Electrical properties of materials
Free electron theory
Only kinetic energy considered
Independent electron approximation
Free electron approximation
Pauli principle
Fermi

Dirac statistics
1
1
)
(
)
(
T
k
E
B
e
E
f
Chemical potential
•
Ohm’s law
•
Hall effect
•
Thermal conductivity (metals)
E
E
m
ne
j
2
j
electrical current
,
,
,
m
e
n
electron density, charge,
mass, relaxation time
E
electric field
Energy Bands
Periodic potential due to nuclei in the solid
Bragg diffraction of electron waves
Forbidden energy gap opens up in the energy bands
E
k
E
0
k
a
a
Energy gap
Free electrons
Electrons in a periodic potential
ne
Electrical conductivity,
,
,
e
n
= number density, charge, mobility
of current carriers
Metals
Free charge carriers
High conductivity
Conductivity decreases with increasing temperature
mobility of charge carriers decreases
Metal
(
洩
慴′0
o
C
Silver
1.59
×
10

8
Copper
1.68
×
10

8
Gold
2.44
×
10

8
Resitivity of gold
Resitivity (10

8
洩
0 2 4 6 8
0 200 400 600 800 1000
Temperature (K)
Courtsey: http://hypertextbook.com/facts/2004/JennelleBaptiste.shtml
Semiconductors
Activated charge carriers
Conductivity increases with increasing temperature
number of charge carriers increases
mobility of charge carriers decreases
Semiconductor
(
洩
慴′0
o
C
Silicon
6.4
×
10
2
Germanium
4.6
×
10

1
GaAs
5
×
10

7

10
×
10

3
T
ln
T
/
1
Slope = E
g
Extrinsic (Impurity) semiconductors
holes
V
B
C
B
p

type
acceptor level
V
B
C
B
n

type
donor level
electrons
thermistors, photoconductors
p

n junction
摩潤敳d
瑲慮t楳瑯牳
Superconductors
Kamerlingh Onnes, 26 October 1911
Critical temperature (T
c
)
zero resistance, persistent current
perfect diamagnetism (Meissner effect)
critical field (H
c
)
Electron

phonon coupling (BCS theory)
Examples: Hg [T
c
~ 4 K], Pb [T
c
= 8 K], Nb
3
Sn [T
c
~ 23 K]
High

T
c
superconductors (YBa
2
Cu
3
O
7

x
[T
c
= 90 K]
High field magnets
SQUID
Magnetic levitation
Problem Set
1.
At 0 K,
=
F
for free electrons in a metal. Demonstrate this using Fermi

Dirac statistics.
2.
Heat capacity of free electron gas is about 1% of that expected on the basis of the law of
equipartition
of energy. Why ?
3.
What are the basic assumptions of free electron theory ? What are the phenomena explained by it ? And what were its
main failures ?
4.
Show that the Hall voltage,
E
y
=

eB
E
x
/mc in a rectangular bar sample when
B
z
= B,
B
x
= B
y
= 0 and current is only in the
x direction.
5.
The
Weidemann

Franz law is known to fail at low temperatures. Suggest a possible explanation.
6.
Given the Bloch function,
k
(r) =
u
k
(r).
e
ikr
, obtain the
eigenvalue
of the crystal translation operation T (
ie
.
translation
through a lattice vector, T).
7.
For a simple cubic lattice, show that the kinetic energy of the free electron at a corner of the first
Brillouin
zone (
ie
.
having
wave vector at this point) is higher than that of an electron at the midpoint of a face of the zone. What bearing does this
have on the conductivity of divalent metals ?
8.
MnO
is experimentally found to be a semiconductor. Draw the
qualitative
band diagram for
MnO
; is it expected to be a
semiconductor on the basis of this band picture ? Suggest a possible explanation if there is a conflict.
9.
Slater
antiferromagnetic
ordering which causes metal

insulator transition, induces the doubling of the unit cell. What
experimental technique would be appropriate to detect such a transition ?
10.
Calculate the number density of conduction electrons and holes in pure
Ge
at 300 K (assume, m
e
=
m
h
;
E
g
= 0.67
eV
).
11.
Qualitatively explain the origin of isotope effect in superconductors.
12.
Give qualitative explanations for the entropy (S) and free energy (G) variations with temperature of the superconducting (S)
and normal (N) states (see figure below). Suggest how the data for the normal states could be obtained below T
C
.
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