Mutual stabilization of charge-density-wave and monoclinic distortion in sulfur at high pressures

The charge-density-wave (CDW) amplitude mode of the high-pressure sulfur-IV phase is observed between 83 and 146 $\mathrm{G}\mathrm{Pa}$ using Raman spectroscopy. The energy of this excitation softens with pressure yet remains finite at ${\nu }_{\mathrm{CDW}}>100\mathrm{c}{\mathrm{m}}^{-1}$ up to the critical pressure, which is indicative of a weakly first-order transition. Our ab initio calculations show that the finite energy of the excitation originates from the coupling and mutual stabilization of the CDW modulation and a monoclinic lattice distortion. At the critical pressure, both the CDW modulation and lattice distortion disappear simultaneously. Due to the prevalence of CDW phases, this coupling between the CDW modulation and lattice distortion is expected to be relevant for a wide variety of elements and compounds.

Figure 1

(a) Raman spectra of sulfur between 55 and 154 GPa normalized to the maximum intensity at a given pressure. Solid colored lines are fits to the individual excitations. Solid black lines correspond to the total fitted spectra. (b) Experimental (symbols) and calculated (lines) energies of Raman-active excitations labeled according to the symmetries identified in DFT calculations and whether they are stretching (s) or bending (b) modes. The five known characteristic regions in the studied pressure range are labeled and indicated by background color. The pressures determined from DFT calculations have been linearly scaled (see Appendix pp4). (c) The unnormalized spectral weights of the excitations associated with the S-IV phase. The dashed line is a guide to the eye showing the evolution of the CDW excitation.

Figure 2

Crystal structures of sulfur and their enthalpy difference per atom, relative to the S-V $R\overline{3}m$ structure at $P_{\mathrm{DFT}}=70\mathrm{GPa}$, which corresponds to an experimental pressure of 129 GPa and is within S-IV. Bonds are colored to indicate symmetrically equivalent bonds. Arrows show the direction of displacement of the atoms in the CDW. Shaded atoms show the S-V structure. Atomic displacements are exaggerated for visibility. The difference between the $P{2}_1/m$ structure with and without the monoclinic distortion (MD) is visible from the nonequivalent bonds.

Figure 3

Contour plots of potential enthalpy surfaces at $P_{\text{DFT}}$ = 70, 75, and 80 GPa around the S-V (at $x=0, y=0$) and S-IV (at $x\approx \pm 0.05, y\approx 0.9$) modulated bcm structures. Enthalpy contours are relative to the global minimum in meV/atom. Enthalpies above $1\mathrm{meV}/\mathrm{atom}$ are saturated on this scale. Orange gradient lines begin at $y=0$ and blue gradient lines begin at $y=1$.

Figure 4

(a) Ambient pressure sulfur sample on the $50\mu \mathrm{m}$ culet with electrodes for transport measurements. (b) The same sulfur sample (outlined in red) at 155 GPa with the laser incident on the center of the sample and culet. The applied pressure caused the sulfur to extrude. (c) Spectra of the diamond phonon under pressure from SOM9. In black is the ambient pressure diamond phonon at 1334 ${\mathrm{cm}}^{-1}$; high-pressure spectra have been normalized to the intensity of the ambient pressure phonon. The high-frequency edge of the diamond is indicated by black arrows and corresponds to a minimum in the derivative (as shown in the inset).

Figure 5

(a) Data from SIO1 across the measured pressure range without background subtraction. Inset: The region demonstrating the large suppression of the CDW amplitude mode from where it is most intense at 104 to 154 $\mathrm{G}\mathrm{Pa}$, where the excitation has been completely suppressed. (b) Background-subtracted spectra from SOM9. (c) Background-subtracted spectra from SIO3 within the S-III phase at 53 GPa. Two polarizers are used in parallel ($0^{\circ }$) and perpendicular (${90}^{\circ }$) orientations. The modes are labeled with the symmetry allocations of the main text. In these spectra, the $E_g^s$ mode is visible as a weak shoulder. (d) Peak energies vs pressure for all three sulfur samples and compared to calculations. Data from SIO1 and calculations (shown in main text) are faded to emphasize the data from SOM9 and SIO3. (e) Normalized $S_W$ of the CDW excitation as a function of pressure for SIO1 and SOM9. (f) The unnormalized spectral weights ($S_W$) of the modes purely associated with the S-III phase from SIO1. Solid lines are linear fits to the data, with the $x$ intercept giving the upper critical pressure of the S-III/S-IV coexistence region as $104\pm 7$ GPa.

Figure 6

Calculated displacement vectors of the Raman-active modes at $P_{\text{DFT}}=60$ GPa for S-IV in the $q=3/4$ approximation. The effective wave vector of each displacement vector is given.

Figure 7

(a) Calculated phonon dispersion of the S-V structure along the [100] direction. Colors correspond to different pressures, in steps of 10 GPa from $P_{\mathrm{DFT}}=50$ GPa (red) to 110 GPa (pink), and $P_{\mathrm{DFT}}=140$ GPa (gray). Vertical lines represent the $q_{\text{CDW}}$ vector with the minimum in the phonon instability toward the CDW S-IV structure. The frequencies of the two positive branches at $q_{\text{CDW}}$ are used to estimate S-IV modes. (b) Calculated Raman-active modes for S-IV using S-V dispersions (magenta) and using the commensurate approximation supercells (black). Open black symbols are Raman-active modes that are unlikely to appear in a real incommensurate structure because they either result from back-folding or are above $P_1$, the pressure at which the unmodulated structure becomes the global minimum. Filled black symbols are modes associated with the $q=3/4$ CDW approximation. $P_2$ is the pressure at which the modulated structure is no longer dynamically stable. $P_3$ is the pressure at which the $R\overline{3}m$ structure has only stable phonon modes. The dashed horizontal line shows the pressure scale used for comparison to experimental measurements.

Figure 8

Phonon-dispersion curves at (a) 50 GPa, (b) 100 GPa, and (c) 130 GPa with varying electronic $k$-point grids compared to previous DFT calculations from Ref. [7].

Figure 9

(a) Calculated displacement vectors of the Raman-active modes at $P_{\text{DFT}}=50$ GPa for S-III. (b) Calculated Raman intensities for S-III at $P_{\text{DFT}}=50$ GPa. The $E_g$ modes are doubly degenerate.

Figure 10

Calculated Raman-active modes for S-III. Symbol size corresponds to calculated mode intensity.

Figure 11

Enthalpy plot of competing sulfur phases. The S-IV line ends when the CDW can no longer be stabilized in geometry optimization.