Quantum field theory of interacting plasmon–photon–phonon system

Since a plasmon is a quasiparticle appearing as a resonance in the collective oscillation of the interacting electron gas in solids, the starting point is the total action functional of

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Quantum field theory of interacting plasmon–photon–phonon system

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2015 Adv Nat Sci: Nanosci Nanotechnol 6 035003

(http://iopscience.iop.org/2043-6262/6/3/035003)

Quantum field theory of interacting

1

Advanced Center of Physics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau

Giay District, Hanoi, Vietnam

2

University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay

District, Hanoi, Vietnam

E-mail:nvhieu@iop.vast.ac.vnandbichha@iop.vast.ac.vn

Received 2 March 2015

Accepted for publication 18 March 2015

Published 14 April 2015

Abstract

This work is devoted to the construction of the quantumfield theory of the interacting system of

plasmons, photons and phonons on the basis of general fundamental principles of

electrodynamics and quantumfield theory of many-body systems Since a plasmon is a

quasiparticle appearing as a resonance in the collective oscillation of the interacting electron gas

in solids, the starting point is the total action functional of the interacting system comprising

electron gas, electromagneticfield and phonon fields By means of the powerful functional

integral technique, this original total action is transformed into that of the system of the quantum

fields describing plasmons, transverse photons, acoustic as well as optic longitudinal and

transverse phonons The collective oscillations of the electron gas is characterized by a real

scalarfield φ(x) called the collective oscillation field This field is split into the static background

field φ0(x) and the fluctuation field ζ(x) The longitudinal phonon fieldsQal( ),x Qol( )x are also

split into the backgroundfieldsQal0( ),x Qol0( )x and dynamicalfieldsqal( ),x qol( )x while the

transverse phononfieldsQat( ),x Qot( )x themselves are dynamicalfieldsqat( ),x qot( )x without

backgroundfields After the canonical quantization procedure, the background fields φ0(x),

x

Qal0( ),Qol0( )x remain the classicalfields, while the fluctuation fields ζ(x) and dynamical phonon

fieldsqal( ),x qat( ),x qol( ),x qot( )x become quantumfields In quantum theory, a plasmon is the

quantum of Hermitian scalarfield σ(x) called the plasmon field, longitudinal phonons as complex

spinless quasiparticles are the quanta of the effective longitudinal phonon Hermitian scalarfields

θ a( ),x θ x0( ),while transverse phonons are the quanta of the original Hermitian transverse

phonon vectorfieldsqat( ),x qot( ).x By means of the functional integral technique the original

action functional of the interacting system comprising electron gas, electromagneticfield and

phononfields is transformed into the total action functional of the resultant system comprising

plasmon scalar quantumfield σ(x), longitudinal phonon effective scalar quantum fields θ x a( ),

θ x0( )and transverse phonon vector quantumfieldsqat( ),x qot( )x

Keywords: functional integral, collective oscillations,fluctuation, plasmon, action functional

Classification numbers: 2.09, 3.00

1 Introduction

Since the early works on the collective motion of charged

particles in plasma, including the interacting electron gas in

solids, it was shown that there exists a resonance of the

col-lective oscillations at some frequency called the plasma

fre-quency This resonance phenomenon was interpreted as the

appearance of an elementary excitation—a complex quasi-particle called a plasmon—and the plasma frequency was also called plasmon frequency (the references on early works on plasmons can be found in the literature [1–3]) In the physical processes with the presence of plasmon the plasmon–photon interaction plays the main role Moreover, in the electron gas

of solids there always exists the electron–phonon interaction

| Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv Nat Sci.: Nanosci Nanotechnol 6 (2015) 035003 (11pp) doi:10.1088/2043-6262/6/3/035003

2043-6262/15/035003+11$33.00 1 © 2015 Vietnam Academy of Science & Technology

Original content from this work may be used under the terms of the Creative Commons Attribution 3.0

licence Any further distribution of this work must maintain attribution to the author(s) and the title

leading to the effective plasmon–phonon interaction

There-fore the knowledge on the mutual interaction of plasmon,

photon and phonons is necessary for both theoretical and

experimental studies on the physical processes and

phenom-ena involving plasmon The present work is devoted to the

elaboration of the quantum field theory of the plasmon–

photon–phonon interacting system by applying the functional

integral technique [4–7] The assumptions comprise only the

fundamental principles of electrodynamics and quantum

the-ory of many-body system

For the application of mathematical tools of functional

integral technique, the physical content of the theory of

phonons in solids must be presented in the languages of the

quantum field theory This will be done in section 2 Here

there is a distinction between longitudinal and transverse

phonons While the transverse phonons are described by the

transverse phonon vector fields as other transverse vector

fields in the theory of the elementary particles, for simplifying

the presentation of the formulae related to longitudinal

pho-nons we propose to describe them by some effective scalar

fields similar to the quantum fields of spinless particles

Moreover, because the interaction of longitudinal phonons

with electron is much stronger than that of transverse ones, in

the study of physical phenomena and processes with the

dominant competition of longitudinal phonons we can neglect

the contribution of transverse phonons Thus the transverse

phonon fields will be retained only in the particular cases

when they play the essential role

Section 3 is devoted to the establishment of the

expres-sion of total action functional of the interacting plasmon–

photon–phonons system It contains all three types of fields:

(i) collective oscillation field φ(x); (ii) transverse

electro-magnetic vector field A(x) and (iii) all phonon fields, both

acoustic and optic phonon fields Qa μ( ),x Qo μ( ),x index μ

labeling the phonon branches By grouping suitable terms

from the formula of total action functional of the whole

system it is possible to derive expressions of action functional

of different subsystems of related fields The fundamental

subsystem is the collective oscillation field φ(x) A short

review of the results of previous works related to thisfield in

the harmonic approximation is presented

The construction of quantum fields of interacting

plas-mon–phonon system is the content of section 4 In the

har-monic approximation with respect to the collective oscillation

field as well as to the fields of both acoustic and optic

pho-nons, the action functional of the subsystem comprising

interacting collective oscillation field φ(x) as well as both

acoustic and optic phonon fields Qaμ( )x and Qo μ( )x is

derived Each of longitudinal phononfieldsQal( )x andQol( )x

is split into two parts, backgroundfieldQal0( )x orQol0( )x and

dynamical field qat( )x or qot( ),x while transverse phonon

fields Qat( ),x Qot( )x themselves are dynamical ones qat( )x

andqot( ).x The dynamical phononfields generate the physical

phonons playing the role of dynamical quasiparticles in

physical phenomena and processes

The construction of the quantum fields of the whole

interacting plasmon–photon–phonon system is the content of

the section 5 The expression of total action functional of this whole system, described by backgroundfields φ0(x),Q0al( )x

and Qol0( ),x fluctuation field ζ(x), electromagnetic field A(x) and dynamical phononfieldsqal( ),x qat( )x andqol( ),x qot( ),x

is derived in the harmonic approximation with respect to each

of three types of fields: (i) fluctuation field, (ii) electro-magnetic field and (iii) all dynamical phonon fields The characterizing features of different subsystems of the whole system are briefly investigated From the obtained expression

of total action functional of the whole system it is possible to derive the expressions of the action functional of different interaction vertices The conclusion and discussions are pre-sented in section 6

2 Phonon quantumfields For using in the study of the interaction of phonons with other quasiparticles in solids by means of the functional integral technique let us construct the quantum fields of phonons There exist many types of phonons with various character-istics in different materials [8] In the present work we limit to the frequently investigated solids: elastic media [3] and crystalline lattices [3,9] The quantumfields of acoustic and optic phonons will be constructed separately For simplifying formulae we use the notations proposed in our previous works [4–6] and the unit system with ℏ =c=1

Consider first the acoustic phonons In both above-mentioned types of solids there exist one longitudinal and two transverse acoustic phonon branches Denote Qa μ( )x their quantum fields, where μ = 1, 2 for transverse phonons and

μ = 3 for longitudinal one For a definite μth branch between angular frequency ω and wave vector k at small values of

=

k k there exists a linear relation

ω=υ μ k.

We assume that this formula is the dispersion law of the acoustic phonon in general It looks like that of a massless relativistic particle, except for the scaling of spatial coordinates

μ

μ

On the basis of the analogy with the freefield of relati-vistic massless particles we have following Lagrange function and action functional of the acoustic phonon inμth branch

∑

υ

∂

∂

∂

∂

∂

μ μ

μ μ

μ

=

1

a a

i a

i

a

i

0

2 2 1

3 2

⎛

⎝

⎠

⎟

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎝

⎠

⎟ ⎤

⎦

⎥

⎥ and

∫

∂

∂

( )

i

⎛

⎝

⎠

⎟ Now we consider the optic phonons In a crystalline lattice with s non-equivalent ions per a primitive cell,s≠1,

beside three acoustic phonon branches there exist 3(s-1)

branches of optic phonons with non-vanishing limiting

angular frequency Ωμ at k = 0 Denote Qo μ( )x the optic

phononfield in the μth branch Since k-dependent terms in

the dispersion law of optic phonon are very small in

com-parison with the constant termsΩμ, let us neglect them Then

the optic phonon fieldQo μ( )x has following Lagrange

func-tion and acfunc-tion funcfunc-tional

Ω

∂

∂

∂

μ μ

μ( μ)

2

(3)

o o

i

o

o

0

2

2 2

⎛

⎝

⎠

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥ and

∫

∂

∂

( )

i

⎛

⎝

⎠

⎟

In the special case of isotropic crystals with s = 2

non-equivalent ions per a primitive cell, there exist one

long-itudinal and two degenerate transverse optic phonon branches

with limiting angular frequenciesΩlandΩtatk → 0 Between

Ωland Ωtthere exists following relation

and

Ω Ω

ε ε

=

∞

l t

2 2 0

whereε0is the static dielectric constant of the medium and ε∞

is the square of the refractive index of the medium at optical

frequencies

In solids there always exists the electron-phonon

inter-action In most cases the interaction of longitudinal acoustic

or optic phonons with electron is much stronger than that of

transverse acoustic or optic phonon, respectively In these

cases the longitudinal phonons play a much more important

role than the corresponding transverse phonons do, so that the

interaction between longitudinal phonons and electron has

been intensively studied during a long time It was shown that

for various solids the Hamiltonians of the interaction between

electron and longitudinal acoustic and optic phonons have

following expression [3,9]



=

Hintal g a dx ¯ ( ) ( )x x Qal( )x (7)

and



=

H ol g o dx ¯ ( ) ( )x x Qol( ),x (8)

int

where ψ x( ) is the electron field operator, ψ x¯ ( ) is its

Hermitian conjugate The coupling constants ga and go

depend on the crystalline and electronic structures of solids

Meanwhile, the interaction between electron and

trans-verse phonons was much less known Let us consider the

simple case of the lattice with 2 non-equivalent ions per a

primitive cell, s = 2 Then beside the two degenerate acoustic

transverse phonon branches with wave functionQat( )x there

exist also only two degenerate optic phonon branches with

wave function Qot( ).x Since the physical origin of the appearance of phonons is the oscillation of ions in solids and the coupling of phonons with electron is caused by the photon exchange between ion and electron, it is natural to believe that the Hamiltonian of the interaction between transverse pho-nons with electron have the expressions similar to the elec-tron-photon interaction Hamiltonian in the transverse gauge Therefore we assume following expressions of the transverse phonon–electron interaction Hamiltonians:

∂⃖

∂

at

int

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥ for acoustic transverse phonon and

∂⃖

∂

ot

int

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥ for optic transverse phonon

The interaction of ions in the lattice with the electro-magnetic wave, in principle, can also generate the direct coupling of electron with transverse acoustic and optic pho-nons In the transverse gauge the effective interaction Hamiltonians have the expressions

∫

=

γ γ

Hinta g a dx Qat( ) ( )x A x (11) for acoustic phonon and

∫

=

γ γ

int0

for optic phonon

3 Total functional integral

As the extension of total functional integral of the interacting plasmon–photon system studied in the previous work [14] we have following total functional integral of the plasmon –pho-ton–phonon system

∫

ψ ψ

∂

×

μ

[ ]

[ ]

o

⎜ ⎟

⎛

⎝

⎞

⎠

where I tot⎡⎣ψ ψ, ¯ ; A Q; a μ,Qo μ⎤⎦ is the total action functional

of this system:

ψ ψ

[ ]

0 0

int int int⎡⎣ ⎤⎦ int⎡⎣ ⎤⎦

γ

I0[ ]A is the action functional of the transverse free electromagneticfield in the transverse gauge

∂

γ

2

( )

0

⎛

⎝

⎜

⎜

⎡

⎣

⎦

⎠

⎟

⎟

ε0being the static dielectric constant of the medium,I a Qa μ

0⎡⎣ ⎤⎦

and I o Qo μ

0⎡⎣ ⎤⎦ are the action functional of two systems of all

acoustic phonon fields and all optic phonon fields,

respec-tively

∑

∑

=

=

μ μ

μ μ

μ μ

μ μ

=

=

,

0

1

3 0

0

1

3 0

μ μ

I a Qa

0 ⎡⎣ ⎤⎦ and I o μ Qo μ

0 ⎡⎣ ⎤⎦ being determined by formulae (1)–

(4), I e⎡⎣ψ ψ, ¯⎤⎦ is the action functional of the system of

electrons mutually interacting through the Coulomb

repul-sion.I e⎡⎣ψ ψ, ¯⎤⎦ consists of two parts

I e[ , ¯ ] I e[ , ¯ ] I e [ , ¯ ], (17)

0 int

where I0e⎡⎣ψ ψ, ¯⎤⎦ is the action functional of free electron

moving in the electrostatic field of ions in the crystalline

lattice

∫

∂

∂

e

0

⎡

⎣

⎦

⎥

− ∂∂

H i x,x is the quantum mechanical Hamiltonian of single

electron

∂

2

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

m is the effective mass of electron, Iinte ⎡⎣ψ ψ, ¯⎤⎦ is the action

functional of electron-electron Coulomb interaction

∫ ∫

×

e

int

δ

u x( y) (x0 y u x0) ( y), (21)

ε

−

2 0

−e is the electron charge It is straightforward to show that

∫

∫

∂⃖

∂

−

γ

e

A

A

, ¯ ;

(23)

e

int

2

2

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥

According to formulae (7)–(10) for the electron–phonon

interaction Hamiltonians we have



∫

∫

= −

∂⃖

∂

μ

a

al

int

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥



∫

∫

= −

∂⃖

∂

μ

o

ol

int

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥ From expression (11) and (12) of the Hamiltonians describing the coupling of transverse phonons with photon it follows that

∫

= −

I a A; Qat g dx Q ( ) ( )x A x (26)

a

at

int⎡⎣ ⎤⎦

and

∫

= −

I o A; Qat g dx Q ( ) ( ).x A x (27)

o

ot

int⎡⎣ ⎤⎦

The Coulomb interaction functional (20) is bilinear with

respect to the electron density ψ¯ ( ) ( ) This expression canx ψ x

be linearized by means of the Hubbard-Stratonovich trans-formation

∫ ∫

∫ ∫

φ

i

i

exp

1 [ ] exp

(28) where

φ

as this was proposed in references [4, 5] The bosonic real integration variable φ(x) describing collective oscillations of electron gas was called the collective oscillationfield Using formulae (14), (17) and (28), we rewrite the total functional integral (13) in the new form

∫

δ

φ

∂

×

×

×

φ

μ μ

[ ]

i

F

[ ] exp

(30)

tot e

a

a o

0

0

int int

⎜ ⎟

⎛

⎝

⎞

⎠

where

Z0e [D ] D¯ exp iI0e[ , ¯ ] (31)

∫

∫ ∫

ψ ψ

=

×

μ

(32)

F

iI

A Q Q

Q

e

e

0

0

Expanding four last exponential functions in rhs of

rela-tion (32) into power series, neglecting the very small terms

proportional to 1 m−2and performing the functional integration

over the Grassmann variables, after lengthy but standard

cal-culations we obtain following expression of the functional (32)

F ;A Q; a ,Qo exp iW ;A Q; a ,Qo , (33)

whereW[ ;φ A Q; a μ,Qo μ]is a functional power series ofφ(x),

Aμ(x),Qaμ(x),Qoμ(x) as the functional variables:

∑ ∑ ∑ ∑

φ

φ

=

μ μ

μ μ

=

∞

=

∞

=

∞

=

∞

W

W

a o

m n p q

m n p q a o

0 0 0 0

( , , , )

the term W( , , , )m n p q [ ;φ A Q; a μ,Qo μ] being a homogeneous

functional polynome of mth, nth, pth, qth orders with respect

to the functions φ(x), A(x), Qa μ(x), Qo μ(x), respectively.

Substituting expression (33) of functionalF[ ;φ A Q; a μ,Qo μ]

into rhs of formula (30), we transform the total functional

integral Ztot of the system of four interacting fields φ(x),

Aμ(x),Qa μ(x),Qo μ(x) into the form

∫

φ

∂

×

φ

μ

[ ]

tot

e

a

o

0 ⎜⎛ ⎟

⎝

⎞

⎠

where the total action functionalI tot[ ;φ A Q; a μ,Qo μ]of this

system has following expression

∫ ∫

φ

μ μ

(36)

0 0 0 int

int

Since the functionalW[ ;φ A Q; a μ,Qo μ]is a series of the

form (34), the total action functional of the interacting system

of four fields φ(x), Aμ(x), Qa μ(x) and Qo μ(x) has following

expression

∫ ∫

∑ ∑ ∑ ∑

φ

=

∞

=

∞

=

∞

=

∞

(37)

int

0 0 0 0

( , , , )

By grouping suitable terms from the expression in rhs of

formula (37), we can derive the expression of total action

functional of any subsystem of above-mentioned interacting system of fourfields φ(x), Aμ(x),Qa μ(x) andQo μ(x).

Thefirst subsystem is the collective oscillation field φ(x)

In references [4,5] it was shown that thisfield is split into two parts

φ( )x =φ0( )x +ζ( ),x (38) where φ0(x) is the static background field, φ0(x) =φ0(x, t) =φ0(x) corresponding to the extreme value of the action functional I0[φ] of this field in the harmonic approximation

∫ ∫

W

0 (1,0,0,0)

(2,0,0,0)

and ζ(x) is the field of small fluctuations around background field φ0(x) We call ζ(x) the fluctuation field In terms of φ0(x) and ζ(x) the action functional I0[φ] has the expression

I0⎡⎣ 0 ⎤⎦ I0[ 0] I eff[ ], (40) where

∫ ∫

eff

Π(x−y)= −iG x( −y G y) ( −x), (43) G(x− y) is the two-point Green function of free electron

Denote ζ˜[ , ] andk ω K k˜ [ , ] the Fourier transforms of theω

field ζ(x) and the kernel K(x − y) It was known that in the case

of a homogeneous electron gas

2

0 2 2

2

0 2 2

whereω0is the plasma frequency of the electron gas

ε

m

4

0

2 0 0

n0is the electron density and

m

3

F

2 2 2

pFis the electron momentum at the Fermi surface In terms of

the Fourier transforms ζ˜[ , ] andk ω K k˜ [ , ] formula (73)ω

becomes

ζ

=

×

K

(2 )

1 2

1

2 ˜( , )*

Setting

k

2

0 2 2

and introducing the scalar field σ( , )with the Fourierx t

transforms σ˜( ,k ω), we rewriteI eff[ ]ζ in the new form



∫ ∫

σ

t

x x

1

2

( , )

(49)

eff 0pl

2

2 2

0 2

⎪

⎪

⎪

⎪

⎧

⎨

⎩

⎡

⎣⎢

⎤

⎦⎥

⎫

⎬

⎭

Functional (49) is the action functional of free plasmon

field It has the form similar to the action functional of the

Klein–Gordon real scalar field in relativistic quantum field

theory [10–13], except for a scaling factor γ at the spatial

coordinates After the canonical quantization procedure, real

scalar field σ( , ) becomes a Hermitian quantumx t field,

whose quantum is plasmon: the quantum plasmonfield The

expression of σ( , ) in terms of the destruction and creationx t

operators of plasmon was known

Thus the quantum plasmonfield based on the study of

collective oscillationfield φ(x) as the fundamental subsystem

has been constructed Another important subsystem is that of

phonon fields developed in the preceding section 2 The

quantumfields of interacting plasmon–photon subsystem were

also constructed in reference [14] The quantumfield theory of

plasmon–phonon subsystem is the subject of the next section

4 Quantumfields of interacting plasmon–phonon

system

Now we study in detail the interacting plasmon–phonon

system In order to avoid lengthy expression we limit to the

harmonic approximation with respect to two types offields:

(i) the collective oscillation field and (ii) both acoustic and

optic phononfields Moreover, since the interaction of

long-itudinal phonons with electron is much stronger than that of

transverse phonons, we can neglect the contribution of

transverse phonons in the phenomena and processes in which

there exists the competition of longitudinal phonons, and

retain the electron–transverse phonon interaction only when

the transverse phonons play the essential role In particular,

the contribution of transverse phonons must be taken into

account when we consider the phenomena and processes with

the participation of the transverse electromagnetic field, as

this will be performed in the next section

First we note that the electron–phonon interaction leads

to the interaction of phonons with the collective oscillation

field The action functional of the interaction between the

fields φ(x), Qaμ(x),Qoμ(x) has the expression of the form

∑ ∑

∑ ∑

∑ ∑∑

φ φ

=

+

+

μ μ

μ μ

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

I

I

a o

m p

m p o a o

m q

m o q a o

m p q

m p q a o

int

1 1 ( , , )

1 1

( , , )

1 1 1

( , , )

where

I( , , )m p q ;Qa ,Qo W( , , , )m o p q ;A Q; a ,Qo (51)

Because the plasmon is the quasiparticle generated by the fluctuation ζ(x) around the background field φ0(x), the state

φ0(x) must be considered as the physical vacuum of the plasmon field Therefore the termIint[φ0;Qa μ,Qo μ]must be included into the total action functional of the subsystem comprising only phononfieldsQa μ andQo μ:

φ

(52)

tot

0 0 int⎡⎣ 0 ⎤⎦

In order to avoid the lengthy and complicated formulae and as the simple example, let us limit to the first order approximation (m = 1) with respect to the field φ0(x) in the expression (50) of I [φ ;Qa μ,Qo μ]

approximation with respect to the phonon fields (p + q ⩽ 2)

we have following action functional of the subsystem of phonon fieldsQa μ and Qo μ interacting with the background field φ0(x) of the collective oscillations of the interacting electron gas





∫

∫

∫

∫ ∫ ∫

∫ ∫

∫ ∫ ∫ ∫

∑

υ Ω

φ Π

Π

φ Π

=

μ μ

μ

μ

μ

μ

μ μ

=

Q

1

1

(53)

o

i

i a

o

o

a al

o ol

a al

o ol

2

2 1

3

2 2

0

0

⎡

⎣⎢

⎤

⎦

⎥

⎥

⎡

⎣⎢

⎤

⎤

⎦

⎡

⎡

⎡

⎡

⎡

where

It consists of two parts

μ μ

I ph Qa ,Qo I t Qat,Qot I l Qal,Qol , (55)

0 0⎡⎣ ⎤⎦ 0⎡⎣ ⎤⎦

where thefirst part

∫

∫

∑

υ

Ω

=

1

t i

ot

0

2 2 1

3

2

2

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎤

⎦

is the action functional of the free transverse phonons, and

∫

∫

∫

∫ ∫ ∫

∫ ∫

∫ ∫ ∫ ∫

∑

υ

Ω

Π

φ Π

=

1

1

1

al

l i

ol

2 2 1

3

2

2

2 2 0

0

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎦

⎥

⎡

⎡

⎡

⎡

⎡

⎡

is the total action functional of the acoustic as well as optic

longitudinal phonons interacting with the background field

φ0(x) of collective oscillationfield φ(x) The interaction action

functional in this expression leads to the mixing between

acoustic and optic phonons

From the extreme action principle

δ

δ

δ δ

I

x

I x

Q

Q

, ( )

,

o

l al ol

al

o

l al ol

ol

it follows the system of differential-integral equations for the

background phononfieldsQo( )x

al

andQo( )x

ol

corresponding to the extreme value of the action functional (57):



∫ ∫

∫

∫

∫ ∫ ∫

υ

Π Π

Π φ

∂

( )

(59)

x

Q

Q Q Q

Q

( )

al

al

2

0

2

2 2

0

0 2

0

0 0













∫ ∫

∫

∫

∫ ∫ ∫

∫ ∫ ∫

Ω

Π Π Π φ Π φ

∂

x

Q

Q Q

Q

Q

( )

ol

l ol o

ol

ol

al

2 0 2

2 0

0 2

0 0 2

Denote qal( )x and qol( )x the small fluctuations of the longitudinal phonon fields Qal( )x and Qol( )x around the background characterized by the functionsQal0( ),x Qol0( )x and considered as the physical vacuum of the system

0 0

It can be shown that in terms of Qal0( ),x Qol0( )x and

x

qal( ),qol( )x the action functional (57) has expression

I l Qal,Qol I l Qal,Qol I q ,q (62)

eff l al ol

0⎡⎣ ⎤⎦ 0⎡⎣ 0 0⎤⎦ ⎡⎣ ⎤⎦

where I eff[q ,q ]

l al ol is the effective action functional of the fluctuation longitudinal phonon fieldsqal( )x andqol( ).x It has following quadratic from



∫

∫

∫ ∫

∫ ∫ ∫ ∫

∑

υ Ω

Π

φ Π

φ

=

1

1

1

(63)

eff

l i

i al

ol

l ol

2 2

1

3

2

2 2 2

0

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎡

⎡

⎡

⎡

Since the fluctuation of longitudinal phonon fields gen-erates the quasiparticles participating in various dynamical processes, the fluctuating longitudinal phonon fields qal( )x

and qol( )x will be called dynamical longitudinal phonon fields

In order to exhibit the property of these fields to be

longitudinal let us use following modified Fourier expansion

∫

i

k

(2 )

˜ ( )

(2 )

1 2

ikx t a o

, ,

4

,

3

( ) ,

so that

=

(2 )

al ol, ikx a o

4

,

In terms of the modified Fourier transforms θ˜a o, ( )k

the action functional (63) becomes

∫

+

}

(66)

k

k l

(2 )

1

1

(2 )

1 (2 )

1

2˜ ( )* ˜( )

a

l a

o

l o

a a o o

a a o o

a

a

o o

a a o o

2 2

0 2

2

⎡

⎡

where Π k ˜ ( ) and Π l k˜ ( , ) are the Fourier transforms of

functions Π(x−y ) and Π(x− y x, −z):

∫

Π

(2 )4 ik x y( ) ˜ ( ), (67)

∫

∫

Π

π

×

− −

−

(2 ) 1

il x y

ik x z

4

( )

4

( )

For evidently demonstrating main features of the total

action functional (66) let us consider the case of the

homo-geneous electron gas with a constant electron density

n(x) = n0= const It can be shown that in this case

m

˜ ( ) ˜( ) (2 ) ( )1

F

2

where pFis the electron momentum at the Fermi level, and

formula (66) of the total action functional becomes

∫

∫

π

⁎

⁎

⁎

(2 )

1

(2 ) 1

˜ ( ) 1

(70)

eff l al ol

a

l a

o

l o

a

a

o o

F

a a

o o

2 2

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

Thus the total action functional of longitudinal phonon fields in the harmonic approximation consists of two parts

I eff l ⎡⎣qal,qol⎤⎦ I0l⎡⎣qal,qol⎤⎦ Iintl ⎡⎣qal,qol⎤⎦, (71) where

∫

∫

∑

θ

θ

∂

∂

=

t

x x

2

1 2

( )

( )

(72)

l i

a

i

o

l

0

2 2 1

3 2

2

2 0 2

⎣

⎢

⎢

⎛

⎝

⎝

⎠

⎟⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎛

⎝

⎠

⎦

⎥

⎥

can be considered as the action functional of a system of two scalarfields θ a( )x = θ a( , )x t and θ o( )x =θ o( , )x t describing longitudinal acoustic and optic phonons, whose Fourier

transforms are the functions θ k a( )and θ k o( )in formula (70),

∫

θ

=

(2 )

a o, ikx a o

4

,

and

l al ol

a a

o o

a a o o

int⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

with

∫

6 ˜ ( , )

(75)

4

( )

2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

is the interacting functional describing the elastic scattering of phonons in the effective potentialfield V(x − y) as well as the mixing between longitudinal acoustic and optic phonons Note that the effective potential V(x− y) is both non-local and non-instantaneous

Since plasmons are generated by the fluctuation ζ(x) of the collective oscillation field φ(x) around its static back-ground φ0(x), in order to study the plasmon–phonon inter-action first we derive the expression of the action functional

I fl[ ;Qa ,Qo ]

int of the interaction between the fluctuation field and phonon fields We have

Iintfl ;Qa ,Qo Iint⎡⎣ 0 ;Qa ,Qo ⎤⎦, (76) where I [ ;φ Qa μ,Qo μ]

(51) Because the terms containing transverse phonon fields are very small, we discard them and retain only the terms containing longitudinalfields According to formula (61) each

of them consists of two parts: the background longitudinal fieldQo al( )x orQo ol( )x and the dynamical longitudinal phonon field qal( )x or qol( ).x Consider again the case of homo-geneous electron gas Then in the harmonic approximation with respect to both types of fields (fluctuation field and phononfields) the action functional can be represented in the

general form as follows:

∫ ∫

∑

∑

∑∑

∑∑

ζ

=

+

+

×

+

+

+

×

+

+

=

=

= =

= =

( ) ( )

( )

( ) ( )

(77)

fl al ol

i

a i i al o i i ol

i

i j

aa i j i al j al

ao i j i al j ol

oo i j i ol j ol

i j

aa

i j

i al j al

ao

i j

i al j ol

oo

i j

i ol j ol

int

1

3

( ) ( )

1 2 1 2

1

3

( )

1 2 ( ) 1 2

1 2

1

3

1

3

( , )

1 2 1 2

( , )

1 2 1 2

( , )

1 2 1 2

1 2 1 2 1 2

1

3

1

3

( , )

1 2 1 2 1 2

( , )

1 2 1 2 1 2

( , )

1 2 1 2 1 2

⎡⎣

⎤⎦

⎡⎣

⎤⎦

The Fourier transform ζ˜( , ) of thek ω fluctuation field

ζ x ( ) is expressed in terms of the Fourier transform σ˜( ,k ω) of

thefluctuation field σ x( ) according to formula (48), i.e

e

˜( , )

0 2 2 2

Therefore between ζ x ( )and σ x( ) there exists following

linear functional relation

∫

ζ( )x = dyT x( −y) ( ),σ y (79)

where

∫ ∫

ε ω π

ω

− − −

k e

k

(2 )

i k x y x y

4

( ) ( )

0 2 2 2

0 0

⎡⎣ ⎤⎦

Similarly, according to formula (64), between the

dyna-mical longitudinal phononfieldsqal ol, ( )x and the scalarfields

θ a o, ( )x there exists following linear functional relation

qal ol, ( ) R( ) a o, ( ) , (81)

where

∫ ∫

k

i k x y x y

4

( ) ( 0 0)

⎡⎣

It is straightforward to derive the expression of the action functional of the interaction of plasmon with longitudinal phonons from formulae (61), (77), (79) and (81)

5 Quantumfields of interacting plasmon–photon– phonon system

On the basis of the results obtained in preceding sections we consider now the whole system of interacting plasmon, photon and phonons The total action functional of this sys-tem has the expression (37) The collective oscillation scalar filed φ(x) is split into two parts: the static background field

φ0(x) and the fluctuation field ζ(x) Each longitudinal phonon field Qal( ),x Qol( )x is also split into two parts: the static background field Qo al( ),x Qo ol( )x and the dynamical long-itudinal phonon fields qal( ),x qol( ),x while the transverse phonon fields Qat( ),x Qot( )x themselves are the dynamical ones without the background fields: Qat( )x =qat( )x and

=

Qot( ) qot( ) Plasmon is the quantum of a Hermitian scalar field σ(x) called plasmon field The longitudinal phonons can be con-sidered as the spinless quasiparticles The scalarfields θ x a( )

and θ x0( )having these spinless quasiparticles as their quanta are called effective longitudinal phonon fields In order to shorten lengthy formulae, they are introduced and used instead of the original longitudinal vector fields qal( )x and

x

qol( ).Meanwhile the transverse acoustic and optic phonon are the quanta of the original transverse vector fieldsqat( )x

andqot( ).x Above-mentioned quantizedfields have following expansions in terms of the destruction and creation operators

of the corresponding quasiparticles:

∫

σ

=

k

(2 )

1

2 ( ) ( ) ikx k t ( ) ikx k t , (83)

3 [ ( ) [ ( )

where c(k) and c(k)+ are the destruction and creation operators of the plasmon with momentum k, and ω(k) is its energy,

∫

θ

=

k

(2 )

1

2 ( )

a

a l

l i kx k t l i kx k t

3

( ) ( )

⎡

and

∫

θ

=

k

(2 )

1

2 ( )

o

o l

l i kx k t l i kx k t

3

( ) ( )

⎡

where al(k) or bl(k) and al(k)+ or bl(k)+ are the destruction and creation operators, respectively, of the longitudinal acoustic or optic phonon with momentum k and energy