Transcript Part VI

Drude’s Classical Model of Metallic Conduction
Contrasted with
Sommerfeld’s Quantum Mechanical Theory
Drude’s Classical Model of Metals
(Beautifully explained in depth in Ashcroft and Mermin, Ch. 1)
•Modern condensed matter physics
was BORN with the discovery of the
electron by J.J. Thompson in 1897.
• Soon afterwards (1900) Drude used the
new concept to postulate
A theory of metallic conductivity
Drude’s Assumptions
1. Matter consists of light negatively charged
electrons which are mobile, & heavy, static,
positively charged ions.
2. 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.
• The neglect of the electron-ion interactions is
The Free Electron Approximation.
3
Drude’s Assumptions Continued
3. Electron-ion collisions are assumed to dominate. 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/τ, where
1/τ  The Electron 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. See the figure.
Ion
The mean time between
collisions is t.
Trajectory of a mobile electron
4
Drude Conductivity: Ohm’s “Law” V = IR
• The Resistance R is a property of a 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 = rj or j = sE (1)
E = Electric Field, j = Current Density,
r = The resistivity & s = The conductivity of the material.
• Consider a wire of cross sectional area A, with a current flowing in
it. The current consists of n electrons per unit volume, all moving
in the direction of the current with velocity v:
A
j
The number of electrons crossing area A in time dt is nAvdt
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 j = sE gives the Drude
conductivity:
2
ne t
s=
m
• As is often true for physics models, this result for
σ has been obtained using some very simple
assumptions, which surely cannot be correct in
reality! How can this result be tested?
• First ask: Does the Drude assumption of
scattering from ions seem reasonable?
• Check it experimentally by measuring s
for a series of known metals, and, using sensible
estimates for n, e and m, estimate t.
• Results show that,at least for “simple metals”
t  10-14 s at room temperature.
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• For many cases, 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 consider the average
electron velocity. This should not be vdrift, which
is the electron velocity due to the electric field.
• Instead, it should be vrandom, the velocity
associated with the intrinsic thermal energy of
the electrons.
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Lets estimate vrandom by treating the electrons
as a classical ideal gas and using the result
from classical statistical physics:
The Equipartition Theorem
½(m)v2random = (3/2)(kB)T
(For a simple derivation, see any Physics I textbook!)
Results:
The mean free path is
l = vrandomt  1-10 Å
This is of the order of interatomic distances,
so it is reasonable!
A Very Important Experimental Result for Metals:
“The Wiedemann-Franz Law”
Since 1853, it has been known that one of the most universal
properties of metals is an experimentally well verified relationship
between the thermal and the electrical conductivities. As we just
discussed, in the presence of an external electric field E, the current
density j is given by Ohm’s “Law”:
j = sE
ΔV
In simple one dimensional
geometry as in the figure,
j
The thermal analogue of this is
jq
j = sE = sdV/dx
jq = -kdT/dx.
k  Thermal Conductivity.
-ΔT
Drude’s assumption was that k in metals is dominated by
the electronic contribution. A result from elementary
kinetic theory is:
k = (1/3)vrandomlcel
(4)
where cel is the electronic specific heat per unit volume.
Each electron has an energy (3/2)kBT so, for n electrons per
unit volume:
Etot = (3/2)kBT,
cel = dEtot/dT = (3/2) nkBT
Recall that l = vrandomt, & divide k by σ = (ne2t / m) giving:
k 1 2
kB 3  kB 
= mvrandom 2 =   T
s 2
e
2 e 
2
Dividing by T gives the simple result that
k
3  kB 
=  
sT 2  e 
2
This is a very nice result!! All the parameters that might be
regarded in some way as poorly known have dropped out,
leaving what looks like it might be a universal quantity.
Experimentally, this is indeed the case.
The measured number is a factor of two different from this
Drude result, but in his original work, a numerical error
made the agreement appear to be exact!
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. 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 (discussed later).
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
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contribution of this size.
The solution to these issues and the other unpleasant
difficulties like the existence of insulators would not have
come in a thousand years of hard work, if we had been
restricted to classical physics.
“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).
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Summerfeld’s Quantum Mechanical Model of
Electron Conduction in Metals
The Free Electron Gas
A Non-trivial Quantum Fluid
Bohr, de Broglie, Schrödinger, Heisenberg, Pauli, Fermi,
Dirac….. With the development of quantum mechanics, a
natural step was to formulate a quantum theory of electrons
in metals. This was first done by Sommerfeld.
Assumptions
Most are very similar to those of Drude. Free & independent
electrons, but no assumptions about the nature of the scattering.
Starting point: time-independent Schrödinger equation
2 2
  = 
2m
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Note that no other potential terms are included; hence we
can solve for a single, independent electron and then
investigate the consequences of putting in many electrons.
To solve the Schrödinger equation we need appropriate
boundary conditions for a metal. Standard ‘particle in a
box’: set ψ = 0 at boundaries. This is not a good
representation of a solid, however.
1. It says that the surface is important in determining
the physical properties, which is known not to be the case.
2. It implies that the surfaces of a large but not
infinite sample are perfectly reflecting for electrons, which
would make it impossible to probe the metallic state by, for
example, passing a current through it.
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The most appropriate boundary condition for solid
state physics is the periodic boundary condition :
x  L, y  L, z  L = x, y, z 
Consider a cube of side L for mathematical
convenience; a different choice of sample shape
would have no physical consequence at the end of
the calculation. Solving then gives allowed
wavefunctions:
2 p
 k  x, y , z  = 1 / 2 e
, kx =
, p integer, etc. (9)
V
L
Here V = L3 and the V-1/2 factor ensures that normalisation
is correct, i.e. that the probability of finding the electron
somewhere in the cube is 1.
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1
i(kx xk y ykz z )
What is the physical meaning of these eigenstates?
 2k 2
k =
2m
First, note energy eigenvalues:
(10)
Then, note that k is also an eigenstate of the
momentum operator

pˆ = 
i
, with eigenvalue p = k.
The state k is just the de Broglie formulation of a free
particle! It has a definite momentum k.
Then we see the close analogy with a well-known
classical result:
 2k 2 p 2
k =
2m
=
2m
(11)
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It thus also has a velocity v = k/m.
How does the spectrum of allowed states look?
Cubic grid of points in k-space, separated by 2/L;
volume per point (2/L)3.
We have just done a quantum calculation of a free
particle spectrum, and seen close analogies with that of
classical free particles. Answer: now we have to
consider how to populate these states with a
macroscopic number of electrons, subject to the rules
of quantum mechanics.
Sommerfeld’s great contribution: to apply Pauli’s
exclusion principle to the states of this system, not just
to an individual atom.
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Each k state can hold only two electrons (spin up and
down). Make up the ground (T = 0) state by filling the
grid so as to minimise its total energy.
Result: At T = 0, get a sudden demarkation between filled and
empty states, which (for large N), has the geometry of a sphere.
Fermi
wavenumber
kF
ky
Filled
states
Fermi surface
. . . . . .
. . . . . .
kz
State volume
(2/L)3
kx
Empty
states
State separation
2/L
20
We set out to do a quantum Drude model, and did not explicitly
include any direct interactions due to the Coulomb force, but we
ended up with something very different. The Pauli principle
plays the role of a quantum mechanical particle-particle
interaction The quantum-mechanical ‘free electron gas’ is a
non-trivial quantum fluid!
.Is everything OK here - doesn’t kF appear to depend on the
arbitrary cube size L?
No -
4 3 N  2 
k F =  
3
2 L 
3
1/ 3
 2 N
 k F =  3

V

(12)
Quantities of interest depend on the carrier number per
unit volume; the sample dimensions drop out neatly.
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How can we scale these quantum mechanical effects against
something we are more familiar with?
Calculate numerical values for the parameters. Use
potassium
Result: kF  0.75 Å-1
vF  1 x 106 ms-1
F  2 eV
TF  25000 K ( recall kBT at room T  1/40 eV)
This is a huge effect: zero point motion so large that a
Drude gas of electrons would have to be at 25000 K for
the electrons to have this much energy!
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A couple of much-used graphs relating to the
Sommerfeld model:
a) The free electron
dispersion
b) The T = 0 state occupation
function.

Probability
of state
occupation
1
kF
k
0
, k
F or
kF
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The specific heat of the quantum fermion gas
The T=0 occupation discussed previously is a limit of the
Fermi-Dirac distribution function for fermions:
f (, T ) =
1
where the chemical potential   F. (13)
e (  -  ) / k BT  1
At finite T:
As expected, T is a minor
player when it comes to
changing things.
~ 2kBT
f()
  F

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The Fermi function gives us the probability of a state of
energy  being occupied. To proceed to a calculation of
the specific heat, we need to know the number of states
per unit volume of a given energy  that are occupied per
unit energy range at a given T.
n(, T ) = g () f (, T )
(14)
Then internal energy Etot(T) can be calculated from

Etot (T ) =   n( , T )d
(15)
0
and the specific heat cel from dEtot/dT as before.
Our next task, then, is to derive a quantity of high and
general importance, the density of states g().
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ky
dk
. . . . . .
. . . . . .
kz
State volume
(2/L)3
kx
State separation
2/L
Number of allowed states per unit volume per shell thickness dk:
2 Vol. of shell at k 2 4k 2 dk
g (k )dk = 3
= 3
L
Vol. per k
L  2  3
 
 L 
spin
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Convert to density of states per unit volume per unit  (the quantity
usually meant by the loose term ‘density of states’):
md
 2m 
dk = 2 ; k =  2 
 k
  
1/ 2
1/ 2
2m m   

4 2 2 
   2m 
g ()d = 2
(2) 3
2


2m 
g () =

2 2  3
(16a, b)
d
3 / 2 1/ 2
(17)
Very important result, but note that  dependence is different
for different dimension (see tutorial question 5).
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Evaluating integral (15) is complicated due to the slight movement of
the chemical potential  with T (see Hook and Hall and for details
Ashcroft and Mermin). However, we can ignore the subtleties and
give an approximate treatment for F >> kBT:
g(
n(,T)
F
Movement of
electrons in
energy at finite T
2kBT

[Etot(T) - Etot(0)]/V  1/2g(F). kBT.2kBT = g(F). (kBT)2
(18)
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Differentiating with respect to T gives our estimate of the specific
heat capacity:
cel = 2g(F). kB2T
(19)
The exact calculation gives the important general result that
cel = (2/3g(F). kB2T
(20)
How does this compare with the classical prediction of the Drude
model? Combining g(F) from (17) with the expression for F
derived in tutorial question 4 gives, after a little rearrangement (do
this for yourselves as an exercise):
 k BT 
2

cel = nk B 
2
 F 
(21)
c.f. Drude:
3
nk B
2
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A remarkable result: Even though our quantum mechanical
interaction leads to highly energetic states at F, it also gives a system
that is easy to heat, because you can only excite a highly restricted
number of states by applying energy kBT.
The quantum fermion gas is in some senses like a rigid
fluid, and its thermal properties are defined by the
behavior of its excitations.
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What about the response to external fields or
temperature gradients?
To treat these simply, should introduce another vital and wideranging concept, the Semi-Classical Effective Model.
Faced with wave-particle duality and a natural tendency to be more
comfortable thinking of particles, physicists often adopt effective
models in which quantum behaviour is conceptualised in terms of
‘classical’ particles obeying rules modified by the true quantum
situation.
In this case, the procedure is to think in terms of wave packets
centered on each k state as particles. Each particle is classified by a k
label and a velocity v.
Velocity is given by the group velocity of the wave packet:
v = dw/dk = -1d/dk = k/m for free particles like those we are
concerned with at present.
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Assumption of the above: we cannot localise our ‘particles’ to better
than about 10 lattice spacings. The uncertainty principle tells us that if
we try to do that, we would have to use states more than 10% of our
full available range (defined roughly by kF).
Not, however, a particularly heavy restriction, since it is unlikely that
we would want to apply external fields which vary on such a short
length scale.
In the absence of scattering, we then use the following ‘classical’
equation of motion in applied E and/or B fields:
mdv/dt = dk/dt= -eE - ev  B
This equation would produce continuous acceleration, which we know
cannot occur in the presence of scattering.
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Include scattering by modifying (22) to
m(dv/dt + v/t = -eE - ev  B
This is just the equation of motion for classical particles subject to
‘damped acceleration’. If the fields are turned off, the velocity that
they have acquired will decay away exponentially to zero. This
reveals their ‘conjuring trick’. The physical meaning of v in (23)
must therefore be the ‘extra’ or ‘drift’ velocity that the particles
acquire due to the external fields, not the group velocity that they
introduced in their (3.22).
In fact, this is formally identical to the process that we discussed in
deriving equation (3) when we discussed the Drude model!
It is no surprise, then, that it leads to the same expression for the
electrical conductivity:
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Set B to zero and stress that the relevant velocity is vdrift ; (23)
becomes
m(dvdrift/dt + vdrift/t = -eE
Steady state solution (dvdrift/dt = 0) is just
vdrift = -(et/m)E
Following the procedure from Kittel gives us the Drude expression
(3):
ne 2t
s=
m
If you give this some thought, it should concern you. What
happened to our new quantum picture?
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To understand, consider physical meaning of the process:
ky
ky
kz
kz
kx
E=0
kx
dk = -1mvdrift= -eEτ/
Fermi surface is shifted along the kx axis by an E field along x. The
‘quasi-Drude’ derivation assumes that every electron state in the
sphere is shifted by dk. This is ‘mathematically correct’, but
physically entirely the wrong picture.
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Which states can ‘interact with the
outside world’?
ky
kz
In the quantum model, only those
within kBT of F, i.e. those very
near the Fermi surface.
kx
dk
Pauli principle: only those states can
scatter, so only processes involving them
can relax the Fermi surface. So how does
the ‘wrong’ picture work out?
Consider amount of extra velocity/momentum acquired in equilibrium:
Drude-  2  -3 4
3

k


like
F .dk
L  3
picture: 
# of states
mom.
gain
Quantum
picture:
-3
2
2

4

k


F dk 2
. k F (24)


2
3
 L 
# of states
mom.
(1/2 FS area) gain
x comp.
only
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So the two pictures, one of which is conceptually incorrect, give the
same answer, because of a cancellation between a large number of
particles acquiring a small extra velocity and a small number of
particles acquiring a large extra velocity.
However, this is only the case for a sphere. As we shall see later,
Fermi surfaces in solids are not always spherical. In this case, the
Drude-like picture is simply wrong, and the conductivity must be
calculated using a Fermi surface integral.
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What about thermal conductivity?
Recall (4) from Drude model: k = 1/3vrandomlcel
Here, vrandom can clearly be identified with vF, and l = vFt.
Provided that t is the same for both electrical and thermal conduction
(basically true at low temperatures but not at high temperatures; see
Hook and Hall Ch. 3 after we have covered phonons), we can now
revisit the Wiedemann-Franz law using (21) for the specific heat:
 k BT 
k
1 m 1 2 
 =
=
vF t nk B 
2
sT T ne t 3
2
 F 
2
  kB 
 
3  e 
2
2
(25)
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The approximate factor of two error from the Drude model has been
corrected (2/3 in quantum model cf. 3/2 in Drude model).
Real question - how on earth was the Drude model so close?
Answer: Because a severe overestimate of the electronic specific
heat was cancelled by a severe underestimate of the characteristic
random velocity.
Thinking for the more committed (i.e. non-examinable): Would all
quantum gas models give the same result for the Wiedemann-Franz
law as the quantum fermion gas?
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The modern conceptualisation of the quantum free electron gas:
Make an analogy with quantum electrodynamics (QED).
Filled Fermi sea at T = 0 is inert, so it is the vacuum. Temperature and /
or external fields excite special particle-antiparticle pairs. The role of
the positron is played by the holes (vacancies in the filled sea with an
effective positive charge).
Thermal excitation: All particles
with k  kF, but sum over k = 0.
ky
Electrical excitation: All particles
with k  kF, but sum over k = 2 kF/3.
ky
kz
kz
kx
kx
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dk
dk
Scorecard so far; achievements and failures of the quantum
Fermi gas model
1. Successful prediction of basic thermal properties of metals.
2. Successful prediction of conductivity, as long as we don’t ask
about the microscopic origins of the scattering time t - why is the
mean free path so long in metals at low temperatures? What
happened to electron-ion and electron-electron scattering?
3. Failure to predict a positive Hall coefficient.
4. No understanding whatever of insulators. ‘… So insulators, which
cannot carry a current, must contain electrons too. In a metal they
must be free to move, and in an insulator they must be stuck. I asked
my tutor why this was so - and he told me that it was not understood.
It was good to know the limits of knowledge at the time’ Quote from
Sir Nevill Mott, writing about his time as a student in Cambridge,
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around 1925.
Classical Drude gas
Quantum Sommerfeld gas: do
wave mechanics and then think in
an ‘equivalent particle’ picture
Random velocity purely
thermal: 3k BT / m
Random velocity dominantly
quantum (due to Pauli principle):
1/ 3
 2 N
vF = k F / m =  3

V

3
nk B
Specific heat cel =
2
Large number of particles
moving slowly.
/m
 k BT 
2

cel = nk B 
2
 F 
Small effective number of
particles moving very fast, due to
special quantum mechanical
constraints.
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