- Lattice vibrations and phonons
- Understanding phonon band structures
- Calculating harmonic phonons
- Dealing with anharmonicity
- Detailed derivations
- Perturbation theory
- Electron-phonon coupling
- Group theory
- Matter doesn’t sit still, ever
- Here’s a gas of NaK molecules at 500 nK: https://doi.org/10.1103/PhysRevLett.114.205302
- Authors claimed to reach a “rovibrational ground state”
- i.e. it’s still spinning and vibrating
- (but hyperfine stuff is happening so it’s complicated)
- By contrast, total energies from ab initio calculations
generally don’t include motion
$$E_\mathrm{DFT} = E_\mathrm{kin,elec} + E_\mathrm{pot,nuc-nuc} + E_\mathrm{pot,elec-elec} + E_\mathrm{pot,nuc-elec}$$ - Maybe we include some electronic excitations
- This is not the ground state! The real ground state includes zero-point movements.
- All of these models relate to crystals in which there is a repeating unit of nominal atom sites
- These methods cannot be applied to liquids/gases, and are difficult to apply to amorphous systems
- For a wider range of systems we can use molecular dynamics
- but MD results converge slowly over many samples
- whereas lattice models give exact solutions
Consider transverse waves in a finite 1D chain
- Harmonic series of vibrational states
- States characterised by frequency and wave vector
- At high wave vectors, displacements are indistinguishable from low wave vector
- So we can limit our sampling to a restricted range of wavevectors…
- … the Brillouin zone!
- The lowest obtainable frequency and wave vector depend on the system size
- Infinite crystal → zero wave vector, zero vibrational energy
Putting it all together:
And if we have different force constants in different directions:
These are the “acoustic” phonon branches
- Body-centered cubic structure
- 1-atom primitive cell
Heiming et al. (1991) Phys. Rev. B **43** 10948
- Near-linear dispersion about Γ
- Three branches (transverse/longitudinal), degenerate in some directions
- Approaches zero at 2/3(1,1,1)
- Ordered stacking in 111 direction: metastable 𝜔-phase
- Degenerate “acoustic” branches based on collective behaviour of cell
- “Optic” branches based on interactions within cell
- So-named because they are active at long wave vector lengths (i.e. near Γ)
- This is requirement for detection with IR, Raman spectroscopy
- Degenerate “acoustic” branches based on collective behaviour of cell
- “Optic” branches based on interactions within cell
- So-named because they are active at long wave vector lengths (i.e. near Γ)
- This is requirement for detection with IR, Raman spectroscopy
- A more realistic ZnS model
- LO-TO splitting due to long-range polarisation effects
- (Longitudinal Optic-Transverse Optic)
- In calculations we account for this using Born effective charges
- LO-TO splitting due to long-range polarisation effects
Shibuya et al. (2016) APL Mater. **4** 104809
- LO-TO splitting can lead to discontinuities at Γ
- Hexagonal structure
- 2-atom primitive cell
Composite from Almqvist, L. & Stedman, R. (1971) J. Phys. F.: Met. Phys. **1** 312
- Linear dispersion assumed about Γ
- Six branches, degeneracy in A direction
- Optical modes appear as “reflection” in A of acoustic modes
- In the harmonic approximation, movements are based on Hooke’s law
$$\mathbf{F} = - k \mathbf{u}$$ where$\mathbf{u}$ is a displacement - The force constant matrix
$\mathbf{Φ}$ collects these “spring constants” in each direction between each atom in the system - The dynamical matrix
$\mathbf{D}$ is a Fourier transform of$\mathbf{Φ}$ that also accounts for mass - The eigenvalue problem
$$ω^2(\mathbf{q})ε(\mathbf{q}) = \mathbf{D}(\mathbf{q})ε(\mathbf{q})$$ can then be solved (where$ε$ is a mass-weighted collective displacement) to yield frequencies and eigenvectors - Vibrational movement is treated as a linear combination of these orthogonal “modes”
- Sometimes referred to as the “supercell” or “finite displacement” method
- Assemble the force constant matrix directly using Hooke’s law
- Make finite diplacements in each direction and compute force with ab initio methods.
- Use a supercell to get meaningful interactions with periodic images
- Use symmetry to reduce number of displacements required
- In principle this is “the clever way”
- DFT reformulated to obtain response to perturbations in ionic position
- obtain force constants (second derivative in energy)
- calculation cost increases with each perturbation considered
- Implementation is complex. Legends of DFPT are
- Stefano Baroni (SISSA, Quantum Espresso)
- Xavier Gonze (UCLouvain, Abinit)
| Direct method | DFPT |
|---|---|
| Linear scaling for large cells | Lower cost for small cells |
| More tolerant of imperfect structure | No need to converge displacements |
| Exact force constants for qpts commensurate with supercell | Large supercells very expensive |
| Displacement calculations can be distributed over many jobs | Restart unavailable in VASP |
- Both methods can be managed with Phonopy
- Rather than sampling displacements individually with ab initio calculations, a model can be fitted from experimental data and/or calculations
- Interatomic forcefield parameters can be fitted to data and then used with the “direct” method (e.g. using GULP)
- Alternatively the force constant matrix can be fitted directly by
sampling from an ensemble of displacements
- This method is implemented in TDEP
- Sometimes you will find “negative” frequencies on plot. What does this mean?
-
$ω$ is actually an imaginary number! $ω^2 < 0$ - Restoring force along the mode is negative
- Structure “wants” to move away from initial lattice
-
- At Γ, imaginary mode usually means your structure is not correctly optimised
- Displacement managed to lower energy
Imaginary modes close to Γ are generally associated with calculation accuracy problems.
- Basis-set / k-point convergence
- Supercell size
- FFT grids
Jonathan Skelton has some helpful slides on these issues https://www.slideshare.net/jmskelton/phonons-phonopy-pro-tips-2015
- Indicate ordering over multiple unit cells
Whalley et al. (2017) J. Chem. Phys. **146** 220901
- Build a supercell which is commensurate with the wave vector
- e.g. if soft mode is around (0.5 0 0.5), you need a
$2×1×2$ supercell
- e.g. if soft mode is around (0.5 0 0.5), you need a
- Give it a “nudge” along the mode and perform local optimisation from there
- Phonopy’s MODULATION feature is helpful for setting these calculations up
- Compare the energy of the new supercell to your original structure
- Try calculating phonons again with the new structure…
- The phonon DOS is a distribution of accessible vibrational states
- Occupied with a thermal distribution of quantised excitations: “phonons”
- Like electronic DOS, which follows Fermi-Dirac statistics
- (electrons are fermions)
- Phonon DOS is occupied by Bose-Einstein distribution
- (phonons are bosons)
- Like electronic DOS, which follows Fermi-Dirac statistics
- Projected DOS (PDOS) can also be constructed
- weighting assigned using mode eigenvectors
- tells us what is moving in which frequency range
- Less informative than dispersion plot and inspecting eigenvectors
- but much easier to interpret in busy system
- This distribution of states is used in thermodynamic partition functions
- vibrational energy as a function of temperature
- related by calculus to heat capacity, entropy, free energy
Zinc blende
Wurtzite
- Vibrational properties of zinc blende and wurtzite phases of ZnS
are different
- Wurtzite phase is higher in energy
- Compare vib entropy, free energy relationship with T
- We can predict phase transition by plotting free energy vs temperature, including difference in formation energy
- In this case, it doesn’t seem sufficient to drive the phase transition!
- Real potential energy surfaces are not symmetric
- As atoms move more, their average location moves to the shallow side of the well
- This drives thermal expansion
- In the quasi-harmonic approximation (QHA), we consider the effect of thermal expansion on the otherwise harmonic vibration model
- Force constants are recalculated at different volumes
- Helmholtz free energy (
$A$ ) is computed vs temperature for each volume - At a given temperature, the structure should minimise
$A$ at equilibrium- Competing volumes are just like competing phases (with easier kinetics!)
- Interpolate between calculated volumes to obtain equation of state
- The quasi-harmonic approximation (QHA) tends to improve the accuracy of thermodynamic property calculations
- QHA also provides very useful qualitative information about how phase transitions relate to thermal expansion
- In practice it seems to improve accuracy of calculated frequencies
- However, it ultimately works by representing anharmonic wells with softer harmonic wells. It will break down if atoms are truly moving in interesting asymmetric ways.
- An alternative approach to anharmonicity is to consider the harmonic approximation as the first part of a Taylor expansion
- In this case, the higher-order terms represent interactions between the lower-order terms
- In the limit of a complete series, this would completely cover
anharmonicity
- It is not guaranteed that using a truncated series will help
- We have to truncate the series for practical calculations
- Phonon-phonon interactions are generally computed when we are
interested in their scattering effects
- e.g. when predicting thermal conductivity
- Lattice dynamics describe collective movements in a crystal
- This determines:
- Measurable IR/Raman frequencies
- Thermochemistry (Temperature-dependent potentials)
- Dynamic stability / phase transitions
- And opens the way to
- Thermal conductivity (need higher-order terms)
- Raman intensities (need mode-dependent polarisability)
- A range of calculation methods are available
- Very sensitive; need precise of forces and optimisation
- If forcefield available, GULP is cheap!
- Generally start with harmonic approximation, Phonopy
- Supercell method or DFPT, depending on unit cell size
- QHA improves accuracy for ~ factor 10 in cost
- Phono3py gives more information for MUCH greater cost
- Still experimenting with TDEP