
Research Article


Binding Energies of Donor States in GaAsGaAlAs Quantum Wells Under Hydrostatic Pressure


H. Panahi
and
M. Maleki


ABSTRACT

The binding energies of donor states (1s, 2s, 2p_{x}, 3p_{x}) in GaAsGa_{0.7}Al_{0.3}As quantum wells are investigated with a variational method under hydrostatic pressure. In the calculation, we take into account the electronic effective mass, dielectric constant and conduction band offset between the well and barriers varying with pressure. Results obtained show that the donor binding energy variation with the well width and the position of impurity under pressure is similar to that without pressure. Also the donor binding energy increases linearly with pressure for all of states in direct gap regime.





INTRODUCTION
The binding energy of a hydrogenic impurity with in an infinite well
has received much attention in Quantum Well (QW) systems (Bastard, 1981;
Fraizzoli et al., 1990; Ferreyra and Proetto, 1991; Cen and Bajaj,
1992; Chaudhuri and Bajaj, 1994; Redinski and Janko, 2005). Similar studies
(PorrasMontenegero and PerezMarchancano, 1992, PorrasMontenegero et
al., 1993; Ban and Liang, 2001; Aktas et al., 2005) have been
extended for structures with lower dimensionality such as Quantum Well
Wires (QWW), Quantum Dots (QD) and also for various geometries. For a
number of reasons, most of the studies on these semiconductor systems
have been carried out on IIIV semiconductor heterostructure systems and
in particular, in GaAsGa_{1x}Al_{x}As semiconductor
systems. We emphasize that the optical properties of these heterostructure
systems are of significant importance for device applications and in this
sense, impurity states play a relevant role. It is well known that Coulombbound
states may be significantly modified by quantum confinement, applied external
fields and hydrostatic pressure and much more experimental and theoretical
studies have been devoted to the understanding of the physical properties
of impurity in lowdimensional semiconductor heterostructures (Morales
et al., 2002, 2003; Lopez et al., 2003a, b; Oyoko et
al., 2001; Correa et al., 2004; Adachi, 1985; Beneditctal et
al., 1993).
Effects of hydrostatic pressure modify the semiconductor band structure
and lead to shift effectively the energy levels without altering the crystal
symmetry of these heterostructure systems also the masses of carriers,
the height barriers of the heterostructures and the ΓX band crossover
on the ground state as well as of some low lying excited states are affected
by hydrostatic pressure (Elabsy, 1994; Raigoza et al., 2005; Neethiulagarajan
and Balasubramanian, 1993; Nithiananthi and Jayakumar, 2006).
In this study, the hydrostatic pressure dependence of the ground state
1s and the 2s, 2p_{x}, 3p_{x} like states of a shallowdonor
in GaAsGa_{0.7}Al_{0.3}As QW with finite barriers are
calculated using a variational procedure with in the effective mass approximation.
Results are calculated for different well widths, shallowdonor positions
and hydrostatic pressure. The pressure effects of the electron effective
mass, the dielectric constant and band offsets between the well and barrier
materials are considered in calculations by restricting ourselves to range
of pressure where there is no ΓX crossover.
MATERIALS AND METHODS
Theoretical framework: In the effectivemass approximation, the
Hamiltonian for a hydrogenic shallow donor impurity in a single GaAsG_{1x}Al_{x}As
semiconductor QW under the effect of a hydrostatic pressure (P) and the
temperature (T) is given by (Lopez et al., 2005).
Where, r (= [x^{2}+y^{2}+ (z−z_{i})^{2}]^{1/2}
is the carrier impurity distance and subscripts w and b stand for the
quantum well layer and barrier layer materials, respectively. The pressure
dependent potential energy V (z, T, P), which confines the donor electron
in the well layer regions, is given by:
Where, V_{0} (P, T) is the pressure dependent barrier height
(Elabsy, 1994) and the pressure dependent width of well layer L (P), can
be obtained by using of fractional changes in volume of the structure
(Kasapoglu et al., 2005):
Where, S_{11} (= 1.16x10^{3} kbar^{1}) and
S_{12} (= 3.7x10^{4} kbar^{1}) are the elastic
constants of the GaAs (Elabsy, 1994) and L(0) is the original width of
the electron confinement potentials in the zdirection without hydrostatic
pressure. In Eq. 1, z_{i} is the impurity position
with respect to the well center that it is chosen as the coordinate origin.
The parabolic conduction effectivemasses m*_{w,b} are given by
(Elabsy, 1994)
Where, m_{0} is freeelectron mass. In Eq. 2,
E_{g} (P,T) is the pressure dependent energy band gap for the
GaAs semiconductor at the Γ point and at low temperature. This is
given by Elabsy (1994):
We want to emphasize that for single quantum wells larger than 50 Å,
the nonparabolic effective mass effects are lower than 5% (Chaudhuri and
Bajaj, 1994). In the above expression ε_{w,b} (P,T) are the
corresponding static dielectric constants of well layer and barrier layer,
where at T = 4 K, the GaAs static dielectric constant with respect to
pressure is given by Lopez et al. (2005)
ε_{w}(P, 4 K) = 12.83 exp (1.67x10^{3}
P) 
(7) 
In present calculations we use x = 0.3 and due to the fact that the binding
energy changes occur for small well width and high aluminum concentration
(Elabsy, 1992), the image potential in QWs can be neglected and the charge
image effects have not been considered. This means that in the Hamiltonian
in Eq. 1, we take ε_{w} (P,T) = ε_{b}
(P,T) (Duque et al., 1997).
We use a variational approach for the calculation of the binding energies
(Elabsy, 1994; Bastard, 1988) and the trial envelope wave functions
are thus taken as products of the hydrogenic functions Γ_{nl},
of the nth state and with l symmetry, with the ground state wave
functions of the quantum well f(z) (Carneiro et al., 1995):
Where, N_{nl} are the normalization constants and f(z) obtained
via the Hamiltonian of Eq. 1, without the impurity term.
The hydrogenic variational wave functions are taken as:
Where, r = [ρ^{2} +(zz_{i})^{2}]^{½}
and {λ_{nl}, β_{nl}} are variational parameters
obtained in such a way that E_{nl} (P,T) = (Ψ_{nl}HΨ_{nl})/(Ψ_{nl}Ψ_{nl})
is minimized, with the requirement that Γ_{nl} form a set
of orthogonal functions (Carneiro et al., 1995).
The hydrostatic pressure dependent donor binding energy is calculated
as:
Where, E_{0} (P,T) is the electron ground state energy without
donor.
RESULTS AND DISCUSSION
Figure 1 shows the binding energies of a donor impurity for
ground state and 2s, 2p_{x}, 3p_{x}like hydrogenic excited
states, as a function of quantum well widths, for oncenter and onedge donors
in a GaAsGa_{0.7}Al_{0.3}As QW, for two given pressures, p
= 0 and 10 kbar. It can be observed that the binding energy increases with the
applied pressure for all of states and the binding energy of a donor oncenter
is more than of a donor onedge. This fact is true for all of states and result
for the ground state is in agreement with reported by Zhao et al. (2003).
This increment of the binding energy reflects the geometrical confinement due
to effective diminishing of the well width and the height of the barrier due
to the applied pressure. In Fig. 1, it is shown that the two
curves of the donor binding energy versus width, both oncenter and onedge,
for pressure p = 0 and 10 kbar are almost parallel. One can see that the pressure
effects for different pressures are qualitatively similar. Therefore, that the
character of the binding energy variation with the well width under pressure
is similar to that without pressure is expected When the well width is much
larger than the effective Bohr radii of the well, the electronic wave function
almost does not penetrate into the barriers and the donor binding energy decreases
with increasing the well width and finally (L→∞) tends to the GaAs bulk
limit. With decreasing the well width, the size of the quantum confinement in
zdirection reduces and the donor binding energy increases, however, for finite
barriers, the electronic wave function cannot be confined completely in the
well and partly penetrates into the barriers when the well width is getting
narrower than the effective Bohr radii of impurity state.


Fig. 1: 
Binding energies of (a) 1s, (b) 2s, (c) 2p_{x}, (d) 3p_{x}
like donor states as functions of the GaAsGa_{0.7}Al_{0.3}As
QW width for donor oncenter and onedge with p = 0, 10 kbar 
Thus, because of competition between this quantum confinement and donor
energy, the donor binding energies variation with well width, are non
monotonic.
Figure 2 shows our theoretical results for the binding
energies of the 1s, 2s, 2p_{x}, 3p_{x}like donor states
for GaAsGa_{0.7}Al_{0.3}As QW as function of impurity
position along the growth direction in a QW with 210 Å in width
and two different pressure p = 0 and 10 kbar. The donor binding energy
firstly increases and then decreases with moving the impurity position
from z_{i} = L to z_{i} = +L. The maximum binding energy
is obtained for impurity located at the center of QW. It is obvious that
the donor binding energy decreases when the distance of donor from the
well center increases. As the pressure increase, the well width and dielectric
constant decrease, the effective mass of electron increases, leading to
more confinement in the well in zdirection of the impurity electron and
so the donor binding energy increase for all impurity positions. The binding
energy for the impurity positions closed to the barriers is lower than
for oncenter, since Coulomb interaction between the electron and impurity
decreases. We note that for donors located at the well center, the binding
energy deference between two different pressure p = 0 and 10 kbar, is
more than donors located at the well edge. In other word, the hydrostatic
pressure raises the binding energy mainly for oncenter impurity than
for onedge ones. This result for the ground state is in good agreement
with that reported by Lopez et al. (2005). The combined effects
of hydrostatic pressure and the impurity position are really not so simple,
in particular for higher pressures (indirect gap regime), the rate at
which the binding energy increases is lower and bend down to smaller values.
One can see this result for other well widths in Fig. 1.


Fig. 2: 
Binding energy of a donor as a function of the growthdirection
impurity position in GaAsGa_{0.7}Al_{0.3}As QW with
p = 0, 10 kbar for (a) 1s, (b) 2s, (c) 2p_{x}, (d) 3p_{x}
donor states 

Fig. 3: 
Binding energies of 1s, 2s, 2p_{x}, 3p_{x} like
donor states in a GaAsGa_{0.7}Al_{0.3}As QW as functions
of pressure for L = 100 Å and donor in center 
In Fig. 3 one may notice that, for pressure up to 13.5
kbar (direct gap regime), the donor binding energy increases linearly
with pressure for ground state (Lopez et al., 2005) and excited
states. This is due to the increment of the barrier and well effective
mass as well as to the decrease of the dielectric constant with pressure,
in other word, in this pressure regime, the ΓX crossover for Ga_{1x}Al_{x}As
layer is not considered and as a consequence the barrier height that confines
the electrons in GaAs layer remains constant. Also the increasing rate
of binding energy with pressure for all excited states is less than 1s
like state.
CONCLUSION
We have studied theoretically the effects of the applied hydrostatic
pressure on the ground state and 2s, 2p_{x}, 3p_{x}, excited
states donor binding energy in a GaAsGa_{0.7}Al_{0.3}As
single quantum well using a variational scheme within the effective mass
approximation. The results show that the donor binding energy increase
almost linearly with the pressure in direct gap regime and the binding
energy variation with pressure for 2s, 2p_{x}, 3p_{x},
excited states is similar to ground state. We observe that the donor binding
energy variation for all of states with the width, both for donor oncenter
and onedge, are almost parallel for two different pressure and the pressure
effects are qualitatively similar. We have also shown that the donor binding
energy, without pressure, decreases for all of states when the distance
of donor from the well center increases, furthermore for donors located
at the well center, the donor binding energy deference between two different
pressure, is more than the donors located at well edge.

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