Notes on the electron-phonon interaction

Министерство образования и науки Российской Федерации Федеральное государственное бюджетное образовательное учреждение высшего образования «Московский педагогический государственный университет»


С. А. Рябчун





                NOTES ON THE ELECTRON-PHONON INTERACTION






Учебное пособие










МПГУ
Москва • 2017

УДК 537.5(075.8)
ББК 22.373.1я73
     Р985


        Рецензент:

Г. М. Чулкова, доктор физико-математических наук, профессор кафедры общей и экспериментальной физики и информационных технологий МПГУ


        Рябчун, Сергей Александрович.

Р985 Notes on the electron-phonon interaction : учебное пособие / С. А. Рябчун. - Москва : МПГУ, 2017. - 18 с. : 2 ил.

    ISBN 978-5-4263-0579-3
    These notes are a result of a series of lectures given to the MS and PhD students of the Department of Physics, Moscow State Pedagogical University. They deal with the subject of electron-phonon interaction in pure threedimensional metals. The goal was to show how one could calculate the temperature dependence of the electron-phonon-interaction time from first principles within a simple model. Students wishing to expand their knowledge of the subject of condensed matter are invited to study any book on solid-state physics (for example by J.M. Ziman, or N.W. Ashcroft and N.D. Mermin.



        УДК 537.5(075.8)
        ББК 22.373.1я73


ISBN 978-5-4263-0579-3

© МПГУ, 2017
© Рябчун С. А., 2017

        CONTENTS


Preface ................................................................4
1. Lattice vibrations ..................................................5
2. Phonons..............................................................6
3. Electron-phonon interaction..........................................7
4. Electron-phonon-interaction time .................................. 10
  4.1. Electron-phonon-interaction time through the transition probability ........................................................ 11
  4.2. Electron-phonon-interaction time from the kinetic equation .... 14

3

Preface


     These notes are a result of a series of lectures given to the MS and PhD students of the Department of Physics, Moscow State Pedagogical University. They deal with the subject of electron-phonon interaction in pure threedimensional metals. The goal was to show how one could calculate the temperature dependence of the electron-phonon-interaction time from first principles within a simple model. Students wishing to expand their knowledge of the subject of condensed matter are invited to study any book on solid-state physics (for example by J.M. Ziman, or N.W. Ashcroft and N.D. Mermin.

     This course was taught within the project No. 14.B25.31.0007 funded by the Ministry of Education and Science of the Russian Federation. The publication of these notes was made possible by the support of the Russian Science Foundation (project No. 17-72-30036).


4

1. Lattice vibrations


     To keep the discussion simple we start with the one-dimensional case of a continuous system. Let x be the coordinate and £(x, t) measure the displacement of an infinitesimal element of the material located at x from its equilibrium position. Then we can construct the action
5 = fdtf dx£;£ = 1-p (Xj) - M^-) .                      ⁽¹⁾
2 \dt/ 2 \Xx/

Here p is the mass density of the material and Y is the Young modulus. For small vibrations, which we are considering, these two material parameters are constant both in space and time, equal to their respective equilibrium values. The Euler-Lagrange equation reads

                   X X£ X X£ _X£ xtx?xi)⁺xxx(xi)=x? leading to the wave equation

^-^=0,;= I Xt² ; Xx² U,; p
V

(2)

(3)

      Trying a plane-wave solution-(x, t) ~ exp[-i(^t - qx)]produces a dispersion relation

f>„ = c|q|.                                  (4)
The general solution is then a superposition of plane-wave solutions:
-(x,t) = f dqAq[aqe⁻ⁱ⁽^4t⁻qx⁾ + ajel⁽^£-q^⁾\,               (5)
where aq are arbitrary coefficients and the overall factor Aq will be determined presently.
     To quantise the system, first, define the canonical momentum
     ■Ф, t) = -pjj = PXt = -ipf dqAquq[aqe⁻ⁱ⁽“«t⁻qx⁾ -aqeⁱ⁽^t⁻qx⁾]         (6)

and then construct the Hamiltonian
// = pdx L|| - £.| = f dx ^ +1I f^⁻) .                      ⁽⁷⁾
Xt          [2p 2 \дх/
     It is convenient in (5) and (6) to merge the time-dependent phase factors exp(± i о qt) with their respective coefficientsaq and ajand also change the sign on q in the second term in the brackets, so that the expressions for the coordinate £, and momentum n assume more compact forms:

5

С. А. Рябчун

^(x,t) = f dqAq[aq(t) + a!q(t)]eⁱqx, п(х, t) = —ipf dqAqMq[aq(t) - atq(t)]eⁱqx,


where we have explicitly shown the time dependence of the coefficients. We now declare £, and n, and alsoaqand aoperators, and impose equal-time commutation relations


[^(х, t), n(x', t)] = ih3(x — x‘), [aq(t),aqf,(t)]=5qq,.


(9)

It is left as an exercise to the reader to show that these commutation relations

are consistent if we take Aq

=ll

h npijjq'

Thus we get

ᵣ dq
!(x,t) = f-±
У2я

²"qP

(aq + atq~)eⁱqx,

(10)

ᵣ dq ⁿ(x,t) = f ^⁽—ⁱ⁾

phmq
2

⁽aq —aLq')eⁱqX.

Here we have suppressed the explicit time dependence of the operators. If we now insert these expressions for the displacement and the momentum into the Hamiltonian (7), we shall get after the normal ordering procedure

                            H = f dqh^qa^aq.






(11)

2. Phonons


      The Hamiltonian (1 1) is identical to the Hamiltonian of the simple harmonic oscillator, except for the zero-energy term which we have removed by the procedure of normal ordering. Its eigenstates can be built up as follows. Starting with the vacuum state, we allow the creation operators to act on it to produce any desired number of excitations:


6

NOTES ON THE ELECTRON-PHONON INTERACTION

gJi0> = |nq = 1>,|nq> =

(aJ)”Q .
  ,--7 , ¹ ⁿo >ⁿp ,■■■ /
^ⁿq
(aq')ⁿ4(gp')ⁿp ■■. ^nq\nₚ\...

(12)

Within the context of the lattice vibrations these excitations are called

phonons. Thus in the last state of (12) there are nq phonons of type q, nₚ phonons of type p etc.
     In the simplest case of a three-dimensional lattice we have atoms arranged in a regular pattern. Let R mark the position of a particular atom in equilibrium. Then the equations (10) should be modified accordingly:

f(R,t)

s

d³q ~ h (2^)571 eq s jT^P

⁽gq, s + g-q, sk'q'¹' ,

n(R, t)

s

  d³q , ..      рЬша s ,
(2я)³/² ⁽⁻ⁱ⁾eq, s J—~ ⁽gq, s

g-q, sk'q'¹' ,

(13)



where s is the polarisation index, andeq, ₛis the unit vector in the direction of the displacement of the q-th mode of polarisation s.






3. Electron-phonon interaction


     The potential energy of an electron located at position x because of the presence of ions at shifted positions R + £(R, t) can be written as follows:

P.E.(x) = ^V(x
           R

- R -ft «^ V(x - R) -^ -7 ^ V(x - R),

(14)

R

R

The first term is the potential energy of the electron in the static lattice. It is responsible for the band structure and of no interest here. The second one describes interaction of the electron with the vibrations of the lattice, i.e. electron-phonon interaction. So, we can immediately write down the Hamiltonian of the electron-phonon interaction

^e-ph

= f d³x^l(x) -£ - 7^ V (x - R) ^(x) = f d³x^l(x)yₑ-ph(x)^(x),

(15)

R

where^f(x)and^(x)are operators creating or destroying an electron at a position x, respectively. To proceed further it is more convenient to go over to the momentum representation:

7

С. А. Рябчун

^e-ph = ^<k IK-ph I k'KV                             (16)
                              k, k'
We approximate the state of the electron with a definite momentum with a plane wave, upon which the matrix elements ivₑ₋ₚₕ i k'>is the Fourier transform of the potential energy describing the electron-phonon interaction.


We have


         <k IK-ph I k'>


-£f(R,t)^(x- R)
R

f dW^*

-^£(R,t) • f d³xeⁱ⁽k-k'⁾'x7R(x - R)
                                  R
-^mo4f d³x7eⁱ⁽k⁻kW(x - R)
                                  R
                                  -j(k - k')J d³xe‘⁽k⁻k'kxV(x - R)]
-^f(R,t) • [f dSeⁱ⁽k⁻k'>W(x - R)
R
                                  -j(k - k')J d³xe‘⁽k⁻k'kxV(x - R)]
j(k - k')^ f (R,t)f d³xe‘⁽k⁻k'kxk(x - R)
R
j(k - k')^f (R,t)e‘⁽k⁻k'kRfd³xe‘⁽k⁻k'kxk(x) R

(17)

In calculating the matrix element we have integrated by parts and used periodic boundary conditions to eliminate the surface integral, and to go to the last line we have changed the variables. Thus we need to compute the Fourier transform of the potential energy of an electron in a lattice.
     The potential that an electron feels is created by ions with charges Ze and by the other electrons. The potential of an ion with charge Ze surrounded by electrons satisfies the Poisson equation

         d                      d d³k
7²0(x) = -4njo(x) = 4rcZ<?<5(x)- 4m?f ——f(6k,$(x)), (2 я)³

where

1
                 f(6ₖ,0(x)) =  ------7—7
                 '    ¹ " exp ek⁻^⁽x⁾


(18)

(19)

is the distribution function of the electrons in the presence of a potential. If the potential is not too large, the distribution function can be expanded:
d/o
/(eₖ,0(x)) « /o(ek) + ^(x)^,                              (20)

where/o(eₖ)is the usual Fermi distribution function. Putting the expansion (20) into (18) we obtain

8

NOTES ON THE ELECTRON-PHONON INTERACTION

Е²Ф(х)

                   ᵣ d'k    _         ₂c , , df₀
4rcZe<5(x)- 4nef (2^)lfo(ek) + 4ne²fdeD(e) —

(21)

4nZe6(x~( — 4nen₀ — 4ne²D(p^(x~),

where we have introduced the number density n₀ of the electrons in the absence of a potential, the density of statesD(e)and used the fact that at low temperatures (which we consider here) the derivative of the Fermi distribution function with respect to the energy is very closely equal to the Dirac delta-function centred at the chemical potential. The uniform charge distribution with the density en ₀ does not produce any physically reasonable potential (the one that goes to zero at infinity). Besides, it is of no interest to our problem. What we should essentially like to find is the potential of a “dressed” ion, i.e. the ion that is surrounded by a cloud of electrons initially uniform. It is reasonable to assume that the potential is spherically symmetric; therefore we the Poisson equation we need to solve is
                 Id²
                 —т-7[гФ(г)] = 4nZe8(x( — 4ne²D(u)<&(r\                      ⁽²²⁾
                 r dr²
which forx ^0 simplifies to
                      Id²               „
                      —— [гФ]г)] = —e-ne(D(ii(<P(r)                          ⁽²³⁾
                      r dr²
with the solution
Ф(г) =—exp(—r/r₀(,                                (24)
                                  r

where the constant C will be determined shortly, and the characteristic lengthscale r₀ is equal to

                            _ 1
                           ⁰ ^4ne²D(^(

(25)

This characteristic length-scale is of the order of 10⁻⁸ cm, roughly the interatomic distance in metals. The constant C can be determined if we note that at

small r the term with the delta-function dominates, and we need to find the potential of a point charge Ze. Finally, the solution to (22) is
                      Ф(г) =—exp(—r/r₀(,                         (26)
                            r
     Thus, the Fourier transform of the screened Coulomb potential that enters the matrix element of the electron-phonon Hamiltonian is
4nZe²
F-TMx»=(k—k^,.-,.                            ⁽²⁷⁾
The matrix element itself is equal to

9

С. А. Рябчун

<k IK-ph (x)l k')

j(k' - k)-I^(R)e-                4wZe2           
R                     (k'-k)-R            c      
If “ q [Ve‘(k-        (k - k')2 +r0-2            
(2я)3/2 Z_i           , , .,, 1       4rcZe2     
                      k+q)-R ___________________ 
                      (k - k')2 +r0-2            
  x£(k' - k)- eq, s   h (a +а+ )                 
s                   A 2p<4|, s (aq, s l    q, s).

(28)

Because we are dealing with a lattice, the sum in the square brackets is only non-zero if k' = k - qmodG,where G is a reciprocal-lattice vector:

                      ^r- k k-q'R =N8k', k - q + g:
R

(29)

where N is the number of primitive cells (equal to the number of ions since we are dealing with a Bravais lattice).
     As one can observe by examining the matrix element of the electronphonon interaction, electrons in our model are only coupled to longitudinal phonons. This can be understood by noting that only longitudinal waves lead to density variations and thus to variations of the positive-charge density. We can now write down the Hamitonian of the electron-phonon interaction:

He-ph

= I
   k, k', q, G

Mk, k' (aq + atq)^ Cₖ<', k - q + G ,

Mk, k'

IN
У

h      4nZe²
2p"k-k' (k - k')² + r₀~²

(k'-k)- ek-k.

(30)

4. Electron-phonon-interaction time

     To proceed further we make the assumption that at sufficiently low temperature we can neglect Lapp processes. This is reasonable since at low temperatures an electron cannot have its momentum changed so that it ends up in a neighbouring Brillouin zone. This allows us to simplify the Hamiltonian (30):

10

NOTES ON THE ELECTRON-PHONON INTERACTION

                            ^e-ph   ' Mq⁽^q + ^-q)^k+q^k,
k, q

Mq = iN

h 4nZe² ~ 2P"q Ц² +r₀⁻² ⁴ eq.

(31)

The transition processes described by this Hamiltonian can be represented with the Feynman diagrams in Fig. 1. Above the diagrams we indicate the corresponding term in the Hamiltonian, and below the energy conservation relation.

     We shall calculate the electron-phonon-interaction time in two ways. One way to do so is to calculate the probability per unit time that the electron initially in a stateik)will make a transition to a state |k + q). Since we are not interested in the final state, we shall be summing over all possible final states, i.e. over all possible values of q. The other way is to solve the Boltzmann kinetic equation in the relaxation time approximation.

Fig. 1. Transition processes described by the Hamiltonian of the electron-phonon interaction



            4.1. Electron-phonon-interaction time through the transition probability


     Using the Fermi golden rule we write
            1 = 2r'|Mq|²[(1 - fk+q )V(£k+q -eq + h^q)
            Tk     h Д                                              (32)

+⁽1 ⁻ fk+q⁾⁽l + ⁿ-q⁾⁵⁽ek+q ⁻eq ⁻h^q⁾],


11