DSpace at VNU: INVESTIGATION OF THE HALL EFFECT IN RECTANGULAR QUANTUM WELLS WITH A PERPENDICULAR MAGNETIC FIELD IN THE PRESENCE OF A HIGH-FREQUENCY ELECTROMAGNETIC WAVE

3 2014 1450001 14 pagesc World Scientific Publishing Company DOI: 10.1142/S0217979214500015 INVESTIGATION OF THE HALL EFFECT IN RECTANGULAR QUANTUM WELLS WITH A PERPENDICULAR MAGNETIC F

Vol 28, No 3 (2014) 1450001 ( 14 pages)

c World Scientific Publishing Company

DOI: 10.1142/S0217979214500015

INVESTIGATION OF THE HALL EFFECT IN RECTANGULAR QUANTUM WELLS WITH A PERPENDICULAR MAGNETIC FIELD IN THE PRESENCE OF A HIGH-FREQUENCY

ELECTROMAGNETIC WAVE

NGUYEN QUANG BAU ∗ Department of Physics, University of Natural Sciences, Vietnam National University in Hanoi, 334 – Nguyen Trai St.,

Thanh Xuan District, Hanoi, Viet Nam nguyenquangbau54@gmail.com BUI DINH HOI Department of Physics, National University Civil Engineering,

55 – Giai Phong St., Hai Ba Trung District, Hanoi, Viet Nam

hoibd@nuce.edu.vn Received 24 June 2013 Revised 6 October 2013 Accepted 8 October 2013 Published 1 November 2013 The Hall effect is theoretically studied in a rectangular quantum well (RQW) with in-finite barriers subjected to a crossed dc electric field and magnetic field (the magnetic field is oriented perpendicularly to the barriers) in the presence of a high-frequency electromagnetic wave (EMW) By using the quantum kinetic equation for electrons in-teracting with acoustic phonons at low temperatures, we obtain analytical expressions for the conductivity tensor as well as the Hall coefficient (HC) Numerical results for the AlGaN/GaN RQW show the Shubnikov–de Haas (SdH) oscillations in the magnetoresis-tance (MR) whose period does not depend on the temperature and amplitude decreases with increasing temperature In the presence of the EMW, the MR shows maxima at Ω/ω c = 1, 2, 3, and minima at Ω/ω c = 3/2, 5/2, 7/2, (Ω and ω c are the EMW and the cyclotron frequencies, respectively), and with increasing of the EMW amplitude the

MR approaches zero Obtained results are in accordance with recent experimental data and in good agreement with other theories in two-dimensional (2D) electron systems The results for the HC show a saturation of the HC as the magnetic field or the EMW frequency increases Furthermore, in the region of large magnetic field the HC depends weakly on the well-width.

Keywords: Hall coefficient; SdH oscillations; quantum kinetic equation; rectangular quantum wells; electron–phonon interaction.

PACS number: 72.20.My, 73.21.Fg, 78.67.De

∗ Corresponding author.

1 Introduction

The propagation of an electromagnetic wave (EMW) in materials leads to the change in the probability of scattering of carriers, and thus, leads to their unusual properties in comparison to the case of absence of the EMW The problem related

to the propagation of an EMW in semiconductors has attracted much attention

in recent times because the presence of EMWs has been stimulated by the possi-bility of their use as a powerful tool for studying the electronic properties of the surfaces and thin layers of solids, such as the microwave-modulated Shubnikov–de Haas (SdH) oscillations, the optically-detected electrophonon and magnetophonon effects, the cyclotron resonance effect, and so on

The Hall effect in bulk semiconductors in the presence of an EMW has been studied in much details.1 5 In Refs 1 and 2, the odd magnetoresistance (MR) was calculated when the nonlinear semiconductors were subjected to a magnetic field and an EMW with low frequency, the nonlinearity was explained by the nonparabol-icity of distribution functions of carriers In Refs 3 and 4, the MR was derived in the presence of an intense high-frequency EMW (laser field) for two cases: the dc magnetic field and the electric field vector of the EMW were perpendicular,3 and were parallel.4The existence of the odd MR was found to be caused by the effect of the EMW on the probability of collision, i.e., the collision integral depended on the amplitude and frequency of the EMW This problem was also studied in the pres-ence of both low-frequency and high-frequency EMW.5 Moreover, the dependence

of MR as well as magnetoconductivity on the relative angle of applied fields was also considered in detail.1 5 However, almost these results obtained by using the Boltz-mann kinetic equation, and are, thus, limited to the case of weak magnetic field re-gion and high temperatures In cases of quantized magnetic fields (strong magnetic fields) and low temperatures, the Boltzmann kinetic equation is invalid Therefore,

we need to use quantum theory to investigate both the weak magnetic field and the quantized magnetic field regions The quantum kinetic equation was used to calculate the nonlinear absorption coefficients of an intense EMW (laser radiation)

in bulk semiconductors,6in quantum wells and compositional semiconductor super-lattices,7 in doped semiconductor superlattices8 and in quantum wires.9 Recently,

we have used this method to calculate the quantum acoustomagnetoelectric field

in a parabolic quantum well subjected to a crossed electric field and magnetic field

in the presence of a sound wave.10 The acoustomagnetoelectric field is similar to the Hall field in bulk semiconductors where the sound flux plays the role of the electric current Throughout these problems, the quantum kinetic equation method has been seen to be a powerful tool

In two-dimensional (2D) electron systems, the Hall effect has been studied in many aspects (see Refs.11and12for recent reviews) However, most of the previ-ous works only considered the case an EMW was absent and at temperatures that electron–electron and electron–impurity interactions were dominant (conditions for the integer and fractional quantum Hall effect) In recent works, we have used the

quantum kinetic equation method to study the influence of a high-frequency EMW

on the Hall coefficient (HC) in parabolic quantum wells,13 in doped semiconductor superlattices14 with an in-plane magnetic field The influence of a high-frequency EMW on the Hall effect in low-dimensional systems, especially in 2D semiconductor systems with different directions of external fields, still remains to be a problem to study, especially by analytical and computational methods Therefore, in this work,

by using the quantum kinetic equation for the distribution function of electrons in-teracting with phonons, we study the Hall effect in a rectangular quantum well (RQW) with infinite barriers subjected to a crossed dc electric field and magnetic field in the presence of a high-frequency EMW, the magnetic field is perpendicular

to the plane of 2D electron gas The main purpose of this work is to make a com-parison between our calculation and other experiments and theories Specially, we investigate the influence of an EMW on the effect by comparing dependencies of the MR and the HC between the absence and the presence of an EMW The paper

is organized as follows In Sec 2, we briefly describe the model of the problem and the derivation of the quantum kinetic equation for electrons The calculation of the

MR and the HC is presented briefly in Sec.3 Numerical results and discussion are given in Sec.4 Finally, remarks and conclusions are shown in Sec.5

2 Hamiltonian of Electron–Phonon System and Quantum Kinetic Equation for Electrons in a RQW

We consider a RQW structure of well-width Lzwith an infinite confinement poten-tial assumed to be in the z-direction Due to the confinement potenpoten-tial, the motion

of electrons in the z-direction is quantized into discrete energy levels called sub-bands while the motion in x–y plane is free If this RQW is subjected to a crossed dc electric field E1= (E1, 0, 0) and magnetic field B = (0, 0, B) (B is applied perpen-dicularly to the x–y plane), the free motion of the 2D electron gas in the x–y plane

is further quantized into Landau orbits with discrete energy levels, called Landau levels If we choose a vector potential A = (0, Bx, 0) to describe the applied dc magnetic field, then the single-particle wave function and its total eigenenergy are given by15 , 16

Ψ(r) ≡ |N, n, kyi = 1

pLy

φN(x − x0)eiky yφn(z) , (1)

εN,n(ky) =



N +1 2



~ωc+ εn− ~vdky+1

2mv

2

d, N = 0, 1, 2, , (2) where N is the Landau level index and n being the subband index; ky = (0, ky, 0) and Ly are the wave vector and the normalization length in the y-direction, re-spectively; ωc = eB/m being the cyclotron frequency (determining the distance between two neighboring Landau levels); e and m are, respectively, the charge and the effective mass of an electron; and vd= E1/B being its drift velocity Also, φN

represents harmonic oscillator wavefunctions, centered at x0 = −ℓ2

B(ky− mvd/~)

where ℓB = p~/(mωc) is the radius of the Landau orbit in the x-y plane; φn(z) and εn are the wavefunctions and the subband energy values due to the infinite confinement potential in the z-direction, respectively, given by

φn(z) ≡ |ni =r 2

Lzsin nπz

Lz



εn =~

2π2n2

When a high-frequency EMW is applied to the system in the z direction with the electric field vector E = (0, E0sin Ωt, 0) (E0and Ω are the amplitude and frequency

of the EMW, respectively), the Hamiltonian of the electron–phonon system in the above mentioned RQW, in the second quantization representation, can be written as

H0= X

N,n,k y

εN,n



ky− e

~cA(t)



a+N,n,kyaN,n,k y+X

q

~ωqb+qbq, (6)

N,N ′

X

n,n ′

X

q,k y

DN,n,N ′ ,n ′(q)a+N′ ,n ′ ,k y +q yaN,n,k y(bq+ b+−q) , (7) where |N, n, kyi and |N′, n′, ky+ qyi are electron states before and after scattering;

~ωq is the energy of phonon with the wave vector q = (qx, qy, qz); a+N,n,ky and

aN,n,k y(b+

q and bq) are the creation and annihilation operators of electron (phonon), respectively; A(t) being the vector potential of the EMW; and,15 , 17

|DN,n,N ′ ,n ′(q)|2= |Cq|2|In,n ′(±qz)|2|JN,N ′(u)|2, (8) where Cq is the electron–phonon interaction constant which depends on the scat-tering mechanism; In,n ′(±qz) is the form factor of electron, given by15

In,n ′(±qz) =

Z +∞

−∞

e±iqz zφ⋆n(z)φn ′(z)dz , (9) and,

|JN,N ′(u)|2= (N′

!/N !)e−uuN′−N[LNN′−N(u)]2 (10) with LN

M(x) is the associated Laguerre polynomial, u = ℓ2

Bq2

⊥/2, q2

x+ q2

y

It is seen that the coefficient DN,n,N ′ ,n ′(q) of the interaction Hamiltonian U , in this case, is more complicated than the case of an in-plane magnetic field.13 , 14We will use above mentioned Hamiltonian to derive the quantum kinetic equation for electrons in a RQW

The quantum kinetic equation for electrons in the single (constant) relaxation time approximation takes the form6 9

i~∂fN,n,ky

∂t = h[a

+

where fN,n,ky = ha+N,n,kyaN,n,kyit is the particle number operator or the electron distribution function perturbed by external fields, hΨit denotes a statistical aver-age value at the moment t; hΨit = Tr( ˆΨ ˆW ) ( ˆW is the density matrix operator) Inserting Eq (5) into Eq (11) and realizing operator algebraic calculations as in the previous works,10,13,14 we find

−(eE1+ ωc[ky∧ h])∂fN,n,ky

~∂ky

= −fN,n,ky− f0

2π

~ X

N ′ ,n ′ ,q

|DN,n,N ′ ,n ′(q)|2

∞

X

s=−∞

Js2 λ Ω



× {[fN ′ ,n ′ ,k y +q y(Nq+ 1) − fN,n,k yNq]

× δ(εN ′ ,n ′(ky+ qy) − εN,n(ky) − ~ωq− s~Ω) + [fN ′ ,n ′ ,k y −qyNq− fN,n,k y(Nq+ 1)]

× δ(εN ′ ,n ′(ky− qy) − εN,n(ky) + ~ωq− s~Ω)} , (12) where λ = eE0qy/(mΩ), h = B/B is the unit vector along the direction of magnetic field, the notation “∧” represents the cross product (vector product), f0 (Nq) is the equilibrium distribution function of electrons (phonons), Js(x) is the sth-order Bessel function of argument x, and τ is the electron momentum relaxation time, which is assumed to be constant in this calculation

Equation (12) is the quantum kinetic equation for electrons interacting with phonons It is fairly general and can be applied for arbitrary (both weak and strong regions) magnetic field and any mechanism of interaction In the following, we will use this equation to derive analytical expressions of the Hall conductivity as well

as the HC in the RQW

3 Analytical Results for the Conductivity Tensor and the Hall

Coefficient

For simplicity, we limit the problem to the cases of s = −1, 0, 1 This means that the processes with more than one photon are ignored By the same way in the previous works,10 , 13 , 14 we obtain an expression for the partial current density jN,n,N ′ ,n ′(ε) (the current caused by electrons that have energy of ε):

jN,n,N ′ ,n ′(ε) = τ

1 + ω2

cτ2{(QN,n(ε) + SN,n,N ′ ,n ′(ε))

− ωcτ ([h ∧ QN,n(ε)] + [h ∧ SN,n,N ′ ,n ′(ε)]) + ω2cτ2(QN,n(ε)h + SN,n,N ′ ,n ′(ε)h)h} , (13) where

QN,n(ε) = −e

m X

N,n,k y

ky



F∂fN,n,k y

~∂ky

 δ(ε − εN,n(ky)) , F= eE1, (14)

SN,n,N′ ,n ′(ε) = 2πe

m~

X

N ′ ,n ′

X

N,n

X

q,ky

|DN,n,N ′ ,n ′(q)|2Nqky

 [fN ′ ,n ′ ,k y +q y − fN,n,k y]

×



1 − λ

2

2Ω2

 δ(εN ′ ,n ′(ky+ qy) − εN,n(ky) − ~ωq)

+ λ

2

4Ω2δ(εN ′ ,n ′(ky+ qy) − εN,n(ky) − ~ωq+ ~Ω) + λ

2

4Ω2δ(εN ′ ,n ′(ky+ qy) − εN,n(ky) − ~ωq− ~Ω)



+ [fN ′ ,n ′ ,k y −qy− fN,n,k y]



1 − λ

2

2Ω2



× δ(εN ′ ,n ′(ky− qy) − εN,n(ky) + ~ωq) + λ

2

4Ω2δ(εN ′ ,n ′(ky− qy) − εN,n(ky) + ~ωq+ ~Ω) + λ

2

4Ω2δ(εN ′ ,n ′(ky− qy) − εN,n(ky) + ~ωq− ~Ω)



In the limit of Lz → ∞, i.e., the confinement vanishes, Eq (13) becomes the same result obtained in bulk semiconductors for acoustic phonon interaction.3The total current density is given by

J=

Z ∞ 0

We now use this general result to derive expressions for the conductivity tensor and the HC by considering the electron–acoustic phonon interaction in RQWs

Acoustic phonons are important at low temperatures If the temperature is low enough, the electrons system is degenerate and the distribution function has the form of Heaviside step function For the electron–acoustic phonon interaction,

~ωq= ~vsq, Nq= kBT /~ωq= (β~vsq)−1 with β = (kBT )−1 and

|Cq|2= ξ

2q

where kB, vs, ξ, ρ and V0 are the Boltzmann constant, the sound velocity, the acoustic deformation potential, the mass density and the normalization volume of specimen, respectively

If the scattering is elastic, the acoustic phonon energy in Eq (15) can be ne-glected.18 Inserting Eq (13) into Eq (16) and performing some manipulation, we

obtain the expression for the conductivity as

σim= τ

1 + ω2

cτ2[δij− ωcτ ǫijkhk+ ωc2τ2hihj]X

N

X

n,n ′

×



−e

2βvdLyI

8π2βmωcvs~2α2ℓ2

BLz

× (2 + δn,n ′)(εN,n− εF) eB ¯ℓ

~



1 −θ 2

 eB ¯ℓ

~

2

×



1 + 2

+∞

X

s=1

(−1)se−2πsΓN /~ω c

cos (2πs¯n1)



+θ 4

 eB ¯ℓ

~

3

1 + 2

+∞

X

s=1

(−1)se−2πsΓN /~ω ccos (2πs¯n2)



+θ 4

 eB ¯ℓ

~

3

1 + 2

+∞

X

s=1

(−1)se−2πsΓN /~ω c

cos (2πs¯n3)



1 + ω2

cτ2δjl[δlm− ωcτ ǫlmphp+ ω2cτ2hlhm]



where δij is the Kronecker delta, ǫijk being 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, εF is the Fermi level, α = ~vd, θ = e2E2/(m2Ω4), C = ξ2/(2ρvs),

I = a1

αβ(e

αβa 1+ eαβa1) − 1

(αβ)2(eαβa1− eαβa1) , a1= Lx/2ℓ2B, (19)

εN,n= (N + 1/2)~ωc+ n2ε1+ mv2

d/2 , ε1= π2~2/(2mL2) , (20)

¯

n1= (n

2− n′2)ε1+ eE1ℓ¯

¯

n2= ¯n1− Ω

ωc

, n¯3= ¯n1+ Ω

ωc

¯

ℓ = (pN + 1/2 + pN + 1 + 1/2)ℓB/2 , (23) and ΓN is the damping factor associated with the momentum relaxation time, τ ,

by ΓN ≈ ~/τ 18The appearance of the parameter ¯ℓ is due to the replacement of qy

by eB ¯ℓ/~, where ¯ℓ is a constant of the order of ℓB The purpose is to a simplicity

in performing the integral over q⊥ This has been used in Ref 14 and is equivalent

to assuming an effective phonon momentum: evdqy ≈ eE1ℓ.¯

The component ρxxof the resistance (the MR) is given by19

ρxx= σxx

σ2

xx+ σ2 yx

and the HC is

RH= ρyx

B = −

1 B

σyx

σ2

xx+ σ2 yx

where σxxand σyx are given by Eq (18)

The above results show the dependencies of the Hall conductivity tensor (resis-tance) and the HC on the external fields, including the EMW They are obtained for arbitrary values of the indices N , n, N′

, n′

and appear very involved If the well potential is not rectangular (such as parabolic or triangle), the wavefunctions

φn(z) and corresponding energies εn in the confinement direction (the z-direction) change Consequently, the form factor In,n ′(±qz) has a different form This leads

to considerable changes of the analytical expressions and numerical value of the re-sults For example, if the well potential is parabolic,10εnhas the form of a quantum oscillator’s energy and the wavefunctions φn(z) contain the Hermite polynomials,

so the integral (9) become more complicated and cannot be done analytically In the following, we will give physical conclusions to above results by carrying out a numerical evaluation and a graphic consideration using a computational method

4 Numerical Results and Discussions

To have a deeper insight of above analytical results, in this section, we present detailed numerical calculations of the MR and the HC in a RQW subjected to the uniform crossed magnetic and electric fields in both the absence and the pres-ence of an EMW For numerical evaluation, we consider the model of a RQW of AlGaN/GaN with the wurtzite (hexagonal) structure The parameters used for the computation are:20 – 25 ξ = 9.2 eV, ρ = 6150 kg · m− 3, vs = 6560 m · s− 1,

m = 0.22 × m0 (m0is the mass of free electron) For the sake of simplicity we take

τ = 10− 12 s, Lx= Ly= 100 nm and only consider the transitions: N = 0, N′

= 1,

n = 0, n′= 1 (the lowest and the first-excited levels)

Figures1and2, respectively, show the dependencies of the MR on the magnetic field and its inversion at different values of the temperature We can see clearly the appearance of the typical SdH oscillations with the period is in 1/B and does not depend on the temperature This type of oscillation is well-known to be controlled

by the ratio of the Fermi energy (or generally chemical potential) and the cyclotron energy Because in GaN materials the Fermi energy is one order larger and electrons are much heavier than they are in GaAs, so it is easy to explain the fact that the magnetic field for the observation of corresponding oscillations in this case is much larger than it is in GaAs.29 – 32It is also seen from the figures that as the temperature increases, the amplitude of SdH oscillations decreases as we expect To estimate the damping of these oscillations with the temperature and make a comparison to other available works, we use a computational program to evaluate the relative amplitude

of these oscillations Denoting A(T, Bn) and A(T0, Bn) respectively, are amplitudes

of the oscillation peaks observed at a magnetic field Bn and at temperatures T and T0 In Fig 3, the relative amplitude A(T, Bn)/A(T0, Bn) is computationally

4 6 8 10 0

0.5 1 1.5 2 2.5

B (T)

ρ xx

T=3K T=4K T=5K

Fig 1 The MR ρ xx as functions of the magnetic field at the different values of the temperature Here, E 1 = 5 × 10 2

V/m, E 0 = 0 and L z = 8 nm.

0.1 0.15 0.2 0.25 0.3 0.35 0

0.5 1 1.5 2 2.5

1/B (T −1 )

ρ xx

T=3K T=4K T=5K

Fig 2 The MR ρ xx as functions of the inversion of magnetic field at the different values of temperature The other parameters used in the computation are the same as in Fig 1

evaluated and plotted versus the temperature at T0= 2 K and Bn = 3 T We can see from the figure that this dependence is in accordance with experimental data taken by Tiras and his coworkers recently in the AlGaN/AlN/GaN heterostructures using temperature-dependent classical Hall effect measurements.20 Theoretically, the relative amplitude versus temperature is also given by26 – 28

A(T, Bn) A(T0, Bn)=

T sinh(2π2kBmT0/~eBn)

T0sinh(2π2kBmT /~eBn). (26) This relation is also plotted in Fig 3 and it is seen that there is a good agree-ment between our calculation and Eq (26) So far, it can be concluded that the quantum kinetic equation method has described well this kind of problems The dependence of the MR on the well-width is shown in Fig 4 where we plot ρxx

versus B at different values of the well-width From the figure we can see that the SdH oscillations become less pronounced as the well-width increases and vanish at very large well-widths This is in accordance with the fact that these oscillations

0 2 4 6 8 10 12 0

0.2 0.4 0.6 0.8 1

T (K)

present calculation

Eq (26) Expt data from Ref 19

Fig 3 The relative amplitude A(T, B n )/A(T 0 , B n ) versus temperature The full squares are our calculation, the full circles are experimental measurements for Al 0.25 Ga 0.75 N/AlN/GaN het-erostructures from Ref 19 and the dashed curve shows the relation given by Eq ( 26 ).

0 5 10 15 20

B (T)

ρ xx

L

z =8nm L

z =30nm L

z =80nm

Fig 4 Dependencies of ρ xx on B at different values of the well-width Here, E 1 = 5 × 10 2

V/m,

E 0 = 0 and T = 4 K.

only can be observed in 2D semiconductor systems, when the well-width increases, the confinement decreases and if the well-width is very large the system becomes a three-dimensional electron system

To show the influence of the EMW on the effect, in Fig.5 the MR is plotted versus ratio Ω/ωc with a fixed Ω for two cases: absence of the EMW (E0 → 0) and presence of the high-frequency EMW (E0= 4 × 105 V/m) It is seen that the oscillation amplitude changes evidently in some region of the magnetic field in the presence of the EMW There occurs the beat phenomenon This is similar to one observed in a GaAS-based 2D electron gas in the presence of a microwave at high frequencies (Ω/(2π) ∼ 280 GHz).29In Fig.6, the MR is shown as a function of Ω/ωc

at a fixed ωc We can see very clearly the maxima are at Ω/ωc= 1, 2, 3, and the minima are at Ω/ωc = 3/2, 5/2, 7/2, Also, as the EMW amplitude (radiation intensity) increases, the minimum of the MR approach zero These behaviors are