The linear dc magnetoconductivity in the$(x,y)$ plane of a parabolic quantum well, with a magnetic field$\vec B = B\vec e_z$ applied, is evaluated for electron - opticalphonon interaction. For nonpolar optical and polar optical phonons,the magneto-conductivity oscillates as a function of the magneticfield with resonances occurring when $P\omega_c=\omega_0$, where$\omega_c$ and $\omega_0$ are cyclotron frequency and optical phononfrequency, respectively, and where $P$ is an integer. The analyticresults are numerically evaluated to show explicitly the dependenceof magneto-conductivity on the magnetic field, the confinementfrequency in $z$ direction, and the temperature of the system

Hoi, Bui Dinh

Lam, Vo Thanh

Phong, Tran Cong

English

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 20, No. 3 (2010), pp. 241-247
MAGNETOPHONON RESONANCE IN QUANTUM WELLS
WITH PARABOLIC POTENTIAL
BUI DINH HOI
National University of Civil Engineering
VO THANH LAM
Sai Gon University
TRAN CONG PHONG
Hue University of Education
Abstract. The linear dc magnetoconductivity in the (x, y) plane of a parabolic quantum well,
with a magnetic field ~B = B~ez applied, is evaluated for electron - optical phonon interaction.
For nonpolar optical and polar optical phonons, the magneto-conductivity oscillates as a function
of the magnetic field with resonances occurring when Pωc = ω0, where ωc and ω0 are cyclotron
frequency and optical phonon frequency, respectively, and where P is an integer. The analytic
results are numerically evaluated to show explicitly the dependence of magneto-conductivity on the
magnetic field, the confinement frequency in z direction, and the temperature of the system.
I. INTRODUCTION
Over the past few decades, many studies of magnetophonon resonance (MPR) effects
in bulk semiconductor systems [1–5], quasi-two-dimensional (Q2D) quantum well systems
[6–12], and quasi-one-dimensional (Q1D) quantum wire systems [13–15] have been reported
in considerable details from both experimental and theoretical points of view. The MPR
effects arise from an electron resonant scattering due to absorption and emission of phonons
when energy separation between two Landau levels equals the longitudinal-optical (LO)
phonon energy. Warmenbol et al. [9] studied the hot electrons in two-dimensional system
(formed in a single heterojunction) theoretically on the basis of the momentum-balance
equation. Mori et al. [10] also studied the same system using Kubo formula and the Fang-
Howard trial function. For Q2D quantum well (superlattice) structures, Vasilopoulos et
al. [6, 7] studied the hot-electron MPR and the magneto-impurity resonance for various
scattering mechanisms, including optical phonon and impurity scattering, utilizing the
method adopted by Barker [11]. For optical phonons in the quantum well with square
confining potential, Vasilopoulos [6] showed that the MPR occurring when ωL = Pωc,
where P is an integer and ωL and ωc are the optical phonon frequency and cyclotron
frequency, respectively.
The MPR effects are powerful spectroscopic tools for investigating the transport
properties of semiconductor , such as the carrier relaxation mechanism, the damping of
oscillation due to electron-phonon interaction, the measurement of the effective mass, and
the determination of the energy difference between two adjacent electric subbands. So the
242 BUI DINH HOI, VO THANH LAM, AND TRAN CONG PHONG
purpose of present work is to apply the magnetoconductivity formula in [17] to evaluate the
conductivity tensor component σxx in a quantum well with parabolic confining potential
in the presence of a magnetic field in z direction (the confining direction) ~B = B~ez . We
consider only the scattering of electrons by longitudinal-optical (LO) phonons (nonpolar
and polar optical phonons) in the deformation potential model. The results show an
explicit oscillatory behavior of the conductivity as a function of the magnetic field. For
optical phonons the resonance condition is ωL = Pωc. Since most of the results involve the
Landau level widths, we evaluate the relevant inverse scattering rates as well. These rates
then are used to extract the level widths. The dependence of the results on the magnetic
field, the confinement frequency, and the temperature of sample is shown explicitly. The
present paper is organized as follows: In Sec. II we present briefly the basic formulas for
the calculation. In the next section, we briefly describe the simple model of a quantum
well and evaluate analytically the dc magnetoconductivity for two scattering mechanism.
Numerical results and discussion are given in Sec. IV. Finally, remarks and conclusions
are shown briefly in Sec. V.
II. BASIC FORMULAS
We consider a many body system with a Hamiltonian given by
H = H0 + λV −AF (t) (1)
H0 is the largest part of H which can be diagonalized, λV is a binary type interaction,
assumed nondiagonal, and −AF (t) is the external field Hamiltonian with A being an
operator and F (t) a generalized force.
In [16] the Hamiltonian given by Eq. (1) was inserted into von Neumann’s equation
for the density operator ρ which was split into a diagonal (ρd) and a nondiagonal (ρnd) part.
Then by means of projection operators and for linear response, two inhomogeneous master
equations (diagonal and nondiagonal) [16] as well as a diagonal [16] and a nondiagonal
quantum Boltzmann’s equation were obtained. They are valid in van Hove limit, λ →
0, 1/τ → ∞, λ2t =finite, where τ is the time for a transition between two eigenstates of
H0 to take place, τ ' ~/δε (this equivalent to the first Born approximation).
For Landau states, the diagonal dc conductivity tensor is given by [6, 7, 17]
σdµν(0) =
βe2
Ω
∑
ξ,ξ
′
< nξ >eq (1− < nξ′ >eq)wξξ′(Rνξ −Rνξ′ )Rµξ′ , (2)
where Ω is the normalization volume of sample; e the charge of the carriers;
∑
i(~ri− <
~ri >eq) =
∑
i ~αi; ~ri is the position of carrier i, and < ~ri >eq is its equilibrium position
prior to the switching on of the field; Rµξ ≡ αµξ =< ξ|µ|ξ >; µ, ν = x, y, z; β = 1/(kBT )
with kB being Boltzmann’s constant, and T the temperature; < nξ >eq and < nξ′ >eq are
the average occupancies of the eigenstates | ξ〉 and | ξ′〉, respectively, given by Fermi-Dirac
distribution functions; wξξ′ is the binary transition rate (transition probability) between
two corresponding eigenstates | ξ〉 and | ξ′〉, given by ”golden rule”.
Formula (2), first derived by Argyres and Roth [18], for µ = ν = x, is valid for both
elastic (< nξ >eq=< nξ′ >eq) and inelastic (< nξ >eq 6=< nξ′ >eq) scattering.
MAGNETOPHONON RESONANCE IN QUANTUM WELLS ... 243
The above formula is fairly general and not tied to a ~k space description. In the case
of electron-phonon interaction (we assume that the phonons remain at equilibrium), H0,
in Eq. (1), contains an electron part and a phonon part; λV is the usual electron-phonon
interaction. In this case the transition probabilities wξξ′ are given by [6, 7, 17]
wξξ′ =
∑
~q
[Q(ξ, ~q→ ξ′) < N~q >eq +Q(ξ → ξ
′
, ~q)(1+ < N~q >eq)], (3)
where
Q(ξ, ~q → ξ′) = 2pi
~
|C(~q)|2| < ξ′ |ei~q~r|ξ > |2δ(εξ − εξ′ + ~ω~q), (4)
Q(ξ → ξ′ , ~q) = 2pi
~
|C(~q)|2| < ξ′ |e−i~q~r|ξ > |2δ(εξ − εξ′ − ~ω~q). (5)
The first and second term of Eq. (3) stand for the absorption and emission of
a phonon of wave vector ~q with energy ~ω~q, respectively. The notation (ξ, ~q → ξ′) in
the first term means that an electron at state |ξ〉 absorbs a phonon of wave vector ~q to
change its state to |ξ′〉, and in the second term, the notation (ξ → ξ′ , ~q) expresses the
transition of an electron from state |ξ〉 to state |ξ′〉 by emitting a phonon of wave vector
~q . < N~q >eq is the average number of phonons, given by Bose-Einstein distribution
function. We consider only LO-phonons as treated by the deformation potential model,
also we assume that optical phonons are not dispersion, i. e., ~ω~q ' const, this means
that < N~q >eq does not depend on wave vector ~q of phonons, so we can set < N~q >eq= N0
(depends only on the temperature of system) for short writing.
III. APPLICATIONS FOR MODEL OF A PARABOLIC
QUANTUM WELL
We consider the transport of an electron gas in a quantum well structure, in which
a one-dimensional electron gas is confined in a heterostructure by a parabolic potential
U(z) along the z direction, and electron motions are free along two directions (the (x, y)
plane). A static magnetic field ~B is applied in the z direction. Then, the one-electron
Hamiltonian (H0 =
∑
h0), its normalized eigenfunctions (|ξ〉), and the eigenvalues (εξ) in
the Landau gauge for the vector potential ~A = (0, Bx, 0) are, respectively, given by [8]
h0 = (~p+ e ~A)2/2m? + U(z), (6)
|ξ〉 ≡ |N, n, ky〉 = 1√
Ly
exp(ikyy)φN(x− x0)ψn(z), (7)
εξ ≡ εN,n,ky = (N +
1
2
)~ωc + εn; N = 0, 1, 2, . . . , (8)
where m? is the effective mass of a conduction electron with electron charge −e (e > 0),
~p is the momentum operator of a conduction electron, N is the Landau level index and
n denotes level quantization in z direction, and ωc = eB/m
? is the cyclotron frequency.
Also, φN represents harmonic oscillator wave functions, centered at x0 = −~ky/m?ωc.
Here ky is the wave vector in the y direction and Ly is the y-directional normalization
length. The radius of the orbit in the (x, y) plane is lB = (~/m
?ωc)
1/2. For a parabolic
244 BUI DINH HOI, VO THANH LAM, AND TRAN CONG PHONG
well given by U(z) = m?ωzz
2/2 with the characteristic frequency ωz of the confinement
potential, the one-electron normalized eigenfunctions and the corresponding eigenvalues
in the conduction band are, respectively, given by [8]
ψn(z) =
( 1
2nn!
√
pilz
)1/2
exp
(
− z
2
2l2z
)
Hn
( z
lz
)
, (9)
εn =
(
n+
1
2
)
~ωz, n = 0, 1, 2, . . . (10)
where Hn(x) is the n
th Hermite polynomial and lz is
√
~/m?ωz. For the calculation of
this paper, we need the following matrix elements [6, 7, 17]
〈ξ|x|ξ′〉 = x0δN,N ′δk,k′ + (l′/ωc)[(N + 1)1/2δN ′,N+1 −N 1/2δN ′,N−1]δkk′ (11)
|〈ξ|e±i~q~r|ξ′〉|2 = |In,n′(qz)|2|JN,N ′(u)|2δky ,k′y±qy , (12)
|JN,N ′(u)|2 = (N ′!/N !)e−uuN
′
−N [LN
′
−N
N ]
2, (13)
In,n′(±qz) =
1√
2nn!2n
′
n′!
√
pilz
∫
∞
−∞
e±iqzzHn
( z
lz
)
Hn′
( z
lz
)
dz, (14)
where l
′
= (l/
√
2)ωc, δkk′ = δkyk′y
δnn′ , u = l
2(q2x+ q
2
y)/2 = l
2q2
⊥
/2, and where LMN (u) is an
associated Laguerre polynomial.
For the σdxx components of magnetoconductivity, from (2) we have
σxx ≡ σdxx(0) =
βe2
Ω
∑
ξ,ξ
′
< nξ >eq (1− < nξ′ >eq)wξξ′(Xξ −Xξ′ )2, (15)
where we have denoted Xξ ≡ Rxξ = 〈ξ|x|ξ〉. Inserting (3) and Eqs. (11-14) into (15), we
derive the following form for magnetoconductivity
σxx =
βe2A0
2pi3~l2B
∑
N,N ′,n,n′
fN,n(1− fN ′,n′)I(n, n′)
∫ +∞
0
|C(q)|2|JNN ′(u)|2udu
×
[
N0δ
(− P~ωc + (n− n′)~ωz + ~ω~q)+ (1 +N0)δ(P~ωc + (n− n′)~ωz − ~ω~q)
]
, (16)
where we have set I(n, n′) ≡ ∫ +∞
−∞
|In,n′(±qz)|2dqz, 〈nξ〉eq ≡ fN,n, and A0 is the surface
area. To derive (16), we have transformed the sum over ~q into integral form in the
usual way using cylindrical coordinates, and set N
′ −N = −P in the emission term and
N
′ −N = P in the absorption term. Expression (16) for magnetoconductivity is general
and can be applied for various electron-phonon scattering mechanism. We now assume
the case where only intraelectric subband transitions (n → n′ = n) are allowed to take
place and consider two kinds of phonons.
MAGNETOPHONON RESONANCE IN QUANTUM WELLS ... 245
III.1. Nonpolar Optical Phonon Scattering
For nonpolar-LO-phonon scattering, we take ~ω~q ' ~ωL ' const (ωL is the LO-
phonon frequency), and
∣∣C(q)∣∣2 = ~D2/2ΩρωL = D′/Ω, where ρ is the density of material
and D
′
is the constant of the nonpolar interaction. Also, for nondegenerate semiconduc-
tors, the distribution function of electrons can be replaced by the Boltzmann distribution,
i. e., fN,n = exp [β(εF − εN,n,ky)], εF is the Fermi level. From (16), after performing
the integral over u and the simple calculation, we obtain the following expression for
magnetoconductivity
σxx =
βe2D′
2pi3l2B~
2ωclz
∑
P,N,n
I(n, n) exp
[− β(εN,n,ky − εF )]
×
[
(2N + 1)(2N0+ 1)− P
]
δ
(
P − ωL
ωc
). (17)
III.2. Polar Optical Phonon Scattering
The only difference from above section is that
∣∣C(q)∣∣2 = A/Ωq2 = A/Ω(q2
⊥
+ q2z ) ≈
Al2B/2Ωu, where A is a constant of the polar interaction. We assumed that q⊥  qz for
transport in the (x, y) plane. This approximation allows us to do separately the integrals
over qz and u in (16). Proceeding with (16) as in the case of nonpolar optical phonon
scattering, we have
σxx =
βe2A
4pi3~2ωclz
∑
P,N,n
I(n, n) exp
[− β(εN,n,ky − εF )](2N0 + 1)δ(P − ωLωc
)
. (18)
Equations (17) and (18) show δ-function singularities of conductivity at resonance,
Pωc = ωL. The sum over P can be performed with the help of Poisson’s summation
formula as follow [6, 7, 16]
∞∑
P=0
δ(P − ωL/ωc) ≈ 1 + 2
∞∑
s=1
e−2pis(ΓN /~ωc) cos[2pis(ωL/ωc)], (19)
where ΓN is the damping factor associated with the energy relaxation time τ by ΓN ≈ ~/τ ,
1/τ =
∑
ξ
′ wξξ′ =
∑
N
′
,n
′
,k
′
y
wNnkyN ′n′k′y
. The equation for ΓN has a complicated form
and will be solved numerically with the help of computer program. The numerical results
are shown in the next section.
IV. NUMERICAL RESULTS AND DISCUSSION
In order to clarify the mechanism of magnetophonon resonance effect in parabolic
quantum wells, in this section, we numerically evaluate, plot the component σxx of magne-
toconductivity tensors for a specific parabolic quantum well. The characteristic parameters
of sample are: εF = 50 meV, ~ωL = 36.25 meV, ρ = 5320 kgm
−3, m = 0.067m0 (m0 is
mass of free electron).
Figures 1 and 2 show the dependence of magnetoconductivity σxx on magnetic field
B in cases of nonpolar optical phonon scattering and polar optical phonon scattering,
respectively, at T = 300 K. For both cases, we can see that each curve has two maximum
246 BUI DINH HOI, VO THANH LAM, AND TRAN CONG PHONG
4 6 8 10 12 14 16 18
0
1
2
3
4
5
6
7
Magnetic field (T)
σ
xx
 
(ar
b. 
un
its
)
P=1
P=2
6 8 10 12 14 16 18
0
20
40
60
80
100
Magnetic field (T)
σ
xx
 
(ar
b. 
un
its
)
P=1
P = 2
Fig. 1. The dependence of σxx on B at three
different values of ωz (nonpolar optical phonon
scattering): ωz = 0.1 × ωL (solid line), ωz =
1.0 × ωL (dashed line), and ωz = 2.0 × ωL
(dotted line).
Fig. 2. The dependence of σxx on B at three
different values of ωz (polar optical phonon
scattering): ωz = 0.3 × ωL (solid line), ωz =
0.5 × ωL (dashed line), and ωz = 0.7 × ωL
(dotted line).
peaks. These maxima appear at the values of magnetic B satisfying the resonant condition
Pωc = ωL. If we take ~ωL = 36.25 meV, from the relation ωc = eB/m
?, it is very easy to
conclude that only P = 1 and P = 2 satisfy this condition. The fairly difference here is
for the case of nonpolar optical phonon, the value of σxx at P = 1 is smaller than it is at
P = 2. This is contrast with the case of polar optical phonon. In addition, the values of
P are remained at different values of ωz. This is, of course, because of the absence of ωz
in the resonant conditions.
2 4 6 8 10 12
x 1013
0
2
4
6
8
10
12
14
Confinement frequency (s−1)
σ
xx
 
(ar
b. 
un
its
)
2 4 6 8 10
x 1013
0
2
4
6
8
Confinement frequency (s−1)
σ
xx
 
(ar
b. 
un
its
)
Fig. 3. The dependence of σxx on ωz at three
different values of T (nonpolar optical phonon
scattering): T = 100 K (solid line), T = 150
K (dashed line), and T = 200 K (dotted line).
B = 10 T.
Fig. 4. The dependence of σxx on ωz at three
different values of T (polar optical phonon
scattering): T = 100 K (solid line), T = 150
K (dashed line), and T = 200 K (dotted line).
B = 10 T.
In figures 3 and 4, we plot the magnetoconductivity versus the confinement fre-
quency associated with nonpolar optical phonon and polar optical phonon scattering at
MAGNETOPHONON RESONANCE IN QUANTUM WELLS ... 247
three different values of temperature. It can be seen that every curve has a maximum
value. When the temperature T increases, this maxima shifts to the region corresponding
to the larger values of ωz. Also, the larger the value of temperature is, the larger the value
of magnetoconductivity is. For both cases, σxx decreases when the confinement frequency
increases. In case of polar optical phonon (Fig. 4), however, it diminishes more rapidly
than it does in case of nonpolar optical phonon (Fig. 3).
V. CONCLUSION
In this paper, we have studied the MPR effect in a parabolic quantum well char-
acterized by the confinement potential, assumed to be in z direction, in the presence of
a magnetic field ~B = B~ez . By using projection operators method, we have calculated
the component σxx of magnetoconductivity tensor, as well as the damping factor ΓN , in
the case of electron-optical phonon scattering with two types of optical phonon: nonpolar
optical phonon and polar optical phonon. For both nonpolar- and polar-optical phonon
scattering show the same resonant behaviors at Pωc = ωL (P is an integer). If we take
~ωL = 36.25 meV, the possible values for P , that satisfy the resonant conditions, are 1
and 2. Furthermore, the dependence of σxx on the confinement frequency is shown clearly
for both cases of scattering mechanisms.
Finally, we notice that, with regard to the results for σxx, the usual divergences
of the Born approximation have been avoided by replacing, as usual, the δ function by
Lorentzians (or Gaussians). The relevant damping factors have been estimated from the
inverse scattering rates but they have complicated forms and have been numerically eval-
uated in this work, with the help of computer, whereas they have been previously treated
only at the resonances as in references [6]- [12].
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Received 30 September 2009.


Magnetophonon Resonance in Quantum Wells with Parabolic Potential

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Magnetophonon Resonance in Quantum Wells with Parabolic Potential

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