EEE 315 - Electrical Properties of Materials

Lecture 6

Band Theory of Solids

In isolated atoms the electrons are arranged in energy levels

In solids the outer electron energy levels become smeared out to form bands

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Band Theory of Solids

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• The highest occupied band is called the VALENCE band. This is full.

• For conduction of electrical energy there must be electrons in the CONDUCTION band. Electrons are free to move in this band.

Band Theory of Solids

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Insulators : There is a big energy gap between the valence and conduction band. Examples are plastics, papers…

Conductors : There is an overlap between the valence and conduction band hence electrons are free to move about. Examples are copper, lead ….

Semiconductors : There is a small energy gap between the two bands. Thermal excitation is sufficient to move electrons from the valence to conduction band. Examples are silicon ,germanium….

“Realistic” Potential in SolidsFor one dimensional case where atoms

(ions) are separated by distance d, we can write condition of periodicity as

)()( ndxUxU

“Realistic” Potential in Solids

Multi-electron atomic potentials are complex

Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex

So we use a model one-dimensional periodic potential to get insight into the problem

Bloch’s TheoremBloch’s Theorem states that for a

particle moving in the periodic potential, the wavefunctions ψ(x) are of the form

uk(x) is a periodic function with the periodicity of the potential

)()(

function periodic a is )( ,)()(

dxuxu

xuwhereexux

kk

kikx

k

Bloch’s Theorem

What is probability density of finding particle at coordinate x?

2

**

*

*2

)()(

)()()()()(

])([])([)(

)()()()(

xuxP

xuxueexuxuxP

exuexuxP

xxxxP

k

kkikxikx

kk

ikxk

ikxk

• But |uk(x)|2 is periodic, so P(x) is as well

Bloch’s Theorem

The probability of finding an electron at any atom in the solid is the same!!!

The most common example of Bloch's theorem is describing electrons in a crystal. Each electron in a crystalline solid “belongs” to each and every atom forming the solid

)()( dxPxP

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Consider initially the known wave functions of two hydrogen atoms far enough apart so that they do not interact.

Band Theory of Solids

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Band Theory of Solids Interaction of the wave functions occurs as the atoms get

closer:

An atom in the symmetric state has a nonzero probability of

being halfway between the two atoms, while an electron in the anti-symmetric state has a zero probability of being at that location.

When more atoms are added (as in a real solid), with a large number of atoms, the levels are split into nearly continuous energy bands, with each band consisting of a number of closely spaced energy levels.

Symmetric Anti-symmetric

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Kronig-Penney Model

An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms.

R. de L. Kronig and W. G. Penney developed a useful one-dimensional model of the electron lattice interaction in 1931.

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Kronig-Penney Model Kronig and Penney assumed that an electron

experiences an infinite one-dimensional array of finite potential wells.

Each potential well models attraction to an atom in the lattice, so the size of the wells must correspond roughly to the lattice spacing.

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Kronig-Penney Model Since the electrons are not free their energies

are less than the height V0 of each of the potentials, but the electron is essentially free in the gap 0 < x < a, where it has a wave function of the form

where the wave number k is given by the usual relation:

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Tunneling In the region between a < x < a + b the

electron can tunnel through and the wave function loses its oscillatory solution and becomes exponential:

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Kronig-Penney Model Matching solutions at the boundary, Kronig

and Penney find

Here K is another wave number.

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The left-hand side is limited to values between +1 and −1 for all values of K.

Plotting this it is observed there exist restricted (shaded) forbidden zones for solutions.

Kronig-Penney Model

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Important differences between the Kronig-Penney model and the single potential well

1) For an infinite lattice the allowed energies within each band are continuous rather than discrete. In a real crystal the lattice is not infinite, but even if chains are thousands of atoms long, the allowed energies are nearly continuous.

2) In a real three-dimensional crystal it is appropriate to speak of a wave vector . A wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation.

k

k

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And…

3) In a real crystal the potential function is more complicated than the Kronig-Penney squares. Thus, the energy gaps are by no means uniform in size. The gap sizes may be changed by the introduction of impurities or imperfections of the lattice.

These facts concerning the energy gaps are of paramount importance in understanding the electronic behavior of semiconductors.

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Band Theory and Conductivity

Band theory helps us understand what makes a conductor, insulator, or semiconductor.

1) Good conductors like copper can be understood using the free electron

2) It is also possible to make a conductor using a material with its highest band filled, in which case no electron in that band can be considered free.

3) If this filled band overlaps with the next higher band, however (so that effectively there is no gap between these two bands) then an applied electric field can make an electron from the filled band jump to the higher level.

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Valence and Conduction Bands

The band structures of insulators and semiconductors resemble each other qualitatively. Normally there exists in both insulators and semiconductors a filled energy band (referred to as the valence band) separated from the next higher band (referred to as the conduction band) by an energy gap.

If this gap is at least several electron volts, the material is an insulator. It is too difficult for an applied field to overcome that large an energy gap, and thermal excitations lack the energy to promote sufficient numbers of electrons to the conduction band.

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For energy gaps smaller than about 1 electron volt, it is possible for enough electrons to be excited thermally into the conduction band, so that an applied electric field can produce a modest current.

The result is a semiconductor.

Smaller energy gaps create semiconductors

The magnitude of the band gap determines the differences between insulators, s/c‘s and metals.

The excitation mechanism of thermal is not a useful way to promote an electron to CB even the melting temperature is reached in an insulator.

Even very high electric fields is also unable to promote electrons across the band gap in an insulator.

Insulators :Insulators :

CB (completely empty)

VB (completely full)

Eg~several electron voltsEg~several electron volts

Wide band gaps between VB and CB

Metals :Metals :

CB

VB

CB

VB

No gap between valance bandvalance band and conduction bandconduction band

Touching VB and CB Overlapping VB and CB

These two bands looks like as if partly filled bands and it is known that partly filled bands conducts well.

This is the reason why metals have high conductivity.

The Concept of Effective Mass :The Concept of Effective Mass :

Comparing Comparing Free e- in vacuum

An e- in a crystal

In an electric field mo =9.1 x 10-31

Free electron mass

In an electric field

In a crystal m = ?

m* effective mass

If the same magnitude of electric field is applied to both electrons in vacuum and inside the crystal, the electrons will accelerate at a different rate from each other due to the existence of different potentials inside the crystal.

The electron inside the crystal has to try to make its own way.

So the electrons inside the crystal will have a different mass than that of the electron in vacuum.

This altered mass is called as an effective-effective-mass.mass.

To find effective mass , mm**

We will take the derivative of energyenergy with respect to k ;k ;

2

2 2

2

2 2

2*

dE k

dk m

d E

mdk

md E dk

Change m*m* instead of mm

This formula is the effective masseffective mass of an electron inside the crystal.

- m*m* is determined by the curvature of the E-k curve

- m*m* is inversely proportional to the curvature

momentumk

Energy

How do Electrons and Holes Populate the Bands?

dEEgc )(The number of conduction band

states/cm3 lying in the energy

range between E and E + dE

(if E Ec).

The number of valence band

states/cm3 lying in the energy

range between E and E + dE

(if E Ev).

dEEgv )(

Density of States Concept

General energy dependence of

gc (E) and gv (E) near the band edges.

How do Electrons and Holes Populate the Bands?

Density of States Concept

Quantum Mechanics tells us that the number of available states

in a cm3 per unit of energy, the density of states, is given by:

Density of States in Conduction Band

Density of States in Valence Band

How do electrons and holes populate the bands?

Probability of Occupation (Fermi Function) Concept

Now that we know the number of available states at each energy, then how do the electrons occupy these states?

We need to know how the electrons are “distributed in energy”.

Again, Quantum Mechanics tells us that the electrons follow the

“Fermi-distribution function”.

Ef ≡ Fermi energy (average energy in the crystal)k ≡ Boltzmann constant (k=8.61710-5eV/K)T ≡Temperature in Kelvin (K)

f(E) is the probability that a state at energy E is occupied. 1-f(E) is the probability that a state at energy E is unoccupied.

kTEE feEf /)(

1

1)(

Fermi function applies only under equilibrium conditions, however, is universal in the sense that it applies with all materials-insulators, semiconductors, and metals.

ASSIGNMENT

Write down an essay on the band theory of solids Use necessary examples and illustrations The essay should be based on your learning from the

class, search online for more ideas, theories, examples and figures

Write up to maximum 5 Pages (cover page excluded). Use standard font, font size and color

Submission dead line is 24 November, 2013 Submit to- maruf@stamforduniversity.edu.bd DO NOT FORGET TO WRITE YOUR ID IN THE COVER PAGE

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