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NNSE 508 EM Lecture #10 1 Lecture contents • Density of states • Statistics • Metals: transport
NNSE 508 EM Lecture #10
1
Lecture contents
• Density of states
• Statistics
• Metals: transport
NNSE 508 EM Lecture #10
2 Density of states
How to fill the states in almost free electron band structure ?
1. Calculate number of states per unit energy per unit volume
2. Use Pauli exclusion principle and distribution function to fill the bands
zz
z
yy
y
xx
x
nL
k
nL
k
nL
k
2
2
2
• Electrons are waves !
• Large 3D box (L is large, n is large) with
Born-von Karman boundary Conditions:
ikrAe
3D : 2D : 1D : 32
VN
• Free electron approximation:
( , , ) ( , , )
same for y and z
xx L y z x y z
0 Lx
V(x)
x
V0
zyx
zyx
zyx
nnnLLL
kkk
32
22
SN
2
kLN
• Number of states:
NNSE 508 EM Lecture #10
3 Density of states
m
kVE
2
22
0
In the interval k to k+k
number of states :
2
kLN
3D :
2102
1VEmk
In the interval E to E+dE number
of states per unit “volume” (spin
included):
2D :
1D :
m
kkE
2 32
VN
22
2
2
kLkN
332
2
4
kLkN
2)(
mEN
210
212)(
VE
mEN
E
N
VEN
1)(
21032
232)( VE
mEN
NNSE 508 EM Lecture #10
4 Density of states and dimensionality
From Singh, 2003
NNSE 508 EM Lecture #10
5
Density of states in 3D and DOS effective mass
Valence band density of states for Si (calculations)
Effective mass density of states
21
032
23*2)( VE
mEN
3D density of states
**
cdos mm
Conduction band DOS mass in
G point:
31*
3
*
2
*
1
32* mmmm cdos
Conduction band DOS mass in
indirect gap semiconductors:
Valence band DOS mass : 322/3*2/3**
lhhhdos mmm
NNSE 508 EM Lecture #10
6
Filling the empty bands: Distribution function
( ) ( ) ( )n E N E f E• Electron concentration at the energy E (Density of
states) x (distribution function):
• Pauli Exclusion Principle: No two electrons
(fermions) can have identical quantum numbers.
• Electrons follow Fermi-Dirac statistics.
• Fermi-Dirac distribution function:
1
1)(
Tk
EEFD
B
F
e
Ef
TkEE BF In the non-degenerate case (electron energies
are far from EF ):
Boltzmann distribution function may be used:
TkEEB
BFeEf
)(
NNSE 508 EM Lecture #10
7
Filling parabolic empty bands: Fermi energy
• Fermi energy is obtained by solving:
• if n is concentration of electrons in the band:
• The Fermi energy is found:
• And Fermi surface (sphere in this case)
for Na with n = 2.65x1022 cm-3
• What is happening if the Fermi surface is not
entirely within the Brillouin zone?
0
3 2 3 2* *1 2 3 2
0 02 3 2 3
2 2 2
3
FE
F
V
m mn E V dE E V
F
o o
E
V V
n N E f E dE N E dE
If DOS changes slowly at EF
2
3 22
0 *3
2FE V n
m
2
32
*3
2Fk n
m
2 2
02
FF
kE V
m since
0 3.22FE V eV 10.92Fk A
81 10 cm/sFv
NNSE 508 EM Lecture #10
8 Nearly free electrons: Fermi surfaces in 2D
(square crystal)
• Fermi level is within the first band
One electron per unit cell Two electron per unit cell
From Hummel, 2000
Fermi level
Fermi surface
• Fermi level is in two bands
NNSE 508 EM Lecture #10
9 Consequences of band model:
Metals, dielectrics and semiconductors
Pauli Exclusion Principle controls filling of the band structure
• Insulators – highest filled band is completely occupied.
• Metals with one valence electron – half band occupied
• Bivalent metals – have s-p overlap – bands partially occupied
• Semiconductors – most common has intermixed s-p states with completely occupied one
of the sp sub-bands
NNSE 508 EM Lecture #10
10 Nearly free electrons: band structure of Cu Band structure of Copper (fcc) (from Segal, 1962)
From Hummel, 2000
Fermi-surface for Cu
L-point
X-point
4s- and 3d-bands of Cu (11 electrons) and Ni (10 electrons)
NNSE 508 EM Lecture #10
11
From Seeger, 1973
Band-structures of Si and Ge: Fermi level is in the bandgap !
(in pure materials)
NNSE 508 EM Lecture #10
12
Conductivity of metals – Quantum mechanical considerations
• Let’s consider parabolic band with minimum in the
center of Brillouin zone.
• In metals the conduction band is filled up to Fermi
energy (within kT):
• If electric field is applied, the distribution of velocities
is displaced by drift velocity v.
• Only electrons close to Fermi surface participate in
current transport.
• In one dimension:
compare with classical form:
• From definition of velocity
• And “drift” momentum
• If accurate 3D averaging is applied:
• Conductivity:
From Hummel, 2000
class dJ qnv
F FJ qv N E E
FE v k
ek
2 2
F FJ q v N E
2 21
3F FJ q v N E
2 21
3F Fq v N E
compare with classical form: 2q
nm
NNSE 508 EM Lecture #10
13
Conductivity of metals – examples
From Hummel, 2000
2 21
3F Fq v N E • Conductivity of metals depends mainly on
scattering (quite expected) and density of
states at Fermi level
• Conductivity is high in monovalent metals:
Cu, Ag, Au
• Conductivity is lower in bivalent metals
• Conductivity can be controlled in
semiconductors by filling the bands with
doping
• In metals, temperature dependence of
resistivity is linear (phonon scattering),
reaching residual value at low temperatures
(imperfections scattering).
NNSE 508 EM Lecture #10
14
Conductivity of alloys – examples
From Hummel, 2000
2 21
3F Fq v N E
• Resistivity of dilute single-phase
alloys increases with the square of
the valence difference (Linde’s rule)
• Scattering on local lattice
imperfections and local charge
differences
• Shift of Fermi level position
• Usually resistivity has maximum at
50% solute content
• If ordered phase forms, the resistivity
drops
NNSE 508 EM Lecture #10
15
Hall effect: carrier charge
1
1
y
H
x z
for n typeE ne
RJ B
for p typepe
Hall coefficient:
Dimensionless Hall coefficient for metals:
(= 1 in Drude theory) 1
HR
Ne
V
Materials with >1 electrons per unit cell can have:
• Complex Fermi surface
• Fermi energies close to discontinuities in the E vs. k
• Almost full bands where the carriers behave as positively charged (holes)
holes
