Computational Solid State Physics 計算物性学特論 第４回 4. Electronic structure of crystals

Single electron Schroedinger equation Expansion by base functions Φ n : overlap integral m: electron mass V(r): potential energy h: Planck constant

: matrix element of Hamiltonian :algebraic equation

:expression of algebraic equation by matrixes and vectors

: ortho-normalized bases : unit matrix eigenvalue equation condition of existence of inverse matrix of secular equation Solution (1)

Solution (2)

Potential energy in crystals a,b,c: primitive vectors of the crystal n.l.m: integers G: reciprocal lattice vectors :periodic potential Fourier transform of the periodic potential energy

Primitive reciprocal lattice vectors : volume of a unit cell Volume of 1st Brilloluin zone Properties of primitive reciprocal lattice vectors

Bloch ’ s theorem for wavefunctions in crystal (1) (2) k is wave vectors in the 1 st Brillouin zone. Equations (1) and (2) are equivalent.

Plane wave expansion of Bloch functions G : reciprocal lattice vectors

Normalized plane wave basis set :satisfies the Bloch’s theorem V : volume of crystal

Schroedinger equation for single electron in crystals : Bragg reflection : potential energy in crystal : secular equation to obtain the energy eigenvalue at k.

Energy band structure of metals

Zincblende structure a b c

Brillouin zone for the zincblende lattice

Empirical pseudopotential method Energy band of Si, Ge and Sn Empirical pseudopotential method Si Ge Sn

Tight-binding approximation i-th atomic wavefunction at (n,l,m)-lattice sites Linear Combination of Atomic Orbits (LCAO) satisfies the Bloch theorem.

1-dimensional lattice (1) a S(n-m)

:Schroedinger equation 1-dimensional lattice (2)

1-dimensional lattice (3) ε 0 =H 00 : site energy t=H 10 =H -10 : transfer energy ka ε(k)/-t t < 0 Energy dispersion relation 1 st Brillouin zone

Valence orbits for III-V compounds 4 bonds

Matrix elements of Hamiltonian between atomic orbits

Matrix element of Hamiltonian between atomic orbit Bloch functions

Calculation of Hamiltonian matrix element

Matrix element between atomic orbits

Hamiltonian matrix for the zincblende structure

1-fold Bottom of conduction band: s-orbit Top of valence band: p-orbit Energy at Gamma point (k=0) 3-fold

Energy band of Germanium

Energy band of GaAs, ZnSe, InSb, CdTe

Spin-orbit splitting at band edge

Efficiency and color of LED Periodic table B C N Al Si P Ga Ge As In Sn Sb PL energy is determined by the energy gap of direct gap semiconductors.

Bond picture (1): sp 3 hybridization [111] [-1-1-1] [-11-1] [-1-11]

Bond picture (2) Hamiltonian for two hybridized orbits : hybridized orbit energy : transfer energy bonding and anti-bonding states Successive transformations of linear Combinations of atomic orbitals, beginning with atomic s and p orbitals and proceeding to Sp3 hybrids, to bond orbitals, and finally to band states. The band states represent exact solution of the LCAO problem.

Problems 4 Calculate the free electron dispersion relation within the 1 st Brillouin zone for diamond structure. Calculate the energy dispersion relation for a graphen sheet, using a tight-binding approximation. Calculate the dispersion relation for a graphen sheet, using pane wave bases.