mena2000 phonons: lectures 5-6 (week 10)...this law of dulong and petit (1819) is approximately...

MENA2000 Phonons: Lectures 5-6 (week 10)

Recap of last week lectures

Born – von Karman boundary conditions;

Phonon density of states (DOS) in 1-D;

Calculation of the lattice heat capacity

Dulong-Petit theory for heat capacity;

Einstien model

Debye model

Comparison of different models

Thermal conductivity and thermal expansion

Temperature dependence of the thermal conductivity

Thermal expamsion

Summary: phonons in a nut-shell

Calculating phonon density of states – DOS – in 1-D

This sets a condition on allowed k

values: ...,3,2,12

2 nNa

nknkNa

So the separation between allowed

solutions (k values) is:

independent of k, so the

density of modes in k-space is

uniform

Since atoms s and s+N have the same displacement, we can write:

Nss uu ))(()( taNskitksai ueue ikNae1

Nan

Nak

22

Thus, in 1-D: 22

1 LNa

kspacekofinterval

modesof#

Energy level diagram for a chain of atoms with one

atom per unit cell and a lengt of N unit cells

Energy level diagram for

one harmonic oscillator







Einstien model

Debye model




Thermal expamsion


Temperature dependence of experimentally measured heat capacity

Classical (Dulong-Petit) theory for heat capacity

For a solid composed of N such atomic oscillators:

Giving a total energy per mole of sample:

TNkENE B31

RTTkNn

TNk

n

EBA

B 333

So the heat capacity at constant volume

per mole is: KmolJ

V

V Rn

E

dT

dC 253

This law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( > 300 K).

Einstein model for heat capacity accounting for quantum properties of oscillators

constituting a solid

Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies

...,2,1,0 nnEn

[later showed is actually correct] 21 nEn

...,2,1,0 nnEn

Einstein (1907): model a solid as a collection of 3N independent 1-D oscillators, all with constant , and

use Planck’s equation for energy levels

occupation of energy level n:

(probability of oscillator being in level

n)

0

/

/

)(

n

kTE

kTE

n

n

n

e

eEf classical physics (Boltzmann

factor)

Average total energy

of solid:

0

/

0

/

0

3)(3

n

kTE

n

kTE

n

n

nn

n

n

e

eE

NEEfNUE

Boltzmann factor is a weighting factor that determines the relative probability of a state i

in a multi-state system in thermodynamic equilibrium at tempetarure T.

Where kB is Boltzmann’s constant and Ei is the energy of state i. The ratio of the

probabilities of two states is given by the ratio of their Boltzmann factors.

kTEie/

Boltzmann factor determines Planck distribution


constituting a solid

0

/

0

/

3

n

kTn

n

kTn

e

en

NU

Using Planck’s equation: Now let

kTx

0

03

n

nx

n

nx

e

en

NU

0

0

0

0 33

n

nx

n

nx

n

nx

n

nx

e

edx

d

N

e

edx

d

NU Which can be

rewritten:

Now we can use the

infinite sum: 1

1

1

0

xforx

xn

n

1

3

1

3

1

13

/

kTx

x

x

x

x

e

N

e

N

e

e

e

e

dx

d

NU

To give: 11

1

0

x

x

xn

nx

e

e

ee

So we obtain:


constituting solids

Differentiating:

Now it is traditional to define an

“Einstein temperature”:

Using our previous definition:

So we obtain the prediction:

1

3/ kT

A

V

Ve

N

dT

d

n

U

dT

dC

2/

/2

2/

/

1

3

1

3 2

kT

kT

kT

kT

kT

kT

A

V

e

eR

e

eNC

kE

2/

/2

1

3)(

T

T

TV

E

EE

e

eRTC


constituting solids

Low T limit:

These predictions are qualitatively correct: CV

3R for large T and CV 0 as T 0:

High T limit: 1T

E

RR

TC

T

TTV

E

EE

311

13)(

2

2

1T

E

T

TT

T

TV

EE

E

EE

eRe

eRTC

/2

2/

/2

33

)(

3R

CV

T/E





High T limit: 1T

E

Low T limit: 1T

E

Correlation with energy level diagram for a harmonic oscillator

Problem of Einstein model to reproduce the rate of heat capacity decrease at low

temperatures High T behavior:

Reasonable agreement with

experiment

Low T behavior: CV 0

too quickly as T 0 !

More careful consideration of phonon occupancy modes as a way to improve

the agreement with experiment

Debye’s model of a solid:

• 3N normal modes (patterns) of oscillations

• Spectrum of frequencies from = 0 to max

• Treat solid as continuous elastic medium (ignore details of atomic structure)

This changes the expression for CV because each

mode of oscillation contributes a frequency-

dependent heat capacity and we now have to

integrate over all :

dTCDTC EV ),()()(max

0

# of oscillators per unit

, i.e. DOS

Distribution function

Debye model

3

3

3

4

2k

LNk

3

126

V

N

k

v

k

B

BD

Density of states of acoustic phonos for 1 polarization

Debye temperature θ

32

3

6 v

VN D

N: number of unit cell

Nk: Allowed number of k points in a

sphere with a radius k

/vk

32

3

3

33

63

4

2)(

v

V

v

LN

32

2

2

)()(

v

V

d

dND

Thermal energy U and lattice heat capacity CV : Debye model

D

D

D

x

x

x

BV

B

B

BV

V

B

e

exdx

TNkC

Tk

Tkd

Tkv

V

T

UC

Tkv

VdnDdU

0

2

43

0

2

4

232

2

0

32

2

)1(9

]1)/[exp(

)/exp(

2

3

1)/exp(23)()(3

3 polarizations for acoustic modes

Debye model

Universal behavior for all

solids!

Debye temperature is

related to “stiffness” of

solid, as expected

Better agreement than

Einstein model at low T

Debye model

Quite impressive

agreement with predicted

CV T3 dependence for

Ar! (noble gas solid)



More careful consideration of phonon occupancy modes

as a way to improve the agreement with experiment



Ensemble of 3N independent harmonic oscillators modeling vibrations in a solid

Quantum

oscillators

Classical

oscillators En

erg

y

TNkENE B31

Any energy state is accessible for any oscillator in

form of kBT, i.e. no distribution function is applied

and the total energy is

Any energy state is accessible for any

oscillator in form of kBT, i.e. no distribution

function is necessary, so that Energy level diagram for a chain of atoms with one



Quantum

oscillators

Classical

oscillators En

erg

y

TNkENE B31

Any energy state is accessible for any oscillator in

form of kBT, i.e. no distribution function is applied

and the total energy is

Not all energies are accessible, but only those in quants of ħωn,

and Planck distribution is employed to calculate the occupancy

at temperature T, so that

nNE 3

1

133)(3

/

0

/

0

/

0Tk

n

TkE

n

TkE

n

n

nnB

Bn

Bn

eN

e

eE

NEEfNE


Dulong-Petit model is valid only at high

temperatures

Einstein model is in a good agreement with the

experiment, except for that at low temperatures





nNE 3

max

min

)(3 nDdE







Einstien model

Debye model




Thermal expamsion


When thermal energy propagates through a solid, it is carried by lattice waves or phonons. If the atomic

potential energy function is harmonic, lattice waves obey the superposition principle; that is, they can pass

through each other without affecting each other. In such a case, propagating lattice waves would never

decay, and thermal energy would be carried with no resistance (infinite conductivity!). So…thermal

resistance has its origins in an anharmonic terms of the lattice energy.

Classical definition of thermal

conductivity vCV

3

1

VC

wave velocity

heat capacity per unit volume

mean free path of scattering (would be if

no anharmonicity)

v

high T low T

dx

dTJ

Thermal energy

flux (J/m2s)

Phenomenological description of thermal conductivity

Temperature dependence of thermal conductivity in terms of phonon prperties

Mechanisms to affect the mean free pass (Λ) of phonons in periodic crystals:

2. Collision with sample boundaries (surfaces)

3. Collision with other phonons deviation from harmonic

behavior

1. Interaction with impurities, defects, and/or isotopes

VC 11 / kT

ph

en

ThighR

TlowT

3

3

ThighkT

Tlow

To understand the experimental dependence , consider limiting values of and (since

does not vary much with T). VC v)(T

deviation from

translation symmetry

1) Please note, that the temperature dependence of T-1 for Λ at the high temperature limit results from considering nph , which is the

total phonon occupancy, from 0 to ωD. However, already intuitively, we may anticipate that low energy phonons, i.e. those with low k-

numbers in the vicinity of the center of the 1st BZ may have quite different appearence conparing with those having bigger k-numbers

close to the edges of the 1st BZ.

1)

Thus, considering defect free, isotopically clean sample having limited size D

CV

low T

T3

nph 0, so

, but then

D (size)

T3

high T

3R

1/T

1/T

How well does this match experimental results?

Temperature dependence of thermal conductivity in terms of phonon properties

T3

However, T-1 estimation for κ in the high

temperature limit has a problem. Indeed, κ

drops much faster – see the data – and the

origin of this disagreement is because – when

estimating Λ – we accounted for all excited

phonons, while a more correct

approximation would be to consider “high”

energetic phonons only. But what is “high” in

this context?

T-1 ?

T3 estimation for κ

the low

temperature limit

is fine!


Experimental (T)

NaNak

2121

aNa

NkN

22

NaNak

4222

𝑬𝟐 𝑬𝑫 𝑬𝟏

«significant» modes «insignificant» modes

The fact that «low energetic phonons» having k-values << 𝝅/𝒂 do not participate in the energy transfer, can

be understood by considering so called N- and U-phonon collisions readily visualized in the reciprocal

space. Anyhow, we account for modes having energy E1/2 = (1/2)ħωD or higher. Using the definition of θD =

ħωD/kB, E1/2 can be rewritten as kBθD/2. Ignoring more complex statistics, but using Boltzman factor only,

the propability of E1/2 would of the order of exp(- kB θD/2 kB T) or exp(-θD/2T), resulting in Λ exp(θD/2T).

𝝎𝟐 𝝎𝑫 𝝎𝟏 1/2

Explanation for κ exp(θD/2T) at high temperature limit

CV

low T

T3

nph 0, so

, but then

D (size)

T3

high T

3R

exp(θD/2T)

exp(θD/2T)



Lecture 9: Thermal conductivity and thermal expansion

•We understood phonon DOS and occupancy as a function of temperature, but what about transport properties?

• Thermal expansion

• Phenomenological description of thermal conductivity

• Temperature dependence of thermal conductivity in terms of phonon properties

• Phonon collisions: N and U processes

• Comparison of temperature dependence of κ in crystalline and amorphous solids

Phonon collisions: N and U processes

How exactly do phonon collisions limit the flow of heat?

2-D lattice 1st BZ in k-space:

1q

2q

3q

a2

a2

321 qqq

No resistance to heat flow

(N process; phonon momentum conserved)

Predominates at low T << D since and q will be

small

What if the phonon wavevectors are a bit larger?

2-D lattice 1st BZ in k-space:

1q

2qa

2

a2

Gqqq

321

Two phonons combine to give a net phonon with an

opposite momentum! This causes resistance to heat flow.

(U process; phonon momentum “lost” in units of ħG.)

More likely at high T >> D since and q will be larger

21 qq

G

3q

Umklapp = “flipping over” of wavevector!

Phonon collisions: N and U processes


11 / kT

ph

en

ThighT

Tlow

1

The temperature dependence of T-1 for Λ results

from considering the total phonon occupancy, from

0 to ωD. However, interactions of low energy

phonons, i.e. those with low k-numbers in the

vicinity of the center the 1st BZ, are not changing

energy. These are so called N-processes having little

impact on Λ.

vCV3

1

1q

2q

3q

a2

a2

1q

2qa

2

a2

21 qq

G

3q

U-process , i.e. to turn over the wavevector by G,

from a German word umklappen.

A more correct approximation for Λ (in high temperature limit) would be to consider “high” energetic phonons

only, i.e those participating in U- processes.


NaNak

2121

aNa

NkN

22

NaNak

4222

𝑬𝟐 𝑬𝑫 𝑬𝟏

«significant» modes «insignificant» modes

The fact that «low energetic phonons» having k-values << 𝝅/𝒂 do not participate in the energy transfer, can

be understood by considering so called N- and U-phonon collisions readily visualized in the reciprocal

space. Anyhow, we account for modes having energy E1/2 = (1/2)ħωD or higher. Using the definition of θD =

ħωD/kB, E1/2 can be rewritten as kBθD/2. Ignoring more complex statistics, but using Boltzman factor only,

the propability of E1/2 would of the order of exp(- kB θD/2 kB T) or exp(-θD/2T), resulting in Λ exp(θD/2T).

𝝎𝟐 𝝎𝑫 𝝎𝟏 1/2



CV

low T

T3

nph 0, so

, but then

D (size)

T3

high T

3R

exp(θD/2T)

exp(θD/2T)


dxe

dxxe

xkTxU

kTxU

/)(

/)(

In a 1-D lattice where each atom experiences the same potential energy function U(x), we can calculate the

average displacement of an atom from its equilibrium position:

Thermal expansion

I

Thermal Expansion in 1-D

Evaluating this for the harmonic potential energy function U(x) = cx2 gives:

dxe

dxxe

xkTcx

kTcx

/

/

2

2

Thus any nonzero <x> must come from terms in U(x) that go beyond x2. For HW you will evaluate the

approximate value of <x> for the model function

The numerator is zero!

!0x independent of T !

),0,,(...)( 43432 kTfxgxandfgcfxgxcxxU

Why this form? On the next slide you can see that this function is a reasonable model for the kind of U(r)

we have discussed for molecules and solids.

Lattice Constant of Ar Crystal vs. Temperature

Above about 40 K, we see: TxaTa )0()(

Usually we write: 00 1 TTLL = thermal expansion coefficient







Einstien model

Debye model




Thermal expamsion


MENA2000

Faste materialers fundamentale oppbygning

https://www.uio.no/studier/emner/matnat/fys/MENA2000/

krystallers atomære oppbygging, hvordan krystaller klassifiseres, symmetri kjemiske bindinger og molekylorbitalteoriteori, og hvordan en periodisk struktur gir opphav til fenomener som gittervibrasjoner og elektroniske båndstrukturer.

Module I – Crystallography and crystal structures

Module II – Phonons

Module III – Molecular orbital therory

Module IV – Electrons

MENA2000 : Fundamental structure of solid materials - Lecture Plan

calender week

Må 14/1 10-12 no lecture 3

Ti 15/1 14-16 Introduction to the course

Module I – Crystallography and crystal structures (H, Fjellvåg, föreläsningskompendium)

Må 21/1 10-12 Basis and unit cells, Bravais lattices (2D and 3D), density, etc 4

Ti 22/1 14-16 Point symmetry; stereographical projections; crystal planes and directions

Må 28/1 10-12 Symmetry of molecules; introduction to groups; Crystal structures; concepts, digital tools 5

Ti 29/1 14-16 continued

Må 04/2 10-12 Translation symmetry, space groups, international tables for crystallography 6

Ti 05/2 14-16 continued

Må 11/2 10-12 Crystal structures of functional materials II; defects, solid solutions 7

Ti 12/2 14-16 Reciprocal lattice; Diffraction, XRD fingerprint analysis

Module II – Phonons C.Kittel’s Introduction to Solid State Physics; chapters 3 (pp73-85) 4, 5, and 18 (pp.557-561)

Må 18/2 10-12 Crystals as diffraction grids; Laue condition; Bragg plains and Brillouin zones; 8

Ti 19/2 14-16 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D

Må 25/2 10-12 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS) 9

Ti 26/2 14-16 Effect of temperature - Planck distribution; lattice heat capacity: Dulong-Petit, Einstein, and Debye models

Må 04/3 10-12 Comparison of different lattice heat capacity models 10

Ti 05/3 14-16 Thermal conductivity and thermal expansion

Module III – Molecular orbital therory (H.Sønsteby, Molecular Symmetry in Solid-State Chemistry, föreläsningskompendium)

Må 11/3 10-12 Introduction to symmetry and molecular symmetry 11

Ti 12/3 14-16 Group theory for material scientists

Må 18/3 10-12 Character tables 12

Ti 19/3 14-16 Bonding from a symmetry perspective

lecture-free week 13

Må 01/4 10-12 The role of symmetry in vibrational spectroscopy 14

Ti 02/4 14-16 Crystal field theory and ligand field theory

Må 08/4 10-12 Interplay between MO-theory and band structure in 1D 15

Ti 09/4 14-16 Band structure and density of states (DOS) in 3D from MO-theory

Easter 16

Module IV – Electrons C.Kittel’s Introduction to Solid State Physics; chapters 6, 7, 11 (pp 315-317) 18 (pp. 528-530) , and Appendix D

Ti 23/4 14-16 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 17

Må 29/4 10-12 DOS of FEFG in 1D and 3D at ground state (T=0) 18

Ti 30/4 14-16 Effect of temperature – Fermi-Dirac distribution; Fermi energy and Fermi level; heat capacity of FEFG

Må 06/5 10-12 Transport properties of electrons – examples for thermal, electric and magnetic fields 19

Ti 07/5 14-16 DOS of FEFG in 2D - quantum wells, DOS in 1D – quantum wires, and in 0D – quantum dots

Må 13/5 10-12 Origin of the band gap; Bloch theorem; Nearly free electron model 20

Ti 14/5 14-16 Kronig-Penney model; Empty lattice approximation; number of orbitals in a band

Må 20/5 10-12 Effective mass method; electrons and holes; 21

Ti 21/5 14-16 Effective mass method for calculating localized energy levels for defects in crystals

Summary and repetition

To 23/5 14-16 Repetition - course in a nutshell

The "physics" part of MENA2000 in terms of Modules II And IV

Waves in Periodic Lattices

Electron waves in lattices

Free electrons

Electron DOS

Fermi-Dirac distribution

Elastic waves in lattices

Vibrations

Phonon DOS

Planck distribution

Elecronic properties: Electron concentration and transport,

contribution to the heat capacity

Thermal properties: heat capacity and conductance,

thermal expansion

Strating point for advanced material science courses

Module IV Module II

SiC ZnO

ITO Si

Cu2O

• Solar cells

• Detectors

• IC’s

• Power electronics

• High temperature sensors • LED’s

• Transparent electronics

• Displays

• Electronic ink

• Optical cavities

• Multi-junction solar cells

• Thermoelectric generators

• Thermoelectric cooling

Thermo-

electric

materials

Semiconductor physics at UiO

Semiconductor physics at UiO

NEC ion implantor

HRXRD

SIMS

ZnO MOCVD

UiO clean room area

Labs

temperature/time

resolved PL

DLTS

6 Professors

4 Adm/technical staff

~ 10 Post docs

~ 15 PhD students and ~ 10 Msc students

Micro- and Nanotechnology Laboratory (MiNaLab)

Halvlederfysikk ved UiO / MiNa-Lab

mena2000 phonons: lectures 5-6 (week 10)...this law of dulong and petit (1819) is approximately...

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