Thermal conductivity A material's ability to conduct heat.

area

Ejt sec

jt vn

Thermal current density

= Energy per particle

v = velocity

n = N/V

Electric current density

Heat current density

Fourier's Law for heat conduction.

(je = I/A)

2l

Temperature

Thermal conductivityHeat current density

Heat Current Density jtot through the plane: jtot = jright - jleft

About half the particles are moving right, half to left.

x

jt vn

Limit as l gets small:

Thermal conductivity

x

v v

v

Thermal conductivity (expanding to 3d)

Heat current density

Tx

T

22222 3 xzyx vvvvv

Tcvj vt

2

3

1vcv 2

3

1

How does / depend on temperature?

x

1/3 cvv2= =

1/3 cvmv2

ne2ne2/m

Drude applied kinetic theory of gases ½ mv2 = 3/2 kBT

= cvkBT

ne2

The book jumps through claiming a value for cv

Thermal conductivity

Classical Theory of Heat CapacityWhen the solid is heated, the atoms vibrate around their sites like harmonic oscillators.

The average energy for a 1D oscillator is ½ kT. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3/2 kBT, and the total energy density is 3/2 nkBT where n is the conduction electron density and kB is Boltzmann constant.

Differentiation w.r.t temperature gives heat capacity 3/2 n kB

=

ne2

3/2 n kB2T = 3kB

2T / 2e2

Thermal conductivity optimization

To maximize thermal conductivity, there are several options: Provide as many conduction electrons as possible

free electrons conduct heat more efficiently than phonons (=lattice vibrations).

Make crystalline instead of amorphous irregular atomic positions in amorphous materials scatter

phonons and diminish thermal conductivity Remove grain boundaries

gb’s scatter electrons and phonons that carry heat Remove pores (air is a terrible conductor of heat)

What happens?

The Seebeck EffectA temperature gradient generates an electric field E = QT,where Q is known as the thermopower = -cv / 3ne

Seebeck and the reverse (Peltier) Effects

The Seebeck effect is the conversion of temperature differences directly into electricity.

Applications: Temperature measurement via thermocouples; thermoelectric power generators;

thermoelectric refrigerators; recovering waste heatDemo: https://www.youtube.com/watch?v=bt5o_rn0FmU

~ millivolts/K for (Pb,Bi)Te

Many open questions: Why does the Drude model work relatively

well when many of its assumptions seem so wrong? In particular, the electrons don’t seem to be scattered by each other. Why?

Why is the actual heat capacity of metals much smaller than predicted?

From Wikipedia: "The simple classical Drude model provides a very good explanation of DC and AC conductivity in metals, the Hall effect, and thermal conductivity (due to electrons) in metals. The model also explains the Wiedemann-Franz law of 1853.

"However, the Drude model greatly overestimates the electronic heat capacities of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature.“ It also does not explain the positive charge carriers from the Hall effect.

Objectives

By the end of this section you should be able to:

Apply Sch. Equation to a metalApply periodic boundary conditionsStart to understand k spaceDetermine the density of states and Fermi

energyFind the Fermi temperature, velocity, etc.

Improvement to the Drude Model

Sommerfield recognized we needed to utilize Pauli’s exclusion principle

Typically, this is the only difference

Electrons cannot all be in the lowest energy state, since this would violate the Pauli Principle.

Number of electrons per unit volume f(v)

Maxwell-Boltzmann

Fermi-Dirac

Normalization condition solves for constants

= N/V

Another common way to write is f(E)

Sommerfield still assumes the

free electron approximation

U(r)

U(r)

Neglect periodic potential & scattering

Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

What does this remind you of?

The Quantum Analogy These conduction electrons can be considered as

moving independently in a square well and the edges of well corresponds to the edges of the sample. (ignores periodic potential from atoms)

A metal with a shape of cube with edge length of L

Inside U=0, for 3 dimensions:

)(

2 2

22

xUdx

d

m

Cube V=L3in 1D

0 L

U

How do we go about solving this?

0 L

U

Possible Boundary conditions 1. Common: Ψ(0)=0 and Ψ(L)=0

Standing waves. Wells aren’t really infinite2. Periodic: Ψ(x,y,z)= Ψ(x+L,y,z)

rkieV

r

1)(

m

k

2

22Eigenstates with eigenvalues

Known as a running wave

To Compare, Let’s Remind Ourselves of the Standing Wave

Solution

0 L

U

Boundary conditions Ψ(0)=0 and Ψ(L)=0

)sin()( kxAx m

kE

2

22Eigenstates with eigenvalues

L

nk

where

How to find A?

Similar idea for running waves:

Or in 3D:L

nk

L

nk

L

nk z

zy

yx

x

,,

Where nx, ny and nz are integers

LkiLkiLki zyx eee

1

19

Wavefunctions: Ideal Quantum Well

1D

In your group, write the wavefunction for the lowest three energies.

)(x

standing waves

)sin()( kxAx

L

nk

Semiconductor Quantum Well

More about this diagram later today

In

h

e

h

InGaAsGaAs: n ii p

Optical Detection of Spin Polarization in Quantum

WellsCoFe

CoFe

GaAs/InGaAs/GaAs

hote

Aluminatunnelbarrier

external magnetic

field

This is a very simple spin-selective device. Electrons of one angular momentum are favored as they travel past the Schottky barrier due to the external magnetic field and spin filtering in the CoFe. They then fall into the quantum well and recombine with holes. Emission from the quantum well gives a good probe of spin.

The wave vector k is very important!

To see why, note that is an eigenstate of the momentum operator p

rkieV

r

1)(

m

k

2

22Eigenstates with eigenvalues

k is the wave vector(Will explore more in Ch.5)

rkie

Momentum space or k-space is the set of all wavevectors k, associated with particles - free and bound

All points in a crystal that have an identical

environment are described by one point

in k space.

This allows us to dramatically reduce the

size of many atom systems.

The Density of Levels(Closely related to the density of

states)

As we’ll see next time, we will often need to know the number of allowed levels in k space in some k-space volume

If >>2/L, then the number of states is ~ / (2/L)3 (in 3d)

Or V/ (2)3

Then the density of those levels is N/ or V/83

Summing over k spaceSince the volume of k space is V/83,

summing any smooth function F(k) over k space can be approximated as:

Will show an example later

Let’s find the number of allowed k values inside a spherical shell of k-space of radius kF

The number of allowed values of k

Consider a spherical reason

3

4 3Fk

VkVk

N FF2

3

3

3

6)

8)(

3

4(

Since there are two spin states for each k

2

3

3Fk

V

Nn

The Fermi SphereThe k-space sphere with radius kF is called the Fermi sphere.

m

kFF 2

22

If we convert k-space to energy space, the resulting radius of the energy sphere surface would give us the

cutoff between occupied and unoccupied energy levels.

Warning: The Fermi level will be defined slightly differently for nonmetals.

kz

ky

kx

Fermi surface

kF

The surface of the Fermi sphere represent the boundary between occupied and unoccupied k

states at absolute zero for the free electron gas.

2/32 23

2F

NE

m V

1/323F

Nk

V

2

3

3Fk

V

Nn

28

Definition of the Work Function

fermi levelFE

=work function (3-4eV)

Additional energy above the Fermi level required to remove electrons

from the solid

2222

))(2

(2 oF

o

FF ak

a

e

m

k

)(6.132

2

RyrydbergeVa

e

o

Ground state energy of

hydrogen atom

Fermi Energy in terms of the Bohr radius

529.02

2

0 me

a

Å – Bohr radius

Bohr radius = mean radius of the orbit of an electron around the nucleus of a hydrogen atom at its ground state

Recognize?

Electrons in 3D Infinite Potential Well

Group: What is the ground state configuration of many electrons in the 3D infinite potential well? Consider the case of solid with 34 electrons.

Determine the energy of each electron relative to

. How many electrons are of each energy? Take the ground state to be when n1 = n2 =

n3 = 1

2

22

2mL

hEo

Extra slides we may not have time to cover (just extra examples)

Summing the energy density over k space

Since the volume of k space is V/83, summing any smooth function F(k) over k

space can be approximated as:

An extra factor of 2 because of spin

dk=k2sin dk d d

Energy per Electron E/Nin the ground state

V

E2

3

3Fk

V

Nn

Fm

kh

N

E 5

3

10

3 22

Combining:

FBF TkFermi Temperature

The density of copper is 8.96 gm/cm3, and its atomic weight is 63.5 gm/mole. (assume valence of 1)

(a)Calculate the Fermi energy for copper.(b)Find the classical electron velocity from EF= ½ mvF

2 and Fermi temperature. F B FE k T

2/32 23

2F

NE

m V

Pressure and Compressibility of an Electron Gas

(Skip if time low, result most important, in book)

Pressure is defined as E/ V for constant N, so the pressure on an electron gas is

2/32 23

2F

NE

m V

rs – another measure of electronic density = radius of a sphere whose volume is equal to the volume per electron (mean inter-electron spacing)

nN

Vrs 1

3

4 3

in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm)

529.02

2

0 me

a

Å – Bohr radius

Reminder: Effective Radius

1/323F

Nk

V

sssF rrr

k92.1)

49

()

4

33(

3/1

3/13

2

ossF arr

k/

63.392.1 Å-1

Combine with above:

rs/a0 ~ 2 – 6