Introduction

Semimetals provide a fertile ground for investigating the interplay between topology, superconductivity, and electron-phonon interactions. Starting from the discovery of a Dirac semimetallic state in graphene1,2, the study of semimetals has grown to include ever more exotic flavors of topological states in materials with Dirac or Weyl crossings in their band structures3,4,5. Many topological semimetals are simultaneously found to exhibit remarkable transport properties including extremely large and non-saturating magnetoresistances and ultrahigh carrier mobilities6,7,8. In certain semimetals, these transport properties have been definitively linked with strong electron-phonon interactions, including WP29,10, PtSn411,12, and NbGe213,14, all of which also manifest interesting topological states.

One particularly interesting class of semimetals are those with broken inversion symmetry. In metals, broken inversion symmetry leads to electronic and magnetic couplings that are forbidden in centrosymmetric materials, thereby promoting exotic states. One such example is the effect of antisymmetric SOC on superconductivity. ASOC results in the splitting of spin-degenerate bands, where the energy splitting is proportional to the degree of SOC of the constituent elements. When the ASOC is sufficiently strong, a mixture of spin singlet and spin triplet pairings will occur yielding an unconventional superconducting state15,16,17,18,19.

The rich physics associated with broken inversion symmetry has led to significant interest in the RTX3 (R = rare earth, T = transition metal, \(X={\rm{Ge}},{\rm{Si}},{\rm{Sn}}\)) family of materials, which crystallize in the non-centrosymmetric I4mm space group. Materials in this family have been known to exhibit heavy fermion behavior, complex magnetic states, and unconventional superconductivity20,21,22,23,24,25,26,27,28. They have also attracted attention in light of their ASOC split Fermi-surfaces26. It has been demonstrated in this class of materials that the transition element T plays a crucial role in the splitting of the Fermi-surface, while the R and X elements play a much smaller role26.

Here, we report a comprehensive characterization of one member of the non-centrosymmetric RTX3 family, LaRhGe3, which has significant antisymmetric SOC due to the presence of Rh on the T site. We synthesize single crystals of LaRhGe3 using a self-flux method for the first time and characterize its low temperature electronic properties via transport, specific heat, muon spin relaxation, and Raman spectroscopy. This ensemble of measurements reveals a rich array of properties in LaRhGe3 including semimetallicity, superconductivity, and electron-phonon drag. Our electronic structure calculations further reveal topological features of interest, with the discovery of almost movable Weyl nodal lines. This set of results on LaRhGe3 highlights the important role of electron-phonon coupling and paves the way for further exploration of the I4mm class of non-centrosymmetric materials.

Results and discussion

Non-centrosymmetric structure of LaRhGe3

Single crystals of LaRhGe3 were grown by the metallic self-flux method29. To confirm the structural characteristics and crystalline quality of our samples, we performed powder X-ray diffraction (XRD) and energy dispersive X-ray spectroscopy (EDX). The measured powder XRD pattern, given by the blue circles in Fig. 1(a), is consistent with the previously reported body-centered tetragonal I4mm (no. 107) crystal structure30. Our Rietveld refinement, shown by the black line in Fig. 1(a), confirms the absence of any significant secondary phases. A minor germanium phase (1.3% by weight based on Rietveld refinement), likely originating from the residual flux on the surface of the crystals, is marked by the orange asterisks. No other impurity phases above the detection limit of our instrument were detected. The fitted lattice parameters for LaRhGe3 are a = 4.4187(2) Å and c = 10.0494(5) Å. The full results of the Rietveld refinement are presented in Supplementary Tables 1 and 2.

Fig. 1: Structural characterization of LaRhGe3.
Fig. 1: Structural characterization of LaRhGe3.The alternative text for this image may have been generated using AI.
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a Rietveld refinement of the powder X-ray diffraction pattern of LaRhGe3 measured at room temperature. The Bragg peak positions for the I4mm space group are indicated by the gray crosses. A small germanium impurity (1.3% by weight) peak is marked with an orange asterisk. b The non-centrosymmetric body-centered tetragonal crystal structure of LaRhGe3, where La is shown in green, Rh in red, and the two Ge sites are shown in purple and blue, respectively. The broken inversion symmetry is most easily appreciated by considering the positions of Rh within the unit cell. c Scanning electron microscopy image of a single crystal of LaRhGe3 (top left) and elemental mapping on the same crystal for La (top right), Rh (bottom left), and Ge (bottom right). Intensity of colors in the elemental mapping corresponds to the intensity of peaks related to each element, showing uniform distribution of the elements across the crystal.

The spatial homogeneity of our crystals is confirmed by EDX mapping, which shows a uniform distribution of La, Rh and Ge on the sample surface (Fig. 1c). A ratio for La:Rh:Ge of 1.00:0.98:2.90 is found based on EDX analysis, which is close to the expected 1:1:3 ratio. Minor Ge-rich regions can be attributed to residual flux on the crystal’s surface, as also detected in the powder XRD measurement.

The non-centrosymmetric crystal structure of LaRhGe3 is presented in Fig. 1b. The broken inversion symmetry can be easily discerned by considering the positions of Rh within the unit cell, where Rh sits at the center of the ab-plane with the lower half of the unit cell and at the unit cell edges in the upper half of the unit cell. There are two unique Ge sites in this structure, with the Ge2 site having double the multiplicity of Ge1. The smallest interatomic distances in this structure are between Rh and Ge in a distorted corner-sharing square pyramidal configuration, where the single Rh-Ge1 distance is 2.377(3) Å and the four-fold Rh-Ge2 distance is 2.436(7) Å. We focus next on the implication of the tetragonal I4mm space group with broken inversion symmetry on the electronic structure of LaRhGe3.

Electronic band structure and protected nodal lines in LaRhGe3

We calculate the band structure of LaRhGe3 using density functional theory (DFT). The band structure, which is plotted along the high symmetry directions of the body-centered tetragonal Brillouin zone (Fig. 2), is plotted according to each atomic site (La, Rh, Ge1, and Ge2), where the thickness and intensity of the color represent the weighted atomic contribution. All four atomic sites contribute to the bands closest to the Fermi level. The largest contribution to all the bands that cross the Fermi energy originates from Rh, while Ge1 contributes to the hole bands, and Ge2 contributes to the electron bands. Due to the presence of ASOC, the spin degeneracy of all bands is lifted. The ASOC in LaRhGe3 was thoroughly experimentally characterized as part of a larger study on RTX3 materials crystallizing in the I4mm space group26. The degree of SOC on the transition element T was shown to dictate the degree of splitting of spin-up and spin-down bands, which is enabled by the lack of inversion symmetry in the crystal structure. A significant splitting of the bands is realized in the case of Rh on the T site26, which is confirmed in our de Haas-van Alphen (dHvA) oscillations, shown in Supplimentary Fig. 1. We compare our the splitting in the FFT of the dHvA signal to those previously reported26 in Supplementary Table 3, where we find a consistent degree of band splitting in our measurements that confirms a strong ASOC. We will discuss the implications of this strong ASOC on the other observed electronic properties of LaRhGe3 in the sections ahead.

Fig. 2: Ab-initio electronic band structures in the primitive unit cell of LaRhGe3.
Fig. 2: Ab-initio electronic band structures in the primitive unit cell of LaRhGe3.The alternative text for this image may have been generated using AI.
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The width and intensity of the superimposed colors indicate the contribution to the states from the respective atoms. The dotted horizontal black line marks the Fermi level. An onsite Hubbard interaction of ULDA+U = 14.5 eV was added to the f-orbitals of La to push the (unoccupied) states out of the energy window.

We next highlight aspects of symmetry-enforced band crossings in LaRhGe3. Group theoretical analysis recently predicted almost movable nodal lines in a number of tetragonal space groups including the I4mm group to which LaRhGe3 belongs31. Their specific dispersion relation, connectivity, and proximity to the Fermi level are not fully fixed by symmetry alone and are instead found to be material-dependent. In the case of LaRhGe3, even though the global Kramers’ degeneracy is lifted, the I4mm space group symmetry enforces two types of two-fold degenerate nodal lines throughout the Brillouin zone. Consider the mirror plane that intersects the rotational axis (Γ - Z) and the N-point. The eigenvalues of the mirror symmetry at N are  ± i. Because N is a time-reversal invariant momentum (TRIM) point, the  +i and  −i eigenvalues are paired, enforcing a two-fold degeneracy at this point. Let k1 and k2 be two k-points on the mirror plane that are related by time reversal. Because they are time reversal partners, their mirror eigenvalues have opposite phases. Therefore, along an arbitrary path on the mirror plane that connects k1 and k2 (which does not cross a TRIM point), the eigenvalues must be exchanged. Since this can only happen at band crossings and we have the freedom to choose any path, the intersections must form a nodal line that connects the N-points. That type of degeneracy is called an almost movable nodal line31. It is only pinned at the two TRIMs that it connects and, in contrast to a point-like topological band feature (e.g., a Weyl point), it is extended in both k-space and energy. The same argument holds for the Γ- and Z-point. However, the presence of additional symmetry pins the nodal line to the rotational axis. Hence, there are two ways the almost movable nodal line can connect the N-points: It either intersects the pinned nodal line (Γ - Z) or it does not.

In Fig. 3a, b the two types of connectivity are shown. The intensity of the heatmap corresponds to a quantity similar to a spectral function that has poles at k-points where a chosen pair of bands becomes degenerate. The overlay shows the body-centered primitive Brillouin zone (type 2) and the high-symmetry points that lie within the plane spanned by Γ-Z and Γ-N. The almost movable nodal lines connect the N-points on (seemingly arbitrary) paths that depend on the specific dispersion relation of the chosen pair of bands. Thus, the evolution of the almost movable nodal lines through k-space for different band pairs varies drastically. Figure 3c shows the dispersion relation along the different nodal lines. The pinned and the almost movable nodal lines shown in Fig. 3a are close to the Fermi energy and can potentially be tuned to the Fermi edge by either doping or applying a gate voltage. Due to the dispersive nature of the nodal lines, there is a large energy region in which the degeneracies can be accessed in contrast to point-like topological band crossing, and thus a wider range of parameters in which drumhead surface states can be found32,33. Even though surface states do not necessarily emerge in nodal line systems34, any curve through k-space that encircles a single nodal line has a Berry curvature that is quantized to π by the mirror symmetry. A suitable surface termination can likely be found for which gapless surface states appear at k-points of the two-dimensional surface Brillouin zone that connect projections of the different nodal lines in the bulk.

Fig. 3: Almost movable nodal lines in LaRhGe3.
Fig. 3: Almost movable nodal lines in LaRhGe3.The alternative text for this image may have been generated using AI.
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Pinned (Γ to Z, black) and almost movable nodal lines (N to N, (a) blue and (b) red) in the mirror plane of the Brillouin zone of the type 2 body centered tetragonal primitive unit cell. The projection of the Brillouin zone onto this mirror plane, as well as the high-symmetry points that lie therein are given in the overlay. The connectivity of almost movable nodal lines is not completely determined by the space group symmetry. The two possibilities are both realized in the material for different pairs of bands. a The almost movable nodal line connecting two N-points. b Same as (a), but intersecting the pinned nodal line. c The dispersion along curves \({\mathcal{C}}\) through k-space along the pinned nodal line from (a) (black), the almost movable nodal line from (a) (blue), and the almost movable nodal line from (b) (red). t parameterizes the curve with \({\mathcal{C}}({t}_{i})\) and \({\mathcal{C}}({t}_{f})\) being the initial and final point of the curves respectively. The pinned and the almost movable lines from (a) are close to the Fermi level.

Materials with these gapless and robust surface modes are highly sought-after, due to the variety of proposals in novel device applications3,35,36,37,38,39,40,41. In realistic materials, disorder and defects weakly break the crystal symmetries that protect both the nodal lines and their edge modes. However, as long as the symmetries are approximate and the effect of the perturbation is small compared to the topological gap, the edge modes remain present42,43. Interaction effects might also weakly break symmetries, but more importantly they generically obstruct an exact description of a material’s electronic structure using single-particle states, which the definition of most topological invariants relies on. Nevertheless, it has been shown that an interacting system can always be adiabatically connected to a noninteracting system with the same topology, as long as the interaction does not drive a phase transition44,45,46,47. Even nonlinear perturbations do not generically suppress edge modes48, and their stability in nodal line semimetals in their superconducting state has been theoretically demonstrated49. Therefore, realizing superconducting state, even a conventional one mediated by electron-phonon coupling, in the bulk of a nodal line semimetal can allow for exotic pairing on the surface due to interaction with the topological surface state.

Semimetallicity in LaRhGe3

We next highlight the semimetallic nature of LaRhGe3 experimentally, which, along with our nodal-line predictions, demonstrate that it is a Weyl semimetal. Electrical transport measurements confirm the metallic nature of this material, with a longitudinal resistivity that monotonically decreases with decreasing temperature and resistivity magnitudes in the μΩ-cm range across all temperatures (Fig. 4a). The high residual resistivity ratio, ρ300K/ρ2K = 110, is suggestive of the good crystalline quality of our samples. As a function of increasing applied magnetic field, there is a rapid increase in the low temperature resistivity, below 50 K. For fields larger than 6 T, the resistivity reaches a minimum value before a low temperature upturn and then ultimately plateaus for temperatures below 10 K. In Fig. 4b, we plot the magnetoresistance (MR) of LaRhGe3 at various temperatures, where MR is defined as [ρxx(B) − ρxx(0)]/ρxx(0). The MR remains unsaturated in fields up to 14 T, where it reaches a value of 30 at 1.8 K. The large and unsaturated MR and the field-driven resistivity upturn in LaRhGe3 are characteristic of a compensated semimetal50,51,52. We perform a scaling analysis of the MR over the various temperatures measured, as shown in Supplementary Fig. 2, and find good agreement with Kohler’s rule.

Fig. 4: Semimetallic electronic properties of LaRhGe3.
Fig. 4: Semimetallic electronic properties of LaRhGe3.The alternative text for this image may have been generated using AI.
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a Temperature dependence of the longitudinal resistivity ρxx measured in different magnetic fields, showing sizable magnetoresistance below 50 K. b Magnetic field dependence of the resistance (MR) as defined in text measured at various temperatures, reaching a value of 30 by 14 T at 1.8 K. c Hall resistivity ρxx measured at various temperatures, showing a change of sign upon cooling below 50 K. d Electron and hole carrier densities, (e) their ratio, and (f) carrier mobilities extracted from fits of MR and ρxy to a two-band model53, revealing LaRhGe3 to be a compensated semimetal at low temperatures.

To verify the semimetallicity scenario, we measured the Hall resistivity ρxy of LaRhGe3, shown in Fig. 4c, at the same fields and temperatures as the MR. We find that ρxy(B) is non-linear for all temperatures, and hence it cannot be fitted with a semi-classical one-band model for which ρxx would be independent of field, also in disagreement with our data. This is expected if we examine the band structure of LaRhGe3, shown in Fig. 2, where we find three pairs of spin-split bands crossing the Fermi level, where two pairs are hole bands and one pair is an electron band. While the electron Fermi surfaces are larger than the hole Fermi surfaces, we fitted both the MR and Hall resistivity to a semi-classical two-band model53 that assumes density and mobility of all hole and electron carriers can each be captured with a single term:

$${\rho }_{{\rm{xx}}}=\frac{1}{e}\frac{({n}_{{\rm{h}}}{\mu }_{{\rm{h}}}+{n}_{{\rm{e}}}{\mu }_{{\rm{e}}})+({n}_{{\rm{h}}}{\mu }_{{\rm{e}}}+{n}_{{\rm{e}}}{\mu }_{{\rm{h}}}){\mu }_{{\rm{h}}}{\mu }_{{\rm{e}}}{B}^{2}}{{({n}_{{\rm{h}}}{\mu }_{{\rm{h}}}+{n}_{{\rm{e}}}{\mu }_{{\rm{e}}})}^{2}+{({n}_{{\rm{h}}}-{n}_{{\rm{e}}})}^{2}{\mu }_{{\rm{h}}}^{2}{\mu }_{{\rm{e}}}^{2}{B}^{2}}$$
(1)
$${\rho }_{{\rm{xy}}}=\frac{B}{e}\frac{{n}_{{\rm{h}}}{\mu }_{{\rm{h}}}^{2}-{n}_{{\rm{e}}}{\mu }_{{\rm{e}}}^{2}+({n}_{{\rm{h}}}-{n}_{{\rm{e}}}){\mu }_{{\rm{h}}}^{2}{\mu }_{{\rm{e}}}^{2}{B}^{2}}{{({n}_{{\rm{h}}}{\mu }_{{\rm{h}}}+{n}_{{\rm{e}}}{\mu }_{{\rm{e}}})}^{2}+{({n}_{{\rm{h}}}-{n}_{{\rm{e}}})}^{2}{\mu }_{{\rm{h}}}^{2}{\mu }_{{\rm{e}}}^{2}{B}^{2}}.$$
(2)

The fitting parameters in this model are the carrier densities, ne and nh, and the carrier mobilities, μe and μh, for electrons and holes, respectively. We simultaneously fit ρxx and ρxy to this two-band model to extract ne,h and μe,h as a function of temperature. The carrier densities, plotted in Fig. 4d, are on the order of 1021 cm−3, a high carrier density semimetal, such as HfB254 and LaAlGe55 with similar carrier densities.

At high temperatures, we find that holes are the majority charge carriers but when the temperature falls below 40 K, the carrier densities are essentially compensated with ne/nh = 0.99(1) at 2 K, as can be seen in Fig. 4e. For both holes and electrons, the mobility increases with decreasing temperature, as can be expected due to reduced scattering of charge carriers by phonons. At high temperature the electron mobility is larger than the hole mobility, as can be seen in Fig. 4f. We should note, however, that the error bars in this fitting range are large due to the small magnetoresistance above 100 K and the limitations of the two band model. Upon cooling, the hole mobility increases more rapidly and exceeds the electron mobility for temperatures where the carriers are compensated, below 40 K, explaining the sign change observed in ρxy in the same temperature range. The high hole mobility of 0.3 m2/Vs at low temperature again demonstrates the good crystal quality, which is also relevant to the non-saturating MR of LaRhGe3 and the electron-phonon interactions discussed in “Electron-Phonon Drag in LaRhGe3”. We have experimentally verified the semimetallic nature of LaRhGe3, and in the next section demonstrate the discovery of superconductivity in this Weyl semimetal.

Superconductivity in LaRhGe3

Our first indication of superconductivity in LaRhGe3 comes from low temperature specific heat measurements. We observe a sharp lambda-like anomaly at Tc = 0.39(1) K as can be seen in the electronic part of the specific heat Cel, as shown in Fig. 5a, where the lattice contribution has been subtracted by fitting the zero-field data above the transition. Upon application of small magnetic fields, the transition is suppressed to lower temperatures. We find that the transition remains sharp for applied fields of 0.5 mT and 1 mT, despite the decrease in Tc, which is consistent with the first order field-induced suppression of superconductivity expected for a type-I superconductor56,57,58 and comparable to what has previously been observed in LaRhSi359. Upon application of a 2 mT field, the superconducting transition is fully suppressed in our measurement window, which extends down to 0.1 K. The magnitude of the specific heat jump, ΔC/Cen ~ 1.43, as shown in Supplementary Fig. 3a, is consistent with a BCS model in the weak-coupling limit. CP data before subtraction of the phonon part are shown in Supplementary Fig. 4. Further analysis of this specific heat data, as described in the Supplementary Materials, allows a number of superconducting parameters to be derived, and these are presented in Table 1.

Fig. 5: Superconducting transition in LaRhGe3.
Fig. 5: Superconducting transition in LaRhGe3.The alternative text for this image may have been generated using AI.
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a Temperature dependence of Cel/T measured in varying magnetic fields. The superconducting transition seen at 0.39(1) K in 0 mT is suppressed with increasing magnetic field. b Representative transverse field muon decay asymmetry spectra collected above and below Tc in LaRhGe3 under an applied magnetic field of 1.0 mT. c The oscillating and non-oscillating amplitude of the signal in (b) as a function of temperature measured in zero-field cooling (ZFC) and field cooling (FC) protocol. Reduction in the oscillating amplitude is due to expulsion of magnetic field in the superconducting state, and the ZFC and FC show similar suppression, demonstrating type-I superconductivity. d Field dependence of the Tc values extracted from CP and transverse field μSR. The solid line is a fit to the expression, Eq. 7, in the text to estimate Bc(0) = 2.2 mT. e Zero field μSR spectra collected at 539 mK and 20 mK, with fit using Kubo-Toyabe function for the data. No significant change is seen between the data above and below Tc. Inset shows the resulting relaxation rate Λ. f Plot of Θc against measured Tc for various elemental type-I superconductors, LaRhSi3, and LaRhGe3. LaRhGe3 falls on the black solid line, which represents the expectation based on weak-coupling BCS theory.

Table 1 Superconducting parameters of LaRhGe3
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We next present muon spin relaxation and rotation (μSR) measurements on LaRhGe3. Muons are highly sensitive to their local magnetic environment, making them an excellent probe of field expulsion in superconductors and spontaneous time-reversal symmetry breaking. We begin by investigating the former, where the muon decay asymmetry plotted as a function of time in an applied field of 1 mT (which is below the critical field) is shown in Fig. 5b. In this geometry, the direction of the magnetic field is transverse to the muon spin. Above the superconducting transition, at 385 mK, the muons that land in the sample or the sample holder will couple to the applied field and undergo Larmor precession, yielding an oscillatory decay asymmetry. Whereas, within the superconducting state at 70 mK, the magnetic field is expelled and the muons that land in the sample will not precess, giving rise to a flat decay asymmetry. The remnant oscillatory signal originates from the fraction of muons that land outside the superconducting sample. We fit the asymmetry spectra using a two-term asymmetry function

$$\begin{array}{ll}{G}_{{\rm{TF}}}(t)&=A\left[f\exp \left(-\lambda t\right)\cos \left({\omega }_{1}t+\phi \right)\right.\\ &\left.+\,(1-f)\left(\frac{1}{3}+\frac{2}{3}\left(1-{\sigma }^{2}{t}^{2}\right)\exp \left(-\frac{{\sigma }^{2}{t}^{2}}{2}\right)\right)\right]\end{array}$$
(3)

where the first term captures the signal from muons stopping in the silver sample holder and non-superconducting part of the sample and the second term captures the signal from muons stopping in the superconducting part of the sample. The fraction of signal from muons that land in silver sample holder and non-superconducting part of the sample is given by f, and ω1 is the muon precession frequencies in the background. The A term is the total asymmetry and the ϕ is the initial phase of the muons. The depolarization rates for the sample and the background signals, respectively, are given by σ and λ.

A global fit was used for all variables, but the fraction f was allowed to be refined with temperature. Our fit yields ω1 = 0.13719 MHz, ϕ = 3.59 rad, and λ = 0.0161 μs−1. σ = 0.1272 μs−1 was fixed based on fit to zero-field μSR data. The value of f as a function of temperature under both ZFC and FC conditions is shown in Fig. 5c. The 100% oscillating amplitude corresponds to all muons stopping inside LaRhGe3 above Tc and the non-superconducting silver plate. A sharp drop in the oscillating amplitude is observed at the superconducting transition. This behavior is expected in a superconductor due to the expulsion of the magnetic field from the bulk of the sample in the superconducting state such that muons implanted sufficiently deeply in the sample will not be affected by the applied field and will therefore not precess. We estimate about 40% of the signal amplitude corresponds to the sample based on the sample coverage, so the drop to about 60% corresponds to the expulsion of the field from most of the sample. We show a shift in the amplitude from the oscillating component to the zero-field in Supplementary Fig. 5. From these transverse field μSR measurements, we estimate Tc ~ 0.28 K in an applied field of 1 mT, in good agreement with the Tc measured in specific heat at 1 mT. We find that the suppression of the amplitude is approximately equivalent under ZFC and FC conditions at the lowest temperatures, which further supports the conclusion that LaRhGe3 is a Type-I superconductor.

The temperature and field dependence of Tc extracted from specific heat and transverse field μSR are plotted in Fig. 5d. We estimate the zero-temperature critical field Bc(0) using the conventional relationship, \({B}_{c}(T)={B}_{c}(0)[1-(T/T_{c})^{2}]\,\), where Tc is 0.39(1) K. This fit yields the curve shown by the black line and a Bc(0) of 2.2 mT. The small critical field value makes the presence of a remnant magnetic field during the CP measurement a serious concern that can shift our measured Tc. However, to address this concern we highlight the good agreement with the point on this phase diagram extracted from transverse field μSR measurement, shown in blue. This point is expected to be accurate due to the high-resolution field-zeroing protocol employed in that experiment.

Our attempts to measure the electric resistivity of LaRhGe3 in a dilution refrigerator reveal a drop in the resistivity at a temperature that is commensurate with the onset of superconductivity, as shown in Supplementary Fig. 4b. This resistance drop is partially suppressed by an applied magnetic field of 1.0 mT. However, even in the zero field measurement, we detect a finite resistance in the superconducting state, which we ascribe to the applied current exceeding the critical current threshold of LaRhGe3. The low critical current suggested by our resistivity measurements is not particularly surprising given the extremely low critical field of 2.2 mT.

Finally, we performed zero field μSR measurements to look for any sign of spontaneous magnetic fields and time-reversal symmetry breaking as LaRhGe3 enters its superconducting state. The magnetic field at the sample position was carefully zeroed following the protocol described in the methods. High statistics zero field spectra were collected above and below Tc with representative spectra shown in Fig. 5e. In the absence of magnetism, the muon depolarization is due to randomly oriented nuclear moments and can be described by the Kubo-Toyabe function

$$\begin{array}{l}{G}_{{\rm{KT}}}(t)=\frac{1}{3}+\frac{2}{3}\left(1-{\sigma }^{2}{t}^{2}\right)\exp \left(-\frac{{\sigma }^{2}{t}^{2}}{2}\right)\end{array}$$
(4)

where σ accounts for the field generated by the randomly nuclear moments at the muon site. The relaxation spectra for LaRhGe3 is fitted to the function

$$\begin{array}{l}A(t)={A}_{S}{G}_{{\rm{KT}}}(t)\exp (-\Lambda t)+{A}_{{\rm{BG}}}\end{array}$$
(5)

where AS and ABG represent the sample and background asymmetry, respectively, while Λ accounts for any additional relaxation rate. Our fits reveal no significant change in Λ above and below Tc, as shown in the inset of Fig. 5e, confirming the absence of any significant TRS breaking in LaRhGe3. The contribution of nuclear dipoles to the σ in our fit is 0.1272 μs−1, and our calculated contribution is 0.120 μs−1 based on the muon stopping site (shown in Supplementary Fig. 5) and the abundance of isotopes with nuclear dipoles.

Type-I superconductivity is common among pure elements but relatively rare in compounds, particularly beyond binaries. LaRhGe3 is therefore one of very few59,60,61,62 type-I superconductors containing more than two elements. In Fig. 5f, we compare the calculated critical temperature, denoted as θc, based on the γn and Bc(0) values63, with the measured Tc for various elemental type-I superconductors, LaRhSi359, and LaRhGe3. We find that LaRhGe3, as well as LaRhSi3, lie close to the solid-black line that represents the expectation based on weak-coupling BCS, further confirming that LaRhGe3 is a type-I superconductor in the weak-coupling limit. Nevertheless, we expect topologically protected states on the surface of LaRhGe3, as described in “Electronic Band Structure and Protected Nodal Lines in LaRhGe3” may allow for novel pairing on the surface despite the conventional nature of the bulk. Next we highlight electron-phonon interactions in the normal state of LaRhGe3, which, unlike the superconducting state, hint at strong coupling.

Electron-Phonon Drag in LaRhGe3

Finally, we return to the normal state of LaRhGe3 to investigate the thermal properties of this material. Our first indication of unusual transport behavior comes from zero-field resistivity where we observe a deviation from a standard Fermi liquid behavior as shown in Fig. 6a. Even using a modified Bloch-Grüneisen model ρ = ρ0 + AT2 + BT5, where the T5 term accounts for electron-phonon interactions and the additional T2 term accounts for electron-electron interactions, a poor fit is obtained below 50 K. Instead, we find that the resistivity data are better described by the exponential form of a phonon-drag model, \(\rho ={\rho }_{0}+C{e}^{-{T}_{0}/T}\)13,64, where the activation temperature is T0 = 110 K. We find an equally good agreement with the data using a power law fit ρ = ρ0 + CTn, which yields n = 2.9, as shown in Supplementary Fig. 6. To further analyze the temperature dependence of the resistivity we take the logarithmic derivative after subtracting residual resistivity to quantify the exponent n = [δln(ρ − ρ0)]/[δlnT]. The temperature dependence of n is presented in Fig. 6b for two different samples, and we find at high temperature n ~ 1 as expected in a standard Block-Gruneisen picture. However, we find an upturn towards n ~ 3 at low temperatures, which is consistent with n = 2.9 from our power law fit.

Fig. 6: Phonon drag in LaRhGe3 detected by transport measurements.
Fig. 6: Phonon drag in LaRhGe3 detected by transport measurements.The alternative text for this image may have been generated using AI.
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a Resistivity as a function of temperature measured along ab-plane of LaRhGe3 in zero-field, and different models fitted to the data below 50 K. b n, the exponent of the temperature dependent resistivity determined derivative of ln(ρ − ρ0) vs. ln(T) as a function of temperature. Data for two samples, S1 and S2, are shown. c Temperature dependence of Seebeck coefficient over temperature (Sxx/T) as a function of temperature of LaRhGe3. Inset shows Sxx as a function of temperature. d Temperature dependence of thermal conductivity of LaRhGe3. Inset shows the Lorenz number as a function of temperature. The dashed vertical line in panels (b−d) marks the approximate phonon drag temperature.

Further evidence for the phonon-drag scenario comes from Seebeck coefficient measurements, shown in the inset of Fig. 6c, where we notice a maximum at around  ~50 K signaling the interaction between phonons and electrons is enhanced at low temperature. When scaled by temperature, Sxx/T, plotted in the main panel of Fig. 6c, the peak appears at a temperature of  ~30 K similar to the peak in the derivative of the resistivity. Coincident with these two features, a peak also appears below 50 K in the thermal conductivity, shown in Fig. 6d. Typically, the Lorenz number is evaluated by dividing the electronic part of the thermal conductivity by the electrical conductivity, but separating the phonon and electron parts of the thermal conductivity proved difficult with our current data. Instead, we show the total thermal conductivity divided by electrical conductivity, representing an upper limit of the Lorenz number, in the inset of Fig. 6d, which shows clear deviation at low temperature with a minimum around 30 K. It’s worth noting that no clear deviations in the scaling behavior of the MR is detected at this temperature, as shown in Supplementary Fig. 2.

The peaks in κxx, Sxx/T, and the resistivity derivative all appear close to  ~30 K, which we take as the phonon-drag temperature. Comparing this with the Debye temperature or 340 K, we find Θph-drag/ΘD ~ 10, which has been seen in other materials with phonon-drag. In “Semimetallicity in LaRhGe3”, we noted that the MR is significantly enhanced below 30 K due to electron-hole compensation. At the same temperature, we also observed a crossover of the carrier mobilities with the holes having higher mobilities in the low-temperature limit and both electron and hole mobilities plateauing in this temperature range. Therefore, the low-temperature increase in κxx and Sxx/T can be attributed to and electron-phonon interactions dominating over phonon-phonon interactions.

To confirm this scenario, we measured the Raman spectra of LaRhGe3 to search for signatures of phonon linewidth broadening due to electron-phonon coupling. Due to the lack of inversion symmetry, all 12 optical phonons in LaRhGe3 are Raman active, which we will denote by the irreducible representations of the C4v point group to 8 optical phonons, where 4 of them are doubly degenerate: 3A1 + B1 + 4E65. Using the Raman selection rules and polarization optics66, we identify the symmetry of each peaks at 121, 214, and 249 cm−1 as the three A1 phonons, the peak at 139 cm−1 is a B1 phonon, whereas the peaks at 106, 116, 163, and 262 cm−1 are the phonons with E symmetry (Supplementary Fig. 7). The Raman spectrum measured at 10 K without any analyzing polarizer is shown in Fig. 7a with all phonon modes labeled. We tabulate the measured phonon energies alongside the values calculated from DFT in Supplementary Table 5 to justify our mode assignment. We also notice that there are two unaccounted peaks at 153 and 189 cm−1, which we designate as phonon overtones, i.e., the simultaneous excitation of two phonons carrying opposite momenta. Our assignment is justified by the DFT calculated phonon (Fig. 7e), where the acoustic phonon branch has maximum energy around 80−90 cm−1.

Fig. 7: Electron-phonon drag in LaRhGe3 detected by Raman spectroscopy.
Fig. 7: Electron-phonon drag in LaRhGe3 detected by Raman spectroscopy.The alternative text for this image may have been generated using AI.
Full size image

a Raman spectra of LaRhGe3 measured at 10 K. The phonon mode symmetries are labeled in accordance with our DFT calculations. The strong peak at 300 cm−1 is due to Ge flux on the surface. b Black and white map of Raman scattering intensity plotted as a function of temperature and Raman shift measured from the (101) surface of LaRhGe3 crystal. c Full width at half maximum (FWHM) temperature dependence of \({A}_{1}^{(1)}\), \({A}_{1}^{(2)}\), \({A}_{1}^{(3)}\), and E(1) modes based on fits described in the text. Solid black lines correspond to fits to the temperature dependence of each phonon mode based on Eq. (6). d Percent difference of the phonon energies of A1 and E modes from energies at 10 K as a function of temperature. e Calculated phonon dispersion curves along high-symmetry lines of the tetragonal primitive unit cell of LaRhGe3. Anomalous temperature dependence of the \({A}_{1}^{(1)}\) phonon lifetime in LaRhGe3. f Schematic depicting excitation of electron-hole pairs by optical phonons, where ωa is the energy offest to take into account that the Fermi level is not exactly halfway between the initial state \(\left\vert i\right\rangle \) and final state \(\left\vert f\right\rangle \) and ω0 is the bare phonon energy.

The Raman spectra were measured between 10 and 300 K in 10 K intervals as shown in Fig. 7b and Suppplementary Fig. 8. In order to maximize the signal-to-noise ratio and track all phonon modes simultaneously, the temperature-dependent measurements were done without the analyzing polarizer. The phonon energies and full-width at half maxima (FWHM) were extracted by fitting to Lorentzian line shapes, as shown in Suppplementary Fig. 9 for spectra collected at 10 K and 300 K. The resulting temperature dependence of the FWHM and energy of select phonon modes are presented in Fig. 7c, d, respectively. At a qualitative level, we note that the energy shifts show a conventional shift to lower energies with increasing temperature for all modes. Meanwhile, a more complex behavior is uncovered for the FWHM, with some modes showing an unexpected narrowing with increasing temperature.

In most materials, the phonons acquire finite linewidths through three channels: impurity scattering (γ0), anharmonic phonon-phonon interaction (γ3p(T)), and electron-phonon coupling (γep(T))67,68:

$$\begin{array}{r}\gamma (T)={\gamma }_{0}+{\gamma }_{3p}[1+n({\omega }_{1},T)+n({\omega }_{0}-{\omega }_{1},T)]\\ +{\gamma }_{ep}[f({\omega }_{a}-{\omega }_{0}/2)-f({\omega }_{a}+{\omega }_{0}/2)].\end{array}$$
(6)

The temperature dependence of the FWHM is typically dominated by the anharmonic “three-phonon process” given by the second term in Eq. (6). Here, a q = 0 optical phonon of ω0 decays into two phonons with opposite momenta and energies ω1 and ω0 − ω1, and n(ω, T) is the Bose distribution function. The last term in Eq. (6) accounts for the line width due to the excitation of an electron from state (k, i) to (k, j) by absorbing an optical phonon of energy ω0 (Fig. 7f), where f(ω, T) is the Fermi distribution function. The constant γep is directly related to the electron-phonon coupling constant, \(\lambda =\frac{4}{\pi N(0)}{\sum }_{j}{\gamma }_{ep,j}/{\omega }_{0,j}^{2}\) with N(0) being the density of states at the Fermi energy68. The asymmetric coefficient, ωa is a phenomenological fitting parameter accounting for the fact that the Fermi level may not lie exactly halfway between the electron and hole states, as shown in Fig. 7f. The electron-phonon induced linewidth is typically negligibly small compared to the anharmonic contribution. This is because q = 0 phonons generally only promote vertical interband transitions to conserve both energy and momentum, and therefore have very limited phase space to excite electron-hole pairs.

We have fitted the temperature dependence of each phonon mode in LaRhGe3 to Eq. (6) and the results are tabulated in Supplementary Table 5. The three highest energy phonons (including \({A}_{1}^{(2)}\) and \({A}_{1}^{(3)}\) in Fig. 7c) exhibit a conventional temperature dependence and are well captured by impurity scattering and anharmonic processes alone. However, we find that the FWHM of the five lowest energy phonons deviates from the prediction of the anharmonic decay model, and could only be well described by adding the electron-phonon interaction term. The most extreme disagreement occurs for the \({A}_{1}^{(1)}\) phonon whose width increases markedly with decreasing temperature (Fig. 7c). Similar effects have been observed in graphite and graphene67,69, and more recently in WP2 and NbGe2 as a signature of the phonon drag effect10,13. Among the phonon modes of LaRhGe3, the \({A}_{1}^{(1)}\) phonon also shows the greatest relative change in energy as a function of temperature, as can be seen in Fig. 7d.

Unlike the line width, electron-phonon coupling has little effect on the phonon energy where the temperature dependence is contributed by two major factors: lattice thermal expansion and anharmonic phonon-phonon interaction70:

$$\omega (T)={\omega }_{0}{e}^{-3{\gamma }_{G}{\alpha }_{0}T}-{\omega }_{3p}[1+n({\omega }_{1},T)+n({\omega }_{0}-{\omega }_{1},T)].$$
(7)

The first term accounts for the lattice thermal expansion, where ω0 is the bare phonon energy, γG is the mode Grüneisen parameter, and α0 is the temperature independent linear thermal expansion coefficient. The second term arises from the same three-phonon anharmonic decay process as in Eq. (6). We show the change in relative energy of the A1 and E modes in Fig. 7d and all of the fitting parameters are tabulated in Supplementary Table 5. All phonon energies decrease monotonically with increasing temperature, indicating they are dominated by the three-phonon anharmonic term in Eq. (7). In some materials, it is necessary to include higher-order anharmonic terms to describe the temperature dependence of phonon self-energy67,71. In LaRhGe3, we find it only marginally improves the fit for the phonon energies and has little effect on the line width. This gives us confidence that the observed anomalous temperature dependence of the phonon linewidths cannot be caused by higher order anharmonic terms, and is more likely a signature of electron-phonon drag.

Summary

Our study establishes LaRhGe3 as a remarkable material with many intertwined novel electronic properties. Our transport measurements demonstrate that LaRhGe3 is a compensated semimetal while our electronic structure calculations reveal Weyl nodal-line states due to the I4mm space group. We report the discovery of superconductivity in LaRhGe3 with a Tc of 0.39(1) K and a Bc(0) of 2.2(1) mT. We find that LaRhGe3 is a type-I superconductor and well described by weak-coupling BCS theory with no evidence for TRS breaking in the zero field μSR measurement suggesting a dominant singlet pairing in the measured temperature range, despite the strong antisymmetric SOC.

In the normal state, we find that the temperature dependence of resistivity in zero-field deviates from that of a typical Fermi liquid. We identify a peak in the derivative of the resistivity, the Seebeck coefficient, and in thermal conductivity at  ~30 K, which is consistent with phonon drag in LaRhGe3. We measure the temperature-dependent Raman spectra of LaRhGe3 and find anomalous temperature dependence of phonon line widths that can be explained by assuming electron-phonon coupling. This appears prominent in the temperature dependence of the FWHM of the lowest energy A1 mode.

With the rich literature on superconductivity in materials crystallizing in the RTX3 structure, we draw some comparisons with LaRhGe3 and find it similar to the previously reported LaRhSi3. However, our current work indicating semimetallicity and electron-phonon drag in the normal state highlights a new area of interest in the class of non-centrosymmetric materials crystallizing in the I4mm space group. More work is needed to clarify the pairing state in the superconducting phase of LaRhGe3. Furthermore, the theoretically predicted almost movable nodal line will push further investigation of topology, both theoretically and experimentally, in the RTX3 family of materials.

Methods

Synthesis and Structural Characterization

Single crystals of LaRhGe3 were grown from Rh-Ge self flux. The elements La, Rh, and Ge in the molar ratio 1:2:7 were loaded into an Al2O3 Canfield crucible set and sealed in a quartz tube under 0.3 atm argon pressure. The mixture was then slowly heated to 1200 ∘C and held at this temperature for 6 h followed by cooling to 975 ∘C over 60 h. At 975 ∘C, the crystals were separated from the flux by centrifuging at 2000 rpm for approximately 30 s. The as-grown crystals were shiny with dimensions up to 3 × 2 × 1 mm3, and are stable in air on a time scale of weeks but develop surface oxidation over more extended periods of air exposure. Phase purity and orientation of the crystals were confirmed by XRD using a Bruker D8 Advance with Cu Kα1 radiation (λ = 1.54056 Å). The composition and homogeneity were confirmed through energy EDX using a Philips XL30 scanning electron microscope equipped with a Bruker Quantax 200 energy-dispersion X-ray microanalysis system and an XFlash 6010 SDD detector.

Transport and specific heat

Electrical resistivity measurements were performed with conventional four- and five-probe geometries using a Quantum Design Physical Property Measurement System (PPMS). For these measurements Pt wires (25 μm) were attached to the sample with silver epoxy. Specific heat measurements were performed using a PPMS with a 3He/4He dilution refrigerator insert. Thermal conductivity and Seebeck measurement were performed using a thermal transport option for the PPMS.

μSR measurements

Muon spin relaxation measurements were performed using a dilution refrigerator at the M15 beamline at TRIUMF. Multiple LaRhGe3 crystals were mounted on an Ag plate using GE varnish and placed on the sample holder. Careful canceling of the remnant field was performed using a Si semiconductor reference prior to the zero-field experiment following the method of ref. 72. The transverse-field experiment was performed by applying a magnetic field along the plate direction of the sample in spin-rotated mode. Muon analysis was performed in MUSRFIT73.

Raman measurements

Raman scattering measurements were performed using a Horiba Jobin-Yvon LabRAM monochromator, equipped with 1800 grooves/mm grating and a Peltier cooled CCD camera. The 632.817 nm line of a He-Ne laser was used as the light source, and the spot diameter is about 10 μm on the sample using a 50x microscopic objective for both focusing and collection of light. The spectral resolution is about 1 cm−1 in this study. A Semrock RazorEdge filter was used to block the elastically scattered laser light, where the cutoff energy is 79 cm−1. For measured performed at non-ambient temperatures, we used a Cryovac Micro continuous helium flow cryostat, where the sample is thermally anchored to the copper cold-finger and sits in a vacuum environment better then 1 × 10−6 mbar. Measurements were taken at temperatures ranging between 5 and 300 K with 1 K accuracy.

LaRhGe3 has five atoms in the primitive cell, leading to a total of three acoustic and 12 optical phonon bands. With visible light, Raman scattering only probes a small portion of the Brillouin zone near q = 0, where the symmetries of the 12 optical phonons can be denoted by the irreducible representations of the C4v point group: 3A1 + B1 + 4E65. The samples was mounted on the cryostat with the as-grown (101) surface normal to the optical axis. At room temperature, spectra were obtained with the incident and scattered light polarized in the following combinations in order to identify the symmetries of the studied excitations: (Ei, Es) = (x, x) couples to phonons with A1, B1 and E symmetries, (Ei, Es) = (x, y) couples only to phonons with the E symmetry. Ei and Es are the polarizations of the incident and scattered light, respectively. x and y corresponds to [10\(\bar{1}\)] and [010] crystal axes, respectively. The analyzing polarizer is removed in temperature dependent experiments to collect data from all phonons in the same cooling cycle.

DFT calculations

Electronic structure calculations were performed within the framework of DFT as implemented in the VASP package74,75,76. The generalized gradient approximation with the PBE parameterization for the exchange-correlation functional77,78 was used. The plane-wave-basis set has been cut off at Ecutoff = 287.6 eV and the charge density was converged on a grid of 1792 irreducible k points. The tetragonal crystal structure (I4mm, space group no. 107, a = 4.432 Å, and c = 10.101 Å) was used26, and structural relaxation resulted in a = 4.437 Å, and c = 10.281 Å. The phonon modes were computed in a 2 × 2 × 1 conventional supercell using the phonopy package79,80.