Energy Band Structures in Materials

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Energy Band Structures in

Electronic Materials


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Introduction to Quantum Physics

Light (or any other EM radiation) showswave behavior.

Using wave theory one can explainphenomena like interference, reflection,diffraction and diffusion which lightundergoes.

C


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Light As a Wave


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Introduction to Quantum Physics

Light waves carry energy in small unitscalled photons/energy packets/quanta.

Energy of a quanta,

Photo-electric effect, Compton scattering,etc.. can be explained with this theory.

However light also shows matterbehavior!!

E h


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Exercise

Calculate the energy of a blue photonwhich has a wavelength of 450nm.

h = 6.6 x 10-34 Js

C = 3 x 108 ms-1


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Introduction to Quantum Physics-Wave Behavior of Electrons

Electrons are usually treated as matterwhich obey Newtons laws.

However, they show wave behavior.

Wave behavior is governed bySchrodinger equation.

This provides a relationship between the

electronic wave function, electron mass,potential energy and position of theelectron.


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Interference of Electrons


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Wave-Particle Duality

Photons behave as waves as well asparticles.

Electrons behave as waves as well asparticles.

This is characterized by

where is the momentum. This is the De Broglie hypothesis.

h

p

p


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Electron Distribution in Shells

Electrons are distributed among shells

and sub-shells to minimize the energy.

The attraction force between the nucleus and

electrons and the repulsion force betweenelectron should balance.


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Pauli Exclusion Principle

To have this equilibrium, no two electronsin an atom can have all four quantumnumbers equal.

In other words in a single orbit, amaximum of two electrons can be presentand they orbit in opposite directions.

This is known as Paulis exclusion

principle.


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Occupancy of Electrons

Shell Sub-shell Number ofOrbits

K s 1

L s, p 1,3

M s, p, d 1,3,5

N s, p, d, f 1,3,5,7


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Occupancy of Electrons

When the shells and sub-shells are filledwith electrons, first each electrons occupies a vacant orbit,

when all the orbits are half filled pairingstarts.

This is known as Hunds rule.


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Hunds Rule-Carbon Atom


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Energy Levels in an Atom


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Energy Band Structure

An isolated atom displays the discussed electronlevel structure.

When more atoms are brought together, each

atoms orbit cause some mutual interaction andcause a change of energy in other atoms levels.

In other words creates multiple wave functionsrather than just one.

With Natoms the result is a N-split energy level.


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See Figure 4.8in Electronic Materials andDevices by S.O.Kasap.


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Any energy level is splitted.

However the inner and fully filled energy levelsare not splitted.

Unfilled high energy levels too undergo splitting.

When the number of atoms participatingbecomes too large, the energy split pattern

becomes a continuous band. Now, instead of different filled and unfilled

energy levels, we have energy band.


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Conductance and Valence Bands

Filled outer most energy bands are known asvalance bands.

Unfilled energy bands are known as

conductance bands. Electrons in conductance band participate in

conducting.

At some higher energy (vacuum level), electroncan get free from the solid. This energy level isthe top boundary in the conductance band.


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Fermi Level

The energy level, up to which electrons canoccupy in an absolute zero temperature.

Value of the Fermi level depends on the

reference point which is usually the bottom ofthe valence band.

The energy needed to free an electron is equalto the energy gap between the Fermi level andthe vacuum level. This is the work function.


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Energy Band Structure in Metals

In a metal the lattice of nuclei issubmerged in an electron pool/gas.

When a large number of atoms participate

in the bond, the conduction and valencelevels are split in to bands.

However conduction and valance bands

overlap.Remember: d sub-shells have high energythan the next level s sub-shells


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Energy Band Structure in SemiConductors

In semi conductors covalent bonds exist.

Bonds share electrons.

Small energy gap With the temperature some electrons in

these bonds can absorb energy and move

to the conduction band.


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Energy Band Structure in Insulators

In insulators, the energy gap is large.

Even with high temperatures, electronscannot acquire sufficient energy to crossthe forbidden energy gap.


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Conductivity in Metals Explained

When an electric field is applied, electronsexperience a force which accelerate theelectron.

The increasing velocity increases the kinetic

energy. The increment in energy takes the electron

energy to a higher level in the conductanceband.

Accelerated electrons generates a current flow. The possibility for this increment of energy is

due to availability of free adjacent energy bands.

eE


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Behavior of SemiconductorsExplained

With an applied electric field, the freeelectrons can accelerate

Current flow.


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Insulators Properties Explained

Energy gap is large.

Therefore no possibility of electronacceleration or current flow.


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Thank You

Energy Band Structures in Materials

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