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P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
http://folk.uio.no/ravi/CMP2013
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
Sommerfield Model for Free Electron Theory
1
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for 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
Quantum free electron theory
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 for both
microscopic and macroscopic particles.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
6
Schrödinger Time Independent Wave Equation
The Schrödinger's time independent wave equation is given by
08
2
22
)VE(
h
m
For one-dimensional motion, the above equation becomes
08
2
2
2
2
)VE(h
m
dx
d
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
7
Introducing,
2
h
In the above equation
02
22
2
)VE(m
dx
d
For three dimension,
02
2
2 )VE(m
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
tiV
m
22
2
tiV
m
22
2(or)
EH
where H = Vm
2
2
2
= Hamiltonian operator
ti
= Energy operatorE =
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 Maxwell-
Boltzmann 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 Theory13
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.
22
2m
(7)
Starting point: time-independent Schrödinger equation
Summerfeld’s Quantum Mechanical Model of Electron
Conduction in Metals
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 14
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.
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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory15
(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.)
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.
Solving then gives allowed wavefunctions:
L
pke
Vzyx x
zkykxki
kzyx
2,
1,,
)(
2/1, p integer, etc. (9)
Most appropriate boundary condition for solid state physics: the periodic
boundary condition first introduced by Born and von Karman:
zyxLzLyLx ,,,, (8)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
What is the physical meaning of these eigenstates?
Then, note that k is also an eigenstate of the momentum operator
i
p
ˆ , 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:
m
p
m
kk
22
222
(11)
m
kk
2
22 (10)First, note energy eigenvalues:
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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.
. . . . . .
. . . . . .
ky
kz
kx
State separation
2L
State volume
(2L)3
Filled
states
Empty
statesFermi surface
Fermi
wavenumber
kF
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory19
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?
Quantities of interest depend on the carrier number per unit volume; the
sample dimensions drop out neatly.
No -
3/1
2
3
3 32
23
4
V
Nk
L
Nk FF
(12)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory20
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
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!
TF 25000 K ( recall kBT at room T 1/40 eV)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory21
kkF
A couple of much-used graphs relating to the Sommerfeld model:
a) The free electron
dispersion
Probability
of state
occupation
1
0 , k
F or
kF
b) The T = 0 state occupation
function.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory22
The specific heat of the quantum fermion gas
1
1),(
/)(
TkBe
Tf
The T=0 occupation discussed previously is a limit of the Fermi-Dirac
distribution function for fermions:
where the chemical potential F. (13)
As expected, T is a minor player
when it comes to changing things.
At finite T:
f()
F
~ 2kBT
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory23
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.
Our next task, then, is to derive a quantity of high and general importance, the
density of states g().
and the specific heat cel from dEtot/dT as before.
),()(),( TfgTn (14)
dTnTEtot ),()(0
Then internal energy Etot(T) can be calculated from
(15)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory24
. . . . . .
. . . . . .
ky
kz
kx
State separation
2L
State volume
(2L)3dk
3
2
332
42
.
.2)(
L
dkk
LkperVol
katshellofVol
Ldkkg
Number of allowed states per unit volume per shell thickness dk:
spin
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory25
Convert to density of states per unit volume per unit (the quantity usually
meant by the loose term ‘density of states’):
32
2/12/3
3
2/12
22
2
2)(
)2(
2
24
2)(
mg
dm
mm
dg
Very important result, but note that dependence is different for
different dimension .
2/1
22
2;
mk
k
mddk (16a, b)
(17)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory26
n(,T)
F
g(
Movement of
electrons in
energy at finite T
2kBT
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:
[Etot(T) - Etot(0)]/V 1/2g(F). kBT.2kBT = g(F). (kBT)2 (18)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory27
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 = (23g(F). kB2T (20)
c.f. Drude: Bnk
2
3
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 :
F
BBel
Tknkc
2
2
(21)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory28
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 Theory29
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.
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory30
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
Include scattering by modifying (22) to
m(dv/dt + vt -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 Theory32
Set B to zero and stress that the relevant velocity is vdrift ; (23) becomes
m(dvdrift/dt + vdriftt -eE
Steady state solution (dvdrift/dt = 0) is just
vdrift = -(et/m)E
Following the procedure from Kittel gives us the Drude expression (3):
m
ne t
2
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 Theory33
ky
kz
kx
To understand, consider physical meaning of the process:
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.
ky
kz
kx
dk = -1mvdrift= -eEτ/E = 0
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory34
Which states can ‘interact with the
outside world’?
ky
kz
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-
like
picture:dkk
LF .
3
42 3
3
In the quantum model, only those
within kBT of F, i.e. those very near the
Fermi surface.
# of states mom.
gain
# of states (1/2
FS area)
mom.
gainx comp.
only
Quantum
picture:F
F kdkk
L 3
2.
2
42 23
(24)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 35
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
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:
2222
2 323
11
t
t
k
e
kTknkv
ne
m
TT
B
F
BBF
(25)
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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 Theory38
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).
dk
ky
kz
dk
Thermal excitation: All particles
with k kF, but sum over k = 0.
kx
ky
kz
Electrical excitation: All particles
with k kF, but sum over k = 2 kF/3.
kx
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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 Theory40
Classical Drude gas
Random velocity purely
thermal: mTkB /3
Specific heat cel = Bnk
2
3
Large number of particles
moving slowly.
Quantum Sommerfeld gas: do wave
mechanics and then think in an
‘equivalent particle’ picture
Random velocity dominantly
quantum (due to Pauli principle):
mV
Nmkv FF /3/
3/1
2
F
BBel
Tknkc
2
2
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
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,
- Distribution function
Modification of the Drude model
- the mean free path
- the Wiedemann-Frantz law
- the thermopower
3/12 )3( nkF
tt FT vvl
22
2 )(3
)(2
3
e
k
e
k
T
BB
k
))((62
2
F
BBB
E
Tk
e
k
e
kQ
)2
exp()2
()(2
2/3
Tk
mv
Tk
mnvf
BB
MB
1]/)2
1exp[(
1
4
)/()(
23
3
TkEmv
mvf
BF
FD
B
F
BBvF
BT kn
E
Tkkncv
m
Tkvv )(
22
3,)
3(:
22/1
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
The Sommerfeld theory of metals
the Drude model: electronic velocity distribution
is given by the classical
Maxwell-Boltzmann distribution
the Sommerfeld model: electronic velocity distribution
is given by the quantum
Fermi-Dirac distribution
3/ 22
3
32
0
( ) exp2 2
/ 1( )
14
2exp 1
( )
MB
B B
FD
B
B
m mvf n
k T k T
mf
mv k T
k T
n d f
v
v
v vnormalization
condition T0
Pauli exclusion principle: at most one electron
can occupy any single electron level
43
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
)()(2 2
2
2
2
2
22
rErzyxm
, , , ,
, , , ,
, , , ,
x y z L x y z
x y L z x y z
x L y z x y z
k
m
m
kE
d
eV
i
2
2)(
)(1
1)(
22
2
k
kv
kp
k
rr
r rk
k
22
2
1
2mv
m
pE
electron wave function
associated with a level of energy E
satisfies the Schrodinger equation
consider noninteracting electrons
L
3D:
1D:
periodic
boundary
conditions
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
44
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
1
2 2 2, ,
yx zik Lik L ik L
x x y y z z
e e e
k n k n k nL L L
rk
k r ieV
1)(
, , , ,
, , , ,
, , , ,
x y z L x y z
x y L z x y z
x L y z x y z
apply the boundary conditions
components of k must be
nx, ny, nz integers
3
33
2
2/2
V
V
L
a region of k-space of volume contains
states i.e. allowed values of k
the number of statesper unit volume of k-space, k-space density of states
VL
L
33
2
22
2
the area per point
the volume per point
k-space
45
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
compare to the ~ 107 cm/s at T=300Kclassicalthermal velocity 0 at T=0
2/1
22
/3
/
2/
mTkv
mkv
kp
kET
mkE
k
Bthermal
FF
FF
BFF
FF
F
Fermi wave vector ~108 cm-1
Fermi energy ~1-10 eV
Fermi temperature ~104-105 K
Fermi momentum
Fermi velocity ~108 cm/s
consider T=0
3 3
3 2
3
2
3
2
4
3 62
26
3
F F
F
F
k V kV
kN V
kn
the number of allowed values of
k within the sphere of radius kF
to accommodate N electrons
2 electrons per k-level due to spin
the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF
312
3222
312
3
32
3
nm
v
nm
E
nk
F
F
F
kx
ky
kFFermi sphere
Fermi surface
at energy EF
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
state of the lowest energydensity of states
volume
46
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Total number of states with energy < E
The density of states – number of states per unit energy EmV
dE
dNED
mEVN
23
22
23
22
2
2)(
2
3
Density of states
32
Vk-space density of states – the number of states per unit volume of k-space
The density of states per unit volume or the density of
states
Em
dE
dnED
23
22
2
2
1)(
3
23
VN k
Total number of states with wave vector < k 2 2
2
kE
m
47
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
FF
F
kk
Vei
kk
Em
k
N
E
m
k
m
kd
V
E
dFV
FV
F
V
km
E
F
F
5
3
10
3
10
1
24
1
)(8
)(8
)(
8
22
22
52
2
22
3
3
..0
3
3
22
k
kkkkk
k
k
kk
dkkd
m
kF
2
22
4
2)(
k
k
3
23Fk
N V
Ground state energy of N electrons
Add up the energies of all electron states inside the Fermi sphere
volume of k-space per state
smooth F(k)
The energy density
The energy per electronin the ground state
48
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
In quantum mechanics particles are indistinguishable
systems where particles are exchanged are identical
exchange of identical particles can lead to changing of the system wave function by a phase factor only
repeated particle exchange → e2ia 1
1221 ,, aie
1221 ,,
122121 21212
1, pppp 122121 21212
1, pppp
Antisymmetric wavefunction with respect
to the exchange of particles
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
Fermions: electron, proton, neutron
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
p1, p2 – single particle states
obey Fermi-Dirac statistics Obey Bose-Einstein statistics
system of N=2 particles
1, 2 - coordinates and
spins for each of the
particles
Remarks on statistics I
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
Bosons: photon, Cooper pair, H atom, exciton
symmetric wavefunction with respect
to the exchange of particles
49
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
1( )
exp 1
1( )
exp 1
( ) exp
( ) ( ) ( )
FD
B
BE
B
MB
B
f EE
k T
f EE
k T
Ef E
k T
n dEn E dED E f E
Fermi-Diracdistribution function
Bose-Einsteindistribution function
Maxwell-Boltzmanndistribution function
Distribution function f(E) → probability that a state at energy E
will be occupied at thermal equilibrium
fermionsparticles with half-integer spins
bosonsparticles with integer spins
both fermions and bosons at high Twhen TkE B
degenerate Fermi gas
fFD(k) < 1
degenerate Bose gas
fBE(k) can be any
classicalgas
fMB(k) << 1
=(n,T) – chemical potential
50
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
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.
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
At T < TdB fBE and fFD are strongly different from fMB
At T >> TdB fBE ≈ fFD ≈ fMB
1 22
1 3
22 3
2~
~
2
dB
B
dB
dB
B
h
mk T p
d n
T nmk
Thermal de Broglie
wavelength
Particles become
indistinguishable when
i.e. at temperatures below
remarks on statistics II
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
2xp x
g(k’)
k0
kk’
m v
vg=v
x
x
(x)
2
'
'( , ) ( ')exp '
2
kt g i t
m
k
r k k r
x
51
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
3 2
2 2
1 2( )
2
dn mD E E
dE
Density of states
Distribution function1
( )
exp 1
( ) ( )
B
f EE
k T
n dED E f E
3D
T≠0 the Fermi-Dirac distribution
E
EEfT
0
,1)(lim0
FT
E
0
lim
VdEEfED
VdEED
1)()(
1)( [the number of states in the energy range from E to E + dE]
[the number of filled states in the energy range from E to E + dE]
EF E
Density of
filled states
D(E)f(E,T)
shaded area – filled
states at T=0
Density of
states
D(E)
per unit volume
52
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Specific heat of the degenerate electron gas, estimate
Bv
B
nkc
Tnkmvnu
2
3
2
3
2
1 2
1v
V V
U uc
V T T
Uu
V
Specific heat
U – thermal
kinetic energy
Classical gas
The observed electronic
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
Number of electrons which gain energy with increasing temperature ~
The total electronic thermal kinetic energy
The electronic specific heat1
~ Bv B
V F
U k Tc nk
V T E
EF/kB ~ 104 – 105 K
kBTroom / EF ~ 0.01
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
T ~ 300 K for typical metallic densities
T = 0
B
F
k TN
E
~ BB
F
k TU N k T
E
53
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
0
3
0
3
)()()(4
)()()()(4
EfEdEDEfd
n
EEfEdEDEfEd
u
kk
kkk
)()()()(
)()(6
)(
)()()(6
)(
00
422
0
422
0
FF
E
B
B
EHEdEEHdEEH
TODTkdEEDn
TODDTkdEEEDu
F
correctly to order T2
Specific heat of the degenerate electron gas
The way in which integrals of the form differ from their zero T values
is determined by the form of H(E) near E=
E
nn
nn
EHdE
d
n
EEH )(
!
)()(
0
Replace H(E) by its Taylor expansion about E=
The Sommerfeld expansion
6
44
22
12
12
1
2
)(360
7)(
6)(
)()()()(
TkOHTkHTkEEH
EHdE
daTkdEEHdEEfEH
BBB
En
n
n
n
n
B
dEEfEH )()(
FE
dEEH )(
Successive terms are smaller by O(kBT/)2
For kBT/ << 1 Replace
by T0 = EF
and v
V
uu c
T
54
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
FD statistics depress
cv by a factor of
2
22
0
22
2
2
11
3 2
( )6
( )3
3( )
2
2
3
2
3
BF
F
B F
v B F
F
F
Bv B
F
classical B
B
F
v
k TE
E
u u k T D E
uc k TD E
T
nD E
E
k Tc nk
E
c nk
k T
E
c T
(1)
(2)
2
2 323
2FE n
m
3 2
2 2
1 2( )
2
mD E E
Specific heat of the degenerate electron gas55
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Thermal conductivity
vv
q
lvccv
T
3
1
3
1 2
tk
kj
thermal current density jq – a vector parallel to the direction of heat flow whose magnitude gives the thermal energy per unit time crossing a unite area perpendicular to the flow
Tkmv
nkc
Te
k
ne
mvc
B
Bv
Bv
2
3
2
1
2
3
2
3
3
2
2
2
2
kWiedemann-Franz law (1853)
Lorenz number ~ 2×10-8 watt-ohm/K2
Drude:
application of
classical ideal
gas laws
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
m
ne t
2
the correct at room T
the correct estimate of v2 is vF2 at room T
01.0~/~2
2
FBclassicalvvB
F
Bv ETkccnk
E
Tkc
100~/~22
TkEvv BFclassicalF
22
3
e
k
T
B
k
For
degenerate
Fermi gas of
electrons
56
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Thermopower
ne
cQ v
3
Drude:
application of
classical ideal
gas laws
Bv nkc2
3
e
kQ B
2
Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field
directed opposite to the T gradient
TQE
Thermopower
For
degenerate
Fermi gas of
electrons
the correct at room T
Q/Qclassical ~ 0.01 at room T
01.0~/~2
2
FBclassicalvvB
F
Bv ETkccnk
E
Tkc
F
BB
E
Tk
e
kQ
6
2
high T low T
gradT
E
thermoelectric field
57
P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory
Electrical conductivity and Ohm’s law
t
t
2
2
1
ne
m
m
ne
( ) (0)
avg
avg
avg
d dm e
dt dt
et t
e
e
m m
t
t
v kE
Ek k
Ek
k Ev
Ej
m
ne t2
avgnevj
Equation of motionNewton’s law
In the absence of collisions the Fermi sphere in k-space is displaced as a whole at a uniform rate by a constant applied electric field
Because of collisions the displaced Fermi sphere is maintained in a steady state in an electric field
Ohm’s law
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
kavg << kF for n = 1022 cm-3 and j = 1 A/mm2 vavg = j/ne ~ 0.1 cm/s << vF ~ 108 cm/s
kavg
( ) ( )( ) 0
d t tt
dt
e
t
t t
p pf
p f E
F
Fermi sphere
ky
kx
58