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EEE 315 - Electrical
Properties of Materials
Lecture 6
Band Theory of Solids
 In isolated atoms the electrons are arranged in
energy levels
 In solids the outer electron energy levels become
smeared out to form bands
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Band Theory of Solids
• The highest occupied band is called the VALENCE
band. This is full.
• For conduction of electrical energy there must be
electrons in the CONDUCTION band. Electrons
are free to move in this band.
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Band Theory of Solids
Insulators : There is a big energy gap between the
valence and conduction band. Examples are plastics,
papers…
Conductors : There is an overlap between the valence and
conduction band hence electrons are free to move about.
Examples are copper, lead ….
Semiconductors : There is a small energy gap between
the two bands. Thermal excitation is sufficient to move
electrons from the valence to conduction band. Examples
are silicon ,germanium….
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“Realistic” Potential in Solids
For one dimensional case where atoms (ions)
are separated by distance d, we can write
condition of periodicity as
U( x)  U( x  d  n)
“Realistic” Potential in Solids
Multi-electron atomic potentials are complex
Even for hydrogen atom with a “simple”
Coulomb potential solutions are quite
complex
So we use a model one-dimensional periodic
potential to get insight into the problem
Bloch’s Theorem
Bloch’s Theorem states that for a particle
moving in the periodic potential, the
wavefunctions ψ(x) are of the form
 ( x)  uk ( x)e  ikx , where uk ( x) is a periodic function
u k ( x)  u k ( x  d)
uk(x) is a periodic function with the periodicity
of the potential
Bloch’s Theorem
What is probability density of finding
particle at coordinate x?
P( x)   ( x)   ( x) ( x)
2
P( x)  [uk ( x)e
P( x)  uk ( x)uk ( x)e
*
*
 ikx *
] [uk ( x)e
 ikx  ikx
e
P( x)  uk ( x)
 ikx
]
 uk ( x)uk ( x)
*
2
• But |uk(x)|2 is periodic, so P(x) is as well
Bloch’s Theorem
P( x)  P( x  d)
The probability of finding an electron at
any atom in the solid is the same!!!
The most common example of Bloch's
theorem is describing electrons in a
crystal. Each electron in a crystalline
solid “belongs” to each and every atom
forming the solid
Band Theory of Solids
 Consider initially the known wave functions of two
hydrogen atoms far enough apart so that they do
not interact.
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Band Theory of Solids
 Interaction of the wave functions occurs as the atoms get closer:
Symmetric
Anti-symmetric
 An atom in the symmetric state has a nonzero probability of being
halfway between the two atoms, while an electron in the antisymmetric state has a zero probability of being at that location.
 When more atoms are added (as in a real solid), with a large
number of atoms, the levels are split into nearly continuous energy
bands, with each band consisting of a number of closely spaced
energy levels.
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Kronig-Penney Model
 An effective way to understand the energy gap in
semiconductors is to model the interaction between
the electrons and the lattice of atoms.
 R. de L. Kronig and W. G. Penney developed a
useful one-dimensional model of the electron lattice
interaction in 1931.
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Kronig-Penney Model
 Kronig and Penney assumed that an electron
experiences an infinite one-dimensional array of finite
potential wells.
 Each potential well models attraction to an atom in the
lattice, so the size of the wells must correspond roughly
to the lattice spacing.
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Kronig-Penney Model
 Since the electrons are not free their energies are less
than the height V0 of each of the potentials, but the
electron is essentially free in the gap 0 < x < a,
where it has a wave function of the form
where the wave number k is given by the usual
relation:
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Tunneling
 In the region between a < x < a + b the electron
can tunnel through and the wave function loses its
oscillatory solution and becomes exponential:
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Kronig-Penney Model
 Matching solutions at the boundary, Kronig and
Penney find
Here K is another wave number.
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Kronig-Penney Model
 The left-hand side is limited to values between +1 and
−1 for all values of K.
 Plotting this it is observed there exist restricted (shaded)
forbidden zones for solutions.
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Important differences between the
Kronig-Penney model and the single
potential well
1)
For an infinite lattice the allowed energies within each
band are continuous rather than discrete. In a real crystal
the lattice is not infinite, but even if chains are thousands
of atoms long, the allowed energies are nearly continuous.
2)
In a real three-dimensional crystal it is appropriate to
speak of a wave vector k . A wave vector is a kvector
which helps describe a wave. Like any vector, it has a
magnitude and direction, both of which are important: Its
magnitude is either the wavenumber or angular
wavenumber of the wave (inversely proportional to the
wavelength), and its direction is ordinarily the direction of
wave propagation.
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And…
3) In
a real crystal the potential function is more
complicated than the Kronig-Penney squares. Thus, the
energy gaps are by no means uniform in size. The gap
sizes may be changed by the introduction of impurities
or imperfections of the lattice.
 These facts concerning the energy gaps are of
paramount importance in understanding the electronic
behavior of semiconductors.
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Band Theory and Conductivity
 Band theory helps us understand what makes a
conductor, insulator, or semiconductor.
1) Good conductors like copper can be understood
using the free electron
2) It is also possible to make a conductor using a
material with its highest band filled, in which case
no electron in that band can be considered free.
3) If this filled band overlaps with the next higher
band, however (so that effectively there is no gap
between these two bands) then an applied electric
field can make an electron from the filled band
jump to the higher level.
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Valence and Conduction Bands
 The band structures of insulators and semiconductors
resemble each other qualitatively. Normally there exists in
both insulators and semiconductors a filled energy band
(referred to as the valence band) separated from the
next higher band (referred to as the conduction band)
by an energy gap.
 If this gap is at least several electron volts, the material is
an insulator. It is too difficult for an applied field to
overcome that large an energy gap, and thermal
excitations lack the energy to promote sufficient numbers
of electrons to the conduction band.
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Smaller energy gaps create
semiconductors
 For energy gaps smaller than about 1 electron volt,
it is possible for enough electrons to be excited
thermally into the conduction band, so that an
applied electric field can produce a modest current.
The result is a semiconductor.
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Insulators :
 The magnitude of the band gap
determines the differences between
insulators, s/c‘s and metals.
 The excitation mechanism of thermal
is not a useful way to promote an
electron to CB even the melting
temperature is reached in an
insulator.
 Even very high electric fields is also
unable to promote electrons across
the band gap in an insulator.
CB (completely empty)
Eg~several electron volts
VB (completely full)
Wide band gaps between VB and CB
Metals :

CB
VB
Touching VB and CB
CB
VB
Overlapping VB and CB

These two bands
looks like as if partly
filled bands and it is
known that partly
filled bands conducts
well.
This is the reason
why metals have high
conductivity.
 No gap between valance band and conduction band
The Concept of Effective Mass :
Comparing
Free e- in vacuum
In an electric field
mo =9.1 x 10-31
 If the same magnitude of electric field is applied
to both electrons in vacuum and inside the
crystal, the electrons will accelerate at a different
rate from each other due to the existence of
different potentials inside the crystal.
Free electron mass
An e- in a crystal
In an electric field
In a crystal
 The electron inside the crystal has to try to make
its own way.
 So the electrons inside the crystal will have a
different mass than that of the electron in
vacuum.
m = ?
m*
effective mass
 This altered mass is called as an effective-
mass.
To find effective mass , m*
We will take the derivative of energy with respect to k ;
2
dE
k

dk
m
2
d2E

2
m
dk
Change
m*
m*
instead of

- m* is determined by the curvature of the E-k curve
- m* is inversely proportional to the curvature
Energy
m
2
2
d E dk
2
k
momentum
This formula is the effective mass of
an electron inside the crystal.
How do Electrons and Holes Populate the Bands?
 Density of States Concept
gc ( E ) dE
gv ( E ) dE
General energy dependence of
gc (E) and gv (E) near the band edges.
The number of conduction band
states/cm3 lying in the energy
range between E and E + dE
(if E  Ec).
The number of valence band
states/cm3 lying in the energy
range between E and E + dE
(if E  Ev).
How do Electrons and Holes Populate the Bands?
 Density of States Concept
Quantum Mechanics tells us that the number of available states
in a cm3 per unit of energy, the density of states, is given by:
Density of States
in Conduction Band
Density of States
in Valence Band
How do electrons and holes populate the bands?
 Probability of Occupation (Fermi Function) Concept
 Now that we know the number of available states at each energy,
hen how do the electrons occupy these states?
t
 We need to know how the electrons are “distributed in energy”.
 Again, Quantum Mechanics tells us that the electrons follow the
“Fermi-distribution function”.
f (E) 
1
1 e
 f(E)
( E  E f ) / kT
Ef ≡ Fermi energy (average energy in the crystal)
k ≡ Boltzmann constant (k=8.61710-5eV/K)
T ≡Temperature in Kelvin (K)
is the probability that a state at energy E is occupied.
 1-f(E) is the probability that a state at energy E is unoccupied.
 Fermi function applies only under equilibrium conditions, however, is
universal in the sense that it applies with all materials-insulators,
semiconductors, and metals.
ASSIGNMENT
 Write down an essay on the band theory of solids
 Use necessary examples and illustrations
 The essay should be based on your learning from the
class, search online for more ideas, theories, examples
and figures
 Write up to maximum 5 Pages (cover page excluded).
Use standard font, font size and color
 Submission dead line is 24 November, 2013
 Submit to- [email protected]
 DO NOT FORGET TO WRITE YOUR ID IN THE COVER PAGE
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