The Small Displacement Method for Phonon Calculations Theory and Implementation

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The Small Displacement Method for Phonon
Calculations Theory and Implementation

• Adam Shai
• May 2008

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Quick Review of Linear Response

• Hellmann-Feynman Theorem
• Make a linear perturbation in your system -gt a
  perturbation in the electron density -gt and get a
  second order perturbation in your energy
• We can express the linear perturbation of the
  electron density in terms of the unperturbed
  ground state and get the dynamical matrix
• This requires no supercell

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Small Displacement Method
displacement of atom s in unit cell i
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Algorithm for Small Displacement Method
Calculate Forces with pwscf
Displace atom
Repeat with next atom
Form Force Constant Matrix
Form Dynamical Matrix
Calculate Thermodynamic Properties
Find phonon frequencies
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Thermodynamic Properties

• We will derive an equation for heat capacity as
  an example of how to go from phonons to
  thermodynamic properties
• Heat Capacity
• Total energy of phonons in a crystal is the sum
  of the energy over all of the phonon modes

Thermal equilibrium occupancy of phonons. The
form is given by Planck distribution

• Putting these together gives

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Heat Capacity

• We can replace the summation by an integral over
  the frequencies with the density of states
• Density of states is the number of vibrational
  modes per frequency
• The energy can now be written as an integral

• The heat capacity is

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Thermodynamic Properties
This slide taken from lecture
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Small Displacement Method In Practice

• To calculate force constant matrix we make a
  supercell
• In Quantum Espresso we are using periodic
  boundary conditions in our calculations we have
  to make sure the supercell is big enough so that
  the forces go to zero at the edges of the
  supercell
• Use symmetry to reduce the number of calculations
• At most the number of calculations must be three
  times the atoms in the primitive cell
• For any other symmetry operation which takes one
  atom to the place of another, and the whole
  supercell invariant, than only one of those has
  to be calculated

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Small Displacement Method In Practice

• Also, if there is a point group symmetry
  operation on a displacement which leaves the new
  displacement linearly independent from the
  original one, there is a form for the new force
  field, so both do not have to be done
• So for an hcp crystal
• 2 atom basis, 3 coordinates -gt 6 calculations
• Translation then inversion leaves crystal the
  same
• So displacements are only needed on one atom
• We can displace in x direction
• Then rotate that vector 120 degrees to get an
  independent direction, only 2 displacements are
  needed

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The PHON code

• Dario Alfe, University College London
• PHON makes supercell, takes force information and
  finds phonon frequencies
• Use PHON to make a supercell gt do calculations
  with pw.x to find forces gt Use PHON to find
  phonon frequencies

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Aluminum

• Start by finding equilibrium lattice parameter
• Make supercell
• Consider symmetry and find minimum amount of
  displacements to be made
• Displace atoms one at a time and build force
  constant matrix
• Do eigenvalue problem on dynamical matrix to get
  phonons

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Aluminum
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Small Displacement vs. Experiment
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Making the Displacement Large
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Where Did We Go Wrong

• DFT is an approximation
• Force Constant Matrix is for the infinite lattice
• Supercells
• K Point Grid

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Convergence by K Point
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Convergence by K Point
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Convergence with Respect to Supercell Size

• Tried to make supercell size larger
• Even 3x3x3 took very long
• This is a major drawback of this method!

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Small Displacement vs. Linear Response
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Thermodynamic Properties
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Magnesium Oxide
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Thermodynamics of MgO
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Silicon Dioxide
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Conclusion

• Main advantages
• For very simple cases, relatively fast
• More intuitive, easy to understand
• Can be coupled with any DFT program
• Works

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Conclusion

• Disadvantages
• Has trouble with polar materials
• Supercells
• Requires multiple scf computations
• Can be extremely computationally expensive
• Though works to get thermodynamic properties, can
  give bad results for phonon dispersion