In comparison with the case of unconfined phonons, different dependence of the absorption coefficient on the electromagnetic wave frequency ω, the doping concentration nD, the number N o
Trang 1THE EFFECT OF CONFINED PHONONS ON THE ABSORPTION COEFFICIENT
OF A WEAK ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS
IN DOPED SUPERLATTICES Luong Van Tung, Le Thai Hung, Nguyen Quang Bau
Department of Physics, Ha Noi National University
334 Nguyen Trai, Thanh Xuan, Ha Noi E-mail: lthung@vnu.edu.vn or hunglethai191182@yahoo.com.vn
ABSTRACT
The effect of confined phonons on the absorption
coefficient of a weak electromagnetic wave by confined
electrons in doped surperlattices is theoretically studied
by using the Kubo-Mori method In comparison with
the case of unconfined phonons, different dependence
of the absorption coefficient on the electromagnetic
wave frequency (ω), the doping concentration (nD), the
number (N) of period, the temperature (T) of the
system is obtained The analytic expressions are
numerically evaluated, plotted, and disscussed for a
specific doping of the n-GaAs/p-GaAs superlattice The
results show that confined phonons cause some unusual
effects There are two resonant peaks of the absorption
coeffient and one of the mass operator The mass
operator’s values are larger and the absorption
coeffient’s values are smaller than they are in the case
of unconfined phonons
Keywords: Doped superlatices, Absorption coefficient,
Conductivity tensor, Confined phonons
INTRODUCTION
Recently, there are more and more interests in
studying and discovering the behavior of
low-dimensional system, in particular two-dimensional
systems, such as semiconductor superlattices, quantum
wells and doped superlattices (DSLs) The confinement
of electrons in low-dimensional systems considerably
enhances the electron mobility and leads to unusual
behaviors under external stimuli Many papers have
appeared dealing with these behaviors, for examples,
electron-phonon interaction and scattering rates [1-3]
and dc electrical conductivity [4, 5] The problems of
the absorption coefficient for a weak electromagnetic
wave (EMW) in semiconductor superlattices [6, 7], in
quantum wells [8] and in doped superlatices [9] have
also been investigated and resulted
In this paper, we study the high-frequency
conductivity tensor and the absorption coefficient of a
EMW due to confined electrons in a DSL with the
influence of confined phonons The electron-optical
phonon scattering mechanism is assumed to be
dominant We shall asume that the EMW has a high frequency Then, we estimate numerical values for a specific doping of the n-GaAs/p-GaAs superlattice to clarify our results
Using Kubo’s formula for the conductivity tensor [10] and Mori’s projection operator method [11] in the second-order approximation of the interaction, we obtain the following formula for the components of the conductivity tensor [7, 9, 12, 13]:
+ +
−
→
1 2
lim )
δ
h
0
, , ,
−
∞ +
−
× ∫e iωt δt U Jµ U Jν I dt (1)
,
h (2) here, [ U , Jν]I is the operator [ U , Jν] in the
interaction picture, and U is the energy of the electron-phonon interaction The averaging of the operators in Eqs.(1-2) is implemented with the non-interaction Hamiltonian of the electron-phonon system
The structure of DSL also modifies the dispersion relation of optical phonons, which leads to interface modes and confined modes [2] However, the contribution from these two modes can be approximated well by calculations with bulk phonons [3] Thus, in this paper, we will deal with bulk confined phonons (2-dimensional) and consider compensated n-p DSL with equal thicknesses
dn = dp=d/2 of the n-doping and p-doping layer and equal constant doping concentrations nD = nA in the respective layers
THE ABSORPTION COEFFICIENT WITH CASE
OF CONFINED PHONONS
It is well known that the motion of an electron is confined in each layer of the DSL and that it’s energy spectrum is quantized into discrete levels In this paper,
we assume that the quantization direction is the z direction The Hamiltonian of the electron-optical
Trang 2phonon system in a DSL in the second quantization
representation can be written as:
U
H
H = 0+ (3)
m m m
n n n
E
, 0 ,
, ,
,
⊥
⊥
⊥
⊥
⊥
+
q k k k
n n m
nna a b b I
C
U
, , , , '
, ' '
+ +
⊥ ⊥
⊥
⊥
q k q k q k
(5)
where H0 is the non-interaction Hamiltonian of the
electron-phonon system, n (n = 1, 2, 3, ) denotes the
quantization of the energy spectrum in the z direction,
( n , k⊥)and( n , k⊥+ q⊥)are electron states before
and after scattering,( k ,⊥ q⊥) is the in-plane (x, y)
wave vector of the electron (phonon), a ,n, a ,n
⊥
⊥
+
k k
(b ,m, b ,m
⊥
⊥
+
q
q ) are the creation and the annihilation
operators of the electron (phonon), respectively, and
0
ω
h is the energy of the optical phonon The electron
energy εk⊥,n in doped superlatices takes the simple
form [14]:
2
2
,
2 2
1
⊥
+
+
=
k
m
n
n
h ε
+
=
2
1
4 1/2 0
2
n m
n
e D
κ
π
Here, m and e are the effective mass and the
charge of the electron, respectively, κ0 is the
electronic constant, nD is the doping concentration and
q
C is the electron-phonon interaction potential In the
case of the electron-optical phonon interaction and
confined phonons, we assume that the quantization
direction is the z direction, Cq[1]is:
2 2
0 0
2
+
−
=
⊥
∞
d
m V
e
C
π χ
χ
ω π
q q
h
(8)
where V is the normalization volume, m=1, 2, …, N
and N is the number of period of DSLs, χ0 and χ∞
are the static and the high-frequency dielectric constant,
respectively, and [3, 5]:
∑∫
=
− Φ
− Φ
=
N
m
d
n n
z iq m
1 0
'
where d is the period of DSL, Φn( z − md ) is the
eigenfunction for a single potential well [15] The interaction of the system, which is described by
Eqs.(3-5) with a EMW E(t) = E 0cos(ωt), is determined
by the Hamiltonian:
∑
−
=
j
t j
H1 r E0 cos ω δ (10)
where r j is the radius vector of jth electron
Using the Kubo-Mori’s method, we obtain the following formula for the transverse component of the high-frequency conductivity tensor:
( ) [ ]1
0 − + ( ) −
ω
σxx i G (11) where γ0 = ( Jx, Jx) ,
1 0
1 0 2
lim
−
∞ +
−
−
ω
h
(12)
Knowing the high-frequency conductivity tensor, the absorption coefficient can be found by using the common relation:
ρ
π ω
c Re
4
= (13) Here, ρ is the refraction index and c is the light velocity
Since the EMW has a high frequency, using Eqs
(3-14) and noting that in compensated n-p DSLs, the bare ionized impurities make the main contribution to the superlattice potential, we obtain:
4
2 2
0
ω ω
ω γ ρ
π ω α
+
=
G
G c
where
4
2 1
2
2 3 / 2
−
βε βε
πβ
γ V e eβ µ ε
( ) ω +( ) ω −( ) ω
+
G (16)
−
=
∞
−
±
o o
e V G
χ χ πβ
γ
1 4 3 / 1
h
o o
e e
ω
ω β
h
h 1 2
1 2
± +
×
β ε β
2 2
1 2
I
n m
with λ± = ( n ' − n ) ε ± h ωo− h ω (18)
No is the equilibrium distribution of optical phonons, µ
is the chemical potential The signs (±) in the
Trang 3superscript of ±( ) ω
G and in the lower-script of the
function λ± correspond to the signs (±) in Eq.(17) and
Eq.(18) The upper sign (+) corresponds to phonon
absorption and the lower sign (−) corresponds to a
phonon emission in the absorption process From
Eqs.(11,14), we can easily see that G ( ) ω plays the
role of the well-known mass operator of the electron in
the Born approximation in the case of the absence of a
magnetic field
NUMERICAL CALCULATION AND
DISCUSSION
In order to clarify the different behaviors of a
quasi-two-dimensional electron gas confined in a DSL
with respect to a bulk electron gas, in this section, we
numerically evaluate the analytic formulae in Eqs
(14-18) for a compensated n-GaAs/p-GaAs DSL The
characteristic parameters of the GaAs layer of the DSL
are χ∞ =10.9, χ0 =12.9, nD= 1017cm−3, d = 2dn=
2dp= 80 nm, µ = 0.01 meV, m = 0.067m0, and h ωo=
36.1 meV, (m0 is the mass of free electron) The system
is assumed to be at room temperature T = 293 K
Then, we compare with the results in the case of
unconfined phonons [9]
Fig.1 Dependence of the operator G ( ) ω (1/s) on
the frequency ω of EMW and the period number N
with the case of confined phonons, n=n’=11
Figure (1) shows the mass operater G ( ) ω as a
function of the frequency ω of the EMW and the period
number N, in case of confined phonons It is seen that
( ) ω
G depends very strongly on the frequency of the
EMW, they are greater when the frequency of EMW
increases This figure shows that there is a resonant
peak of G ( ) ω in the region of the values of N from
N=5 to N=20 on the number of periods axis but the
values of G ( ) ω with case of confined phonons are
much larger than their values with the case of
unconfined phonons [9] Namely, the resonant peak’s
value of G ( ) ω is about 120s-1 in case of unconfined phonons [9] but is 850s-1 in case of the confined phonons at the same the value of N It means that the confined phonons make the life-span of an electron is
so much shorter This figure exactly shows that there is
a resonant peak of G ( ) ω when h ω ≈ 50 meV ( )
Fig.2 Dependence of the absorption coe fficient of
EMW (cm −1 ) on the frequency ω of EMW and the period number N with the case of confined phonons, n=n’=11
Figure (2) shows the absorption coefficient
( ) ω
αxx as a function of the frequency ω of the EMW and the period number N, with case of confined phonons This figure shows the resonant regions in the absorption spectra of the absorption coefficient
( ) ω
αxx as the EWM is high frequency ωτ >> 1, the resonant regions of the absorption coefficient appear when the values of G(ω) are greater There is one more resonant peak of αxx( ) ω and the values of αxx( ) ω
are much smaller than those with the case of unconfined phonons [9] Namely, the resonant peak’s value of αxx( ) ω approximates 0.85 cm-1 in case of confined phonons but 65 cm-1 in case of unconfined phonons [9]
In the figure (3-4), the graphs show that the absorption coefficient of EMW as a function of temperature T, the doping concentration nD Their values are much smaller than those in the case of uncofined phonons [9] Namely, the value of αxx( ) ω
is about 60 (cm-1) in case of unconfined phonons [9] but is 0.15 (cm-1) in the case of confined phonons at T=120K; the value of αxx( ) ω is about 63 (cm-1) in case of unconfined phonons [9] but is 2.3 (cm-1) in the case of confined phonons at nD=0.1.1016.In figure (3), the graph shows that in the case of confined phonons did not appear a resonant peak of the absorption coefficient of EMW but in the case of unconfined phonons did [9]
Trang 4Fig.3 Dependence of the absorption coe fficient of
EMW (cm −1 ) on the Temperature T with the case of
confined phonons, n=n’=11
Fig.4 Dependence of the absorption coe fficient of
EMW (cm −1 ) on the doping concentration n D with
case of confined phonon, n=n’=11
CONCLUSIONS
In this paper, we have presented out the analytical
formulae for the transverse components of the
absorption coefficient of a weak EMW due to free
carriers in a DSL for the case of confined phonon,
Eqs (14)-(18) by using Kubo-Mori method The
numerical evaluations of these formulae for
compensated n-p doped superlattices show that the
confinement of phonons in the doping superlattices not
only leads to different dependences of the
high-frequency conductivity tensor and the absorption
coefficient on the EMW frequency ω, the temperature
of system T, doping concentration nD, the number of
periods N in comparison with normal semiconductors
[6,7] and quantum wells [8] but also creates many
significant differences in the absorption coefficient of a
EMW from the case of unconfined phonons in a DSL
[9]
All results show that the confined phonon
influences very powerfully on the absorption
coefficient of a weak electromagnetic wave by confined
electrons in doped superlattices It’s found that the confined phonons have made the values of the operator G ( ) ω increase and the values of the absorption coefficient αxx( ) ω of EMW decrease very much In other words, the confined phonons cause the life-span of an electrons is shorter and the absorption coefficient’s values are much smaller
Acknowledgments
This work is completed with financial support from the program of Basic research in Natural Science, 405906
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