Journal of Computational Electronics 1: 179–183, 2002c© 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Empirical Pseudopotential Method for the Band Structure Calculationof Strained-Silicon Germanium Materials

SALVADOR GONZALEZ AND DRAGICA VASILESKADepartment of Electrical Engineering and Center for Solid State Electronics Research,

Arizona State University, Tempe, AZ 85287-5706, USA

ALEXANDER A. DEMKOVPhysical Sciences Research Labs, Motorola, Inc., 7700 S. River Parkway, Tempe, AZ 85284, USA

Abstract. The band structure of strained-silicon germanium (Si1−x Gex ) is calculated as a preliminary step indeveloping a full band Monte Carlo (FBMC) simulator. The band structure for the alloy is calculated using theempirical pseudopotential method (EPM) within the virtual crystal approximation (VCA). Spin-orbit interaction isincluded into the calculation via the Lowdin quasi-degenerate perturbation theory, which significantly reduces thecomputation time. Furthermore, strain is included by utilizing basic elastic theory. Ultimately, the band structurefor strained Si1−x Gex is calculated at various germanium concentrations.

Keywords: strained-silicon germanium, band structure calculation, empirical pseudopotential method, virtualcrystal approximation

1. Introduction

Current market forces demand that the semiconductorindustry produce faster integrated circuits (ICs) withhigh functionality at a low cost. One way of achievingthis trend is to scale the device geometry. The industryis quickly reaching the physical limitations of small de-vices, however. In metal-oxide semiconductor (MOS)transistors, for example, thin oxides give way to highgate leakage currents. Increased short-channel effects(SCE) also impede performance improvements. Onesolution that replaces device scaling is the introductionof new materials.

For this purpose, strained-silicon or strained-silicongermanium (Si1−x Gex ) material systems have receivedmuch attention as possible candidates for improvingperformance of existing Si technology (Iyer et al. 1989,Harame et al. 1995a, b, Cressler 1995). This trend hasbeen made possible via recent innovations in molecularbeam epitaxy (MBE) growth techniques that allow forrelatively easy growth of Si1−x Gex on Si1−yGey sub-strates. Furthermore, Si1−x Gex can be integrated into

existing Si technology without the need for significantfactory retooling.

For strained-Si1−x Gex material systems, the fullband structure is required in order to capture the bandsplitting and warping, especially near the valenceband maximum at the zone center (�). To this end, thefull band structure of strained-silicon germanium iscalculated using the EPM with spin-orbit interactionincluded.

2. Empirical Pseudopotential Method

The pseudopotential method is based on the Phillips-Kleinman cancellation theorem (Phillips and Kleinman1959), which provides justification why the electronicstructure can be described using a nearly-free electronmodel and weak potentials. For this purpose, the pseu-dopotential Hamiltonian can be written as

H = −(h2/2m)∇2 + VP (r ), (1)

where VP (r ) is the smoothly-varying pseudopotential(Cohen and Bergstresser 1966). Because the crystal

180 Gonzalez

potential is periodic, the pseudopotential is also aperiodic function and can be expanded into a Fourierseries over the reciprocal lattice to obtain

VP (r) =∑

G

S(G)V f f (G) eiG·r , (2)

where S(G) is the structure factor and V f f (G) isthe pseudopotential form factor, which is definedas twice the inverse Fourier transform of the atompotential. For diamond-lattice materials, the struc-ture factor is defined as S(G) = cos(G · τ ), whereτ = a(1/8, 1/8, 1/8) is the atomic basis vector definedin terms of the lattice constant a when the coordinateorigin is taken to be halfway between the basis atoms.

Because the pseudopotential in a crystal lattice isperiodic, it follows that the pseudo-wave function cor-responding to (1) is also periodic and can be expressedas a Bloch function, which consists of a plane-wavepart and a cell periodic part. The cell periodic part, inturn, can be expanded into a Fourier series over the re-ciprocal lattice. By substituting the expanded pseudo-wave function and the pseudopotential defined by (2)into the Schrodinger wave equation, the Hamiltonianmatrix results and is defined as

Hi, j =

h2

2m|k + Gi |2, i = j

V f f (|Gi − G j |) cos[(Gi − G j ) · τ ], i �= j

(3)

where G is a reciprocal lattice vector and k is a wavevector lying within the first Brillouin zone. The so-lution to the energy eigenvalues and correspondingeigenvectors can then be found by diagonalizing theHamiltonian matrix. For this work, 137 plane waves,each corresponding to reciprocal lattice vectors up toand including the 10th-nearest neighbor from the ori-gin, were used to expand the pseudopotential. The pub-licly available eigenvalue solver LAPACK was used todiagonalize the Hamiltonian matrix.

3. Spin-Orbit Interaction

To develop a more refined picture of the energy bands,the spin-orbit interaction must be included into thepseudopotential calculation. In the context of electronicstructure theory, the spin-orbit interaction serves tosplit degenerate energy levels. This influence is mostpronounced for the valence band maxima near theBrillouin zone center.

For an electron orbiting a nucleus, which producesa spherically symmetric potential V , the spin-orbit in-teraction is calculated using Einstein’s special theoryof relativity to obtain

HSO = h

4m2c2

[1

r

∂V

∂r

]L · σ, (4)

where h is the reduced Planck constant, m is the elec-tron’s rest mass, c is the speed of light, L is the elec-tron’s orbital angular momentum and σ is the Pauli spintensor.

It may be tempting to add the Hso term from (4) di-rectly to (1) and obtain the solution by diagonalizing thetotal Hamiltonian. This would not be the correct way toproceed, however, given that the pseudo-wave functioncorresponding to (1) is a spinless quantity. When spinis included into the problem, the crystal wave functionbecomes a (2×1)-spinor. By using shorthand subscriptnotation for spin, the spin Hamiltonian is given by

Hm ′k′σ ′;mkσ = E0mkδm ′σ ′;mσ +〈m ′k′σ ′ |HSO |mkσ 〉, (5)

where m ′ is the row index and m is the column indexand σ = ±1 is the Pauli spin index corresponding ei-ther to the spin up or spin down state. In this way, thespin Hamiltonian can be constructed using the spin-less eigenvalues as the diagonal elements and includ-ing the spin-orbit interaction as a perturbation. It hasbeen shown (Saravia and Brust 1968), however, thatfor states containing l-symmetry already included inthe core states (2p core states for Si and 3p core statesfor Ge), the perturbation in (5) can be written as a dou-ble summation over the reciprocal lattice vectors

〈m ′k′σ ′ |HSO|mkσ 〉= − iλp

∑Gi ,G j

a∗m ′k′ (Gi )amk(G j )S(G j − Gi )

× Fp(k + Gi )Fp(k + G j )

× [e(k + Gi )e(k + G j )] ·σσ ′σ (6)

where S(G) is the structure factor, λp is a free parameterused to adjust the energy splitting, Fp is a functionassociated with p-core states, e(k) is a unit vector in thek direction and σ is related to the Pauli spin matrices.

Including the spin-orbit interaction serves to doublethe size of the Hamiltonian matrix. In addition, eachspin-orbit matrix element is calculated as the doublesummation over the reciprocal lattice vectors Gi andG j , as seen in (6). As a result, the Hamiltonian is

Empirical Pseudopotential Method 181

computationally expensive to claculate, especiallysince there are 137 reciprocal lattice vectors employedin the EPM. To minimize the computational cost,Lowdin’s quasi-degenerate perturbation theory isapplied.

Lowdin’s perturbation technique serves to reduce thesize of the eigenvalue problem by “concentrating” theinformation in the initial Hamiltonian matrix to obtaina smaller matrix (Lowdin 1951). Lowdin uses the vari-ational principle to arrive at a perturbation formula,which gives the influence of the higher-lying (class B)states on the lower-lying (class A) states. The class Bstates are eliminated through a process of iteration toobtain

A∑n

(Umn − Eδmn)cn = 0, (7)

where

Umn = Hmn +B∑α

H ′mα H ′

αn

E − Hαα

+B∑α

B∑β

H ′mα H ′

αβ H ′βn

(E − Hαα)(E − Hββ)+ · · · (8)

The first term Hmn in (8) is a matrix element, whichcorresponds to an A-class state, in the initial matrix.The subsequent terms correspond to the influence ofthe B-class states, which are treated as a perturbationhere, on the A-class states. For this work, the first twoterms in (8) are included in the calculation of Umn.

The benefits of using the quasi-degenerate perturba-tion theory are: (a) one does not need to solve an eigen-value problem of size 2N when spin is included into theproblem and (b) degenerate and non-degenerate statesare treated on an equal footing, which means that thereis no need to first lift the degeneracy before applyingthe perturbative approach. Within this scheme, the de-generacy of the states is lifted via the introduction ofthe effective matrix element Umn .

Since (8) is calculated through a process of iteration,the value E is introduced into the expression. For thiswork, E is estimated to be the average energy of theclass-A states. Furthermore, 60 class-A states are usedto achieve an eigenvalue convergence within 5 meV forstates near the valence band maxima. For the case thatthe problem is solved exactly, each k-point requires ap-proximately 10.5 sec of central processing unit (CPU)time on a 500 MHz Pentium III microprocessor. Us-ing the Lowdin perturbation technique with 60 class-Astates only 1 sec of CPU time is required to solve forthe energy spectrum at each k-point.

Finally, the spin-orbit parameter λp in (6) that pro-duces the appropriate spin-orbit splitting, i.e. 44 meVfor Si and 300 meV for Ge, is determined by linear in-terpolation. The value for Si is λSi

p = 0.00156 eV-cm3,and the value for Ge is λGe

p = 0.0112 eV-cm3.

4. Silicon Germanium Alloy

The elemental semiconductors silicon (Si) and germa-nium (Ge) are isoelectronic. As a result, their chemicaland electronic properties are similar. Si and Ge are theonly group-IV elements that are completely miscible. Itis thus possible to form a solid solution of one elementin the other to obtain a silicon germanium (Si1−x Gex )alloy. The material properties vary gradually over theentire range. The lattice constant, for example, variesnearly linearly over the range of x . This fact is quanti-fied by Vegard’s rule (Vegard 1921) which states thatthe the bulk lattice constant is given by

aSi1−x Gex (x) = aSi(1 − x) + aGex . (9)

Like its constituent elements, the bulk silicon germa-nium alloy crystallizes in a diamond lattice, which ischaracterized by face-centered cubic (FCC) symmetry.From the definition of alloy, Ge atoms substitute for Siatoms randomly throughout the crystal, in proportionto the Ge concentration, x .

Because the material properties vary gradually overthe range of Ge concentrations, it is possible to ap-ply the virtual crystal approximation (VCA) to includealloy information. A silicon germanium alloy can beapproximated as a FCC lattice of “hybrid” atoms. Itthen follows from the VCA that all the alloy parame-ters in the EPM can be interpolated with respect to theGe concentration, x .

5. Strained-Silicon Germanium Alloy

To obtain a strained-silicon germanium alloy, Si1−x Gex

is pseudomorphically grown on top of a Si1−yGey sub-strate. The in-plane lattice constant of the growth layerconforms to the substrate, making the in-plane latticeconstant different that its bulk value. From elastic the-ory it follows that the growth layer experiences biaxialstrain in the direction of the growth plane. The in-planestrain condition can be expressed as

ε// = aSi1−y Gey − aSi1−x Gex

//

aSi1−x Gex

//

, (10)

182 Gonzalez

which is a relative change in lattice constants due tothe stress.

Elastic theory predicts that the growth layer will re-spond in the direction normal to the growth interfaceplane in order to minimize its elastic energy (Rieger andVogl 1993). To satisfy minimum energy, the transversestrain ε⊥ is given by

ε⊥ = −2c12

c11ε//, (11)

where c12 and c11 are the elastic constants of Si1−x Gex ,also calculated within the VCA. It then follows that thestrain of the Si1−x Gex growth layer is given by thefollowing second-rank tensor

↔ε=

ε// 0 0

0 ε// 0

0 0 ε⊥

, (12)

for which the elements are defined by (10) and (11).The vanishing off-diagonal elements in (12) indicatethat there is no shear strain in the system. The sys-tem undergoes a deformation along the principle axesonly.

The key elements used in the implementation of theEPM are the reciprocal lattice vectors. To include straininto the EPM, it is necessary to apply strain to the recip-rocal lattice vectors. To do this, strain is first applied tothe primitive lattice vectors of the direct space to obtainthe strained lattice vector

a′α = (

↔1 + ↔

ε )aα, (13)

where↔1 is the unit tensor and aα is an unstrained lattice

vector. The atomic basis vector, τ , is also transformedunder (13). Once the strained direct lattice vectors arecalculated, the strained reciprocal lattice vectors G′

α arecalculated as

G′α = 2π

a′β × a′

γ

a′1 · (a′

2 × a′3)

, (14)

The pseudopotential is then expanded over the strainedreciprocal lattice vectors to include strain into the EPM.

6. Results

The band structure is calculated for strained-Si1−x Gex (x = 40%) on a Si substrate (Fig. 1). Thepseudopotential form factors for Si and Ge are takenfrom Chelikowsky and Cohen (1974) and Saravia andBrust (1968), respectively. A key feature in the band

Figure 1. Band structure of strained-Si1−x Gex , x = 40%.

Figure 2. Zoom-in of valence band maximum in Fig. 1.

structure is the splitting of the heavy hole (HH) andlight hole (LH) bands at the valence band maximum,which is located at the � point (Fig. 1). At x = 40%,the splitting is calculated to be approximately 80 meV.Furthermore, strain also serves to warp the valencebands near the �-point. In addition, the spin splittingis enhanced with strain. The value indicated in Fig. 2is approximately 400 meV, which is larger than that ofpure Ge (�SO = 300 meV).

7. Conclusion

In summary, the band structure for strained-Si1−x Gex

was calculated using the empirical pseudopotential

Empirical Pseudopotential Method 183

method within the virtual crystal approximation. Alloyinformation is included into the calculation via theLowdin quasi-degenerate perturbation theory, andstrain is included via elastic theory. Strain serves tosplit the degeneracy of the HH and LH bands at the�-point. Furthermore, band warping results from thestrain. Finally, applying the Lowdin quasi-degenerateperturbation theory serves to reduce the computationtime by a factor of 10.

Acknowledgments

The authors wish to acknowledge the Office of NavalResearch and the National Science Foundation forsupport of this research.

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