Transcript Conductivity

CONDUCTIVITY
 Conductivity
 Superconductivity
Electronic Properties
Robert M Rose, Lawrence A Shepart, John Wulff
Wiley Eastern Limited, New Delhi (1987)
Resistivity range in Ohm m  25 orders of magnitude
Semi-conductors
Metallic materials
10-9
10-7
10-5
Ag
Cu Al
Au
Ni
Pb
Sb Bi
Graphite
L
R
A
10-3
10-1
Ge
(doped)
10-1
Ge
103
Si
Insulators
105
Window
glass
Ionic
conductiv
ity
107
109
1011
1013
Bakelite
Porcelain
Diamond
Lucite
Rubber
Mica
Polyethyl
ene
1015
1017
PVC
SiO2
(pure)
Metals
Classification
based on
Conductivity
Semi-metals
Semi-conductors
Insulators
Free Electron Theory
 Outermost electrons of the atoms take part in conduction
 These electrons are assumed to be free to move through the whole solid
 Free electron cloud / gas, Fermi gas
 Potential field due to ion-cores is assumed constant
 potential energy of electrons is not a function of the position
(constant negative potential)
 The kinetic energy of the electron is much lower than that of bound
electrons in an isolated atom
Wave particle duality of electrons
h

mv
  → de Broglie wavelength
 v → velocity of the electrons
 h → Planck’s constant
6.62 x 10 34 J s
7.27 x10 4


m
31
v
9.109 x 10 kg v


Wave number vector (k)
 2
k 

h2k 2
E
8 2 m
1
E  mv 2
2
Non relativistic
↑→ k↓→E↓
E →
h2k 2
E
8 2 m
Discrete energy levels
(Pauli’s exclusion principle)
k →
Electron in an 1D box
L
If the length of the box is L
n

n L k
L
2
2
2
n h
E
8mL2
Number of electrons moving from left to right
equals the number in the opposite direction
n → integer (quantum number)
Quantization of Energy
levels
In 3D

h2
2
2
2
E
n

n

n
x
y
z
2
8mL

 Each combination of the quantum numbers nx , ny , nz corresponds to
to a distinct quantum state
 Many such quantum states have the same energy and said to be degenerate
 The probability of finding an electron at any point in box is proportional
to the square of the amplitude  there are peaks and valleys within L
 If the electron wave is considered as a travelling wave the amplitude will be
constant
Fermi level
 At zero K the highest filled energy level (EF) is called the Fermi level
 If EF is independent of temperature (valid for usual temperatures)
► Fermi level is that level which has 50% probability of occupation
by an electron
T>0K
0K
P(E) →
1
1
P( E ) 
 E  EF 
1  exp 

 kT 
0
E →
EF
Conduction by free electrons
 If there are empty energy states above the Fermi level then in the presence
of an electric field there is a redistribution of the electron occupation
of the energy levels
E →
Field
 

Electric
Field
EF
k →
EF
k →
Force experienced by an electron

F  ma  Ee
 m → mass of an electron
 E → applied electric field
 In the presence of the field the electron velocity increases by an amount
(above its usual velocity) by an amount called the drift velocity
 The velocity is lost on collision with obstacles
Collisions
vd
 vd  
F  m   Ee
 
Velocity →
 vd → Drift velocity
  → Average collision time

Ee
vd 
m

time →
The flux due to flow of electrons → Current density (Je)

ne  E
J e  n e vd 
m
2
 n → number of free electrons
Flux (J e )
Je   E
Conductivity (  ) 
unit potential gradient (E)
n e2 

m
~ Ohm’s law
 Amp   1   V 
 m 2    Ohm m   m 
V  IR
 V 
 Ohm    Amp
 Amp   V 1 
 m 2    Ohm m 2 
Mean free path (MFP) (l) of an electron
 l = vd 
 The mean distance travelled by an electron between successive collisions
 For an ideal crystal with no imperfections (or impurities) the MFP
at 0 K is 
 Ideal crystal  there are no collisions and the conductivity is 
 Scattering centres → MFP↓ , ↓  ↓ , ↑
Scattering centres
Thermal vibration → Phonons
Sources of
Electron Scattering
Solute / impurity atoms
Defects
Dislocations
Grain boundaries
Etc.
Thermal scattering
 At T > 0K → atomic vibration scatters electrons → Phonon scattering
T↑→↓→↑
 Low T
 MFP  1 / T3
   1 / T3
 High T
 MFP  1 / T
   1 / T
Impurity scattering
 Resistivity of the alloy is higher than that of the pure metal at all T
 The increase in resistivity is  the amount of alloying element added !
Cu-Ni alloy
Resistivity () [x 10-8 Ohm m] →
Increased phonon scattering
5
Cu-3%Ni
4
Cu-2%Ni
3
2
1
Impurity scattering (r)
Pure Cu
100
200
300
T (K) →
→ 0 as T→ 0K
With low density of
imperfections
Mattheissen rule
 = T + r
Net resistivity =
Thermal resistivity + Resistivity due to impurity scattering
Conductors
 Power transmission lines → low I2R loss → large cross sectional area
 Al used for long distance distribution lines
(Elastic ModulusAl increased by steel reinforcement)
 OFHC (Oxygen Free High Conductivity) Cu (more expensive) is used for
distribution lines and busbars.
► Fe, P, As in Cu degrade conductivity drastically
Electrical contacts
 Electrical contacts in switches, brushes and relays
 Properties:
► High electrical conductivity
► High thermal conductivity → heat dissipation
►High melting point → accidental overheating
► Good oxidation resistance
 Cu and Ag used
 Ag strengthened by dispersion strengthening by CdO
■ CdO
► Strengthens Ag
► Improves wear resistance
► If arcing occurs → decomposes (At MP of Ag) to
absorb the heat
Resistor
 Properties:
► Uniform resistivity → homogenous alloy
► Stable resistance → Avoid aging / stress relaxation / phase change
► Small T coefficient of resistance (R) → minimizes error in measurement
► Low thermoelectric potential wrt Cu
1 dR
► Good corrosion resistance
R 
R dT
 Manganin (87% Cu, 13% Mn, R = 20 x
/ K) and
Constantan (60% Cu, 40% Ni) are good as resistor materials
[R (Cu) = 4000 x 106 / K]
106
 Low thermoelectric potential wrt to contact material (usually Cu) reduces
error due to temperature difference between junctions. For high
precision dissimilar junctions should be maintained at same
temperature
 Ballast resistors are used in maintaining constant current →
I↑→T↑→R↑  I↓
Requriement: high R (71% Fe, 29% Ni → R = 4500 x 106 / K)
Heating elements
 Properties:
► High melting point
► High resistivity
► Good oxidation resistance
► Good creep strength
► Resistance to thermal fatigue
 low elastic modulus
 low coefficient of thermal expansion
 ■ Upto 1300oC
Nichrome (80% Ni, 20% Cr), Kanthal (69% Fe, 23% Cr, 6% Al, 2% Co)
■ Upto 1700oC: SiC & MoSi2
■ Upto 1800oC: Graphite
 Mo and Ta need protective atmosphere at high T
 W (MP = 3410oC) is used is used as filament in light bulbs → creep
resistance above 1500oC improved by dispersion hardening with ThO2
 Resistance thermometers: ► High temperature coefficient of resistivity
► Pure Pt
SUPERCONDUCTIVITY
10
Resistivity () [x 10-11 Ohm m] →
Resistivity () [x 10-11 Ohm m] →
Superconducting transition
Ag
5
?
0
10
T (K) →
20
20
Sn
10
0
Tc
5
10
T (K) →
Superconducting transition temperature
Current carrying capacity
 The maximum current a superconductor can carry is limited by the
magnetic field that it produces at the surface of the superconductor
Hc / Jc
Jc [Amp / m2] →
0 Hc [Wb / m2] →
Normal
Superconducting
T (K) →
Tc
Meissner effect
 A superconductor is a perfect diamagnet (magnetic suceptibility  = 1)
 Flux lines of the magnetic field are excluded out of the superconductor
 Meissner effect
Normal
Superconducting
Theory of low temperature superconductivityBardeen-Cooper-Schreiffer (BCS) theory
 Three way interaction between an two electron and a phonon
 Phonon scattering due to lattice vibrations felt by one electron in the
Cooper pair is nullified by the other electron in the pair
 the electron pair moves through the lattice without
getting scattered by the lattice vibrations
 The force of attraction between the electrons in the Cooper pair is stronger
than the repulsive force between the electrons when T < Tc
Type I and Type II superconductors
Type I (Ideal) superconductors
 Type I SC placed in a magnetic field totally repels the flux lines till the
magnetic field attains the critical value Hc
Type I
M →
 H  H  H c
M 
0  H  H c
Normal
Superconducting
H →
Hc
Type II (Hard) superconductors
 Type II SC has three regions
 H

M    H
 0

M →
H  H c1
 H  (H c1 , H c2 )

H  H c2
Vortex
Region
Gradual penetration of the
magnetic flux lines
Type I
Super
conducting
Vortex
H →
Hc
Hc1

Normal
Hc2
 As type II SC can carry high current densities (Jc) they are of great practical
importance
 The penetration characteristics of the magnetic flux lines
(between Hc1 and Hc2) is a function of the microstructure of the
material  presence of pinning centres in the material
 Pinning centres:
 Cell walls of high dislocation density
(cold worked/recovery annealed)
 Grain boundaries
(Fine grained material)
 Precipitates
(Dispersion of very fine precipitates with interparticle spacing ~ 300 Å)
 Jc ↑ as Hc2 ↑
Nb – 40%Ti alloy, T = 4.2 K, Magnetic field strength = 0.9 Hc2
Microsctructure
Jc (A / m2)
Recrystallized
105
Cold worked and recovery annealed
107
Cold worked and precipitation hardened
108
Potential Applications
 Strong magnetic fields → 50 Tesla
(without heating, without large power input)
 Logic and storage functions in computers
Josephson junction → fast switching times (~ 10 ps)
 Magnetic levitation (arising from Meissner effect)
 Power transmission
High Tc superconductivity
Compound
Tc
Comments
Nb3Ge
23 K
Till 1986
La-Ba-Cu-O
34 K
Bednorz and Mueller (1986)
YBa2Cu3O7-x
90 K
> Boiling point of Liquid N2
Tl (Bi)-Ba(Sr)-Ca-Cu-O
125 K
Manufacture of YBa2Cu3O7-x
Please read from text book
Crystal structure of YBa2Cu3O7x
Y
Cu
O
Ba