the quantum theory of solids
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The Quantum Theory of Solids. Allowed and forbidden energy bands Pauli Exclusion Principle In any given system, no two electrons can occupy the same state Application Two interacting hydrogen atoms. The Quantum Theory of Solids. Allowed and forbidden energy bands - PowerPoint PPT PresentationTRANSCRIPT
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
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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
€
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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⎝ ⎜
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The Quantum Theory of SolidsStatistical mechanics(cont.)
The Quantum Theory of SolidsStatistical mechanics(cont.)
Boltzmann approximation
€
fF (E) ≈ exp−(E − EF )
kT
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