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"Explore the electronic structure of a Si crystal through Density Functional Theory, which optimizes single-electron crystal orbitals in a periodic potential. Discover the impact of impurities on band states and the localization of donor and acceptor orbitals. Dive into the de-localization of valence and conduction band orbitals in a P-doped or B-doped Si crystal lattice."
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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
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.
energy quantum number pure silicon
energy quantum number pure silicon conduction “band”
energy quantum number pure silicon conduction “band” valence “band”
energy quantum number pure silicon conduction “band” “band” gap valence “band”
energy quantum number pure silicon conduction “band” “band” gap valence “band”
energy quantum number pure silicon conduction “band” “band” gap valence “band”
energy quantum number pure silicon conduction “band” “band” gap valence “band”
donor state energy quantum number P-doped silicon conduction “band” “band” gap valence “band”
energy quantum number P-doped silicon conduction “band” donor state “band” gap valence “band”
energy acceptor state quantum number B-doped silicon conduction “band” valence “band”
energy quantum number B-doped silicon conduction “band” acceptor state valence “band”
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
