Lattice Dynamics

Physical properties of solids

determined by electronic structurerelated to movement of atomsabout their equilibrium positions

•Sound velocity •Thermal properties: -specific heat -thermal expansion -thermal conductivity (for semiconductors)

•Hardness of perfect single crystals (without defects)

Reminder to the physics of oscillations and waves:

Harmonic oscillator in classical mechanics:

Example: spring pendulum

Hooke’s law

2

2

1xDEpot

x

springFxm

Equation of motion:

0 xDxm or 0 x~m

Dx~

where ))t(x~Re()t(x

Solution with tieA~)t(x~

)tcos(A)t(x

where m

D

X=A sin ωt

X

Dx

m

D

Traveling plane waves: )kxt(cosA)t(y

X0

Y

X=0: tcosA)t(y

t=0: kxcosA)x(y

Particular state of oscillation Y=const

0 in particular

or )kxt(ieA~)t(y~

)kxt(cosA)t(y

travels according

0 .constdt

dkxt

dt

d

kvx

/2

2v

)kxt(ieA~)t(y~ 2

2

2

2

2

1

x

y

t

y

v

solves wave equation

Transverse wave

Longitudinal wave

Standing wave

)tkx(ieA~y~ 1

)tkx(ieA~y~ 2

)tkx(i)tkx(is eeA~y~y~y~ 21

titiikx eeeA~ tcoseA~ ikx 2

Re( ) 2 cos coss sy y A kx t

Large wavelength λ 02

k

Crystal can be viewed as a continuous medium: good for m810

λ>10-8m

10-10m

Speed of longitudinal wave:

sBv where Bs: bulk modulus with

compressibilityBs determines elastic deformation energy density 2

2

1 sBU

dilation V

V

(ignoring anisotropy of the crystal) sB

1

sB

v

E.g.: Steel

Bs=160 109N/m2

ρ=7860kg/m3 s

m

m/kg

m/Nv 4512

7860

101603

29

(click for details in thermodynamic context)

< interatomic spacing continuum approach fails

In addition: phononsvibrational modes quantized

Vibrational Modes of a Monatomic Lattice

Linear chain:

Remember: two coupled harmonic oscillators

Superposition of normal modes

Symmetric mode Anti-symmetric mode

generalization Infinite linear chain

How to derive the equation of motion in the harmonic approximation ?n n+1 n+2n-1n-2

un un+1 un+2 un-1un-2

un un+1 un+2 un-1un-2

fixed

D

1 nnln uuDF

1 nnrn uuDF

a

Total force driving atom n back to equilibrium

11 nnnnn uuDuuDF

n n nnn uuuD 211

equation of motion nn Fum

nnnn uuum

Du 211

Solution of continuous wave equation )tkx(ieAu

approach for linear chain )tkna(in eAu

)tkna(in eAu 2 ika)tkna(i

n eeAu 1 ika)tkna(i

n eeAu 1, ,

? Let us try!

22 ikaika eem

D kacosm

D 122

)/kasin(m

D22

)/kasin(m

D22

Continuum limit of acoustic waves:

m

D2

k

02

k

.../ka/kasin 22 kam

D a

m

Dv

k

Note: here pictures of transversal wavesalthough calculation for the longitudinal case

k

)t)k(nak(ieAnu

ahkk

2

)k()k(

)tnak(ieA

, here h=1

)tna)a

hk((ieA

2nhie)tnak(ieA 2 )tnak(ieA

12 nhie

))k(,k(nu))k(,k(nu

ahkk

2 1-dim. reciprocal

lattice vector Gh

ak

a

Region is called

first Brillouin zone

Brillouin zones

We saw: all required information contained in a particular volume in reciprocal space

first Brillouin zone 1d:a

xeannr xea

hhG

2

mnrhG 2 where m=hn integer

a

2

1st Brillouin zone

In general: first Brillouin zone Wigner-Seitz cell of the reciprocal lattice

Vibrational Spectrum for structures with 2 or more atoms/primitive basis

Linear diatomic chain:

2n 2n+1 2n+22n-12n-2

u2n u2n+1 u2n+2 u2n-1u2n-2

D a

2a

nununum

Dnu 2212122 Equation of motion for atoms on even positions:

Equation of motion for atoms on even positions: 12222212 nununuM

Dnu

)tkna(ieAnu 22Solution with:

)tka)n((ieBnu 12

12and

A)ikaeikae(B

m

DA 22

B)ikaeikae(A

M

DB 22

kacosBm

D

m

DA 222

kacosAM

D

M

DB 222

22

2

mD

kacosB

m

DA

kacosMm

D

M

D

m

D 22

42222

kacosMm

D

m

D

M

D

Mm

D 22

4422222

4

0212

4224

kacos

Mm

D

M

D

m

D

kasin2

Mm

kasin

MmD

MmD

24211112

1 12D

m M

22

M1

m1

DM1

m1

D

mD

2 , MD

2

mD

2

MD

2

2 2

•Click on the picture to start the animation M->m note wrong axis in the movie

:a

k2

Ato

mic

Dis

plac

emen

t

Optic Mode

M

mkA

B0

Ato

mic

Dis

plac

emen

t

Acoustic Mode10 kA

B

Click for animations

Dispersion curves of 3D crystals

•Every additional atom of the primitive basis

•3D crystal: clear separation into longitudinal and transverse mode only possible in particular symmetry directions

•Every crystal has 3 acoustic branches sound waves of elastic theory1 longitudinal

2 transverseacoustic

further 3 optical branches

again 2 transvers 1 longitudinal

p atoms/primitive unit cell ( primitive basis of p atoms):

3 acoustic branches + 3(p-1) optical branches = 3p branches

1LA +2TA (p-1)LO +2(p-1)TO

Intuitive picture: 1atom 3 translational degrees of freedom

3+3=6 degrees of freedom=3 translations+2rotations

+1vibraton

Solid: p N atoms

no translations, no rotations

3p N vibrations

x

yz

# of primitive unit cells

# atomsin primitivebasis

diamond lattice: fcc lattice with basis

(0,0,0)),,(4

1

4

1

4

1

Longitudinal Acoustic

Longitudinal Optical

Transversal Acoustic

degenerated

Part of the phonon dispersion relation of diamond

Transversal Opticaldegenerated

P=2

2x3=6 branches expected

2 fcc sublattices vibrate against one anotherHowever, identical atoms no dipole moment

Phonon spectroscopy

Inelastic interaction of light and particle waves with phonons

Constrains: conservation law of

momentum energy

Condition for elastic scattering

hklGkk 0

in

± q

incoming wave scattered wave

Reciprocal lattice

vector

phonon wave vector

hklGqkk 0

00 )q(

elastic sattering in

“quasimomentum”

02

20

2

2

22 )q(

nM

k

nM

k

for neutrons

for photonscattering

0

)q(0k

k

q

Triple axis neutron spectrometer

@ ILL in Grenoble, France

Lonely scientist in the reactor hall

Very expensive and involved experiments

Table top alternatives ?

Yes, infra-red absorption and inelastic light scattering (Raman and Brillouin)

However only 0q accessible

see homework #8