Download Study notes for electronic materials and more Study notes Engineering Physics in PDF only on Docsity!
Semiconductor Physics (BTPH104-18)
Study Material prepared by Dr. Arvind Sharma
Course Objectives: The aim and objective of the course on Semiconductor Physics is to introduce the students
of B. Tech. class to the formal structure of semiconductor physics so that they can use these in Engineering as
per their requirement.
Course Outcomes: At the end of the course, the student will be able to
CO1: Understand and explain the fundamental principles and properties of electronic materials and
semiconductors.
CO2: Understand and describe the interaction of light with semiconductors in terms of fermi golden rule.
CO3: Understand and describe the impact of solid-state device capabilities and limitations on electronic circuit
performance.
CO4: Understand the design, fabrication, and characterization techniques of engineered semiconductor
materials.
CO5: Develop the basic tools with which they can study and test the newly developed devices and other
semiconductor applications.
Detailed Syllabus
PART-A
UNIT 1: Electronic materials (10 lectures)
Free electron theory of metals, Density of states in 1D, 2D, and 3D, Bloch’s theorem for particles in a periodic
potential, Energy band diagrams, Kronig-Penny model (to introduce origin of band gap), Energy bands in
solids, E-k diagram, Direct and indirect band gaps, Types of electronic materials: metals, semiconductors, and
insulators, Occupation probability, Fermi level, Effective mass.
UNIT II: Semiconductors (10 lectures)
Intrinsic and extrinsic semiconductors, Dependence of Fermi level on carrier-concentration and
temperature (equilibrium carrier statistics), Carrier generation and recombination, Carrier transport:
diffusion and drift, p-n junction, Metal-semiconductor junction (Ohmic and Schottky), Semiconductor
materials of interest for optoelectronic devices.
PART-B
UNIT III: Light-semiconductor interaction (10 lectures)
Optical transitions in bulk semiconductors: absorption, spontaneous emission, and stimulated emission; Einstein
coefficients, Population inversion, application in semiconductor Lasers; Joint density of states, Density of states
for phonons, Transition rates (Fermi's golden rule), Optical loss and gain; Photovoltaic effect, Exciton, Drude
model.
UNIT IV: Measurement Techniques (10 lectures)
Measurement for divergence and wavelength using a semiconductor laser, Measurements for carrier density,
resistivity, hall mobility using Four-point probe and vander Pauw method, Hot-point probe measurement,
capacitance-voltage measurements, parameter extraction from diode I-V characteristics.
Reference books and suggested reading:
- J. Singh: Semiconductor Optoelectronics: Physics and Technology, McGraw-Hill Inc. (1995).
- B. E. A. Saleh and M. C. Teich: Fundamentals of Photonics, John Wiley & Sons, Inc., (2007).
- S. M. Sze: Semiconductor Devices: Physics and Technology, Wiley (2008).
- A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, Oxford University Press,
New York (2007).
- P. Bhattacharya: Semiconductor Optoelectronic Devices, Prentice Hall of India (1997).
- Ben G. Streetman: Solid State Electronics Devices, Pearson Prentice Hall.
semiconductors from each other. There are some other differences, which will become clear as we go
along in this chapter.
What is & Why: The Electron Theory of Metals
A solid is formed as a result of bonding among huge number of atoms.
The entities responsible for the bonding are the electrons.
The physical and chemical properties of a given solid are decided by how the
constituent atoms are bonded through the interaction of their electrons
among themselves and with the potentials of the ions. 4
The electron theory of metals covers properties of electrons responsible for the bonding of
solids and electron transport properties manifested in the presence of external
fields or a temperature gradient.
Studies of the electron theory of metals are also important from the point of view
of application-oriented research and play a vital role in the development of new
functional materials.
Recent progress in semiconducting devices like the IC (Integrated Circuit) or
LSI (Large Scale Integrated circuit), as well as developments in
magnetic and superconducting materials, certainly owe much to the successful
application of the electron theory of metals.
As another unique example, we may refer to amorphous metals and semiconductors,
having no long-range order in their atomic arrangement. Amorphous Si is
now widely used as a solar-operated battery for small calculators.
Some Basic Terminologies
Free electrons:
Electrons which move freely or randomly in all directions in the absence of an external field
are known as free electrons.
Bound Electrons:
All the electrons other than valence electrons, in an isolated atom, that are bound to their
parent nuclei are called as bound electrons.
Electric Field ( E ):
The electric field E of a conductor having uniform cross section is defined as the potential drop
( V ) per unit length ( L ) i.e.
E = V / L V/m
Current density ( J ):
It is defined as the current ( I ) per unit area of cross section ( A ) of an imaginary plane
normal to the direction of the flow of current in a current carrying conductor i.e.
J = I / A Am-^2
Electron Theory of Metals
The electron theory of metals explains the following concepts:
o Structural, electrical and thermal properties of materials.
o Elasticity, cohesive force and binding in solids.
o Behavior of conductors, semiconductors, insulators etc.
So far three electron theories have been proposed.
1. Classical Free electron theory:
It is a macroscopic theory.
Proposed by Drude and Loretz in 1900.
It explains the free electrons in lattice.
It obeys the laws of classical mechanics.
2. Quantum Free electron theory:
It is a microscopic theory.
Proposed by Sommerfeld in 1928.
It explains that the electrons move in a constant potential.
It obeys the Quantum laws.
Hence, the average velocity of the free electrons with which they move towards the positive terminal
under the influence of the electrical field.
Mobility
It is defined as the drift velocity of the charge carrier per unit applied electric field.
Collision time
The average time taken by a free electron between two successive collisions is called collision time.
Mean free path
The average distance travelled by a free electron between two successive collisions is called mean free
path.
Relaxation time
It is defined as the time taken by a free electron to reach its equilibrium position from the disturbed
position in the presence of an electric field.
Merits of Classical Free Electron Theory
- It explains the Ohm’s law.
- It explains the electrical and thermal conductivities of metals.
- It derives Wiedemann–Franz law.
- It explains the optical properties of metals.
Verification of Ohm’s Law
We know that the current density,
A
I
J or I JA (1)
Here I is the current flowing per unit cross-sectional area A of the conductor. As, the microscopic form
of Ohm’s law is given by
J ∝ E or J E (2)
Here, is known as electrical conductivity of material. Putting (2) in (1), we get
A
l
V
I EA
(as
l
V
E )
or V L
A V l
A I (^)
or R
V
I (as A
l R
Here, ‘ R ’ is the resistance of metal piece with length ‘ l ’ and area ‘ A ’ and 𝜌 is the ‘resistivity ’ of the metal
and it defines the properties of metal. Now, from above expression we can write,
V IR
Hence, Ohm’s Law is verified.
Electrical conductivity
Electrical conductivity is defined as the rate of charge flow across unit area in a conductor per unit
potential (voltage) gradient.
E
J
Its unit is Ω
Expression for the Electrical Conductivity
When an electrical field (E) is applied to an electron of charge ‘e’ of a metallic rod, the electron moves in
opposite direction to the applied field with a velocity vd. This velocity is known as drift velocity.
Lorentz force acting on the electron F = eE (1)
This force is known as the driving force of the electron.
Due to this force, the electron gains acceleration ‘a’.
From Newton’s second law of motion,
F= ma (2)
From the equation (1) and (2), ma = eE
or m
eE a (3)
m
e
mne
ne
ne
2 ( )
2
m
ne
Thermal conductivity
Thermal conductivity K is defined as the amount of heat flowing per unit time through the material having
unit area of cross-section per unit temperature gradient,
dx
dT
Q
K
Expression for Thermal Conductivity of a Metal
Consider two cross-sections A and B in a uniform metallic rod AB separated by a distance . Let A at a high
temperature ( T ) and B at low temperature ( T-dT ). Now heat conduction takes place from A and B by the
electrons. The conduction electron per unit volume is n and average velocity of these electrons is v. During the
movement of electrons in the rod, collision takes place. Hence, the electrons near A lose their kinetic energy
while electrons near B gain kinetic energy.
At A, average kinetic energy of an electron, E 1 = kT 2
At B, average kinetic energy of the electron, E 2 = ( ) 2
k T dT (2)
Here, k is the Boltzmann constant.
The excess of kinetic energy carried by the electron from A to B is, E 1 - E 2 = kT 2
k T dT
= kdT 2
Let the electrons move in all 6 directions with equal probability. If n is the free electron density and v is the
thermal velocity then,
Number of electrons crossing per unit area per time from A and B = nv 6
The excess of energy carried form (A to B) per unit area in unit time is,
E = nv kdT nvkdT 4
Similarly, the deficient of energy carried from B to A per unit area per unit time is,
E’ = nvkdT 4
Hence, the net amount of energy transferred from A to B per unit area per unit time is,
Q = nvkdT 2
But from the basic definition of thermal conductivity, the amount of heat conducted per unit area
per unit time is,
dx
dT
Q
K i.e.
dx
dT Q K
If we take dx , the mean free path of electrons, the we can write,
dT Q K (7)
Putting the value of Q from (6) into (7), we get
nvkdT 2
dT K
Hence, K nvk 2
We know that for the metals i.e. v
c
or v (9)
Substituting the equation (8) in equation (7), we have
K nv k
2
2
This is the required expression for thermal conductivity. As seen, thermal conductivity of a metal is
directly proportional to the concentration of free electrons and mean free path of electrons.
Wiedemann – Franz Law
The law states that the ratio of thermal conductivity to electrical conductivity of the metal is
directly proportional to the absolute temperature of the metal.
T
K
i.e. LT
K
Where L is the constant of proportionality and is known as Lorentz number.
- It does not explain the electrical conductivity of semiconductors and insulators.
- It gives T
K
= constant for all temperatures. But, it is so only at low temperatures.
- It does not explain the ferromagnetism.
- The long mean free paths (more than one cm) of the free electrons at low temperatures cannot be
explained on the basis of the classical theory.
8. This theory predicted that resistivity varies as T , whereas actually it is found to vary linearly with
temperature.
- Experimental results show that para magnetism of metals is independent of temperature which deviates
from classical result that paramagnetic susceptibility is inversely proportional to the temperature.
- The resistivity of metals increases with increasing impurity concentration. On the other hand, in
semiconductors even a very small amount of impurity causes a drastic decrease in their resistivity. The
above feature cannot be explained by the classical theory.
Quantum Free Electron Theory of Metals
The drawbacks of the classical free electron theory were removed by Sommerfeld in 1928. He applied
the Schrodinger’s wave equation and de Broglie’s concept of matter waves to obtain the expression for
electron energies. Sommerfeld treated the problem quantum mechanically using the Fermi–Dirac
statistics rather than the classical Maxwell–Boltzmann statistics. The important assumptions made by
Sommerfeld are given below:
- The free electrons move in a constant potential inside the metal and are confined within defined
boundaries.
The eigen values of the conduction electron are quantized.
The electrons are considered to posses’ wave nature.
In the various allowed energy levels, distribution of electrons takes place according to Pauli’s exclusion
principle.
- Mutual attraction between electrons and lattice ions and the repulsion between individual electrons may
be ignored.
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 de-Broglie wave concept , he assumed that a moving electron behaves as if it were a system
of waves. (called matter waves i.e. the 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).
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.
Assumptions of Quantum Free Electron Theory along with those which are
applicable from Classical Free Electron Theory also
Similarities between the two theories:
The valence electrons are treated as though they constitute an ideal gas.
Valence electrons can move freely throughout the body of the solid.
The mutual collisions between the electrons and the force of attraction between the electrons and ions
are considered insignificant.
Difference between the two theories:
According to classical free electron theory:
The free electrons which constitute the electron gas can have continuous energy values.
It is possible that many electrons possess same energy.
The pattern of distribution of energy among the free electron obeys Maxwell-Boltzmann statistics.
Here p is the momentum of the electron. From (1), we can write
p 2 meV 2 mE^ (2)
The expression for de-Broglie wavelength is given by,
mE
h
meV
h
p
h
mv
h
2 2
Wave Function
A variable quantity which characterizes de-Broglie waves is known as Wave function and is
denoted by the symbol 𝛙.
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.
Physical significance of 𝛙
The wave function 𝛙 enables all possible information about the particle. 𝛙 is a complex
quantity and has no direct physical meaning. It is only a mathematical tool in order to
represent the variable physical quantities in quantum mechanics.
Born suggested that, the value of wave function associated with a moving particle at the
position co-ordinates ( x,y,z ) in space, and at the time instant ‘ t ’ is related in finding the particle
at certain location and certain period of time ‘t’.
If 𝛙 represents the probability of finding the particle, then it can have two cases.
Case 1: certainty of its Presence: +ve probability
Case 2: certainty of its absence: - ve probability, but – ve probability is meaningless.
Hence the wave function 𝛙 is complex number and is of the form a+ib
Even though 𝛙 has no physical meaning, the square of its absolute magnitude |𝛙|
2 gives a
definite meaning and is obtained by multiplying the complex number with its complex
conjugate then |𝛙|^2 represents the probability density ‘ P ’ of locating the particle at a place at a
given instant of time. And has real and positive solutions.
P = 𝛙𝛙∗ =| 𝛙 |
2
2
2
2 =− 1
where ‘ P ’ is called the probability density of the wave function.
If the particle is moving in a volume ‘ V ’, then the probability of finding the particle in a
volume element dv , surrounding the point x,y,z and at instant ‘ t ’ is Pdv
∫|𝛙|^2 𝑑𝑣=1 𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙e 𝑖𝑠 𝑝𝑟𝑒𝑠𝑒𝑛𝑡
∫|𝛙2|𝑑𝑣 = 0 if particle does not exist
This is called normalization condition.
Schrödinger Wave Equations
Schrödinger describes the wave nature of a particle in mathematical form and is known as
Schrödinger wave equation. Schrödinger wave equation plays the role of Newton’s laws and
conservation of energy in classical mechanics i.e. it predicts the future behavior of a dynamic
system. It is a wave equation in terms of wave function which predicts analytically and preciously
the probability of events or outcomes. Schrödinger wave equation are of two types:
Time dependent wave equation and
Time independent wave equation.
To obtain these two equations, Schrödinger connected the expression of de-Broglie wavelength into
classical wave equation for a moving particle. The obtained equations are applicable for both
microscopic and macroscopic particles.
Schrödinger Time Dependent Wave Equation
To explain the wave function, let us consider a particle of mass m moving along the positive x-direction
having accurately known momentum p and total energy E. The position of the particle is completely
undetermined.
Let wave associated with such a particle be a plane, continuous harmonic wave travelling in the positive x-
direction. The wavelength of the wave is
p
h
mv
h
k
h h p
2 . 2
where
k
h
Now,
Now, Kinetic Energy m
p
2
2
Equation (5) in terms of wave function 𝛙 can be written as,
V
m
p E
2 (6)
Putting the values of E and p from (4) and (5), we have
V
t i x m
i (^)
2
V t m x
i
2
2 2
2
(7)
Equation (7) is called Schrödinger time dependent wave equation in one-dimension. The
Schrödinger time dependent wave equation in three-dimensional form is written as,
V
t m x y z
i (^)
2
2
2
2
2
2 2
or
V t m
i
(^2)
2
2
(9)
z
k y
j x
i^ and
2
2
2
2
2 2
x y z
or t
V i m
2
Equation (10) contains time and hence is called time dependent Schrödinger wave equation.
The operator
V
m
2
2
is called Hamiltonian and is represented by H.
Schrödinger Time Independent Wave Equation
Again consider equation (3), the we have
iEt px iEt ipx Ae Ae e
( )
or
Et
i
e
Where
px
i
0 Ae
Differentiate (11) partially w.r.t. t , we get
Et
i
e
iE
t
Differentiate (11) partially w.r.t. x twice, we get
iEt e x x
2
0
2
2
2
Putting equations (11), (12) and (13) in equation (7)
V
t m x
i
2
2 2
, we get
iEt iEt iEt
e V e m x
e
iE i ^
0
2 2
0 2
or 20 0
2 2
0 2
V m x
E
or 0
2 2 2 0
0
2
E V
m
x
This is time independent Schrödinger wave equation in one-dimension.
In three-dimensional, it will be of the form as
0
0 2 0
2 E V
m
where,
2
2
2
2
2 2
x y z
is called the Laplacian operator.
