Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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374 12 Many Body Interactions: Green’s Function Method

the polarization part. This RPA approximation for W is exactly equivalent to 256

V (q)

ε(q,ω)

,whereε(q,ω) is the Lindhard dielectric function. The key role of the 257

electron self-energy in studying electron–electron interactions in a degenerate 258

electron gas was initially emphasized by Quinn and Ferrell.

In their paper, 259

the simplest GW approximation to Σ was used. G

(0)

was used for the Green’s 260

function and

V (q)

ε(q,ω)

, the RPA screened interaction (equivalent to Lindhard 261

screened interaction) was used for W. 262

12.5 Green’s Function Approach 263

to the Electron–Phonon Interaction 264

In this section we apply the Green’s function formalism to the electron–phonon 265

interaction. The Hamiltonian H is divided into three parts: 266

H = H

+ H

, (12.63)

where

267







U(r

− R

)



, (12.64)

268







l>m

V (R

− R

)



, (12.65)

and

269



i>j

−



i,l

·∇U(r

− R

). (12.66)

Here, U(r

−R

)andV (R

−R

) represent the interaction between an elec- 270

tron at r

and an ion at R

and the interaction potential of the ions with 271

each other, respectively. Let us write R

= R

+ u

for an ion where R

is 272

the equilibrium position of the ion and u

is its atomic displacement. The 273

electronic Hamiltonian H

is simply a sum of one-electron operators, whose 274

eigenfunctions and eigenvalues are the object of considerable investigation for 275

energy band theorists. To keep the calculations simple, we shall assume that 276

the eﬀect of periodic potential can be approximated to suﬃcient accuracy for 277

our purpose by the introduction of an eﬀective mass. The nuclear or ionic 278

Hamiltonian H

has already been analyzed in normal modes in earlier chap- 279

ters. It should be pointed out that the normal modes of (12.65) are not the 280

usual sound waves. The reason for this is that V (R

− R

) is a “bare” ion– 281

ion interaction, for a pair of ions sitting in a uniform background of negative 282

charge, not the true interaction which is screened by the conduction electrons. 283

We can express (12.64)–(12.66) in the usual second quantized notation as 284

J.J. Quinn, R.A. Ferrell, Phys. Rev 112, 812 (1958).

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12.5 Green’s Function Approach to the Electron–Phonon Interaction 375



¯h

∗

†

, (12.67)

285



¯hω



†



, (12.68)

and

286



k,k



4πe

Ωq

†

k+q

†



−q





k,α,G

γ(α, G)(b

− b

†

−α

†

k+q+G

. (12.69)

The c

and b

are the destruction operators for an electron in state k and a 287

phonon in state α =(q,μ), respectively.

The creation and annihilation oper- 288

ators satisfy the usual commutation (phonon operators) or anticommutation 289

(electron operators) rules. The coupling constant

4πe

Ωq

is simply the Fourier 290

transform of the Coulomb interation, and γ(α, G)isgivenby 291

γ(α, G)=−i(q + G)ε



¯hN

2Mω



1/2

U(q + G), (12.70)

where U(q + G) is the Fourier transform of U(r −R). For simplicity we shall

292

limit ourselves to normal processes (i.e., G = 0), and take U(r − R)asthe 293

Coulomb interaction −

|r−R|

between an electron of charge −e and an ion of 294

charge Ze. With these simpliﬁcations γ(α, G) reduces to 295

γ(q)=i

4πZe



¯hN

2Mω



1/2

(12.71)

for the interaction of electrons with a longitudinal wave, and zero for interac-

296

tion with a transverse wave. Furthermore, when we make these assumptions, 297

the longitudinal modes of the “bare” ions all have the frequency 298

= Ω

, (12.72)

where Ω



4πZ



1/2

is the plasma frequency of the ions. 299

We want to treat H

as a perturbation. The brute-force application of 300

perturbation theory is plagued by divergence diﬃculties. The divergence arise 301

from the long range of the Coulomb interaction, and are reﬂected in the behav- 302

ior of the coupling constants as q tends to zero. We know that in the solid, the 303

Coulomb ﬁeld of a given electron is screened because of the response of all the 304

other electrons in the medium. This screening can be taken into account by 305

We should really be careful to include the spin state in describing the electrons.

We will omit the spin index for simplicity of notation, but the state k should

actually be understood to represent a given wave vector and spin: k ≡ (k,σ).

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376 12 Many Body Interactions: Green’s Function Method

pertubation theory, but it requires summing certain classes of terms to inﬁnite 306

order. This is not very diﬃcult to do if one makes use of Green’s functions 307

and Feynman diagrams. Before we discuss these, we would like to give a very 308

quanlitative sketch of why a straightforward perturbation approach must be 309

summed to inﬁnite order. 310

Suppose we introduce a static positive point charge in a degenerate electron 311

gas. In vacuum the point charge would set up a potential Φ

. In the electron 312

gas the point charge attracts electrons, and the electron cloud around it con- 313

tributes to the potential set up in the medium. Suppose that we can deﬁne 314

a polarizability factor α such that a potential Φ acting on the electron gas 315

will distort the electron distribution in such a way that the potential set up 316

by the distortion is αΦ. We can then apply a perturbation approach to the 317

potential Φ

. Φ

distorts the electron gas: the distortion sets up a potential 318

= αΦ

.ButΦ

further distorts the medium and this further distortion sets 319

up a potential Φ

= αΦ

, etc. such that Φ

n+1

= αΦ

. The total potential Φ 320

set up by the point charge in the electron gas is 321

Φ = Φ

+ Φ

+ ···= Φ

(1 + α + α

+ ···)

= Φ

(1 − α)

−1

. (12.73)

We see that we must sum the straightforward perturbation theory to inﬁnite

322

order. It is usually much simpler to apply “self-consistent” perturbation the- 323

ory. In this approach one simply says that Φ

will ultimately set up some 324

self-consistent ﬁeld Φ. Now, the ﬁeld acting on the electron gas and polariz- 325

ing it is not Φ

but the full self-consistent ﬁeld Φ. Therefore, the polarization 326

contribution to the full potential should be αΦ:thisgives 327

Φ = Φ

+ αΦ, (12.74)

which is the same result obtained by summing the inﬁnite set of perturbation

328

contributions in (12.73). 329

We want to use some simple Feynman propagation functions or Green’s 330

functions, so we will give a very quick deﬁnition of what we must know to use 331

them. If we have the Schr¨odinger equation 332

i¯h

∂Ψ

∂t

= HΨ, (12.75)

and we know Ψ(t

), we can determine Ψ at a later time from the equation 333

Ψ(x



; x

)Ψ(x

). (12.76)

By substitution, one can show that G

satisﬁes the diﬀerential equation 334



i¯h

∂

∂t

− H(x

)



(2, 1) = i¯hδ(t

− t

)δ(x

− x

), (12.77)

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12.5 Green’s Function Approach to the Electron–Phonon Interaction 377

where (2, 1) denotes (x

; x

). One can easily show that G

(2, 1) can be 335

expressed in terms of the stationary states of H.Thatis,if 336

= E

, (12.78)

then G

(2, 1) can be shown to be 337

(2, 1) =



∗

−iE

/¯h

, if t

> 0,

=0 otherwise. (12.79)

If we are considering a system of many Fermions, we can take into account

338

the exclusion principle in a very simple way. We simply subtract from (12.79) 339

the summation over all states of energy less than the Fermi energy E

. 340

(2, 1) =



∗

−iE

/¯h

, if t

> 0,

= −



∗

−iE

/¯h

, if t

< 0. (12.80)

We always represent a Fermion propagator by a directed solid line. A negative

341

(relative to the last ﬁlled state E

) energy Fermion propagates backward in 342

time. This corresponds to the propagation of a hole in a normally ﬁlled state. 343

For free electrons the functions u

(x) are plane waves. We are often interested 344

in G

(q, ω), the Fourier transform of G

(2, 1): 345

(2, 1) =



qdω

(2π)

(q, ω)e

iq·x

−iωt

. (12.81)

The single particle propagator G

(q, ω) for a system of free electrons is 346

(q, ω)=

ω −E(q)(1 − iδ)

, (12.82)

where E(q)=

¯h

−k

) is the energy measured relative to the Fermi energy 347

and takes on both positive and negative values. k

is the Fermi wave number, 348

and δ is a positive inﬁnitesimal. 349

In the language of second quantization G

(2, 1) can also be expressed as 350

the ground state expectation value of the time ordered product of two electron 351

ﬁeld operators 352

(2, 1) = GS | T {Ψ(2)Ψ

†

(1)}|GS. (12.83)

353

In this expression, Ψ(2) = Ψ(x

) is the electron ﬁeld operator and Ψ

†

(2) 354

is its conjugate. These operators satisfy the usual Fermion anticommutation 355

relations. T is the chronological operator. It should be pointed out that people 356

often deﬁne G

(2, 1) with an additional factor of i on the right hand side of 357

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378 12 Many Body Interactions: Green’s Function Method

(12.83). This arbitrariness in deﬁning the propagation functions is compen- 358

sated for by slight diﬀerences in the rules for calculating the amplitudes of 359

the Feynman diagrams which appear in perturbation theory. 360

We can also deﬁne a propagation function for the instantaneous Coulomb 361

interaction

δ(t

) between electrons at two points in space time. We shall 362

use i times the Fourier transform of

δ(t

) as the bare Coulomb propagator 363

V (q, ω) 364

V (q, ω)=

4πe

Ωq

. (12.84)

If we deﬁne the phonon ﬁeld operator Φ(x) by the equation

365

Φ(x)=



γ(q)e

iq·x

†

− b

−q

) (12.85)

then we can deﬁne the space time representation for the phonon propagator

366

in the usual way 367

(2, 1) = iGS | T {Φ(x

)Φ

†

)}|GS (12.86)

368

where Φ(x

)=e

−iHt

Φ(x

iHt

. The Fourier transform of (12.86) is 369

(q, ω), the wave vector frequency space representation of P

. For the phonon 370

system of (12.71) and (12.72), P

(q, ω)isgivenby 371

(q, ω)=

2iΩ

(q)

− Ω

. (12.87)

It is quite convenient to use Feynman diagrams to keep track of the various

372

terms in perturbation theory. The rules for constructing diagrams are quite 373

simple.Eachelectroninanexcitedstate is represented by a solid line directed 374

upward. Each hole in a normally ﬁlled state is represented by a solid line 375

directed downward. The instantaneous Coulomb interaction is represented by 376

a horizontal dotted line connecting the two particles undergoing a virtual 377

scattering, and propagation of a phonon is represented by a wavy line. 378

Consider the scattering of two electrons. In vacuum they can scatter by the 379

exchange of one virtual photon (Coulomb line) in only one way, which is shown 380

in Fig. 12.5. Now consider the Coulomb interaction in the medium. The set of 381

Fig. 12.5. Diagrammatic expression of the exchange of virtual photon

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12.5 Green’s Function Approach to the Electron–Phonon Interaction 379

(a) (b) (c)

Fig. 12.6. Diagrammatic expressions of representative polarization parts in pair

approximation

diagrams, of which Fig. 12.6a and b are representative, are additional processes 382

which can not occur in the absence of the polarizable medium. In Fig. 12.6c, 383

the circle represents any possible part of a diagram which is connected to 384

the remainder by two Coulomb interaction lines only. All such parts of a gen- 385

eral diagram are called polarizable parts , because they obviously represent the 386

response or screening of the polarizable medium. The eﬀective Coulomb inter- 387

action between two particles should be the sum of all the possible polarization 388

parts (the bare intraction can be thought of as the zeroth order polarization 389

part). Actually we can not sum all the possible polarization parts, but we can 390

sum the class of which Fig. 12.6a and b are representative, that is, the chain 391

of bubbles. The approximation of replacing the eﬀective interaction by the 392

sum of all bubble graph is called the pair approximation. Before looking at 393

the sum we will write down the rules for calculating the amplitude associated 394

with a given Feynman diagram which appears in perturbation theory. The 395

amplitude for a given diagram contains a product of 396

(i) a propagation function G

(k, ω) for each internal electron–hole line of 397

wave vector k and frequency ω. 398

(ii) a propagation function P

(k, ω) for each phonon line of wave vector q 399

and frequency ω. 400

(iii) a propagation function V (q, ω) for each Coulomb line of wave vector q. 401

(iv) afactor(−1) for each closed loop. 402

(v) (−i/¯h)

for the nth-order term in perturbation theory. 403

(vi) delta functions conserving energy, momentum, and spin at each vertex. 404

(vii) Finally, we must integrate over the wave vectors and frequencies of all 405

internal lines. 406

The set of diagrams we would like to sum in order to obtain the eﬀec- 407

tive Coulomb propagator W (q,ω) can easily be seen to be the solution of the 408

equation given pictorially by Fig. 12.7. This equation can be written 409

W (q, ω)=V (q, ω) − V (q, ω)Q

(q, ω)W (q, ω), (12.88)

410

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380 12 Many Body Interactions: Green’s Function Method

Fig. 12.7. Diagrammatic expression of Dyson equation for the eﬀective Coulomb

propagator W = V + V ΠW

where 411

(q, ω)=2(−1)



dω

(2π)

,ω

)δ(ω

− ω

+ ω)

×δ(k

− k

+ q) (12.89)

is the propagation function for the electron–hole pair. The factor of two is

412

introduced to account for the two possible spin orientations. Using the electron 413

propagation functions deﬁned by (12.82) and integrating gives 414

(q, ω)=

−2i

(2π)



k<k



E(k + q)−E(k)+ω

E(k + q)−E(k) − ω



(12.90)

The solution of (12.88) is simply

415

W =

1+VQ

(12.91)

416

and using (12.90) one can easily see that 1 + VQ

is just the Lindhard dielec- 417

tric function ε(q, ω). This dielectric function is discussed at some length in 418

the previous chapter, and the reader is referred to Lindhard’s paper

for a 419

complete treatment. For our purposes we must note two things: ﬁrst ε(q, ω) 420

is complex, the imaginary part being proportional to the number of electrons 421

which can be excited to an unoccupied state by addition of a momentum 422

¯hq whose energy change is equal to ¯hω. The second point is that for zero 423

frequency ε(q, 0) is given by 424

ε(q)=1+F





, (12.92)

where k

is the Fermi–Thomas screening parameter and k

is the Fermi wave 425

number. The function F





is the function sketched in Fig. 11.12. F (x)is

426

equal to unity for x equal to zero, approaches zero as x approaches inﬁnity, 427

and has logarithmic singularity in slope at x =1. 428

Now, let us return to our “model solid” which contains longitudinal 429

phonons as well as electrons. Two electrons can scatter via the virtual 430

exchange of phonons. In fact, anywhere a Coulomb interaction line has 431

J. Lindhard, Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 28, No. 8 (1954);

ibid., 27, No. 15 (1953).

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12.5 Green’s Function Approach to the Electron–Phonon Interaction 381

appeared previously, a phonon line may equally well appear. If we call D

(q,ω) 432

the sum of P

(q,ω)andV (q,ω) we can just replace V and W by D

and D 433

in Eq.(12.88). D(q,ω) then represents the propagation function for the total 434

interaction, i.e., the sum of the Coulomb interaction and the interaction due 435

to virtual exchange of phonons. It is apparent that 436

D(q,ω)=

(q,ω)

1+D

(q,ω)Q

(q,ω)

. (12.93)

437

By substituting into (12.92) the expressions for the propagation functions and 438

using the fact that 1 + VQ

= ε(q, ω), one can obtain 439

D(q,ω)=

4πe



ε(q, ω) − Ω

/ω



. (12.94)

The propagation function D as a pole at

440

ε(q, ω)

. (12.95)

The solutions of this equation are our “renormalized” phonons. From the long

441

wavelength, zero frequency dielectric constant we get the approximate solution 442

q. (12.96)

For most metals,

is within about 15–20% of the velocity of longitudinal 443

sound waves. If we look at the derivative of ω

with respect to q we see a 444

logarithmic singularity at q =2k

. This is responsible for the Kohn eﬀect, 445

which has been observed by neutron scattering. If we take account of the 446

imaginary as well as the real part of the dielectric constant, the solution of 447

(12.95) has both real and imaginary part. If we write ω = ω

+iω

,thenω

448

turns out to be 449

≈

+ q

, (12.97)

where c

is the velocity of sound and v

the Fermi velocity. The coeﬃcient 450

of attenuation of the sound wave (due to excitation of conduction electrons) 451

is simply

. This result agrees with the more standard calculations of the 452

attenuation coeﬃcient. 453

Finally, if we wish to deﬁne the eﬀective interaction between electrons due 454

to virtual exchange of phonons, or the eﬀective phonon propagator ,wecan 455

simply subtract from D(q, ω) that part which contains no phonons, namely 456

W (q, ω). If we call the resultant eﬀective phonon propagator P (q, ω), we 457

obtain 458

P (q, ω)=

2iω

| γ

eﬀ

− ω

, (12.98)

459

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382 12 Many Body Interactions: Green’s Function Method

where ω

is given by (12.96) and 460

| γ

eﬀ

4πZe

qε(q, ω

)



2Mω



1/2

. (12.99)

Replacing ε(q, ω

) by its long wavelength, zero frequency limit





461

reduces (12.99) to the result of Bardeen and Pines

for the eﬀective electron– 462

electron interaction. 463

12.6 Electron Self Energy 464

The Dyson equation for the Green’s function can be written 465

G(k, ω)=G

(0)

(k, ω)+G

(0)

(k, ω)Σ(k, ω)G(k, ω). (12.100)

Dividing by GG

(0)

gives 466

Σ(k, ω)=[G

(0)

(k, ω)]

−1

− [G(k, ω)]

−1

. (12.101)

The energy of a quasiparticle can be written

467

= ε

+Σ(p, ω) |

ω=E

. (12.102)

468

and ε

, the kinetic energy, are usually measured relative to E

,theFermi 469

energy. Knowing how Σ(p, ω) depends on p, ω,andr

allows one to calculate 470

almost all the properties of an electron gas that are of interest. Some results 471

of interest are worth mentioning. 472

(1) Σ(p, E

) has both a real and an imaginary part. 473

Σ(p, E

)=Σ

(p, E

)+iΣ

(p, E

). (12.103)

474

The imaginary part is related to the lifetime of the quasiparticle excitation. 475

476

(2) The spectral function A(p, ω) is deﬁned by 477

A(p, ω)=

−2Σ

(p, ω)

[ω −ε

− Σ

(p, ω)]

+[Σ

(p, ω)]

. (12.104)

For noninteracting electrons A(p, ω)hasaδ function singularity at ω = ε

, 478

the energy of the excitation. The δ function is broadened by Σ

(p, ω). If 479

A(p, ω) has a pole in a region where Σ

(p, ω) is zero, the strength of the 480

pole is decreased by a renormalization factor Z(p). 481

A(p, ω)=2πZ(p)δ (ω − ε

− Σ

(p, ω)) (12.105)

John Bardeen and David Pines, Phys. Rev. 99, 1140 (1955).

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12.7 Quasiparticle Interactions and Fermi Liquid Theory 383

and 482

Z(p)=



1 −

∂

∂ω

(p, ω)



ω=E

. (12.106)

(3) The quasiparticle excitations at the Fermi surface have an eﬀective mass

483

∗

given by 484

∗



∂Σ

(k,ω)

∂ε

1 −

∂Σ

(k,ω)

∂ω



k=k

ω=0

. (12.107)

(4) Properties like the spin susceptibility, the speciﬁc heat, the compressibil-

485

ity, and the ground state energy can be evaluated from a knowledge of 486

Σ(k, ω). But, we do not have time to go through these in any detail. 487

(5) The self-energy approach leads very naturally to an understanding of the 488

Landau theory of a Fermi liquid. We will describe a very brief and intuitive 489

explanation of the theory. 490

12.7 Quasiparticle Interactions and Fermi Liquid Theory 491

Instead of describing the interacting ground state and excited states of an 492

electron gas, we can think of simply describing how many quasiparticles are 493

present in some excited state. Let us start by noting that if we begin with a 494

ﬁlled Fermi sphere of noninteracting electrons (i.e., the Sommerfeld model) 495

and adiabatically turn on the electron–electron interaction, we will generate at 496

t = 0 the exact interacting ground state. Now consider the noninteracting state 497

described by a ﬁlled Fermi sphere plus one electron of momentum p outside 498

the Fermi sphere (or one hole of momentum p inside the Fermi sphere). When 499

interactions are adiabatically turned on, this is a single quasiparticle state. 500

The energy of this quasielectron (or quasihole) is written by 501

= ε

+Σ(p,E

). (12.108)

If E

is much larger than Σ

(p,E

), then the quasiparticles have long life- 502

times. It is much simpler to describe a state by saying how many quasielectrons 503

and quasiholes are present. Then, the energy of the state can be written as 504

E = E



pσ

δn

pσ



p, p



σ, σ



σσ



(p, p



)δn

nσ

δn



. (12.109)

505

The ﬁrst term on the right is the ground state energy, the second is the quasi- 506

particle energy E

pσ

multiplied by the quasiparticle distribution function, and 507

the third represents the interactions of the quasiparticles with one another. 508

Σ(p,E

) represents the interaction of a quasiparticle of momentum p with 509

the ground state of the interacting electron gas. But if the electron gas is 510