Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

Vibrations

of a

Quantum—Mechanical

Lattice

353

|>.

Jï-R'

5t „-i*-J?l „,

;Λ

â. _

;Α

ηΜω

Ιν

î*

±=T\he

*+P\e-

^ ί

^*^

= -'ν-2

^^·

(13

43b)

*!/

From the commutation relation [Ρ', Λ']

—ifô, one finds for the creation and

annihilation operators the commutation relation

\ä

= \. (13.44)

In moving between Eqs. (13.42) and (13.43), the frequencies

UJ^

and the unit

vectors e-^ may in principle be chosen arbitrarily, just so long as the three unit

vectors e^ at every value of k are orthonormal. However, the only choice one

would sensibly make for these quantities is the one that diagonalizes the Hamil-

tonian (13.41), which means choosing

e-r.

to be the eigenvectors of the dynamical

matrix Φ and choosing

ω^

to be related to the eigenvalues by Eq. (13.18). Also

notice from Eq. (13.17) that Φ(ϋ) =

Φ(—k),

because one is related to the other just

by changing order of summation, and therefore

<*„

"-*»■

(13.45)

Making use of

Eq.

(13.45) gives

τ-^

hcor

[âr

+ âl âr 1

£_

y-_^

L ku

kv kv

kui

(1346a)

^ 4 Ì -[âr âr er ·€ r +4 âl S

· e*

] f

^°

ku y y ku

kv ku

-ku

ku ku ku -ku

)

,,, „

,—. Hutr

I [âr âl +âl âr

ίΰ

Φ"ΰ'=Υ—^<

» ku

ku ku\ i.(13.46b)

.„

4^ 4 |

+ [âr

âr

·? r

+ âl ô!

■

e* -

//'

*„ (

<■

ku ku

-ku

ku ku ku -ku'

)

In order to arrive at Eqs. (13.46), one employs the facts that eï

e-^, = δ

νν

' and that

~e_~

vanishes unless v = v'. Although one has the freedom in defining e^

to request that ë|

—

e_-

such a choice is very confusing for longitudinal modes,

where ë3

—e

is more natural, so no relation between ei and e

will be

-ku ku -kv

assumed.

Summing Eqs. (13.46a) and (13.46b) gives for the Hamiltonian of (13.41) that

A = E^M,

4ÂJ = EH,(«L^ +

5)·

UseE

>· (13.47)

For a lattice with a basis, Eq. (13.47) still holds true, but the summation ac-

quires an additional index corresponding to the branch of the phonon mode, which

ranges over the number of atoms in each unit

cell.

If the number of indices becomes

too cumbersome, one can adopt the abbreviated notation

5ΐ

2_^ ^W/(«

(

-+ 2)· The number operator

«,

=â/â;. (13.48)

where the sum over reciprocal vectors k, polarizations i/, and modes is subsumed

into the single index i.

354

Chapter 13. Phonons

Time Evolution. In the study

neutron scattering,

will also be necessary

know how the operators

and

evolve

time.

the Heisenberg picture,

the

annihilation operators evolve according to

{t) = ê™l

™l

(13.49)

so that

^Ι^

j™/*i[ÔÎ,

â^e-^'K/n

(13.50)

^à

(t)

(13.52)

As for the time evolution of û

, Eq. (13.43a) generalizes to

"'(0

Σ^/^'^'

ûle-

'^}.

(13.53)

13.3.1 Phonon Specific Heat

The specific heat of solids was one of the "19th century clouds" over 19th-century

physics described by Kelvin (1904). Solid objects were clearly built by connect-

ing small massive objects together with some form

spring. According

the

equipartition theorem

statistical mechanics, the specific heat must be

ke/2

per

degree

freedom. Each mass point must have three kinetic and three potential

degrees of freedom, and therefore a specific heat of Cy = 3Nke, the law of Dulong

and Petit. Not many data were available to test this law thoroughly, but by 1907,

specific heat had been measured down

around 50

K in

diamond. Instead

remaining constant, as shown in Figure 13.8,

declines precipitously at low tem-

peratures. For some reason, oscillators within the solid refused to take in energy at

low temperatures.

Einstein conceived the remarkable hypothesis that one might employ Planck's

radiation law, which stated that for black-body radiation the probability of exciting

a mode of frequency ω varied as

/(exp(hßu>)

—

1). This formula was still purely

empirical,

there was not

yet the possibility

deriving the specific heat of

solids from firm underlying principles. However, one could guess that

the solid

were built of N oscillators of frequency

ωο,

then the mean energy at temperature T

would be given by

3ΝΗωο

Ηβω

_ ]

(13.54)

d£

3N(hLü

)

Hßu}

o/{k

ßf

lv-

ighßwo

112

V=T^

lv= rSI TT5 ■

(13.55)

A particular strength

this formula was the fact that one was not condemned

fit

ωο from the specific heat data alone.

the solid were truly built

oscil-

lators concentrated

at a

certain frequency, then

should absorb radiation

this

Vibrations of a Quantum-Mechanical Lattice 355

frequency. Diamond was known to have a "residual ray," ( Section 22.3.1) at 11

μνη

ωο = 1.71 · 10

s"'

;

placing this value into Eq. (13.55) gave the theoretical

result shown in Figure 13.8. Despite the promise of this initial comparison, fur-

ther experimental measurements drew details of the calculation into question. The

theory dropped off far too fast at very low temperatures. The resolution of this

dif-

ficulty was soon found to lie in more accurate computations of vibrational spectra.

In order to explain how they entered, one must turn to the formal computation of

specific heats.

IO"

io-

Temperature T (K)

Figure 13.8. Specific heat formula Eq. (13.55) with

U>Q

= 1.71 · 10

s~' compared with

data for diamond available to Einstein (1907) and more recent data of Touloukian and

Buyco (1970b). At temperatures below 100 K, the Einstein formula falls below the data.

Debye's formula, Eq. (13.75), is a clear improvement.

In order to calculate the partition function for the Hamiltonian (13.48), one

must begin by identifying all its eigenstates. The states of a single harmonic os-

cillator are indexed by a single integer / ranging from zero to infinity, producing

energies

Ηω(1

+ |), and therefore the states of a large collection of harmonic os-

cillators are indexed by many integers

/,·,

each ranging from zero to infinity, and

producing energy

£ = ÇAU;

+ I). (13.56)

The partition function and ensuing thermodynamic quantities are therefore

oc oo

=Σ Σ···

βΣίΛωί{1ί+ι/2)

)

/l=0 /

'ΰ

356 Chapter 13. Phonons

OC I OO

Π Σ^^'

(/+

/2)

Π3.58)

ι=1

l/=0

= n(^

expW

} 03.59)

=>J=-yt

rinZ = ^ ^+Α:

Γ1η(1-ί>-^

') (13.60)

" ί' I

with 1

/ΐ

;

· = This expression defines the (13.62)

gßhwj _ ] Bose-Einstein factor m.

=►

v=§

iv =

Σ «^-

(13

63)

Phonon Density of States. Just as the density of electronic states plays a crucial

role in determining the thermal properties of the electron gas, similarly the density

of phonon states encodes the information needed to deduce thermal properties of

lattices. This density of states is defined by

' kv

and one may use it, for example, to write

f°° d Hui

= V duDUjj)- s= . (13.65)

,/o

' dT

β?

ηω

- 1

Before presenting the simple models frequently used to represent densities of

states and thereby extract thermodynamic quantities, it is best to present some ac-

tual data derived from experiment. Figure 13.9 presents representative data for

silicon.

Two qualitative lessons are apparent in these data.

The characteristic upper frequency at which the phonon modes terminate is

16 teraherz; this is the characteristic frequency of the optical modes and is the

upper limit of the acoustic modes.

The densities of states are littered with cusps. These cusps are precisely the

van Hove singularities discussed in Section 7.2.5. They arise at any point in

frequency space where one of the phonon modes passes through an extremal

value. In three-dimensional solids the result is a cusp, in two-dimensional

solids it is a logarithmic divergence, and for one-dimensional chains it is a

square root divergence.

Vibrations

of a Quantum-Mechanical Lattice 357

ΙΛ

4-H

*—>

Experiment

Model /

..-■■y

.A-^K/i'Si

4 8 12

Frequency ^ (THz)

Figure 13.9. Experimental data for the phonon density of states in silicon, compared with

results of model calculation that produced Figure 13.7. [Source: Dolling and Cowley

(1966),

p. 469.]

For temperatures kßT greater than all the energies /ϊω, of the vibrating lattice,

the specific heat takes the form

C-O = NflCR Take ^Ί- (13.62) for small β, and differentiate with respect to T. Nf gives M 3 foß\

number of terms in Eq. (13.63); for monatomic lattice in 3-d, Nf = 3/V.

This relation is the law of Dulong and Petit, a result of the classical theorem of

equipartition of

energy.

From a classical perspective, it is impossible to understand

obtaining any result for specific heat which differs from Eq. (13.66).

The quantum-mechanical expressions for specific heat do however, differ from

Eq. (13.66), and at low temperatures they differ substantially. Low temperatures

bring several simplifications. Any phonon mode whose energy

Ηω

is much greater

than kgT contributes nothing of note to the specific heat. At a temperature of 10

K, one needs therefore to focus upon frequencies of order 10 kß/h K=0.208 THz.

Sound speeds in solids are of order 1000 m s

-1

or more, so such frequencies corre-

spond to wavelengths of

Â, a distance comfortably larger than interatomic spac-

ing. The densities of states in Figure 13.9 are settling into the low-frequency limit

for such frequencies. Either argument leads to the conclusion that for temperatures

of order 10 K, it should be possible to use the long-wavelength, low-frequency

limit ω = ck for the dispersion relation of the phonons. In this limit, Eq. (13.64)

becomes

358 Chapter 13. Phonoris

3ω

2π

with

= V

d 3(k

dT2n

(cH)

V—k

Σ/

ί/Σ 1

dx-

The sum on u is over the three

acoustic modes, approximately /1

-3

f.n\

one longitudinal and two ^ ' '

transverse. However, the sound

speed depends generally upon

orientation; hence dependence

upon k.

The integral is a surface integral (13.68)

over orientations of k.

x =

β%ω.

(13.69)

The integral in Eq. ( 13.69) was (13.70)

evaluated in Eq. (6.67).

13.3.2 Einstein and Debye Models

Before the theory of lattice vibrations made detailed calculations of the density

of states D(uj) possible, simple guesses for the form of Ό{ω) were employed to

obtain an estimate for the consequence of quantum mechanics for specific heats.

To recover Einstein's theory of specific heats, take the density of states to be

3iV

D(u)) = δ(ίϋ —

U>o),

3iV

gives the total number of oscillatory modes

V in a solid responsible for the specific heat.

(13.71)

resulting immediately in Eq. (13.55).

The Debye model improves upon the Einstein model. Debye's calculation cor-

rectly reproduces Eq. (13.70) at low temperatures and correctly reproduces the law

0.5 1.0 1.5

Temperature Τ/θο

2.0

Figure 13.10. Specific heat in the Debye approximation, scaled by the prediction of Du-

long and Petit.

Vibrations of a Quantum-Mechanical Lattice

359

150

100

Sodium

o Silver

200

Figure

13.11.

The specific heats of sodium and silver are displayed, illustrating the validity

the

form (13.76). The intercepts of regression lines accurately reproduce the experimen-

tal values given in Table 6.2. [Source: Touloukian et al. (1975).]

of Dulong and Petit at high temperatures. The device to achieve this aim is to

approximate the density of states by

3a;

D(uj) = θ(ϋϋη — w) See Eq. (13.68). Use the low temperature

27r

form for the density of states everywhere, but

cut it off to give the right total number of

modes.

(13.72)

where the sharp cutoff at

UJD,

the Debye frequency, is chosen so that the total

number of modes equals the number of vibrational degrees of freedom:

roo

3N = V dujD{uj)^n-

67T

n is the number of ions per volume.

(13.73)

The characteristic Debye temperature Θο at which all phonons become thermally

active in this approximation is

®D = hu

The specific heat in this approximation is

T \

/-θο/Γ v-V

(13.74)

(13.75)

360 Chapter 13. Phonons

Table 13.1. Debye temperatures of elements

El.

C(gr)

C(dia)

121

227

433

282

162

111

1481

120

1480

412

2250

229

210

179

460

606

40.5

347

183

188

El.

118

All

325

373

182

122

34-108

252

190

109

112

420

91.1

71.9

150

344

183

403

409

423

El.

157

276

163

74.6

477

259

467

185

105

271

152

237

206

56.5

416

512

555

220

346

153

645

El.

Sm 169

Sn 199

Sr 147

Ta 245

Tb 176

Te 152

Th 160

Ti 420

TI 78.5

Tm 200

U 248

V 399

W 383

Xe 64.0

Y 248

Yb 118

Zn 329

Zr 290

Not available for all elements. Source: Stewart (1983).

and is plotted in Figures 13.8 and 13.10.

Because the low-temperature lattice contribution to the specific heat in

Eq. (13.70) varies as Γ

, it should disappear in metals beneath the electronic con-

tribution, which is linear in temperature. However, putting in numbers, one finds

that very low temperatures, on the order of a few degrees kelvin, are needed before

the electronic contribution stands out. The reason is that the natural temperature

scale for the electrons is the Fermi temperature, which is on the order of 20000

K, and the electronic contribution goes as T/Tf. However, the natural tempera-

ture scale for the phonons is the Debye temperature, given by Θο =

Hujo/kß-

The

contribution of the phonons to the specific heat goes as (Γ/θο)

. So the electron

and phonon contributions are roughly equal when T = θο

^Θο/Tf.

Debye tem-

peratures are tabulated in Table 13.1. Because they are typically on the order of a

few hundred degrees, the electronic specific heat has barely become visible even

at 10 K. Adding together electron and phonon contributions, the low temperature

specific heat at constant pressure

behaves quite accurately as

Cp ~ ^Τ + βΤ The constant 7 is the Sommerfeld parameter. ( 13.76)

Vibrations

of a Quantum-Mechanical Lattice 361

a claim best checked experimentally by plotting Cp/T versus T

, as in Figure

13.11.

Debye temperatures are usually 30% to 50% percent of the melting tempera-

ture of the element, so that by the time one gets to temperatures high enough to

see the fully classical specific heat of \kßT per degree of freedom, the harmonic

approximation for phonons is beginning to break down.

13.3.3 Thermal Expansion

Figure 13.12. Thermal expansion of

molecule.

The change of objects' size and shape as they heat and cool often poses an

unpleasant challenge for engineers. Road sections swell and buckle in the summer;

engine parts designed to work at high temperatures barely mesh when they are cool.

Yet it is a curious fact that the general framework allowing calculation of so many

other mechanical properties of condensed matter fails completely to predict the

possibility of thermal expansion. A solid whose energy changes only to quadratic

order when its atoms move does not change size or shape with temperature at all.

The reason for this somewhat unexpected result is most easily seen in the small-

est possible solid, one consisting only of two identical atoms (see Figure 13.12).

In a reference frame tied to their center of

mass,

the energy of the two atoms is

„ 1„ 2

Λ:

= «i

—

«2 is the difference of the deviation

~Ζ·^Χ · of the two atoms from equilibrium,

some (13. / /)

•^ spring constant.

In any thermal average over the locations of the atoms, positive and negative

values of x occur with equal frequency, and the mean distance between the two

atoms does not change no matter how much the molecule may be heated. If there

were any solid in which atomic interactions were really only present to quadratic

order, it would similarly refuse to change size in response to temperature.

Suppose now that the interaction energy of the molecule is a general function

1 -,

ε(χ) = ε

+ -Χχ

. . (13.78)

and ask under what conditions the mean size x of the molecule changes. The ther-

mal average of x is

f dxe-W*) dA V J

A can be viewed as a purely

formal quantity, but also

can be interpreted as an (13.79)

A=0 external force acting on the

system.

/ dxe-P^x) dA

In /

dxe

Ax-ßS.(xa)-ß£.'(xo)(x-xo)-ßZ"(xo)(x-xo)

/2\

A=Q

(13.80)

dÄ

362

Chapter 13. Phonons

According to the technique of steepest descents (Appendix B.4),

should be cho-

sen so that the linear term inside the exponential in Eq. (13.80) vanishes;

determined by

A = ß E '

(XQ

)

= ß%XQ Maximizing the integrand; A can be (13.81)

taken very small because it is on the

verge of being set to zero, and

should be viewed as a function of

./^Zl_

A*ö-/3£(*,)

ßV'ixo)

T d

% dxç,

kßT δω

In W

2π

ßx4/2-0£o

Λ=Ο

(13.82)

,ο=ο (13.83)

U=o

with μω

(χ) =

E"(x).

ß is the reduced mass of the (13.84)

OCui dx ' molecule

The possibility of thermal expansion therefore rests upon a nonzero third derivative

of the energy £ about equilibrium, or equivalently upon the change of vibrational

frequency ω if the molecule is forced to expand or contract.

General Theory. The general theory of thermal expansion begins with a collection

of thermodynamic identities. One wishes to calculate

„,~ „ ,„ I An identity from thermodynamics concern-

er) I

θΓ/ΟΙ\·γ

ing partial derivatives; the minus sign is cor- ,..„->

*ΡΤ = O-JÏ \p ilo /imi rect. Do not confuse the coefficient of volume (,lJ.oJj

Ol -Or/OV\T expansion/3

with

\/k

V d

= The bulk modulus B =

-VdP/dV.

(13 86)

so the formal aim can be achieved by calculating derivatives of the free energy

with respect to temperature and volume. Using Eq. (13.60) for the free energy of

a collection of interacting ions and using the definition of the Bose-Einstein factor

in Eq. (13.62), one has

5 v-^ dtii δΗω,

dVdT 4- dT OV

(13.88)

Comparing with Eq. (13.63) for the specific heat, define the Grüneisen parameter

'( ν

9Ηω

A/y.

_ The Griincisen constant is an average over the (13.89)

OYli volume rate of change of the frequencies of

3 7rC

the phonon modes, weighted by the contribu-

tion of each mode to the specific heat.

The denominator of Eq. (13.89) is just the (13.90)

d*V0T V specific heat; see Eq. (13.63).

JT C-

β^

_ΊΤ V

Combining Eqs. (13.90) and Eq.

(

13.86).

(13.91)