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The Electron-Phonon Interaction in Normal and Superconducting Metals

1

LE O P . KA D A N O F F

Department of Physics, University of Illinois, Urbana, Illinois

I. Correlation Function Approach

This series of lectures has two purposes:

(a) To report on recent developments in the theory of the electron- phonon interaction in metals.

(b) To serve as an introduction to the use of Green's function methods in many particle physics.2

We begin by noticing that one of the very simplest things you can do to a many-particle system is add a particle to it or pull one out. For this reason, it is very convenient to describe the properties of the system in terms of the creation and annihilation operators:

CP+(t') = creation operator Cp(t) = annihilation operator.

When these operators act to the right on a state of the system, they respectively add a particle with momentum ρ to this state at the time t' and pull one out at the time t.

1 A series of lectures given at the 1963 Spring School of Physics at Ravello, Italy.

This work was supported in part by the U.S. Army Ordinance DA-ARO(D)-31- 124-G340.

2 The Green's function methods that 1 shall employ can be traced back to the work of Martin and Schwinger (/). These methods are discussed in detail in a book by Baym and Kadanoff (2).

77

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78 LEO P. K A D A N O F F

W e shall also m a k e use o f the wave field operators tp+(r\ t') and

^(r, t) which respectively add and substract a particle at the space-time points r', t' and r, t. If one quantizes in a b o x o f unit v o l u m e with periodic boundary conditions the t w o sets o f operators are related by

W+(r\ tf) =

Σ

exp ( - ip · r') C / ( f )

(1.1) ψ(τ, t) = ^ e xP (Φ * R) CP W

where the m o m e n t u m sums run over all the allowed m o m e n t a in the b o x . ( W e use units in which h = 1.)

These are all Heisenberg representation operators and, as such, they have the time depencence

A(t) = exp (iH't) A(0) exp (— iH't) (1.2)

Of course, we m a y c h o o s e the zero point o f energy at our convenience.

W e c h o o s e our zero point by using in (1.2)

Η' = Η — μΝ (1.3)

where Η is the standard Hamiltonian, μ the chemical potential, and Ν the number operator to b e the basic operator which gives the time de­

pendence of our operators.

The t w o basic correlation functions w h i c h we shall discuss are G> (r, t; r', t') = (ip(r, t)y>+(r,' *')>

(1.4)

G<(r

9

t;

r \ t') = <>+(r', t')y>(r, φ

where < > stands for b o t h a quantum-mechanical average and a sta­

tistical average according to the averaging procedure defined by the grand canonical ensemble of statistical mechanics.

Because of the equilibrium nature of the system all physical quan­

tities are independent o f the time. Furthermore, our periodic boundary condition in space guarentees the spatial h o m o g e n e i t y of the system.

Therefore, G> and G< cannot depend u p o n r and r' or t and t' individ­

ually but only u p o n the difference variables r — r', t —1 \ This fact

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THE E L E C T R O N - P H O N O N INTERACTION I N METALS 79 is conveniently included by using the Fourier representations of these quantities as

G> 0 , t; r', t') = G>{r — r'\ t — t') (1.5a)

Σ /

dcu exp [ip • (r — r') — iw(t — t')]G' (p, to) at

= Σ exp [ip · (r - r')] <Cp(i) C/(?')>

Ρ

while

G<(r — r'\ t — t') = ^ exp [i> · (r - r')] < C/ ( i ' ) C,(i)> (1.5b)

Σ

= | ^ e x p [ip · (r - r') - io:{t - i')] G< (ρ, ω).

G> (/?, ω) and G< (/?, ω) have a very direct physical interpretation.

Consider G< (ρ, ω). We know that Cp(t) is an operator which removes a particle with momentum at the time t. In the usual quantum-mechanical way, the time Fourier transform of Cp(t) removes a particle of momentum ρ and energy ω. But, if you are going to remove such a particle, there must be a particle with this energy and momentum present initially.

For this reason,

G< (/?, ω) = density of particles with momentum ρ and energy ω. (1.6)

Because the time Fourier transform of Cp+(t') adds a particle with momentum ρ and energy ω, we can see that G> (ρ, ω) measures the sys­

tem's ability to accept a particle with this energy and momentum.

That is,

G> (/?, ω) = effective density of states in /?, ω (1.7) or, if you prefer, this may be termed a density of holes.

These statements (1.6) and (1.7) will be very important for our future discussions. Therefore, we examine them in a little detail. From (1.5b), the total number of particles with momentum p, Np= Cp+ Cp, is given by

<Np} = < C / CP> = j d£~ G< (/>, ω) (1.8)

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80 LEO P . K A D A N O F F

which is the integral over all energies of the density as function of energy.

Also the total effective density of states is, from (1.5b)

J ΊΪΓ

G> ( A Ω) = <CR

>

CP+>

'

However, the creation and annihilation operators satisfy the equal time commutation or anticommutation relation

Cp(t)Cp+(t) Τ Cp+(t)Cp(t) = 1 (1.9)

where the upper sign here, and in what follows, is appropriate when the particle obey Bose statistics and the lower sign is for Fermi statistics.

Therefore, the total density of states is

S ^ r

G > ( P 9 U J ) = 1 ±<N

*

>

'

( L 1 0 )

For fermions as <iVp> increases, the effective density of states decreases while for bosons, which are very gregarious, the larger the occupation of a state, the larger is the effective density of states.

To illustrate the meaning of G> and G<, let us consider a tunneling experiment. In its simplest form this is an experiment in which two

Insulator

Metal A Metal Β

- M i l l

FIG. 1

conductors are separated by a thin insulating layer and a D C voltage is applied across the gap. The induced current can then be measured as a function of voltage.

Electrons cross the gap via a quantum-mechanical tunneling process.

Following many recent authors (3; 4) (see the chapter by R. Prange in

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THE E L E C T R O N - P H O N O N I N T E R A C T I O N I N METALS 81 this volume) we can describe the tunneling by a tunneling amplitude T(p',p), which is the probability amplitude for an electron with mo­

mentum ρ on one side of the gap reappearing with momentum p' on the other. (This is essentially the overlap between the wave functions on opposite sides of the insulator.)

We calculate the tunneling rate from A to Β by making use of the golden rule, which states that a transition rate is proportional to a matrix element squared [here, | Τ(ρ,ρ') |2] , an energy conserving ^-function, a density of initial and final states, summed over all initial and all final states. Thus,

dN Γ doj Γ άω'

ι ^ ^ |

2

A^B PP 'J 2 π J 2 π (1.11a) 2πδ(ω — ω' + eV) GA< (ρ, ω) GB > (/>', ω') .

Here the δ-ίunction requires that the energy change on traversing the barrier be eV. GA< and GB >, of course, represent the density of par­

ticles in A and the density of states in B.

The rate of tunneling in the opposite sense is given by a result identical to (1.11a) except for the appearance of the density of particles in Β and the density of states in A, that is

dN

Σ ί — f

dco C dco'γ - \T(p, , P')\2

l n (11.1b)

dt

2πδ{ω — ω' + eV) GA> (/?, ω) GB< (/?', ω') .

To obtain the net tunneling rate we substract (1.11b) from (1.11a) to find

dN — Γ dco r dco' ,

net — = Σ H t " I Άρ, ρ') |2 2πδ(ω - ω' + eV) dt ~ J 2π J 2π ^ ^

[GA< (ρ, ω) GB> (p\ ω') — GA> (ρ, ω) GB< (p\ ω)].

Equation (1.12) may be used to predict the tunneling rate in all cases except that in which both conductors are in a superconducting state.

In that case, some very exceptional behavior results as first predicted by Josephson (5) and experimentally verified by Anderson and Rowell (6).

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82 LEO P. K A D A N O F F

This kind of tunneling is discussed in detail by P. W. Anderson in this volume.

In all but this exceptional case, one expects that, as V—> 0, the tunneling current should vanish. However, if A and Β are different materials so that GA< φ Gb<, Eq. (1.12) does not appear to predict the vanishing of the current at F — • ( ) .

In order to see how this vanishing does in fact occur, we must make use of a kind of detailed balancing condition which relates the density of particles and the density of states. It is a characteristic feature of a system in equilibrium at temperature Τ that the relative occupation of a state with energy ω is proportional to exp (— βω), where β = (kT)'1, and k is the Boltzmann constant. This characteristic feature shows up as a restriction upon G< and G> that, for a system in equilibrium,

(density of particles) = exp (— βω) (density of states).

We shall prove this condition in a moment. For now let us see its con­

sequences in the tunneling rate given by (1.12). We use (1.13) to eliminate

GA< in favor of GA > and GB < in favor of GB >. Then (1.12) becomes

GA>(p,to) GB>(p',co') [exp(— βω) — exp (— βω')].

Clearly now as V->0, so does the current.

Equation (1.13) will be quite crucial for us so we consider its proof in some detail. In the ground canonical ensemble, the expectation value of any operator is given by

where the sum runs over all possible states with any number of particles and where

G< (ρ, ω) = exp (—βω) G> (ρ, ω) (1.13)

<XY = Σ

e xP 1 —β(Εξ—ΜΝξ)] <£\Χ\

?>IZ G

(1.14)

= Tr [exp (-βΗ')Χ]/Ζ0

ZG = Σ e xP 1 —β(Εξ — ΜΝξ)] = Tr [exp (-βΗ')}

is the grand partition function.

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THE E L E C T R O N - P H O N O N INTERACTION I N METALS 83 Now consider the two functions

F>(t) = <A(t) X}

(1.15) F<(t) = (X A(t)} .

If A(t) is a Heisenberg representation operator with no explicit time dependence

A(t) = exp (iH't) A(0) exp (— iH't). (1.2)

Equation (1.2) can be used to define A(t) for complex values of the time.

Now consider, in particular,

^ ( 0 U~ifi = <exp ( + βΗ') A(0) exp (-βΗ') X}

= Tr [exp (—βΗ') exp (βΗ') A(0) exp (—βΗ') X]/ZG

= TT [A(0) exp (-βΗ') X]/ZG

= Tr [exp (-βΗ')Χ A(0)\IZG = <X A(0)> . Consequently,

F>(t)\t^-lP= F<(t)\t=0. (1.16)

Let us apply this theorem to G> and G<. Equation (1.16) implies that

G>(r,/;r',fU--

l V

, = G

<

(r, t\ r', t') \

t=0

dco

— exp [— i(o(t — t') G*(p, co) \t__,p (1.17)

L7X

= J exp [— ico(t — t')} G<(p, ω) \ t==0

In

or

Γ αω Γ dco

exp (iwt') exp (—βω) G>(p, co) = exp (icut') G<(/?, ω),

J In J In

which immediately leads us back to (1.13).

The functions G>(p, ω) and G<(p, ω) contain a tremendous amount of useful information. They describe, as we have seen, the density of

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84 LEO P . K A D A N O F F

particles and the density of states, and they describe all the thermody­

namic properties of the system. Consequently, it is quite worthwhile to know these functions.

In actual calculations, it is very convenient to make use of the spec­

tral weight function

A(p9 w) = G>(p, w) ψ G<(p, w) (1.18)

= Fourier transform <Cp(t)Cp+(t') =F C+(t')Cp(t)y.

The usefulness of A(p9 w) is derived from the equal time commutation relation (1.7), which implies the sum rule

ί

— Α(ρ,ω)=1. (1.19) dw

By making use of the detailed balancing condition (1.13) together with the definition (1.17) we can express G> and G< in terms of A as

G>(p9 w) = [1 ±f(w)} A(p9 w) (1.20) G<(p9 w) =f(w) A(p9 w)

where ^

f(f») = ,Λ , (1-21)

exp ( ρ ω ) =p 1

is the a priori probability of observing a particle with energy w in the grand canonical ensemble.

Now we must face the hard job of determining G> and GK. T o see how this goes consider just the trivial case of free particles. In general, w represents the total energy of a particle added to the system, its kinetic energy plus the interaction energy that it aims through its interaction with all the other particles in the system. However, for noninteracting par­

ticles w must be just the kinetic energy

ερ = ρ2/2ηι—μ. (1.22)

Therefore G>9 G<, and A are all proportional to 3(w — ep). However, for A(p9w)9 we have a sum rule (1.18) which tells us the constant of proportionality, so that

A(jp9 w) = 2π(5(ω — ep) . (1.23)

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THE E L E C T R O N - P H O N O N INTERACTION I N METALS 85

e x p [ ^ / 2 m - / / ) ] T l '

the familiar result for a system of noninteracting fermions or bosons.

In generalizing these free particle results, it is convenient for us to define a function of a complex variable z:

Γ dw A(p, w)

<***> = JV

ζ-co • 0·24) For the free-particle system:

G(p,z)= — . (1.25)

Z ~ £ P

In the general case,

<KP, Z) = L-— . (1.26)

ζ — ερ—Σ(ρ, ζ)

Since the variable w represents the total energy of an added particle we can interpret Σ (/?, w) [that is, Σ (ρ, ζ) for real ζ] as the extra interac­

tion energy that would be produced by the addition of a hypothetical particle of momentum ρ and energy ω. For this reason Σ is termed the self-energy. If the hypothetical energy is to be a realizable energy dif­

ference w must satisfy the dispersion relation

w = ep+ Σ (ρ, ω). (1.27)

Notice that these energy levels give the position of the poles of G(p, z) and hence of (5-function singularities in A(p9 w).

To illustrate how Σ (/?, w) may be calculated, let us consider the in­

teraction of electrons with longitudinal phonons. In the simplest model, Using (1.22), we can easily calculate the density of particles with mo­

mentum ρ as

Γ dw Γ dw

< N P > = J ~ * Γ G < ( P ' W ) = J / (Ω ) Α ( Ρ' Ω )

=

/(*,)

1

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86 LEO P. K A D A N O F F

this interaction may be represented by the interaction Hamiltonian Hep =

Σ

lvQaQd r e xP * r) ν+(ΓΜ Ο

q J (1.28)

dr exp (— iq · r) y>+(r)yj(r)].

Here aq+ and aq are phonon creation and annihilation operators, and vg is a matrix element which measures the coupling strength. We can rewrite (1.27) as

HeP =

Σ (

ν

Λ

CP+q+CP + v , V CP i/ Cp) (1.29)

qp

The first term in (1.29) describes a process in which an electron with momentum ρ absorbs a phonon with momentum q and hence scatters into the momentum state ρ + q. The second term describes a similar process in which a phonon is emitted.

As a preliminary to the calculation of Σ(/?, ω), we calculate (Hep} in second order perturbation theory. As usual, the' second order per­

turbation theory gives a sum over all initial and the final states of a ma­

trix element squared divided by an energy denominator which is the difference in energy between the initial and the final state. In this case

(1.30) Γ dco Γ dco __ ,

Γ ^

+ 1

"

|_ ω + cog — ω' ω — coQ — ω' \q = | p_p, (.

Here coq is the energy of a phonon with wave vector q, and 1

TV =

9 exp (βω9) — 1

is the equilibrium number of phonons with wave vector q. The first term in the square bracket of (1.30) describes a process in which an electron with (ρ, ω) absorbs a phonon and scatters into the state (//, ω').

The energy difference between the initial state an the final state is ω + coQ — ω'. The density of initial states is given by the density of

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THE E L E C T R O N - P H O N O N INTERACTION I N METALS 87 electrons, G<(p9 ω) times the density of phonons, Nq9 and the density of final states if G>(p'9 ω'). In just the same way, the other term in the square bracket gives the energy shift due to phonon emission processes.

FIG. 2

We calculate the self-energy as the change in the interaction energy which occurs upon the addition to the system of a particle with momen­

tum ρ and energy ω. This addition may be represented by changing the density of particles according t o

(?<(/?', ω') - * (/<(/?', ω') + 2πδ(ω — ω')δρ^. (1.31a)

The added particle also changes the density of states. Since an electron is a fermion, the change is a reduction in the density of states according t o

G>(p'9 ω') — (?>(/?', ω') — 2πδ(ω — ω') δΡιΡ,. (1.31b)

The self-energy is then defined as

Σ(ρ9ω) =δ(Ηερ> (1.32)

Γ _ , , Γ Γ Να N0+l 1

= h r - Σ

Ν ?

^ > ' ) Ι — τ -

1

— , - + — - Γ

J 2π

^7 I |_

ω

+

ω9 — ω' ω — ως — ω'_\

+ G<(p'9 ω') ! Γ + -

|_ ω — ω? — ω ω + ω $ — ω JJi =|P_p'|.

This result can again be understood in terms of second order perturba­

tion theory.3

3 The argument given here may be extended, but only with considerable care, to obtain Σ as a variational derivative of an object related to < He„ > with respect to G.

A rigorous version of this argument is presented by Baym (7).

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88 LEO P . K A D A N O F F

Before we go any further, we must face a serious difficulty. Σ(ρ,ω) is, strictly speaking, undefined at a fantastic number of different points on the real axis: all those points at which the energy denominator may vanish. The exact position of these singularities depends in great detail upon the exact size and shape of the system. We seek a method of elimi­

nating this unwanted detail and retaining only that part of the informa­

tion contained in (1.32) which pertains to the properties of a very large system.

There is a very simple method for accomplishing this elimination. We simply replace the ω which appears in (1.32) by the complex variable z.

Then (1.32) becomes

(1.33) G>(p\ ω') ——? - + < T

{ \_z + ως — ω ζ

ojq — ω

J

Γ N0 N0 + 1 Τ)

\_z — cog — (o ζ + ωα — ω J J ^ , . By taking ζ to have an imaginary part we stand far enough away from the singularities so that we may consider them to be essentially contin­

uously distributed along the real axis. In fact, we may allow them to be­

come continuously distributed by taking the limit as the volume of the system goes to infinity and

Then, Σ(/?, ζ) has the form:

Γ άω Γ(ρ,ω) Σ(ρ, ζ) =

J 2π ζ — (1.35)

with Γ(ρ, ω) now being a continuous function of ω.

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THE E L E C T R O N - P H O N O N I N T E R A C T I O N I N METALS 89

II. Green's Function Approach

At this point, we shall consider how the results we have just obtained may be rederived by using a more formal Green's function technique.

To do this, we define

G(r, f; r'9 t') = - i - <Τ[ψ(τ9 ί)ψ*{τ\ *')]> (2.1) ι

where the time variables are limited to be on the imaginary axis with 0<ιί<β; 0<it' <β.

Here Γ is a Wick time ordering symbol which tells you to order the operators according to the sense of the " t i m e " it: the operator with the larger value of it appearing on the left. Also, for fermions, / contains a factor of (— i) for each permutation of the operators from their stan­

dard order. Thus,

t 1

\ — G>(r, t; r'9 t') for it > if

G(r9t9r'9t') = : J (2.2)

J

± — G<(r, t\ r'9 t') for it < it'.

The main reason for defining G for imaginary times lies in the fact that we can relate the values of G at the two end points of its region of de­

finition by using Eq. (1.16). This implies

G(r91\ r't') |,_o = ± G(r, /; r'91') \t =.i 0. (2.3) The boundary condition (2.3) is most conveniently represented by writ­

ing G as a Fourier series in its time variable as

r *

P ι ^ ( 2 ·4 )

G(r91; r'91') = -JL -— £ exp [ip · (r-r')-izv(t - t')] G2(p) . J (2π)3 —ιβ 2

This form for G will necessarily satisfy the boundary condition if

πν

[ even integer for bosons I odd integer for fermions.

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90 LEO P. K A D A N O F F

By inverting the Fourier series, one can easily show that the Fourier coefficient is

Γ dco A(p, ω)

Gy(p) = G(p,z,) = ^ - L . (2.5)

J 2π zv

ω

A useful technique for evaluating G involves the equation of motion of the annihilation operator:

dw(r, t) at

For free particles this equation of motion is

" d |72

2m + μ ip(r9 0 = 0 .

In calculating the time derivative of G(r, r', tf), we use Eq. (2.4) to­

gether with the time derivative of the discontinuity at it = it' produced by the time ordering. This discontinuity is

4 - <ΙΦ, t)ip+(r'91) Τ W+(r\ t)tp(r9 φ = - <3(r - r').

/ ι Thus for free particles G obeys:

[

f

+ + ^]

G ( r

'

1; r

'' ^ =

d(r

-

r

'

} 6{ti

-

i t y

·

( 2

'

6 )

To find G(/7, z) from (2.6), we multiply by exp [— ip · (r — r') + iz(t — t')]

and integrate over all space and all time in the interval [0, — //?]. Then, we find

ν — ερ] G(p9zv)= 1 or

λ - x αω Α(ρ9ω)

>,*,.) = = ί

Zv — £ p J 2n z„ — ω

There exists a theorem which states that if an equation that is F(z) and G(z) are two analytic functions of ζ such that F(z) = G(z) holds for all

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THE E L E C T R O N - P H O N O N INTERACTION I N METALS 91 zv = {nvj— ίβ) for even or odd integral ν and if F(z) and G(z) have no essential singularity at infinity then

F(z) = G(z)

for all z. Therefore (2.6) implies

dco A(p9 ω) 1

2π ζ — ω ζ — ερ for all ζ and we find

A(p9 ω) = 2πδ(ω — ερ) just as before.

To illustrate the application of Green's function techniques, we con­

sider the interaction of electrons with longitudinal phonon, which may be represented by the Hamiltonian

H' = He + Hp+ Hep (2.7a)

He = jdr ip+(r) ^ -//J ip(r) (2.7b)

HP = ΣW QaQ+aQ ( 2'7C)

Q

He p =

Σ

VQaQd t e XP i Q ' r)

Ψ

+

^ΜΉ

q

J

+ Σ

vQ *aQ+ d r e xP ( IA ' r )

V

+

(

R

M

R

) ·

q J

(2.7d)

Here y>+{r) and ip(r) are electron wave field creation and annihilation operators, a* and aq are phonon creation and annihilation operators, wQ is the phonon energy. For simplicity, we ignore Umklapp processes, the possibility of several types of phonons, and the spin of the electrons.

We see immediately that the various operators obey the equations of motion

Γ d

F

2 Ί

L

f

- a r

+

2 ^

+ / /

J ^ °

(2

-

8a)

= Σ t W O e xP ( ^ * r)+vg*aq+(t) exp(—iq · r)] ip(r, t)

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92 LEO P . K A D A N O F F

f d

i- dt

<o)j (ψ) = J

d r v

*

e x

P (to ·') ψ*(τ> 0 y*r> 0 (

2

-

8 b

)

( ' + W ?) Α«+ ( Ί ) = ~~ J D R V«E X P ( — Ί ? ' R ) ^+ ( Ί' R ) ^ ? ) · ( 2-8 C ) It is quite simple to compute the equation of motion of the electronic Green's function. We have:

V d . P2 "1

ί — + — + μ G(r, t; r\ t') = <5(r - r') <5(f - ί')

+ 4- < Γ { Σ Μ ' ) « φ ( ί ? · ' )

1 « (2.9)

+ W W e xP (— i(l · r)]

V<r, ί) ν+( Λ *')}>·

Therefore, in order to find G(l, /') we need to know

<Γ[α?)ν<1)ν+(1')]>

and

< Π α / ( 0 Κ ΐ) ν+( ΐ ' ) ] >

where 1 stands for r, t, and Γ for /·', t'. These functions are not known, but we can use the equation of motion for at(t) and ajt') to eliminate the phonon variables from these expressions. If we are to make use of these equations of motion, we must allow the time in at(t) and ae+(t') to be different from tu so that we may differentiate with respect to t.

For this reason, we define:

Fjt, 1,1') = <J[ajt) ψ(1) y+(l)]> (2.10a) Ρ jit, 1,1') = <T[a+(t) ψ(1) y>+(l')]> . (2.10b)

From (2.3b) and (2.3c) we find

\ _ * Ύ ί— ψ · ( ί , 1*1 ') ( 2 , l l a )

= ν* <Γ[ j dr exp ( - iq • r) ψψ, t) f(r, t) ψ{\) y;+(l')]>

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THE E L E C T R O N - P H O N O N INTERACTION I N METALS 93 and

^ - ^ + ω ^ ^ , 1 , Γ ) (2.11b)

= — vq <T[ J dr exp (iq · f) ψ+(Ρ, t) ψ(Ρ, i) ψ(\) ψ+(1')]> .

There are no discontinuity terms coming from differentiating Τ because the equal time commutator of aq and ag+ with ψ and ψ+ vanishes.

Equations (2.11) are to be solved with the standard 0 to — ϊβ boundary condition:

Fq(t,hV)\t=o = Fq(t,l9r)\t = _i0 (1.16)

and similarly for F. In obtaining the solution to (2.11) it is convenient to use the function d0)(t, t')9 which is defined by the differential equation

\i - o J dWq(t9 t') = d(t -t') for 0 < it < β

L a t

J o<*r</?

and the boundary condition:

da><i(0,

/') = <(-*'/?, ο.

The solution to this equation for d0) may be written down at once.

We split d0) into two parts in just the same way as we divided G(l, Γ) into two parts by writing

( d>(tl912) for itx > it2

idWq(h, t2) = \ (2.12)

( d<q(tl912) for itx < it2.

We can see immediately that d>q and d<q are exactly analogous to

^>( P j h — ^ ) a nd G<(p9 tx12) for free bosons if the replacement

P2

2m is made. Therefore, we have

dlq(h — t2) = exp [— iwq(tx — t2)] (Nq + 1)

(2.13a) diq(h — t2) = exp [— ico^/i — t2)] Nq

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94 LEO P . K A D A N O F F

where Nq is the equilibrium number of phonons with wave vector q.

Ν = — 1

9 exp (βω0) — 1 Also, we see

d> — t2) = — exp [ιω^ίχ — t2)] Nq

(2.13b) d<aq(tx — t2) = — exp [ico^t, — t2)] (Nq + 1)

since

1 1 = 1 = — (1 + N0) .

exp (— βωα) — 1 exp (βω)0 — 1

We can see immediately that solution to Eq. (2.11) with the bound­

ary condition (1.16) may be expressed in terms of d as:

Fq{U 1, 1')

= J

dr £ 0 di v* exp ( - iq · f) f)

< 7 V ( f , /) V(r,

0 KD V

+

0')]>

(2.14a)

Fe(f, 1, Γ) = — J dr β dt vq exp (/? · f) dWQ(t, t)

<r[y+(r, f) y(f, /) y>(l) y>+(l')]> · (2.14b) Notice the great utility of the boundary condition and the [0, — ίβ]

approach. This approach enables us to eliminate all reference to the phonon variables in expressions like Fq and Fq and reduce these to expres­

sions which involve only electron operators. Of course, we have to pay a price for this elimination. This price may be seen from the result of substituting (2.14) and back into the equation of motion of (2.6):

Γ d V* Ί

L'-aT + lSr + ' L J G (1 '1 , )E I ( 1 - R )

[exp (iq • (r — F)) daq(t, t)

— exp (— iq · (P — P)) dm(J, I)

<Τ[Ψ(1)]ψ+(1') ψ+(Ρ,Ι)Ψ(Ρ,ϊ)]).

(2.15)

(19)

THE E L E C T R O N - P H O N O N INTERACTION I N METALS 95 It is convenient to rewrite this expression in terms of

(2.16) as

+ μ (7(1, Γ) = (5(1 — Γ) (2.17)

± ι

Γ

c/r2

f

" ώ , F(l — 2) G2(l, 2; 1', 2)

where J o

V(l - 2) = Σ I rf f I2 [exp [if · (rx - r2)] d0)q(tl912)

q (2.18)

exp [— iq · (rx — r2)] </_ω (Λ, >2) ] .

The reason that Eq. (2.17) is so instructive is that it is quite similar in structure to the equation of motion which would emerge from an ordinary electron-electron interaction. If the ordinary interaction can be represented by a central potential v(\ rx — r21 ) , then the equation of motion for G again takes the form (2.13) except for this case:

Therefore the phonons act to produce an affective electron-electron potential. This potential may be considered to result from the fact that an electron at one point in space and time may change the phonon field by emitting or absorbing phonons. This change in the phonon field can at a later time, scatter electrons at another point in space so that the scattering looks like it is produced by a retarded interaction between electrons.

To find G from Eq. (2.17), we must make some approximation for the G2( l , 2; Γ , 2') which appears in that equation. This two-particle Green's function, of course, describes the correlated motion of two electrons ad­

ded to the system at Γ and 2' and removed at 1 and 2. The simplest possible approximation for this G2 is to come from the assumption that the added electrons move quite independently of one another. This assumption may be expressed mathematically as

V(l,2) = d(t1-t2) K| Γ χ - r . (2.19)

G2( l , 2; 1', 20 » G(l, 1') G(2, 2') — G(l, 2') G(2, 1'). (2.20)

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96 LEO P. K A D A N O F F

The appearance of two terms in this expressions is a result of Fermi statistics obeyed by the electrons which requires that

G2( l , 2 ; l ' , 2 ' ) = - G2( 2 , 1; Γ, 2').

When we substitute (2.20) in (2.17) we find

Γ d Ε2 Ί

Γ" ~βΓ + ~im + μ \ °i h V) = 3(1 ~ 10 (2'21)

±i[JV(l—2) G(l, 2) i/2 G(l, Γ) + / Jd2K(l-2)G(l,2)G(2,l')

the middle term on the right-hand side of (2.21) should be neglected for two reasons:

(a) This term is proportional to

J dl V(\ — 2) <«(2)> = J dr jdt V(r, t).

After the r2 integral is performed, this expression becomes proportional to I ν |2 I

I vg I ltf-o ·

However, a ^ = 0 phonon is pure nonsense. This " p h o n o n " represents the effect of picking up and bodily displacing the whole crystal. There­

fore this whole term is physically meaningless.

(b) This term is proportional to the average potential field produced by all the electrons in the system. As such, it has actually been included in the original definition of the band structure which underlies our original Hamiltonian. Therefore, this term has already been counted and it must now be left out.

After this term is thrown out Eq. (2.21) may be rewritten as:

\_*~ΊΪ'

+ - ^ - + ^ ]G( 1'1 ,) = < J(1 — ^ 2 Z ( 1 , 2 ) G ( 2 , Γ) (2.22) where the self-energy Σ is

Σ(1, Γ) = iV(\ — Γ) G(l, Γ ) . (2.23)

(21)

THE E L E C T R O N - P H O N O N INTERACTION I N METALS 97 It is a trivial matter to verify that Σ ( 1 , Γ) obeys the same boundary conditions as G, i.e.,

Σ ( 1 , 1 ' ) |ί ι_0 = - Σ ( 1 , 1') I (2.24) since V(l, Γ) obeys the boson boundary condition

V(l,V)\ti=0 = V(l,V)\li=_itl.

Since Σ obeys the same boundary condition as G, its formal prop­

erties are very similar to the formal properties of G. For example, when Σ ( 1 , Γ) is split into two parts as

Σ>(1 — 1') = f — - , J (2π 1 · exn tin · O,

/ Σ ( 1 , 1 ' ) =

dzp dco (2.25)

)3 2n

exp [ip · (rx — r2) — ico(tx — f/)] Σ>(/?, ω) for ίίλ > it ι dzp dto

— Σ<(1 — Γ) =

— Γ—-

r)

3 2n

' · exp [ip · (rj — r / ) — ιω(ϊχ — Σ<( ^ , ω) for < if/

then Σ>(ρ, ω) and Σ<(/?, ω) obey the detailed balancing relation Σ<(ρ, ω) = exp (—βω) Σ<(ρ, ω) ;

also, Σ may be written as a Fourier series

2 ( M ' ) = J 7 ^ exp [«>·(,·, - / · / ) ]

</3/> 1 (2.26)

• 2 exp [ - κ , ( ί ι - ί ι ' ) ] Σ , ( ρ )

ι»

with the Fourier coefficient given by

dto Σ>(ρ,ω)+ Σκ9ω) f dco Γ(ρ,ω) 2n z„ — ω Σ„(/?) = Σ(/7, ζ„) = — =

J 2η ζν — ω J

. (2.27) This Fourier series representation of Σ ( 1 , Γ) is very convenient be­

cause it enables us to solve the differential Equation (2.22) quite directly.

(22)

98 LEO P . K A D A N O F F

If we multiply this equation by exp [—ip · (RX— /·/) + IZX^ — TI)]

and then integrate over all rx and all T X in the interval 0 < ITX < β9 then (2.22) becomes

[z„ - ε ρ - Σ(ρ, ζ,,)] G(p9 ZP) = 1 . (2.28) Equation (2.28) is a relation between the analytic functions

Γ dco Σ>(ρ9ω) + Σ<(ρ9ω) [dco Γ(ρ9ω)

Σ(ρ, ζ) = = — (2.29a) J 2π ζ — co J 2π ζ — ω

Γ dco G>(p9co) + G<(p9w) Γ dco Α(ρ9ω)

G(P,z) =

-Τ = -Ζ (

2

-

29B

)

J 2π ζ

ω J 2π ζ

ω which is known to hold on all the points

ζ = zv = ν = odd integer. πν

— ιβ

From the fact that this relation holds on this limited set of points and the fact that neither Σ(ρ9 ζ) nor G(p9 z) has an essential singularity at o o it follows that (2.28) holds for all z , i.e., that

G(p, z ) = 1 — . (2.30)

ζ — ερ — Σ(ρ9 ζ)

In order to make use of Eq. (2.30) we use (2.18) and (2.23) to write Σ>(1, Γ) = K>(1, Γ) G>(\9 1')

= Σ I

V

J

2 e xP

· ('ι - < ( > ι

- T,')

u (2.31)

- exp [ - iq · (

Γ ι

- r/)] d^

Q

(T

X

- //)] G>(1, Γ ) .

The Fourier transform of (2.31) is

Γ d3p' dco' , ,

( P'0 J ) = J

~DTF

G>(P

'

9 A)L) {| VQ |2[NQ2ΠΔ(Ω + A)Q

~

Ω Ί

+ (Nq + 1) 2πδ (co — coq — co')]} q =-- \ ρ — p ' | . (2.32a)

(23)

THE E L E C T R O N - P H O N O N INTERACTION I N METALS 99 A similar evaluation gives

cPp1 dw' Σ<(ρ, ω )

J

(2π> )3 2ττ

+ Ν92πδ(ω-ω0-ω')]}

G<(p', ω') {| vq |« [(TV, + 1)2π<5(ω + ω , - ω ' )

q = \ p - p ' \ . (2.32b)

The substitution of (2.32) into (2.29a) leads back to our earlier result (1.33):

J

J

(2π

?>(/>', ω ' ) |~

ζ + ω . — ω '

+ <?<(/>', ω ' ) ^

_ ζ — ωγ -

+ +

Nq + i 1

Nt+l ζ

+

ωα

ω'

(2.33)

Ρ—Ρ |·

It is now quite simple to derive an expression for A(p, ω ) since iA(p, ω ) is the discontinuity in G(p, z) as ζ crosses the real axis. From (2.30),

A(p, ω ) = Γ(ρ, ω )

[ ω - ερ - Re Σ ( ρ , + [/Χ/>, ω)/2]* (2.34) Here Re Σ(/?, ω ) is the part of Σ(/?, ζ) which is continuous as ζ crosses the real axis. It is given by an expression identical with (2.33) except that

1 1

ζ

db

coq

ω' ω

±

coQ

ω'

where Ρ stands for principal value. Γ(ρ9 ω ) is the discontinuity in (2.33).

In Γ9 the energy denominators in (2.33) are replaced by (5-functions:

1

ζ ±ω0 — ω' 2πδ(ω ± ω — ω ' ) .

Since Α(ρ9 ω ) is defined in terms of G> and G< by (2.29b) and Σ(ρ9 ζ) is defined in terms of G> and G< by (2.33), Eq. (2.34) gives us one rela­

tion between G> and G<. The boundary condition

G<(p, ω) = exp (— βω) G>(p9 ω) (2.35)

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100 LEO P . K A D A N O F F

provides another relation between G> and G<. Thus, we have two equa­

tions in two unknowns and we may solve for G> and G<.

At this point, I would like to quote two rather surprising facts about the approximation described above:

(a) This approximation is exact in a normal (nonsuperconducting) metal if the correct phonon energy spectrum is employed and the elec­

tron-electron interactions are neglected.

(b) The highly nonlinear equations we have written down are exactly soluble for a metal.

These statements have been proved by Midgal (8). Before outlining Midgal's proof, I should point out that these statements are very closely related to the well-known fact that the adiabatic approximation (9) gives a correct description of the electron-phonon interaction in metals Therefore, I and II really tell us that our Green's function approximation is a correct restatement of the adiabatic approximation.

In order to see the essence of Migdal's arguments, let us consider the diagrammatic expansion for the self-energy in a power series in G and V. The terms we have considered in the self-energy correspond to the diagrams

III. Specialization to a Metal

FIG. 3

Σ ^ Ι , Γ) = (5(1 - Γ) [ - ι d2V(l - 2) G(2, 2)] (3.1) and

FIG. 4

Σ2( 1 , Y) = iV{\, Y)G(\, 1'). (3.2)

(25)

THE E L E C T R O N - P H O N O N INTERACTION I N METALS 101 Here the —*— line stands for a G and for the interaction V.

The first of these terms vanishes and the next is the one we have taken into account.

The next order terms, which we have neglected are:

FIG. 5

Σ3( 1 , Γ) = i6V(h 1') 0 ( 1 , 1 ' ) (3.3)

6V(l, Γ) = — ι

f

K(l, 2) G(2, 20 G(T, 2) Κ(2', Γ) ί/2' dl

and J

FIG. 6

Σ4( 1 , 10 = i2

J

dl dl! V(l, 2) G(l, 20 G(T9 2) G(2, Γ) V(l9 1'). (3.4) The substance of point I in the Migdal paper is that terms like Σ4 are negligible while terms like Σ3 serve to shift the phonon energies in the K(l, Γ) of Σ2 from the bare phonon energies to the energies of the fully interacting phonon system. The manner in which Σ3 serves to shift the energies is, I think, clear. However, some further discussion is nec­

essary in order to see why Σ4 is small.

A crucial fact in this discussion is that only electrons with very small values of ερρ<^μ) play any role in the electron-phonon interaction.

The reason for this limitation lies in the fact that the phonon energies are all quite small, in fact,

(3.5) where m is the electronic mass and Μ the ionic mass. Therefore, an elec­

tron which initially lies quite close to the Fermi surface must always

(26)

102 LEO P. K A D A N O F F

remain quite close to ερ ^ 0. If all electronic momenta must be very close to

This puts considerable restrictions on the momentum integrals which are to be done in evaluating Σ4. In fact, this phase space restriction leads to Σ4 being smaller than Σ2 by a factor of order -γ/ m/M. (We assume that the dimensionless coupling constant [| vq\2 mpF/coq] is of order unity.)

In this way, Migdal proved that the approximation that we discussed in the last two sections was an exact perturbation theoretic expansion of the self-energy to order (mjM)112. Migdal apparently did not publish this proof for several years because it obviously contained a grave defect—it appears to prove that the electron-phonon interaction has nothing to do with superconductivity. The trouble is that, for a super­

conductor, perturbation theory does not converge. Hence this whole perturbation theoretic argument is quite wrong for a superconductor.

We shall return to the discussion of the superconductor later. For now let us consider Migdal's second point. We can solve (2.34) for A(p9 ω). To effect this solution we examine (2.33) and notice that Σ(/?, ζ) is a very slowly varying function of p, but a rapidly varying function of z. The rate of variation in ζ is measured by coq but the rate of variation in ερ = (p2/2m — / / ) , is measured by μ. Because of his slow variation in p, we can simply set ρ = pF in Eq. (2.33) and write (2.34) as

Ρ = PF =

A(p, ω) = Γ(ω)

(3.6)

-ερ-ΚεΣ(ω)]2+ [Γ(ω)/2] |2 with

Re Σ(ω) = Re Σ ( / ^ , ω) Γ(ω) = ΚεΓ(ρΓ9 ω).

Let us observe that (3.6) implies

(3.7) Equation (3.7) is crucially important because a huge variety of physical quantities depend upon the integral (3.7). Because this integral is in-

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