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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Maxwell-Boltzmann approximation

when E - EF >> kT

This equation is called Maxwell-Boltzmann approximation or Boltzmann

approximation to Fermi-Dirac function

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3- Energy Band Theory