Phonon origin and lattice evolution in charge density wave states

Metallic transition metal dichalcogenides, such as tantalum diselenide $\left(\mathrm{TaS}{\mathrm{e}}_2\right)$, display quantum correlated phenomena of superconductivity and charge density waves (CDWs) at low temperatures. Here, the photophysics of 2H-$\mathrm{TaS}{\mathrm{e}}_2$ during CDW transitions is revealed by combining temperature-dependent, low-frequency Raman spectroscopy and density functional theory (DFT). The spectra contain amplitude, phase, and zone-folded modes that are assigned to specific phonons and lattice restructuring predicted by DFT calculations with superb agreement. The noninvasive and efficient optical methodology detailed here demonstrates an essential link between atomic-scale and microscopic quantum phenomena.

Figure 1

(Top) 300 K (blue curve) and 5 K (red curve) Raman spectrum of $\mathrm{TaS}{\mathrm{e}}_2$ ($\lambda =515\mathrm{nm}$). The 300 K spectrum (multiplied by a factor of 3 for clarity) consists of three modes: the two-phonon mode at $151\mathrm{c}{\mathrm{m}}^{-1}$, the $E_{2g}^1$ mode at $212\mathrm{c}{\mathrm{m}}^{-1}$, and the $A_{1g}$ mode at $241\mathrm{c}{\mathrm{m}}^{-1}$. The vibrations are shown schematically from a side view of the lattice. The 5 K spectrum consists of the $E_{2g}^1$ and $A_{1g}$ modes, as well as additional modes due to the CDW, most notably the intense peaks below $100\mathrm{c}{\mathrm{m}}^{-1}$ and several midfrequency peaks between $175\mathrm{c}{\mathrm{m}}^{-1}$ and $200\mathrm{c}{\mathrm{m}}^{-1}$. (Bottom) DFT-calculated Raman mode frequencies for electronic temperatures of 10 K (red lines) and 300 K (blue lines). The gray arrows indicate that the 10 and 300 K lines are the same mode.

Figure 2

(a) Normal Raman phonon spectra of $\mathrm{TaS}{\mathrm{e}}_2$ using 633-nm excitation. Transition temperatures to the incommensurate (ICM) at 122 K and commensurate (CM) at 90 K CDW states are indicated in the legend by blue and red text, respectively. (b) In the top (bottom) plot, the change in the experimental and DFT-calculated frequency of the $A_{1g}$ ($E_{2g}^1$) mode is plotted as a function of temperature through the CDW transitions. Transition temperatures for the ICM- (CM-) CDW are indicated by a dark (light) gray dashed line. The $A_{1g}$ and $E_{2g}^1$ DFT-calculated frequencies are plotted (gray spheres) as a function of electronic temperature using the same temperature axis. (c) DFT-calculated $E_{2g}^1$ vibrations as viewed from above for 300 and 10 K electronic temperatures. At both temperatures, all the selenium and tantalum atoms vibrate parallel to one another in the plane, but the direction of those vibrations at 10 K occurs ${30}^{\circ }$ relative to the vibrations at 300 K yet remains in the plane.

Figure 3

(a) The calculated phonon dispersion at 90 K. Green dotted lines represent optical branches, whereas acoustic and QA modes are in black (with the exception of the one QA mode involved in the two-phonon mode that is in red). (b) Calculated evolution of the two-phonon band as a function of smearing temperature for the unit cell of 2H-$\mathrm{TaS}{\mathrm{e}}_2$. (c) UC, supercell, and experimental results for the frequency shift as a function of temperature.

Figure 4

(a) DFT-calculated atomic rearrangement driven by the formation of the CDW phase, viewed from above. Ta-Ta bonds (blue lines) are drawn for Ta atom separation less than 0.340 nm. The breaking of hexagonal symmetry is visible for $T=122\mathrm{K}$ and $T=90\mathrm{K}$. The final configuration $(T=10\mathrm{K})$ shows the clear formation of a 1D stripe structure for the Ta atoms. (b) Side views at select temperatures showing the gradual formation of out-of-plane Se-Se bonding within the layer; bonds are drawn for atom separation $<0.325\mathrm{nm}$. At $T=122\mathrm{K}$, no Se-Se bonds are observed. However, at $T=90\mathrm{K}$, Se-Se bonds begin to form at random locations within the layer. Finally, at $T=10\mathrm{K}$, there are Se-Se bonds for all Se atoms between the Ta stripes and no other Se atoms.

Figure 5

(a) Raman spectra at 515-nm excitation show the emergence of modes located at 175 and $181\mathrm{c}{\mathrm{m}}^{-1}$, and a band of modes centered around $196\mathrm{c}{\mathrm{m}}^{-1}$. The transition temperature for the ICM (CM) phase is indicated in the legend by blue (red) text. (b) Comparison of experimental (solid lines) and DFT-calculated (dotted lines) frequencies in the $170–200\mathrm{c}{\mathrm{m}}^{-1}$ range as a function of temperature. (c) Top view of the DFT-calculated vibrations for the mode at $194.9\mathrm{c}{\mathrm{m}}^{-1}$, which emerges in the ICM-CDW phase. (d) Top view of the DFT-calculated vibrations of the Ta and (e) Se atoms for the mode at $176.6\mathrm{c}{\mathrm{m}}^{-1}$, which emerges in the CM-CDW phase. Interestingly, both Ta and Se vibrations are in plane and circular with opposite helicities.

Figure 6

(a) LF Raman spectra using 515-nm excitation. Four modes emerge at low temperatures: Amp 1, Amp 2, P1, and P2. Amp 1 and Amp 2, indicated by red and blue arrows, respectively, are the CDW amplitude modes, and P1 and P2 are the phase modes. The transition temperature for the ICM (CM) phase is indicated in the legend by the blue (red) star. (b) Temperature dependence of the frequencies and FWHM of Amp 1, Amp 2, P1, and P2. The DFT-calculated frequencies are plotted as a function of electronic temperature using the same temperature axis. Solid lines guide the eye. The transition temperature for the ICM- (CM-) CDW is indicated by a dark (light) gray dashed line. (c) DFT-calculated vibrations for the phase (side view) and amplitude (top view) modes.