Transcript Without electric field

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4
Chapter 7 in the textbook
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7.1 Introduction and Survey
j  ,
V  IR
V

L
V / cm
Current density:
, j  N e,
I
j ,
A
R
L
A
,

# of electrons/cm 2  sec
amphere( A)  coulomb(C ) / sec.
(# electrons )  coulomb / cm 2  sec  A / cm 2
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
.
 ( 1   cm)
7.2 Conductivity-Classical Electron Theory
Understanding of electrical conduction
As Drude did,
A free electron gas or plasma ; valence electrons of individual
atoms in a crystal
For a monovalent metal,
What is plasma?
Na
N0
N
Na  0 ,
M
: # of atoms / cm3
: Avogadro constant (#/mole)
 : density (gram/cm3)
M : atomic mass of element (gram/mole)
One calculates about 1022 to 1023 atoms per cubic centimeter, i.e., 1022
to 1023 free electrons per cm3 for a monovalent metal.
Without electric field:
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vg 


k
the electrons move randomly
so that no net velocity results. when, E 
Thermal velocity:
1
2
(
E
)
k
k2
2m*
vg 
k
k

(for free electrons)
*
m
m
If E is applied
For a free electron
d
F  ma  m
 e,
dt
(a)
1) Acceleration,
2) Constant velocity after the field is removed
Free electron model should be modified.
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In a crystal,
Friction force (or damping force)

collision
Drift velocity
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Equation of Motion
d
m
   eE ,
dt
For the steady state
  f , i.e., d / dt  0
d
m
   e,
dt
f  e,
m

d eE
   eE.
dt f
e
f
.
Final drift velocity

  eE
  f 1  exp   
  m f

 
 
t   .
 
 
Where,  is a relaxation time: average time between two consecutive collisions
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
 t
  f 1  exp   
 

 
  .
 
m f
 
,
eE
Final drift velocity
mf

,
e
f 
j  Nff e  .
Nf: number of free electrons
 e
m
.
Nf e2
 
.
m
Mean free path between two consecutive collisions
l   .
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(  f )
  N f e 

e
m
N f e 2
m
7.3 Conductivity-Quantum Mechanical
Considerations
Without field
Fermi
velocity
At equilibrium, no net velocity
The maximum velocity that the electrons can
have is the Fermi velocity (i.e., the velocity of
electrons at the Fermi energy)
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With an electric field
What difference between Class. and QM
Only specific electrons participate in conduction and
that these electrons drift with a high velocity (vF)
We now calculate the conductivity by quantum mechanical means and apply,
as before, Ohm’s law.
j   eN .
j  FeN .
N  N
N   N  EF  E
j   F eN  EF  E   F eN  EF 
E
dE
k .
dk
2
2m
k 2.
2
2
m F
dE
p
 k

 F
dk m
m
m
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Fig.7.4 Population
density
j  F2 eN  EF  k.
d d  m  dp
dk
F m



 e,
dt
dt
dt
dt
e
e
e
t   ,
dk 
dt , k 
j  F2 e2 N  EF   .
Population density at EF
j  e N  EF   
2
 e 2 N  EF  
 / 2
F cos  
 / 2
2

 e 2 N  EF   F


d
 / 2
 1

sin
2


,


 4
2   / 2
2
F
1 2
j  e N  EF   F2 .
2
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2
Fig.7.5
 / 2

 /2
cos 2  d
For a spherical Fermi surface,
1 2 2
j  e  F N  EF  .
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Fermi velocity, relaxation
time, population density at
Fermi energy
Since,  = j/E
1 2 2
  e  F N  E F  .
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vs
Fig.7.6
N f e2

.
m
Contains the information that not all free
electrons are responsible for conduction, i.e.,
the conductivity in metals depends to a
large extent on the population density of
the electrons near the Fermi surface.
EM: monovalent metals
EB: bivalent metals
EI: Insulators
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Classical expression
7.4 Experimental Results and Their Interpretation
7.4.1 For Pure Metals
2  1 1   T2  T1   ,
Linear temperature coefficient of resistivity
Matthiessen’s rule
   th  imp  def   th   res .
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 th
Ideal resistivity
res  f (T )
Residual resistivity
7.4.2 For alloys
For dilute single-phase alloys
Linde’s rule ∝(valence electrons)1/2
Atoms of different size cause a variation in the lattice parameters local charge
valence alter the position of the Fermi energy.
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7.4.3 Ordering vs Disordering
res  X A  A  X B  B  CX A X B
Two phase mixture:
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(Nordheim’s rule)
7.5 Superconductivity
High Tc superconductors (Tc>77K)
77K: boiling point of liquid nitrogen
20K: boiling point of liquid hydrogen
4K: boiling point of liquid helium
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7.5.1 Experimental Results
ma  Tc  const.,
ma: atomic mass
: material constant
Ellimination of the superconducting state also occurs by
subjecting the material to a strong magnetic field.
 T2
H c  H 0 1  2
 Tc

,

Ceramic superconductors usually have a smaller Hc than metallic super conductors,
i.e., they are more vulnerable to lose superconductivity by a moderate magnetic field.
Type I superconductors
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Type II superconductors:
Transition metals and alloys consisting of Nb, Al, Si, V
Pb, Sn, Ti, Nb3Sn, Nb-Ti, ceramic superconductors
The interval represents a state in which superconducting and normal conducting
areas are mixed in the solid.
Contains small circular regions called vortices or fluxoids, which are in the normal state
with a mixed superconducting and normal conducting area.
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Flux quantum:
0 
h
 2.07 1015 (T  m 2 ).
2e
7.5.2 Theory
Postulate superelectrons that experience no scattering, having zero entropy (perfect order),
and have long coherence lengths.
BCS theory: Cooper pair (pair of electrons that has a lower energy than two individual electrons)
Electrons on the Fermi surface having
opposite momentum and opposite spin forms
cooper-pair
These electrons form a cloud of Cooper pairs
which drift cooperatively through the crystal.
The superconducting state is an ordered state
of the conduction electrons.
10-4 eV
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7.6 Thermoelectric Phenomena
Thermoelectric power, or Seebeck coefficient
V
S
T
Contact potential: metals with different EF
Explain the band diagram of metals to
explain the contact potential (based on
workfunction difference)
Peltier effect: a direct electric current that flows through the junctions
made of different materials causes one junction to be cooled and the other
to heat up.
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