# Discovery of Superconductivity and Electron-Phonon Drag in
the Non-Centrosymmetric Semimetal LaRhGe3

Mohamed Oudah<EMAIL_ADDRESS>Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, BC V6T 1Z4, Canada  Hsiang-Hsi Kung
Stewart Blusson Quantum Matter Institute, University of British Columbia,
Vancouver, BC V6T 1Z4, Canada  Samikshya Sahu  Stewart Blusson Quantum Matter
Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
Department of Physics & Astronomy, University of British Columbia, Vancouver,
BC V6T 1Z1, Canada  Niclas Heinsdorf  Stewart Blusson Quantum Matter
Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
Department of Physics & Astronomy, University of British Columbia, Vancouver,
BC V6T 1Z1, Canada  Max Planck Institute for Solid State Research,
Heisenbergstrasse 1, 70569 Stuttgart, Germany  Armin Schulz  Max Planck
Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart,
Germany  Kai Philippi  Max Planck Institute for Solid State Research,
Heisenbergstrasse 1, 70569 Stuttgart, Germany  Marta-Villa De Toro Sanchez
TRIUMF, Vancouver, British Columbia, V6T 2A3 Canada  Yipeng Cai  TRIUMF,
Vancouver, British Columbia, V6T 2A3 Canada  Kenji Kojima  Stewart Blusson
Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T
1Z4, Canada  TRIUMF, Vancouver, British Columbia, V6T 2A3 Canada  Andreas P.
Schnyder  Max Planck Institute for Solid State Research, Heisenbergstrasse 1,
70569 Stuttgart, Germany  Hidenori Takagi  Max Planck Institute for Solid
State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany  Bernhard Keimer
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569
Stuttgart, Germany  Doug A. Bonn  Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, BC V6T 1Z4, Canada  Department of
Physics & Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1,
Canada  Alannah M. Hallas<EMAIL_ADDRESS>Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4,
Canada  Department of Physics & Astronomy, University of British Columbia,
Vancouver, BC V6T 1Z1, Canada  Canadian Institute for Advanced Research,
Toronto, Ontario, Canada M5G 1M1

###### Abstract

We present a comprehensive study of the non-centrosymmetric semimetal LaRhGe3.
Our transport measurements reveal evidence for electron-hole compensation at
low temperatures, resulting in a large magnetoresistance of 3000% at 1.8 K and
14 T. The carrier concentration is on the order of $10^{21}\rm{/cm}^{3}$,
higher than typical semimetals. We predict theoretically the existence of
almost movable Weyl nodal lines that are protected by the tetragonal space
group. We discover superconductivity for the first time in this compound with
a $T_{\text{c}}$ of 0.39(1) K and $B_{\rm{c}}(0)$ of 2.1(1) mT, with evidence
from specific heat and transverse-field muon spin relaxation ($\mu\rm{SR}$).
LaRhGe3 is a weakly-coupled type-I superconductor, and we find no evidence for
time-reversal symmetry breaking in our zero-field $\mu\rm{SR}$. We study the
electrical transport in the normal state and find an unusual $\sim T^{3}$
dependence at low temperature while Seebeck coefficient and thermal
conductivity measurements reveal a peak in the same temperature range. We
conclude that the transport properties of LaRhGe3 in its normal state are
strongly influenced by electron-phonon interactions. Furthermore, we examine
the temperature dependent Raman spectra of LaRhGe3 and find that the lifetime
of the lowest energy $A_{1}$ phonon is dominated by phonon-electron scattering
instead of anharmonic decay.

semimetal, superconductor, electron-phonon drag

##  I 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 graphene Novoselov  _et al._
(2005); Geim and Novoselov (2007), 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 structures Burkov (2016); Yan and Felser
(2017); Armitage  _et al._ (2018). Many topological semimetals are
simultaneously found to exhibit remarkable transport properties including
extremely large and non-saturating magnetoresistances and ultrahigh carrier
mobilities Ali  _et al._ (2014); Shekhar  _et al._ (2015); Kumar  _et al._
(2017). In certain semimetals, these transport properties have been
definitively linked with strong electron-phonon interactions, including WP2
Yao  _et al._ (2019); Osterhoudt  _et al._ (2021), PtSn4 Wu  _et al._ (2016);
Fu  _et al._ (2020), and NbGe2 Yang  _et al._ (2021); Emmanouilidou  _et al._
(2020), all of which also manifest interesting topological states.

One particularly interesting class of semimetals are those with broken
inversion symmetry. In metals, the breaking of inversion symmetry leads to
symmetry allowed electronic and magnetic couplings that are forbidden in
centrosymmetric materials, thereby promoting exotic states. One such example
is antisymmetric spin-orbit coupling (SOC), which can significantly impact any
superconducting state that arises. When the antisymmetric SOC is sufficiently
strong, a mixture of spin singlet and spin triplet pairings will occur
yielding an unconventional superconducting state Bauer  _et al._ (2004); Bauer
and Sigrist (2012); Smidman  _et al._ (2017); Yip (2014); Schnyder and Brydon
(2015). Similarly, asymmetric exchange in magnetic metals, namely the
Dzyaloshinskii–Moriya interaction, can lead to complex magnetic states
including chiral magnetic order Bode  _et al._ (2007) and skyrmion phases
Nagaosa and Tokura (2013).

The rich physics associated with broken inversion symmetry has led to
significant interest in the $RTX_{3}$ ($R$ = rare earth, $T$ = transition
metal, $X=\rm{Ge,Si,Sn}$) 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
superconductivity Kimura  _et al._ (2005); Sugitani  _et al._ (2006); Okuda
_et al._ (2007); Kimura  _et al._ (2007); Settai  _et al._ (2007); Méasson
_et al._ (2009); Kawai  _et al._ (2008); Eguchi  _et al._ (2011); Isobe  _et
al._ (2014). In particular, they have attracted attention in light of their
antisymmetric SOC split Fermi-surfaces Kawai  _et al._ (2008). 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 role Kawai  _et al._ (2008).

Figure 1: Structural characterization of LaRhGe3. (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 grey
crosses. A small germanium metal 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.

Here, we report a comprehensive characterization of one member of the non-
centrosymmetric $RTX_{3}$ family, LaRhGe3, which has significant antisymmetric
SOC due to the presence of Rh on the $T$ site. We synthesize large 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 unique set of results
on LaRhGe3 paves the way for further exploration of the $I4mm$ class of non-
centrosymmetric materials.

##  II Results and Discussion

###  II.1 Non-Centrosymmetric Structure of LaRhGe3

Large single crystals of LaRhGe3 were grown by the metallic self-flux method.
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 structure Venturini  _et al._ (1985). Our
Rietveld refinement, shown by the black line in Fig. 1(a), confirms the
absence of any significant secondary phases. A minor germanium metal phase
phase (1.3% by weight based on Rietveld refinement), likely originating from
residual flux on the surface of the crystals, is marked by by the orange
asterisks. 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
Tables S1 and S2.

The non-centrosymmetric crystal structure of LaRhGe3 is presented in Fig.
1(b). 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.

Figure 2: Semimetallic electronic properties of LaRhGe3. (a) Temperature
dependence of the longitudinal resistivity $\rho_{\mathrm{xx}}$ 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 $\rho_{\mathrm{xx}}$ 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 $\rho_{\mathrm{xy}}$ to a two-band model Pippard (1989), revealing
LaRhGe3 to be a compensated semimetal at low temperatures.

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.
1(c)). 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.

###  II.2 Semimetallicity in LaRhGe3

We begin our investigation of the electronic properties of LaRhGe3 in its
normal state. 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 $\mu\Omega$-cm
range across all temperature (Fig. 2(a)). The high residual resistivity ratio,
$\rho_{\mathrm{300\rm{K}}}/\rho_{\mathrm{2\rm{K}}}=110$, 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 $10~{}\rm{K}$. In Fig. 2(b), we plot the magnetoresistance
(MR) of LaRhGe3 at various temperatures, where MR is defined as
$[\rho_{\mathrm{xx}}(B)-\rho_{\mathrm{xx}}(0)]/\rho_{\mathrm{xx}}(0)$. 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 semimetal Wang  _et al._ (2016,
2017a); Bannies  _et al._ (2021).

To verify the semimetallicity scenario, we measured the Hall resistivity
$\rho_{\mathrm{xy}}$ of LaRhGe3, shown in Fig. 2(c), at the same fields and
temperatures as the MR. We find that $\rho_{\mathrm{xy}}(B)$ is non-linear for
all temperatures, and hence it cannot be fitted with a semi-classical one-band
model for which $\rho_{\mathrm{xx}}$ would be independent of field, also in
disagreement with our data. Instead, we fitted both the MR and Hall
resistivity to a semi-classical two-band model Pippard (1989):

$\rho_{\mathrm{xx}}=\frac{1}{e}\frac{(n_{\mathrm{h}}\mu_{\mathrm{h}}+n_{\mathrm{e}}\mu_{\mathrm{e}})+(n_{\mathrm{h}}\mu_{\mathrm{e}}+n_{\mathrm{e}}\mu_{\mathrm{h}})\mu_{\mathrm{h}}\mu_{\mathrm{e}}B^{2}}{(n_{\mathrm{h}}\mu_{\mathrm{h}}+n_{\mathrm{e}}\mu_{\mathrm{e}})^{2}+(n_{\mathrm{h}}-n_{\mathrm{e}})^{2}\mu_{\mathrm{h}}^{2}\mu_{\mathrm{e}}^{2}B^{2}}$
(1)

$\rho_{\mathrm{xy}}=\frac{B}{e}\frac{n_{\mathrm{h}}\mu_{\mathrm{h}}^{2}-n_{\mathrm{e}}\mu_{\mathrm{e}}^{2}+(n_{\mathrm{h}}-n_{\mathrm{e}})\mu_{\mathrm{h}}^{2}\mu_{\mathrm{e}}^{2}B^{2}}{(n_{\mathrm{h}}\mu_{\mathrm{h}}+n_{\mathrm{e}}\mu_{\mathrm{e}})^{2}+(n_{\mathrm{h}}-n_{\mathrm{e}})^{2}\mu_{\mathrm{h}}^{2}\mu_{\mathrm{e}}^{2}B^{2}}.$
(2)

The fitting parameters in this model are the carrier densities,
$n_{\mathrm{e}}$ and $n_{\mathrm{h}}$, and the carrier mobilities,
$\mu_{\mathrm{e}}$ and $\mu_{\mathrm{h}}$, for electrons and holes,
respectively. We simultaneously fit $\rho_{\mathrm{xx}}$ and
$\rho_{\mathrm{xy}}$ to this two-band model to extract $n_{\mathrm{e,h}}$ and
$\mu_{\mathrm{e,h}}$ as a function of temperature. The carrier densities,
plotted in Fig. 2(d), are on the order of $10^{21}$ cm-3, as expected for a
semimetal. At high temperature, we find that holes are the majority charge
carriers but when the temperature falls below 40 K, the carrier densities are
essentially compensated with $n_{\mathrm{e}}/n_{\mathrm{h}}=0.99(1)$ at 2 K,
as can be seen in Fig. 2(e). 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. 2(f). 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 $\rho_{xy}$ 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.

Figure 3: Ab-initio electronic band structures in the primitive unit cell of
LaRhGe3. The width and intensity of the superimposed colors indicate the
contribution to the states from the respective atoms. The red horizontal line
marks the Fermi level. An onsite Hubbard interaction of $U_{LDA+U}=14.5$ eV
was added to the $f$-orbitals of La to push the (unoccupied) states out of the
energy window.

To further investigate the electronic properties of LaRhGe3 we calculate its
band structure using density functional theory (DFT). The band structure,
which is plotted along the high symmetry directions of the body-centered
tetragonal Brillouin zone (Fig. 3), 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. Due to the presence of
antisymmetric SOC, the spin degeneracy of all bands is lifted, as has been
previously reported Kawai  _et al._ (2008). There are three pairs of spin-
split bands crossing the Fermi level, where two pairs are hole bands and one
pair is an electron band. 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. We note that our extracted
de Haas-van Alphen (dHvA) oscillation frequencies, as shown in Supplemental
Fig. S1, are consistent with previous studies and the calculated band
structure. There are three pairs of bands, one electron and two hole, crossing
the Fermi level, where the pairs are split in energy due to the antisymmetric
SOC Kawai  _et al._ (2008).

###  II.3 Protected Nodal Lines in LaRhGe3

Next we 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
belongs Hirschmann  _et al._ (2021). Their specific dispersion relation and
proximity to the Fermi level, as well as their connectivity is not fully fixed
by symmetry alone, and in general depends on the particular material. 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 ($\Gamma$ \- $Z$) and the $N$-point. The eigenvalues of
the mirror symmetry at $N$ are $\pm i$. 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 $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}$ 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 phase. Therefore, along an arbitrary path on the
mirror plane that connects $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ (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 line Hirschmann  _et al._
(2021). 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 $\Gamma$\- 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
($\Gamma$ \- $Z$) or it does not.

In Figure 4(a) and 4(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 $\Gamma$-$Z$ and
$\Gamma$-$N$. 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
4(c) shows the dispersion relation along the different nodal lines. The pinned
and the almost movable nodal lines shown in panel (a) 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 found Weng  _et al._
(2015); Chan  _et al._ (2016). Even though surface states do not necessarily
emerge in nodal line systems Chiu  _et al._ (2018), any curve through
$k$-space that encircles a single nodal line has a Berry curvature that is
quantized to $\pi$ by the mirror symmetry. It is very likely that a suitable
surface termination can 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. Materials with these
gapless and robust surface modes are highly sought-after, due to the variety
of proposals in novel device applications Liu  _et al._ (2019a); Burkov
(2016); Tian  _et al._ (2015, 2017); Li  _et al._ (2018); Kim  _et al._
(2015); Lundgren  _et al._ (2014); Wang  _et al._ (2017b). 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 present Guo  _et al._
(2022); Shang  _et al._ (2023). 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 transition Gurarie (2011); Wang and Zhang
(2012); Song and Schnyder (2017); Iraola  _et al._ (2021). Even nonlinear
perturbations do not generically suppress edge modes Chaunsali  _et al._
(2021), and their stability in nodal line semimetals in their superconducting
state has been theoretically demonstrated Wang and Nandkishore (2017).

Figure 4: Almost movable nodal lines in LaRhGe3. (a,b) Pinned ($\Gamma$ 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) 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 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.

Exploring the evolution of these almost movable nodal lines in other materials
with the $I4mm$ space group but different elements should be the subject of
future studies. Our calculations have established the compensated semimetal
state in LaRhGe3 with an antisymmetric SOC-split Fermi surface and almost
movable nodal lines; we next highlight our discovery of superconductivity in
LaRhGe3.

###  II.4 Superconductivity in LaRhGe3

As our electrical transport measurements do not extend to sufficiently low
temperature, our first indication of superconductivity in LaRhGe3 came from
low temperature specific heat measurements, as shown in Figure 5(a), where a
sharp anomaly is observed at $T_{\text{c}}$ $=0.39(1)$ K. 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 $T_{\text{c}}$, which is consistent with a first order
transition as expected for a type-I superconductor. Upon application of a 2 mT
field the superconducting transition is fully suppressed in our measurement
window, which extends down to 0.1 K.

Figure 5: Superconducting transition in LaRhGe3. (a) Temperature dependence of
$C_{\rm{el}}$/$T$ 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 $T_{\text{c}}$ in LaRhGe3 under an applied magnetic
field of 1.0 mT. (c) The amplitude of the signal in (b) as a function of
temperature measured in zero-field cooling (ZFC) and field cooling (FC)
protocol. Reduction in 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
$T_{\text{c}}$ values extracted from $C_{P}$ and transverse field
$\mu\rm{SR}$. The solid line is a fit to the data used to estimate
$B_{\rm{c}}(0)$ $=2.1$ mT.

To further investigate the superconducting state of LaRhGe3, we performed muon
spin relaxation ($\mu$SR) measurements. The muon decay asymmetry plotted as a
function of time in an applied transverse field (TF) of 1 mT (which is below
the critical field) is shown in Fig. 5(b). The field was applied perpendicular
to the plate-like crystals and the muon spin is paralel to the plate of the
crystal, such that the muon spin is perpendicular to the field applied. The
muons precess in the applied field, yielding an oscillating decay asymmetry.
The measurement above $T_{\text{c}}$ is shown in red while the measurements
below $T_{\text{c}}$ under zero-field cooled (ZFC) and field cooled (FC)
conditions are shown in blue and green, respectively. We observe an increase
in relaxation below $T_{\text{c}}$ under both the field-cooled and the zero-
field-cooled conditions, which results in a smaller amplitude of the
oscillations. We fit the asymmetry spectra using a two term sinusoidal
decaying function

$\displaystyle G_{\mathrm{TF}}(t)$
$\displaystyle=A\left[f\exp\left(\frac{-\sigma^{2}t^{2}}{2}\right)\cos\left(\omega_{1}t+\phi\right)\right.$
$\displaystyle\left.+(1-f)\exp(-\lambda
t)\cos\left(\omega_{2}t+\phi\right)\right]$

where the first term captures the signal from muons stopping in the sample and
the second term captures the signal from muons stopping in the silver sample
holder. The fraction of signal from muons that land in the sample is given by
$f$, while $\omega_{1}$ and $\omega_{2}$ are the muon precession frequencies
in the sample and the background, respectively. The $A$ term is the total
asymmetry and the $\phi$ is the initial phase of the muons. The depolarization
rates for the sample and the background signals, respectively, are given by
$\sigma$ and $\lambda$.

The fitted amplitude of the oscillating component as a function of temperature
under both ZFC and FC conditions is shown in Fig. 5(c). The amplitude
corresponds to all muons stopping inside the superconducting LaRhGe3 and the
non-superconducting silver plate. A sharp drop in amplitude is observed at the
superconducting transition. This behavior is expected in a superconductor due
to the expulsion of magnetic field from the bulk of the sample in the
superconducting state such that muons implanted sufficiently deeply in the
sample will not feel the applied field and will therefore not precess. From
these transverse field $\mu$SR measurements, we estimate $T_{\text{c}}$$\sim
0.28$ K in an applied field of 1 mT, in good agreement with the $T_{\text{c}}$
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 indicates that LaRhGe3 is a Type-I superconductor.
The temperature and field dependence of $T_{\text{c}}$ extracted from specific
heat and transverse field $\mu$SR are plotted in Fig. 5(d).

To further characterize the superconductivity of LaRhGe3, we next turn to
modelling its specific heat. Above $T_{\text{c}}$ $=0.39(1)$ K the $C_{P}$
data are well described by

$C_{p}=\gamma_{n}T+\beta T^{3}\ $ (3)

where fitting the data yields the Sommerfeld coefficient $\gamma_{n}=4.92(4)$
mJ mol-1 K-2 and $\beta=0.25(1)$ mJ mol-1 K-4. The phonon contribution is
represented by the cubic term $\beta T^{3}$ and the Debye temperature
$\Theta_{D}$ is then calculated according to

$\Theta_{D}=\left(\frac{12\pi^{4}}{5\beta}nR\right)^{1/3}\ $ (4)

where $n$, the number of atoms per formula unit, is 5 and $R$ is the gas
constant 8.314 J mol-1 K-1, giving $\Theta_{D}=341(10)\,K$. Next, taking
$\Theta_{D}$ and $T_{\text{c}}$, we calculate the superconducting parameter
$\lambda_{ep}$ using the inverted McMillan McMillan (1968) equation

$\lambda_{\text{ep}}=\frac{1.04+\mu^{*}ln\left(\frac{\Theta_{D}}{1.45T_{c}}\right)}{(1-0.62\mu^{*})\text{ln}\left(\frac{\Theta_{D}}{1.45T_{c}}\right)-1.04}\
$ (5)

where $\mu\rm{*}=0.10$ and $T_{\text{c}}$ $=0.39(1)$ K. We find
$\lambda_{ep}=0.34(2)$ for LaRhGe3, suggesting it is a weak coupling
superconductor. Using $\lambda_{ep}$, $\gamma$, and the Boltzmann constant
$k_{\rm{B}}$, we calculate the electronic density of states (DOS) at the Fermi
energy $N(E_{\rm{F}})$ with the equation

$N(E_{F})=\frac{3\gamma}{\pi^{2}k_{B}^{2}(1+\lambda_{\text{ep}})}\ $ (6)

$N(E_{\rm{F}})$ was estimated to be 1.56 states eV-1 per formula unit of
LaRhGe3. From our DFT calculations we find the DOS at the Fermi energy
$N(E_{\rm{F}})^{calc}$ of 3.24 states eV-1 per formula unit, and we evaluate
the effective mass based on specific heat to be
$m^{*}/m_{e}=N(E_{\rm{F}})/N(E_{\rm{F}})^{calc}=0.54$. This reduced effective
mass compared with the free electron mass is consistent with LaRhGe3 being a
semimetal, and the value of $m^{*}/m_{e}=0.54$ falls within the cyclotron
effective masses, $m^{*}/m_{e}\sim$ 0.23-1.04, reported previously by Kawai
_et al._ Kawai  _et al._ (2008).

Figure 6: Type-I BCS nature of the superconducting state in LaRhGe3. (a)
Superconducting transition of LaRhGe3 in the electronic part of the specific
heat $C_{\rm{el}}$/$T$ as a function of $T$ in 0 T. The dashed black line is a
fit for a $T_{\text{c}}$ of 0.39(1) K for weak-coupling BCS behavior. (b) Plot
of $\Theta_{c}$ against measured $T_{\text{c}}$ 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. (c) Zero
field $\mu\rm{SR}$ 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 $T_{\text{c}}$. Inset shows the resulting relaxation rate
$\sigma$. (d) Uemura plot showing the superconducting transition temperature
($T_{\text{c}}$) vs the Fermi temperature ($T_{\rm{F}}$) for various
superconductors Uemura (2004); Poole  _et al._ (2007); Carbotte (1990);
Peabody and Meservey (1972); Adroja  _et al._ (2018); Albedah  _et al._
(2017); Bhattacharyya  _et al._ (2018); Sonier  _et al._ (2007); Uemura
(1991); Talantsev and Tallon (2015); Uemura  _et al._ (1989); Lin  _et al._
(2013), where LaRhGe3 falls within the region of conventional superconductors.
$T=T_{\rm{B}}$ line in purple represents the Bose-Einstein condensation
temperature of an ideal three-dimensional boson gas, and $T=T_{\rm{F}}$ line
is shown in black.

The applied magnetic fields plotted against $T_{\text{c}}$ as extracted from
$C_{P}$ measurements and transverse-field $\mu\rm{SR}$ are shown in Fig. 5(d).
We estimate the zero-temperature critical field $B_{\rm{c}}(0)$ using the
conventional relationship below

$B_{c}(T)=B_{c}(0)\left[1-\left(\frac{T}{T_{c}}\right)^{2}\right]\ $ (7)

where $T_{\text{c}}$ is $0.39(1)$ K we obtain the curve in red to the data
extracted from $C_{P}$. This fit yields a $B_{\rm{c}}(0)$ of 2.1 mT. We have
been unsuccessful so far in measuring the drop in resistivity to zero below
$T_{\text{c}}$, which is most likely due to the low critical current of
LaRhGe3, consistent with its low critical field. Our current capabilities for
measuring resistivity with the dilution fridge do not allow us to apply a low
enough current to perform the measurement. The small critical field makes the
presence of a small remnant field during the $C_{P}$ measurement a serious
concern that can shift our measured $T_{\text{c}}$. However, to address this
concern we highlight the point on this phase diagram extracted from transverse
field $\mu\rm{SR}$ measurement, where magnetic field for this point is
extracted from the fit to transverse field $\mu\rm{SR}$ experiment, which is
expected to be accurate and confirms the fit of $B_{\rm{c}}(0)$ based on the
$C_{P}$ data.

Table 1: Superconducting parameters of LaRhGe3. Parameter |  Value |  Units
---|---|---
$T_{\text{c}}$ |  0.39(1) |  K
$B_{\rm{c}}(0)$ |  2.1(1) |  mT
$\gamma_{n}$ |  4.92(4) |  mJ mol-1 K-2
$\beta$ |  0.25(1) |  mJ mol-1 K-4
$\theta_{D}$ |  341(10) |  K
$\lambda_{ep}$ |  0.34(2) |  -

The magnitude of the specific heat jump $\Delta C/C_{en}\sim 1.43$, $C_{en}$
being the electronic specific heat in the normal state, is consistent with
that expected for a BCS model in the weak-coupling limit. The BCS model, shown
in black in Fig. 3(a), fits the data well just below $T_{\text{c}}$, but some
deviations appears at lower temperatures. With the current $C_{P}$ we cannot
conclude whether the gap structure of LaRhGe3 is an isotropic $s$-wave gap or
whether the gap structure has some nodes, and this will be the subject of
future experiments going to lower temperatures.

Table 2: Superconductivity in LaRhGe3 compared with other superconductors in $I4mm$ space group. Compound |  $T_{\text{c}}$ (K) |  $\gamma_{n}$ (mJ mol-1 K-2) |  $B_{\rm{c2}}(0)$ (T) | Reference
---|---|---|---|---
Ba(Pt,Pd)Si3 |  2.3–2.8 |  4.9–5.7 |  0.05–0.10 | Bauer  _et al._ (2009); Kneidinger  _et al._ (2014)
La(Rh,Pt Pd,Ir)Si3 |  0.7–2.7 |  4.4–6 |  Type I/0.053 | Anand  _et al._ (2011); Kitagawa  _et al._ (1997); Smidman  _et al._ (2014); Okuda  _et al._ (2007); Anand  _et al._ (2014)
Ca(Pt,Ir)Si3 |  2.3–3.6 |  4.0–5.8 |  0.15–0.27 | Eguchi  _et al._ (2011); Singh  _et al._ (2014)
Sr(Ni,Pd,Pt)Si3 |  1.0–3.0 |  3.9–5.3 |  0.039–0.174 | Kneidinger  _et al._ (2014)
Sr(Pd,Pt)Ge3 |  1.0–1.5 |  4.0–5.0 |  0.03–0.05 | Kneidinger  _et al._ (2014); Miliyanchuk  _et al._ (2011)
LaRhGe3 |  0.39(1) |  4.92(4) |  Type I/0.0021(1) | This work

To check whether the superconductivity in LaRhGe3 breaks TRS we perform muon
measurements in zero-field, since muons are a sensitive probe to magnetic
fields in the sample. We check for any spontaneous magnetic fields inside the
superconducting state by performing the measurement above and below
$T_{\text{c}}$, as shown in Fig 6(c). In the absence of magnetism, the muon
depolarization is due to randomly oriented nuclear moments, and can be
described with Kubo-Toyabe function below

$G_{\mathrm{KT}}(t)=\frac{1}{3}+\frac{2}{3}\left(1-\sigma^{2}t^{2}\right)\exp\left(-\frac{\sigma^{2}t^{2}}{2}\right)$

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

$A(t)=A_{S}G_{\mathrm{KT}}(t)\exp(-\Lambda t)+A_{\mathrm{BG}}$

where $A_{S}$, $A_{BG}$ represent the sample and background asymmetry,
respectively, and the $\Lambda$ accounts for any additional relaxation rate.
By fitting the data to the equation above, we find no significant change in
$\Lambda$ above and below $T_{\text{c}}$; the results of the fitted value are
shown in the inset of Fig. 6(c). This indicates the absence of any significant
TRS breaking in LaRhGe3.

While pure elements tend to host type-I superconductivity rather than type-II,
most compound superconductors are type-II. Only a few binary compounds exhibit
type-I superconductivity, and even fewer ternary compounds. LaRhGe3 is one of
very few type-I superconductors containing more than two elements. Other
ternary type-I superconductors include $\rm{La}\rm{Pd}_{2}\rm{Ge}_{2}$ Hull
_et al._ (1981), $\rm{(Y,La,Lu)}\rm{Pd}_{2}\rm{Si}_{2}$ Palstra  _et al._
(1986), $\rm{La}\rm{Rh}_{2}\rm{Si}_{2}$ Palstra  _et al._ (1986),
$\rm{Ag}_{5}\rm{Pb}_{2}\rm{O}_{6}$ Yonezawa and Maeno (2005), and
$\rm{La}\rm{Rh}\rm{Si}_{3}$ Anand  _et al._ (2011). We compare the
$T_{\text{c}}$, $\gamma_{n}$, and $B_{\rm{c}}(0)$ values for various
superconductors in the $I4mm$ space group, including LaRhGe3 and
$\rm{La}\rm{Rh}\rm{Si}_{3}$. Despite the comparable $\gamma_{n}$ values for
all of these materials, we have varying superconducting $T_{\text{c}}$ and
critical fields. Whether this difference is due to details of electronic band
structure or atomic size effects is unclear at the moment, but should be
further investigated in future studies.

In Fig. 6(b), we compare the expected critical temperature denoted as
$\theta_{c}$ based on the $\gamma_{n}$ and $T_{\text{c}}$ values with the
following equation Marques  _et al._ (2020):

$\Theta_{c}=\frac{2B_{\rm{c}}(0)}{\sqrt{1.43\mu_{0}\gamma_{n}}}$ (8)

with the measured $T_{\text{c}}$ for various elemental type-I superconductors,
$\rm{La}\rm{Rh}\rm{Si}_{3}$ Anand  _et al._ (2011), and LaRhGe3. We find that
LaRhGe3, as well as $\rm{La}\rm{Rh}\rm{Si}_{3}$ Anand  _et al._ (2011), lie
close to the solid-black line that represents the expectation based on weak-
coupling BCS. This is consistent with our analysis of the jump in $C_{P}$ and
further confirms that LaRhGe3 is a type-I superconductor in the weak-coupling
limit. LaRhGe3 and LaRhSi3 have similar degrees of antisymmetric SOC Kimura
_et al._ (2007); Terashima  _et al._ (2008), and we find no sign of
unconventional superconductivity in the silicide Anand  _et al._ (2011) nor in
our bulk measurements on LaRhGe3.

Figure 7: Electron-phonon drag in LaRhGe3. (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 $ln(\rho-\rho_{0}$ 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 ($S_{xx}/T$ as
a function of temperature of LaRhGe3. Inset shows $S_{xx}$ as a function of
temperature. (d) Temperature dependence of thermal conductivity of LaRhGe3.
Inset shows the Lorentz number as a function of temperature.

We evaluate the Fermi temperature $T_{\rm{F}}$ for a 3D system via the
relation $T_{F}=(\hbar^{2}/2)(3\pi^{2})^{2/3}n^{2/3}/k_{B}m^{*}$ Uemura
(2004), where $n$ is the density of quasiparticles per unit volume. We use the
low temperature carrier density from our Hall measurements and the effective
mass estimated from specific heat to obtain $T_{\rm{F}}=28340$ K and
$T_{\rm{c}}/T_{\rm{F}}=1.36\times 10^{-5}$, which places LaRhGe3 in a region
with other conventional superconductors. This further confirms that the
ternary compound LaRhGe3 can be described in the bulk as a BCS superconductor
in the weak-coupling limit. Next we highlight electron-phonon interactions in
the normal state of LaRhGe3.

###  II.5 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 came from zero-field resistivity where we observe a deviation from a
standard Fermi liquid behaviour as shown in Fig. 7(a). Using the typical
$\rho=\rho_{0}+AT^{2}+BT^{5}$ to account for electron-electron interactions
with the $T^{2}$ term and electron-phonon interactions with the $T^{5}$ term
fits the data poorly below 50 K, and the extracted coefficients
$A=6.24^{-4}~{}\mu\Omega\rm{cm~{}K}^{-2}$ and $B=1.425\times
10^{-8}~{}\mu\Omega\rm{cm~{}K}^{-5}$ are physically unreasonable. If we use
the $A$-coefficient along with the Sommerfeld coefficient from specific heat
we evaluate a Kadowaki-Woods ratio of
$R_{KW}=\frac{A}{\gamma^{2}}=12.4~{}\mu\Omega\,\rm{cm}\,\rm{mol}^{2}\rm{K}^{2}\rm{J}^{-2}$,
which is comparable with heavy fermions. However, we have no evidence of heavy
Fermion behaviour in LaRhGe3 considering the absence of f-electrons and the
low effective mass established from dHvA measurements Kawai  _et al._ (2008).
Instead, we find that the resistivity data fits well to a phonon-drag model
$\rho=\rho_{0}+Ce^{-\theta/T}$. The exponential fit yields $\theta=110$ K,
which is approximately one-third of the Debye temperature of $\theta_{D}\sim
341$ K determined from specific heat measurements. Also, we find an equally
good agreement the data with a power law fit $\rho=\rho_{0}+T^{n}$, which
yields $n=2.9$. A value of $n=3$ can arise due to selective phonon mode
softening associated with electron-phonon drag in materials Kos (1972).

Figure 8: Phonon spectrum of LaRhGe3. (a) Raman spectra of LaRhGe3 measured at
10 K. The phonon mode symmetries are labelled in accordance with our DFT
calculations. The strong peak at 300 $\rm{cm}^{-1}$ is likely due to Ge flux
on the surface. (b) Calculated phonon dispersion curves along high-symmetry
lines of the tetragonal primitive unit cell of LaRhGe3

Furthermore, to analyze the temperature dependence of the resistivity we take
the logarithmic derivative after subtracting residual resistivity to quantify
the exponent $n={[\delta ln(\rho-\rho_{0})]/[\delta lnT]}$. This analysis has
been applied to extract the exponent in antimony, heavy fermions, cuprates,
and strontium titanate Jaoui  _et al._ (2022). The temperature dependence of
$n$ is presented in Fig. 7(b) for two different samples, and we find at high
temperature $n\sim 1$ as expected in a standard Block-Gruneisen picture.
However, we find an upturn towards $n\sim 3$ at low temperature, which is
consistent with $n=2.9$ from our power law fit above. The lowest temperature
data is too noisy to identify any $n$ value.

To further explore the phonon-drag scenario, we measure the Seebeck
coefficient at low temperature, shown in Fig. 7(c), and notice a peak at
around $\sim 50$ K signaling the interaction between phonons and electrons is
enhanced at low temperature. The Seebeck coefficient over temperature, inset
of Fig. 7(c), pushes the peak to an even lower temperature of $\sim 30$ K
similar to the peak in the derivative of the resistivity. A peak below 50 K in
the thermal conductivity, shown in Fig. 7(d), provides another proof for the
electron-phonon drag scenario. Typically, the Lorentz 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 Lorentz number, in the inset of Fig. 7(d), which shows
clear deviation at low temperature with a minimum at around 30 K. A Kohler
plot can be utilized to scale the MR over the various temperatures measured,
and in our current measurements we find no clear change around 30 K as shown
in the supplemental materials Fig. S5.

We measure the $C_{P}$ up to 300 K, shown in Fig. S6(c), and confirm
consistency with the Dulong-Petit limit at high temperature and find no
anomalies at low temperature. The Debye temperature of 341 K estimated in Sec.
II.4 is consistent with our high temperature data. The peaks in $\kappa$, $S$,
and derivative of the resistivity appear at $\sim 30$ K, which represent a
phonon-drag temperature. Comparing this with the Debye temperature or 340 K,
we find $\Theta_{\rm{ph-drag}}/\Theta_{\rm{D}}\sim 10$, which has been seen in
other materials with phonon-drag. We notice in the MR data that a significant
enhancement appears at around 30 K that we attribute in Sec. II.2 to electron-
hole compensation, but the connection with electron-phonon interactions is
unclear at the moment and should be the subject of future studies.

To confirm the electron-phonon drag model, we measure the Raman spectra
between 300 and 10 K to search for signatures of phonon line width broadening
due to electron-phonon coupling, previously observed in some conventional
superconductors Ponosov and Streltsov (2017); Girotto and Novko (2023), and
more recently reported in WP2 and NbGe2 Osterhoudt  _et al._ (2021); Yang  _et
al._ (2021). Due to the lack of inversion symmetry, all 12 optical phonons are
Raman active in LaRhGe3, which we will denote by the irreducible
representations of the $C_{4v}$ point group: $3A_{1}+B_{1}+4E$ Kroumova  _et
al._ (2003). Using the Raman selection rules and polarization optics Hayes and
Loudon (1978), we identify the peaks at 121, 214, and 249 cm-1 as the 3
$A_{1}$ phonons, the peak at 139 cm-1 is a $B_{1}$ phonon, whereas the peaks
at 106, 116, 163, and 262 cm-1 are the phonons with $E$ symmetry (Fig. S2).
The Raman spectrum measured at 10 K without any analyzing polarizer is shown
in Fig. 8(a) with all phonon modes labeled. We tabulate the measured phonon
energies along side the values calculated from DFT in Table S4 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 dispersion, where the acoustic phonon
branch has maximum energy around 80-90 cm-1.

The Raman spectra were measured between 300 and 10 K in 10 K intervals, shown
in Fig. S3. 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, such as
shown in Fig. S4 for spectra collected at 10 K and 300 K.

In most materials, the phonons could acquire finite line-width through 3
channels, i.e., impurity scattering ($\gamma_{0}$), anharmonic phonon-phonon
interaction ($\gamma_{3p}(T)$), and electron-phonon coupling
($\gamma_{ep}(T)$) Bonini  _et al._ (2007); Allen and Silberglitt (1974).

$\begin{multlined}\gamma(T)=\gamma_{0}+\gamma_{3p}[1+n(\omega_{1},T)+n(\omega_{0}-\omega_{1},T)]\\\
+\gamma_{ep}[f(\omega_{a}-\omega_{0}/2)-f(\omega_{a}+\omega_{0}/2)]\end{multlined}\gamma(T)=\gamma_{0}+\gamma_{3p}[1+n(\omega_{1},T)+n(\omega_{0}-\omega_{1},T)]\\\
+\gamma_{ep}[f(\omega_{a}-\omega_{0}/2)-f(\omega_{a}+\omega_{0}/2)]$ (9)

The temperature dependence of FWHM in most materials is dominated by the
anharmonic “3-phonon process”, i.e. second term in Eq. 9, where a $q=0$
optical phonon of $\omega_{0}$ decays into 2 phonons with opposite momenta and
energies $\omega_{1}$ and $\omega_{0}-\omega_{1}$, and
$n(\omega,T)=1/(\exp(\hbar\omega/k_{B}T)-1)$ is the Bose distribution
function. The strength of the anharmonicity is determined by
$\gamma_{3p}=\frac{18\pi}{\hbar^{2}}\sum_{q,j_{1},j_{2}}|V(j,0;j_{1},q;j_{2},-q)|^{2}$,
where $V(j,0;j_{1},q;j_{2},-q)$ is the matrix element for the $j$-th phonon at
$q=0$ decaying into $j_{1}$-th phonon at $q$ and $j_{2}$-th phonon at $-q$.
The last term in Eq. 9 accounts for the line width due to excitation of an
electron from state $(k,i)$ to $(k,j)$ by absorbing an optical phonon of
energy $\omega_{0}$ (Fig. 9(c)). The constant $\gamma_{ep}$ 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
state at Fermi energy Allen and Silberglitt (1974).
$f(\omega,T)=1/(\exp(\hbar\omega/k_{B}T)+1)$ is the Fermi distribution
function. The asymmetric coefficient, $\omega_{a}$ is a phenomenological
fitting parameter accounting for the fact that the Fermi level may not lie
exactly half-way between the electron and hole states, as shown in Fig. 9(c).

Figure 9: Anomalous temperature dependence of the $A_{1}^{(1)}$ phonon
lifetime in LaRhGe3. (a) 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. (b) Percent difference of the phonon
energies of $A_{1}$ and $E$ modes from energies at 10 K as a function of
temperature. (c) Schematic depicting excitation of electron-hole pairs by
optical phonons, where $\omega_{a}$ is the energy offest to take into account
that the Fermi level is not exactly halfway between the initial state
$\left|i\right\rangle$ and final state $\left|f\right\rangle$ and $\omega_{0}$
is the bare phonon energy.

We note that the electron-phonon induced line width 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. Indeed this is the case with the temperature dependence of the 3
highest energy phonons’ FWHM in LaRhGe3, _e.g._ the data of $A_{1}^{(3)}$ in
Fig. 9(a). However, we found the FWHM of the 4 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. In particular, the
$A_{1}^{(1)}$ phonon width increases with decreasing temperature, which cannot
be explained by the anharmonic model alone even after adding higher order
terms (Fig. 9(a)). Similar effects have been observed in graphite and graphene
Bonini  _et al._ (2007); Liu  _et al._ (2019b), and more recently in WP2 and
NbGe2 as a signature of the phonon drag effect Osterhoudt  _et al._ (2021);
Yang  _et al._ (2021). Here, the strong SOC and the lack of inversion symmetry
generates several spin-split Fermi surfaces closely separated in energy,
splitting for our LaRhGe3 sample in Fig. S1 and Table S3 is consistent with
previous report Kawai  _et al._ (2008), allowing excitation of electron-hole
pairs by absorbing $q=0$ optical phonons. Furthermore, the almost movable
nodal lines near the Fermi level as we discussed in Sec. II.3 could also be
favorable for phonon absorption. We compare the electron-phonon coupling
estimated from superconductivity data in Sec. II.4, $\lambda_{ep}=0.33$, with
the electron-phonon coupling estimated from the $T$-dependent data for various
Raman modes in Table.S4. We find the total contribution to $\lambda_{ep}$ from
the optical phonons to be around 0.0018, which is a negligibly small fraction.

Unlike the line width, electron-phonon coupling has little effect on the
phonon energy where the temperature dependence is contributed by two major
factors, i.e. lattice thermal expansion and anharmonic phonon-phonon
interaction Menéndez and Cardona (1984).

$\omega(T)=\omega_{0}\exp(-3\gamma_{G}\alpha_{0}T)-\omega_{3p}[1+n(\omega_{1},T)+n(\omega_{0}-\omega_{1},T)]$
(10)

The first term accounts for the lattice thermal expansion, where $\omega_{0}$
is the bare phonon energy, $\gamma_{G}$ is the mode Grüneisen parameter, and
$\alpha_{0}$ is the temperature independent linear thermal expansion
coefficient. The second term arises from the same 3-phonon anharmonic decay
process as in Eq. 9. We show the change in relative energy of the $A_{1}$ and
$E$ modes in Fig. 9(b). All phonon energies decrease monotonically with
increasing temperature dominated by the 3-phonon anharmonic term in Eq. 10. In
some materials, it is necessary to include higher order anharmonic terms to
describe the temperature dependence of phonon self-energy Bonini  _et al._
(2007); Ye  _et al._ (2019). 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
phonon line width cannot be caused by higher order anharmonic terms, and is
more likely a signature of electron-phonon drag. All of the fitting parameters
are tabulated in the Supplemental Material Table S4.

##  III 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 $T_{\text{c}}$ of 0.39(1) K
and a $B_{\rm{c}}(0)$ of 2.1(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 $\mu\rm{SR}$ 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 $\sim 30$ K, which is consistent with electron phonon drag in
LaRhGe3. We measure the Raman spectra of LaRhGe3 between 10 and 300 K, 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 $A_{1}^{(1)}$ mode, the lowest
energy $A_{1}$ mode.

With the rich literature on superconductivity in materials crystallizing in
the $RTX_{3}$ 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 $RTX_{3}$ family of materials.

##  IV Methods

###  IV.1 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 $3\times 2\times 1$
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 X-ray diffraction (XRD) using a
Bruker D8 Advance with Cu K$\alpha_{1}$ radiation ($\lambda=1.54056$ Å). The
composition and homogeneity were confirmed through energy dispersive X-ray
spectroscopy (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.

###  IV.2 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 $\mu$m) 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.

###  IV.3 $\mu\rm{SR}$ 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 cancelling of
remnant field was performed using a Si semiconductor reference prior to the
zero-field experiment. 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 musrfit Suter and Wojek (2012).

###  IV.4 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 $\mu$m 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 $1\times
10^{-6}$ mbar. Measurements were taken at temperatures ranging between 5 to
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 $C_{4v}$ point group: $3A_{1}+B_{1}+4E$ Kroumova  _et
al._ (2003). 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:
$(E_{i},E_{s})=(x,x)$ couples to phonons with $A_{1}$, $B_{1}$ and $E$
symmetries, $(E_{i},E_{s})=(x,y)$ couples only to phonons with the $E$
symmetry. $E_{i}$ and $E_{s}$ 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.

###  IV.5 DFT Calculations

Electronic structure calculations were performed within the framework of
density functional theory (DFT) as implemented in the VASP package Kresse and
Hafner (1993); Kresse and Furthmüller (1996, 1996). The generalized gradient
approximation with the PBE parameterization for the exchange-correlation
functional Perdew  _et al._ (1996a, b) was used. The plane-wave-basis set has
been cut off at $E_{\text{cutoff}}=287.6$ 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
used Kawai  _et al._ (2008), and structural relaxation resulted in a = 4.437
Å, and c = 10.281 Å. The phonon modes were computed in a $2\times 2\times 1$
conventional supercell using the phonopy packageTogo and Tanaka (2015); Togo
(2023).

###### Acknowledgements.

This research was undertaken thanks in part to funding from the Max Planck-
UBC-UTokyo Centre for Quantum Materials and the Canada First Research
Excellence Fund, Quantum Materials and Future Technologies Program. This work
was also supported by the Natural Sciences and Engineering Research Council of
Canada (NSERC), the CIFAR Azrieli Global Scholars program, and the Sloan
Research Fellowships program. We thank TRIUMF staff for their technical
support during muon experiment. We thank K. Behnia, E. Benckiser, M. Minola,
S. Dierker, and M. Sigrist for valuable discussion.

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Supplemental Materials:
Discovery of Superconductivity and Electron-Phonon Drag in
the Non-Centrosymmetric Semimetal LaRhGe3

Table S1: LaRhGe3 crystals structure refinement fit results. Parameter | Value
---|---
Chemical Formula | LaRhGe3
Formula weight | 459.73
Temperature (K) | 300
Crystal System | Tetragonal
Space Group | $I4mm$
$a$, ( Å) | 4.4187(2)
$c$, ( Å) | 10.0494(5)
Volume ( Å3) | 196.21(6)
Z | 2
$\rho_{calc}$ (g/cm3) | 7.781
Radiation | Cu${}_{k\alpha 1}=1.5406$ Å
$R_{\rm{Bragg}}$ | 1.544
Table S2: Refined atomic positions of LaRhGe3. Atom | x | y | z
---|---|---|---
La | 0.0 | 0.0 | 0.0
Rh | 0.0 | 0.0 | 0.65762
Ge1 | 0.0 | 0.0 | 0.42106
Ge2 | 0.0 | 0.5 | 0.25990
Figure S1: (a) Oscillatory component $\Delta M$ corresponding to the de Haas-van Alphen (dHvA) oscillations in LaRhGe3 measured at 1.8 K. (b) Fast Fourier Transform spectrum of the dHvA oscillations in (a). Table S3: Comparing de Haas-van Alphen (dHvA) oscillations Frequency in our LaRhGe3 samples with Literature Kawai  _et al._ (2008). Branch | F ($\times 10^{7}$ Oe) |  Fliterature ($\times 10^{7}$ Oe) Kawai  _et al._ (2008)
---|---|---
$\epsilon$ (1st) | 3.09 | 3.06
$\epsilon$ (2nd) | 3.06 | 3.04
$\theta$ (1st) | 0.43 | 0.44
$\theta$ (2nd) | 0.38 | 0.38
$\eta$ (1st) | 0.082 | 0.08
$\eta$ (2nd) | 0.065 | 0.06
Figure S2: Raman spectra collected with fixed linearly polarized laser, and polarization analyzer at different orientations. The blue curve shows data without the analyzer (mixed parallel and perpendicular); orange curve is the analyzer polarization parallel to the laser (xx geometry); red curve is the analyzer 90∘ rotated with respect to laser polarization (xy geometry). Three phonon modes that are strongly suppressed were highlighted with arrows, demonstrating that these are $A_{1}$ modes of LaRhGe3. Figure S3: (a) Raman spectra of LaRhGe3 measured along different directions of the crystal at room temperature. (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. Unless otherwise specified, all of the Raman data in this work were taken from this sample. (c) Black and white map of Raman scattering intensity measured from the same sample as panel (b), but from a different spot. Since this data was not reproduced by other crystals that we measured, this spot is likely a crystallite with different orientation and possibly even different stoichiometry as the host crystal. However, we highlight the mode at 144 cm-1 at 10 K where the energy softens drastically with increasing temperature, to 120 cm-1 at 300 K. Such large temperature dependence is highly unusual and is difficult to reconcile with the moderate anharmonicity found in LaRhGe3. The energy of this mode is in the vicinity of the the $A_{1}^{(1)}$ mode that shows strong electron-phonon coupling, raising the possibility of electron-phonon induced self-energy effect. However, more study is required on this crystallite before any conclusion can be drawn. In this paper, our discussion is focused on data in panel (b) which is reproduced across samples. Figure S4: Raman spectra of LaRhGe3 measured at 300 K (a) and 10 K (b), sliced from Fig. S3(b) to showcase the extraction of phonon energy and width through fitting. Measured data shown as points and fitted curves for each phonon mode shown individually in yellow and total fit in black.  Table S4: Calculated energies and symmetries of the phonon modes for LaRhGe3 and comparison with the fitting parameters of Eq. 10 based on our Raman spectra in Fig. S3. All units are in cm-1. The errors represent one standard deviation of the least-squared fit. DFT |  $\omega$(10 K) | $\omega_{0}$ | $\omega_{1}$ | $\omega_{3p}$ | $\gamma_{0}$ | $\gamma_{3p}$ | $\gamma_{ep}$ | $\omega_{a}$
---|---|---|---|---|---|---|---|---
$E^{(1)}$ 98.1 |  107.0(1) | 107.3(1) | $\omega_{0}/2$ | 0.18(8) | 1.8(4) | 0.27(6) | 2.3(7) | 47(4)
$E^{(2)}$ 108.6 |  116.3(1) | 116.6(1) | $\omega_{0}/2$ | 0.14(19) | 1.6(5) | 0.36(8) | 2.0(9) | 39(19)
$A_{1}^{(1)}$ 112.8 |  124.9(1) | 125.2(1) | $\omega_{0}/2$ | 0.60(2) | 0.9(8) | 0.59(14) | 9.9(15) | 61(1)
$B_{1}^{(1)}$ 159.1 |  141.6(6) | 142.6(4) | $\omega_{0}/2$ | 0.50(60) | 0(17) | 1.0(30) | 24(29) | 71(5)
$E^{(3)}$ 162.0 |  166.2(2) | 167.2(1) | $\omega_{0}/2$ | 0.86(2) | 3(10) | 0.7(18) | 2.1(77) | -14(645)
$A_{1}^{(2)}$ 207.9 |  219.9(0) | 221.1(1) | $\omega_{0}/2$ | 1.85(5) | 8.2(1) | 0.22(3) | – | –
$A_{1}^{(3)}$ 240.8 |  255.2(0) | 256.6(4) | 52(6) | 1.4(4) | 5.1(1) | 0.84(4) | – | –
$E^{(4)}$ 252.7 |  267.1(3) | 268.8(15) | 61(22) | 1.6(15) | 1.4(9) | 2.4(3) | – | –
Figure S5: Kohler scaling analysis of MR data shown in main text (Fig. 7). Any
change of slope related to the electron phonon drag in the current data is
unclear. Figure S6: (a) Temperature dependence of $C_{P}$ measured between
2-300 K, which shows that the high temperature region approaches the Dulong-
Petit limit. (b) Temperature dependence of $C_{P}$ measured in different
applied magnetic fields, showing the superconducting transition,
$T_{\text{c}}$$0.39(1)$ K in zero-field, that gets suppressed with applied
field. Fit to the data above $T_{\text{c}}$ to the form $\gamma$ and $\beta$
taking into account the electronic and the phononic contributions. The
phononic contribution was subtracted to obtain the $C_{\rm{el}}$ in the main
text.