MO 201: Structure of Materials
Electronic Conduction
Dr. Bratindranath Mukherjee
How do we quantify the electrical behaviour of materials?
Ohm's law
How do we quantify the amount of light reflected at an interface?
How do we quantify the thermal properties of materials?
What determines the increase in temperature of a material when it is heated?
Is there a relationship between the electrical and the optical properties of a
metal?
The Hagen-Rubens law
Is there a relationship between the electrical and thermal properties of a metal?
The Wiedemann-Franz law
The Dulong-Petit law
The continuum model offers no explanations.
The de Broglie wave group associated with a moving
body travels with the same velocity as the body.
PARTICLE IN A BOX: Why the energy of a trapped
particle is quantized
1. Find the de Broglie wavelength of a 1.0-mg grain of sand blown by the wind at a
speed of 20 m/s.
4. Find the de Broglie wavelength of the 100-keV electrons used in a certain electron
microscope. Find velocity of the electrons?
5. A 10-g marble is in a box 10 cm across. Find its permitted energies.
Schrödinger Equation
Time-dependent Schrödinger
equation in one dimension
Time-dependent in three dimension
OPERATORS
Another way to find expectation values
SCHRÖDINGER’S EQUATION: STEADY-STATE FORM
Steady-state Schrödinger equation in three dimension
Only wave functions with all the following properties can yield physically meaningful
results when used in calculations, so only such “well-behaved” wave functions are
admissible as mathematical representations of real bodies.
These rules are not always obeyed by the wave functions of particles in model
situations that only approximate actual ones. For instance, the wave functions of a particle
in a box with infinitely hard walls do not have continuous derivatives at the walls,
since 0 outside the box But in the real world, where walls are never
infinitely hard, there is no sharp change in at the walls and the derivatives are
continuous.
PARTICLE IN A BOX
How boundary conditions and normalization determine wave functions
FINITE POTENTIAL WELL
The wave function penetrates the walls, which lowers the energy levels
In regions I and III
Schrödinger’s steady-state
equation is
TUNNEL EFFECT
A particle without the energy to pass over a potential barrier may
still tunnel through it
Although the walls of the potential well of were of finite height, they were
assumed to be infinitely thick. As a result the particle was trapped forever even though
it could penetrate the walls.
Particle in a one-dimensional lattice
https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice
Simple square well potential representation
Spatially we separate the regions V = V0 for —b < x < 0, and
V = 0 for 0 < x < a - b, and furthermore, V(x + a) = V(x).
The Schrôdinger equation for the one-dimensional, time-independent case is,
Kronig-Penney approximation
General results from Kronig-Penney model:
- if potential barrier between wells is strong, energy bands are narrowed and spaced far
apart
(Corresponds to crystals in which electrons are tightly bond to ion cores, and
wavefunctions do not overlap much with adjacent cores. Also true for lowest energy
bands)
- if potential barrier between wells is weak, energy bands are wide and spaced close
together (this is typically situation for metals with weakly bond electrons – e.g. alkali
metals. Here the “nearly free” electron model works well.)
Widening of the sharp energy levels into bands
and finally into a quasi continuous energy
region with decreasing interatomic distance, a,
for a metal
Electron diffraction and E-k diagram
Nλ = 2dsinθ
How does quantum mechanics affect the distribution ofelectron
energies?
Unlike classical statistical mechanics, which allows any energy state to be occupied
by any number of electrons, quantum mechanics imposes restrictions on the
number of electrons which can occupy a given energy level.
The Pauli principle states that no two electrons can have the same set of quantum
numbers, and therefore cannot occupy identically the same energy level in our
solid. We are ignoring electron spin for the moment.
With a finite number of allowed electron states in a material, how do the
electrons arrange themselves?
What other information is needed to describe the electron
distribution?
We can describe the number of available electron states as a function of energy E by the
density of available states D(E). This density of states is independent of the available
electrons to fill the states; it is simply an expression of what energy values are allowed.
The occupational density of states N(E) describes the number of electron states
per unit energy interval as a function of energy. This is related to the density of
available states D(E) through the probability of occupancy f(E) by the equation
N(E) = 2f(E)Z(E),
Density of states
How does the number of available energy states for electrons vary with energy of
the electrons in the free electron approximation?
The number of different k states that is possible up to a given value kn can be calculated easily in the
free electron limit by noting that the values nx, ny and nz must all be positive. Therefore the number
of available states is simply the number of unit cubes in the positive quadrant of w-space of radius n.
In other words it is one eighth of the volume of a sphere in «-space:
Z(E)
N(E) is called the (electron) population density
N(E) = 2F(E)Z(E),
Population density N(E) within a band for free
electrons. dN* is the number of electrons in
the energy interval dE.
Calculating Fermi energy
Consequences of the Band Model
because of the Pauli principle, each s-band of a crystal, consisting of N atoms, has space for 2N electrons, i.e., for two
electrons per atom. If the highest filled s-band of a crystal is occupied by two electrons per atom, i.e., if the band is
completely filled, we would expect that the electrons cannot drift through the crystal when an external electric field is
applied. An electron has to absorb energy in order to move. Keep in mind that for a completely occupied band higher
energy states are not allowed. (We exclude the possibility of electron jumps into higher bands.) Solids in which the
highest filled band is completely occupied by electrons are, therefore, insulators
In solids with one valence electron per atom (e.g., alkali metals) the valence band is essentially half-filled. An
electron drift upon application of an external field is possible; the crystal shows metallic behavior.
Bivalent metals should be insulators according to this consideration, which is not the case. The reason for this lies in
the fact that the upper bands partially overlap, which occurs due to the weak binding forces of the valence electrons
on their atomic. If such an overlapping of bands occurs, the valence electrons flow in the lower portion of the next
higher band, because the electrons tend to assume the lowest potential energy. As a result, bivalent solids may also
possess partially filled bands. Thus, they are also conductors.
Conductivity : Quantum Mechanical Approach
This quantum mechanical equation reveals that the conductivity depends on the Fermi
velocity, the relaxation time, and the population density (per unit volume). The latter is,
as we know, proportional to the density of states.
Equation is more meaningful than the expression derived from the classical electron
theory It contains the information that not all free electrons Nr are responsible for
conduction, i.e., the conductivity in metals depends to a large extent on the popUlation
density of the electrons near the Fermi surface.
For example, monovalent metals (such as copper, silver, or gold) have partially filled
valence bands, Their electron population densities near their Fermi energy are
high, which results in a large conductivity.
Bivalent metals, on the other hand, are distinguished by an overlapping of the upper bands
and by a small electron concentration near the bottom of the valence band.As a
consequence, the electron population near the Fermi energy is small , which leads to a
comparatively low conductivity.
Finally, insulators and semiconductors have, under certain conditions, completely filled
electron bands, which results in a virtually zero population density near the top of the
valence band . Thus, the conductivity in these materials is extremely small.
Conductivity : Classical Approach
The electrons move randomly (in all possible directions) so that their individual velocities in
the absence of an electric field cancel and no net velocity results. This situation changes when
an electric field is applied. The electrons are then accelerated with a force towards the
anode and a net drift of the electrons results, which can be expressed by a form of Newton's
law (F = m.a)
We postulate that the resistance in metals and
alloys is due to interactions of the drifting
electrons with some lattice atoms, i.e.,
essentially with the imperfections in the crystal
lattice (such as impurity atoms, vacancies, grain
boundaries, dislocations, etc.). The second term
is a damping or friction force which contains the
drift velocity, v, of the electrons. The electrons
are thought to be accelerated until a final drift
velocity Vf is reached (see Fig. 7.2(b)). At that
time the electric field force and the friction
force are equal in magnitude.
The magnitude of the current, I, that results from electron drift is
defined as the amount of charge that
The force exerted on an electron by an electric field, E, is given by eE,
and the acceleration, a, imposed on an electron
current will decay to zero in a time t after the voltage is turned off,
where t is the time between successive collisions, called the relaxation
time. The mean free path of the electron, which is the length of the
path between successive collisions
he conductivity is often written in terms of another variable, the
mobility of the electrons, µe, defined as the drift velocity gained
per unit electric field,
The resistivity of alloys increases with increasing amount of solute content.
The slopes of the individual p versus T lines remain, however, essentially
constant. Small additions of solute cause a linear shift of the p versus T curves
to higher resistivity values in accordance with Matthiessen's rule. This
resistivity increase has its origin in several mechanisms.
• First, atoms of different size cause a variation in the lattice parameter and,
thus, in electron scattering.
• Second, atoms having different valences introduce a local charge
difference that also increases the scattering probability.
• Third, solutes which have a different electron concentration compared to
the host element alter the position of the Fermi energy. This, in tum,
changes the population density N(E) and thus the conductivity
Classification of Materials in terms of band gap
Origin of band gap
Approach 1 : Tight Binding Approximation
Direct and indirect bandgap
Intrinsic semiconductor
Extrinsic semiconductor
Advantages of Si over Ge in VLSI processes
Property Si Ge
Energy Gap 1.11 eV 0.66eV
Upper temperature limit 150 oC 100 oC
Junction leakage current Less More
Breakdown strength Higher Lower
Oxide quality Excellent Water soluble
and unstable
Relative cost of electronic grade 1 10
What are the advantages of gallium arsenide over silicon and
germanium for fabrication of devices?
These include:
(i) GaAs circuits are faster and operate at equal, or lower, power than silicon circuits.
(ii) The separation between the conduction and valence bands is more easily controlled
in GaAs and related compounds than in silicon.
(iii) GaAs can radiate and detect near infrared and visible red radiation dependingon its
band gap.
(iv) GaAs can support optoelectronic functions while silicon cannot.
p-n Junction
(i) When two solids are in contact, charge transfer occurs until their Fermi energies are
the same.
(ii) In n- and p-type semiconductors, the Fermi level lies approximately at thedonor
and acceptor levels, respectively
What happens if a voltage is applied across the junction to reduce the
potential difference across the junction?
What happens if a voltage is applied to increase the potential difference
across the junction?
i- V characteristics
Photo detector
Light emmiter
3- level LASER
MO 201: Materials Science
Dielectrics
Dr. Bratindranath Mukherjee
What is a dielectric material ?
Solids that appeared not to conduct electricity were known as dielectrics or
insulators. Many oxides and most polymers fall into this category
Electrons in these materials are localised in strong bonds if the material is
considered to be a covalent compound, or else are restricted to the region close to
an atomic nucleus if the compound is supposed to be ionic. In either case, these
electrons are trapped and cannot move from one region to another.
Basics of capacitors
Dipole moment and polarization
The induced dipoles on the internal
constituents add together, with a
result that opposite surfaces of the
solid become positively and negatively
charged and the solid becomes
polarised. The polarisation of the
dielectric, P, is a vector quantity,
defined as the electric dipole moment
per unit constituent
In Isotropic solids
Clausius-Mossotti relation
Origin of Polarisation
Electronic Polarization
Electronic polarization may be induced to one degree or another in all atoms. It
results from a displacement of the center of the negatively charged electron cloud
relative to the positive nucleus of an atom by the electric field .This polarization type is found in
all dielectric materials and exists only while an electric field is present.
Ionic Polarization
Ionic polarization occurs only in materials that are ionic. An applied field acts to displace
cations in one direction and anions in the opposite direction, which gives rise to a net dipole
moment.. The magnitude of the dipole moment for each ion pair pi is equal to the product of
the relative displacement di and the charge on each ion, or
pi = qdi
Orientation Polarization
The third type, orientation polarization, is found only in substances that possess permanent
dipole moments. Polarization results from a rotation of the permanent moments into the
direction of the applied field,. This alignment tendency is counteracted by the thermal
vibrations of the atoms, such that polarization
decreases with increasing temperature.
Space charge polarization
P = Pe + Pi + Po + Ps
Frequency dependence
tan δ : loss component
Effect of Temperature
Properties of common Dielectric materials
Why on melting dielectric constant abruptly decreases for
solid HCl and increases for Nitrobenzene ?
Breakdown of Dielectric Materials
Dielectric strength = Voltage/ unit thickness
➢ Intrinsic Breakdown: Excitation of electrons and subsequent creation of
avalanche of electron in presence of strong electric field
➢ Thermal Breakdown: cumulative accumulation of heat leading to
melting
➢ Defect Breakdown: moisture from atmosphere or gas collected on
surface discontinuities, pores and cracks leading to shorting or gas discharge
➢ If polarisation, P, changes with applied electric field, E, we have a dielectric
➢ In some dielectrics the polarisation, P, can arise from mechanical stress, s, to give a
piezoelectric.
➢ In some piezoelectrics, there is a spontaneous polarisation, Ps, when the applied
electric field, E, and the stress, s, are zero, that changes with temperature,
T, to give pyroelectrics.
➢ In some pyroelectrics the direction of the spontaneous polarisation, Ps, is easily
switched in anelectric field, to give a ferroelectric.
With increasing temperature the thermal motion of the atoms will increase,
and eventually this alone can overcome the energy barrier separating the
various orientations. At high temperatures the distribution of atoms becomes
statistical and the crystal behaves as a normal dielectric. It is no longer a polar
material but is in the paraelectric state. The temperature at which this occurs is
known as the transition temperature, Curie temperature or Curie point, TC.
Curie-Weiss Law
