QUANTUM THEORY OF THE NONLINEAR ABSORPTION COEFFICIENT OF A
STRONG ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN
CYLINDRICAL QUANTUM WIRE
Dr. Hoang Dinh Trien
Faculty of physics, Danang University of Education, The University of Danang
Abstract. Quantum theory of the nonlinear absorption coefficient of a strong
electromagnetic wave by confined electrons in cylindrical quantum wires with infinite
potential is studied by using the quantum kinetic equation for electrons. The analytic
expressions of the nonlinear absorption coefficient of a strong electromagnetic wave in
the presence of a magnetic field parallel to axis of wire for the case electron-optical
phonon scattering are obtained. The dependence of the nonlinear absorption coefficient
on the intensity E0 and frequency ω of the external strong electromagnetic wave, the
cyclotron frequency ωc, the temperature T of the system and the radius R of wires are
shown explicitly that it is nonlinear. The results are numerically calculated and
discussed for GaAs/GaAsAl cylindrical quantum wires.
Keywords: nonlinear absorption, cylindrical quantum wires, electromagnetic wave.
1.
Introduction
In quantum wires, the motion of electrons is restricted in two dimensions, so that can flow
freely in one dimension. The confinement of electron in these systems has changed the electron
mobility remarkably. This has resulted in a number of new phenomena, which concern a reduction
of sample dimensions. These effects differ from those in bulk semiconductors, for example,
electron-phonon interaction and scattering rates [1, 2] and dc electrical conductivity [3, 4].
The problem of optical properties in bulk semiconductors, as well as low dimensional
systems has also been investigated [5-9]. However, in those papers, the linear absorption of a weak
electromagnetic wave has been only considered in low dimensional systems [6-9]; the nonlinear
absorption of a strong electromagnetic wave (EMW) has been considered in bulk [5], but in
quantum wires, it still opens for studying. In this paper, we use the quantum kinetic equation for
electrons to theoretically study influence of magnetic field on the nonlinear absorption coefficient
of a strong EMW by confined electrons in a cylindrical quantum wire (CQW) for the case electronoptical phonon scattering. Numerical calculations are carried out with a specific GaAs/GaAsAl
cylindrical quantum wires.
1
2
2.
The nonlinear absorption coefficient of a strong EMW in the presence of a magnetic field
in a CQW
We consider a wire of GaAs with a circular cross section with radius R and length Lz
embedded in AlAs. The carries (electrons) are assumed to be confined by infinite potential barriers
and free along the wire's axis (Oz). A constant magnetic field with the magnitude B is applied
parallel to the axis of wire. In the case, the Hamiltonian is given by [1,7] (in this paper, we select
 =1)
H =   , p a, p a , p    0 bq bq 

 ,p

q
C
 
 , ' , p , q

q
I
n , , n ' , '
J
(u )a '
N ,N '
 
 , pq
a , p (bq  bq )
(1)
Where the sets of quantum numbers ( N , n, ) and ( N ' , n ' ,  ' ) , characterizing the states of electron in
'
the quantum wire before and after scattering with phonon, are designated by  and  ,
respectively; a, p ( a , p ) is the creation (annihilation) operator of an electron, p = (0,0, pz ) is the
electron wave vector (along the wire's z axis), bq ( bq ) is the creation operator (annihilation)

operator of an phonon for state have wave vector q , 0 is the frequency of optical phonon, the
electron form factor I n ,l ,n,l  (q ) can be written as [9]
I
n , , n' '
(q ) =
2 R
J ' (q R) *' ' (r ) n, (q R)rdr
n,
R2 0 |nn |
(2)
according to Ref. [11], it can be calculated for ground states of electrons by applying the
approximate expression for the wave function and for the energies of states: namely,
I 01,0,1 (q ) = 24
J 3 (q R)
(q R)3
,
I 1,1,0,1 (q ) = 48
J 4 (q R)
,
(q R)3
(3)
The electron-optical phonon interaction constants can be taken as [6-8]. | Cq |2 = e20 1/ 1/0 /2 0q 2V
here V is the normalization volume,  0 is the permittivity of free space,

and
 0 are the high
and low-frequency dielectric constants, respectively, and J ' (u ) takes the form
N ,N
J

N ,N '
(u ) =  dr ' (r  ac2 ( p z  q  ))e

N
iq p z
 N (r  ac2 p z )
(4)
here u = ac q2 /2 , r and ac = c/eB is position and radius of electron in the (x,y) plane, e is the
electron charge, c is the light velocity.
When the magnetic field is strong and the radius R of wires is very bigger than cyclotron
radius  c , the electron energy spectra have the form
3
  , p =
p z2
n  1
 c ( N    )
2m
2 2 2
(5)
where c = eB/m is the cyclotron frequency, m is the effective mass of electron, N is the
Landau level index (N=0,1,2,...), N (x) represents the harmonic wave function.

The carrier current density j (t ) and the nonlinear absorption coefficient of a strong
electromagnetic wave  take the form [6]

 e 
e
j (t ) = ( p  A(t )) n , p (t );
m p
c
 =
8
c   E02
(6)


 j (t ) E0 sin t  t
where n , p (t ) is electron distribution function,
 X t
(7)
means the usual thermodynamic average of X


at moment t,   is the high-frequency dielectric constants, A(t ) = c E0 cos (t ) is the vector


potential, E0 and  is the intensity and frequency of EMW.
In order to establish analytical expressions for the nonlinear absorption coefficient of a
strong EMW by confined electrons in CQW, we use the quantum kinetic equation for particle
number operator of electron n , p (t ) :
i
n , p (t )
t
=i
a, p a , p  t
t
= [a, p a , p , H ] t
(8)
From Eq.(8), using Hamiltonian in Eq.(1) and realizing calculations, we obtain quantum kinetic
equation for confined electrons in CQW. Using the first order tautology approximation method to
solve this equation, we obtain the expression of electron distribution function n , p (t ) . We insert the


expression of n , p (t ) into the expression of j (t ) and then insert the expression of j (t ) into the
k
expression of  in Eq.(4). Using property of Bessel function J k 1 ( x)  J k 1 ( x) = J k ( x) , and replacing
x
delta functions by Lorentzians (collision broadening of Landau levels) [10], we obtain the analytic
expression of the nonlinear absorption coefficient of a strong EMW by confined electrons in CQWs
with infinite potential in the presence of a magnetic field
=
e 4 n0c kbT

1
1
n  1
(
 ) | I ' ' |2 [exp{ c [ N    ]}
2 3
n , , n 


k
T
2 2 2
2 0c   mac 

0  '
b
 exp{
A| M |
 c ' n '  ' 1
3e 2 E 2
[ N    ]}]  [1  2 2 0 4 ] 
kbT
2 2 2
8ac m 
M (  0  Mc )  A
(9)
4
2
where M = N  N '  (n  n' )/2  (   ' )/2 ; A = e kbT | I ' ' |2 (1/   1/ 0 )
n , , n 
2
From analytic expressions of the nonlinear absorption coefficient of a strong EMW by
confined electrons in CQWs with infinite potential in the presence of a magnetic field (Eq.9), we
can see that when the intensity of EMW E0=0, the nonlinear result will turn back to the linear result
which was calculated by using Kubo - Mori method.
3.
Numerical results and discussions
In order to clarify the results that have been obtained, in this section, we numerically
calculate the nonlinear absorption coefficient of a strong EMW for a GaAs/GaAsAl RQW. The
nonlinear absorption coefficient is considered as a function of the intensity E0=0 and energy of
strong EMW, the temperature T of the system , and the parameters of RQW. The parameters used in
the numerical calculations [5,11] are  0 =12.5,   = 10.9 ,  0 = 13.1 , m = 0.066m0 , m0 being the
mass
of
free
electron,
0 = 36.25meV ,
kb = 1.3807 10 23 j/K ,
n0 = 10 23 m 3 ,
e = 1.60219 10 19 C ,  = 1.05459 1034 j.s , q = 2 105 m1 .
Figure 1: Dependence of  on T and E0
Figure 2: Dependence of  on radius of
wire
Figure 1 shows the dependence of the nonlinear absorption coefficient of a strong EMW on
the temperature T of the system and intensity E0 of EMW. It can be seen from this figure that 
depends strongly and nonlinearly on T and E0.  is stronger at large values of the intensity E0 of
EMW and the temperature T.
Figure 2 presents the dependence of the nonlinear absorption coefficient of a strong EMW
on the radius of wire at different values of the temperature T of the system. It can be seen from this
figure that  depends strongly and nonlinear on the radius of wire but it does not have the
5
maximum value (peak), the absorption increases when R is reduced. This is different from the case
of the absence of a magnetic field.
Figure 3: Dependence of  on 
Figure 4: Dependence of  on c
Figure 3 the dependence of the nonlinear absorption coefficient of a strong EMW on the
EMW energy at different values of the temperature T of the system. It is seen that  has the same
maximum values (sharp peaks) The EMW energy at which  has a maximum are not changed as
the temperature T of the system is varied. These sharp peaks are demonstrated that the nonlinear
absorption coefficient only significant when there is the condition. This means that  depends
strongly on the frequency  of the EMW.
Figure 4 presents the dependence of the nonlinear absorption coefficient of a strong EMW
on the cyclotron energy (  c ) of the magnetic field. It can be seen from this figure that there are
same resonance peaks at different values of cyclotron frequency c . The nonlinear absorption
coefficient only significant at these resonance peaks. Based on this result we make the following
remarks: The index of Landau level N' which electrons can move to after absorption. Only at these
peaks, strong EMW is absorbed strongly. In addition, the density of resonance peaks is very high in
the region where c <  , corresponding to the weak magnetic field B, but this density is low when
B increases. These resonance peaks, reflect the effect of quantum magnetic field on the quantum
wire. When the magnetic field is stronger, the peaks is more discrete, the influence of the magnetic
field is shown more clearly.
4. Conclusion
In the present paper, we have obtained analytical expression of the nonlinear absorption of a
strong EMW by confined electrons in CQWs in the presence of a magnetic field for the case
electron-optical phonon scattering. The dependence of  on the intensity E0 and frequency  of
the external strong EMW, the temperature T of the system, the radius R of wires and cyclotron
6
frequency is complex and nonlinear. From the analytic results, we see that when the intensity of
EMW E0=0, the nonlinear result will turn back to the linear result which was calculated by using
Kubo - Mori method
The numerical results obtained for a GaAs/GaAsAl CQW show that  depends strongly
and nonlinearly on the intensity E0 and frequency  of the external strong EMW, the temperature
T of the system and the radius R of wires and cyclotron frequency c and there are different from
the case of the absence of a magnetic field. When the magnetic field is stronger, the influence of the
magnetic field is shown more clearly.
References
[1]
N. Mori and T. Ando, Phys. Rev.B, 40 (1989) 6175.
[2]
J. Pozela and V. Juciene, Sov. Phys. Tech. Semicond, 29 (1995) 459.
[3]
P. Vasilopoulos, M. Charbonneau and C. N. Van Vlier, Phys. Rev.B, 35 (1987) 1334.
[4]
A. Suzuki, Phys. Rev.B, 45 (1992) 6731.
[5]
V. V. Pavlovich and E. M. Epshtein, Sov. Phys. Solid State, 19 (1977) 1760.
[6]
G. M. Shmelev, L. A. Chaikovskii and N. Q. Bau, Soc. Phys. Tech. Semicond, 12 (1978) 1932.
[7]
N. Q. Bau and T. C. Phong, J.Phys. Soc. Japan, 67 (1998) 3875
[8]
N. Q. Bau, N. V. Nhan and T. C. Phong, J. Korean. Phys. Soc, 41 (2002) 149.
[9]
N. Q. Bau, L. Dinh and T. C. Phong, J. Korean. Phys. Soc, 51 (2007) 1325.
[10]
[11]
M. P. Chaubey and C. M. Van Viliet, Phys. Rev.B, 33 (1986) 5617
A. Gold and A. Ghazali, Phys. Rev. B 41, 7626 (1990).