Marder M.P. Condensed Matter Physics

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Point Defects and Color Centers

683

oscillator wavefunction,

>D,

SeeSchiff(1968),p.67. (22.71)

(22.72)

hMw?

This limit corresponds to weak coupling of the ion to the center, or high

LO\,

which

means that the ion can respond rapidly to the photon. The functions in Eq. (22.70)

overlap best when n = 0, which means that the dominant transition is one that goes

straight from the ground state at A to the lowest excited state C in Figure 22.12.

The opposite limit is

2)I>JCO, (22.73)

which corresponds to weak coupling or to slowly responding ions. Now the max-

imum overlap in Eq. (22.70) occurs for a large value of

The integral appearing

there can be evaluated exactly, and defining x =

X/XQ

gives

dxMx)Mx

V\)

(

)

Use

the standard

, /

gX£>iAo-(I>i/*o)

/2('_ \\

-—

ressions

for

harmonic (22 75)

Tr2.

oscillator wave

^ ' '

functions—for

example,

Schiff

(1968) p.

71,

Landau

and Lifshitz

(1977),

p. 70.

/ dx \ (—-Ye^'/^'^/^^e"*

Integrating by parts n times.(22.76)

J y

7r2"rc!

\/^(-)

V(1V

)2/4

y 2

xo '

As a function of

(22.77) is largest when

« = i(3V*o)

(

)

Just as drawn in the path from A to B in Figure 22.12, the most likely final energy

of the system is a quadratic function of the displacement coordinate T)\. If the

excited state is long-lived, the ion will slowly relax down to C, and emission will

take place from C to D.

22.4.6 Urbach Tails

If the optical gap of a solid is 8,

, then one should expect no measurable absorp-

tion for incoming light with

hco

< £.

. However, some absorption does occur, and

in a wide range of solids including ionic crystals, insulators with F centers, and

amorphous semiconductors, optical absorption takes the form

a ocexp

(22.79)

684

Chapter 22. Optical Properties of Insulators

(/)

10"

200 220 240 260 280 300 320 340

Wavelength (fim)

Figure 22.13. Attenuation constant versus wavelength for radiation incident upon a KI

crystal at a variety of temperatures. The range of exponential decay spans six decades.

[Source: Haupt (1959), p. 239.]

In the range where Eq. (22.79) is valid, a can vary by six orders of magnitude

or more, as first found by Urbach (1953). Data from high-quality KI crystals are

displayed in Figure

22.13.

Problems

Clausius-Mossotti relation

(a) Using elementary electrostatics, verify Eq. (22.3).

(b) Verify that (22.4) vanishes for dipoles arranged in any structure with cubic

symmetry.

Faraday rotation: In a Faraday rotation experiment, infrared radiation passes

through a thin sample, along the direction of an applied magnetic field. If the

light is linearly polarized, then the angle of polarization rotates. The Faraday

rotation 8 per unit length along the sample is defined by

- l

= 2c

—

n+]

(22.80)

where n+ and n_ are the indices of refraction for right and left circularly

polarized light.

(a) Consider a population of electrons obeying

r+-

— rxB

-Ee

rrr

(22.81)

Take the magnetic field to point along z. Find the current, and therefore the

conductivities a

, for right- and left-polarized fields E.

References 685

(b) Assume that the frequency

is large, which means

LOT

^> 1

and eB/m*c

UJ.

Expand the conductivities to leading order in

and

jr.

same quantities. Assume

^> u;

, and find an expression for 6.

Polaron decay: Assume that the energy h

/2m* of an electron in a po-

lar solid is greater than the energy

HLOL

of a longitudinal optical phonon.

Employing Fermi's Golden Rule, Eq. (13.99), show that the rate at which

an electron of wave number q decays is given by Eq. (22.57). Perform the

angular integral over cos 9 before the momentum integral, and recall that

cosh"

x =

In [x

± \fx

— 1

Franck-Condon effect:

Consider an F center sitting in the excited state at position C in Figure 22.12,

described by the Hamiltonian

3<F

+ 3iion+3^int, (22.82)

where the three portions of the Hamiltonian are described by Eqs. (22.59),

(22.60),

and (22.61).

In the case where the distance

Mcof

is large compared to the quantum zero-point fluctuations of the ion coordi-

nate jtn, find the quantum number n characterizing the most likely final state

indicated in Figure 22.12 by position D.

Urbach tails: Theoretical calculations tend not to reproduce Eq. (22.79)

exactly, but to find functional forms that fall off more rapidly as

HLO

falls below

. As an illustration, one can use the configuration coordinate model for the

Franck-Condon effect.

(a) Write down Eq. (20.70) for the real part of the conductivity, and relate it to

the optical absorption.

(b) Take the initial state to be the state of energy £o,„ from Eq. (22.67), occupied

with probability /„ = exp[—/?£o,

], and take the final state to be the one with

energy

S^Q:

The electron is excited to state |Fi), but the ionic coordinate is

relaxed. Rewrite Eq. (20.70), assuming that

n\P\Fi, 0) = <F

|P|fi)(Fo, n\F

0). (22.84)

the temperature dependence of the result is correct, but that the absorption

falls off too quickly as a function of £

—

HLO.

686

Chapter 22. Optical Properties of Insulators

References

W. S. Baer (1967), Faraday rotation in ZnO: Determination of the electron effective mass., Physical

Review, 154, 785-789.

M. Balkanski (1972), Photon-phonon interactions in solids, in Optical Properties of Solids,

Abelès,

ed., pp. 529-652, North-Holland, Amsterdam.

Bendow (1978), Multiphonon infrared absorption in the highly transparent frequency regime of

solids, Solid State Physics: Advances in Research and Applications, 33, 249-316.

Berlincourt and H. H. A. Krueger (1959), Domain processes in lead titanate zirconate and barium

titanate ceramics, Journal of Applied Physics, 30, 1804-1810.

A. Einstein (1907), Planck's theory of radiation and the theory of specific heat, Annalen der Physik

(Leipzig),

22, 180-190. In German.

A. Einstein (1911), A relation between the elastic constants and specific heat of solid bodies made

from atoms of a single type, Annalen der Physik (Leipzig), pp. 170-174. In German.

Emin (1982), Small polarons, Physics

Today,

35(6), 34-40.

G. Feher (1959), Observation of nuclear magnetic resonances via the electron spin resonance line,

Physical Review, 103, 834-835.

R. P. Feynman (1972), Statistical Mechanics, Addison-Wesley, Redwood City, CA.

C. P. Flynn (1972), Point Defects and Diffusion, Clarendon Press, Oxford.

W. B. Fowler, ed. (1968), Physics of Color Centers, Academic Press, New York.

H. Fröhlich (1952), Interaction of electrons with lattice vibrations, Proceedings of the Royal Society

of London, A215, 291-298.

Gerlach and H. Lowen (1991), Analytical properties of polaron systems or: Do polaronic phase

transitions exist or not?, Reviews of Modern Physics, 63, 63-90.

U. Haupt (1959), On the temperature dependence and form of the long wavelength excitation band

in KI crystals, Zeitschrift für

Physik,

157, 232-246.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su (1988), Solitons in conducting polymers,

Reviews of Modern Physics, 60, 781-850.

J. W. Hodby (1971), Cyclotron resonance of the polaron in the alkali and silver halides. iii, Journal

of Physics C, 4,

L8-L11.

E. Kartheuser

(

1972),

Dielectric properties of polar crystals, in Polarons in Ionic Crystals and Polar

Semiconductors, J. T. Devreese, ed., North Holland, Amsterdam.

R. D. Kingsmith and D. Vanderbilt (1993), Theory of Polarization of Crystalline Solids, Physical

Review B, 47(3), 1651-1654.

L. D. Landau and E. M. Lifshitz (1977), Quantum Mechanics (Non-relativistic Theory), 3rd ed.,

Pergamon Press, Oxford.

H. Landolt and R. Börnstein (New Series), Numerical Data and Functional Relationships in Science

and

Technology,

New Series, Group III, Springer-Verlag, Berlin.

M. E. Lines and A. M. Glass (1977), Principles and Applications of Ferroelectrics and Related

Materials, Clarendon Press, Oxford.

Liity (1961), Electronic transitions in color centers, Halbleiter

Probleme,

6, 238-278.

E. Mollwo (1933), On the color centers of alkali halide crystals, Zeitschrift für

Physik,

85, 56-67. In

German.

F. Mott and R. W. Gurney

(

1940),

Electronic Processes in Ionic Crystals, Oxford University Press,

Oxford.

M. Peeters and J. T. Devreese (1984), Theory of polaron mobility, Solid State Physics: Advances

in Research and Applications, 38, 81-133.

H. Pick (1972), Structure of trapped electron and trapped hole centers in alkali halides: "Color

centers", in Optical Properties of Solids,

Abelès, ed., pp. 654-747, North-Holland, Amsterdam.

R. Resta (1994), Macroscopic polarization in crystalline dielectrics: The geometric phase approach,

References 687

Reviews of Modern Physics, 66, 899-915.

R. Resta, M. Posternak, and A. Baldereschi (1993), Towards a quantum theory of polarization in

ferroelectrics: The case of knbo3, Physical Review Letters, 70, 1010-1013.

W. C. Röntgen (1921), On the electrical conductivity of single crystals, and on the influence of

radiation, Annalen der

Physik,

64,

1-195.

In German.

G. Sâghi-Szabô, R. E. Cohen, and H. Krakauer (1999), First-principles study of piezoelectricity in

tetragonal pbtio3 and pbzr\/2ti\/2o3, Physical Review B, 59(20),

771-12776.

G. A. Samara and P. S. Peercy (1981), The study of soft-mode transitions at high pressure, Solid

State Physics: Advances in Research and Applications, 36,

1-118.

L. Schiff (1968), Quantum Mechanics, 3rd ed., McGraw-Hill, New York.

H. Seidel and H. C. Wolf (1968), ESR and ENDOR spectroscopy of color centers in alkali halide

crystals, in Physics of Color Centers, W. B. Fowler, ed., pp. 537-624, Academic Press, New York.

Urbach (1953), The long-wavelength edge of photographic sensitivity and the electronic absorp-

tion of

solids,

Physical Review, 92, 1324.

E. VugmeisterandM. D. Glinchuk(1990), Dipole glass and ferroelectricity in random-site electric

dipole systems, Reviews of Modern Physics, 62, 993-1026.

R. Wahl, D. Vogtenhuber, and G. Kresse (2008), Srtio[sub 3] and batio[sub 3] revisited using the

projector augmented wave method: Performance of hybrid and semilocal functionals, Physical

Review B, 78, 104116.

I. S. Zheludev (1971), Ferroelectricity and symmetry, Solid State Physics: Advances in Research and

Applications, 26, 429^164.

23.

Optical Properties of

Metals

and

Inelastic Scattering

23.1 Introduction

The interaction of electromagnetic radiation with metals divides between low and

high frequencies. Low frequency refers to frequencies too small to induce transi-

tions from one Bloch band to another. The energies involved in hopping between

bands are characteristically around

eV, so the dividing line is around

1 eV^>u;~ 10'

Hz^A~l /an. (23.1)

For frequencies below this point, the absence of transitions between bands means

that the behavior of electrons is semiclassical and can be treated by the formalism

of Section 17.2. This is not to say that there are no quantum-mechanical effects; the

response of electrons even to static electric fields has many quantum-mechanical

features, but classical notions of electrons wiggling back and forth in response to

oscillating fields provide the right starting point. At frequencies above the cutoff

(23.1),

the language and mode of thought in discussing the problem begin to alter.

Although the incoming light will still be treated classically, and there is no need

to quantize the electromagnetic field, the effects of the light waves are most easily

understood as a beam of photons whose absorption is able to induce transitions

between various quantum mechanical states. For energies of 10-100 eV, light has

become the most powerful probe of electronic structure in metals and semicon-

ductors, making possible in some cases a direct experimental investigation of the

one-electron picture, including quantitative measurements of the energy bands.

23.1.1 Plasma Frequency

In the Drude model, a metal consists of a gas of noninteracting mobile electrons.

The response of such electrons to external electric fields was described in

Eqs.

(20.4)

and (20.31), which showed that

e(uj) = 1 To compare v

LO(cj + Î/T) often must be

p=M

/^ = 5.64.10

689

ith experiment, the constant 1

eplaced by e°°.

1/2

cm-

(23.2)

(23.3)

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

690 Chapter 23. Optical Properties of Metals and Inelastic Scattering

Charge motion

-^ ►

\A-

Figure

23.1.

Plasma oscillations result from the bulk motion of charge in a solid.

The definition of the plasma frequency u

is motivated by the limit

» 1,

which leads to

(23.4)

««> =

'-(Ï)'

It has a very simple physical interpretation, illustrated in Figure

23.1.

Consider

a slab of electrons sitting on top a slab of neutralizing positive charge. If the whole

slab of electrons is moved a distance 8, then positive charge develops on one side

of the slab, negative charge on the other. The charge, say, on the positive side is

enA8, where A is the area of the slab, and n the density of charged particles. A

uniform electric field E = Airenö develops between two such slabs of charge. If the

total volume of the system is V, then the force on all of the electrons is

enVE = 4Trn

VS.

(23.5)

Setting this force equal to mass, mnV, times acceleration 8 gives

•• 4-7rae

o = o.

(23.6)

So the frequency of plasma oscillations is

A-Kne

w„

This result is correct only for free electrons. In the

presence of a periodic potential, the plasma fre-

quency is modified. Conventionally, the electron

mass m is replaced by an effective mass called the

optical mass; see Eq. (23.27).

(23.7)

Introduction

691

0.0 0.5 1.0 1.5 2.0

Frequency

LÜ/U>

Figure 23.2. Index of refraction n and extinction coefficient

for metal obeying Eq. (23.2),

with

ujpT

= 100, showing absorbing, reflecting, and transmitting frequency ranges.

Three Regimes for Metallic Response. The product LÜ

T can be estimated from

Eqs.

(16.7) and (23.3) and can range from 10 to 100. Therefore, according to

Eq. (23.2), the optical absorption of metals should pass through three phases,

shown in Figure 23.2.

Absorbing: For 0 <

LOT

< 1 the metal absorbs the incident radiation, because

e «

+ /r-^

( 1

/WT)

^n^Kïx

JTLOJ/2LU.

(23.8)

The index of refraction n and extinction coefficient

were defined in Eqs. (20.18)

and (20.20).

Reflecting: For 1 < LOT < LO

T the real part of e is negative, equaling approxi-

mately

=-^- (23.9)

LÜ

and resulting in

692 Chapter

23.

Optical Properties of Metals and Inelastic Scattering

According to Eq. (20.43), because the index of refraction n is small in com-

parison with the extinction coefficient K, almost all radiation incident upon

the metal in this range of frequencies will be reflected.

Transparent: For

the real part of the dielectric constant first vanishes and

then becomes positive as

increases. The metal becomes nearly transparent

to the incoming radiation, with index of refraction and extinction coefficient

given by

l-u^/uP At«

(23.11)

2TLO

IUP — uP

23.2 Metals at Low Frequencies

When the frequency of radiation is low enough that the semiclassical approach is

valid, a restriction that was the subject of Problem 20.2, the population of electrons

is described by the Boltzmann equation (Section 17.2), which, in the absence of

thermal gradients reads

dg - dg g

— = -V-Vg-eE -V- . Usedf/dk = d£/dkdf/8EN = -Hvdf/dfj,, (23.12)

Ot Oll T starting withEq. (17.10) in the relaxation time

approximation.

Suppose the electric field to have the form

Ê = E(q, *)<&*-**. (23.13)

Then Eq. (23.12) is easily solved by taking

gjf(<7,

Uj)e

r lwt

The label k describes the Bloch index of an (23.14)

electron, while the label q describes the wave

vector of the electromagnetic wave.

so that

- df

8k(<i,

üj)[-iuj] = \-iv-q- l/r]gj(^, u)-eE-v— (23.15)

i-> ^f E(q,uj)-v

=^ gr{q, LO)

=—e————.

-: _ _. . Remember that v is a function of k (23.16)

Oß\/T — l[fjJ — q-v) according to Eq. (7.59).

The current associated with the distribution function (23.16) is determined by an

integral over k, and from it can be obtained the conductivity a through

j = —e [dk]vg-£

[dk]

defined in Eq. (6.15). (23.17)

f[dk]^J

[

(23.18)

\/T

—

i{uj

—

q-v)