Computational Solid State Physics 6 Pseudopotential Potential energy

Computational Solid State Physics 計算物性学特論　第６回 6. Pseudopotential

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

Summation over ionic potentials Zj: atomic number : position of j-th atom in (n, l, m) unit cell

Bragg reflection : position of the j-th atom in a unit cell Assume all the atoms in a unit cell are the same kind. : structure factor The Bragg reflection disappears when SG vanishes.

Valence states We are interested in behavior of valence electrons, since it determines main electronic properties of solids. ØValence states must be orthogonal to core states. ØCore states are localized near atoms in crystals and they are described well by the tight-binding approximation. Which kinds of base set is appropriate to describe the valence state?

Orthogonalized Plane Wave (OPW) OPW : plane wave : core Bloch function

Core Bloch function ・Tight-binding approximation

Inner product of OPW

Expansion of valence state by OPW : Extra term due to OPW base set orthogonalization of valence Bloch functions to core functions

Pseudo-potential: OPW method Fc(r’) generalized pseudo-potential

Generalized pseudopotential : pseudo wave function : real wave function

Empty core model Core region completeness

Empty core pseudopotential (r<rc) (r>rc) Ω: volume of a unit cell

Screening effect by free electrons dielectric susceptibility for metals n: free electron concentration εF: Fermi energy

Screening effect by free electrons ・screening length in metals ・Debye screening length 　in semiconductors

Empty core pseudopotential and screened empty core pseudopotential

Brillouin zone for fcc lattice

Pseudopotential for Al

Energy band structure of metals

Merits of pseudopotential n The valence states become orthogonal to the core states. n The singularity of the Coulomb potential disappears for pseudopotential. n Pseudopotential changes smoothly and the Fourier transform approaches to zero more rapidly at large wave vectors.

The first-principles norm-conserving pseudopotential (1) : Norm conservation First order energy dependence of the scattering logarithmic derivative

The first-principle normconserving pseudopotential (2) : spherical harmonics

The first-principle norm conserving pseudo-potential(3)

The first-principles normconserving pseudopotential (4) n Pseudo wave function has no nodes, while the true wave function has nodes within core region. n Pseudo wave function coincides with the true wave function beyond core region. n Pseudo wave function has the same energy eigenvalue and the same first energy derivative of the logarithmic derivative as the true wave function.

Flow chart describing the construction of an ionic pseudopotential

First-principles pseudopotential and pseudo wave function Pseudopotential of Au

Pseudopotential of Si

Pseudo wave function of Si(1)

Pseudo wave function of Si(2)

Lattice constant vs. total energy of Si

Energy band of Si

Problems 6 n Calculate Fourier transform of Coulomb potential and obtain inverse Fourier transform of the screened Coulomb potential. n Calculate both the Bloch functions and the energies of the first and second bands of Al crystal at X point in the Brillouin zone, considering the Bragg reflection for free electrons. n Calculate the structure factor SG for silicon and show which Bragg reflections disappear.