Download Prof.P. Ravindran, Sommerfield Model for Free Electron Theory

1
Sommerfield Model for 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 16 July: Sommerfield Model for Free Electron Theory
Quantum free electron theory
deBroglie wave concepts
The universe is made of Radiation(light) and
matter(Particles).The light exhibits the dual nature(i.e.,) it can
behave s both as a wave [interference, diffraction phenomenon]
and as a particle[Compton effect, photo-electric effect etc.,].
Since the nature loves symmetry was suggested by Louis
deBroglie. He also suggests an electron or any other material
particle must exhibit wave like properties in addition to particle
nature
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
In mechanics, the principle of least action states” that a
moving particle always chooses its path for which the action is a
minimum”. This is very much analogous to Fermat’s principle of
optics, which states that light always chooses a path for which the
time of transit is a minimum.
de Broglie suggested that an electron or any other material
particle must exhibit wave like properties in addition to particle
nature. The waves associated with a moving material particle are
called matter waves, pilot waves or de Broglie waves.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Wave function
A variable quantity which characterizes de-Broglie waves
is known as Wave function and is denoted by the symbol .
The value of the wave function associated with a moving
particle at a point (x, y, z) and at a time ‘t’ gives the probability of
finding the particle at that time and at that point.
de Broglie wavelength
deBroglie formulated an equation relating the momentum
(p) of the electron and the wavelength () associated with it, called
de-Broglie wave equation.
 hp
where h - is the planck’s constant.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Schrödinger Wave Equation
Schrödinger describes the wave nature of a particle in
mathematical form and is known as Schrödinger wave equation.
They are ,
1. Time dependent wave equation and
2. Time independent wave equation.
To obtain these two equations, Schrödinger connected the
expression of deBroglie wavelength into classical wave equation
for a moving particle.
The obtained
equations are applicable
microscopic and macroscopic particles.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
for both
6
Schrödinger Time Independent Wave Equation
The Schrödinger's time independent wave equation is given by
2
8

m
2
 
( E  V )  0
2
h
For one-dimensional motion, the above equation becomes
d 
8 2 m
 2 ( E  V )  0
2
dx
h
2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
7
Introducing,
h
 
2
In the above equation
d 2
2m

( E  V )  0
2
2
dx

For three dimension,
2m
   2 ( E  V )  0

2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
8
Schrödinger time dependent wave equation
The Schrödinger time dependent wave equation is
2

2

    V  i
2m
t

2
2


 V

2
m



  i

t

(or)
H  E
where H =
 2 2
 V
2m

E = i
t
= Hamiltonian operator
= Energy operator
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
9
The salient features of quantum free electron theory
Sommerfeld proposed this theory in 1928 retaining the concept of free
electrons moving in a uniform potential within the metal as in the classical
theory, but treated the electrons as obeying the laws of quantum mechanics.
Based on the deBroglie wave concept, he assumed that a moving electron
behaves as if it were a system of waves. (called matter waves-waves associated
with a moving particle).
According to quantum mechanics, the energy of an electron in a metal is
quantized.The electrons are filled in a given energy level according to Pauli’s
exclusion principle. (i.e. No two electrons will have the same set of four
quantum numbers.)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
10
Each Energy level can provide only two states namely, one with spin
up and other with spin down and hence only two electrons can be
occupied in a given energy level.
So, it is assumed that the permissible energy levels of a free electron are
determined.
It is assumed that the valance electrons travel in constant potential
inside the metal but they are prevented from escaping the crystal by
very high potential barriers at the ends of the crystal.
In this theory, though the energy levels of the electrons are discrete, the
spacing between consecutive energy levels is very less and thus the
distribution of energy levels seems to be continuous.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
11
Success of quantum free electron theory
According to classical theory, which follows MaxwellBoltzmann statistics, all the free electrons gain energy. So it
leads to much larger predicted quantities than that is actually
observed. But according to quantum mechanics only one percent
of the free electrons can absorb energy. So the resulting specific
heat and paramagnetic susceptibility values are in much better
agreement with experimental values.
According to quantum free electron theory, both experimental
and theoretical values of Lorentz number are in good agreement
with each other.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
12
Drawbacks of quantum free electron theory
It is incapable of explaining why some crystals have metallic
properties and others do not have.
It fails to explain why the atomic arrays in crystals including
metals should prefer certain structures and not others
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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….. The
development of the new theory of quantum mechanics.
A natural step was to formulate a quantum theory of electrons in metals.
First done by Sommerfeld.
Assumptions
Most are very similar to those of Drude. Free and independent electrons, but
no assumptions about the nature of the scattering.
Starting point: time-independent Schrödinger equation
2 2

   
2m
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
(7)
13
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 (7), 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.
a) It says that the surface is important in determining the
physical properties, which is known not to be the case.
b) 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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
14
Most appropriate boundary condition for solid state physics: the periodic
boundary condition first introduced by Born and von Karman:
x  L, y  L, z  L   x, y, z 
(8)
(We 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:
 k  x, y , z  
1
V
1/ 2
e
i (kx xk y y kz z )
2 p
,
, kx 
L
p integer, etc. (9)
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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
15
What is the physical meaning of these eigenstates?
First, note energy eigenvalues:
 2k 2
k 
2m
(10)
Then, note that k is also an eigenstate of the momentum operator
 , with eigenvalue p = k.
pˆ  
i
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
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
(11)
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.
So, why have we come anywhere here? 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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
19
How can we scale these quantum mechanical effects against something we are
more familiar with?
Calculate numerical values for the parameters. Use potassium (tutorial
question 4).
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!
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
20
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
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
21
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
e
(    ) / k BT
1
where the chemical potential   F. (13)
At finite T:
~ 2kBT
As expected, T is a minor player
when it comes to changing things.
f()
  F

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
22
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().
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
23
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
 
spin
 L
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
24
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

(16a, b)
d

2m 
g () 

2 2  3
3 / 2 1/ 2
(17)
Very important result, but note that  dependence is different for
different dimension .
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
25
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)
Movement of
electrons in
energy at finite T
F

2kBT
[Etot(T) - Etot(0)]/V  1/2g(F). kBT.2kBT = g(F). (kBT)2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
(18)
26
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 :
 k BT 
2

cel 
nkB 
2
 F 
(21)
c.f. Drude:
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
3
nkB
2
27
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 behaviour of its excitations.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
28
What about the response to external fields or temperature gradients?
To treat these simply, should introduce another vital and wide-ranging 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 centred 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.
29
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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
(22)
This equation would produce continuous acceleration, which we know cannot
occur in the presence of scattering.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
30
Include scattering by modifying (22) to
m(dv/dt + vt  -eE - ev  B
(23)
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 when we discussed the Drude model!
It is no surprise, then, that it leads to the same expression for the electrical
conductivity:
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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

m
If you give this some thought, it should concern you. What happened to our
new quantum picture?
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
32
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 ‘quasiDrude’ derivation assumes that every electron state in the sphere is shifted by
dk. This is ‘mathematically correct’, but physically entirely the wrong picture.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
33
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
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?
dk
Consider amount of extra velocity/momentum acquired in equilibrium:
Drudelike
picture:
3
 2  4 3
k F .dk
 
 L  3
# of states
mom.
gain
Quantum
picture:
3
2
2

4

k
 
F dk 2
. kF
 
2
3
 L 
# of states (1/2
FS area)
mom.
gain
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
x comp.
only
34
(24)
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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
35
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 nkB 
2
T T ne t 3
2
 F 
2

3
2
 kB 
 
 e 
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
2
(25)
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?
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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
dk
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
dk
38
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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Classical Drude gas
Quantum Sommerfeld gas: do wave
mechanics and then think in an
‘equivalent particle’ picture
Random velocity purely
thermal:
Random velocity dominantly
quantum (due to Pauli principle):
3k BT / m
1/ 3
Specific heat cel =
3
nkB
2
Large number of particles
moving slowly.
 2 N
vF  k F / m   3

V

 k BT 
2

cel 
nkB 
2
 F 
Small effective number of particles
moving very fast, due to special
quantum mechanical constraints.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
40
/m
41
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
The Sommerfeld Model
Electrons are fermions.
- Ground state: Fermi sphere,
k F  (3 2 n)1 / 3
- Distribution function
m
mv 2
) 3 / 2 exp( 
)
2 k B T
2k B T
(m / ) 3
1
f FD (v) 
3
1
4
exp[( mv 2  E F ) / k B T ]  1
2
f MB (v)  n(
Modification of the Drude model
3k B T 1 / 2
3
 2 k BT
v : vT  (
)  v F , cv  n k B 
(
)nk B
m
2
2 EF
- the mean free path l  vT t  v Ft
- the Wiedemann-Frantz law
- the thermopower
Q
k
3 k
 2 kB 2
 ( B )2 
( )
T 2 e
3 e
kB
 2 k B k BT

( )(
)
2e
6 e EF
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
43
The Sommerfeld theory of metals
the Drude model: electronic velocity distribution
is given by the classical
Maxwell-Boltzmann distribution
 m 
f MB ( v )  n 

2

k
T
B 

the Sommerfeld model: electronic velocity distributionf
FD
is given by the quantum
Fermi-Dirac distribution
m / 
( v) 
3
4 3
3/ 2
 mv 2 
exp  

2
k
T
B 

1
1 2

mv

k
T
B
0


exp  2
1

k BT




Pauli exclusion principle: at most one electron
can occupy any single electron level
n   dvf ( v )
normalization
condition
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
T0
44
consider noninteracting electrons
electron wave function
associated with a level of energy E
satisfies the Schrodinger equation
2   2
2
2 
 2  2  2  ( r )  E ( r )

2m  x
y
z 
  x, y , z  L     x, y , z 
periodic
boundary
conditions
  x, y  L, z     x, y, z 
  x  L, y, z     x, y, z 
a solution neglecting
the boundary conditions
normalization constant: probability
of finding the electron somewhere
in the whole volume V is unity
energy
momentum
velocity
wave vector
de Broglie
wavelength
 k (r ) 
1 ikr
e
V
1   dr (r )
 2k 2
E (k ) 
2m
p  k
k
v
m
k
2

k
3D:
1D:
L
2
p2 1 2
E
 mv
2m 2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
45
  x, y , z  L     x, y , z 
  x, y  L, z     x, y, z 
1 ikr
 k (r ) 
e
V
  x  L, y, z     x, y, z 
 e  e 1
2
2
2
kx 
nx , k y 
ny , kz 
nz
L
L
L
nx, ny, nz integers
apply the boundary conditions
components of k must be
e
ik x L
ik y L
ik z L

V

2 / L 3 2 3
V
the number of states
per unit volume of k-space, 2 3
a region of k-space of
volume  contains
the area
per point
 2 
 
 L 
2
the volume  2 3 2 3
  
per point
V
 L 
states
i.e. allowed
values of k
k-space density of states
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
k-space
consider T=0
the Pauli exclusion principle postulates that only one electron can occupy a single state
therefore, as electrons are added to a system, they will fill the states in a system
like water fills a bucket – first the lower energy states and then the higher energy states
46
the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF
state of the lowest energy
volume
density
of states
the number of allowed values of
k within the sphere of radius kF
 4 k F 3  V
kF 3
 2V
 3 
3
6

  2 
to accommodate N electrons
2 electrons per k-level due to spin
kF 3
N 2 2V
6
kF 3
n 2
3
Fermi wave vector k F
Fermi energy
~108 cm-1
E F   2 k F / 2m
2
Fermi temperature TF  EF k B
Fermi momentum
pF  k F
~1-10 eV
~104-105 K
ky
Fermi sphere
kF
kx
Fermi surface
at energy EF
k F  3 2n 
13
23
2

EF 
3 2n 
2m
13

vF  3 2n 
m
vF  k F / m
Fermi velocity
~108 cm/s
1/ 2
compare to the
vthermal  3k BT / m  ~ 107 cm/s at T=300K
classical
P.Ravindran,
PHY075- Condensed Matter Physics, Spring 2013 16 July: 0
Sommerfield Model
for Free Electron Theory
thermal
velocity
at T=0
47
Density of states
V 3
k
3 2
Total number of states with wave vector < k
N
Total number of states with energy < E
V  2mE 
N 2 2 
3   
E
32
The density of states – number of states per unit energyD ( E )  dN  V  2m 
dE 2 2   2 
The density of states per unit volume or the density of
states
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
k
2m
32
dn
1  2m 
D( E ) 
 2 2 
dE 2   
k-space density of states – the number of states per unit volume of k-space
2 2
E
32
E
V
2 3
48
Ground state energy of N electrons
2 2
E  2 k
Add up the energies of all electron
k  k F 2m
states inside the Fermi sphere
volume of k-space per state
 smooth F(k)
The energy density
The energy per electron
in the ground state
k  8 3 V
 F (k ) 
k
V
V
k 0i . e.V 
F
(
k
)

k






F ( k )dk
3 
3 
8 k
8
5
E
1
 2k 2
1  2k F
 3  dk
 2
V 4 k k F
2m  10m
2
E
3  2k F
3

 EF
N 10 m
5
 2k 2
F (k ) 
2m
dk  4k 2 dk
kF 3
N  2V
3
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Remarks on statistics I
In quantum mechanics particles are indistinguishable
49
systems where particles are exchanged are identical
exchange of identical particles can lead to changing
ia




,


e
 2 , 1 
1 2
of the system wave function by a phase factor only
 1, 2    2 , 1 
repeated particle exchange → e2ia  1
system of N=2 particles
1, 2 - coordinates and
spins for each of the
particles
Antisymmetric wavefunction with respect
to the exchange of particles
1
 1, 2  
 p1 1  p2 2   p1 2  p2 1 
2
symmetric wavefunction with respect
to the exchange of particles
1
 1, 2  
 p1 1  p2 2   p1 2  p2 1 
2
Fermions are particles which have half-integer spin
the wavefunction which describes a collection
of Fermions must be antisymmetric with respect
to the exchange of identical particles
Bosons are particles which have integer spin
the wavefunction which describes a collection
of bosons must be symmetric with respect
to the exchange of identical particles
Fermions: electron, proton, neutron
Bosons: photon, Cooper pair, H atom, exciton




p1, p2 – single particle states
if p1 = p2   0
→ at most one fermion can occupy
any single particle state – Pauli principle
Unlimited number of bosons can occupy
a single particle state
obey Fermi-Dirac statistics
Obey Bose-Einstein statistics
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Distribution function f(E) → probability that a state at energy E
will be occupied at thermal equilibrium
50
fermions
particles with
half-integer spins
Fermi-Dirac
distribution
function
f FD ( E ) 
1
E
exp 
1

 k BT 
degenerate
Fermi gas
fFD(k) < 1
bosons
particles with
integer spins
Bose-Einstein
distribution
function
f BE ( E ) 
1
E
exp 
 1
k
T
 B 
degenerate
Bose gas
fBE(k) can be any
both fermions and
bosons at high T
when E    k BT
Maxwell-Boltzmann
E
f
(
E
)

exp
distribution


MB
 k BT 
function
n   dEn( E )   dED( E ) f ( E )
=(n,T) – chemical potential
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
classical
gas
fMB(k) << 1
remarks on statistics II
BE and FD distributions differ from the classical MB distribution
because the particles they describe are indistinguishable.
Particles are considered to be indistinguishable if their wave packets
overlap significantly.
Two particles can be considered to be distinguishable
if their separation is large compared to their de Broglie wavelength.
m v
(x)
12
 2 
h
dB  
~

p
 mk BT 
Particles become
indistinguishable when
dB ~ d  n
2
1 3
2 2 2 3
n
i.e. at temperatures below TdB 
mk B
At T < TdB fBE and fFD are strongly different from fMB
At T >> TdB fBE ≈ fFD ≈ fMB
Electron gas in metals:
n = 1022 cm-3, m = me → TdB ~ 3×104 K
Gas of Rb atoms:
n = 1015 cm-3, matom = 105me → TdB ~ 5×10-6 K
Excitons in GaAs QW
n = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K
x
vg=v
x
x
k
g(k’)
Thermal de Broglie
wavelength
51
k’
k0
 
k '2  
 (r, t )   g (k ') exp i  k ' r 
t 
2
m
k'



A particle is represented by a
wave group or wave packets
of limited spatial extent,
which is a superposition of many matter
waves with a spread of wavelengths
centered on 0=h/p
The wave group moves
with a speed vg – the group speed,
which is identical to the classical
particle speed
Heisenberg uncertainty principle, 1927:
If a measurement of position is made with
precision x and a simultaneous
measurement of momentum in the x
direction is made with precision px,
then
p x 
x Theory
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron
2
T≠0
the Fermi-Dirac distribution
3D
dn
1  2m 
Density of states
D( E ) 
 2 2 
dE 2 

1
Distribution function f ( E ) 
E
exp 
 1
k
T
 B 
n   dED ( E ) f ( E )  
32
E
lim f ( E )  1, E  
T 0
0 E
lim   EF
T 0
1
 [the number of states in the energy range from E to E + dE]
V
1
D( E ) f ( E )dE   [the number of filled states in the energy range from E to E + dE]
V
D( E )dE 
Density of
states
D(E)
per unit volume
Density of
filled states
D(E)f(E,T)
shaded area – filled
states at T=0
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
EF
E
52
Specific heat of the degenerate electron gas, estimate
1  U 
 u 





V  T V  T V
U
U – thermal
u
kinetic energy
V
Specific heat
53
T ~ 300 K for typical metallic densities
T=0
cv 
f(E) at T ≠ 0 differs from f(E) at T=0
only in a region of order kBT about 
because electrons just below
EF have been excited to levels just above EF
1
3
Classical gas u  n mv 2  nkBT
2
2
3
cv  nkB
The observed electronic
2
contribution at room T is
usually 0.01 of this value
Classical gas: with increasing T all electron gain an energy ~ kBT
Fermi gas: with increasing T only those electrons in states within
an energy range kBT of the Fermi level gain an energy ~ kBT
k T
Number of electrons which gain energy with increasing temperature ~ N B
EF
 k BT 
The total electronic thermal kinetic energy U ~  N
 k BT
E
F 

EF/kB ~ 104 – 105 K
1  U 
k BT
kBTroom / EF ~ 0.01
The electronic specific heat cv  
~
nk
B

V  T V Model for Free
EF Electron Theory
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield
Specific heat of the degenerate electron gas

dk
u   3 E (k ) f E (k )    dED( E ) Ef ( E )
4
0
 u 

and
u

c

v



dk
 T V
n   3 f E ( k )    dED( E ) f ( E )
4
0

54
The way in which integrals of the form  H ( E ) f ( E )dE differ from their zero T values H ( E )dE


is determined by the form of H(E) near E=

( E   )n d n
Replace H(E) by its Taylor expansion about E= H ( E )  
H ( E ) E 
n
n
!
dE
n 0
EF


 H ( E )dE   k T 
The Sommerfeld expansion  H ( E ) f ( E )dE 



n 1

B
d 2 n 1
an
H ( E ) E 
2 n 1
dE
7 4
k BT  H (  ) 
kBT 4 H (  )  O kBT 
  H (E )E 
6
360
  

Successive terms are
smaller by O(kBT/)2
2
2n

u   ED ( E )dE 
2
0
For kBT/ << 1

n   D ( E )dE 
0

6
2
6
k BT 2 D(  )  D(  )  O (T 4 )
k BT 2 D(  )  O (T 4 )
EF
 H ( E )dE   H ( E )dE  (   E
F
) H ( EF )
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
0
0
6
2
Replace 
by T0 = EF
correctly to order T2
Specific heat of the degenerate electron gas
 1   k T 2 
  EF 1   B  
 3  2 E F  
u  u0 
1  2m 
D( E )  2  2 
2 

EF 
2
 3 n 
2m
2
23
32
E
6
 k BT 
2
D( EF )
u  2 2
cv 

k B TD ( E F )
T
3
3 n
D( EF ) 
2 EF
cv 
(1)
2
 2 k BT
2 EF
cclassical 
nk B
3
nk B
2
FD statistics depress  k BT
cv by a factor of
3 EF
2
(2)
cv  T
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
55
Thermal conductivity
56
thermal current density jq – a vector parallel to the direction of heat flow
whose magnitude gives the thermal energy per unit time
jq  kT
crossing a unite area perpendicular to the flow
1
1
k  v 2tcv  lvcv
3
3
ne2t

m
k cv mv 2 3  k B 

  T
 3ne2 2  e 
Drude:
application of
classical ideal
gas laws
cv 
2
3
nkB
2
1 2 3
mv  k BT
2
2
Wiedemann-Franz law (1853)
Lorenz number ~ 2×10-8 watt-ohm/K2
success of the Drude model is due to the cancellation
of two errors: at room T the actual electronic cv is 100
times smaller than the classical prediction, but v is 100
times larger
 2 k BT
For
cv 
nkB cv cv classical ~ k BT / EF ~ 0.01 at room T
the correct
degenerate
2 EF
Fermi gas of
2
2
the correct estimate of v2 is vF2
vF vclassical ~ EF / k BT ~ 100 at room T
electrons
k  2  kB 

 
T
3  e 
2
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Thermopower
Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field
directed opposite to the T gradient
E  QT
high T
low T
thermoelectric field
gradT
E
Drude:
application of
classical ideal
gas laws
For
degenerate
Fermi gas of
electrons
the correct
Thermopower
cv 
3
nkB
2
cv 
 2 k BT
2 EF
Q
57
cv
3ne
Q
kB
2e
nkB
cv cv classical ~ k BT / EF ~ 0.01 at room T
Q/Qclassical ~ 0.01 at room T
Q
 2 k B  k BT 


6 e  EF 
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Electrical conductivity and Ohm’s law
Equation of motion
dv
dk
Newton’s law
m

  eE
dt
dt
In the absence of collisions the Fermi sphere in
k-space is displaced as a whole at a uniform rate k (t )  k (0)   eE t
by a constant applied electric field
eE
Because of collisions the displaced Fermi sphere
k


t
avg
is maintained in a steady state in an electric field
ky
k
eE
F
v avg 
Fermi sphere
kx
avg
m

m
58
dp ( t )
p( t )

 f (t )  0
dt
t
p  ft   eEt
Ohm’s law
t
j  nev avg
kavg
the mean free path l = vFt
because all collisions involve only electrons near the Fermi surface
vF ~ 108 cm/s for pure Cu:
at T=300 K
t ~ 10-14 s
l ~ 10-6 cm = 100 Å
at T=4 K
t ~ 10-9 s
l ~ 0.1 cm
 ne2t 
E
j  
 m 
ne2t

m
1
m
  2
 ne t
kavg << kF for n = 1022 cm-3 and j = 1 A/mm2 vavg = j/ne ~ 0.1 cm/s << vF ~ 108 cm/s
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory