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Phonons: the vibrations of silicon

Phonons are the vibrations of the atoms in a crystal. Their frequencies along a path through the Brillouin zone (the phonon dispersion) show how the crystal responds to small displacements, and they set its heat capacity and its zero-point energy. In this tutorial you calculate the phonon dispersion of silicon with a workflow that SiestaStudio exports and runs in three steps:

  • relax the structure,
  • compute the force constants: displace each atom a little in a supercell and record the forces on all the others (SIESTA, with the fcbuild utility building the supercell),
  • turn the force constants into phonon frequencies along a k-path (the vibra utility).

It takes about 20 minutes; the workflow runs in about 4 minutes on a laptop.

Before you start

  • A SiestaStudio account and SIESTA 5 on your computer, with its utilities fcbuild and vibra (they come with SIESTA).
  • Your first calculation explains the downloads and Analyze File; Relaxing a structure explains relaxations.

1. Build the silicon primitive cell

Create a project Si phonons, open the studio, and click Edit Structure. In Generate Structure choose Diamond Cubic, set a to 5.43 Å and the element to Si, and click Generate Structure (8 atoms). Then open Symmetry and click Primitive Cell: the status bar shows 2 atoms.

The primitive cell

The force constants are computed in a supercell built from this cell, so the smaller the cell, the cheaper the calculation. Click Apply to Studio and set the System Name to Si.

2. Set up the calculation

Open the wizard and choose Bulk Crystal, Semiconductor, Geometry optimization and Quick, then Apply All. Quick (SZP basis, mesh cutoff 150 Ry) keeps the force-constant run short; use Standard or better for numbers you want to publish.

Wizard

3. The k-path for the phonon bands

vibra computes the frequencies along the same k-path as an electronic band structure. In DOS & Band Structure, switch on Enable Band Structure Calculation, click Open BZ Editor, and click Apply: the editor recognises the face-centred cubic cell and writes the standard path Γ-X-W-K-Γ-L-U-W-L-K-U-X.

Brillouin zone editor

4. Check, download the pseudopotentials and save

Diagnostics reports All parameters look good! Download the pseudopotentials (Pseudopotentials, Download All (ZIP)) and save the calculation as Si.

5. The phonon workflow

Click Workflow Builder in the header. Under Built-in Templates, choose Phonon dispersion (relax → FC → vibra).

Workflow templates

The workflow has three steps: Relaxation, Force Constants and Vibra (phonon dispersion). Select a step to see its settings; Edit in Studio on the vibra step opens the vibra panel, where the supercell size and a k-path of its own can be set. The defaults are what this tutorial uses: a supercell of 3 × 3 × 3 primitive cells (SuperCell_1/2/3 = 1), and the k-path of the calculation.

The phonon template

Under Job Scheduler choose Direct, set CPU Cores to 1, and click Generate. You get workflow_si.zip.

Workflow builder

6. Run the workflow

Unzip it, put the pseudopotentials in pseudos/ (or let the script download them), and run:

sh download_pseudos.sh
bash submit_workflow.sh

The script runs the steps in order and stops with a message if a step did not produce its output:

  1. 01_relaxation: SIESTA relaxes the cell's atoms (for perfect silicon the forces are already zero, so it stops after one step).
  2. 02_fc: the script first rewrites the geometry from the relaxed siesta.XV, then runs fcbuild, which writes FC.fdf, a supercell of 54 atoms. SIESTA displaces each of the 2 atoms of the original cell by 0.04 Bohr in each direction (12 calculations) and writes the force constants to siesta.FC. This is the long step, about 3 minutes.
  3. 03_vibra: vibra reads siesta.FC and writes siesta.bands, the phonon frequencies along the path in cm⁻¹, and siesta.vectors, the vibration patterns.

7. The phonon dispersion

Click Analyze File and drop 03_vibra/siesta.bands. SiestaStudio recognises the phonon file and shows Phonon dispersion — 6 branches × 252 k-points: two atoms give six branches, three acoustic and three optical.

Phonon dispersion

Read it from left to right:

  • At Γ the three acoustic branches start at 0: moving the whole crystal costs no energy. This is the first check of a phonon calculation; values far from zero would mean the force constants are not converged.
  • The three optical branches at Γ are at 554–555 cm⁻¹. The measured value for silicon is 517 cm⁻¹; the Quick basis gives a slightly stiffer crystal.
  • No frequency is negative anywhere on the path, so the structure is stable.
  • At X the longitudinal branches meet at 430 cm⁻¹ (measured 410) and the transverse optical branch is at 508–511 cm⁻¹ (measured 463).
  • The transverse acoustic branch is too soft near X and L: 37–43 cm⁻¹ at X against 150 cm⁻¹ measured. Its frequencies there depend on force constants between atoms further apart than a 3 × 3 × 3 supercell holds. A larger supercell (SuperCell_i = 2, 250 atoms) improves it, at a much higher cost.

Below the plot, Phonon DOS counts the frequencies along the path, and Zero-point energy gives ½ Σ ħω averaged over the path points: 126.6 meV per cell here. Both are estimates from the path; a dense sampling of the whole Brillouin zone gives the exact values.

8. Save the analysis

Click Save Analysis and choose the project Si phonons.

What next