Computational Solid State Physics 6. Pseudopotential

Potential energy in crystals a,b,c: primitive vectors of the crystaln,l,m: integersG: reciprocal lattice vectors:periodic potentialFourier transform of the periodic potential energy

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

Bragg reflectionAssume all the atoms in a unit cell are the same kind.:structure factor The Bragg reflection disappears when SG vanishes.

Valence statesValence states must be orthogonal to core states.

Core states are localized near atoms in crystals and they are described well by the tight-binding approximation.

We are interested in behavior of valence electrons, since it determines main electronic properties of solids.Which kinds of base set is appropriate to describe the valence state?

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

Core Bloch functionTight-binding approximation

Inner product of OPW

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

Pseudo-potential: OPW methodgeneralized pseudo-potential Fc(r)

Generalizedpseudopotential:pseudo wave function:real wave function

Empty core modelCore regioncompleteness

Empty core pseudopotential(rrc)

Screening effect by free electronsdielectric susceptibility for metals n: free electron concentration F: Fermi energy

Screening effect by free electronsscreening length in metalsDebye screening lengthin semiconductors

Empty core pseudopotential and screened empty core pseudopotential

Brillouin zone for fcc lattice

Pseudopotential for Al

Energy band structure of metals

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

The first-principles norm-conserving pseudopotential (1): Norm conservationFirst order energy dependence of the scatteringlogarithmic derivative

The first-principle norm- conserving pseudopotential (2): spherical harmonics

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

The first-principles norm-conserving pseudopotential (4)Pseudo wave function has no nodes, while the true wave function has nodes within core region.Pseudo wave function coincides with the true wave function beyond core region. 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 functionPseudopotential of Au

Pseudopotential of Si

Pseudo wave function of Si(1)

Pseudo wave function of Si(2)

Si=Total Energy2EXC()2ATOM Energy()Total Energy = -0.891698734009E+01 [HR]EXC = -0.497155935945E+00 [HR]ATOM TOTAL = -3.76224991 [HT]Si = 0.068 [eV]

5.4515[]5.429 []+0.42%5.3495[eV/atom]4.63[eV/atom]+15.5399%0.925[Mbar]0.99 [Mbar]-7.1%

0.665[eV]1.12[eV]-40.625%

Lattice constant vs. total energy of Si

Energy band of Si

Problems 6 Calculate Fourier transform of Coulomb potential and obtain inverse Fourier transform of the screened Coulomb potential.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. Calculate the structure factor SG for silicon and show which Bragg reflections disappear.