Lattice dynamics of the tin sulphides SnS2, SnS and Sn2S3: vibrational spectra and thermal transport

First-principles lattice-dynamics calculations are used to model and compare the vibrational spectra and thermal transport of four bulk tin-sulphide materials.


Peak tables for SnS2, Pnma and π-cubic SnS and Sn2S3
Tables S5-S8 present a list of the calculated Γ-point phonon frequencies of SnS2, Pnma and π-cubic SnS and Sn2S3 together with the irreducible representations of the mode eigenvectors, the calculated infrared (IR) and Raman activities, and the calculated linewidths at 10, 150 and 300 K. This is the data used to model the spectra in Fig. 3 in the text. The frequencies obtained from both the density-functional perturbation theory (DFPT) routines in the VASP code 1 and using the Phonopy code 2, 3 are shown for comparison ( DFPT and FD , respectively), and typically agree to within 1 cm -1 . When determining the irreducible representations, doubly-and triply-degenerate modes are identified based on their frequencies being identical, and for degenerate modes the spectroscopic intensities and linewidths listed are the average of the modes in the set.

Table S7
Spectroscopic data for π-cubic SnS. The first and second columns compare the frequencies obtained from density-functional perturbation theory (DFPT) and finite-displacement (FD) calculations. The doubly-and triply-degenerate modes are identified based on the phonon frequencies, and the intensities and linewidths given are the average of the modes in each set.

Mode eigenvectors of the Pnma and π-cubic SnS and Sn2S3
Figure S1 Eigenvectors of the 24 Γ-point phonon modes of Pnma SnS with frequencies as marked (cm -1 ). The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4

Figure S1
Eigenvectors of the 24 Γ-point phonon modes of Pnma SnS with frequencies as marked (cm -1 ). The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4
The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4

Figure S3
Eigenvectors of the 60 Γ-point phonon modes of Sn2S3 with frequencies as marked (cm -1 ). The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4 -Page 23 -

Figure S3
Eigenvectors of the 60 Γ-point phonon modes of Sn2S3 with frequencies as marked (cm -1 ). The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4

Figure S3
Eigenvectors of the 60 Γ-point phonon modes of Sn2S3 with frequencies as marked (cm -1 ). The Sn and S atoms are coloured green and yellow, respectively. The three acoustic modes, which correspond to rigid translations of the crystal lattice, necessarily have zero frequency and are spectroscopically inactive, and so the frequencies of these modes are not shown. These images were generated using the ascii-phonons software. 4

Dependence of the calculated Raman intensities on the displacement-step size
In principle, the calculated Raman intensities obtained by performing small steps along the mode eigenvectors may be sensitive to the choice of the step size used to compute the polarisability derivatives. This can be a particular issue for systems with small bandgaps at the generalised-gradient approximation (GGA) level of theory used in these calculations, for which certain atomic displacements (or combinations of displacements) could lead to disproportionately large changes in the electronic structure. 5 To check this, we computed the Raman intensities of the four compounds using three different step sizes, viz. 0.005, 0.01 and 0.015 amu 1 2 Å (Tables S9-S12). For Sn2S3, we also calculated the intensities for the 0.01 amu 1 2 Å step using a denser k-point mesh (Table S12). As noted in the text, the Raman intensities were found for these systems to be relatively insensitive to the choice of the step size, with the various parameters yielding quantitatively very similar intensities.
We note that, due to the number of modes and the large size of the π-cubic model, when assessing the effect of the step size on the calculated mode intensities in this system we only performed calculations for a subset of the modes, viz. every 5 th mode plus the ten modes with the largest intensities (49 modes in total; see Table S11). -

Convergence of the -point spectral linewidths with respect to the lifetime-sampling mesh
The computational cost of post processing the third-order calculations to calculate the phonon lifetimes, spectral linewidths and lattice thermal conductivity scales unfavourably with the size of the reciprocal-space mesh used to evaluate the Brillouin-zone integrals.
The phonon linewidths of even "simple" binary compounds have been shown to vary non-smoothly with respect to phonon wavevector, 6 and it is therefore important explicitly to check convergence of these properties with respect to the Brillouin-zone sampling.
We therefore calculated the linewidths of the Γ-point phonon modes with systematically-increasing lifetime-sampling meshes (a subset of the post processing required to calculate the full thermal-conductivity tensors, and therefore a less computationally-demanding task). We also tested two different methods of interpolating between -points during the Brillouin-zone integration, viz. the linear tetrahedron method and a Gaussian smearing scheme with a width, , of 0.1 THz. The tetrahedron method is, in principle, more accurate, but Gaussian smearing typically converges faster with respect to the size of the sampling mesh. Interestingly, this slow convergence does not seem to manifest in the calculated thermal-conductivity curves (see Section 8, Fig. S19), perhaps because in most cases the broad linewidths (short lifetimes) would mean that these modes would invariably make a negligible contribution to the overall thermal conductivity.
Despite the non-smooth convergence of the Pnma SnS and Sn2S3 linewidths, these tests do indicate that the mesh sizes used to generate the simulated spectra in Fig. 3 in the text are well converged, at least within the limitations of other technical parameters (e.g. the supercell size used to calculate the second-and third-order force constants).
We note that, for -cubic SnS, we were only able to perform tests with the tetrahedron method. This is due to a performance optimisation in the Phono3py code, 6 which makes the Gaussian smearing considerably more expensive for systems with large numbers of phonon bands. However, for the other systems the two integration methods appear to give similar results, and, as noted above, the convergence of the linewidths with respect to mesh sampling appears to be relatively rapid in this system.    b RTA data on CZTS/Se from Ref. 8 .

Figure S11
Isotropically-averaged lattice thermal conductivity ( latt ) of SnS2 as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects.

Figure S12
Isotropically-averaged lattice thermal conductivity ( latt ) of Pnma SnS as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects.

Figure S13
Isotropically-averaged lattice thermal conductivity ( latt ) of π-cubic SnS as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects.

Figure S14
Isotropically-averaged lattice thermal conductivity ( latt ) of Sn2S3 as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects.

Figure S15
Isotropically-averaged lattice thermal conductivity ( latt ) of Pnma SnSe as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects. The RTA data was taken from Ref. 7 .

Figure S16
Isotropically-averaged lattice thermal conductivity ( latt ) of Cu2ZnSnS4 (CZTS) as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects. The RTA data was taken from Ref. 8 .

Figure S17
Isotropically-averaged lattice thermal conductivity ( latt ) of Cu2ZnSnSe4 (CZTSe) as a function of temperature, calculated within the relaxation-time approximation (RTA) with (blue triangles) and without (black circles) isotope effects. The RTA data was taken from Ref. 8 .

Convergence of the lattice thermal conductivity with respect to the lifetime-sampling mesh
Figure S18 Isotropically-averaged lattice thermal conductivity ( latt ) of SnS2 computed with various lifetimesampling -point meshes.