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

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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