FYS3410 - Vår 2015 (Kondenserte fasers fysikk) · 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h 24/2 Effect of temperature – Fermi- Dirac distribution FEFG

FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v15/index.html

Pensum: Introduction to Solid State Physics

by Charles Kittel (Chapters 1-9 and 17, 18, 20)

Andrej Kuznetsov

delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO

Tel: +47-22857762,

e-post: [email protected]

visiting address: MiNaLab, Gaustadaleen 23a

http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html


FYS3410 Lectures (based on C.Kittel’s Introduction to SSP, chapters 1-9, 17,18,20)

Module I – Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h

27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment

Bragg planes and Brillouin zones 4h

28/1 Elastic strain and structural defects in crystals 2h

30/1 Atomic diffusion and summary of Module I 2h

Module II – Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h

10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models

Comparison of different models 4h

11/2 Thermal conductivity 2h

13/2 Thermal expansion and summary of Module II 2h

Module III – Electrons (Chapters 6, 7 and 18) 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h

24/2 Effect of temperature – Fermi- Dirac distribution

FEFG in 2D and 1D, and DOS in nanostructures 4h

25/2 Origin of the band gap and nearly free electron model 2h

27/2 Number of orbitals in a band and general form of the electronic states 2h

Module IV – Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Energy bands; metals versus isolators 2h

10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier

generation 4h

12/3 Carrier statistics 2h

13/3 p-n junctions and optoelectronic devices 2h

Lecture 6: Vibrations and phonons

• Examples of phonon-assisted processes

• Infinite 1D lattice with one or two atoms in the basis;

• Examples of dispersion relations in 3D;

• Finite chain of atoms, Born – von Karman boundary conditions;

• Phonon density of states in 1-D;

• Collective crystal vibrations – phonons;








Ghkl

k′

-k

k

Diffraction

k K k G

Photoluminescence

CB

VB

ED

EA hn

hn

hn

EXCITATION •Photo generation •Electrical injection

Eg

Photons

Photoluminescence








longitudinal wave transverse wave

Vibrations of crystals with monatomic basis


a

Spring constant, g Mass, m

xn xn+1xn-1

Equilibrium Position

Deformed Position

us: displacement of the sth atom from its equilibrium position

us-1 us us+1

M

1 1s s s s sF C u u C u u Force on sth plane =

Equation of motion: 2

1 122s

s s s

d uM C u u u

dt

i t

s su t u e → 2

1 1 2s s s sM u C u u u

0

iK as

su u e → 2 2i K a i K aM C e e 2 21 cos

CKa

M

Dispersion relation

2 24 1sin

2

CKa

M

4 1sin

2

CKa

M

(only neighboring planes interact )


G

dv

d K

vG = 0 at zone boundaries

2 1cos

2

CaKa

M

g K

v

1-D:

Group velocity:

4 1sin

2

CKa

M


2

1 12

2

2 12

2

2

ss s s

ss s s

d uM C v v u

dt

d vM C u u v

dt

i sK a i t

su ue

i sK a i t

sv ve →

2

1

2

2

1 2

1 2

i K a

i K a

M u Cv e Cu

M v Cu e Cv

2

1

2

2

2 10

1 2

i K a

i K a

C M C e

C e C M

Vibrations of crystals with two atoms per basis

Ka → π:

(M1 >M2 )

22

1

2 /

2 /

C M optical

acousticalC M

4 2 2

1 2 1 22 2 1 cos 0M M C M M C Ka

=

=

1 22

2 2

1 2

1 12

2

C opticalM M

CK a acoustical

M M

Ka → 0:







• Phonon density of states in 1-D


p atoms in primitive cell → d p branches of dispersion.

d = 3 → 3 acoustical : 1 LA + 2 TA

(3p –3) optical: (p–1) LO + 2(p–1) TO

E.g., Ge or KBr: p = 2 → 1 LA + 2 TA + 1 LO + 2 TO branches

Ge KBr

Number of allowed K in 1st BZ = N


Phonon dispersion in real crystals: aluminium FCC lattice with 1

atom in the basis

In a 3-D atomic lattice we

expect to observe 3 different

branches of the dispersion

relation, since there are two

mutually perpendicular

transverse wave patterns in

addition to the longitudinal

pattern we have considered.

Along different directions in

the reciprocal lattice the

shape of the dispersion

relation is different. But

note the resemblance to the

simple 1-D result we found.

Phonon dispersion in real crystals: FCC lattice with 1 (Al) and 2

(Diamond) atoms in the basis

Characteristic points of the reciprocal space – Γ, X, K, and L points are

introduced at the center and bounduries of the first Brillouin zone








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

A vibrational mode is a vibration of a given wave vector (and thus ),

frequency , and energy . How many modes are found in the

interval between and ?

E

k

),,( kE

),,( kdkdEEd

# modes kdkNdEENdNdN

3)()()(

We will first find N(k) by examining allowed values of k. Then we will be

able to calculate N() and evaluate CV in the Debye model.

First step: simplify problem by using periodic boundary conditions for the

linear chain of atoms:

x = sa x = (s+N)a

L = Na

s

s+N-1

s+1

s+2

We assume atoms s

and s+N have the

same displacement—

the lattice has periodic

behavior, where N is

very large.








• Thermal equilibrium occupancy of phonons – Planck distribution.


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#






• Phonon density of states in 1-, 2-, and 3-D;


• Thermal equilibrium occupancy of phonons – Planck distribution.

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




Andrej Kuznetsov


Tel: +47-22857762,











Module II – Phonons (Chapters 4, 5 and 18) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h





Module III – Electrons (Chapters 6 and 7) 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h







generation 4h



Lecture 7: Lattice heat capacity: Dulong-Petit, Einstien and Debye models

• Repetition of phonon DOS

• Classical theory for heat capacity of solids treating atoms as classical harmonic

oscillators - Dulong-Petit model – success and problems

• Einstein model for heat capacity considering quantum properties of oscillators

constituting a solid – success and problems

• Debye model

• Comparison of different models







• Debye model



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 one

harmonic oscillator







• Debye model


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).


Now for a 3-D lattice we can apply periodic boundary

conditions to a sample of N1 x N2 x N3 atoms:

N1a N2b

N3c

)(8222 3

321 kNVcNbNaN

spacekofvolume

modesof#

Now we know from before

that we can write the

differential # of modes as:

kdkNdNdN

3)()( kdV

3

38

We carry out the integration

in k-space by using a

“volume” element made up

of a constant surface with

thickness dk:

dkdSdkareasurfacekd )(3


A very similar result holds for N(E) using constant energy surfaces for the

density of electron states in a periodic lattice!

dkdSV

dNdN

38

)(Rewriting the differential

number of modes in an interval:

We get the result: k

dSV

d

dkdS

VN

1

88)(

33

Temperature dependence of experimentally measured heat capacity







• Debye model


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


properties of oscillators 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:


properties of oscillators 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


properties of oscillators 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




Energy level

diagram for one

harmonic oscillator

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 !







• Debye model


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







• Debye model





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





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




Energy level

diagram for one

harmonic oscillator

nNE 3

max

min

)(3 nDdE




Andrej Kuznetsov


Tel: +47-22857762,























generation 4h



Lecture 8: Thermal conductivity

• We understood phonon DOS and occupancy as a function of temperature, but what

about transport properties?

• 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








Understanding phonons as «harmonic waves» can not explain thermal

restance since harmonic wafes perfectly move one through another








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

potential 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

VC v

To understand the experimental dependence , consider limiting values

of and (since does not vary much with T).

)(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-naumbers 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?


phonon prperties

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?

Experimental (T)

T-1 ?

T3 estimation

for κ the low

temperature

limit is fine!


phonon prperties

Better estimation for Λ in high temperature limit

NaNak

2121

aNa

NkN

22

NaNak

4222

𝝎𝟐 𝝎𝑫 𝝎𝟏

1/2

«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).

estimate in terms of affecting Λ!

CV

low T

T3

nph 0, so

, but then

D (size)

T3

high T

3R

exp(θD/2T)

exp(θD/2T)


phonon prperties









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

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

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 the 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

𝝎𝟐 𝝎𝑫 𝝎𝟏

1/2

«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).

estimate in terms of affecting Λ!



phonon prperties

CV

low T

T3

nph 0, so

, but then

D (size)

T3

high T

3R

exp(θD/2T)

exp(θD/2T)









Comparison of temperature dependence of κ in crystalline and amorphous solids




Andrej Kuznetsov


Tel: +47-22857762,























generation 4h



Lecture 9: Thermal expansion and repetition of Module II

• Thermal expansion

• Repetition

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.

Potential Energy of Anharmonic Oscillator

(c = 1 g = c/10 f = c/100)

0

2

4

6

8

10

12

14

16

-5 -3 -1 1 3 5

Displacement x (arbitrary units)

Po

ten

tia

l E

ner

gy

U (

arb

. u

nit

s)

U = cx2 - gx3 - fx4 U = cx2

Lattice Constant of Ar Crystal vs. Temperature

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

Usually we write: 00 1 TTLL = thermal expansion coefficient

Documents

FYS3410 - Vår 2015 (Kondenserte fasers fysikk) · 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h 24/2 Effect of temperature – Fermi- Dirac distribution FEFG