DSpace at VNU: Negative absorption coefficient of a weak electromagnetic wave caused by electrons confined in rectangular quantum wires in the presence of laser radiation

Journal of the Korean Physical Society, Vol. 64, No. 4, February 2014, pp. 572∼578

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 Thanh Nhan
Department of Physics, College of Natural Sciences, Hanoi National University, Hanoi, Vietnam

Nguyen Vu Nhan
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 intensity E01 and the frequency
Ω1 of the external laser radiation, the intensity E02 and the frequency Ω2 of the weak EMW, the
temperature T of the system and the size L (Lx and Ly ) of the RQWs is obtained. The results
are numerically calculated and discussed for GaAs/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

semiconductor that is located in a weak EMW. The influence of laser radiation on the absorption of a weak
EMW in normal bulk semiconductors has been investigated [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 results 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 different 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 wavelength of the carriers. We can use this effect as one of
the criteria for quantum-wire fabrication technology.

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 onedimensional electron gas. The confinement of electrons
in one-dimensional systems remarkably affects many of
the physical properties of the material, including its optical properties, and those properties are very different
from the properties of normal bulk semiconductors [1–5].
Among the optical properties, the absorption of electromagnetic waves by matter is very interesting and has
been developed in both theory and experiment. The linear absorption of a weak electromagnetic wave (EMW)
and the nonlinear absorption of a strong EMW in lowdimensional 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:

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Negative Absorption Coefficient of a Weak Electromagnetic Wave· · · – Nguyen Quang Bau et al.

II. ABSORPTION COEFFICIENT OF A
WEAK EMW IN THE PRESENCE OF A
LASER RADIATION FIELD IN A RQW

where n and (n, =1, 2, 3, ...) denote the quantization
of the energy spectrum in the x and the y directions,
respectively, pz = (0, 0, pz ) is the wave vector of an electron 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 E02 and a frequency
Ω2 :

We consider a wire of GaAs with a rectangular cross
section (Lx ×Ly ) and a length Lz , 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]

⎧
ipz z
⎪
⎪
⎨ √2e
ψn,

,pz

=

sin nπx
Lx sin

Lx Ly Lz

⎪
⎪
⎩

E(t) = E01 sin (Ω1 t + ϕ1 ) + E02 sin (Ω2 t) .

0

(3)

The vector potential of that field of the two EMWs is

0 ≤ x ≤ Lx
0 ≤ y ≤ Ly

πy
Ly

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A(t) =

otherwise

c
c
E01 cos(Ω1 t + ϕ1 ) +
E02 cos(Ω2 t). (4)
Ω1
Ω2

(1)

εn, (pz ) =

2 2
pz
2m∗

H=

2 2

+

n, ,pz

+
n, ,n , ,pz ,q

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

2

n
+ 2
L2x
Ly

π
2m∗

pz −

εn,

2

,

e
Az (t) a+

n,
c

(2)

,pz an, ,pz

ω q b+
q bq

+
q

Cq In, ,n , (q⊥ )a+
n , ,pz +qz an, ,pz (bq

+ b+
−q ),

2

2

In,

,n ,

(q⊥ ) =

×

4

n+n

1 − (−1)

cos(qx Lx )

2

2

(qx Lx ) − 2π 2 (qx Lx ) (n2 + n 2 ) + π 4 (n2 − n 2 )
32π 4 (qy Ly
4

2

)

1 − (−1)
2

(qy Ly ) − 2π 2 (qy Ly ) (

2

+

+
2)

cos(qy Ly )
+ π4 (

Because the motion of the electrons is confined in the

2

e ω0
1
1
[7–9], |Cq | = 2ε
2
χ∞ − χ0 , where V and ε0 are
0V q
the normalization volume and the electronic constant,
and χ0 and χ∞ are the static and the high-frequency dielectric constants, respectively. In,l,n ,l (q⊥ ) is the electron form factor (which characterizes the confinement of
electrons in a RQW). This form factor can be written as
[20]

where e is the elemental charge, c is the velocity of light,
ωq ≈ ω0 is the frequency of an optical phonon, (n, , pz )
and (n , , pz + qz ) are the electron states before and after scattering, respectively, a+
n, ,pz (an, ,pz ) is the creation
(annihilation) operator of an electron, bq+ (bq ) is the creation (annihilation) operator of an phonon for a state
having wave vector q = (qx , qy , qz ), and qz = (0, 0, qz ).
Cq is the electron - optical phonon interaction constant

32π 4 (qx Lx nn )

(5)

2

−

2 )2

2.

2

(6)

xOy plane, we only consider the current density vector


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Journal of the Korean Physical Society, Vol. 64, No. 4, February 2014

of electrons along the z direction in the RQW, which has
the form
jz (t) =

e
m∗

pz −
n, ,pz

e
Az (t) nn,
c

,pz (t).

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[16]:
tion) nn, ,pz (t) = a+
n, ,pz an, ,pz

(7)

t

The ACF of a weak EMW caused by the confined electrons in the presence of laser radiation in the RQW takes
the form [16]
8π
α= √
jz (t)E02 sin ωt .
2
c χ∞ E02
t

i

2

|Cq | |In,

,n ,

a+
n,

,pz an, ,pz , H

t

.

(9)

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:

(8)

In order to establish the quantum kinetic equations for

∂nn, ,pz (t)
1
=− 2
∂t

∂nn, ,pz (t)
=
∂t

+∞

2

(q⊥ )|

Ju (a1z qz )Js (a1z qz )Jm (a2z qz )Jf (a2z qz )
u,s,m,f =−∞

n , ,q

× exp {i {[(s − u)Ω1 + (m − f )Ω2 − iδ] t + (s − u)ϕ1 }}
t

×

dt2 {[nn,

,pz (t2 )Nq

− nn ,

,pz +qz (t2 )(Nq

+ 1)]

−∞

× exp
+ [nn,
× exp
− [nn ,
× exp
− [nn ,
× exp

i

[εn , (pz + qz ) − εn, (pz ) − ωq − s Ω1 − m Ω2 + i δ] (t − t2 )

,pz (t2 )(Nq

i

+ 1) − nn ,

[εn , (pz + qz ) − εn, (pz ) + ωq − s Ω1 − m Ω2 + i δ] (t − t2 )

,pz −qz (t2 )Nq

i

− nn,

,pz (t2 )(Nq

+ 1) − nn,

,pz (t2 )Nq ]

[εn, (pz ) − εn , (pz − qz ) + ωq − s Ω1 − m Ω2 + i δ] (t − t2 ) ,

where a1z and a2z are the z-components of a1 =
eE02
,
m∗ Ω22

+ 1)]

[εn, (pz ) − εn , (pz − qz ) − ωq − s Ω1 − m Ω2 + i δ] (t − t2 )

,pz −qz (t2 )(Nq

i

,pz +qz (t2 )Nq ]

eE01
m∗ Ω21

and a2 =
respectively. Nq is the balanced distribution function of phonons, ϕ1 is the phase difference
between the two electromagnetic waves, and Jk (x) is the
Bessel function.
In Eq. (10), the quantum numbers n and characterize the quantum wire. These indices are not present in

the previously-published quantum kinetic equation for

(10)

the electrons in a similar problem, but in normal bulk
semiconductors [17]. The first - order tautology approximation method is used to solve this equation [16–19].
The initial approximation of nn, ,pz (t) is chosen as
¯ n, ,pz , n0n, ,pz +qz (t2 ) = n
¯ n, ,pz +qz ,
n0n, ,pz (t2 ) = n
0
nn, ,pz −qz (t2 ) = n
¯ n, ,pz −qz .
As a result, the expression for the unbalanced electron
distribution function nn, ,pz (t) can be obtained:


Negative Absorption Coefficient of a Weak Electromagnetic Wave· · · – Nguyen Quang Bau et al.

nn,

,pz (t)

=n
¯ n,

,pz

1

−

2

|Cq | |In,

,n ,

-575-

+∞

2

(q⊥ )|

Js (a1z qz )Jk+s (a1z qz )Jm (a2z qz )Jr+m (a2z qz )
k,s,r,m=−∞

n , ,q

n
¯ n , ,pz −qz Nq − n
¯ n, ,pz (Nq + 1)
exp {−i {[kΩ1 + rΩ2 + iδ] t + kϕ1 }}
×
kΩ1 + rΩ2 + iδ
εn, (pz ) − εn , (pz − qz ) − ωq − s Ω1 − m Ω2 + i δ
n
¯ n, ,pz Nq − n

¯ n, ,pz Nq
¯ n , ,pz +qz (Nq + 1)
n
¯ n , ,pz −qz (Nq + 1) − n
−
+
εn, (pz ) − εn , (pz − qz ) + ωq − s Ω1 − m Ω2 + i δ
εn , (pz + qz ) − εn, (pz ) − ωq − s Ω1 − m Ω2 + i δ
n
¯ n, ,pz (Nq + 1) − n
¯ n , ,pz +qz Nq
,
(11)
−
εn , (pz + qz ) − εn, (pz ) + ωq − s Ω1 − m Ω2 + i δ
×

where n
¯ n, ,pz is the balanced distribution function of electrons, and the quantity δ is an infinitesimal and appears
due to the assumption of an adiabatic interaction of the
EMW.

e4 n0 ω0
√
α=
2πε0 c 2πχ∞ m∗ kB T m∗ Ω32 Z1 Z2

Substituting nn, ,pz (t) into the expression for jz (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

1
1
−
χ∞
χ0

+∞

2

cos α2

IIn,

,n ,

n, ,n , =1

1
3
1
(H0,1 − H0,−1 ) +
(G0,1 − G0,−1 ) + (H−1,1 − H−1,−1 + H1,1 − H1,−1 )
2
32
4

× (D0,1 − D0,−1 ) −

1
1
(G−1,1 − G−1,−1 + G1,1 − G1,−1 ) + (G−2,1 − G−2,−1
16
64

−

+G2,1 − G2,−1 ) ,
where

ξ

− 2ks,mT

Ds,m = e

B

K0

|ξs,m |
2kB T
ξ

− 2ks,mT

Hs,m = a21 cos2 α1 e

B

ξ

− 2ks,mT

Gs,m = a41 cos4 α1 e
2

2

π
2m∗

n
L2x

eE01
,
m∗ Ω21

Z1 =

εn, =
a1 =

2

+

B

2

(12)

ε

− kn,T

e

B

2

2
4m∗ ξs,m

2

n=1

B

2

π
2m∗
x

+∞

n
L2x

, Z2 =

ξs,m = εn , − εn, + ω0 − s Ω1 − m
+∞

2

,n ,

=

ε

− kn,T

+

2

L2y

e

B

ε

e

B

, Nω0 =

2 π2 2
+∞ −
2m∗ k T L2

e

=1
Ω2 , with

B

y

Nω0 ,
−

(Nω0 + 1) − e
−

(Nω0 + 1) − e
1

ε

−ξs,m
n ,
kB T

Nω0 ,

ε

−ξs,m
n ,
kB T

Nω0 ,

,

ω0

e kB T −1

,

s=-2, -1, 0, 1, 2, and m=-1, 1.

2
dqx
dqy |In, ,n , (q⊥ )| = [A1 (1

−∞
−∞
(n2 +n 2 )
5(n2 +n 2 )
,
where A1 = L1x π3 + 2π(n
2 −n 2 )2 + 2πn2 n 2
1
3π
105
1
3π
105
B1 = Lx 2 + 16πn2 , B2 = Ly 2 + 16π 2 .

IIn,

−ξs,m
n ,
kB T

− kn,T

|ξs,m |
2kB T

K2

4

ε

|ξs,m |
2kB T

K1

4

2 π 2 n2
+∞ −
2m∗ k T L2

e

1/
2

2

2
4m∗ ξs,m

, εn , =

L2y

−

(Nω0 + 1) − e

− δn,n ) + B1 δn,n ] [A2 (1 − δ
A2 =

α1 is the angle between the vector E01 and the positive
direction of the Oz axis, and α2 is the angle between the

1
Ly

π
3

+

( 2+ 2)
2π( 2 − 2 )2

,

+

) + B2 δ
5( 2 +
2π 2

2
2

)

,

],

,

vector E02 and the positive direction of the Oz axis.
Equation (12) is the expression for the ACF of a weak


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Journal of the Korean Physical Society, Vol. 64, No. 4, February 2014

Fig. 1. (Color online) Dependence of α on T .

Fig. 2. (Color online) Dependence of α on Ω1 .

EMW in the presence of external laser radiation in a
RQW. As one can see, the ACF of a weak EMW is independent of E02 ; it depends only on E01 , Ω1 , Ω2 , T, Lx ,
and Ly . 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 temperature T for five different values of E01 , with Ω1 = 3×1013
Hz, Ω2 = 1013 Hz, Lx = 24 nm, and Ly = 26 nm. Figure 1 shows that when the temperature 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 Ω1 of the laser radiation for three different
values of T , with Ω2 = 1013 Hz, Lx = 24 nm, Ly = 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 Ω2 of the weak EMW for three different values of T , with Ω1 = 3 × 1013 Hz, Lx = 24 nm,

Fig. 3. (Color online) Dependence of α on Ω2 .

Ly = 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 frequencies 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 × 1013 Hz, Ω2 = 3 × 1013 Hz, Lx = 24 nm,

and Ly = 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 Lx for three different
values of T , with Ω1 = 3 × 1013 Hz, Ω2 = 7 × 1013
Hz, Ly = 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 different from the similar problem in bulk semiconductors and


Negative Absorption Coefficient of a Weak Electromagnetic Wave· · · – Nguyen Quang Bau et al.

Fig. 4. (Color online) Dependence of α on E01 .

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The distance between consecutive energy levels has to
be significantly greater than the thermal energy of the
kB T , where ε1 and ε2 are two concarriers: ε2 − ε1
secutive energy levels of the electrons in a quantum wire,
and kB is the Boltzmann constant. If the electron gas
is degenerate and has a Fermi energy level ζ, the following 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 relative 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 ,
Lx , and Ly 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

Fig. 5. (Color online) Dependence of α on Lx .

from the case without laser radiation. The main scientific reason leading to the negative ACF of weak EMW in
the presence of laser radiation in low-dimensional semiconductors in general and in quantum wires in particular 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 properties 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 entirely for bulk semiconductors (not low-dimensional systems); 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:

In this research, we investigated the negative absorption coefficient of a weak EMW caused by electrons confined 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 expression shows that the ACF of a weak EMW is independent of E02 and depends only on E01 , Ω1 , Ω2 , T ,
Lx , and Ly . 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 resonant 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 semiconductors and from the case without laser radiation. We can
use this effect as one of the criteria for quantum-wire


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Journal of the Korean Physical Society, Vol. 64, No. 4, February 2014

fabrication technology.

ACKNOWLEDGMENTS
This research was done with financial support from
Vietnam NAFOSTED (number 103.01-2011.18).

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