Level anticrossing effect on electron properties of coupled quantum wells under an in-plane
magnetic field
A. Herna
´
ndez-Cabrera* and P. Aceituno
Departamento de Fı ´sica Ba ´sica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain
F. T. Vasko
Institute of Semiconductor Physics, NAS Ukraine, Pr. Nauki 45, Kiev, 252650, Ukraine
Received 22 March 1999
The influence of an in-plane magnetic field on the energy spectrum and zero-temperature equilibrium
properties of tunnel-coupled double and triple quantum wells is studied. Both the appearance of the gap due to
anticrossing of two energy branches and the peculiarities of the third-order crossing point for symmetric triple
quantum well case are discussed. As results, magnetization of two-dimensional electrons in double and triple
quantum wells is modified essentially if the Fermi level is localized near such peculiarities. Another effect
under consideration is the interlevel charge redistribution between quantum wells and the transverse voltage
induced by the in-plane magnetic field. Self-consistent numerical calculations for double and triple quantum
wells, which take into account the modifications of energy spectra under gate voltage, are presented.
S0163-18299909431-X
I. INTRODUCTION
Currently noticeable interest is focused on the electronic
properties of double and triple quantum wells DQW’s and
TQW’s and on the transport or optical phenomena in such
semiconductor structures. Such tunnel-coupled two-
dimensional 2D electron systems, when subjected to a per-
pendicular or parallel in-plane magnetic field, exhibit a set
of new physical phenomena. While the effect of a perpen-
dicular magnetic field is due to Landau quantization, the in-
fluence of an in-plane magnetic field appears due to different
displacements of the energy dispersion parabolas of the dif-
ferent quantum wells QW’s. The sketches of the energy
spectrum modifications under an in-plane magnetic field are
shown in Fig. 1. It is clear that different types of cross points
between dispersion parabolas which are independent for
tunnel-uncoupled wells are possible for DQW’s and TQW’s
the type of peculiarity is determined both by the strength of
the magnetic field H and by the parameters of the tunnel-
coupled structure.
Such in-plane magnetic-field-induced modifications of the
energy spectra in DQW’s and TQW’s change the in-plane
conductivity of these systems see experimental data for
DQW’s in Refs. 1–6 and first measurements for TQW’s in
Ref. 7; theoretical results for DQW’s are discussed in Ref. 8
and the photoluminescence spectra.
9
Two reasons for con-
ductivity changes were found: the modification of the resis-
tance resonance peak these results are reviewed in Ref. 10
and the formation of density of states singularity
11
due to the
anticrossing effect presented in Figs. 1b and 1d. The pho-
toluminescence line shape depends on the modification of
hole states and is due to the many-particle interaction.
12
In
the recent years, a weak perpendicular magnetic field has
been employed as a probe in order to study the effects of
in-plane field on tunnel-coupled states of electrons
Shubnikov—de Haas oscillations
5
and cyclotron resonance
absorption
13
have been measured. Note that all aforemen-
tioned papers deal with the transport properties when the
peculiarities of the energy spectra together with other factors
scattering processes, transformation of the hole states are
essentials. The aim of this paper is the description of the
equilibrium electron properties of tunnel coupled structures
when collisions do not determine the character of the re-
sponse. Both the magnetization of tunnel-coupled QW’s un-
der in-plane field and magnetoinduced transverse voltage al-
low the direct collisionless investigation of the energy
spectrum peculiarities.
Our calculations are based on the one-electron Hamil-
tonian for the usual effective mass approximation
H
¯
=
p-e A z / c
2
2 m
-
2
2 m
d
2
dz
2
+U
CQW
z +U
sc
z 1
FIG. 1. Sketches of the energy spectra of symmetric tunnel-
coupled QW showing the types of cross-points: a ‘‘vertical’’
crossing in DQW’s; b ‘‘horizontal’’ crossing in DQW’s; c two
‘‘vertical’’ cross points for TQW’s; d triple-cross point in TQW’s.
The anticrossing effect is shown by the dashed lines.
PHYSICAL REVIEW B 15 AUGUST 1999-II VOLUME 60, NUMBER 8
PRB 60 0163-1829/99/608/56987/$15.00 5698 ©1999 The American Physical Society