R?ssler U. Solid State Theory: An Introduction

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8.2 Coupling Mechanisms 239

where μ is the reduced ion mass. The continuum limit M(r) is obtained by

replacing the discrete lattice vectors R

by the local position vector r.Thus,

we can write

∇·P (r)=iη



¯h

2Nμω

q · q

iq·r



†

(−q)+a

(q)



. (8.28)

The two volume integrals in (

8.23) are p erformed by taking the Fourier

transform of 1/|r − r

′

| (see Appendix) to arrive at



′

iq·r

′

·r

′

|r − r

′

4π

q,−q

′

. (8.29)

Finally, we take the dynamical eﬀective charge from (

3.115) and use the

Lyddane–Sachs–Teller relation (

3.114) to replace the transverse optical phonon

frequency by the longitudinal one

η =

μV ε



∞

−

ε(0)



1/2

∞

(8.30)

to write E

int

as the electron–phonon interaction for the Fr¨ohlich c oupling

el−ph



k,q



†

(−q)+a

(q)



†

k+q

(8.31)

with the interaction matrix element



¯hω

2Vε

∞

−

ε(0)

!

1/2

. (8.32)

The strength of the interaction is frequently q uantiﬁed by the Fr¨ohlich

coupling constant

8πε

¯hω

∗

¯h

1/2

∞

−

ε(0)

, (8.33)

with the eﬀective electron mass m

∗

. It is deﬁned in analogy with the ﬁne

structure constant, which is the corresponding coupling co nstant for the

electron–photon interaction and shall ﬁnd an obvious meaning when deal-

ing with the polaron in Sect.

8.4. Depending on the polarity, determined by

the diﬀerence between ε

∞

and ε(0) of the solid, on the eﬀective mass and

on the energy of the longitudinal optical phonon, α

takes val ues between

0.07 in GaAs, 0.39 in CdTe, and 6 .6 for RbBr (compared with the ﬁne struc-

ture constant of 1/137). For the weak coupling regime (α

< 1) the Fr¨ohlich

interaction can be treated by perturbation theory (see Sect.

8.4), while special

concepts have been developed for the strong coupling regime (α

> 1)[

239].

240 8 Electron–Phonon Interaction

Piezoelectric Coupling (Acoustic Phonons):

In crystals without inversion symmetry, a homogeneous strain causes a dielec-

tric polarization P ,knownasthepiezoelectric eﬀect. It is quantiﬁed by the

relation (double index summation understood)

= e

ijk

, (8.34)

where e

ijk

is the piezoelectric a nd ǫ

the strain tensor. Due to symmetry

considerations, a third rank tensor has nonvanishing elements only for crystals

lacking inversion symmetry [

74, 235]. For the particular case of zinc blende,

the piezoelectric tensor takes the form e

ijk

= e

|ε

ijk

| with the Levi–Civita

symbol ε

ijk

and the piezoelectric constant e

(written in Voigt notation, see

Sect.

3.4). The piezoelectric eﬀect, originally related to static strain, applies

as well to the dynamical case of strain ﬁelds connected with acoustic phonons

and gives rise to the piezoelectric electron–phonon coupling. For this case,

the classical interaction energy is to be formulated with the p olarization ﬁeld

P (r) according to (

8.34) with the strain ﬁeld

(r)=

+ q

iq·r

(8.35)

connected with the displacement ﬁeld u(r)=u exp(iq · r). For acoustic

phonons, u can be written as

u =



(q), withe

(q)=



(q). (8.36)

Tak in g the corresponding displacement ﬁeld P (r), the piezoelectric electron–

phonon inter a ction can be derived in analogy with the Fr¨ohlich coupling

(Problem 8.4). The result is

el−ph



s,k,q



†

(−q)+a

(q)



†

k+q

(8.37)

with the coupling matrix element

= −

2ee

∞

¯hN

2Mv

1/2

(q)+c.p.

, (8.38)

where v

is the sound velocity of the phonon branch s.

8.3 Scattering Processes: Lifetime, Relaxation

The electron–phonon interaction represents a link between the electron and

phonon systems, which in the previous chapters have been investig ated sepa-

rately. In this Section, we consider it as a perturbation of the electron system

by evaluating the scattering processes depicted in Fig.

8.1: phonon emission

8.3 Scattering Processes: Lifetime, Relaxation 241

and phonon absorption by a single electron. According to the time-dependence

of the phonon operators, the electron–phonon interaction is periodic in time

and Fermi’s Golden Rule applies. In general, the scattering rate between Bloch

electron states with energies ǫ

and ǫ

′

under a perturbation H

′

nk,n

′

2π

¯h

|n

′

|nk|

δ(ǫ

− ǫ

′

− ∆ǫ), (8.39)

where ∆ǫ is the energy change in the case of inelastic scattering. Considering

scattering-out from the Bloch state with n, k to all other possible Bloch states

′

, k

′

, one ﬁ n ds the inverse single-par ticle or carrier lifetime



′

nk,n

′

(1 − f(n

′

)) , (8.40)

where f(n

′

) is the distribution function. It vanishes if we consider a single

electron in an otherwise empty band. In the language of Green functions, this

lifetime is related with the imaginary part of a self-energy contribution ¯h/τ

which represents a level broadening.

Taking now H

el−ph

as the perturbation H

′

,wehavetoevaluatethematrix

element with the electron and phonon states in the occupation number rep-

resentation. This can formally be done, bu t th e result is written immediately

by inspection of the graphs for phonon absorption and emission processes (see

Fig.

8.1). In the one-band approximation adopted in Sect. 8.2, we can drop the

band index and know from momentum conservation that k

′

= k + q for nor-

mal processes, to which we can restrict ourselves here. The matrix elements

of the phonon operators yield

n

(q) − 1|a

(q)|n

(q) =

(q) for phonon absorption (8.41)

n

(q)+1|a

†

(q)|n

(q) =

(q) + 1 for phonon emission (8.42)

with the phonon occupation numbers n

(q), and the inv erse lifetime is

expressed by

2π

¯h



s,q

(1 − f(k + q))

(q)+

∓

δ(ǫ

− ǫ

k+q

∓ ¯hω

(q)). (8.43)

Here, the upper(lower) sign refers to phonon absorption(emission). Phonon

absorption is possible only if the occupation facto r n

(q) diﬀers from zero.

Energy conservation in the scattering process (depicted in Fig.

8.4) is expressed

by the δ-function. For conduction electrons in semiconductors with ǫ

≫

¯hω

(q), the scattering with acoustic phonons is almost elastic, while scatter-

ing with optical phonons is connected with a substantial change in energy.

242 8 Electron–Phonon Interaction

hω

Fig. 8.4. Scattering of conduction electrons in a semiconductor with acoustic and

optical phonons

The emission processes describe energy dissipation from the electrons to the

lattice, which serves as a heat sink. The rate of energy transfer between the

electron and phonon systems is given for this process by an expression similar

to (

8.44) but with an additional factor ¯hω

(q)/ǫ

under the sum. For high

energy transfer rates, the phonon system will be heated up and one has to

consider phenomena related with hot phonons.

The general result of (

8.44) can be speciﬁed for the diﬀerent mechanisms of

the electron–phonon interaction with their particular matrix elements, which

diﬀer with r espect to their dep endence on the wave vector q of the emitted

or abso rbed phonon:

LAq

∼ q (deformation potential coupling) (8.44)

LOq

∼ q

−2

(Fr¨ohlich coupling) (8.45)

∼ q

−1

(piezoelectric coupling). (8.46)

This has a consequence when evaluating the sum over q as an integral with

the thermal phonon and electron occupations. This is done by substitutions

(see Appendix) which lead to diﬀerent dependencies of these scattering mech-

anisms, their contribution to electron lifetime and energy dissipation, on the

temperature.

The dynamical processes connected with electron–phonon scattering can

be studied for two diﬀerent scenarios. The ﬁrst applies to semiconductors and

insulators, where electrons in the conduction band can be created by optical

excitation fr om the valence band. These electrons thermalize due to carrier–

carrier interaction on a very short time scale and form a hot carrier system

(with a temperature higher than the lattice temperature), which relaxes due

to phonon emission (energy relaxation) and ﬁnally equilibrates with the lattice

before the electrons and holes recombine [

240–242]. These processes can be

investigated in time-resolved spectroscopy [

95, 243].

8.3 Scattering Processes: Lifetime, Relaxation 243

The second scenario is that of transport, which applies as well to met-

als as to doped semiconductor s. The ensemble of carriers, which in a n

external electric ﬁeld are accelerated, is described by a nonequilibr ium distri-

bution function f (k,T). The carriers dissipate their excess energy by emitting

phonons. A stationary situation is achieved if the rate of energy gain of the

carriers in the electric ﬁeld equals the rate of energy dissipation by phonon

emission. In a homogeneous system, this situation is accounted for by the

Boltzmann equation (or Boltzmann’s stationarity condition) [

64, 244–246]

−

¯h

E ·∇

f(k,T)=

∂f (k,T)

∂t

coll

, (8.47)

where the lhs accounts for the rate of energy gain by the carriers in the ﬁeld E.

The rhs of this relation, the scattering or c ollision term, can be formulated in

terms of the single-particle scattering rates r

k,k

′

and occupation factors

∂f (k,T)

∂t

coll



′

(

(1 − f(k,T)) f(k

′

,T)r

k,k

′

−f(k,T)(1−f (k

′

,T)) r

′

)

, (8.48)

where the ﬁrst term under the sum on the rhs represents the scattering pro-

cesses into the state with k while the second term represents scattering-out

from this state. For isotropic scattering rates, r

k,k

′

= r

′

, we ﬁnd

∂f (k,T)

∂t

coll



′

k,k

′

(f(k

′

,T) −f (k,T)) . (8.49)

Instead of these microscopic expressions, the collision term is frequently

treated in the relaxation time approximation

∂f (k,T)

∂t

coll

= −

f(k,T) − f

(k,T)

(k)

, (8.50)

which describes the evolution of the nonequilibrium distribution f(k,T)into

the equilibrium distribution f

(k,T) in the characteristic transport relaxation

time τ

(k). Equations (

8.47)and(8.50) can be solved by iteration

f(k,T)=f

(k,T)+

¯h

(k)E ·∇

f(k,T)

= f

(k,T)+

¯h

(k)E ·∇

(k,T)+.... (8.51)

For small deviations from the equilibrium distribution, it is suﬃcient to take

into acco unt only the l owest order correction (linear in E), which deﬁnes the

regime of linear transport, while the higher order terms describe the nonlinear

transport [

247].

244 8 Electron–Phonon Interaction

The physical observable to quantify the carrier transport, the electric

current density, can be expressed as

j = −



v(k)f(k ,T). (8.52)

Here, the carrier velocity v(k) is the group velocity of electrons in an energy

band ǫ

v(k)=

¯h

∇

, (8.53)

which for a simple parabolic band equals ¯hk/m

∗

with the eﬀective mass m

∗

Consider now one component of the current density with the nonequilibrium

distribution function from (

8.51)

= −



(k)

⎧

⎨

⎩

(k,T)+

eτ

(k)

¯h



∂f

(k,T)

∂k

⎫

⎬

⎭

. (8.54)

The ﬁrst term on the rhs vanishes because the system in equilibrium does not

carry a current. By writing the distribution function as f(ǫ

,T), the derivative

with respect to k

in the second term becomes

∂f

(k,T)

∂k

∂f

(ǫ, T )

∂ǫ

∂k

∂f

(ǫ, T )

∂ǫ

¯hv

(k), (8.55)

and we ﬁnd Ohm’s law in the form



αβ

(8.56)

with the electric conductivity

αβ

= −



(k)

∂f

(ǫ, T )

∂ǫ

(k)v

(k). (8.57)

This second rank tensor resembles the form derived in Chap.

2 within the

concepts of linear response (see Problem 2.4), and we may recognize the cor-

relation between the two components of the velocity. On the other hand, here

the appearance of the transport relaxation time is a new aspect. It accounts

for the dissipa tion of energy in collisions, which is essential for obtaining a

ﬁnite conductiv ity.

Comparing (

8.49)and(8.50)wemayidentify

f(k,T) − f

(k,T)

(k)



′

k,k

′

(f(k ,T) − f (k

′

,T)) . (8.58)

and see that in general the transp ort relaxation time depends on the distribu-

tion function, which limits the validity of the relaxation time approximation.

8.3 Scattering Processes: Lifetime, Relaxation 245

However, this is not the case for an isotropic dispersion ǫ

= ǫ

and elastic

scattering for which we may write

f(k)=f

(k)+k · EC(k) (8.59)

and similar for f (k

′

) with the electric ﬁeld E and a scalar function C(k)=

C(k

′

). In polar coordinates with respect to the direction of E, one has

k · E = kE cos ϑ and k · k

′

= k

cos ϑ

′

, (8.60)

where ϑ

′

is the angle between k and k

′

,and

′

· E = kE (cos ϑ cos ϑ

′

+sinϑ sin ϑ

′

cos ϕ

′

). (8.61)

As a consequence, the diﬀerence f(k) − f (k

′

) can b e expressed as

f(k) − f (k

′

)=kE C(k)



cos ϑ(1 − cos ϑ

′

) − sin ϑ sin ϑ

′

cos ϕ

′



(8.62)

where the second term on the rhs vanishes by integration over ϕ

′

(note that

for isotropic scattering r

k,k

′

depends on k − k

′

)whichgiveswithdΩ

′

dϕ

′

sin ϑ

′

dϑ

′



dΩ

′



f(k) − f (k

′

)



= kEC(k)cosϑ



dΩ

′

(1 − cos ϑ

′

) . (8.63)

With (

8.59), we identify the factor in front of the integral on the rhs as

f(k) − f

(k) and obtain, using (

8.39), for the relaxation rate

(k)

4π

¯h



|k|H

′

|

δ(ǫ

− ǫ

′

)(1 − cos ϑ

′

(8.64)

which is independent of the distribution function. The factor 1 −cos ϑ

′

under

the integral tells us that collisions by which the propagation direction of the

charge carrier is reversed (ϑ

′

≃ π, back scattering) contribute much to the

relaxation rate, while the eﬀect of forward scattering with ϑ

′

≪ π is only

small.

For further evaluation of σ

αβ

(

8.57), let us assume the electric ﬁeld in

the z direction. Then, we have σ

as the only nonvanishing component of

the conductivity tensor. The sum over k can be performed in spherical polar

coordinates. By making use of the dispersion relation ǫ

=¯h

/2m

∗

,wecan

write with v

(k)=¯hk

∗

and k

= k cos θ

(k)=

∗

ǫ cos

θ (8.65)

and formulate the remaining integration over k as an energy integral

= −

∗

¯h

3/2



3/2

(ǫ)

∂f

(ǫ, T )

∂ǫ

dǫ. (8.66)

246 8 Electron–Phonon Interaction

The transport relaxation time is a power function of the carrier energy ǫ.

Integrals of this type can be evaluated by using the fact that ∂f

(ǫ, T )/∂ǫ is

a symmetric function with a pronounced maximum at ǫ = μ (see Appendix).

Depen di n g on the temperature, two limiting cases shall be discussed here.

For a degenerate electron system with E

≫ k

T , the derivative of the Fermi

distribution function can be written as ∂f

/∂ǫ ≃−δ(ǫ − E

) thus giving

nτ(E

)

∗

(8.67)

with the degenerate carrier density

n =

3π

∗

¯h

3/2

. (8.68)

Formally, this result is identical with that obtained by the classical Drude the-

ory [

12, 29, 31]. However, here, the quantum statistical nature of the degener-

ate electron system becomes relevant in the d ependence on the Fermi energy

and the transport relaxation time can be traced back to the microscopic

scattering processes.

For the nondegenerate car rier system, k

T ≫ ǫ, the derivative of the

distribution function takes the form ∂f

/∂ǫ ≃−exp (−ǫ/k

T )/k

T and we

may write

nτ(ǫ)

∗

(8.69)

with the nondegenerate carrier density

n =

2π

∗

¯h

3/2



dǫǫ

1/2

−ǫ/k

(8.70)

and the averaged transport relaxation time

τ(ǫ) =

dǫǫ

3/2

τ(ǫ)e

−ǫ/k

dǫǫ

1/2

−ǫ/k

. (8.71)

The eﬀects of the band structure (here the eﬀective mass) and of the

scattering processes are comprised in the mobility

μ =

eτ

∗

(8.72)

with the temperature-dependent transport relaxation time τ.Thus,thecon-

ductivity is written in the form σ = enμ. The mobility is frequently measured

in magneto-transp ort experiments making use of the Hall eﬀect and then

called Hall mobility. An example is shown in Fig.

8.5 for n-doped GaAs. The

measured values are compared with the calculated mobilities for diﬀerent scat-

tering mechanisms (Problem 8.5). Besides scattering with phonons, which

8.4 The Fr¨ohlich Polaron 247

11010

Temperature T [K]

Half mobility m

[cm

–1

]

GaAs

neutral impurity

ionized impurity

piezoelectric

deformation

potential

polar

Fig. 8.5. Hall mobility versus temperature for n-doped GaAs, showing exp erimental

data together with the contributions from diﬀerent scattering mechanisms, after [

89]

limits the mobility at high temperatures, sca t tering also with impurities (see

Chap.

9) is considered. The latter dominates the mobility at low temperature,

when the phonons are frozen out. For degenerate electron systems, the relax-

ation time can be converted into a mean free path l

mfp

= v

τ,whichisthe

average distance between collisions covered by a carrier with Fermi velocity v

Note that the momentum selection rulerepresentsageometrical con-

straint depending on the system dimension. Thus, compared with the three-

dimensional case, electron–phonon interaction is modiﬁed in electron systems

with reduced di mensionality (e.g., for electrons conﬁned at semiconductor het-

erostructures or in quantum wells or quantum wires). Moreover, by remote

doping, the ionized impurities can be separated from the mobile carriers to

suppress the scattering. This leads to an enormous increase of the mobil-

ity at low temperatures. Thus, carrier conﬁnement ta kes inﬂuence on the

lifetime [

242, 243, 248]. At the same time, the mean-free-path, the distance

between successive scattering events, can become comparable or larger than

the system dimension, as is typical for mesoscopic systems. In this case, the

carrier passes through the sample without being scattere d and the transport

is called ballistic.

8.4 The Fr¨ohlich Polaron

In contrast with the single-particle approximation in Chaps. 4 and 5,where

a free electron o r a Bloch state was aneigenstateoftheelectron Hamilto-

nian, these states are now perturbed by the electr on–phonon interaction. We

248 8 Electron–Phonon Interaction

encounter a similar situation as for interacting electrons: As the electron–

electron interaction gives rise to formation of quasi-particles consisting of the

bare particle and a cloud of virtual collective excitations, so does the electron–

phonon interaction. B u t now, instead of charge or spin density excitations, the

electron (or hole) will be dressed by a cloud of virtual phonons. This composite

particle is called polaron with particular reference to the lattice deformation

in a polar semiconductor or insulator by a charge c arrier, which repels the ions

with the same charge but attracts those with the opposite charge thus gener-

ating a distortion typical for optical phonons. The interaction to be considered

in this case is the Fr¨ohlich coupling.

We follow here [

4, 14] and adopt the concept of lowest order perturbation

theory to calculate the change of the single-electron state |k and its energy by

the coupling with longitudinal optical phonons. We consider the state |k, 0



as the eigenstate of H

+ H

characterizing an electron in the undistorted

harmonic lattice. Due to electron–phonon interaction, this state gets modiﬁed

by contributions from states with phonons. The lowest order correction is due

to one-phonon contributions

|k, 0



(1)

= |k, 0

 +



|k − q, 1



k − q, 1

el−ph

|k, 0



− ǫ

k−q

− ¯hω

. (8.73)

The lattice distortion caused by the carrier can be quantiﬁed by the expec-

tation value of the phonon num ber operator which due to the ﬁrst order

correction is

¯n =



|k − q, 1

el−ph

|k, 0

|

(ǫ

− ǫ

k−q

− ¯hω

)

. (8.74)

This expression is still not speciﬁed to one of the coupling mechanisms and

yields diﬀerent results for the sum over q according to the q dependence of the

interaction matrix element. For the Fr¨ohlich interaction, we have ¯hω

→ ¯hω

and may write with

− ǫ

k−q

− ¯hω

¯h

∗



2k · q −q

− q



, (8.75)

where q

=2m

∗

/¯h,thesumoverq as an integral over the Brill ouin zone

¯n = |C|

∗

¯h

2π



(2k · q − q

− q

)

q, (8.76)

where |C|

= e

¯hω

/2Vε

(1/ε

∞

− 1/ε(0)). Let us consider the state with

k = 0, representing an electron at rest. Then the integral can b e performed in

spherical polar coordi n ates over a sphere with radius q

equal to the volume

of the Brillouin zone



...d

q =4π

∗

¯h



+ q

)

. (8.77)