 86


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Furthermore, it is the largest-gap material that is still comonly considered to be a semiconductor. The absorpmeasurements of Yimet al.383 and Perry and Rutz384 indicatethat the energy gap in wurtzite AlN varies from 6.28 eV aK to 6.2 eV at room temperature. Varshni parameters oa51.799 meV/K andb51462 K were reported by Guo anYoshida, who also found the low-temperature gap to be 6eV.385 A similar energy gap was reported by Visputeet al.386

With the aid of cathodoluminescence experiments, Taet al.387 resolved at 300 K what they believed to be the fror shallow-impurity-bound exciton, at an energy of 6.11 eWe recommend an intermediate value of 6.23 eV forlow-temperature band gap, in conjunction with the Varsparameters of Guo and Yoshida.385 Although Brunneret al.318 also reported Varshni parameters, the finding ofsignificant differences from GaN for the entire compositirange of the AlGaN alloy may indicate that their results asomewhat less reliable.

The crystal-field splitting in AlN is believed to be negtive, which implies that the topmost valence band is cryshole-like. Suzuki et al.327 calculated Dcr5258 meV,whereas Wei and Zunger326 obtainedDcr52217 meV. Pughet al.371 cited values of2104 and2169 meV from first-principles and semiempirical pseudopotential calculatiorespectively, and Kimet al.388 obtained Dcr52215 meV.Averaging all of the available theoretical crystal-field spltings, we obtain our recommended value ofDcr52164 meV. Spin-orbit splittings ranging from 11371 to 20meV327 have been cited in the literature. We adopt the vaof 19 meV suggested by Wei and Zunger.326 Again, it isimportant to emphasize that our recommendations forcrystal-field and spin-orbit splittings in AlN have only provisional status, since it appears that no experimental dexist.

A number of calculations are available for the electreffective mass in AlN.327,371,388A greater anisotropy than inwurtzite GaN is predicted.327 The recommended values ome

'50.28m0 and mei50.32m0 were obtained by averagin

all available theoretical masses, although it is again nothat experimental studies are needed to verify these calctions. We recommend the valence-band effective massrameters of Suzukiet al.327 An alternative set ofA param-eters was recently published by Dugdaleet al.347 Theapparent disagreement in signs in various papers forA5 andA6 is ignored, since only absolute values of these parameenter the Hamiltonian.371,388

The hydrostatic deformation potential for wurtzite Alis believed to lie in the range between27.1 and 29.5eV.319,348 We select a median value of29.0 eV, which isconsistent with the observation that the band gap prescoefficients in AlGaN alloys have little dependencecomposition.389 Theoretical values are also available forfew of the valence-band deformation potentials~D3

59.6 eV,D4524.8 eV!.348 The elastic constants in wurtzitAlN were measured by Tsubouchiet al.390 and McNeilet al.391 We recommend the valuesC115396 GPa, C12

5137 GPa, C135108 GPa, C335373 GPa, and C44

5116 GPa suggested by Wright, who provides a detadiscussion of their expected accuracy.351


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Several early measurements392,393 of the piezoelectriccoefficient in AlN were compiled in Ref. 357. That referenobtainedd3355.6 pm/V andd13522.8 pm/V, which wererather similar to the previous determinations. The availacalculations360,361,394are in reasonable agreement with texperimental values. Two rigorous calculations360,362 of thespontaneous polarization in AlN have been performed wthe reported results of Psp520.081 C/m2 and Psp

520.12 C/m2. The effect of the spontaneous polarizationthe optical properties of GaN/AlGaN quantum wells was oserved by Lerouxet al.395,396However, it was found that theresults were consistent with a lower value for the spontaous polarization in AlN(20.051,Psp,20.036 C/m2). Astudy of the charging of GaN/AlGaN field-effect transistoled to similar conclusions.397 Hogg et al. were able to fittheir luminescence data by assuming negligible spontanepolarization.398 Park and Chuang399 required Psp

520.040 C/m2 to reproduce their GaN/AlGaN quantumwell data. On the other hand, Cingolaniet al. reported goodagreement with experiment using the original Bernardet al.360 calculation.400 The magnitude of the estimated spotaneous polarization is dependent on the assumed piezotric coefficients. It may be possible to explain the remainidiscrepancy between the majority of the experimental invtigations and the Bernardiniet al. calculation if a linear in-terpolation of Psp is invalid for the AlGaN alloy, due toeither bowing or long-range ordering. At this juncture, wrecommend the calculated value and note that the conversy will likely be resolved in future work. The recommended band structure parameters for wurtzite AlN are copiled in Table X.

Zinc blende AlN is projected to be an indirect-band gmaterial, withX-, G-, andL-valley gaps of 4.9, 6.0, and 9.eV, respectively.319,369,371The spin-orbit splitting is believedto be the same as in wurtzite AlN~19 meV!.326 Averagingthe theoretical results from different sources,369,371,378,379,388

we obtain the recommendedG-valley effective mass of0.25m0 . The longitudinal and transverse masses for theXvalley are predicted to be 0.53m0 and 0.31m0 ,respectively.369 The same procedure employed for Gayields the recommended Luttingeparameters:369,371,378,379,388 g151.92, g250.47, and g3

50.85 (mso50.47m0). The momentum matrix element itaken to be an average of the reported values:371,379 EP

527.1 eV (F50.76). Hydrostatic deformation potentials o29.0 eV319 and29.8 eV369 have been reported. Our recommended values for the deformation potentials area529.4 eV, av523.4 eV,42,369 b521.9 eV,369,381 and d5210.0 eV.348,381 The elastic constants ofC115304 GPa,C125160 GPa, andC445193 GPa are adopted from the caculations of Wright.351 Similar sets were quoted in other theoretical works.362,382,401 The recommended band structuparameters for zinc blende AlN are compiled in Table XI

L. InN

Although InN is rarely if ever used in devices in itbinary form, it forms an alloy with GaN that is at the corethe blue diode laser.293 Especially since some degree of se


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regation is believed to occur when that alloy is grown, itimportant to understand the properties of bulk InN inwurtzite phase. Osamuraet al.402 measured the energy gap78 K to be 2.0 eV~although for the purposes of the quadrafit for GaInN, a different gap was stated in the abstractthat article!, which became approximately 60 meV lowerroom temperature. The absorption measurements of Pchevrier and Menoret403 on polycrystalline InN indicatedgaps of 2.21 and 2.09 eV at 77 and 300 K, respectivAnother set of absorption measurements by Tyagaiet al.404

yielded Eg52.05 eV at 300 K. The result of Tansley anFoley,405 Eg51.89 eV at 300 K for high-purity InN thinfilms, is often quoted in the literature. It is judged moreliable than earlier experiments performed on samples whigh electron densities. An even lower gap was obtainedWestra and Brett,406 although in their case a high electrodensity was also present. Varshni parameters ofa50.245 meV/K andb5624 K were reported by Guo anYoshida385 for wurtzite InN, along with low-temperature anroom-temperature gaps of 1.994 and 1.97 eV, respectivThese values, which closely resemble a previous result fthe same group,407 represent our recommended temperatdependence. The recommended crystal-field and spin-osplittings of 41 and 1 meV, respectively, are taken fromcalculation of Wei and Zunger.326

There appear to be only two measurements of the etron mass in InN, which found values of 0.11m0

404 and0.12m0 .408 We recommend the latter, since it closematches the theoretical projection.321 Valence-bandeffective-mass parameters were calculated by Yeoet al.323

using the empirical pseudopotential method, by Puet al.371 and Dugdaleet al.347 using essentially the samtechniques. The results of the first two studies are quite slar, and we recommend the parameters derived by Pet al.371

Christensen and Gorczyca predicted a hydrostatic demation potential of24.1 eV for wurtzite InN,319 although asmaller value of22.8 eV was calculated by Kimet al.348 Werecommend the average ofa523.5 eV. Since apparentlythere have been no calculations of the valence-band demation potentials, we recommend appropriating thespecified above for GaN. While elastic constants were msured by Sheleg and Savastenko,352 we recommend the improved set of Wright:351 C115223 GPa,C125115 GPa,C13

592 GPa,C335224 GPa, andC44548 GPa. The piezoelectric coefficients and spontaneous polarization for InNtaken from the calculation by Bernardiniet al.360 The recom-mended band structure parameters for wurtzite InN are cpiled in Table X.

Although the growth of zinc blende InN has beereported,409 only theoretical estimates are available for aof its band parameters. It is predicted to be a direct-gapterial, with G-, X-, andL-valley gaps of 1.94, 2.51, and 5.8eV, respectively.370 The spin-orbit splitting is projected to b6 meV.326 We recommend an electron effective mass idecal to that in wurtzite InN, 0.12m0 , which is in the middle ofthe range 0.10– 0.14m0 that has been calculated.370,371,379

The longitudinal and transverse masses for theX valley arepredicted to be 0.48m0 and 0.27m0 , respectively.370 The rec-


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ommended Luttinger parameter set:g153.72,g251.26, andg351.63 is from the results of Pughet al.,371 and the split-off mass is chosen to bemso* 50.3m0 .370,371 For the hydro-static deformation potential, an average of23.35 eV fromthe theoretical319,348,370range of22.2 to24.85 eV is recom-mended. Valence-band deformation potentials are taken fa combination of the calculations of Wei and Zunger,42 Kimet al.,370 and Van de Walle and Neugebauer:381 av521.5 eV,b521.2 eV, andd529.3 eV. Elastic constantsof C115187 GPa, C125125 GPa, andC44586 GPa havebeen adopted from the calculations of Wright.351 Similar setswere derived from other calculations.362,382 The recom-mended band structure parameters for zinc blende InNcompiled in Table XI.

IV. TERNARY ALLOYS

For all of the ternary alloys discussed below, the depdence of the energy gap on alloy composition is assumefit a simple quadratic form:410

Eg~A12xBx!5~12x!Eg~A!1xEg~B!2x~12x!C,~4.1!

where the so-called bowing parameterC accounts for thedeviation from a linear interpolation~virtual-crystal approxi-mation! between the two binariesA andB. The bowing pa-rameter for III–V alloys is typically positive~i.e., the alloyband gap is smaller than the linear interpolation result! andcan in principle be a function of temperature. The physiorigin of the band gap bowing can be traced to disoreffects created by the presence of different catio~anions!.410 A rough proportionality to the lattice mismatcbetween the end-point binaries has also been noted.201

In what follows, the bowing concept has been generized to include quadratic terms in the alloy-compositionries expansions for several other band parameters aswhich in some cases may be attributable to specific physmechanisms but in others simply represent empirical fitsthe experimental data. We will employ the above functionform for all parameters and, with minor exceptions, neglhigher-order terms in the expansions. Since full seconsistency has been imposed upon all of the recommenparameter sets, we will give only bowing parameters foralloy properties, and note that the end points may be founthe tables corresponding to the relevant binaries.

We also point out that since theG-valley electron massme* can be obtained from Eq.~2.15! in conjunction with thespecified values forEg , EP , Dso, andF, it is not an inde-pendent quantity. In compiling the tables for binaries,have assured that the values given for the mass and theparameters are consistent with Eq.~2.15!. For alloys, thesuggested approach is to:~1! interpolate linearly theEP andF parameters,115 ~2! use the bowing parameter specified fthe alloy to deriveEg(T) andDso(T) from Eq.~4.1!, and~3!obtain the temperature-dependent electron mass in the afrom Eq. ~2.15!. While this procedure yields a complex dependence of the effective mass on composition, it assuself-consistency and in all of the cases that we are awareappears to be reliable. Simpler approximations are natur


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sometimes possible. In the tables, we attempt to give elecmass bowing parameters that are consistent with the abprocedure.

Linear interpolations are suggested for the electmasses in theX and L valleys, split-off hole mass, anheavy-hole and light-hole masses along the@001#direction.146 In order to estimate the valence-band warpinthe suggested procedure is to interpolate theg3–g2 differ-ence. Direct interpolation of the individual Luttinger parameters is not recommended. Lattice constants and elasmoduli may also be linearly interpolated.

A. Arsenides

1. AlGaAs

AlGaAs is the most important and the most studiIII–V semiconductor alloy. Its key role in a variety of transistor and optoelectronic devices has necessitated a prknowledge of the fundamental energy gap as well asalignment of the three main conduction-band valleys. Invtigations are complicated by the fact that whereas GaAsdirect-gap material withG –L –X valley ordering. AlAs is anindirect material with exactly the reverse ordering. Particuattention has been devoted to the crossover point, at wthe G andX valley minima have the same energies.

Bowing parameters from 0.14 to 0.66 eV have been pposed for theG-valley energy gap when Eq.~4.1! is used forall compositions of AlxGa12xAs.8,92,140,142,143,411Casey andPanish suggested a linearx dependence in the range 0,x,0.45 ~also supported by other data!412–415and a quadraticdependence when 0.45,x,1.2 Using spectroscopic ellipsometry to derive the positions of the critical points, Aspnet al. obtained a composition-dependent bowing parameC5(20.12711.310x) eV.416 A small bowing parametewas favored on theoretical grounds,410 although a more re-cent treatment by Magri and Zunger417,418 gave a complexfourth-order dependence reminiscent of the Aspneset al. re-sult. Although the bowing parameters that we recommenall other cases~with the exception of the direct gap in AlGaSb! are not a function of composition, in the present cait appears that much more accurate results can be obtaineusing the cubic form of Aspneset al. There is insufficientdata to determine the temperature dependence of the boparameter, since most of the determinations were perforeither at room temperature or at liquid-helium temperatuVarshni parameters for AlxGa12xAs have been obtainefrom photoluminescence, photoreflectance, and specscopic ellipsometry studies.90,138,419These are in reasonabagreement with our recommended assumption that the bing parameter is independent of temperature.

Bowing parameters for theX-valley andL-valley gaps inAlGaAs were determined using electrical measurementcombination with a theoretical model by Leeet al.140 andSaxena141 as well as empirically by Casey and Panis2

These results and photoluminescence excitation spectrosdata420 support aC(Eg

L) almost equal to zero. We seleC(Eg

X)50.055 eV obtained from photoluminescenmeasurements,134,139which is near the bottom of the earlierange of values but is the most recent and seems the


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reliable. This result implies aG –X crossover composition ox50.38 at low temperatures~and 0.39 at 300 K!, whichagrees with the trend in the table compiled by Adachi.3 Com-position dependences for all three of the direct and indirgaps in AlxGa12xAs are plotted in Fig. 5. Most studies finthat the split-off gap can be fit quite well by lineainterpolation.101,138,143A value of C(Dso)50.147 eV derivedby Aubelet al.142 cannot be considered fully reliable, sincewas based on data points in a rather narrow composirange.

Several studies of the composition dependence ofG-valley electron mass have been reportedx,0.33.146,147,421 The points have been fit to a quadratdependence,421 although owing to the narrow compositiorange and spread in the data points, it is difficult to judgeaccuracy of such a scheme. From other reports, it appthat the linear approximation gives adequate results for smx.3,104,140,146,147Since the effective mass in AlAs was chosto be consistent with the results in AlGaAs, a zero bowiparameter is recommended and gives good agreementthe interpolation procedures discussed at the beginningSec. V. The same procedures should be used to obtainX-valley andL-valley electron masses, Luttinger parameteand hole masses.

Qianget al. have derived a hydrostatic deformation ptential of a5210.6 to210.85 eV forx50.22.422 However,the same authors obtained a much smaller value oa528.6 eV for x50.27. The latter result is in good agrement with a linear interpolation between the GaAs and Alvalues~a528.3 eV, in this case!. A linear dependence othe shear deformation potentials on composition was sgested by the ellipsometry study of exciton splittings a

FIG. 5. G-, X-, andL-valley gaps for the AlGaAs alloy atT50 K ~solid,dotted, and dashed curves, respectively!.

TABLE XII. Nonzero bowing parameters for AlGaAs.

Parameters~eV! Recommended values Range

EgG 20.12711.310x 20.127–1.183

EgX 0.055 0.055–0.245

EgL 0 0–0.055

Dso 0 0–0.147


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ultsen-


shifts by Logothetidiset al.,153 although their best fit wasobtained for a slightly smaller value ofb in GaAs than werecommend in Table I. The relative paucity of deformatiopotential data for this important alloy is due in no small pto the very property that is one of its greatest attractionamely its excellent lattice match to GaAs that renders steffects on the band structure rather insignificant.

The recommended bowing parameters for AlGaAscollected in Table XII. In those cases where no valuelisted, linear variation should be assumed.

2. GaInAs

The GaInAs alloy is a key component in the activegions of high-speed electronic devices,423 infrared lasers,424

and long-wavelength quantum cascade lasers.425 It remains adirect-gap material over its entire composition range. Whbowing parameters spanning the wide range 0.32–0.6have been reported,101,166,201,410,426–434the most recent andseemingly most reliable values lie within the more restricvicinity of 0.45–0.5 eV. It has also been proposed thatbowing depends on temperature, being almost flat belowK and decreasing rapidly at higher temperatures.434 Owing tothe spread in values, we have fixed our recommenGa12xInxAs bowing parameter by emphasizing the fit at timportantx50.53 alloy that is lattice matched to InP. Thfundamental energy gap of Ga0.47In0.53As has been studiedextensively, e.g., by absorption,435,436magnetoabsorption,437

photoluminescence,430,438,439 and photoconductivityexperiments.440 On the basis of low-temperature results raning from 810 to 821 meV, we choose a composite averagEg(T50)5816 meV, which in turn implies a bowing parameterC50.477 eV. This is quite close to the recent detminations of Paulet al. ~0.475 eV!,431 Karachevtsevaet al.~0.486 eV!,434 Kim et al. ~0.479 eV!,433 and Jensenet al.441

This bowing parameter also agrees well with photorefltance measurements on GaAs-rich GaInAs by Hanget al.442

We choose to assume that the bowing is temperaindependent,431 in disagreement with Karachevtsevaet al.,434

because the room-temperature gap implied by thatC(T) ishigher than nearly all experimental values.438,439,443 Thecomposition dependence of the Varshni parameters obtain this manner is in good agreement with the functional foof Karachevtsevaet al.434

Bowing parameters for the indirect energy gaps wcalculated by Porod and Ferry428 using a modified virtualcrystal approximation. That study predicts large bowingrameters for bothEg

X(1.4 eV) andEgL(0.72 eV).166 Tiwari

and Frank432 give a much different set:C(EgX)50.08 eV and

C(EgL)50.5 eV. The only reliable experimental result a

pears to be a determination ofEgL in Ga0.47In0.53As.444 That

study supports a lower value for the bowing parameter. Texperimental and theoretical bowing parameters for the soff gap have been reported by Vishnubhatlaet al.,426 VanVechtenet al.,445 and Beroloet al.446 These are in reasonably good agreement with the electroreflectance measment of Pereaet al.443 for Ga0.47In0.53As, and implyC(Dso)50.15 eV.

The electron effective mass in GaInAs has been studboth theoretically and experimentally.171,172,446,447As in the


-ts,in

es

-

eV

de0

d

-of

-

-

re

ed

e

-

eit-

e-

d

case of the energy gap, the most reliable data areGa0.47In0.53As lattice matched to InP. While early studieproposed me* 50.041m0 .443,448–452 and even smallervalues,453 more recent experiments in strong magnetic fiehave suggested that the polaron effective mass is inhigher.440 That agrees with modeling of the diamagnetic shof the exciton absorption peaks in Ga0.47In0.53As/InP quan-tum wells.454,455 In recent years, evidence has accumulafavoring a low-temperature value of me*50.043m0 .447,456–458 This result implies the presence obowing if the electron mass is interpolated directly usingexpression similar to Eq.~4.1!. Application of the more gen-eral approach using Eq.~2.15! in conjunction with interpo-lated values for the interband matrix element and theF pa-rameter is discussed below. Since no reliable data onbowing parameters for theX- and L-valley electron masseappear to exist, we suggest linear interpolation.

For Ga0.47In0.53As, Alavi et al.437 suggested the set oLuttinger parameters:g1511.01,g254.18,g354.84. Thosevalues are in good agreement with the light-hole massesrived from spin-polarized photoluminescence measuremby Hermann and Pearsall,459 and with cyclotron resonancexperiments.460 On the other hand, Sugawaraet al.455 ob-tained a much larger light-hole mass, although with conserable spread in the results. We suggest that the bowingrameters for the hole effective masses should be consiswith the results of Alaviet al.437 The split-off hole massshould be interpolated linearly.

Most studies have employed an interband matrix ement ofEP525.3 eV for Ga0.47In0.53As437,448,453,459althoughZielinski et al.436 derived a much smaller value from aanalysis of absorption spectra. The former value is mumore consistent with the matrix elements employedGaAs and InAs~see above!, and implies only a smallEP

bowing parameter. The bowing ofF is then obtained usingthe already-derived relations for the energy gap and thefective mass.

The hydrostatic deformation potential in Ga0.47In0.53Aswas measured by Peopleet al.461 to bea527.79 eV. A re-duction in a was also observed by Wilkinsonet al.462 in astudy of GaInAs strained to a GaAs substrate. These resindicate some bowing in the hydrostatic deformation pot

TABLE XIII. Nonzero bowing parameters for GaInAs.


EgG ~eV! 0.477 0.32–0.46

EgX ~eV! 1.4 0.08–1.4

EgL ~eV! 0.33 0.33–0.72

Dso ~eV! 0.15 0.15–0.20me* (G) 0.0091 ¯

mhh* (001) 20.145 ¯

mlh* (001) 0.0202 ¯

g32g2 0.481 ¯

EP ~eV! 21.48 ¯

F 1.77 ¯

VBO ~eV! 20.38 ¯

ac ~eV! 2.61 ¯


on

fo

nisa

-t

da

ine

f

t

ta

pa

litre

en

ss-h

-

eth

indeha

ngalve

thedea-dn-

for

psre,

apodeofat

ndam-

fa-

ing-nd

y, ame-ers

lat-

100eVmi-

--

ta

rly

oar


tial, which we ascribe to a nonlinear shift of the conductiband edge.4

The recommended nonzero bowing parametersGaInAs are collected in Table XIII.

3. AlInAs

AlInAs serves as the barrier layer in the importaGa0.47In0.53As/Al0.48In0.52As heterostructure system thatlattice matched to InP. For this reason, the most precise bgap determinations are available for Al12xInxAs with the x50.52 composition. While Matyas reported aG-valley bow-ing parameter of 0.24 eV forx,0.7 on the basis of absorption measurements,463 theoretical work indicated that ishould be larger.201,418Wakefieldet al.464 reported a value of0.74 eV based on cathodoluminescence spectroscopySimilar results were recently obtained by Kopfet al.,465 andwere also recommended in several compilations of bowparameters.101,432 With the inclusion of strain effects, thtemperature-dependent band gap of Al0.46In0.54As on InP re-ported by Abrahamet al. implies a bowing parameter o'0.66 eV.466 We select a composite value ofC50.70 eV toreflect the majority of these results.

In contrast to GaInAs, theX conduction valley is lowerthan theG valley in AlAs-rich AlxIn12xAs. Linear interpo-lations are usually employed to obtain theX-valley andL-valley minima in AlInAs. That approximation implies thathe X and G valleys should cross at a composition ofx50.64, which is slightly lower than the early experimenresult of x50.68 by Lorenz and Onton.467 Since the largercrossover composition would require a negative bowingrameter for theX-valley gap, we recommendC50. Krijn101

gave a bowing parameter of 0.15 eV for the spin-orbit spting, which is equal to the GaInAs value adopted in the pvious subsection.

Optically detected cyclotron resonance measuremhave yieldedme* 50.1060.01m0 for Al0.48In0.52As.468 Onthe other hand, Curyet al.457 obtained a much smaller masof 0.069m0 , which is only a little lower than extrapolationfrom GaInAs-rich AlGaInAs.456,465The smaller result is supported by the cyclotron-resonance measurements of Cet al.469 Calculations by Shen and Fan470 also indicate aneffective mass of'0.075m0 . The reasonably good agreement of all but one result allows us to suggest aG-valleyeffective mass bowing parameter for AlInAs. In view of thlack of hard data, we recommend linear interpolation forother masses in the AlInAs alloy.

In order to explain the electron effective massAlGaInAs quaternaries, Fan and Chen introduced a disorinduced conduction-valence band mixing. They found tan interband matrix element of 22.5 eV was necessaryaccount for the experimental results. This value ofEP re-quiresF520.63 for consistency. We have derived bowiparameters for AlInAs employing this system of values,though it should be noted that the results depend sensition the electron effective mass adopted for Al0.48In0.52As.458

The hydrostatic deformation potential forx50.52 wasmeasured by Fergusonet al.471 to be a526.7 eV, whichfalls between the values adopted for InAs~26.1 eV! andAlAs ~28.1 eV!. On the other hand, Yehet al.472 obtained a


r

t

nd

ta.

g

l

-

--

ts

en

e

r-t

to

-ly

valence-band deformation potential opposite in sign fromtrend predicted by the model-solid theory of VanWalle.129 That result would imply a considerable bowing prameter even within theav theory presented by Wei anZunger.42 Pending further confirmation, we recommend liear interpolation.

The recommended nonzero bowing parametersAlInAs are collected in Table XIV.

B. Phosphides

1. GaInP

The GaInP alloy exhibits some of the largest direct gaamong the non-nitride III–V semiconductors. FurthermoGa0.51In0.49P ~Eg51.9 eV at 300 K! is lattice matched toGaAs, which makes it an attractive material for wide-gGaAs-based quantum well devices such as red dilasers.473 This application has spurred extensive studiesthe band structure characteristics of GaInP, which arepresent rather well known.

For theG-valley band gap, early photoluminescence acathodoluminescence determinations yielded bowing pareters ranging from 0.39 to 0.76 eV.467,474–478Subsequentelectroreflectance and modulation spectroscopy studiesvored a value near the higher end of that range.479–481The-oretical studies have also produced a wide range of bowparameters,101,201,410,418,482with the most reliable results clustered around 0.5–0.75 eV. By analogy to GaInAs aAlInAs, it is useful to consider the Ga0.51In0.49P alloy forwhich the most extensive data are available. Unfortunatelprecise measurement for this lattice-matched alloy is sowhat complicated by its proximity to the indirect crossovpoint, and by long-range ordering of the group-III atomwhich can take the form of a monolayer InP–GaP supertice along the@111# direction.483–486 The ordering-inducedreduction of the direct energy gap can be on the order ofmeV.487 Low-temperature band gaps of 1.969–2.018have been reported for random GaInP alloys that are nonally lattice matched to GaAs.488–494 Using the result ofEmanuelssonet al.,495 corrected for the exciton binding energy of 8 meV,493 we obtain a recommended bowing parameter ofC50.65 eV. That value is consistent with recent dafor nonlattice-matched compositions.496,497

The X-valley gap energies in InP and GaP are neaequal~2.38 and 2.35 eV, respectively, at 0 K!, and theG –Xcrossover composition in GaInP is believed to be close tx50.7.479 Although early work usually assumed a line

TABLE XIV. Nonzero bowing parameters for AlInAs.


EgG ~eV! 0.70 0.24–0.74

EgX ~eV! 0 20.5–0

Dso ~eV! 0.15 ¯

me* (G) 0.049 ¯

EP ~eV! 24.81 ¯

F 24.44 ¯

VBO ~eV! 20.64 ¯

ac ~eV! 21.4 ¯


-tim

.thb

omn

lla

-

r-

r-

olin-

r

iely

fo

V

ge tor,

er-

rly

he

r

3.63-a

alns

the-

ter,ng

ri-ap-

InP

p

ep-

es-


variation of theX-valley gap, Auvergneet al. have morerecently suggestedC(Eg

X)50.147 eV on the basis of piezoreflectance spectroscopy in the composition range nearcrossover point. Somewhat larger bowing parameters areplied by the pressure experiments of Goniet al.496 and Me-ney et al.498 Our recommended value ofC(Eg

X)50.2 eVagrees with other experimental and theoretical results482

Early experimental and theoretical determinations ofbowing parameter for the L-valley gap were summarizedBugajskiet al.482 TheG –L crossover most likely occurs atxslightly smaller than theG –X crossover point,480 whichmakesL the lowest conduction valley forx greater than'0.67. Krutogolovet al.499 suggestedC(Eg

L)50.71 eV, al-though that article assumedL-valley indirect band gaps inthe end-point binaries that are considerably different frthe ones adopted here. Modeling of the ellipsometric athermoreflectance data of Ozakiet al.500 yielded Eg

L

52.25 eV in Ga0.5In0.5P at 300 K, which favors a smaL-valley bowing. In deriving our recommended bowing prameter, we employed theG –L crossover point ofx50.67,482,499 which was confirmed recently by Interholzingeret al.501 The bowing of the spin-orbit splitting in GaInPis known to be very small,446,476,479with recent results502

implying a linear interpolation to within experimental uncetainty.

The electron effective mass in a random alloy withx50.5 was measured by Emanuelssonet al.495 to be me*50.092m0 , which is somewhat lower than the linearly intepolated values quoted in other papers.473,490 Similar resultswere obtained by Wonget al.503 In the absence of reliabledata for the other electron and hole masses, linear interption is advised using the scheme outlined above. With aearly interpolated value ofEP526 eV for the interband matrix element in Ga0.5In0.5P, we derive anF parameter of21.48, and from it the corresponding bowing.

The shear deformation potentiald was measured foGaInP grown on GaAs~111!B substrates.504 While the resultsimply a bowing parameter of 2.4 eV, the large uncertaintin existing determinations make it difficult to conclusiveprefer this value over a linear interpolation.

The recommended nonzero bowing parametersGaInP are collected in Table XV.

2. AlInP

Al xIn12xP has a direct energy gap forx,0.44, and at thecrossover composition theG-valley value ofEg

G'2.4 eV isthe largest of any non-nitride direct-gap III–

TABLE XV. Nonzero bowing parameters for GalnP.


EgG ~eV! 0.65 0.3920.76

EgX ~eV! 0.20 0–0.35

EgL ~eV! 1.03 0.23–0.86

Dso ~eV! 0 20.05–0me* ~G! 0.051 ¯

F 0.78 ¯

d ~eV! 0 0–2.4


he-

ey

d

-

la--

s

r

semiconductor.2 For a precise determination of the bowinparameter, it is convenient to consider compositions closAl0.52In0.48P, which is lattice-matched to GaAs. Howevethe inherent difficulty of experimentally determiningEg

G inclose proximity to the indirect-gap transition caused undestimates by some workers. Bouret al.203 obtained 0.38 eVfor the direct-gap bowing parameter from relatively eaelectroreflectance measurements. Mowbrayet al.505 reporteda low-temperature direct excitonic gap of 2.680 eV for tlattice-matched composition, which implies a negativeC.This result was supported by the work of Dawsonet al.506,507

and by the ellipsometry experiments of Adachiet al.502 andSchubert et al.508 In fact, a direct gap of 2.69 eV foAl0.52In0.48P at 0 K~corrected for the exciton binding energy!must be considered a better-established value than theeV gap for AlP~see above!. However, instead of extrapolating the AlInP gap to the AlP binary, we recommend usingnegative bowing parameter ofC520.48 eV, while notingthat considerable uncertainty exists for largex.

For Al0.52In0.48P, indirectX-valley gaps of 2.34–2.36 eVhave been deduced from a variety of opticmeasurements.500,505,507Taking into account the correctiofor the exciton binding energy~a rough estimate insofar aprecise values have not been calculated!, a bowing parameterC(Eg

X)50.38 eV is deduced. Few data are available forposition of theL-valley minimum in AlInP, although a roomtemperature value ofEg

L52.7 eV is given by Ozakiet al.500

Since that implies at most a very small bowing paramelinear interpolation is recommended. The spin-orbit splittiin Al0.5In0.5P is thought to be 135 meV,502,509 which trans-lates into an upward bowing of 0.19 eV. No direct expemental determinations of the effective masses in AlInPpear available, although various estimates put theG-valleyelectron mass close to 0.11m0 .509,510

The recommended nonzero bowing parameters for Alare collected in Table XVI.

3. AlGaP

The ‘‘exotic’’ AlGaP alloy has an indirect band gathroughout its composition range.202 The lowest-energy op-tical transitions are typically associated with donor or acctor impurities. Linear interpolation of the indirectX-valleygap was found to give good agreement with photolumin

TABLE XVI. Nonzero bowing parameters for AllnP.


EgG ~eV! 20.48 20.48–0.38

EgX ~eV! 0.38 ¯

Dso ~eV! 20.19 ¯

me* ~G! 0.22 ¯

TABLE XVII. Nonzero bowing parameters for AlGaP.


EgG 0 0.0–0.49

EgX 0.13 0–0.13


io

ezu-atistoe

rt

r a

toho

A

TheeVer

g

thhe

eul

on

e,n

te

apdemthhe0.

n

r

thela-ole

l ofd, its.

rjonticho-

for

ntnds.am-

alith

t

elyblythepre-

me-ter-

po-ing.

ro.


cence spectra, once impurity and phonon band transitwere carefully separated from the interband transitions.202,511

The direct gap in AlGaP was studied by Rodriguet al.512,513 Using a limited number of data points, the athors concluded that either a linear variation or a quadrvariation with a bowing parameter of 0.49 eV were constent with the data. As in the case of AlInP, extrapolationAlP gives a value forEg

G that is somewhat higher than thvalue listed in Table V for the binary~also based on limiteddata!. Recent cathodoluminescence experiments supposmall bowing of 0.13 eV for the lowestX-valley gap inAlGaP.514

We recommend that a linear variation be assumed foother band structure parameters of AlGaP~see Table XVII!.

C. Antimonides

1. GaInSb

Although GaInSb cannot by itself be lattice matchedany of the readily available substrates, it serves as thequantum well material in type-II infrared lasers155 andphotodetectors515 with strain-balanced active regions.number of experimental243,260,426and theoretical410,516studiesof the direct band gap in GaInSb have been published.different reports, which are generally in excellent agreemwith each other, support a bowing parameter of 0.415Negligible dependence of the bowing parameter on tempture was obtained using the photovoltaic effect,259 whereasthe pseudopotential calculations of Bouarissa and Aoura517

suggested a slow variation from 0.43 at 0 K to 0.415 at roomtemperature.

Bowing parameters for theX- and L-valley gaps wereestimated by Adachi166 and Glissonet al.201 The method em-ployed by Adachi is rather indirect, in that an average ofbowing for two direct critical points is used to represent tbowing of the indirect gap. Nonetheless, in the absenceexperimental data for theX-valley gap, it appears to be thbest available approximation. The pseudopotential calctions of Bouarissaet al.516 yielded very weak bowing of thetwo indirect gaps. This finding is in apparent contradictiwith the limited data of Lorenzet al.,518 which suggest asmallerL-valley gap for GaInSb than that for GaSb~no es-timate of theL-valley gap in InSb was available at the timso the article assumed a linear variation with compositio!.Also, the data presented in clear form by Zitouniet al.,519

which cover only part of the composition range, indicaappreciable bowing for both theX and L valleys. We con-clude that the considerable uncertainty for the indirect gin GaSb and especially InSb translates into a poor unstanding of the indirect-gap bowing in GaInSb. We recomend using bowing parameters of 0.33 and 0.4 eV forX-valley andL-valley gaps, respectively, which are near ttop of the reported range. A small bowing parameter ofeV was found for the spin-orbit splitting in GaInSb.243,260

A small bowing of the electron effective mass in theGvalley has been determined both experimentally atheoretically.260,446Roth and Fortin243 compiled results froma number of references, from which Levinshteinet al.11 de-duced a bowing parameter of 0.0092m0 . We assume a linea


ns

ic-

a

ll

le

ent.

a-

e

of

a-

sr--e

1

d

variation of the interband matrix element and determineF bowing parameter from that assumption. Linear interpotion is also suggested for the heavy-hole and split-off hmasses. Levinshteinet al.11 give a large bowing of the light-hole mass, in agreement with the band structure modeAuvergneet al.260 Since the light-hole masses in InSb anGaSb are only a little larger than the electron massesfollows that the bowing should be similar in the two caseWhile considerably larger masses were reported by Baet al.520 on the basis of a model of their galvanomagnemeasurements, the bowing parameter for light holes is csen to be consistent with the results of Roth and Fortin.243

The recommended nonzero bowing parametersGaInSb are collected in Table XVIII.

2. AlInSb

While not widely used, AlInSb provides a conveniestrain-compensating barrier material for mid-IR interbacascade lasers521 and other antimonide device structureEarly absorption measurements of Agaev and Bekmedov522

yielded a linear variation of the direct energy gap with coposition. However, that result was discounted1,2 in favor ofthe electroreflectance determination ofC50.43 eV by Iso-muraet al.,523 which is in good agreement with the empiriccurve charting the increase of the bowing parameter wlattice mismatch between the binary constituents.201 WhileDai et al.524 recently found a linear variation of the direcenergy gap with alloy lattice constant~i.e., composition! forInSb-rich AlInSb, those results were confined to a relativsmall range of compositions and as such were conceivanot sensitive enough to the quadratic bowing term. Sinceelectroreflectance measurements should have been morecise than the absorption experiments of Agaev and Bekdova, we recommend use of the bowing parameter demined by Isomuraet al.523

The only other band structure parameter whose comsition dependence has been studied is the spin-orbit splittIsomuraet al.523 deduced a relatively strong bowing ofC50.25 eV in the split-off gap. The recommended nonzebowing parameters for AlInSb are collected in Table XIX

TABLE XVIII. Nonzero bowing parameters for GaInSb.


EgG ~eV! 0.415 0.36–0.43

EgX ~eV! 0.33 20.14–0.33

EgL ~eV! 0.4 0.093–0.6

Dso ~eV! 0.1 0.06–0.72me* ~G! 0.0092 ¯

mlh* ~001! 0.011 ¯

F 26.84 ¯

TABLE XIX. Nonzero bowing parameters for AlInSb.


EgG 0.43 0–0.43

Dso 0.25 ¯


e

bAa

m

e

nelow

t

ner

f

ie

e,teleis

-th

eowd

conono

th0.

ise

onth

ingin

for

al-

ngeers

that

d-pa-or-

for

inis.ent-eeV,nalSb

me

the

,ntaling

.2

-n

e

tive


3. AlGaSb

AlGaSb is an important material employed in high-speelectronic525 and infrared optoelectronic526 devices. Apartfrom a considerably larger lattice mismatch, the GaSAlGaSb heterostructure is a lower-gap analog of the GaAlGaAs material system. The band structure in AlGaSb woriginally studied by piezomodulation, and bowing paraeters of 0.69 and 0.48 eV were proposed for theG-valley andX-valley gaps, respectively.251 A much more comprehensivinvestigation was carried out by Alibertet al.,236 who incor-porated the results of many other reports. They obtaidirect-gap bowing parameters of 0.48 and 0.47 eV attemperatures and room temperature, respectively, valueshave been used by many subsequent workers.255,527 How-ever, recently Bignazziet al.254 obtained a better fit to theabsorption spectra by assuming a linear band gap variatiolow Al mole fractions. Those results were confirmed by thmoreflectance spectroscopy performed by Bellaniet al.528

Consequently, a cubic band gap variation was proposedthe direct gap of AlxGa12xSb:C520.04411.22x, whichproduces little deviation from linearity at smallx but alsobowing parameters in the range suggested by Mathet al.251 and Alibertet al.236 in the middle of the compositionrange~with the inflection point atx50.35!. In spite of thefact that onlyx,0.5 alloys were investigated in that articlit must be considered to be the most reliable study to da

Although there was an early indication of appreciabX-valley bowing,251 later reports have established that itquite small or nonexistent.2,236 L-valley gap bowing parameters ranging from 0.21 to 0.75 eV are encountered inliterature.2,236,527,528TheG –L crossover composition is quitsensitive to the exact choice of the bowing parameters,ing to the proximity of the two valleys in both GaSb anAlSb. Crossover compositions ofx50.27 andx50.23 weredetermined on the basis of photoluminescenmeasurements529 and from wavelength-modulated absorptispectra,2 respectively. If those crossover points are to be csistent with our choice for the direct gap’s dependencecomposition, we must choose a very weakL-valley bowing.Therefore, we recommendC(Eg

L)50 and note that with thischoice the bowing in AlGaSb becomes rather similar tocase of AlGaAs covered above. A bowing parameter ofeV was deduced by Alibertet al.236 for the spin-orbit split-ting.

For electron effective masses in the AlGaSb alloy, itrecommended that the procedure outlined in the AlGaAs stion be followed. That is, any nonlinearity in the compositidependence of the effective mass stems entirely frombowing of the energy gap.251,527 This is expected to give

TABLE XX. Nonzero bowing parameters for AlGaSb.


EgG 20.04411.22x 21.18–0.69

EgX 0 0–0.48

EgL 0 0.21–0.754

Dso 0.3 ¯


d

/s/s-

d

hat

at-

or

u

.

e

-

e

-n

e3

c-

e

better results than using the Landolt–Bornstein bowparameter1 or interpolating linearly between the massesGaSb and AlSb.

The recommended nonzero bowing parametersAlGaSb are collected in Table XX.

D. Arsenides antimonides

1. GaAsSb

GaAs12xSbx is most often encountered with thex50.5composition that matches the lattice constant of InP,though it should also be noted that atx50.91 is latticematched to InAs. Direct-gap bowing parameters in the ra1.0–1.2 eV have been determined by a number of workfrom absorption measurements.530–535 It was noted in thefirst study536 of epitaxial GaAs0.5Sb0.5 on InP in 1984 that theapparent band gap of 0.804–0.807 eV was smaller thanimplied by previously determined bowing parameters.537–539

Recently, a low-temperatureEgG of 0.813 eV was measure

by Merkel et al.540 and Huet al.,541 and its temperature dependence was obtained. Those results implied a bowingrameter of 1.42–1.44 eV, although it was suggested thatdering effects may have reduced the band gap.537 A low-temperature bowing parameter of 1.3 eV was determinedGaAs0.09Sb0.91 lattice matched to InAs,542 and quite recentlya bowing parameter of 1.41 eV was obtained by Ferret al.543 from ellipsometry and photoreflectance studieSince different growth temperatures were employed in recstudies of GaAs12xSbx with a rather small range of compositions nearx'0.5, the evidence for ordering is inconclusivat present. We recommend a bowing parameter of 1.43although this value should be revised downward if additioinvestigations substantiating the partial ordering in GaAswith compositions close to a lattice match with InP becoavailable.

Rough estimates of the bowing parameters forX-valley andL-valley gaps~both 1.09 eV! have been pub-lished by Adachi.166 We recommend slightly higher valuesin order to assure consistency with both the experimecrossover points and the larger adopted direct-gap bowparameter.L-gap andX-gap bowing parameters of 1.1–1eV are consistent with the reported measurements.11 On thebasis of rather limited data, Maniet al.542 suggested a bowing parameter of 0.1 eV for the spin-orbit splitting iGaAsSb. On the other hand, theoretical studies101,166 havederived a considerably larger value of 0.6 eV, which wrecommend.

The composition dependence of the GaAsSb effecmass was determined by Delvinet al.544 Although they pro-

TABLE XXI. Nonzero bowing parameters for GaAsSb.


EgG 1.43 1.0–1.44

EgX 1.2 ¯

EgL 1.2 ¯

Dso 0.6 0.1–0.61VBO 21.06 ¯


nnslivfo

aarid-

.6rado

tuetin

ki

,e.In

l

raf

e

sm

iooler

ela

at

l

ghesane

ent-wellal-g aing

sis-be

for

anreasrgyg

cert

intylcu-w-n tolec-een

rl

for

-


posed a bowing parameter of 0.0252m0 , the room-temperature GaSb mass obtained in that study was sigcantly higher than our recommendation based on a conseof other data. A safer procedure is to take the mass nonearity to arise from the band gap bowing as outlined abo

The recommended nonzero bowing parametersGaAsSb are collected in Table XXI.

2. InAsSb

The InAsSb alloy has the lowest band gap amongIII–V semiconductors, with values as small as 0.1 eVroom temperature. For that reason, it is an important matefor a variety of mid-infrared optoelectronic devices, incluing lasers545 and photodetectors.546 Initial reports put thedirect-gap bowing parameter in InAsSb at 0.58–0eV.160,161,426,547Those studies were performed at tempetures above or near 100 K, and gaps were extrapolatelower temperatures in a linear fashion. It is now understothat this resulted in an overestimate of the low-temperaenergy gap and an underestimate of the bowing paramTheoretical considerations led to a higher projected bowparameter of 0.7 eV,410 which is recommended by Rogalsand Jozwikowski.548 This estimate was revised toC50.65 eV by a more accurate pseudopotential calculation549

while Bouarissaet al.516 computed an even smaller valuMore recent photoluminescence studies on MBE-grownAsSb obtainedC50.67– 0.69 eV.164,261,550 Similar results~e.g.,C50.64 eV) were obtained by Gonget al.551,552 frommeasurements on a sequence of InAs-rich samples. Eet al.553 obtained Varshni parameters for InAs0.91Sb0.09, andsuggested a temperature dependence of the bowing paeter based on those results. The possible importance odering has been discussed in several recent works~Wei andZunger,554 Kurtz et al.,555 Marciniak et al.556!. Smaller thanexpected band gaps were obtained for compositions closthe lattice-matching condition on GaSb (x50.09).557 On thebasis of all these investigations, we recommend a compobowing parameter of 0.67 eV. Currently, the bowing paraeters for both theX-valley and L-valley gaps are boththought to be'0.6 eV.166,201,516

Experimental and theoretical data for the compositdependence of the spin-orbit splitting in InAsSb were clected by Beroloet al.,446 who suggest a bowing parametof 1.1–1.2 eV. The estimated558 spin-orbit splitting of 0.325eV in InAs0.91Sb0.09 ~lattice matched to GaSb! implies a simi-lar bowing parameter of 1.26 eV.

Plasma reflectance and other measurements of thetron effective mass in InAsSb were summarized by Thomand Woolley.171 The best fit to the data collected in tharticle favored a mass of 0.0103m0 for the alloy with thesmallest direct gap (InAs0.4Sb0.6). That result agreed welwith the theory developed by Beroloet al.446 Whilemagneto-optical measurements by Kucharet al. indicated anextrapolated band edge electron mass of 0.0088m0 forInAs0.145Sb0.855, which corresponds to a larger mass bowinthat finding is inconsistent with a linear interpolation of tinterband matrix elements andF parameters between InAand InSb, and also disagrees with the model of RogalskiJozwikowski.548 A recent report of the electron effectiv


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mass in InAsSb is from a cyclotron resonance measuremby Stradlinget al.184 for two alloy compositions. Their measurements appear to support a smaller mass bowing asas negative bowing for the interband matrix element,though a great deal of uncertainty is inherent in assigninvalue based on two data points. Our recommended bowparameter for the electron effective mass is 0.035m0 , whichis near the bottom of the reported range and roughly content with the results of assuming the mass bowing tocaused entirely by band gap bowing.

The recommended nonzero bowing parametersInAsSb are collected in Table XXII.

3. AlAsSb

AlAsSb is a versatile large-gap barrier material that cbe lattice matched to InP, InAs, or GaSb substrates. Whemany workers assume a linear variation of the direct enegap in AlAsSb,559 theoretical projections indicate a bowinparameter in the 0.72–0.84 eV range.101,201The little experi-mental information that is available on AlAsSb alloys lattimatched to GaSb560 and InP561 can be interpreted to suppoeither a small~'0.25 eV!560 or a large~'0.8 eV!561 value.We recommend the latter, but also note that the uncertamay not greatly affect most quantum heterostructures calations in view of the large absolute value of the gap. Boing parameters for the two indirect gaps are both chosebe 0.28 eV in accordance with photoluminescence and etroreflectance measurements, the results of which have bsummarized by Ait Kaciet al.560 The bowing parameter fothe spin-orbit splitting~0.15 eV! is taken from the theoreticaestimate of Krijn.101

The recommended nonzero bowing parametersAlAsSb are collected in Table XXIII.

E. Arsenides phosphides

1. GaAsP

GaAs12xPx is a wide-band gap alloy that is often employed in red LEDs.562 The alloy becomes indirect forx.0.45~at 0 K!482 when theX valley minimum crosses below

TABLE XXII. Nonzero bowing parameters for InAsSb.


EgG ~eV! 0.67 0.58–0.7

EgX ~eV! 0.6 ¯

EgL ~eV! 0.6 0.55–0.8

Dso ~eV! 1.2 ¯

me* (G) 0.035 0.03–0.055

TABLE XXIII. Nonzero bowing parameters for AlAsSb.


EgG 0.8 0–0.84

EgX 0.28 ¯

EgL 0.28 ¯

Dso 0.15 ¯

VBO 21.71 ¯


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the G valley minimum. Most experimental results for thdirect-gap bowing parameter lie within a relatively narrorange, 0.175–0.21 eV.93,426,467,482,562–565A small bowing pa-rameter is also expected on theoretical grounds, for bothdirect and indirect gaps.191 However, it has been noted thathe bowing parameter more than doubles if CuPt-like ording sets in.566 While a recent ellipsometry study of disodered GaAsP alloys produced a direct-gap bowing paramof 0.54 eV,567 that outlying result lacks other verification anrelied on only two data points with intermediatex. We there-fore assume that either it was anomalous or some ordedid occur. Since a meaningful temperature dependencenot be extracted from the existing data, we recommendC50.19 eV for the direct gap at all temperatures. Thissomewhat higher than the value suggested by Aspne93

partly because we employ a larger band gap for GaP at 7The X-valley gap bowing parameter was studied by

number of workers.93,482,562,568,569Again, there is not muchcontroversy since the crossover composition is rather westablished. Our recommended value (C50.24 eV) lies ap-proximately in the middle of the reported range of 0.20–0eV. Both experimental and theoretical results for theL-valleybowing parameter implyC50.16 eV.93,482 The spin-orbitsplitting was found to vary linearly with composition.563

A linear variation of the electron effective mass withx inGaAsP was reported by Wetzelet al.190 A k"P calculationwith explicit inclusion of the higher conduction bands aslightly different band parameters predicts a small bowparameter of 0.0086m0 .191 We recommend following thegeneral procedure tying the mass bowing to the direct-bowing as outlined above, which yields reasonable agment with that theory and experiment.

The deformation potentials in GaAsP were studiedGonzalezet al.570 The shear deformation potentialb found inthat work lies between the recommended values for G~22.0 eV! and GaP~21.7 eV!. Although the hydrostaticdeformation potentials were found to be somewhat smathan either of the binary values, that may have been anfact of the measurements rather than an alloy-specific perty.

The recommended nonzero bowing parametersGaAsP are collected in Table XXIV.

2. InAsP

InAsP spans an interesting IR wavelength range~0.87–3mm!, retains a direct gap throughout, and has a high elecmobility. However, it has found far less practical use thanquaternary cousin GalnAsP because except at the endpthe InAsxP12x lattice constant does not match any of tbinary III–V substrate materials.

TABLE XXIV. Nonzero bowing parameters for GaAsP.


EgG 0.19 0.174–0.21

EgX 0.24 0.20–0.28

EgL 0.16 0.16–0.25


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The dependence of the energy gap on compositionoriginally studied by Vishnubhatlaet al.426 and Antypas andYep.571 Whereas the first group reported a bowing parameof 0.27 eV at 300 K, the second group obtained a similaCat T577 K but a much smaller value~0.10 eV! at 300 K.Bodnaret al. later reported the opposite trend for the temperature dependence ofC,158 and Nicholaset al. suggestedsimilar bowing parameters of 0.32–0.36 eV throughoutentire temperature range.572 A recent fitting of the absorptionspectra of InAsP/InP strained quantum wells also yieldeweak temperature dependence, but with values betweenand 0.12 eV.573 Similar results were reported by Wadet al.,574 who used a combination of PL, x-ray diffractionand absorption measurements. We recommend a valueC50.10 eV, which is consistent with the latest experimenworks and is slightly lower than the theoretical estimate0.23 eV.410

The bowing parameters for the indirect gaps were emated by Adachi166 and Glissonet al.201 Using their projec-tions, we recommendC50.27 eV for bothX andL valleys.A bowing parameter of 0.16 eV was determined for the sporbit splitting.446

The electron effective mass in InAsP alloys was fiinvestigated by Kesamanlyet al.173 This work and also thesubsequent magnetophonon experiments of Nichoet al.572,575found a nearly linear dependence of the masscomposition. Although the authors conjectured a reductionthe interband matrix element in the alloy, their quantitaticonclusions are in doubt since they overestimated the dirgap bowing parameter as well as the interband matrixment for InAs. With these corrections, it is unclear whethenonlinear term needs to be introduced forEP . The effectivemass for InAs-rich alloys was determined by Kruzhaevet al.from tunneling magnetospectroscopy576 and for three differ-ent compositions by Sotomayor Torres and Stradling usfar-IR magneto-optics.577 Again, due to the experimental uncertainty it is difficult to determine whether the standard pcedure needs to be supplemented. In fact, the most reexperimental results are in very good agreement withassumption that both the interband matrix element and thFparameter vary linearly with composition.

TABLE XXV. Nonzero bowing parameters for InAsP.


EgG 0.10 0.09–0.38

EgX 0.27 ¯

EgL 0.27 ¯

Dso 0.16 ¯

TABLE XXVI. Nonzero bowing parameters for AlAsP.


EgG 0.22 ¯

EgX 0.22 ¯

EgL 0.22 ¯


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The recommended nonzero bowing parameters forAsP are collected in Table XXV.

3. AlAsP

If the exotic AlAsP alloy has ever been grown, it waapparently not reported. Glissonet al. surmised that thedirect-gap bowing parameter in AlAsP would be quite sm~0.22 eV!.201 Using the criterion relating to the bond-lengdifference in the endpoint binaries, one would not expectbowing parameter to exceed those in GaAsP~0.19 eV! andInAsP ~0.22 eV! ~see Table XXVI!.

F. Phosphides antimonides

1. GaPSb

The growth of GaPSb was first reported by Jet al.578,579The primary object was to determine whether tenergy gap in the GaP0.68Sb0.32 alloy, which is latticematched to GaAs, is direct or indirect. Strong PL was oserved, which led the authors to conclude that the energyis direct. Large bowing parameters of 3.8 and 2.7 eV~lowerbound! were derived for theG-valley andX-valley gaps, re-spectively. Since these greatly exceed the available theocal estimates,201,580it cannot be ruled out that ordering sustantially reduced the energy gaps in these and perhapsother investigated GaPSb alloys. Subsequently, Loualet al.581 studied GaSb0.65P0.35, which is lattice matched to anInP substrate. Since at this composition the direct naturthe energy gap is not in question, it may be argued thatanalysis of their data should yield a more reliable valuethe direct-gap bowing parameter (C52.7 eV). The room-temperature PL measurements of Shimomuraet al.582 pro-duced approximately the same result, which is our recomended value. The sameC is also recommended for theXandL-valley gaps, since no studies have been reported~seeTable XXVII!.

2. InPSb

The energy gap in InPSb remains direct at all compotions. Bowing parameters in the 1.2–2.0 eV range have breported for the direct gap.101,166,201,579,580,583–586The studyby Jouet al.579 of alloys with a range of compositions ap

TABLE XXVII. Nonzero bowing parameters for GaPSb.


EgG 2.7 2.7–3.8

EgX 2.7 ¯

EgL 2.7 ¯

TABLE XXVIII. Nonzero bowing parameters for InPSb.


EgG 1.9 1.2–2.0

EgX 1.9 ¯

EgL 1.9 ¯

Dso 0.75 ¯


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pears to be the most useful. A recent experiment587 obtainedan energy gap of 0.48 eV for InP0.69Sb0.31 lattice matched toan InAs substrate, which implies an even larger value ofbowing parameter. Although further work will be neededestablish complete confidence, our recommended directbowing parameter isC51.9 eV. Since the indirect gaps iInPSb have not been studied, it is not unreasonable tosume the same values for their bowing. The bowing paraeter for the spin-orbit splitting has been calculated to be 0eV ~see Table XXVIII!.101

3. AlPSb

The successful growth of AlP0.40Sb0.60 lattice matched toInP has been reported.582 An early projection of Glissonet al.201 wasC51.2 eV for the direct-gap bowing parameteHowever, considering the trends in the common groupalloys, it is likely that the gaps corresponding to the thrmajor valleys have bowing parameters that are not tooferent from those in GaPSb. We therefore recommendC52.7 eV ~see Table XXIX!.

G. Nitrides

1. GaInN

GaInN quantum wells represent a key constituent inactive regions of blue diode lasers and LEDs.293 This tech-nological significance justifies the quest for a thorough uderstanding of the bulk properties of wurtzite GaInN alloyUnfortunately, however, there is still considerable disagrment over such fundamental parameters as the bowing oenergy gap. A~partial! phase decomposition of the GaInquantum wells employed in blue and green LEDs is believto occur.588 Nearly pure InN quantum dots are formed, whicact as efficient radiative recombination centers. Since inot yet clear whether this phase segregation has been c

TABLE XXIX. Nonzero bowing parameters for AlPSb.


EgG 2.7 1.2–2.7

EgX 2.7 ¯

EgL 2.7 ¯

TABLE XXX. Energy-gap bowing parameters for nitride ternaries. Fother information available for these compounds, see Table XXXI andtext.

Materials Recommended values

Wurtzite GaInN 3.0Zinc blende GaInN 3.0Wurtzite AlGaN 1.0Zinc blende AlGaN 0Wurtzite AlInN 16 – 9.1xZinc blende AlInN 16 – 9.1xZinc blende GaAsN 120.4 – 100xZinc blende GaPN 3.9Zinc blende InPN 15Zinc blende InAsN 4.22


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pletely avoided in thicker layers of GaInN, interpretationthe reported bowing parameters requires great care~seeTables XXX and XXXI!.

Early studies suggested an energy-gap bowing paramC of between 0 and 1.0 eV.402,589,590Wright and Nelsonmore recently derivedC51 eV for zinc blende GaInN,591

which is relevant because of the common expectationthe zinc blende and wurtzite alloys should have appromately the same bowing parameters. Nakamura foundthis bowing parameter produced a good fit to the resultsPL measurements for low In compositions.592 A slightlylarger bowing parameter~1.6 eV if our values for the energgaps of GaN and InN are employed! was found by Liet al.,on the basis of PL from GaInN/GaN superlattices.593 A simi-lar bowing parameter of 1.4 eV was reported for zinc blenGaInN; however, the results were given for relatively thGaInN layers, the strain in which cannot be considered furelaxed.594 Bellaiche and Zunger595 investigated the effectsof short-range atomic ordering in GaInN, and establishthat a large reduction in the band gap should be expectedthat scenario. Although all of these studies are consiswith a bowing parameter of'1 eV for the random GaInNalloy, several more recent investigations of GaInN laywith small In fractions arrived at significantly larger valueThe experimental band gap results of McCluskeyet al. forGa12xInxN epilayers withx,0.12 were found to be consistent with bowing parameters as large as 3.5 eV.596 Further-more, first-principles calculations performed by thosethors showed that the bowing parameter itself may bstrong function of composition, at least for small In fractionSimilarly large bowing parameters~in the range 2.4–4.5 eV!were obtained in a large number of subsequent studiesdifferent groups.597–602 A bowing parameter of 2.7 eV caalso be inferred from a recent investigation of zinc blenGaInN.603 Some of those works tentatively attributed the pvious reports of small bowing to erroneous estimates ofalloy composition. On the other hand, Shanet al.599 sug-gested that the more recent PL emission may have resufrom local fluctuations in the In fraction, which could leadoverestimates of the bowing parameter. Since the mostportant practical applications of GaInN alloys require onlysmall In fraction, we suggest a bowing parameter of 3 eVboth the wurtzite and zinc-blende phases, although wephasize that at present there exists no verification that thCapplies equally well to higher In compositions. There is aa strong possibility that the alloys studied in recent worksnot truly random. It should be emphasized that the physunderstanding of these materials is far from complete,that the parameters recommended in this review are pr

TABLE XXXI. Bowing parameters for quantities other than the energy gof nitride ternaries.

Parameters~eV! Materials Recommended values

EgX Zinc blende GaInN 0.38

EgX Zinc blende AlGaN 0.61

EgL Zinc blende AlGaN 0.80

Dso Zinc blende GaAsN 0


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sional quantities that will almost surely be revised in tcourse of future work. A further comment which appliesall of the nitride alloys as well as other systems affectedunknown degrees and types of segregation, is that the paeter set most relevant to a given theoretical comparison wdata may not be that representing the ideal materials,rather the nonideal properties resulting from specific growconditions of interest.

There is very little information on the other band paraeters for GaInN. Tight-binding calculations321 provide somesupport for our recommended standard procedure of uthe band-gap bowing parameter to derive the compositiovariation of the electron effective mass and interpolatingrest of the quantities linearly. For theX-valley gap in zincblende GaInN, a small bowing parameter ofC50.38 eV wasestimated from first principles.591

2. AlGaN

AlGaN is often used as the barrier material for nitridelectronic and optoelectronic devices. Initial studies ofcompositional dependence of the energy gap repodownward,604 upward,605 and negligible606 bowing. Subse-quent early PL607 and absorption608 measurements foundbowing parameter of 1.0 eV, which continues to be wideused in band structure calculations even though a numbemore recent investigations question the conclusions ofearly work. Several studies597,609,610found negligible bow-ing, and it has been suggested that the other values resfrom an incomplete relaxation of strain in the AlGaN thfilms.609 This statement is supported to some extent bylarge bowing parameter of 1.78 eV reported for highstrained layers grown on SiC.611 Other workers quite re-cently calculated612 and measured613 smaller bowing param-eters~0.25–0.6 eV depending on the measurement metand assumed binary end points!. Brunneret al.318 reportedC51.3 eV and the data of Huang and Harris614 imply aneven larger bowing parameter for AlGaN epilayers grownpulsed laser deposition, although in both cases residual sdue to the differing lattice and thermal expansion coefficieof AlGaN and sapphire could have affected the resuCathodoluminescence measurements for AlGaN epitaxigrown on Si~111! suggestC51.5 eV.615 We recommendcontinued use of the accepted bowing parameter of 1.0until a broader consensus is reached on the effects of sand other issues.

Theory projects that theG-valley bowing parameter inzinc blende AlGaN is small.369,591This is consistent with theexperimental report of a linear variation of PL energiesthin films of cubic AlGaN.616 We therefore recommendzero bowing parameter for this case. Recommended vafor the X-valley ~0.61 eV! and L-valley ~0.80 eV! bowingparameters are taken from the empirical pseudopotenmethod calculations of Fanet al.369

3. AlInN

Al xIn12xN is drawing attention because atx50.83 it canbe lattice matched to GaN. The first experimental studyserved such a strong bowing that the band gap for the latt


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matched composition was found to be smaller than thaGaN.617 Furthermore, the standard quadratic expressionnot fit the compositional variation of the band gap very weGuoet al.618 and Kimet al.619 subsequently presented resufor InN-rich and AlN-rich AlInN, respectively, which indi-cated somewhat weaker bowing. Penget al.620 gave a cubicexpression for the energy gap, based on results over thetire range of compositions. A similarly large bowing waobserved by Yamaguchiet al.621 On the theoretical side, afirst-principles calculation for zinc blende AlInN yieldedbowing parameter of 2.53 eV, which was assumed toequal to that in the wurtzite alloy.622 Of these, the most trustworthy quantitative report appears to be that of Penget al.However, the suggested expression,Eg(300 K)51.9711.968x26.9x219.1x3 eV, has been corrected to reflect orecommendations for the binary end points. In the absencfurther information, we recommend that the temperaturependence be incorporated by using the recommendedpoints to fix the constant and linear terms at eachT, and thenuse the Penget al. quadratic and cubic terms at all tempertures. The same approach is recommended for zinc bleAlInN.

4. GaAsN

Although it has been known for a long time that smquantities of nitrogen form deep-level impurities in GaAand GaP, growth of the GaAsN alloy with appreciable~closeto 1%! N fractions has been reported only recently.623,624Thesomewhat unexpected discovery of a giant bowing paramin this and other AB12xNx alloys in principle opens prospects for growing direct-gap III–V semiconductors wiband gaps in the near-IR onto Si substrates.

As a general rule, the non-nitride constituents doeasily take the wurtzite form. It is therefore expected tGaAsN and the other analogous alloys will crystallize inzinc blende lattice, and that a large miscibility gap will mait difficult to prepare alloys with large N fractions. Phaseparation has indeed been observed in GaN-rich alloy625

Furthermore, the substantial differences between the proties of the light-atom constituent~e.g., GaN! and the heavy-atom constituent~e.g., GaAs! call for a close scrutiny of theusual quadratic relations for the alloy band parameters, sone expects the alloy concentration dependence to be hinonlinear.

An early theoretical study predicted a bowing parameof 25 eV for the direct energy gap of GaAsN.626 C518 eVwas obtained from PL measurements of GaAsN with N frtions of no more than 1.5%.627 The same result was inferrefrom studies of the GaAsN near the two binary limits, withe decrease in the energy gap being linear for N fractionhigh as 3%.628 A quadratic form withC'11 eV fit the resultsof an ellipsometry study forx,3.3% fairly well.629 Otherstudies of dilute GaAsN indicated bowing parameterslarge as 22 eV.630–632A series of first-principles calculationexamined various aspects of the band structure for GaAincluding ordering effects.566,633–638Those studies found thathe band-edge wave functions in GaAsN tend to be localiimpurity-like states, with the conduction-band wave functistrongly localized on the As sublattice and the valence-b


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wave function on the N sublattice. The projected bowiparameters were large and composition-dependent~varyingin the range from 7 to 16 eV between GaAs0.5N0.5 andGaAs0.875N0.125! whenever the dilute alloy displayed a locaized deep impurity level in the gap.566,635Recent experimen-tal work on GaAsN with N fractions as large as 15% cofirmed the strong composition dependence of the bowiwith values ranging from 20 eV for dilute alloys to 5 eV foconcentrated alloys.639,640On the other hand, Uesugiet al.641

found a more gradual reduction of the bowing paramewith composition, and attributed the discrepancy to differestrain conditions in the different studies. On the basis ofmost complete available data set,639 a compromise cubicform for the band-gap dependence on composition~at alltemperatures! can be derived with the following nonlineaterms: 20.4x2– 100x3 eV. This expression avoids the earsemimetallic transition~predicted by usingC520 eV! thatwas not observed in the single relevant experimenstudy.639 A note of caution is that this expression is expectto be valid only in the regionx,0.15, beyond which noexperimental data are available. This description may in fbe a rather crude approximation of the band-gap dependover the entire range of compositions.

An alternative description of the energy gap in GaArich GaAsN in terms of the level repulsion model has bedeveloped recently.642 The transformation from N acting aan isoelectronic impurity to band formation was foundoccur atx50.2%. The energy gap is sublinear with compsition for N fractions as small as a few percent. No upplimit on the composition, for which this description is validis available at this moment and the results have been veronly for x,3%. Nonetheless, in view of the strong evidenfor its correctness, it may be advisable to use that descriprather than the nonlinear bowing approximation for lowcompositions, which are of interest for most device applitions. A potentially more accurate four-level repulsion modwas recently introduced by Gil.643

The Varshni parameters measured by Malikovaet al.644

for two GaAsN alloys were found to lie between thoseGaAs and zinc-blende GaN. The temperature dependencthe energy gap in a few GaAsN alloys was found to be muweaker~60%! than that in GaAs for N fractions as small a1%.645 The spin-orbit splitting obtained from electrorefletance measurements646 was found to vary approximately linearly in GaAsN. The interband matrix element of GaAs~and other alloys such as GaPN! is predicted647 to be stronglyreduced relative to the virtual crystal approximation. Ththeoretical result, which is important primarily at largecompositions, has yet to be verified experimentally. Furthmore, an abrupt increase in the effective mass in GaAGaAs quantum wells with 1.2% and 2% N has bereported.648,649 The effective mass is larger than that prdicted on the basis of theoretical calculations.638,650

The valence-band shear deformation potentialb in diluteGaAsN epilayers was studied by Zhanget al.651 using elec-troreflectance measurements to determine the splittingtween the heavy-hole and light-hole bands. The deforma


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potential did not follow a simple linear interpolation betweGaAs and GaN. The cause is not quite clear, since N inporation is expected to affect primarily the conduction baof GaAsN, and further studies over a wider compositirange are required to pin down the~possibly complicated! Nfraction dependence.

5. GaPN

Most of the band properties of GaPN are expected toanalogous to those of GaAsN, based on the simple obsetion that the N impurity behaves in a similar manner in boGaAs and GaP. The main complication is that GaP isindirect-gap semiconductor. Both theG-valley andX-valleygaps in GaPN were predicted to have bowing parameter14 eV,652 and subsequent experimental data agreed wthose projections.652–654Sakaiet al. used the same theoretcal description and obtained similar results.626 Recently, Biand Tu were able to observe the energy-gap variationalloys incorporating up to 16% N.639 Their results are consistent with the predictions of another model, which yiebowing parameters of 10 and 3.9 eV for theX-valley andG-valley gaps, respectively. These results are quite closthe recent tight-binding calculations of Miyoshiet al.655 andappear to be the best available values for interpolatingtween GaP and zinc blende GaN using the conventiomodel. Pseudopotential calculations indicate that ethough both GaP and GaN have substantial matrix elemethe momentum matrix element in GaPN is very smallalmost any concentrated GaPN alloy.647 On the other handXin et al.656 recently observed intense PL from GaPN allowith a N fraction larger than 0.43% and demonstrated aLED based on GaP0.989N0.011.

657 These results are quite consistent with the picture developed by Shanet al.,658 whichchallenges the once accepted wisdom that GaPN retainindirect gap for very small N concentrations~'1%!. Theseauthors formulated a model based on the interaction betwhighly localized nitrogen states and extended states at thGconduction-band minimum. According to the so-call‘‘band anticrossing’’ picture, incorporation of even smaamounts of N into GaP changes the nature of the fundamtal optical transition from indirect to direct. The optical trasition energy is smaller than the nitrogen level in GaP~2.18eV! by a term linear inx for small x and varying asx forlarger x. Another piece of the evidence supporting stroG –X mixing is the recent measurement of a heavy (0.9m0)electron mass in GaP0.975N0.025/GaP quantum wells.659 Pro-vided the accuracy and validity limits of the ‘‘band anticrosing’’ picture are conclusively established, the theoreticalproach summarized in Ref. 658 should be employeddetermine the optical transition energies in GaPN. Simconsiderations apply to other low-N-fraction-containingloys ~see Sec. IV G 4!. In this review, we refrain from ex-ploring the consequences of the band anticrossing modemore detail.

6. InPN

InPN with less than 1% N has been studied by Bi aTu.660 In spite of the low solubility of N in InP, they, found


r-d

ea-

n

ofth

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s

to

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d

an

en

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--or

-

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d

a significant band gap reduction that corresponded to a bing parameter in the range 13–17 eV. We recommendaverage value ofC515 eV, and refer the reader to SeIV G 4 for the necessary caveats.

7. InAsN

A theoretical calculation of the band structure of InAshas been reported,661 and one recent experimental studythe growth of this interesting ternary has been made.662 Theprimary importance of InAsN is that it serves as one of tend points of the GaInAsN quaternary, which is promisifor 1.55 mm semiconductor lasers on GaAs substratestight-binding calculation661 deduced a bowing parameter o4.22 eV, which was obtained assuming a particular va~3.79 eV! for the valence-band discontinuity between InAand InN. This is our recommended value, although it mustnoted that the quadratic fit in Ref. 661 was rather poor. Aother recent tight-binding study663 showed that the band gavariation for small N fractions depends on the degree ofdering present in the material and that the maximaN-clustered alloy has a lower energy gap than the Aclustered alloy. It remains to be determined which configration can be realized experimentally.

The recommended bowing parameters for all the nitrternary alloys are collected in Tables XXX and XXXI.

V. QUATERNARY ALLOYS

The capabilities of III–V quantum well devices can frquently be expanded by introducing quaternary layers todesign. This increased flexibility does, however, come atexpense of a more difficult growth coupled with the needmultiple tedious calibration runs to accurately fix the coposition. Furthermore, extensive miscibility gaps limit thrange of stable compositions, since in thermal equilibriuthe components often tend to segregate into inhomogenmixtures of binaries and ternaries. While the nonequilibriuMBE growth process can extend the miscibility boundarconsiderably, inaccessible composition gaps remain for soof the III–V quaternary systems.

General methods for deriving quaternary alloy bandrameters from those of the underlying binary and ternmaterials have been summarized by a numberauthors.101,166,201While no single approach guarantees goresults in all cases, the interpolation procedure introducedGlisson et al.201 usually provides a reasonable approximtion. It is applicable to the most commonly encountered qternaries of theAxB12xCyD12y type, that are made up otwo group-III and two group-V elements. Using the notatifrom Eq. ~4.1!, a given band parameter for the ternaA12xBxC is given by

GABC8 ~x!5~12x!GAC1xGBC2x~12x!CABC , ~5.1!

whereGAC andGBC are the values at the binary end poinandCABC is the appropriate bowing. The corresponding baparameter in the quaternaryAxB12xCyD12y is then ex-pressed as a weighted sum of the related ternary values



GABCD9 ~x,y!5x~12x!@~12y!GABD8 ~x!1yGABC8 ~x!#1y~12y!@xGACD8 ~y!1~12x!GBCD8 ~y!#

x~12x!1y~12y!. ~5.2!

This approach can be extended to treat quaternary alloys of theABxCyD12x2y andBxCyD12x2yA types664

GABCD8 ~x,y!5xyGABD8 ~u!1y~12x2y!GBCD8 ~v !1~12x2y!xGACD8 ~w!

xy1y~12x2y!1~12x2y!x, ~5.3!

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where u[(12x2y)/2, v[(22x22y)/2, and w[(222x2y)/2. The approach of Eqs.~5.2! and ~5.3! tends to givebetter agreement with experiment than an alternative trment of Moonet al.,665 which is known to overestimate thquaternary bowing.201,666 Krijn101 gives polynomial expan-sions of Q(x,y) derived from Eqs.~5.2! and ~5.3! for theenergy gaps and spin-orbit splittings of several III–V quatnaries.

Generally speaking, a given quaternary comprises atwo-dimensional space of compositions,@x,y#. In practice,however, most experiments have focused on the odimensional subsets,@x,y(x)# that are~approximately! lat-tice matched to one of the common binary substrate maals ~GaAs, InP, InAs, or GaSb!. The quaternary may then brepresented as a combination of two lattice-matched cstituents, one of which must be a ternary while the other mbe either a binary or a ternary. The treatment is considerasimplified by the usual absence of any strong bowing ofband parameters for such an alloy, which is expectedtheoretical grounds because the two constituents have itical lattice constants.

In the following, we will assume two lattice matchebinary or ternary end points,a ~e.g.,A12xBxC! andb ~e.g.,AC12yDy!, which are combined with arbitrary compositionzto form the lattice-matched quaternary alloya12zbz . Wethen employ the expression

Gab9 ~z!5~12z!Ga81zGb82z~12z!Cab , ~5.4!

whereGa8 andGb8 are the values at the end points andCab isthe additional bowing associated with combining the two epoint materials to form a quaternary. For lattice-matchquaternaries, using Eq.~5.4! with the available experimentaevidence to determineCab for each property should lead tobetter representation than either the procedure of Eqs.~5.2!and ~5.3! or simple linear interpolation.

A. Lattice matched to GaAs

1. AlGaInP

(Al zGa12z)0.51In0.49P @or, more accurately(Al0.52In0.48P)z/~Ga0.51In0.49P!12z# lattice matched to GaAs isoften employed as the barrier and cladding materialGaInP/AlGaInP red diode lasers.473,498 Early studies oflattice-matched AlGaInP, which is a quaternary of the sond type~with one group-V element!, found that the bandgap variation between Ga0.51In0.49P and Al0.51In0.49P is in-deed nearly linear, as expected.494,505,667 However, subse-quent PL, PL excitation~PLE!,507 electroreflectance,502

thermoreflectance,500 and ellipsometry508 measurements indicated that a small additional bowing parameter between 0and 0.18 eV is needed to fully account for the data. UsC50.18 eV in conjunction with the recommended expresions for the temperature-dependent direct gaps of GaInP


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ri-

n-ylyenn-

dd

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-

11g-nd

AlInP, Eq. ~5.4! can be used to find the lattice-matched AGaInP gaps for allz andT. For example, atT50 we [email protected](12z)12.691z20.18z(12z)# eV.

Using linear interpolation for theX-valley energy gap inAlGaInP ~supported by the PL data of Najdaet al.507!, wefind that the direct-to-indirect crossover~at 0 K! should oc-cur at z50.55, which is in good agreement witexperiment.494,507 The composition-dependent variationthe 300 K energy gap in the lattice-matched quatern(Al zGa12z)0.51In0.49P is plotted by Fig. 6, in which theG-valley gap is given by a solid curve and theX-valley gapby a dashed curve. Linear interpolation between the lattmatched alloys is also suggested for theL-valley gap and thespin-orbit splitting.502 There appears to be only one cyclotroresonance study of the electron mass in Al0.15Ga0.35In0.5P,495

in which ordering andL-valley interactions could have distorted the reported electron mass of 0.14m0 . It is thereforesuggested that the standard procedure be followed, exthat electron masses in the lattice-matched ternaries shbe used as the end points in place of binaries. In this partlar example, linear interpolation of the end point ternamasses should also give adequate results.

2. GaInAsP

Not very much is known about the band structuof GaInAsP lattice matched to GaAs @or(GaAs)12z~Ga0.51In0.49P!z#. There has been little motivationto pursue this quaternary, since its direct band gap sproughly the same range as direct-gap AlGaAs, which is csiderably easier to grow.

However, there has been some work668,669 on lattice-matched and strained alloys~in the vicinity of z50.67!, for

FIG. 6. Lowest energy gaps as a function of composition(Al0.52In0.48P)z /(Ga0.51In0.49P)12z and (Ga0.51In0.49P)z /(GaAs)12z quater-nary alloys, lattice matched to GaAs, atT5300 K. The region of AlGaInPfor which theX-valley gap is lowest is indicated with a dashed line.


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which the quaternary constituents may be rewrit(GaP)x~InAs!12x ~with x'z for the case of lattice matchinto GaAs!. Absorption measurements669 yieldedxc50.73 forthe direct-to-indirect crossover point. Since the authgreatly overestimated theX valley gap in InAs, their corre-sponding bowing parameter cannot be considered reliaInstead we adopt their direct-gap bowing parameter of 0eV and estimateCX520.28 eV from the crossover composition ~note that these bowings are in terms ofx with GaPand InAs as end points, notz with GaAs and Ga0.51In0.49P asend points!. It should be noted that results for the GaP-riquaternary do not necessarily support upward bowiwhereas a linear variation does produce a reasonably gfit.

Using the results for (GaP)x~InAs!12x discussed abovea straightforward evaluation of the bowing parameters forlattice-matched quaternary (GaAs)12z~Ga0.51In0.49P!z givesCX50.53 eV and CG520.62 eV. The compositiondependent variation of the 300 K energy gap in GaInAlattice matched to GaAs is plotted in Fig. 6. The recomended parameters imply a direct band gap at all comptions z.

3. AlGaInAs

Band gaps for strained AlGaInAs on GaAs~with low Aland In fractions! have been reported by Jensenet al.,441 onthe basis of PL measurements.

4. GaInAsN

The GaInAsN quaternary has drawn considerable attion recently, since the addition of a small N fraction compensates the compressive strain that limits the critical thness of GaInAs layers grown on GaAs substrates. In termthe band structure properties, the alloy may be thought o(GaAs)12z~InAs0.62N0.38)z and represents a quaternary anlog of the nitrogen-containing zinc blende ternaries discusin the previous section. Unfortunately, the properties ofAsN have not been measured, and such experiments mainto difficulty in the future owing to the expected miscibilitgap. An alternative approach proposed by Chowet al.670 onthe basis of the experimental result of Joneset al.671 is toderive the energy gap from the GaInAs alloy with the saIn fraction and thenreduceit by ~53D«! eV, whereD« is thedifference between the in-plane strains computed for thecontaining ternary and quaternary~we have revised the coefficient in order to fit our band-gap scheme!. The results ofabsorption672 and photomodulation spectroscopy673 are in ac-cord with the PL measurement of Joneset al.671 Pseudopo-tential calculations674 are in good agreement with this approach for low N fractions. Calculations and measuremefor GaInAsN lattice matched to InP have also bereported.674,675Recently, a large increase in the electron min GaInAsN with a small N fraction was observed via refletivity measurements.676 This result appears to invalidate ansimple interpolation scheme and favor the authors’ propomodel of the interaction between localized N states andextended states of the semiconductor matrix as discusseSec. IV G 4.


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B. Lattice matched to InP

1. GaInAsP

GaInAsP lattice matched to InP@(InP)12z~Ga0.47In0.53As!z , which is GaxIn12xAszP12z withx50.47z,# is an extremely important quaternary alloy. Itcurrently employed in commercial optoelectronic~especiallysemiconductor lasers emitting at 1.3 and 1.55mm! and elec-tronic ~especially high-electron-mobility transistor! devices.This alloy has been the subject of numerous reviarticles5,6,429,666,677and at least one book.4

Numerous measurements of the direct energy gapGaInAsP have been reported.427,443,678–680Pearsall summa-rized the experimental results available by 1982 inreview,677 and deduced a bowing parameter of 0.149 eV. Fthe most part, a small spread in the experimental datafound, and it was pointed out that Pereaet al.443 (C50.25 eV! underestimated the gaps somewhat in analyztheir electroreflectance data. Further electroreflectanceperiments of Lahtinen and Tuomi680 indicated a substantiallysmaller bowing parameter of 0.038 eV. An early tighbinding calculation also yielded a bowing parameter of 0eV,428 with error bounds sufficient to include almost all othe experimental data sets. We recommend using Eq.~5.4!with a bowing parameter ofC50.13 at all temperatureswhich is consistent with the relations currently employedestimate device characteristics.4 This procedure yieldsEg

[email protected](12z)10.816z10.13z2# eV at 0 K and Eg

[email protected](12z)10.737z10.13z2# eV at 300 K. Thecomposition-dependent variation of the 300 K energy gapthe lattice-matched quaternary (InP)12z~Ga0.47In0.53As!z isplotted in Fig. 7.

Two studies have reported Varshni parametersGaInAsP.681,682Although the band gaps obtained in the twstudies were similar, the resulting Varshni parameters wvery different. Deducing the temperature variation of tquaternary band gap from the bulk Varshni parameters giabove is therefore probably a safer procedure~and gives rea-sonable agreement with the results of Satzkeet al.682!. Alinear variation is usually assumed for the indirect band g

FIG. 7. Energy gaps as a function of composition f(InP)z(Ga0.47In0.53As)12z , (Al0.48In0.52As)z(Ga0.47In0.53As)12z , and(GaAs0.5Sb0.5)z(Ga0.47In0.53As)12z quaternary alloys, lattice matched to InPat T5300 K.


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between the two end point materials,4 although that dependence has apparently not been verified experimentally.

Electroreflectance studies have concluded that the bing parameter for the spin-orbit splitting is either closezero678,680 or negative.443,679 This upward bowing was explained by considering the interband and intraband contrtions to disorder-induced mixing of conduction-band avalence-band states.683 We derive our recommended valueC(Dso)520.06 eV, which implies [email protected](12z)10.33z20.06z2# eV, by averaging all reported spin-orbbowing parameters.

The electron mass in GaInAsP has been measuredvariety of approaches such as cyclotron resonance, matophonon resonance, Shubnikov–de Haas oscillations,shallow donor photoconductivity.448–451,453,684Some studiesfound roughly zero bowing,448,449,684while others obtainedappreciable downward bowing.450,451,453In the former case,it was claimed that the lack of effective mass bowing wdue to the band gap bowing being compensated bdisorder-induced bowing of the interband matrix element677

However, from a consideration of all available experimendata there is no obvious reason to deviate from our stanmass-bowing procedure~e.g., see Fig. 6.7 in Ref. 4!. Usingour adopted band gap variation and interpolating the inband matrix element and theF parameter linearly, we findthe totality of the data to be consistent with that proceduThe resulting masses are slightly lower than those in R448, 449, and 684, but not nearly as small as in Ref. 4The light-hole effective mass in GaInAsP lattice matchedInP was measured by Hermann and Pearsall.459 On the basisof those data, Adachi suggested a bowing paramete0.03m0 .4 However, we recommend using a slightly modifieversion of the Adachi expression,mlh* 5(0.120820.099z10.0302z2) eV, in order to assure consistency with the epoint values in Ga0.47In0.53As and InP. Using linear interpolation for the heavy-hole mass and the valence-band anropy, the split-off mass can be determined using Eqs.~5.4!and ~2.18!.

By measuring the pressure dependence of the stimulemission in a buried heterostructure near-IR diode lasehydrostatic deformation potential ofa525.7 eV was deter-mined for GaxIn12xAszP12z with y50.6.685 This representsa smaller absolute value than in either InP~26.6 eV! orGa0.47In0.53As(27.79 eV!, and requires a rather large bowinparameter ofC526.7 eV. While we recommend this valuin the absence of other information, additional experimeare necessary to confirm this strong bowing.

2. AlGaInAs

Another important quaternary lattice matched to InPAlGaInAs @or (Al0.48In0.52As)z~Ga0.47In0.53As!12z#, whichcombines two lattice-matched ternary alloys. Initial charterization of this quaternary was performed by Oleet al.,456 who found a direct-gap bowing parameter of 0.eV @when cast into the form of Eq.~5.4!#. Subsequent PLmeasurements of Kopfet al.465 and Curyet al.457 implied alinear variation of the energy gap, whereas the lotemperature PL results of Bohreret al.686 suggested a largebowing parameter of 0.68 eV. Fan and Chen obtaine


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-

-

a

value of 0.225 eV between those two extremes,458 whileShen and Fan calculated a rather small bowing usingtight-binding method with the virtual-crystaapproximation.470 Averaging the various experimental results, we obtain a composite direct-gap bowing paramete0.22 eV, which is close to the values of Olegoet al.456 andFan and Chen.458 The composition-dependent variationthe 300 K energy gap in the lattice-matched quatern(Al0.48In0.52As)z~Ga0.47In0.53As!12z is plotted in Fig. 7.

The only other AlGaInAs band structure parameter wa validated nonlinear compositional variation is the electeffective mass, for which there is strong evidence for a sliupward bowing.457,469 This can be explained by disordeinduced band mixing,447,458 whose effect on the interbanmatrix element is described by expressions in Refs. 447458. Whereas the evidence for a similar mechanism in GaAsP was inconclusive, here we suggest an effective-mbowing parameter of20.016m0 . This is consistent with theother parameters ifEP is assumed to have a quadratic depedence, withC525.68 eV adjusted to provide the correresult for AlGaInAs withx50.5. TheF parameter is theninterpolated linearly~between22.89 in Ga0.47In0.53As and20.63 in Al0.48In0.52As! in that scheme.

3. GaInAsSb

Despite the presence of a miscibility gap, the growthmetastable GaInAsSb lattice matched to InP@or(Ga0.47In0.53As)z~GaAs0.5Sb0.5)12z# has been reported ovethe entire composition range.687,688However, the usefulnesof this quaternary is limited because both end points hnearly the same energy gap, even though the cutoff walengths are close to the important 1.55mm low-loss windowfor optical fiber communications.

The first reported growth and characterization studythis quaternary obtained little bowing,687 whereas more re-cent PL measurements on bulk Ga0.64In0.36As0.84Sb0.16 at 8 Kimplied a gap of 0.7654 eV,688 as compared to'0.81 eV atthe two end points. Based on this data point we recomma composition dependence ofEg(T50)[email protected](12z)10.816z20.22z(12z)# eV. Further study is clearly desirable. The recommended composition-dependent variatiothe 300 K energy gap in the lattice-matched quatern(Ga0.47In0.53As)z~GaAs0.5Sb0.5)12z is plotted in Fig. 7. Theapproximate location of the miscibility gap according to rcent calculations689 is indicated with the dotted line. Theprecise extent of the miscibility gap depends on the growprocedure and temperature, and the growth of metastmaterials inside the gap is not ruled out.

C. Lattice matched to InAs

1. GaInAsSb

GAInAsSb lattice matched to InAs @or(InAs)12z~GaAs0.08Sb0.92)z# has been studied by a numberauthors.165,690,691The growth of slightly strained GaInAsSlayers has also been reported by Shinet al.692 Although datafor the composition dependence of the direct band gapsomewhat sparse, the available results are consistent witrelatively large bowing parameter of'0.6 eV suggested by


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Levinshteinet al.11 ~see also Fig. 3 of Ref. 691!. This is ourrecommended value, which impliesEg(T50)[email protected](12z)20.6z(12z)# eV. In the following section itwill be seen that this value is similar to the recommendbowing parameter for GaInAsSb on GaSb, which is not sprising in view of the near equality of the InAs and GaSlattice constants. The empirical relation suggested in R691 also appears to produce good agreement with the dalthough it cannot be recast in the form of Eq.~5.4! with aconstant bowing parameter. While in one study Varshnirameters have been deduced for GaInAsSb on InAs,165 it isrecommended that our usual procedure be followed to dethe temperature dependence of the energy gap in this qunary. The composition-dependent variation of the 300 Kergy gap in GaInAsSb lattice matched to InAs is plottedFig. 8. The approximate location of the miscibility gapindicated by the dotted line. Bouarissa693 obtainsX-valleyand L-valley bowing parameters of 0.15 and 0.60 eV fropseudopotential calculations accommodating the effectsalloy disorder.

2. AlGaAsSb

The direct and indirect energy gaps in AlGaAsSb lattmatched to InAs@or (GaAs0.08Sb0.92)12z(AlAs0.16Sb0.84)z#were studied theoretically by Adachi,166 and Anwar andWebster.559 They respectively employed expressions sugested by Glissonet al.201 and Moon et al.665 ~who com-bined the direct and indirect-gap composition dependenc!.Abid et al.694 also treated AlGaAsSb, using the empiricpseudopotential method with the virtual crystal approximtion. That little bowing was found for any of the energy gamay be an artifact of neglecting the disorder potential. Wtherefore recommend using the bowing parameters specbelow for AlGaAsSb on GaSb.

3. InAsSbP

InAs12x2ySbxPy is the only quaternary with thregroup-V elements that has been studied in the literaturespite of some miscibility-gap problems, it has been succe

FIG. 8. Energy gaps as a function of composition f(InAs)12z(InSb0.31P0.69)z and (InAs)12z(GaAs0.08Sb0.92)z quaternary alloys,lattice matched to InAs, atT5300 K. The approximate locations of miscbility gaps are indicated by dotted lines.


dr-

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-

eer--

of

-

sl-

eed

Ins-

fully grown on InAs substrates@(InAs)12z~InSb0.31P0.69!z ,wherez5x1y# by several groups.583,695–697Because of thelarge uncertainty in the InSb0.31P0.69 energy gap, the firststudy tried to use the quaternary data, assuming a linvariation between the InAs and InP0.69Sb0.31 end points, todeduce the bowing parameter for the ternary.583 Subsequentstudies similarly derived a vanishing bowing parameterthe quaternary.695–697 Since with that assumption the morecent data696,697 are consistent with our adopted band gfor InP0.69Sb0.31, we recommendC50. However, Adachi166

suggested the possibility of upward bowing based ongeneral quaternary relations of Glissonet al.,201 and the dataof Voroninaet al.695 can possibly be explained in that maner. The recommended composition-dependent variationthe 300 K energy gap in InAsSbP lattice matched to InAsplotted in Fig. 8. The approximate location of the miscibiligap is indicated with a dotted line.

The spin-orbit splitting in InAsSbP has been found tolarger than in either InAs or InP0.69Sb0.31,

696 although onlytheoretical estimates are available for the latter. The msurements imply the form: Dso50.39(12z)10.166z10.75z(12z), where the upward bowing parameterlarger than in Ref. 696 owing to a different value assumfor InPSb. While an electron effective mass of 0.027m0

determined698 for InAs0.62Sb0.12P0.26 is lower than the valueof 0.0288m0 derived from a linear interpolation of the inteband matrix element and theF parameter, the latter is nonetheless quite close to the band edge mass found fromband structure fits to the carrier density dependenceformed in the same reference. We therefore recommendploying the usual procedure.

D. Lattice matched to GaSb

1. GaInAsSb

GaInAsSb lattice matched to GaSb @or(GaSb)12z(InAs0.91Sb0.09)z# is particularly well studied,699 inpart because it is an important active-region constituendiode lasers emitting atl52 mm.700 Early work687,690,701onthe direct band gap in GaSb-rich GaInAsSb was summarby Karoutaet al.,558 who suggested a bowing parameter0.6 eV. At the other extreme, data for InAs0.91Sb0.09-rich al-loys were found to be consistent with a slightly higher boing parameter of 0.65–0.73 eV.695,702Photoreflectance studies by Herrera-Perezet al.703 and spectral ellipsometryresults of Munozet al.704 support an even higher value forC.Similarly, recent reports of GaInAsSb grown by liquid-phaepitaxy tend toward higher bowing parameters.705,706 Ourrecommended value ofC50.75 eV is a composite obtaineby averaging all of the available results. The dependencecomposition is then:[email protected](12z)10.346z20.75z(12z)# eV at 0 K [email protected](12z)10.283z20.75z(12z)# eV at 300 K. The data of Karoutaet al.558 suggestC520.26 eV for the spin-orbit splitting, which implies:D0

[email protected](12z)10.33z10.26z(12z)# eV. However, this ex-pression is at variance with the recent ellipsometric dataMunozet al., which suggests downward bowing of the spiorbit splitting with C50.25 eV.704 The latter result was obtained only for compositions ofz50.14– 0.15. Both the


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downward bowing of the energy gap and the upward bowof the spin-orbit splitting are well described699 quantitativelyby the expression of Moonet al.,665 although it should beremarked that in performing such calculations some authused different bowing parameters for the related ternalloys.695 Energy gaps for certain strained GaInAsSb comsitions have also been reported.707–709 The composition-dependent variation of the 300 K energy gap in GaInAslattice matched to GaSb is plotted in Fig. 9. The approximlocation of the miscibility gap is indicated by the dotted lin

Pseudopotential results of Bouarissa710 predict bowingparameters of 0.85 and 0.43 eV for theL-valley andX-valleygaps, respectively. The direct-gap bowing parameter quin that reference is in reasonably good agreement withexperimental results.

2. AlGaAsSb

AlGaAsSb lattice matched to GaSb @or(GaSb)12z(AlAs0.08Sb0.92)z# is a natural barrier and claddinmaterial for mid-infrared semiconductor lasers. Relationsthe direct and indirect energy gaps were calculatedAdachi,166 and the experimental results have been sumrized by Ait Kaciet al.560 The quoted direct-gap bowing parameter of 0.47 eV agreed well with the pseudopotential cculation of Abid et al.694 and the photoreflectancmeasurements of Herrera-Perezet al.703 Based on all of thesedata points, we recommend a composite result ofC50.48 eV. The band gap at 300 K is then:[email protected](12z)12.297z20.48z(12z)# eV. This composition-dependent variation of the direct energy gap in AlGaAslattice matched to GaSb is plotted in Fig. 9.

Although general considerations imply that the bowiof the L-valley gap should be similar while theX-valley gapshould display little bowing,166 pseudopotentiacalculations694 have suggested bowing parameters of 0.8and 1.454 eV, respectively. A consequence of the laX-valley bowing would be a decrease of the predicted direto-indirect crossover composition toz50.14.

FIG. 9. Energy gaps as a function of composition f(GaSb)12z(AlAs0.08Sb0.92)z and (InAs0.91Sb0.09)12z(GaSb)z quaternary al-loys, lattice matched to GaSb, atT5300 K. The approximate location of amiscibility gap for InGaAsSb is indicated by a dotted line.


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1. AlGaAsP

AlGaAsP has been grown on commercially availabGaAs0.61P0.39 substrates.711–713 It can be considered ascombination of GaAsP and AlAsP alloys with phosphorfractions similar to the substrate. A careful investigationthe pump intensity dependence of the PL peaks revealedimpurity and band-to-band transitions.711 Those results implya surprisingly large direct-gap bowing parameter of 1.3for the quaternary. However, that value would be reducconsiderably if it turns out that the theoretical projection0.22 eV for the bowing parameter in AlAsP~see above! istoo small.

F. Nitride quaternaries

The growth of AlGaInN has been reported,602,621 al-though little is known about its band structure propertiRecent results indicate a nearly linear band gap reductionsmall In compositions~,2%!.714 The cutoff wavelengths ofAlGaInN ~lattice matched to GaN! ultraviolet photodetectorswere also generally consistent with a linear interpolation715

Improved crystal quality in comparison with AlGaN has albeen noted.716 Until more precise data become available, wsuggest employing the usual quaternary expressions in oto estimate the parameters of this poorly explored mater

VI. HETEROSTRUCTURE BAND OFFSETS

The preceding sections have discussed the bulk proties of III–V semiconductors and their alloys in isolation.this section, we turn to a consideration of the conduction avalence band alignments that result when the materialsjoined to form heterojunctions in various combinations. Ftunately, it is usually a good approximation to view the vlence band position as a bulk parameter for each individmaterial, which can then be subtracted to determine the rtive band alignment at a given heterojunction. The interfadipole contribution, that is particular to each material cobination tends to be small, since it is largely screened ewhen the interface bonding configurations are vastly diffent. However, it will be seen below that at least for the caof an InAs/GaSb interface, which has no common anioncation, there is reliable experimental evidence for a smdependence of the offset on the interface bond type.

Our discussion will build upon the previous summaryYu et al.717 who comprehensively reviewed the understaning of band offsets as of 1991. That work, which mayconsidered an update of earlier reviews by Kroemer,718,719

also provided an excellent overview of the methods comonly used in experimental band offset determinations.

Some of the existing theories, such as the model stheory of Van de Walle,129 assert that very little bowing othe valence band offset should be expected~although somebowing may arise if there is a strong nonlinearity in tspin-orbit splitting!. However, if we are to maintain consistency with the most reliable experimental results for a variof heterojunctions~e.g., GaAs/AlAs and lattice-matcheGa0.47In0.53As/Al0.48In0.52As!, a bowing parameter must bassigned to some of the ternary alloys. Since there are alm


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no reports of temperature variations that exceed the expmental uncertainties, in all cases we will take the valenband offsets to be independent ofT.

Recommended valence band offsets~VBOs! ~openpoints! and conduction band offsets~CBOs! ~filled points!for all 12 of the binaries are summarized in Fig. 10. Textent of the energy gap for each material is indicated byvertical line. Relative positions of the CBOs and VBOsthis figure may be compared to determine the offset forgiven heterojunction combination.

The recommended VBOs for materials lattice matchto the common substrate materials of GaAs, InP, InAs,GaSb are shown in Fig. 11. The points indicate offsetsbinary and ternary compounds, while the vertical lines snify VBO ranges that are available using lattice-matchquaternaries. Since a linear variation with composition issumed for all quaternaries, simple interpolation between

FIG. 10. Conduction~filled! and valence~open! band offsets for the 12binaries. TheG-valley energy gap for a given binary corresponds todifference between the conduction and valence band positions, i.e.length of the vertical line connecting the filled and open points. Similathe conduction~valence! band offset between two distinct binaries corrsponds to the energy difference between their respective conduction olence band positions on the absolute energy scale of the figure.

FIG. 11. Valence band offset as a function of lattice constant. The offfor binaries and lattice-matched ternaries are indicated by points, ovariations with composition for lattice-mismatched ternaries~not includingstrain effects! are given by dashed curves, and the VBO ranges for quanary alloys lattice matched to a particular substrate material~GaAs, InP,InAs, or GaSb! are given by the vertical solid lines.


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end point offsets yields the VBO for any desired quaternalloy. The dashed lines illustrate VBO variations in a numbof important lattice-mismatched ternary alloys. Correspoing conduction-band offsets are given in Fig. 12. Althoustrain effects are neglected in these plots, they are relatistrong and must be included to determine the correct CBNote also that since the band gap variation with compositis in general nonlinear~and sometimes double valued!, agiven point on one of the vertical lines in Fig. 12 does nnecessarily map to a single, distinct quaternary alloy comsition.

A. GaAs ÕAlAs

The GaAs/AlAs heterojunction, which is unique amonthe III–V semiconductors in terms of growth quality anlattice match, is also the one that has received the mostensive investigation over the years. Although the early wby Dingleet al.720,721suggested that nearly all of the discotinuity was in the conduction band, later measurementstablished the well-known 65:35 split between the conductand valence bands, respectively.717 Batey and Wright exam-ined the full range of AlGaAs compositions and found thalinear variation of the band offset with Al fraction fitted thresults quite well.722 Although other reports have impliedslight deviation from linearity,723 we will take the small bow-ing to occur entirely in the CBO. With this assumption, waverage the results obtained for GaAs/AlGaAs by variomeasurement methods717,724–743to obtain the relative VBObetween GaAs and AlAs. The result isDEv50.53 eV50.34DEg , which is the best known value for all of thIII–V semiconductors and is well within the uncertainty limits of most experiments.

Ref. 717 presented a compilation of early theoreticalsults for the GaAs/AlAs band offset. While our composexperimental value agrees quite well with the predictionsfirst-principles calculations by Christensen,744 Lambrechtet al.,745 and Wei and Zunger,746 the model-solid theory ofVan de Walle129 obtained a slightly larger offset ofDEv50.59 eV and the transition-metal impurity theory of Langet al. yielded a smaller value of 0.453 eV.747 Similarly smallvalues were also predicted by the dielectric midgap enemodels of Cardona and Christensen748 and Lambrecht and

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Segall.749 Offsets ranging from 0.36 to 0.54 eV were otained by Wanget al. from the alignment of the averagbonding–antibonding energy on the two sides ofheterojunction.750 The interface dipole theory of Ohleret al.yielded the lowest of the recently reported values,DEv50.38 eV.751 Considering the uncertainties involved in reably calculating band offsets, it must be concluded thatbulk of the theoretical work agrees reasonably well withexperiments.

B. GaInAs ÕAlInAs ÕInP

The other well-studied group of heterojunctionsGa0.47In0.53As/Al0.48In0.52As/InP. This system is especiallinteresting in that it provides a direct test of the degreeoffset bowing, since the lattice-matched GaInAs and AlInalloys have nearly identical InAs fractions. In light of thwell-established GaAs/AlAs result, the expected VBO in tabsence of any bowing would beDEv50.25 eV ~with DEc

50.46– 0.465 eV!. However, both the early evidence summarized by Yuet al.717 and subsequent data have constently favored a CBO in the 0.50–0.53 eV range. For eample, CBO determinations include 0.52 eV by Welchet al.from low-temperature PL studies,752 0.505 eV by Satzkeet al. from electroabsorption data,753 0.53 eV by Morriset al.from Schottky diode transport,754 0.51–0.52 eV by Baltaget al. from photoreflectance data on single quantuwells,755,756 0.49 eV by Huang and Chang from an admtance spectroscopy technique,757 and 0.5 eV by Lugandet al.from photocurrent spectroscopy.758 By carefully modelingthe results of PL measurements, Bohreret al. suggestedDEc50.504 eV,759 and a similar result was reported bHuang and Chang on the basis of capacitance–vol(C–V) and current–voltage–temperature data.760 From anXPS study of the Ga0.47In0.53As/Al0.48In0.52As junction, Wal-drop et al. obtained 0.22 eV for the VBO,761 while Tanakaet al. measured the same result using photocurrspectroscopy.762 Hybertsen derived a VBO of 0.17 eV fromfirst-principles calculations,763 which was in good agreemenwith earlier theoretical determinations~0.14–0.21 eV!.129,748

A CBO of 0.516 eV was determined from tight-binding caculations of Shen and Fan.470 While Seidelet al. observednontransitivity for the GaInAs/AlInAs CBO using a combnation of internal photoemission, current–voltage (I –V)measurements, and PLE spectroscopy~values of 0.5 and 0.64eV were obtained for the two possible growth sequences764!,further investigations of that effect are called for.

Based on this broad and remarkably consistent varietexperimental and theoretical determinations, we recomma composite result of 0.52 eV~0.19 eV! for the CBO~VBO!at the Ga0.47In0.53As/Al0.48In0.52As heterojunction. Makinguse of the already determined GaAs/AlAs VBO, we can aderive thedifferencebetween the offset bowing parametefor the two alloys, which is found to be 0.26 eV. Thishigher than the value of 0.048 eV suggested by the averbond-theory results of Zhenget al.765 It will be seen belowthat by correlating with the data for other heterojunctions,can further determine how this bowing is distributed betwethe two alloys.


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Using the Ga0.47In0.53As/Al0.48In0.52As results, we canalso extract reliable band offsets for the Ga0.47In0.53As/InPand Al0.48In0.52As/InP heterojunctions. On the basis of eperimental reports up to 1991, Yuet al.717 suggested that theCBO for Ga0.47In0.53As/InP constituted approximately 40%of the total band gap discontinuity of 0.608–0.616 eV. Thassignment is in excellent agreement with the later workBohreret al.686,759Slightly smaller CBOs of 0.22 and 0.2 eVwere obtained by Lee and Forrest766 and Guillot et al.,767

respectively, usingC–V techniques. A much larger CBO o0.41 eV was reported from the results of absorption spectcopy by Koteles,732 and the XPS measurements of Waldret al. yielded a VBO of 0.34 eV~corresponding toDEc

50.27 eV!.761 Theoretical work by Van de Walle129 andHybertsen763 indicated CBOs in the 0.2–0.26 eV range.

A number of works have reported nontransitivity of thGa0.47In0.53As/InP band offset. Landesmanet al. used ultra-violet transmission spectroscopy to study the junction afound a considerable~180 meV! difference between the offsets resulting from the two possible growth sequences~withdifferent interface bond types!.768 The average VBO of 0.35eV obtained in that study is in good agreement with tresults given in other reports. A smaller noncommutativity86 meV was also reported by Seidelet al.,764 whose averageCBO of 0.23 eV once again agreed reasonably well withexperiments that did not find transitivity violations. The issof noncommutativity at heterojunctions with no common aion across the interface is controversial from the theoretpoint of view as well.763,769,770A first-principles pseudopo-tential calculation by Hybertsen763 found the GaInAs/InPoffset to be transitive to within 10 meV, whereas the seconsistent tight-binding model of Foulon and Priester769

yielded a 60 meV difference, depending on whethergrowth sequence employed InAs-like or GaInP-like interfabonds. Since the evidence for appreciable noncommutatiis inconclusive at this point, we do not specify an interfacbond-type dependence of the band offset in this reviewfuture work confirms noncommutativity for a particular heerojunction, our recommended offset values can still be uas long as they are considered to be an average for thebond-type combinations.

Our recommended composite VBO for thGa0.47In0.53As/InP heterojunction is 0.345 eV, irrespectivethe growth sequence, which corresponds to a CBO of 0.eV. With the assumption of transitivity, we also recommea VBO of 0.155 eV for the staggered Al0.48In0.52As/InPinterface. This value is near the lower end of t0.11–0.31 eV range of values reported in tliterature.466,472,748,763,759,769–774One group reported noncommutativity with a 53 meV band-offset difference for thheterojunction.764,775

C. Strained InAs ÕGaAs ÕInP and related ternaries

In~Ga!As/GaAs is another important heterojunction this used in a wide variety of electronic and optoelectrodevices. However, direct measurement of its band offseconsiderably complicated by the high degree of strain twas not present in the more straightforward GaAs/AlAs a


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The x-ray photoemission spectroscopy measuremenHwanget al.776 and Kowalczyket al.777 predicted VBOs of0.1160.05 and 0.1760.07 eV, respectively, for the bulk, unstrained InAs/GaAs junction. Similar measurements by Wdropet al.778,779implied a consistent offset value of 0.12 eValthough it is somewhat questionable whether strain wcompletely relieved by dislocations in all of those studies777

Correlating the results of Hirakawaet al.780 for strainedInAs/GaAs heterojunctions with our assumed deformatpotentials, we obtain an average VBO of 0.365 eV. A simistudy by Ohleret al.781 claimed good agreement with thmodel-solid theory of Van de Walle,129 yielding an offset of0.28 eV.

Early results for strained InGaAs/GaAs were summrized by Yuet al.717 Although an explicit value for the banoffset was not derived in most works, Menendezet al. ex-trapolated DEv50.49 eV for the InAs/GaAs junction.782

More recently, Hrivnak pointed out the consistency of seeral experiments with an unstrained conduction~valence!band offset of 0.69~0.38! eV.783 Numerous reports of banoffsets in InGaAs/GaAs quantum wells have bepublished.732,784–801Since many of those values were founnot to be a constant fraction of the band gap, extrapolatinInAs/GaAs is of doubtful validity. Relative CBOs~with re-spect to the gap difference! tended to be in the 0.57–0.9range. Variation of the offset with growth direction was alreported.802

InAs/GaAs band offsets have also been determined foptical measurements on very thin InAs layers imbeddedGaAs, although strain effects must again be subtracted fthe offsets that were actually measured for heavy and lholes.803 The recent results of Brubachet al. are consistentwith an unstrained VBO of approximately 0.22 eV,804 and atheoretical fit of PL data indicated an even smaller value0.08 eV.805 A combination ofC–V and deep level transienspectroscopy~DLTS! measurements yielded 0.69 eV for thstrained CBO.806 On the other hand, another DLTS studyInAs/GaAs self-organized quantum dots found a CBO0.341 eV.807 A study of InAs/AlAs superlattices produced aunstrained VBO of 0.5 eV~implying a nearly null VBO forInAs/GaAs!, although the error bounds were rather largethat result.808 I –V measurements on relaxed InAs/GaAsterfaces yielded 0.34 eV for the VBO.809

Considering the wide spread in the experimental offsfor In~Ga!As/GaAs, it is useful to compare with theoreticfindings. The Schottky-barrier arguments of Tersoff led tooffset of 0.2 eV,810 while the transition-metal impurity theorof Langeret al. yieldedDEv50.33 eV.747 Using midgap en-ergy levels as a point of reference, Menendez811 predicted a


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VBO of 0.13 eV. The model-solid theory of Van de Walle129

predictedDEv50.28 eV, and Ohleret al. obtained a slightlylarger result of 0.33 eV.751 The common-anion rule,717 asreformulated by Wei and Zunger for the Ga/In cation papredicts a much smaller VBO of 0.06 eV.746

Weighing the large number of experimental and theorical findings together, we conclude that the VBO for tunstrained InAs/GaAs heterojunction is most likely in t0.1–0.35 eV range. We recommend a composite value0.21 eV.

X-ray photoemission spectroscopy has yielded 0.31for the VBO at the InAs/InP heterojunction.779 A somewhatsmaller offset of 0.27 eV was reported on the basis of optdata.812 Similar offsets were deduced from InAsP/InP quatum well measurements,573,813–816although a considerablylarger VBO was found in one study.817 An unstrained VBOof 0.39 eV was determined from PL measurements by Dseix et al.,818 and 0.41 eV was obtained from fits to the Pdata for very thin InAs/InP quantum wells.819 The model-solid theory of Van de Walle129 predicted a VBO of 0.44 eV,whereas Ohleret al.751 and Tersoft810 calculated smaller val-ues of 0.26–0.27 eV. In order to be consistent with the mjority of investigations, an intermediate VBO value of 0.3eV is recommended for the unstrained InAs/InP heterojution.

Assuming transitivity, the discussion above implies ththe VBO for the unstrained GaAs/InP heterojunction shobe 0.14 eV, which is roughly consistent with the resultx-ray photoemission spectroscopy~0.19 eV!.779 With theseresults in hand, we can calculate VBO bowing parametersthe GaInAs and AlInAs alloys, and find20.38 and20.64eV, respectively. Note that the negative sign for the VBbowing parameter is consistent with the reduction of theergy gap below the virtual-crystal approximation in alloys

Accurate measurements of the VBO at heterojunctiocombining GaAsSb with GaInAs and AlInAs have also beperformed.537,541,820–822In Ref. 823, the CBO appears relable, but the VBO was determined incorrectly, the accurresult being 0.48 eV. These data are reasonably well cverged for the case of GaAsSb lattice matched to InP,imply a large VBO bowing parameter of21.06 eV. GaPSblattice matched to InP has also been studied and tentatifound to exhibit a VBO of 0.5 eV, which is roughly consistent with a linear variation of the VBO in this alloy~nobowing!.581 A VBO of 0.28 eV has been reported for thInAlAs/AlAsSb heterojunction lattice matched to InP~with atype-II staggered alignment!.561 On the basis of this resultwe tentatively assign a large bowing parameter of21.71 eVto the AlAsSb alloy.

D. GaInPÕAlInP ÕGaAs

The band gap for the lattice-matched alloy Ga0.51In0.49Pis 0.49 eV higher than that of GaAs in the temperature raup to 300 K. An early Shubnikov–de Haas experimeyielded a CBO of 0.39 eV.824 Watanabe and Ohba measura smaller CBO of 0.19 eV using the conduction–voltage pfiling technique,825 which is in good agreement with similastudies by Raoet al.,826 Leeet al.,827 and Fenget al.,828 and


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with the DLTS measurements of Biswaset al.829 Otherexperiments found that the VBO consitutes an even larfraction of the band gap discontinuity (DEv50.32– 0.46 eV).830–837 One study obtained a small CBOfor the GaInP/AlGaAs heterojunction in the low-Al-fractiolimit,838 while other recent reports have found values intmediate between the two limits.839–841Considering the per-sisting disagreement, we have averaged all of the availdata points to obtain a composite recommended VBO of 0eV for Ga0.51In0.49P. This value is near the median of threported range, and is consistent with the latest studies.lon et al. predicted noncommutativity for the band offsetthe GaInP/GaAs interface.842

The important GaInP/Al~Ga!InP heterojunction forwhich both constituent materials are lattice matched to Ghas also been investigated extensively. The earlieststudy843 assigned 57% of the total GaInP/AlInP band-gdiscontinuity of 0.68 eV to the VBO. Subsequent PL aPLE studies489,844 found the offset to be 35% and 25%, rspectively. The smaller values were confirmed by the widcited report of Dawson and Duggan who found a VBO33%,493 as well as Kowalskiet al. ~35%!,845 Interholzingeret al. ~31%!,501 and Zhanget al. ~32%!.846 Whereas all ofthose authors considered a quaternary on one side ofheterojunction, the GaInP/AlInP junction was also invesgated and found to exhibit506,847 DEv50.24 eV in goodagreement with the other reports. Two studies498,848consid-ered the full composition range of the quaternary and fouan offset of 0.22 eV in the AlInP limit, with evidence forVBO bowing parameter of 0.157 eV. On the other hanphotoemission measurements at a considerably larger nber of compositions yieldedDEv50.305 eV and no substantial deviation from linearity.849 The VBO obtained fromDLTS ~0.36 eV!850 was 53% of the total band gap disconnuity. On the other hand, the combined PL, PLE, and ptoreflectance~PR! investigations of Ishitaniet al.851 gaveonly 25% ~0.17 eV!. It is unclear why these latter studieappear to disagree with the converging PL data on whichbase our recommended value ofDEv50.24 eV ~35% ofDEg!. Two available studies of the GaAs/AlInP heterojuntion put the VBO at 0.62–0.63 eV.825,839Transitivity impliesa GaInP/AlInP VBO of 0.31–0.32 eV, although the disagrement with the smaller recommended result is nearly witthe stated error bounds. We recommend that the unconfirVBO bowing for the AlGaInP quaternary be disregarded.

E. GaP and AlP

Both experimental and theoretical VBOs have beenported for the nearly lattice-matched GaP/AlP heterojution. C–V profiling,852 x-ray photoemission spectroscopy,853

and PL854 indicatedDEv50.41, 0.43, and 0.55 eV, respetively. A wide variety of theoreticalcalculations129,746,748–750,770,855,856produced results fallingmostly in the 0.34–0.69 eV range. These values can be cpared directly with the GaInP/AlInP VBO, since the Infraction in the latter does not change across the interfaLinear extrapolation yields an offset of 0.47 eV for the GaAlP interface. In view of the experimental uncertainty a


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the small strain correction, we recommend this value. Fther inference is that the VBO bowing parameters for Gaand AlInP are nominally equal~the simplest assumption ithat they both vanish!. A linear variation of the VBO may betaken for the AlGaP alloy.852,855

Experimental and theoretical studies of the GaAs/Gband alignment are also available. Some extrapolated fthe GaAsP alloy with the assumption that there is no offbowing. Katnani and Margaritondo used photoemissiondetermine an unstrained VBO of 0.63 eV,857 while Gourleyand Biefeld obtained 0.6 eV from PL/PLE measurements858

The assumption by Pistolet al.859,860 that nearly all of thelow-temperature band gap discontinuity of 0.54 eV is in tvalence band, while theX-valley conduction minima line upin the two unstrained materials, was corroborated bymeasurements. Another recent optical study861 obtainedDEv50.6 eV. On the other hand, optical experimentsGaAs/GaAsP quantum wells found a much smaller VBO0.38–0.39 eV.862,863 Large CBOs were derived by many othe same authors for GaAsP/GaP quantum wells.864 A smallextrapolated VBO of 0.28 eV was also proposed by Shet al. on the basis of pressure-optical measurements865

Pseudopotential calculations in conjunction with PL mesurements performed by Neffet al.866 imply a VBO of 0.41eV, which is similar to the value calculated by Di Ventet al.867 Experimental and calculated band offsets for tGaAsP/AlGaAs heterojunction with small P and Al fractiohave also been reported,868,869although extrapolation to GaPmay be problematic in that case. Varioutheories129,746,870–873found values in the 0.19–0.63 eV rangAveraging the reported experimental values we obtain aommended GaAs/GaP VBO of 0.47 eV, which is also in tmiddle of the theoretical range and equal to recent fiprinciples calculations.746 This value is also fully consistenwith an independent report of 0.6 eV for the VBO at thGaP/InP heterojunction.874 In view of the GaAs/InAs, InAs/InP, and GaAs/GaInP offsets derived above, it is appathat the VBO bowing for GaInP~and by implication AlInP!can be assumed to vanish.

F. GaSbÕInAs ÕAlSb

Since the InAs/GaAs band offset was already establisabove, we can use the alignment for the nearly lattimatched InAs/GaSb and InAs/AlSb heterojunctions to foa link between the antimonides and the GaAs-based andbased systems.

We first focus on the GaSb/AlSb heterojunction, whihas a type-I band alignment. Early results suggesting a sVBO were reviewed by Yuet al.717 However, Tejedoret al.875 deducedDEv.0.27 eV from a comparison of resonant Raman scattering experiments with tight-binditheory. Using x-ray photoemission spectroscopy, Gualtet al.876 reportedDEv50.40 eV with relatively large~38%!error bounds. While that finding was later disputedLey,877 who suggested a revision to 0.27 eV, the former athors responded by insisting on the accuracy of thresult.878 Menendezet al. obtainedDEv50.45 eV using alight-scattering method.879 Cebullaet al. found 0.35 eV from


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absorption and excitation spectroscopy,880 Yu et al. mea-sured 0.39 eV using x-ray photoelectron spectroscopy,881 andShen et al. obtained 0.23 eV from PR measurementsGaSb/AlSb quantum wells.882 Chenet al. deduced a VBO of0.48 eV from a comparison of band structure calculatiowith tunneling measurements on InAs/AlSb/GaSb singbarrier interband diodes.883 Leroux and Massies884 derivedan unstrained offset of 0.360.075 eV from fits to the PLspectra for GaSb/AlGaSb quantum wells. Yuet al. notedpreviously717 that most of the later experimental studies coverge on an average value of 0.38 eV, which is our recomendation as well. More recent theoretical studies haveproduced results in the 0.30–0.49 erange.129,745,746,749,750,770,870,885,886Theory predicts a lineavariation of the offset in the AlGaSb ternary.887

The InAs/GaSb junction has a type-II broken-gap lineuwith the bottom of the conduction band in InAs being lowthan the top of the valence band in GaSb. This unusual balignment has stimulated numerous investigations of typsuperlattices and quantum wells. For example, the juncexhibits semimetallic properties as long as quantum confiment and field-induced energy shifts are not too strong.

There have been many theoretical calculations of thelence band offset at the InAs/GaSinterface,129,717,744–746,748–750,769,770,810,888,889the most reliabletending to yield results in the 0.43–0.59 eV range. Sevemodels predicted that the VBO should be larger for the Inlike interface bond type than for GaAs-like interfaces. Folon et al.,769 Dandreaet al.,890 and Montanariet al.888 re-ported an average difference of 35 meV.Ab initio moleculardynamics calculations of Hemstreetet al. yielded a differ-ence as large as 150 meV due to varying interface chadistributions,891 later refinements in the calculation reducthis difference to 40 meV.892

Experimentally, energy gaps were measured to be 25meV lower in InAs/GaSb superlattices with InSb-like inteface bonds than in structures with nominally identical laythicknesses and GaAs-like bonds.893–895Based on a magnetotransport study the Naval Research Laboratory~NRL! con-cluded that the valence band offset is 14 meV largerInSb-like bonds.896 The Oxford University group obtained 4meV897 and 30 meV898 differences in two studies, whereaWanget al. did not observe any variation of the VBO witbond type.899 It is possible that the offset may dependdetails of the interface structure produced by particugrowth conditions. Since most of the experimental and toretical studies have found a small but real dependencebond type, we recommend that the ‘‘average’’ InAs/GaVBO specified below should be applied only to cases whno particular bond type was forced in the growth. For InSlike bonds 15 meV should be added to that result, whileGaAs-like bonds 15 meV should be subtracted.

Experimental band offset determinations for this marial system fall into two main classes:~1! direct determina-tions, and~2! fits to the measured optical transitions in semconducting superlattices and quantum wells~i.e., in whichthere is enough quantum confinement to induce an engap!. Among the former class of results, Gualtieriet al. ob-tained a VBO of 0.5160.1 eV using x-ray photoemissio


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spectroscopy.900 Whereas Srivastavaet al.901 measuredDEv50.67 eV fromC–V profiling of a GaSb/InAs0.95Sb0.05

heterojunction, Mebarkiet al.902 obtained 0.36 eV for aGaSb/InAs0.89Sb0.11 interface using essentially the samtechnique. X-ray photoelectron spectroscopy yielded 0eV.903 For an InAs/GaSb long-period superlattice at 4 K, tOxford group measured band overlap energies of 100–meV, depending on the interface bond type.897 That thosevalues increased by 30 meV at room temperature impliesaverage VBO of 0.52 eV, with little temperature variatio~,8 meV!. Another study by the same group producedaverage offset of 0.54 eV.898 Wang et al. found a 90 meVvariation of the VBO based on growth sequence~InAs onGaSb versus GaSb on InAs!.899

Next, we discuss a previously unpublished applicationthe second approach. Experimental optical transition engies for a variety of semiconducting type-II superlattices aquantum wells893,904–934have been fit to theory, in this casan eight-band finite-element methodk"P algorithm.57 Theimportant strain effects are included and the recommenband structure parameters in Tables III, VII, and XVIII aemployed, with the only fitting parameter being the InAGa~In!Sb interface VBO that is taken to be consistent wthe recommended GaSb/InSb VBO discussed below. Mosthe data are from PL and mid-IR laser experiments at teperatures in the 4.2–300 K range, and energies on the oof kT have been subtracted from the PL emission peaks.

Results for the fitted VBO, plotted as a function of eergy gap for the given structure, are presented in Fig. 13.various types of points correspond to data for structugrown by different groups, which are identified in the cation. The range of fitted VBO values is surprisingly largspanning more than 200 meV. Especially noticeable aresubstantial systematic differences between the offsetsrived using data from different groups~or sometimes within

FIG. 13. InAs/GaSb valence-band offsets derived from fits to energy gmeasured for various type-II quantum well and superlattice structures8-bandk"p finite-element algorithm was used to calculate the offset corsponding to each energy gap when all other band structure parametesumed their recommended values. The data are taken from Refs. 904~solidsquares!, 905~solid circles!, 906~solid upright triangles!, 911, 912, 931, 933~solid inverted triangles!, 907, 908, 910, 913, 914, 916, 918, 920, 922, 92926 ~crosses!, 915, 921~multiplication signs!, 909 ~diamond!, 917 ~openinverted triangle!, 924, 925, 927, 929~open circles!, 930 ~open uprighttriangles!, 932 ~open squares!, and 919~star!.


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the same group for different types of structures or for strtures grown during different time intervals!. Possible impli-cations include:~1! the growth conditions actually influencthe offset,~2! the growth-dependent nonideality of the struture, e.g., related to interface roughness or anion crosstamination, has a larger-than-expected effect on the engap, or~3! the layer-thickness calibrations have large unctainties. For example, while the energy gaps correspondto growths at the Fraunhofer Institute915,921 and HRLLaboratories907,908,910,913,914,916,918,920,922yield an averageVBO of 0.53 eV and structures grown at the UniversityHouston924,925,927,929yield 0.54 eV~all with fluctuations of atleast 100 meV!, the band gaps corresponding to NRgrowths911,912,933imply a much larger average VBO of 0.6eV. In some cases the differences are as large as 100for structures that are nominally quite similar. Such diffeences are well outside the usual bounds of experimentalcertainties in the spectral data and layer thicknesses. A rtively strong dependence of the energy gap on grotemperature for nominally identical structures has also breported.935 Although the primary mechanism has not beisolated, subsequent TEM measurements showed that thing dislocations are present in the samples grown aboveoptimal growth temperature, and cross-sectional STM incated greater interface roughness.936

Averaging all of the band offsets displayed in Fig. 1along with the direct determinations summarized above,obtain a composite InAs/GaSb VBO of 0.56 eV. This valis somewhat larger than the earlier suggestion of 0.51 eV717

and is at the high end of the 0.51–0.56 eV range thausually employed in modeling antimonide materials. Sinthe fluctuations in reported values are so large for this sysat its current level of understanding, it is advisable to accofor the source and type of the structure in deciding whoffset to use. Again, the recommended 0.56 eV represent‘‘average’’ value that should be corrected for the interfabond type.

The transitivity rule, applied in conjunction with the reommended InAs/GaSb and GaSb/AlSb band offsets, impan ‘‘average’’ VBO of 0.18 eV for the staggered InAs/AlSheterojunction. Bearing in mind that the transitivity assumtion may be of questionable validity when applied to systewith more than one anion,769 we can examine whether thavalue is consistent with the direct experimental and theoical evidence for this interface. Nakagawaet al.937 employedC–V profiling to obtain a CBO of 1.35 eV between thG-valley minimum in InAs and theX-valley minimum inAlSb. Using our recommended band gap parameters,corresponds to a VBO of 0.09 eV. More recently, that wowas extended to AlAsSb barriers.938 The findings are in reasonable agreement with the previously assumed large V~;2 eV! in AlAsSb. A PL study by Yanget al. found thatthe band alignment between InAs12xSbx /AlSb quantumwells becomes type I whenx.0.15.939 In that study, InAsSbwas realized as a digital superlattice. If all the offsetsreferenced to the InSb valence band maximum, this impthat the AlSb VBO is no more than 85% of the InAs VBOUsing x-ray photoemission spectroscopy to study interfawith InSb-like and AlAs-like bonds, Waldropet al. obtained


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an average VBO of 0.18–0.19 eV and derived a difference60 meV between the two bond types.940 However, a first-principles calculation by Dandrea and Duke941 found no de-pendence of the VBO on bond type, and suggested that Wdrop et al.940 misinterpreted their data by discarding thsmaller offset obtained for an AlAs-like bond in one expement. Other calculations129,717,745,746,749,750,770,810,871,873,8

have foundDEv spanning the wide range between 0 and 0eV, with the most reliable values clustered around 0.05–0eV. The combined experimental and theoretical evidencepear to offer no compelling reason to reject the transitivrule when applied to the antimonide heterojunctions. We arecommend neglecting any dependence of the InAs/AVBO on interface bond type.

G. GaSbÕInSb and InAs ÕInSb

The corrected common-anion rule predicts a negligiband offset for the unstrained GaSb/InSb heterojunction746

Other theories have produced values in the range20.08 to0.16 eV129,810,872,873,889,942,943~where we take the sign to bpositive if the valence band maximum is higher in InSb!. Inexperimental practice, this offset must be derived from msurements on heterojunctions such as GaSb/Ga12xInxSb andInAs12ySby /InAs12xSbx due to the large lattice mismatcbetween the two binaries. For an InAs12ySby /InAs12xSbx

junction in the Sb-rich limit~largex andy51!, initial opticalmeasurements supported an unstrained VBO(0.36– 0.41)(12x) eV.944,945 In the As-rich limit ~small xandy50!, on the other hand, Liet al.concluded from fits tomagneto-optical spectra that the top of the InAsSb valeband appeared lower than that in InAs (20.84x eV).946,947

That result implied a substantial CBO and negative VBOsmall x.948,949 Wei and Zunger pointed out that a negatiVBO for InAs/InSb directly contradicts other theoretical anexperimental evidence.549 Other experiments950–952 are inmuch better agreement with the usual theoretical findingan essentially null CBO, which yields a type-I or type-staggered alignment depending on details of the strainordering conditions. All of the above works found that thunstrained VBO in InAs12xSbx depends almost linearly oncompositionx. While an unstrained VBO of 1.1 eV waderived for ultrathin InSb embedded in InP,953 one cannotplace high confidence in that result owing to the difficultyprecisely accounting for the very large strain effects.

A few groups954–956have used PL, electroluminescencand photoconductivity to measure VBOs for GaInSb/Gaquantum wells. Those studies assumed a linear variatiothe Ga12xInxSb VBO with alloy composition, although inone study957 the PL peak energies could not be fit usingreasonable dependence. Finally, a recent study of exctransitions in InSb/AlInSb strained quantum wells producheavy-hole offsets that were 38% of the total band gapcontinuity for Al fractions below 12%.958 Unfortunately, ex-trapolation to the InSb/AlSb VBO from the results of thwork may not be justified.

Accounting for the effects of strain and averaging tvarious results, we recommend a GaSb/InSb VBO0.03 eV.


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H. Quaternaries

Band offsets involving lattice-matched quaternaries halso been studied both experimentally and theoretically959

See Sec. V A 1 for a discussion of the AlGaInP/GaAs hetejunction.

Cho et al. considered the GaInP/GaInAsP interface ltice matched to GaAs, and suggested a linear variation ofCBO with quaternary composition.960 Since the direct GaIn-AsP band gap appears to be nearly independent of comption, this implies a linear variation of the VBO as well. However, note that the value employed for the absolute gapsomewhat different from our recommendation. The samethors also investigated the band offset transitivity in AlGaAGaInP/GaInAsP heterostructures lattice matched to GaA961

The results are in good agreement with our suggested offprovided a linear variation is assumed for each constituThe small direct-gap bowing parameter and the resultscussed above also imply that any bowing in the VBO forAlGaInP alloy should be small.

Sugawara has theoretically predicted a slight bowingthe position of the valence-band maximum for GaInAsPInP.962 Experimentally, a VBO of 0.08 eV was measured fGa0.13In0.87As0.29P0.71/InP, as compared to 0.10 eV from linear interpolation.963 Although error bounds were not citedthis discrepancy is probably within the experimental unctainty. Forrestet al.964 found evidence that the CBO bowinis at most small, which, in combination with the small bagap bowing, implies a negligible VBO bowing. Soucailet al.found a small bowing parameter of 0.09 eV, which is barlarger than the estimated experimental uncertainty.965

The band offsets for AlGaInAs on InP have also bestudied both experimentally and theoretically.470,686,966–968

There is no clear agreement as to whether most of the bgap bowing, which is rather small in any event, shouldassigned to the CBO or the VBO. We recommend a lininterpolation between the dependence for GaInAsAlInAs, both of which exhibit appreciable VBO bowing.

Recently, a series of three articles reported determtions of the offsets for GaInAsSb and AlInAsSb, latticmatched to InP, from fits to low-temperature Pmeasurements.688,969,970All of the results agree with lineainterpolations of our recommended values~with appropriateVBO bowing parameters! to within the cited experimentauncertainty.

Calculations of the band offsets for AlGaAsSb quaterries lattice matched to various substrates have breported.559 In particular, Polyakovet al. obtained a VBO of0.15 eV for AlGaAsSb on GaSb.971 Rather than using thisvalue to derive the bowing parameter for AlGaAsSb, weterpret it as reflecting the uncertainty in the bowing paraeter for AlAsSb. The study on the quaternary implies a boing parameter of 2.4 eV, which is slightly larger than tassumed value of 1.7 eV. We recommend the VBOs forGaAsSb obtained by linear interpolation between the ternrelations.

Mikhailova and Titkov have reviewed band offset resufor the GaInAsSb quaternary on InAs and GaSb~up to1994!.699 A theoretical work by Nakaoet al. predicted some


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bowing in the VBO.959 Results of the direct experimentastudy of GaInAsSb/GaSb by Mebarkiet al.972 are in goodagreement with a linear interpolation of our recommendoffsets for InAsSb and GaSb.C–V measurements of Polyakov et al. are also consistent with that procedure.973 The lin-ear interpolation approach differed appreciably only with tfindings of Baranovet al.974 and Afrailovet al.,702 for whichthe discrepancies are opposite in sign. Considering all ofreported experimental results, we see no compelling reato introduce bowing. The VBOs have also been reportedthe GaInAsSb/AlGaAsSb heterojunctions lattice matchedGaSb.975 The results for both In-rich and In-poor junctionindicated reasonably good agreement with a linear interption. Band offset measurements have also been reportedthe strained GaInAsSb/AlGaAsSb system.976,977 Except forthe case of GaInAsSb, a tight-binding calculation959 supportsour conclusion that the bowing in these quaternaries is nligible.

I. GaN, InN, and AlN

We emphasize at the outset that the band offsets recmended below for the nitride system should be understoohave large uncertainties. This is primarily because theported results have rarely included the full effects of residstrain, spontaneous polarization, and piezoelectric fieldscompletely satisfying manner. In particular, the presencemacroscopic polarization can render the notion of referebulk energy levels somewhat ambiguous. The most signcant future advancements in the understanding of nitridesets will probably be connected with that issue.

Since the nitride materials typically crystallize in thwurtzite form, an obvious question is how the valence baedge of that phase lines up with the zinc blende phase ofsame compounds. The issue was addressed by Murayand Nakayama using a first-principles pseudopotencalculation,978 which projected VBOs for the two phases thdiffered by only 34 meV in GaN and 56 meV in AlN~noresults were quoted for InN!. Those VBOs are much smallethan the error bounds for current experimental determitions of the nitride offsets. Wei and Zunger326 calculated asimilar difference of 30 meV between the wurtzite and ziblende forms of the GaN/AlN interface, and also a differenon the order of 0.2 eV for GaN/InN and AlN/InN heterojuntions. While the effects of macroscopic polarization were nincluded, these results provisionally allow us to align twurtzite and zinc blende materials on an absolute enescale. Furthermore, study of the GaAs/GaN~zinc blende!heterojunction has provided a tentative connection withother III–V materials. X-ray photoemission spectroscoyielded a VBO of 1.8460.1 eV,979 which is the only avail-able direct measurement and our recommended value.implication of a staggered alignment withDEc50.03 eVstrongly disagrees with the finding ofDEv50.5 eV and alarge positive CBO by Martinet al.980 and Huanget al.981

from electrical measurements. The discrepancy may belated to strain and other uncertainties in the electrical studThe investigations of the VBO in the GaAsN/GaN heterjunctions yielded a range of different results, and the N fr


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tion was too low to extrapolate the GaAs/GaN VBO wiany confidence.982–984The observed VBO was found to bquite small, with the question of type-I versus type-II aligment remaining somewhat controversial.

Theoretically, the GaAs/GaN VBO was first estimatby Harrison and Tersoff.985 Their result of 2.21 eV is quitesimilar to the value of 2.18–2.28 eV recently suggestedBellaiche et al.636,746 A first-principles linear-muffin-tin-orbital calculation by Agrawalet al. yielded 1.86 eV for thefree-standing ~i.e., strain-free! GaAs/GaN superlattice.986

Those authors also predicted a strong dependence on thterface properties. In view of the present uncertainties,agreement between experiment and theory should be coered quite good~apart from the electrical studies!.

The valence-band discontinuity at the~zinc blende!GaN/AlN interface was first probed experimentally by Siet al.,365 who obtained 1.4 eV from fits to optical measurments on GaN/AlN superlattices. Subsequently, Bauret al.found a VBO of 0.5 eV by measuring the difference betwethe acceptor levels of iron in each material.987 X-ray photo-emission spectroscopy yielded a VBO of 0.8 eV at the wuite GaN/AlN junction,988 which was revised to 0.70 eV inlater article by the same authors.989 Using the same approach, Waldrop and Grant found a considerably differvalue of 1.36 eV.990 Those authors also reported a nealinear VBO variation in the AlGaN alloy, with a positivebowing parameter of 0.59 eV.991 Using x-ray and ultravioletphotoelectron spectroscopy, Kinget al. found that the GaN/AlN VBO ranged from 0.5 to 0.8 eV, depending on thgrowth temperature.992 They surmised that the differencearose from strain, defects, and film stoichiometry effectsVBO in the 0.15–0.4 eV range was reported by Rizet al.,993 who pointed out that the Ga 3d core level, whichhas been used as a reference in GaN, is in fact hybridwith other valence bands.

On the theoretical front, calculations were performedAlbanesiet al.994 and Chenet al.995 for the zinc blende AlN/GaN interface, and by Sattaet al.,996 Ke et al.,997 and Weiand Zunger746 for the corresponding wurtzite junction. All othese works obtained VBOs in a rather narrow range fr0.7 to 0.85 eV. The last two articles found almost no diffeence between offsets for the cubic and hexagonal versionthe junction. A slightly lower value of 0.6 eV was calculateby Monch.998 The importance of strain was studied bBinggeli et al. and Nardelli et al. The former foundDEv50.94 eV for the relaxed zinc blende interface,999 while thelatter obtained 0.44–0.73 eV for the strained zinc bleninterface ~depending on the substrate lattice constant! and0.57 eV for the strained wurtzite interface.1000Recently, Ber-nardini and Fiorentini1001 studied macroscopic bulk polarization effects on the interface-charge contribution to the offat the wurtzite heterojunction. They found VBO values0.20 and 0.85 eV corresponding to the underlying GaNAlN lattice constants. However, most of the reported diffence was from band-edge shifts in the bulk band structwith only 0.18 eV due to the interface-charge contributioWe recommend using an unstrained band offset of 0.8 eVboth wurtzite and zinc blende interfaces. This representsmedian of the reported values, and is also closest to the l


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est number of experimental and theoretical results. Howewe caution again that the effects of spontaneous polarizaand piezoelectric fields must be very carefully accountedwhen applying this recommendation to the modeling of rGaN/AlN interfaces.

In comparison with GaN/AlN, the GaN/InN and AlNInN junctions have strongly mismatched lattice constanTheoretical values for the unstrained GaN/InN VBO are 0eV381 and 0.26 eV326 for the zinc blende phase and 0.48 efor the wurtzite phase.326 The valence band in the technologcally important GaInN alloy may be obtained from a lineinterpolation. VBOs of 0.70 and 1.37 eV were calculaterespectively, for the wurtzite GaN/InN and AlN/InN junctions strained to AlN.1000 Strain-induced piezoelectric fieldswhich can depend on the growth sequence, significacomplicate any experimental determination of the GaN/Iand AlN/InN band offsets. Only one experimental studythese two interfaces has been reported to date.989 The VBOresults of 1.05 and 1.81 eV were roughly corrected forezoelectric fields due to residual strains, and it was ccluded that transitivity for the GaN/AlN/InN system iobeyed to within experimental precision. Provisionally, wadopt a VBO of 1.05 eV for wurtzite GaN/InN. The largdisagreement with the intuitive expectation of a small offfor this common-anion heterojunction remains to besolved. Using transitivity, we derive 1.85 eV for the VBO othe wurtzite AlN/InN junction. For the zinc blende GaN/Inand AlN/InN interfaces, we recommend using the resultsWei and Zunger326 ~slightly modified to ensure transitivity!:0.26 eV and 1.06 eV, respectively.

VII. SUMMARY

We have reviewed the available information about bastructure parameters for 12 technologically important III–semiconductors and their ternary and quaternary alloys aclose of the 20th century. Whereas numerous reports of sparameters may be found in the literature, nocompleteandfully consistentset has been published previously. Earlreviews were either restricted to particular material systedid not address all parameters of interest, or were biasethe results of a specific group of investigators. On the othand, our goal has been to provide a comprehensiveeven-handed reference source, as free of internal contrations and significant omissions as is practically possible iwork of this scope. We have also illustrated how the pposed parameters fit into the band structure computation

The semiconductor band parameter knowledge basetinues to expand as new reports are published daily. Wthe most fundamental parameters such as the energyand effective masses are by now fairly well establishedmost of the III–V materials, we anticipate many new avances, particularly concerning band offsets and otherrameters for the less mature systems such as the nitridesantimonides. Furthermore, novel ternary, quaternary,even quinternary alloys continue to be introduced andproved with an eye toward achieving greater flexibility in tdesign of quantum heterostructure devices. In view ofrapid ongoing progress on a broad front, this review sho


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be regarded as a snapshot of the status at a given mometime rather than the final word on III–V semiconductor baparameters. The goal has been to provide a one-stop smary of the present understanding, which will be revisrefined, and augmented by future experimental and theocal investigations.

ACKNOWLEDGMENTS

The authors wish to thank Quantum Semiconductorgorithms for the use of their finite element band structsoftware that was employed to test many of the input nubers. One of the authors~L.R.R.! wishes to thank Dr. TomReinecke and Dr. Ben Shanabrook at the Naval ReseLaboratory, Washington, DC, and Dr. John Loehr andTom Nelson at the Air Force Research Laboratory, DaytOhio, for their hospitality during his sabbatical visits ther

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