H. UNLU and A. NUSSBAUM : Position-Dependent Theory of Heterojunctions 687

phys. stat. sol. (a) 94, 687 (1986)

Subject classification: 73.40; 73.30

Department of Electrical Engineering, University of Minnesota, Minneapolis')

Position-Dependent Theory of Heterojunctions

BY H. UNLU and A. NUSSBAUM

Heterojunction band discontinuities determined by any of the traditional rigid-band models show discrepencies between theory and experiment of a t least &4 kT. I n contrast, a new posi- tion-dependent model based on a more extensive study, (described briefly here) gives agreement with direct measurements to within & 1kT. I n addition, there is a strong correlation with the discontinuity values obtained from Schottky barrier height measurements, although - as ex- pected - this correlation becomes poorer as the lattice or dielectric constant mismatch incieases.

Banddiskontinuitaten von Heteroiibergangen, die mit irgendeinem der traditionellen starren Bandmodelle bestimmt werden, ergeben Diskrepanzen zwischen Theorie und Experiment von wenigsten &4 kT. Im Gegensatz dazu ergibt ein neues positionsabhangiges Model1 (das hier kurz beschrieben wird) auf der Grundlage einer ausfiihrlicheren Untersuchung Ubereinstimmung mit direkten Messungen innerhalb f 1kT. Zusatzlich existiert eine strenge Korrelation mit den Dis- Irontinuititswerten, die aus Schottkybarrierenhohenmessungen erhalten werden, obwohl - wie erwartet - diese Korrelation schwacher wird, wenn die Gitter- oder Dielektrizitatskonstanten- fehlanpassung wachst.

1. Introduction

The currently-accepted models of heterojunctions, based directly on the Shockley rigid-band approach [I to 31 or involving ab initio calculations [4 to 61, do not give agreement with the various kinds of experiments (X-ray photoelectric spectroscopy, thermionic emission, optical absorption, etc.) t o an accuracy any better than + 4 k T , and then only for a small number of structures. (A complete comparison of theory and experiment for several dozen systems is being incorporated in a more extensive report [7].) The theory to be given here (taken from the same report) replaces these six prior models with one whose genesis can be regarded as the James I S ] and Bardeen-Shockley [9] deformation-potential concept, as extended to heterojunctions by Kroemer [lo]. He proposed originally that non-uniformity in impurities, spatial variatioii in composi- tion or elastic strain due to the mismatch between the lattice constants of the two parts of a heterojunction would create effective potential potentials whose gradients would then represent additional forces. More recently, this idea was extended [12] t o perfectly matched heterojunctions, since these position-dependent forces can arise from the differences in electron bonding structures in a very thin (a few Angstroms) region surrounding the ideal, abrupt interface.

2. Formulation of the Model

The effective potential of the electrons in the conduction band of a heterojunction in equilibrium can be expressed as

688 H. UNLU and A. NUSSBATJN

The conduction-band edge E,(x) is position-dependent in the heterojunction transition region (as are all the other variables in (1)) and the electrostatic potential is required to be continuous (but - unlike previous theories [ l t o 31 - remains otherwise un- determined). The term Eexc(x) is the electron exchange-correlation energy, described by Stern and Das Sarma [ l l a] as representing effects beyond those of the self-consist- ent field and explicitly given by them as

(2) (Nc(”))1/3 Eexc(z) = -2.5 x lO-’[l + 0 . 7 7 3 4 ~ ~ log (1 + y;’)] -_--,

& (2)

where

The image force term Ei,(x) has the classical form

and

as applied to a heterojunction composed of homogeneous materials A and B, and assuming a (Ill)-interface, with the image charges about one-fourth of the bond length d from this interface. For this case, d = )/3a/4, where a is the lattice constant; a (100)-plane requires an additional factor of a /4 and a (110)-interface has a factor of VFa/4. The final term in (1) is the interface dipole bond energy, due to interaction of dangling bonds, and expressed as

1 e2 E A = ____ (A-side)

4Z&o&Ad~ and (4)

1 e2 EB = __ (B-side) .

4ZE0 E B d ,

Here we should note that these expressions given by (4) should be multiplied by 1/4, 1/2/4, and )/3/4for (loo), (110), and (111) planes, respectively.

This idea in a different form was originally proposed by Harrison [13]. Similar condi- tions apply to the valence band, with

(5 ) where Ev(x) is the position-dependent valence band edge, the last two terms are the same as above, and the hole exchange correlation potential is

E : f f ( ~ ) = E,(z) + e V ( 4 + E h d x ) + E i m ( x ) + Eb(x) ,

Ehxc(x) = 2.5 x lO-’[l + 0.7734yhlog (1 + y $ ) ] ( N m ) -~ ll3 ,

4%) where

-113 . Here we note that the limited form of (1) and (5) was also used by Asbeck et al. [ll b] (excluding the microscopic effects). Let us temporarily specify that materials A and B

Position-Dependent Theory of Heterojunctions

are both undoped. Then the equilibrium Fermi levels are respectively

689

and

where the first term represents the geometric center of the forbidden gap, the second term is the shift due to the difference in effective masses (as expressed in terms of effective densities of states), and the remaining terms take care of the corresponding shifts due to the microscopic forces introduced above. As one can see from (6) and (7) the equilibrium Fermi level E , is not located at exactly the middle of the energy gap of a real semiconductor. These two Fermi levels must have a common value for an equilibrium heterojunction, so that

(8) and this when combined with (6) and (7) gives

B E$ = Ey = E ,

Using this in conjunction with the definition AE, + AEv == AEc

provides a pair of equations which can be solved for AE, and AE,, the conduction and valence band discontinuities, respectively. I n using (lo),- it is necessary to consider algebraic signs, as determined by the specific way in which the two bands overlap: straddling, staggered, or broken gap, following the nomenclature of Kroemer [12]) These relations have been derived in an alternative way [7] through the use of a modi- fied law of the junction and they are valid for arbitrary non-degenerate impurity con- centrations. Equation (9) implies that the magnitudes of the band discontinuities are concentration-independent, as confirmed by many experiments.

3. Experimental Confirmation Table 1 shows a comparison of the predictions of (9) and (10) with experiments per- formed on systems that have been cited by Kroemer [12] or Wang and Stern [14] as having good lattice matching. The parameters were taken from Casey and Panish, and

- - -

-

I I I I I I I I Fig. 1. AE, (theory) versus AE, (experimental) 0 04 08 72 16

L1 Ev ,,, (eV) - 44 physica (a) 94/2

690 H. UNLU and A. NUSSBAUM

T a b l e 1 Valence-band discontinuities at selected [12, 141 heterojunctions (eV) and their comparisons with metal-semiconductor Schottky barrier heights difference &pbp [6] to EL common metal, Au

heterojunction

GaAs/AlAs( 110)

GaAs/AlAs( 100) GaAs/Ge( 110) AlAs/Ge( 110) ZnSe/GaAs( 110) ZnSe/Ge( 110) InAs/GaSb(llO) AlSb/GaSb( 100) InP/CdS( 11 1 ) Si/Ge(ll l) HgTe/CdTe( 110) GaP/Si( l l l )

experiment

0.54 [16], 0.50 [17]

=0.50 [14] 0.53 [12] 0.95 [18] 0.96 [I21 1.52 [12] 0.51 [12] 0.48 [14] 1.63 1121 0.40 [19]

0.60 [all 1.10 [20]

theory

0.48

0.51 0.51 1.03

20.94 1.54 0.52, 0.52*) 0.49 1.61 0.38 1.12

20.59

0.44

0.48 [14] 0.45 0.89 - -

0.52*) 0.48**)

0.25

0.62

-

-

*) See [22] for discussion; it reports plbn(GaSb/Au) = 0.75 eV at 77 OK which gives qbp = 0.05 eV (using table 5-22 of Casey and Panish [15] and qbp(InAs/Au) =0.47 eV a t 77 OK. Since InAs/GaSb has a broken-gap structure [lZ] &bp = 0.47 0.05 = = 0.52 eV. Assuming the dielectric constants be the same (approximately) as a t 300 O K

we get AE, = 0.52 eV. **) S. M. Sze [15].

S. &I. See [15].2) In addition, in Fig. 1 we compare our theory with the results of Schottky diode experiments,3) where band discontinuities are determined by taking the difference in barrier heights as measured for two different semiconductors with respect t o a common metal. It seems quite obvious that the combination of the posi- tion-dependent idea with the microscopic effects listed leads to a heterojuntion model which can be experimentally substantiated.

Acknowledgement

We wish to acknowledge support for this work by the Microelectronics and Information Sciences Center of the University of Minnesota.

References [l] R. L. ANDERSON, Solid State Electronics 6, 341 (1962). [2] M. J. ADAMS and A. NUSSBAUM, Solid State Electronics 22, 783 (1979). [3] 0. VON Roos, Solid State Electronics 23, 1069 (1980). [a] W. FRENSLEY and H. KROEMER, Phys. Rev. B 16,2642 (1977). [5] W. A. HARRISON, J. Vacuum Sci. Technol. 14, 1016 (1977). [6] J. TERSOFF, Phys. Rev. B 30,4874 (1984).

2, The effective masses for AlAs/GaAs system were taken from Miller e t al. [23]. a) Although [6] and [13] suggest that metal--semiconductor Schottky barrier height differences

may provide a good test of theories (for elemental and 111-V systems), after the comparison with a variety of heterojunctions we find that this is only valid for lattice matched 111-V systems whose dielectric constants are very similar. This is still under investigation and a related paper will be published later.

Position-Dependent Theory of Heterojunctions 691

[7] H. UNLU, Ph.D. Thesis, University of Minnesota (in preparation). [S] H. M. JAMES, Phys. Rev. 76, 1611 (1949). [9] W. SHOCKLEY and J. BARDEEN, Phys. Rev. 77,407 (1950).

[lo] H. KROEMER, RCA Rev. 332 (1957). [ll] a) F. STERN and S. DAS SARMA, Phys. Rev. B 30, 840 (1984).

b) P. M. ASBECK, D. L. MILLER, R. ASATOURIAN, and C. G. KIRKPATRICK, IEEE Electron Device Letters, 3,403 (1982).

[ 121 H. KROEMER, Heterojunction Device Physics: Band Discontinuities as Device Design Para- meters, in: VLSI Electronics: Microstructure Science, Vol. 10, Ed. L. EISENPRUCH, Academic Press, 1985.

[13] W. A. HARRISON, J. Vacuum Sci. Technol. B3, 1231 (1985). [14] W. I. WANG and F. STERN, J. Vacuum Sci. Technol. B3,1280 (1985). [15] H. C. CASEY, JR. and M. B. PANISH, Heterostructure Lasers Part B: Materials and Operating

Characteristics, Academic Press, New York 1978. S. M. SZE, Physics of Semiconductor Devices, Wiley, New York 1981.

29 May 1985. [16] J. BATEY and S. L. WRIGHT, Research Report No. 5029, IBM Watson Research Center,

[17] A. D. KATNANI and R. 5. BAUER, J. Vacuum Sci. Technol. B 3,1239 (1985). [ lS] M. K. KELLEY, D. W. NIELS, E. COLAVITA, and G. MARGARITONDO, Appl. Phys. Letters 46,

[19] P. H. MAHOWALD, R. S. LIST, and W. E. SPICER, J. Vacuum Sci. Technol. B 3, 1252 (1985). [20] T. N. CASSELMAN, A. SHEEL, J. SILBEECMAN, W. E. SPICER, and A. B. COREN, J. Vacuum Sci.

[21] A. D. KATNAMI and G. MAROARITONDO, Phys. Rev. B 29,1944 (1983). [22] C. A. MEAD, Solid State Electronics 9, 1023 (1966). [23] R. C. MILLER, A. C. GOSSARD, D. A. KLEINMAN, and A. D. MAINTEANU, Phys. Rev. B 29,

(Received October 30, 1985)

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3740 (19784).

44 *