the quantum theory of solids

The Quantum Theory of Solids

Allowed and forbidden energy bandsPauli Exclusion Principle

In any given system, no two electrons can occupy the same state

Application Two interacting hydrogen atoms

The Quantum Theory of SolidsAllowed and forbidden energy bands

Application to Siliconn l m spin

3s 3 0 0 1/2 or -1/2

3p 3 1 -1,0,1 1/2 or -1/2 n - principal; l - angular (0..n-1); m- magnetic (-l..+l)

The Quantum Theory of SolidsAllowed and forbidden energy bands

The k-space diagram

where

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f (αa) = P'sinαa

αa+ cosαa = coska = cos

p

ha

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P'=mV0ba

h2

The Quantum Theory of SolidsAllowed and forbidden energy bands

The k-space diagram (cont.)

€

α 2 = 2mE /h2

The Quantum Theory of SolidsElectrical Conduction in Solids

Silicon at T = 0KAll valence electrons are in the valence band

The Quantum Theory of SolidsElectrical Conduction in Solids

Silicon at T > 0KSome electrons have moved into the conduction band

The Quantum Theory of SolidsEffective Mass

but, difficult to know Fint, therefore

where m* takes into account internal forces

Analogy

a force acting on ball in air vs. ball in oil

Given concept of m*, we can determine acceleration

in normal way

F= -eE = m*a, therefore a = -eE/m*

€

Ftotal = Fext + Fint = ma

€

Fext = m*a

The Quantum Theory of SolidsConcept of a Hole

Current flow may be thought of as the movement of valence electrons elevated to the conduction band

Alternatively, can think of this process as the flow of (positively-charged) holes

The Quantum Theory of SolidsMetals, Insulators, and Semiconductors

Insulator Semiconductor Conductor

non-overlapping overlapping

The Quantum Theory of SolidsExtension to three dimensions

direct bandgap indirect bandgap

The Quantum Theory of SolidsDensity of states

Ec - lowest energy of the conduction band

Ev - highest energy of the valence band

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gc (E) =4π (2mn

* )3 / 2

h3 E − Ec

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gv (E) =4π (2mp

* )3 / 2

h3Ev − E

The Quantum Theory of SolidsStatistical mechanics

Number of possibilities for N particles in g states

For n energy levels

Most probable distribution

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g!

N!(g−N)!

€

W =gi!

N i!(gi −N i)i=1

n

∏

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fF (E) =1

1+ expE − EFkT

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The Quantum Theory of SolidsStatistical mechanics(cont.)

The Quantum Theory of SolidsStatistical mechanics(cont.)

Boltzmann approximation

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fF (E) ≈ exp−(E − EF )

kT

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