Substantial contribution of effective mass variation to electron-
acoustic phonon interaction via deformation potential
in semiconductor nanostructures
V. I. Pipa and V. V. Mitin
a)
Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202
M. Stroscio
U.S. Army Research Office, P.O. Box 12211, Research Triangle Park, New Jersey 27709
͑Received 5 October 1998; accepted for publication 14 January 1999͒
Using the approach of deformed ions and the tight binding, we have demonstrated that the
interaction of electrons confined in a nanostructure with acoustic phonons in a cubic crystal is
described by a deformation potential tensor ͑DPT͒ whose symmetry is determined by the geometry
of the nanostructure. Here in, we present additional contribution to the DPT which is caused by the
deformation dependence of the electron effective mass and it increases as L
Ϫ2
when the
characteristic size of a nanostructure, L, decreases. For narrow GaAs-based quantum wells, this
contribution is comparable with and can overcome that from the usual deformation potential
coupling. © 1999 American Institute of Physics. ͓S0003-6951͑99͒00911-0͔
Interaction via the deformation potential ͑DP͒ is one of
the most important mechanisms of the coupling between
electrons and acoustic vibrations in semiconductors. For a
homogeneous semiconductor of the cubic crystal symmetry
with a conduction band minimum at the ⌫ point, the interac-
tion Hamiltonian is given by
1
H
int
ϭD div u, ͑1͒
where D is the deformation potential constant, and the diver-
gence of the lattice displacement, u, determines the relative
change of crystal volume due to deformation. In the frame-
work of the effective-mass approximation, the theory of DP
has been shown to be valid for slowly varying ͑on the scale
of the lattice period͒ deformations.
At low temperatures, DP interaction plays an important
role in the kinetics of confined electrons. It determines the
phonon limited mobility and energy losses of a two-
dimensional ͑2D͒ electron gas
2
and acoustic-phonon emis-
sion may be the only process responsible for electron energy
relaxation in dot structures.
3
The accepted procedure in the
calculation of the deformation interaction in semiconductor
heterostructures is to use the bulk interaction Hamiltonian of
Eq. ͑1͒. It is assumed that the parameter D does not depend
on a size and a shape of the quantum heterostructure. Only
acoustic field modifications which result from the difference
in the elastic properties of materials forming the heterojunc-
tions are taken into account.
4
Recently, a new mechanism of
electron-acoustic phonon interaction, which is intrinsic to
semiconductors that have interfaces, and is additional to
usual deformation potential coupling of Eq. ͑1͒, has been
considered.
5,6
It arises when acoustic waves cause the inter-
face spacing to change and thereby to perturb the electron
states. This additional interaction,
5
known as ‘‘macroscopic
deformation potential’’ ͑MDP͒, for electrons occupying the
lowest subband of a rectangular quantum well ͑QW͒ has
been written in the form
H
int
MDP
ϭ
2E
1
L
͓
u
z
͑
x,y,0
͒
Ϫu
z
͑
x,y,L
͒
͔
, ͑2͒
where E
n
ϭ
2
ប
2
n
2
/2m
*
L
2
, m
*
is the isotropic effective
mass, and the planes z ϭ0 and z ϭL are at the positions of
the interfaces. An interaction of the same nature, known as
the ‘‘ripple mechanism’’ ͑RM͒, which is applicable for all
nanostructure geometries, was introduced by
6
H
int
RM
ϭu
͑
r
͒
ٌU
͑
r
͒
, ͑3͒
where U(r) is a confinement potential. In contrast to the case
of the bulk mechanism of Eq. ͑1͒, which allows only the
interaction with longitudinal phonons in isotropic media,
transverse acoustic phonons contribute to the interactions of
Eqs. ͑2͒ and ͑3͒ as well. The MDP interaction is weak com-
pared with the bulk interaction of Eq. ͑1͒ as 2E
1
/D Ӷ1. For
small dot sizes, it was found
6
that the RM contribution to the
electron scattering rates can be larger than that from Eq. ͑1͒.
The finite-size effect in quantum heterostructures is not
limited to the MDP ͑or RM͒ interaction. The deformation
variation of the electron effective mass also gives rise to a
size-dependent contribution. Let us consider a change of
electron energy in a rectangular QW, under an applied
uniaxial strain: the displacement u(z) ϭϪ
␥
z with
␥
ϭconst. Pressure is accommodated by shifting of the zone-
center energy of the bulk semiconductor according to Eq.
͑1͒, and by shifting of the energy levels in the QW. Taking
into account the change of E
1
both due to L(
␥
)ϭL(0)(1
Ϫ
␥
) and m
*
(
␥
)ϭm
*
(0)(1ϩ
␥
), where
is a phenom-
enological parameter, in the first order of
␥
we get the overall
energy shift
␦
Eϭ
␥
͓
ϪD ϩ
͑
2Ϫ
͒
E
1
͔
. ͑4͒
Here the term 2
␥
E
1
coincides with H
int
MDP
. The sign of this
shift corresponds to that of the relative change of QW width:
a͒
Electronic mail: mitin@ece6.eng.wayne.edu
APPLIED PHYSICS LETTERS VOLUME 74, NUMBER 11 15 MARCH 1999
15850003-6951/99/74(11)/1585/3/$15.00 © 1999 American Institute of Physics