Full text of DOI:10.1002/1521-3951(200007)220:1<159::AID-PSSB159>3.0.CO;2-H

M. E. Mora-Ramos and C. A. Duque: Polaron Free Energy in GaAs QWs
159
phys. stat. sol. ꢀb) 220, 159 ꢀ2000)
Subject classification: 71.38.+i;73.61.Ey;S7.12
Polaron Free Energy in GaAs Quantum Wells
1
M. E. Mora-Ramos † and C. A. Duque
ꢀ
a) Facultad de Ciencias, Universidad Aut o noma del Estado de Morelos,
Ave. Universidad 1001, CP. 62210, Cuernavaca, MOR, M e xico
ꢀ
b) Departamento de F Âõ sica, Universidad de Antioquia, A.A. 1226, Medell Âõ n, Colombia
Received November 1, 1999)
ꢀ
The polaron free energy in a GaAs/AlAs quantum well is calculated as a function of temperature
and well width. The interaction considered is that between the conduction electrons and the
GaAs-like polar optical oscillations. The calculation follows a perturbative scheme in the coupling
constant for the thermodynamical potential in the one-particle limit. Phonons are described with
the use of a long-wavelength dispersive electroelastic continuum model.
Zero temperature polaron properties in semiconducting heterostructures have been sys-
tematically investigated with the use of different model interaction Hamiltonians since
the second half of the past decade ꢀsee, for instance [1 to 4]). However, the finite tem-
perature polaron free energy does not seem to have been so far widely calculated in low-
dimensional heterostructures may be due to the explicit difficulties for the application of
the traditional scheme based on the Feynman path integral procedure [5 to 7] in systems
bearing interfaces. An alternative way which is the use of the perturbative method com-
ing from a Green`s function approach to the thermodynamical potential appears to be
appropriate in the case of semiconducting materials like the III±V and II±VI compounds,
where the magnitudes of the electron±phonon coupling constants are small enough to
allow for a perturbative treatment usually restricted ±± at most ±± to the second-order
corrections ꢀsee, for instance [8]). Therefore, such a calculation process can be carried out
as well in the case of semiconducting heterostructures made from those materials.
In this work, we are going to deal with a particular system, namely, a GaAs/AlAs
double heterostructure ꢀDHS), for which a Fr oÈ hlich-like interaction Hamiltonian has
been recently put forward in the framework of a dispersive electroelastic continuum
model ꢀDECM) [9]. In all Fr oÈ hlich-like problems, the expansion for the electron±phonon
correction to the electronic gas thermodynamical potential can be viewed as a series in
the electron±phonon coupling constant a, in such a way that the n-th order term in the
expansion for the one-electron Green`s function is proportional to the n-th power of a
[10]. This feature allows us to write the expression for the first-order correction ꢀwhich is
the usually evaluated in III±V arsenides) ±± equivalent to take into account the first
`
`bubbleº diagram in the expansion for the thermodynamical potential ‰10Š ±± in the form
a
ꢁ
ꢂ
ꢀ
ꢀ
ꢀ
1†
0†
1
da P
G
G
ꢀk; z; t†
ꢀk; z; t†
DW ˆ b
À 1
;
ꢀ1†
a
k; z
0
1)
e-mail: memora@servm.fc.uaem.mx

160
M. E. Mora-Ramos and C. A. Duque
ꢀ
1†
ꢀ0†
where G ꢀk; z; t† and G ꢀk; z; t† are the first-order and unperturbed one-electron
Green's functions, respectively, in the imaginary time representation, associated to elec-
tron states in the conduction band with effective mass m . k is the in-plane electron
*
wavevector, and z is the quantum number labelling the energy states of the motion
along the growth direction of the heterostructure, z. b ˆ 1=kBT. We shall assume the
non-degenerate situation and write, in the one-particle limit, the relation for the chemi-
cal potential m as
p ꢀh 2b
ꢃ
ꢄÀ1
2
P
bm
ÀbEz
e
ˆ
e
:
ꢀ2†
S m*
z
In our case, the conduction band electronic states in the GaAs/AlAs DHS are calcu-
lated using a finite barrier quantum-well model, explicitly considering the difference
between the effective masses in both materials ꢀsee [1]). It will be also assumed that
the electron is in the G-point even when really X is the lowest conduction band. This is
done mostly in order to avoid the extra complication coming from the consideration of
an indirect gap. The barrier height is taken to be V0 ˆ 915 meV. For that reason, we
shall limit to consider only the discrete energy levels in the above expression ꢀz ˆ l),
avoiding the more intractable situation of the continuum part of the spectrum. On the
other hand, we follow the usual procedure and use the so-called leading term approxi-
mation ꢀLTA) [1], which means to consider only the intrasubband matrix elements cor-
responding to l ˆ 1, the ground level.
The GaAs-like phonon modes in the DECM are neither purely confined slab modes
nor interface modes [9]. There is a mixed character and certain modes are rather more
interface-like than others. Furthermore, we only can say that the modes may be predo-
minantly ªlongitudinalº ꢀquasi-L) in some cases or predominantly ªtransversalº ꢀquasi-T)
in others because the polarization also exhibits a mixed character. The expression for
m
the LTA electron±phonon matrix elements I ꢀq† within the DECM is reported in [4].
1
1
q is the phonon two-dimensional wavenumber, and m is a discrete index labelling the
different modes appearing in the system.
With all this, it is possible to write for the LTA first-order polaron free energy correc-
tion in the DHS, the following expression [11]:
1
t=2
ꢀ
ꢀ
Àbe1
m
F
1†ꢀt† ˆ À 4
a ꢀh wLOR
P
e
P
dq
qI ꢀq†
11
dx Dqmꢀx† e
2
2
x
t
;
ꢀ3†
ꢀ
Àq R xꢀ1À †
Àbel
d
e
wqm
m
l
0
0
where t ˆ ꢀh wLOb is a dimensionless temperature-dependent variable. wLO is the bulk
GaAs longitudinal optical phonon frequancy, wqm ˆ wmꢀq†=wLO, wmꢀq† being the dis-
1
=2
*
persion relation of the m-th mode [4, 9], R ˆ ꢀ ꢀh =2m wLO† is the bulk polaron radius.
The function Dmqꢀx† is
wqmx
Àwmqx
e
e
Dqmꢀx† ˆ
‡
:
ꢀ4†
ewqmt À 1 1 À eÀwqmt
À1
*
In the calculations, the values wLO ˆ 291:9 cm , m ˆ 0:0665m0, and a ˆ 0:068 were
*
taken for GaAs. For the well barrier effective mass we used m ꢀAlAs) = 0.15m0, m0
being the bare electron mass.
In Fig. 1 the relative polaron free-energy Fr is shown as a function of the quantum well
ꢀ
1†
ꢀ
1†
width d for different values of the temperature. Fr is defined as the ratio F ꢀt†=F3Dꢀt†;

Polaron Free Energy in GaAs QWs
161
Fig. 1. Relative polaron free energy as a function of the GaAs quantum well width. Curves ꢀin
decreasing order) correspond to 613, 420, 300, and 0 K, respectively
where F3ꢀ D1 †ꢀt†, the polaron free-energy of first order for bulk GaAs is given by
t=2
ꢀ
p
a ꢀh wLO
Dꢀx†
ꢀ
1†ꢀt† ˆ À p
F
t
dx p ;
ꢀ5†
ꢀ6†
3
D
p
xꢀt À x†
0
where
ex
Àx
e
Dꢀx† ˆ
‡
:
e À 1 1 À eÀt
t
In the limit T ˆ 0 K ꢀt ˆ 1), Eq. ꢀ5) gives precisely the bulk polaron binding energy
value Àa ꢀh wLO.
The three upper curves in the figure correspond, from top to bottom, to the Debye
temperature of GaAs ꢀTD ˆ 613 K), the LO-phonon temperature of the material
ꢀ
Tph ˆ ꢀh wLO=kB ˆ 420 K), and to T ˆ 300 K, respectively. Besides the room tempera-
ture value, the other two were chosen because they constitute significative temperatures
for the material. For the sake of comparison, a lowest curve, corresponding to the
T ˆ 0 K polaron relative binding energy has been included. This curve is obtained from
the expression resulting when taking the limit t ! 1 of Eq. ꢀ3), which precisely coin-
cides with the Rayleigh-Schr oÈ dinger-perturbation-theory equation presented in [4].
As expected, the polaron free energy raises with increasing quantum well width. The
rate of increment varies for different temperatures indicating that the way by which the
GaAs-like phonon modes contribute depends on the temperature as well. Nevertheless,
it can be observed a tendency to change the monotony, which reverses towards a limit-
ing bulk-value for d sufficiently large. This decreasing behavior will then be stronger at
high temperatures ±± as it is shown in the curve corresponding to TD ±± while is much
more subtle for T ˆ 0 K. Hence, the range of well width values selected represents the
region of greater interest for the effect here studied. Furthermore, there exists another
reason, this time of practical character: for higher values of d, the number of eigen-
modes associated to the characterictics equations increases very rapidly, and the calcula-
tion process becomes really tedious [4, 9].

162
M. E. Mora-Ramos and C. A. Duque: Polaron Free Energy in GaAs QWs
The relative polaron free energy is not reported for d below 2 nm because that
region a macroscopic continuum model for the GaAs long-wavelength oscillations
would certainly not work well, and its validity is doubtful. When d ! 0, only the bulk
AlAs is present. However, in this study we have not considered so far the contribution
to the electron±phonon interaction coming from the electric potential of the barrier
modes. If taken into account, it will significantly change ꢀby increasing) the polaronic
correction for the smallest values of d;but in the case of well width around 10 nm and
beyond the main contribution should come from GaAs-like polar optical modes.
The results obtained within this model indicate that the effect of the modification of
the GaAs phonon spectrum due to the existence of interfaces is relevant only for layer
ꢁ
thickness not larger than 100 A. Significant differences with the 3D values are obtained
for a GaAs-layer width of a few tens of angstroms. For wide enough wells, the use of
the bulk Fr oÈ hlich Hamiltonian would provide correct results.
Acknowledgement M.E.M.R. acknowledges partial support from CONACYT ꢀMex)
through grant 32270-E.
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