Polaron in a quantum disk

Iqltlla ELSEVIER

Physica B 245 (1998) 92 102

Polaron in a quantum disk Chuan-Yu

C h e n a'b'*, W a i - S a n g

L i b, X i a o - Y i n g T e n g b'c, S h i - D o n g

Liang d

aDepartment of Physics, Guangzhou Teacher's College, Guangzhou 510400, People "s Republic of China bDepartment of Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong cDepartment of Physics, Neimenggu University of Technology, Huhehote 010062, People "s Republic of China dDepartment of Physics, Guangzhou Teacher's College, Guangzhou 510400, People's Republic of China Received 25 May 1996; received in revised form 26 February 1997

Abstract

The eigenwavefunction and eigenenergy of an electron in a quantum disk and the polaronic correction to its energy levels are investigated in the present work. Both the bulk longitudinal-optical (LO) and the surface optical (SO) phonon modes in a slab are adopted in considering the interaction between an electron and a phonon in the quantum disk. The second-order energy levels correction due to either mode of the phonon have been derived as a function of the disk radius and its thickness. The numerical computation for the GaAs quantum disk have shown that the energy correction by SO phonon for the total polaronic energy shift was very significant and was even dominant when the thickness is small. © 1998 Elsevier Science B.V. All rights reserved.

PACS: 71.38 + i; 73.20.Dx; 63.20.Kr Keywords- Polaron; Quantum disk; Eigenwavefunction; Polaronic energy shift

1. Introduction

During the last decade, a great deal of work on the study of the electron properties in semiconductor nanostructures has been carried out. In the theoretical investigation of these microstructures, the effect of the phonon on the electron, hole and exciton is one of the important subjects and has been of increasing interest [1 11]. For example, Das Sarma and Stopa have calculated the bulk LO phonon effect on the binding energy and effective mass of an electron in a GaAs/AlxGal_xAs heterostructure [1]. Hai et al. [2, 3] reported a detailed study of polaron properties in a quantum well. Chen et al. I-4] have investigated the strongcoupling polaron theory in quasi-two-dimensional system, exciton-phonon coupling system in a quantum well within an electric field 1-5] and the size effect on exciton-phonon scattering in quantum wires [6]. Wang and Lei [11] have investigated the electron-phonon interaction when calculating the linear and nonlinear

* Corresponding author. Tel.: 8666 3804 ext. 3270; fax: + 8 667 8160; e-mail: [email protected]. 0921-4526/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 4 9 1-2

c - Y . Chen et al. / Physica B 245 (1998) 92-102

93

electron transport in cylindrical quantum wires. In many research papers it has been shown that the electron-phonon interaction becomes more significant with decrease in dimension. Recently, several advanced technological developments have made the growth of not only the quantum wires but also the quantum dots (QDs) feasible. Using technologies such as molecular beam epitaxy [12], metal-organic vapor-phase epitaxy [13] and metal-organic chemical vapor deposition [14], the self-organized growth of InxGal-xAs QDs has been obtained. Especially, QDs with radius (R = )10nm [15] to 6.5 nm [16] with thickness (2d -~ ) 2.5 nm have been realized. Theoretically, most of the studies on QDs are focused on spherical QDs and only some of them are on rectangular and cylindrical QDs or on quantum disks. In fact, whereas the microcrystallites are approximately spherical, QDs are better described by thin disk [17] or cylinders [18, 19]. In the theoretical research on disk-shaped quantum structures, Kayanuma [20] has studied Wannier excitons for infinite potential barriers by using the variational method. Le Goff and St6b+ [21] have also researched the same problem for cylindrical QDs with finite potential. Guy Lamouche and Lepine [22] have obtained the ground state of an electron or a hole confined within a quantum disk by the effective-index method. Recently, Peeters and Schweifert [23] have investigated the energy levels of a two-electron quantum disk with finite potential under magnetic field. However, it has not been considered that the electron-phonon interaction above works on quantum disks. In the present work, we have investigated the eigenwavefunction, eigenenergy and the polaronic effect on energy levels of an electron in a quantum disk by using the perturbation theory. Interaction of electron both with the bulk longitudinal-optical (LO) phonons and with the surface-optical (SO) phonons are included. Considering the shape of a quantum disk whose radius is very large compared to its thickness, phonon modes which we adopted here are the slab phonon modes [24]. The second-order energy levels correction due to either mode of the phonon has been derived as function of disk radius and its thickness. The numerical computation for the GaAs quantum disk has shown that the SO phonon energy correction to the electron either because of the ground state or the first excited state in the quantum disk is much larger than that of the LO phonon. And the SO phonon energy correction to the electron in the quantum disk increases with decreasing thickness, but this decreases in the case of the LO phonon.

2.

Theory

We consider an electron moving in a disk of polar crystal with thickness 2d and radius R. In cylindrical coordinates (p, q~,z), let the disk occupy the space for Dzl <~d and p ~< R; while [z[ > d or p > R the space is vacuum. For simplicity, we assume that the isotropic effective mass approximation is valid within the disk and that the inner potential is null and the outer potential is approximately infinite. Then the Hamiltonian of the electron interacting with the longitudinal optical phonon can be written as H = Ho + Hph nt- Hc-ph.

(1)

The first term in Eq. (1) is the Hamiltonian of the electron

h2

He

-- -~----~,V2, 2m

[z I ~
~/z V2 + V0, otherwise 2mo where m* is the effective mass of the electron and mo is the free-electron rest mass, Vo is the height of the potential barrier at the surface and for an infinite approximation Vo --, oc. The second term in Eq. (1) is the phonon Hamiltonian. We note that in the case of disk, R >>d, the slab optical-phonon modes can be used here

C-Y. Chen et al. / Physica B 245 (1998) 92 102

94

approximately. Thus, ~(DLOal, m ak, m + Z

H p h = HLO + HSO = 2

k,m

Q,p

h°)sp

bte.p be,,

(3)

where a~,,n(at,,m) creates (annihilates) a confined LO phonon with frequency OgLo and wave vector K = (k, mTz/2d), b~,p(be,p) is the creation (annihilation) operator for the SO phonon with frequency Ogspand wave vector Q, subscripts m and p denote the phonon modes, m with respect to the z component of the wave vector K. It should be noticed that the K is limited by the Brillouin-zone boundary condition, i.e., mn/2d <~7t/2a (a is the lattice constant). So m can be any integer in the range 1 <~m <~d/a. p (p = + or - ) stands for the parity. For even parity, p is + , and for odd parity, p is - . The frequencies of LO phonon and SO phonon can be expressed in terms of the TO phonon frequency as

COCo= (8o/8~o)m~o,

(4)

2 = [(eo + 1) -Y-(Eo - 1)e -2aa 7 2 e)s± L(s~ + 1)+ ~ ~ ~ ] o ) T O ,

(5)

eo and eo~ are the static and optical dielectric constants, respectively. The last one in Eq. (1) is the electron-phonon interaction Hamiltonian which is given by [24-1 H~_ph=H¢ L o + H ~ - s o =~

t B*e [ i*'p k

-c°s(m~z/2d) - - -+- -

~

L m = l , 3 ....

+ a L--Q---3

t + a,,,,

Z m=2,4 ....

e-e" {C*e~a'PEG+(Q'z)bte'+ + G-(Q'z)bb'-] + H.C.},

here k is the component of the wave vector K in the

B*

s i n ( m ~ z / 2 d ) ]a1,r. } + I-I.C.

X - Y plane. Also

[41te2 /1 1 ) ] 1/2, = i/---z-;-~hCOLo[-- --

C* = "[-2/ze2

(6)

(7)

7 1/2

(8)

,

here, A and V are the surface area and volume of the disk. The functions G± are given by

ch(Qz)/ch(Qd)

[(~o~ + 1 ) - (e~o- 1)e-2°al '/4 G+(Q,z) = (E~ + 1 ) - (coo - 1)e-ZaaL(eo + 1) ( T o - 1)e---g=TNJ '

sh(Qz)/sh(Qd) [(e~ + 1) + (s~ - 1)e-aaa] G_(Q,z) = (e~ + 1) + (Coo- 1)e-2aa[_ (Co + 1) + (-~o- 1)e--e-:T~ J

(9)

1/4

(10)

In the case of electron-phonon weak coupling, the He_ph is generally small and we shall treat it as a perturbation. In the following, we proceed to calculate the energy of the polaron by the perturbation theory. The first two terms in Eq. (1) as the unperturbed Hamiltonian. First, we try to find the unperturbed wave function of the system. Since the Hamiltonians of electron and phonons in unperturbed Hamiltonian are separated, the unperturbed wave function of the system can be written as I7j) = ~O(r)I N*,m,No,p),

(11)

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C.-Y. Chen et al. / Physica B 245 (1998) 92-102

where the ~O(r)is the wave function for an electron within the disk. INk,.,, N~,) is the phonon state in particle number representation. In the low-temperature limit, only few phonons are excited. As the general approximation, we assume that no real phonon exists in the phonon ground state (i.e. phonon vacuum state) and denote it as 10,0) which satisfies

ak,mlO,O) = bq,pl0,0) = 0.

(12)

We now emphasise on finding the electron wave function which must satisfy the Schr6dinger equation

He~O(r) = EchO(r),

(13)

which can be written in cylindrical coordinates as follows:

\'~v 0(~-~2+ p~pl0 + ~ O~P 0 2 + ~z2 + -~-Ee)$(r) 2m* "X

= 0.

(14)

Because of the rotational symmetry of the ~o, the 9-dependence of the wave function possess the form e "*. Then the Schr6dinger equation can be rewritten as

(02

10

+ pop

/2

2m*

02) "

p2 + - ~ E e + ~z 2 ea~c~(p, Z) = O.

(15)

Since the radial and z-dependent parts are decoupling in He, the wave function 4~(p,z) is taken to be in product form

qS(p, z) = f(p)Z(z).

(16)

By substituting Eq. (16) into Eq. (15), one may get 1 [d2f(p) + 1 d/(p)]

f(p)Ldp~

{2m*Ee

pdp-p3+ \ - ~

l2 ~

y j-

1 d2Z(z)

(17)

Z(z) dz~

The condition that two sides in Eq. (17) are equal is that there must be a number which is neither pdependent nor z-dependent and this is denoted 22. Thus Eq. (17) becomes the following two equations:

Z"(z) + 2ZZ(z) = 0

(18)

and

if(p) + if(p) + ~--~-Ep -- -~ f(p) = 0

(19)

where

Ep = Ee - h222/2m *.

(20)

It is easy to solve Eq. (18) and get

Z(z) = A sin(2z + fl).

(21)

With the boundary conditions Z(d) = Z( - d) = 0, solution of Eq. (18) can be written as

Z(z) = A sin nrC(z,+ d)

za

(22)

C-Y. Chen et a l . / Physica B 245 (1998) 92 102

96

The quantum number n in Eq. (22) is a positive integer and is limited by the band width, that is n <~2d/a = D, where D is the number of monolayers in the disk. If we let 2/2/2~*Ep ,

x = ~/~-p

(23)

then Eq. (19) can be rewritten as

Id

ld (

+xdx + 1 - ~

f(x)=O.

(24)

This is just the Bessel equation, so the solution of Eq. (24) is

with the boundary condition

Jlt~/-~R)=O.

(26)

Assuming that x~, is the tth zero of the/th order Bessel function J~(x), one may get E, = 2m*\R ) "

(27)

Putting Eqs. (22) and (25) together, we have the eigenwavefunction of the electron in quantum disk •

~.,(r) = / l e

i,~ • n~(z + d ) . / x , , sln~a,t~-p).

(28)

Therefore, the unperturbed wave function of the system IgS) : AueiWsinmZ(2d

d!Jt(~p)iNk,mNQ,p),

(29)

with the normalization constant _

1

2

A,, J,+7(x,,)X/~?"

(30)

The unperturbed energy of the system is

= 2m*L\R ) + \2d)

J + k,m ~ Nk.mhCOLO+ ~ No.phmsp, Q,p

(31)

where, Nk.mand NQ,pare the numbers of LO and SO phonons, respectively. Second, we calculate the perturbation energy due to the electron-phonon interaction of this system. For simplicity we assume that no real phonon exists (i.e. the phonon vacuum state, Eq. (12)) in the polaron state which is valid in the low-temperature limit. Because electrons in QDs occupy only the ground subband

(0D

C.-Y. Chert et al. / Physica B 245 (1998) 92-102

97

and the intersubband transitions can be neglected Ell, 25]. Thus it is sufficient to consider l = 0. F r o m Eq. (31) it is easy to write the unperturbed wave function of the polaron within q u a n t u m disk

1 ~./Xo, ~ . ,,=(z + , 0 In't'O'O>-Jfxo,)~/Va°~RP)Sln - - 2d

(32)

IO, O > ~

and its eigenvalue

E,,,,o = 2m'\ R /

(33)

+ 2~m*\~/ "

It is obvious that the first-order perturbation energy is null as all the terms of the e l e c t r o n - p h o n o n interaction Hamiltonian, Eq. (6), are nondiagonal. In this paper, we calculate the second-order perturbation energy. The calculation of all the nonvanishing matrix elements between n, t, p h o n o n vacuum state and n', t', one-phonon state are lengthy but straightforward. F o r simplicity, here we only have the results:

=

4nBdAotAoc ~ , 1 / m ~ 2q ,/2Lin + m ) 2

+- F 2

-

(n' - m) 2

n2

-

n2

t +t )l x o Jo(kp)Jo(---~p Jo

p p dp,

(34)

where m = 1, 3, 5,..., n' - n = 0, ___2, + 4 , . . . , and + ( - ) sign which appears in the right side of Eq. (34) is applicable to the case when (m + 1)/2 is odd (even);

4nBdAotAoc [i 1 (n,t,O, OIHe-phln',t',k , 0 > = -I- Ik2 + \2d/l(mrt~211/2J n' + m ) 2 - n

×

1 2

(n' -

m) 2 -

] n2

L"Jo(k )Jo ( ) ( Jo)

where m = 2, 4, 6 .... , n' - n = + 1, _ 3, + 5 . . . . . and + ( - ) sign which appears in the right side of Eq. (35) is applicable to the case of m/2 is even (odd); (n,t,O, OlHe-phln',t',O,Q + > =

4C xdJx(xot)J,(xoc)x/[(Xot + Xo,,)z - (QR)Z][(QR) 2 - (Xot - Xo,,)2]

ch(Qd)

2 + [ ( f f L n 2)~d]2

Q2 + [(n~+ n 2~d

x [(eo~ + 1) - (eo~ - 1)e-2qa]3/4[(eo + 1) -- (,% -- 1)e-2aall/4'

(36)

98

C.-Y. Chen et al. /Physica B 245 (1998) 92-102

where n' - n -- O, + 2, _+ 4 . . . . . and 4C
ndJl(xot)Jx(xoc)x/[(Xot

x

~- Xot') 2 - -

e -O-a cth(Qd)

(QR)2][(QR) 2 - ( X o ,

Q2 +

( n ' - - n~2d

-- Xot')21

Q2 +

(n' +

1 x [(e~ + 1) + (eo~ - 1)e-2ed]3/4[(eo + 1) + (eo -- 1)e-2ea] 1/4'

n~2d (37)

wheren'-n= ___1,___3,+_5,.... F r o m Eqs. (34)-(37) and using the second-order perturbation theory, we obtain the second-order correction to the p o l a r o n energy. The calculating process is still lengthy but straightforward. Here we give the final results: A E = AELo + AEso.

(38)

Here AELo and AEso are the L O and SO p h o n o n contributions for the correction to the p o l a r o n energy, respectively. They are written as the following. The first part in Eq. (38) is (39)

AELo = AEk+ + AEk with AEk+ = -

t',n',m,k

I12 , 4096n2dSul En',t',k+ -- En,t,o = -- 0~LOF/(DLO 7z484

(.'-m)2-. x

2 2 2 Z j2(Xo,)JZ1(Xo,,)[(n,2 _ n2)lr2 + I Xo,, - Xo,l(2d/R) + (2du,) z] t~,n'~rn

(40) w h e r e n' - n = O, +_ 2, ___ 4 , . . .

AEk =-

a n d m -- 1, 3, 5,...

~ Kn't'O'O[He-k-ln"t"k-'O>12= --~XLohmL040~ 5uzt',n',m,k

En',C,k -- En,t,o i n' + m) 2 -- n 2

X

E

2

2

,,,,,,m J l(xo,)J l(xo,,)[(n

t2

(n' -- m) 2 -- n 2

2 - n2)n 2 + I x,,,, - x2o, l (2d/R) 2 + (2du~)2]

><{F,In[fl_(~dlt/a)2]+Fzln[l /~

(2dlt)Zl+F31n[l+ 7 a2 I

(2rtd)21"[ za2 J J '

(41)

where n' - n = + 1, + 3, + 5 .... and m = 2,4,6 . . . . . In Eqs. (40) and (41), we have defined the functions 1 E l = (fl -t'- "C)(fl -- ~)'

1 F 2 -- (fl -- ~))('~ + "g)'

fl = [(Xo, + xo,,)2d/R] 2,

-1 F3 -- (fl + Z)(3' + z)'

~ = [(Xo,, -- Xo,)2d/R] 2,

z = (taro 2,

(42) (43)

C.-Y. Chen et al. / P h y s i c a B 245 (1998) 9 2 - 1 0 2

99

and the electron-LO-phonon coupling constant ~LO = m * e 2 f L

(44)

where the wave number u~ is given by (45)

h2u~ /2m" = he)Lo .

The second part in Eq. (38) is (46)

A E s o = A E a+ + AEQ with

I(n, t,O,O I He-Q+I n', t',O, Q+)I 2

AEQ+ = t',n',Q

256duze~2e~/2 =

En,,t,,Q + - - En,t, o

~2dTt/a~

- - ~LohC.OLo

1r2

x Z e - X shxthZ(x/2)

× n',t'E JO [J2(xot)J2(Xot')[(n'2

__

2 /12)7/:2+ I Xo,' -- Xot2 l(2d/R) 2 + (2dus+) 2]

1

1

"X2 -~- ( n t - - /'1)2~ 2

X 2 + (rl' + n)27z 2

12

x [(eo~ + 1) - ( ~ - 1)e-x]a/z[(eo + 1) - (eo - 1)e-X] 1/z .dx,

(47)

where n' - n = 0, _+ 2, _+ 4,... AEe

= -

~

I(n't'O'OIH*-e-

t',n',Q

l n " t " O ' Q - )12 = _ C~LohOgLo 256dute~2e~/2

En',t',Q- -- En,t,o

1r'2

~2a,/aj" x 2 e - Xshxcth2(x/2) × tl',t" E Jo [j2(Xo,)j2(Xot')[(n'2 -- n2) n2 + [ X o2" -- xo,2 [(2d/R)Z + (2dus_)2] I ×

1 X 2 -~- (n' -

]2 (n' + n)2n ~ 1

n)2ff 2 -

x 2 +

x [(e~ + 1) + (e~ - 1)e-~3/2[(eo + 1) + (eo - 1)e-X] 1/2 .dx,

(48)

where n' - n = + 1, + 3, + 5 ..... In Eqs. (47) and (48), Us+ and us_ are the wave numbers of even- and odd-parity SO phonon and are defined as h2u2+/2m" = htos+,

(49)

hZu 2/2m* = ho)s .

(50)

100

C.-E Chen et al. / Physica B 245 (1998) 92-102 40

30

20 t

10

5

10

15 D

20

25

30

Fig. 1. The polaronic energy corrections of an electron in a GaAS quantum disk with radius R = 500 A versus thickness D. The solid line and the dashed line represent the results in ground state (n = 1) and first excited state (n = 2), respectively.

3. Results and discussion In the above section, we obtained the eigenwavefunction and eigenenergy of an electron in a quantum disk by solving the Schr6dinger equation in cylindrical coordinate, as well as the second-order energy correction due to the coupling of electron-phonon by the perturbation theory which is valid for the weak elect r o n - p h o n o n coupling. In general, the method of perturbation theory may be used for the electron-phonon coupling constant ~ < 1. Now, use the above theory to calculate numerically the energy correction for the ground state (n = 1) and the first excited state (n = 2) of an electron in a GaAs quantum disk. The parameters of the conduction electron used in the computation are taken from Ref. [26]: eo = 13.18, e~ = 10.89, hco = 36.25 meV, a = 0.068, m = 0.067mo and the lattice constant a = 5.6533 A. The results of the energy shifts which is caused by phonon on the ground state and the first excited state are plotted as a function of thickness D for the disk radius R = 500 and 1000 ~, in Figs. 1 and 2, respectively. In order to see the difference between the electron energy corrections caused by the L O and SO phonon, we tabulate their absolute values in the ground-state and first excited-state for two different radii R = 500 ~,and 1000 k i n Table 1. We take the thickness D = 10 and 40 for comparison. In all, firstly, the results of Figs. 1 and 2 show that the energy shift caused by the phonon in either the ground state or the first excited state decreases quickly with increasing thickness and the curves flatten at large D. Secondly, the calculation results show that the L O phonon energy correction to the electron in either the ground state or the first excited state increases with increasing thickness while decreases in the case of the SO phonon. We notice that the SO phonon energy correction to either the ground state or the first excited state of the electron in quantum disk is much larger than that of the L O phonon energy correction. This can be observed

101

C.-Y. Chen et a l . / P h y s i c a B 245 (1998) 92 102

20

15

lO

5

0

I

I

I

I

k

5

10

15

20

25

30

D Fig. 2. The polaronic energy corrections of an electron in a GaAs q u a n t u m disk with raidus R = 1000 ,~ versus the thickness D. The solid line and the dashed line represent the results in ground state (n = 1) and first excited state (n = 2), respectively.

Table 1 Polaronic effects on the electron ground state (n = 1) and first excited state (n = 2) energy, where AELo and A E s o (the unit of them is meV) are the energy shifts caused by the LO phonon and SO phonon, respectively, for two values of the disk thickness D. The disk radius is R1 = 500 A or R2 = 1000 D = 10 - AELo

R1 R2

n n n n

= = = =

1 2 1 2

0.0784 0.0892 0.0418 0.0495

D = 40 - AEso

6.7790 6.3350 4.723 4.0301

AELo/AE(%)

1.14 1.39 0.88 1.21

-- AELo

0.2132 0.3893 0.1409 0.2740

-- AEso

0.9399 1.1250 1.1271 1.2001

AELo/AE(%)

18.48 25.70 11.11 18.59

from Table 1. The percentage effects on the total energy correction by the LO phonon within the range from about 1% when the disk thickness is small to about 26% when the disk thickness is large. Therefore the self-energy contribution by SO phonon is rather significant and is even dominant at small thickness. Thus, SO phonon-electron coupling effects on the polaron in quantum disk is very significant and it must be taken into account in the investigation of polaron properties within disk-shaped quantum dots. In conclusion, we have studied the eigenwavefunction and eigenenergy of an electron in a quantum disk and the polaronic correction to its energy levels in the present paper. Both the bulk LO phonon and the SO phonon modes in a slab are used in the consideration of the interaction between an electron and a phonon in a quantum disk. The energy correction due to either mode of phonon is calculated as a function of radius and

102

C.-Y. Chen et aL / Physica B 245 (1998) 92-102

thickness of the disk. Our numerical computations for the GaAs quantum disk have shown that the electron energy correction due to the LO phonon in quantum disk increases with increasing thickness but decreases in the case of SO phonon energy correction. However, the energy correction by SO phonon is rather significant and is even dominant when the thickness is small.

Acknowledgements This work is supported by the RGC and UGC of Hong Kong. Two of authors (C.Y. Chen and S.D. Liang) are also supported by the Natural Science Foundation of Guangdong Province of China.

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