Modern Theory of Solids

Transcription

1 Quantum Mechanical Approach to the Energy Bandgap Knowlton 1

2 Quantum Mechanical Approach to the Energy Bandgap a+ b = a o = d-spacing of 1D lattice (or plane in 3D) Knowlton 2

3 Symmetric vs- Asymmetric wavefunctions in a periodic potential i x i x a a i x ( ) e e 2Cos a i x i x i x a a ( ) e e 2iSin a Knowlton 3

4 Quantum Mechanical Approach to the Energy Bandgap E k diagram: 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) Knowlton 4

5 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) Knowlton 5

6 Quantum Mechanical Approach to the Energy Bandgap 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) Knowlton 6

7 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 7

8 Ch. 4: Examples of E-k diagrams: Bandstructure of GaAs Knowlton Blakemore, SSP (1985) 8

9 Ch. 4: Examples of E-k diagrams: Bandstructure of Ge & Si Blakemore, SSP (1985) Knowlton Harrison, Electronic Structure & the Properties of Solics (1989) 9

10 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> <1 1 0> 1 st Brillioun Zone Knowlton 0 /4a o /2a o 3 /4a o /a o Γ a o = d-spacing of plane of X or L or K Jones & March, Theoretical SSP, Vol. 1 (Dover Press, 1973) X or L or K k 10

11 Ch. 4: Examples of E-k diagrams: Knowlton Harrison, Electronic Structure & the Properties of Solics (1989) 11

12 Ch. 4: Use Band diagrams to classify materials based on their electrical properties: Knowlton McKelvey, SSP for Engineering & Matls Sci. (1993) 12

13 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 (1 st 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 1 st principles Knowlton 13

14 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 e V Red E3 ; Green E2; & Blue E1 ; Vo 1eV k wavenumber E k diagram: Knowlton 14

15 MSE 310/ECE 340 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

16 Other Quantum Mechanical Models Density Functional Theory (DFT) o ab initio QMs (1 st 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

17 Other Quantum Mechanical Models Density Functional Theory (DFT) o ab initio QMs (1 st 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)