Higgs-mode radiance and charge-density-wave order in $2H-{\mathrm{NbSe}}_2$

Despite being usually considered two competing phenomena, charge-density wave and superconductivity coexist in few systems, the most emblematic one being the transition-metal dichalcogenide $2H-{\mathrm{NbSe}}_2$. This unusual condition is responsible for specific Raman signatures across the two phase transitions in this compound. While the appearance of a soft phonon mode is a well-established fingerprint of the charge-density-wave order, the nature of the sharp subgap mode emerging below the superconducting temperature is still under debate. In this work we use external pressure as a knob to unveil the delicate interplay between the two orders, and consequently the nature of the superconducting mode. Thanks to an advanced extreme-conditions Raman technique, we are able to follow the pressure evolution and the simultaneous collapse of the two intertwined charge-density-wave and superconducting modes. The comparison with microscopic calculations in a model system supports the Higgs-type nature of the superconducting mode and suggests that charge-density wave and superconductivity in $2H-{\mathrm{NbSe}}_2$ involve mutual electronic degrees of freedom. These findings fill the knowledge gap on the electronic mechanisms at play in transition-metal dichalcogenides, a crucial step to fully exploit their properties in few-layer systems optimized for device applications.

Figure 1

Raman scattering under hydrostatic pressure and at low temperature of $2H-{\mathrm{NbSe}}_2$. (a) $(P,T)$ phase diagram of $2H-{\mathrm{NbSe}}_2$ drawn from resistivity measurements [26, 36]. The incommensurate CDW collapses at a critical pressure of $\sim 4$ GPa; a pure SC state persists up to at least 10 GPa. Large colored circles mark the $(P,T)$ positions of the experimental spectra reported in Fig. 2. (b) Membrane diamond-anvil cell designed for a large numerical aperture collection (green cone), low Raman signal from the environment, and access to low temperature ($\sim 3$ K). The $350-\mu \mathrm{m}$-diameter pressure chamber containing the freshly cleaved $2H-{\mathrm{NbSe}}_2$ sample and rubies as pressure gauge is depicted below. (a) Thermal link between the metallic gasket and the cold finger of the cryostat made of high-conductivity copper wires. (c) Raman spectra of $2H-{\mathrm{NbSe}}_2$ in the $E_{2g}$ symmetry at 2 GPa and 3 K in the coexisting region of charge-density-wave and superconducting orders. The $★$ designs a CDW-phonon mode. Consistently with the theory, the two collective excitations at 14 and 40 ${\mathrm{cm}}^{-1}$ at 2 GPa are assigned to the superconducting Higgs mode and the charge-density-wave amplitudon mode.

Figure 2

Collapse of the superconducting Higgs mode in the pure superconducting state of $2H-{\mathrm{NbSe}}_2$, measurements, and theoretical predictions. (a) Raman spectra in the $E_{2g}$ symmetry measured at various $(P,T)$ positions as identified in Fig. 1: in the coexisting SC+CDW (green) and pure CDW (blue) states at ambient pressure and in the pure SC (brown) and paramagnetic (red) states at high pressure. Both the CDW amplitudon and the SC Higgs modes disappear at high pressure. A small Cooper-pair- breaking peak remains at $2{\mathrm{\Delta }}_{\text{SC}}$. $2{\mathrm{\Delta }}_{\text{SC}}$ is marked by the gray band ranging from $2{\mathrm{\Delta }}_{\text{SC}}$ measured by STM [39] at ambient pressure to the value we extrapolate at high pressure accordingly to the increase of $T_c$ with pressure [26, 36]. Inset: Raman spectra in the pure SC state of $2H-{\mathrm{NbSe}}_2$ (above 4 GPa) and non-CDW ${\mathrm{NbS}}_2$ (0 GPa) versus the Raman shift normalized to $2{\mathrm{\Delta }}_{\text{SC}}$ [39, 40]. (b) Theoretical Raman responses calculated in a microscopic model (see text and the Appendix pp1) in the four phases (SC+CDW, CDW, SC, paramagnetic (PM)) for comparison with the experimental spectra in (a). $t$ is the hopping term. The parameters are ${\mathrm{\Delta }}_{\text{SC}}/t=0.025, g_{\text{CDW}}/t=0.14$ and 0.12 in the SC+CDW and SC phases, respectively. The spectra are well reproduced in all phases. (c) Raman response of $2H-{\mathrm{NbSe}}_2$ in the pure superconducting phase above $P_c$ in the $E_{2g}$ and $A_{1g}$ symmetries. The black line is the theoretical Raman response of a Cooper-pair-breaking peak in a two-gap (or anisotropic gap) $s$-wave superconductor in the BCS regime with an additional electronic background $\beta$. The form of $\beta$ is $\beta \left(\omega \right)=a\omega /\sqrt{b+c{\omega }^2\phantom{{}_0^1}}$. It barely affects the shape of the Cooper-pair-breaking peak.

Figure 3

Pressure dependence of the Raman spectra of $2H-{\mathrm{NbSe}}_2$ in the coexisting superconducting and charge-density-wave phases, experiments, and theory. Raman response at 8 K [(b),(c)] in the CDW state and at 3 K [(e),(f)] in the SC+CDW state for various pressures up to the critical pressure. The spectra are normalized to the $E_{2g}$ phonon and consecutively shifted up. (a),(d) Theoretical Raman response computed microscopically, with frequency given in units of the hopping parameter $t$, which sets the energy scale. The pressure dependence is simulated by suppressing the CDW coupling $g_{\text{CDW}}$ with respect to its value $g_{\text{CDW}}^0$ at ambient pressure, with $\alpha =2(g_{\text{CDW}}^0-g_{\text{CDW}})/g_{\text{CDW}}^0$. The experiments at a given $P/P_c$ are compared to calculations at $\alpha /{\alpha }_c$, where ${\alpha }_c=0.30$ is the critical coupling at which CDW order disappears.

Figure 4

(a) Pressure evolution of the energy of the amplitudon in the $E_{2g}$ symmetry above and below $T_c$ compared to the evolution of the amplitudon in the microscopic model. Inset: Evolution of the energy of the folded CDW phonon, denoted with * (190 ${\mathrm{cm}}^{-1}$) in the spectra of Fig. 1, measured at 8 K. (b) Pressure evolution of the width of the amplitudon in the $E_{2g}$ symmetry above and below $T_c$ compared to the evolution of the amplitudon in the microscopic model. (c) Pressure dependence of the energy of the Higgs mode normalized to its value at zero pressure in the two $A_{1g}$ and $E_{2g}$ channels, and in the microscopic model. From 0 to $P_c=4\mathrm{GPa}$, the $A_{1g}$ mode softens, qualitatively following the behavior of the Higgs mode in the microscopic model, whereas the $E_{2g}$ one is constant, showing strong symmetry-dependent behavior. Inset: Pressure dependence of the Higgs mode in both symmetries compared to the pair-breaking threshold $2{\mathrm{\Delta }}_{\text{SC}}$, with ${\mathrm{\Delta }}_{\text{SC}}$ extrapolated from STM measurements [39] at ambient pressure scaled with the pressure evolution of $T_c$ [26, 36].

Figure 5

Spectral weight transfer between the superconducting mode and the charge-density-wave mode. (a) Raman spectra of $2H-{\mathrm{NbSe}}_2$ at 3.2 GPa above and below $T_c$ in the $E_{2g}$ symmetry. The spectral weight transfer is depicted in green. (b),(c) Difference (in percent) of the total spectral weight of both modes between 3 and 8 K (below/above $T_c$) as a function of pressure in $E_{2g}$ (b) and $A_{1g}$ (c) symmetries. At every pressure and in both symmetries, the total spectral weight of the Higgs mode and the amplitudon is relatively conserved ($\sim 20\%$) within the large error bars.

Figure 6

Change in the phonon spectral function in the transition from the CDW (solid red line) to the superconducting state with (solid blue line) and without (dashed blue line) coupling to the Higgs. The bare phonon energy is taken at ${\omega }_0=0.16t=3.2\left(2{\mathrm{\Delta }}_{\text{SC}}\right)$, so the softening of the phonon frequency from ${\omega }_0$ to a value ${\mathrm{\Omega }}_0$ of the order of $2{\mathrm{\Delta }}_{\text{SC}}$ is due to the coupling to the CDW amplitude fluctuations, encoded in self-energy (A7). Below $T_c$ the bare self-energy (A7) is weakly affected (dashed blue line) by the superconducting gap opening, due to the fact that the electronic excitations were already gapped by the CDW gap. However, the coupling to the Higgs encoded in the full self-energy (A9) leads to a sharp additional subgap peak, and to a hardening and broadening of the phonon spectral function.