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CHAPTER ONE
INTRODUCTION
1.0 ELECTRICAL CONDUCTION
A well known experimental evidence in electrical conduction is presented as ohm’s
law:
J = E , (1.1)
relating the electric field, E , to current density, J.
In its simple form, the constant of proportionality, the conductivity of material is a
scalar. Furthermore, the ohm’s law holds good, if not in entire range of electric field, at
least in a narrow range.
This material property, the electrical conductivity, is probably the most fantastic of
all properties, in the sense it ranges over 25 orders of magnitude at room temperature
alone.
Figure 1: Conductivity (in Ωm) of materials(from Hummel)
What in the material leads to electrical conduction? In metals, one view is that
electrons from the outer shell of an atomic configuration contribute to electrical
conduction. The electrons under normal circumstances move around randomly with no
net flow in any direction. However, under electric field, the driving force – q E on
electrons results in a net electron flow
a .
Thinking of an electron as a particle, we are tempted to write the Newton’s law of
motion for an electron as:
E
v q dt
d me ^ ,^ (1.2)
which would suggest that an electron will continue to accelerate, resulting in ever
increasing current density, J = – nq v , where n is number density of electron.
For sure, we know this does not happen. Then what provides resistance to the
electron flow? Could it be interaction with lattice atoms, impurities, imperfections (grain
boundaries, dislocations etc.)?
Whatever the reason, in light of our experience with conduction, we would like
modify the force equation on electron by including a damping parameter with the
velocity.
a q is fundamental charge, 1.6x
2
v E
v q dt
d me , (1.3)
All this discussion above makes use of particulate nature of the electron. An
alternative view is to see an electron as a wave. In an ideal crystal, the electron waves are
scattered coherently. But, the incoherent scattering, for example, in presence of
impurities explains the resistance in a material. In the following, we fully develop the
idea of electron as a particle and thus compute material’s conductivity. Then we view an
electron as a wave and use quantum mechanical approach to study the conduction due to
“free electrons”.
4
3
1
s 4 n
r (^)
We find in comparison to Bohr radius ao = 5.29 x 10
- 7 cm, which is a measure of
size of a ground state hydrogen atom, the volume taken up by conduction electrons is 3- 6
times even for alkali metals (rs/ao for Li=3.25, Na=3.93, K=4.86, Cs=5.62).
These densities are much greater than those in gases where kinetic theory is
applied. Yet Drude applied the same theory with some modification.
1.1 Assumptions of Drude Theory
- The electrons are both independent and free. An independent electron assumption
implies that between two collisions, an electron does not interact with another
electron. Therefore, in absence of an applied field, an electron moves in a straight
line, and in presence of a field, the electron trajectory is decided by the field; but in
both cases, the field due to other electrons plays no role. Similarly, in free electron
approximation, the electron–ion interaction is ignored. Thus field produced by both
electrons and ions is neglected.
- Drude’s electron collide with ion cores and instantaneously change speed and
direction. But, electron-electron scattering is neglected, that is electron–electron
collision is neglected.
- Probability that an electron experiences a collision per unit time is 1/. is known
as relaxation time or mean free time. It means, on average, an electron picked
randomly will experience next collision after time .
- Electrons reach thermal equilibrium by collision. That is the velocity of an electron
just after a collision is determined by surrounding temperature, and is in a random
direction. It is in no way related to its velocity before collision.
2.0 DEMONSTRATION OF DRUDE THEORY
We will discuss conduction in three different conditions. First, for a metal in a
static electric field (DC conductivity) and then also with spatially uniform static magnetic
field (Hall Effect) and spatially uniform, time varying electric with no magnetic field (AC
conductivity).
2.1 D. C. Conductivity
In absence of an E field, the average velocity of electrons, directed randomly, is
zero. In the presence of an E field, an electron emerges from a collision with velocity vo
(say). Even this velocity is in a random direction and completely determined by local
thermal equilibrium, according to the assumption of Drude theory. Thus, this velocity
does not contribute to the average electron velocity. But, at time t, after a collision, an
electron has also acquired a velocity – q E t/me, which does have an average component.
Since average of t is ,
e
avg m
q
E
v^ (2.3)
Therefore, the current flux, which is J = - nq v, on average is
5
e
2
m
nq E J
Thus, we get the definition of the electrical conductivity by comparing equations (1.1)
and (2.4):
e
2
m
nq (2.5)
What have we gained by doing this? Can we estimate ?
We can estimate only if we knew . So all we have accomplished is that we have
replaced an observable/measurable quantity , whose estimate we want, by a non-
measurable or difficult to measure quantity. But this new quantity is a physical entity-
the time.
Based on measured resistivity of various metals at room temperature, we now
estimate For resistivity of metals on order of a ohm-cm, varies between 10
s. Furthermore, according to Drude’s assumption, speed with which an electron comes
out of a collision is exclusively determined by thermal equilibrium. Therefore, following
the ideas inherent in kinetic theory, it is reasonable to estimate the speed of an electron by
k T 2
mv 2
B
2 e o^. This speed at room temperature is on order of 10
7 cm/s. Thus, the
mean free path = vo turns out to be 1-10 Å, a distance comparable to inter-atomic
spacing. In effect, the original assertion of Drude that resistance to electron transport
comes from electrons bumping into large heavy ions may seem true. This, indirectly, is a
validation of Drude theory.
However, because now we have tools of modern quantum theory, we know that
estimate of vo by equipartition of energy is an order of magnitude too small, while is an
order of magnitude larger at room temperature. The two errors cancel each other to make
Drude theory seem correct.
In absence of any good classical estimate of , we now seek predictions that can be
made by Drude’s model and are also independent of so that they can be verified.
2.2 Hall Effect and AC Conductivity
Before discussing these two cases separately let us build the mathematical frame
work. Regard the average electron velocity at any time t as p (t)/me , where p is total
momentum per electron. Therefore,
m
nq p (t) J
Now we seek p (t+Δt), that is momentum at time t+Δt. The fraction of electrons that have
suffered a collision in duration dt, according to the definition of relaxation time is, Δt/.
Thus the electrons that do not collide evolve under the force f (t) due to spatially uniform
field, and acquire additional momentum, of f (t)Δt + O(Δt 2 ). Therefore, contribution to
total momentum per electron due to electron that did not undergo collision is
) (t) (t) t O(dt )
t ( 1
2
p f (2.7)
7
Pursuing Equation (2.11) further, with force on electrons f ( t)q E v x B ,
(t) q x dt
d p E v B
p (2.12)
which in steady state requires 0
dt
d
p
. Then writing x- and y- components of Equation
0
p q E v B
x x y
(2.13a)
0
p q E v B
y y x
(2.13b)
Since in steady state we require that no current flows in y direction, therefore vy=0, py=0.
Then the Hall field Ey is determined from Equation (2.13b),
nq
J B
E v B x y x ^ , and after using the definition of Hall coefficient,
nq
R H ^ (2.14)
Clearly the measure of Hall coefficient is independent of relaxation time.
Therefore, we can validate Drude theory, if we measure the Hall coefficient, as per its
definition, and also estimate it according to Equation (2.14) under the assumption that
only the valence electrons contribute to electrical conduction (see Equation 2.1).
It turns out that alkali metals (Li, Na, K, Rb, Cs) obey Drude’s model well and
noble metals Cu, Ag, Au to an lesser extent. But, for Be, Mg, In, Al, RH is actually
positive! Here we encounter first failure of Drude’s view of conduction.
Also Equation (2.14a) yields known Drude’s results x
e
2
x E m
nq J
, consistent with
Equation (2.4). Therefore,
2
e MR nq
m
. Although, the magnetoresistance is not
relaxation time independent, however, it is indeed field independent. But it turns out, for
many metals this is not so, indicating another failure of Drude theory.
2.2.2 AC Electrical Conductivity
Field E is uniform spatially, but is time varying. That is, (t ) Re[ ( )e ]
where E
() is the field in phasor notation. Accordingly, in phasor notation, we seek
momentum per electron in form (t ) Re[ ( )e ]
jt
p p** from Equation (2.11). In phasor
notation, Equation (2.11) becomes
q ( )
( ) j ( )
E**
p p , or (2.15)
1 j
q ( )
p E**^ (2.16)
8
Noting that **J
(), which satisfies (t ) Re[ ( )e ]
e
m
nq ()
p , from Equation
(2.16), we have
m 1 j
nq ( ) e
2
J E** (2.17)
Hence, the AC conductivity is defined as
1 j
o ac
where
e
2
o m
nq .
We shall now see application of this result in propagation of electromagnetic
waves(EM) in metals. But before we can do that, we have to resolve two issues. First, in
EM radiation, E field is associated with a magnetic field H (or magnetic flux density B ),
which we have not considered in writing Equation 2.18. Second, the electric field varies
both in time and space, whereas we have derived the AC-conductivity only for a uniform
E field.
The first issue does not pose much of a problem because in expression for force
experienced by an electron, f = - ( E + v x B ), the second term is small for electromagnetic
waves in medium we consider. In fact, it is smaller than the first term by a factor v/c
b .
With current densities as large as 10
6 A/m
2 , the average electron speed is only 0.1 cm/s.
Hence, the second term is significantly smaller.
The second issue, however, is more serious. Nonetheless, we make an
approximation that the electric field is spatially constant when the wavelength of the EM
radiation is much longer than the mean free path of electrons in the metal. As an
example, consider visible light with wavelength between 10
3
4 Å. Recognizing that an
electron is accelerated by the electrical force only during and that the corresponding
mean free path is typically 1-10 Å, the electric field can be considered uniform over such
short distances. Thus, our estimates are valid, only when the mean free path is much
greater than wavelength of the EM radiation. It must be noted, however, the mean free
path at low temperatures and in high purity metals can be rather large.
Below we write the well known Maxwell equations for transmission in a medium
with no free charge in both time and frequency domain/phasor notation:
b For example, consider transverse electromagnetic waves, in which the magnitude of the magnetic field is
related to that of electric field byH* E*
, where is permittivity and is permeability of the
medium. Then with speed of light in the medium as
c 1 , E* c
B* ^1.
10
Table 2.1: Comparison of theoretical and observed plasma frequencies
Theoretical , 10 3 Å Observed , 10 3 Å
Li 1.5 2.
Na 2.0 2.
K 2.8 3.
Rb 3.1 3.
Cs 3.5 4.
A good agreement with experiments is shown for alkali metals, although only for
our good fortune; the actual dielectric constant of metal is much more complicated, with
contributions from many other sources which cannot be represented in the framework of
Drude’s theory. The discussion on these other sources we will present later, in context of
dielectric material.
Before closing the chapter on Drude theory of metals, let us summarize what we
have studied so far. For his time, Drude presented an elegant theory for conduction in
metals. We have seen that theory yields excellent predictions in so many cases, which is
why any discussion on conduction starts with Drude theory. But in other cases, such as
Hall effect in aluminum, the theory fails. Also, we are far from understanding
conduction in non-metals, such as semiconductors and dielectrics.
These problems are resolved by taking a quantum mechanical approach. In fact, in
most cases where Drude theory gives correct predictions, it is only for our good fortune,
as we will see in subsequent chapters.
First notable improvement in Drude theory is presented as Sommerfeld theory of
metals.
3.0 SOMMERFELD THEORY OF METALS
Number density of gas atoms/molecules follows the Maxwell-Boltzman
distribution. Prior to formulation of Quantum theory, therefore, it was reasonable for
Drude to assume a similar distribution for electron density, that is, for number of
electrons per unit volume with velocity in range v and v +d v as fMB( v )d v ,
2 k T
mv 2
3
B
e MB
B
2
e 2 k T
m f ( ) n
v (2.21)
However, with the advent of quantum theory and realization that electrons are
bound by Pauli’s exclusion principle, the distribution instead is Fermi- Dirac
mv k T )/k T 1 2
exp (
m / f( )
B o B
2
3
3
e
v^ (2.22)
where To is obtained by normalizationn (^) f( v )d v
At temperatures of interest, below 10 3 K, the two distributions are remarkably
different, as shown in Figure 2.2.
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Figure 2.2: Comparison of Maxwell-Boltzmann and Fermi Dirac distributions at room
temperature (from Ashcroft & Mermin)
Therefore Sommerfeld’s contribution in this regard is nothing more than employing
Fermi-Dirac distribution instead of Maxwell-Boltzman distribution in analysis of Drude.
However, how one can apply a quantum distribution in otherwise a classical theory
requires an explanation, which therefore leads us into studying Free Electron theory with
a quantum mechanical approach.
13
it cannot be detected and hence de Broglie wavelength is not of much value. On the other
hand an electron of mass 9.11x
- 31 kg moving at 1% of speed of light has of 2.4 A
0 .
This wavelength corresponds to the atomic dimensions and the calculation above
corresponds to wavelength of x-rays. So in example above, we suggest an electron can
act like x-rays.
Then, in 1926, G. P. Thompson (son of J. J. Thompson) et al. showed electron
diffraction a clear demonstration of wave nature of light experimentally, proving de
Broglie's proposition.
So we must treat both light and matter either as wave or as a particle. But, this
approach causes a problem if you wish to locate an electron in space within x because
the light you must use should be of wavelength at least as small. To observe the electron,
the light photon will interact with the electron and in process change its momentum.
Now if you try to locate the electron more accurately, then you will use light of even
smaller wavelength and hence greater momentum
h p. But now the momentum
transferred to the electron will be greater than that in the previous case. In short, a more
accurate determination of location of an electron leads to a greater uncertainty in
momentum. This led Heisenberg to postulate x phand then he went on to develop
quantum mechanics based on a matrix method.
Quantum mechanics is based on probabilistic nature of events. Thus it is not
appropriate to speak of exact values of such quantities as position, momentum and
energy. Instead, what matters is only expectation values of these quantities, and that
implies uncertainty.
1.1 Heisenberg’s Uncertainty Principle
We have already discussed the reasons of uncertainty. Now we succinctly write the
principle as
x (p) h , or
E (t) h
This means, instead of finding a position for an electron, we must talk about
probability of finding an electron at a position. Mathematically, this implies that we
formulate probability density function and then find expectation values.
Where are we heading? While Heisenberg developed quantum mechanics on
matrix method, independently, Erwin Schrodinger formulated the quantum theory based
on partial differential equations (PDEs) which was shown to be equivalent to the method
of Heisenberg. We are setting ourselves up to deal with quantum mechanics in
Schrodinger's formulation.
1.2 Schrodinger’s Approach
We begin with basic postulates:
1. Each particle (electron) is described by a wave function ( r ,t), which specifies all
information about a particle, except the spin. The wave function follows a property
that
- 3 d r gives the probability of finding a particle in a volume. This also
implies d 1 all
r.
14
- For every observable property in classical mechanics, such as position, energy,
momentum, there is an abstract, linear Hermitian operator. Thus the classical
quantities are replaced by abstract quantum mechanical operators given in Table
3.1.
Table 3.1: Observables and corresponding quantum-mechanical operator
Classical variable Quantum operator
r r
f( r ) f( r )
p ( r ) j
j t
- When the observable corresponding to an operator is measured, only values
observed are the eigen values o of Oop^ ^ o .
- Average value of a variable O is computed using its corresponding operator
Oop as = all
3 op
O d r. So, once is known, one can compute an average
value for position, energy and momentum.
2.0 SCHRODINGER WAVE EQUATION
Having given the basic postulates of the quantum mechanics, now we are all set for
the Schrodinger equation, as fundamental to quantum mechanics as Newton laws are to
classical mechanics. In fact, there can’t be a derivation of Schrodinger equation;
sometimes one may start with an alternative formulation and arrive after mathematical
manipulations to the familiar form of the equation. That should not be taken for a
derivation of the equation.
Now I am also going to play the trick and “derive” Schrodinger equation.
2.1 “Derivation” of Schrodinger Wave Equation from Classical Mechanics
The derivation is rather simple. We start with equation of classical mechanics
p V E 2 m
and simply look-up Table 3.1 to replace classical observables with corresponding
operators. And we have the Schrodinger wave equation as
t
(,t)
j
( ,t) V( ) (,t) 2 m
2
2
r r r r
All we will work with is time independent Schrodinger equation. The space and
time dependent Schrodinger equations can be separated, at least, whenever we can write
the wave function as ( r ,t) ( r )(t).
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CHAPTER FOUR
FREE ELECTRON GAS
Consider an electron, a free electron, by which we mean both free and independent
electron approximation. That is, our electron remains unaffected by other electrons
(independent electron) as well as the ion cores (free electron). In mathematical terms,
what this implies is that potential V=0 in Equation (3.5), which, therefore becomes
(r) (r) 2 m
2
e
2
The wave function here and the spin (1/2 or – 1/2) an electron possesses describes
the state of an electron completely.
Let us confine this free electron in a cube of edge L= V
1/ ; that the electron is
confined in a cube, as we will see, has no impact on conclusions we will make. There are
two ways we can confine the electron in V. One, requiring that ( r ) vanishes at the
boundary results in standing waves. But to discuss transport of energy, it is more
convenient to deal with running waves. Hence we apply Born-Von Karman periodic
boundary conditions. That is,
(x L,y,z)(x,y,z) (4.2a)
(x ,yL,z)(x,y,z) (4.2b)
(x ,y,zL)(x,y,z)^ (4.2c)
It will be easy to depict these boundary conditions in one dimension by drawing the
dimension as a circle. That is, the wave function is the same at any point as after going
around full the circumference. Similarly, the boundary conditions mentioned in Equation
(4.2) mean the same in three-dimensional space.
We seek a solution to the schrodinger Equation (4.1) in form of a travelling plane
wave
kr r
j k e
V
for wave vector
k k (^) x xˆkyyˆkzzˆ (4.4)
and with energy eigen values
k k e
2
2 m
(k )
where V
in solution is picked so that normalization condition on wave function,
| (r)| d 1
2 3 V
r , is satisfied.
The wave vector k introduces an important concept. We will see, it expresses the
state of an electron, minus the information on spin. If the allowed values of k are
continuous, clearly the energy values an electron can attain is also continuous. We know
that allowed energy values for an isolated atom, such as hydrogen, are discrete. How will
our electron behave here?
17
To answer the question, we now apply the boundary conditions in (4.2). From
(4.2a),
j k (^) x (x L) kyy kzz jk (^) xx kyy kz z e V
e V
which is satisfied whene 1
jk (^) xL (^) or L
2 n k
x x
(^) for integer nx. Similarly, when all
boundary conditions in (4.2) are exhausted, we get,
L
2 n k
x x
^ ,
L
2 n k
y y
L
2 n k
z z
where nx, ny and nz are all integers.
These requirements also mean allowed states, k , are discrete, as are the energy
values. Yet, shortly we will see how we begin to represent the states and energies as
quasi-continuous entities.
1.0 WHAT’S THE MEANING OF VECTOR k
The vector k appearing in Equation (4.3), clearly, is a wave vector. What else does
it mean? To answer this question, let us examine eigen values of the momentum
operator.
The momentum operator is given by j
, as we have seen before. If p is the
momentum eigen value, then it is obtained according to ( )
j
p k ( r ) k r
, which after
using Equation (4.3) becomes p k ( r ) k k( r ).
This means, k ( r ) is eigen state of the momentum operator also with an eigen
value
p k (4.7).
This has an important implication. Not only k is a wave vector, but it also defines the
momentum of an electron.
In passing, we would also mention that later, when we go beyond the free electron
assumption, we will have to modify our definition and k then would only be crystal
momentum. Meaning, it would not be the momentum of an electron, rather will only
appear to be like momentum in presence of external forces.
2.0 OCCUPATION OF ELECTRONS IN k STATES
In the following, our goal is to figure out how the N electrons in volume V will fill
up the energy levels, and hence what will be the total energy and velocity distribution of
these electrons.
We have already established that nx, ny, nz, are integers and that the k vector is
accordingly quantized. Thus, the components of k are the quantum numbers for the free
electrons; in addition one more quantum number for spin up or down completely
describes the electron. Furthermore, if we examine Equation (4.3), we find that the
dimension of vector k is reciprocal length. This we will re-state by saying that vector k
belongs to a space called reciprocal space.
19
^3
1 2
e
F 3 n m
v
Like before, using Equation (2.2) and expressing n in terms of
o
s
a
r ,
8
s o
F x^10 r a
v cm/s. Remember, we are talking about ground state properties, that is,
at T=0K. Yet, the Fermi velocity is about 1% of the velocity of light, while all classical
particles have speed = 0 at T=0. This is first difference compared to Drude theory.
Note, also, if we define a temperature TF as TF= , k (^) B
F
with fermi energy typical in
eV, the value of TF is about 10 4 K. This Fermi temperature is, of course, not the
temperature of the electron but it just says a classical gas with energy k T
2
B per electron
has to have temperature of about 10 4 K to have same energy as F.
Before we move on, it is worth while to define density of available states; this is a
concept, we will use in later more extensively.
2.1 Density of Electron States
Let us first discuss the meaning of density of states, g(). It simply means that
g()d are the number of electron states available to be occupied in energy range [,
+d]. It, however, does not mean that all these states are indeed filled. Also note that
this density would be twice the density of k-states, because we have already considered
that each k-state can take two electrons.
If we are interested in knowing what is the number of electrons actually occupying
the states in [, +d], we will also have to specify the probability of an electron
occupying a state. And if we call this probability f(), the population of electron in [,
+d] is g()f()d, or the population density N(), is g()f().
With the definitions in place, we start with Equation (4.9) to derive an expression
for free electron density of states. A modified form of Equation (4.9)
2
3 2
3
2
e 2
2 m
n
would read n is the number of electrons (per unit volume) that occupy levels up to energy
. Remember, however, it is not important that whether we have n electrons or not; only
that if we had n electrons, then they will fill up to energy . This also means, irrespective
of availability of electrons, at least the electron states are there. With this view, the
density of states, signifying availability of states, is simply
^2
1 2
3
2
e 2
2 m
d
dn g
20
Now let us estimate electrical conductivity in the frame work we have been
discussing in this chapter.
3.0 ELECTRICAL CONDUCTIVITY: FREE ELECTRON VERSION
When no electric field is applied, the electrons lie in sphere of radius kF with center
at origin, as in Fig. 4.1 (a) and have velocity v= me
k , for all allowed values of k. For
every vector k in this sphere, there is also a vector – k. As a result velocity vectors cancel
out and the net velocity is zero, as expected; without electric field there could not be a net
velocity, or else we would have current even in the absence of electric field!
Schematically, this situation is shown in Figure 4.1a, in two dimensions, where Fermi
sphere reduces to a circle. However, when electric field E is applied, the electrons
experience a force – q E. Thus, the equation of motion for electron is
E
p q dt
d
and because Equation (4.7) identifies k as the momentum, the equation of motion
becomes
E
k q dt
d ^.^ (4.14)
Thus in time t, the Fermi sphere shifts by
t
q (t ) ( 0 )
E
k k . (4.15)
kx
ky
E =0, t=
kx
ky
E = -|E| , t= xˆ
kx
(a) (b)
Figure 4.1: (a) With no field until t=0, sum over all k vectors in Fermi sphere is zero. (b)
With applied field E Exˆ, during time , the Fermi sphere shifts to the right; electrons
only near the fermi surface can conduct as sum of all remaining k vectors is zero. The
shifting of sphere in the figure is a bit exaggerated, but if you imagine a small shift, then
clearly on those k vectors, leading to edge of fermi sphere do not cancel out.
