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Dependence of the Hall Coefficient

on Doping Concentration in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation

Nguyen Quang Baua & Bui Dinh Hoiab a

Department of Physics, College of Natural Science, Vietnam National University, Hanoi, Viet Nam

b Department of Physics, National University of Civil Engineering, Hanoi, Viet Nam

Published online: 23 May 2014

To cite this article: Nguyen Quang Bau & Bui Dinh Hoi (2014) Dependence of the Hall Coefficient

on Doping Concentration in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation, Integrated Ferroelectrics: An International Journal, 155:1, 39-44, DOI: 10.1080/10584587.2014.905109

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ISSN: 1058-4587 print / 1607-8489 online

DOI: 10.1080/10584587.2014.905109

Dependence of the Hall Coefficient on Doping Concentration in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation

1Department of Physics, College of Natural Science, Vietnam National University, Hanoi, Viet Nam

2Department of Physics, National University of Civil Engineering, Hanoi, Viet Nam

The dependence of the Hall coefficient on doping concentration in doped semiconductor superlattices (DSSLs) under a crossed dc electric field and magnetic field in the presence

of a laser radiation, is investigated by using a quantum kinetic equation for electrons Analytical results for the resistance and the Hall coefficient (HC) are computationally evaluated and graphically plotted for the GaAs:Si/GaAs:Be DSSL Numerical results for the magnetoresistance are in accordance with available theories The dependence

of the HC on the doping concentration shows an oscillation whose phase is strongly affected by the laser radiation.

Keywords Hall coefficient, SdH oscillation, doped superlattice, quantum kinetic equation

1 Introduction

The propagation of an electromagnetic wave (EMW) in materials leads to changes in probability of scattering of carriers, and thus, leads to their unusual properties in comparison

to the case of absence of the EMW There have been many papers dealing with problems related to the incidence of EMWs in semiconductor systems such as the calculations of the linear and nonlinear absorption coefficients in low-dimensional semiconductor systems [1–5], the studies of the Hall effect in bulk semiconductors in the presence of an EMW

by using quantum kinetic equation [6–10] In a recent work, we have used the quantum kinetic equation method to study the influence of an intense EMW on the Hall coefficient

in parabolic quantum wells with an in-plane magnetic field [11] In this work, by using this method we study the Hall effect in doped semiconductor superlattices (DSSLs), subjected

to a crossed dc electric field and magnetic field (the magnetic field is applied along the DSSL axis), in the presence of a laser radiation (intense EMW) We only consider the case in which the electron-acoustic phonon interaction is assumed to be dominant and electron gas is degenerate at low temperatures We derive analytical expressions for the conductivity tensor and the Hall coefficient (HC) taking account of arbitrary transitions

Received July 23, 2013; in final form January 12, 2014

∗Corresponding author E-mail: nguyenquangbau54@gmail.com

39

40 N Q Bau and B D Hoi

between Landau levels and between subbands The paper is organized as follows In the next section, we briefly describe a regime of the problem and present basic formulae of the calculation Numerical results and discussion are also given Remarks and conclusions are shown briefly in Sec 3

2 Hall Effect in a DSSL under the Influence of a Laser Radiation

We consider a simple model of a DSSL (n-i-p-i superlattice), in which electron gas is

confined by an additional potential along the z direction and free in the (x-y) plane The

motion of an electron is confined in each layer of the system and its energy spectrum is

quantized into discrete levels in the z direction If the DSSL is subjected to a crossed electric

field E1= (E1, 0, 0) and magnetic field  B = (0, 0, B), the single-particle wave function

and its eigenenergy are given by [12, 13]

 (r) =

 1

L y φ N (x − x0) eik y y φ n (z) , (1)

ε N,n



k y



=



N +1

2



ωc+



n + 1

2



ωp− vdk y+1

2mv2d;N, n = 0, 1, 2 , (2)

where m and vd= E1/B are the effective mass and the drift velocity of a conduction

electron, respectively, k y being its wave vector in the y direction, ωp and ωc= eB/m are the plasma and the cyclotron frequencies, respectively, ωp=e2nD/κ0m1/2

with κ0is the

electronic constant and nDis the doping concentration, x0= −k y /(mωc) and L y are the

center of Landau orbits and the normalization length in the y direction, respectively, N denotes the Landau level index and n being the quantization index of energy levels in the z direction due to the DSSL potential, φ N (x) and φ n (z) are the harmonic wave functions.

By using above wave function and energy spectrum, we can write out the Hamiltonian

of electrons and phonons system and obtain the quantum kinetic equation for electrons

in the presence of a laser radiation with electric field vector E = (0, E0sin (t) , 0) (E0

and  are the amplitude and the frequency of the EMW, respectively), utilizing the same

procedures as in Ref 11 Then by considering the electron - acoustic phonon interaction,

we obtain the expression for the conductivity tensor after some manipulation:

1+ ω2

c τ2

δ ij − ω c τ ε ij k h k + ω2

c τ2h i h j

aδ j m+be

m

τ



1+ ω2

c τ2δ j

δ m − ω c τ ε mp h p + ω2

c τ2h  h m

where δ ij is the Kronecker delta, ε ij kbeing the antisymmetric Levi - Civita tensor, the Latin

symbols i, j, k, l, m, p stand for the components x, y, z of the Cartesian coordinates,

a = eL y

2πmα



ε N,n − εF



εFis the Fermi level, and

b = 4πe m N,n,n

{b1+ b2+ b3+ b4}, (5)

Dependence of the Hall Coefficient on Doping 41

b1 = −γ



eB



 

1+ 2 ∞

s=1

(−1)se−2πs/(ωc )

cos (2πs ¯n1)



,

b2 = γ θ 2



eB



3

1+ 2

∞

s=1

(−1)se−2πs/(ωc )

cos (2πs ¯n1)



,

¯n1 = n − n

ωp+ eE1 / (ωc) ,

b3 = −γ θ

4



eB



3

1+ 2 ∞

s=1

(−1)se−2πs/(ωc )

cos (2πs ¯n2)



,

¯n2 = n − n

ωp+ eE1 −  / (ωc) ,

b4 = −γ θ

4



eB



3

1+ 2

∞

s=1

(−1)se−2πs/(ωc )

cos (2πs ¯n3)



,

¯n3 = n − n

ωp+ eE1 +  / (ωc) ,

θ = e2E02/

m24

, γ = AL y kBT



ε N,n − εF



I

n, n

(2π)3vsωc4v2d2B

, =

N + 1/2 +

N + 1 + 1/2



B/2,  = /τ

(τ is the relaxation time), B=√/(mωc), ε N,n=N +12

ωc+n + 12

ωp+

1

2mv2d, I (n, n)=+π/d

−π/d







s0



j =1 φ n (z − j d)| e ±iq z z |φ n(z − j d)







2

dq z , d and s0are the period

and the number of periods, respectively, T is the temperature, kB being the Boltzmann

constant,A = ξ2/ (2ρvs) where vs, ξ and ρ are the sound velocity, the deformation potential

constant and the mass density, respectively

The magnetoresistance ρ xx and the Hall coefficient RHare determined by [14]:

ρ xx= σ xx

σ2

xx + σ2

yx

, RH= −1

B

σ yx

σ2

xx + σ2

yx

(6)

where σ yx and σ xxare given by Eq (3) In the following, we will give a deeper insight into these results by carrying out a numerical evaluation with the help of a computer program For the numerical evaluation, we consider the n-i-p-i DSSL of GaAs:Si/GaAs:Be with the

following parameters [4, 5]: ξ = 13.5 eV, ρ = 5.32 gcm−3, vs= 5378 ms−1, εF= 50 meV,

m = 0.067m0(m0is the mass of a free electron), τ = 10−12s, L y = 100 nm, d = 2 nm.

We also consider transitions N = 0, N= 1, n = 0, n= 1

Figure 1(a) shows the dependences of the magnetoresistance ρ xxon the magnetic field

at different values of the temperature We can see the appearance of the typical

Shubnikov-de Haas oscillations with the period does not Shubnikov-depend on the temperature Our results are similar to those (for the type of oscillations) obtained experimently in a two-dimensional electron system [15] Moreover, the figure also shows that the amplitude of these oscillations

at a fixed magnetic field decreases when the temperature increases Denoting A (B n T ) and

A (B n T0), respectively, are amplitudes of the oscillation peaks observed at a magnetic field

42 N Q Bau and B D Hoi

Figure 1 (a) The magnetoresistance as functions of the magnetic field at different values of the

temperature in the absence of the EMW (b) The relative amplitude versus temperature Here, E1=

5× 102V/m, and nD= 1023m−3

B n and at temperatures T and T0 The relative amplitudes versus temperature have been shown to be [15]

A (B n T )

A (B n T0) =T sinh



2π2k B mT0/eB n



T0sinh

2π2k B mT /eB n

This relation is also plotted in Fig 1(b) for T0= 2 K, B n= 3 T, and it is seen that there is a good agreement between our calculation and Eq (7)

In Fig 2(a), the magnetoresistance is plotted as a function of the magnetic field for two cases: the absence and the presence of the EMW There occurs the beat phenomenon in the case of the presence of the EMW This property has been observed in some two-dimesional electron systems (see Ref 16 and references therein)

Figure 2(b) shows the dependences of the HC on the doping concentration for two cases: the absence and the presence of the EMW The HC can be seen to oscillate and decrease with increasing the doping concentration It is also seen that the presence of the

Figure 2 The dependences of the magnetoresistance on the magnetic field (a), and the dependences

of the HC on the doping concentration (b) for two cases: the absence and the presence of an EMW

Here, B = 3T, E1= 5 × 102V/m, and T= 4 K

Dependence of the Hall Coefficient on Doping 43

EMW does not change the HC value considerably but causes the change in the phase of the oscillated HC

3 Conclusion

So far, we have studied the Hall effect in DSSLs subjected to crossed dc electric and magnetic fields under the influence of a laser radiation (intense EMW) The electron-acoustic phonon interaction is taken into account We obtain the expressions for the magne-toresistance as well as the HC The analytical results are numerically evaluated and plotted

for the GaAs:Si/GaAs:Be DSSL The dependence of the magnetoresistance ρ xx on the magnetic field shows Shubnikov–de Haas oscillations with periods do not depend on the temperature and amplitudes decrease with increasing the temperature The HC oscillates and decreases with increasing the doping concentration The presence of the EMW does not affect the HC value considerably but causes the change in the phase of oscillations

Funding

This research is funded by the National Foundation for Science and Technology Devel-opment of Vietnam (NAFOSTED) (Grant No.: 103.01-2011.18) and Vietnam National University (Grant No.: QGTD.12.01)

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Figure The dependences of the magnetoresistance on the magnetic field (a) , and the dependences