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Lecture V
METALS
dr hab. Ewa Popko
Metals and insulators
Measured resistivities range over more than 30 orders of magnitude
Material
Resistivity
(Ωm) (295K)
Resistivity
(Ωm) (4K)
10-5
10-12
Copper
2  10-6
10-10
SemiConductors
Ge (pure)
5  102
1012
Insulators
Diamond
1014
1014
1020
1020
Potassium
“Pure”
Metals
Polytetrafluoroethylene
(P.T.F.E)
Metals, insulators & semiconductors?
Pure metals: resistivity
increases rapidly with
increasing temperature.
Diamond
Resistivity (Ωm)
At low temperatures all
materials are insulators
or metals.
1020-
1010Germanium
100 Copper
10-100
100
200
Temperature (K)
300
Semiconductors: resistivity decreases rapidly with increasing
temperature.
Semiconductors have resistivities intermediate between metals
and insulators at room temperature.
Core and Valence Electrons
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
Simple picture. Metal have CORE electrons that are bound to the
nuclei, and VALENCE electrons that can move through the metal.
Metallic bond
Atoms in group IA-IIB let electrons to roam in
a crystal. Free electrons glue the crystal
Attract
eAttract
Repel
Repel
Na+
Na+
Attract
Attract
e-
Additional binding due to interaction of partially filled d – electron shells
takes place in transitional metals: IIIB - VIIIB
Bound States in atoms
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
F6
F7
F8
F9
00
-1
V(r)
E2
Increasing
Binding
E1
Energy
-2
E0
-3
-4
 qe
V( r ) =
4  o r
2
-5
-8
-6
-4
-2
0
r
r
2
4
6
8
Bound and “free” states in solids
The 1D potential energy
of an electron due to an 0
0
array of nuclei of charge
q separated by a distance
-1
R is
2
V(r ) = 
n
 qe
4 o r  nR
-2
Where n = 0, +/-1, +/-2 etc. -3
V(r)
E2
E1
E0
V(r)
Solid
-3
This is shown as the
black line in the figure.
-4
V(r) lower in solid (work
function).
-5
-8
-6
-4
-2
Naive picture: lowest
binding energy states can +
+
become free to move
Nuclear positions
throughout crystal
00
r2
+r
4
+
R
6
8
+
Energy Levels and Bands
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
Band of allowed energy states.
E
+
position
+
+
+
Electron level similar to
that of an isolated atom
+
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, classical
Maxwell-Boltzmann distribution
independent electrons
free electrons
scattering from ion cores (relaxation time approx.)
A. Sommerfeld: 1928
Fermi-Dirac statistics
F. Bloch’s theorem: 1928
Bloch electrons
L.D. Landau: 1957
Interacting electrons (Fermi liquid theory)
Free classical electrons:Assumptions
We 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
distinguishable, p~exp(-E/kT)
CLASSICAL
PARTICLES:
(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
n free electrons per m3 with charge –e ( e = +1.6x10-19 Coulombs )
L
V = E/L = IR
(Volts)
Area A
Resistance R = rL/A (Ohms)
Resistivity r = AR/L
(Ohm m)
E = V/L = rI/A = rj
(Volts m-1)
Electric field E
Conductivity s = 1/r (low magnetic field) Force on electron F
Drift velocity vd
j = sE
(Amps m-2)
I = dQ/dt
(Coulomb s-1)
Current density j = I/A
Force on electrons F = -eE results in a
constant electron drift velocity, vd.
dx
Area A
Charge in volume element dQ = -enAdx
1 dQ
dx
j=
= en = env d
A dt
dt
vd
Relaxation time approximation
At 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/t.
t is called the ‘scattering time’, or ‘momentum relaxation time’.
3/ t 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
F1
electron momentum is
p(t = 0) = mevd(t = 0)
dp/p(t) = - dt/tp ; dp/dt = -p(t)/tp
integrating from t=0 to t then gives
p(t) = p(0)exp(-t/tp )
p(t)/p(t=0)
In a time interval dt the fractional
change in the average electron
momentum due to collisions is
1.0
0.8
0.6
0.4
0.2
0.0
-1
0
t/tp1
2
3
tp is the characteristic momentum or drift velocity relaxation time.
If, in a particular conductor, the average time between scattering
events is ts and it average takes 3 scattering event to randomise
the momentum. Then the momentum relaxation time is tp = 3ts.
4
Electrical Conductivity
In the absence of collisions, the average momentum of free
electrons subject to an electric field E would be given by
 dp 
= F = eE
 
 dt  Field
The rate of change of the momentum due to collisions is
 dp 
= p / t p
 
dt
 Collisions
At equilibrium
Now
 dp 
 dp 

=0
 
 
 dt  Field  dt Collisions
So p = - etΕ
j = -nevd = -nep/me = (ne2tp /me) E
So the conductivity is s = j/E = ne2tp /me
The electron mobility, m, is defined as the drift velocity per unit
applied electric field
m = vd / E = etp /me (units m2V-1s-1)
The Hall Effect
Ex, jx
Ey
Bz
vd = vx
An electric field Ex causes a current jx to flow.
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 .
F = -e (E + v  B). In equilibrium jy = 0 so Fy = -e (Ey - vxBz) = 0
Therefore Ey = +vxBz
jx = -nevx
so
The Hall resistivity is rH
For a
general vx.
vx+ve or -ve
Ey = -jxBz/ne
= Ey/jx
= -B/ne
The Hall coefficient is RH = Ey/jxBz = -1/ne
j
E
The Hall Effect
Bz
The Hall coefficient RH = Ey/jxBz = -1/ne
j=jx
vd = vx
Ey
The Hall angle is given by tan f = Ey/Ex = rH/r
Ex
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.
Sign of Hall Effect
Hall Effect for free particles
with charge +e ( “holes” )
Ex, jx
Ex, jx
Bz
Hall Effect for free particles
with charge -e ( electrons )
Ey
vd
Bz
Ey
vd
Ey = +vxBz = vd Bz
Ey = +vxBz = - vd Bz
jx = nevx = ne vd
jx = -nevx = ne vd
Ey = jxBz/ne
Ey = -jxBz/ne
RH = Ey/jxBz = 1/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
 k (r ) =  0 e
U(r)

ik  r
=  0e
  
ik ( r  L )

k = 2 (nx / Lx , n y / Ly , nz / Lz )
  2 2


  U  = E
 2m



2

Ek =
k
2m
2
Ek
|k|
k- space
y
py
Free Classical Electrons states
Defined by position (x,y,z) and
momentum (px, py, pz)
z
pz
Free Quantum Electrons states
ky
Uniquely determined by the
wavevector, k. Or equivalently by
(px, py, pz) = (kx, ky, kz).
Equal probability of electron
being anywhere in conductor.
Electron state defined by
a point in k-space
px
x
kx
kz
Eigenstates & energies

k = 2 (nx / Lx , n y / Ly , nz / Lz )
 2 (n  n  n ) 

Ek =
k =
2m
2mL
2
2
x
2
y
2
z
2
2
2
3D analog of energy levels for a particle in a box!!
The allowed values of nx, ny, and nz, are positive integers
for the electron states in the free electron gas model.
Density of states
The 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 :
n 2rs = nx2  ny2  nz2
Density of states
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:
4
3
nrs
3
and for integer numbers are positive –only 1/8 the total volume:
14 3
1 3
nrs = nrs
83
6
Including spin, the number of allowed electron states is equal to:
nrs3
1 3
N = 2 nrs =
6
3
Density of states (DOS)
N =
n


E = Ek =
2
2mL
2
rs
n
3
rs
3
3
2
(2m) VE
N =
2 2  3
3
2
3
2
2
2
V = L3
1
2
(2m) VE
dN =
dE
2 3
2 
dN
m
g (E) =
= 2 3 2mE
dE  
g(E)
E
Fermi-Dirac distribution function
The Density of States tells us what states are available.
We now wish to know the occupancy of these states.
For T=0 all states are occupied
up to an energy EF, called the
Fermi energy, and all states
above EF are empty.
1
f(E)
The probability of occupation of a
particular state of energy E is
given by the Fermi-Dirac
distribution function, f(E).
f(E)
Electrons obey the Pauli exclusion principle. So we may only
have two electrons (one spin-up and one spin-down) in any
energy state.
0
EF
Fermi-Dirac function for T=0.
E
The Fermi Energy
The number of occupied states per unit volume in the energy
range E to E+dE is
n(E)dE = g(E)f(E)dE
n(E)dE
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.
F
o n(E)dE = n
i.e.
EF

0
3/2
1  2m 
1
2 dE = n


E
2 2   2 
This gives
2
 (3 2 n)
=

EF
2m
2
n(E) at T = 0
3
Free Electron Fermi Surface
Metals 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 l = 2/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.
The effects of
temperature
At a temperature T the
probability that a state is
occupied is given by the
Fermi-Dirac function
where μ is the chemical
potential. For kBT << EF μ is
almost exactly equal to EF.
The finite temperature only
changes the occupation of
available electron states in
a range ~kBT about EF.
Fermi-Dirac function for a Fermi temperature
TF =50,000K, about right for Copper.
N(E) dE
n(E)dE


 E -m 
f(E) =  exp 
 + 1 
 kB T 


-1
kBT
T=0
T>0
EF
E
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.
The total energy of the electrons per
m3 in a metal can be written as
E = Eo(T=0) + DE(T).
Where Eo(T=0) is the value at T=0.
n(E)dE
N(E) dE
Electronic Specific heat
kBT
T=0
At a temperature T only those
electrons within ~ kBT of EF have a
greater energy that they had at T=0.
T>0
The number of electrons that increase their energy is
~n(kBT/EF)
where n is the number of electrons per m3
Each of these increases its energy by ~kBT
2
kBT
nk
 DE  n
.k B T = B T 2
EF
EF
EF
E
The electronic specific
heat is therefore
dE d( DE) 2 nk 2B
=

.T
C el =
dT
dT
EF
 nk B
=
.T
Cel
2 EF
2
A full calculation gives
(Kittel p151-155)
2
( JK 1m 3 )
(JK 1 m 3 )
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.
Low Temperature Specific Heat
Predicted electronic
specific heat
 nk .T = T
2
C el =
2
B
2 EF
( JK 1 m 3 )
At low temperature one finds that
CV = T +  T3
where  and  are constants
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.
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
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-wt) ]
The wavefunction extends throughout the conductor.
Can construct localise wavefunction i.e. a wave packets
(r) = k A k exp[i(k.r - wt )]
The velocity of the wave packet is
the group velocity of the waves
v=
dw 1 dE
=
dk  dk
for E = 2k2/2me
v=
k p
=
me me
The expectation value of the momentum of the wave packet responds
to a force according to F = d<p>/dt
(Ehrenfest’s Theorem)
Free quantum electrons have free electron dynamics
Conductivity & Hall effect
Free 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
s = j/E = ne2tp /me
Mobility
m = vd / E = etp /me
Hall coefficient
RH = Ey/jxBz = -1/ne
Electronic Thermal Conductivity
Treat conduction electrons as a gas. From kinetic theory the thermal
conductivity, K, of a gas is given by
K = vCv/3
where v is the root mean square electron speed,  is
the electron mean free path & Cv is the electron heat capacity per m3
For T<< TF we can set v = vF and Λ = vFtp and Cv = (2/2) nkB (T/TF)
So
K = 2nk2BtpT/3m
now
s = ne2tp/m
Therefore K /sT = (2/3)(kB/e)2 = 2.45 x 10-8 WWK-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 WWK-2 (very good agreement)
Free electron model: Successes
Introduces 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: Failures
Electronic 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.
Energy band
theory
2 atoms
6 atoms
Solid state
N~1023 atoms/cm3
Metal – energy band theory
Insulator -energy band theory
diamond
semiconductors
Intrinsic conductivity
ln(s)
1/T
s s = s 0s e
 Eg / 2kT
Extrinsic conductivity – n – type semiconductor
ln(s)
s d = s 0d e
1/T
 Ed / kT
Extrinsic conductivity – p – type semiconductor
Conductivity vs temperature
ln(s)
s s = s 0s e
 Eg / 2kT
s d = s 0d e  Ed / kT
1/T
Actinium
Aluminium (Aluminum)
Americium
Antimony
Argon
Arsenic
Astatine
Barium
Berkelium
Beryllium
Bismuth
Bohrium
Boron
Bromine
Cadmium
Caesium (Cesium)
Calcium
Californium
Carbon
Cerium
Chlorine
Chromium
Cobalt
Copper
Curium
Darmstadtium
Dubnium
Dysprosium
Einsteinium
Erbium
Europium
Fermium
Fluorine
Francium
Gadolinium
Gallium
Germanium
Gold
Hafnium
Hassium
Helium
Holmium
Hydrogen
Indium
Iodine
Iridium
Iron
Krypton
Lanthanum
Lawrencium
Lead
Lithium
Lutetium
Magnesium
Manganese
Meitnerium
Mendelevium
Mercury
Molybdenum
Neodymium
Neon
Neptunium
Nickel
Niobium
Nitrogen
Nobelium
Osmium
Oxygen
Palladium
Phosphorus
Platinum
Plutonium
Polonium
Potassium
Praseodymium
Promethium
Protactinium
Radium
Radon
Rhenium
Rhodium
Rubidium
Ruthenium
Rutherfordium
Samarium
Scandium
Seaborgium
Selenium
Silicon
Silver
Sodium
Strontium
Sulfur (Sulphur)
Tantalum
Technetium
Tellurium
Terbium
Thallium
Thorium
Thulium
Tin
Titanium
Tungsten
Ununbium
Ununhexium
Ununoctium
Ununpentium
Ununquadium
Ununseptium
Ununtrium
Uranium
Vanadium
Xenon
Ytterbium
Yttrium
Zinc
Zirconium