Introducing another RESCU-DFPT simulator feature: phonon spectra of elemental crystals

Our atomistic tool RESCU is a powerful KS-DFT solver (Kohn-Sham), which solves the KS equation on real space grids by using the Chebyshev filtered subspace iteration (CFSI) method. CFSI scales efficiently with the number of atoms and computational resources which empowers RESCU to simulate large systems which are beyond the reach of conventional solvers.

In our previous LinkedIn post, we introduced our density functional perturbation theory (DFPT) toolbox and the so-called perturbed CFSI (PCFSI) method, as our unique solution for large scale DFPT simulations. We demonstrated how it can compute the ion-clamped (or static) dielectric constant as precisely as state-of-the-art DFPT techniques, while being more efficient and allowing us to simulate systems up to several hundreds of atoms. As a reminder or for those who have missed our previous postings, the current set of features of DFPT toolbox is summarized in the following diagram (Figure 1):

Figure 1 – List of our core RESCU DFPT features.

In this article, we focus on atomic displacements as perturbation factors and show how DFPT together with PCFSI can be used to precisely compute phonon spectra of materials, comprising hundreds of atoms. Let’s begin by demonstrating the accuracy of the DFPT formalism by comparing the phonon band-structure of two of the most important elemental semiconductors: diamond and silicon.

Figure (1) illustrates the simulated phonon band-structure of diamond (using the GGA exchange–correlation functional) compared with experimental data (blue circles). We can see how simulation results closely follow the experimental data points:

Figure 2 – Face centered cubic diamond phonon band structure: experiments Vs our simulation as a very close match.

We also observe the same accuracy in comparing the simulated phonon band-structure of silicon (in the diamond lattice phase and using GGA exchange-correlation functional) with experimental data (blue circles):

Figure 3 – Face centered cubic silicon phonon band structure: experiments Vs our simulation as a very close match too!

The experimental data was extracted from [1, 3] using a plot digitizer [2]. Overall, we see that DFPT provides quantitative predictions of the phonon spectrum of various materials. In the present case, we note a slight underestimation of the most energetic optical phonon energies (of the order of 20 cm-1).

Next, we demonstrate the efficiency of PCFSI method in dealing with heftier systems, by computing the phonons density of states (DOS) of a supercell of diamond including 216 atoms. Since it is computationally very expensive (if not impossible due to memory requirements and long computational time) to generate the same results with a conventional method for comparison, we take a different approach to show the accuracy of our simulation. We simulate an 8 atom (conventional unit cell) of diamond at high q-sampling with a conventional solver and 64 (8 time larger) and 216 (27 times larger) supercells with PCFSI method. We readily see that the 216-atom supercell’s DOS sits right on top of the high-sampling conventional cell’s DOS. In other words, we obtain the same results for simulations with converged computational parameters (here, high q-sampling for the conventional cell and large unit cell for the supercell) representing the same physical reality. In contrast, the 64-atom supercell’s DOS displays minor differences as it is not quite large enough to resolve all long-ranged interatomic forces if only Γ-sampled:

Figure 4 –Diamond phonon density of states: 216 atoms are necessary to recover the high q-sampling limit (of a conventional unit cell).

In terms of efficiency, the 64-atom supercell would have been challenging on a single node and 216-atom system impossible because of memory requirements, but it posed no problem to our PCFSI implementation. While it only takes around 30 minutes to simulate an atomic displacement with PCFSI on small number of CPUs (48), it would take an order of magnitude longer (5 hours or more) to perform the same simulation with a conventional solver. Generally, for a system comprised of N atoms, we need to perform 3xN such calculations per each q-point (phonon wave-vector) to obtain the full phonon spectra, which emphasizes the importance of PCFSI solver for supercell calculations. Following table lists some timing regarding our simulations on diamond systems:

Table 1 – Computational timings for diamond supercell calculations.

In this article, we have shown that RESCU can precisely compute phonon spectra of materials comprising hundreds of atoms. Phonon band-structure in polar materials is affected by coupling of the electric field with longitudinal vibrations which is realized through introduction of Born effective charges: in a forthcoming publication here on LinkedIn, we will explain how to compute Born effective charges and correct the phonon band structure of polar materials which leads to so called LO-TO splitting.

We hope you enjoyed the read, thank you and feel free to comment and ask us specific information about our atomistic tools, we will be more than happy to give additional insights.

More simulation science content to come very soon!

[1] Gonze, X., Rignanese, G., & Caracas, R. (2005). First-principle studies of the lattice dynamics of crystals, and related properties. Zeitschrift für Kristallographie – Crystalline Materials, 220, 458 – 472.

[2] https://apps.automeris.io/wpd/

[3] Kulda, Strauch, Pavone, & Ishii (1994). Inelastic-neutron-scattering study of phonon eigenvectors and frequencies in Si. Physical review. B, Condensed matter, 50 18, 13347-13354.