- Email: [email protected]
Polynomial kinetic energy approximation for direct-indirect heterostructures
Superlattices and Microstructures, Vol. 3, No. 2, 1987
167
POLYNOMIAL KINETIC ENERGY APPROXIMATION FOR DIRECT-INDIRECT HETEROSTRUCTURES E. S. Hellman and J. S. Harris, Jr. Stanford Electronics laboratory Stanford University Stanford, CA 94305 USA Received 18 August 1986
The effective mass approximation, in which the band structure of a semiconductor is replaced by a simple parabolic dispersion relation for electrons, has worked surprisingly well for quantum calculations of electron eigenenergies and eigenstates in semiconductor heterostructures. It can be extended to systems with spacially varying effective mass by requiring wavefunction and particle flux continuity. However, for indirect heterostructures which include materials with electron bands of different symmetry, it fails to incorporate enough physics to give correct answers. An important example where effective mass calculations are inapplicable is the AlAs/GaAs system, in which the conduction band minima occur at the F and X points, respectively. The mixture of these two types of electrons in A1As/GaAs supedattices has only been calculated using tight-binding or pseudopotential methods, which are difficult to apply to a wide range of heterostructures. We have extended the spirit of effective mass calculations to a method applicable to indirect heterostructures. To do this, we write a SchriSdinger equation in which the Hamiltonian is a n th degree polynomial in the gradient operator, V. For any energy, there exist n (complex) plane wave solutions. For spatially varying band structures, we can write a probability conserving Schrtdinger equation which has a flux operator consistent with the usual interpretation of plane wave group velocities. The requirements imposed by this Schrodinger equation on the wavefunction and its derivatives allow matching of the plane wave solutions across heterojunctions. We have applied this method to AlAs/GaAs double heterostructures, where we see interesting resonance and anti-resonance behaviors. The computational speed of our method will allow complicated structures, including compositional grading and electric fields, to be modeled on microcomputers.
Introduction The lattice match between AlAs and GaAs has made this alloy system extremely useful for the construction of single crystal semiconductor heterostructures. Usually, however, 30-40% aluminum mole fraction AlGa.As is used with GaAs because the conduction band discontinuity between the two materials is largest for this composition. Above about 40% aluminum, A1GaAs becomes an indirect semiconductor, as the L and X minima move below the F minimum. The physics of beterostructures made with indirect A1GaAs or AlAs is poorly understood. For example, structures grown along (100) directions can allow transfer of energetic F valley conduction electrons in GaAs to one of the X valleys in AlAs, because symmetry is broken along the growth axis. To calculate the behavior of this transfer has required knowledge about the molecular structure of the interface, which is not easily accessible experimentally. Calculations using multi-band tigbt-bindine methods 1 and more rece n tly empirical pseudopotentials2,3 and single-band generalized Wannier functions,4 have been done mainly for superlattices and for single and double heterojuctions. More complicated structures, with graded compositions or applied electric fields, are usually analyzed using simple, but inadequate, approximations such as the use of an effective mass. In the effective mass approximation the effect of the crystal potential is modeled by replacing the kinetic energy part of the Schr0dinger equation with a term quadratic in the crystal momentum k ----iV so as to match the energy and curvature of the band near its minimum. Given the simplicity of this
0 7 4 9 - 6 0 3 6 / 8 7 / 0 2 0 1 6 7 + 0 3 $02.00/0
approximation, it is remarkable that it works as well as it does. For example, the bound state energies in quantum wells arc fit remarkably well even for wells just a few atomic layers thick. 5 In any case, the effective mass approximation cannot hope to predict direct-indir~t transport properties because it contains nothing about where the minimum is in momentum space. Spatially varying effective masses have been treated by a number of workers. Morrow and Brownstein6 have showed that the kinetic energy T which describes such a system must be a Hermitian operator of the form T = (-h2/2) m "~ V m "(1"2a) V m "~
(1)
for a sensible theory to result even when the effective mass changes abruptly across the heterojunction. Zhu and Kroemer7 have argued from tight binding theory that or= 1/'2 is the best value. Most practitioners of effective mass theory have used the assumption that ct =0, because it results in the usual matching conditions of wavefunction and quasiparticale flux continuity across an abrupt beterojunction. In this paper we will show how the effective mass approximation can be extended to include the effects of secondary conduction band minima. Instead of the usual quadratic approximation to the kinetic energy, we use nigher order polynomials to fit the band structure over the full range of crystal momentum along a given symmetry axis. From our choice of the kinetic energy, we then derive matching conditions for abrupt GaAs-AIAs heterojunctions. We present initial results for transport through a single AlAs barrier.
© 1987 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vol. 3, No, 2, 1987
168 1.25
100
................. ",,, 1.00 . . . . . . . . . . . . ,,,,, >
0.75
¢
0.50
- - GaAs ........... A As
AlAs Barrier Thickness: 5 a o
10-1
/3~
~
10-2
10 0
IJJ
0.25
10-1
X
10-2 0.00
0
7t/2a Wave Vector
~/a
.m
10`3 ~
o3
Figure 1: Polynomial approximations used in this paper for the conduction bands along the (~00) direction of GaAs and AlAs. The polynomial is 5th order in k '~. The coefficients of the polynomials are obtained by fitting the effective masses at the F and X points. The first and third derivatives with respect to k are set to zero at the X point.
E ¢/)
10 4
10"5 t~ L-t100
AlAs Barner Thickness 10 a o
Theory We look for a kinetic energy of the form
10`3
N
E (k)= Z
an k2n
F 7'
(2)
t1=0
where k is the wave vector, or eigenvalue of the operator k, and the a n are a set of coefficients.To model the conduction band structure along the (100) axis for AlAs and GaAs requires at least a 4th order polynomial in k 2. The 5 free parameters allow the fitting of the zone center effective mass, the X valley effective mass, the energies of the 2 minima, and the momentum at the X minimum. We fred that we obtain a much better fit if we use the extra free parameter in a 5th order polynomial to set the 3rd derivative of E with respect to k at the X minimum to 0. The resulting approximate band structures are shown in figure 1. We use effective masses of 0.068m 0 and 0.15 m 0 for the F valley in GaAs and AlAs, respectively. The X (longitudinal) masses are 1.3m 0 and l . l m 0, and the minima are assumed to occur at the zone edge, k=rda, where a is the lattice constant. The X minima is 0.48 eV above the F minimum in GaAs. These parameters are given in recent review articles. 8,9 We assume a 60:40 splitting of the direct band gao discontinuity between the conduction band and the v al ence band. 10 The approximation breaks down rapidly beyond the zone edge. Inaccuracies may result from the fact that the polynomial band has two distinct minima at k=+Tda, and that there are no repeated zone F minima at k=+2x/a. For band stuctures which vary spacially, as in a heterostructure, the kinetic energy must be written as a Hermitian operator. In this case, the coefficients for the kinetic energy found in equation (2) will be functions of position. If the junctions are abrupt, then we can use the argument of Morrow and Brownstein to limit the possible Hamiltonians to a form similar to equation (1).
10-4 10`5
'
10-6 0.0
0.1
~
0.2 0.3 0.4 Incident Energy (eV)
0.5
0.6
Figure 2: Calculated transmissivity versus energy of a single AlAs layer in GaAs. The AlAs thicknesses are 5a (top), 10a (middle), and 20a (bottom), where a is the lattice constant. The incident elecuon is in the GaAs F valley. For energies higher than the X minimum, the transmitted elecla'on can be in the F or X valley, as indicated by the arrows. The Hamiltonian we use is written simply in terms of the polynomial expansion coefficients
H . . . . V( V ( V a N V - aN.l) V + aN.2 ) V... (3). We have put all the an'S inside the V's, which corresponds to using ot = 0 in equation 1. Using this in a Schr6dinger equation defines a particle flux which corresponds to the group velocity obtained from the slope of the E vs. k curve for eigenstates of k. Note that this is not true for all the possible ways to write a Hamiltonian. The generalization of the Zhu and Kroemer rule (put all the an's outside the V's) does not satisfy this criterion.
Superlattices and Microstructures, Vol. 3, No. 2, 1987
transmissivity peaks to those seen for double barrier resonant tunneling diodes suggests that such structures might be useful for negative differential resistance applications. Another striking feature of the curves is the presence of very sharp dips. These "anti-resonances" are not seen in the usual double barrier structures. For energies above the GaAs X minimum at 0.48 eV, reflection and transmission by the barrier to the GaAs X valleys can occur. The GaAs X valleys are strongly coupled to the AlAs X states, so most of the transmitted current is converted from F to X. Figure 3 shows a plot of the transmissivity as a function of the barrier thickness for an incident electron at 0.2eV, which is above the AlAs X minimum, so that propagating wave can exist in the barrier. The periodic peaks in the transmissivity indicate that quantum interference is occurring in the barrier.
100 incident energy: 0.2 eV 10 -1
10 -2
/
I-10 -3
10-4 0
10a
20a
169
30a
AlAs Barrier Thickness
Figure 3: Calculated transmissivity versus layer thickness of a single AlAs layer in GaAs. The incident electron energy is 0.2 eV. The periodic resonances indicate quantum interference phenomena. Once we have written the Hamiltonian as in equation (3), the conditions for propagating the wavefunction across a heterojunction are well defined if we require the Hamiltonian to be bounded at all points through the heterojunction. Each gradient operator must act on a piecewise continuous function of position, and there are thus N independent conditions on the wavefunction. Each condition is needed because the Nth order polynomial in k has N roots, and thus N plane wave solutions, on each side of the junction. Although none of the conditions correspond to continuity of particle flux, the conditions together guarantee continuity of particle flux, and thus conservation of probability density. The matching conditions are linear in the wavefunction and its derivatives, so we can construct a transfer matrix which connects the k eigenstates on either side of the heterojunction. Applications Once the transfer matrix has been calculated for a given energy, it is straightforward to apply the polynomial kinetic energy approximation to real heterostructures. Figure 2 shows the calculated transmissivity of three AlAs barriers of different thicknesses, as a function of energy. Each 120 point scan took less than 2 minutes using FORTRAN with complex double precision arithmetic on an Htx)000 minicomputer. The roots of the 10th order polynomial were computed numerically. After transfer matrices were calculated for the two heterojunctions, the product matrix was transformed to a matrix connecting inward and outward states. The inward state was taken to be a F plane wave in GaAs, and the evanescent states increasing to infinity away from the origin were set to zero. The transmissivity is taken to be the transmitted current divided by the incident current. Several features of the transmissivity curves are prominent in figure 2. For energies less than the AlAs X minimum at 0.1 leV, the transport through the barrier is through evanescent states. There is no abrupt change at the AlAs X energy, where propagating X states in the AlAs become allowed. In the energy range between the AlAs X minimum and the GaAs X minimum, there are sharp peaks in the transmissivity which correspond to resonances in the AlAs barrier. They are caused by quantum interference analogous to Fabry- Perot interference in thin dielectric layers. Their sharpness results from the weak coupling of GaAs F states with AlAs X states. Such resonances have been predicted by tight-binding calculations. 11 The similarity of the
Conclusions We have demonstrated the use of a polynomial approximation to the kinetic energy to model electron transport in heterostructures composed of multi-valleyed semiconductors. Application of this simple model to transmission through single AlAs barriers has yielded insight into the physical processes occurring in these structures, including resonances and "anti-resonances". The simplicity and computational speed of this model will allow realistic calculations of quantum transport in complicated heterostructures, including the effects of graded composition and electric fields. For approximations as simple as this one, it is important to have experimental and theoretical validation of the results. Experimental measurement of the resonance ~ergies by techniques such as hot electron spectroscopy • ~ are essential for this purpose. Quantitative comparison of our results with comprehensive theoretical calculations will indicate which physics has been left out of the polynomial model. Sophisticated extensions of the polynomial approximation such as the use of sinusoidal operators to include the full effect of lattice periodicity should greatly improve the accuracy of the method. Acknowledgements- We would like to thank G. Yoffe for drawing our attention to this problem. The computer used for the calculations was donated by the Hewlett-Packard Corporation. E.S.H. would like to thank IBM Corp. for fellowship support. This work is sponsored by the Joint Services Electronics Program under contract No. DAAG-84-K-0047. References 1 G. C. Osbourne and D. L. Smith, Physical Review B19, (1979),2124. '~ M. Jaros and K. B. Wong, Journal of Physics C: Solid State Physics 17, (1984) L765. A. C. Marsh and J. C. Inkson, Journal of Physics C: Solid State Physics 17, (1984) 6561. 4 p. Roblin and M. W. Muller, PhysicalReview B32, (1985)5222. R. Dingle, in FestkSperprobleme, ed. by H. J. Queisser, Advances in Solid State Physics, Vol. XV, (Pergamon, New York, ~975), p. 21. u R. A. Morrow and K. R. Brownstein, Physical Review B30, (1984) 678. t Q.-G. Zhu and H. Kroemer, PhysicalReview B27, (1983)o3519. o j. S. Blakemore, Journal of Applied Physics 53, (1982) R123. 9 S. Adachi, Journal of Applied Physics 58, (1985) R1. 10 j. Batey and S. L. Wright, Journal of Applied Physics 59, (1986) 200. 11 C. Mailhiot, T. C. McGill and J. N. Schulman, Journal of Vacuum Sciece and Technology BI , (1983) 439. 12 J. R. Hayes, A. F . J. Levi and W. W'egmann, 1 Physical Review Letters 54, (1985) 1570.
167
POLYNOMIAL KINETIC ENERGY APPROXIMATION FOR DIRECT-INDIRECT HETEROSTRUCTURES E. S. Hellman and J. S. Harris, Jr. Stanford Electronics laboratory Stanford University Stanford, CA 94305 USA Received 18 August 1986
The effective mass approximation, in which the band structure of a semiconductor is replaced by a simple parabolic dispersion relation for electrons, has worked surprisingly well for quantum calculations of electron eigenenergies and eigenstates in semiconductor heterostructures. It can be extended to systems with spacially varying effective mass by requiring wavefunction and particle flux continuity. However, for indirect heterostructures which include materials with electron bands of different symmetry, it fails to incorporate enough physics to give correct answers. An important example where effective mass calculations are inapplicable is the AlAs/GaAs system, in which the conduction band minima occur at the F and X points, respectively. The mixture of these two types of electrons in A1As/GaAs supedattices has only been calculated using tight-binding or pseudopotential methods, which are difficult to apply to a wide range of heterostructures. We have extended the spirit of effective mass calculations to a method applicable to indirect heterostructures. To do this, we write a SchriSdinger equation in which the Hamiltonian is a n th degree polynomial in the gradient operator, V. For any energy, there exist n (complex) plane wave solutions. For spatially varying band structures, we can write a probability conserving Schrtdinger equation which has a flux operator consistent with the usual interpretation of plane wave group velocities. The requirements imposed by this Schrodinger equation on the wavefunction and its derivatives allow matching of the plane wave solutions across heterojunctions. We have applied this method to AlAs/GaAs double heterostructures, where we see interesting resonance and anti-resonance behaviors. The computational speed of our method will allow complicated structures, including compositional grading and electric fields, to be modeled on microcomputers.
Introduction The lattice match between AlAs and GaAs has made this alloy system extremely useful for the construction of single crystal semiconductor heterostructures. Usually, however, 30-40% aluminum mole fraction AlGa.As is used with GaAs because the conduction band discontinuity between the two materials is largest for this composition. Above about 40% aluminum, A1GaAs becomes an indirect semiconductor, as the L and X minima move below the F minimum. The physics of beterostructures made with indirect A1GaAs or AlAs is poorly understood. For example, structures grown along (100) directions can allow transfer of energetic F valley conduction electrons in GaAs to one of the X valleys in AlAs, because symmetry is broken along the growth axis. To calculate the behavior of this transfer has required knowledge about the molecular structure of the interface, which is not easily accessible experimentally. Calculations using multi-band tigbt-bindine methods 1 and more rece n tly empirical pseudopotentials2,3 and single-band generalized Wannier functions,4 have been done mainly for superlattices and for single and double heterojuctions. More complicated structures, with graded compositions or applied electric fields, are usually analyzed using simple, but inadequate, approximations such as the use of an effective mass. In the effective mass approximation the effect of the crystal potential is modeled by replacing the kinetic energy part of the Schr0dinger equation with a term quadratic in the crystal momentum k ----iV so as to match the energy and curvature of the band near its minimum. Given the simplicity of this
0 7 4 9 - 6 0 3 6 / 8 7 / 0 2 0 1 6 7 + 0 3 $02.00/0
approximation, it is remarkable that it works as well as it does. For example, the bound state energies in quantum wells arc fit remarkably well even for wells just a few atomic layers thick. 5 In any case, the effective mass approximation cannot hope to predict direct-indir~t transport properties because it contains nothing about where the minimum is in momentum space. Spatially varying effective masses have been treated by a number of workers. Morrow and Brownstein6 have showed that the kinetic energy T which describes such a system must be a Hermitian operator of the form T = (-h2/2) m "~ V m "(1"2a) V m "~
(1)
for a sensible theory to result even when the effective mass changes abruptly across the heterojunction. Zhu and Kroemer7 have argued from tight binding theory that or= 1/'2 is the best value. Most practitioners of effective mass theory have used the assumption that ct =0, because it results in the usual matching conditions of wavefunction and quasiparticale flux continuity across an abrupt beterojunction. In this paper we will show how the effective mass approximation can be extended to include the effects of secondary conduction band minima. Instead of the usual quadratic approximation to the kinetic energy, we use nigher order polynomials to fit the band structure over the full range of crystal momentum along a given symmetry axis. From our choice of the kinetic energy, we then derive matching conditions for abrupt GaAs-AIAs heterojunctions. We present initial results for transport through a single AlAs barrier.
© 1987 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vol. 3, No, 2, 1987
168 1.25
100
................. ",,, 1.00 . . . . . . . . . . . . ,,,,, >
0.75
¢
0.50
- - GaAs ........... A As
AlAs Barrier Thickness: 5 a o
10-1
/3~
~
10-2
10 0
IJJ
0.25
10-1
X
10-2 0.00
0
7t/2a Wave Vector
~/a
.m
10`3 ~
o3
Figure 1: Polynomial approximations used in this paper for the conduction bands along the (~00) direction of GaAs and AlAs. The polynomial is 5th order in k '~. The coefficients of the polynomials are obtained by fitting the effective masses at the F and X points. The first and third derivatives with respect to k are set to zero at the X point.
E ¢/)
10 4
10"5 t~ L-t100
AlAs Barner Thickness 10 a o
Theory We look for a kinetic energy of the form
10`3
N
E (k)= Z
an k2n
F 7'
(2)
t1=0
where k is the wave vector, or eigenvalue of the operator k, and the a n are a set of coefficients.To model the conduction band structure along the (100) axis for AlAs and GaAs requires at least a 4th order polynomial in k 2. The 5 free parameters allow the fitting of the zone center effective mass, the X valley effective mass, the energies of the 2 minima, and the momentum at the X minimum. We fred that we obtain a much better fit if we use the extra free parameter in a 5th order polynomial to set the 3rd derivative of E with respect to k at the X minimum to 0. The resulting approximate band structures are shown in figure 1. We use effective masses of 0.068m 0 and 0.15 m 0 for the F valley in GaAs and AlAs, respectively. The X (longitudinal) masses are 1.3m 0 and l . l m 0, and the minima are assumed to occur at the zone edge, k=rda, where a is the lattice constant. The X minima is 0.48 eV above the F minimum in GaAs. These parameters are given in recent review articles. 8,9 We assume a 60:40 splitting of the direct band gao discontinuity between the conduction band and the v al ence band. 10 The approximation breaks down rapidly beyond the zone edge. Inaccuracies may result from the fact that the polynomial band has two distinct minima at k=+Tda, and that there are no repeated zone F minima at k=+2x/a. For band stuctures which vary spacially, as in a heterostructure, the kinetic energy must be written as a Hermitian operator. In this case, the coefficients for the kinetic energy found in equation (2) will be functions of position. If the junctions are abrupt, then we can use the argument of Morrow and Brownstein to limit the possible Hamiltonians to a form similar to equation (1).
10-4 10`5
'
10-6 0.0
0.1
~
0.2 0.3 0.4 Incident Energy (eV)
0.5
0.6
Figure 2: Calculated transmissivity versus energy of a single AlAs layer in GaAs. The AlAs thicknesses are 5a (top), 10a (middle), and 20a (bottom), where a is the lattice constant. The incident elecuon is in the GaAs F valley. For energies higher than the X minimum, the transmitted elecla'on can be in the F or X valley, as indicated by the arrows. The Hamiltonian we use is written simply in terms of the polynomial expansion coefficients
H . . . . V( V ( V a N V - aN.l) V + aN.2 ) V... (3). We have put all the an'S inside the V's, which corresponds to using ot = 0 in equation 1. Using this in a Schr6dinger equation defines a particle flux which corresponds to the group velocity obtained from the slope of the E vs. k curve for eigenstates of k. Note that this is not true for all the possible ways to write a Hamiltonian. The generalization of the Zhu and Kroemer rule (put all the an's outside the V's) does not satisfy this criterion.
Superlattices and Microstructures, Vol. 3, No. 2, 1987
transmissivity peaks to those seen for double barrier resonant tunneling diodes suggests that such structures might be useful for negative differential resistance applications. Another striking feature of the curves is the presence of very sharp dips. These "anti-resonances" are not seen in the usual double barrier structures. For energies above the GaAs X minimum at 0.48 eV, reflection and transmission by the barrier to the GaAs X valleys can occur. The GaAs X valleys are strongly coupled to the AlAs X states, so most of the transmitted current is converted from F to X. Figure 3 shows a plot of the transmissivity as a function of the barrier thickness for an incident electron at 0.2eV, which is above the AlAs X minimum, so that propagating wave can exist in the barrier. The periodic peaks in the transmissivity indicate that quantum interference is occurring in the barrier.
100 incident energy: 0.2 eV 10 -1
10 -2
/
I-10 -3
10-4 0
10a
20a
169
30a
AlAs Barrier Thickness
Figure 3: Calculated transmissivity versus layer thickness of a single AlAs layer in GaAs. The incident electron energy is 0.2 eV. The periodic resonances indicate quantum interference phenomena. Once we have written the Hamiltonian as in equation (3), the conditions for propagating the wavefunction across a heterojunction are well defined if we require the Hamiltonian to be bounded at all points through the heterojunction. Each gradient operator must act on a piecewise continuous function of position, and there are thus N independent conditions on the wavefunction. Each condition is needed because the Nth order polynomial in k has N roots, and thus N plane wave solutions, on each side of the junction. Although none of the conditions correspond to continuity of particle flux, the conditions together guarantee continuity of particle flux, and thus conservation of probability density. The matching conditions are linear in the wavefunction and its derivatives, so we can construct a transfer matrix which connects the k eigenstates on either side of the heterojunction. Applications Once the transfer matrix has been calculated for a given energy, it is straightforward to apply the polynomial kinetic energy approximation to real heterostructures. Figure 2 shows the calculated transmissivity of three AlAs barriers of different thicknesses, as a function of energy. Each 120 point scan took less than 2 minutes using FORTRAN with complex double precision arithmetic on an Htx)000 minicomputer. The roots of the 10th order polynomial were computed numerically. After transfer matrices were calculated for the two heterojunctions, the product matrix was transformed to a matrix connecting inward and outward states. The inward state was taken to be a F plane wave in GaAs, and the evanescent states increasing to infinity away from the origin were set to zero. The transmissivity is taken to be the transmitted current divided by the incident current. Several features of the transmissivity curves are prominent in figure 2. For energies less than the AlAs X minimum at 0.1 leV, the transport through the barrier is through evanescent states. There is no abrupt change at the AlAs X energy, where propagating X states in the AlAs become allowed. In the energy range between the AlAs X minimum and the GaAs X minimum, there are sharp peaks in the transmissivity which correspond to resonances in the AlAs barrier. They are caused by quantum interference analogous to Fabry- Perot interference in thin dielectric layers. Their sharpness results from the weak coupling of GaAs F states with AlAs X states. Such resonances have been predicted by tight-binding calculations. 11 The similarity of the
Conclusions We have demonstrated the use of a polynomial approximation to the kinetic energy to model electron transport in heterostructures composed of multi-valleyed semiconductors. Application of this simple model to transmission through single AlAs barriers has yielded insight into the physical processes occurring in these structures, including resonances and "anti-resonances". The simplicity and computational speed of this model will allow realistic calculations of quantum transport in complicated heterostructures, including the effects of graded composition and electric fields. For approximations as simple as this one, it is important to have experimental and theoretical validation of the results. Experimental measurement of the resonance ~ergies by techniques such as hot electron spectroscopy • ~ are essential for this purpose. Quantitative comparison of our results with comprehensive theoretical calculations will indicate which physics has been left out of the polynomial model. Sophisticated extensions of the polynomial approximation such as the use of sinusoidal operators to include the full effect of lattice periodicity should greatly improve the accuracy of the method. Acknowledgements- We would like to thank G. Yoffe for drawing our attention to this problem. The computer used for the calculations was donated by the Hewlett-Packard Corporation. E.S.H. would like to thank IBM Corp. for fellowship support. This work is sponsored by the Joint Services Electronics Program under contract No. DAAG-84-K-0047. References 1 G. C. Osbourne and D. L. Smith, Physical Review B19, (1979),2124. '~ M. Jaros and K. B. Wong, Journal of Physics C: Solid State Physics 17, (1984) L765. A. C. Marsh and J. C. Inkson, Journal of Physics C: Solid State Physics 17, (1984) 6561. 4 p. Roblin and M. W. Muller, PhysicalReview B32, (1985)5222. R. Dingle, in FestkSperprobleme, ed. by H. J. Queisser, Advances in Solid State Physics, Vol. XV, (Pergamon, New York, ~975), p. 21. u R. A. Morrow and K. R. Brownstein, Physical Review B30, (1984) 678. t Q.-G. Zhu and H. Kroemer, PhysicalReview B27, (1983)o3519. o j. S. Blakemore, Journal of Applied Physics 53, (1982) R123. 9 S. Adachi, Journal of Applied Physics 58, (1985) R1. 10 j. Batey and S. L. Wright, Journal of Applied Physics 59, (1986) 200. 11 C. Mailhiot, T. C. McGill and J. N. Schulman, Journal of Vacuum Sciece and Technology BI , (1983) 439. 12 J. R. Hayes, A. F . J. Levi and W. W'egmann, 1 Physical Review Letters 54, (1985) 1570.