CONDUCTIVITY

Conductivity

Superconductivity

Electronic PropertiesRobert M Rose, Lawrence A Shepart, John Wulff

Wiley Eastern Limited, New Delhi (1987)

Resistivity range in Ohm m 25 orders of magnitude

10-9 10-7 10-5 10-3 10-1 10-1 103

Ag

Cu Al

Au

Ni

Pb

Sb Bi Graphite

Ge(doped)

Ge Si

105 107 109 1011 1013 1015 1017

WindowglassIonicconductivity

Bakelite

Porcelain

Diamond

Rubber

Polyethylene

Lucite

MicaPVC

SiO2

(pure)

Metallic materials

Insulators

Semi-conductors

A

LR

Classificationbased on

Conductivity

Semi-metals

Semi-conductors

Metals

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

mv

h

→ de Broglie wavelength v → velocity of the electrons h → Planck’s constant

mv

x

v kg x

s J x 4

31

34 1027.7

10109.9

1062.6

Wave number vector (k)

2

k

2

2

1mvE Non relativistic

m

khE

2

22

8

m

khE

2

22

8

↑ → k ↓ → E ↓

E →k →

Discrete energy levels(Pauli’s exclusion principle)

If the length of the box is L

Ln 2

n → integer (quantum number)

2

22

8mL

hnE

L

nk

Number of electrons moving from left to right equals the number in the opposite direction

Electron in an 1D boxL

Quantization of Energylevels

2222

2

8 zyx nnnmL

hE

In 3D

Each combination of the quantum numbers nx , ny , nz corresponds toto 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 > 0 K

kT

EEEP

Fexp1

1)(

P(E

) →

E →

1

FE

Incr

easin

g T

0K

0

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 occupationof the energy levels

E →

k → k →

Field

EF EFElectric

Field

eEmaF

Force experienced by an electron

m → mass of an electron E → applied electric field

Vel

ocit

y →

time →

vd

Collisions

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

eEv

mF d

vd → Drift velocity → Average collision time

m

eEvd

The flux due to flow of electrons → Current density (Je)

m

E e nv e nJ de

2

n → number of free electrons

(E) gradient potential unit

(J Flux)(ty Conductivi e ) E J e

m

e n 2

m

V

m Ohm

1

m

Amp2

IRV AmpOhm

V

2

1

mOhm

V

m

Amp2

~ Ohm’s law

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

Sources ofElectron Scattering

Solute / impurity atoms

Defects

Thermal vibration → Phonons

Grain boundaries

Dislocations

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 !

Res

isti

vity

()

[x

10-8 O

hm m

] →

T (K) →

Cu-Ni alloy

100 200 300

1

2

3

4

5

Cu-2%Ni

Cu-3%Ni

→ 0 as T→ 0K

With low density ofimperfections

Pure Cu

Increased phonon scattering

Impurity scattering (r)

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 fordistribution 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 ► Good corrosion resistance

Manganin (87% Cu, 13% Mn, R = 20 x 106 / K) and Constantan (60% Cu, 40% Ni) are good as resistor materials [R (Cu) = 4000 x 106 / K]

Low thermoelectric potential wrt to contact material (usually Cu) reduceserror due to temperature difference between junctions. For highprecision 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 106 / K)

dT

dR

RR

1

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

Res

isti

vity

()

[x

10-1

1 Ohm

m]

→

T (K) →10 20

5

10 Ag Sn

Res

isti

vity

()

[x

10-1

1 Ohm

m]

→

T (K) →5 10

10

20

00 Tc

Superconducting transition temperature

Superconducting transition

?

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

0 H

c [W

b / m

2 ] →

T (K) → Tc

Hc / Jc

Normal

Superconducting

J c [A

mp

/ m2 ]

→

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 superconductivity- Bardeen-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

M →

H → Hc

NormalSuperconducting

Type I

Type I (Ideal) superconductors

Type I SC placed in a magnetic field totally repels the flux lines till themagnetic field attains the critical value Hc

c

c

HH

HH HM

0

M →

H → Hc

Normal

Type I

Type II (Hard) superconductors

Type II SC has three regions

c2

c2c1

c1

HH 0

)H,(HHH

HHH

M

Vortex

VortexRegion

Gradual penetration of the magnetic flux lines

Superconducting

Hc1 Hc2

As type II SC can carry high current densities (Jc) they are of great practicalimportance

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 computersJosephson 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 YBa2Cu3O7x

Y

Ba

Cu

O