1. isolated Si atoms

2. Si crystal

3. Ĥ=E Density Functional Theory (DFT)
Replaces the exact multi-electron Schrȍdinger equation with an approximate single-electron Schrȍdinger equation.
Electron moves in a periodic potential determined by the background electron density and the symmetry of the crystal lattice.
Numerically optimizes energy of single electron crystal orbitals.
DFT is applicable to systems with a relatively large number of electrons. (Not practical using conventional Hartree-Fock methods.)
Ĥ=E

4. Real-space calculation of electronic structure
27 unit cells (3x3x3)
276 Si atoms
Passivated [100]-oriented crystallite with 240 surface H atoms
4104 electrons
4404 electronic states including core, surface, and band states
Geometry of pure Si crystallite was optimized using a gradient corrected potential with the “HCTH” exchange functional.

5. pure silicon
energy
quantum number

6. pure silicon
conduction “band”
energy
quantum number

7. pure silicon
conduction “band”
energy
valence “band”
quantum number

8. pure silicon
conduction “band”
energy
“band” gap
valence “band”
quantum number

9. pure silicon
conduction “band”
energy
“band” gap
valence “band”
quantum number

10. pure silicon
conduction “band”
energy
“band” gap
valence “band”
quantum number

11. pure silicon
conduction “band”
energy
“band” gap
valence “band”
quantum number

12. P-doped silicon

13. P-doped silicon conduction “band” “band” gap energy valence “band”
donor state
“band” gap
valence “band”
quantum number

14. P-doped silicon conduction “band” “band” gap energy valence “band”
donor state
“band” gap
valence “band”
quantum number

15. B-doped silicon

16. B-doped silicon conduction “band” energy valence “band” quantum number
acceptor state
valence “band”
quantum number

17. B-doped silicon conduction “band” energy valence “band” quantum number
acceptor state
valence “band”
quantum number

18. Density Functional Theory (DFT)
Confirms that valence and conduction band orbitals are de-localized (in agreement with classical electron gas in a periodic background potential).
Confirms that doping introduces donor and acceptor orbitals at the band edges.
Confirms that donor and acceptor orbitals are localized with dopant atoms and in addition exhibit the tetrhedral symmetry charcteristic of the crystal lattice