Danhong Huang and M. O. Manasreh. Phillips Laboratory (PL/VTRP), 3550 Aberdeen Avenue SE, Building 426, Kirtland Air Force Base, New Mexico 87117.
PHYSICAL REVIEW B
VOLUME 54, NUMBER 3
15 JULY 1996-I
Effects of the screened exchange interaction on the tunneling and Landau gaps in double quantum wells Danhong Huang and M. O. Manasreh Phillips Laboratory (PL/VTRP), 3550 Aberdeen Avenue SE, Building 426, Kirtland Air Force Base, New Mexico 87117 ~Received 2 January 1996! A self-consistent screened Hartree-Fock calculation, combined with the Landau quantization of in-plane electron motion, is performed to find the eigenstates and eigenenergies of electrons in double quantum wells. This theory is applicable to both the low and strong magnetic-field cases. The screened exchange interaction is calculated by using a generalized Thomas-Fermi screening model. The approximately linear increase of the tunneling gap at low magnetic fields (B,9 T! and the switching of the ground state between the tunnelingsplit first Landau levels are seen and explained as a result of the increase of screening effects on the exchange interaction when both tunneling-split first Landau levels are filled. @S0163-1829~96!00623-6#
There have long been both theoretical and experimental interests in double-quantum-well structures in the presence of an external magnetic field. The destruction of integer quantum Hall states with odd filling factors in doublequantum-well structures and in a single wide quantum well was observed recently.1–3 Several theoretical explanations4–6 were given for these observations. Also, the collapse of the fractional quantum Hall states in the single wide quantum well7 and half integer fractional quantum Hall state in the double-quantum-well structure8,9 was reported. The theoretical explanations to these experimental findings were proposed10–12 for both single wide quantum-well and doublequantum-well structures. Some theoretical calculations on the magnetoroton excitation spectrum13–15 and the suppression of the spin-density excitation16,17 were shown. More recently, the effect of an in-plane magnetic field on the electron tunneling was studied both experimentally18–22 and theoretically.23,24 The electron tunneling behavior in the presence of a perpendicular magnetic field was also found very interesting and nontrivial,25,26 where instead of a AB dependence an approximately linear B dependence of the tunneling gap from the dc measurement was observed. This behavior was first explained by Yang and MacDonald as a result of the ‘‘Coulomb gap’’27 in the density of states when the filling factor is less than unity. Unfortunately, this explanation only produced a AB dependence on the tunneling gap, which is not supported by a recent experiment.26 Besides, the screening effects on the exchange interaction is neglected in their calculation.27 The magnetoplasmon excitation in the doublequantum-well structure was also calculated.28 However, the effect of the Landau quantization in the presence of a magnetic field was not included in the self-consistent Hartree calculation of the eigenstates and eigenenergies in the z direction. This limits the application of their theory most to the very week magnetic-field case. In this paper, we have performed a self-consistent screened Hartree-Fock calculation combined with the Landau quantization of the in-plane electron motion to find the eigenstates and eigenenergies of electrons in a doublequantum-well structure. This theory is expected to be appli0163-1829/96/54~3!/2044~5!/$10.00
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cable to both low (B,9 T! and strong magnetic-field (B.9 T! cases. In our model, the screened exchange interaction is calculated using a generalized Thomas-Fermi screening model. From our numerical computation, we find the approximately linear increase of the tunneling gap at low magnetic fields (B,9 T! and the switching of the ground state between the tunneling-split Landau levels, which are explained as a result of the increase of screening effects on the exchange interaction when both the tunneling-split first Landau levels are filled. In our calculation, we concentrate on the strong-coupling region with thin barrier, which is out of the Coulomb gap region with a thick barrier. The predicted switching of the ground state below 9 T in this paper is different and, to our knowledge, the first reported in this system that can be observed in the cyclotron resonance experiment as an anticrossing of two resonance peaks. The model we consider is a doped symmetric doublequantum-well ~DQW! structure that contains two wells separated by a middle barrier. An external magnetic field is applied perpendicular to the planes of quantum wells. In the self-consistent screened Hartree-Fock approximation, the vertical electron motion in the z direction perpendicular to the DQW is described by the Schro¨dinger equation containing both the DQW potential V DQW(z) and the self-consistent Hartree potentials V H (z), 28
F
2
S
D
G
1 d \2 d 1V DQW~ z ! 1V H ~ z ! f j ~ z ! 2 dz m * ~ z ! dz
5E j f j ~ z ! ,
~1!
where j51, 2, . . . is the subband index and m * (z) is the position-dependent effective mass of electrons that varies from well to barrier materials. Meanwhile, the in-plane electron motion in each subband in the DQW is Landau quantized in the presence of a magnetic field B perpendicular to the DQW. In the Landau gauge, we can assign a plane wave for the electron motion in the y direction and the electron motion in the x direction is determined from another Schro¨dinger equation11 2044
© 1996 The American Physical Society
54
EFFECTS OF THE SCREENED EXCHANGE . . .
F
2
G
\2
d2 1 v 2 ~ x2X 0 ! 2 j nk y ~ x ! 2 1 m* 2 j cj 2m *j dx
S D
5 n2
1 \ v c j j nk y ~ x ! , 2
P j5 ~2!
where n51, 2, . . . is the Landau quantum number, m *j is the electron effective mass in the jth subband, which will be given below, v c j 5eB/m *j is the cyclotron frequency in the jth subband, X 0 52k y l 2H is the orbital guiding center in the plane, k y is the electron wave number in the y direction, and
l H 5 A\/eB is the magnetic length. In Eq. ~1!, the shape of a symmetric DQW potential is chosen as
V DQW~ z ! 0, 2d W 2d B /2,z,2d B /2 or d B /2,z,d B /21d W 5 V 0 otherwise, ~3!
H
where d W is the individual well width, d B is the middlebarrier width, and V 0 is the middle-barrier height. The Hartree potential can be directly calculated from the Poisson equation28
F
G
d d e2 e b ~ z ! V H ~ z ! 5 @ n ~ z ! 2N im ~ z !# . dz dz e0
~4!
In Eq. ~4!, e b (z) is the dielectric constant of the well or barrier bulk materials, which varies with z. n(z) is the electron-density function n~ z !5
1
(
p l 2H n, j
H
1 exp$ @ E j 1 ~ n2 ! \ v c j 2 m c # /k B T % 11 1 2
3 u f j~ z !u 2,
J ~5!
where T is the electron temperature and m c is the chemical potential at finite T. N im (z) is the donor doping density profile N im ~ z ! 5N 3D im ~ z ! 1
(i N 2D im ~ i ! d ~ z2z i ! ,
P j 12 P j 5 1 , mW mB m *j
2d B /2
2d B /22d W
~6!
~7!
which depends on the subband index. Here m W and m B are the electron effective masses in the well and barrier materials, respectively. In Eq. ~7!, the electron quantum-well dwelling probability is calculated as
dz u f j ~ z ! u 2 1
E
d B /21d W
d B /2
dz u f j ~ z ! u 2 .
~8!
The charge neutrality condition for the completely ionized donors leads us to
E
1`
2`
dz n ~ z ! 5n 2D5
E
1`
2`
dz N 3D im ~ z ! 1
(i N 2D im ~ i ! ,
~9!
where n 2D is the areal electron density. Equation ~9! can be used to determine m c in a self-consistent way. The total electron energy including both the in-plane and vertical electron motions can be written as E n j 5 ~ n2 12 ! \ v c j 1E j 1V Fn j ,
~10!
where V Fn j is the exchange energy and will be given below. From the self-consistent equations ~1!–~9!, we are able to calculate the eigenstate f j (z) and total eigenenergy E n j in each step. Consequently, the eigenstate in each step is found to be
c n jk y ~ r! 5
exp~ ik y y !
AL y
F
3exp
A
1
p 2
1/2 n21
~ x2X 0 ! 2
2l 2H
n!l H
G S D H n21
x2X 0 f j ~ z ! , ~11! lH
where H n (x) is the nth-order Hermite polynomial and L y is the size of the sample in the y direction.11 Based on the calculated self-consistent wave function c n jk y (r) and subband edge E n j in each step, we can compute the screened exchange energy that is needed to finish the self-consistent Hartree-Fock calculation. A straightforward calculation brings us to the result29 V Fn j 52 3
which includes both the selective doping and d -doping contributions, where N 2D im (i) is the sheet density for d doping in the ith layer at z5z i . Because the electron effective masses in the well and barrier materials are different and by using the first-order perturbation theory, we find the ‘‘average’’ effective mass, used in Eq. ~1!, for the electrons in the jth subband to be28 1
E
2045
1 2 p n8, j8
(
H
E
1`
dq xy q xy
0
1 exp@~ E n 8 j 8 2 m c ! /k B T # 11
J
F
3 u An 8 n ~ q xy ! u 2 V j j 8 ~ q xy ! ,
~12!
where q xy 5 Aq 2x 1q 2y is the module of a two-dimensional wave vector qxy 5(q x ,q y ) and the form factor due to Landau quantization is u An 8 n ~ q xy ! u 2 5
S D S DF
n , ! q 2xy l 2H n .! 2 3exp 2
n . 2n ,
q 2xy l 2H 2
~ n . 2n , !
Ln
,
S DG q 2xy l 2H 2
2
. ~13!
In Eq. ~13!, n , 5min(n,n8), n . 5max(n,n8), and L(n) m (z) is the generalized Laguerre polynomial. By using a generalized Thomas-Fermi screening model, we find
DANHONG HUANG AND M. O. MANASREH
2046
F
V j j 8 ~ q xy ! 5
e2 2 e 0 e b q xy 3
E
1`
2`
54
dz (m « 21 jm ~ q xy ! E 2` 1`
*~ z ! f j 8~ z ! dz 8 f m
3exp~ 2q xy u z2z 8 u ! f *j 8 ~ z 8 ! f m ~ z 8 ! , ~14! where e b is the average background dielectric constant and « 21 jm (q xy ) is the inverse of the static dielectric-function matrix30 « i j ~ q xy ! 5 d i j 1
q TF j F ~ q !. q xy i j xy
~15!
In Eq. ~15!, we have defined the notations Fi j ~ q xy ! 5
E E 1`
2`
dz
1`
2`
FIG. 1. Landau levels as a function of the magnetic field B. The Landau-level label E(n, j) corresponds to the notation E n j in the text, where n is the Landau label and j is the subband index. The dashed line is for the chemical potential m c .
dz 8 u f i ~ z ! u 2
3exp~ 2q xy u z2z 8 u ! u f j ~ z 8 ! u 2 ,
~16!
and the inverse of the Thomas-Fermi screening length q TF j 5
e2
1
( A2 p G 2
2 p l 2H e 0 e b n 3exp
H
j
2 @ m c 2E j 2 ~ n2 21 ! \ v c j # 2 2G 2j
J
,
~17!
where we have assumed a Gaussian form of the Landau-level broadening. The uniform broadening of Landau level superposed on the jth subband in Eq. ~17! is taken as31 G 2j 5
1 \ \vcj , 2p t
~18!
where t is the relaxation time of electrons by scattering and can be determined from the sample mobility. This is a result of the Born approximation. In our numerical calculation, we have chosen the well material as GaAs and the barrier material as Al x Ga 12x As. The sample parameters are set as x50.3, T577 K, d W 5140 Å, d B 530 Å, V 0 50.83x, m W 50.0665, m B 50.066510.0835x, n 2D54.231011 cm 22 , e W 512.02, e B 512.0222.925x, and t 52.8310211 sec. In Fig. 1 the calculated landau levels E n j are shown as a function of magnetic field B. The labels n and j are for the Landau quantum number and subband index, respectively. The dashed line is for the chemical potential m c from which the population and depopulation of each Landau level can be seen. Because the Landau-level broadening is proportional to B, the density of states becomes nonzero far away from the center of Landau level ~inside the Landau gap! when B.9 T. This makes the chemical potential m c deviate downward from E 11 . Similar reasoning can explain the deviation of chemical potential upward from E 1,2 when B
