Electrical properties of materials

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15 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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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



(




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





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



(




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
.