R?ssler U. Solid State Theory: An Introduction

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228 7 Correlated Electrons

For m = 3 i t is shown in Fig.

7.13 (solid line) together with the radial dis-

tribution function for the Wigner crystal (dashed line) calculated in the HF

approximation. It has been found that the energy of the Laughlin wave func-

tion is always lower than that of the Wigner crystal. The radial distribution

function indicates that in contrast with the Wigner crystal, the Laughlin

state does not exhibit a long-range order and can be identiﬁed as a liquid.

A more detailed investigation shows also that the excitation sp ectrum out of

the Laughlin state has an energy gap, classifying the fractional quantum Hall

states as incompressible quantum liquids. The excitation spectra uncover sev-

eral unexpected properties of these states, such as the hierarchy, the fractional

charge, and the composite fermion concept, for which the reader is referred

to the literature [

202–204, 229, 230].

The geometry of a Hall eﬀect measurement is always connected with the

ﬁniteness of the sample with the edge of the sample representing a potential

barrier for the electrons. Consequently, the Landau levels, which are constant

except for ﬂuctuations due to disorder (see Chap.

9), bend upwards towards

the edg e of the sample where they cross the Fermi energy. Thus, along the

sample boundary, each Landau level represents a one-dimensional electron

system, the so-called edge channel. In a classical picture, the edge channels

correspond to skippin g cyclotron orbits along the edge. The quantum Hall

eﬀect can be understood as transmission between the diﬀerent pro bes of the

Hall bar along these edge-channels [

231, 232]. In the fractional QH regime,

the edge-channels are seen also as a realization of one-dimensional systems of

interacting electrons to which the Tomonaga–Luttinger model applies [

204].

Problems

7.1 Calculate the comm utators of c

kσ

and c

†

kσ

with the interaction part H

int

of the Hamiltonian and verify (

7.20).

7.2 Derive the Dyson equation (

7.25) by making use of the Fourier transform

of (

7.23) in the equation of motion for the Green function.

7.3 Evaluate the commutators of n

i−σ

iσ

with the single-particle and with

the interaction term of the Hubbard Hamiltonian (

7.45)fortheone-

dimensional case to verify (

7.54)and(7.55).

7.4 Calculate the s pectral weight Z =(1−∂Σ/∂E)

−1

for the lower and upper

Hubbard band, especially for the center of the band at k = π/2a.

7.5 Consider the Hubbard Hamiltonian for the case of half-ﬁlling and weak

hopping or strong correlation. Show by applying perturbation theory in

second order and by introdu cing spin operators



iσ

= c

†

iσ

i−σ

(7.143)

Problems 229

with z

= ±1, that the Hubbard Hamiltonian can be mapped onto an

operator of the Heisenberg type. Identify the sign of the exchange coupling

and the magnetic order of the ground state.

7.6 Set up the Hamiltonian for a system with localized electrons and on-site

correlation (e.g., d or f electrons) overlapping with delocalized electrons

deriving e.g., from atomic s states! This Hamiltonian is known in the

literature as the Anderson Hamiltonian.

7.7 Calculate the momentum distribution n

at T = 0 by evaluating (

7.87)

in order to quantify the quasi-particle jump.

7.8 Evaluate the real part of Lindhard function (

4.136)atT =0fortheone-

dimensional and the three-dimensional electron systems and discuss the

behavior for q → k

at ω = 0. While for d = 1 the calculation can easily

be carried out for ﬁnite ω andbediscussedforω → 0 it is advantageous

to do the calculation for d = 3 from the beginning for ω =0.

Electron–Phonon Interaction

Based on the Born–Oppenheimer (or adiabatic) approximation (Chap.

2), the

dynamics of the heavy and light constituents of a solid, the ions and the

electrons, respectively, have been presented in the previous chapters as those

of independent systems. For the lattice dyna mics (Chap.

3), the electrons were

considered only by their contribution to the binding forces which determine

the dynamical matrix. For the electron systems, their energy spectrum and

excitations (Chaps.

4–7), the position of the ions was kept ﬁxed in the periodic

conﬁguration of a lattice, while the chemical nature of the ions determined the

material speciﬁc properties. Releasing the Born–Oppenheimer approximation

enables the two subsystems to communicate with each other by exchanging

energy. This leads to a variety of eﬀects, which are not restricted to solids but

are foun d in all types of condensed matter including macromolecula r systems

in chemistry and biology.

The electrons experience the moving lattice as a perturbation of the peri-

odic po tential, which can be understood as scattering between electrons and

phonons. Four scenarios will be considered here:

1. An excited electron gets rid of its excess energy by emitting phonons and

at the same time changes its momentum. This leads to a ﬁnite lifetime of

the carrier in its single-particle state.

2. An electron system in a solid driven by an external ﬁeld (electric ﬁeld,

temperature gradient) adopts a nonequilibrium state. When the ﬁeld is

switched oﬀ, the electrons emit ph onons and the system relaxes into an

equilibrium state. This process of electron–lattice relaxation is character-

ized by a transp ort r elaxation time.(Inthesamewayanexcitedspin

system can transfer its excess energy to the ion system and equilibrate

by spin–lattice relaxation.) In thermodynamic terminology, the lattice

is a heat bath serving as an energy sink or a reservoir. In exchanging

energy with this bath, the relaxation processes change the phase of the

individual electron wave function which makes these processes incoher-

ent. By discussing these phenomena, electrical transport will become a

U. R¨ossler, Solid State Theory,

DOI 10.1007/978-3-540-92762-4

 Springer-Verlag Berlin Heidelberg 2009

232 8 Electron–Phonon Interaction

topic of this Section. Diﬀerent physical scattering mechanisms, depend-

ing on the phonons involved, determine by their speciﬁc contributions the

temperature-dependence of the electric conductivity.

3. Recalling the eﬀect of electron–electron interaction (Chap.

4), the electron–

phonon interaction also can lead to a new ground state of the system. Due

to its charge, an electron creates a polarization cloud in a polar lattice,

which moves around with the electron and changes its dynamic proper-

ties. Thus, electron–phonon interaction leads to a new quasi-particle, the

electron and its polarization cloud, called polaron because of the polar

electron–phonon coupling causing the interaction.

4. Electron–phonon interaction can result also in an attractive electron–

electron interaction if the p honon emitted by one electron is absorbed

by another one within their lifetimes (virtual phonon exchange). This

phonon-mediated electron–electron interaction favors the formation of

pairing of electrons, which is one o f the basic mechanisms of supercon-

ductivity.

The diﬀerent aspects o f electron–phonon interaction are subjects of most

textbo oks in Solid State Theory. For complementary reading, we refer here

to [

10, 13, 14, 89, 95, 233, 234].

8.1 Preliminaries

Let us recall from Sect.

2.1 the separation of the Hamiltonian (2.1)

= H

+ H

ion

+ H

el−ion

. (8.1)

The electron–ion interaction with the general form

el−ion



l=1



n,τ

v(r

− R

nτ

) (8.2)

has b een considered in the electronic energy E

({R

nτ

}) as a contribution to

the adiabatic p otential U({ R

nτ

}) deﬁned in (

2.26), which was the starting

point for the lattice dynamics in Chap.

3. This energy, obtained as the eigen-

value of the electron problem (

2.25) in a static conﬁguration of the ions, was

assumed later to be that of the equilibrium conﬁguration {R

nτ

}. Here, we

have to take into account the moving lattice with time-dependent positions

nτ

(t)=R

nτ

+ u

nτ

(t) and do this by expanding around the equilibrium

positions

el−ion



l,n,τ

v(r

− R

nτ

) −



l,n,τ

∇

v(r

− R

nτ

)

nτ

· u

nτ

+ .... (8.3)

The ﬁrst term has become part of the eﬀective periodic single-particle poten-

tial of the band structure problem (Chap.

5). In the second term, linear in

8.1 Preliminaries 233

the displacements u

nτ

, ∇

means the derivative with respect to the position

vector of the lth electron. This term will be considered in the following as the

linear electron–phonon coupling H

el−ph

. The higher order terms, indicated

by dots, will be neglected. The linear approximation is suﬃcient for displace-

ments that are small compared to the lattice spacing, as will be assumed

throughout this chapter.

It is advantageous for the illustration and for the evaluation of the inter-

action to make use of the occupation number representation and write it in

terms of creation and annihilation operators. The lattice displacements can

be formulated as (see Chap.

3,(3.23, 3.24) together with (3.39))

nτ

√



s,q

(q)e

iq·R



s,q

¯h

2NM

(q)

(q)e

iq·R



†

(−q)+a

(q)



(8.4)

with the boson operators a

†

(q)anda

(q) of phonons with frequency ω

(q)

and eigenvectors e

(q). The time-dependence is not indicated here to simplify

notation.

The gradient of the potential is a single-particle term, which according to

(

4.76) can be written in terms of fermion operators for Bloch states (Prob-

lem 8.1). We want to simplify here the electron–phonon interaction within a

single band, for which we adopt the eﬀective mass approximation. For this

case, we may write, with the help of the Fourier transform of the potential,



∇

v(r

− R

nτ

)

nτ



′

v(q

′

·(r

−R

nτ

)

(8.5)

and use



′

·r



†

k+q

′

(8.6)

with the fermion operators c

†

k+q

′

and c

for free electrons. Making use of (

8.4)

together with (

8.5), we can perform the lattice sum in the second term of (8.3)

with



i(q−q

′

)·R

= N



q−q

′

(8.7)

and write the linear electron–phonon interaction in the convenient operator

form

el−ph



s,k,q,G

(q − G)



†

(−q)+a

(q)



†

k+q−G

(8.8)

with the coupling matrix element

(q − G)=−i



N¯h

(q)

(q − G) · e

(q)e

−i(q−G)·τ

v(q − G), (8.9)

234 8 Electron–Phonon Interaction

k–q

k+q

sq sq

Fig. 8.1. Graphical representation of the electron–phonon interaction: phonon

emission (left) and phonon absorption (right)

k+q

k+q–G

normal umklapp

Fig. 8.2. Kinematics of the electron–phonon interaction: normal process (left)and

Umklapp process (right), the thin solid lines mark the boundary of the ﬁrst Brillouin

zone

which depends on the coupling mechanism as will be outlined below.

The operator part of H

el−ph

tells us about the kinematics as determined

by the wave vectors. For this, we illustrate the interaction in graphical fo rm

(Fig.

8.1) and the kinematics in k space (Fig. 8.2). The graphs contain the

fermion operators as straight lines and the boson op erator as a wavy line.

el−ph

contains only a single phonon operator: a phonon is created or anni-

hilated, and two electron operators: a creation and a n annihilation operator.

In the interaction process, the electron is scattered b etween two Bloch states

while a phonon is emitted or absorbed (see Fig.

8.1). The total momentum

is conserved in the scattering (the sum of the wave vectors of the creation

operators equals that of the annihilation operators, up to a recipro cal lat-

tice vector). We note in passing that (

8.8), although derived here for the

electron–phonon system, is the standard form of a fermion–boson interaction.

It applies as well to the coupling of electrons with photons or magnons and

also to similar problems in nuclear physics.

Two characteristic scenarios are shown in Fig.

8.2 to demonstrate the kine-

matics of electron–phonon interacion. If the wave vectors of the electron before

and after the scattering are within the 1st Brillouin zone and the momentum

transfer is small (k and k + q are almost parallel), the scattering does not

strongly change the direction of the moving electron. This is called the normal

process. If the same small momentum transfer shifts the wave vector across

8.2 Coupling Mechanisms 235

the Brillouin zone boundary, we have to bring it back by subtracting a recip-

rocal lattice vector, which almost inverts the direction of the moving electron

(k and k + q −G are almost antiparallel). This is called the Umklapp process.

It is intuitively clear th at the Umklapp processes ta ke a stronger inﬂuence on

the transport properties of electrons than the normal processes.

Formally, the energy balance of the scattering process will be considered by

treating the electron–phonon interaction as a time-dependent perturbation,

which leads to self-energy corrections changing the energy of the electrons and

giving them a ﬁnite lifetime. But, it can be made clear from the graphs: the

total energy is conserved and the energy of an emitted (absorbed) phonon

has to b e provided (is carried away) by the electron. The diﬀerent coupling

mechanisms, which depend on the lattice properties of the solid and exhibit

characteristic dependencies on q, will be presented in the following Section.

8.2 Coupling Mechanisms

In Chap. 3, we have demonstrated the physical properties of phonons in dif-

ferent branches. The long-wavelength acoustic phonons were recognized as

causing local lattice compression or dilation, while long-wavelength optical

phonons have been identiﬁed with electrical dipole vibrations. These pic-

tures are helpful in deriving the corresponding mechanisms of electron–phonon

coupling.

Deformation Potential Coupling (Acoustic Phonons):

The lo cal homogeneous compression or dilation ∆(r)causedbyacoustic

phonons in the long-wavelength limit can be described as a local relative

volume change (Fig.

8.3)

∆(r)=

∆V

. (8.10)

It is experienced by an electron as a local change of the lattice constant which

shifts the single particle energy. If we consider a simple nondegenerate band

∆V

Fig. 8.3. Sketch of the homogeneous dilation caused by a longitudinal acoustic

phonon

236 8 Electron–Phonon Interaction

(deriving from atomic s states as e.g., the conduction band in a normal metal

or semiconductor), the electron–phonon interaction can be written as a local

potential seen by the electrons

el−ph

= −



D ∆(r

), (8.11)

where D is the deformation potential corresponding to the energy shift for a

relative volume change ∆V/V = 1. Deformation p o tentials are of the order

of a few eV (see the data collection of Landolt–B¨ornstein [

1]). The relative

volume change caused by phonons, being much smaller than 1, is related with

the ﬂux of the continuous lattice displacement ﬁeld u(r) through the area

enclosing the volume V as depicted in Fig.

8.3. By Gauss’ theorem, we have

∆V =



u · dA =



∇·u dV (8.12)

which for a homogeneous lattice distortion in the volume V (assumed to have

a small linear extension compared with the wavelength of the phonon) can

be written also as V ∇·u(r). Note that ∇·u(r) ca n be expressed as the

trace of the strain tensor ﬁeld, Trǫ(r) (Problem 8.2). The displacement ﬁeld

u(r) is obtained in the limit of long wavelengths from the expression for the

displacement u

nτ

(

3.23) by r epl acing R

→ r and using the appropriate

expression for the eigenvector e

(q). As the electron–phonon interaction is

determined by ∇·u(r), only the longitudinal acoustic phonons with e

(q)=

/M q/q contribute, where M is the total mass of the ions in the Wigner–

Seitz cell. Thus, we can write

el−ph

= −D



∇

· u(r

)=−iD



l,q

LAq

√

q · q

iq·r

(8.13)

and obtain with

LAq

¯h

2ω

(q)



†

(−q)+a

(q)



(8.14)

and (

8.6) the Hamiltonian for the deformation potential coupling

el−ph



k,q



†

(−q)+a

(q)



†

k+q

. (8.15)

The cou p l i n g matrix element reads

= −iD

¯h

2NMc

1/2

(8.16)

where the use was made of the dispersion relation ω

(q)=c

8.2 Coupling Mechanisms 237

Supplement: Deformation potential coupling of electrons in a p band

In deriving (

8.15), we have assumed the electrons to be in a simple s band. For

energy bands originating from atomic orbitals with higher angular momentum, we

have to consider the orbital degeneracy, according to which the Hamiltonian for the

electron–phonon interaction becomes a matrix in the Bloch (or angular momentum)

representation. Here, we face the same problem as in the eﬀective-mass approxima-

tion for the p type valence band in semiconductors (see Sect.

5.5), which resulted in

a3×3 matrix Hamiltonian with bilinear expressions in the components of the wave

vector as matrix elements and three-material speciﬁc parameters (L, M, N ), which

deﬁne the curvature of the bands in diﬀerent directions in k space. This matrix can

be expressed also in terms of angular momentum matrices for I =1,whichinthe

basis |x, |y, |z read

⎛

⎝

00 0

00−i

0i 0

⎞

⎠

⎛

⎝

00 i

000

−i00

⎞

⎠

⎛

⎝

0 −i0

i00

000

⎞

⎠

. (8.17)

As it turns out, the k · p matrix M (

5.104) can be decomposed according to

M = Ak

1 + B



−

− 2N



α<β

, (8.18)

where in the ﬁrst term k

= k

+ k

and 1 is a 3 × 3 unit matrix (which is

proportional to I

+ I

)and{I

} =(I

+ I

)/2, while the constants

are related by A =(L +2M)/3andB = −L + M.

The individual terms of M are invariant tensor products under the point group

and can be formulated on group theoretical grounds [

235, 236]. The symmetry prop-

erties of the symmetric second-rank strain tensor with the components ǫ

αβ

are the

same as those of the tensor formed by k

. Therefore, in the same degeneracy space

(here the 3-fold space of the p states), the Hamiltonian of the deformation potential

coupling has the form

el−ph

= D

Trǫ + D



−

αα

+2D



α,β

}ǫ

αβ

. (8.19)

It consists of three invariant contributions connected with three deformation poten-

tials. D

is recognized as the deformation potential of hydrostatic strain (as in

(

8.11)), while D

and D

correspond to shear strain in (001) and (111) direc-

tion, respectively. This strain Hamiltonian, originally derived for homogeneous static

strain, is used here to describe the deformation potential coupling. The components

of the strain tensor ﬁeld ǫ

αβ

(r), which are symmetrized derivatives of the displace-

ment ﬁeld (see (

3.76)),canbeexpressedintermsofphonon operators as before and

one obtains the matrix Hamiltonian for the deformation potential coupling of elec-

trons in a p band. The complexity of this interaction Hamiltonian allows not only

for the coupling to b e longitudinal but also to transverse phonons (Problem 8.3).

The same concept can be applied to energy bands with other orbital degeneracies

and to coupling between diﬀerent bands.

238 8 Electron–Phonon Interaction

Polar Coupling with Optical Phonons or Fr¨ohlich

Coupling

[

237, 238]:

In the long-wavelength limit, longitudinal optical phonons cause a macroscopic

polarization ﬁeld

P (r)=

M(r)

(8.20)

withtheelectricdipoleﬁeldM (r). A charge densit y ρ(r) placed in this

polarization ﬁeld gives rise to the interaction energy

int

= −



P (r) · E(r)d

r (8.21)

with

E(r)=−∇



ρ(r

′

)

4πε

∞

|r − r

′

, (8.22)

where ε

∞

is the background (or high frequency) dielectric constant. By partial

integration, this can also be written as

int

= −





∇

· P (r)





′

ρ(r

′

)

4πε

∞

|r − r

′

(8.23)

with the charge distribution given by

ρ(r

′

)=−



l,q

′

·(r

′

−r

)

= −



′

·r

′



†

k−q

′

. (8.24)

The lattice displacement for longitudinal optical phonons in the long-

wavelength limit reads in operator form

nτ



¯h

2NM

(q)e

iq·R



†

(−q)+a

(q)



. (8.25)

Here we assume a crystal with two ions in the Wigner–Seitz cell with masses

, M

and M = M

+ M

for which the phonon eigenvectors are (see

Problem 3.1)

(q)=

, e

(q)=−

. (8.26)

The moving ions, carrying the dynamical eﬀective charge η

= −η

= η

(introduced in Sect.

3.5), provide a macroscopic dipole moment

M =



n,τ

= η



n,q

¯h

2Nμω

iq·R



†

(−q)+a

(q)



(8.27)

Herbert Fr¨ohlich, 1905 – 1991