Level quantization in the narrow-gap-semiconductor quantum well in a parallel magnetic field
V. K. Dugaev and V. I. Litvinov*
Chernovtsy Branch of the Institute of Materials Science Problems, National Academy of Sciences of Ukraine,
I. Vilde 5, 58001 Chernovtsy, Ukraine
W. Dobrowolski
Institute of Physics, Polish Academy of Sciences, Al. Lotniko ´w 32/46, 02-668 Warsaw, Poland
Received 19 January 2000
An analytical approach is developed for the energy levels in Group-IV–VI narrow-gap semiconductor
quantum wells in a parallel magnetic field. Asymptotic closed-form expressions are obtained. Numerical
calculations are done for the energy spectrum of the EuS/PbS/EuS quantum well in the magnetic field of an
arbitrary strength. The effect of an anisotropy of the electron valleys is also discussed.
I. INTRODUCTION
The Group-IV–VI narrow-gap-semiconductors find their
applications in infrared detectors, lasers, and thermoelectric
devices.
1
Further progress in the Group-IV–VI midinfrared
optoelectronics is expected from quantum wells QW’s, su-
perlattices, and nanoparticles that enable the engineering of
their electric and magnetic properties.
2,3
The energy spec-
trum of Group-IV–VI-based quantum-size structures is a key
element of the device modelling and analysis of the experi-
mental data such as photoluminescence and interband
magnetoabsorption.
4
In the case of bulk Group-IV–VI semiconductors, the ex-
perimental and theoretical study of a spectrum in a magnetic
field with detailed numerical calculations have been done.
5,6
At present, known Group-IV–VI QW spectra have been ob-
tained only as numerical solutions of Schro
¨
dinger’s equation
even in the absence of the magnetic field.
7,3
Numerical cal-
culations have also been done for PbTe/Pb
1 -x
Sn
x
Te super-
lattices in magnetic fields both parallel and perpendicular to
layers.
8
The analytical approaches to an energy spectrum in
Group-IV–VI-based QW, with and without perpendicular
magnetic field, have been developed in Refs. 9–12. The
problem with a perpendicular magnetic field is more easy to
analyze because the magnetic quantization concerns the in-
plane electron motion and is completely decoupled from size
quantization, which concerns a motion in the growth
direction.
10
The problem in a parallel field is more complex
due to an interference of size- and magnetic-field
quantizations—they cannot be separated in any way. The
spectrum of Group-IV–VI quantum wells in parallel mag-
netic field has been studied analytically in recent works of
Ghatak et al.
13,14
within the approach that takes into account
the nonparabolicity in a one-band approximation. This ap-
proach neglects the electron spin and thus makes impossible
to account for the spin-orbit interaction.
In this paper we calculate the energy spectrum of a
Group-IV–VI QW in parallel magnetic field. The interband
coupling and spin-orbit interaction both play a significant
role forming the electron spectrum of Group-IV–VI com-
pounds. For this reason we use the 4 4 Hamiltonian that is
adequate enough to account for all these effects. In the lim-
iting cases of weak and strong magnetic field, specified be-
low, we obtain closed-form analytical expressions for the
field-dependent spectrum. We apply the obtained results to
EuS/PbS/EuS QW. This QW is an active region of Group-
IV–VI QW lasers, besides, it has the ferromagnetic/
nonmagnetic semiconductor interfaces making it an interest-
ing object that combines properties of magnetic and confined
systems.
15–18
II. MODEL AND EQUATIONS
We consider rectangular quantum well of width 2 L as
shown in Fig. 1. In the case of Group-IV-VI type semicon-
ductors, the Hamiltonian of electrons in parallel magnetic
field with H along axis x is a Dirac-like type
1
we take units
with =1)
H =
c
z
v
• k
-eA
/ c
v
• k
-eA
/ c
-
v
z
, 1
where
c
( z ) =
0
for | z | L , and
c
( z ) =
1
for | z | L ,
while
v
( z ) =
0
for | z | L , and
v
( z ) =
2
for | z | L Fig.
1, v is the band-coupling parameter, A
is the vector poten
FIG. 1. The model of quantum well.
PHYSICAL REVIEW B 15 JULY 2000-I VOLUME 62, NUMBER 3
PRB 62 0163-1829/2000/623/19057/$15.00 1905 ©2000 The American Physical Society