Impact of confined lo phonons on the hall effect in doped semiconductor superlattices

Original articlesemiconductor superlattices Faculty of Physics, Hanoi University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f

Original article

semiconductor superlattices

Faculty of Physics, Hanoi University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Received 3 June 2016

Received in revised form

10 June 2016

Accepted 10 June 2016

Available online 18 June 2016

Keywords:

Confined LO-phonons

The Hall effect

The magnetoresistance

Doped superlattices

Semiconductor

a b s t r a c t Based on the quantum kinetic equation method, the Hall effect in doped semiconductor superlattices (DSSL) has been theoretically studied under the influence of confined LO-phonons and the laser radia-tion The analytical expression of the Hall conductivity tensor, the magnetoresistance and the Hall co-efficient of a GaAs:Si/GaAs:Be DSSL is obtained in terms of the external fields, lattice period and doping concentration The quantum numbers N, n, m were varied in order to characterize the effect of electron and LO-phonon confinement Numerical evaluations showed that LO-phonon confinement enhanced the probability of electron scattering, thus increasing the number of resonance peaks in the Hall conductivity tensor and decreasing the magnitude of the magnetoresistance as well as the Hall coefficient when compared to the case of bulk phonons The nearly linear increase of the magnetoresistance with tem-perature was found to be in good agreement with experiment

© 2016 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)

1 Introduction

It is wellknown that the effect of phonon confinement in

low-dimensional semiconductor systems leads to a change in the

probability of carrier scattering, thus creating new behaviours of

materials in comparison to the case of unconfined phonons[1,2]

Consequently, there have been many published works dealing with

the influence of confined phonons on the optical, electrical, and

magnetic properties of low-dimensional semiconductor systems

such as the influence of confined phonons on the absorption

co-efficient of strong electromagnetic waves[3]and carrier capture

processes[4], as well as the resonant quasiconfined optical

pho-nons in semiconductor superlattices[5] In semiconductors

sys-tems, the optical phonons branches do not overlap and it can be

considered to be confined, the wave vector of confined optical

phonon contained the quantized component and the in-plane one

[1,6] The different boundary conditions placed on the electrostatic

potential or vibrational amplitude of the phonons, lead to be

distinct confined phonon models such as the guided mode model,

the slab mode model and the Huang-Zhu model[7] In the previous

work[8], we have studied the Hall effect in doped semiconductor

superlattices (DSSL) with bulk phonons Through the works of

[2,3,6], the contribution of phonon confinement is shown to be

important in the properties of low-dimensional semiconductor systems and should not be neglected Thus, in this work, we continue studying the impact of the confined LO-phonons on the Hall effect in DSSLs subjected to a dc electricfield, a perpendicular magneticfield and varying laser radiation The analytical expres-sions of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient (HC) in DSSLs under the influence of confined LO-phonons are obtained by using the quantum kinetic equation method[3,8] This article is organized as follows: we outline the effects of confined electrons and confined LO-phonons in doped semiconductor superlattices and present the basic formulae for the calculations in Sec.2 Numerical results and discussion for the GaAs:Si/GaAs:Be doped semiconductor superlattices are given in the Sec.3 Finally, Sec.4 shows remarks and conclusions

2 The Hall effect in DSSLs under the impact of confined LO-phonons

Consider a simple model for doped semiconductor superlattices

in which the motion of the electrons is restricted along the z axis due to the DSSL confinement potential and free in the x
y plane The thicknesses and concentrations of the n-doping and p-doping layer of the DSSL are assumed to be equal: da¼ dp¼ d/2 and

na¼ np¼ nD, here d, nDare the period and the doping concentra-tions of the DSSL, respectively A dc electricfield E!1¼ ðE1; 0; 0Þ; a magneticfield B!¼ ð0; 0; BÞ and laser radiation E!0¼ ð0; E sinUt; 0Þ was applied to the DSSL Under the influence of the material

* Corresponding author.

E-mail address: nguyenquangbau54@gmail.com (N.Q Bau).

Peer review under responsibility of Vietnam National University, Hanoi.

Contents lists available atScienceDirect

Journal of Science: Advanced Materials and Devices

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j s a m d

http://dx.doi.org/10.1016/j.jsamd.2016.06.010

2468-2179/© 2016 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license

Journal of Science: Advanced Materials and Devices 1 (2016) 209e213

confinement potential and these external fields, the single-wave

function of an electron and its discrete energy now becomes[8,9]:

L

p



k

!

y



¼



2





2



(2)

where N, n are the Landau level index and the subband index,

respectively;Z is the Planck constant; meis the effective mass of an

electron; ky, Ly being the wave vector of the electron and the

normalization length along the y direction;up¼



4pe 2 n D

ε 0 m e

1=2

is the plasma frequency;ε0is the electric constant;FNis the harmonic

oscillator wave function, here x0¼
[2ðky
meyd=ZÞ with

[B¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Z=meuc

p

is the radius of the Landau orbit in the x
y plane;

fnðzÞ being the electron subband wave functions due to the

ma-terial confinement potential: fnðzÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

2 n n ! pffiffiffip

[ z

q exp
z 2

2[ 2 z

!

Hn



z [ z

 with [z¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Z=meup

p

and Hn(z) is the Hermite polynomial of n-th order;uc¼ eB=meis the cyclotron frequency;yd¼ E1=B being the

drift velocity of the electron

The quantized frequency of confined LO-phonon and its wave

vector are given by[7]:

q

u2

m;!q

⊥

q

⊥þ q2 m



wherenis velocity parameter and m being the quantum number

characterizing the LO-phonon confinement

Also, the matrix element for confined electron e confined

LO-phonon interaction in doped semiconductor superlattices

Dm

N ;n;N 0 ;n 0 ;!q

⊥

¼ C

m;!q

⊥

Im n;n 0JN;N0ðuÞ now becomes[7,10]:



⊥



 1

c∞
1

c0

 1 q

⊥þ q2 m

;

0
Nh

;

k¼1

ffiffiffi

2

d

0



d

d



whereε0is the electric constant; V0is the normalization volume of

specimen; c0 and c∞ are the static and the high frequency

dielectric constants; LM

NðuÞ is the associated Laguerre polynomial,

u¼ [2!q2

⊥=2; Nd being the the number of periods of the DSSL;

h(m)¼ 1 if m is even,h(m)¼ 0 if m is odd; fn(z) and fn 0ðzÞ are the

electron sub-band wave functions in the initial and final states

The effect of LO-phonon confinement and these external fields

change the probability of electron scattering, thus modifying the

Hamiltonian of the confined electron e confined phonon system in

the DSSL This leads the quantum kinetic equation for electron distribution to now become:

1þ Zuc

h k

!

y∧ h!i;

vf N;n;!k y

!

¼

f N;n;!k y

t

N 0 ;n 0 ; m;!q

⊥



⊥



n;n 02JN;N0ðuÞ2 Xþ∞

s¼
∞

!

!



 f

N 0 ;n 0 ;!k

yþ!q y

 N m;!q

⊥

f N;n;!k y N m;!q

⊥

 k

!

yþ q!y

 k

! y



þ

f

N 0 ;n 0 ;!k y
!q y N m;!q

⊥

f N;n;!k y

 N m;!q

⊥

þ 1

 k

!

y
q!y

y



(5)

where h!

¼ B!

B is the unit vector along the magneticfield; the no-tation “∧” represents the vector product; f0 is the equilibrium electron distribution function,tis the momentum relaxation time

of electron, which is assumed to be a constant, Js(x) is the sth-order Bessel function of argument x and d(X) being the Dirac delta function

The electron distribution function is now non-equilibrium and the current density is nonlinear as a result Let us consider that the electron gas is non-degenerate, f0¼ n0expfb½εF
εN;nð k!yÞg; where εF is the Fermi level, b¼ 1/kBT and kB is the Boltzmann constant For simplicity, we limit the problem to the cases of

s¼
1,0,1, meaning the processes with more than one photon are ignored After some manipulation, the expression for the conduc-tivity tensor is obtained:

sip¼ t

ct2



dik
uctεijkhjþu2

ct2hihk

8

>

>

>

>

N;n

b εF
εN;n

dkp
uctεklphlþu2

ct2hkhp



N 0 ;n 0 ; N;n;m

t

ct2

pe2Zu0A

 1

c∞
1

c0



b εF
εN;n

n;n 02

9

>

>

>

>

;

(6)

where symbols i, j, k, l, p correspond the components x, y, z of the Cartesian coordinates,dikis the Kronecker delta andεijkbeing the antisymmetric Levie Civita tensor The terms b1, b2,…, b8are given below:

N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices 1 (2016) 209e213 210

Z



N!

M

"

m 2

#

dðX1Þ;



eB[

Z

N!

1 M

"

m 2

#

dðX2Þ;



eB[

Z

N!

1 M

"

m 2

#

dðX3Þ;

Z



N!

1

M

"

m 2

#

dðX4Þ;



eB[

Z

N!

1 M

"

m 2

#

dðX5Þ;



eB[

Z

N!

1 M

"

m 2

#

dðX6Þ;

where



2





2



a¼ Lx=2[2;

here the appearance of[ ¼ ðpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNþ 1=2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Nþ 1 þ 1=2

p

Þ[B=2 as-sumes an effective phonon momentum eydqyzeE1[, which

sim-plifies the summation of q!⊥ [9] The delta functions are also

replaced bydðXÞ ¼1

X 2 þG2whereG¼ Z=tis the damping factor, to avoid divergence[9,11]

The componentrxxof the magnetoresistance and the Hall

co-efficient are given by[9]:

rxx¼ sxx

s2

xxþs2 yx

B

syx

s2

xxþs2 yx

wheresyxandsxxare derived byformula (6) Through equations(6)e(8), the impact of confined LO-phonons

on the Hall effect is interpreted by the dependence of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient

on the quantum number m characterizing the LO-phonon confinement and the other parameters of the external fields as well as the DSSL The different form of the confined LO-phonon wave vector and frequency lead to considerable changes of the theoretical results in comparison with the bulk phonons from the previous study[8] When m goes to zero, we obtain results as the case of bulk phonon in doped semiconductor superlattices

3 Numerical results and discussion

To clarify the obtained theoretical results, in this section, we present in detail the numerical evaluation of the Hall conductivity, the magnetoresistance and the Hall coefficient for the GaAs:Si/ GaAs:Be doped semiconductor superlattices Parameters used in this calculation are as follows: me¼ 0.067 m0, (m0is the free mass of

an electron), c∞¼ 10:9, c0¼ 12:9, εF ¼ 50meV, t¼ 10
12s,

n¼ 8:73  104ms
1, Zu0¼ 36:6meV, U¼ 4:1012s
1, T¼ 290K,

E1¼ 2:102V=m, E ¼ 105V=m, Lx¼ Ly¼ 100nm, Nd¼ 3, N ¼ 0,

N0¼ 2, n ¼ 0, n0¼ 0/1 (the transition between the lowest and the first excited level of an electron)

As we can see inFig 1, there are multiple resonance peaks of the Hall conductivity tensor sxx These peaks correspond to the condition:

q

which is called the intersubband magnetophonon resonance (MPR) condition [12e15] From the left to the right, in Fig 1a, resonance peaks of the conductivity tensor in case of bulk phonons correspond to the conditions 2Zuc¼ Zu0
ðn0
nÞZup
ZU;

Zu0
ðn0
nÞZup; Zu0
ðn0
nÞZupþ ZU; Zu0
ZU; Zu0; Zu0þ

ZU; Zu0þ ðn0
nÞZup
ZU; Zu0þ ðn0
nÞZup; Zu0þ ðn0
nÞZup

þZU; here eE1[≪Zu0 and it should be neglected for simplicity When phonons are confined, in this case we have m ¼ 1 / 2, the LO-phonon frequency is now modified tou1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2
n2p2=med2

q and u2¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2
4n2p2=med2

q

, thus, giving the additional reso-nance peaks of the conductivity tensor It is easy to see that the parameters of the superlattices have important roles on the MPR condition Indeed, the small value of the doping concentration nD leads to a weak material confinement effect Thus, inFig 1b, the resonance peaks, which are associated with the effect of confined electrons, have mostly disappeared with only the center peaks being observed With increasing period d of a DSSL, the contri-bution from LO-phonon confinement decreases thus correspond-ing resonance peaks will also be difficult to detect Therefore, the confinement of LO-phonons, as well as doped superlattice pa-rameters, make a remarkable impact on the magneto-phonon resonance condition

Fig 2shows the dependence of the magnetoresistance on the temperature at different values of quantum number m which characterizes the LO-phonon confinement It can be seen that the

N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices 1 (2016) 209e213 211

magnetoresistance increases nearly linear at high temperatures.

This result is in accordance with that obtained in experiment[16]at

the same range of the temperature.Fig 2also shows that the

in-crease of quantum number m leads to a dein-crease of the

magneto-resistance The mechanism behind this decrease is likely the

increase in the probability of electron scattering due to the increase

in LO-phonon confinement The current density rises with electron scattering, thus, the magnetoresistance decreases as a result Fig 3shows the Hall coefficient plotted as a function of laser amplitude at different values of quantum number m It can be seen that the HC decreases nonlinearly to a near-zero saturating value as the laser amplitude is increased It has been seen that the HC de-creases nonlinearly to the saturation value as the raising of the laser amplitude In addition, the increasing of quantum number m leads

to a faster HC decline Hence, there are new behaviours of the Hall effect in doped semiconductor superlattices due to the effect of LO-phonon confinement

4 Conclusions

So far, the influence of confined LO-phonons on the Hall effect in doped semiconductor superlattices GaAs:Si/GaAs:Be has been studied The analytical expressions for the Hall conductivity tensor, the magnetoresitance and the Hall coefficient are obtained base on quantum kinetic equation method Theoretical results are very different from previous one [8] because of the considerable contribution of the confined LO-phonon The effect of LO-phonon confinement enhances the probability of electron scattering The magnetoresistance, as well as the Hall coefficient, thus, decreases as

a result In addition, the MPR condition in doped semiconductor superlattices under the influence of external fields and the effect of confined LO-phonon now contains new terms It was found that the increase of LO-phonon confinement leads to a decrease in the HC and the magnetoresistance When increasing the laser amplitude, the HC declined in magnitude to a saturation value near zero Furthermore, the near linear increase in the magnetoresistance with temperature has good agreement with experimental data[16] This study shows that confined LO-phonons create new properties and behaviours of the Hall effect in doped semiconductor superlattices

Acknowledgements This paper is dedicated to the memory of Dr P.E Brommer - a founding editor of the Journal of Science: Advanced Materials and Devices This work was completed withfinancial support from the National Foundation for Science and Technology Development of

0

100

200

300

400

500

600

Cyclotron energy (meV)

σ xx

bulk phonon confined LO−phonon

n

d=12 nm (a)

(b)

0

50

100

150

200

250

300

350

400

450

Cyclotron energy (meV)

σ xx

bulk phonon confined LO−phonon

nD=1018 m−3 d=15 nm

Fig 1 The dependence of the conductivity tensorsxx on the cyclotron energy for

confined phonon (solid curve) and bulk phonon (dashed curve), here

nD ¼ 3.5  10 20 m
3, d¼ 12 nm ( Fig 1 a) and nD ¼ 10 18 m
3, d ¼ 15 nm( Fig 1 b).

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Temperature (K)

ρxx

m=0 m=1 m=2

Fig 2 The dependence of the magnetoresistancerxx on the temperature T for bulk

phonon m ¼ 0 (dotted curve) and confined phonon m ¼ 1 (dashed curve), m ¼ 1 / 2

(solid curve), here B ¼ 2.5 T, nD¼ 3.10 20 m
3and d¼ 12 nm.

x 105

−18

−16

−14

−12

−10

−8

−6

−4

−2

0x 10

−4

The laser amplitude (V/m)

m=0 m=1 m=2

Fig 3 The dependence of the Hall coefficient on the laser amplitude for bulk phonon

m ¼ 0 (dotted curve) and confined phonon m ¼ 1 (dashed curve), m ¼ 1 / 2 (solid curve), here B ¼ 2.5 T, nD¼ 3.10 20 m
3and d ¼ 12 nm.

N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices 1 (2016) 209e213 212

Vietnam (Nafosted 103.01e2015.22) and Vietnam International

Education Development (Project 911)

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4 Conclusions

So far, the in? ??uence of confined LO- phonons on the Hall effect in. .. equations(6)e(8), the impact of confined LO- phonons

on the Hall effect is interpreted by the dependence of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient

on. .. base on quantum kinetic equation method Theoretical results are very different from previous one [8] because of the considerable contribution of the confined LO- phonon The effect of LO- phonon confinement