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MSE 310/ECE 340Elec Props of Matls
Knowlton 1
Modern Theory of Solids Quantum Mechanical Approach to the Energy Bandgap
MSE 310/ECE 340Elec Props of Matls
Knowlton 2
Modern Theory of Solids Quantum Mechanical Approach to the Energy Bandgap
a+ b = ao = d-spacing of 1D lattice (or plane in 3D)
MSE 310/ECE 340Elec Props of Matls
Knowlton 3
Modern Theory of Solids
Symmetric –vs- Asymmetric wavefunctions in a periodic potential
( ) 2
( ) 2
i x i xa a
i x i xa a
i xe e Cos
a
i xe e iSin
a
MSE 310/ECE 340Elec Props of Matls
Knowlton 4
Modern Theory of Solids Quantum Mechanical Approach to the Energy Bandgap
E – k diagram:
Rob
ert F. P
ierret, "A
dvan
ced S
emicon
du
ctor Fu
nd
amen
tals", 2n
d E
d., V
ol. 4 of Mod
ular S
eries on S
olid S
tate Devices, E
ditors G
. Neu
deck, R
. Pierret (P
rentice H
all, 2003)
MSE 310/ECE 340Elec Props of Matls
Knowlton 5
Modern Theory of Solids Quantum Mechanical Approach to the Energy Bandgap
Bound states
Unbound states
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)
MSE 310/ECE 340Elec Props of Matls
Knowlton 6
Modern Theory of Solids Quantum Mechanical Approach to the Energy Bandgap
Rob
ert F. P
ierret, "A
dvan
ced S
emicon
du
ctor Fu
nd
amen
tals", 2n
d E
d., V
ol. 4 of Mod
ular S
eries on S
olid S
tate Devices, E
ditors G
. Neu
deck, R
. Pierret (P
rentice H
all, 2003)
MSE 310/ECE 340Elec Props of Matls
Knowlton 7
Modern Theory of Solids Quantum Mechanical Approach to the Energy Bandgap
Another example from Levi.
A.F
.J. Levi, "
Ap
plied
Qu
antu
m M
echan
ics", 2nd
Ed
., (Cam
brid
ge Un
iv. Press, 2006)
MSE 310/ECE 340Elec Props of Matls
Knowlton 8
Ch. 4: Modern Theory of Solids
Examples of E-k diagrams:
Bandstructure of GaAs
Blakemore, SSP (1985)
MSE 310/ECE 340Elec Props of Matls
Knowlton 9
Ch. 4: Modern Theory of Solids Examples of E-k diagrams:
Bandstructure of Ge & Si
Harrison, Electronic Structure & the Properties of Solics (1989)
Blakemore, SSP (1985)
MSE 310/ECE 340Elec Props of Matls
Knowlton
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
10Jones & March, Theoretical SSP, Vol. 1 (Dover Press, 1973)
<0 1 0>
<1 1 1>
<1 1 0>
Chem 584 Notes, U. Illinois
Γ X or L or Kk
0 ∏/ao∏/2ao 3∏/4ao∏/4ao
1st
BrilliounZone
ao= d-spacing of plane of X or L or K
MSE 310/ECE 340Elec Props of Matls
Knowlton 11
Ch. 4: Modern Theory of Solids Examples of E-k diagrams:
Harrison, Electronic Structure & the Properties of Solics (1989)
MSE 310/ECE 340Elec Props of Matls
Knowlton 12
Ch. 4: Modern Theory of Solids
Use Band diagrams to classify materials based on their electrical properties:
McKelvey, SSP for Engineering & Matls Sci. (1993)
MSE 310/ECE 340Elec Props of Matls
Knowlton 13
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 Methodo 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 Dynamicso Time dependent SE
o Not quite 1st principles
MSE 310/ECE 340Elec Props of Matls
Knowlton 14
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
E – k diagram:
-3 -2 -1 0 1 2 3k wavenumber
0
1
2
3
4
5
E
Ve
Red E3; Green E2; & Blue E1; Vo1eV
MSE 310/ECE 340Elec Props of Matls
Knowlton 15
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
MSE 310/ECE 340Elec Props of Matls
Knowlton 16
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
MSE 310/ECE 340Elec Props of Matls
Knowlton 17
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
Marzari, MRS Bulletin, 31(9) 2006
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