Lecture V

METALS

dr hab. Ewa Popko

Measured resistivities range over more than 30 orders of magnitude

Material Resistivity (Ωm) (295K)

Resistivity (Ωm) (4K)

10-12

“Pure”Metals

Copper

10-5

 

 

Semi-Conductors

Ge (pure) 5 102 1012

 

 

Insulators Diamond 1014  

Polytetrafluoroethylene (P.T.F.E)

1020  

1014

1020

Potassium

2 10-6 10-10

Metals and insulators

Metals, insulators & semiconductors?

At low temperatures all materials are insulators or metals.

Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.

Pure metals: resistivity increases rapidly with increasing temperature.

1020-

1010-

100 -

10-10-R

esis

tivi

ty (

Ωm

)

100 200 3000Temperature (K)

Diamond

Germanium

Copper

Core and Valence Electrons

Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal.

Most metals are formed from atoms with partially filled atomic orbitals.

e.g. Na, and Cu which have the electronic structure

Na 1s2 2s2 2p6 3s1

Cu 1s2 2s2 2p6 3s23p63d104s1

Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases

He 1s2 Ne 1s2 2s2 2p6

Or form close shells by covalent bonding i.e. Diamond

Note orbital filling in Cu does not follow normal rule

Metallic bond

Atoms in group IA-IIB let electrons to roam ina crystal. Free electrons glue the crystal

Na+ Na+

e-

e-

Attract

Attract

Attract

AttractRepelRepel

Additional binding due to interaction of partially filled d – electron shells takes place in transitional metals: IIIB - VIIIB

Bound States in atoms

r4

qe = )r(V

o

2

Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. .

The potential energy of an electron a distance r from a positively charge nucleus of charge q is

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

V(r)E2

E1

E0

r

0

Increasing Binding Energy

Bound and “free” states in solids

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

V(r)E2

E1

E0

The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is

Where n = 0, +/-1, +/-2 etc.

This is shown as the black line in the figure.

n o

2

nRr4

qe = )r(V

r

0

0

+ + + + +RNuclear positions

V(r) lower in solid (work function).

Naive picture: lowest binding energy states can become free to move throughout crystal

V(r)Solid

Energy Levels and Bands

+E

+ + + +position Electron level similar to

that of an isolated atom

Band of allowed energy states.

In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms.

Lower binding electron states become bands of allowed states.

We will find that only partial filled band conduct

Why are metals good conductors?Consider a metallic Sodium crystal to comprise of a lattice of Na+ ions, containing the 10 electrons which occupy the 1s, 2s and 2p shells, while the 3s valence electrons move throughout the crystal.

The valence electrons form a very dense ‘electron gas’.

_

_

_

_

_ _ __

_

_

__

_

_

__

+ +

+

+

+

+ +

+

+ + +

+

+

+

+ +

+

+

+ + + + + +

Na+ ions:Nucleus plus 10 core electrons

We might expect the negatively charged electrons to interact very strongly with the lattice of positive ions and with each other.

In fact the valence electrons interact weakly with each other & electrons in a perfect lattice are not scattered by the positive ions.

Electrons in metals

P. Drude: 1900 kinetic gas theory of electrons, classicalMaxwell-Boltzmann distributionindependent electronsfree electronsscattering from ion cores (relaxation time approx.)

A. Sommerfeld: 1928Fermi-Dirac statistics

F. Bloch’s theorem: 1928Bloch electrons

L.D. Landau: 1957Interacting electrons (Fermi liquid theory)

Free classical electrons:AssumptionsWe will first consider a gas of free classical electrons subject to external electric and magnetic fields. Expressions obtained will be useful when considering real conductors

(i) FREE ELECTRONS: The valence electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electrons from a `gas' of non-interacting electrons. They behave as INDEPENDENT ELECTRONS; they do not show any `collective' behaviour. (iii) ELECTRONS ARE CLASSICAL PARTICLES: distinguishable, p~exp(-E/kT) (iv) ELECTRONS ARE SCATTERED BY DEFECTS IN THE LATTICE: ‘Collisions’ with defects limit the electrical conductivity. This is considered in the relaxation time approximation.

Ohms law and electron drift

V = E/L = IR (Volts)

Resistance R = L/A (Ohms)

Resistivity = AR/L (Ohm m)

E = V/L = = j (Volts m-1)

Conductivity (low magnetic field)

j = E (Amps m-2)

I = dQ/dt (Coulomb s-1)

Area A

dx

vd

denvdt

dxen

dt

dQ

A

1j

L

Area A

Electric field E

Force on electron F

Drift velocity vd

Current density j = I/A

n free electrons per m3 with charge –e ( e = +1.6x10-19 Coulombs )

Force on electrons F = -eE results in a constant electron drift velocity, vd.

Charge in volume element dQ = -enAdx

Relaxation time approximationAt equilibrium, in the presence of an electric field, electrons in a conductor move with a constant drift velocity since scattering produces an effective frictional force.

Assumptions of the relaxation time approximation :

 

1/ Electrons undergo collisions. Each collision randomises the electron momentum i.e. The electron momentum after scattering is independent of the momentum before scattering.

 

2/ Probability of a collision occurring in a time interval dt is dt/ is called the ‘scattering time’, or ‘momentum relaxation time’.

 

3/is independent of the initial electron momentum & energy.

Momentum relaxation Consider electrons, of mass me, moving with a drift velocity vd due to

an electric field E which is switch off at t=0. At t=0 the average electron momentum is

 In a time interval dt the fractional change in the average electronmomentum due to collisions is

integrating from t=0 to t then gives

pis the characteristic momentum or drift velocity relaxation time.

p(t = 0) = mevd(t = 0)

dp/p(t) = - dt/pdp/dt = -p(t)/p

p(t) = p(0)exp(-t/p) -1 0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

p(t)

/p(t

=0)

F1

t/p

If, in a particular conductor, the average time between scattering events is s and it average takes 3 scattering event to randomise

the momentum. Then the momentum relaxation time is p3s.

Electrical ConductivityIn the absence of collisions, the average momentum of free electrons subject to an electric field E would be given by  

 The rate of change of the momentum due to collisions is  

 At equilibrium  

 Now j = -nevd = -nep/me = (ne2p /me) E So the conductivity is = j/E = ne2p /me

Ep

eFdt

d

Field

p/dt

d

Collisions

p

p

Εp

ppe- So0

dt

d

dt

d

CollisionsField

The electron mobility, is defined as the drift velocity per unit applied electric field  

= vd / E = ep /me (units m2V-1s-1)

The Hall Effect

An electric field Ex causes a current jx to flow.

Ex, jxEyBz

vd = vx

The Hall coefficient is RH = Ey/jxBz = -1/ne

The Hall resistivity is = Ey/jx = -B/ne

jx = -nevx so Ey = -jxBz/ne

Therefore Ey = +vxBz

F = -e (E + v B). In equilibrium jy = 0 so Fy = -e (Ey - vxBz) = 0

A magnetic field Bz produces a Lorentz force in the y-direction on

the electrons. Electrons accumulate on one face and positive charge on the other producing a field Ey .

j

For a general vx.

vx+ve or -ve

The Hall Effect

The Hall coefficient RH = Ey/jxBz = -1/ne

The Hall angle is given by tan = Ey/Ex = H

For many metals RH is quiet well described by this expression which is useful for obtaining the electron density, in some cases.

However, the value of n obtained differs from the number of valence electrons in most cases and in some cases the Hall coefficient of ordinary metals, like Pb and Zn, is positive seeming to indicate conduction by positive particles!

This is totally inexplicable within the free electron model.

j=jx

Ey

Bzvd = vx

Ex

E

Sign of Hall Effect

Ex, jxEyBz

vd

Ex, jxEyBz

vd

Hall Effect for free particles with charge +e ( “holes” )

Hall Effect for free particles with charge -e ( electrons )

Ey = +vxBz = - vd Bz

jx = -nevx = ne vd

Ey = -jxBz/ne

RH = Ey/jxBz = -1/ne

Ey = +vxBz = vd Bz

jx = nevx = ne vd

Ey = jxBz/ne

RH = Ey/jxBz = 1/ne

The (Quantum)Free Electron model: Assumptions

(i) FREE ELECTRONS: The valence electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electron from a `gas' of non-interacting electrons. That is they behave as INDEPENDENT ELECTRONS that do not show any `collective' behaviour. (iii) ELECTRONS ARE FERMIONS: The electrons obey Fermi-Dirac statistics. (iv) ‘Collisions’ with imperfections in the lattice limit the electrical conductivity. This is considered in the relaxation time approximation.

Free electron approximation

U(r)U(r)

Neglect periodic potential & scattering (Pauli)

Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

Eigenstates & energies

EUm

2

2

2 2

2

)(00

2

)/,/,/(2

)(

km

E

LnLnLnk

eer

k

zzyyxx

Lrkirkik

Ek

|k|

U(r)

k- space

Free Classical Electrons states

Defined by position (x,y,z) and momentum (px, py, pz)

Electron state defined by a point in k-space

x

z

y

px

py

pz

Free Quantum Electrons states

Uniquely determined by the wavevector, k. Or equivalently by (px, py, pz) = (kx, ky, kz).

Equal probability of electron being anywhere in conductor. kx

ky

kz

2

222222

2

2

)(

2

)/,/,/(2

mL

nnnk

mE

LnLnLnk

zyxk

zzyyxx

3D analog of energy levels for a particle in a box!!

Eigenstates & energies

The allowed values of nx, ny, and nz, are positive integers for the electron states in the free electron gas model.

Density of statesThe number of states that have energies in a given range dE is called the density of states g(E).

Let us think of a 3D space with coordinates nx, ny, and nz.

The radius nrs of the sphere :

2222zyx nnnn rs

Each point with integer coordinates represents one quantum state. Thus the total umber of points with integer coordinates inside the sphere equals the volume of the sphere:

and for integer numbers are positive –only 1/8 the total volume:

Including spin, the number of allowed electron states is equal to:

Density of states

3

34

rsn

33

61

34

81

rsrs nn

361

23

3 rsrs

nnN

3

3rsn

N

Density of states (DOS)

2

222

2mL

nEE rs

k

32

2

3

2

3

2)2(VEm

N 3LV

dEVEm

dN 32

2

1

2

3

2)2(

g(E)

E

mEm

dEdN

Eg 2)( 32

Fermi-Dirac distribution functionThe Density of States tells us what states are available. We now wish to know the occupancy of these states. Electrons obey the Pauli exclusion principle. So we may only have two electrons (one spin-up and one spin-down) in any energy state.

f(E

)

E

1

0EF

f(E

)

Fermi-Dirac function for T=0.

For T=0 all states are occupied

up to an energy EF, called the Fermi energy, and all states

above EF are empty.

The probability of occupation of a particular state of energy E is given by the Fermi-Dirac distribution function, f(E).

The Fermi Energy

Calculated EF for free electrons by equating the sum over all

occupied states at T=0 to the total number of valence electrons per unit volume, n i.e.

 

i.e.

This gives

 

n = n(E)dE F

o

32

n)(3 2m

= E 22

F

g(E)f(E)dE = n(E)dEThe number of occupied states per unit volume in the energy range E to E+dE is

EEFN

(E)

dEn(

E)d

En(E) at T = 0

ndEE2m

2

121

F

2

3/2

2

E

0

Free Electron Fermi SurfaceMetals have a Fermi energy, EF.

Free electrons so EF = 2kF2/2m

At T=0 All the free electron states within a Fermi sphere in k-space are filled up to a Fermi wavevector,kF.

The Fermi wavelength kF

The surface of this sphere is called the Fermi surface.

On the Fermi surface the electrons have a Fermi velocity vF = hkF/me.

The Fermi Temperature,TF, is the temperature at which kBTF = EF.

When the electron are not free a Fermi surface still exists but it is not generally a sphere.

At a temperature T the probability that a state is occupied is given by the Fermi-Dirac function

1 +

Tk

-Eexp = f(E)

B

-1

EEF

N(E

) dE

kBT

T=0

T>0

n(E

)dE

The finite temperature only changes the occupation of available electron states in a range ~kBT about EF.

where μ is the chemical potential. For kBT << EF μ is

almost exactly equal to EF.

Fermi-Dirac function for a Fermi temperature TF =50,000K, about right for Copper.

The effects of temperature

Electronic specific heat capacity Consider a monovalent metal i.e. one in which the number of free electrons is equal to the number of atoms.

If the conducting electrons behaved as a gas of classical particles the electron internal energy at a temperature T would be

U = (kBT/2) x n x (number of degrees of freedom = 3)

So the specific heat at constant volume CV = dU/dT= 3/2nkB.

At room temperature the lattice specific heat, 3nkB ( n harmonic

oscillators with 6 degrees of freedom).

In most metals, at room temperature, CV is very close to 3nkB.

The absence of a measureable contribution to CV was historically the major objection to the free classical electron model.

If electrons are free to carry current why are they not free to absorb heat energy? The answer is that they are Fermions.

Electronic Specific heat The total energy of the electrons per m3 in a metal can be written as

E = Eo(T=0) + E(T).

Where Eo(T=0) is the value at T=0.

T E

nk = Tk.E

Tk n E 2

F

2B

BF

B

EEF

N(E

) dE

kBT

T=0

T>0

n(E

)dE

Each of these increases its energy by ~kBT

The number of electrons that increase their energy is ~n(kBT/EF) where n is the number of electrons per m3

At a temperature T only those electrons within ~ kBT of EF have a greater energy that they had at T=0.

)( 31 mJKT .

E

nk2

dT

E)d( =

dT

dE = C

F

2B

el

)mJK(.T E2nk = C

31

F

2B

2

el

For a typical metals this is ˜ 1% of the value for a classical gas of electrons. E.g. Copper (kT/EF) ~ 300/50,000 = 0.6%

At room temperature the phonon contribution dominates.

The electronic specific heat is therefore

A full calculation gives (Kittel p151-155)

Low Temperature Specific Heat

At low temperature one finds that

The first term is due to the electrons and the second to phonons. The linear T dependent term is observed for virtually all metals. However the magnitude of γ can be very different from the free electron value.

T + T = C 3V

Metal calc. (JK-1 mol-1) expt(JK-1mol-1)

Cu 5 10-7 7 10-7

Pb 1.5 10-6 3 10-6

where and are constants

)( 31 mJKT.T E2nk = C

F

2B

2

el Predicted electronic specific heat

Dynamics of free quantum electrons Classical free electrons F = -e (E + v B) = dp/dt and p =mev .

Quantum free electrons the eigenfunctions are ψ(r) = V-1/2 exp[i(k.r-t) ]

The wavefunction extends throughout the conductor.

Can construct localise wavefunction i.e. a wave packets

The velocity of the wave packet is

the group velocity of the waves

The expectation value of the momentum of the wave packet responds to a force according to F = d<p>/dt (Ehrenfest’s Theorem)

for E = 2k2/2me

)]t -exp[i(k.rA (r)k k

kkv

d

dE

d

d

1

ee mm

pkv

Free quantum electrons have free electron dynamics

Conductivity & Hall effectFree quantum electrons have free electron dynamics

Free electron expressions for the Conductivity, Drift velocity, Mobility, & Hall effect are correct for quantum electrons.

Current Density j = -nevd

Conductivity = j/E = ne2p /me

Mobility = vd / E = ep /me

Hall coefficient RH = Ey/jxBz = -1/ne

Electronic Thermal ConductivityTreat conduction electrons as a gas. From kinetic theory the thermal conductivity, K, of a gas is given by

K = ΛvCv/3 where v is the root mean square electron speed, Λ is the electron mean free path & Cv is the electron heat capacity per m3

For T<< TF we can set v = vF and Λ = vFpand Cv = (2) nkB (T/TF)

So K = 2nk2BpT/3m now = ne2p/m

Therefore K /T = (/3)(kB/e)2 = 2.45 x 10-8 WK-2 (Lorentz number)

The above result is called the Wiedemann-Franz law

Measured values of the Lorentz number at 300K are Cu 2.23, In 2.49, Pb 2.47, Au 2.35 x 10-8 WK-2 (very good agreement)

Free electron model: SuccessesIntroduces useful idea of a momentum relaxation time.

Give the correct temperature dependent of the electronic specific heat

Good agreement with the observed Wiedemann-Franz Law for many metals

Observed magnitudes of the electronic specific heat and Hall coefficients are similar to the predicted values in many metals

Indicates that electrons are much more like free electrons than one might imagine.

Free electron model: FailuresElectronic specific heats are very different from the free electron predictions in some metals

Hall coefficients can have the wrong sign (as if current is carried by positive particles ?!) indicating that the electron dynamics can be far from free.

Masses obtained from cyclotron resonance are often very different from free electron mass and often observe multiple absorptions (masses). More than one type of electron ?!

Does not address the central problem of why some materials are insulators and other metals.

Solid stateN~1023 atoms/cm32 atoms 6 atoms

Energy band theory

Metal – energy band theory

Insulator -energy band theory

diamond

semiconductors

Intrinsic conductivity

kTEss

ge2/

0

ln()

1/T

1/T

ln()

kTEdd

de /0

Extrinsic conductivity – n – type semiconductor

Extrinsic conductivity – p – type semiconductor

Conductivity vs temperature

kTEss

ge2/

0

ln()

kTEdd

de /0

1/T

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