MSE 141 ELECTRONIC MATERIALSElectrical Conduction in MaterialsDr. Benjamin O. ChanAssociate ProfessorJanuary 2012

ELECTRONS IN A CRYSTAL How are the electrons distributed

among the available energy levels? Probabilistic distribution expected

We can never specify the location and kinetic energy of each individual electron (out of ~ 1022 cm-3)

Fermi Distribution Applies to fermions

Pauli Exclusion Principle applies Bosons require Bose-Einstein distribution

FERMI DISTRIBUTION FUNCTION

kB = Boltzmann’s constant = 8.616 x 10-5 eV/K

EF = Fermi energy Metals: Highest occupied energy at T = 0 K Semiconductors: energy level with a 0.5 probability

of occupation chemical potential

T = absolute temperature

1exp

1)(

TkEE

EF

B

F

FERMI DISTRIBUTION FUNCTION (T=0 K)

Ground StateT = 0 KE = EFSatisfactory for MetalsFor semiconductors,

EF is usually in the energy gap: these energies have no chance of being occupied!01

11)(

e

EEF F

1101

11)(

e

EEF F

FERMI DISTRIBUTION FUNCTION (T>0 K) At room temperature,

DE~1% For E >> EF

Boltzmann tail

21

111

11)( 0

e

EF F

TkEEEF

B

Fexp)(

DENSITY OF STATES Distribution of energy levels over a band Consider lower part of valence band (e.g. 4s

for Cu) Electrons essentially free

Assume electrons are confined in a square potential well

where nx, ny and nz are the principal quantum numbers and a is now the length of the crystal

2222

22

2 zyxn nnnma

E

QUANTUM NUMBER SPACE nx, ny and nz specify an

energy state En If we take n to be the

radius of a sphere containing quantum numbers

The surface of the sphere will contain states yielding the same En

2222zyx nnnn

NUMBER OF QUANTUM STATES All points inside sphere represent

states with E < En Number of states with E En is

proportional to the volume of the sphere with radius n

Only the positive octant and integral values of nx, ny and nz are allowed

2323

22

23 2

634

81 Eman

DOS Z(E) WITHIN A BAND Density of states (DOS)

= number of energy states per unit energy

where V = a3 is the volume occupied by the electrons

2/12/3

22

224

)( EmadEdEZ

2/12/3

22

24

)( EmVEZ

SOME NOTES ON DOS E vs Z(E) graph

Not the usual Z vs E ! Parabolic graph Lower end of the band has fewer

available E values compared to higher energies

Area within the curve = number of states with E En Element area d = Z(E)dE

POPULATION DENSITY Pauli exclusion principle

2 electrons of opposite spin per state Number of electrons per unit energy

N(E)

Where F(E) is the Fermi distribution function

N(E) = population density

)()(2)( EFEZEN

12

2)( /

2/12/3

22

TkEE BFe

EmVEN

POPULATION DENSITY N(E) As T0 and E<EF,

N(E)=2Z(E) T0 and E<EF, N(E) is

smeared out Area within the curve =

number of electrons, N*, that have an energy E<En

For an energy interval E to E + dE dEENdN )(*

CALCULATING EF

For T0 and E<EF

Defining N’= number of electrons per unit volume and solving for EF, we get

Note: N* does not change with T

FF EE

dEEmVdEENN0

2/12/3

220

* 22

)(

m

NmV

NEF 23

23

23/22

23/2*2

2/32/3

22* 23 FEmVN

CRYSTAL DOS Z(E) WITHIN A BAND

CRYSTAL Z(E) WITHIN FBZ For low E, constant energy curves are

circlesFree-electron-likeZ(E) is parabolic

For larger energies, Z(E)0 as FBZ boundary is reached

Largest number of energy states available is near the center of a band

SIMPLIFIED ENERGY BANDS

CONSEQUENCES OF BAND MODEL

Solids in which the highest filled band is completely occupied by electrons are insulators Including s-band?

Alkali metals (monovalent) have half-filled valence band

Bivalent metals should be insulatorsClosed s-shell

Semiconductor4 valence electrons: valence band full!

BIVALENT METAL

BIVALENT METAL Is it insulating or conducting?

p band overlaps with s band Weak binding forces of valence electrons on their

atomic nuclei Upper s band electrons empty into low levels of p band Partially filled energy bands: metal!

Semi-metal Are alkali metals more conducting than bivalent

metals?

EFFECTIVE MASS Empirical evidence shows that mass of

electron in a crystal can be larger or smaller compared to the mass of a free electron We’ll call this the effective mass m* of the

electron m*/mo describes deviation from mo

Usually bigger than 1 (so what?) Determine an expression for m*

VELOCITY OF AN ELECTRON IN AN ENERGY BAND

Wave packet velocity given angular velocity w and wavenumber k

Acceleration a

For a free electron

This gives us

dkdE

dkhEd

dkd

dkd

g 1)/2()2(v

w

dtdk

dkEd

dtd

a g2

21v

kp

dtdk

dtdp

EFFECTIVE MASS Substituting,

Applying Newton’s Second Law

Comparing the two previous equations

m* is inversely proportional to the curvature of the energy band!!

*mFa

Fdk

Eddtdp

dkEda 2

2

22

2

2

11

1

2

22*

dkEdm

ENERGY BAND STRUCTURES REVISITED High curvature = low mass Low curvature = high mass Usually at the center or boundary of a BZ

m*/mo can be as low as 1% Multiple curves for same k leads to multiple

values for effective mass More than one type of carrier available!

AL BAND STRUCTURE

CU BAND STRUCTURE

SI BAND STRUCTURE

GAAS BAND STRUCTURE

MORE NOTES ON ENERGY BAND DIAGRAMS

Ideal electron band within FBZ

Calculate first derivative and reciprocal of second derivative

m* is small and positive near center of FBZ and increases for larger values of k

Electrons in the upper part have a negative mass Defect electron/ electronhole Particle travels opposite to an

electron under an applied electric field

Important in semiconductors Effective mass is a tensor!

PROBLEM SET 2Hummel, 3rd editionChapter 6

5, 8, 12Chapter 7

1, 3, 8, 9

Due Jan. 26, 2012

ELECTRICAL CONDUCTION IN METALS AND ALLOYS Electrical Conductivity

The ability of materials to let charge carriers move through it

Units: Siemens/m = 1/Wm Resistivity is the inverse (Wm)

Insulators/Dielectrics Unable to conduct electricity High resistivity

ROOM TEMPERATURE CONDUCTIVITY

CURRENT DENSITY AND ELECTRIC FIELD Current Density

Flow of charge per unit time per unit area Units: A/m2

Electric Field Region of influence of a charge carrier Units: V/m

In general, the relationship between the current density j across a material to the applied electric field e across it is

Where s is called the conductivity of the material This equation does not assume s to be constant!

esj

OHM’S LAW If s is constant, we get the general form of

Ohm’s Law

Hey, nothing changed!! It’s true the equation did not change externally,

but s suffered a transformation, from being a function to becoming constant

esj

OHM’S LAW Consider the material to be a cylinder of

length l and cross-sectional area A

Substituting into Ohm’s law equation,

Solving for I,

A

l

lVAIj ab // e

lV

AI abs

abVlAI s

CONDUCTANCE G Defining the conductance G to be

Ohm’s Law becomes

If we set G=1/R, R being the resistance of the material, we get

Or, the more familiar form is

lAG s

abGVI

abGVI

IRVab

RESISTANCE AND RESISTIVITY The resistance R of the material is

then defined as

with r=1/s, the resistivity of the material R is due to collisions with lattice atoms

Crystal defects tend to increase R More collisions take place!

Particle Picture

Al

AlR r

s

WAVE PICTURE OF R R is due to scattering of charge

carriers by atomsAtoms absorb energy of incoming wave

and re-emit them in the form of spherical waves

Treat atoms as oscillators Periodic structures: waves propagating in the

forward direction are “in-phase” (constructive interference)

electron wave passes without hindrance through crystal (coherent scattering)

Non-periodic structures: incoherent scattering

CAUTION! R, G, r and s (while being constant)

are temperature dependent!

R tends to increase (G decreases) with increasing T

What exceptions can you think of?

CLASSICAL ELECTRICAL CONDUCTION THEORY Pre-QM period Drude: consider the valence electrons

(in a metal) as a gas or plasmaMonovalent atoms contribute one electron

per atom Na=number of atoms=number of charge

carriers=N

Where No is Avogadro’s number, d the density of the material and M is the atomic mass (monatomic crystal)

MNNN o

ad

FREE ELECTRON MOTION Electrons are accelerated by an

applied electric field

For constant e, acceleration is constant v builds up to infinity!

When e =0, v = constant This can happen only in superconductors!

This is not what we observe…

eedtdm v

ELECTRON PATH THROUGH THE CONDUCTOR Electrons suffer intermittent collisions

with lattice ions

EFFECT OF COLLISIONS Drag: let g = damping coefficient

Steady state case dv/dt = 0

where vf = drift velocity

eedtdm γvv

γvv ee

f

ELECTRON VELOCITY Modified Differential Equation

Solution

Average time between collisions t

ee eedtdm

f

vv

v

fmte

f e v1vve

eem fvτ

CONDUCTIVITY AND T Current density

Conductivity

Conductivity is large for large N and t Mean free path l

vNeJ

mNe t2σ eσv eN f

tfvl

EFFECT OF TEMPERATURE AND LATTICE DEFECTS

As T increases, atoms vibrate more about their equilibrium positionst decreases and s

decreases More collision sites

(specially for point and line defects)t decreases and s

decreases

QUANTUM MECHANICAL CONSIDERATIONS

Plot velocities in velocity spaceMomentum spacek-space

Since there is no preferential velocity, the range of velocities form a disk with radius vFvF = Fermi velocity, the maximum velocity

the electrons can assume In 3-D, the velocity distribution forms a

sphere, the surface of which we call a Fermi surface

VELOCITY DISTRIBUTION OF ELECTRONS

MORE ON VELOCITY… At equilibrium, all points inside the

Fermi sphere are occupiedVelocities cancel each other out resulting

in zero net velocity for the whole group Under an applied electric field, the

Fermi sphere is displaced opposite the field directionNet velocity gain resultsGreat majority of electrons still cancel

each other pair-wiseUncompensated electrons cause the

observed current

SO? Not all conduction electrons contribute

to current Their drift velocity is very close to the

Fermi velocity (which is relatively high!)

Classical Mechanics assumes that all conduction electrons contribute to current and move with a modest velocity

How do you compare vf and vF?

POPULATION DENSITY There are more electrons

near EF Only a little energy DE is

needed to raise a substantial number of electrons into slightly higher states

Velocity of excited electrons only slightly higher than vF Mean velocity remains vF

QM CONDUCTIVITY Taking v=vF and N=N’=the number of

displaced electrons as shown by the previous figure, Ohm’s law gives

OrEEeNNej FFF D )(vv

kdkdEEeNj FF D )(v

QM CONDUCTIVITY For free electrons

Substituting into equation for j

We now calculate the displacement Dk of the Fermi sphere in k-space. We note that

eedtdk

dtdp

dtdmF v

kEeNj F D )(v2F

Fmpk

mdkdE v

2

QM CONDUCTIVITY Thus,

And

Where t is the relaxation time. Thus,te)(v 22

F FENej

teeetek DD

dtedke

QM CONDUCTIVITY Assume electric field vector points in the –

v(k)x direction v(k)y components cancel out pairwise Only projections of v(k) on the +v(k)x axis

contribute to the current

2D VELOCITY SPACE

QM CONDUCTIVITY Summing the velocities in the first and fourth

quadrants

t

e dENej FF

22/

2/2 )cosv()(

2/

2/2

22 cos)(

te dvENe F

F

22

v)(2 FFENe te

QM CONDUCTIVITY For a spherical Fermi surface

Since s = j/e

22

v)(3 FFENej te

22

v)(3 FFENe ts

SOME NOTES Conductivity depends on the square of

the Fermi velocity, the relaxation time and the population density at EF

Not all free electrons participate in the conduction processOnly those near EFMonovalent vs. bivalent metal Insulators and semiconductors

N(EF) effectively zero!

QM VS CLASSICAL Nf constant N(E) changes very little with T t decreases with increasing T

Large rate of collisions between drifting electrons and vibrating lattice atoms

s decreases! Both QM and classical models

accurately describe the temperature dependence of resistivity

RESISTIVITY AND TEMPERATURE For metals, r decreases linearly with T

according to

Where a is the linear coefficient of resistivity Oscillating atoms increase incoherent

scattering of electron waves (electron-atom collisions)

Residual resistivity is due to crystal imperfections Impurities, vacancies, GB’s, dislocations Essentially temperature independent

)](1[ 1212 TT arr

RESISTIVITY VS. TEMPERATURE

MATTHIESSEN’S RULE The resistivity arises from independent

scattering sources that are additive

rth is the ideal resistivityrres is due to defects and impurities

Number of impurities usually constant Number of vacancies and GB’s can be changed

via heat treatment

resthdefimpth rrrrrr

RESISTIVITY AND TEMPERATURE Annealing exercise

Annealing near melting temperature + rapid quenching to room temperature increases r due to frozen in vacancies

Room temperature aging or annealing at slightly elevated temperature annihilates some vacancies, decreasing r

Recrystallization and grain growth can also change r

RESISTIVITY OF ALLOYS r increases with increasing amount of

solute (Cu-Ni graph) Slope of r vs T graph essentially

constant Small additions of solute cause linear

shift of r vs T graph to higher rDifference in size alter lattice parameter,

increasing electron scatteringDifference in valence produces local

charge difference, increasing scatteringDifference in electron distribution alters EF

This changes N(E) and thus s

RESISTIVITY VS. SOLUTE TYPE

r VS. SOLUTE TYPELinde’s Rule

r of dilute single phase alloys increases with the square of the valence difference between solute and solvent constituents

RESISTIVITY FOR ORDERED OR DISORDERED ALLOYS

ORDERING Solute atoms are usually

randomly distributed in the solvent resulting in incoherent scattering of electron waves

If they are periodically arranged (e.g., alternately with host atoms for a 50/50 alloy) coherent scattering can occur Dramatic decrease in r results

(increase in mean free path) Only selected alloys show long-

range ordering Cu3Au, CuAu, Au3Mn, …

HOW TO ACHIEVE ORDER AND DISORDER Order

Annealing slightly below order-disorder transition (395 °C for Cu3Au) followed by moderate cooling rate

Slow cooling from above the transition temperature

DisorderQuench rapidly in ice brine from slightly

above transition temperatureAnneal above transition temperature

r FOR ISOMORPHOUS ALLOYS Maxima recorded at around 50%

composition Nordheim’s Rule

Residual resistivity depends on the fractional atomic composition (XA and XB) of the constituents

Where C is a material constantHolds strictly only for a few binary systems

Does not take into account changes in DOS arising from composition (alloys containing transition metal)

BABBAA XCXXX rrr

r FOR TWO-PHASE ALLOYS Mixture rule

Sum of the resistivities of each component taking the volume fractions of each phase into consideration

Crystal structure and the kind of distribution of the phases must also be considered

No maxima observed with respect to composition More like a linear interpolation of the individual

phases

RESISTORS For limiting current flow in an electrical

circuit I = V/R

Usual materialCarbon compositesWire-wound (around a ceramic body)

Alloys of high resistivity (e.g., nichrome) Needs to withstand corrosion and high

temperaturesMetal films on glass or ceramic substrates

JOULE HEATINGHeat dissipation in resistors

Expressed in Watts

Variable resistor = varistorUsually with sliding contact

RIP 2

SUPERCONDUCTIVITY Resistivity becomes immeasurably

small (zero!) below a critical temperature Tc

H.K. Onnes (1911)Hg below 4.15 K

27 elements, numerous alloys, ceramics containing CuO, organic compounds based on Se or S

Tc ranges from 0.01 to 130 KHigh Tc superconductors are hot!

CRITICAL TEMPERATURES

SOME NOTES Superconducting transition is reversible Can be considered a separate state

Apart from solid, liquid, gas Higher degree of order

Entropy = 0 !!

HIGH TC SUPERCONDUCTORS

Cu oxide based ceramics exhibit large Tc (40 K)

1-2-3 (rare earth to alkaline earth to copper ratio)

High Tc>77 K (boiling point of liquid nitrogenUsual coolants: liquid H (20 K) and liquid

He (4 K)Dry Ice (195 K or -79C)

APPLICATIONS

Strong electromagnets MRI, particle accelerators, electric power storage

devices Currently Nb-Ti or Nb3Sn alloy microns in diameter

embedded in a Cu matrix Lossless power transmission

Once a current is induced in a loop, it continues to flow without significant decay

MAGLEV trains More compact and faster computers Switches: cryotrons

Coiled Nb wire around Ta wire immersed in liquid He

SUPERCONDUCTORS.ORG…reports the observation of record high

superconductivity near 254 Kelvin (-19C, -2F). This temperature critical (Tc) is believed accurate +/- 2 degrees, making this the first material to enter a superconductive state at temperatures commonly found in household freezers.         This achievement was accomplished by combining two previously successful structure types: the upper part of a 9212/2212C and the lower part of a 1223. The chemical elements remain the same as those used in the 242K material announced in May 2009. The host compound has the formula (Tl4Ba)Ba2Ca2Cu7O13+ and is believed to attain 254K superconductivity when a 9223 structure forms (shown below left).

TRANSITION TEMPERATURE TC Sharp transition for pure and

structurally perfect superconductorsTransition width ~10-5 K for GaTransition can be as wide as 0.1 K for

alloysCeramics display an even wider spread

Varies with atomic mass, ma

Where a is a materials constantHg: Tc varies from 4.185 K to 4.146 K as ma

changes from 199.5 amu to 293.4 amu

constantcaTma

PURE VS. IMPURE SUPERCONDUCTORS

ELIMINATING SUPERCONDUCTING STATE

Raise T above Tc Raise magnetic field H above critical

magnetic field strength Hc Hc depends on the actual temperature of the

superconductor T

Where Ho is the critical magnetic field strength at 0 K

Ceramics usually have smaller Hc compared to metals (more vulnerable to losing superconductivity)

2

2

1c

oc TTHH

HIGH STRENGTH ELECTROMAGNETS

Large currents required Generates substantial resistive heating (water

cooling required) Heat dissipation occurs anyway

Superconductors are immune to resistive losses But they need to be operated below Tc Costly cooling process

Economic issue Weigh acquisition price and operation cost

LIMITING FACTORS

H must be below Hc Superconducting state is destroyed by its own

magnetic field! (a case of self-destruction!) Critical current Ic above which

superconductivity disappears THI critical space

where superconductivity reigns!

SUPERCONDUCTOR LIMITS

SUPERCONDUCTOR CLASSES

Type ISharp transition for Hc or TcNot used for magnet coils

Type IIGradual transition for Hc or TcSuperconductivity extends to Hc2 which can

be 100 times bigger than the initial Hc1Mainly utilized for superconducting

solenoids

TYPE I SUPERCONDUCTOR

TYPE II SUPERCONDUCTOR

MORE ABOUT TYPE II Supermagnets >> 10T Transition metals and alloys

Nb, Al, Si, V, Pb, Sn, Ti, Nb3Sn, NbTi Ceramics work too! The region between Hc1 and Hc2

represent a “mixed” state of superconducting and normal regionsNormal regions grow as we go from Hc1 to

Hc2

OF VORTICES AND FLUXOIDS Small circular regions in the normal

state carrying the smallest possible unit of magnetic flux (flux quantum)

Surrounded by superconducting regions

Forms a 2-D superlatticeParallel to magnetic field linesRegularly arranged in space

mutual repulsion

)(1005.22

215 mTeh

o

ELECTRONS AND VORTICES Electrons flowing perpendicular to

fluxoids would always find an unobstructed path through the superconducting matrixUnlimited superconductivity!

Lorentz force pushes fluxoids perpendicular to current and magnetic field directionFluxoids become obstacles to drifting

electronsCurrent decreases, resistivity increases

PINNED VORTICES Pinned down by defects

GB’s, dislocations, fine particles of alloying components

Achieved by heat treament and plastic deformationWire drawing

Used for Nb3Sn superconducting magnet

Increased resistivity is avoided!

CERAMIC SUPERCONDUCTORS

YBa2Cu3O7-x (YBCO)Orthorhombic layered

perovskite containing periodic O vacancies

DrawbacksBrittlenessCannot carry high current

densitiesEnvironmental instability

SUPERCONDUCTOR THEORY BCS Theory

Bardeen, Cooper, Schrieffer (1957) Works well for conventional superconductors

Key Idea Existence of Cooper pair

Pair of electrons that has lower energy than two individual electrons

COOPER PAIR FORMATION

Consider electron in metal at 0 K Electron perturbs neighboring lattice sites Electron drifts through crystal generating temporary

perturbations along its path Displaced ion can be set oscillating by perturbation

generating a phonon Phonon interacts quickly with a second electron

lowering its energy due to lattice deformation Second electron emits phonon which interacts with

first electron Phonon is passed back and forth, coupling the

electrons together, bringing them into a lower energy state

COOPER PAIRS AND FERMI SURFACES

All electrons on a Fermi surface having opposite momentum and spin form Cooper pairsCloud of Cooper pairs is formedCooperative drifting through the crystal

proceedsThis is an ordered state of the conduction

electrons!Ordering eliminates scattering by lattice

atoms, resulting in zero resistance!

FERMI ENERGY EF lower in superconducting state Superconducting state is separated

from the normal state by an energy gap Eg

Energy gap stabilizes Cooper pair against small changes of net momentum they won’t break apart!

Eg ~ 10-4eV Observed via IR absorption measurements

at T<Tc

DOS SUPERCONDUCTOR

JOSEPHSON EFFECT

Another way to measure Eg Two pieces of metal

One in superconducting state, the other in normal state

Energy bands in superconductor are raised by an appropriate voltage

If the applied voltage is big enough, filled states in superconductor are opposite empty states in normal conductor Cooper pairs can tunnel through

Eg is determined from the threshold voltage for tunneling

JOSEPHSON JUNCTION

COOPER PAIR BINDING

Electron-phonon binding occurs at very low temperaturesT < 40 KNoble metals = poor superconductors

(coupling is hard to achieve) Excitons (electron-hole pairs) may link

electrons to form Cooper pairsOrganic superconductors

Resonating valence bandsHigh Tc superconductors

THERMOELECTRIC PHENOMENA

ThermocoupleTwo different wires connected togetherCharge carrier concentration gradient at

junction generates emfCharge carrier concentration gradient

usually increases with temperatureTemperature measurement is possibleUse small junctions for quick

measurement!Cu-constantan (Cu-45%Ni)

-180 C to +400 CChromel (Ni-10%Cr)-alumel (Ni-

2%Mn2%Al) Higher temperaures

SEEBECK EFFECT

SEEBECK EFFECT Back to back thermocouples

One cold, the other one hot Potential difference between the

junction pairs due to carrier concentration gradient

Seebeck coefficient S

after T.J. Seebeck, German physicistUsually in mV/K but can go as high as mV/K

TVS

DD

PELTIER EFFECT

PELTIER EFFECT Seebeck Effect in reverse DC flowing through back to back

junctions cause one junction to be colder and the other one hotter

PbTe or BiTe together with metals Up to 70 C gradients has been

achieved with n- and p-type semiconductorsThermoelectric refrigerator

Lower temperatures can be achieved by cascading refrigerators

Each stage acts as heat sink for the next

CONTACT POTENTIAL Two metals in contact with each other

Seebeck: Electrons from material with higher EF go down to material with lower EF until EF equalizes Space charge accumulated produces contact

potential Temperature dependent

Peltier: Current causes electrons with higher EF to transfer energy to material with lower EF Material with higher EF loses energy and becomes

colder! Always a concern in electrical

measurements