Effect of electron-phonon scattering mechanisms on free-carrier absorption in quasi-one-dimensional structures

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Physica B 316–317 (2002) 346–349

Effect of electron–phonon scattering mechanisms on

free-carrier absorption in quasi-one-dimensional structures

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 free-carrier absorption in ultrathin wires fabricated from III–V semiconductors such as n-type InSb has been investigated for the case where the electrons are scattered either by polar optical phonons or acoustic phonons. We study the interaction of longitudinal polar optical phonons with electrons and have neglected the interaction between electrons and transverse optical phonons in solids. The energy band of electrons in semiconductors is assumed to be nonparabolic. The scattering mechanisms of the interaction between electrons and phonons we consider here come from (a) electron-polar–optical-phonon scattering, (b) electron–acoustic–phonon scattering, and (c) piezoelectric scattering in semiconductors. Results are shown that the free-carrier absorption coefficient for the deformation-potential coupling is much larger than that for the piezoelectric coupling. It is also shown that the free-carrier absorption coefficient for the electron-polar–optical-phonon scattering is smaller than that for the electron–acoustic– phonon scattering. However, the free-carrier absorption coefficient increases quite slowly with the photon frequency for the electron–acoustic–phonon scattering. This is not the same result as that for the quasi-two-dimensional semiconducting structures. r 2002 Published by Elsevier Science B.V.

Keywords: Semiconducting wires; Polar optical phonons; Acoustic phonons

1. Introduction

One-dimensional structures such as quantum wires have received a considerable attention in recent years because of their physical properties and their potential device applications [1,2]. The motion of electrons in such semiconducting structures is conﬁned and leads to size quantiza-tion effects which play an important role in determining their optical and electronic properties. In this work, we investigate the intraband optical absorption in quantum-well wire structures due to the absorption of the photons by free electrons in

the system and study the size quantization effect in quasi-one-dimensional structures. We consider a nondegenerate electron gas with a Maxwell– Boltzmann distribution for nondegenerate semi-conductors. Here, it is assumed that the interaction between electrons and phonons originates from

(a) electron-polar–optical-phonon scattering, (b) electron–acoustic–phonon scattering, and (c) piezoelectric scattering in semiconductors.

2. Theory

For a rectangular thin wire of dimensions a and d; the eigenfunctions and eigenvalues for electrons

*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 4 - 5

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in a semiconductor with nonparabolic band structure are given by [3]

CkxcnðrÞ ¼ 2 V1=2bkxcne ikxx_sin pcy a sin pnz d ; ð1Þ Ekxcn¼ 1 2Eg ( 1 1 þ 4 Eg : _ 2 k2_x 2m þp 2_{_}2 2m c 2 a2þ n2 d2 1=2) ; c; n ¼ 1; 2; 3; y; ð2Þ where V is the volume of the thin wire, m is the effective mass of electrons in semiconductors, Eg

is the energy gap between the conduction and valence bands, and bkxcn is the annihilation

operator of electrons, satisfying commutative relations of the Fermi type. The absorption coefﬁcient for the absorption of photons can be expressed as [4] a ¼e 1=2 n0c X i Wifi; ð3Þ

where e is the dielectric constant of the material, n0

is the number of photons in the radiation ﬁeld, fiis

the free-carrier distribution function, and Wiis the

transition probability. The transition probability is given with the Born approximation

Wi¼

2p _

X

f

½j/f jMþjiSj2dðEf Ei_O _oÞ

þ j/f jMjiSj2dðEf Ei_O þ _oÞ; ð4Þ

where Ei and Ef are the initial and ﬁnal electron

energies, _O is the photon energy, and _o is the phonon energy. For the interaction between electrons, photons, and phonons, the transition matrix elements /f jM₇jiS can be calculated from the electron–phonon interaction Hamiltonian and different electron–phonon scattering mechanisms. The transition probabilities for different mechan-isms are given as follows.

2.1. Electron-polar–optical-phonon scattering W_iPOP¼2p 3_e4_{_n} 0nqoðO oÞ2 ee0_ðm_Þ2_OV2 X cf;nf

ðO þ 2oÞ2ð2O þ oÞ2L2_c_i_c_fðqyÞL2ninfðqz

h

Þ dðEf Ei_O _oÞ þ O2ð2O oÞ2

L2 cicfðqyÞL 2 ninfðqzÞdðEf Ei_O þ _oÞ i " 2K1=2ðci;cf; ni; nfÞ L1=2ðci;cf; ni; nfÞ tan12L1=2ðci;cf; ni; nfÞK1=2ðci;cf; ni; nfÞ 2Lðci;cf; ni; nfÞ ðO=vsÞ2 # ; ð5Þ where e0_{¼ ðe}1 N e 1_Þ1

; eN and e are the

high-frequency and static dielectric constants, respec-tively, and nq is the number of phonons in the

mode of a wave vector q in the thermal equili-brium. Moreover,

Lnn0ðqÞ ¼ d_q;pðn0_nÞ=aþ d_q;pðn0_nÞ=a d_q;pðn0_þnÞ=a

d_q;pðn0_þnÞ=a¼ L_nn0ðqÞ; ð6Þ Kðci;cf; ni; nfÞ ¼ O vs 2 p a 2 ðcf ciÞ2 p d 2 ðnf niÞ2; ð7Þ Lðci;cf; ni; nfÞ ¼ p a 2 ðcf ciÞ2 þ p d 2 ðnf niÞ2: ð8Þ

2.2. Electron-acoustic phonon scattering W_iACP¼p 2_e2_n 0E2dkBT ðO oÞ2 3ðm_Þ2 rv2 seV2O X cf;nf K3=2ðci;cf; ni; nfÞ

½ðO þ 2oÞ2ð2O þ oÞ2 L2 cicfðqyÞL 2 ninfðqzÞdðEf Ei_O _oÞ þ O2ð2O oÞ2L2_c_i_c_fðqyÞL2ninfðqzÞ dðEf Ei_O þ_ˇloÞ; ð9Þ

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Fig. 1. (a) Free-carrier absorption coefficient in n-type InSb wires as a function of photon frequency with a ¼ 100 (A and d ¼ 200 (A for electron-polar–optical-phonon scattering. (b) Free-carrier absorption coefficient in n-type InSb wires as a function of photon frequency with a ¼ 100 (A and d ¼ 200 (A for electron–acoustic–phonon scattering. (c) Free-carrier absorption coefficient in n-type InSb wires as a function of photon frequency with a ¼ 100 (A and d ¼ 200 (A for piezoelectric scattering.

C.-C. Wu, C.-J. Lin / Physica B 316–317 (2002) 346–349 348

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where r is the density of material and Ed is the deformation potential. 2.3. Piezoelectric scattering W_iP¼p 2_e4_b2 Pn0kBTðO oÞ2 2e3_ðm_Þ2_rv2 sV2O X cf;nf ðO þ 2oÞ2

ð2O þ oÞ2L2_c_i_c_fðqyÞLninfðqzÞdðEf Ei

_O _oÞ þ O2ð2O oÞL2 cicfðqyÞ L2_n_i_n_fðqzÞdðEf Ei_O þ _oÞ " 2K1=2ðci;cf; ni; nfÞ L1=2ðci;cf; ni; nfÞ: tan12L 1=2_ð_c i;cf; ni; nfÞK1=2ðci;cf; ni; nfÞ 2Lðci;cf; ni; nfÞ ðO=vsÞ2 # ; ð10Þ where b_P is the piezoelectric constant.

3. Numerical analysis

From Eqs. (3)–(6), (9) and (10), we can calculate the free-carrier absorption coefﬁcient for different electron–phonon scattering mechanisms. The re-levant values of physical parameters for n-type InSbultrathin wires are taken to be [4,5] ne

(electron concentration)=1.75 1014cm3, m_¼

0:013m0 (m0 is the mass of free electron), e ¼ 18;

eN¼ 16; Eg¼ 0:2 eV, Ed¼ 4:5 eV, r ¼ 5:8 gm/

cm3, b_P¼ 1:8 104_esu/cm2

, vs¼ 4 105cm/s,

and o ¼ 5:5 1013_{rad/s. In Fig. 1(a), the}

free-carrier absorption coefﬁcient a in n-type InSbfor a quasi-one-dimensional structure is plotted as a function of the photon frequency when the

electron-polar–optical-phonon scattering is domi-nant. It is shown that a decreases monotonically with increasing the photon frequency and in-creases with increasing temperature. This is similar to what is predicted to occur in quasi-two-dimensional structures when the free carriers are scattered by polar optical phonons [5]. In Fig. 1(b), the free-carrier absorption coefﬁcient is plotted as a function of the photon frequency when the electron–acoustic–phonon scattering is dominant. It can be seen that a increases with increasing the photon frequency initially and then increases quite slowly in the high photon frequency region. This is not the same result as that for quasi-two-dimen-sional structures when the electrons are scattered by acoustic phonons [6]. In Fig. 1(c), the free-carrier absorption coefﬁcient is plotted as a function of the photon frequency when the piezo-electric scattering is dominant in semiconductors. It shows that a decreases monotonically with increasing the photon frequency and increases with increasing temperature. However, the numer-ical results of a for the piezoelectric scattering are much smaller than those for other two kinds of phonon scatterings.

References

[1] C.M. Sotmayor Torres, P.D. Wang, W.E. Leitch, H. Benisty, C. Weisbuch, Proceedings of the International Meeting, Giardini Naxos, Italy, 24–27 September 1991. [2] U. R.ossler, Proceedings of a NATO Advanced Study

Institute on Quantum Transport in Ultrasmall Devices, 17– 30 July, 1994, Italy, Plenum, New York, 1995.

[3] C.C. Wu, H.N. Spector, Phys. Rev. B 3 (1971) 3979. [4] H. Adamska, H.N. Spector, J. Appl. Phys. 59 (1986) 619. [5] C.C. Wu, C.J. Lin, J. Appl. Phys. 79 (1996) 781. [6] C.C. Wu, C.J. Lin, Physica B 205 (1995) 183.