The Quantum Theory of Solids
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Transcript of The Quantum Theory of Solids
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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
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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)
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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
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The Quantum Theory of SolidsAllowed and forbidden energy bands
The k-space diagram (cont.)
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α 2 = 2mE /h2
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The Quantum Theory of SolidsElectrical Conduction in Solids
Silicon at T = 0KAll valence electrons are in the valence band
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The Quantum Theory of SolidsElectrical Conduction in Solids
Silicon at T > 0KSome electrons have moved into the conduction band
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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*
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Ftotal = Fext + Fint = ma
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Fext = m*a
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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
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The Quantum Theory of SolidsMetals, Insulators, and Semiconductors
Insulator Semiconductor Conductor
non-overlapping overlapping
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The Quantum Theory of SolidsExtension to three dimensions
direct bandgap indirect bandgap
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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
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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)!
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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.)
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The Quantum Theory of SolidsStatistical mechanics(cont.)
Boltzmann approximation
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fF (E) ≈ exp−(E − EF )
kT
⎡ ⎣ ⎢
⎤ ⎦ ⎥