The relative lineup of the band structures of the two constituents of a semiconductor superlattice can cause charge carriers to be confined. This occurs when the energy of a superlattice state is located in an allowed energy region of one of the constituents (the "well" semiconductor), but in the band gap of the other (the "barrier" semiconductor). A charge carrier will tend to be confined in the layers made from the semiconductor with the allowed region at that energy. It will have an exponentially decaying amplitude to be found in the

semiconductor with a band gap at that energy

McGill, T. C.

Schulman, J. N.

English

Caltech Authors - Main

Summary Abstract: Localization of superlattice electronic 
states and complex bulk band structures 
J. N. Schulman 
University of Hawaii, Honolulu, Hawaii 96822 
T. C. McGill 
Calij(Jrnia Institute of Technology, Pasadena, California 91125 
(Received 14 May 1980; accepted 20 May 1980) 
PACS numbers: 73.40.Lq 
The relative lineup of the band structures of the two con-
stituents of a semiconductor superlattice can cause charge 
carriers to be confined. 1•2 This occurs when the energy of a 
superlattice state is located in an allowed energy region of one 
of the constituents (the "well" semiconductor), but in the band 
gap of the other (the "barrier" semiconductor). A charge 
carrier will tend to be confined in the layers made from the 
semiconductor with the allowed region at that energy. It will 
have an exponentially decaying amplitude to be found in the 
semiconductor with a band gap at that energy. 
The decay length of the state is a measure of the degree to 
which the carrier is confined to the semiconductor with the 
allowed energy region. Short decay lengths in the barrier 
semiconductor indicate that there is only a small overlap be-
tween the well-like wave functions centered in neighboring 
wells. The effective mass in the direction perpendicular to the 
superlattice layers will thus be large. Conversely, long decay 
lengths result in small perpendicular effective masses. 
We report the results of a theoretical calculation which 
determined the decay lengths of superlattice states at the 
conduction and valence band edges for two superlattice sys-
tems: AlAs/GaAs and CdTe/HgTe. A parameterized tight-
binding Hamiltonian was used to calculate bulk band struc-
tures for complex values of the wave vector. The complex bulk 
band structures of the barrier semiconductors in the forbidden 
energy regions gave dispersion curves of energy vs complex 
wave vector. The reciprocal of the imaginary part of the wave 
vector for a state with a specified energy is the decay length 
of that state in the barrier semiconductor.3•4 
The procedure followed to determine the super lattice state 
decay lengths was first to find the superlattice valence and 
conduction band edge energies from a superlattice tight-
binding calculation, and then to find the intersection of these 
energies with the complex bulk band structure of the barrier 
semiconductor. The decay lengths of the bulk states at these 
energies are, to a good approximation, the decay lengths of 
the superlattice states.3•4 The influence of the decay lengths 
on effective masses in the direction perpendicular to the su-
per lattice layers was investigated by comparing calculated 
effective masses vs layer thickness for the two superlattice 
systems. 
In addition, the InAs/GaSb superlattice was studied. This 
system is of particular interest because the conduction band 
edge state is localized in the InAs layers while the valence 
band edge state is localized in the GaSh layers. 2 Thus both 
semiconductors act as barriers. For this system a simple two-
band approximation was used to calculate the complex bulk 
band structures. 4 Only the band gap and bulk effective masses 
were needed in this calculation. The energies of the super-
lattice states, however, cannot be found with this method. Our 
results for all three superlattice systems follow. 
In the case of the AlAs/GaAs superlattice, AlAs is the barrier 
semiconductor. The conduction and valence band edge 
energies of the superlattice consisting of 20 atomic layers of 
AlAs alternating with 10 atomic layers of GaAs were found. 
These two energies intersect the AlAs complex dispersion 
curves at two points each. There are thus two possible decay 
lengths for each superlattiee state. Group theory arguments 
eliminate one of the two possibilities, resulting in a unique 
decay length for each state. The decay length of the valence 
band maximum state is 7.2 A and that of the conduction band 
minimum state is 4.0 A. These values of the decay lengths are 
in excellent agreement with those determined from the su-
perlattice wave function directly. 
The CdTe/HgTe superlattice is more complicated. CdTe 
is the barrier semiconductor.5 Spin-orbit splitting, which was 
ignored when calculating the AlAs complex band structure, 
is large and must be included. The group theory arguments 
no longer apply with the spin-orbit interaction and therefore 
there is no unique decay length. The conduction band su-
perlattice state in the CdTe layers is found to consist of bulk 
states with decay lengths of 9.3 and 18.7 A. 
The longer decay lengths of the CdTe/HgTe superlattice 
states as compared with the AlAs/GaAs superlattice states 
result in more overlap between the welllike wavefunctions 
centered in neighboring wells. The effective masses of the 
CdTe/HgTe superlattice in the perpendicular direction 
would therefore be expected to vary more slowly with CdTe 
layer thickness than would the AlAs/GaAs superlattice ef-
fective masses with AlAs layer thickness. Detailed calculations 
of these effective masses for varying layer thicknesses support 
this conclusion. 
The complex bulk band structure of InAs and GaSh were 
obtained in a two-band approximation. This approximation 
was found to give excellent agreement with the more detailed 
tight-binding calculation when applied to the AlAs/GaAs and 
CdTe/HgTe superlattices.4 Again, the spin-orbit interaction 
in InAs and GaSh is large and, therefore, a unique decay 
length cannot be found. Instead, a superlattice state at a cer-
tain energy would consist of bulk states with two decay 
1118 J. Vac. Sci. Techno!., 17(5), Sept./Oct. 1980 0022-5355/80/051118-02$01.00 © 1980 American Vacuum Society 1118 
................................ · .... ·.·.·.·.·.·.·.·.·.·.·.·.·-:-·.·.··· 
1119 J. N. Schulman and T. C. McGill: localization of superlattice 1119 
lengths, as with the CdTe/HgTe superlattice. Superlattice 
state energies have been determined by others, although with 
some disagreement. 6- 9 We estimated the decay lengths of 
these states using the InAs and GaSh complex band struc-
h!res. 
In summary, we have developed a theoretical technique 
that is useful in determining the degree of confinement of 
charge carriers in semiconductor superlattices. The complex 
band structures of the barrier semiconductor can be investi-
gated to determine the decay lengths of superlattice states and 
thus the carrier confinement This technique should be useful 
in understanding properties which depend on the carrier 
confinement, such as optical absorption2 and transport 
properties perpendicular to the superlattice layers. 
J. Vae. Set Technot, Vol. 17, No.5, Sept./Oct. 1980 
Acknowledgment: Work supported in part by the ARO under 
Contract No. DAAG29-77 -C-0015. 
1R. Dingle, H. L. Stormer, A. C. Gossard, and W. Wiegmann, Appl. Phys. 
Lett. 3:1, 66.5 (H:l78). 
2G. A. Sai-Halasz. L. L. Chang, J. M. Walter. C. A. Chang, and L. Esaki, 
Solid-State Commun. 27,935 (1978). 
3J N. Schulman and T. C. McGill, Solid-State Communun. 34,29 (1980). 
4J. N. Schulman and T. C. McGill (unpublished). 
5J. N. Schulman and T. C. lVlcGill, Appl. Phys. Lett 34, 66.'3 (1979). 
6J. lhm, Pui K. Lam. and Marvin L. Cohen, Phys. Rev. B 20,4120 (1979). 
7G. A. Sai-Halasz, R. Tsu. and L. Esaki, Appl. Phys. Lett. 30, 6!51 (1977). 
8H. Bluyssen, J. C. Maan, P. Wyder. L. Chang, and L. Esaki, Solid-State 
Commun. 31,35 (1979). 
9R N. Nucho and A. Madhukar, J. Vac. Sci. Techno!. 15, 1530 (1978). 


Localization of superlattice electronic states and complex bulk band structures

https://authors.library.caltech.edu/32582/1/SCHUjvst80.pdf

Localization of superlattice electronic states and complex bulk band structures

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main