### Electrons in metals

```Electrons in metals
Jellium model:
Electron “sees” effective smeared potential

Energy E

electrons shield potential to a large
extent
+
+
+
+
+
+
+
Nucleus with
localized core
electrons
+
Spatial coordinate x
Electron in a box
In one dimension:
In three dimensions:


2
  (r )  V (r )  (r )  E  (r )
2m
where
 V 0  const . for 0  x , y , z  L

V ( x, y, z )  
  otherwise

2
E 
 k
2

2m
where
kx 
2
2m

L
k
nx , k y 
h
E 
2
8 mL
and

2
n
2
x
2
x
 ky  kz

L
2
2
ny , kz 
 ny  nz
2
2


L
nz

n x , n y , n z  1, 2, 3,...
2
 (r )   
L 
3/2
sin k x x sin k y y sin k z z
+

+
+
+
+
+
+
+
+
+
+
+
0
x
+
L
Fixed boundary conditions:
+
+
+
Periodic boundary conditions:
(x  0)  0  (x  L )
 ( x  L, y  L, z  L )   ( x, y, z )
“free electron parabola”
2
 kx
2
1
 (r )   
L 
2m
kx 
dE
2
and
dk
# of states
in [E , E  dE ]
x

L
3/2
e
nx , k y 
ik r
2
L
ny , k z 
2
L
nz
n x , n y , n z  0,  1,  2,  3,...
2
Remember the concept of
L
kx
density of states
1. approach
use the technique already applied for phonon density of states
~
D (E ) 

 ( E  E ( k ))
where
k
E1   E
~
D
 (E )dE
E1
D ( E ) :
1 ~
D (E )
V
Density of states per unit volume
E1   E

   (E  E ( k )) dE  1  1  1  1  4
k
E1
 ( E  E ( k 1 ))
E (k 1 ) E (k 2 )
E
E
Because I copy this part of the lecture from my solid state slides, I use E as the single particle
energy.
In our stat. phys. lecture we labeled the single particle energy  to distinguish it from the total
energy of the N-particle system.
Please don’t be confused due to this inconsistency.
~
D (E ) 


 ( E  E ( k ))
k
ky
k

V
2 
d k
3
3
1/ Volume occupied by a state in k-space
2
L
2
2
L
3
Volume(
L
kx
kz
2  
 2 

 
V
 L 
)
3
Independent from
 and 
2
Free electron gas: E 
 k
2
2
 k

2m
k 
Independent from
 and 
2
d k  4  k dk
3
2m
1

dk 
2 mE
2
1
m

2E
dE
k2
D (E ) 
1 ~
1
D (E ) 
4    ( E  E ( k ))
3
V
2  
D (E )  2
1
2
2
2
1/ 2
m

3
3/2
E
dk
2 mE

2
1
m

2E
dE
1  2m 
D (E ) 
 2 
2
2   
3/2
E
Each k-state can be occupied with 2 electrons of spin up/down
2. approach
2
E (k ) 
 k
calculate the volume in k-space enclosed by the spheres
2
2m
 const .
E ( k )  dE  const .
and
ky
2
L
kx
k
2

2 mE

2
dk 
~
# of states between spheres with k and k+dk : D ( k ) dk 
2 spin states
with
1 ~
D (E )  2 D (E )
V
4  k dk
2
1
m

2E
2  / L 3
 2m 
D (E ) 
 2 
2
2   
1
3/2
E
dE
D(E)
D(E)dE =# of states in dE / Volume
E’
E’+dE
E
The Fermi gas at T=0
f(E,T=0)
D(E)
1
EF
E
E
EF0
Fermi energy
depends on T
#of states in [E,E+dE]/volume

n
 D (E ) f (E , T )dE
0
E
0
F
  D ( E ) dE 
0
 2m 
 2 
2
2   
1
0
3 / 2 EF
Electron density
T=0
Probability that state is occupied
0
EF


E dE
0

2
2m
3  n 
2
2/3

E (k )
Energy of the electron gas: U  2 
e
k
E
  E  EF

U 
1
0
F
0
 2m 

 2 
2
2   
3/2
2
5
05/2
EF
1  2m 
 2  2 
   
3/2
5
0
EF
3
U0 
there is an average energy of
3
5
with electron density
n  10
22
1
cm
3
1
5
0
EF
0
EF


e
0
Energy of the electron gas @ T=0: U 0   E D ( E ) dE 
1

 2m 
 2 
2
2   
1
E
D(E )
  E  EF

1
dE
0
3 / 2 EF
E
E dE
0
03/2
EF
2
2m
3  n 
2
2/3
0
n EF
per electron without thermal stimulation
we obtain
E F  4  12 eV  k B T 
0
1
40
eV @ T  300 K
only a few electrons in the vicinity of EF can be scattered by thermal energy
into free states
Specific heat much smaller than expected from classical consideration
Specific Heat of a Degenerate Electron Gas
Density of occupied states
here: strong deviation from classical value
energy of
electron
state
D(E)
#states in [E,E+dE]

U
E
0
D (E )
f ( E , T ) dE
probability of occupation,
average occupation #
2kBT
Before we calculate U let us estimate:
E
EF
increase energy from
EF  kBT
These
1 D (E F )
2
to
EF  kBT
2
2k B T
 U  D ( E F ) k B T  
2
# of electrons
kBT
EF
n kBT
 U  D ( E F ) k B T 
2
C el
π2
2

D (E F ) k B T
3
subsequent more precise calculation

Calculation of Cel from U 
 E D (E ) f (E , T )dE
0
C el
 U 

 
 T  V

f
 E D (E )  T
0

dE

f
 E  E F  D (E )  T
dE
0
Trick:
0  EF
n
T

 EF

0
D (E )
f
T
dE
E EF
f
T

E  EF
kBT
2
e
kB T
 E EF

 kB T

e
1




2
Significant contributions only in the vicinity of EF
C el 
f
 E  E F  D (E )  T
D(E)

dE
0
D (E )  D (E F )

C el  D ( E F )
 E  E F 
0
with
x :
E  EF
and
kBT
f
dE
T
f
dE  k B T dx
T
C el 
D (E F )
2
x e

EF / k B T

C el 
2
kBT
D (E F )



x
e
x

T ex  1

2
decreases rapidly to zero for

2
kBT
E
EF
e
x
2
x
x e
e
x
1
x
1

2
x  

2
dx

2
3
dx
C el 

2
3
2
k B T D (E F )
C el 

2
3
2
kBT
C el 

D (E F )
2
2
n kB
kBT
EF
 2m 
with D ( E F ) 
 2 
2
2   
1
in comparison with
3/2
EF
C el
and
classical

0
EF
3
2


2
2m
3  n 
2
n kB
 1 for relevant temperatures
Heat capacity of a metal:
C   T  AT
electronic contribution
W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964)
3
lattice contribution
@ T<<ӨD
2/3
```