Electrons in metals
Download
Report
Transcript 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