Solid state physics

Lecture 5: Phonons and specific heat

Prof. Dr. U. Pietsch

p

npnpn uuCF )(

Crystal lattice vibration

Hook´s law

longitudinal

transversal

Fn– Force on plane n; Cp – force constant between planes of distance p

lattice vibration of 1D monoatomic chain

Equation of motion

p

npnpn uuCF )(

p

npnpn uuC

dt

udM )(

²

²

Ansatz:

)])[(

0

taqpni

pn euu

a – lattice spacing; q – wave number, - wave vector q

(*)

Solution of 1D wave equation

(*)

Cp= -Cp

Dispersion relation

tinqai

p

qapnitnqain eeeeMudt

udM ][²

²

² ][])[()][(

0

)1(² p

ipqa

p eCM

)]1()1([²0

ipqa

p

p

ipqa

p eCeCM

)2)cos(2()2(²00

p

p

ipqa

p

ipqa

p pqaCeeCM

))cos(1(2

²0

p

p pqaCM

(q) – dispersion – nonlinear function

Small q ²...)²²11(2

)cos(1(2

²21 apq

M

Cpqa

M

C

Only nearest neighbours , p=1

)2

²(sin4

²

)2

²sin2(2

)cos(1(2

²

1

11

qa

M

C

qa

M

Cpa

M

C

~ q

1st Brillouin zone contains all information : (q) is periodic in q*a

)(

][

])1[(

1 tqai

tnqai

tqani

n

n ee

e

u

u

Only in range |qa|< 2p independent values of q

aq

a

pp

Range of 1st Brillouin zone

²²² 1 aqM

C

)2

²(sin4

² 1 qa

M

C

C.Kittel

vs=/q M

C1max 2

=q

110

max 10 ma

qp

aq

22

max

min p

dq

dvg

Group velocity= signal velocity of a wave packet

)cos(²

2

1 qam

Cavg

vs =0 at q=p/a

phase velocity > group velocity

qvp

)/()sin(

²21

21 qaqa

m

Cavp

lattice vibration of 1D bi-atomic chain

)2(²

²

)2(²

²

12

11

ssss

ssss

vuuCdt

vdM

uvvCdt

udM

)(

0

tsqai

s euu )(

0

tsqai

s evv

CveCuvM

CueCvuM

iqa

iqa

2]1[²

2]1[²

2

1

0²2]1[

]1[²2

1

1

MCeC

eCMCiqa

iqa

0)cos1²(2²)(2 21

4

21 qaCMMCMM

Quadratic equation in ²

M1 M2

212121

2 ²sin4)²

11()

11(

MM

qa

MMC

MMC

For q=0 0²1

)11

(2²21

2MM

C

For sin(qa) ≈ qa )²(²21

21

1 qaMM

C

Acoustic branch

Optical branch

1

1

2²

M

C

2

2

2²

M

C 1

v

u

Optical frequency gap

)11

(2²21

2MM

C

v

u

1

2

M

M

v

u

11

11

21

2

21

1

;

;0;2

,)11

(2_

0;2

;0_

20

nn

nn

uu

vvuM

C

M

M

v

u

MMCbranchoptical

vM

Cvubranchacoustic

aqq

p

C

C

C

Equal masses : M1=M2

aq

a

pp

aq

a 22

pp

Phonons in 3D

• 3D crystal of size Pa1, Pa2, Pa3 may contain S different atoms per unit cell • Each atoms has 3SP³=3SN degrees of freedom for displacements • Each displacement results in 1D equations of motion • 3SN differential equation with 3SN solutions • Direction of wave propagation given by vector K • Displacement u= u0 exp(i(K´an-t) ) using

GKK

´ 321 lbkbhbG

p2)(´ 321 maKalbkbhbaKaK nnnn

pp naK

• Valid values in 1st BZ • Number of solutions is 3S • 3S equation give 3S branches j(K), j=1,2,3, corresponds to 1 long + 2 trans modes

• In total 3 acoustic branches and 3(S-1) optical branches

K´ periodic in 3D

Group and phase velocities

Phase velocity

)2

(cos0

qav

dq

dvgr

|2/

)2/sin(|0

qa

qav

qvph

Group velocity = signal velocity

This shows a wave with the group velocity and phase velocity going in different directions. The group velocity is positive, while the phase velocity is negative.

https://en.wikipedia.org/wiki/Phase_velocity#/media/File:Wave_opposite-group-phase-velocity.gif

Phonon dispersion of Cu

Phonon dispersion of GaAs

Ab initio phonon dispersions of transition and noble metals: effects of the exchange and correlation functional Andrea Dal Corso

http://wolf.ifj.edu.pl/phonondb/gaas.htm

Pseudo-momentum of phonons

)exp(1

)exp(1

ika

iNka

dt

duMe

dt

duMu

dt

duMp iSKa

s

C. Kittel p 101: A phonon of wavector K will interact with particles such as photons, neutrons, and electrons as if it had a momentum ℏK. However, a phonon does not carry physical momentum. [...] The true momentum of the whole system always is rigorously conserved.

Phonons have no physical momentum , p= ℏK, because the the atomic displacement is on relative coordinates and the sum over all these atomic displacements is always zero

0 sudt

duMp

Crystal length L=Na k=2pl/Na; with l – integer; 1)2exp()exp( liiNka p

0)exp(1 iNkaBecause of

= 5 A = 3.3meV

elastic scattering

GKK

0

E = E0

K

0K

G

Methods to measure phonons: neutron scattering

De Broglie 1892- 1987

kinEm

h

p

h

n2

Methods to measure phonons: neutron scattering

qGKKK

0´

Inelastic neutron scattering E<>E0

Phonon is created (+) or annihilated (-)

Energy conservation

G

0K

K

q

000

hkl

´0 EE

)(2

´²²

2

² 2

0 qm

K

m

K

Nature Physics 7, 119–124 (2011)

Research Reactor FRM II in Garching

https://www.frm2.tum.de/

Methods to measure phonons: Raman and IR

Sir C. V. Raman 1888 - 1970

])cos()[cos()cos(

)cos())cos((

)cos(

)cos(

00012

1000

0010

10

00

ttEtE

tEtm

t

tEEm

MM

M

M

Stokes line Anti- Stokes line Rayleigh line

Raman spectrum from sulfur

www.spectroscopyonline.com/

Energy of lattice vibration : Phonons

Energy of lattice vibration is quantized, quantum = phonon = bosons, thermally excited lattice vibrations are „thermal phonons“, calculated following black body radiation

total energy of N oscillators

)(21 nEn n=0,1,2…

Harmonic oscillator model

N

n

ntot EE1

Number n of excited phonons?what is mean quantum number <n> ?

)(21 nE

Following Boltzmann

0

1

)/exp(

)(exp(

s

n

n

kTs

kT

N

N

0

0

)/exp(

)(exp(

s

s

kTs

kTss

n

1

1/

kTe

n

Bose-Einstein distribution

For T=0 <n>=0 21 E

For kT 2

1

kTn

kTkTE )/(2

1

2

1

)()(21 qnE

q

Total energy of whole phonon spectrum

For low T

<n> ≈ exp (- ħ/kT)

Classical limit

1D Crystal with length L=Na

L

N

LLLq

pppp;...;

3;

2;

standing waves at

Distance between : L

qp

Number of modes D() d

1D Density of states :

sv

dL

dqd

dLd

d

dqLdD

p

p

p

/)(

Each normal mode has form of standing wave

)sin()exp(0 sqatiuus Propagating waves

In 3D Crystal back and forth wave

Periodic boundary condition:

Distance between :

Volume in q-space containing one wave : )

2(

Lq

p

)( tqsdi

s eu

)()( Lsqdusqdu

L

N

LLLq

pppp ;...;

6;

4;

2;0

³3

4)³

2( q

LN

p

pNumber modes in 3D :

Spectral density function Sj() for each branch j max

0

)(

NdS j

3N branches , 1L + 2T, for p atoms in uc

max

0

3

1

3)()(

NpdSdSp

j

jj

dqL

dqqdS jjp

2

)()( dqd

LS j

/

1

2)(

p In 1D

In 3D ||

)³2

()(p

q

jgrad

dFLS Generally no analytic solution

Phonon spectrum

A all approximated by single E Einstein frequency

B (q) approximated by =vs q - Debye model

A: Einstein model :

E

Ej

forpN

forS

__3

__0)(

B: Debye model : s

jv

dFLS )³

2()(

p p

p 4

²

²4²

svqdF

p

p 4

³

²)³

2()(

s

jv

LS

D

D

j forN

S

__²3

)(3 p

p

4)³(

3 3

2

33

maxs

LD

vN

Debye frequency

Specific heat : impact of Phonons

constppconstVVT

Ec

T

Ec

|)(__|)(

Vp

V

Vpcc

c

cc

410*3

Energy per degree of freedom 0.5 kT for N atoms with 3N degrees of freedom E= 3NkT=3RT/mol

³(exp) TcV

for low T

0 2

1

2

1 ))(()()( nSdqnEq

0 /00 /02

1

1)(

1)()(

kTkT eSdE

eSdSdE

0 /

/

)²1()²)(()(

kT

kT

VVe

e

kTSdk

T

Ec

Depending on

A: Einstein model :

0 /

/

)1(3)²1(

)²)((3kT

pNke

e

kTSdpNkc

kT

kT

EV

E

E

kTE kNp

cV 3 Correct : Dolong-Petit law

kTE kT

VEec

/ Wrong !

B: Debye model :

0 0

4

/

/

³ )²1()³(3

)²1()²(

²33

Dx

x

x

D

kT

kT

D

Ve

exdx

TpNk

e

e

kT

NdpNkc

Debye integral

kTx

TxD

T )³(

TcV Correct !

Universal cv curve

Exp. vibration spectrum of Vanadium, and Debye approximation

All optical modes excited

Only acoustic modes excited

Anharmonic effects: thermal expansion

So far phonons are described in terms of Hook´s law: )²()()( 00 rrArVrV

Cxt

xM

²

²Based on harmonic oscillator model

anharmonic oscillator model

³....²²

²FxGxCx

t

xM

0rrx V(r)

r r0

...³²)()( 4

0 fxgxcxxVxV

)/()(exp(

)/()(exp(

kTxVdx

kTxVdxx

x

p

p

kTc

g

ßc

ßc

g

x²4

34

3 2/3

2/5

kc

g

²4

3 (g)=const;

for ()= (g, f…)

Anharmonic phonon modes: Grüneisen constants

Frequency of lattice vibrations depends on volume

V

T

c

VaB

c

gx

VdV

d

6/

/Grüneisen constant

)()(ln

))((lnq

Vd

qd

Mode Grüneisen constant

)²(sin4

²21 ka

M

C

c

gxcx

²)²( Phonon dispersion

Anharmonic correction

i

Vi

Vii

i

c

c

Thermal conductivity

TJh Heat current density -Thermal conductivity

In solids calculate J as function of mean number of phonons <n>(x) being different at different positions x. Using x=vxt

xrq

x

rq

xhq

nV

rqvnV

J

)(

1),()(

121

,

21

,

,

Jh<>0, if <n> ≠ <n0> x

rq

xh vnnV

J )(1

0

,

,

Transport by phonon diffusion, or single phonons can decompose into two phonons

t

0||

nn

t

n

t

n

t

ndecaydiff

t - mean decay time

vx – group velocity

x

T

T

nv

x

nv

t

nxxdiff

0| x

T

T

nv

VJ

rq

xh

0

,

, ²3

1t

0

,

1n

TVc

rq

V

tvl vlcV3

1using and

l - mean free pathway Note: cv = cv(T); l=l(T)

Stoff

Öl 0,13…0,15

Wasser[19] (0,0 °C) 0,5562[20]

Eis (−10,0 ± 10 °C) 2,33±0,10[21]

Schnee mit ρ = 0,25 g/cm³ 0,16[1]

Kohlenstoff (Graphit) 119…165

Kohlenstoffnanoröhren 6000

Graphen 5300±480

Diamant 2300

Siliciumcarbid 350

Aluminiumnitrid 180…220

Silicium 148

Siliciumdioxid (Quarz) 1,2…1,4

Aluminiumoxid (99,6 % α-Al2O3) 28

Kreide 0,92

Schwefel 0,269

Humus 1,26

Wärmeleitpaste 4…73[22] Aluminium (99,5 %) 236

Aluminiumlegierungen 75…235[13]

Bismut 8,4

Blei 35

Chromstahl 1.400 30

Eisen 80,2

Gold, rein 314

Kalium 135

Korund (Al2O3 ~99%) 41,9[14]

Kupfer, Handelsware 240…380[11]

Kupfer, rein 40

Wärmeleit- fähigkeit λ in W/(m·K)

Wikipedia

Phonon- phonon scattering ; scattering at defects (point defects and surfaces, electron – phonon scattering (in metals and semiconductors)

Scattering processes

....1111

321

llll

Scattering at defects

)(

1

Tnl

„Umklapp“ processes

q1+q2 = q3 ±G

„Normal“ processes

q1+q2 = q3

4

1

defectsnl

T- independent

Ph- Ph- Scattering

Umklapp processes

q1+q2 = q3 ±G

Phonon „flow“

321

321

qqq

Phonon processes

Normal processes

Momentum G transfered to lattice lattice heating

q1+q2 = q3

Momentum conserved

Temperature dependence of ph-ph scattering described in terms of Debye model

D

D

D

T

TforT

Tfore

n

D

__

__2/

DD

D

T

TforT

Tforel

D

__

__2/

)(__³

)(__

)__(__1

2/

scatteringdefectTforT

scatteringPhPhTforeT

scatteringPhPhTforT

D

D

Tn

D

D

Scattering at defects

4

1

defectsnl

T- independent

In summary:

Pure silicon Doped Germanium

NA 1019 /cm

NA 1015 /cm

Phys.Rev.134, 1058(1964)

Proc.Royal. Soc. 238, 502 (1957)

Defect scattering - Experiments