ENERGY_BAND_THEORY.pdf for physics students

Physics

1. 1
ENERGY BAND THEORY

2. 2
Introduction
To develop the current-voltage characteristics of semiconductor
devices, we need to determine the electrical properties of
semiconductor materials.
To accomplish this, we have to:
determine the properties of electrons in a crystal lattice,
determine the statistical characteristics of the very large number of
electrons in a crystal.
We know that electron in a single crystal take discrete values of
energy.
We expand this concept to a band of allowed energies in a crystal.
This energy band theory is a basic principle of semiconductor
material physics.
It can also be used to explain differences in electrical
characteristics between metals, insulators, and semiconductors.
We will introduce electron effective mass which relates quantum
mechanics to classical Newtonian mechanics.

3. 3
Introduction
we will define a new particle in a semiconductor called a hole.
We will develop the statistical behavior of electrons in a crystal
To determine the statistical law of electrons, we note that Pauli
exclusion principle is an important factor.
The resulting probability function will determine the distribution of
electrons among the available energy states.
The energy band theory and the probability function will be used
later to develop the theory of the semiconductor in equilibrium.

4. 4
Probability Density Functions for
One and Two Hydrogen Atoms
Probability density function for the
lowest electron energy state of the
single, noninteracting hydrogen atom
Overlapping probability density
function of two adjacent hydrogen
atoms
The wave functions of the two
atom electrons overlap, which
means that the two electrons will
interact.

5. 5
Energy Level Splitting By Interaction
Between Two Atoms
This interaction or perturbation results in the
discrete quantized energy level splitting into
two discrete energy levels.
The splitting is consistent with Pauli
exclusion principle.
When we push several atoms together to
make close to each other, the initial quantized
energy level will split into a band of discrete
energy levels.
Within the allowed band, the energies are at
discrete levels.
According to Pauli exclusion principle, total
number of quantum states does not change.
However, since no two electrons can have
the same quantum number, the discrete
energy must split into a band of energies in
order that each electron can occupy a distinct
quantum state.
When a large number of atoms get close to
make a crystal, difference between energy
states are very small.
equilibrium interatomic distance

6. 6
Allowed And Forbidden Energy Bands
Two atoms with n=3 energy level
When these two atoms are
brought close together, first the
outermost level (n=3) is split and
then the second level and finally the
first level (n=1).
This energy-band
splitting and the
formation of allowed
and forbidden bands
is the energy-band
theoryof single-
crystal materials

7. 7
Energy Band Formation
As the interatomic distance decreases,
the 3s and 3p states interact and overlap.
At the equilibrium interatomic distance,
the bands have again split.
But now four quantum states per atom
are in the lower bandand four quantum
states per atom are in the upper band.
At absolute zero degrees, electrons are in
the lowest energy state,
So that all states in the lower band (the
valence band) will be full and all states in
the upper band (the conduction band) will
be empty.
The bandgap energy E
g
between the top of the valence band and the bottom of
the conduction hand is the width of the forbidden energy band.

8. 8
Potential Function for Single
Isolated Atom
Consider an one-dimensional array of atoms in a crystalline lattice:
a= lattice constant
The attractive force between an atomic core located at x=0 and electron
situated at an arbitrary point x is:
Allowed energy levels for the electron
2
0
1
( )
4
q
V x
r rπε
=− ∝

9. 9
Potential Functions of
Adjacent Atoms

If we add the attractive force by the
atomic core located at x=a:
r
V(r)
And for the one-dimensional crystalline
lattice:
V(r)
r
The potential functions of adjacent
atoms overlap:
We need this potential function to
use in Schrodinger's wave equation to
model a one-dimensional single-crystal
material.

10. 10
Kronig-Penney Model of
Potential Functions
a: potential well width;
b: potential barrier width
This model is used to represent a one-
dimensional single-crystal lattice for
considering electron behavior in crystalline
lattice.
The Kronig-Penney model is
an idealized periodic potential
representing a one-dimensional
single crystal.
Schrodinger's wave equation in
each region must be solved.
To obtain the solution to Schrodinger's wave equation, we make use Bloch
theorem.

11. 11
Bloch Theorem
The theorem states that:
If V(x+a)=V(x)
Then ψ(x+a)=e
jka
ψ(x)
Or, equivalently
ψ(x)=e
jkx
u(x)
u(x) is unit cell wavefunction and
u(x+a)= u(x)
The parameter k is called a constant of motion
( / )
( , ) ( ) () ( ) .
jkx j E t
xt x t u xe eψ ψ
−
Ψ = =
ℏ
We have
()( )/
( , ) ( )
j kx E t
xt u xe
−
Ψ =
ℏ
ℏThis traveling-wave solution represents the motion of an electron in a single-
crystal material.
ℏThe amplitude of the traveling wave is a periodic function.
ℏk=wave number.

12. 12
The k-Space Diagram
For free electron
and
and
For electron in a infinite potential well
2 2 2
2
2
n
n
E
ma
π
=
ℏ
n
n
p
a
π
=
ℏ
Discrete points lie along the E-p curve
of a free electron.
ℏSince the momentum and wave number
are linearly related, these figures are also
the E versus k curve.

13. 13
Relation between k and E
Time-independent Schrödinger’s Equation:
Assume E < V
o
In region I, 0<x<a, V(x)=0 and using ψ(x)=e
jkx
u(x)
u
1
(x) is the amplitude of wave function in region I
In Region II (-b < x < 0), V(x) = V
o

14. 14
For region I:
( ) ( )
1
( )
j k x j k x
u x Ae Be
α α− − +
= +
For region II
( ) ( )
2
( )
j k x j k x
u x Ce De
β β− − +
= +
Since the potential function V(x) is everywhere finite, both the wave
function ψ(x) and its first derivative ∂ψ(x)/ ∂x must be continuous.
So, at x=0: u
1
(0)=u
2
(0)A+B –C-D=0
On other hand using
1 2
0 0
| |
x x
du du
dx dx
= =
=
We obtain
Also u
1
(a)=u
2
(b) and by applying it:
( ) ( ) ( ) ( )
0
j k a i k a j k b j k b
Ae Be Ce De
α α β β− − + − − +
+ = − =

15. 15
Finally, the boundary condition
gives
By using the resulted equations, we obtain constants and the solution.
The result is:
This equation relates the parameter k to the total energy E (through the
parameter a) and the potential function V
o
(through the parameter β).

16. 16
We have
→If E < V
0
, then βis imaginary quantity.
→Then the equation
can be written as
→The solution of this equation results in a band of allowed energies.
→To obtain a graphical solution for this equation, let the potential
barrier width b →0 and the barrier height V
o
→0, such that the product
bV
o
remains finite.
→We may approximate sinh βb≈βband coshβb≈1
→The equation can be written as:
2 2
(sin )(sinh ) (cos )(cosh ) cos ( )
2
a b a b k a b
β α
α β α β
αβ−
+ = +

17. 17
' 0
2
mV ba
P=
ℏ
If we define
'sin
cos cos( )
a
P a ka
a
α
α
α+ =
ℏIf the left side of this equation is plotted as a function of αa, we have:
ℏSince |coska|ℏ1, the
right side falls between 1,
-1.
ℏTherefore, only the
shaded regions are allowed

18. 18
The k-Space Diagram

19. 19
Consider the equation
'sin
cos cos( )
a
P a ka
a
α
α
α+ =
For cosine we have
where n is a positive integer
We may consider
various segments of the curve can be
displaced by the 2πfactor.
E versus k diagram in the
reduced-zone representation.

20. 20
Forbidden Gap

21. 21
The E versus k diagram of the
conduction and valence bands of a
semiconductor at T > 0 K.
The E versus k diagram of the
conduction and valence bands
of a semiconductor at T = 0 K
The energy states in the valence band
are completely full and the states in the
conduction band are empty.
At T>0
o
K, some electrons have
gained enough energy to jump to the
conduction band and have left
empty states in the valence hand.

22. 22
Electron Effective Mass
The movement of an electron in a lattice will, in general, be different from that
of an electron in free space.
In addition to an externally applied force, there are internalforces in the
crystal due to
positively charged ions or protons and
negatively charged electrons, which will influence the motion of electrons
in the lattice.
We have
F ma=
To take into account internal forces, we can write:*
F ma=

23. 23
Electron Effective Mass
Newton’s second Law of motion,
Electron’s wave-particle duality,
ℏThe first
derivative of E
with respect to k
is related to the
velocity of the
particle.
The second derivative of E with respect to k is inversely proportional to the
mass of the particle.
2 2
2 *
d E
dk m
=
ℏ

24. 24
Example
Consider energy band segment
First derivative
Second derivative
One concludes that m
*
>0 near the band-energy
minimum and m
*
<0 near the band-energy maximum.m
*
is positive near the bottoms of all bands.
m
*
is negative near the tops of all bands.
A negative effective mass simply means that, in
response to an applied force , the electron will
accelerate in a direction opposite to that expected from
purely classical considerations.
In general, the effective mass of an electron is a
function of the electron energy.
Since near the top or the bottom of band edge are
populated by the carriers, the E-k relationship is
parabolic and we may write:
A constant energy-independent
effective mass

25. 25
Concept of Hole
A positively charged "empty state" is created when a valence electron is elevated
into the conduction band.
if a valence electron gains a small amount of thermal energy, it may move into the
empty state.
Hole has a positive charge
Hole has a positive effective mass.
Hole moves in the same direction as an
applied electric field.

26. 26
Conduction Band and Valence Band
For valence band
holes
For conduction
band electrons
Bandgap

27. 27
Band Structure of Insulators

28. 28
Band Structure of Semiconductors

29. 29
Band Structure of Metals
It is easy for the electrons to jump into the empty levels, so metals
have high conductivity.

30. 30
E-k Diagram in 3D
we will extend the allowed and forbidden energy band and effective mass
concepts to three dimensions and to real crystals.
One problem encountered in extending the potential function to a three-
dimensional crystal is that the distance between atoms varies asthe direction
through the crystal changes.
So, Electrons traveling in
different directions
encounter different potential
patterns and
therefore different k-space
boundaries.
The E versus k diagrams
are in general a function of
the k-space directionin a
crystal.

31. 31
The k-Space Diagrams of Si and GaAs
direct bandgap semiconductor indirect bandgap semiconductor
Germanium is also an indirect bandgap material, whose valence band
maximum occurs at k=0 and whose conduction band minimum occurs along
the [111] direction.

32. 32
Carriers for Conductance
The number of carriers that can contribute to current flow is a function of the
number of available energy states or quantum states.
we indicated that the band of allowed energies was actually made up of discrete
energy levels.
We must determine the density of these allowed energy states asa function of
energy in order to calculate the electron and hole concentrations.
Probability of
occupation of
states
X
Number of
available
states
Actual
Population of
Conductance
Band

33. 33
Density of States
For the 3D infinite potential well
For the conductance band
n
k
a
π
=and
For the valence band

34. 34
Plot of Density of States

35. 35
The Distribution Function
Distribution function:the probability that a quantum state at the energy E will be
occupied by an electron.
( )
( )
( )
N E
f E
g E
=
N (E):the number density, i.e., the number of particles per unit energy
per unit volume
g (E):the density of states, i.e., the number of quantum states per unit energy per
unit volume

36. 36
The Fermi-Distribution Function
At equilibrium, the electrons’behavior follows the Fermi (or Fermi-Diract)
distribution function
( )
( )
N E
g E
=
E
F
: the Fermi level or
Fermi energy
energy of the highest
quantum state of
electrons at 0 K
At 0 K,
E<E
F
, f(E)=1
E>E
F
, f(E)=0

37. 37
The Fermi-Distribution Function

38. 38
The Fermi-Distribution Function

39. 39
Maxwell-Boltzmann approximation
when E - E
F
>> kT
This equation is called Maxwell-Boltzmannapproximation or Boltzmann
approximation to Fermi-Diracfunction