Computational Solid State Physics ??????? ?6?

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Computational Solid State Physics ??????? ?6?

• 6. Pseudopotential

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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
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Summation over ionic potentials
Zj atomic number
position of j-th atom in (n,l,m) unit cell
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Bragg reflection
Assume all the atoms in a unit cell are the same
kind.
structure factor The Bragg reflection
disappears when SG vanishes.
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Valence states
We are interested in behavior of valence
electrons, since it determines main electronic
properties of solids.

• Valence 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.

Which kinds of base set is appropriate to
describe the valence state?
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Orthogonalized Plane Wave (OPW)
OPW
plane wave
core Bloch function
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Core Bloch function
Tight-binding approximation
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Inner product of OPW
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Expansion of valence state by OPW
Extra term due to OPW base set
orthogonalization of valence Bloch functions to
core functions
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Pseudo-potential OPW method
Fc(r)
generalized pseudo-potential
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Generalizedpseudopotential
pseudo wave function
real wave function
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Empty core model
Core region
completeness
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Empty core pseudopotential
(rltrc)
(rgtrc)
O volume of a unit cell
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Screening effect by free electrons
dielectric susceptibility for metals
n free electron concentration eF Fermi
energy
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Screening effect by free electrons
screening length in metals
Debye screening length in semiconductors
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Empty core pseudopotential and screened empty
core pseudopotential
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Brillouin zone for fcc lattice
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Pseudopotential for Al
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Energy band structure of metals
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Merits of pseudopotential

• The 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.

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The first-principles norm-conserving
pseudopotential (1)
Norm conservation
First order energy dependence of the scattering
logarithmic derivative
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The first-principle norm- conserving
pseudopotential (2)
spherical harmonics
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The first-principle norm conserving
pseudo-potential(3)
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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.

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Flow chart describing the construction of an
ionic pseudopotential
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First-principles pseudopotential and pseudo wave
function
Pseudopotential of Au
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Pseudopotential of Si
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Pseudo wave function of Si(1)
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Pseudo wave function of Si(2)
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Si?????
??? ??? ??????????
???? 5.4515Å 5.429 Å 0.42
??????? 5.3495eV/atom 4.63eV/atom 15.5399
????? 0.925Mbar 0.99 Mbar -7.1
????????? 0.665eV 1.12eV -40.625
???????Total Energy-2EXC(???????????)-2ATOM
Energy-(??????????)
Total Energy -0.891698734009E01 HR EXC
-0.497155935945E00 HR ATOM TOTAL -3.76224991
HT Si??????????? 0.068 eV
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Lattice constant vs. total energy of Si
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Energy band of Si
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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.