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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory

http://folk.uio.no/ravi/CMP2013

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Sommerfield Model for Free Electron Theory

1


deBroglie wave concepts

The universe is made of Radiation(light) and

matter(Particles).The light exhibits the dual nature(i.e.,) it can

behave s both as a wave [interference, diffraction phenomenon]

and as a particle[Compton effect, photo-electric effect etc.,].

Since the nature loves symmetry was suggested by Louis

deBroglie. He also suggests an electron or any other material

particle must exhibit wave like properties in addition to particle

nature

Quantum free electron theory


In mechanics, the principle of least action states” that a

moving particle always chooses its path for which the action is a

minimum”. This is very much analogous to Fermat’s principle of

optics, which states that light always chooses a path for which the

time of transit is a minimum.

de Broglie suggested that an electron or any other material

particle must exhibit wave like properties in addition to particle

nature. The waves associated with a moving material particle are

called matter waves, pilot waves or de Broglie waves.


Wave function

A variable quantity which characterizes de-Broglie waves

is known as Wave function and is denoted by the symbol .

The value of the wave function associated with a moving

particle at a point (x, y, z) and at a time ‘t’ gives the probability of

finding the particle at that time and at that point.

de Broglie wavelength

deBroglie formulated an equation relating the momentum

(p) of the electron and the wavelength () associated with it, called

de-Broglie wave equation.

h p

where h - is the planck’s constant.


Schrödinger Wave Equation

Schrödinger describes the wave nature of a particle in

mathematical form and is known as Schrödinger wave equation.

They are ,

1. Time dependent wave equation and

2. Time independent wave equation.

To obtain these two equations, Schrödinger connected the

expression of deBroglie wavelength into classical wave equation

for a moving particle.

The obtained equations are applicable for both

microscopic and macroscopic particles.


6

Schrödinger Time Independent Wave Equation

The Schrödinger's time independent wave equation is given by

08

2

22

)VE(

h

m

For one-dimensional motion, the above equation becomes

08

2

2

2

2

)VE(h

m

dx

d


7

Introducing,

2

h

In the above equation

02

22

2

)VE(m

dx

d

For three dimension,

02

2

2 )VE(m


8

Schrödinger time dependent wave equation

The Schrödinger time dependent wave equation is

tiV

m

22

2

tiV

m

22

2(or)

EH

where H = Vm

2

2

2

= Hamiltonian operator

ti

= Energy operatorE =


9

The salient features of quantum free electron theory

Sommerfeld proposed this theory in 1928 retaining the concept of free

electrons moving in a uniform potential within the metal as in the classical

theory, but treated the electrons as obeying the laws of quantum mechanics.

Based on the deBroglie wave concept, he assumed that a moving electron

behaves as if it were a system of waves. (called matter waves-waves associated

with a moving particle).

According to quantum mechanics, the energy of an electron in a metal is

quantized.The electrons are filled in a given energy level according to Pauli’s

exclusion principle. (i.e. No two electrons will have the same set of four

quantum numbers.)


10

Each Energy level can provide only two states namely, one with spin

up and other with spin down and hence only two electrons can be

occupied in a given energy level.

So, it is assumed that the permissible energy levels of a free electron are

determined.

It is assumed that the valance electrons travel in constant potential

inside the metal but they are prevented from escaping the crystal by

very high potential barriers at the ends of the crystal.

In this theory, though the energy levels of the electrons are discrete, the

spacing between consecutive energy levels is very less and thus the

distribution of energy levels seems to be continuous.


11

Success of quantum free electron theory

According to classical theory, which follows Maxwell-

Boltzmann statistics, all the free electrons gain energy. So it

leads to much larger predicted quantities than that is actually

observed. But according to quantum mechanics only one percent

of the free electrons can absorb energy. So the resulting specific

heat and paramagnetic susceptibility values are in much better

agreement with experimental values.

According to quantum free electron theory, both experimental

and theoretical values of Lorentz number are in good agreement

with each other.


12

Drawbacks of quantum free electron theory

It is incapable of explaining why some crystals have metallic

properties and others do not have.

It fails to explain why the atomic arrays in crystals including

metals should prefer certain structures and not others

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory13

The Free Electron Gas: A Non-trivial Quantum Fluid

Bohr, de Broglie, Schrödinger, Heisenberg, Pauli, Fermi, Dirac….. The

development of the new theory of quantum mechanics.

A natural step was to formulate a quantum theory of electrons in metals.

First done by Sommerfeld.

Assumptions

Most are very similar to those of Drude. Free and independent electrons, but

no assumptions about the nature of the scattering.

22

2m

(7)

Starting point: time-independent Schrödinger equation

Summerfeld’s Quantum Mechanical Model of Electron

Conduction in Metals

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 14

To solve (7), we need appropriate boundary conditions for a metal.

Standard ‘particle in a box’: set ψ = 0 at boundaries. This is

not a good representation of a solid, however.

a) It says that the surface is important in determining the

physical properties, which is known not to be the case.

b) It implies that the surfaces of a large but not infinite sample

are perfectly reflecting for electrons, which would make it

impossible to probe the metallic state by, for example, passing a

current through it.

Note that no other potential terms are included; hence we can

solve

for a single, independent electron and then investigate the

consequences of putting in many electrons.


(We consider a cube of side L for mathematical convenience; a different

choice of sample shape would have no physical consequence at the end of

the calculation.)

Here V = L3 and the V-1/2 factor ensures that normalisation is correct, i.e. that

the probability of finding the electron somewhere in the cube is 1.

Solving then gives allowed wavefunctions:

L

pke

Vzyx x

zkykxki

kzyx

2,

1,,

)(

2/1, p integer, etc. (9)

Most appropriate boundary condition for solid state physics: the periodic

boundary condition first introduced by Born and von Karman:

zyxLzLyLx ,,,, (8)


What is the physical meaning of these eigenstates?

Then, note that k is also an eigenstate of the momentum operator

i

p

ˆ , with eigenvalue p = k.

The state k is just the de Broglie formulation of a free particle! It has a

definite momentum k.

Then we see the close analogy with a well-known classical result:

m

p

m

kk

22

222

(11)

m

kk

2

22 (10)First, note energy eigenvalues:


It thus also has a velocity v = k/m.

How does the spectrum of allowed states look?

Cubic grid of points in k-space, separated by 2/L; volume per point (2/L)3.

So, why have we come anywhere here? We have just done a quantum

calculation of a free particle spectrum, and seen close analogies with that of

classical free particles.

Answer: now we have to consider how to populate these states with a

macroscopic number of electrons, subject to the rules of quantum mechanics.

Sommerfeld’s great contribution: to apply Pauli’s exclusion principle to the

states of this system, not just to an individual atom.


Each k state can hold only two electrons (spin up and down). Make up the

ground (T = 0) state by filling the grid so as to minimise its total energy.

Result: At T = 0, get a sudden demarkation between filled and empty states,

which (for large N), has the geometry of a sphere.

. . . . . .

. . . . . .

ky

kz

kx

State separation

2L

State volume

(2L)3

Filled

states

Empty

statesFermi surface

Fermi

wavenumber

kF


We set out to do a quantum Drude model, and did not explicitly include any

direct interactions due to the Coulomb force, but we ended up with something

very different. The Pauli principle plays the role of a quantum mechanical

particle-particle interaction.

The quantum-mechanical ‘free electron gas’ is a non-trivial quantum fluid!

Is everything OK here - doesn’t kF appear to depend on the arbitrary cube

size L?

Quantities of interest depend on the carrier number per unit volume; the

sample dimensions drop out neatly.

No -

3/1

2

3

3 32

23

4

V

Nk

L

Nk FF

(12)


How can we scale these quantum mechanical effects against something we are

more familiar with?

Calculate numerical values for the parameters. Use potassium (tutorial

question 4).

Result: kF 0.75 Å-1

vF 1 x 106 ms-1

F 2 eV

This is a huge effect: zero point motion so large that a Drude gas of

electrons would have to be at 25000 K for the electrons to have this

much energy!

TF 25000 K ( recall kBT at room T 1/40 eV)


kkF

A couple of much-used graphs relating to the Sommerfeld model:

a) The free electron

dispersion

Probability

of state

occupation

1

0 , k

F or

kF

b) The T = 0 state occupation

function.


The specific heat of the quantum fermion gas

1

1),(

/)(

TkBe

Tf

The T=0 occupation discussed previously is a limit of the Fermi-Dirac

distribution function for fermions:

where the chemical potential F. (13)

As expected, T is a minor player

when it comes to changing things.

At finite T:

f()

F

~ 2kBT


The Fermi function gives us the probability of a state of energy being

occupied. To proceed to a calculation of the specific heat, we need to know

the number of states per unit volume of a given energy that are occupied

per unit energy range at a given T.

Our next task, then, is to derive a quantity of high and general importance, the

density of states g().

and the specific heat cel from dEtot/dT as before.

),()(),( TfgTn (14)

dTnTEtot ),()(0

Then internal energy Etot(T) can be calculated from

(15)


. . . . . .

. . . . . .

ky

kz

kx

State separation

2L

State volume

(2L)3dk

3

2

332

42

.

.2)(

L

dkk

LkperVol

katshellofVol

Ldkkg

Number of allowed states per unit volume per shell thickness dk:

spin


Convert to density of states per unit volume per unit (the quantity usually

meant by the loose term ‘density of states’):

32

2/12/3

3

2/12

22

2

2)(

)2(

2

24

2)(

mg

dm

mm

dg

Very important result, but note that dependence is different for

different dimension .

2/1

22

2;

mk

k

mddk (16a, b)

(17)


n(,T)

F

g(

Movement of

electrons in

energy at finite T

2kBT

Evaluating integral (15) is complicated due to the slight movement of the

chemical potential with T (see Hook and Hall and for details Ashcroft and

Mermin). However, we can ignore the subtleties and give an approximate

treatment for F >> kBT:

[Etot(T) - Etot(0)]/V 1/2g(F). kBT.2kBT = g(F). (kBT)2 (18)


Differentiating with respect to T gives our estimate of the specific heat

capacity:

cel = 2g(F). kB2T (19)

The exact calculation gives the important general result that

cel = (23g(F). kB2T (20)

c.f. Drude: Bnk

2

3

How does this compare with the classical prediction of the Drude model?

Combining g(F) from (17) with the expression for F derived in tutorial

question 4 gives, after a little rearrangement :

F

BBel

Tknkc

2

2

(21)


A remarkable result: Even though our quantum mechanical interaction leads to

highly energetic states at F, it also gives a system that is easy to heat, because you

can only excite a highly restricted number of states by applying energy kBT.

The quantum fermion gas is in some senses like a rigid fluid, and its thermal

properties are defined by the behaviour of its excitations.


What about the response to external fields or temperature gradients?

To treat these simply, should introduce another vital and wide-ranging concept,

the Semi-Classical Effective Model.

Faced with wave-particle duality and a natural tendency to be more comfortable

thinking of particles, physicists often adopt effective models in which quantum

behaviour is conceptualised in terms of ‘classical’ particles obeying rules

modified by the true quantum situation.

In this case, the procedure is to think in terms of wave packets centred on each

k state as particles. Each particle is classified by a k label and a velocity v.

Velocity is given by the group velocity of the wave packet:

v = dw/dk = -1d/dk = k/m for free particles like those we are concerned

with at present.


Assumption of the above: we cannot localise our ‘particles’ to better

than about 10 lattice spacings. The uncertainty principle tells us that if we try

to do that, we would have to use states more than 10% of our full available

range (defined roughly by kF).

Not, however, a particularly heavy restriction, since it is unlikely that we would

want to apply external fields which vary on such a short length scale.

In the absence of scattering, we then use the following ‘classical’ equation of

motion in applied E and/or B fields:

mdv/dt = dk/dt= -eE - ev B (22)

This equation would produce continuous acceleration, which we know cannot

occur in the presence of scattering.


Include scattering by modifying (22) to

m(dv/dt + vt -eE - ev B (23)

This is just the equation of motion for classical particles subject to ‘damped

acceleration’. If the fields are turned off, the velocity that they have acquired

will decay away exponentially to zero. This reveals their ‘conjuring trick’.

The physical meaning of v in (23) must therefore be the ‘extra’ or ‘drift’

velocity that the particles acquire due to the external fields, not the group

velocity that they introduced in their (3.22).

In fact, this is formally identical to the process that we discussed in deriving

equation when we discussed the Drude model!

It is no surprise, then, that it leads to the same expression for the electrical

conductivity:


Set B to zero and stress that the relevant velocity is vdrift ; (23) becomes

m(dvdrift/dt + vdriftt -eE

Steady state solution (dvdrift/dt = 0) is just

vdrift = -(et/m)E

Following the procedure from Kittel gives us the Drude expression (3):

m

ne t

2

If you give this some thought, it should concern you. What happened to our

new quantum picture?


ky

kz

kx

To understand, consider physical meaning of the process:

Fermi surface is shifted along the kx axis by an E field along x. The ‘quasi-

Drude’ derivation assumes that every electron state in the sphere is shifted by

dk. This is ‘mathematically correct’, but physically entirely the wrong picture.

ky

kz

kx

dk = -1mvdrift= -eEτ/E = 0


Which states can ‘interact with the

outside world’?

ky

kz

kx

dk

Pauli principle: only those states can scatter, so

only processes involving them can relax the

Fermi surface. So how does the ‘wrong’

picture work out?

Consider amount of extra velocity/momentum acquired in equilibrium:

Drude-

like

picture:dkk

LF .

3

42 3

3

In the quantum model, only those

within kBT of F, i.e. those very near the

Fermi surface.

# of states mom.

gain

# of states (1/2

FS area)

mom.

gainx comp.

only

Quantum

picture:F

F kdkk

L 3

2.

2

42 23

(24)

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 35

So the two pictures, one of which is conceptually incorrect, give the

same answer, because of a cancellation between a large number of

particles acquiring a small extra velocity and a small number of

particles acquiring a large extra velocity.

However, this is only the case for a sphere. As we shall see later,

Fermi surfaces in solids are not always spherical. In this case, the

Drude-like picture is simply wrong, and the conductivity must be

calculated using a Fermi surface integral.


What about thermal conductivity?

Recall (4) from Drude model: k = 1/3vrandomlcel

Here, vrandom can clearly be identified with vF, and l = vFt.

Provided that t is the same for both electrical and thermal conduction

(basically true at low temperatures but not at high temperatures; see Hook

and Hall Ch. 3 after we have covered phonons), we can now revisit the

Wiedemann-Franz law using (21) for the specific heat:

2222

2 323

11

t

t

k

e

kTknkv

ne

m

TT

B

F

BBF

(25)


The approximate factor of two error from the Drude model has been

corrected (2/3 in quantum model cf. 3/2 in Drude model).

Real question - how on earth was the Drude model so close?

Answer: Because a severe overestimate of the electronic specific heat was

cancelled by a severe underestimate of the characteristic random velocity.

Thinking for the more committed (i.e. non-examinable): Would all quantum

gas models give the same result for the Wiedemann-Franz law as the

quantum fermion gas?


The modern conceptualisation of the quantum free electron gas:

Make an analogy with quantum electrodynamics (QED).

Filled Fermi sea at T = 0 is inert, so it is the vacuum. Temperature and / or

external fields excite special particle-antiparticle pairs. The role of the positron is

played by the holes (vacancies in the filled sea with an effective positive charge).

dk

ky

kz

dk

Thermal excitation: All particles

with k kF, but sum over k = 0.

kx

ky

kz

Electrical excitation: All particles

with k kF, but sum over k = 2 kF/3.

kx


Scorecard so far; achievements and failures of the quantum Fermi gas

model

1. Successful prediction of basic thermal properties of metals.

2. Successful prediction of conductivity, as long as we don’t ask about the

microscopic origins of the scattering time t - why is the mean free path so

long in metals at low temperatures? What happened to electron-ion and

electron-electron scattering?

3. Failure to predict a positive Hall coefficient.

4. No understanding whatever of insulators. ‘… So insulators, which cannot

carry a current, must contain electrons too. In a metal they must be free to move,

and in an insulator they must be stuck.


Classical Drude gas

Random velocity purely

thermal: mTkB /3

Specific heat cel = Bnk

2

3

Large number of particles

moving slowly.

Quantum Sommerfeld gas: do wave

mechanics and then think in an

‘equivalent particle’ picture

Random velocity dominantly

quantum (due to Pauli principle):

mV

Nmkv FF /3/

3/1

2

F

BBel

Tknkc

2

2

Small effective number of particles

moving very fast, due to special

quantum mechanical constraints.


41


The Sommerfeld Model

Electrons are fermions.

- Ground state: Fermi sphere,

- Distribution function

Modification of the Drude model

- the mean free path

- the Wiedemann-Frantz law

- the thermopower

3/12 )3( nkF

tt FT vvl

22

2 )(3

)(2

3

e

k

e

k

T

BB

k

))((62

2

F

BBB

E

Tk

e

k

e

kQ

)2

exp()2

()(2

2/3

Tk

mv

Tk

mnvf

BB

MB

1]/)2

1exp[(

1

4

)/()(

23

3

TkEmv

mvf

BF

FD

B

F

BBvF

BT kn

E

Tkkncv

m

Tkvv )(

22

3,)

3(:

22/1


The Sommerfeld theory of metals

the Drude model: electronic velocity distribution

is given by the classical

Maxwell-Boltzmann distribution

the Sommerfeld model: electronic velocity distribution

is given by the quantum

Fermi-Dirac distribution

3/ 22

3

32

0

( ) exp2 2

/ 1( )

14

2exp 1

( )

MB

B B

FD

B

B

m mvf n

k T k T

mf

mv k T

k T

n d f

v

v

v vnormalization

condition T0

Pauli exclusion principle: at most one electron

can occupy any single electron level

43


)()(2 2

2

2

2

2

22

rErzyxm

, , , ,

, , , ,

, , , ,

x y z L x y z

x y L z x y z

x L y z x y z

k

m

m

kE

d

eV

i

2

2)(

)(1

1)(

22

2

k

kv

kp

k

rr

r rk

k

22

2

1

2mv

m

pE

electron wave function

associated with a level of energy E

satisfies the Schrodinger equation

consider noninteracting electrons

L

3D:

1D:

periodic

boundary

conditions

a solution neglecting

the boundary conditions

normalization constant: probability of finding the electron somewhere in the whole volume V is unity

energy

momentum

velocity

wave vector

de Broglie wavelength

44


1

2 2 2, ,

yx zik Lik L ik L

x x y y z z

e e e

k n k n k nL L L

rk

k r ieV

1)(

, , , ,

, , , ,

, , , ,

x y z L x y z

x y L z x y z

x L y z x y z

apply the boundary conditions

components of k must be

nx, ny, nz integers

3

33

2

2/2

V

V

L

a region of k-space of volume contains

states i.e. allowed values of k

the number of statesper unit volume of k-space, k-space density of states

VL

L

33

2

22

2

the area per point

the volume per point

k-space

45


compare to the ~ 107 cm/s at T=300Kclassicalthermal velocity 0 at T=0

2/1

22

/3

/

2/

mTkv

mkv

kp

kET

mkE

k

Bthermal

FF

FF

BFF

FF

F

Fermi wave vector ~108 cm-1

Fermi energy ~1-10 eV

Fermi temperature ~104-105 K

Fermi momentum

Fermi velocity ~108 cm/s

consider T=0

3 3

3 2

3

2

3

2

4

3 62

26

3

F F

F

F

k V kV

kN V

kn

the number of allowed values of

k within the sphere of radius kF

to accommodate N electrons

2 electrons per k-level due to spin

the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF

312

3222

312

3

32

3

nm

v

nm

E

nk

F

F

F

kx

ky

kFFermi sphere

Fermi surface

at energy EF

the Pauli exclusion principle postulates that only one electron can occupy a single state therefore, as electrons are added to a system, they will fill the states in a system like water fills a bucket – first the lower energy states and then the higher energy states

state of the lowest energydensity of states

volume

46


Total number of states with energy < E

The density of states – number of states per unit energy EmV

dE

dNED

mEVN

23

22

23

22

2

2)(

2

3

Density of states

32

Vk-space density of states – the number of states per unit volume of k-space

The density of states per unit volume or the density of

states

Em

dE

dnED

23

22

2

2

1)(

3

23

VN k

Total number of states with wave vector < k 2 2

2

kE

m

47


FF

F

kk

Vei

kk

Em

k

N

E

m

k

m

kd

V

E

dFV

FV

F

V

km

E

F

F

5

3

10

3

10

1

24

1

)(8

)(8

)(

8

22

22

52

2

22

3

3

..0

3

3

22

k

kkkkk

k

k

kk

dkkd

m

kF

2

22

4

2)(

k

k

3

23Fk

N V

Ground state energy of N electrons

Add up the energies of all electron states inside the Fermi sphere

volume of k-space per state

smooth F(k)

The energy density

The energy per electronin the ground state

48


In quantum mechanics particles are indistinguishable

systems where particles are exchanged are identical

exchange of identical particles can lead to changing of the system wave function by a phase factor only

repeated particle exchange → e2ia 1

1221 ,, aie

1221 ,,

122121 21212

1, pppp 122121 21212

1, pppp

Antisymmetric wavefunction with respect

to the exchange of particles

Fermions are particles which have half-integer spin

the wavefunction which describes a collection of Fermions must be antisymmetric with respect

to the exchange of identical particles

Fermions: electron, proton, neutron

if p1 = p2 0

→ at most one fermion can occupy

any single particle state – Pauli principle

Unlimited number of bosons can occupy

a single particle state

p1, p2 – single particle states

obey Fermi-Dirac statistics Obey Bose-Einstein statistics

system of N=2 particles

1, 2 - coordinates and

spins for each of the

particles

Remarks on statistics I

Bosons are particles which have integer spin

the wavefunction which describes a collection of bosons must be symmetric with respect

to the exchange of identical particles

Bosons: photon, Cooper pair, H atom, exciton

symmetric wavefunction with respect

to the exchange of particles

49


1( )

exp 1

1( )

exp 1

( ) exp

( ) ( ) ( )

FD

B

BE

B

MB

B

f EE

k T

f EE

k T

Ef E

k T

n dEn E dED E f E

Fermi-Diracdistribution function

Bose-Einsteindistribution function

Maxwell-Boltzmanndistribution function

Distribution function f(E) → probability that a state at energy E

will be occupied at thermal equilibrium

fermionsparticles with half-integer spins

bosonsparticles with integer spins

both fermions and bosons at high Twhen TkE B

degenerate Fermi gas

fFD(k) < 1

degenerate Bose gas

fBE(k) can be any

classicalgas

fMB(k) << 1

=(n,T) – chemical potential

50


BE and FD distributions differ from the classical MB distribution

because the particles they describe are indistinguishable.

Particles are considered to be indistinguishable if their wave packets

overlap significantly.

Two particles can be considered to be distinguishable

if their separation is large compared to their de Broglie wavelength.

Electron gas in metals:

n = 1022 cm-3, m = me → TdB ~ 3×104 K

Gas of Rb atoms:

n = 1015 cm-3, matom = 105me → TdB ~ 5×10-6 K

Excitons in GaAs QW

n = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K

At T < TdB fBE and fFD are strongly different from fMB

At T >> TdB fBE ≈ fFD ≈ fMB

1 22

1 3

22 3

2~

~

2

dB

B

dB

dB

B

h

mk T p

d n

T nmk

Thermal de Broglie

wavelength

Particles become

indistinguishable when

i.e. at temperatures below

remarks on statistics II

A particle is represented by a

wave group or wave packets

of limited spatial extent,

which is a superposition of many matter

waves with a spread of wavelengths

centered on 0=h/p

The wave group moves

with a speed vg – the group speed,

which is identical to the classical

particle speed

Heisenberg uncertainty principle, 1927:

If a measurement of position is made with

precision x and a simultaneous

measurement of momentum in the x

direction is made with precision px,

then

2xp x

g(k’)

k0

kk’

m v

vg=v

x

x

(x)

2

'

'( , ) ( ')exp '

2

kt g i t

m

k

r k k r

x

51


3 2

2 2

1 2( )

2

dn mD E E

dE

Density of states

Distribution function1

( )

exp 1

( ) ( )

B

f EE

k T

n dED E f E

3D

T≠0 the Fermi-Dirac distribution

E

EEfT

0

,1)(lim0

FT

E

0

lim

VdEEfED

VdEED

1)()(

1)( [the number of states in the energy range from E to E + dE]

[the number of filled states in the energy range from E to E + dE]

EF E

Density of

filled states

D(E)f(E,T)

shaded area – filled

states at T=0

Density of

states

D(E)

per unit volume

52


Specific heat of the degenerate electron gas, estimate

Bv

B

nkc

Tnkmvnu

2

3

2

3

2

1 2

1v

V V

U uc

V T T

Uu

V

Specific heat

U – thermal

kinetic energy

Classical gas

The observed electronic

contribution at room T is

usually 0.01 of this value

Classical gas: with increasing T all electron gain an energy ~ kBT

Fermi gas: with increasing T only those electrons in states within

an energy range kBT of the Fermi level gain an energy ~ kBT

Number of electrons which gain energy with increasing temperature ~

The total electronic thermal kinetic energy

The electronic specific heat1

~ Bv B

V F

U k Tc nk

V T E

EF/kB ~ 104 – 105 K

kBTroom / EF ~ 0.01

f(E) at T ≠ 0 differs from f(E) at T=0

only in a region of order kBT about because electrons just below EF have been excited to levels just above EF

T ~ 300 K for typical metallic densities

T = 0

B

F

k TN

E

~ BB

F

k TU N k T

E

53


0

3

0

3

)()()(4

)()()()(4

EfEdEDEfd

n

EEfEdEDEfEd

u

kk

kkk

)()()()(

)()(6

)(

)()()(6

)(

00

422

0

422

0

FF

E

B

B

EHEdEEHdEEH

TODTkdEEDn

TODDTkdEEEDu

F

correctly to order T2

Specific heat of the degenerate electron gas

The way in which integrals of the form differ from their zero T values

is determined by the form of H(E) near E=

E

nn

nn

EHdE

d

n

EEH )(

!

)()(

0

Replace H(E) by its Taylor expansion about E=

The Sommerfeld expansion

6

44

22

12

12

1

2

)(360

7)(

6)(

)()()()(

TkOHTkHTkEEH

EHdE

daTkdEEHdEEfEH

BBB

En

n

n

n

n

B

dEEfEH )()(

FE

dEEH )(

Successive terms are smaller by O(kBT/)2

For kBT/ << 1 Replace

by T0 = EF

and v

V

uu c

T

54


FD statistics depress

cv by a factor of

2

22

0

22

2

2

11

3 2

( )6

( )3

3( )

2

2

3

2

3

BF

F

B F

v B F

F

F

Bv B

F

classical B

B

F

v

k TE

E

u u k T D E

uc k TD E

T

nD E

E

k Tc nk

E

c nk

k T

E

c T

(1)

(2)

2

2 323

2FE n

m

3 2

2 2

1 2( )

2

mD E E

Specific heat of the degenerate electron gas55


Thermal conductivity

vv

q

lvccv

T

3

1

3

1 2

tk

kj

thermal current density jq – a vector parallel to the direction of heat flow whose magnitude gives the thermal energy per unit time crossing a unite area perpendicular to the flow

Tkmv

nkc

Te

k

ne

mvc

B

Bv

Bv

2

3

2

1

2

3

2

3

3

2

2

2

2

kWiedemann-Franz law (1853)

Lorenz number ~ 2×10-8 watt-ohm/K2

Drude:

application of

classical ideal

gas laws

success of the Drude model is due to the cancellation of two errors: at room T the actual electronic cv is 100 times smaller than the classical prediction, but v is 100 times larger

m

ne t

2

the correct at room T

the correct estimate of v2 is vF2 at room T

01.0~/~2

2

FBclassicalvvB

F

Bv ETkccnk

E

Tkc

100~/~22

TkEvv BFclassicalF

22

3

e

k

T

B

k

For

degenerate

Fermi gas of

electrons

56


Thermopower

ne

cQ v

3

Drude:

application of

classical ideal

gas laws

Bv nkc2

3

e

kQ B

2

Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field

directed opposite to the T gradient

TQE

Thermopower

For

degenerate

Fermi gas of

electrons

the correct at room T

Q/Qclassical ~ 0.01 at room T

01.0~/~2

2

FBclassicalvvB

F

Bv ETkccnk

E

Tkc

F

BB

E

Tk

e

kQ

6

2

high T low T

gradT

E

thermoelectric field

57


Electrical conductivity and Ohm’s law

t

t

2

2

1

ne

m

m

ne

( ) (0)

avg

avg

avg

d dm e

dt dt

et t

e

e

m m

t

t

v kE

Ek k

Ek

k Ev

Ej

m

ne t2

avgnevj

Equation of motionNewton’s law

In the absence of collisions the Fermi sphere in k-space is displaced as a whole at a uniform rate by a constant applied electric field

Because of collisions the displaced Fermi sphere is maintained in a steady state in an electric field

Ohm’s law

the mean free path l = vFt

because all collisions involve only electrons near the Fermi surface

vF ~ 108 cm/s for pure Cu:

at T=300 K t ~ 10-14 s l ~ 10-6 cm = 100 Å

at T=4 K t ~ 10-9 s l ~ 0.1 cm

kavg << kF for n = 1022 cm-3 and j = 1 A/mm2 vavg = j/ne ~ 0.1 cm/s << vF ~ 108 cm/s

kavg

( ) ( )( ) 0

d t tt

dt

e

t

t t

p pf

p f E

F

Fermi sphere

ky

kx

58

Prof.P. Ravindran, - Universitetet i oslofolk.uio.no/ravi/cutn/cmp/SomerfieldModel.pdf · According...

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Transcript of Prof.P. Ravindran, - Universitetet i oslofolk.uio.no/ravi/cutn/cmp/SomerfieldModel.pdf · According...