MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Quantum Mechanical Approach to the Energy Bandgap
Knowlton
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MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Quantum Mechanical Approach to the Energy Bandgap
a+ b = ao = d-spacing of 1D lattice (or plane in 3D)
Knowlton
2
MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Symmetric –vs- Asymmetric wavefunctions in a
periodic potential
 ()  e
 ( )  e
Knowlton
i x
i x
a
a
e
e
 i x
 i x
a
a
 i x 
 2Cos 
 a 
 i x 
 2iSin 
 a 
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Robert F. Pierret, "Advanced Semiconductor Fundamentals", 2nd Ed., Vol. 4 of Modular Series on Solid State Devices, Editors G. Neudeck, R. Pierret (Prentice Hall, 2003)
E – k diagram:
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Knowlton
Modern Theory of Solids
MSE 310/ECE 340
Elec Props of Matls
 Quantum Mechanical Approach to the Energy Bandgap
MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Quantum Mechanical Approach to the Energy Bandgap
Bound states
Unbound states
Knowlton
Robert F. Pierret, "Advanced Semiconductor Fundamentals", 2nd Ed., Vol. 4 of Modular Series on Solid State Devices,
Editors G. Neudeck, R. Pierret (Prentice Hall, 2003)
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Robert F. Pierret, "Advanced Semiconductor Fundamentals", 2nd Ed., Vol. 4 of Modular Series on Solid State Devices, Editors G. Neudeck, R. Pierret (Prentice Hall, 2003)
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Knowlton
Modern Theory of Solids
MSE 310/ECE 340
Elec Props of Matls
 Quantum Mechanical Approach to the Energy Bandgap
MSE 310/ECE 340
Elec Props of Matls

Modern Theory of Solids
Quantum Mechanical Approach to the Energy Bandgap
 Another example from Levi.
A.F.J. Levi, "Applied Quantum Mechanics", 2nd Ed., (Cambridge Univ. Press, 2006)
Knowlton
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MSE 310/ECE 340
Elec Props of Matls
Ch. 4: Modern Theory of Solids
 Examples of E-k diagrams:
Bandstructure of GaAs
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Blakemore, SSP (1985)
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MSE 310/ECE 340
Elec Props of Matls
Ch. 4: Modern Theory of Solids
 Examples of E-k diagrams:
Bandstructure of Ge & Si
Blakemore, SSP (1985)
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Harrison, Electronic Structure & the Properties of Solics (1989)
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MSE 310/ECE 340
Elec Props of Matls
Brillouin Zone - FCC
 Questions: What are the points labeled: Γ, L, X, K,
Λ, Δ, Σ?
 Answer: Lattice directions in reciprocal space
within the first Brillouin zone.
 Example below: Brillouin Zone for FCC
<1 1 1>
Chem 584 Notes, U. Illinois
<0 1 0>
1st
Brillioun
Zone
<1 1 0>
∏/4ao
0
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∏/2ao
3∏/4ao
∏/ao
k
L or K
Γa = d-spacing of plane of X or LXor or
K
o
Jones & March, Theoretical SSP, Vol. 1 (Dover Press, 1973)
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MSE 310/ECE 340
Elec Props of Matls
Ch. 4: Modern Theory of Solids
 Examples of E-k diagrams:
Knowlton
Harrison, Electronic Structure & the Properties of Solics (1989)
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MSE 310/ECE 340
Elec Props of Matls
Ch. 4: Modern Theory of Solids
 Use Band diagrams to classify materials based
on their electrical properties:
Knowlton
McKelvey, SSP for Engineering & Matls Sci. (1993)
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MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Other Quantum Mechanical Models to
Determine Band Theory (& physical properties)
of crystalline solids with periodic potentials
 Tight Binding Method
o Linear combination of atomic orbitals (LCAO)
o Nearest Neighbor interaction
 Wigner-Seitz Method
o Alkali metals
o E-s on ion cores
o Bloch Functions
 Density Functional Theory (DFT)
o ab initio QMs (1st principles QM)
o Pseudopotential method
• Basically, ignore atom potentials
• Reasoning: Core potentials have little effect on
conduction electrons
• Due to screening by core e-’s.
• Thus can use WFs of conduction e-’s
 Molecular Dynamics
o Time dependent SE
o Not quite 1st principles
Knowlton
13
MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Other Quantum Mechanical Models to
Determine Band Theory (& physical properties)
of crystalline solids with periodic potentials
 Tight Binding Method
o Linear combination of atomic orbitals (LCAO)
o Nearest Neighbor interaction
Red  E3; Green  E2; & Blue  E1; Vo 1eV
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E eV
4
3
2
1
0
-3
Knowlton
-2
-1
0
1
k wavenumber 
E – k diagram:
2
3
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MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Other Quantum Mechanical Models Density
Functional Theory (DFT)
o ab initio QMs (1st principles QM)
o Pseudopotential method
• Basically, ignore atom potentials
• Reasoning: Core potentials have little effect on
conduction electrons
• Due to screening by core e-’s.
• Thus can use WFs of conduction e-’s
Knowlton
15
MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Other Quantum Mechanical Models Density
Functional Theory (DFT)
o ab initio QMs (1st principles QM)
o Pseudopotential method
• Basically, ignore atom potentials
• Reasoning: Core potentials have little effect on
conduction electrons
• Due to screening by core e-’s.
• Thus can use WFs of conduction e-’s
Knowlton
16
MSE 310/ECE 340
Elec Props of Matls
Modern Theory of Solids
 Other Quantum Mechanical Models Density
Functional Theory (DFT)
o ab initio QMs (1st principles QM)
o Pseudopotential method
• Basically, ignore atom potentials
• Reasoning: Core potentials have little effect on
conduction electrons
• Due to screening by core e-’s.
• Thus can use WFs of conduction e-’s
Knowlton
Marzari, MRS Bulletin, 31(9) 2006
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