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Commun. Theor. Phys. 60 (2013) 124–128
Vol. 60, No. 1, July 15, 2013
Donor Binding Energy in GaAs/Ga1−x Alx As Quantum Well: the Laser Field and
Temperature Effects∗
WEI Shu-Yi (危书义),† HOU Wen-Xiu (侯文秀), CHEN Xiao-Yang (陈晓阳), and XIA Cong-Xin (夏从新)
Department of Physics, Henan Normal University, Xinxiang 453007, China
(Received December 17, 2012; revised manuscript received March 28, 2013)
Abstract Based on the effective-mass approximation theory and variational method, the laser field and temperature
effects on the ground-state donor binding energy in the GaAs/Ga1−x Alx As quantum well (QW) are investigated. Numerical results show that the donor binding energy depends on the impurity position, laser parameter, temperature, Al
composition, and well width. The donor binding energy is decreased when the laser field and temperature are increased
in the QW for any impurity position and QW parameter case. Moreover, the laser field has an obvious influence on the
donor binding energy of impurity located at the vicinity of the QW center. In addition, our results also show that the
donor binding energy decreases (or increases) as the well width (or Al composition x) increases in the QW.
PACS numbers: 73.21.Fg, 71.55.-I, 71.55.Eq
Key words: laser field, temperature, impurity states
1 Introduction
In recent years, there has much interest in the lowdimensional semiconductor heterostructures due to their
specific physical properties and promising applications
in the optoelectronic devices.[1−3] Moreover, past studies show that impurities and external perturbations (such
as electric field and temperature) can obviously affect the
electronic and optical properties of semiconductor optoelectronic devices.[4−5] Recently, some investigations have
been reported to understand the laser field effects on hydrogenic impurity states in semiconductor quantum structures. Brandi et al.[6−7] presented the theoretical study
of the laser field effects on hydrogenic impurity states
in semiconductor quantum structures. Effects of laser
field and electric field on impurity states in zinc-blende
GaN/AlGaN quantum well (QW) were also studied by Xia
et al.[8] The effects of an intense laser field and hydrostatic
pressure on the binding energy of shallow donor impurities
in the QW was also investigated by Yesilgul et al.[9] Ungan
et al [10] also calculated the effects of an intense laser field
on the bound impurity states in Ga1−x Inx Ny As1−y /GaAs
double QWs. These researches show that the laser field
effects on impurity states are obvious in low dimensional
quantum structures. In addition, the temperature effects
on impurity states were also investigated in low dimensional semiconducting quantum structures. Nithiyananthi et al.[11] have discussed the effects of temperature
on the lower excited impurity states in the QW. Kasapoglu also studied the combined effects of hydrostatic
pressure and temperature on the donor binding energy
in GaAs/Ga1−x Alx As double QWs.[12] More recently, the
hydrostatic pressure and temperature effects on a hydro∗ Supported
genic impurity in a spherical quantum dot were considered
by Liang and Xie.[13] The hydrostatic pressure influences
on the donor binding energy was also studied in a spherical
quantum dots.[14] The high hydrostatic pressure effect on
the shallow donor binding energies in GaAs/Ga1−x Alx As
quantum wires at selected temperatures has been taken
into account by Karki et al.[15] These results show that
temperature and pressure effects play an important role
in impurity states in the low dimensional quantum structures. Thus, it is necessary to investigate external field,
hydrostatic pressure and temperature effects on the hydrostatic impurity states in the low dimensional semiconductor nanostructures.
To our knowledge, there are few studies on the combined effects of the intense laser field and temperature on
impurity states in the semiconducting QWs, in this paper,
we will study the effects of laser field and temperature on
the ground-state binding energy of shallow-donor impurity states in the GaAs/Ga1−x Alx As QW based on the
effective mass approximation theory and the variational
method. The present paper is organized as follows. In
Sec. 2, we give a simple theoretical model to investigate
the laser field and temperature effects on the donor binding energy in GaAs/Ga1−x Alx As QW. Numerical results
for the donor binding energy in the QW are given and
discussed in Sec. 3. Finally, we summarize the main conclusions in Sec. 4.
2 Theoretical Model
In the presence of the laser field, the timedependent Schrodinger equation describing the electron
by the National Natural Science Foundation of China under Grant No. 60906044
[email protected]
c 2013 Chinese Physical Society and IOP Publishing Ltd
°
http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn
† E-mail:
Communications in Theoretical Physics
No. 1
in GaAs/Ga1−x Alx As QW was given as follows[16]
h ~2
i
∂φ(~r, t)
− ∗ ∇2 + V (~r + α
~ (t)) φ(~r, t) = i~
,
(1)
2m
∂t
Here V (~r + α
~ (t)) is the “dressed” potential under the influence of laser field in the well, α
~ (t) = ~eα0 sin(ωt) describes the quiver motion of the classical electron in the
presence of laser field. Here ~e is the unit vector of the
polarization. α0 is the laser-dressing parameter given by
eA0
I 1/2 e ³ 8π ´1/2
α0 = ∗
= 2
,
(2)
m cω
ω m∗ c
where A0 is the amplitude of the vector potential. I, ω,
e, m∗ , and c are the average intensity of the laser, the
frequency of laser field, the charge, effective mass of the
electron and velocity of the light, respectively.
Following the Floquet approach,[17−18] Eq. (1) can be
cast in the equivalent form of a system of coupled timeindependent differential equations for the Floquet components of the wave function φ, containing the quasi-energy
E. An iteration scheme was developed to solve it, for the
zeroth Floquet component φ0 , the system reduces to the
time-independent Schrödinger equation[17−18] (the polarization of the laser radiation is parallel to the z-direction)
h ~2
i
− ∗ ∇2 + V (α0 , z) φ0 = E0 φ0 .
(3)
2m
Using the effective-mass approximation, the Hamiltonian for a hydrogenic donor impurity in a QW under the
combined action of the laser field and temperature can be
given by,
~2
∇2 + V (α0 , z) − Vc (~
α0 , ~r ) , (4)
2m∗w(b) (T )
p
where r = x2 + y 2 + (z - zi )2 is the distance between
the electron and the impurity. x, y, and z are the coordinate of electron and zi is the impurity position along
the z-axial direction in the GaAs/Ga1−x Alx As QW. The
subscript w and b stand for the well and barrier layer,
respectively. m∗w(b) (T ) is the temperature-dependent effective mass of the well or barrier layer. The corresponding electron effective mass in the barrier layer is obtained
from a linear interpolation between the GaAs and AlAs
compounds[19−20]
H=−
m∗b (T ) = m∗w (T ) + 0.083x ,
(5)
where x is the mole fraction of aluminum in the
Ga1−x Alx As layer.
V (α0 , z) is the laser-dressed confinement potential for
electron in the QW, which is given by the following
expression[21−23]
V (α0 , z) =
V0
[Θ(|Z| − (L/2 + α0 ))
2
+ Θ(|Z| − (L/2 − α0 ))] ,
(6)
L is the well width of the QW, Θ is the step function, V0
is the barrier height.
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Vc (~
α0 , ~r ) is “dressed” Coulomb potential, which can
be written as follows:
³
´
e2
1
1
+
,
(7)
Vc (~
α0 , ~r ) = −
2εw(b) (T ) |~r + α
~ 0 | |~r − α
~ 0|
εw(b) (T ) is the temperature-dependent dielectric constant
of well or barrier layer material.
In order to consider the effects of laser field and temperature on the electron effective mass, the band gap, the
dielectric constant and the elecron confinement potential,
the theoretical and the computational methods used in
this paper are the same as Ref. [12].
In order to calculate the ground-state binding energy
of donor impurity in single GaAs/Ga1−x Alx As QW, the
trail wave function may be written as[24−26]
φ = φ0 exp(−λr) ,
(8)
which λ is the variational parameter.
The ground-state energy of the donor impurity in a
GaAs/Ga1−x Alx As QW may be obtained by minimizing,
hφ|Ĥ|φi
.
(9)
hφ|φi
The ground-state donor binding energy Eb can be calculated as follows
E = min
λ
E b = E0 − E ,
(10)
where E0 is the ground-state energy for the Hamiltonian
of Eq. (2).
3 Numerical Results and Discussion
In order to investigate the laser field and temperature
effects on impurity states in the GaAs/Ga1−x Alx As QW,
we have calculated the ground-state donor binding energy
Eb as a function of the impurity position Zi , structural
parameters (Al composition x and well width L), temperature T and laser parameter α0 in the QW.
In Fig. 1, the ground-state donor binding energy is
studied as a function of impurity position Zi in the
GaAs/Ga0.9 Al0.1 As QW with the well widths L = 30 nm
(a) and L = 10 nm (b), considering different laser parameters and temperatures. We can see from Fig. 1 that the
donor binding energy distributes symmetrically with respect to the center of the GaAs/Ga0.9 Al0.1 As QW. The
reason is that the electron wave function shows the symmetrical distribution with respect to the center of the
GaAs/Ga0.9 Al0.1 As QW. And it is clear that as the laser
field increases, the donor binding energy decreases. However, the donor binding energy of the on-center impurity
decreases rapidly than that of other impurity cases, which
agrees with the results of Refs. [27–29], as expected. As
laser parameter α0 increases, the electron wave function
starts to spill over the barrier material, leading to an enhancement of the electron and impurity separation that
weakens the Coulomb interaction. However, for on-edge
impurity, the electron wave function penetrates into the
potential barriers, thus the effect of the confinement potential on the Coulomb interaction is no longer important.
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Communications in Theoretical Physics
Thus, we may find from Fig. 1 that the effect of laser
field on the donor binding energy of impurity is more pronounced in the QW with small size. In addition, Fig. 1
also displays the donor binding energy drops with increasing temperature for any given constant laser parameter
and impurity position.
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an increase of the electron and impurity separation that
weakens the Coulomb potential, thus the donor binding
energy begins to decrease. Moreover, by comparing the
results of Figs. 2(a) and 2(b), laser field has more obvious effect on impurity donor binding energy in small well
width, as expected. In addtion, Figs. 2(a) and 2(b) also
show that with the increase of temperature, the groundstate donor binding energy decreases for impurity located
at different impurity positions.
Fig. 1 (Color online) The ground-state donor binding energy as a function of impurity position Zi in the
GaAs/Ga0.9 Al0.1 As QW for different laser parameter α0
and temperature T : (a) L = 30 nm, (b) L = 10 nm.
In order to investigate the influence of laser field on the
donor binding energy of impurity in GaAs/Ga1−x Alx As
QW, in Fig. 2, we calculate the variation of the
ground-state binding energy of donor impurity in the
GaAs/Ga0.9 Al0.1 As QW as the laser parameter changes
for different impurity positions, temperatures and well
widths. In Figs. 2(a) and 2(b), it is apparent that, for the
impurity near the center of the well, the increase of the
laser field leads to a considerable decrease of Eb . However, the effect of the laser field on the binding energy
becomes less pronounced as the impurity approaches the
boundary of the quantum well, this penomenon further
confirms the conclusion of Fig. 1. The reasons leading to
this phenomenon have been explained in Fig. 1. In particular, in Fig. 2(a), for impurity located at Zi = L/2,
the donor binding energy first slowly increases, reaching
a maximum value, and then decreases. This feature is
due to the large value of the electron and impurity separation at α0 = 0, followed by a decrease of this quantity
towards a minimum at α0 = 10 nm, when the electron
wave function becomes more concentrated near the impurity center. Further increasing laser field, again we find
Fig. 2
The ground-state donor binding energy as a
function of laser parameterα0 in GaAs/Ga0.9 Al0.1 As QW
with well width L = 30 nm (a) and L = 10 nm (b) for
different impurity position Zi and temperature T . The
curves a, b and c represent different impurity positions
Zi = 0, Zi = L/4 and Zi = L/2, respectively.
In order to confirm the temperature effects on
the donor binding energy of impurity located at
GaAl/Ga1−x Alx As QW, we further calculate the variation of the ground-state binding energy of donor impurity
in QW with well width L = 30 nm and L = 10 nm as the
temperature changes for different laser parameters and Al
compositions. We may see from Fig. 3 that as the temperature is increased, the ground-state donor binding energy
drops, This is because an increase in temperature increases
dielectric constant and decreases effective mass, this result
conforms to the result of Ref. [12]. In addition, Figs. 3(a)
and 3(b) display that the donor binding energy decreases
with the increasing laser parameter. Figures 3(a) and 3(b)
also show that the donor binding energy increases as Al
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Communications in Theoretical Physics
composition increases for any laser parameter and well
width.
Fig. 3
(Color online) The ground-state donor
binding energy as a function of temperature T in
GaAs/Ga1−x Alx As QW with the well width L = 30 nm
(a) and L = 10 nm (b) for different laser parameter α0
and Al composition x.
In Fig. 4, the ground-state donor binding energy
is investigated as a function of Al composition in the
GaAs/Ga1−x Alx As QW with well width L = 30 nm and
L = 10 nm for different laser parameters and temperature.
As seen in Fig. 4, the donor binding energy increases when
Al composition increases in the QW. The main reason can
be explained as follows. When Al composition increases,
the barrier height V0 of Eq. (6) increases, quantum confinement is further enhanced, and then the ground-state
electron wave function can be more localized inside the
QW. Thus, the donor binding energy increases as a result of the increase of the Coulomb interaction between
the electron and impurity when Al composition increases
in GaAl/Ga1−x Alx As QW. Moreover, in Fig. 4(b), as Al
composition increases, the laser field effects on the donor
binding energy become weak. In addition, we may also
notice that there is a decrease in donor binding energy
with the increase of temperature, which agrees with the
result in Fig. 3.
Figure 5 presents the variation of the ground-state
donor binding energy as a function of the well width in
the GaAs/Ga0.7 Al0.3 As QW for different laser parameters and temperature. It is displayed from Fig. 5 that for
given laser parameter and temperature, the donor binding energy decreases as the well width increases. This is
because the increase of the well width results in a spreading of the wave function, which lowers the donor binding
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energy. In addition, we also find that for any given temperature, the effect of laser field on donor binding energy
becomes weak when the well width increases. This results
also suggest that the laser field effects on impurity states
are more obvious in the QW with small size.
Fig. 4 (Color online) The ground-state donor binding energy as a function of Al composition x in
GaAs/Ga1−x Alx As QW with the well width L = 30 nm
(a) and L = 10 nm (b) considering different laser parameters and temperature.
Fig. 5 (Color online) The ground-state donor binding energy as a function of the well width in the
GaAs/Ga0.7 Al0.3 As QW for different laser parameters
and temperature.
4 Conclusions
In conclusion, we have investigated the groundstate binding energy of donor impurity in the
GaAs/Ga1−x Alx As QW under the influences of laser field
and temperature using a variational method. Numerical results show that the donor binding energy depends
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Communications in Theoretical Physics
on impurity position, laser parameter, temperature, Al
composition and the well width. For the impurity near
the center of the well, the donor binding energy decreases
rapidly with the increase of the laser parameter. However,
the laser field effect on the donor binding energy becomes
less sensible as the impurity approaches the boundary of
the QW. Moreover, the effect of laser field on the donor
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Vol. 60
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QW. In addition, the donor binding energy increases as
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that these results may be helpful to understand further
the related physics as well as device applications.
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