ModeMap

A set of tools for mapping and analysing potential-energy surfaces along phonon modes.

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Note (26/04/2017):

A bug was recently identified in the ModeMap.py script whereby normal-mode coordinates were scaled internally by 1/sqrt(N_a), leading to U(Q) curves that were inconsistent with the harmonic mode frequencies. This issue has now been fixed.

If you have output generated with earlier versions of the code, you can use the --rescale_na command-line option to the ModeMap_PostProcess.py script to correct for the bug during post processing. If you need to check whether you were using the affected version, you can type ModeMap_PostProcess.py -h and see whether the --rescale_na option appears in the list of command-line arguments.

You can also use the new --harmonic_fit option of ModeMap_PolyFit.py to fit a potential to a harmonic function U(Q) = 1/2 ω² Q² and extract the fitted frequency.

We apologise for any inconvenience this may cause, and we are very grateful to Janine George (RWTH Aachen) for pointing out this issue and suggesting a fix.

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Prerequisites

These tools are based around the Phonopy code. To use them requires a harmonic phonon calculation with Phonopy to have been performed on the system of interest. It is assumed that users are comfortable with these calculations, and with the basics of the underlying theory.

The ModeMap* scripts are written in Python. They were developed and tested on Python 3.x, but should be compatible with Python >= 2.7. ModeMap.py uses routines from the Phonopy Python package to generate structures modulated along phonon modes. Phonopy therefore needs to be installed and "importable" (e.g. the library directories added to PYTHONPATH). ModeMap_PostProcess.py and ModeMap_PolyFit.py require the NumPy, SciPy and Matplotlib packages.

Users wishing to use the 1D Schrödinger solver provided for post processing will need to compile it using a Fortran 90 compiler such as GFortran.

Usage

A typical usage will consist of three or four steps, implemented in a series of short programs:

1. Prepare a sequence of structures with displacements along one (1D map) or two (2D map) phonon modes over a range of amplitudes "frozen in" (ModeMap.py).

2. Run single-point energy calculations on these structures, and extract the total energies. A basic script for the Vienna ab initio Simulation Package (VASP) code is provided (ExtractTotalEnergies.py).

3. Post-process the calculation results to generate the potential-energy surface maps (ModeMap_PostProcess.py).

4. If desired, fit the calculated potential-energy surfaces to polynomial functions for further analysis (ModeMap_PolynomialFit.py).

5. Again, if desired, carry out further post processing by using the fitted potential-surface(s) as input to a 1D Schrödinger solver. A Fortran code written by J. Buckeridge (UCL) is provided in 1DSchrodingerSolver. This uses the Fourier method to obtain the eigenvalues and eigenvectors in the anharmonic potential, and to determine an effective renormalised (harmonic) frequency for the mode that reproduces its contribution to the thermodynamic partition function at a given temperature. The TISH code by J. M. Frost, which was written to study bandgap-deformation potentials, can also work with polynomial fits.

Limitations

The expected use case for this code is to study phonon modes along which the potential-energy surface is anharmonic (e.g. modes representing double-well structural instabilities).

By explicitly mapping the potential energy along the mode eigenvector, the energetic minima along imaginary modes can be located. By sloving a 1D Schrödinger equation for the potential, anharmonic eigenvalues (energy levels) can be recovered. Further analyses can be performed using this as a base, e.g. to study the effect of the soft-mode instabilities on the electronic structure.

Users should bear in mind, however, that this technique makes several assumptions, in particular:

• That the potential-energy surfaces along the phonon modes are independent; and
• That the mode eigenvectors obtained within the harmonic approximation are a reasonable representation of the anharmonic motion.

A common case where these assumptions may not hold is when modes are degenerate. In such cases, linear combinations of the mode eigenvectors are valid solutions to the harmonic problem. In the case of degenerate double-well modes, this means that the eigenvectors may not "point" directly at the energetic minima, and the minima instead lie at linear combinations of both eigenvectors.

Within this approximation, the analysis of an N-fold degenerate set of modes should technically be approached by mapping the N-dimensional potential-energy surface and solving the corresponding N-dimensional Schroödinger equation. This is presently beyond the scope of these codes, although they may provide a basis for further development.

Examples

The following examples are provided to illustrate some of the applications of these codes:

• Cmcm SnSe Reproduces some of the calculations and analysis in Ref. 1.

Further Information

The basic theory and its application to the high-temperature Cmcm phase of SnSe are described in detail in Ref. 1 in the References section.

The mapping codes were used to map the potential-energy surfaces along the imaginary phonon modes in the cubic phase of methylammonium lead iodide ((CH₃)(NH₃)PbI₃, MAPbI₃) in Ref. 2.

An extension of the method to study the effect of the phonon instabilities in MAPbI₃ on the electronic structure is described in Ref. 3.

This method implemented in these codes is very similar in principle to the "Decoupled Anharmonic Mode Approximation" technique developed in Ref. 4, albeit with the simplification that in this implementation the potential-surface mapping and analysis would normally only be performed for selected modes.

Citation

If you use these codes in your own work, please cite Skelton et al., Phys. Rev. Lett. 117, 075502 (2016) (Ref. 1 in the References section).

References

1. J. M. Skelton, L. A. Burton, S. C. Parker, A. Walsh, C.-E. Kim, A. Soon, J. Buckeridge, A. A. Sokol, C. R. A. Catlow, A. Togo and I. Tanaka, "Anharmonicity in the High-Temperature Cmcm Phase of SnSe: Soft Modes and Three-Phonon Interactions", Physical Review Letters 117, 075502 (2016), DOI: 10.1103/PhysRevLett.117.075502

2. A. N. Beecher, O. E. Semonin, J. M. Skelton, J. M. Frost, M. W. Terban, H. Zhai, A. Alatas, J. S. Owen, A. Walsh and S. J. L. Billinge, "Direct Observation of Dynamic Symmetry Breaking above Room Temperature in Methylammonium Lead Iodide Perovskite", ACS Energy Letters 1 (4), 880 (2016), DOI: 10.1021/acsenergylett.6b00381

3. L. D. Whalley, J. M. Skelton, J. M. Frost and A. Walsh, "Phonon anharmonicity, lifetimes, and thermal transport in CH₃NH₃PbI₃ from many-body perturbation theory", Physical Review B 94, 220301(R) (2016), DOI: 10.1103/PhysRevB.94.220301

4. D. J. Adams and D. Passerone, "Insight into structural phase transitions from the decoupled anharmonic mode approximation", Journal of Physics: Condensed Matter 28 (30), 305401 (2016), DOI: 10.1088/0953-8984/28/30/305401