phonons pt1

22
Lattice Vibrations, Lattice Vibrations, Part I Part I Solid State Physics Solid State Physics 355 355

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Page 1: Phonons Pt1

Lattice Vibrations, Part ILattice Vibrations, Part I

Solid State PhysicsSolid State Physics355355

Page 2: Phonons Pt1

IntroductionIntroduction Unlike the Unlike the static lattice modelstatic lattice model, which deals , which deals

with average positions of atoms in a crystal, with average positions of atoms in a crystal, lattice dynamics lattice dynamics extends the concept of extends the concept of crystal lattice to an array of atoms with finite crystal lattice to an array of atoms with finite masses that are capable of motion.masses that are capable of motion.

This motion is not random but is a This motion is not random but is a superposition of vibrations of atoms around superposition of vibrations of atoms around their equilibrium sites due to interactions with their equilibrium sites due to interactions with neighboring atoms.neighboring atoms.

A collective vibration of atoms in the crystal A collective vibration of atoms in the crystal forms a wave of allowed forms a wave of allowed wavelengths wavelengths and and amplitudesamplitudes. .

Page 3: Phonons Pt1

ApplicationsApplications• Lattice contribution to specific heat• Lattice contribution to thermal conductivity• Elastic properties• Structural phase transitions• Particle Scattering Effects: electrons, photons, neutrons, etc.• BCS theory of superconductivity

Page 4: Phonons Pt1

Normal ModesNormal Modes

x1 x2 x3 x4 x5

u1 u2 u3 u4 u5

Page 5: Phonons Pt1

x1 x2 x3

u1 u2 u3

Consider this simplified system...

Suppose that only nearest-neighbor interactions are significant, then the force of atom 2 on atom 1 is proportional to the difference in the displacements of those atoms from their equilibrium positions.

)(and

)(

12112

21121

uuCF

uuCF

)(and

)(

32223

23232

uuCF

uuCF

Net Forces on these atoms...

)()()(

)(

2323

3221212

2111

uuCFuuCuuCF

uuCF

Page 6: Phonons Pt1

Normal ModesNormal Modes

)(

)()(

)(

23223

2

32212122

2

21121

2

uuCdt

udm

uuCuuCdt

udm

uuCdt

udm

Mr. Newton...

To find normal mode solutions, assume that each displacement has the samesinusoidal dependence in time.

tiii euu 0

Page 7: Phonons Pt1

0)(

0)(

0 )(

32

222

3222

2111

2112

1

umCuC

uCumCCuC

uCumC

Normal ModesNormal Modes

00

0

222

22

211

12

1

mCCCmCCC

CmC

0 3)(2 1212

21422 uCCmCCmm

Page 8: Phonons Pt1

Normal ModesNormal Modes

2/12/1

2122

21213

2/12/121

22

21212

1

)()(1

)()(10

CCCCCCm

CCCCCCm

Page 9: Phonons Pt1

Longitudinal Wave

q

Page 10: Phonons Pt1

Transverse Wave

q

Page 11: Phonons Pt1

22

1 12

1 1

( ) ( )

( 2 )

n n n n

n n n

d um C u u C u u

dtC u u u

tiii euu 0

1inqa iqa

nu ue e

2 ( 1) ( 1)

2

2cos

2

[ 2 ]

[ 2]

2 1 cos

inqa i n qa i n qa inqa

iqa iqa

qa

m ue C e e e

m C e e

C qam

Traveling wave solutions

Dispersion Relation

Page 12: Phonons Pt1

Dispersion RelationDispersion Relation

q

mC /4 0.6

qamC 2

1sin/4

Page 13: Phonons Pt1

First Brillouin ZoneFirst Brillouin ZoneWhat range of q’s is physically significant for elastic waves?

iqan ueu

1

iqainqa

qani

n

n eue

ueu

u

)1( 1

The range to + for the phase qa covers all possible values of the exponential. So, only values in the first Brillouin zone are significant.

Page 14: Phonons Pt1

First Brillouin ZoneFirst Brillouin Zone

There is no point in saying that two adjacent atoms are out of phase by more than . A relative phase of 1.2 is physically the same as a phase of 0.8 .

Page 15: Phonons Pt1

First Brillouin ZoneFirst Brillouin Zone

At the boundaries q = ± /a, the solution

Does not represent a traveling wave, but rather a standing wave. At the zone boundaries, we have

Alternate atoms oscillate in opposite phases and the wave can move neither left nor right.

inqan ueu

ninn ueu )1(

Page 16: Phonons Pt1

][sin 421 qa

mC

Group VelocityGroup Velocity

The transmission velocity of a wave packet is the group velocity, defined as

)q(or

qg

g

v

dqdv

]cos1[ 22 qamC

][cos 21

2qa

mCa

dqdvg

Page 17: Phonons Pt1

q

][cos 21

2qa

mCa

dqdvg

Group VelocityGroup Velocity

Page 18: Phonons Pt1

The The phase velocityphase velocity of a wave is the rate at which the phase of the of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. The phase velocity is would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector given in terms of the wave's angular frequency ω and wave vector kk by by

Note that the phase velocity is not necessarily the same as the Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in group velocity of the wave, which is the rate that changes in amplitude (known as the amplitude (known as the envelopeenvelope of the wave) will propagate. of the wave) will propagate.

Phase VelocityPhase Velocity

Pvk

Page 19: Phonons Pt1

Long Wavelength LimitLong Wavelength Limit

When qa << 1, we can expandso the dispersion relation becomes

The result is that the frequency is directly proportional to the wavevector in the long wavelength limit.

This means that the velocity of sound in the solid is independent of frequency.

221 )(1cos qaqa

22 ][ qamC

qv ω

Page 20: Phonons Pt1

Force ConstantsForce Constants]cos1[ 22 pqaC

mp

p

aC

dqrqapqaCdqrqamp

a

apa

a

2

)cos(]cos1[ 2 )cos(0

2

rqacosand integrate

The integral vanishes except for p = r. So, the force constant at range pa is

for a structure that has a monatomic basis.

a

ap dqpqaπ

maC

)cos(

22

Page 21: Phonons Pt1

Diatomic CoupledDiatomic CoupledHarmonic OscillatorsHarmonic Oscillators

q

)2(

)2(

12

2

2

12

2

1

nnnn

nnnn

vuuCdt

vdm

uvvCdt

udm

Page 22: Phonons Pt1

For each q value there are two values of ω.

These “branches” are referred to as “acoustic”and “optical” branches. Only one branchbehaves like sound waves ( ω/q → const. For q→0).For the optical branch, the atoms are oscillatingin antiphase. In an ionic crystal, these chargeoscillations (magnetic dipole moment) couple toelectromagnetic radiation (optical waves).

Definition: All branches that have a frequencyat q = 0 are optical.

Diatomic CoupledDiatomic CoupledHarmonic OscillatorsHarmonic Oscillators

qammmm

mmCmmmmC cos12

21

2

21

21

21

212

q