DSpace at VNU: The quantum acoustomagnetoelectric field in a quantum well with a parabolic potential

DSpace at VNU: The quantum acoustomagnetoelectric field in a quantum well with a parabolic potential tài liệu, giáo án,...

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The quantum acoustomagnetoelectric ﬁeld in a quantum well with a parabolic potential

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 17 March 2012

Received in revised form 7 July 2012

Accepted 30 July 2012

Available online 5 August 2012

Keywords:

Parabolic quantum well

Quantum acoustomagnetoelectric ﬁeld

Electron-external phonon interaction

Quantum kinetic equation

a b s t r a c t

The acoustomagnetoelectric (AME) field in a quantum well with a parabolic potential (QWPP) has been studied in the presence of an external magnetic field The analytic expression for the AME field

in the QWPP is obtained by using the quantum kinetic equation for the distribution function of electrons interacting with external phonons The dependence of the AME field on the temperature T of the system, the wavenumber q of the acoustic wave and external magnetic field B for the specific AlAs/GaAs/AlAs is achieved by using a numerical method The problem is considered for both cases: The weak magnetic field region and the quantized magnetic field region The results are compared with those for normal bulk semiconductor and superlattices to show the differences, and we use the quantum theory to calculate the AME field in the QWPP

1 Introduction

The acoustic waves propagate along the stress-free surface of an elastic medium has attracted much attention in the past two decades because of their utilization in acoustoelectronics Considerable interest in such waves has also been stimulated by the possibility of their use as a powerful tool for studying the electronic properties of the surfaces and thin layers of solids

It is well known that the propagation of the acoustic wave in conductors is accompanied by the transfer of the energy and momentum to conduction electrons which may give rise to a current usu-ally called the acoustoelectric current, in the case of an open circuit called acoustoelectri ﬁeld Pres-ently this effect has been studied in detail both theoretically and experimentally and has been found in wide application in radioelectrionic systems[1–8] The presence of an external magnetic ﬁeld

⇑Corresponding author.

E-mail address: nguyenvanhieu@gmail.com (N Van Hieu).

Contents lists available atSciVerse ScienceDirect

Superlattices and Microstructures

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 / s u p e r l a t t i c e s

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applied perpendicularly to the direction of the sound wave propagation in a conductor can induce an-other field, the so-called AME field It was predicted by Galperin and Kagan[9]and observed in bis-muth by Yamada[10] Calculations of the AME field in bulk semiconductor[11–14]and the Kane semiconductor[15]in both cases The weak and the quantized magnetic field regions have been inves-tigated In recent years, the AME field in low-dimensional structures have been extensively studied

[16,17] So far, however, almost all these works obtained by using the Boltzmann kinetic equation method, and are, thus, limited to the case of the weak magnetic ﬁeld region and the high temperature,

in the case of the quantized magnetic field (strong magnetic field) region and the low temperature using the Boltzmann kinetic equation is invalid Therefore, we use quantum theory to investigate both the weak magnetic field and the quantized magnetic field region The AME field is similar to the Hall field in the bulk semiconductor where the sound fluxUplays the role of electric current~j The essence

of the AME effect is due to the existence of partial current generated by the different energy groups of electrons, when the total acoustoelectric (longitudinal) current in specimen is equal to zero When this happens, the energy dependence of the electron momentum relaxation time causes average mobilities

of the electrons in the partial current, in general, to differ, if an external magnetic ﬁeld is perpendicular

to the direction of the sound flux, the Hall currents generated by these groups will not compensate one another, and a non-zero AME effect will result Here it must be emphasized that the direction of the AME field depends on the carrier scattering mechanism The direction of the AME field is opposite depending upon whether the deformation potential scattering or the ionized impurity scattering is dominant

In the present paper, we study the AME ﬁeld in a QWPP by using the quantum kinetic equation for the distribution function of electrons interacting with external phonons We restricted our consider-ation to the case of specular reﬂection of electron at the surface of bulk crystal We assumed the defor-mation mechanism of electron-acoustic phonon interaction We also supposed that the mechanism that limits the electron mean free path is scattered on randomly distributed point defects (impurities)

in the bulk of the crystal Within the framework of this model we analyzed the magnetic field depen-dence of the AME field in the weak magnetic field region (Xc kBT; Xcg), and for the quantized magnetic field region (Xc kBT; Xcg), (Xcis the cyclotron frequency;gis the frequency of the electron collisions and in this paper, we select h ¼ 1) Numerical calculations are carried out for a spe-cific quantum well AlAs/GaAs/AlAs to clarify our results This paper is organized as follows In Section

2, we calculate the AME field in a QWPP, in Section3we find analytic expression for the AME field in the QWPP, in Section4we discuss the results, and in Section5we come to a conclusion

2 The AME Field in a QWPP

2.1 Electronic Structure in a QWPP

When the magnetic ﬁeld is applied in the x-direction, in that case the vector potential is chosen as:

A ¼ Ay¼ zB If the conﬁnement potetial is assumed to take the form VðzÞ ¼ mx2z2=2 the eigenfunc-tion of an unperturbed electron is expressed as

LxLy

where Lxand Lyare the normalization length in the x and y direction, respectively, /Nðz z0Þ is the oscillator wavefunction centred at z0¼ pyXc=½mðX2cþx2Þ; m is the effective mass of a conduction electron,x andXc are the characteristic frequency of the potential and the cyclotron frequency, respectively, N ¼ 0; 1; 2 is the azimuthal quantum number; ~p ¼ ðpx;py;0Þ is the electron momen-tum vector The electron energy spectrum takes the form

2 x

p2 y

2m

x X

withX¼ ðX2þx2Þ1=2andX ¼ eB=m

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2.2 General expression for the AME Field in a QWPP

Let us suppose that the acoustic wave of frequencyxqis propagated along the z QWPP axis (along the z direction, the energy spectrum of electron is quantizied or the motive direction of electron is lim-ited) and the magnetic ﬁeld is oriented along the x axis We consider the most realistic case from the point of view of a low-temperature experiment, when

where csis the velocity of the acoustic wave and q is the modulus of the acoustic wave vector and l is the electron mean free path The compatibility of these conditions is provided by the smallness of the sound velocity in comparison with the characteristic velocity of the Fermi electrons We also suppose that inequalities(3)hold, i.e the quantization of the electron motion in the magnetic ﬁeld is essential

If the conditions(3)are satisﬁed, a macroscopic approach to the description of the acoustoelectric ef-fect is inapplicable and the problem should be treated by using quantum mechanical methods The acoustic wave will be considered as a packet of coherent phonons with the delta-like distribution function N~ k¼ ð2pÞ3Udð~k ~qÞ=xqcsin the wavevector ~k space,Uis the sound ﬂux density The Ham-iltonian describing the interaction of the electron-phonon system in the QWPP, which can be written

in the secondary quantization representation as

N;~ k

eNð~kÞaþ

N;~ k;N 0 ;~ q

C~ qUN;N 0ð~qÞaþ

with C~ qis the electron-phonon interaction factor and takes the form[11]

C~ q¼ iKc2

lðhx3

~

q=2q0NSÞ1=2; N¼ q 1 þr2

l

rl

rt 2

t

2rt

s=c2

l

s=c2 t

Kis the deformation potential constant; aþ

N;~ kand aN;~kare the creation and the annihilation operators of the electron, respectively; b~ qis the annihilation operator of the external phonon jN;~ki and jN0

;~k þ ~qi are electron states before and after interaction, UN;N0ð~qÞ is the matrix element of the operator

U ¼ expðiqy klzÞ; kl¼ q2x2

q=c2 l

is the spatial attenuation factor of the potential part the dis-placement ﬁeld; cland ctare the velocities of the longitudinal and the transverse bulk acoustic wave;

q0is the mass density of the medium and S ¼ LxLyis the surface area Using expression2it is straight-forward to evaluate the matrix elements of the operator U We obtain

UN;N0ð~qÞ ¼ 2

LxLy

ð2pÞ2LNN 0

N

k2 l

!

2 l

!

dk0

y ;k y þqdk0

x ;k xdN;N0; ð8Þ

dis the Kronecker delta symbol and LNN 0

N ðxÞ is the associated Laguerre polynomials The quantum ki-netic equation for electrons in the single (constant) scattering time approximation takes the form:

@fN;~ p

@t e~E þXc½~h;~p

fN;~ p f0

where ~h ¼ ~B=B is the unit vector along the direction of the external magnetic field, f0is the equilibrium electron distribution function, fN;~ pis an unknown distribution function perturbed due to the external fields, andsis the electron momentum relaxation time In order to find fN;~p, we use the quantum equa-tion for the particle number operator or the electron distribuequa-tion funcequa-tion fN;~ p¼ haþ

N;~ paN;~ pit:

i@fN;~p

þ

N.Q Bau et al / Superlattices and Microstructures 52 (2012) 921–930

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where hWitdenotes a statistical average value at the moment t; hWit¼ TrðcW bWÞ (cW is the density ma-trix operator)

From Eq.(10), using the Hamiltonian in Eqs.(4) and (5)and realizing operator algebraic calcula-tions, we ﬁnd

@fN;~ p

1

dt1

X

N 0 ;~ k

jC~ kj2jUN;N 0j2½fN0 ;~ pþ~ kðN~ kþ 1Þ fN;~ pN~ k exp iðh eN0 ;~ pþ~ keN;~ px~ kÞðt t1Þi

þ fhN0 ;~ p~ kN~ k fN;~ pðN~ kþ 1Þ exp iðh eN0 ;~ p~ keN;~ pþx~ kÞðt t1Þi

substituting Eq.(11)into Eq.(9)and realizing calculations, we obtained the basic equation of the prob-lem which is that equation for the distribution function of electrons interacting with external phonons

in the presence of an external magnetic ﬁelds in QWPP:

e~E þXc½~h;~p@fN;~p

@p ¼

fN;~p f0

X

N 0 ;~ k

jC~ kj2jUN;N 0j2 ½f N0 ;~ pþ~ kðN~ kþ 1Þ fN;~ pN~ kdðeN0 ;~ pþ~ keN;~px~ kÞþ þ½fN0 ;~ p~ kN~ k fN;~ pðN~ kþ 1ÞdðeN0 ;~ p~ keN;~ pþx~ kÞ

Eq.(12)is fairly general and can be applied for any mechanism of interaction In the limit ofx! 0, i.e., the electron conﬁnement vanishes, it gives the same results as these obtained in bulk semiconductor

[13,14] Multiply both sides of Eq.(12)by ðe=mÞ~pdðeeN;~pÞ and carry out the summation over N and ~p,

we have the equation for the partial current density ~RN;N0ðeÞ (the current caused by electrons which have energy ofe):

~RN;N 0ðeÞ

where

~

N;~ p

e~p

@fN 0 ;~ p

@~p

dðeeN;~ pÞ;

~SN;N0ðeÞ ¼ð2pÞ3jC~ qj2U

x~ qcs

X

N;N 0 ;~ p;~ k

jUN;N 0j2~p

mdðeeN;~ pÞdð~k ~qÞ

ðf N0 ;~ pþ~ k fN;~ pÞdðeN0 ;~ pþ~ keN;~ px~ kÞ þ ðfN0 ;~ p~ k fN;~ pÞdðeN0 ;~ p~ keN;~ pþx~ kÞ

: Solving the Eq.(13), we obtained the partial current ~RN;N 0ðeÞ

~RN;N 0ðeÞ ¼ sðeÞ

1 þX2cs2ðeÞ

~

QNðeÞ þ ~SN;N 0ðeÞ

sðeÞ XcsðeÞ ½~h; ~QNðeÞ þ ½~h;~SN;N 0ðeÞ n

þX2cs2ðeÞ ~QNðeÞ þ ~SN;N 0ðeÞ;~h~

ho

the total current density is generally expressed as

~j ¼

0

we ﬁnd the current density

whereaijand bijare the electrical conductivity and the acoustic conductivity tensors, respectively

aij¼e

2n0

m a1dijXca2ijkhkþX2ca3hihj

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here ijk is the unit antisymmetric tensor of third order, n0 is the carrier concentration, and al;bl

(l ¼ 1; 2; 3) are given as

2

pn0

0

slðeÞ

1 þX2cs2ðeÞðeXðN þ 1=2ÞÞ@f0

@ede;

bl¼

0

slðeÞ

1 þX2cs2ðeÞ

@f0

@ede;

A ¼8ep3jCqj2

xqcs

X

N;N 0

4

ðLyLxÞ2ð2pÞ4 L0

N

k2l

!

q

k2l

2 c

!

dðN0 NÞXxq

d ðN 0 NÞXþxq

:

We considered a situation whereby the sound is propagating along the x axis and the magnetic ﬁeld

B is parallel to the z axis and we assume that the sample is opened in all directions, so that ji¼ 0 Therefore, from Eq.(16) we obtained the expression of the AME ﬁeld EAME, which appeared along the y axis of the sample

Ey¼ EAME¼bzzayz byzayy

a2

yyþa2 yz

Eq.(19)is the general expression to calculate the AME ﬁeld in a QWPP in the case of the relaxation time of carriersdependent on carrier energy

3 Analytic expression for the AME ﬁeld in the QWPP

We can see that Eq.(19)in the general case is very complicated, so that we only examined the relaxation time of carriersdepending on carrier energy as follow:

s¼s0

e

kBT

by using the Eqs.(17) and (18)and carrying out manipulations, we derived the expression for the AME ﬁeld as follows:

EAME¼p XcAs0U

e2mkBT : F2m;2mFmþ1;2m Fm;2mF2mþ1;2m

kBT Fm;2m

cs2 F2mþ1;2mXðN þ 1=2Þ

kBT F2m;2m

where

Fm;m0¼

0

xm

1 þX2cs2xm0

@f0

The Eq.(21)is the AME field in the QWPP in the case of the external magnetic field We can see that the dependence of the AME field on the external magnetic field and the frequencyx~ qis nonlinear

We will carry out further analysis of the Eq.(21)separately for the two limiting cases: the weak magnetic ﬁeld region and the case of quantized one

3.1 The case of weak magnetic ﬁeld region

In the case of the weak magnetic ﬁeld

in this case, the expression of E in the Eq.(20)takes the form

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EAME¼p XcAs0U

e2mkBT F2m;2mFmþ1;2m Fm;2mF2mþ1;2m

F2

mþ1;2mþX2cs2F2

2mþ1;2m

we calculated for f0¼ ð1 expðx zÞÞ1which is the Fermi-Dirac distribution function, x ¼e=kBT; z ¼

eF=kBT, and by carrying out a few manipulation we obtained an analysis expression for the AME ﬁeld

as follows:

EAME¼ð2pÞ5UqX4jCqj2

exqkBTcsk3lXc

2 l

4mX2klX

2

cq

mX3

!

N;N 0

L0N k

2 l

!

dðN0 NÞXxq

d ðN 0 NÞXþxq

2

ðN þ 1=2Þs0

kBT

X cs0

X cso

8

<

:

9

=

;

0

@

1 A

1

; ð24Þ with

L0

NðyÞ

22NN!

k¼0

ð2kÞ!ð2N 2kÞ!L02kðyÞ

and

k¼1

ð1Þky2k1

k¼1

ð1Þky2k

where L0

2kðyÞ is the associated Laguerre polynomials

3.2 The case of quantized magnetic ﬁeld region

In the case of quantized magnetic ﬁeld region

in this case, the expression of EAMEin Eq.(20)takes the form

EAME¼ p XcAs0kBTU

e2mX2ðN þ 1=2Þ2 F2m;2mFmþ1;2m Fm;2mF2mþ1;2m

Fn 2m;2mþX2cs2F22m;2mo1

by carrying out a few manipulation we obtained an analysis expression for the AME ﬁeld as follow:

EAME¼ð2pÞ5UqX2jCqj2

exqcsk3lXc

2 l

4mX2klX

2

cq

mX3

!

N;N 0

L0 N

k2 l

!

d ðN0 NÞXxq

d ðN0 NÞXþxq

ðN þ 1=2Þ

2

s0ðN þ 1=2Þ

X cs0

X cso

8

<

:

9

=

;

0

@

1 A

1

: ð27Þ From Eq.(24)and Eq.(27), we see that in both cases, the weak magnetic field and the quantized mag-netic field, the dependence of AME field on external magmag-netic field B is nonlinear These results are different from those obtained in bulk semiconductor[11–14]and the Kane semiconductor[15]

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4 Numerical results and discussion

To clarify the results that have been obtained, in this section, we considered the AME field in two limited cases weak magnetic field region and the case of quantized magnetic field region in QWPP This quantity is considered as a function of an external magnetic field B, the frequencyxq of ultra-sound, the temperature T of system, and the parameters of the AlAs/GaAs/AlAs quantum well The

104W m2; m ¼ 0:067m0;m0 being the mass of free electron, q0¼ 5320 kg m3; cl¼ 2

103ms1;ct¼ 18 102ms1;cs¼ 8 102ms1; K¼ 13:5 eV;xq¼ 109s1

4.1 The case of weak magnetic ﬁeld region

Fig 1shows the dependence of AME field on the magnetic field for the case of the weak magnetic field at different values of the lattice temperature The dependence of the AME field on the magnetic field at different values of the temperature shows that when magnetic field rises up, the AME field in-creases monotonically However, it reached a maximum value at B is about 0.08 T, and decreased again above 0.08 T The AME field is very small, approximates 2:5 106V=m On the other hand, EAME in-creases nonlinearly with the magnetic field This result is different from those for bulk semiconductor

[11–14]and the Kane semiconductor[15]under condition ql 1, and the weak magnetic ﬁeld region Because the AME ﬁeld expression EAMEin bulk semiconductor[11–14]and the Kane semiconductor

[15]is proportional to B In other words, EAME increases linearly with the magnetic ﬁeld Our result indicates that the dominant mechanism for such a behaviour is attributed to the electron conﬁnement

in the QWPP FromFig 1, we see that the EAMEdepends signiﬁcantly on the lattice temperature See

Fig 2, in the limit ofx! 0, i.e., the electron conﬁnement vanishes, EAMEincreases linearly with the magnetic ﬁeld, it gives the same results as these obtained in bulk semiconductor[11–14]

Fig 3investigated the dependence of AME field on the magnetic field in the quantized magnetic field region which have many distinct maxima The result showed the different behaviour from results

in bulk semiconductor[11–14]and the Kane semiconductor[15] Different from the bulk semiconduc-tor, these peaks in this case are much sharper According to the result in the bulk semiconductor

[11–14]and the Kane semiconductor[15]in the case of strong magnetic ﬁeld EAMEwhich is propor-tional to1 There are two reasons for the difference between our result and other results: one is that

0 0.5 1 1.5 2 2.5

−6

Magnetic field B(T)

T=250K T=270K

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in the presence of the quantum magnetic field, the electron energy spectrum was affected by quan-tized magnetic field and the other is the effect of the electrons confinement in the QWPP, that means above B > 1:8T and below 4K, carriers in the samples satisfy the quantum limit conditions:Xc kBT andXcs 1, and in the QWPP the energy spectrum of electron is quantized Also, the result is differ-ent from those in superlattice[16,17] In[16,17]by using the Boltzmann kinetic equation, AME field is proportional to B with all regions of temperature By using the quantum kinetic equation method, our result indicate that it is only linear to B in case of the weak magnetic field and higher temperature,

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−7

Magnetic field B(T)

Fig 2 The dependence of AME field on the magnetic field B for the case of the weak magnetic field and high temperature T=270K, in the limit ofx ! 0

0 0.5 1 1.5 2 2.5 3

−3

Magnetic field B(T)

Fig 3 The dependence of AME field on the magnetic field B for the case of the strong magnetic field and low temperature;

T ¼ 4K.

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while in case of the strong magnetic ﬁeld and low temperature AME ﬁeld is not proportional to B, but there are many peaks inFig 3 This is our new development

5 Conclusions

In this paper, we have obtained analytical expressions for the AME ﬁeld in a QWPP for both the case

of quantized magnetic field and the weak magnetic field region There is a strong dependence of AME field on the cyclotron frequencyXcof the magnetic field,x~ qof the acoustic wave, and the temperature

T of system The result showed that the cause of the AME effect is the existence of partial current gen-erated by the different energy groups of electrons, and the dependence of the electron energy due to the momentum relaxation time In addition, the absorption of acoustic quanta by electron is accom-panied by the electrons conﬁnement and quantized magnetic ﬁeld which led to the increase of the AME effect

The numerical result obtained for AlAs/GaAs/AlAs QWPP shows that in the quantized magnetic ﬁeld region, the dependence of AME ﬁeld on the magnetic B is nonlinear, and there are many distinct maxima This dependence has differences in comparison with that in normal bulk semiconductors

[12–14]and the Kane semiconductor[15] The AME field in the QWPP is bigger The results show a geometrical dependence of AME field due to the electrons confinement in the QWPP In the limit of

x! 0, i.e., the electron confinement vanishes, EAMEincreases linearly with the magnetic field, it gives the same results as these obtained in bulk semiconductor[12–14] In addition, this results are quite interesting as a similar result in the superlattice[16,17]for the case of the weak magnetic field and the higher temperature, but in the case of the strong magnetic field and the low temperature the result

EAME¼ 2:5 106V=m at T=170K, B=0.08(T) (in the case of the weak magnetic ﬁeld) and

EAME¼ 3:2 103V=m at T=4K, B=1.9(T) (in the case of quantized magnetic ﬁeld region), Which are small but should be possible to measure experimentally

Acknowledgement

This work is completed with ﬁnancial support from the Vietnam NAFOSTED (No 103.01-2011.18) Appendix A Appendix

Here we add some brief explanations about deriving AME ﬁeld in Section3 The electrical conduc-tivity tensorsaijand the acoustic conductivity tensors bijhave the form

ayy¼e

2n0

Substituting Eqs.(28, 29)into Eq.(19)we obtain

e2n0

b2a1 b1a2

a2þX2ca2

!

here

2

pn0

0

sðeÞ

1 þX2cs2ðeÞ eX N þ1

2

@f0

2

pn0

0

s2ðeÞ

1 þX2cs2ðeÞ eX N þ1

2

@f0

0

sðeÞ

1 þX2s2ðeÞ

@f0

@ede; b2¼

0

s2ðeÞ

1 þX2s2ðeÞ

@f0

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substitutings¼s0 e

k B T

m into Eqs.(31)–(33), with x ¼ e

k B T

a 1 ¼m

2

pn0

Z 1

0

s0 e

kBT

m

1 þX2

cs2

0 e

k B T

2 m e N þ12

@f0

@ede¼

m 2

pn0

Z 1 0

s0 k B Txmþ1

1 þX2

cs2

0 x 2 m @f0

@ dx X N þ12

0

s0 x m

1 þX2

cs2

0 x 2 m @f0

@ dx

!

; ð34Þ

a 2 ¼m

2

pn 0

Z 1

0

s0 e

k B T

2 m

1 þX2

cs2

0 e

kBT

2 m e N þ12

@f0

@ede¼

m 2

pn 0

Z 1 0

s2

0 k B Tx 2 m þ1

1 þX2

cs2

0 x 2 m @f0

@ dx X N þ12

0

s2

0 x 2 m

1 þX2

cs2

0 x 2 m @f0

@ dx

!

; ð35Þ

b1¼

0

s0ðe

k B TÞm

1 þX2cs2ðe

kBTÞ2m

@f0

@ede¼

0

s0 xm

1 þX2cs2x2m@f0

b2¼

0

s0ðkBeTÞ2m

1 þX2cs2ðkBeTÞ2m

@f0

@ede¼

0

s2 x2m

1 þX2cs2x2m@f0

We used following notations

Fm;m0¼

0

xm

1 þX2cs2xm0

@f0

We obtain

2

pn0

s0kBT Fmþ1;2mXðN þ 1=2Þs0 Fm;2m

2

pn0

s2kBT F2mþ1;2mXðN þ 1=2Þs2 F2m;2m

Substituting Eqs.(38)–(40)into Eq.(30)and realizing calculations, we obtain Eq.(21)

EAME¼mXcAU

e2n0

s2F2m;2mm

2

pn0

s0kBTFmþ1;2mXðN þ 1=2Þs0Fm;2m

2

pn0

s2kBTF2mþ1;2mXðN þ 1=2Þs2F2m;2m

pn0

kBT Fm;2m

(

þX2cs2 F2mþ1;2mXðN þ 1=2Þ

kBT F2m;2m

¼p XcAs0U

e2mkBT F2m;2mFmþ1;2m Fm;2mF2mþ1;2m

kBT Fm;2m

þX2cs2 F2mþ1;2mXðN þ 1=2Þ

kBT F2m;2m

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2

pn0

s2kBTF2mỵ1;2mXN ỵ 1=2ịs2F2m;2m

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