R?ssler U. Solid State Theory: An Introduction

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290 9 Defects, Disorder, and Localization

and we identify the dimensionless conductance g(L)=σL

d−2

,wherethe

conductivity σ characterizes the material property and does not depend on L.

Using this relation, the scaling argument (

9.96) can be discussed with respect

to its dependence on the system dimension d (see Problem 9.5). As it turns out,

the universal function β(g) passes through zero only for a three-dimensional

system. For a two -dimensional system, it approaches zero asymptotically for

large g and is always negative for d = 1. This means tha t a disorder-driven

metal–insulator transitio n should be possible only for d>2. Detailed investi-

gations of two-dimensional electron systems seem to disprove this conclusion

[

278]. The critical discussion in the light of these data leads to the result,

that possibly disorder is not the only mechanism but that electron–electron

interaction may contribute, which is not in clud ed in the scaling theory.

Problems

9.1 Verify the eigenvalue equation (

9.27) for the additional poles of the full

Green function G(E)(

9.20) by making use of the block-diagonal form of

the short-range defect matrix.

9.2 To demonstrate self-averaging, consider the classical resistance R =



i=1

of N resistances r

,i =1...N in a row. Make sure that the

average r exists and that R = Nr. Calculate the relative variance of

R and show that it tends to zero with increasing N or system size!

9.3 Calculate the current–current response χ(0,ω)of(

9.83) with the current

operator

of (

9.86) to obtain the conductivity σ(0,ω)(9.88).

9.4 Show that the double sum expression under the integral of σ

(

9.89)can

be cast into the form Tr(ˆvImG(E

)ˆvImG(E

)).

9.5 Discuss the asymptotics of β(g)forg →∞(using σ independent of L)

and for g → 0 (assuming g(L) ∼ exp (−L/λ)) in dependence on the sys-

tem dimension d. Show that β(g) < 0 for all g and d ≤ 2 while it changes

sign for d>2.

Light–Matter Interaction

The investigation of condensed matter systems using light as a probe is a ver-

satile and powerfull concept, which provides various kinds of information not

only about the structure but also of the electronic and lattice excitations and

their dynamics. Besides linear optical processes, li ke absorption and emission

(including photoemission), light-scattering, two- (or three-)photon absorption,

photo-luminescence and high-excitation spectroscopy also belong to the meth-

o ds to gain information about the single-particle and collective excitations of a

solid. In s emiconductor samples with a proper desig n , coherent light emission

is possible and used in solid-state lasers. These methods, w hi ch allow studies

in the frequency or time domain, are all based on the interaction of light with

matter. In Sect.

3.5 we have already treated the optical excitation of lattice

vibrations as an example. In this chapter, we shall present a systematic the-

oretical description of light–matter interaction, which can b e found in one or

the other representation in standard textbooks[

7, 14, 89, 95, 282, 283].

10.1 Preliminaries

Let us start with the microscopic Maxwell equations

∇·E =

, ∇×E = −

∂B

∂t

∇·B =0, ∇×B = μ

j +

∂E

∂t

. (10.1)

Here, E(r,t)andB(r,t) are the space and time dependent electric ﬁeld and

magnetic induction, respectively. The latter can be replaced with B = μ

by the magnetic ﬁeld H(r,t). ε

and μ

are the vacuum constants and

c =1/

√

is the vacuum velocity of light. In this microscopic formula-

tion, ρ(r,t)andj(r ,t) are the charge and current densities, respectively, of

James Clark Maxwell 1831–1879.

U. R¨ossler, Solid State Theory,

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

10,

 Springer-Verlag Berlin Heidelberg 2009

292 10 Light–Matter Interaction

all charged particles in the system (nuclei and electrons). We could adopt

here the rea sonable concept of the previous chapters, according to which we

understand the solid as composed of ions and valence electrons. In this case

ρ(r,t)andj(r,t) would be connected with the latter, while the electrons in

closed shells would belong to the ions, which essentially do not contribute to

the material pr operties of the solid. We come back to this point later in this

Chapter.

For what follows, instead of the ﬁelds E(r,t)andB(r,t), it is convenient

to introduce the potentials A(r,t)andφ(r,t), which are related to the ﬁelds

B = ∇×A , E = −∇φ −

∂A

∂t

. (10.2)

Replacing E and B in Maxwell’s equations by the potentials A and φ,we

may write (using ∇×∇×A = ∇(∇·A) −∆A) the inhomogeneous equations

in the form

∆A −

∂

∂t

−∇

∇·A +

∂φ

∂t

= −μ

j (10.3)

and

∆φ +

∂

∂t

(∇·A)=−

. (10.4)

The potentials are free to a gauge transformation with the scalar gauge ﬁeld

χ(r,t)

A → A + ∇χ, φ → φ −

∂χ

∂t

, (10.5)

which leaves (

10.2), i.e., the ﬁelds E and B, invariant.

Among the commonly used gauges, the transverse or Coulomb gauge, ∇·

A = 0, is the appropriate one for the interaction of light with matter in the

form of solids. With this choice, the third term on the lhs of (

10.3)vanishes

while (

10.4) becomes Poisson’s equation. Thus, we obt ain the potential form

of Maxwell’s equations in the form of an inhomogeneous wave equation for

the vector potential

∆A −

∂

∂t

= −μ

j. (10.6)

The inhomogeneity is connected with the nuclei and electrons (or ions and

valence electrons) and vanishes outside the solid, where (

10.6) describes the

free propagation of electro-magnetic waves in vacuum.

Consider now the solid interacting with the electro-magnetic ﬁeld of the

light. In the most general case, it is described by the Schroedinger equation

with the Hamiltonian of the solid (

2.1), including the electro-magnetic ﬁeld.

After having dealt already with the excitation of optical phonons, which are

connected with the moving ions (see Chap.

3.), we would like to consider here

only the electrons as movable charged particles in a rigid periodic arrangement

10.1 Preliminaries 293

of nuclei or ions. Thus, we restrict in this chapter to the interaction of light

with the electrons of the solid and start accordingly with the Hamiltonian

(

5.2) for the electrons. To include the electro-magnetic ﬁeld, we have to take

into account its energy and to fulﬁll the requirement, that the time-dependent

Schroedinger equation

=i¯h

∂Ψ

∂t

(10.7)

has to be invariant under the gauge transformation (

10.5), together with a

simultaneous gauge transformation of the time-dependent N-electron wave

function Ψ

...r

,t)

Ψ(r

...r

,t) → exp



−i

¯h

{χ(r

,t)+···+ χ(r

,t)}



Ψ(r

...r

,t) .

(10.8)

This is achieved by th e replacement

→ p

+ eA(r

,t) . (10.9)

For the rest of this chapter the following Hamiltonian will be the starting

point:



l=1

+ eA(r

))



k,l=1

k=l

4πε

− r



n,l

v(r

− R

)



V (R

− R





+ c



. (10.10)

In the last term, describing the energy of the radiation ﬁeld, E and B can b e

replaced with (

10.2) by the vector potential to yield

rad



∂

∂t

A(r,t)

+ c

(∇×A(r,t))

. (10.11)

Having in mind linear response theory (Sect.

2.4), we may separate H

into

a Hamiltonian for the uncoupled solid and electro-magnetic ﬁeld, H

+ H

rad

and an interaction term H

el−rad

for the light–matter coupling

= H

+ H

el−rad

+ H

rad

, (10.12)

where H

describes the unperturbed electron system of the solid, (

5.2), and

(by making use of the transverse gauge)

el−rad



l=1

A(r

,t) · p

(A(r

,t))

(10.13)

294 10 Light–Matter Interaction

is the time-dependent perturbation by the radiation ﬁeld, which can be id en-

tiﬁed as the external potential of (

2.37). Using j = −e



/m, the ﬁrst term

can also be written as −j · A (see Sects.

2.4 and 9.4).

10.2 Single-Particle Approximation

In Chap.

5 we have invoked density-functional theory (DFT) to reduce the

many-body problem with H

to a single-particle problem, by accounting for

the electron–electron interaction in an eﬀective s i n gle-particle potential V

eﬀ

(r)

(see (

5.30)), which is the same for all electrons. This can now be used to write

the many-body Hamiltonian H

as a sum of single-particle terms. Leaving

aside the energy of the radiation ﬁeld as a constant, the single-particle Hamil-

tonian of an electron interacting with the radiation ﬁeld takes the form

H = −

¯h

∆ + V

eﬀ

(r)+

A · p +

. (10.14)

The eigenstates of the ﬁrst term are the Bloch states of Chap.

5.Thelasttwo

terms represent a p eriodic time-dependent interaction o f Bloch electrons with

the radiation ﬁeld. For suﬃciently weak amplitudes of the vector potential

they can be treated as perturbations of ﬁrst an d second order. Sometimes

the Ap -term is also expressed in the form −eE · r,where−er is the dipole

operator (Problem 10.1).

LetusﬁrsttreatthetermlinearinA. For monochromatic light, its space

and time dependence is given by

A(r,t)=A

e e

i(κ·r−ωt)

+ c.c. , (10.15)

with a scalar amplitude A

and the unit vector e for transverse polarization,

e ⊥ κ. The frequency follows the linear dispersion relation ω = c|κ| of light

in vacuum.

Because of the periodic time-dependence of the perturbation, Fermi’s

Golden Rule applies and we can immediately write the rate for the transition

probability between Bloch states

nk,n

′

2π

¯h

|nk|H

′

|

δ (E

(k) − E

′

) − ¯hω) , (10.16)

where H

′

= eA

exp(iκ · r)e · p/m is the perturbation without the time-

dependent exponential. Together with the time-dependent pha se factors of the

stationary Bloch functions, it results in the δ-function, which expresses the

energy conservation: the energy quantum ¯hω of the radiation ﬁeld equals

the energy diﬀerence between the Bloch states. Depending on the sign of this

diﬀerence, the transition between the Bloch states can be an absorption or

emission process.

10.2 Single-Particle Approximation 295

The transition matrix element can be transformed from an integral over

the crystal volume V

nk|H

′

 =



∗

(r)



iκ·r

e · p



′

(r)d

r (10.17)

to an integral over a Wigner-Seitz cell by shifting r by lattice vectors R

and

extracting the corresponding phase factors from the Bloch functions

nk|H

′

 =



WSC

...



i(κ+k

′

−k)·R

r . (10.18)

Here the dots stand for the integrand of the previous integral. The lattice sum

results in a Kronecker δ

k,k

′

+κ

, which represents the momentum conservation

of the excitation process. The diameter of the Wigner–Seitz cell is about the

lattice constant of a few

A, which is much smaller than the wavelength of the

light corresp onding to the typical excitation energy ¯hω of a few eV . Therefore,

the exponential with κ in the integral is almost constant over the cell and

we may safely replace it by 1. This is the dipole approximation. Finally, the

transition matrix element can be written (see also Problem 10.2)

nk|H

′

 =



WSC

∗

(r)



e · p



′

(r)d

r δ

k,k

′

nk|e · p|n

′

kδ

k,k

′

. (10.19)

Thus, one obtains a ﬁnite transition probability

nk,n

′

(ω)=

2π

¯h

|nk|e ·p|n

′

k|

δ (E

(k) − E

′

(k) − ¯hω) (10.20)

only for dire ct transitions (or vertical transitions), in which the wave vector

of the Bloch state is not changed.

The absorption of light passing through a solid o f thickness d determines

the damping of the light intensity according to

I(d)=I

−αd

. (10.21)

On t he other side, the absorption coeﬃcient α(ω) is connected with the real

) and imaginary (n

) parts of the complex index of refraction accordin g to

α =

ω, (10.22)

where 2n

= ε

is the imaginary part of the complex dielectric function

ε(ω). Microscopically, α(ω) is determined here by the power loss or the rate

by which energy of the radiation ﬁeld is converted into excitations between

296 10 Light–Matter Interaction

Bloch states[

14, 89]. Thus,

α(ω)=

2¯h

cωA

W (ω) , where W (ω)=



′

nk,n

′

(ω), (10.23)

and the imaginary part of the dielectric function is given by

(ω)=

4πe



′

|nk|e · p|n

′

k|

δ (E

(k) − E

′

(k) − ¯hω) . (10.24)

The real part of ε

(ω) can be obtained using the Kramers–Kronig relation

(

2.80) formulated for susceptibilities χ(ω). Con sidering the relation ε(ω)=

1+χ(ω)/ε

,wehavetouse

(ω)=1+



∞

′

(ω

′

)

′

− ω

dω

′

, (10.25)

leading to

(ω)=1+



′

(k)

′

(k)

− ω

(10.26)

with the oscillator str ength

′

(k)=

2 |nk|e · p|n

′

k|

m¯hω

′

(k)

(10.27)

for the transition between Bloch states with the energy diﬀerence ¯hω

′

(k)=

(k) − E

′

(k).

In cubic solids, the dipole matrix elements do not depend on the polariza-

tion dir ection e and usually their dependence on the wave vector is not very

strong. Thus, we can write

nk|e · p|n

′

k≃P

′

(10.28)

with P

′

independent of k and ε

(ω) becomes

(ω)=

4π



mω





′

(¯hω) , (10.29)

with the combined density of states

′

(¯hω)=



δ (E

(k) − E

′

(k) − ¯hω) (10.30)

10.2 Single-Particle Approximation 297

Fig. 10.1. (a) Band structure of Ge with vertical transitions at critical points and

(b) comparison of measured and calculated ε

(ω), showing van Hove singularities

at these critical points (from Landolt-B¨ornstein [1])

of the pair of bands n, n

′

. As in Sect.

3.3,thek-sum over the Brillouin zone

can be converted into an integral

′

(¯hω)=

(2π)



δ (E

(k) − E

′

(k) − ¯hω)d

k , (10.31)

which, due to the δ-function, reduces to an integral over the surface S

,for

which the energy diﬀerence of the Bloch states equa l s ¯hω

′

(¯hω)=

(2π)



|∇

(k) − E

(k))|

. (10.32)

The points, at which |∇

(k) − E

(k))| vanishes (critical p oints), give rise

to van Hove singularities in the combined density of states, by which they can

be identiﬁed.

As an example, we show in Fig.

10.1 the band structure of Ge in the funda-

mental band gap region with marked direct transitions at critical p oints and

the measured and calculated imaginary part of the dielectric function, ε

(ω).

In spite of discrepancies in the critical point energies, there is a clear corre-

spondence between the experimental and theoretical curves. It turns out, that

the optical spectra of semiconducto rs with tetrahedral coordination resemble

each other [

89]. Their band structures have the same topology determined by

the common crystal structure but diﬀer with respect to their critical point

energies E

′

,.... Investigations of the op t i cal spectra have p aved the road

to the quantitative understanding of semiconductor band structures.

Light–matter interaction stood at the beginning of quantum mechanics

with Einstein’s explanation of the photo-electric eﬀect, which earned him the

Nobel prize in 1922. It is based on the assumption of a quantized radiation

ﬁeld. Photoemission can take place if the photon energy suﬃces to excite an

298 10 Light–Matter Interaction

electron beyond the energy barrier at the surface of the solid so that it can

be detected ou tside the solid as free particle. Einstein found (in accordance

with experimental data for metals) the linear dependence of the maximum

kinetic energy of the emitted electrons on the photon energy, which is ruled

by the ratio ¯h/e. Later, this eﬀect has been exploited by Siegbahn

to develop

the concept of photo-electron spectr oscopy (PES). It is based on analyzing not

only the maximum energy but also the energy spectrum of the photo-emitted

electrons.

The minimum energy required to free an electron from a metal, is the

diﬀerence between the vacuum level and the Fermi energy, known as the work

function. In a semiconductor, with the Fermi energy somewhere in the gap,

this energy would not be suﬃcient for photoemission, because the highest

occupied state, in the intrinsic case, is the top of the valence band. Instead

it is the ionization energy that deﬁnes the threshold for photoemission. The

photoemissive yield is roughly proportional to the density of the initial states.

PES is applied at diﬀerent photon energies: ultraviolet light (UPS: ultraviolet

PES) for the investigation of valence electron structures and X-rays (XPS)

for the study of the more tightly bound core electrons[

65, 287]. Note, that

for the latter case one has to treat the core electrons in the same way as the

valence electrons.

The photo-emitted electrons can be also analy zed with respect to their

momentum parallel to the surface, which does not change when the electron

leaves the solid. This is done in angular resolved photoemission (ARPES)

experiments, which give information about the energy bands E

(k), as shown

in Fig.

10.2 for Cu as an example. The experimental data points map out all

details of the energy bands, which we already know from Chap.

Further variants of photo-electron spectroscopy are spin-polarized UPS

(SPUPS) and the inverse photoemission, in which an electron of known energy

is injected and the emitted photon is detected. The former method allows to

study the bands of minority and majority spins in ferromagnets (see Fig.

6.1),

and the latter yields information about the unoccupied states above the Fermi

energy.

10.3 Excitons

The basic assumption of the single-particle approximation in Sect.

10.2 is the

same eﬀective single-particle potential V

eﬀ

(r) for electrons in all ban d s. This

assumption is correct for the ground state of the solid, which, for a semicon-

ductor, is characterized by ﬁlled valence and empty conduction bands. But

for the excited state an electron in the valence band is missing, because it

is in the conduction band. Intuitively, this is the two-particle problem of an

Kai Manne B¨orje Siegbahn 1918–2007, Nobel prize in Physics 1981, together with

N. Bloemb ergen and A.L. Schawlow.

10.3 Excitons 299

wave vector

energy (eV)

Fig. 10.2. Calculated energy bands of Cu and data from angular resolved

photoemission, after [

31]

electron–hole pair, if we consider the remaining N −1 electrons in the valence

band as the hole and the missing contribution to the eﬀective potential as

the attractive Coulomb interaction between the electron in the conduction

band and the hole in the valence band. In eﬀective-mass approximation, the

Hamitonian for the electron-hole pair can be written

e−h

= E

−

κ|r

− r

. (10.33)

Note, that the kinetic e nergy of the hole increases in the downward bent

parabola for the valence band. If we only had the two bands (one valence

and one conduction band) in vacuum, then κ =4πε

. However, the correct

description has to include all the other valence and conduction bands and the

possible excitations between these bands, i.e., the two -band model for electron

and hole is embedded in a dielectric with material constant ε and, therefore,

κ =4πε

ε, i.e., the Coulomb interaction is screened. We would like to mention

here, that the electron–hole interaction is just the diﬀerence of the Hartree

potentials in the excited and ground states, but there should also be exchange

and correlation contributions, which we leave aside for the moment.

In relative and center-of-mass co ordinates, using for p

=¯h k

,i= e, h,

= k +

Q , k

= k −

Q , (10.34)