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Failure of the classical gas model

heat capacity

The free-electron Fermi gas

confinement within crystal

density of states function

Paul exclusion principle

Fermi energy

Fermi distribution function

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Free electron gas model: application to a metal

Fermi energy

Heat capacity

Electrical conductivity

Successes and failures

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Fermi energy estimation

The total number of states from E = 0 to E = EF must be equal tothe total number of electrons in the system:

For a 3D metal, there is typically one free electron for every

atom, or in other words one electron for every (31010)3 m3 of

volume (typical volume of a primitive unit cell)

Ne =V

322mE F

2

3 2

EF =

2

2m32

Ne

V

2 3

Ne

V

1

31010( )

3= 4 10

28m3

EF = 4 eV

Ne =A

2

2mE

2

EF=

2Ne

Am

3D:

2D:

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Fermi energy estimation

Comments:

over 100 times kBT at room temperature

EF = 4 eV

EF = kBTF TF 50,000 K

often useful to talk about the Fermi temperature

Electrons at EF have velocities ~ 106 to 107 ms1

step-like T=0 behaviour of Fermi function a

good approximation

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When is a thin film 2 or 3 dimensional?Lx

Ly

Lz

E=2kx

2+ ky

2+ kz

2( )2m

kx =nx

Lx,ky =

ny

Ly,kz =

nz

Lz

The energy levels associated with different nx

and ny

values are more

closely spaced than those associated with different nz values

Electrons will fill levels with increasing nx

and ny

values while nz

= 1

D(E)

E

nz

= 2

nz

= 3

The nz

= 2 level begins to be

occupied when either the

number of electrons or the

temperature is increased

sufficiently

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Heat capacity

In the classical gas every particle acquires extra thermal

energy as the temperature increased, so every particle

contributes to the heat capacity:

C =3

2NekB

(T/TF~1/100 at room

temperature)C =

3

2NekB

T

TF

In the Fermi gas, only electrons within kBT of EF acquire

extra thermal energy

That is roughly a fraction T/TF of the total number

therefore expect

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Electrical conductivity

In equilibrium we now have a

picture of the electrons in a

solid filling up allowed states in

k space up to an energy EF:

ky

kx

Apply an electric field:

Equation of motion is

or

since

dp

dt= eE

dkdt = eEp = k

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Electrical conductivity

In equilibrium we now have a

picture of the electrons in a

solid filling up allowed states in

k space up to an energy EF:

ky

kx

Apply an electric field:

After a time t, an electron with

wavevector k will acquire an extra

k from the field: k = eEt

ky

kx

E

k

Electric field causes all electrons

to transfer from k-state to k-state,

in unison

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Electrical conductivity

If the electrons were in a vacuum, or an infinite, perfect

crystal at absolute zero the Fermi sphere would continue to

shift in this way forever as the electrons accelerated

indefinitely

ky

kx

E

In a real solid it is assumed that

the electrons scatter, on

average, after a time scattering

k = eE

... so each electron acquires, on

average

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Electrical conductivity

Now electrical conductivity is defined by the equation

j=E

Here, vd is the drift velocity the extra velocity due to the

field, which is related to k by:

vd =p

m

=

k

m

=

eE

m hence j=

nee2E

m

=nee

2

m

j= neqvdand (note: ne = Ne/V)

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Electrical conductivity

This is exactly the same formula as for the classical gas

model

BUT the mean free path will be different:

=nee

2

m

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Electrical conductivity

In the classical model, the mean free path was calculated

from

mfp = vthermal electrons were assumed to move at the

thermal velocity

=nee

2

m

1

2mvthermal

2=

3

2k BT

vthermal =3kBT

m

1 2

105 ms1

Remember this gave a mean free path ~ a lattice constantusing a typical room temperature value of

Note: the drift velocity caused by the electric field is very

small compared with this a few mm s1

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Electrical conductivity

In the free-electron fermi-gas model, the velocity to use is

the Fermi velocity

mfp = vF where

=nee

2

m

Consequently, the value of obtained from measurement of

electrical conductivity gives a mean free path ~ 10 100

lattice constants.

1

2mvF

2= EF

vF =2E F

m

1 2

106 ms1

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Furthermore, if we measure the low-temperature

conductivity in very pure crystals, we get mfp ~ size of

crystal scattering is from edges of crystal!

Electrical conductivity

In the free-electron fermi-gas model, the velocity to use is

the Fermi velocity

mfp = vF where

=nee

2

m

1

2mvF

2= EF

vF =2E F

m

1 2

106 ms1

WHAT HAPPENED TO SCATTERING FROM THE LATTICE?

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Consider the effect of applying a magnetic field B to theelectron gas. The electron has magnetic moment B and

so the electrons gain additional energy BB depending

upon whether their spin lies parallel or anti-parallel to B.

EF

D E( )

D E( )

EF

D E( )

D E( )

B

EF

D E( )

D E( )

B

2BB

D EF( )2

m = 2B BB( )D EF( )

2

= m /B = B

2D EF( )

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Successes and failures

explains the small heat capacity with linear T dependence

as observed

predicts small contribution to paramagnetic susceptibility

("Pauli paramagnetism") in metals

gives same (successful) formula for electrical conductivity

as did the classical theory

BUT unexpectedly long mean free path (turns out to be

correct though!)

no explanation of insulators:

insulators have no free electrons BUT

WHY?

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Free electron gas model: application to a metal

Fermi energy

Heat capacity

Electrical conductivity

Successes and failures