Collision Integral For The Interaction Of Electrons With Long Wavelength LO Phonons In Spatially Inhomogeneous Medium

Collision Integral For The Interaction Of Electrons With Long Wavelength LO Phonons In Spatially Inhomogeneous Medium

S. K. Suman1* and V. S. Giri2

1Department of Pure and Applied Physics, Guru Ghasidas University (A Central University ), Bilaspur , Chhattisgarh , India.

2University Department of Physics, Ranchi University, Ranchi- 834008, India.

Abstract : We have evaluated the total mechanical energy of the long wavelength longitudinal optical (LO) phonons. Then we have calculated the transition probability for the interaction of electrons with long wavelength longitudinal optical (LO)phonons in the spatially inhomogeneous medium. For the evaluation of transition probability we have calculated the advanced Greens function and hence the retarded Greens function. finally we have used the transition probability to find the expression for the collision integral for the interaction of electrons with long wavelength LO phonons in spatially inhomogeneous medium.

Key words : Phonons, Greens function, transition probability, collision integral.

1. Introduction :

   There are various types of electron-phonon interaction[1], such as interaction of the electrons with longitudinal optical (LO) phonons known as Fro hlich interaction, deformation potential interaction, interaction of the electrons with optical phonons due to deformation of the lattice which is important only in the non-ionic crystals where the Fro elich interaction is not present, and interaction of the electrons with acoustic phonons due to piezoelectric fields which is often weaker than the deformation-potential interaction. We have to mainly consider only the interaction of long wavelength LO phonons with electrons.

2. Energy of LO phonon

   In case of ionic crystals, the optical phonons produce a dipolar field when the oppositely charged ions oscillate as a whole with respect to each other [2]. This dipolar field polarizes the electrons and leads to the coupling of electrons and phonons. Consider a biatomic ionic crystal, under the optical vibrations, the centre of mass of the cell remains at rest, and the atoms of the cell oscillate in antiphase. Here we have made the approximation that the displacements uns (where elementary crystal cell is numbered by the integer vector n and the index s is the number of atoms in the cell) from the equilibrium position of the ions, do not depend on n, and the vibrations are described by the two variables u+ and u- corresponding to the two ions having atomic masses m+ and m- and effective charges +e and e respectively. Now as the centre of mass of the cell remains at rest, and the atoms of the cell oscillate in antiphase m+ u+ + m- u- = 0 therefore both u+ and u- may be expressed in terms of the relative ionic displacement u = u+

   – u- so there is only one independent variable u . In such vibrations each elementary cell will have a dipole moment e.u , and a significant contribution to the interatomic forces comes from the long range dipole-dipole interaction. The short-range part of the force is proportional to the relative shift u , and the long- range part is given by e.E. The longitudinal field E induces the polarization (N/V)E, where is the polarizability of the cell and N/V is the number of cells per unit volume. Therefore total polarization is P = (N/V)[ e.u + E] .

   when the polarization induced by longitudinal modes is expressed in terms of the effective dielectric constant * we get

   P = LO . ..(i)

   P = LO . ..(i)

   2

   4

   The potential energy of interaction of electrons with the field induced by long wavelength optical phonons is obtained from the poisson equation

   U = . …(ii)

   e

   Only longitudinal optical(LO) phonons produce the long range electric field. The mechanical energy of the long wavelength longitudinal optical vibration in an ionic biatomic crystal is given by the expression[3]

   ELO = (v) d

   (2 . + 2) .(iii)

   LO

   LO

   Where = (Nm/V)1/2 u, and m is the reduced mass given by the expression m = (m+ m-)/ ( m+ + m- ).

   LO

   LO

   Therefore using (ii), (iii) and (iv) we can write,

   ELO = 2 (v) d

   ( 22 + 2) (iv)

   ELO

   1

   =

   =

   8e2

   (v) d

   [(U )22 + (U)2] .(v)

   LO

   LO

   In case of inhomogeneous medium the dielectric constant =(r) and the longitudinal optical phonon frequency LO = LO (r).

3. Calculation of retarded Greens function

   Using the equation (v), we can write the equation for Greens function[4] corresponding to the Lagrange equation of motion for U as follows,

   2

   2

   [ ( i0)2

   LO

   – ] (, ) =4e2 ( ) …(vi)

   The equation for the retarded Greens function differs from this equation by the factor ( + i0) in place of ( i0). These functions are expressed through each

   other as[3] , = (, ).

   We are going to define the spatial in-homogeneity in such a way that the spatial in-homogeneity is along the z-axis only and the medium is homogeneous along x- axis and y-axis. i.e. the total mechanical energy for the long wavelength longitudinal optical phonon ELO will depend on the z- coordinate only. Now to simplify the problem we define

   LO 1

   LO 1

   () (z)

   LO

   () = LO

   (z) = 1

   for z > 0 .(vii)

   LO 2

   LO 2

   () (z)

   LO

   () = LO

   (z) = 2

   for z < 0 .(viii)

   and () LO ()

   changes abruptly at z = 0.

   Where , , LO 1 and LO 2 are constants.

   1 2

   Equation (vi) can be solved using the boundary conditions (vii) and (viii).

   To find the transition probability K (q z, z) , first of all we shall evaluate the advanced Greens function and hence the retarded Greens function using the equations (vi) , (vii) and (viii).

   The advanced Greens function can be written in the (q,z) representation as

   (q z, z) . After simplifying the equation (vi) we get

   [ 2

   4e2 2 ()

   2 LO 1

   2 LO 1

   q ] (q z, z) =

   for z > 0

   z2

   [ i0 2 2 ]

   2

   1 LO 1

   4e2 2 ()

   ..(ix)

   [z2

   q2] (q z, z) =

   LO 2

   [ i0 2 2 ]

   for z < 0

   2 LO 2

   ….(x)

   Now we solve the equation (ix) and (x) under the circumstances that (a) lim± (q z, z) = finite.

   (b) (q z, z) is continuous at z = 0.

   LO

   LO

   (c) [ i0 2 1] . (q z, z) is continuous at z = 0.

   2

   After solving the equation (ix) we get

   2e22

   , + ,

   (q z, z) =

   LO1 ( qzz +

   q[ i0 22 ]

   + +

   qzz )

   1 LO1

   for z > 0 (xi)

   and after solving the equation (x) we get

   2e22

   , + ,

   (q z, z) =

   LO2 ( qzz

   q[ i0 22 ]

   + +

   qzz )

   2 LO2

   for z < 0 .(xii)

   Where we have defined the parameters + and as

   = [ i0 2 1] and = [ i0 2 1]

   + 1 2

   LO 1

   2 2

   LO 2

   Therefore from equations (xi) and (xii) we can fid (q z, z) using the

   relation (q z, z) = (q z, z)

4. Transition probability

   The transition probability[5],[6] from the electron state z for the electron with momentum p to the state z with momentum due to the interaction of an electron with long wavelength LO phonon is given by

   1. For the emission process

      2

      2

      W(, ) = 1

      L

      d K [(p p ) / ]. [N() +1]. ( ) .(xiii)

   2. For the absorption process

   2

   2

   W( , ) = 1

   L

   d K [(p p ) / ]. N() . ( ) .(xiv)

   Where N() = [exp(/kT) – 1]-1 is planck distribution function. k is Boltzmanns constant, and L2 is the normalized area in the x-y plane.

   and K (q) = dz d z z 2 . z 2 . K (q z, z) . …(xv)

   To find K (q) in the above integral, we consider a rectangular potential well in the hard-wall model[6] , with z = 2/d cos [(z z0 )/d], where z0 is the distance of the centre of the well from the surface.

   And the function K (q z, z) is given by the equation[3]

   K (q z, z) = i[ (q z, z) – (q z, z)]

   = -2. Imaginary part of (q z, z) .(xvi) Hence we can write from equation (xv) and (xvi)

   K (q) = dz d z z 2 . z 2 . K (q z, z)

   = -2. dz d z . 2/d cos [(z z0 )/d] 2 .

   2/d cos [(z z0 )/d] 2.Im (q z, z)

   2

   2

   = – 8

   d

   dz d z . cos [(z z0 )/d] 2 .

   cos [(z z0 )/d] 2.Im (q z, z) …(xvii)

   From (xiii), (xiv) and (xvii) we can write

   W(, ) = W( , ). exp[( )/kT] ….(xviii)

   Finally we get the collision integral[6] corresponding to the kinetic equation for the electrons interacting with long wavelength LO phonons in spatially inhomogeneous medium as follows

   Je, LO phonon =

   [W( , ). fr t (1- frpt) – W(, ). frpt(1 – fr t)]

   .(xix)

   Where frpt is Wigner distribution function of electrons for the case of weakly inhomogeneous system.

From the equation (xix) for the collision integral it is obvious that, as a special case if we assume Fermi distribution function in place of Wigner distribution function we will get the zero value (as it should be) for the collision integral. The equation (xix) may also be used to find the quantum kinetic equation[7] for the electrons interacting with long wavelength LO phonons in a spatially non-homogeneous medium.

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