Transcript Slide 1
The Quantized Free Electron Theory
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
V0 const. f or 0 x, y, z L
V( x, y, z )
otherw ise
2
2k
2 2
E
k x k 2y k 2z
2m
2m
L
L
L
where k x nx , k y n y , k z nz
h2
E
n2x n2y n2z
2
8mL
and
nx , ny , nz 1, 2, 3,...
2
(r )
L
3/ 2
sin k x x sin k y y sin k z z
+
+
+
+
+
+
+
+
+
+
+
+
0
+
x
+
L
+
+
Periodic boundary conditions:
Fixed boundary conditions: ( x 0) 0 ( x L)
( x L, y L, z L) ( x, y, z)
“free electron parabola”
2k x
2m
2
1
(r )
L
kx
dE
3/ 2
eikr
2
2
2
nx , k y
ny , k z
nz
L
L
L
and nx , ny , nz 0, 1, 2, 3,...
dk x
# of states
in [E, E dE]
2
L
Remember the concept of
kx
density of states
1. approach
use the technique already applied for phonon density of states
~
D(E) (E E(k ))
where D(E) :
k
E1 E
~
D
(E)dE
E1
k
1~
D(E)
V
Density of states per unit volume
E1 E
(E E(k )) dE 1 1 1 1 4
E1
(E E(k1 ))
E(k1 ) E(k 2 )
E
E
~
D(E) (E E(k ))
k
ky
k
2
3
2
L
2
2
)
V
L
3
Volume(
kx
kz
3
d
k
1/ Volume occupied by a state in k-space
2
L
2
L
V
3
Independent from
and
Independent from
and
2
2k
2k 2
Free electron gas: E
2m
2m
k
d3 k 4k 2dk
1
2mE
dk
1 m
dE
2E
k2
dk
1~
1
2mE 1 m
D(E) D(E)
4 (E E(k ))
dE
3
2
V
2E
2
D(E) 2
1/ 2
3/ 2
1 2 m
2 2
3
E
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
calculate the volume in k-space enclosed by the spheres
2k 2
E(k )
const.
2m
E(k ) dE const.
and
ky
2
L
kx
k2
2mE
2
4k 2 dk
~
# of states between spheres with k and k+dk : D(k )dk
2 / L 3
dk
1 m
dE
2E
2 spin states
with
1~
D(E) 2 D(E)
V
1
D(E)
22
2m
2
3/ 2
E
D(E)
D(E)dE =# of states in dE / Volume
E’
E’+dE
E
Statistics of the electrons (fermions)
Fermions are indistinguishable particles which obey the Pauli exclusion principle
T=0
E
Let us distribute
4 electrons spin
4 electrons spin
En=6
En=5
EF
En=4
1
f(E,T=0)
En=3
En=2
En=1
Probability that a qm state is occupied
Occupation
number 0
for state
E En4
Occupation
number 1
for states
E En4 : EF
of a given spin
x
Fermi Dirac distribution function at T>0
1
f (E, T )
e
E
kBT
here chemical
potential
1
EF
Fermi energy
With accuracy sufficient for many estimations: f(E,T) linearized at EF
More detailed approach to Fermi statistics
The grand canonical ensemble
Particle reservoir
U En n
Average energy
n
N Nn n Average particle #
System
n
1 n
n
Normalized
probabilities
T=const.
Heat Reservoir R
E i ni
Now we consider independent particles
i
U E i ni
Total energy of
N fermion system
occupation # ni=0,1
of single particle state i with energy i
i
N ni ni
i
i
where n j
( n1 ,n2 ,...)
n j (n1, n2 ,...)
average occupation of state j is given by
U
F
G
N S,V N T,V N T,P
Chemical potential
n j
( i ) ni
n
e
j
For details see
& additional info see
{ni }
( i ) ni
e
(n1 , n2 ,...)
{ni }
where the summation
...
n j
( i ) ni
n
e
j
means
n j
( i ) ni
e
{ni }
...
Repeat this step
( n1 , n2 ,...)
{ni }
e
n1
e
n1
1 n1
n je
n2 , n3 ,...
1 n1
e
n je
i 2
i ni
i 2
...
n j 0
e
nj
n2 , n3 ,...
n j f ( j , T )
1
i ni
1
e ( i ) 1
j n j
j n j
The Fermi gas at T=0
f(E,T=0)
D(E)
1
EF
E
EF0
#of states in [E,E+dE]/volume
E0
F
1
n D(E)f (E, T )dE D(E) dE
2
2
0
0
Electron density
T=0
Probability that state is occupied
E
Fermi energy
depends on T
2m
2
0
3 / 2 EF
E dE
0
2
0
2 2/3
EF
3 n
2m
E0
F
1
Energy of the electron gas @ T=0: U0 E D(E) dE
2
2
0
1 2m
2 2
2
3/ 2
2m
2
0
3 / 2 EF
E
E dE
0
2 0 5 / 2 1 2m 3 / 2 1 0 0 3 / 2
EF
EF EF
2 2
5
5
2
0
2 2/3
EF
3 n
2m
3
U0 n EF0
5
there is an average energy of 3 EF0 per electron without thermal stimulation
5
with electron density n 1022
1
cm3
we obtain EF0 4 12 eV k B T
1
eV @ T 300K
40
Click for a table of Fermi energies,
Fermi temperatures and Fermi velocities
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
D(E)
energy of
electron
state
#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:
EF
E
increase energy from EF kBT to EF kBT
These
1 D(EF )
2k B T
2
2
U D(EF ) kBT
2
# of electrons
kBT
n kBT
EF
π2
Cel
D(EF ) kB2 T
3
U D(EF ) kBT
2
subsequent more precise calculation
Calculation of Cel from U E D(E)f (E, T )dE
0
f
U
Cel
dE
E D(E)
T
T V 0
f
E EF D(E) T dE
0
Trick:
n
0 EF
EF
T
0
D(E)
f
dE
T
EEF
kB T
f E EF
e
2
T k BT 2 EEF
e kBT 1
Significant contributions only in the vicinity of EF
D(E)
f
Cel E EF D(E)
dE
T
0
D(E) D(EF )
Cel D(EF )
f
E
E
F
T
0
with
dE
f
x
ex
T T e x 1 2
E EF
x :
and dE kBT dx
k BT
Cel
D(EF )
x 2e x
EF / kBT
Cel
k B2 T
D(EF )
decreases rapidly to zero for x
k B2 T
E
EF
e
x
x 2e x
1
e x 12
2
dx
2
3
dx
2 2
Cel
k BT D(EF )
3
2 2
Cel
k BT D(EF )
3
2
k T
Cel n kB B
2
EF
1 2m
with D(EF )
22 2
in comparison with
1
3/ 2
EF
Cel
2
0
2 2/3
3 n
and EF
2m
classical
3
n kB
2
for relevant temperatures
Heat capacity of a metal:
C T AT
3
electronic contribution
W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964)
lattice contribution
@ T<<ӨD
Selected phenomena which don’t require detailed knowledge of the band structure
Temperature dependence of the electrical resistance
T
T5
residual
T
Impurities: temperature independent imperfection scattering
phonon scattering
Scattering of electrons: deviations from a perfect periodic potential
Matthiessen’s rule:
(T) residual phon (T)
Simple approach to understand
phon T
for T>>ӨD
e2 n
Remember Drude expression:
m
1 1
scattering rate
scattering cross section
1
N
V
vF
u2
scattering cross section
Fermi velocity of electrons:
u u0 cos t
1
2
u u0
2
vF 2EF / m
#of scattering centers/volume
u
2
1
3N
D
u
2
D()d
0
9N
D
3
1
3N
D
0
3
1 0 2 9N 2
u
d
3
3
M
2
D
D
2
compare lecture notes:
Thermal Properties of Crystal Lattices
E( )
2
1
M2u0
2
D
0
1
1
d
2
kB T
e
1
1
1
2
e kB T 1
u2
u
3
3
D M
2
D
0
1
1
d
2
e kBT 1
kBT 2
3
MD
3
D / T
0
with x
kBT
and d
kBT
dx
1
1
x x
dx
2 e 1
Let us consider the high temperature limit: T>>ӨD
u
2
kBT 2
3
MD
3
D / T
0
Note:
T5 –low temperature dependence
not described by this simple approach
3 2 T
1
1
x x
dx
M kB D 2
2 e 1
1
1
x x
1
2
e 1 x0
Lindemann melting temperature TM: u
2
TM
1 1
u2 T
emperical value
3 2 TM
where
M k B D 2
u2
0.1
average atomic
spacing
xm rs
2
TM
Thermionic Emission
Finite barrier height of the potential
E
Evac
: Evac EF work function
EF
2k 2x
EF
2m
Current density
for homogeneous velocity
jx q n v x
x
generalized
q
jx
V
v x (k)
k
Current density
for k-dependent velocity
jx
q
Again:
k
Since
V
2
3
2
3
3
d
k
Spin degeneracy
2q
3
v x (k)d kk 23 m
Occupied and
2 2
x
2m
EF
dk ydk z min
dk xk x f (E(k, T))
kx
kBT
Fermi distribution approximated by Maxwell Boltzmann distribution
1
f (E, T)
e
EEF
kBT
approximated
f (E, T ) e
EF E
kBT
e
EF
kB T
2 ( k 2x k 2y k 2
z)
e
2m kBT
1
2q
jx
23 m
2k 2y
2m kBT
dk
e
y
dk ze
2k 2
z
2m kBT
mindk xk xe
kx
2k 2x
2m kBT
e
EF
kBT
Let us investigate the integral
mindk xk x e
2k 2x
2m k B T
e
EF
kBT
kx
1
2
d(k x ) e
2
2k 2x
2m k B T
2m(EF ) / 2
e
EF
kB T
mk B T
e
2
kB T
2k 2x
EF
2m
Remember integrals of the type:
dk ze
2k 2
z
2m kBT
x
dk z
2m k B T
k z
2m k B T
2m kBT
dx
e
x2
2m k BT
dx
2 q mk BT 2m k BT kBT
jx
e
3
2
2
2 m
jx
4 m e
2
(
k
T
)
e
B
3
h
Richardson-Dushman
kBT
Vacuum tube
Typical value of
Tungsten:
4.5 eV
A (1-r)=0.72 X 106 A/m2 K
Universal constant: A=1.2 X 106 A/m2K
Reflection at the potential step
jx
4 m e 2
2
k
(
1
r
)
T
e
B
3
h
kB T
Richardson Constant
Owen Willans Richardson
Nobel prize in 1928
"for his work on the thermionic phenomenon and
especially for the discovery of the law named after him".
Field-Aided Emission
Image potential
E
Evac
Electric field in x-direction
V( x) x Ex
EF
x
@ very high electric fields of 108 V / m tunneling through thin barrier
cold emission