Electron-acoustic phonon field induced tunnel scattering

Electron-acoustic phonon field induced tunnel scattering

Abstract

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  K are presented. It is established that there is no fracture on the curve of electron scattering time dependence on the electron energy. \keywordstilted band semiconductor, electron-acoustic phonon scattering, transition probability, scattering time \pacs72.10-d, 63.20.kd

Abstract

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.

\keywords

напiвпровiдник з нахиленою зоною, електронноакустичне фононне розсiяння, ймовiрнiсть переходу, час розсiяння

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201518223702 \doinumber10.5488/CMP.18.23702 \addressDepartment of Physics of Semiconductors & Microelectronics, Yerevan State University, 1 Alex Manoogian St., 0025 Yerevan, Armenia

1 Introduction

Current carrier (electron) mobility is an important parameter characterizing many transport phenomena in semiconductors under electric field . Electron mobility is determined as [1, 2], where is the electron quasi-momentum relaxation time, is the electron charge magnitude, is the electron effective mass, is the symbol of averaging over conduction zone quantum states. Relaxation time is determined by the electron scattering by various dynamic and static imperfections of a crystall 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 , a flat-band semiconductor model is used, as a rule [1, 2, 3, 4]. In this case, in low electric field region (, where is a characteristic field), the electron relaxation time and, therefore, the mobility are field-independent quantities [1, 4, 5]. Particularly, the relaxation time related to electron-acoustic single phonon elastic scattering in non-degenerate n-type semiconductor with a spherical parabolic conduction band is given by [1, 2, 3, 4]

 1/τk,ac=D2ac(2m)3/2kBT2πℏ4ρrν20√εk. (1)

Here, is a reduced mass density of a crystal, is the acoustic deformation potential constant, is the Boltzmann constant, is temperature, is the long-wavelength longitudinal acoustic phonon velocity, is the electron energy, boldface is the electron wave vector (herein below, the magnitudes of vector quantities are denoted by non-boldface symbols).

At high electric fields (), the time and, therefore, the mobility depend on the applied electric field [4]. Thus, at electron scattering by an acoustic phonon, electron mobility decreases with an increase of electric field above [5]. The magnitude of characteristic field depends on semiconductor parameters such as crystallographic directions, impurity concentration, temperature, etc. It is of an order of  V/cm at  K, e.g., for pure Si  V/cm, for pure Ge  V/cm, for high purity GaAs  V/cm [5]. The dependence 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 transition 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 probability and evaluate the relaxation time. The task of the transition probability calculation is solved based on the perturbation theory (see, e.g. [1, 2, 3, 4]). According to this theory, the probability per unit time of quantum system transition from state to state, to the first order in the perturbation, is determined as [1, 2]

 Missing or unrecognized delimiter for \left (2)

Here, is the perturbation matrix element:

 aλ′,λ(t)=∫VdRψ∗λ′(R,t)^W(R,t)ψλ(R,t), (3)

, is the set of quantum numbers characterizing different states of a non-perturbed system, and are the wave function and energy eigenvalues of stationary state of a non-perturbed quantum system, respectively, is the perturbation operator, is the set of the quantum system coordinates, 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 description 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, should be replaced by an electron quasi-wave vector and by the phonon occupation numbers of all possible states. The electron-phonon interaction Hamiltonian is taken as perturbation ; the radius vector and the normal coordinates of lattice vibration are taken as a quantum system coordinate . 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 to a final state , the transition probability per unit time is evaluated based on the Bloch functions [1, 2, 3, 4]. To simplify the calculations, a plane wave (where , and are the sizes of a semiconductor, ) is used sometimes as an electron wave function [1, 2, 3, 4] (nearly free electron approximation). Calculations of the probability based on the Bloch function or plane wave are well known and reported in detail in numerous publications (see, for example, [1, 2, 3, 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 is recalculated following the above-mentioned general approach. It is assumed that the perturbation Hamiltonian , which is a harmonic function of time (harmonic perturbation), is determined using the deformation potential theory [1, 2, 3, 4] of electron-phonon interaction. The results of our calculations show that:

• the probability of electron transition with phonon absorption can be presented as follows:

 Wa(k,k′)=D2ac2ρrVℏddt∣∣ ∣∣∑qiq√nqωqt∫0dt\reiεk′−εk−ℏωqℏt∫Vdrψ∗k′(r)\reiqrψk(r)∣∣ ∣∣2, (4)
• the probability of electron transition with phonon emission can be presented as follows:

 We(k,k′)=D2ac2ρrVℏddt∣∣ ∣∣∑qiq√nq+1ωqt∫0dt\reiεk′−εk+ℏωqℏt∫Vdrψ∗k′(r)\re−iqrψk(r)∣∣ ∣∣2. (5)

Here, , is the conduction electron wave function of stationary state , is the phonon angular frequency, 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 , (4) and (5) are expressed as follows:

 Wa,e(k,k′)=D2acℏ2ρ%rVddt∣∣ ∣ ∣∣∑qq√ωq√nq+12∓12\reiεk′−εk∓ℏωqℏt−1εk′−εk∓ℏωq∫Vdrψ∗k′(r)\re±iqrψk(r)∣∣ ∣ ∣∣2. (7)

Here and bellow, upper and lower symbols of double signs refer to electron transitions with the phonon absorption and emission, respectively.

In the presence of a uniform electric field (which is parallel to the -axis), the electron wave-function is determined by the stationary Schrödinger equation which in the effective mass approximation is expressed as follows:

 [−ℏ22m(d2dx2+d2dy2+d2dz2)+eFz−εk]ψk(r)=0. (8)

Here and below, the index of is omitted to simplify the expressions, so we can substitute and in further expressions.

The solution of (8) is given by (see, e.g., [1, 7, 8])

 ψk(r)=1√LxLy\rei(kxx+kyy)χn(z). (9)

Inserting (9) into (8), the following equation is obtained [1, 7, 8]

 [−ℏ22md2dz2+eFz−εn]χn(z)=0. (10)

The solution of this equation is [7] as follows:

 χn(z)=CnAi[zl−(knl)2], (11)

where is the normalization constant, , is the Airy function [8]:

 Ai(s)=1π∞∫0ducos(u33+us). (12)

Electron energy eigenvalues are determined as follows:

 ε=ℏ2k2⊥2m+εn, (13)

where , the index identifies the electron energy eigenvalues, , is the electron wave-vector perpendicular to the electric field. Energy eigenvalues (or ) are determined from the boundary conditions (see below) for the wave function .

For a semiconductor of length in -direction (i.e., ) from (9) and relation [1, 3]

 1LzLz/2∫−Lz/2dz\rei(qz−q1,z)z=δqz,q1,z (14)

it follows

 ∫dr3ψ∗k′(r)ψk(r)\re±iqr=δk′x,kx±qxδk′y,ky±qyLz/2∫−Lz/2dzχ∗n′(z)χn(z)\re±iqzz. (15)

Inserting (15) into (7) yields

 Missing or unrecognized delimiter for \right (16)

The Kronecker in this equation expresses the laws of conservation of perpendicular to the electric field quasi-momentum , components of the scattered particles. Here, only the normal N-processes of scattering are considered.

After summation with respect to and with the help of the Kronecker , (16) becomes as follows:

 Wa,e(k,k′) = D2acℏ2ρrVddt∣∣ ∣ ∣ ∣∣∑qz⎡⎢ ⎢⎣q√ωq√nq+12∓12eiεk′−εk∓ℏωqℏt−1εk′−εk∓ℏωq⎤⎥ ⎥⎦qx=±(k′x−kx),qy=±(k′y−ky) (17) ×Lz/2∫−Lz/2dzχ∗n′(z)χn(z)e±iqzz∣∣ ∣ ∣∣2.

Taking into account that is an even function on (and particularly on ), from (17) one obtains

 Wa,e(k,k′) = 2D2acℏρrVddt∣∣ ∣ ∣ ∣∣∑qz⩾0⎡⎢ ⎢⎣q√ωq√nq+12∓12eiεk′−εk∓ℏωqℏt−1εk′−εk∓ℏωq⎤⎥ ⎥⎦qx=±(k′x−kx),qy=±(k′y−ky) (18) ×ReLz/2∫−Lz/2dzχ∗n′(z)χn(z)e±iqzz∣∣ ∣ ∣∣2.

Using the following formal transformations

 ddt∣∣ ∣∣∑qcq(t)∣∣ ∣∣2=ddt∑q,q1cq(t)c∗q1(t)=2Re∑q,q1dcq(t)dtc∗q1(t), (19)

(18) may be written as follows:

 Wa,e(k,k′) = 4D2acρrV∑qz⩾0q1,z⩾0Re⎡⎢ ⎢⎣\rei(∓ωq±ωq1)t−\reiεk′−εk∓ℏωqℏtεk′−εk∓ℏωq1iqq1√ωqωq1√nq+12∓12√nq1+12∓12 (20) Unknown environment '% ×ReLz/2∫−Lz/2dzχ∗n′(z)χn(z)\re±iqzzReLz/2∫−Lz/2dz1χ∗n′(z1)χn(z1)\re±iq1,zz1.

For a sufficiently long time , when the relation can be used [1, 2], (20) is expressed as follows:

 Wa,e(k,k′) = 4D2acρrV∑qz⩾0q1,z⩾0{qq1√ωqωq1√nq+12∓12√nq1+12∓12 (21) ×[−π(±ωq∓ωq1)δ(ωq−ωq1)εk′−εk∓ℏωq1+πδ(εk′−εk∓ℏωq)εk′−εk∓ℏωqεk′−εk∓ℏωq1]√nq+12∓12}qx=±(k′x−kx),qy=±(k′y−ky) ×ReLz/2∫−Lz/2dzχ∗n′(z)χn(z)\re±iqzzReLz/2∫−Lz/2dz1χ∗n′(z1)χn(z1)\re±iq1,zz1.

In this expression, the summation with respect to is non-zero only when . Then, electron transition probability can be written as follows:

 Wa,e(k,k′) = 4πD2acρrV∑qx,qy,qz⩾0q2ωq(nq+12∓12)δ(εk′−εk∓ℏωq) (22) ×δk′x,kx±qxδk′y,ky±qy⎡⎢ ⎢⎣ReLz/2∫−Lz/2dzχ∗n′(z)χn(z)\re±iqzz⎤⎥ ⎥⎦2.

Here, the Dirac -function indicates the energy conservation law.

Note, if in (22) the plane wave () or Bloch function is taken as an electron wave function, then the well-known classical expression of probability [1, 2, 3, 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 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, ) with the assumption that

 nq+1≈nq≅kBTℏωq=kBTℏν0q≫1. (23)

Here, 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 and , (22) takes the following simpler form:

 Wa,e(k,k′)=4πD2ackBTρrVℏν20δ(εk′−εk)∑qz⩾0⎡⎢ ⎢⎣ReLz/2∫−Lz/2dzχ∗n′(z)χn(z)\re±%iqzz⎤⎥ ⎥⎦2. (24)

In what follows, we use the transformation

 ∑qz⩾0⎡⎢ ⎢⎣ReLz/2∫−Lz/2dzχ∗n′(z)χn(z)\re±iqzz⎤⎥ ⎥⎦2 (25) =∑qz⩾0⎡⎢ ⎢⎣12Lz/2∫−Lz/2dz[χ∗n′(z)χn(z)\re±iqzz+χn′(z)χ∗n(z)\re∓iqzz]⎤⎥ ⎥⎦2 =12Re∑qz⩾0Lz/2∫−Lz/2Lz/2∫−Lz/2dzdz1[χ∗n′(z)χn(z)χn′(z1)χ∗n(z1)\re±iqzz∓iqzz1 +χ∗n′(z)χn(z)χ∗n′(z1)χn(z1)\re±iqzz±iqzz1]

and the relation [3]

 ∑qz\reiqz(z−z1)=Lzδ(z−z1). (26)

The result is as follows:

 Wa,e(k,k′) = πD2ackBTLzρrVℏν20δ(εk′−εk)ReLz/2∫−Lz/2Lz/2∫−Lz/2%dzdz1[χ∗n′(z)χn(z)χn′(z1)χ∗n(z1)δ(z−z1) (27) +χ∗n′(z)χn(z)χ∗n′(z1)χn(z1)δ(z+z1)].

Delta-integration over yields:

 Wa,e(k,k′) = πD2ackBTLzρrVℏν20δ(εk′−εk) (28) ×ReLz/2∫−Lz/2dz{|χn′(z)|2|χn(z)|2+χ∗n′(z)χn(z)χ∗n′(−z)χn(−z)}.

Electron energy eigenvalues are determined from boundary conditions to (10). Note, there is some difference between the peculiarities of the movement of an electron under electric field in vacuum and an electron in a semiconductor. Conduction band gap 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 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 -direction. Then, the allowed values of (or ) are computed from the boundary condition as (see, [7] and figure 1)

 Lz2l+(knl)2=−an, (29)

where are the zeros of the Airy function which are located in the negative part of the real axis [8]. They are well approximated as , where . Inserting this relation into (29) we obtain

 Lz2l+(knl)2=[3π2(n−14)]2/3. (30)

Solving (30) for and inserting the result into (13), one obtains energy eigenvalues [7]:

 ε=ℏ22m{k2⊥−Lz2l3+1l2[3π2(n−1/4)]2/3}. (31)

The values in (31) are obtained as a function of the electron state quantum numbers. The quantities , (or ) and (or ) are a set of quantum numbers which determine the conduction electron state in the presence of an electric field.

Normalization constant in (11) is determined as follows:

 C−2n=Lz/2∫−Lz/2dz∣∣∣Ai(zl−(knl)2)∣∣∣2. (32)

It is known [7, 8] that

 ∫dsAi2(s)=sAi2(s)−Ai′2(s), (33)

where is the derivative of the Airy function.

Consequently, normalization constant can be presented as follows:

 C−2n=l[sAi2(s)−Ai′2(s)]∣∣Lz/l+anan. (34)

For a semiconductor with large length one has

 C−2n=lAi′2(an). (35)

Here, we used the fact that ; functions and exponentially vanish for positive large argument [8]. On the other hand, the value is well approximated as [8]

 Ai′(an)≅(−1)n−11√π(−an)1/4. (36)

Insertion of (36) into (35) yields:

 C2n=π/l√−an. (37)

Thus, using (11), (29) and taking into account that is the real function [see, (11), (12)] the expression (28) can be presented as follows:

 Wa,e(k,k′) = πD2ackBTLzlρrVℏν20δ(εk′−εk)C2n′C2n⎧⎪ ⎪⎨⎪ ⎪⎩Lz/2l∫−Lz/2ldu∣∣Ai(u−(kn′l)2)∣∣2∣∣Ai(u−(knl)2)∣∣2 (38) +Lz/2l∫−Lz/2ldu[Ai(u−(kn′l)2)Ai(−u−(kn′l)2)Ai(u−(knl)2)Ai(−u−(knl)2)]⎫⎪ ⎪⎬⎪ ⎪⎭,

where 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 symmetric function: .

3 Electron-acoustic phonon FIT scattering time

The scattering time is defined by [3]:

 τ−1k,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: . Then, (39) can be written as follows:

 τ−1k,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 in (40), we replace the sums over and by integrals over and , respectively. The transition from the sum to integral with the help of the relation can be presented as follows:

 ∑k′⟶2LxLy(2π)2∫BZdk′xdk′y∑n′=LxLy2π∫BZdk′2⊥∑n′. (41)

Here, coefficient 2 in the numerator is related to the electron spin.

From (30) it follows that . The distance between and depends on and it is small for large . Therefore, at low-field regime, one can change the summation over by an integral over :

 ∑n′⟶∫dkn′d% n′dkn′. (42)

The derivative can be evaluated based on (30). The solution of (30) for is given the following expression for :

 n′=23π(Lz2l+(kn′l)2)3/2+14. (43)

Therefore,

 dn′dkn′=2l2kn′π√Lz2l+(kn′l)2. (44)

Based on (29), the derivative can be presented as follows:

 dn′dkn′=2l2kn′π√−an′. (45)

Thus, transition (41) can be presented as follows:

 ∑n′⟶LxLyl22π2k′2⊥,max∫k′2⊥,mink2n′,max∫k2n′,mindk′2⊥dk2n′√−an′. (46)

Inserting (38) into (40), simultaneously taking into account (37) and transition (46), for the scattering time one obtains:

 τ−1k,sc = D2ackBTC2nρrℏν20k′2⊥,max∫k′2⊥,mins′max∫s′mindk′2⊥ds′δ(εk′−εk)⎧⎪ ⎪⎨⎪ ⎪⎩Lz/2l∫−Lz/2ldu∣∣Ai(u−s′)∣∣2∣∣Ai(u−(knl)2)∣∣2 (47) +Lz/2l∫−Lz/2lduAi(u−s′)Ai(−u−s′)Ai(−u−(knl)2)Ai(u−(knl)2)⎫⎪ ⎪⎬⎪ ⎪⎭.

Here, is the dimensionless variable of integration, Brillouin zone is replaced by the infinite range: . The limits of integration over are determined by the -function

 δ(εk′−εk)=2mℏ2δ(k′2⊥+k2n