Solid state physics for Nano

Lecture 5: Specific heat andanharmonicity

Prof. Dr. U. Pietsch

Safety instructions: 13.5.19 10 am

Additional excercise: 10.5. 12:30B019

Pseudo-momentum of phonons

)exp(1)exp(1

ikaiNka

dtduMe

dtduMu

dtduMp 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= ℏΚ, because the the atomicdisplacement is on relative coordinates and the sum over all these atomicdisplacements is always zero

0== ∑ sudtduMp

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

0)exp(1 =− iNkaBecause of

BUT: Phonon can exchange momentum by interaction with otherquasi particles : electron – phonon intercation, phonon-phonon interaction

λ= 5 A = 3.3meV

elasticscattering

GKK

=− 0

E = E0

K

0K

G

Methods to measure phonons: neutron scattering

De Broglie 1892- 1987

kinEmh

ph

n2==λ

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

ω±= ´0 EE

)(2

´²²2² 2

0 qmK

mK ω

±=

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. Raman1888 - 1970

])cos()[cos()cos()cos())cos((

)cos()cos(

000121

000

0010

10

00

ttEtEtEtm

ttEEm

MM

M

M

ωωωωαωα

ωωααωααα

ωαα

++−+=

+=+===

Stokes line Anti- Stokes lineRayleigh line

Raman spectrum from sulfur

www.spectroscopyonline.com/

electr. dipole moment, mpolarizability, α

Energy of lattice vibration : Phonons

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

total energy of N oscillators

ω)( 21+= nEn n=0,1,2…

Harmonic oscillator model

∑=

=N

nntot EE

1

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

ω)( 21+><>=< nE

Following Boltzmann

∑∞

=

+

−

−=

0

1

)/exp(

)(exp(

s

n

n

kTs

kTN

N

ω

ω

∑

∑∞

=

∞

=

−

−>=<

0

0

)/exp(

)(exp(

s

s

kTs

kTssn

ω

ω

11

/ −>=< kTe

n ωBose-Einstein distribution

For T=0 <n>=0 ω21>=< E

For ω>kT 21−>=<

ωkTn

kTkTE =+−>=< )/( 21

21ωω

)()( 21 qnE

qω∑ +><>=<

Total energy of whole phonon spectrum

For low T

<n> ≈ exp (- ħω/kT)

Classical limit

1D Crystal with length L=Na

LN

LLLq ππππ ;...;3;2;=standing waves at Distance :

Lq π

=∆

Number of modes D(ω) dω

1D Density of states :sv

dLdqd

dLdddqLdD ω

πωω

πω

ωπωω ===

/)(

Each normal mode has form of standing wave

)sin()exp(0 sqatiuus ω−=Propagating waves

Standing waves in 3D³

34)³

2( qLN π

π=Number modes in 3D :

Spectral density function Sj(ω) for each branch j ∫ =max

0

)(ω

ωω NdS j

∫∫=||

)³2

()(ωπ

ωq

j graddFLS∫ ∑

=

==max

0

3

13)()(

ω

ωωωω NpdSdSp

jjj

3N branches , 1L + 2T, for p atoms in unit cell 3D spectral Density of states

Approxomations of the Phonon spectrum

A all ω approximated by single ωΕ Einstein frequency

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

A: Einstein model :

=<>

=E

Ej forpN

forSωω

ωωω__3

__0)(

B: Debye model : ∫∫=s

j vdFLS )³

2()(

πω πωπ 4

²²4²

svqdF == πω

πω 4

³²)³

2()(

sj v

LS =

DD

j forNS ωωωω

ω <= __²3)( 3 πωω

π 4)³(3 3

2

33max

sLD

vN==

Debye frequency

Specific heat : impact of Phonons

constppconstVV TEc

TEc == ∂

∂=

∂∂

= |)(__|)( VpV

Vp ccc

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

121 ))(()()( ωωωω nSdqnE

q

∫∫∫∞∞∞

−+=

−+>=<

0 /00 /021

1)(

1)()( kTkT e

SdEe

SdSdE ωω

ωωωωωωωωω

∫∞

−=

∂∂

=0 /

/

)²1()²)(()( kT

kT

VV ee

kTSdk

TEc ω

ωωωω

Depending on ω

A: Einstein model : ∫∞

+≈−

=0 /

/

)1(3)²1(

)²)((3kT

pNke

ekT

SdpNkc kT

kTE

V E

E ωωωω ω

ω

kTE <ω kNpcV 3= Correct : Dolong-Petit law

kTE >ωkT

VEec /ω−≈ Wrong !

B: Debye model : ∫ ∫∞

−=

−=

0 0

4

/

/

³ )²1()³(3

)²1()²(²33 Dx

x

x

DkT

kT

DV e

exdxTpNke

ekT

NdpNkcθ

ωω

ωω ω

ω

Debye integral

kTx ω

=T

xDθ

=θ<T )³(θTcV ≈ 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 −+=

CxtxM =

∂∂

²²Based on harmonic

oscillator model

anharmonic oscillatormodel

³....²²

² FxGxCxtxM −−=

∂∂

0rrx −= V(r)

rr0

...³²)()( 40 ++++= fxgxcxxVxV

)/()(exp(

)/()(exp(

kTxVdx

kTxVdxxx

−

−>=<

∫

∫∞

∞−

∞

∞− Τ==>=<

−

απ

π

kTcg

ßc

ßcg

x²4

343 2/3

2/5

kcg²4

3=α α(g)=const;

for α(Τ)= α(g, f…)

Anharmonic phonon modes: Grüneisen constants

Frequency of lattice vibrations depends on volume

γωω−===

V

T

cVaB

cgx

VdVd

6//

Grüneisen constant

)()(ln))((ln q

Vdqd γω

−= Mode Grüneisen constant

)²(sin4² 21 ka

MC

=ω cgxcx +

= ²)²( ωωPhonon dispersion Anharmoniccorrection

∑∑

=

iVi

Viii

c

cγγ

Thermal conductivityTJ h ∇−= κHeat current density κ -Thermal conductivity

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

xrqx

rqxh q

nV

rqvnV

J∂∂

+><=+><= ∑∑ ωωω )(1),()(121

,21

,,

Jh<>0, if <n> ≠ <n0> xrq

xh vnnV

J )(10

,, ><−><= ∑ ω

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

τ0||

nntn

tn

tn

decaydiff

−−=

∂∂

+∂

∂=

∂∂ τ - mean decay time

vx – group velocity

xT

Tn

vxn

vtn

xxdiff ∂∂

∂∂

−=∂

∂−=

∂∂ 0| x

TTn

vV

Jrq

xh ∂∂

∂∂

−= ∑ 0

,, ²

31 ωτ

0,

1 nTV

crq

V ∂∂

= ∑ ω τvl = vlcV31=κusing and

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

Pure silicon Doped Germanium

NA 1019 /cm

NA 1015 /cm

Phys.Rev.134, 1058(1964)

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

Defect scattering - Experiments