3- Energy Band Theory
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Transcript of 3- Energy Band Theory
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ENERGY BAND THEORY
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Introduction
To develop the current-voltage characteristics of semiconductordevices, we need to determine the electrical properties ofsemiconductor 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 ofenergy.
We expand this concept to a band of allowed energies in a crystal.
This energy band theory is a basic principle of semiconductormaterial 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.
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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 usedlater to develop the theory of the semiconductor in equilibrium.
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Probability Density Functions forOne and Two Hydrogen Atoms
Probability density function for the
lowest electron energy state of thesingle, noninteracting hydrogen atom
Overlapping probability densityfunction of two adjacent hydrogenatomsThe wave functions of the twoatom electrons overlap, which
means that the two electrons willinteract.
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Energy Level Splitting By InteractionBetween Two Atoms
This interaction or perturbation results in thediscrete quantized energy level splitting intotwo discrete energy levels.The splitting is consistent with Pauli
exclusion principle.When we push several atoms together tomake close to each other, the initial quantizedenergy level will split into a band of discreteenergy levels.
Within the allowed band, the energies are atdiscrete levels.
According to Pauli exclusion principle, totalnumber of quantum states does not change.However, since no two electrons can have
the same quantum number, the discreteenergy must split into a band of energies inorder that each electron can occupy a distinct
quantum state.When a large number of atoms get close to
make a crystal, difference between energystates are very small.
equilibrium interatomic distance
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Allowed And Forbidden Energy Bands
Two atoms with n=3 energy levelWhen these two atoms are
brought close together, first theoutermost level (n=3) is split and
then the second level and finally thefirst level (n=1).
This energy-bandsplitting and theformation of allowed
and forbidden bands
is the energy-bandtheory of single-crystal materials
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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 atomare in the lower band and 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 (thevalence band) will be full and all states in
the upper band (the conduction band) willbe empty.
The bandgap energy Eg between the top of the valence band and the bottom ofthe conduction hand is the width of the forbidden energy band.
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Potential Function for SingleIsolated 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 electronsituated at an arbitrary point x is:
Allowed energy levels for the electron
2
0
1( )
4
qV x
r r=
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Potential Functions ofAdjacent Atoms
If we add the attractive force by theatomic core located at x=a:
rV(r)
And for the one-dimensional crystalline
lattice: V(r)
r
The potential functions of adjacentatoms overlap:
We need this potential function touse in Schrodinger's wave equation to
model a one-dimensional single-crystalmaterial.
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Kronig-Penney Model ofPotential 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 crystallinelattice.
The Kronig-Penney model isan idealized periodic potential
representing a one-dimensionalsingle crystal.
Schrodinger's wave equation ineach region must be solved.
To obtain the solution to Schrodinger's wave equation, we make use Blochtheorem.
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Bloch TheoremThe theorem states that:
If V(x+a)=V(x)
Then (x+a)=ejka (x)
Or, equivalently
(x)=ejkx u(x) u(x) is unit cell wavefunction andu(x+a)= u(x)
The parameter k is called a constant of motion
( / )( , ) ( ) ( ) ( ) .jkx j E tx t x t u x e e = = We have
( )( )/( , ) ( )
j kx E t x t u x e
=
This traveling-wave solution represents the motion of an electron in a single-crystal material.
The amplitude of the traveling wave is a periodic function.
k=wave number.
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The k-Space Diagram
For free electron
and
and
For electron in a infinite potential well2 2 2
2
2
n
nE
m a
=
n
np
a
=
Discrete points lie along the E-p curve
of a free electron.
Since the momentum and wave number
are linearly related, these figures are alsothe E versus k curve.
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Relation between k and ETime-independent Schrdingers Equation:
Assume E < VoIn region I, 0
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For region I:
( ) ( )
1( )j k x j k x
u x Ae Be +
= +For region II
( ) ( )
2
( )j k x j k x
u x Ce De += +
Since the potential function V(x) is everywhere finite, both the wave
function (x) and its first derivative (x)/x must be continuous.So, at x=0: u1(0)=u2(0) A+B C-D=0
On other hand using 1 20 0| |x xdu du
dx dx= ==
We obtain
Also u1(a)=u2(b) and by applying it:
( ) ( ) ( ) ( )
0j k a i k a j k b j k b
Ae Be Ce De + +
+ = =
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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 theparameter a) and the potential function Vo (through the parameter ).
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We have
If E < V0, then is 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 Vo 0, such that the productbVo remains finite.We may approximate sinhb b and coshb 1
The equation can be written as:
2 2
(sin )(sinh ) (cos )(cosh ) cos ( )2
a b a b k a b + = +
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' 0
2
m V b a
P =
If we define
' s in
c o s c o s ( )
a
P a k aa
+ =
If the left side of this equation is plotted as a function ofa, we have:
Since |cos ka| 1, the
right side falls between 1,
-1.
Therefore, only the
shaded regions are allowed
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The k-Space Diagram
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Consider the equation' s in c o s c o s ( )
aP a k a
a
+ =
For cosine we have
where n is a positive integer
We may consider
various segments of the curve can be
displaced by the 2 factor. E versus k diagram in thereduced-zone representation.
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Forbidden Gap
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The E versus k diagram of the
conduction and valence bands of asemiconductor at T > 0 K.
The E versus k diagram of the
conduction and valence bandsof 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>0oK, some electrons have
gained enough energy to jump to the
conduction band and have left
empty states in the valence hand.
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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 internal forces 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 m a=
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Electron Effective Mass
Newtons second Law of motion,
Electrons wave-particle duality,
The first
derivative of E
with respect to k
is related to thevelocity of the
particle.
The second derivative of E with respect to k is inversely proportional to themass of the particle.
2 2
2 *d Ed k m
=
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Example
Consider energy band segment
First derivative
Second derivative
One concludes that m*>0 near the band-energy
minimum and m*
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Concept of HoleA 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 chargeHole has a positive effective mass.
Hole moves in the same direction as an
applied electric field.
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Conduction Band and Valence Band
For valence band
holes
For conduction
band electrons
Bandgap
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Band Structure of Insulators
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Band Structure of Semiconductors
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Band Structure of Metals
It is easy for the electrons to jump into the empty levels, so metals
have high conductivity.
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E-k Diagram in 3Dwe 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 as the 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 direction in acrystal.
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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.
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Carriers for Conductance
The number of carriers that can contribute to current flow is a function of thenumber 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 as a function of
energy in order to calculate the electron and hole concentrations.
Probability ofoccupation of
statesX
Number of
available
states
Actual
Population ofConductance
Band
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Density of States
For the 3D infinite potential well
For the conductance band
nk
a
=and
For the valence band
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Plot of Density of States
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The Distribution Function
Distribution function: the probability that a quantum state at the energy E will beoccupied 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 volumeg (E): the density of states, i.e., the number of quantum states per unit energy per
unit volume
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The Fermi-Distribution Function
At equilibrium, the electrons behavior follows the Fermi (or Fermi-Diract)distribution function
( )
( )
N E
g E=
EF: the Fermi level or
Fermi energy
energy of the highestquantum state of
electrons at 0 K
At 0 K,EEF, f(E)=0
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The Fermi-Distribution Function
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The Fermi-Distribution Function
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Maxwell-Boltzmann approximation
when E - EF >> kT
This equation is called Maxwell-Boltzmann approximation or Boltzmann
approximation to Fermi-Dirac function