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Conductivity
Semiconductors & Metals
Chemistry 754
Solid State Chemistry
Lecture #20
May 14, 2003
References – Conductivity
There are many references that describe electronic
conductivity in metals and semiconductors. I used
primarily the following texts to develop this lecture.
“The Electronic Structure and Chemistry of
Solids”
P.A. Cox, Oxford University Press, Oxford (1987).
“Solid State Physics”
H. Ibach and H. Luth, Springer-Verlag, Berlin (1991).
“Physical Properties of Semiconductors”
C.M. Wolfe, N. Holonyak, Jr., G.E. Stillman, Prentice Hall,
Englewood Cliffs, NJ (1989).
Resistivities of Real Materials
Compound Resistivity (-cm) Compound Resistivity (-cm)
Ca
3.9  10-6
Si
~ 0.1
Ti
42  10-6
Ge
~ 0.05
Mn
185  10-6
ReO3
36  10-6
Zn
5.9  10-6
Fe3O4
52  10-6
Cu
1.7  10-6
TiO2
9  104
Ag
1.6  10-6
ZrO2
1  109
Pb
21  10-6
Al2O3
1  1019
Most semiconductors in their pure form are not good
conductors, they need to be doped to become conducting.
Not all so called “ionic” materials like oxides are insulators.
Microscopic Conductivity
We can relate the conductivity, s, of a material to microscopic
parameters that describe the motion of the electrons (or other
charge carrying particles such as holes or ions).
s = ne(et/m*)
m = et/m*
s = nem
where
n = the carrier concentration (cm-3)
e = the charge of an electron = 1.602  10-19 C
t = the relaxation time (s) {the time between collisions}
m* = the effective mass of the electron (kg)
m = the electron mobility (cm2/V-s)
Metals, Semiconductors & Insulators
DOS
Metal
EF
DOS
Semimetal
Energy
Energy
Energy
EF
Conduction
Band
EF
Valence
Band
DOS
Semiconductor
/Insulator
In a metal the Fermi level cuts through a band to produce a partially filled
band. In a semiconductor/insulator there is an energy gap between the filled
bands and the empty bands. The distinction between a semiconductor and an
insulator is artificial, but as the gap becomes large the material usually
becomes a poor conductor of electricity. A semimetal results when the band
gap goes to zero.
Resistivity and Carrier Concentration
The resistivities of real materials span nearly 25 orders of magnitude.
This is due to differences in carrier concentration (n) and mobility (m).
Let’s first consider carrier concentration.
•The carrier concentration only includes electrons which can easily be
excited from occupied states into empty states. The remaining electrons
are localized.
•In the absence of external excitations (light, voltage, etc.) the
excitation must be thermal, this is on the order of kT (~ 0.03 eV at RT)
•Only electrons whose energies are within a few kT of EF can contribute
to the electrical conductivity.
Generally this means that EF should cut a band to
achieve appreciable carrier concentration. Alternatively
impurities/defects are introduced to partially populate a
band.
Fermi-Dirac Function
The Fermi-Dirac function gives the fraction of allowed states, f(E), at
an energy level E, that are populated at a given temperature.
f(E) = 1/[1 + exp{(E-EF)/kT}]
where the Fermi Energy, EF, is defined as the energy where f(E) = 1/2.
That is to say one half of the available states are occupied. T is the
temperature (in K) and k is the Boltzman constant (k = 8.62  10-5
eV/K)
As an example consider f(E) for T = 300 K and a state 0.1 eV above EF:
f(E) = 1/[1 + exp{(0.1 eV)/((300K)(8.62  10-5 eV/K)}]
f(E) = 0.02 = 2%
Consider a band gap of 1 eV.
f(1 eV) = 1.6  10-17
See that for even a moderate band gap (Silicon has a band gap of 1.1
eV) the intrinsic concentration of electrons that can be thermally
excited to move about the crystal is tiny. Thus pure Silicon (if you
could make it) would be quite insulating.
Fermi Dirac Function
Metals and Semiconductors
f(E) as determined
experimentally for
Ru metal (note the
energy scale)
f(E) for a
semiconductor
Carrier Mobility
Recall the expression for carrier mobility:
m = et/m*
where,
e = electronic charge
m* = the effective mass
t = the relaxation time between scattering events
What factors determine the effective mass?
• m* depends upon the band width, which in turn depends
upon orbital overlap.
What entities scatter the carriers and reduce the
mobility?
• A defect or impurity (t increases as purity increases)
• Lattice vibrations, phonons (t decreases as temp.
increases)
What is the meaning of k?
In our development of the electronic band structure
from a linear combination of atomic orbitals the variable
k was used to determine the phase of the orbitals.
What exactly is k?
Wavevector – It tells us the how the phases of the
orbitals change when translational symmetry is applied.
Quantum Number – Identifies a particular electronic
wavefunction (that can hold 2 electrons with opposite
spin).
Crystal Momentum – In free electron theory k is
proportional to the momentum of the electron in the kth
wavefunction.
Crystal Momentum
To better understand the meaning of k, consider an electron at
the outer edge of the Brillouin zone, where k = p/a. The phase of
the electronic wavefunction changes sign every unit cell (similar to
a p-orbital changing phase at its nodal plane)
l
a
l = 2a  a = l/2
k = p/a  a = p/k
Combining these two relationships gives:
l/2 = p/k
k = 2p/l
l = 2p/k
The wavelength of the wavefunction is inversely
proportional to k.
Crystal Momentum
Now consider the DeBroglie relationship (wave-particle duality of
matter)
l = h/p
p = h/l
p = hk/2p
where.,
• p is the momentum of the wavepacket,
• h is Planck’s constant, 6.626  10-34 J-s
The momentum of an electron is directly proportional to k.
k is a measure of the “crystal” momentum of an electron in the
yK wavefunction.
From the ideas on the previous 2 slides one can derive the
following relationships to describe the properties of a conduction
electron:
Velocity  v = hk/2pm = (2p/h)(dE/dk)
Energy  E = (h/2p)(k2/2m*)
Effective Mass  m* = (2p/h)2 (1/{d2E/dk2})
dE/dk  The first derivative of the E vs. k curve.
d2E/dk2  The second derivative of the E vs. k curve.
Quantity
dE/dk
m
Velocity
m*
Wide Band
Large
High
Fast
Light
Narrow Band
Small
Low
Slow
Heavy
Wide (disperse) bands are better for conductivity.
Bandstructure & DOS for Cu
EF cuts the very wide (disperse) s band, giving rise to a
large carrier concentration, along with high mobility.
This combination gives rise to high conductivity.
Temperature Dependence-Metals
Recall that
s = ne2t/m*
In Metals
– The carrier concentration, n, changes very
slowly with temperature.
– t is inversely proportional to temperature
(t a 1/T), due to scattering by lattice vibrations
(phonons).
– Therefore, a plot of s vs. 1/T (or r vs. T) is
essentially linear.
– Conductivity goes down as temperature
increases.
Scattering by Impurities and Phonons
Phonon scattering
•Proportional to temperature
Impurity scattering
•Independent of temperature
•Proportional to impurity
concentration
Bandstructure for Ge
CB minimum
VB maximum
p-bands
s-band
No mixing
at G.
EF falls in the (0.67 eV) band gap. Carrier concentration and
conductivity are small.
Ge is an indirect gap semiconductor, because the uppermost VB energy
and the lowest CB energy occur at different locations in k-space.
Direct & Indirect Gap Semiconductors
Ge
Si
GaAs
Figure taken from
“Fundamentals of
Semiconductor
Theory and Device
Physics”, by S. Wang
Direct Gap Semiconductor: Maximum of the valence band and minimum of the
conduction band fall at the same place in k-space.
a a (hn-Eg)1/2
Indirect Gap Semiconductor: Maximum of the valence band and minimum of
the conduction band fall different points in k-space. A lattice vibration
(phonon) is involved in electronic excitations, this decreases the absorption
efficiency.
a a (hn-Eg)2
Doping Semiconductors
The Fermi-Dirac function shows that a pure semiconductor with a
band gap of more than a few tenths of an eV would have a very small
concentration of carriers. Therefore, impurities are added to
introduce carriers.
n-doping  Replacing a lattice atom
with an impurity (donor) atom that
contains 1 additional valence electron
(i.e. P in Si). This e- can easily be
donated to the conduction band.
p-doping  Replacing a lattice atom
with an impurity (acceptor) atom that
contains 1 less valence electron (i.e. Al
in Si). This atom can easily accept an
e- from the VB creating a hole.
Conduction Band
Conduction Band
e-
EF
EF
e-
Valence Band
Valence Band
Common Semiconductor Structures
Diamond
Fd-3m (Z=8)
C, Si, Ge, Sn
Sphalerite
F-43m (Z=4)
GaAs, ZnS, InSb
Chalcopyrite
I-42d (Z=4)
CuFeS2, ZnSiAs2
Properties of Semiconductors
Compound
Structure
Bandgap
(eV)
e- mobility
(cm2/V-s)
h+ mobility
(cm2/V-s)
Si
Diamond
1.11 (I)
1,350
480
Ge
Diamond
0.67 (I)
3,900
1,900
AlP
Sphalerite
2.43 (I)
80
---
GaAs
Sphalerite
1.43 (D)
8,500
400
InSb
Sphalerite
0.18 (D)
100,000
1,700
AlAs
Sphalerite
2.16 (I)
1,000
180
GaN
Wurtzite
3.4 (D)
300
---
Temperature Dependence-Semiconductors
Recall that
s = ne2t/m*
In Semiconductors
– The carrier concentration increases as temperature
goes up, due to excitations across the band gap, Eg.
– n is proportional to exp{-Eg/2kT}.
– t is inversely proportional to temperature
– The exponential dependence of n dominates,
therefore, a plot of ln s vs. 1/T is essentially linear.
– Conductivity increases as temperature increases.
p-n Junctions
In the middle of the
junction EF falls midway
between the VB & CB as it
would in an intrinsic
semiconductor.
When a p-type and an n-type semiconductor are brought into contact
electrons flow from the n-doped semiconductor into the p-doped
semiconductor until the Fermi levels equalize (like two reservoirs of
water coming into equilibrium). This causes the conduction and valence
bands to bend as shown above.
Applications of p-n Junctions
Rectifier:
Reverse Bias
LED
MOSFET Transistor
Photovoltaic Cell