- Email: [email protected]
Ellipsoidal valley donor binding energies in a quantum well
0038-1098/9255.00+ .00 Pergamon Press Ltd
State Communications, Vol. 84, No. 9, pp. 885-888, 1992. ~)'Solid Printed in Great Britain.
E L L I P S O I D A L V A L L E Y D O N O R B I N D I N G E N E R G I E S I N ,4, Q U A N T U M ~NELL S. R. Parihar and S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA (Received August 31, 1992 by A. Pinczuk)
We describe a modification of the usual separable wave function approach to donor electrons confined in heterostructures which reflects the intersubband Coulomb coupling. The donor Coulomb potential is split into effective normal and in-plane components which are then used to solve for the corresponding wave functions. The procedure is carried out self consistently which can be shown to result in the best donor wave function that can be expressed in separable form. As an example the method is applied to calculate a few of the lower lying levels of a donor in a quantum well in a material with a single ellipsoidal valley conduction band.
growth axis coordinate 4,5. The result is an effective two dimensional (2D) hydrogenic problem associated with each quantized subband. The wave function is simply a product of the in-plane hydrogenlc wave function and the quantized subband of the heterostructure in the absence of the donor potential. Such an approach works best for narrow wells and shallow donors where the subband energy dominates the Coulomb binding energy. W h e n the two energies become comparable, several subbands are mixed together by the donor potential and the method loses its validity. Still, the approach is attractive because the solution procedure is straightforward regardless of the symmetry in the problem, and the separable form of the wave function makes it easy to visualize and work with. In this communication we describe a modification which extends the range of validity of the separable approach by taking into account the Coulomb potential of the donor in calculating the quantlzed subbands. The effective potentials for the in-plane motion and motion along the growth axis are determined self consistently. The wave function calculated from these potentials is the optimal one satisfying a product form. Further, we show that the ground state calculated in this manner gives a rigorous upper bound on the true ground state energy. The geometry of the problem is illustrated in Fig. 1 . We choose the long axis of the conduction band ellipsoid with mass m I along the growth axis while the transverse mass, mr, lies in the QW plane. The Schrodinger equation in the effective mass approximation is given in cylindrical coordinates by
Calculations of donor and acceptor states in semiconductor heterostructures have been actively pursued following Bastard's original paper on the effects of confinement on a hydrogenic impurity in an infinite quantum well (QW) 1. A commonly used technique is to perform a variational calculation using a basis set of the approprlate symmetry with Gaussian type orbitals 2'3. This approach yields accurate energies and gives a rigorous upper bound as well. However, variational wave functions are not as accurate as the energies they produce; moreover there are a number of steps involved in setting up and solving the problem this way. First, to avoid unnecessarily large basis sets one has to incorporate the symmetry of the problem into the basis functions. This automatically means t h a t situations where the symmetry is lowered (such as the donor located off center) entail larger basis sets and longer computation times. Second, after diagonalizing the matrix equation one must still vary shape and shift parameters, typically in the exponential part of the basis functions, to minimize the energy. Problems with multiple minima at widely separated values of these parameters can arise casting doubt onto the accuracy of the resulting wave functions. Finally, a variational solution yields the successively higher states of a given symmetry with increasingly poorer bounds. This is a problem when one wishes to solve for hydrogenlc states attached to excited subbands of a narrow heterostructure 4 for then there are many intervening states. Given the above, one is led to consider alternative methods of solution. One such approach, which lends itself naturally to layered structures, is to make the approximation that the wave function is separable in the in-plane coordinates versus the heterostructure
H~=885
+
~ p Op p2 0¢~
886
VALLEY DONOR BINDING ENERGIES - ~
2
¢
Vol. 84, No. 9
Z
+ VB(z)~b = E ¢ ,
where "7-- m r is the transverse to longitudinal mass raml rio and Vs(z ) is the QW potential
/
vB('.)-I" I> ~2
T
/
Kz
quantum well
a)
The units of energy and length used are Ry*-- mte4 ,
potentlai energy
~s ~'2
and aO=m---~2, respectively where Q is the static dielectric constant. We have neglected the dielectric and effective mass differences between the well and the barrier materials. Following Chang s we write the donor wave function in separable form
¢ffif.(z)¢(p,¢),
(2)
where fn(z) is the n th QW subband satisfying { " 7 5 + V B ( Z ) }fn ffi e~fn •
O
i
_
Lw 0 2
L__w 2
z
b) FIG 1. a) Relative alignment of quantum well and conduction band ellipsoid, b) Quantum well potential profile.
(3)
Substituting (2) into (1), using (3) and projecting the final equation onto fn(z) yields an effective in-plane equation for ~(p,~b)
is accomplished by projecting (1) onto ~b and using (4a,b) to give 02fn
~ - ~ - 2 +VB(~)f.+V~(z)fo = ~f., 21r oo
VeZ~(zl------f0 fo
vNp)=- f 2fi(,-)f~(,.)d,..
(4b)
(s)
,
24 ¢
~pdpd*,
2W
e~=E--eP+ f f ~*~V~pdpd~b. Thus for each subband the problem is reduced to finding the bound states of an effective 2D central potential V ~ p ) . For electrons with small effective mass as is typical for III-V compounds equation (4) is a satisfactory approximation in dealing with a QW of width smaller than a few hundred A . Under these conditions the Coulomb binding energy (a few meV) is much smaller than the QW subband spacing and little subband mixing occum. However, for heavier masses such as occur in the X valley, the subband energy and the Coulomb energy become comparable to each other. In that case one cannot assume the QW subbands remain unaffected as equation (3) implies. In order to get a physical picture we first look at V~(p). It can be interpreted as the in-plane Coulomb potential with the electron smeared out into a line segment along z because of the finite extent of the QW subband. In a llke fashion, having solved (4a) for the in-plane wave function &(p,¢) we can now modify (3) to include a Coulomb potential along z resulting from smearing out the electron into a disk of charge given by ~*~. This
0
0
In this way the effect of the Coulomb potential can be introduced into the equation for the QW subbands. By iterating (4) and (5) to self consistency we obtain the best solution to the problem possible in a separable form. 6 It is worth noting that Zhu7 had previously introduced a two parameter V shaped potential into (3) at the donor position to improve the blnding energies. However, the assumption of linear variation of the potential is true only near the donor ion. With large binding energies especially, the small extent in plane of the wave function makes V~(z) quickly deviate from the simple V shape as is evident from Fig. 4. As Fig. 1 shows the problem is rotationally invariant about the z axis. Therefore the z component of angular momentum is conserved and we may write for a state with Lzffiffiml~ ~ffiffifn(z)Rm(P)eim¢ .
(6)
We solve equations (4) and {5) numerically for the functions fn(Z) and Rm(P) respectively. The Numerov method is used to find QW subbands along the lines
VALLEY DONOR BINDING ENERGIES
Voh 84: No. 9
"given in Ref. 8 while fourth order Runge-Kutta is employed for the in-plane radial equation. The effective potentials, Veel~p) and Vega(z), are calculated at a fixed number of points and fitted with cubic splines. Near the donor position V~t~p) becomes logarithmically singular while VeZll(z) remains finite with a constant slope. The number of iterations necessary to achieve convergence depends on both the bulk binding energy of the donor and the well width. Thus, while no iteration is necessary for the calculation of silicon in GaAs at typical well widths (100 .~ ), five or more iterations are needed when using the mass parameters of AlAs or Si X valleys. In general as the well width increases more iterations are needed because the Coulomb energy is no longer small relative to the subband spacing. Calculated results are shown in Fig. 2. We use parameters appropriate for an AlAs Q W with AlGaAs(x=0.5) barriers ie. mt==0.1gm o, m l = l . l m o, Q----10.06,and VB=169meV (m o is the free electron mass) 9. The donor is at the well center and since we consider only a single valley, valley-orbit effects which would split the ground state are neglected. The energies of three states, the ls, 2po, and 2p=~ are shown as functions of well width. The 2Po and 2p~. are low lying excited states having a single node in the wave function at z=0 and p=0 respectively corresponding to the 2pz and 2px,y bulk hydrogenic states. Their behaviour is similar to t h a t of donors with isotroplc effective mass except that here the 2po level lies below the 2p~ level in the bulk limit because of the heavier longitudinal mass m. As the well width decreases the 2p o state rises rapidly in energy because of its association by symmetry with the first excited Q W subband 11, crossing the 2p~ level at about 60 A well width. We have carried out a variational calculation along the lines of Ref. 3 to check the accuracy of the separable approach. The difference in the p state energies is indistinguishable on the scale of Fig. 2 while the variational ground state energy is at most only 2 meV
1 O0
887
lower at the larger well width. The fact t h a t the ground state energy in the bulk limit is accurate to within 5 percent is remarkable given the fact we are using separable wave functions. In addition, by taking the diagonal matrix elements of the full Hamiltonian, H, of equation (1) in the self consistent state (2) one obtains after using equations (4) and (5) < ~ [H I ¢ > = < f n
I--'/-~2 +V~(z)+VB(z)If,>
+e-<¢
(7)
Ivt~(p)I¢>, 2~
=,~+,'- f 0
co
f'~*¢V:@dpd¢, 0
which is identical to the self consistent energy of that state. Since the three states in Fig. 2 are the lowest of their respective symmetry types, the variational principle ensures the obtained energies are rigorous upper bounds. The case of the off center donor at position zd is easily handled by the substitution
in the expressions for Vega(z) and V•ll(p ). Fig. 3 shows the effects of moving the donor from the center of the well to the edge. As expected the energies decrease because the Q W barrier pushes the wave function away from the donor. One appealing feature of the separable approach is t h a t it is easy to visualize how the Coulomb potential appears algng the z axis in relation to the heterojunetion barriers. This is demonstrated in Fig. 4 where the effective potential energy.and wave funetlon of the ground subband of a 100 A well are plotted along z for the ease of the donor at the center of the well or at well edge. In summary we have presented a self consistent version of the 2D effective potential approach to donors in QW's. It has the advantage of maintaining the wave function in a convenient separable form
0
•
|
|
|
|
2p..
-10
2P o
50
-20 E (meV)
Y
E (meV) -30
0
ls
-40 -50
. . . . . 0
'
'
'
50 well width
'
' . . . . 1 O0
150
(A)
FIG 2. Variation of Is,2p0,2p± levels with well width for donor in the center of the well. Zero of energy is at the bottom of well in the absence of the donor ion.
-50
|
O
10
|
!
Z0 30 donor position ( A )
|
40
50
F I G 3. Variation of the Is,2p0,2P± levels as the donor is moved from the center of a I00 A well to the edge. Donor position is measured from well center.
VALLEY DONOR BINDING ENERGIES
888 200
*
100 E (meV) 0
-100 l
-200
i
-1 O0
I
,
l
|
200
1 O0
Oo
z (A) a)
while allowing the donor Coulomb potential to be in-" cluded in the equation for the QW subbands. Since the wave functions are computed directly from differential equations as opposed to linear combinations found by variational methods, it is possible that they will give a better representation of the true donor state. As an application of the method, donor states in an AlAs QW were calculated and the effect of the donor position in the well on both the energies and wave functions demonstrated. Acknowledgements - This work was supported by NASA through the Jet Propulsion Laboratory under subcontract :~--958997 and by the Advanced Technology Center for Photonics and Optoelectronic Materials established by the State of New Jersey and Princeton University. REFERENCES
200
1. 2.
100
E (meV) 0
-100 ,
-200
Vol. 84", No. 9
l
-1 O0
,
l
0
,
I
I O0
,
200
z (~) b) FIG 4. Effective potential of the donor and quantum well, V~dz) + VB(Z), for motion in the z direction in a 100 A well is shown alongwith the function f0(z) of the ls state for two cases a) donor at center of well b) donor at edge of well.
G. Bastard, Phys. Rev. B 24, 4714 (1981). C. Mailhiot, Y. C. Chang, and T. C. McGill, Phys. Rev. B 26, 4449 (1982). 3. R . L . Greene and K. K. Bajaj, Solid State Communications 45, 825 (1983). 4. C. Priester, G. Allan and M. Lannoo, Phys. Rev. B 28, 7194 (1983); C. Priester, G. Allan and M. Lannoo, Phys. Rev. B 29, 3408 (1984); 5. Y.C. Chang, Physica 146B, 137 (1987). 6. The proof is similar to that used to derive the Hartree equations for two particles. See, for example, 0. Madelung, Introduction to Solid State Theory, p. 11. Springer-Verlag, Berlin(1081). 7. Jia-Lin Zhu, Phys. Rev. B 40, 10529 (1980). 8. J.M. Blatt, J. Comp. Phys 1, 382 (1967). 9. S. Adachi, d. Appl. Phys. 58, R1 (1985). 10. R . A . Faulkner, Phys. Rev. 184, 713 (1969). 11. R. L. Greene and K. K. Bajaj, Phys. Rev. B 31,
4oo6 (19s5).
State Communications, Vol. 84, No. 9, pp. 885-888, 1992. ~)'Solid Printed in Great Britain.
E L L I P S O I D A L V A L L E Y D O N O R B I N D I N G E N E R G I E S I N ,4, Q U A N T U M ~NELL S. R. Parihar and S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA (Received August 31, 1992 by A. Pinczuk)
We describe a modification of the usual separable wave function approach to donor electrons confined in heterostructures which reflects the intersubband Coulomb coupling. The donor Coulomb potential is split into effective normal and in-plane components which are then used to solve for the corresponding wave functions. The procedure is carried out self consistently which can be shown to result in the best donor wave function that can be expressed in separable form. As an example the method is applied to calculate a few of the lower lying levels of a donor in a quantum well in a material with a single ellipsoidal valley conduction band.
growth axis coordinate 4,5. The result is an effective two dimensional (2D) hydrogenic problem associated with each quantized subband. The wave function is simply a product of the in-plane hydrogenlc wave function and the quantized subband of the heterostructure in the absence of the donor potential. Such an approach works best for narrow wells and shallow donors where the subband energy dominates the Coulomb binding energy. W h e n the two energies become comparable, several subbands are mixed together by the donor potential and the method loses its validity. Still, the approach is attractive because the solution procedure is straightforward regardless of the symmetry in the problem, and the separable form of the wave function makes it easy to visualize and work with. In this communication we describe a modification which extends the range of validity of the separable approach by taking into account the Coulomb potential of the donor in calculating the quantlzed subbands. The effective potentials for the in-plane motion and motion along the growth axis are determined self consistently. The wave function calculated from these potentials is the optimal one satisfying a product form. Further, we show that the ground state calculated in this manner gives a rigorous upper bound on the true ground state energy. The geometry of the problem is illustrated in Fig. 1 . We choose the long axis of the conduction band ellipsoid with mass m I along the growth axis while the transverse mass, mr, lies in the QW plane. The Schrodinger equation in the effective mass approximation is given in cylindrical coordinates by
Calculations of donor and acceptor states in semiconductor heterostructures have been actively pursued following Bastard's original paper on the effects of confinement on a hydrogenic impurity in an infinite quantum well (QW) 1. A commonly used technique is to perform a variational calculation using a basis set of the approprlate symmetry with Gaussian type orbitals 2'3. This approach yields accurate energies and gives a rigorous upper bound as well. However, variational wave functions are not as accurate as the energies they produce; moreover there are a number of steps involved in setting up and solving the problem this way. First, to avoid unnecessarily large basis sets one has to incorporate the symmetry of the problem into the basis functions. This automatically means t h a t situations where the symmetry is lowered (such as the donor located off center) entail larger basis sets and longer computation times. Second, after diagonalizing the matrix equation one must still vary shape and shift parameters, typically in the exponential part of the basis functions, to minimize the energy. Problems with multiple minima at widely separated values of these parameters can arise casting doubt onto the accuracy of the resulting wave functions. Finally, a variational solution yields the successively higher states of a given symmetry with increasingly poorer bounds. This is a problem when one wishes to solve for hydrogenlc states attached to excited subbands of a narrow heterostructure 4 for then there are many intervening states. Given the above, one is led to consider alternative methods of solution. One such approach, which lends itself naturally to layered structures, is to make the approximation that the wave function is separable in the in-plane coordinates versus the heterostructure
H~=885
+
~ p Op p2 0¢~
886
VALLEY DONOR BINDING ENERGIES - ~
2
¢
Vol. 84, No. 9
Z
+ VB(z)~b = E ¢ ,
where "7-- m r is the transverse to longitudinal mass raml rio and Vs(z ) is the QW potential
/
vB('.)-I" I> ~2
T
/
Kz
quantum well
a)
The units of energy and length used are Ry*-- mte4 ,
potentlai energy
~s ~'2
and aO=m---~2, respectively where Q is the static dielectric constant. We have neglected the dielectric and effective mass differences between the well and the barrier materials. Following Chang s we write the donor wave function in separable form
¢ffif.(z)¢(p,¢),
(2)
where fn(z) is the n th QW subband satisfying { " 7 5 + V B ( Z ) }fn ffi e~fn •
O
i
_
Lw 0 2
L__w 2
z
b) FIG 1. a) Relative alignment of quantum well and conduction band ellipsoid, b) Quantum well potential profile.
(3)
Substituting (2) into (1), using (3) and projecting the final equation onto fn(z) yields an effective in-plane equation for ~(p,~b)
is accomplished by projecting (1) onto ~b and using (4a,b) to give 02fn
~ - ~ - 2 +VB(~)f.+V~(z)fo = ~f., 21r oo
VeZ~(zl------f0 fo
vNp)=- f 2fi(,-)f~(,.)d,..
(4b)
(s)
,
24 ¢
~pdpd*,
2W
e~=E--eP+ f f ~*~V~pdpd~b. Thus for each subband the problem is reduced to finding the bound states of an effective 2D central potential V ~ p ) . For electrons with small effective mass as is typical for III-V compounds equation (4) is a satisfactory approximation in dealing with a QW of width smaller than a few hundred A . Under these conditions the Coulomb binding energy (a few meV) is much smaller than the QW subband spacing and little subband mixing occum. However, for heavier masses such as occur in the X valley, the subband energy and the Coulomb energy become comparable to each other. In that case one cannot assume the QW subbands remain unaffected as equation (3) implies. In order to get a physical picture we first look at V~(p). It can be interpreted as the in-plane Coulomb potential with the electron smeared out into a line segment along z because of the finite extent of the QW subband. In a llke fashion, having solved (4a) for the in-plane wave function &(p,¢) we can now modify (3) to include a Coulomb potential along z resulting from smearing out the electron into a disk of charge given by ~*~. This
0
0
In this way the effect of the Coulomb potential can be introduced into the equation for the QW subbands. By iterating (4) and (5) to self consistency we obtain the best solution to the problem possible in a separable form. 6 It is worth noting that Zhu7 had previously introduced a two parameter V shaped potential into (3) at the donor position to improve the blnding energies. However, the assumption of linear variation of the potential is true only near the donor ion. With large binding energies especially, the small extent in plane of the wave function makes V~(z) quickly deviate from the simple V shape as is evident from Fig. 4. As Fig. 1 shows the problem is rotationally invariant about the z axis. Therefore the z component of angular momentum is conserved and we may write for a state with Lzffiffiml~ ~ffiffifn(z)Rm(P)eim¢ .
(6)
We solve equations (4) and {5) numerically for the functions fn(Z) and Rm(P) respectively. The Numerov method is used to find QW subbands along the lines
VALLEY DONOR BINDING ENERGIES
Voh 84: No. 9
"given in Ref. 8 while fourth order Runge-Kutta is employed for the in-plane radial equation. The effective potentials, Veel~p) and Vega(z), are calculated at a fixed number of points and fitted with cubic splines. Near the donor position V~t~p) becomes logarithmically singular while VeZll(z) remains finite with a constant slope. The number of iterations necessary to achieve convergence depends on both the bulk binding energy of the donor and the well width. Thus, while no iteration is necessary for the calculation of silicon in GaAs at typical well widths (100 .~ ), five or more iterations are needed when using the mass parameters of AlAs or Si X valleys. In general as the well width increases more iterations are needed because the Coulomb energy is no longer small relative to the subband spacing. Calculated results are shown in Fig. 2. We use parameters appropriate for an AlAs Q W with AlGaAs(x=0.5) barriers ie. mt==0.1gm o, m l = l . l m o, Q----10.06,and VB=169meV (m o is the free electron mass) 9. The donor is at the well center and since we consider only a single valley, valley-orbit effects which would split the ground state are neglected. The energies of three states, the ls, 2po, and 2p=~ are shown as functions of well width. The 2Po and 2p~. are low lying excited states having a single node in the wave function at z=0 and p=0 respectively corresponding to the 2pz and 2px,y bulk hydrogenic states. Their behaviour is similar to t h a t of donors with isotroplc effective mass except that here the 2po level lies below the 2p~ level in the bulk limit because of the heavier longitudinal mass m. As the well width decreases the 2p o state rises rapidly in energy because of its association by symmetry with the first excited Q W subband 11, crossing the 2p~ level at about 60 A well width. We have carried out a variational calculation along the lines of Ref. 3 to check the accuracy of the separable approach. The difference in the p state energies is indistinguishable on the scale of Fig. 2 while the variational ground state energy is at most only 2 meV
1 O0
887
lower at the larger well width. The fact t h a t the ground state energy in the bulk limit is accurate to within 5 percent is remarkable given the fact we are using separable wave functions. In addition, by taking the diagonal matrix elements of the full Hamiltonian, H, of equation (1) in the self consistent state (2) one obtains after using equations (4) and (5) < ~ [H I ¢ > = < f n
I--'/-~2 +V~(z)+VB(z)If,>
+e-<¢
(7)
Ivt~(p)I¢>, 2~
=,~+,'- f 0
co
f'~*¢V:@dpd¢, 0
which is identical to the self consistent energy of that state. Since the three states in Fig. 2 are the lowest of their respective symmetry types, the variational principle ensures the obtained energies are rigorous upper bounds. The case of the off center donor at position zd is easily handled by the substitution
in the expressions for Vega(z) and V•ll(p ). Fig. 3 shows the effects of moving the donor from the center of the well to the edge. As expected the energies decrease because the Q W barrier pushes the wave function away from the donor. One appealing feature of the separable approach is t h a t it is easy to visualize how the Coulomb potential appears algng the z axis in relation to the heterojunetion barriers. This is demonstrated in Fig. 4 where the effective potential energy.and wave funetlon of the ground subband of a 100 A well are plotted along z for the ease of the donor at the center of the well or at well edge. In summary we have presented a self consistent version of the 2D effective potential approach to donors in QW's. It has the advantage of maintaining the wave function in a convenient separable form
0
•
|
|
|
|
2p..
-10
2P o
50
-20 E (meV)
Y
E (meV) -30
0
ls
-40 -50
. . . . . 0
'
'
'
50 well width
'
' . . . . 1 O0
150
(A)
FIG 2. Variation of Is,2p0,2p± levels with well width for donor in the center of the well. Zero of energy is at the bottom of well in the absence of the donor ion.
-50
|
O
10
|
!
Z0 30 donor position ( A )
|
40
50
F I G 3. Variation of the Is,2p0,2P± levels as the donor is moved from the center of a I00 A well to the edge. Donor position is measured from well center.
VALLEY DONOR BINDING ENERGIES
888 200
*
100 E (meV) 0
-100 l
-200
i
-1 O0
I
,
l
|
200
1 O0
Oo
z (A) a)
while allowing the donor Coulomb potential to be in-" cluded in the equation for the QW subbands. Since the wave functions are computed directly from differential equations as opposed to linear combinations found by variational methods, it is possible that they will give a better representation of the true donor state. As an application of the method, donor states in an AlAs QW were calculated and the effect of the donor position in the well on both the energies and wave functions demonstrated. Acknowledgements - This work was supported by NASA through the Jet Propulsion Laboratory under subcontract :~--958997 and by the Advanced Technology Center for Photonics and Optoelectronic Materials established by the State of New Jersey and Princeton University. REFERENCES
200
1. 2.
100
E (meV) 0
-100 ,
-200
Vol. 84", No. 9
l
-1 O0
,
l
0
,
I
I O0
,
200
z (~) b) FIG 4. Effective potential of the donor and quantum well, V~dz) + VB(Z), for motion in the z direction in a 100 A well is shown alongwith the function f0(z) of the ls state for two cases a) donor at center of well b) donor at edge of well.
G. Bastard, Phys. Rev. B 24, 4714 (1981). C. Mailhiot, Y. C. Chang, and T. C. McGill, Phys. Rev. B 26, 4449 (1982). 3. R . L . Greene and K. K. Bajaj, Solid State Communications 45, 825 (1983). 4. C. Priester, G. Allan and M. Lannoo, Phys. Rev. B 28, 7194 (1983); C. Priester, G. Allan and M. Lannoo, Phys. Rev. B 29, 3408 (1984); 5. Y.C. Chang, Physica 146B, 137 (1987). 6. The proof is similar to that used to derive the Hartree equations for two particles. See, for example, 0. Madelung, Introduction to Solid State Theory, p. 11. Springer-Verlag, Berlin(1081). 7. Jia-Lin Zhu, Phys. Rev. B 40, 10529 (1980). 8. J.M. Blatt, J. Comp. Phys 1, 382 (1967). 9. S. Adachi, d. Appl. Phys. 58, R1 (1985). 10. R . A . Faulkner, Phys. Rev. 184, 713 (1969). 11. R. L. Greene and K. K. Bajaj, Phys. Rev. B 31,
4oo6 (19s5).