Negative Absorption Coefficient of a Weak Electromagnetic Wave Caused by Electrons Confined in Rectangular Quantum Wires in the Presence of Laser Radiation Nguyen Quang Bau∗ and Nguyen Thi
Trang 1Negative Absorption Coefficient of a Weak Electromagnetic Wave Caused by Electrons Confined in Rectangular Quantum Wires in the Presence of Laser
Radiation
Nguyen Quang Bau∗ and Nguyen Thi Thanh Nhan
Department of Physics, College of Natural Sciences, Hanoi National University, Hanoi, Vietnam
Department of Physics, Academy of Defence Force - Air Force, Hanoi, Vietnam
(Received 17 January 2013, in final form 22 October 2013)
Analytic expressions for the absorption coefficient (ACF) of a weak electromagnetic wave (EMW)
caused by electrons confined in rectangular quantum wires (RQWs) in the presence of laser radiation
are calculated using the quantum kinetic equation for electrons in the case of electron-optical phonon
scattering The dependence of the ACF of a weak EMW on the intensityE01 and the frequency
Ω1 of the external laser radiation, the intensityE02and the frequency Ω2 of the weak EMW, the
temperatureT of the system and the size L (L x andL y) of the RQWs is obtained The results
are numerically calculated and discussed forGaAs/GaAsAl RQWs The numerical results show
that the ACF of a weak EMW in RQWs can have negative values Thus, in the presence of laser
radiation, under proper conditions, a weak EMW is increased This is different from the similar
problem in bulk semiconductors and from the case without laser radiation
PACS numbers: 78.67.Lt, 78.67.-n
Keywords: Rectangular quantum wires, Absorption coefficient, Electron-phonon interaction, Laser radiation
DOI: 10.3938/jkps.64.572
I INTRODUCTION
Quantum wires are one-dimensional semiconductor
structures In quantum wires, the motion of electrons
is restricted in two dimensions, so they can only flow
freely in one dimension Hence, the energy spectrum of
the electrons becomes discrete in two dimensions, and a
system of electrons in a quantum wire is similar to a
one-dimensional electron gas The confinement of electrons
in one-dimensional systems remarkably affects many of
the physical properties of the material, including its
op-tical properties, and those properties are very different
from the properties of normal bulk semiconductors [1–5]
Among the optical properties, the absorption of
electro-magnetic waves by matter is very interesting and has
been developed in both theory and experiment The
lin-ear absorption of a weak electromagnetic wave (EMW)
and the nonlinear absorption of a strong EMW in
low-dimensional systems have been studied [6–15]
Experimentally, measuring the absorption coefficient
(ACF) of a strong EMW directly is very difficult, so
in an experiment, one usually studies the influence of
the strong EMW (laser radiation) on the electrons in a
∗E-mail: nguyenquangbau54@gmail.com
semiconductor that is located in a weak EMW The in-fluence of laser radiation on the absorption of a weak EMW in normal bulk semiconductors has been investi-gated [16–19] However, in that problem, the ACF of a weak EMW has only positive values Similar studies on low-dimensional systems, in particular, RQWs, have not been done Therefore, in this paper, we use the quantum kinetic equation for electrons to theoretically calculate the ACF of a weak EMW caused by electrons confined
in a RQW in the presence of laser radiation The re-sults are numerically calculated for the specific case of a
GaAs/GaAsAl RQW We show that the ACF of a weak
EMW in a RQW can have negative values This is dif-ferent from the similar problem in bulk semiconductors and from the case without laser radiation Thus, for a RQW, in the presence of laser radiation, under proper conditions, the weak EMW is increased The nature of this effect is due to our system being low-dimensional;
i.e., the system has a size around the De Broglie
wave-length of the carriers We can use this effect as one of the criteria for quantum-wire fabrication technology
Trang 2
-572-II ABSORPTION COEFFICIENT OF A
WEAK EMW IN THE PRESENCE OF A
LASER RADIATION FIELD IN A RQW
We consider a wire of GaAs with a rectangular cross
section (L x ×L y ) and a length L z , embedded in GaAsAl.
The carriers (electron gas) are assumed to be confined by
infinite potential barriers in the xOy plane and to be free
along the wire’s axis (the Oz-axis), where O is the origin
The EMW is assumed to be planar and monochromatic,
to have a high frequency, and to propagate along the x
direction In a RQW, the state and the electron energy
spectrum have the forms [20]
ψ n,,p z =
⎧
⎪
⎪
2e ipzz
√
L x L y L zsinnπx
L x sinπy
L y
0≤ x ≤ L x
0≤ y ≤ L y
(1)
ε n, (p z) =2p2
2m ∗ +
2π2
2m ∗
n2
L2x +
2
L2
, (2)
where n and (n, =1, 2, 3, ) denote the quantization
of the energy spectrum in the x and the y directions,
respectively, p z = (0, 0, p z) is the wave vector of an
elec-tron along the wire’s z axis, and m ∗is the effective mass
of an electron
We consider a field of two EMWs: laser radiation as a
strong EMW with an intensity E01 and a frequency Ω1,
and a weak EMW with an intensity E02and a frequency
Ω2:
E(t) = E01sin (Ω1t + ϕ1) + E02sin (Ω2t) (3) The vector potential of that field of the two EMWs is
A(t) = Ωc
1
E01cos(Ω1t + ϕ1) + c
Ω2
E02cos(Ω2t) (4)
The Hamiltonian of the electron-optical phonon system
in the RQW in that field of two EMWs in the second quantization representation can be written as
H =
n,,p z
ε n,
p z − e
c A z (t) a+n,,p z a n,,p z+
q
ω q b+q b q
n,,n , ,p z ,q
C q I n,,n , (q ⊥ )a+n , ,p z +q z a n,,p z (b q + b+−q ), (5)
where e is the elemental charge, c is the velocity of light,
ω q ≈ ω0 is the frequency of an optical phonon, (n, , p z)
and (n , , p z + q z) are the electron states before and
af-ter scataf-tering, respectively, a+n,,p z (a n,,p z) is the creation
(annihilation) operator of an electron, b+q (b q) is the
cre-ation (annihilcre-ation) operator of an phonon for a state
having wave vector q = (q x , q y , q z ), and q z = (0, 0, q z).
C q is the electron - optical phonon interaction constant
[7–9], |C q |2 = e2ω0
2ε0V q2
1
χ ∞ − 1
χ0 , where V and ε0 are
the normalization volume and the electronic constant,
and χ0and χ ∞are the static and the high-frequency
di-electric constants, respectively I n,l,n ,l (q ⊥) is the
elec-tron form factor (which characterizes the confinement of electrons in a RQW) This form factor can be written as [20]
I n,,n , (q ⊥) =
32π4(q x L x nn )2
1− (−1) n+n cos(q x L x)
(q x L x)4− 2π2(q x L x)2(n2+ n 2 ) + π4(n2− n 2)2 2
× 32π
4(q y L y )2
1− (−1) + cos(q y L y)
(q y L y)4− 2π2(q y L y)2(2+ 2 ) + π4(2− 2)2 2. (6)
Because the motion of the electrons is confined in the xOy plane, we only consider the current density vector
Trang 3of electrons along the z direction in the RQW, which has
the form
j z (t) = e
m ∗
n,,p z
p z − c e A z (t) n n,,p
z (t). (7)
The ACF of a weak EMW caused by the confined
elec-trons in the presence of laser radiation in the RQW takes
the form [16]
α = 8π
c √
χ ∞ E022 j z (t) E02sin ωt
In order to establish the quantum kinetic equations for
the electrons in a RQW, we use the general quantum equation for the statistical average value of the electron particle number operator (or electron distribution
func-tion) n n,,p z (t) = a+n,,p z a n,,p z
t [16]:
i ∂n n,,p z (t)
∂t = a
+
n,,p z a n,,p z , H
Using the Hamiltonian in Eq (5) and the commutative relations of the creation and the annihilation operators,
we obtain the quantum kinetic equation for electrons in the RQW:
∂n n,,p z (t)
∂t =−12
n , ,q
|C q |2|I n,,n , (q ⊥)|2 +∞
u,s,m,f=−∞
J u (a 1z q z )J s (a 1z q z )J m (a 2z q z )J f (a 2z q z)
× exp {i {[(s − u)Ω1+ (m − f )Ω2− iδ] t + (s − u)ϕ1}}
×
t
−∞
dt2{[n n,,p z (t2)N q − n n , ,p z +q z (t2)(N q+ 1)]
× exp
i
[ε n , (p z + q z)− ε n, (p z)− ω q − sΩ1− mΩ2+ iδ] (t − t2)
+ [n n,,p z (t2)(N q+ 1)− n n , ,p z +q z (t2)N q]
× exp
i
[ε n , (p z + q z)− ε n, (p z) +ω q − sΩ1− mΩ2+ iδ] (t − t2)
− [n n , ,p z −q z (t2)N q − n n,,p z (t2)(N q+ 1)]
× exp
i
[ε n, (p z)− ε n , (p z − q z)− ω q − sΩ1− mΩ2+ iδ] (t − t2)
− [n n , ,p z −q z (t2)(N q+ 1)− n n,,p z (t2)N q]
× exp
i
[ε n, (p z)− ε n , (p z − q z) +ω q − sΩ1− mΩ2+ iδ] (t − t2)
where a 1z and a 2z are the z-components of a1 = m e E ∗Ω012
and a2 = m e E ∗02Ω 2, respectively N q is the balanced
distri-bution function of phonons, ϕ1 is the phase difference
between the two electromagnetic waves, and J k (x) is the
Bessel function
In Eq (10), the quantum numbers n and
character-ize the quantum wire These indices are not present in
the previously-published quantum kinetic equation for
the electrons in a similar problem, but in normal bulk semiconductors [17] The first - order tautology approx-imation method is used to solve this equation [16–19]
The initial approximation of n n,,p z (t) is chosen as
n0n,,p z (t2) = ¯n n,,p z, n0n,,p z +q z (t2) = n¯n,,p z +q z,
n0n,,p z −q z (t2) = ¯n n,,p z −q z
As a result, the expression for the unbalanced electron
distribution function n n,,p z (t) can be obtained:
Trang 4n n,,p z (t) = ¯ n n,,p z −1
n , ,q
|C q |2|I n,,n , (q ⊥)|2 +∞
k,s,r,m=−∞
J s (a 1z q z )J k+s (a 1z q z )J m (a 2z q z )J r+m (a 2z q z)
×exp{−i {[kΩ1+ rΩ2+ iδ] t + kϕ1}}
kΩ1+ rΩ2+ iδ ×
¯
n n , ,p z −q z N q − ¯n n,,p z (N q+ 1)
ε n, (p z)− ε n , (p z − q z)− ω q − sΩ1− mΩ2+ iδ
+ n¯n , ,p z −q z (N q+ 1)− ¯n n,,p z N q
ε n, (p z)− ε n , (p z − q z) +ω q − sΩ1− mΩ2+ iδ − n¯n,,p z N q − ¯n n , ,p z +q z (N q+ 1)
ε n , (p z + q z)− ε n, (p z)− ω q − sΩ1− mΩ2+ iδ
− n¯n,,p z (N q+ 1)− ¯n n , ,p z +q z N q
ε n , (p z + q z)− ε n, (p z) +ω q − sΩ1− mΩ2+ iδ
where ¯n n,,p zis the balanced distribution function of
elec-trons, and the quantity δ is an infinitesimal and appears
due to the assumption of an adiabatic interaction of the
EMW
Substituting n n,,p z (t) into the expression for j z (t), we
calculate the ACF of the weak EMW by using Eq (8) The resulting ACF of a weak EMW in the presence of laser radiation in a RQW can be written as
4n0ω0
2πε0c √
2πχ ∞ m ∗ k B T m ∗Ω3Z1Z2
1
χ ∞ − 1
χ0
cos2α2
+∞
n,,n , =1
II n,,n ,
×
(D 0,1 − D 0,−1)−1
2(H 0,1 − H 0,−1) +323 (G 0,1 − G 0,−1) +14(H −1,1 − H −1,−1 + H 1,1 − H 1,−1)
−1
16(G −1,1 − G −1,−1 + G 1,1 − G 1,−1) + 1
64(G −2,1 − G −2,−1
+G 2,1 − G 2,−1)
where
D s,m = e − 2kBT ξs,m K0
|ξ s,m |
2k B T e − kBT εn, (N ω0+ 1)− e − εn, −ξs,m kBT N ω0
,
H s,m = a21cos2α1e − 2kBT ξs,m
4m ∗2 ξ2
s,m
4
1/2
K1
|ξ s,m |
2k B T e − kBT εn, (N ω0+ 1)− e − εn, −ξs,m kBT N ω0
,
G s,m = a4cos4α1e − 2kBT ξs,m
4m ∗2 ξ2
s,m
4
K2
|ξ s,m |
2k B T e − kBT εn, (N ω0+ 1)− e − εn, −ξs,m kBT N ω0
,
ε n,= 2m2π ∗2
n2
L2
x+ 2
L2 , ε n , = 2π2
2m ∗
n 2
L2
x + 2
L2 , N ω0 = 1
e kBT ω0 −1
,
a1= m eE ∗01Ω 2, Z1=
+∞
n=1 e − 2m∗kBT L2x 2π2n2 , Z2=
+∞
=1 e −
2π22 2m∗kBT L2y,
ξ s,m = ε n , − ε n,+ω0− sΩ1− mΩ2, with s=-2, -1, 0, 1, 2, and m=-1, 1.
II n,,n , = +∞
−∞ dq x
+∞
−∞ dq y |I n,,n , (q ⊥)|2= [A1(1− δ n,n ) + B1δ n,n ] [A2(1− δ , ) + B2δ , ],
where A1=L1x
π
3 + (n
2+n 2)
2π(n2−n 2) 2 +5(n2+n 2)
2πn2n 2
, A2=L1y
π
3 + (
2+ 2)
2π(2− 2) 2 +5(2+ 2)
2π2 2
,
B1=L1x3π
2 +16πn1052
, B2=L1y 3π
2 +16π1052
α1 is the angle between the vector E01 and the positive
direction of the Oz axis, and α2is the angle between the
vector E02 and the positive direction of the Oz axis.
Equation (12) is the expression for the ACF of a weak
Trang 5Fig 1 (Color online) Dependence ofα on T
EMW in the presence of external laser radiation in a
RQW As one can see, the ACF of a weak EMW is
in-dependent of E02; it depends only on E01, Ω1, Ω2, T, L x,
and L y This expression is different from that in the
normal bulk semiconductors [17] From Eq (12), when
we set E01 = 0, we will obtain the expression for the
ACF of a weak EMW in the absence of laser radiation in
a RQW In Section III, we will show clearly that under
the influence of laser radiation, the ACF of a weak EMW
in a RQW can have negative values
III NUMERICAL RESULTS AND
DISCUSSION
In order to clarify the analytical expression for the
ACF of a weak EMW in the presence of laser radiation
in a RQW and to show clearly that the ACF can have
negative values, in this section, we numerically calculated
the ACF for the specific case of a GaAs/GaAsAl RQW.
The parameters used in the calculations are as follows
[8, 21]: χ ∞ = 10.9, χ0 = 13.1, m = 0.066m0, m0 being
the mass of free electron, n0 = 1023 m −3, ω0 = 36.25
meV , α1= π3, and α2= π6.
Figure 1 describes the dependence of α on the
temper-ature T for five different values of E01, with Ω1= 3×1013
Hz, Ω2= 1013 Hz, L x = 24 nm, and L y = 26 nm
Fig-ure 1 shows that when the temperatFig-ure T of the system
rises from 20 K to 400 K, the curves have a maximum and
a minimum Figure 2 describes the dependence of α on
the frequency Ω1of the laser radiation for three different
values of T , with Ω2= 1013 Hz, L x = 24 nm, L y = 26
nm, and E01 = 11× 105 V /m This figure shows that
the curves can have a maximum or no maximum in the
investigated interval Figure 3 describes the dependence
of α on the frequency Ω2of the weak EMW for three
dif-ferent values of T , with Ω1= 3× 1013Hz, L x = 24 nm,
Fig 2 (Color online) Dependence ofα on Ω1.
Fig 3 (Color online) Dependence ofα on Ω2.
L y = 26 nm, and E01= 15× 106 V /m From Fig 3, we
see that the curves have a maximum (peak) at Ω2= ω0
and smaller maxima (peaks) at Ω2 = ω0 The
frequen-cies Ω2 of the weak EMW at which ACF has maxima
(peaks) do not change as the temperature T is varied.
Figure 4 shows the ACF as a function of the intensity
E01 of the laser radiation for three different values of T ,
with Ω1= 6× 1013Hz, Ω2= 3× 1013Hz, L x = 24 nm, and L y = 26 nm From the figure, we see that the curves
have a maximum in the investigated interval Figure 5
describes the dependence of α on L x for three different
values of T , with Ω1 = 3× 1013 Hz, Ω2 = 7× 1013
Hz, L y = 26 nm, and E01 = 15× 106 V /m From this
figure, we also see that the curves have many maxima (peaks) These figures show that under influence of laser radiation, under proper conditions, the ACF of a weak EMW in a RQW can have negative values This is differ-ent from the similar problem in bulk semiconductors and
Trang 6Fig 4 (Color online) Dependence ofα on E01.
Fig 5 (Color online) Dependence ofα on L x.
from the case without laser radiation The main
scien-tific reason leading to the negative ACF of weak EMW in
the presence of laser radiation in low-dimensional
semi-conductors in general and in quantum wires in
partic-ular is that the systems are low-dimensional Namely,
when the size of the system is reduced down to around
the De Broglie wavelength of carriers, the quantum laws
markedly appear, leading to new properties of the system
appearing, the so-called size effect One of those
proper-ties is that the ACF of a weak EMW in the presence of
laser radiation can have a negative value; i.e., the weak
EMW is increased This property does not appear
en-tirely for bulk semiconductors (not low-dimensional
sys-tems);i.e., no increase in the weak EMW occurs in bulk
semiconductors, even when the parameters are adjusted
[17,18]
However, if we want to observe quantum effects, the
quantum wires must satisfy the following conditions:
The distance between consecutive energy levels has to
be significantly greater than the thermal energy of the
carriers: ε2− ε1 k B T , where ε1 and ε2 are two
con-secutive energy levels of the electrons in a quantum wire,
and k B is the Boltzmann constant If the electron gas
is degenerate and has a Fermi energy level ζ, the follow-ing condition is needed: ε2 > ζ > ε1 In the opposite
case, when ζ ε2− ε1, in principle, we can observe
quantization effects due to size reduction, but the rela-tive amplitudes are very small In addition, the distance between consecutive energy levels has to be significantly
greater than the error in the energy: ε2− ε1
τ, where
τ is the average lifetime of the carrier in a quantum state
with a set of determined quantum numbers
If a quantum wire satisfies the above conditions [22,
23], when we change parameters such as E01, Ω1, Ω2, T ,
L x , and L y in a proper way, a negative ACF of a weak
EMW in the presence of laser radiation will be observed The negative ACF effect is an important characteristic that only low-dimensional systems have Thus, it can
be used as one of the criteria to check the fabrication of low-dimensional systems in general and quantum wires in particular If a quantum wire is fabricated successfully, when we change the parameters in a proper way, the ACF of a weak EMW in a quantum wire in the presence
of laser radiation will have a negative value; if this effect does not appear, the fabrication has failed
IV CONCLUSIONS
In this research, we investigated the negative absorp-tion coefficient of a weak EMW caused by electrons con-fined in RQWs in the presence of laser radiation We obtained an analytical expression for the ACF of a weak EMW in the presence of laser radiation in a RQW for the case of electron-optical phonon scattering The ex-pression shows that the ACF of a weak EMW is
inde-pendent of E02 and depends only on E01, Ω1, Ω2, T ,
L x , and L y This expression is different from that in
normal bulk semiconductors From this expression, the ACF of a weak EMW in the absence of laser radiation
in a RQW can be obtained by setting E01 = 0 The
ACF is numerically calculated for the specific case of a
GaAs/GaAsAl RQW Computational results show that
the dependence of the ACF on various physical factors
of the system is complex Figure 3 shows that a resonant peak appears for Ω2 = ω0 and that many smaller
reso-nant peaks appear for Ω2= ω0 Figure 5 shows that the ACF of a weak EMW has many maxima (peaks) These results show that under the influence of laser radiation, the ACF of a weak EMW in a RQW can have negative values Thus, in the presence of a strong EMW, under proper conditions, a weak EMW is increased This is different from the similar problem in bulk semiconduc-tors and from the case without laser radiation We can use this effect as one of the criteria for quantum-wire
Trang 7fabrication technology.
ACKNOWLEDGMENTS
This research was done with financial support from
Vietnam NAFOSTED (number 103.01-2011.18)
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... weak EMW caused by electrons con-fined in RQWs in the presence of laser radiation We obtained an analytical expression for the ACF of a weak EMW in the presence of laser radiation in a RQW for the. .. the ACF of a weak EMW has many maxima (peaks) These results show that under the in? ??uence of laser radiation, the ACF of a weak EMW in a RQW can have negative values Thus, in the presence of a strong... wires in particular If a quantum wire is fabricated successfully, when we change the parameters in a proper way, the ACF of a weak EMW in a quantum wire in the presenceof laser radiation