This article was downloaded by: [University of Tennessee At Martin]On: 06 October 2014, At: 06:19 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Numbe
Trang 1This article was downloaded by: [University of Tennessee At Martin]
On: 06 October 2014, At: 06:19
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Integrated Ferroelectrics: An International Journal
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/ginf20
The Dependence of the Magnetoresistance in Quantum Wells with Parabolic Potential on Some Quantities under the Influence of Electromagnetic Wave
Nguyen Dinh Nama, Do Tuan Longa & Nguyen Vu Nhanb a
Department of Physics, College of Natural Science, Viet Nam National University, Ha Noi, Viet Nam
b Department of Physics, Academy of Defence force-Air force, Ha Noi, Viet Nam
Published online: 23 May 2014
To cite this article: Nguyen Dinh Nam, Do Tuan Long & Nguyen Vu Nhan (2014) The Dependence of the
Magnetoresistance in Quantum Wells with Parabolic Potential on Some Quantities under the Influence
of Electromagnetic Wave, Integrated Ferroelectrics: An International Journal, 155:1, 45-51, DOI: 10.1080/10584587.2014.905122
To link to this article: http://dx.doi.org/10.1080/10584587.2014.905122
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the
“Content”) contained in the publications on our platform However, Taylor & Francis,
our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors,
and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources
of information Taylor and Francis shall not be liable for any losses, actions, claims,
proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content
This article may be used for research, teaching, and private study purposes Any
substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,
systematic supply, or distribution in any form to anyone is expressly forbidden Terms &
Trang 2Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Trang 3Copyright © Taylor & Francis Group, LLC
ISSN: 1058-4587 print / 1607-8489 online
DOI: 10.1080/10584587.2014.905122
The Dependence of the Magnetoresistance
in Quantum Wells with Parabolic Potential
on Some Quantities under the Influence of
Electromagnetic Wave
NGUYEN DINH NAM,1,*DO TUAN LONG,1
AND NGUYEN VU NHAN2
1Department of Physics, College of Natural Science, Viet Nam National University, Ha Noi, Viet Nam
2Department of Physics, Academy of Defence force-Air force, Ha Noi, Viet Nam
The magnetoresistance is one of the important properties of semiconductors Start-ing from the hamiltonian for the electron-acoustic phonon system, we obtained the expression for the electron distribution function and especially the expression for the magnetoresistance in quantum wells with parabolic potential (QWPP) under the influ-ence of electromagnetic wave (EMW) in the presinflu-ence of magnetic field We estimated numerical values and graphed for a GaAs/GaAsAl quantum well to see the nonlinear dependence of the magnetoresistance on the temperature of the system T, the ampli-tude E0 and the frequency of the electromagnetic waves, the magnetic field B, the parameters of the quantum well and the momentum relaxation time τ clearly.
Keywords Dependence of magnetoresistance
I Introduction
In the past few years, there have been many scientific works related to the properties of the low-dimensional systems such as the optical, magnetic and electrical properties [1–9] These results show us that there are some differences between the low-semiconductor and the bulk semiconductor that the previous work studied
The magnetoresistance is also interested However, it has not been resolved in the quantum wells with parabolic potential under the influence of electromagnetic wave The calculation of the magnetoresistance in the QWPP in the presence of magnetic field under the influence of EMW is done by using the quantum kinetic equation method that brings the high accuracy and the high efficiency [1–4] Comparing the results in this case with in the case of the bulk semiconductors, we also see some differences
II The Magnetoresistance in Quantum Wells with Parabolic Potential under the Influence of Electromagnetic Wave in the Presence of Magnetic Field
It is well known that in the quantum wells, the motion of electrons is restricted in one dimension, so that they can flow freely in two dimensions [1, 2, 7] We consider a quantum
Received July 23, 2013; in final form January 12, 2014
∗Corresponding author E-mail: dinhnamt2@yahoo.com
45
Trang 446 N D Nam et al.
well with parabolic potential of GaAs embedded in AlAs It subjected to a crossed electric field −→
E1= (0, 0, E1)and magnetic field −→
B = (0, B, 0) In the presence of an EMW with
electric field vector −→
E =−→
E0sin t (where E0and are the amplitude and the frequency
of the EMW, respectively), the Hamiltonian of the electron-acoustic phonon system in the above-mentioned QWPP in the second quantization presentation can be written as:
N,−→
k x
ε N−→
k x − e
c−→A (t)
a+
N,−→
k x
a
N,−→
k x +
−→q ω−→q b−→q +b−→q +
N,N,−→
k x ,−→q C N,N
(−→q )a+
N,−→
k x +−→q
x
+ a
N,−→
k x
b−→q + b+−−→q
+
−
→q φ(−
→q )a+
N,−→
k x +−→q x a N,−→
where N is the Landau level index (N= 0,1 ,2 ), a+
N,−→
k x
and a
N,−→
k x
, (b−→+q and b−→q ) are the creation and the annihilation operators of the electron (phonon), |N, −→k x > and
|N,−→
k x + −→q x > are electron states before and after scattering, ω−→q is the energy of an
acoustic phonon; C N,N(−→q ) = C−→
q I N,N(q z ), where C−→q is the electron-phonon
interac-tion constant and I N,N(q z ) is the electron form factor [4], φ−→q
is the scalar potential
of a crossed electric field −→
E1, −→
A (t) is the vector potential of an external electromagnetic
wave −→
A (t) = e−→
E0sin(t)/ If the confinement potential is assumed to take the form
V (z) = mω2(z − z0)2/2, then the single-particle wave function and its energy are given by
[1, 2]:
ψ(−→r ) = 1
2π e
i−→
k⊥−→
ε N (k x)= ω p
N +1
2
+ 1
2m∗
2
k2x−
k x ω c + eE1
ω p
2
Here, m∗and e are the effective mass and charge of conduction electron, respectively,
k⊥= (k x , k y ) is its wave vector in the (x,y) plan, z0= (k x ω c + eE1) /mω2
p , ω2
p = ω2
0+ω2
c,
ω0and ω care the confinement and the cyclotron frequencies, respectively, and
ψ m (z − z0)= H m (z − z0) exp
−(z − z0)2/2
with H m (z)being the Hermite polynomial of m thorder
From the quantum kinetic equation for electron in single scattering time approximation and the electron distribution function, using the Hamiltonian in the Eq 1, we find:
∂f
N,−→
k x
e−→E
1
+ ω c −→
k x ,−→
−
→
k x
∂−→
k x
= 2π
N,−→q
C N,N (−→q )2
2N−→q +1
+∞
l=∞
J l2(αq x)×
f
N,−→
k x +−→q
x − f
N,−→
k x δ
ε N(k x + q x)− ε N (k x)− l. (5)
Trang 5Dependence of Magnetoresistance in QWPP 47
where −→
k x = (k x , 0, 0),−→
h =−→
B /B is the unit vector in the direction of magnetic field, f
N,−→
k x is an unknown distribution function perturbed due to the external fields, J l (x) is the
l th - order Bessel function of argument x and N−→q is the time-independent component of the electron distribution function
For simplicity, we limit the problem to case of l = −1, 0, 1 If we multiply both sides
of the Eq 5 by (e/m∗)−→
k x δ(ε − ε N (k x )), carry out the summation over N and k x and use
J2
0(αq x)≈ 1 − (αq x)2/2, we obtain:
−
→
R (ε)
τ (ε) + ω c −→
h ,−→
R (ε) = −→Q (ε) +−→
where
−
→
R (ε) =
N,−→
k x
e
m∗
−
→
k x f
N,−→
k x δ (ε − ε N (k x )) , (7)
−
→
S (ε) = −4m2πe∗
N,−→q
C N,N(q)2
(2N−→q + 1)(αq x)2
N,−→
k x
f
N,−→
k x +−→q
x − f
N,−→
k x
−
→
k x
×2δ
ε N(k x + q x)− ε N (k x)
− δε N(k x + q x)− ε N (k x)−
−δε N(k x + q x)− ε N (k x)+ , (8)
−
→
Q (ε) = − m e∗
N,−→
k x
−
→
k x
−
→
F ,
∂f
N,−→
k x
∂−→
k x
δ (ε − ε N (k x )), (9)
with
−
→
F = e−→
E1− ∇ε F −ε − ε N (k x)
Finding −→
R (ε) in term of−→
Q (ε),−→
S (ε)and through some computation steps, we obtain
the expression for conductivity tensor:
σ im= e
m∗
τ (ε F) 1+ω2
c τ2(ε F)
c0δ ik +d0d1 τ (ε F)
1+ω2
c τ2(ε F)
δ ik −ω c τ (ε F ) ε ikl h l + ω2
c τ2(ε F ) h i h k
+ d0d2 τ (ε F − )
1+ ω2
c τ2(ε F − )
δ ik − ω c τ (ε F − ) ε ikl h l + ω2
c τ2(ε F − ) h i h k
+ d0d3 τ (ε F + )
1+ ω2
c τ2(ε F + )
δ ik − ω c τ (ε F + ) ε ikl h l + ω2
c τ2(ε F + ) h i h k
×δ km − ω c τ (ε F ) ε kmn h n + ω2
where
c0=
N
eL x π
Trang 6
48 N D Nam et al.
d0=
N,N
eL x
4π2m∗
ξ2k B T
ηυ2
e2E02
4ω4
eE1ω c
ω2 0
d1 =
N,N
4
0 1
0 2
−
0 3
0 3 0 3)+ 2√20 21
0 1
d2= √21 24
4 1
d3= √21 25
5 1
0 =
eE1ω c
ω2 0
2
−
2m∗ω3
p
N +12
− e2E2
1− 2m∗ω2
p ε F
2ω02
≈ 2m∗ω
2
p
2ω20
ε F − ω p
N +1
2
1= 2m∗ω
2
p
2ω2
ε F − ω p
N+1 2
2= 2m∗ω
2
p
2ω2
ε F + − ω p
N+1 2
3 = 2m∗ω
2
p
2ω20
ε F − − ω p
N+1 2
,
4 = 2m∗ω
2
p
2ω2 0
ε F − − ω p
N +1
2
,
5 = 2m∗ω
2
p
2ω2
ε F + − ω p
N +1
2
where L x ξ, η, υ, k B , T , ε Fare the x-directional normalization lengths, the deformation potential constant, the density, the acoustic velocity, the Boltzmann constant, the tempera-ture of system and the Fermi energy, respectively
In this work, we consider the case of electron-acoustic phonon scattering and the presence of electric field −→
E1 Comparing with the case of electron-optical phonon scattering
Trang 7Dependence of Magnetoresistance in QWPP 49
and no electric field −→
E1that was studied previously [10], we see some differences in the expression of conductivity tensor and also in the expression of magnetoresistance The magnetoresistance is given by the formula:
ρ = σ zz (H ) σ zz(0)
σ2
zz (H ) + σ2
Using the Eq 11, we obtain the explicit formula of the magnetoresistance as following:
N,N
e
m∗
τ (ε F)
1+ τ2(ε F)
c0+ d0d1τ (ε F)
1+ ω2
c τ2(ε F)
1− ω2
c τ2(ε F )h2
+ d0d2τ (ε F − )
1+ ω2
c τ2(ε F − )
1− ω2
c τ (ε F )τ (ε F − )h2
+ d0d3τ (ε F + )
1+ ω2
c τ2(ε F + )
×1− ω2
c τ (ε F )τ (ε F + )h2
×
N,N
e2τ (ε F )L x
m∗π2
1+ τ2(ε F)
×
2m∗
ε F − ω0
N +1
2
θ
ε F − ω0
N +1
2
×
⎧
⎨
⎩
⎧
⎨
⎩
N,N
e
m∗
τ (ε F)
1+ τ2(ε F)
c0+ d0d1τ (ε F)
1+ ω2
c τ2(ε F)
1− ω2
c τ2(ε F )h2
+ d0d2τ (ε F − )
1+ ω2
c τ2(ε F − )×
1− ω2
c τ (ε F )τ (ε F − )h2
+ d0d3τ (ε F + )
1+ ω2
c τ2(ε F + )
1− ω2
c τ (ε F )τ (ε F + )h2 2 +
N,N
e
m∗
τ (ε F)
1+ ω2
c τ2(ε F)
c0ω c τ (ε F )h + d0d1τ (ε F )2ω c τ (ε F )h
2
1+ ω2
c τ2(ε F) + d0d2τ (ε F − )
1+ ω2
c τ2(ε F − ) × ω c[τ (ε F)+ τ(ε F − )] h2
+ d0d3τ (ε F + )
1+ ω2
c τ2(ε F + ) ω c[τ (ε F)+ τ(ε F + )] h2
2 −1
− 1 (22)
Eq 22 is the analytical expression of the magnetoresistance in the QWPP It shows the dependence of the magnetoresistance on the external fields, including the EMW In the next section, we will give a deeper insight into this dependence by carrying out a numerical evaluation In Eq 22, we can see that the formula of the magnetoresistance is easy to come
back to the case of bulk semiconductor when ω0reaches to zero [11, 12]
III Numerical Results and Discussion
For the numerical evaluation, we consider the model of a quantum well of GaAs/GaAsAl
with the following parameters: ε F = 50 meV , k B = 1.3807 × 10−23J K−1, υ = 5220 m/s
Trang 850 N D Nam et al.
Figure 1 The dependence of the magnetoresistance on the temperature.
and m∗ = 0.0067m0with m0is the mass of a free electron For the sake of simplicity, we
also choose N = 0, N= 1, τ = 10−12s [1, 2].
Figure 1 shows the magnetoresistance as a function of the temperature The value of the magnetoresistance increases sharply when the temperature is low, after that it decreases
steadily With the different values of the electric field E1, we get the resonant peaks at the different points of temperature
Figure 2 shows us the dependence of the magnetoresistance on the amplitude of the EMW The higher the amplitude of the EMW is, the faster the magnetoresistance grows
up The line of the dependence of the magnetoresistance on the amplitude E0of EMW also
changes when we change the value of the frequency of the EMW We see that there are
some differences in the dependence of the magnetoresistance on the temperature and the amplitude from the case of electron-optical phonon scattering [10] We also get the same
graphs as in the case of bulk semiconductor [11, 12] when the confinement frequency ω0
reaches to zero
Figure 2 The dependence of the magnetoresistance on the amplitude of the EMW.
Trang 9Dependence of Magnetoresistance in QWPP 51
IV Conclusions
In this paper, we obtain the analytical expression of the magnetoresistance in QWPP under the influence of EMW in the presence of magnetic field We see that the magnetoresistance
in this case depends on some quantities such as: the magnetic field B, the temperature T, the
parameters of QWPP, the momentum relaxation time τ , the amplitude E0and the frequency
of EMW Estimating numerical values and graph for a GaAs/GaAsAl quantum well to
see this dependence clearly Looking at the graph, we see that the magnetoresistance gets the negative values and the dependence of the magnetoresistance on the temperature, the
amplitude and the frequency of the EMW are nonlinear When ω0reaches to zero, we obtain the results as the case of bulk semiconductor that was studied [11, 12]
Funding
This research is completed with financial support from Vietnam NAFOSTED (103.01-2011.18) and TN13-04
References
1 T C Phong and N Q Bau, Parametric resonance or acoustic and optical phonons in a quantum
well J Korean Phys Soc 42, 647–651 (2003).
2 N Q Bau, L T Hung, and N D Nam, The nonlinear apsorption coefficient of strong electromag-netic waves by confined electrons in quantum wells under the influences of confined phonons
JEWA 13, 1751–1761 (2010).
3 N Q Bau, L Dinh, and T C Phong, Absorption Coefficient of Weak Electromagnetic Waves
Caused by Confined Electrons in Quantum Wires J Korean Phys Soc 51, 1325–1330 (2007).
4 N Q Bau and H D Trien, The nonlinear absorption coefficient of strong electromagnetic waves
caused by electrons confined in quantum wires J Korean Phys Soc 56, 120–127 (2010).
5 S G Yua, K W Kim, M A Stroscio, G J Iafrate, and A Ballato, Electron interaction with
confined acoustic phonons in cylindrical quantum wires via deformation potential J.Appl Phys.
80, 2815–2822 (1996).
6 N Nishiguchi, Resonant acoustic-phonon modes in quantum wire Phys Rev B 52, 5279–5288
(1995)
7 P Zhao, Phonon amplification by absorption of an intense laser field in a quantum well of polar
material Phys Rev B 49, 13589–13599 (1994).
8 E M Epstein, Parametric resonance of acoustic and optical phonons in semiconductors Sov
Phys Semicond 10, 1164 (1976).
9 M V Vyazovskii and V A Yakovlev, Parametric resonance of acoustic and optical phonons in
impurity semiconductors in low temperature Sov Phys Semicond 11, 809 (1977).
10 S C Lee, Optically detected magnetophonon resonances in quantum wells J Korean Phys Soc.
51, 1979–1986 (2007).
11 V L Manlevich and E M Epshtein, Photostimulated odd magnetoresistance of semiconductors
Sov Phys Semicond 18, 739–741 (1976).
12 V L Manlevich and E M Epshtein, Photostimulated kinetic effects in semiconductors J Sov
Phys 19, 230–237 (1976).
...Eq 22 is the analytical expression of the magnetoresistance in the QWPP It shows the dependence of the magnetoresistance on the external fields, including the EMW In the next section, we will... change the value of the frequency of the EMW We see that there are
some differences in the dependence of the magnetoresistance on the temperature and the amplitude from the case of electron-optical... the EMW The higher the amplitude of the EMW is, the faster the magnetoresistance grows
up The line of the dependence of the magnetoresistance on the amplitude E0of