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Free Electron Gas
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Transcript of Free Electron Gas
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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
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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
( )
( ) /
)( )
( ) /=
+=
= =>