Nonlinear effects of energy band structures on optical transitions in quantum dots

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Physica B 316–317 (2002) 342–345

Nonlinear effects of energy band structures on optical

transitions in quantum dots

Chhi-Chong Wu

a,

*, Chau-Jy Lin

b

a

Institute of Electronics, National Chiao Tung University, Hsinchu, Taiwan

b

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

Abstract

The quantum theory of nonlinear effects for optical transitions of electrons in quasi-zero dimensional (Q0D) quantum dots fabricated from n-type III–V compound semiconductor materials such as n-type GaAs has been studied due to the nonparabolicity of energy band structures. We use the effective mass approximation for carriers in the quantum dots. Most realistic quantum dot systems contain the box with a thickness c and the lateral width (a; b). Using the time-independent perturbation theory, the first order correction of the eigenfunctions and eigenvalues for the system has been calculated. And the free-carrier absorption coefficient may be calculated for Q0D quantum dots from n-type GaAs where the polar optical phonon scattering is dominant. Our results show that the free-carrier absorption coefficient in Q0D quantum dots increases quite rapidly with increasing temperature in the region of low temperatures. When temperatures are larger than 100 K, the free-carrier absorption coefficient increases slowly with temperature. This shows that the nonlinear property of energy band structures due to the nonparabolicity plays an important role in low temperatures. The discussion about the dot size effect of the quantum confinement region in n-type GaAs quantum dots has also been given. r 2002 Published by Elsevier Science B.V.

Keywords: Optical transition; Quantum dot system; Polar optical phonon scattering

1. Introduction

Low-dimensional quantum nanostructures such as quantum wires and quantum dots have attracted considerable attention in view of their basic physics and potential device applications [1– 3]. Especially, the interesting things in the fabrica-tion of low-dimensional semiconductor structures are the modiﬁed electronic and optical properties of these structures, which are controllable to a certain degree through the ﬂexibility in the structure design. These features make quantum

confined semiconductors very promising for pos-sible device applications in microelectronics, non-linear optics, and many other fields. The electron confinement in low-dimensional structures modi-fies the density of states and enhances the Coulomb interaction of electrons. In this work, electrons in a quantum dot are confined by a heterostructure of compound semiconductors such as Al0:45Ga0:55As=GaAs:

2. Theory

As a quantum dot is assumed to be a well with inﬁnitely high-energy barriers, the effective

*Corresponding author. Fax: +886-35-724361. E-mail address:[email protected] (C.-C. Wu).

0921-4526/02/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 0 3 - 3

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Hamiltonian for the nonparabolic energy band can be expressed as Heff¼ H 1 þ H Eg ¼X i p2_i 2mþ V0ðriÞ þ1 2 X iaj e2 ejri rjj ; ð1Þ

where Egis the energy gapbetween the conduction

and valence bands, and V0ðriÞ is an inﬁnite

potential deﬁned as follows: V0¼

0 for jxjoa; jyjob; jzjoc; N for jxj > a; jyj > b; jzj > c: (

ð2Þ Using the time-independent perturbation theory, the eigenfunctions and eigenvalues for the system are given by C_cmnðrÞ ¼ 8 abc 1=2 sin pcx a sin pmy b sin pnz c þ 2 ffiffiffi 2 p e2 p2_ðabcÞ3=2_eE g X ðc0_;m0_;n0_Þ aðc;m;nÞ Pðc0_{c; m}0_{m; n}0_nÞ Qðc; m; n; c0; m0; n0Þ sin pc 0_x a sin pm 0_y b sin pn 0_z c ð3Þ and E_cmn¼ 1 2 1 1 þ 2p2_{_}2 m_E g c2 a2þ m2 b2 þ n2 c2 1=2 ( ) ; ð4Þ where Pðc; m; nÞ and Qðc; m; n; c0; m0_{; n}0_{Þ are}

func-tions of c; m; n and c0; m0_{; n}0 _{with parameters a; b;}

and c:

The electron-polar–optical-phonon interaction potential is given by [4,5] Us¼ ieo 4p e0_V 1=2_X q q #eq jqj2 ½expðiq rÞQ n q expðiq rÞQq; ð5Þ

where q is the wave vector of the polar optical phonons, #eq is their polarization vector,

and e0¼ ð1=eN 1=eÞ 1

: Here eN and e are

the high-frequency and static dielectric con-stants of semiconductors, respectively, Qn

q and

Qq are creation and annihilation operators

for the optical phonons. The matrix element of the electron–photon interaction Hamil-tonian is /c0 m0n0jHradjcmnS ¼ ffiffiffi 2 p p3=2_e_{_}3=2 m n0 eOabc 1=2 c 2 a2þ m2 b2 þ n2 c2 1=2 d_c0_;_cd_m0_;md_n0_;n; ð6Þ

where n0 is the number of photons in the

radiation ﬁeld and O is the photon fre-quency. The electron distribution function in a nondegenerate semiconductor can be expre-ssed as f_cmn¼ neeEcmn=kBT X cmn eEcmn=kBT !1 ; ð7Þ

where neis the concentration of electrons in solids.

The free-carrier absorption coefﬁcient can be obtained as a ¼ 8p 3_e4 e1=2_e0_m_abcv cO exp _o kB 1 1 X c;m;n exp Ecmn kBT " #1 X cf>ci¼1 X mf>mi¼1 X nf>ni¼1 eEcimini=kBT c 2 f a2þ m2 f b2 þ n2 f c2 ! ( ðO þ 2oÞ2þ O2 þ c 2 i a2þ m2_i b2 þ n2_i c2

ð2O þ oÞ2þ ð2O oÞ2

þ 2 c 2 i a2þ m2 i b2 þ n2 i c2 1=2 _c2 f a2þ m2 f b2 þ n2 f c2 !1=2 1

ðO þ 2oÞð2O þ oÞþ 1 Oð2O oÞ

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ðcf ciÞ 2 a2 þ ðmf miÞ2 b2 þ ðnf niÞ2 c2 " # ( 1 ðcf þciÞ 2 a2 þ ðmf þ miÞ2 b2 þ ðnf þ niÞ2 c2 " #19₌ ;; ð8Þ where vcis the velocity of light.

3. Numerical results

As a numerical example, we consider the free-carrier absorption coefﬁcient in n-type GaAs with the optical phonon frequency o ¼ 5:5 1013rad/s. The relevant values of physical parameters are taken to be [6] ne¼ 1:73 1015cm3, m¼ 0:07m0

(m0 is the mass of free electrons), e ¼ 12:9; eN¼

11:1 and Eg ¼ 1:51 eV. We plot the free-carrier

absorption coefﬁcient a as a function of lateral width a ¼ b as shown in Fig. 1 with the photon frequency O ¼ 53:4 THz (or 5.6 mm wavelength of a CO2 laser). It is shown that a decreases with

increasing the lateral width a ¼ b or the ﬁlm thickness c: It can also be seen that a increases with temperature. This is the same property as that for the acoustic phonon scattering [7] and for the polar optical phonon scattering [6] in quasi-two-dimensional structures. In Fig. 2, we plot a as a function of the photon frequency O with the lateral width a ¼ b ¼ 20 nm. It shows that a decreases monotonically with increasing photon frequency. In Fig. 3, a is plotted as a function of temperature with O ¼ 53:4 THz and the ﬁlm thickness c ¼ 10 nm. It shows that a increases rapidly with

Fig. 1. Free-carrier absorption coefﬁcient as a function of lateral width. Solid curves: c ¼ 10 nm; broken curves: c ¼ 20 nm.

Fig. 2. Free-carrier absorption coefﬁcient as a function of photon frequency. Solid curves: c ¼ 10 nm; broken curves: c ¼ 20 nm.

C.-C. Wu, C.-J. Lin / Physica B 316–317 (2002) 342–345 344

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temperature in the lower-temperature region, and then increases slowly with temperature in the higher-temperature region.

Acknowledgements

This study was supported by the National Science Council, Republic of China, under con-tract number NSC 89-2215-E-009-061.

References

[1] H. Sakaki, Jpn. J. Appl. Phys. 19 (1980) L735.

[2] Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939. [3] H. Gotoh, H. Ando, H. Kanbe, Appl. Phys. Lett. 68 (1996)

2123.

[4] S. Das Sarma, A. Madhukar, Phys. Rev. B 22 (1980) 2823. [5] P.J. Price, Ann. Phys. 133 (1981) 217.

[6] C.C. Wu, C.J. Lin, J. Appl. Phys. 79 (1996) 781. [7] C.C. Wu, C.J. Lin, Physica B 219&220 (1996) 65.

Fig. 3. Free-carrier absorption coefﬁcient as a function of temperature with O ¼ 53:4 THz and c ¼ 10 nm.