1 Crystal Lattice Vibrations: Phonons Introduction to Solid State Physics...

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Crystal Lattice Vibrations: Phonons

Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html

PHYS 624: Crystal Lattice Vibrations: Phonons 2

Lattice dynamics above T=0

•Crystal lattices at zero temperature posses long range order – translational symmetry (e.g., generates sharp diffraction pattern, Bloch states, …).

•At T>0 ions vibrate with an amplitude that depends on temperature – because of lattice symmetries, thermal vibrations can be analyzed in terms of collective motion of ions which can be populated and excited just like electrons – unlike electrons, phonons are bosons (no Pauli principle, phonon number is not conserved). Thermal lattice vibrations are responsible for:

→ Thermal conductivity of insulators is due to dispersive lattice vibrations (e.g., thermal conductivity of diamond is 6 times larger than that of metallic copper). → They reduce intensities of diffraction spots and allow for inellastic scattering where the energy of the scatter (e.g., neutron) changes due to absorption or creation of a phonon in the target. → Electron-phonon interactions renormalize the properties of electrons (electrons become heavier). → Superconductivity (conventional BCS) arises from multiple electron-phonon scattering between time-reversed electrons.


Vibrations of small amplitude: 1D chain

4 1 1 4

21 1 2 2

22 2 3 3

4 3 3 4

1

2

3

4

0

0

0

0

2,

K K K K

K K K Kd UU

K K K Kdt

K K K K

u

u VU U U

u m

u

K

Classical Theory: Normal Modes

Quantum Theory: Linear Harmonic Oscillator for each Normal Mode

3

4

2

1

1 2 3 4

0 1 2 3

1 1 0 1

1 0 1 11 1 1 10 , 1 , 2 , 3

1 1 0 12 22 2

1 0 1 1

0, 2, 2, 2

cos( )

K K K K

A t

2

2 22 2

2

1ˆˆ ˆ, ,2 2

1 1ˆ ( ) ( ), , ( )2 2 !

m x

n n n n n nn

x x p p i H H m xm

mH x x n x e h x

n


Normal modes of 4-atom chain in pictures

0 00 ( )u v t

2

1

3


Adiabatic theory of thermal lattice vibrations

•Born-Oppenheimer adiabatic approximation:

•Electrons react instantaneously to slow motion of lattice, while remaining in essentially electronic ground state → small electron-phonon interaction can be treated as a perturbation with small parameter:

11;electron

electron ionion F Debye

m

M

orelectron ion D Fm M

1 11. ions : , , ( , , )N electron NER R R R

1 12. potential for ions : ( , , ) ( , , ) ion-ion interactionN electron NE R R R R 2

11

3. Hamiltonian for ions : ( , , )2

Ni

Ni

PH

M

R R


Adiabatic formalism: Two Schrödinger equations (for electrons and ions)

01 1 1 1 1

ˆ ˆ ( , , ; , , ) ( , , ) ( , , ; , , )e e

n nelectron electron ion electron N N n N electron N NH H E

r r R R R R r r R R

( , ) ( ) ( ; )n ncrystal ion electron

n

r R R r R

01 1 1

ˆ ( , , ) ( , , ) ( , , )p pion p N ion N ion N pH E E Q R R R R R R

2* 2 * *

,

( ; ) ( ) ( ; ) 2 ( ) ( ; )2

p n n n np electron ion i electron i ion i electron

n i

Q dM

r r R R r R R r R

The non-adiabatic term can be neglected at T<100K!0pQ


Newton (classical) equations of motion

2

,

( ) ( )1( ) ( )

2n i n i n i n i

n i n i n i n i n i n i m jn i n i m jn i n i m j

r s r sr s r s s s s

r r r

•Lattice vibrations involve small displacement from the equilibrium ion position: 0.1Å and smaller → harmonic (linear) approximation

2

2

,

( )

( ) 1 1

2 2

m j n i n in i

n i m j

m jn i n in i n i n i n i m j

n i n i m jn i

m jn i n i m j

m j

r s

r r

r sF H M s s s

s

M s s

•N unit cells, each with r atoms → 3Nr Newton’s equations of motion0

0


Properties of quasielastic force coefficients

are analogous to elastic coefficients

m jn i n i m j

m j

m jn i

M s s mx kx

k

( )0 from translational invariance

0

m j n jn i m i

m j m n jn i i

m jn i

m


Solving equations of motion: Fourier Series

( )1( ) ( ) ( )ni t i

n i i n i n is u e T s e sM

qr qaaq q q

( )2

( )( )0

1( ) ( )

1 1

dynamical matrix (does not depend on )

n m

pn m

im ji n i j

m j

iij m j p ji n i i

m p

ji n

u e uM M

D e eM M M M

D

q r r

q rq r r

q q

r

2 2( ) ( ) ( ) 0j j ji i j i i j

j j

u D u D u

q q q

2 2det 0 for each : eigenvalues ( )sd r D(q) I q q


Example: 1D chain with 2 atoms per unit cell

1 1 2( )

0

11 2

24 2 2

1 2 1 2

21

1

21

1 1 42 sin 0

2

p

iqa

ip

iqap

f fe

M M MD e

f fM M eMM M

f qaf

M M M M

q r

22

1 2 2 1,1

1 2 2 1 1,2 1,11 2 1 2 1 2

1

2

2 ;

n n n nn

n n n n n nn n n n n n

f s s s s

f f


1D Example: Eigenfrequencies of chain

2

2 2

1 2 1 2 1 2

1 1 1 1 4( ) sin

2

qaq f f

M M M M M M

optical mode

acoustic mode

1 2

01 2

2 ( )lim ( )q

f M Mq

M M

01 2

lim ( )2( )q

fq qa

M M

+ (0)

( ) 2BvK: ( ) ( ) 2i qna t

n N n

ms s u q e q n N a qna m q

Na


1D Example: Eigenmodes of chain at q=0

Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with a

1 1 21 2

1 2

21 2

2 2

2 ( )( 0) , ( 0)

2 2

f f

M M Mf M Mq q

f fM M

MM M

D

1 2 1

1 1 2 1 2 12 21 2

1 2 1 2 1

2 1 21 2 2

22 2

( )0 ,

22 2

( )

n

n

fM Mf f uM M M M M sM M

u ufM Mf f M s M

M M MM M u

Center of the unit cell is not moving!


2D Example: Normal modes of chain in 2D space

2 2 21( ) 2 cos( )r r r

M q q a

2 21ˆ( )

2 r j i ij j iij

s s r s s

•Constant force model (analog of TBH) : bond stretching and bond bending

( )0 0

( )1

( )0 r

( )1 r


3D Example: Normal modes of Silicon

L — longitudinal

T — transverse

O — optical

A — acoustic

8.828THz

2.245THz

r

Si

Si

M

M


Symmetry constraints

( ) *1( ) , ( ) ( ), ( ) ( )ni t i i

n i i j js u e t t D DM

qrq q q q q q q

( ) ( )*0 0

( ) ( )* 00

* † 2

1 1

1 1

is Hermitian matrix

p p

p p

i ij p j p ji i i

i ij i p i ii p j j j

T

D e eM M M M

D e e DM M M M

q r q r

q r q r

D D D

→Relevant symmetries: Translational invariance of the lattice and its reciprocal lattice, Point group symmetry of the lattice and its reciprocal lattice, Time-reversal invariance.

2 2 *

*

( ) ( ) ( ) ( ) ( ) ( ) 0

( ) ( )

j j j ji i j i i j

j j

j j

D u D u

u u

q q q q q q

q q

( ) ( )

( ) ( )

( ) ( )

j ji i

j j

D D

u u

q G q

q G q G

q G q + G


Acoustic vs. Optical crystal lattice normal modes

→All harmonic lattices, in which the energy is invariant under a rigid translation of the entire lattice, must have at least one acoustic mode (sound waves)

( ) q q

2 2

10

1(0) (0)

j m ji n i

m j

jm j m ji n i j i n i

j m m j m

q DM M

uu u u M

M M M

21

22

23

(0) 0, (0) 0

0 (0) 0, (0) 0

(0) 0, (0) 0

x

i y

z

u M

u M u M

u M

3 3 optical modes which at 0 behave as :

( 0) ( 0) 0i i

r

u q M M s q M

q

←3 acoustic modes (in 3D crystal)


Normal coordinates

→The most general solution for displacement is a sum over the eigenvectors of the dynamical matrix: *

,

1( , ) ( ) , ( , ) ( , )nis

n i s i s sBZ s

s Q t e Q t Q tM N

qr

q

q q q q

2

, , ,

2( ),

,

1 1( , ) ( ) ( , ) ( )

2 2

1 1; ( ) ( ) ( , )

2

n n

n

i ir skinetic n i r i s i

n i n i BZ r s

i k q r r sk q i i rs kinetic r

n i r

E M s Q t e Q t e

e E Q tN

qr kr

q k

q

q q k k

q k q

22

,

1( ) ( , )

2potential s ss

E Q t q

q q2 22

,

* *

2* *

1( , ) ( ) ( , )

2

( ) ( )( )

0 ( ) ( ) ( ) 0( ) ( )

kinetic potential s s ss

s ss

s s ss s

L E E Q t Q t

LP Q

Q

d L LQ Q

dt Q Q

q

q q q

q qq

q q qq q

( , )sQ tq

•In normal coordinates Newton equations describe dynamics of 3rN independent harmonic oscillators!


Quantum theory of small amplitude lattice vibrations: First quantization of LHO

→First Quantization:

ˆ ˆ( ) ( ), ( ) ( )s s s sQ Q P P q q q q

222

,

1 ˆˆ ˆ( ( ), ( )) ( ) ( ) ( )2s s kinetic potential s s s

s

H Q P E E H P Q q

q q q q q

ˆ ˆ( ), ( ) ( ), ( )r s rs r s rsPoissonQ P Q P i kq kqk q k q

, ,, ,

ˆ , Syms s

s s

H E E E q qq q


Second quantization representation: Fock-Dirac formalism

**

*

( , ) ( ) ( )

( , )ˆ ( , ) ;

ˆ( ) ( )

t a t

a atH t i i H a i H a

t t t

H H d

k kk

k kkk k kk k

k k

kk k k

r r

rr

r r r

** *

*ˆ, , ( , ) ( , )

H Ha ai i H t H t d a H a

t i a t a

k kk k k kk k

kkk k

r r r

*

* † †

"generalized coordinate"; "generalized momentum"

ˆ ˆ ˆ ˆ ˆ, ; , 0, ,

i a a

a a a a a a a a

k k

k k k k k k k k kk

† † †ˆ ˆ ˆ ˆˆ ˆ( , ) ( ) ( ), ( , ) ( ) ( ); ( , ), ( , ) ( )t a t t a t t t k k k kk k

r r r r r r r - r

† †ˆH H a a O O a a

kk k k kk k kkk kk


Quantum theory of small amplitude lattice vibrations: Second quantization of LHO

→Second Quantization applied to system of Linear Harmonic Oscillators:

†

†

†

† † †

ˆ ˆ ˆ ˆ( ), ( ) ( ), ( )

ˆ ˆ ˆ( ) ( ) ( )2 ( )

( )ˆ ˆ ˆ( ) ( ) ( )2

ˆ ˆ ˆ ˆ ˆ ˆcanonical transformation: ( ), ( ) , ( ), ( ) ( ), ( ) 0

s s s s

s s ss

ss s s

s r sr s r s r

Q P a a

Q a a

P i a a

a a a a a a

kq

q q q q

q q qq

qq q q

k q k q k q

→Hamiltonian is a sum of 3rN independent LHO – each of which is a refered to as a phonon mode! The number of phonons in state is described by an operator:

†

,

1ˆ ˆ ˆ( ) ( ) ( )2s s s

s

H a a

q

q q q

( , )sq

†ˆ ˆ ˆ( ) ( ) ( )s s sn a aq q q


Phonons: Example of quantized collective excitations

†ˆ ( ) ( ) ( ) 1 ( ) 1

ˆ ( ) ( ) ( ) ( ) 1

s s s s

s s s s

a n n n

a n n n

q q q q

q q q q

→Creating and destroying phonons:

( )†

, ,

1ˆ( ) ( ) 0

( )!sn

s ss ss

n an

q

q q

q qq

→Lattice displacement expressed via phonon excitations – zero point motion!

†

,

1ˆ ˆ( ) ( ) ( )

2 ( )nis

n i s s is s

s a a eM N

qr

q

q q qq

2

00

Ts

→Arbitrary number of phonons can be excited in each mode → phonons are bosons:


Quasiparticles in solids

•Electron: Quasiparticle consisting of a real electron and the exchange-correlation hole (a cloud of effective charge of opposite sign due to exchange and correlation effects arising from interaction with all other electrons).

•Hole: Quasiparticle like electron, but of opposite charge; it corresponds to the absence of an electron from a single-particle state which lies just below the Fermi level. The notion of a hole is particularly convenient when the reference state consists of quasiparticle states that are fully occupied and are separated by an energy gap from the unoccupied states. Perturbations with respective to this reference state, such as missing electrons, are conveniently discussed in terms of holes (e.g., p-doped semiconductor crystals).

•Polaron: In polar crystals motion of negatively charged electron distorts the lattice of positive and negative ions around it. Electron + Polarization cloud (electron excites longitudinal EM modes, while pushing the charges out of its way) = Polaron (has different mass than electron).

6 15eV, 10 msF F F ev k m


Collective excitation in solids

In contrast to quasiparticles, collective excitations are bosons, and they bear no resemblance to constituent particles of real system. They involve collective (i.e., coherent) motion of many physical particles.

•Phonon: Corresponds to coherent motion of all the atoms in a solid — quantized lattice vibrations with typical energy scale of

•Exciton: Bound state of an electron and a hole with binding energy

•Plasmon: Collective excitation of an entire electron gas relative to the lattice of ions; its existence is a manifestation of the long-range nature of the Coulomb interaction. The energy scale of plasmons is

•Magnon: Collective excitation of the spin degrees of freedom on the crystalline lattice. It corresponds to a spin wave, with an energy scale of

0.1eV 2 0.1eVe a

2 5 20eVene m

0.001 0.1eV


Classical theory of neutron scattering

2

0 0

( ( ) )

( ( ( )) )

( , ) , ,

1( ( )) ( ( )), ( ) ( )

( , )

n

n n

i tn n n

n

i t t

n

I

r t t t eM

dt e

qr q

K r s

K K = k - k

r r r r u q

K

( ( )2small amplitude: ( ) , ( , ) (1 ( ) )ni t

nn

t a dt e i ta

K rs K K Ks

0 0first term non-zero: , 0 K = k - k G

0 0second term non-zero: ; ( ) ( ) 0s s K q = k - k q q q

Bragg or Laue conditions for elastic scattering!


Classical vs. quantum inelastic neutron scattering in pictures

•Lattice vibrations are inherently quantum in nature → quantum theory is needed to account for correct temperature dependence and zero-point motion effects.

Phonon absorption is allowed only at finite temperatures where a real phonon be excited:

0 ( ) 0sT n K

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