Electron-acoustic phonon field induced tunnel scattering

Theory of electron-acoustic single phonon scattering has been reconsidered. It is assumed that the non-degenerate semiconductor has a spherical parabolic band structure. In the basis of the reconsideration there is a phenomenon of the tilting of semiconductor bands by the perturbing potential of an...

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Datum:	2015
Hauptverfasser:	Melkonyan, S.V., Harutyunyan, A.L., Zalinyan, T.A.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2015
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spelling	irk-123456789-1535842019-06-15T01:29:27Z Electron-acoustic phonon field induced tunnel scattering Melkonyan, S.V. Harutyunyan, A.L. Zalinyan, T.A. Theory of electron-acoustic single phonon scattering has been reconsidered. It is assumed that the non-degenerate semiconductor has a spherical parabolic band structure. In the basis of the reconsideration there is a phenomenon of the tilting of semiconductor bands by the perturbing potential of an electric field. In this case, electron eigenfunctions are not plane waves or Bloch functions. In low-field regime, the expressions for electron intraband transition probability and scattering time are obtained under elastic collision approximation. Dependencies of scattering time on electron energy and uniform electric field are analyzed. The results of corresponding numerical computations for n-Si at 300 K are presented. It is established that there is no fracture on the curve of electron scattering time on the electron energy dependence. У статтi подано новий погляд на теорiю електронноакустичного розсiювання одного фонона. При цьому припускається, що невироджений напiвпровiдник має сферичну параболiчну зонну структуру. В основу перегляду теорiї покладено ефект нахилу напiвпровiдникових зон при накладаннi збурюючого потенцiалу електричного поля. У цьому випадку власнi функцiї електрона вже не є плоскими хвилями чи функцiями Блоха. В режимi слабких полiв отримано вирази для ймовiрностi електронних внутрiшньозонних переходiв i для часу розсiяння в наближеннi пружнiх зiткнень. Також проаналiзовано залежнiть часу розсiяння вiд енергiї електрона та напруженостi однорiдного електричного поля. Представлено результати вiдповiдних числових обчислень для n-Si при температурi 300 K. Встановлено вiдсутнiсть зламу на кривiй залежностi часу розсiяння електрона вiд енергiї електрона. 2015 Article Electron-acoustic phonon field induced tunnel scattering / S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan // Condensed Matter Physics. — 2015. — Т. 18, № 2. — С. 23702: 1–12. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 72.10-d, 63.20.kd DOI:10.5488/CMP.18.23702 arXiv:1506.03971 http://dspace.nbuv.gov.ua/handle/123456789/153584 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Theory of electron-acoustic single phonon scattering has been reconsidered. It is assumed that the non-degenerate semiconductor has a spherical parabolic band structure. In the basis of the reconsideration there is a phenomenon of the tilting of semiconductor bands by the perturbing potential of an electric field. In this case, electron eigenfunctions are not plane waves or Bloch functions. In low-field regime, the expressions for electron intraband transition probability and scattering time are obtained under elastic collision approximation. Dependencies of scattering time on electron energy and uniform electric field are analyzed. The results of corresponding numerical computations for n-Si at 300 K are presented. It is established that there is no fracture on the curve of electron scattering time on the electron energy dependence.
format	Article
author	Melkonyan, S.V. Harutyunyan, A.L. Zalinyan, T.A.
spellingShingle	Melkonyan, S.V. Harutyunyan, A.L. Zalinyan, T.A. Electron-acoustic phonon field induced tunnel scattering Condensed Matter Physics
author_facet	Melkonyan, S.V. Harutyunyan, A.L. Zalinyan, T.A.
author_sort	Melkonyan, S.V.
title	Electron-acoustic phonon field induced tunnel scattering
title_short	Electron-acoustic phonon field induced tunnel scattering
title_full	Electron-acoustic phonon field induced tunnel scattering
title_fullStr	Electron-acoustic phonon field induced tunnel scattering
title_full_unstemmed	Electron-acoustic phonon field induced tunnel scattering
title_sort	electron-acoustic phonon field induced tunnel scattering
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/153584
citation_txt	Electron-acoustic phonon field induced tunnel scattering / S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan // Condensed Matter Physics. — 2015. — Т. 18, № 2. — С. 23702: 1–12. — Бібліогр.: 10 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT melkonyansv electronacousticphononfieldinducedtunnelscattering AT harutyunyanal electronacousticphononfieldinducedtunnelscattering AT zalinyanta electronacousticphononfieldinducedtunnelscattering
first_indexed	2025-07-14T05:02:11Z
last_indexed	2025-07-14T05:02:11Z
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fulltext	Condensed Matter Physics, 2015, Vol. 18, No 2, 23702: 1–12 DOI: 10.5488/CMP.18.23702 http://www.icmp.lviv.ua/journal Electron-acoustic phonon field induced tunnel scattering S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan Department of Physics of Semiconductors & Microelectronics, Yerevan State University, 1 Alex Manoogian St., 0025 Yerevan, Armenia Received October 16, 2014 Theory of electron-acoustic single phonon scattering has been reconsidered. It is assumed that the non-dege- nerate semiconductor has a spherical parabolic band structure. In the basis of the reconsideration there is a phenomenon of the tilting of semiconductor bands by the perturbing potential of an electric field. In this case, electron eigenfunctions are not plane waves or Bloch functions. In low-field regime, the expressions for elec- tron intraband transition probability and scattering time are obtained under elastic collision approximation. Dependencies of scattering time on electron energy and uniform electric field are analyzed. The results of cor- responding numerical computations for n-Si at 300 K are presented. It is established that there is no fracture on the curve of electron scattering time dependence on the electron energy. Key words: tilted band semiconductor, electron-acoustic phonon scattering, transition probability, scattering time PACS: 72.10-d, 63.20.kd 1. Introduction Current carrier (electron) mobility µ is an important parameter characterizing many transport phe- nomena in semiconductors under electric field F . Electron mobility is determined as µ = 〈eτk/m〉 [1, 2], where τk is the electron quasi-momentum relaxation time, e is the electron charge magnitude, m is the electron effective mass, 〈· · · 〉 is the symbol of averaging over conduction zone quantum states. Relaxation time τk is determined by the electron scattering by various dynamic and static imperfections of a crys- tall lattice such as lattice vibrations (optic and acoustic phonons), ionized and neutral impurity atoms, vacancies, etc. For theoretical consideration of scattering probability and relaxation time τk, a flat-band semiconductor model is used, as a rule [1–4]. In this case, in low electric field region (F < Fc, where Fc is a characteristic field), the electron relaxation time τk and, therefore, the mobility µ are field-independent quantities [1, 4, 5]. Particularly, the relaxation time related to electron-acoustic single phonon elastic scat- tering in non-degenerate n-type semiconductor with a spherical parabolic conduction band is given by [1–4] 1/τk,ac = D2 ac(2m)3/2kBT 2πħ4ρrν 2 0 p εk . (1) Here, ρr is a reducedmass density of a crystal, Dac is the acoustic deformation potential constant, kB is the Boltzmann constant, T is temperature, ν0 is the long-wavelength longitudinal acoustic phonon velocity, εk =ħ2k2/2m is the electron energy, boldface k is the electronwave vector (herein below, themagnitudes of vector quantities are denoted by non-boldface symbols). At high electric fields (F > Fc), the time τk and, therefore, the mobility µ depend on the applied elec- tric field [4]. Thus, at electron scattering by an acoustic phonon, electron mobility decreases with an increase of electric field above Fc [5]. The magnitude of characteristic field Fc depends on semiconductor parameters such as crystallographic directions, impurity concentration, temperature, etc. It is of an or- der of 103 V/cm at 300 K, e.g., for pure Si Fc ≈ 103 V/cm, for pure Ge Fc ≈ 600 V/cm, for high purity GaAs © S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan, 2015 23702-1 http://dx.doi.org/10.5488/CMP.18.23702 http://www.icmp.lviv.ua/journal S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan Fc ≈ 2.8 ·103 V/cm [5]. The dependence µ(F ) is explained by the phenomenon of electron gas heating-up under the effect of a high electric filed [1, 4]. However, in recent work [6], a new mechanism of electron lattice scattering, referred to as electron-phonon FIT (field induced tunnel) scattering, is observed. In the basis of the FIT scattering there lies a phenomenon of tilting of semiconductor bands by the perturbing potential of an electric field. The effect of the electron-phonon FIT scattering is explained in terms of penetration of an electron wave function into a semiconductor band gap in the presence of an electric field. Contrary to a flat-band semiconductor, in a tilted-band semiconductor, a conduction electron tran- sition in the band gap region is allowed. In [6], reconsidering the electron-phonon interaction theory, the case of electron intraband FIT scattering by non-polar optical phonon is analyzed. In the present work, electron-acoustic phonon FIT intraband scattering is considered. It is assumed that the non-degenerate n-type semiconductor has a parabolic conduction band. 2. Electron-acoustic phonon FIT transition probability To theoretically characterize the carrier scattering, it is necessary to consider the scattering probabil- ity and evaluate the relaxation time. The task of the transition probability calculation is solved based on the perturbation theory (see, e.g. [1–4]). According to this theory, the probability per unit time of quantum system transition from λ state to λ′ state, to the first order in the perturbation, is determined as [1, 2] W (λ,λ′) = 1 ħ2 d dt ∣ ∣ ∣ ∣ ∣ ∣ t ∫ 0 aλ′,λ(t)dt ∣ ∣ ∣ ∣ ∣ ∣ 2 . (2) Here, aλ′,λ(t) is the perturbation matrix element: aλ′,λ(t) = ∫ V dRψ∗ λ′ (R, t)Ŵ (R, t)ψλ(R, t), (3) ψλ(R, t) =ψλ(R)exp(−iEλt/ħ), λ is the set of quantum numbers characterizing different states of a non- perturbed system, ψλ(R) and Eλ are the wave function and energy eigenvalues of stationary state of a non-perturbed quantum system, respectively, Ŵ (R, t) is the perturbation operator, R is the set of the quantum system coordinates, V is the volume, ‘∗’ is the complex conjugate symbol. Consideration of the electron scattering by phonons is based on (2) and (3) as well. In brief, the de- scription of the probability calculation is as follows. In the present case, the quantum system consists of a conduction electron in the crystall periodic field and lattice normal vibrations. Then, λ should be re- placed by an electron quasi-wave vector k and by the phonon occupation numbers of all possible states. The electron-phonon interaction Hamiltonian He-ph is taken as perturbation Ŵ (R, t); the r radius vector and the normal coordinates of lattice vibration are taken as a quantum system coordinate R. The wave function of non-perturbed state of an electron-phonon system is expressed as a product of one-electron wave function and harmonic oscillator wave functions. In flat-band semiconductors, an electron state is described by Bloch functions. Therefore, for an electron scattered from an initial state \|k〉 to a final state \|k′〉, the transition probability per unit time W (k,k′) is evaluated based on the Bloch functions [1–4]. To simplify the calculations, a plane wave ψk(r) = eikr/ √ Lx Ly Lz (where Lx , Ly and Lz are the sizes of a semiconductor, V = Lx Ly Lz ) is used sometimes as an electron wave function [1–4] (nearly free electron approximation). Calculations of the probability W (k,k′) based on the Bloch function or plane wave are well known and reported in detail in numerous publications (see, for example, [1–4]). However, in the present work, a semiconductor, whose bands are tilted by the perturbing potential of a uniform electric field, is of interest. In this case, electron eigenfunctions are not plane waves or Bloch functions [1, 3, 7]. Here, based on such an assumption, the probability W (k,k′) is recalculated following the above-mentioned general approach. It is assumed that the perturbation Hamiltonian He−ph, which is a harmonic function of time (harmonic perturbation), is determined using the deformation potential theory [1–4] of electron-phonon interaction. The results of our calculations show that: 23702-2 Electron-acoustic phonon field induced tunnel scattering • the probability of electron transition with phonon absorption can be presented as follows: Wa(k,k′) = D2 ac 2ρrV ħ d dt ∣ ∣ ∣ ∣ ∣ ∣ ∑ q iq √ nq ωq t ∫ 0 dt ei ε k′ −εk−ħωq ħ t ∫ V drψ∗ k′ (r)eiqrψk(r) ∣ ∣ ∣ ∣ ∣ ∣ 2 , (4) • the probability of electron transition with phonon emission can be presented as follows: We (k,k′) = D2 ac 2ρrV ħ d dt ∣ ∣ ∣ ∣ ∣ ∣ ∑ q iq √ nq +1 ωq t ∫ 0 dt ei ε k′ −εk+ħωq ħ t ∫ V drψ∗ k′(r)e−iqrψk(r) ∣ ∣ ∣ ∣ ∣ ∣ 2 . (5) Here, ψk(r, t) =ψk(r)exp(−iεkt/ħ), ψk(r) is the conduction electron wave function of stationary state k, ωq is the phonon angular frequency, nq is the occupation number of the equilibrium phonons, which is given by the Bose-Einstein distribution: nq = 1 /[ exp(ħωq/kBT )−1 ] . (6) The summations in (4) and (5) should be carried out in the range of the first Brillouin zone (BZ). After integration over t , (4) and (5) are expressed as follows: Wa,e (k,k′) = D2 acħ 2ρrV d dt ∣ ∣ ∣ ∣ ∣ ∣ ∑ q q p ωq √ nq + 1 2 ∓ 1 2 ei ε k′ −εk∓ħωq ħ t −1 εk′ −εk ∓ħωq ∫ V drψ∗ k′ (r)e±iqrψk(r) ∣ ∣ ∣ ∣ ∣ ∣ 2 . (7) Here and bellow, upper and lower symbols of double signs refer to electron transitions with the q phonon absorption and emission, respectively. In the presence of a uniform electricfield F (which is parallel to the z-axis), the electron wave-function is determined by the stationary Schrödinger equation which in the effective mass approximation is ex- pressed as follows: [ − ħ2 2m ( d2 dx2 + d2 dy2 + d2 dz2 ) +eF z −εk ] ψk(r) = 0. (8) Here and below, the index k of εk is omitted to simplify the expressions, so we can substitute εk → ε and εk′ → ε′ in further expressions. The solution of (8) is given by (see, e.g., [1, 7, 8]) ψk(r) = 1 √ Lx Ly ei(kx x+ky y)χn(z). (9) Inserting (9) into (8), the following equation is obtained [1, 7, 8] [ − ħ2 2m d2 dz2 +eF z −εn ] χn(z) = 0. (10) The solution of this equation is [7] as follows: χn(z)=Cn Ai [z l − (kn l)2 ] , (11) where Cn is the normalization constant, l = (ħ2/2eF m)1/3, Ai is the Airy function [8]: Ai(s)= 1 π ∞ ∫ 0 du cos ( u3 3 +us ) . (12) Electron energy eigenvalues ε are determined as follows: ε= ħ2k2 ⊥ 2m +εn , (13) 23702-3 S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan where εn = ħ2k2 n/2m, the index n identifies the electron energy eigenvalues, k2 ⊥ = k2 x + k2 y , k⊥ is the electron wave-vector perpendicular to the electric field. Energy eigenvalues εn (or kn) are determined from the boundary conditions (see below) for the wave function χn(z). For a semiconductor of length Lz in z-direction (i.e., −Lz /2 É z É Lz /2) from (9) and relation [1, 3] 1 Lz Lz /2 ∫ −Lz /2 dz ei(qz−q1,z )z = δqz ,q1,z (14) it follows ∫ dr3ψ∗ k′ (r)ψk(r)e±iqr = δk ′ x ,kx±qx δk ′ y ,ky±qy Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz z . (15) Inserting (15) into (7) yields Wa,e (k,k′) = D2 acħ 2ρrV d dt ∣ ∣ ∣ ∣ ∣ ∑ q q p ωq √ nq + 1 2 ∓ 1 2 × e i ε k′ −εk∓ħωq ħ t −1 εk′ −εk ∓ħωq δk ′ x ,kx±qx δk ′ y ,ky±qy Lz /2 ∫ −Lz /2 dzχ∗ n′(z)χn(z)e±iqz z ∣ ∣ ∣ ∣ ∣ ∣ 2 . (16) The Kronecker δ in this equation expresses the laws of conservation of perpendicular to the electric field quasi-momentum x, y components of the scattered particles. Here, only the normal N-processes of scattering are considered. After summation with respect to qx and qy with the help of the Kronecker δ, (16) becomes as follows: Wa,e (k,k′) = D2 acħ 2ρrV d dt ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∑ qz   q p ωq √ nq + 1 2 ∓ 1 2 e i ε k′ −εk∓ħωq ħ t −1 εk′ −εk ∓ħωq   qx =±(k′ x −kx ), qy =±(k′ y −ky ) × Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz z ∣ ∣ ∣ ∣ ∣ ∣ 2 . (17) Taking into account that ωq is an even function on q (and particularly on qz ), from (17) one obtains Wa,e (k,k′) = 2D2 acħ ρrV d dt ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∑ qzÊ0   q p ωq √ nq + 1 2 ∓ 1 2 e i ε k′ −εk∓ħωq ħ t −1 εk′ −εk ∓ħωq   qx =±(k′ x −kx ), qy =±(k′ y −ky ) ×Re Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz z ∣ ∣ ∣ ∣ ∣ ∣ 2 . (18) Using the following formal transformations d dt ∣ ∣ ∣ ∣ ∣ ∑ q cq(t) ∣ ∣ ∣ ∣ ∣ 2 = d dt ∑ q,q1 cq(t)c∗q1 (t) = 2Re ∑ q,q1 dcq(t) dt c∗q1 (t), (19) 23702-4 Electron-acoustic phonon field induced tunnel scattering (18) may be written as follows: Wa,e (k,k′) = 4D2 ac ρrV ∑ qz Ê 0 q1,z Ê 0 Re   iqq1p ωqωq1 √ nq + 1 2 ∓ 1 2 √ nq1 + 1 2 ∓ 1 2 ×ei(∓ωq±ωq1 )t −ei ε k′ −εk∓ħωq ħ t εk′ −εk ∓ħωq1   qx =±(k′ x −kx ), qy =±(k′ y −ky ) ×Re Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz zRe Lz /2 ∫ −Lz /2 dz1χ ∗ n′ (z1)χn(z1)e±iq1,z z1 . (20) For a sufficiently long time t , when the relation lim t→∞ sin(at)/a =πδ(a) can be used [1, 2], (20) is expressed as follows: Wa,e (k,k′) = 4D2 ac ρrV ∑ qz Ê 0 q1,z Ê 0 { qq1p ωqωq1 √ nq + 1 2 ∓ 1 2 √ nq1 + 1 2 ∓ 1 2 × [ −π (±ωq ∓ωq1 )δ(ωq −ωq1 ) εk′ −εk ∓ħωq1 +πδ(εk′ −εk ∓ħωq) εk′ −εk ∓ħωq εk′ −εk ∓ħωq1 ] } qx =±(k′ x −kx ), qy =±(k′ y −ky ) ×Re Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz zRe Lz /2 ∫ −Lz /2 dz1χ ∗ n′(z1)χn(z1)e±iq1,z z1 . (21) In this expression, the summation with respect to q1,z is non-zero only when qz = q1,z . Then, electron transition probability can be written as follows: Wa,e (k,k′) = 4πD2 ac ρrV ∑ qx , qy , qz Ê 0 q2 ωq ( nq + 1 2 ∓ 1 2 ) δ ( εk′ −εk ∓ħωq ) ×δk ′ x ,kx±qx δk ′ y ,ky±qy  Re Lz /2 ∫ −Lz /2 dzχ∗ n′(z)χn(z)e±iqz z   2 . (22) Here, the Dirac δ-function indicates the energy conservation law. Note, if in (22) the plane wave (χn(z) = eikz z / p Lz ) or Bloch function is taken as an electron wave function, then the well-known classical expression of probability Wa,e (k,k′) [1–4] (Fermi Golden rule) is derived. In the common case, the calculation of the integral and sum in (22) is complicated. On the other hand, the expression (1) for τk,ac is derived (within the framework of the flat-band semiconductor model) by using the approximation of elasticity of electron-acoustic phonon scattering (elastic collision approximation, ħωq ≪ εk) with the assumption that nq +1 ≈ nq � kBT ħωq = kBT ħν0q ≫ 1. (23) Here, ωq = ν0q dispersion law of the long wavelength longitudinal acoustic phonon is used. To reveal the difference between the results of flat and titled band approaches it is reasonable that these assumptions should be used here as well. Then, after simple summing with respect to qx and qy , (22) takes the following simpler form: Wa,e (k,k′) = 4πD2 ackBT ρrV ħν2 0 δ(εk′ −εk) ∑ qzÊ0  Re Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz z   2 . (24) 23702-5 S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan In what follows, we use the transformation ∑ qzÊ0  Re Lz /2 ∫ −Lz /2 dzχ∗ n′ (z)χn(z)e±iqz z   2 = ∑ qzÊ0   1 2 Lz /2 ∫ −Lz /2 dz [ χ∗ n′ (z)χn(z)e±iqz z +χn′ (z)χ∗ n(z)e∓iqz z ]   2 = 1 2 Re ∑ qzÊ0 Lz /2 ∫ −Lz /2 Lz /2 ∫ −Lz /2 dz dz1 [ χ∗ n′ (z)χn(z)χn′(z1)χ∗ n(z1)e±iqz z∓iqz z1 +χ∗ n′(z)χn(z)χ∗ n′(z1)χn(z1)e±iqz z±iqz z1 ] (25) and the relation [3] ∑ qz eiqz (z−z1) = Lzδ(z − z1). (26) The result is as follows: Wa,e (k,k′) = πD2 ackBT Lz ρrV ħν2 0 δ(εk′ −εk)Re Lz /2 ∫ −Lz /2 Lz /2 ∫ −Lz /2 dz dz1 [ χ∗ n′(z)χn(z)χn′(z1)χ∗ n(z1)δ(z − z1) +χ∗ n′(z)χn(z)χ∗ n′(z1)χn(z1)δ(z + z1) ] . (27) Delta-integration over z1 yields: Wa,e (k,k′) = πD2 ackBT Lz ρrV ħν2 0 δ(εk′ −εk) ×Re Lz /2 ∫ −Lz /2 dz { ∣ ∣χn′(z) ∣ ∣ 2 ∣ ∣χn(z) ∣ ∣ 2 +χ∗ n′(z)χn(z)χ∗ n′(−z)χn(−z) } . (28) Electron energy eigenvalues εn are determined from boundary conditions to (10). Note, there is some difference between the peculiarities of themovement of an electron under electric field in vacuum and an electron in a semiconductor. Conduction band gap ∆Ec of a semiconductor is a finite quantity. Contrary to the vacuum, the movement of a conduction electron in a semiconductor with perfect lattice has an oscillation character, as it is shown in figure 1. Electron ‘el.1’ oscillates between the bottom of the conduction band and semiconductor edge; electron ‘el.2’ periodically reflects from the conduction band top and bottom edges. The behavior of an ‘el.2’ is well-known as a Bloch oscillation. Boundary conditions for Schrödinger equation (10) and, therefore, electron energy eigenvalues in case of Bloch oscillations have been reported, for example, in [9]. Here, for definiteness, neglecting the Bloch oscillations, the case of ‘el.1’ is considered only, i.e., it is assumed that the electric field is low and the magnitude of ∆Ec is very large. In other words, the model of a triangular quantum well with finite sizes is considered. Here, we are interested in a large length semiconductor, particularly, in z-direction. Then, the allowed values of kn (or εn) are computed from the boundary condition χn (z = −Lz /2) = 0 as (see, [7] and figure 1) Lz 2l + (kn l)2 =−an , (29) where an are the zeros of the Airy function which are located in the negative part of the real axis [8]. They are well approximated as an �−(3π[4n−1]/8)2/3 , where n = 1,2, . . . . Inserting this relation into (29) we obtain Lz 2l + (kn l)2 = [ 3π 2 ( n− 1 4 )]2/3 . (30) 23702-6 Electron-acoustic phonon field induced tunnel scattering Figure 1. Semiconductor energy-band diagram in the presence of a uniform electric field F (parallel to the z-axis); el.2 – Bloch oscillations. Solving (30) for kn and inserting the result into (13), one obtains energy eigenvalues [7]: ε= ħ2 2m { k2 ⊥− Lz 2l 3 + 1 l 2 [ 3π 2 (n−1/4) ]2/3 } . (31) The values ε in (31) are obtained as a function of the electron state quantum numbers. The quantities kx , ky (or k⊥) and n (or kn) are a set of quantum numbers which determine the conduction electron state in the presence of an electric field. Normalization constant Cn in (11) is determined as follows: C−2 n = Lz /2 ∫ −Lz /2 dz ∣ ∣ ∣Ai ( z l − (kn l)2 )∣ ∣ ∣ 2 . (32) It is known [7, 8] that ∫ ds Ai2(s) = s Ai2(s)−Ai′2(s), (33) where Ai′ is the derivative of the Airy function. Consequently, normalization constant Cn can be presented as follows: C−2 n = l [ s Ai2(s)−Ai′2(s) ]∣ ∣ Lz /l+an an . (34) For a semiconductor with large length Lz one has C−2 n = l Ai′2(an). (35) Here, we used the fact that Ai(an) = 0; functions Ai(s) and Ai′(s) exponentially vanish for positive large argument s [8]. On the other hand, the value Ai′(an) is well approximated as [8] Ai′(an)� (−1)n−1 1 p π (−an)1/4. (36) Insertion of (36) into (35) yields: C 2 n =π/l p−an . (37) 23702-7 S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan Thus, using (11), (29) and taking into account that χn(z) is the real function [see, (11), (12)] the expression (28) can be presented as follows: Wa,e (k,k′) = πD2 ackBT Lz l ρrV ħν2 0 δ(εk′ −εk)C 2 n′C 2 n    Lz /2l ∫ −Lz /2l du ∣ ∣Ai ( u− (kn′ l)2 )∣ ∣ 2 ∣ ∣Ai ( u− (kn l)2 )∣ ∣ 2 + Lz /2l ∫ −Lz /2l du [ Ai ( u− (kn′ l)2 ) Ai ( −u− (kn′ l)2 ) Ai ( u− (kn l)2 ) Ai ( −u− (kn l)2 )]    , (38) where u = z/l is the dimensionless variable of integration. (38) describes electron transition probability in the presence of an electric field. Transition probability depends on the electric field and it is a symmet- ric function: Wa,e (k,k′) =Wa,e (k′,k). 3. Electron-acoustic phonon FIT scattering time The scattering time is defined by [3]: τ−1 k,sc = ∑ k′ W (k,k′) (39) (38) shows that in the present case, the probabilities of phonon absorption and emission by electron are the same: Wa (k′,k) =We (k,k′). Then, (39) can be written as follows: τ−1 k,sc = 2 ∑ k′ We (k,k′). (40) In the common case, the scattering time (the inverse of the scattering rate) differs from the relaxation time although sometimes both of them are equivalent (for example, the above-mentioned case of (1)) [3]. To calculate τ−1 k,sc in (40), we replace the sums over k ′ x and k ′ y by integrals over k ′ x and k ′ y , respectively. The transition from the sum to integral with the help of the relation dk ′ x dk ′ y =πdk ′2 ⊥ can be presented as follows: ∑ k′ −→ 2Lx Ly (2π)2 ∫ BZ dk ′ x dk ′ y ∑ n′ = Lx Ly 2π ∫ BZ dk ′2 ⊥ ∑ n′ . (41) Here, coefficient 2 in the numerator is related to the electron spin. From (30) it follows that k2 n+1 −k2 n ∼ 1/l 2 . The distance between kn and kn+1 depends on n and it is small for large l . Therefore, at low-field regime, one can change the summation over n′ by an integral over kn′ : ∑ n′ −→ ∫ dkn′ dn′ dkn′ . (42) The derivative dn′/dkn′ can be evaluated based on (30). The solution of (30) for n is given the following expression for n′: n′ = 2 3π ( Lz 2l + (kn′ l)2 )3/2 + 1 4 . (43) Therefore, dn′ dkn′ = 2l 2kn′ π √ Lz 2l + (kn′ l)2 . (44) Based on (29), the derivative dn′/dkn′ can be presented as follows: dn′ dkn′ = 2l 2kn′ π p−an′ . (45) 23702-8 Electron-acoustic phonon field induced tunnel scattering Thus, transition (41) can be presented as follows: ∑ n′ −→ Lx Ly l 2 2π2 k ′2 ⊥,max ∫ k ′2 ⊥,min k2 n′ ,max ∫ k2 n′ ,min dk ′2 ⊥ dk2 n′ p−an′ . (46) Inserting (38) into (40), simultaneously taking into account (37) and transition (46), for the scattering time one obtains: τ−1 k,sc = D2 ackBT C 2 n ρrħν2 0 k ′2 ⊥,max ∫ k ′2 ⊥,min s ′max ∫ s ′ min dk ′2 ⊥ ds′δ(εk′ −εk)    Lz /2l ∫ −Lz /2l du ∣ ∣Ai(u− s′) ∣ ∣ 2 ∣ ∣Ai(u− (kn l)2) ∣ ∣ 2 + Lz /2l ∫ −Lz /2l du Ai(u− s′) Ai(−u− s′) Ai(−u− (kn l)2) Ai(u− (kn l)2)    . (47) Here, s′ = (lkn′ )2 is the dimensionless variable of integration, Brillouin zone is replaced by the infinite range: 0 É k ′2 ⊥ <∞. The limits of integration over s′ are determined by the δ-function δ(εk′ −εk) = 2m ħ2 δ ( k ′2 ⊥ +k2 n′ −k2 ⊥−k2 n ) , (48) as: s′ min =−a1 −Lz /2l , s′max = ( k2 ⊥−k2 n ) l 2, where a1 =−(9π/8)2/3 is the first zero of the Airy function [8]. δ-integration over s′ in (47) yields τ−1 k,sc = 2D2 acmkBT C 2 n ρrħ3ν2 0 ∞ ∫ 0 ds′⊥    Lz /2l ∫ −Lz /2l du ∣ ∣Ai(u+ s′⊥−∆) ∣ ∣ 2 ∣ ∣Ai(u− (kn l)2) ∣ ∣ 2 + Lz /2l ∫ −Lz /2l du Ai(u+ s′⊥−∆) Ai(−u+ s′⊥−∆) Ai(−u− (kn l)2) Ai(u− (kn l)2)    . (49) Here, s′⊥ = k ′2 ⊥ l 2 is the dimensionless variable of integration, ∆ ≡ k2 ⊥l 2 +k2 n l 2 = 2mεl 2/ħ2 is the dimen- sionless energy of an electron. In (49), evaluations of the first and second the integrals over s′⊥ can be carried out based on (33) and following the integral of the product of two Airy functions [8], respectively, ∞ ∫ 0 ds Ai(b + s) Ai(c + s) = 1 b −c [ Ai(b)Ai′(c)−Ai′(b)Ai(c) ] . (50) The result is as follows: τ−1 k,sc = D2 acmkBT C 2 n ρrħ3ν2 0 [I1 + I2], (51) where I1 = Lz /2l ∫ −Lz /2l du { (∆−u)Ai2(u−∆)+Ai′2(u−∆) }∣ ∣Ai(u− (kn l)2) ∣ ∣ 2 , (52) I2 = Lz /2l ∫ −Lz /2l du 1 2u [ Ai(u−∆)Ai′(−u−∆)−Ai′(u−∆)Ai(−u−∆) ] Ai ( −u− (kn l)2 ) Ai ( u− (kn l)2 ) . (53) 23702-9 S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan The functional analyses of the sub-integral expression show that the main contribution in integrals (52) and (53) is given by the range near u ∼ 0. An approximate estimation of the integral (53) is carried out with the help of L’Hopital’s rule. The result is as follows: I1 + I2 ≈ 1 lC 2 n [ ∆Ai2(−∆)+Ai′2(−∆) ] . (54) Therefore, the electron scattering time can be presented as follows: τ−1 k,sc = D2 acmkBT ρrħ3ν2 0l [ ∆Ai2(−∆)+Ai′2(−∆) ] . (55) 4. Summary The electron-acoustic phonon scattering theory has been reconsidered. In semiconductors, whose bands are tilted under uniform electric field, the time of electron scattering by acoustic phonon is deter- mined by (55). Scattering time depends on the electron energy ε. It depends on the electric field as well, because ∆∼ εl 2; l ∼ F−1/3. Those dependencies are determined by the Airy function properties [8]. Thus, for negative arguments, the Airy function oscillates. From the asymptotic series of the Airy function Ai(∆) and of their derivative Ai′(∆) it follows that for large negative argument [8] \|∆\|Ai2(−\|∆\|)+Ai′2(−\|∆\|) = \|∆\|1/2/π . (56) The Airy function decays exponentially for positive arguments. The first terms of asymptotic series of the Airy function and of their derivative for positive arguments are as follows: Ai(\|∆\|) = 1 2 p π \|∆\|1/4 exp ( −2\|∆\|3/2/3 ) , Ai′(\|∆\|) =−\|∆\|1/4 2 p π exp ( −2\|∆\|3/2/3 ) . (57) Then, from (55), (56) and (57) it follows: • for positive large ∆ τ−1 k,sc = D2 acmkBT \|∆\|1/2 πρrħ3ν2 0l , (58) • for negative large ∆ τ−1 k,sc = D2 acmkBT \|∆\|1/2 πρrħ3ν2 0l 1 2 exp ( −4\|∆\|3/2/3 ) . (59) Insertion ∆ and l into (59) yields: τ−1 k,sc = τ−1 k,ac 1 2 exp ( −4 √ 2m\|ε\|3/ħ2 /3eF ) . (60) General expression (55) for the electron-acoustic phonon scattering time can be modified by inserting ∆ into (55). Then, one has τ−1 k,sc = D2 acmkBT ρrħ3ν2 0l [ 2mεl 2 ħ2 Ai ( −2mεl 2 ħ2 ) +Ai′2 ( −2mεl 2 ħ2 )] . (61) It is easy to establish that (58) and (1) are the same. As equation (60) shows, the scattering time of low energy electron depends on the electric field. The dependence has exponential character. This effect has been explained in terms of the penetration of the electron wave function into a band gap of a semicon- ductor [6]. As a result, for low energy electrons, which are located near the bottom edge of the conduction band, the transition with phonon emission becomes allowable in the region below the conduction band edge. Note, at flat-bands approach, there is a threshold of a phonon emission by a low energy electron, for detailes see [10]. All these peculiarities are well displayed in figure 2, where the dependencies of τ−1 k,ac 23702-10 Electron-acoustic phonon field induced tunnel scattering and τ−1 k,sc on electron dimensionless energy ε/kBT are plotted for n-Si at T = 300 K with the following parameters [5]: m = 0.32m0 , ρr = 2329 kg/m3, ν0 = 8.43·103 m/s, Dac = 9 eV. The dependence τ−1 k,sc (ε/kBT ) (curve ‘a’) is calculated at F = 800 V/cm based on (61). The dependence τ−1 k,ac (ε/kBT ) (curve ‘b’) is cal- culated based on (1). As shown in figure 2, the curve ‘a’ has a character of light oscillations around the curve ‘b’. At a low field regime F <∼ 400 V/cm, the curves ‘a’ and ‘b’ practically coincide in the range of positive energy. Other important peculiarities are as follows: on the curve ‘b’ there is a fracture, i.e., dτ−1 k,ac /dε ∣ ∣ ∣ ε=0 =∞ [see, equation (1)]; on the curve ‘a’ there is no fracture [see, (61)]. Figure 2. (Color online) The dependencies of τ−1 k,sc (curve ‘a’ – F = 800 V/cm) and τ−1 k,ac (curve ‘b’ – F = 0) on electron dimensionless energy ε/kBT for n-Si at T = 300 K. Taking into acount the problem solution reported in [6] it can be stated that the results of the present study can have a principial effect on the mobility fluctuation theory, especially. It should be noted that in the above-presented electron-acoustic phonon FIT scattering study, the electron quasi-momentum relax- ation time and its relation to the scattering time is not included. It requires a separate consideration. References 1. Anselm A.I., Introduction to Semiconductor Theory, Prentice Hall Publ., New Jersey, 1982. 2. Askerov B.M., Electron Transport Phenomena in Semiconductors, World Scientific, Singapore, 1994. 3. Hamaguchi Ch., Basic Semiconductor Physics, Springer, Berlin, 2010. 4. Jacoboni C., Theory of Electron Transport in Semiconductors, Springer, Berlin, 2010. 5. Dargys A., Kundrotas J., Handbook on Physical Properties of Ge, Si, GaAs and InP, Science and Encyclopedia Publishers, Vilnius, Lithuania, 1994. 6. Melkonyan S.V., Physica B, 2012, 407, 4804; doi:10.1016/j.physb.2012.09.007. 7. Haug H., Koch S.W., Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scien- tific, Singapore, 2004. 8. Vallee O., Soares M., Airy Functions and Applications to Physics, Imperial College Press, London, 2004. 9. Hartmann T., Keck F., Korsch H. J., Mossmann S., New Journal of Physics, 2004, 6, 2; doi:10.1088/1367-2630/6/1/002. 10. Gantmakher V.F., Levinson I.B., Carrier Scattering in Metals and Semiconductors, Elsevier, Amsterdam, 1987. 23702-11 http://dx.doi.org/10.1016/j.physb.2012.09.007 http://dx.doi.org/10.1088/1367-2630/6/1/002 S.V. Melkonyan, A.L. Harutyunyan, T.A. Zalinyan Iндуковане електронноакустичним фононним полем тунельне розсiяння С.В. Мелконян, А.Л. Харатюнян, Т.А. Залiнян Факультет фiзики напiвпровiдникiв i мiкроелектронiки, Єреванський державний унiверситет, 0025 Єреван, Вiрменiя У статтi подано новий погляд на теорiю електронноакустичного розсiювання одного фонона. При цьому припускається, що невироджений напiвпровiдник має сферичну параболiчну зонну структуру. В основу перегляду теорiї покладено ефект нахилу напiвпровiдникових зон при накладаннi збурюючого потенцi- алу електричного поля. У цьому випадку власнi функцiї електрона вже не є плоскими хвилями чи фун- кцiями Блоха. В режимi слабких полiв отримано вирази для ймовiрностi електронних внутрiшньозонних переходiв i для часу розсiяння в наближеннi пружнiх зiткнень. Також проаналiзовано залежнiть часу роз- сiяння вiд енергiї електрона та напруженостi однорiдного електричного поля. Представлено результати вiдповiдних числових обчислень для n-Si при температурi 300 K. Встановлено вiдсутнiсть зламу на кривiй залежностi часу розсiяння електрона вiд енергiї електрона. Ключовi слова: напiвпровiдник з нахиленою зоною, електронноакустичне фононне розсiяння, ймовiрнiсть переходу, час розсiяння 23702-12 Introduction Electron-acoustic phonon FIT transition probability Electron-acoustic phonon FIT scattering time Summary

Electron-acoustic phonon field induced tunnel scattering

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