Free Electron Gas

8/3/2019 Free Electron Gas

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Free electron Fermi gas (Sommerfeld, 1928)M.C. Chang

Dept of Phys

counting of states

Fermi energy, Fermi surface

thermal property: specific heat transport property

electrical conductivity, Hall effect

thermal conductivity

In the free electron model, there are neither lattice, nor

electron-electron interaction, but it gives good result on electronspecific heat, electric and thermal conductivities etc.

Free electron model is most accurate for alkali metals.


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Energy levels in an empty 1-dim box

Plane wave solution

Box BC

k =/L, 2/L, 3/L

( ) , ( ) / x Ae Be k k mikx ikx= + = h2 2 2

Periodic BC

k =2/L,4/L,6/L


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Energy levels in an empty 3-dim box

box BC periodic BC

Different BCs give the same Fermi wave vector and the same energy

box BC periodic BC

Each point can have 2 electrons (because of spin). After filling in N

electrons, the result is a spherical sea of electrons called the Fermisphere. Its radius is called the Fermi wave vector, and the energy of the

outermost electron is called the Fermi energy.


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Connection between electron density and Fermi energy

For K, the electron density N/V=1.41028 m-3,therefore

Fermi energy and fermi wave vector for typical metals

eVJF 12.21040.319 == 10.746Fk A

=

E

m

N

V

k Tm

vF B F F =F

HG

I

KJ = =

h2 2

2 3

2

2

3

2

/

eF is roughly the order of the atomic binding energy.

kFa is of the order of 1.

The Fermi temperature defined by EF=kTFis of theorder of 10000 K


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Density of states (for electrons)

The number of states within the shell bounded by the constant energy

surfaces with energy Eand E+dEis D(E)dE

Free electron DOS

and dimension

E

D(E)

3D

2D

1D

3

3( ) 2

(2 / )shell

d kD E dE

L=

r

Whats the DOS for free

electron gas in 3 dim?

D E dE k dk

L

E kk

m

D EV m

E

( )

( / )

( )

( )

/

=

=

= FHG

IKJ

24

2

2

2

2

2

3

2 2

2 2

3 2

r h

h


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counting of states

Fermi energy, Fermi surface

thermal property: specific heat

transport property

electrical conductivity, Hall effect

thermal conductivity


7/17

3

32

(2 / )( , )( )dED E d EkN

Lf T

=

r

3

3( ) 2 ( ) ( ) ( , )

(2 / )

d kU T E k dED E Ef E T

L=

rr

Thermal distribution of electrons (fermions)

Combine DOS D(E) and thermal dist f(E,T)

Recall that 2 f E d kk

f E dED E f E k

k

kk

( ) ( ) ( ) ( )rr

r

r

r

r

3 0

3

32 = zz

Hotel rooms

tourists

money

D(e) D(e)

where

and is the energy that makes

(in 3- dim

f Ee

f

T K ET K E

E kT

F

F

( )

( ) /

)( )

( ) /=

+=

= =>

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Free Electron Gas