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Drude-Lorenz Free Electron Theory
Prof.P. Ravindran,
Department of Physics, Central University of Tamil
Nadu, India
http://folk.uio.no/ravi/CMP2013
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
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Drude-Lorentz Model: 1900-1904
Paul Drude
(1863-1906)
Hendrik Lorentz
(1853-1928)
Ashcroft-Mermin Chapter 1:
- Based on the discovery of electrons by J. J.
Thomson (1897)
- Electrons as particles (Newton’s equation)
- Electron “gas” (Maxwell-Boltzmann statistics)
- Worked well
- Explained Wiedemann-Franz law (1853) k/sT
~ 2-3 x 10-8 (W-ohm/K2) (by double mistakes)
- Could not account for:
- T-dependence of k alone
- T-dependence of s alone
- Electronic specific heat Ce (too big)
- Magnetic susceptibility cm (too big)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory


1.
2.
Drude’s Assumptions:
3
Drude assumed that the compensating positive charge was attached to
much heavier particles, so it is immobile.
In Drude model, when atoms of a metallic element are brought together
to form a metal, the valence electrons from each atom become detached and
wander freely through the metal, while the metallic ions remain intact and
play the role of the immobile positive particles.
Matter consists of light negatively charged electrons which are mobile, &
heavy, static, positively charged ions.
The only interactions are electron-ion collisions, which take place in a
very short time t.
 The neglect of the electron-electron interactions is the INDEPENDENT
ELECTRON APPROXIMATION.
 Neglect of the electron-ion interactions is the FREE ELECTRON
APPROXIMATION.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
3. Electron-ion collisions are assumed to dominate, as these will
abruptly alter the electron velocity & maintain thermal
equilibrium.
4. The probability of an electron suffering a collision in a short time
dt is dt/t, where 1/t  the scattering rate. Electrons emerge from
each collision with both the direction & magnitude of their
velocity changed; the magnitude is changed due to the local
temperature at the collision point. 1/t is often an adjustable
parameter.
Ion
Mean time between
collisions is t.
Trajectory of mobile electron
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
•
•
In a single isolated atom of the metallic element has a
nucleus of charge e Za as shown in Figure below.
Figure represents Arrangement of atoms in a metal
•
where Za - is the atomic number and
•
e - is the magnitude of the electronic charge
•
[e = 1.6 X 10-19 coulomb] surrounding the nucleus,
there are Za electrons of the total charge –eZa.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
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
Some of these electrons ‘Z’, are the relatively weakly bound
valence electrons. The remaining (Za-Z) electrons are relatively
tightly bound to the nucleus and are known as the core electrons.

These isolated atoms condense to form the metallic ion, and
the valence electrons are allowed to wander far away from their
parent atoms. They are called `conduction electron gas’ or
`conduction electron cloud’.

Due to kinetic theory of gas Drude assumed, conduction
electrons of mass ‘m’ move against a background of heavy
immobile ions.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The density of the electron gas is calculated as follows. A
metallic element contains 6.023X1023 atoms per mole
(Avogadro’s number) and ρm/A moles per m3
Here ρm is the mass density (in kg per cubic metre) and
‘A’ is the atomic mass of the element.
Each atom contributes ‘Z’ electrons, the number of
m Z
electrons per cubic metre.
N
.
n
V
A
The conduction electron densities are of the order of 1028
conduction electrons for cubic metre, varying from 0.91X1028 for
cesium upto 24.7X1028 for beryllium.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory

These densities are typically a thousand times greater
than those of a classical gas at normal temperature and
pressures.

Due to strong electron-electron and electron-ion
electromagnetic interactions, the Drude model boldly treats
the dense metallic electron gas by the methods of the kinetic
theory of a neutral dilute gas.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
BASIC ASSUMPTION FOR KINETIC THEORY OF
A NEUTRAL DILUTE GAS
1.
In the absence of an externally applied electromagnetic fields, each electron
is taken to move freely here and there and it collides with other free electrons or
positive ion cores. This collision is known as elastic collision.
2.
The neglect of electron–electron interaction between collisions is known as the
“independent electron approximation”.
3. In the presence of externally applied electromagnetic fields, the electrons acquire
some amount of energy from the field and are directed to move towards higher
potential. As a result, the electrons acquire a constant velocity known as drift
velocity.
4.
In Drude model, due to kinetic theory of collision, that abruptly alter the
velocity of an electron. Drude attributed the electrons bouncing off the
impenetrable ion cores.
5.
Let us assume an electron experiences a collision with a probability per unit
time 1/τ . That means the probability of an electron undergoing collision in any
infinitesimal time interval of length ds is just ds/τ.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The time ‘t’ is known as the relaxation time and it is defined as the
time taken by an electron between two successive collisions. That
relaxation time is also called mean free time [or] collision time.
Electrons are assumed to achieve thermal equilibrium with their
surroundings only through collision. These collisions are assumed to
maintain local thermodynamic equilibrium in a particularly simple way.
Trajectory of a conduction electron
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Success of classical free electron theory
 It is used to verify ohm’s law.
 It is used to explain the electrical and thermal conductivities
of metals.
 It is used to explain the optical properties of metals.
 Ductility and malleability of metals can be explained by this
model.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drawbacks of classical free
electron theory
 From the classical free electron theory the value of specific heat of metals is given by
4.5R, where ‘R’ is called the universal gas constant. But the experimental value of specific
heat is nearly equal to 3R.

With help of this model we can’t explain the electrical conductivity of
semiconductors or insulators.

The theoretical value of paramagnetic susceptibility is greater than the experimental
value.

Ferromagnetism cannot be explained by this theory.

At low temperature, the electrical conductivity and the thermal conductivity vary in
different ways. Therefore K/σT is not a constant. But in classical free electron theory, it
is a constant in all temperature.

The photoelectric effect, Compton effect and the black body radiation cannot be
explained by the classical free electron theory.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude’s classical theory


Theory by Paul Drude in 1900, only three years after
the electron was discovered.
Drude treated the (free) electrons as a classical
ideal gas but the electrons should collide with the
stationary ions, not with each other. average rms speed
so at room temp.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude’s classical theory
relaxation time
scattering probability per unit time
mean free path
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
...this must surely be wrong....



The electrons should strongly interact with each other. Why
don’t they?
The electrons should strongly interact with the lattice ions.
Why don’t they?
Using classical statistics for the electrons cannot be right.
This is easy to see:
de Broglie wavelength of an electron:
for RT
condition for using classical statistics
is some Å
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical conductivity
we apply an electric field. The equation of motion is
integration gives
and if
is the average time between collisions then the average
drift speed is
for
we get
remember:
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical conductivity
number of electrons passing in unit time
current of negatively charged electrons
current density
Ohm’s law
and with
we get
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical conductivity
Ohm’s law
and we can define
the conductivity
and the
resistivity
and the
mobility
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Validity of Ohm’s law



Valid for metals.
also valid for homogeneous semiconductors
not valid for inhomogeneous semiconductors
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical conductivity



Drude’s theory gives a reasonable picture for the
phenomenon of resistance.
Drude’s theory gives qualitatively Ohm’s law (linear relation
between electric field and current density).
It also gives reasonable quantitative values for resistivity, at
least at room temperature.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The Wiedemann-Franz law
k
 constant
s


Wiedemann and Franz found in 1853 that the ratio of
thermal and electrical conductivity for ALL METLALS is
constant at a given temperature (for room temperature and
above). Later it was found by L. Lorenz that this constant is
proportional to the temperature.
Let’s try to reproduce the linear behaviour and to calculate L
here.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The Wiedemann Franz law
Estimated thermal conductivity
(from a classical ideal gas)
the actual quantum mechanical result is
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Many open questions:




Why does the Drude model work so relatively well when
many of its assumptions seem so wrong? In particular, the
electrons don’t seem to be scattered by each other. Why?
How do the electrons sneak by the atoms of the lattice?
Why do the electrons not seem to contribute to the heat
capacity?
Why is the resistance of an disordered alloy so high?
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
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Density of conduction electrons in metals ~1022 – 1023 cm-3
rs – measure of electronic density
rs is radius of a sphere whose volume is equal to the volume per
1/ 3
3
3
1
4rs electron
V 1


 
rs  
~

3
N n
n1/ 3
 4n 
mean inter-electron spacing
in metals rs ~ 1 – 3 Å (1 Å= 10-8 cm)
rs/a0 ~ 2 – 6
2
a0 
 0.529 Å – Bohr radius
me 2
● Electron densities are thousands times greater than those of a gas at normal conditions
● There are strong electron-electron and electron-ion electromagnetic interactions
In spite of this the Drude theory treats the electron gas by the methods of
the kinetic theory of a neutral dilute gas
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
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t
m
ne2
At room temperatures
resistivities of metals are typically of the order of microohm centimeters (mohm-cm)
and t is typically 10-14 – 10-15 s
Mean free path l=v0t
v0 – the average electron speed
l measures the average distance an electron travels between collisions
2
estimate for v0 at Drude’s time 1 2 mv0  3 2 k BT → v0~107 cm/s → l ~ 1 – 10 Å
consistent with Drude’s view that collisions are due to electron bumping into ions
At low temperatures very long mean free path can be achieved
l > 1 cm ~ 108 interatomic spacings!
the electrons do not simply bump off the ions!
The Drude model can be applied where
a precise understanding of the scattering mechanism is not required
Particular cases: electric conductivity in spatially uniform static magnetic field
and in spatially uniform time-dependent electric field
Very disordered metals and semiconductors
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The Drude Model - Results
Main Results
1) DC Conductivity ( f  e E )
ne 2t
so 
m
- mean free path l  vT t  (1 ~ 10)  at RT
v
2) Hall Coefficient ( f  e( E   H ) )
c
1
R
nec
3) AC Conductivity ( f  e E ( ) exp(i t ) )
so
s ( ) 
1  it
4 ne 2
2
- plasma frequency  p 
m
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The Drude Model - results
B. Main Results
4) Thermal Conductivity ( j q  k T )
1
k  v 2t cv
1 2
3
v t cv
3k B T
k 3
3
2

,
v

,
c

nk B
- Wiedemann-Franz law s
T
v
2
m
2
ne t
m
=
3 kB 2
( ) T
2 e
5) Thermopower ( E  QT )
Q
=
cv
,
3ne

cv 
3
nk B
2
kB
2e
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude Conductivity
Ohm’s “Law”: V = IR
The Resistance R is a property of the conductor (e.g. a wire) which
depends on its dimensions, V is a voltage drop & I is a current. In
microscopic physics, it is more common to express Ohm’s “Law” in
terms of a dimension-independent conductivity (or resistivity) which
is intrinsic to the material the wire is made from.
In this notation, Ohm’s “Law” is written
E = j or j = sE (1)
Here, E = the electric field, j = the current density,   the resistivity
& s  the conductivity of the material.
Consider n electrons per unit volume, all moving in the direction of
the current with velocity v.
The number of electrons crossing area
A in time dt is nAvdt
A
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The charge crossing A in dt is -nevAdt, so
j = -nev. (2)
In the real material, we expect the electrons to be moving
randomly even in zero electric field due to thermal energy.
However, they will have an average, or drift velocity along the
field direction.
vdrift = -eEt / m
(3)
This comes from integrating Newton’s 2nd Law over time t.
This is the velocity that must be related to j.
Combining (2) & (3) gives
j = (ne2t / m)E.
Comparison of this with (1) gives the Drude conductivity:
ne2t
s
m
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
(3)
As is often the case for physics models, this result for σ has been
obtained using some very simple assumptions, which surely cannot
be correct! How can we test this result?
First - does it in any way self-justify its assumptions?
Does the Drude assumption of scattering from ions seem sensible?
Check it by measuring s for a series of known metals, and, using
sensible estimates for n, e and m, estimate t.
Result - t  1014 s at room temperature.
Instead of a an average scattering time t, it’s often necessary to formulate a
theory of conductivity in terms of an average distance between collisions. This
distance is called the mean free path between collisions. To do this, we have to
also consider the average electron velocity. This should not be vdrift, which is
the electron velocity in the presence of a field. Instead, it should be vrandom, the
velocity associated with the intrinsic thermal energy of the electrons.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
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Estimate vrandom by treating the electrons as a classical gas and
using the well-known result:
½(m)v2random = (3/2)(kB)T
(for revision see for example Halliday and Resnick, Physics (Wiley)).
Result: The mean free path is l = vrandomt  110 Å
Good news: this is of the order of interatomic distances.
Clearly, however, you would need more to convince a skeptical world.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
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Go after one of the most striking experimental results on metals,
the Wiedemann-Franz law.
Since 1853, it had been known that one of the most universal
properties of metals concerns a relationship between thermal and
electrical conductivity.
Recall electrical result j = sE; in simple geometry j = sE = sdV/dx
Thermal equivalent jq = -kdT/dx. k is the thermal conductivity.
ΔV
j
jq
-ΔT
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude’s assumption (with some experimental backup) was that k in metals is
dominated by the electronic contribution.
A result from elementary kinetic theory:
k = (1/3)vrandomlCel
(4)
where Cel is the electronic specific heat per unit volume. If each electron has an
energy (3/2)kBT, Cel = dEtot/VdT = (3/2) nkBT .
Recalling that l = vrandomt, and dividing k by σ gives:
k 1 2
kB 3  kB 
 mvrandom 2    T
s 2
e
2 e 
2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
(5)
Dividing by T gives the simple result that
k
3  kB 
  
sT 2  e 
2
(6)
This is exactly the kind of result that we always like to reach. All
the parameters that might be regarded in some way as suspect
have dropped out, leaving what looks like it might be a universal
quantity.
Experimentally, this is indeed the case. The true number is a
factor of two different to the Drude result, but in his original
work, a numerical error made the agreement appear to be exact!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
So, Drude’s model appeared to be reasonably self-consistent in
identifying electron-ion collisions as the main scattering mechanism,
and had a triumph regarding the most universal known property of
metals.
As would happen today, this was enough to set it up as the main theory
of metals for two decades. However, fundamental problems began to
emerge:
1. It could not explain the observation of positive Hall coefficients
in many metals.
2. As more became known about metals at low temperatures, it was
obvious that since the conductivity increased sharply, l was far too
long to be explained by simple electron-ion scattering.
3. A vital part of the thermal conductivity analysis is the use of the
kinetic theory value of 3/2nkB for the electronic specific heat.
Measurements gave no evidence for a contribution of this size.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The physics of solids is deeply quantum mechanical; indeed
condensed matter is arguably the best ‘laboratory’ for studying
subtle quantum mechanical effects in the 21st century.
Advanced general interest reading on this issue (probably more
suitable some time later in the year unless you have already read quite
a bit about quantum mechanics):
‘The theory of everything’, R.B. Laughlin and D. Pines, Proc. Nat.
Acad. Sci. 97, 28 (2000).
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Gas of classical charged particles, electrons, moves through immobile
heavy ions arranged in a lattice, vrms from equipartition theorem
(which is of course derived from Boltzmann statistics)
_2
1
3
me v  k BT
2
2
_
vrms
2
3k BT
 v 
me
Between collisions, there is
a mean free path length: L
= vrms τ
and a mean free time τ (tau)
Figure 12.11 (a) Random successive displacements of an electron in a metal without an applied electric field.
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
If there is an electric
field E, there is also a
drift speed vd (108 times
smaller than vrms) but
proportional to E,
equal for all electrons
eEt
vd 
me
Figure 12.11 (b) A combination of random displacements and displacements produced by
an external electric field. The net effect of the electric field is to add together multiple
displacements of length vd t opposite the field direction. For purposes of illustration, this
figure greatly exaggerates the size of vd compared with vrms.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
38
neAvd dt
J
 nevd
Adt
Substituting for vd
ne t
J
E
me
2
Figure The connection between current density, J, and
drift velocity, vd. The charge that passes through A in time
dt is the charge contained in the small parallelepiped,
neAvd dt.
So the correct form
of Ohm’s law is
predicted by the
Drude model !!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
ne t
s
me
40
2
2
With mean free time τ
= L/vrms
With vrms according
to Maxwell-Boltzmann
statistics
2
ne L
s
3kBTme
ne L
s
mevrms
Proof of the pudding: L should be on the order of magnitude of the interatomic distances, e.g. for Cu 0.26 nm
s
8.49  1022 cm 3 (6.02  1019 C )2  0.26nm
3  1.381 1023 JK 1  300K  9.109  1031 kg
σCu, 300 K = 5.3 106 (Ωm)-1 compare with experimental value 59 106 (Ωm) -1,
something must we wrong with the classical L and vrms
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Result of Drude theory one order of magnitude too small, so L
must be much larger, this is because the electrons are not classical
particles, but wavicals, don’t scatter like particles, in addition, the
vrms from Boltzmann-Maxwell is one order of magnitude smaller
than the vfermi following from Fermi-Dirac statistics
41
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
s
1
3k BTme


2
ne L
0.5
So ρ ~T theory for all
temperatures, but ρ ~T for
reasonably high T , so
Drude’s theory must be
wrong !
Figure 12.13 The resistivity of pure copper as a function of temperature.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
43
Phenomenological similarity conduction of electricity and conduction
of heat, so free electron gas should also be the key to understanding
thermal conductivity
V
J  s
x
Q
T
 K
At
x
1
K  CV vrms L
3
k B nvrms L
K
2
Ohm’s law with Voltage gradient,
thermal energy conducted through area A in
time interval Δt is proportional to
temperature gradient
Using Maxwell-Boltzmann statistics,
equipartion theorem, formulae of Cv for
ideal gas = 3/2 kB n
Classical expression for K
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
_
vrms
2
3k BT
 v 
me
Lets
continue
For 300 K and Cu
23
vrms
1
3  1.381 10 JK 300K

9.109  1031 kg
k B nvrms L
K
2
1.381 1023 JK 1  8.48  1022 cm 3  1.1681 105 ms 1  0.26nm
K
2
1.381 1023 JK 1  8.48  1028 m 3  1.1681 105 ms 1  0.26  109 m
K
2
Ws
K  17.78
Kms
-1 -1
Experimental value for Cu at (300 K) = 390 Wm K , again
one order of magnitude too small, actually roughly 20 times
too small
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
45
ne2 L
s
mevrms
K
With MaxwellBoltzmann
This was also one order of magnitude too
small,
2
e mrs
2
k B nvrms Lmevmrs k B m v


2
s
2ne L
2e
_
vrms
2
3k BT
 v 
me
Lorenz number classical K/σ
K 3k B2
8
2
 2  1.12  10 WK
sT 2e
2
3
k
K  BT
s 2e 2
Wrong only by a
factor of about 2,
Such an agreement
is called fortuitous
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
46
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
47
replace Lfor_a_particle with Lfor_a_wavial and vrms with vfermi,
s classical
2
ne L for _ a _ particle

mevrms
L for _ a _ wavical
_
vrms
2
3k BT
 v 
me
s quantum
ne2 L for _ a _ wavical

mev fermi
mev fermis quantum

ne2
v fermi 
2 EF
me
For Cu (at 300 K), EF = 7.05 eV , Fermi energies have only small temperature dependency,
frequently neglected
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
48
v fermi 
2 EF
me
v fermi ,copper ,300 K
2  7.05  1.602  109 J
6
1


1
.
57

10
ms
9.109  1031 kg
one order of magnitude larger than classical vrms
for ideal gas
L for _ a _ wavical
mev fermis quantum

ne2
L for _ a _ wavical _ cooper 
9.109  1031 kg  1.57  106 ms 1  5.9  107 1m 1
8.49  1028 m 3 (1.602  1019 C )2
L for _ a _ wavical _ cooper  39nm
two orders of magnitude larger than classical
result for particle
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
49
s classical
2
ne L for _ a _ particle

mevrms
So here something two orders of two magnitude too small (L) gets
divided by something one order of magnitude too small (vrms),
i.e. the result for electrical conductivity must be one order of
magnitude too small, which is observed !!
But L for particle is quite reasonable, so replace Vrms with Vfermi and
the conductivity gets one order of magnitude larger, which is close to
the experimental observation, so that one keeps the Drude theory of
electrical conductivity as a classical approximation for room
temperature
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
.
In effect, neither the high vrms of 105 m/s of the electrons
derived from the equipartion theorem or the 10 times higher
Fermi speed do not contribute directly to conducting a current
since each electrons goes in any directions with an equal
likelihood and this speeds averages out to zero charge transport
in the absence of E
Figure 12.11 (a) Random successive displacements of an electron in a metal without an applied
electric field. (b) A combination of random displacements and displacements produced by an
external electric field. The net effect of the electric field is to add together multiple displacements
of length vd t opposite the field direction. For purposes of illustration, this figure greatly
exaggerates the size of vd compared with vrms.
50
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
51
K classical
k B nvrms Lclassical

2
Vrms was too small by one order of magnitude, Lclassical
was too small by two orders of magnitude, the classical
calculations should give a result 3 orders of magnitude
smaller than the observation (which is of course well
described by a quantum statistical treatment)
So there must be something fundamentally wrong with our
ideas on how to calculate K.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
52
Wait a minute, K has something to do with the heat capacity that
we derived from the equipartion theorem
Kclassical
1
 CV _ for _ ideal _ gas vrms L for _ particle
3
We had the result earlier that the contribution of the electron gas
is only about one hundredth of what one would expect from an
ideal gas, Cv for ideal gas is actually two orders or magnitude
larger than for a real electron gas, so that are two orders of
magnitude in excess, with the product of vrms and L for particle
three orders of magnitude too small, we should calculate
classically thermal conductivities that are one order of
magnitude too small, which is observed !!!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
53
2
e mrs
2
k
nv
Lm
v
k
m
v
B
rms
e
mrs
B
K 

2
s
2ne L
2e
K 3k B2
8
2
 2  1.12  10 WK
sT 2e
Fortunately L cancelled, but vrms gets squared, we are indeed
very very very fortuitous to get the right order of magnitude
for the Lorenz number from a classical treatment
(one order of magnitude too small squared is about two orders of
magnitude too small, but this is “compensated” by assuming that the heat
capacity of the free electron gas can be treated classically which in turn
results in a value that is by itself two order of magnitude too large- two
“missing” orders of magnitude times two “excessive orders of magnitudes
levels about out
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
54
K fermi

2
2
B
kT

(
)nLfor _ a _ wavical
3 mev fermi
s quantum
2
ne L for _ a _ wavical

mev fermi
That gives for the Lorenz number in a quantum treatment
K  k
8
2

 2.45  10 WK
sT
3e
2 2
B
2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
55
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
56
Back to the problem of the temperature dependency of
resistivity
Drude’s theory predicted a dependency on square root of T, but at reasonably high
temperatures, the dependency seems to be linear
This is due to Debye’s phonons
(lattice vibrations), which are
bosons and need to be treated
by Bose-Einstein statistics,
electrons scatter on phonons, so
the more phonons, the more
scattering
Number of phonons proportional to Bose-Einstein distribution function
n phonons 
1
e / k BT  1
Which becomes for
reasonably large T
k BT
n phonons 

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
At low temperatures, there are hardly any phonons,
scattering of electrons is due to impurity atoms and
lattice defects, if it were not for them, there would not be
any resistance to the flow of electricity at zero
temperature
Matthiessen’s rule, the resistivity of a metal can be
written as
σ = σlattice defects + σlattice vibrations
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Electric Conduction
 Drude’s model
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
59
Electric Conduction
 Drude’s model (cont’d)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
60
Electric Conduction
 Drude’s model (cont’d)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
61
Exercise 1
−8
Calculate the resistance of a coil of platinum wire with diameter 0.5 mm and length 20 m at 20 C given  =11×10  m. Also determine the resistance at 1000
platinum  =3.93×10
−3
 C, given that for
/°C.
l
20 m
8
R0    (1110   m)
 11 
3
2
A
 [0.5(0.5 10 m)]
To find the resistance at 1000
But
  0 1   T  T0  
 C:
R
l
A
so we have :
R  R0 
1   T  T0  

Where we have assumed l and A are independent of temperature could cause an error of about 1% in the resistance change.
R(1000C)  (11 )[1 (3.93103 C1 )(1000 C  20 C)] = 53 
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
63
Exercise 2
A 1000 W hair dryer manufactured in the USA operates on a 120 V source. Determine the resistance of the hair dryer, and the current it draws.
P
1000 W
I

 8.33A
V
120 V
V  IR 
V 120 V
R

 14.4 
I
8.33A
The hair dryer is taken to the UK where it is turned on with a 240 V source.
What happens?
(V ) 2 (240 V) 2
P

 4000 W
R
14.4 
This is four times the hair dryer’s power rating – BANG and SMOKE!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Chapter 24: Electric Current
Current
 Definition of current
A current is any motion of charge from one region to another.
• Suppose a group of charges move perpendicular
to surface of area A.
• The current is the rate that charge flows through
this area:
I
dQ
; dQ  amount of charge that flows
dt
during the time interval dt
Units: 1 A = 1 ampere = 1 C/s
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
64
65
Current
 Microscopic view of current
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
66
Current
 Microscopic view of current (cont’d)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
67
Current
 Microscopic view of current (cont’d)
• In time t the electrons move a distance
x  d t
• There are n particles per unit volume that carry charge q
• The amount of charge that passes the area A in time t is
Q  q(nAd t )
• The current I is defined by:
dQ
Q
I
 lim
 nq d A

t

0
dt
t
• The current density J is defined by:
I
J   nq d
A


J  nq d
Current per2unit area
Units: A/m
Vector current density
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
68
An Overview
In this chapter,
we will treat conduction ‘e’ in metal as
“free charges” that can be accelerated
by an applied electric field, to explain
the electrical and thermal conduction in
a solid.
Electrical conduction involves the motion of charges in a material
nd under the
Influence of an applied electric field. By applying Newton’s 2 law to ‘e’ motion
& using a concept of “mean free time” between ‘e’ collisions with lattice vibrations,
crystal defects, impurities, etc., we will derive the fundamental equations that
govern electrical conduction in solids.
Thermal conduction,i.e., the conduction of thermal E from higher to lower temperature regions in a metal, involves the
conduction ‘e’ carrying the energy. Therefore, the relationship between the electrical conductivity and thermal conductivity
will be reviewed in this textbook.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.1 Classical theory : The DRUDE model
Goal: To find out the relation between the conductivity (or resistivity) and drift velocity
, and thereby its relation to mean free time and drift mobility, from the description of the
current density
 In a conductor where ‘e’ drift in the presence of an electric field,
current density is defined as the net amount of charge flowing across a unit area per
unit time
q
J
A t
J : current density

q : net quantity of charge flowing through an
area A at Ex
In this system, electrons drift with an average velocity vdx in the x-direction, called the drift velocity.
(Here Ex is the electric field.)
 Drift velocity is defined as
the average velocity of electrons in the x direction at time t, denote by vdx(t)
1
vdx  [vx1  vx 2  vx 3  ...  vxN ]
N
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
vxi : x direction velocity of the ith electrons
N : # of conduction electrons in the metal
[2.1]
2.1.1 Metals and conduction by electrons
 Current density in the x direction can be rewritten as a function of the drift velocity
q enAvdx t
Jx 

At
At
J x (t )  envdx (t )
[2.2]
: In time Δt, the total charge Δq crossing the area A is enAΔx, where Δx=vdxΔt and n is assumed to be the #
of ‘e’ per unit volume in the conductor (n=N/V).
: time dependent current density is useful since the average velocity at one time is not the same as at
another time, due to the change of Ex
 Think of motions of a conduction ‘e’ in metals before calculating Vdx.
(a)
A conduction ‘e’ in the electron gas moves about randomly in a metal (with a
mean speed u) being frequently and randomly scattered by thermal
vibrations of the atoms.
In the absence of an applied field there is no net drift in any direction.
(b)
Ex  Ex (t )
In the presence of an applied field, Ex, there is a net drift along the x-direction. This net drift
along the force of the field is superimposed on the random motion of the electron. After many
scattering events the electron has been displaced by a net distance, Δx, from its initial position
toward the positive terminal
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.1.1 Metals and conduction by electrons
To calculate the drift velocity vdx of the ‘e’ due to applied field Ex, we first consider the
Since
is the acceleration a of the ‘e’eEx
[F=qE=ma],
velocity vxi of the ith ‘e’ in the x direction at t.
vxi in the x direction at t is given by
me
Let uxi be the initial velocity of ‘e’ i in the x direction just after the collision. Vxi is written as the
sum of uxi and the acceleration of the ‘e’ after the collision. Here, we suppose that its last collision
was at time ti; therefore, for time (t-ti), it accelerated free of collisions, as shown in Fig.2.3.
uxi is velocity of ith‘e’ in the x direction after the collision
However, this is only for the ith electron. We need the average
velocity vdx for all such electrons along x as the following eqn.
vdx 
1
eEx
[vx1  vx 2  vx 3  ...  vxN ] 
(t  ti )
N
me
(t-ti) : average free time for N electrons between collision (~ τ = mean
free time or mean scattering time)
Fig 2.3 Velocity gained in the x direction at time t from the electric field ( Ex) for three different electrons.
There will be N electrons to consider in the metal.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.1.1 Metals and conduction by electrons
 Drift mobility (vs. mean free time)
: widely used electronic parameter in semiconductor device physics.
Suppose that τ is the mean free time or mean scattering time. Then, for some electrons, (t-ti) will be greater than ,and for others, it will
be shorter, as shown in Fig 2.3. Averaging (t-ti) for N electrons will be the same as . Thus we can substitute for (t-ti) in the previous
expression to obtain
t
t
t
vdx 
et
Ex
me
[2.3]
Equation 2.3 shows that the drift velocity increases linearly with the applied field. The constant of proportionality
a special name and symbol, called drift mobility
, which is defined as
md
et / me
vdx  m d E x
where m d 
et
me
has been given
[2.4]
[2.5]
which is often called the relaxation time, is directly related to the microscopic processes that cause the scattering of the electrons in
tthe, metal;
that is, lattice vibration, crystal imperfections, and impurities.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.1.1 Metals and conduction by electrons
From the expression for the drift velocity vdx the current density Jx follows immediately… by substituting Equation 2.4 into 2.2, that is,
J x  enm d Ex
Therefore, the current density is proportional to the electric field and the conductivity
s  enm d
 sEx [2.6]
s
term is given by
[2.7]
Then, let’s find out temperature dependence of conductivity (or resistivity) of a metal
by considering the mean time .
t
t
t
The mean time between collisions has further significance. Its 1/ represents the mean frequency of collisions or scattering
events; that is 1/ is the mean probability per unit time that the electron will be scattered. Therefore, during a small time interval
probability of scattering will be
.
t
t / t
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
t
, the
2.2 Temperature dependence of resistivity
To find the temperature dependence of
determines the drift velocity.
t
s
, let’s consider the temperature dependence of the mean free time
, since this
Fig 2.5 scattering of an electron from the thermal vibration of the atoms. The
electron travels a mean distance
between collisions.
Since the scattering cross-sectional area is S,tin the volume Sl there must be at
least one scatterer as
l u
t
Ns Sut   1
volume
t  a1
t
2
1
SuNs
[2.11]
Ns : concentration of scattering centers
When the conduction electrons are only scattered by thermal vibrations of the
metal ion, then in the mobility expression
refers to the mean
time between scattering events by this process.
m d  et m
t
S : cross-sectional area
u : mean speed
a : amplitude of the vibrations
e
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.2 Temperature dependence of resistivity
 Lattice-scattering-limited conductivity
: the resistivity of a pure metal wire increase linearly with the temperature, due to the scattering of conduction electrons by thermal
vibrations of the atoms.  feature of a metal (cf. semiconductors)
The thermal vibrations of the atom can be considered to be simple harmonic motion, much
the same way as that of a mass M attached to a spring. From the kinetic theory of matter,
1
1
Ma 2 w2 (average kinetic energy of the oscillations)  kT
4
2
So a 2  T . This makes sense because raising the T increases atomic vibrations. Thus
t
1
1
C

or
t

a 2 T
T
Since the mean time between scattering events τ is inversely
proportional to the area
that scatters 2the ‘e’,
a
eC
(to show a relation with T)
results in
meT
mT
1
1
T  
 2e
T  AT
[2.12]
sT enm d e nC
substituting for t in md  et / me
So, the resistivity of a metal
m d
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.3 MATTHIESSEN’s and NORDHEIM’s Rules.
2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity
: The theory of conduction that considers scattering from lattice vibrations only works well with pure metal and it fails for metallic alloys.
Their resistivities are weakly T-dependent, and so, different type of scattering mechanism is required for metallic alloys.
Let’s consider a metal alloy that has randomly distributed impurity atoms.
We have two mean free times between collision.
Strained region by impurity exerts a
scattering force F = - d (PE) /dx
tI
t T : scattering from thermal vibration only
t i : scattering from impurity only
In unit time, a net probability of scattering,
1
1
t
t
Two different types of scattering processes involving scattering from
impurities alone and thermal vibrations alone.

1
tT
t
1

ti
is given by
[2.13]
Then, since drift mobility depends on effective
scattering time, effective drift mobility is given by
1
1 1


ud u L u I
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
[2.14]
2.3.1 Matthiessen’s rule and the temperature coefficient
of resistivity
where u L
uI
is the lattice-scattering-limited drift mobility,
is the impurity-scattering-limited drift mobility.
Since effective resistivity

of the material
 is simply
1
1
1


enud enuL enuI
1/ enud
which can be written   T   I
[2.15]
This summation rule of resistivities from different scattering mechanisms is called Matthiessen’s rule.
Furthermore, in a general from, effective resistivity can be given by
  T   R (  R : residual resistivity)
: scattering E of impurities, dislocations, in ternal atom,
vacancies, gain boundaries, etc
Since residual resistivity shows very little T-dependence
whereas ρT = AT .
  AT  B
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
[2.17]
2.3.1 Matthiessen’s rule and the temperature coefficient
of resistivity
 Temperature coefficient of resistivity (TCR)
Eqn. 2.17 indicates that the resistivity of a metal varies with T, with A and B depending on the material. Instead of listing A and B in
resistivity tables, we prefer a temperature coefficient that refers to small, normalized changes around a reference temperature.
0 
1   
0  T  T T0
[2.18]
- temp sensitivity of the resistivity of metals
If the resistivity follows the behavior like in Eqn. 2.17, then
Eqn. 2.18 leads to
  0 1  0 (T  T0 )
[2.19]
where a0 is constant over a
temperature range T0 to T,
    o
&
T  T  To
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Resistivity of various metals vs. T
  AT  B
However
is only an approximation for some metals and not true for all metals.
This is because the origin of the scattering may be different depending on the temperature.
100
T
2000
Inconel-825
NiCr Heating Wire
1000
10
Scattering from vibration
Iron
Resistivity (n m)
Resistivity(n m)
Tungsten
Monel-400
T
Tin
100 Platinum
 (n m)
3.5
0.1
0.01
T
3
  T5
2.5
2
Copper
Nickel
0.001
Silver
0.0001
  T5
1.5
1
0.5  = R
  R
0
0
0.00001
10
100
1000
10000
10
20
40
60
80
100
T (K)
Scattering from impurity
100
1000
10000
Temperature(K)
Temperature (K)
The resistivity of copper from lowest to highest temperatures (near
The resistivity of various metals as a function of temperature above 0 melting temperature, 1358 K) on a log-log plot. Above about 100 K,
°C. Tin melts at 505 K whereas nickel and iron go through a magnetic
5
to non-magnetic (Curie) transformations at about 627 K and 1043 K   T, whereas at low temperatures,   T and at the lowest
respectively. The theoretical behavior ( ~ T) is shown for reference. temperatures  approaches the residual resistivity R . The inset
[Data selectively extracted from various sources including sections in shows the  vs. T behavior below 100 K on a linear plot (  is too
R
Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals
small
on
this
scale).
Park, Ohio,
1991)]
P.Ravindran,
PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.3.2 Solid solution and Nordheim’s rule
How does the resistivity of solid solutions change with alloy composition ?
In an isomorphous alloy of two metals, that is, a binary alloy that forms a solid solution (Ni-Cr alloy), we would expect Eqn 2.15 to
apply, with the temperature-independent impurity contribution
increasing with the concentration of solute atoms.
I
  T   I
[2.15]
This means that as alloy concentration increases, resistivity
increases and becomes less temperature dependent as ρI,
overwhelms ρT, leading to αo << 1/273.
This (temperature independency) is the advantage of alloys in resistive components.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
2.3.2 Solid solution and nordheim’s rule
1500
 Nordheim’s rule for solid solutions: an important semiempirical Eqn.
that can be used to predict the resistivity of an alloy, which relates the
impurity resistivity to the atomic fraction X of solute atoms in a solid
solution, as follows:
Temperature (°C)
How does the concentration of solute atoms affect on ρI ?
US
UID
Q
I
L
US
LID
O
S
1400
LIQUID PHASE
1300
L+
1200
1100
1000
S
SOLID SOLUTION
20
0
40
100% Cu
60
80
at.% Ni
100
100% Ni
(a)
[2.21]
C (Nordheim’s coefficient):
represents effectiveness of the solute atom in
increasing the resistivity.
600
Resistivity (n m)
 I  CX (1  X )
500
Cu-Ni Alloys
400
300
200
100
consistent
0
Nordheim rule is useful for predicting the resistivities of dilute
alloys, particularly in the low-concentration region.
0
20
100% Cu
40
60
at.% Ni
80
100
100% Ni
(b)
(a) Phase diagram of the Cu-Ni alloy system. Above the liquidus line
only the liquid phase exists. In the L + S region, the liquid (L) and solid
(S) phases coexist whereas below the solidus line, only the solid phase (a
%Nordheim’s rule assumes that the solid solution has the solute atoms randomly
solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a function
distributed in the lattice, and these random distributions of impurities cause the ‘e’ to
of Ni content (at.%) at room temperature. [Data extracted from Metals
become scattered as they whiz around the crystal.
Handbook-10th Edition, Vols 2 and 3, ASM, Metals Park, Ohio, 1991 and
Constitution of Binary Alloys, M. Hansen and K. Anderko, McGraw-Hill,
New York, 1958]
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz
Free Electron Theory
2.3.2 Solid solution and Nordheim’s rule
 Combination of Matthiessen and Nordheim rules leads to a general expression for ρ of the solid solution:
  matrix  CX (1  X )
[2.22]
where  matrix  T   R is the resistivity of the matrix due to scattering
from thermal vibrations and from other defect, absence of alloying elements.
160
Quenched
140
Resistivity (n m)
Exception: at some concentrations of certain binary alloys,
Cu and Au atoms are not randomly mixed but occupy
regular sites, which decrease the resistivity. ------------
120
100
80
Annealed
60
40
20
Cu3Au
CuAu
0
0
10 20 30 40 50 60 70 80 90 100
Composition (at.% Au)
Electrical resistivity vs. composition at room temperature in Cu-Au
alloys. The quenched sample (dashed curve) is obtained by quenching
the liquid and has the Cu and Au atoms randomly mixed. The
resistivity obeys the Nordheim rule. On the other hand, when the
quenched sample is annealed or the liquid slowly cooled (solid curve),
certain compositions (Cu3Au and CuAu) result in an ordered
crystalline structure in which Cu and Au atoms are positioned in an
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz
Free Electron
Theory
ordered fashion
in the crystal
and the scattering effect is reduced.
A Model for Electrical
Conduction:
The Drude Model
Atoms
Electrons
A regular array of atoms
surrounded by a “cloud” of
free electrons
Random movement under
zero field
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Random movement
modified by a field
Drude Model
F  qE  mea
qE
a
me
nq 2 E
J  nqvd 
t
me
J  sE
qE
v f  v i  at  v i 
t
me
nq2t
s
me
Now take the average over all times. Then
vi=0 (random movement),
me
 2
nq t
qE
qE
v f  vd 
t
t
me
me
, where t is the mean time between
collisions
l  t  vd
is the mean free path
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Resistivity and Temperature
Resistivity in metals is linear with temperature over a limited range
  0 1   T  T0 
R  R0 1   T  T0 
 : temperature coefficient of resistivity
1 

 0 T
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
-1
(C )
Plasma: A neutral gas of heavy ions and light electrons. Metals and doped
semiconductors can be treated as plasmas because they contain equal numbers of fixed
positive ions and free electrons. Free electrons in this system experience no : restoring
force from the medium when they interact with electromagnetic waves. driven by the
electric field of a light wave.
p
 r ()  1 
,
(2  i)
2
 Ne
 p  
  0 m0
2
where
86
1
2

 .

p: plasma frequency
For a lightly damped system,  = 0, so that
7.1 Plasma reflectivity
Drude-Lorentz model:
Considering the oscillations of a free electron induced by AC electric field E(t) of a light wave
with polarized along the x direction:
2p
 r ()  1  2

~
 n  r ,
 n~ is imaginary for  < p,
positive for for  > p
zero for  = p,
2
d x
dx
 m0 
 eE(t )  eE0e it .
2
dt
at
By substituting
x  x0e it
m0
x(t ) 
eE (t )
.
m0 (2  i)
The reflectivity:
~ 1 2
n
R ~
n 1
The electric displacement:
D   r 0 E  0 E  P
Therefore:
Ne 2 E
 0 E 
m0 (2  i)
Ne 2
 r ()  1 
Reflectivity of an undamped free carrier gas as a function of frequency.
2

m
(


i

)
P.Ravindran, PHY0750
0Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Thus optical measurements of r() are equivalent to AC conductivity measurement of
s().
7.2 Free carrier conductivity
Considering the damping and the electron velocity

v  x , the momentum p = m0v
By splitting



dv
m0
 m0 v  eE ,
dt



dp
p
   eE ,
dt
t
The damping time t = 1/. The shows that the electron is being accelerated by the field,
but loses its momentum-14
in the -13
time t. So t is the momentum scattering time. t is
typically in the range 10 —10 s, hence optical frequency must be used to obtain
information about t.
-it
By substituting v = v0e
,

 et 1 
v (t ) 
E (t )
m0 1  it
The current density:



j   Nev  sE,
where the AC conductivity s():
s0
s() 
,
1  it
where s0 is the DC conductivity.
Ne 2 t
s0 
m0
p
 r ()  1 
2
(  i)
2
into its real and imaginary components:
1  1 
2 
2p t 2
1  2 t 2
2p t
,
(1  2 t 2 )
.
n, k and , the real and imaginary parts of the complex refractive index and the
attenuation coefficient can
1/2be worked out. At very low frequencies that satisfy t << 1 and
2 >>1,, n  k = (2 / 2 ) , thus:
1
2
1
2
 2 t 
2k 2( 2 / 2)
 .


 c

c
c


Ne 2 t
 s0 
  0 2p t,
m0

This gives:
2
p
2
  (2s0m 0 )1/ 2 .
The attenuation coefficient is proportional to the square root of the DC conductivity and the
frequency.
Define the skin depth :
1/ 2

2  2 


  s0m 0 
Ne 2
is()
This implies that AC field can only penetrate a short distance into a conductor such as a
 r ()  1 

1

2
metal.
 0 m0 (PHY075  Condensed
i) Matter Physics,
 0  Spring 2013 : Drude-Lorenz
P.Ravindran,
Free Electron Theory
87
7.3
88
Metals
7.3.1 The Drude model
The valence electrons is free. The density N is equal to the density of metal atoms multiplied
by their valency;
The characteristic scattering time t can be determined by the measurement of s.
Experimental reflectivity of Al as a function of photon energy. The experimental data is
compared to predictions of the free electron model with h =-15
15.8 eV. The dotted line is
calculated with no damping. The dashed line with t = 8.010
s, which is the value deduced
from the DC conductivity.
All metals will become transmitting if  > p ( UV transparency of metals)
Free
and plasma properties of some metals. The values of N are in the range
28 electron
29 density
-3
10 —10 m . the very large values of N lead to plasma high electrical and thermal
conductivities and plasma frequency in the UV region.
The figure shows that the reflectivity of Al is over 80% up
to 15 eV, and then drops off to zero at higher frequencies.
From this figure, one can see that the model accounts for
the general shape of the spectrum, but there are some
important detials that are not explained.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
7.3.2 Interband transitions in metals
Interband absorption is important in metals because the EM penetrate a short distance
into the surface, and if there is a significant probability for interband absorption, the
reflectivity will be reduced from the free carrier value. The interband absorption spectra of
metals are determined by their complicated band structures and Fermi surfaces.
Furthermore, one needs to consider transitions at frequencies in which the free carrier
properties are also important.
2 1
89
Electronic configuration: [Ne]3s 3p with three valence
electrons; the first Brillouin zone is completely full, and the valence electrons spread into the second,
third and slightly into the fourth zones. The bands are filled up to the Fermi energy EF, and direct
transitions can take place from any the states below the Fermi level to unoccupied bands directly above
them on the E—k diagram. “parallel band effect” corresponding to the dip in the reflectivity at 1.5 eV
originates the high density of states between the two parallel bands. Moreover, there are further
transition at a whole range of photon energies greater than 1.5 eV. The density of states for these
transition will be lower than at 1.5 eV because the bands are not parallel, however, the absorption rate is
still significant, and accounts for the reduction of the reflectivity predicted by the Drude model..
Aluminium
Copper
10 1
[Ar]3d 4s ,
The wide outer 4s band (1),
Approximately free
states
2 electron
2
Dispersion : E = h k /2m0;
The narrow 3d band (10)
More tightly bound
Relatively dispersionless
The Fermi energy lies in the middle of the 4s band above the
3d band
A well-defined threshold for interband transitions from the 3d
to the 4s.
Band diagram of Al at the W and K points that are responsible for the reflectivity dip at
1.5 eV are labelled
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
90
7.3.2 Interband transitions in metals
Copper
Gold and silver
In gold the interband absorption threshold occurs
at a slightly higher energy than copper.
In silver the interband absorption edge is around
4 e, the frequency is in the ultraviolet, and so the
reflectivity remains high throughout the whole
visible spectrum.
The 3d electrons lie in relatively bands with very high densities of states, while the 4s are much broader with a low density of
states. The Fermi energy lies in the middle of the 4s band above the 3d band. Interband transition are possible from the 3d band
below EF. The lowest energy transitions are marked on the band diagram. The transition energy is 2.2 eV which corresponds to a
wavelength of 560 nm.
The measured reflectivity of copper. Based on the plasma frequency, one would expect near-100% reflectivity for photon energies
below 10.8 eV (115nm). However, the experimental reflectivity falls off sharply above 2 eV due to the interband absorption edge.
The explain why copper has a reddish colour.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
91
A model for electrical
conduction

(a)
(b)
A classical model of electrical
conduction in metals: Drude model in
1900
– In the absence of an electric field, the
conduction electrons move in random
directions through the conductor with
average speeds v ~ 106 m/s. The drift
velocity of the free electrons is zero.
There is no current in the conductor since
there is no net flow of charge.
– When an electric field is applied, in
addition to the random motion, the free
electrons drift slowly (vd ~ 10-4 m/s) in a
direction opposite that of the electric
field.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
92
Derivation of the drift
velocity, vd, using Drude
model
Electric force on an electron: F = qE, where q = -e.



Acceleration of the electron: a = F/me = qE/me.
Define the following:
– t = 0: the instant just after one collision has occurred;
– t : the instant just before the next collision occurs;
– vi: velocity of the electron at t = 0; vf: velocity of the
electron at time t.


Apply Newton’s 2nd law: vf = vi + at = vi + (qE/me)t.
Average vf over all possible values of vi and collision
time t: v  v  qE t  qE t ; so, v  qE t
f
i
me
me
d
me
– t: average time interval between successive collisions =
mean free time = relaxation time.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
93
Conductivity and resistivity in
terms of microscopic quantities

According to Drude model:
– Conductivity s = (nq2t)/me.
– Resistivity:  = 1/s = me/(nq2t).
– n: charge carrier density = the number of charge carriers
per unit volume.
– q: the charge on each carrier. For electrons, q=-e.
– me: electronic mass.
– t: mean free time or relaxation time


According to Drude model, conductivity and resistivity
do not depend on the strength of the electric field, a
feature characteristic of a conductor obeying Ohm’s
law.
Mean free path = average distance between
collisions: l  v t
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
94
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
95
Sommerfeld Model: 1928
Ashcroft-Mermin Chapter 2:
- Systematically recast Drude-Lorentz theory in terms of FD
statistics rather than MB statistics
- Wiedemann-Franz law (still) came out right
- Estimated specific heat right
- Difficulties remained:
- Sign of Hall coefficient
- Magneto-resistance
- What determines the scattering time t?
- What determine the density n?
- Why are some elements non-metals?
Arnold Sommerfeld
(1868-1951)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: why are metals
shiny?

Drude’s theory gives an explanation of why
metals do not transmit light and rather reflect
it.
96
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Some relations from basic
optics
wave propagation in matter
plane wave
complex index
of refraction
Maxwell relation
97
all the interesting physics in in the dielectric function!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Free-electron dielectric
function
one electron in
time-dependent field
we write
and get
the dipole moment
for one electron is
and for a unit volume
of98solid it is
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Free-electron dielectric
function
we use
to get
so the final
result is
is called
the plasma frequency
99
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Meaning of the plasma
frequency
the dielectric function in the Drude model is
with
remember
ε real and negative, no wave propagation
metal is opaque
ε real and positive, propagating waves
metal is transparent
100
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
plasma frequency: simple
interpretation
values for the plasma energy

101
longitudinal collective mode of the
whole electron gas
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
the Wiedemann-Franz law
k
 constant
s

Wiedemann and Franz found in 1853 that the
ratio of thermal and electrical conductivity for
ALL METALS is constant at a given
temperature (for room temperature and
above). Later it was found by L. Lorenz that
this constant is proportional to the
102
102
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
The Wiedemann Franz law
estimated thermal conductivity
(from a classical ideal gas)
the actual quantum mechanical result is
103
this is 3, more or less....
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Comparison of the Lorenz
number to experimental data
at 273 K
metal
10-8 Watt Ω K-2
Ag
2.31
Au
2.35
Cd
2.42
Cu
2.23
Mo
2.61
Pb
2.47
Pt
2.51
Sn
2.52
W
3.04
Zn
2.31
L = 2.45 10
-8
-2
Watt Ω K
104
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Failures of the Drude model

Despite of this and many other correct
predictions, there are some serious problems
with the Drude model.
105
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical
conductivity
line
106
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Failures of the Drude model:
he mean free path
A very good example is
http://www.sciencemag.org/cgi/content/full/319/5867/1226/FIG1

It seems that the electrons manage to sneak
past the (close packed) atoms and by all the
other electrons. How do they do this?
107
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Failures of the Drude model:
electrical conductivity of an alloy
•
•
The resistivity of an alloy should be between those of its components, or at least similar to them.
It can be much higher than that of either component.
108
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Failures of the Drude model:
heat capacity
consider the classical energy for one mole of solid in a heat bath: each degree of freedom contributes with
energy
heat capacity
monovalent
divalent
trivalent
el. transl.

109
ions vib.
Experimentally, one finds a value of about
at room temperature, independent of the
number of valence electrons (rule of Dulong
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 109
: Drude-Lorenz Free Electron Theory
Many open questions:




110
Why does the Drude model work so relatively
well when many of its assumptions seem so
wrong? In particular, the electrons don’t
seem to be scattered by each other. Why?
How do the electrons sneak by the atoms of
the lattice?
Why do the electrons not seem to contribute
to the heat capacity?
Why is the resistance of an disordered alloy
so high?
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
111
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Electrical properties of metals:
Classical approach (Drude theory)
at the end of this lecture you should understand....
 Basic assumptions of the
classical theory
 DC electrical conductivity in the Drude model
 Hall effect
 Plasma resonance / why do metals look shiny?
 thermal conduction / Wiedemann-Franz law
 Shortcomings of the Drude model: heat capacity...
112
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Some relations from basic
optics
wave propagation in matter
plane wave
complex index
of refraction
Maxwell relation
113
all the interesting physics in in the dielectric function!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Free-electron dielectric
function
one electron in
time-dependent field
we write
and get
the dipole moment
for one electron is
and for a unit volume
of114
solid it is
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Free-electron dielectric
function
we use
to get
so the final
result is
is called
the plasma frequency
115
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
11
6
The free electron theory of metals
The Drude theory of metals
Paul Drude (1900): theory of electrical and thermal conduction in a metal
application of the kinetic theory of gases to a metal,
which is considered as a gas of electrons
mobile negatively charged electrons are confined in a
to immobile positively charged ions
metal by attraction
isolated atom
nucleus charge eZa
Z valence electrons are weakly bound to the nucleus (participate in chemical reactions)
Za – Z core electrons are tightly bound to the nucleus (play much less of a role in chemical reactions)
in a metal – the core electrons remain bound to the nucleus to form the metallic ion
the valence electrons wander far away from their parent atoms
called conduction electrons or electrons
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
in a metal
Doping of Semiconductors
C, Si, Ge, are valence IV , Diamond fcc structure. Valence band is full
Substitute a Si (Ge) with P. One extra electron donated to conduction band
N-type semiconductor
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
11
7
11
8
The basic assumptions of the Drude model
1. between collisions the interaction of a given electron
with the other electrons is neglected
and with the ions is neglected
independent electron approximation
free electron approximation
2. collisions are instantaneous events
Drude considered electron scattering off
the impenetrable ion cores
the specific mechanism of the electron scattering is not considered below
3. an electron experiences a collision with a probability per unit time 1/τ
dt/τ – probability to undergo a collision within small time dt
randomly picked electron travels for a time τ before the next collision
τ is known as the relaxation time, the collision time, or the mean free time
τ is independent of an electron position and velocity
4. after each collision an electron emerges with a velocity that is randomly directed and
with a speed appropriate to the local temperature
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
11
9
DC electrical conductivity of a metal
V = RI Ohm’s low
the Drude model provides an estimate for the resistance
introduce characteristics of the metal which are independent on the shape of the wire
E  j
j=I/A – the current density
 – the resistivity
R=L/A – the resistance
s = 1/  the conductivity
j  sE
L
A
j  env
v is the average electron velocity
eE
v t
m
j  sE
 ne2t
j  
 m

E

ne2t
s
m
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
12
0
motion under the influence of the force f(t) due to spatially uniform
electric and/or magnetic fields
average
momentum
equation of motion
for the momentum per electron
dp(t )
p( t )

 f (t )
electron collisions introduce a frictional damping termdt
for the momentum
t per electron
Derivation:
dt
 dt 
p(t  dt )   1     p(t )  f (t )dt    0
t 
t

scattered
part
fraction of electrons that
does not experience scattering
p(t  dt )  p(t )  f (t )dt 
p( t )
t
total loss of momentum
after scattering
dt  O  dt 2 
p(t  dt )  p(t )
p( t )
 f (t ) 
 O  dt 
dt
t
dp(t )
p( t )
 f (t ) 
dt
t
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
average
velocity
p(t )  mv(t )
1. The Drude Model
A. Assumptions
1) Free Electron Model
2) Independent Electron Model
3) Relaxation Time Approximation
- Equation of motion
dv
v
m
 f m
dt
t
- Elementary kinetic theory
4) Maxwell Boltzmann Statistics
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Electrical properties of metals:
Classical approach (Drude theory)
at the end of this lecture you should understand....
 Basic assumptions of the
classical theory
 DC electrical conductivity in the Drude model
 Hall effect
 Plasma resonance / why do metals look shiny?
 thermal conduction / Wiedemann-Franz law
 Shortcomings of the Drude model: heat capacity...
122
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude’s
classical
theory
Theory by Paul
Drude in 1900,
only


three years after the electron was
discovered.
Drude treated the (free) electrons as a
classical ideal gas but the electrons
should collide with the stationary ions,
not with each other.
average rms speed
so at room temp.
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude’s classical theory
relaxation time
(average time between scattering events)
mean free path
124
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
onduction electron Density n
#atoms
per
volume
calculate as
#valence
electrons
per atom
density
atomic
mass
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
...this must surely be wrong....



The electrons should strongly interact with
each other. Why don’t they?
The electrons should strongly interact with
the lattice ions. Why don’t they?
Using classical statistics for the electrons
cannot be right. This is easy to see:
condition for using classical statistics
is some Å
de Broglie wavelength of an electron:
for RT
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
but:
In a theory which gives results like this, there must certainly be a great deal of truth.
Hendrik Antoon Lorentz
So what are these results?
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical
conductivity
we apply an electric field. The equation of motion is
integration gives
and if
is the average time between collisions then the average drift speed is
for
128
we get
remember:
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical
conductivity
number of electrons passing in unit time
current of negatively charged electrons
current density
Ohm’s law
and with
we get
129
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical
conductivity
Ohm’s law
define
ivity
130
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Ohm’s law



valid for metals
valid for
homogeneous
semiconductors
not valid for
inhomogeneous
semiconductors
131
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Electrical conductivity of materials
132
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
How to measure the
conductivity / resistivity

A two-point probe can be used but the
contact ore wire resistance can be a
problem, especially if the sample has a small
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
133
How to measure the
conductivity / resistivity

The problem of contact resistance can be
overcome by using a four point probe.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
134
Drude theory: electrical conductivity
line
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
Drude theory: electrical
conductivity



Drude’s theory gives a reasonable picture for
the phenomenon of resistance.
Drude’s theory gives qualitatively Ohm’s law
(linear relation between electric field and
current density).
It also gives reasonable quantitative values,
at least at room temperature.
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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
To explain the high conductivities and the trend we need to have a model for both thermal and electrical conductivity, that model should be able to explain
Ohm’s law, empirical for many metals and insulators, ohmic solids
J = σ E current density is proportional to applied electric field
U
ρl
R = / I for a wire R =
/A
Conductivity , resistivity is its reciprocal value
2
J: current density A/m
σ: electrical conductivity Ω
-1
-1
m , reciprocal value of electrical resistivity
E: electric field V/m
Also definition of σ: a single constant that does depend on the material and temperature but not on
applied electric field represents connection between I and U
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 : Drude-Lorenz Free Electron Theory
137