- Email: [email protected]
Binding energy of a double donor in a parabolic quantum well
30 January 1995 PHYSICS LETTERS A
Physics Letters A 197 (1995) 330-334
El ~qEVIER
Binding energy of a double donor in a parabolic quantum well Ecaterina C. Niculescu Department of Physics, "'Politehnica'" Universityof Bucharest, 77206Bucharest, Romania
Received 3 October 1994;accepted for publication 22 November 1994 Communicated by J. Flouquet
Abstract We have calculated the binding energies of the ground state of a double donor associated with the first subband in a parabolic quantum well. Taking into account the nonparabolicity of the conduction band, we have considered the two cases of positive donor center and neutral donor center. The results have been obtained as a function of the parabolicity parameter by using a variational approach. It is found that the nonparabolicity causes an increase in the binding energy; also, this energy increases with the increasing of the parabolicity parameter a,
1. Introduction With modern semiconductor crystal growth techniques, quantum wells can be made in different shapes, such as the square well (SW), the triangular well ( T W ) , and the parabolic well ( P W ) [ 1 ]. In recent years, the problem o f shallow impurities in these structures has been studied extensively. The energy levels o f a hydrogenic impurity in a SW without a magnetic field [ 2 ] or With a magnetic field applied [ 3 ] have been calculated. Lane and Greene [4 ] and Helm et al. [ 5 ] have generalized this calculation to a superlattice in the absence o f an applied magnetic field and recently, Nguyen et al. [ 6 ] have studied the ground and first excited states of a hydrogenic donor placed in a finite superlattice in a magnetic field. For PWs, Luna-Acosta [ 7 ] and Leavitt [ 8 ] calculated the binding energy of a hydrogenic impurity using variational methods. The effect o f a magnetic field on donor states in a PW was examined by Zang and Rustgi
[9]. The present work deals with the study o f the ground state o f a two-electron impurity center confined by a onedimensional parabolic potential well. We follow a variational approach, in which a trial wave function containing only a single variational parameter (as in Ref. [7] ) is used for both electrons. In addition, we take into account the effect of the nonparabolicity of the conduction band on the binding energy of the impurity.
2. Theory We consider impurities which may be described by the effective-mass approximation (EMA). A detailed discussion on the validity of EMA in bulk materials has been given by Pantelides and Sah [ 10 ]. Contrary to 0375-9601/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved SSD10375-9601 ( 94 )00957-0
E.C. Niculescu / Physics Letters A 197 (1995) 330-334
331
traditional understanding, the method is shown to be applicable for both shallow and deep levels but only for the cases when the impurity atom is substitutional and it belongs to the same row of the Periodic Table as the host atom. For this special case, the impurity potential is very nearly equal to the Coulomb potential of n point charges [ 10], with n = 1 for a single donor (hydrogenic-type impurity) and n = 2 for a double donor (heliumtype impurity), and, therefore, the EMA is valid [ 10 ]. Finally, especially for deep levels, it is necessary to take into account the nonparabolicity of the conduction band. This effect is included in our work, as in Ref. [ ! 1 ], through an energy-dependent effective mass m* given by m*= m y ( E ) ,
(1)
where m is the effective mass at the band extremum. For GaAs, with the minimum of the conduction band at g = 0 , ~,(E) is[ 11 ] y(E) = 1 + 0.6556E+ 3.4586E 2 - 2.2105E 3 ,
(2)
with E expressed in eV. For the two-electron centers, we can write the Hamiltonian as H = H ~ +H2r
(3)
Uee,
where He is the Hamiltonian for an electron in a Coulomb field of a two point charge located at the bottom of a PW, He=-
h2 V 2 - 2 e 2 + 2 a 2 z 2 . 2m* Eor
(4)
~o is the dielectric constant of the medium and a is the parabolicity parameter [9]. In (3), Uee is the electronelectron interaction term, e2
U ~ = Eolr,-rzl "
(5)
If the donor center binds one electron, so that it is positively charged, the Hamiltonian is given by expression (4). For a variational estimate of the ground state for the one-electron system, we use a trial wave-function [ 7 ] of the form ~v(p, z ) = N e x p ( - z 2 / 2 2) e x p ( - ~ / a )
,
(6)
where N is the normalization constant [ 7 ], a is the variational parameter and 22=
h a [ m T ( E ) ] I/2 "
E is the lowest subband energy and can be computed by solving the transcendental equation E=h
2a [~,(E)m]l/2 .
(7)
The expectation value of H, is minimized with respect to a, and the binding energy for this system is obtained as
E(b+)=E-(ae)min,
(8)
where ( H e ) -= ( ~PIH, I ~v). When two electrons are hound at a donor center, so that we are dealing with a neutral system, in the ground
332
E.C. Niculescu / Physics Letters A 197 (1995) 330-334
state both electrons occupy the I So state with opposite spins, a situation analogous to the helium atom. If we use the simple trial function (6) for both electrons, the total energy of the two electron system is given by ( H ) - ( ~(Pl, zl ) ~(P2, Z2) IHI ~U(p,, z~ ) ~(P2, Zz) ) = 2 ( ~[ He [ 7u) + ( ~(p, , z, ) ~(Pz, z2 ) I Ueo I ~U(p~,z, ) ~(P2, zz) >,
(9)
where (
Uee)
~-- ( ~ / ( P l , -71 ) ~'/(P2, 22 ) I Uee ] ~/(P, , -71 ) ~t(P2, z2 ) ) ,
is the expectation value of the electron-electron interaction term. The most difficult part of the variational calculation is the evaluation of this last term. In cylindrical coordinates we have e2 Uee= e0[p2 +p~ --2p,p2 COS(0, --02) + (Z, - - Z 2 ' z ] ,/2"
(10)
In order to simplify the calculation, in a zero order approximation, we will replace, as in Ref. [12], p 2 +p z2 _ 2p tPz cos ( 0, - 02) with its average value P~v in the ground state,
p2v = ( ~U(p,, Zl ) ~V(p2, Zu ) ]p2 +p2 _ 2pipe cos (01 - 02 ) ] ~(Pl, Zl ) 7J(p2, Zu) ) .
( 11 )
This quantity can be expressed in terms of the complementary error function erfc as follows, nN 2 Ply = T [ - 2 4 a + 3a322+ x / c ~ exp(2Z/2aZ) erfc(2/x/~a)(2s-Z23aZ+ 32a4) ] " Thus, in the zero order approximation, the electron-electron interaction term is U(O)_
82
(12)
ee -- (:0[p2v..] = (Z 1 __72)2 ] ,/2"
We will consider the quantity e2
up = Eo[p~ +p~ + (z, - z 2 ) 2 ] ~/2 - u-(~
(13)
as a perturbation. (Within this term, we will replace only the quantity PiP2 cos(01- 02) with its expectation value.) For a given 2, ( U~~ ) is dependent on the variational parameter a; its expression, after the integrals over 01, Oz, p, andp2 are performed, is (U,~= 16nZN4a 4 ii o o (~a+zl 1)(~aaZ2 + 1 ) e x p (
2(Zla+Z2)- ) (f~ + f 2 ) e x p ( - 2(zz~2+zz)X)dz,dz2,
(14,
where fl = [p~v(a) + (zl - z 2 ) a ] -1/2,
f2 = [p~v(a) + (zt +z2)21 -,/2
In the zero order approximation, the total energy ( Ho ) = 2 (He) + ( U~(~ is then minimized with respect to a. Let am be the value of a corresponding to the minimized ( H o ) . With the perturbation term defined by Eq. ( 13 ), in the first approximation of perturbation theory, for the ground state energy of the neutral center, we obtain (H)min = (Ho(am)) + (Up(a~) ) .
(15)
The first binding energy for the double donor is obtained as Eb = E - ( (H)min -- (He)min)
"
(16)
E.C Niculescu / Physics Letters A 197 (1995) 330-334
333
7.5 40
7.0 6.5 *
3,5
9"o 6.0
r~
cr
3,0 5.5 5.0
25
4.5 2.0 4.0 0
1
2
3
4
0
2
1
3
4
5
OC
Fig. 1. Binding energy (in effective Rydberg units Rd*) as a function of the parabolicity parameter tx for a positively charged donor center. Solid curve: parabolic band model; dashed curve: nonparabolicity included. Fig. 2. Binding energy (in effective Rydberg units Rd*) as a function of the parabolicity parameter c~ for a neutral donor center. The effect of the nonparabolicity of the conduction band is included.
Table I Ground state energies (in Rd*) for a double donor in a PW c~
Energy of a double donor positive center
Energy of a single donor neutral center
(He> min (Ho > rain
0.1 0.4 1 2 4 5
--3.9874 " --3.9843 --3.8215 " - - 3.8082 --3.2097 " --3.1692 -- 1.8705 a - - 1.7640 1.1855 a 1.5059 2.7756 ~ 3.2454
--6.2542 --6.2503 --5.8555 -- 5.8308 --4.5394 --4.4637 -- 1.7605 -- 1.5591 4.4834 5.0947 7.7164 8.6137
( H > rain
"
--5.9540 a
~
--5.4431 ~
"
--4.0372 ~
~
-- 1.1437 "
"
5.2623 a
"
8.5617 ~
--0.9529 --0.9520 --0.5937 -- 0.5892 0.3589 0.3765 2.1091 2.1654 5.7317 5.9328 7.5452 7.8547
~ ~ " " " "
a Results of calculation with the nonparabolicity of the conduction band included.
3.
Results
We apply the method
to the calculation of the binding energy of a helium-type
AlxGa~_xAS. We shall use the effective Rydberg
Rd* (= 5.8 meV)
impurity
in PWs in GaAs/
as the energy unit and the effective Bohr
radius a* ( =99.6 A) as the length unit. Table 1 shows the ground state energies for the two point charges when the parabolicity parameter
a assumes
334
E.C. Niculescu / Physics Letters A 197 (1995) 330-334
various values. For comparison, the corresponding values for a single point charge are given. We observe a decrease o f the ground state energy due to the nonparabolicity effect, especially for high values of a, in analogy to the results obtained for a hydrogenic donor in square q u a n t u m well [ 11 ]. In the studied range of the parabolicity parameter, the m a x i m u m decrease in the ground state energy is about 20% for (He)rain, 12% for (Ho)mjn and only 4% for the hydrogenic impurity. Note that as a goes to zero, the correct 3D limit for (He)rain is obtained, but not for the quantity (Ho)min ( ~ - 5.70 Rd [ 13 ] ), which shows that the zero order approximation is insufficient for the evaluation o f (Uee). Table 1 also shows the ground state energies obtained with Eq. ( 15 ) (in the first approximation o f perturbation theory), for various values of the parabolicity parameter. The effect of the nonparabolicity o f the conduction band is included. The agreement with the value o f the He atom ground state energy (the limit for o~--,0 o f ( H ) min) is better than in the zero order approximation. The main source of errors in the calculation o f (H)min is the approximate evaluation o f the electron-electron interaction term. Figs. 1 and 2 show the binding energy Eb as function o f the parabolicity parameter a for the positively charged center and for the neutral center, respectively. In both cases, this quantity increases with a, consistent with the previous results [ 7-9 ]. Note that as c~ goes to zero, the binding energies for the two states determined by doubledonor impurities are approximately in a ratio 2: 1, as in bulk materials [ 10 ].
4. Conclusions We present a variational calculation o f the ground state energy for a two-electron impurity center, located at the bottom o f a one-dimensional parabolic well. The effect of the nonparabolicity o f the conduction band is included for both the positively charged center and the neutral center. An application to a d o n o r in a G a A s / AlxGa~ _xAs quantum well is presented and reasonable agreement is obtained for the 3D limit, in both cases.
References [ 1] P. Yuh and K.L. Wang, Phys. Rev. B 38 (1988) 13307. [2] H. Chen and S. Zhou, Phys. Rev. B 36 (1987) 9581. [3] R.L. Greene and K.K. Bajaj, Phys. Rev. B 31 (1985) 913. [4] P. Lane and R.L. Greene, Phys. Rev. B 33 (1986) 5871. [5] M. Helm, F.M. Peeters, F. Derosa, E. Colas, J.P. Harbison and L.T. Florez, Phys. Rev. B 43 ( 1991 ) 13983. [6 ] N. Nguyen, J.X. Zang, R. Ranganathan, B.D. McCombe and M.L. Rustgi, Phys. Rev. B 48 ( 1993) 14226. [7] G.A. Luna-Acosta,Solid State Commun. 55 ( 1985 ) 5. [8] R.P. Leavitt, Phys. Rev. B 36 (1987) 7650. [ 9 ] J.X. Zang and M.L. Rustgi, Phys. Rev. B 48 (1993) 2465. [10] S.T. Pantelides and C.T. Sah, Phys. Rev. B 10 (1974) 621. [ 11 ] S. Chaudhuri and K.K. Bajaj, Phys. Rev. B 29 (1984) 1803. [ 12 ] Wei-DongSheng and Shi-Wei Gu, Phys. Star. Sol. (b) 178 ( 1993) 377. [ 13] L. Landau and E. Lifshitz, Mecanique quantique (Mir, Moscow, 1966) p. 286.
Physics Letters A 197 (1995) 330-334
El ~qEVIER
Binding energy of a double donor in a parabolic quantum well Ecaterina C. Niculescu Department of Physics, "'Politehnica'" Universityof Bucharest, 77206Bucharest, Romania
Received 3 October 1994;accepted for publication 22 November 1994 Communicated by J. Flouquet
Abstract We have calculated the binding energies of the ground state of a double donor associated with the first subband in a parabolic quantum well. Taking into account the nonparabolicity of the conduction band, we have considered the two cases of positive donor center and neutral donor center. The results have been obtained as a function of the parabolicity parameter by using a variational approach. It is found that the nonparabolicity causes an increase in the binding energy; also, this energy increases with the increasing of the parabolicity parameter a,
1. Introduction With modern semiconductor crystal growth techniques, quantum wells can be made in different shapes, such as the square well (SW), the triangular well ( T W ) , and the parabolic well ( P W ) [ 1 ]. In recent years, the problem o f shallow impurities in these structures has been studied extensively. The energy levels o f a hydrogenic impurity in a SW without a magnetic field [ 2 ] or With a magnetic field applied [ 3 ] have been calculated. Lane and Greene [4 ] and Helm et al. [ 5 ] have generalized this calculation to a superlattice in the absence o f an applied magnetic field and recently, Nguyen et al. [ 6 ] have studied the ground and first excited states of a hydrogenic donor placed in a finite superlattice in a magnetic field. For PWs, Luna-Acosta [ 7 ] and Leavitt [ 8 ] calculated the binding energy of a hydrogenic impurity using variational methods. The effect o f a magnetic field on donor states in a PW was examined by Zang and Rustgi
[9]. The present work deals with the study o f the ground state o f a two-electron impurity center confined by a onedimensional parabolic potential well. We follow a variational approach, in which a trial wave function containing only a single variational parameter (as in Ref. [7] ) is used for both electrons. In addition, we take into account the effect of the nonparabolicity of the conduction band on the binding energy of the impurity.
2. Theory We consider impurities which may be described by the effective-mass approximation (EMA). A detailed discussion on the validity of EMA in bulk materials has been given by Pantelides and Sah [ 10 ]. Contrary to 0375-9601/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved SSD10375-9601 ( 94 )00957-0
E.C. Niculescu / Physics Letters A 197 (1995) 330-334
331
traditional understanding, the method is shown to be applicable for both shallow and deep levels but only for the cases when the impurity atom is substitutional and it belongs to the same row of the Periodic Table as the host atom. For this special case, the impurity potential is very nearly equal to the Coulomb potential of n point charges [ 10], with n = 1 for a single donor (hydrogenic-type impurity) and n = 2 for a double donor (heliumtype impurity), and, therefore, the EMA is valid [ 10 ]. Finally, especially for deep levels, it is necessary to take into account the nonparabolicity of the conduction band. This effect is included in our work, as in Ref. [ ! 1 ], through an energy-dependent effective mass m* given by m*= m y ( E ) ,
(1)
where m is the effective mass at the band extremum. For GaAs, with the minimum of the conduction band at g = 0 , ~,(E) is[ 11 ] y(E) = 1 + 0.6556E+ 3.4586E 2 - 2.2105E 3 ,
(2)
with E expressed in eV. For the two-electron centers, we can write the Hamiltonian as H = H ~ +H2r
(3)
Uee,
where He is the Hamiltonian for an electron in a Coulomb field of a two point charge located at the bottom of a PW, He=-
h2 V 2 - 2 e 2 + 2 a 2 z 2 . 2m* Eor
(4)
~o is the dielectric constant of the medium and a is the parabolicity parameter [9]. In (3), Uee is the electronelectron interaction term, e2
U ~ = Eolr,-rzl "
(5)
If the donor center binds one electron, so that it is positively charged, the Hamiltonian is given by expression (4). For a variational estimate of the ground state for the one-electron system, we use a trial wave-function [ 7 ] of the form ~v(p, z ) = N e x p ( - z 2 / 2 2) e x p ( - ~ / a )
,
(6)
where N is the normalization constant [ 7 ], a is the variational parameter and 22=
h a [ m T ( E ) ] I/2 "
E is the lowest subband energy and can be computed by solving the transcendental equation E=h
2a [~,(E)m]l/2 .
(7)
The expectation value of H, is minimized with respect to a, and the binding energy for this system is obtained as
E(b+)=E-(ae)min,
(8)
where ( H e ) -= ( ~PIH, I ~v). When two electrons are hound at a donor center, so that we are dealing with a neutral system, in the ground
332
E.C. Niculescu / Physics Letters A 197 (1995) 330-334
state both electrons occupy the I So state with opposite spins, a situation analogous to the helium atom. If we use the simple trial function (6) for both electrons, the total energy of the two electron system is given by ( H ) - ( ~(Pl, zl ) ~(P2, Z2) IHI ~U(p,, z~ ) ~(P2, Zz) ) = 2 ( ~[ He [ 7u) + ( ~(p, , z, ) ~(Pz, z2 ) I Ueo I ~U(p~,z, ) ~(P2, zz) >,
(9)
where (
Uee)
~-- ( ~ / ( P l , -71 ) ~'/(P2, 22 ) I Uee ] ~/(P, , -71 ) ~t(P2, z2 ) ) ,
is the expectation value of the electron-electron interaction term. The most difficult part of the variational calculation is the evaluation of this last term. In cylindrical coordinates we have e2 Uee= e0[p2 +p~ --2p,p2 COS(0, --02) + (Z, - - Z 2 ' z ] ,/2"
(10)
In order to simplify the calculation, in a zero order approximation, we will replace, as in Ref. [12], p 2 +p z2 _ 2p tPz cos ( 0, - 02) with its average value P~v in the ground state,
p2v = ( ~U(p,, Zl ) ~V(p2, Zu ) ]p2 +p2 _ 2pipe cos (01 - 02 ) ] ~(Pl, Zl ) 7J(p2, Zu) ) .
( 11 )
This quantity can be expressed in terms of the complementary error function erfc as follows, nN 2 Ply = T [ - 2 4 a + 3a322+ x / c ~ exp(2Z/2aZ) erfc(2/x/~a)(2s-Z23aZ+ 32a4) ] " Thus, in the zero order approximation, the electron-electron interaction term is U(O)_
82
(12)
ee -- (:0[p2v..] = (Z 1 __72)2 ] ,/2"
We will consider the quantity e2
up = Eo[p~ +p~ + (z, - z 2 ) 2 ] ~/2 - u-(~
(13)
as a perturbation. (Within this term, we will replace only the quantity PiP2 cos(01- 02) with its expectation value.) For a given 2, ( U~~ ) is dependent on the variational parameter a; its expression, after the integrals over 01, Oz, p, andp2 are performed, is (U,~= 16nZN4a 4 ii o o (~a+zl 1)(~aaZ2 + 1 ) e x p (
2(Zla+Z2)- ) (f~ + f 2 ) e x p ( - 2(zz~2+zz)X)dz,dz2,
(14,
where fl = [p~v(a) + (zl - z 2 ) a ] -1/2,
f2 = [p~v(a) + (zt +z2)21 -,/2
In the zero order approximation, the total energy ( Ho ) = 2 (He) + ( U~(~ is then minimized with respect to a. Let am be the value of a corresponding to the minimized ( H o ) . With the perturbation term defined by Eq. ( 13 ), in the first approximation of perturbation theory, for the ground state energy of the neutral center, we obtain (H)min = (Ho(am)) + (Up(a~) ) .
(15)
The first binding energy for the double donor is obtained as Eb = E - ( (H)min -- (He)min)
"
(16)
E.C Niculescu / Physics Letters A 197 (1995) 330-334
333
7.5 40
7.0 6.5 *
3,5
9"o 6.0
r~
cr
3,0 5.5 5.0
25
4.5 2.0 4.0 0
1
2
3
4
0
2
1
3
4
5
OC
Fig. 1. Binding energy (in effective Rydberg units Rd*) as a function of the parabolicity parameter tx for a positively charged donor center. Solid curve: parabolic band model; dashed curve: nonparabolicity included. Fig. 2. Binding energy (in effective Rydberg units Rd*) as a function of the parabolicity parameter c~ for a neutral donor center. The effect of the nonparabolicity of the conduction band is included.
Table I Ground state energies (in Rd*) for a double donor in a PW c~
Energy of a double donor positive center
Energy of a single donor neutral center
(He> min (Ho > rain
0.1 0.4 1 2 4 5
--3.9874 " --3.9843 --3.8215 " - - 3.8082 --3.2097 " --3.1692 -- 1.8705 a - - 1.7640 1.1855 a 1.5059 2.7756 ~ 3.2454
--6.2542 --6.2503 --5.8555 -- 5.8308 --4.5394 --4.4637 -- 1.7605 -- 1.5591 4.4834 5.0947 7.7164 8.6137
( H > rain
"
--5.9540 a
~
--5.4431 ~
"
--4.0372 ~
~
-- 1.1437 "
"
5.2623 a
"
8.5617 ~
--0.9529 --0.9520 --0.5937 -- 0.5892 0.3589 0.3765 2.1091 2.1654 5.7317 5.9328 7.5452 7.8547
~ ~ " " " "
a Results of calculation with the nonparabolicity of the conduction band included.
3.
Results
We apply the method
to the calculation of the binding energy of a helium-type
AlxGa~_xAS. We shall use the effective Rydberg
Rd* (= 5.8 meV)
impurity
in PWs in GaAs/
as the energy unit and the effective Bohr
radius a* ( =99.6 A) as the length unit. Table 1 shows the ground state energies for the two point charges when the parabolicity parameter
a assumes
334
E.C. Niculescu / Physics Letters A 197 (1995) 330-334
various values. For comparison, the corresponding values for a single point charge are given. We observe a decrease o f the ground state energy due to the nonparabolicity effect, especially for high values of a, in analogy to the results obtained for a hydrogenic donor in square q u a n t u m well [ 11 ]. In the studied range of the parabolicity parameter, the m a x i m u m decrease in the ground state energy is about 20% for (He)rain, 12% for (Ho)mjn and only 4% for the hydrogenic impurity. Note that as a goes to zero, the correct 3D limit for (He)rain is obtained, but not for the quantity (Ho)min ( ~ - 5.70 Rd [ 13 ] ), which shows that the zero order approximation is insufficient for the evaluation o f (Uee). Table 1 also shows the ground state energies obtained with Eq. ( 15 ) (in the first approximation o f perturbation theory), for various values of the parabolicity parameter. The effect of the nonparabolicity o f the conduction band is included. The agreement with the value o f the He atom ground state energy (the limit for o~--,0 o f ( H ) min) is better than in the zero order approximation. The main source of errors in the calculation o f (H)min is the approximate evaluation o f the electron-electron interaction term. Figs. 1 and 2 show the binding energy Eb as function o f the parabolicity parameter a for the positively charged center and for the neutral center, respectively. In both cases, this quantity increases with a, consistent with the previous results [ 7-9 ]. Note that as c~ goes to zero, the binding energies for the two states determined by doubledonor impurities are approximately in a ratio 2: 1, as in bulk materials [ 10 ].
4. Conclusions We present a variational calculation o f the ground state energy for a two-electron impurity center, located at the bottom o f a one-dimensional parabolic well. The effect of the nonparabolicity o f the conduction band is included for both the positively charged center and the neutral center. An application to a d o n o r in a G a A s / AlxGa~ _xAs quantum well is presented and reasonable agreement is obtained for the 3D limit, in both cases.
References [ 1] P. Yuh and K.L. Wang, Phys. Rev. B 38 (1988) 13307. [2] H. Chen and S. Zhou, Phys. Rev. B 36 (1987) 9581. [3] R.L. Greene and K.K. Bajaj, Phys. Rev. B 31 (1985) 913. [4] P. Lane and R.L. Greene, Phys. Rev. B 33 (1986) 5871. [5] M. Helm, F.M. Peeters, F. Derosa, E. Colas, J.P. Harbison and L.T. Florez, Phys. Rev. B 43 ( 1991 ) 13983. [6 ] N. Nguyen, J.X. Zang, R. Ranganathan, B.D. McCombe and M.L. Rustgi, Phys. Rev. B 48 ( 1993) 14226. [7] G.A. Luna-Acosta,Solid State Commun. 55 ( 1985 ) 5. [8] R.P. Leavitt, Phys. Rev. B 36 (1987) 7650. [ 9 ] J.X. Zang and M.L. Rustgi, Phys. Rev. B 48 (1993) 2465. [10] S.T. Pantelides and C.T. Sah, Phys. Rev. B 10 (1974) 621. [ 11 ] S. Chaudhuri and K.K. Bajaj, Phys. Rev. B 29 (1984) 1803. [ 12 ] Wei-DongSheng and Shi-Wei Gu, Phys. Star. Sol. (b) 178 ( 1993) 377. [ 13] L. Landau and E. Lifshitz, Mecanique quantique (Mir, Moscow, 1966) p. 286.