Evidence of a coupled electron-phonon liquid in NbGe2

Whereas electron-phonon scattering relaxes the electron’s momentum in metals, a perpetual exchange of momentum between phonons and electrons may conserve total momentum and lead to a coupled electron-phonon liquid. Such a phase of matter could be a platform for observing electron hydrodynamics. Here we present evidence of an electron-phonon liquid in the transition metal ditetrelide, NbGe2, from three different experiments. First, quantum oscillations reveal an enhanced quasiparticle mass, which is unexpected in NbGe2 with weak electron-electron correlations, hence pointing at electron-phonon interactions. Second, resistivity measurements exhibit a discrepancy between the experimental data and standard Fermi liquid calculations. Third, Raman scattering shows anomalous temperature dependences of the phonon linewidths that fit an empirical model based on phonon-electron coupling. We discuss structural factors, such as chiral symmetry, short metallic bonds, and a low-symmetry coordination environment as potential design principles for materials with coupled electron-phonon liquid.

obtain the FFT spectra in Supplementary Fig. 1b,c. NbGe 2 has both large orbits with frequencies more than 2 kT and small orbits with frequencies less than 1 kT. The thermal damping of FFT peaks fit to a Lifshitz-Kosevich (LK) formula 2      The resistivity data in the main Fig. 2 text and Supplementary Fig. 3 were obtained from single crystals with hand-made contacts. Thus, there is an uncertainty in the geometric factor that converts electrical resistance to resistivity (width×height/length). To ensure this uncertainty is not the source of discrepancy between theory and experiment, we have also measured resistivity on mesoscopic devices made by FIB technology (Supplementary Fig. 4) with geometric errors less than 5% (i.e. ∆L/L, ∆W/W , ∆t/t <0.05). The comparison between the data from conventional contacts and mesoscopic device in Supplementary Fig. 4 shows an agreement within the error bars and confirms the discrepancy between the theoretical results of Ref. 5 and our experiments.

Supplementary Note 4: Phonon drag and heat capacity
In the main text (Fig. 2c), we showed that a phonon-drag model gives the best fit to the resistivity data. This is confirmed in Supplementary Fig. 5a which is a semilog (Arrhenius) plot of the resistivity versus temperature. We also explained in the main text that the coefficients of a power-law fit, ρ xx = ρ 0 + AT 2 + BT 5 , do not make physical sense when combined with the Sommerfeld coefficient γ from the heat capacity data to compute the Kadowaki-Woods ratio R KW = A γ 2 . Figure 5b shows the results of our heat capacity measurements. By fitting the low-temperature data to a Sommerfeld-Debye model C/T = γ + βT 2 (inset of Supplementary Fig. 5b), we evaluate the Sommerfeld coefficient γ = 6.18 mJ mol −1 K −2 and Debye temperature Θ D = 433 K (us- . A small γ is consistent with NbGe 2 being a non-magnetic system with weak electronic correlations. Using k B Θ D =hck D , we also evaluate the sound velocity c = 5292 m/s.

Supplementary Note 6: Carrier Concentration and Mobility
By performing a multiband fit to the longitudinal and transverse resistivity data (ρ xx and ρ xy ), we estimate the electron and hole concentrations in excess of 10 21 cm −3 in NbGe 2 (Supplementary Table 3). Thus, NbGe 2 is classified as a metal, not a semimetal as mentioned in previous works 5 .
To extract phenomenological carrier concentrations and mobilities of NbGe 2 , we fit a multiband model with the following expressions to the transport data 6 : where σ i = n i eµ i is the conductivity of the band i, n i is carrier concentration, µ i is mobility, and the summation i runs over all the bands considered. We are assuming a minimal model with two electron bands and one hole band, and fit both ρ xx and ρ xy data to the above expressions simultaneously. Figure 6 shows a decent agreement between the model and experimental data, and the high carrier concentrations and mobilities (Supplementary Table 3.) are consistent with the Supplementary which are fitting coefficients of the three-band model ( Supplementary Fig. 6), are reported here.

Supplementary Note 7: Sample quality
NbGe 2 crystals were grown using a chemical vapor transport (CVT) method. We have improved the sample quality iteratively by changing the amount of transport agent (iodine), the tube length, the heating sequence, and the magnitude and direction of temperature gradient. The resistivity data from 9 samples are presented in (Supplementary Fig. 7) with increasingly larger residual resistivity ratio (RRR) from S1 to S9. Although several parameters can be tuned during a CVT growth, we found that the most efficient way of improving sample quality was by reducing the temperature gradient to be less than 10 • C while the furnace hot zone was at 900 • C or higher. For the best NbGe 2 sample (S9) used in this study, the details of the growth is as follows: Nb and Ge powders were mixed and ground with the ratio Nb:Ge=1:2. 500 mg of the powder mixture was transferred to a 3.5-inch long, small-sized silica tube. 10 mg of I 2 was added, and then the tube was sealed under vacuum. The tube was placed in a box furnace, heated up to 900 • C at 3 • C/min, and dwelled at that temperature for 1 month.
We performed energy dispersive X-ray spectroscopy (EDX) on most of the samples in Supplementary Fig. 7 and summarized the results in Supplementary  Figure 7: Sample quality. Normalized resistance R/R(300K) is plotted as a function of temperature for 9 samples grown under slightly different conditions. The residual resistance is systematically reduced by improving sample quality. This is reflected in the increasing residual resistivity ratio (RRR) from S1 to S9.
tary Fig. 7 are due to varying degrees of extended defects (such as dislocations) instead of local defects (e.g. vacanies and inter-site disorder). Supplementary