# Role of electronic correlations in the Kagome lattice superconductor LaRh3B2

Savita Chaudhary  Department of Physical Sciences, Indian Institute of Science
Education and Research (IISER) Mohali, Knowledge City, Sector 81, Mohali
140306, India.  Shama  Department of Physical Sciences, Indian Institute of
Science Education and Research (IISER) Mohali, Knowledge City, Sector 81,
Mohali 140306, India.  Jaskaran Singh  Department of Physical Sciences, Indian
Institute of Science Education and Research (IISER) Mohali, Knowledge City,
Sector 81, Mohali 140306, India.  Department of Physics, Punjabi University,
Patiala, 147002, India.  Armando Consiglio  Institut für Theoretische Physik
und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat,
Universität Würzburg, 97074 Würzburg, Germany  Domenico Di Sante  Department
of Physics and Astronomy, Alma Mater Studiorum, University of Bologna, 40127
Bologna, Italy  Center for Computational Quantum Physics, Flatiron Institute,
162 5th Avenue, New York, New York 10010, USA  Ronny Thomale  Institut für
Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence
ct.qmat, Universität Würzburg, 97074 Würzburg, Germany  Yogesh Singh
Department of Physical Sciences, Indian Institute of Science Education and
Research (IISER) Mohali, Knowledge City, Sector 81, Mohali 140306, India.

###### Abstract

LaRh3B2 crystallizes in a layered structure where Rh atoms form a perfect
Kagome lattice. The material shows superconductivity at $T_{c}\approx 2.6$ K
and no signature for density wave instabilities. We report our measurements of
electronic transport, magnetization, and heat capacity in the normal and
superconducting state, and derive normal and superconducting parameters. From
first principles calculations of the electronic band structure, we identify
all features of Kagome bands predominantly formed by the Rh $d$ orbitals: a
flat band, Dirac cones, and van Hove singularities. The calculation of the
phonon dispersions and electron-phonon coupling suggests a strong similarity
between LaRh3B2 and AV3Sb5 (A=K,Cs,Rb). For LaRh3B2, it matches quantitatively
with the observed $T_{c}$, supporting a conventional phonon mediated pairing
mechanism. By comparison to the $A$V3Sb5 family, we conjecture a reduced
importance of electron correlations in LaRh3B2.

##  I Introduction

The kagome lattice has long been a playground for novel physics in condensed
matter. Insulating kagome lattice realizations with localized magnetic moments
are platforms to explore the effects of geometric magnetic frustration. The
quantum spin liquid (QSL) ground state in the mineral Herbertsmithite
ZnCu3(OH)Cl2 is a prime example of this behaviour Fu2015 ; Han2012 .
Insulating quantum magnets with a kagome network in higher dimensions have
also shown novel frustrated magnetism and QSL behaviour Balz ; Okamoto2007 ;
Singh2013 . More recently metallic kagome lattice materials have been brought
into focus due to the prediction that the electronic structure of electrons on
a kagome lattice might allow to access correlated Dirac cones or van Hove
singularities near the Fermi energyKiesel2013 ; Wang2013 ; Mazin2014 . The
two-dimensional kagome lattice has features in its band structure which
provide, even still in the itinerant limit, the opportunity of marrying non-
trivial topology and strong electron correlations. Search for realizations of
a material with an ideal isolated kagome lattice is therefore a fundamentally
important quest. In recent years a few families of metallic materials
possessing a kagome lattice have indeed been reported or theoretically
predicted. These include the Herbertsmithite related material Ga/ScCu3(OH)6Cl2
Mazin2014 , the magnetic kagome metals CoSn and FeSn Mingu2020 ; Ming2020 ,
and the ferromagnetic kagome metal YMn6Sn6 Li2021 . Most recently, the
$A$V3Sb5 ($A=~{}$K, Rb, Cs) family of materials have been discovered and shown
to host a perfect kagome network of V ions Ortiz2019 ; Neupert2022 . Evidence
for electron correlations and non trivial topology in these materials emerges
from the discovery of charge density waves, superconductivity, anomalous Hall
effect, and multiple van Hove singularities nearby the Fermi energy Ortiz2019
; Ortiz2020 ; Ortiz2021 ; Yang2020 .

Another family of materials possessing the kagome lattice is $R$T${}_{3}X_{2}$
($R=$ Lanthanide, $T=$ $4$d or $5$d transition metal, $X=$ Si, B). These
materials were discovered in the 1980s Ku1980 ; Barz ; Vandenberg  and several
of them were reported to show superconductivity with $T_{\rm c}$s between $1$
K to $\sim 7$ K Ku1980 ; Barz ; Vandenberg ; Malik ; Athreya ; Rauchschwalbe .
However, most of these studies were not made in the context of the connection
of properties with the underlying kagome lattice. Only recently LaRu3Si2,
which has the highest $T_{\rm c}=7$ K in this family of materials, has been
studied in relation to the kagome lattice, and several unconventional
properties have been reported possibly arising from electron correlations from
the flat bands Li2011 ; Li2012 ; Li2016 ; Mielke2021 . Materials in the
$R$T${}_{3}X_{2}$ family thus form another promising platform to study the
kagome related features in the band structure, and their interplay with
superconductivity.

We report on the electronic structure, phonon profile, and superconducting
properties of LaRh3B2 which has previously been reported to show
superconductivity at low temperatures, where the reported superconducting
$T_{\rm c}$ ranges from $<1.2$ to $2.8$ K Ku1980 ; Malik . Our electronic band
structure calculations reveal a flat band above the Fermi energy, and van Hove
singularities and Dirac cones at several locations in the Brillouin zone
including close to the Fermi energy $E_{\rm F}$. We find that the $E_{\rm F}$
is located at the top of a sharp peak in the density of states (DOS). We use
this to address the extreme sample dependence of the superconducting $T_{\rm
c}$. The superconductivity is found to be of conventional weak coupling type.
This is supported by estimations of the $T_{c}$ from phonon calculations and
the estimate of electron-phonon coupling. The van Hove singularities in the
band structure are found to be located a few eVs away from $E_{\rm F}$, which
is large against the characteristic ordering scales and thus explains why
these materials do not show signals of correlation-induced phenomena such as
charge density waves or other instabilities. This is also supported by the
phonon calculations which stress the absence of any imaginary frequency mode.
In addition, we observe anomalous temperature dependencies of the magnetic
susceptibility and heat capacity, and a slightly enhanced Sommerfeld
coefficient, which we argue to arise from the narrow band which is part of the
DOS. In comparison with the $A$V3Sb5 materials, our results point to a reduced
importance of electronic-correlations in the LaRh3B2 material.

##  II Methods

Polycrystalline samples of LaRh3B2 were synthesized by arc-melting
stoichiometric ratios of La (3N, Alfa Aesar), Rh (5N, Alfa Aesar) and B (6N,
Alfa Aesar). The melted buttons were flipped over and melted $5$–$10$ times to
promote homogeneity. Powder X-ray diffraction (PXRD) on a Bruker D8 Advance
diffractometer system with Cu-K$\alpha$ radiation was used to determine the
phase purity of the arc-melted LaRh3B2 sample. The relative stoichiometry of
La and Rh was confirmed using energy dispersive spectroscopy using a scanning
electron microscope. The dc magnetic susceptibility $\chi$, heat capacity $C$,
and electrical transport were measured using a Quantum Design Physical
Property Measurement System equipped with a He3 insert. To theoretically
simulate the electronic structure of LaRh3B2, we performed first-principles
density functional theory (DFT) calculations using the Vienna Ab initio
simulation package (VASP) Kresse93 ; Kresse94 ; Kresse96a ; Kresse96b . We
considered the projector-augmented wave (PAW) pseudo potential with exchange-
correlation functional of generalized gradient approximation (GGA) of Perdew-
Burke-Ernzerhof Kresse ; Perdew . Starting with the experimental structure,
the lattice relaxation was performed to optimize the crystal structure by
using variable cell relaxation. We adopted a $12\times 12\times 12$ k mesh for
the first Brillouin zone. We have used an energy cut-off of $450$ eV for the
plane wave basis. The convergence criteria for energy and force are set to
$10^{-6}$ eV and $0.02$ eV/Å, respectively. In the DFT calculation, spin-orbit
coupling was not included. However, we have used scalar relativistic potential
which takes scalar relativistic effects into account. Phonon calculations have
been performed using density functional perturbation theory, as implemented in
Quantum Espresso Giannozzi2020 ; Giannozzi2009 ; Giannozzi . Exchange and
correlation effects were included using the generalized gradient approximation
(GGA) with the Perdew-Burke-Ernzerhof (PBE) functional Perdew ; the
pseudopotentials are norm-conserving, with core correction, and scalar
relativistic Hamann .
Self-consistent calculations of the previously relaxed unit cell have been
performed with a 8$\times$8$\times$12 $k$-grid. The kinetic energy cutoff for
the wavefunctions is equal to 100 Ry, while the cutoff for charge density is
400 Ry.
Convergence threshold for ionic minimization and electronic self-consistency
are set to be 1.0D-15.
The self-consistency threshold for phonon calculations is 1.0D-15 as well,
with a $q$-grid of 4$\times$4$\times$2\.
Non-self consistent calculations for the density of states have been performed
with a 60$\times$60$\times$48 $k$-grid.
Finally, the electron-phonon interaction is computed via an interpolation over
the Brillouin Zone Wierzbowska .

Figure 1: (Color online) Powder x-ray diffraction and results of refinement.
Figure 2: (Color online) (a) A schematic of the crystal structure of LaRh3B2
viewed perpendicular to the crystallographic $c$-axis showing the layered
nature of the structure with Rh atomic planes separated along the $c$-axis by
planes made up of La and B atoms. (b) Viewed along the $c$-axis, the Rh atoms
form an undistorted kagome lattice.

##  III Structure

LaRh3B2 crystallizes in a honeycomb structure with space group $P6/mmm$. There
are no variable parameters in the structure apart from the unit cell size. The
powder x-ray diffraction (PXRD) is shown in Fig. 1. The PXRD pattern confirmed
that the synthesized material is single phase and a refinement, shown in Fig.
1, of the powder pattern gave lattice parameters $a=5.486$Å and $c=3.136$Å. In
literature, a range of values for the lattice parameters have been reported
and our values fall within this range of values Ku1980 . We will make a
connection of unit cell parameters with the electronic properties later. A
schematic of the crystal structure of LaRh3B2 is shown in Fig. 2. The
structure is made up of layers of Rh planes separated by planes of La and B
stacked along the $c$-axis, as shown in Fig. 2(a). The arrangement of the Rh
atoms within the Rh-planes is a perfect kagome lattice as shown in Fig. 2(b).
These materials therefore have the structural ingredients to show electronic
structure features expected for a kagome metal. It must be noted however, that
the short $c$-axis necessarily means that coupling between kagome planes may
be significant.

##  IV Results

###  IV.1 Electronic Band Structure

Figure 3: (Color online) (a) The electronic structure of LaRh3B2 along high
symmetry directions in $k$-space. The boxes are to highlight some interesting
features as discussed in the main text (b) The total and partial density of
states as a function of energy measured from the Fermi energy.

Figure 3(a) shows the electronic band structure for LaRh3B2 along some high
symmetry directions in the Brillouin zone. It is evident that several bands
cross the Fermi level, confirming that LaRh3B2 is a metal. The total and
partial density of states (DOS) are shown in Fig. 3(b). The Fermi level
($E_{F}$) is situated near the top of a very narrow band resulting in a fairly
large DOS at $E_{F}$ of $6.6$ states/eV. From the partial DOS it is clear that
the majority contribution to the total DOS comes from Rh $4$d orbitals and
both La and B contribute very small amounts to the total DOS at $E_{F}$. The
narrow band at $E_{F}$ leads to a strong sensitivity of the superconducting
$T_{c}$ on the unit cell size and to other anomalous physical properties as we
will discuss later.

We now turn to the novel features of the band structure arising from the
kagome Rh planes. As can be seen in Fig. 3(a), we observe a flat band (FB) in
the $\Gamma-M-K-\Gamma$ direction about $0.4$ eV above $E_{F}$. This flat band
is separate from any other bands. Another series of disconnected flat bands
are observed along the $\Gamma-A$ direction about $0.75$ eV above $E_{F}$. In
addition to these flat portions of the electronic dispersion, Dirac cones (DC)
are observed at several locations in the band structure. There are Dirac bands
$140$ meV below and $2.75$ eV above $E_{F}$ at $H$ and a Dirac cone about $1$
eV below $E_{F}$ along the $M-K$ direction in the BZ. We also identify van
Hove (VH) singularities located symmetrically above and below the Dirac cone
at 2.75 eV. Thus the band structure of LaRh3B2 possesses the predicted
features of the kagome lattice band structure near $E_{F}$ with modifications
arising most likely from the three-dimensional nature of the material.

###  IV.2 Physical Properties

Figure 4 shows the electrical, magnetic, and thermal properties of LaRh3B2 in
the normal and superconducting states. Figure 4 (a) shows the magnetic
susceptibility $\chi$ versus temperature $T$ between $2$ K and $300$ K in an
applied magnetic field of $H=2$ T. At low temperatures, small amounts of
magnetic impurities lead to a Curie like upturn. The $\chi$ is found to be
temperature dependent in the whole temperature range. This is not what is
expected for a Pauli paramagnetic metal where a $T$ independent $\chi$ is
expected. This $T$ dependent $\chi$ arises due to the $E_{F}$ being situated
on a narrow peak in the DOS. The change in temperature results in a change in
the DOS at $E_{F}$ leading to a $T$ dependent Pauli paramagnetic
susceptibility. To support this idea the $\chi(T)$ in the full temperature
range was fit with the expression
$\chi(T)=\chi_{o}[1-(T/T_{E})^{2}]+C/(T-\theta)$, where the first term
represents the $T$ dependent Pauli paramagnetic susceptibility and the second
term represents the contribution from the small amounts of magnetic impurities
which give rise to the Curie like upturn in $\chi(T)$ at the lowest
temperatures. The fitting parameters are $\chi_{o}$ the temperature
independent average Pauli paramagnetic susceptibility, $T_{E}$ which is a
phenomenological parameter related to the Fermi energy, $C$ which is the Curie
constant of the impurities, and $\theta$ which is the Weiss temperature
representing any interactions between the magnetic impurities. A very good fit
with the above expression was obtained and is shown as the solid curve through
the data in Fig. 4 (a). The fit parameters obtained were
$\chi_{o}=11.8(2)\times 10^{-5}$ G cm3/mol, $T_{E}=860(7)$ K, $C=0.0010(4)$ G
cm3 K/mol, and $\theta=-5.5(1)$ K. This value of $C$ is equivalent to $0.25\%$
of $S=1/2$ impurities, which is quite small.

Figure 4: (Color online) (a) The normal state magnetic susceptibility at $H=2$
T, (b) Resistivity versus temperature at zero field. Inset shows the
superconducting transition at various fields, (c) Dimensionless magnetic
susceptibility ($4\pi\chi$) in the superconducting state at various fields,
and (d) the electronic contribution to the zero field specific heat $C_{\rm
el}/T$ versus $T$.

Figure 4 (b) shows the electrical resistivity $\rho$ in zero field between $2$
K and $300$ K. We observe metallic behaviour with a residual resistivity ratio
RRR $=\rho(300{\rm K})/\rho(2{\rm K})\approx 5$. The inset in Fig. 4 (b) shows
the $\rho(T)$ data below $T=4$ K measured in various applied fields. The sharp
drop to zero resistance below $T_{c}\sim 2.6$ K in zero field signals the
onset of superconductivity in LaRh3B2. Further evidence of a superconducting
state is obtained from the diamagnetism observed in magnetic measurements
shown in Fig. 4 (c) and (d). From Fig. 4 (c) we can see that the value of
$4\pi\chi$ in the superconducting state is greater than $-1$ suggesting
demagnetization factors due to the irregular shape of the sample. The
magnetization data in Fig. 4 (d) show behaviour typical of a Type-II
superconductor. The bulk nature of superconductivity is confirmed from heat
capacity measurements. Figure 4 (e) shows the electronic heat capacity $C_{\rm
el}$ divided by $T$ versus $T$. The $C_{\rm el}$ is obtained by subtracting a
lattice term ($\sim T^{3}$) from the total heat capacity. It is of interest to
note that the $C_{\rm el}/T$ in the normal state would be expected to be $T$
independent. However, this is true only for $T\leq 4.5$ K while there is a
strong $T$ dependence of $C_{\rm el}/T$ above these temperatures. This
suggests that the lattice contribution may not just have a $T^{3}$ term but
an-harmonic terms may also contribute to the lattice heat capacity. A sharp
anomaly at the onset of the superconducting transition can clearly be seen in
Fig. 4 (e). To evaluate the magnitude of the jump in heat capacity at the
transition and to obtain an alternate estimate of the bulk superconducting
$T_{c}$, we use an equal entropy construction. The result is shown as the
solid curve through the data near $T_{c}$. This gives a value $T_{c}\approx
2.5$ K. The normal state data above $T_{c}$ can be extrapolated to $T=0$ to
give an estimate of the Sommerfeld coefficient $\gamma_{\rm n}=11.8$ mJ/mol
K2. With this we obtain the jump height at $T_{c}$ to be $\Delta C_{\rm
el}/\gamma_{\rm n}T_{c}\approx 1.2$, which is smaller than the value $1.43$
expected for a weak coupling single gap superconductor. The $C_{\rm el}$ data
at the lowest temperatures seem to extrapolate to a finite value suggesting
some residual contribution from normal electrons. The $C_{\rm el}$ data below
$T=0.8$ K were fit by the expression $C_{\rm el}=\gamma_{\rm res}T+$
Aexp${}^{-\Delta/k_{B}T}$ assuming a fully gapped superconducting state. The
$\gamma_{\rm res}T$ term represents the contribution from any non-
superconducting fraction of electrons. An excellent fit, shown in Fig. 4 (e),
was obtained with the following values for the parameters $\gamma_{\rm
res}=1.7$ mJ/mol K2 and $\Delta=6$ K. This value of $\gamma_{\rm res}$
suggests that $\approx 14\%$ electrons do not participate in the
superconductivity. So we must revise our estimate of $\Delta C_{\rm el}/\gamma
T_{c}$ using $\gamma=\gamma_{n}-\gamma_{\rm res}$. This gives the value
$\Delta C_{\rm el}/\gamma T_{c}\approx 1.44$ which is close to the value
expected from a weak coupling single gap superconductor. These results suggest
that LaRh3B2 is a weak coupling single-gap Type-II superconductor.

From various measurements in finite magnetic field we can track the $T_{c}$ as
a function of the field $H$. The $H$-$T$ diagram obtained from the various
measurements is shown in Fig. 4 (f) where both the lower critical field
$H_{C1}$ and the upper critical field $H_{C2}$ are shown. We observe that the
$H_{C2}$ data from all measurements except the resistivity measurements agree
with each other, while the critical field measured from $\rho$ are
consistently higher than values measured from other bulk probes like
magnetization and heat capacity. Such observations have been reported
previously in some materials and have been linked to surface superconductivity
Zeinali ; Pradip . It has been shown that the critical field for surface
superconductivity is $\approx 1.69H_{c2}$, where $H_{c2}$ is the bulk critical
field. We also plot in Fig. 4 (f) the $H_{C2}$ obtained from resistivity
measurements divided by $1.69$. The critical field so obtained matches the
critical field values obtained from other bulk measurements. So we will treat
the scaled critical field from resistivity measurements as the true bulk
critical field $H_{c2}$. The $H_{c2}$ vs $T$ plot shows an upward curvature in
the whole temperature range. This is unusual and inconsistent with
observations for most conventional superconductors.

To learn about the strength of the electron-phonon coupling we make an
estimate of the electron-phonon coupling constant $\lambda_{ep}$ using
McMillan’s formula, which relates the superconducting transition temperature
$T_{c}$ to $\lambda_{ep}$, the Debye temperature $\theta_{D}$, and the Coulomb
repulsion constant $\mu^{*}$

$T_{c}=\frac{\Theta_{D}}{1.45}\exp\left[-\frac{1.04\left(1+\lambda_{\mathrm{ep}}\right)}{\lambda_{\mathrm{ep}}-\mu^{*}\left(1+0.62\lambda_{\mathrm{ep}}\right)}\right]$

which can be inverted to give $\lambda_{ep}$ in terms of $T_{c}$, $\theta_{D}$
and $\mu^{*}$ as

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

From the heat capacity measurements, we had obtained $\theta_{D}$ = 518 K and
using $T_{c}$ = 2.5 K, we get $\lambda_{ep}=0.43$ and 0.52 for $\mu^{*}=0.10$
and 0.15, respectively. These values of $\lambda_{ep}$ suggest moderate
electron-phonon coupling in LaRh3B2. This is supported by the estimates of the
$\lambda_{ep}$ made from our phonon calculations which will be discussed
later.

We now present our estimation of various superconducting parameters using
expressions previously collected in Refs. Singh2007, ; Singh2010, . An
estimate of the $T=0$ upper critical field $H_{c2}(0)$ was made by first
making a linear extrapolation of the data near $T_{c}$ to give the slope
${dH_{c2}\over dT}|_{T_{c}}=-511$ Oe/K. This linear slope can then be used to
get an estimate of $H_{c2}(0)$ using the Werthamer-Helfand-Hohenberg (WHH)
formula for the clean limit $H_{c2}(0)=-0.693T_{c}{dH_{c2}\over
dT}|_{T_{c}}=920$Oe. From the value of $H_{c2}$ we can now estimate the value
of the coherence length $\xi$ by the expression $H_{c2}=\phi_{0}/2\pi\xi^{2}$,
where $\phi_{0}=hc/2e=2.068\times 10^{-7}$ G cm2 is the flux quantum. Using
$H_{c2}(0)=920$ Oe obtained above, we estimate $\xi(0)=60$ nm. At $T=2.3$ K
near $T_{c}$ where $H_{c2}=250$ Oe we get $\xi(0)=114$ nm. We have collected
the various normal and superconducting state parameters in Table 1.

Table 1: Normal and superconducting state parameters for LaRh3B2. Here $\gamma$ is the Sommerfeld coefficient, $\beta$ is the coefficient of the $T^{3}$ term in the low temperature heat capacity, $\theta_{\rm D}$ is the Debye temperature, $n$ is the electron density, $\xi$ is the superconducting coherence length, $\lambda$ is the penetration depth, $l$ is the electron mean free path, and $v_{\rm F}$ is the Fermi velocity. RRR | $\approx 5$
---|---
$\gamma$ (mJ/mol K2) | 11.8
$\beta$ (mJ/mol K4) | 0.084
$\theta_{\rm D}$ (K) | 520
$n$ (cm${}^{-3})$ | $0.97\times 10^{23}$
TC(K) | 2.6
$\xi_{2{\rm K}}$(nm) | 114
$\xi_{0{\rm K}}$(nm) | 60
$\lambda_{0{\rm K}}$(nm) | 5.4
$l_{4{\rm K}}$(nm) | 43
$v_{{\rm F}}$(cm/s) | $3.5\times 10^{8}$

We now address the large variation in the $T_{C}$ of LaRh3B2 which has been
reported in the literature Ku1980 ; Malik . Figure 5 shows the resistivity of
two samples of LaRh3B2 prepared with the same nominal ratios of starting
materials. A refinement of their powder x-ray pattern gave lattice parameters
that are slightly different. The lattice parameters for sample 1 (S1) are
$a=5.484$ Å and $c=3.139$ Å while those for sample 2 (S2) are $a=5.486$ Å and
$c=3.136$ Å. Thus the S1 has slightly smaller in-plane lattice parameters
while its $c$-axis is longer indicating that in this sample the kagome planes
are shrunk while the separation between the kagome lattice increases. S2 on
the other hand has a larger kagome plane but the planes are separated by a
smaller distance along the $c$-axis. From Fig. 5 we see that the electrical
transport properties are sensitive to these small changes. S1 has a larger
residual resistivity ratio RRR $=10$ but has a larger residual resistivity
$\rho_{0}=26~{}\mu\Omega$cm. S2 on the other hand has a smaller RRR $=5$ but a
smaller residual resistivity $\rho_{0}=15~{}\mu\Omega$cm. From the inset it
can be seen that while S2 has a superconductivity onset at $T_{c}=2.6$ K, for
S1 the onset is $T_{c}=2.05$ K, more than $0.5$ K lower. We also find this
large variation in $T_{\rm c}$ with unit cell size of LaRh3B2 in previous
reports Ku1980 ; Malik . For example, superconductivity with a $T_{\rm c}=2.6$
K was reported for a LaRh3B2 sample with lattice parameters $a=5.480$Å and
$c=3.137$ Å Ku1980 , a $T_{\rm c}=2.2$ K was reported for a sample with
lattice parameters $a=5.483$Å, $c=3.142$ Å while no superconductivity down to
$1.2$ K was found for a sample with lattice parameters $a=5.512$Å and
$c=3.115$ Å Malik .

We can address this variation in $T_{\rm c}$ for different samples using our
DFT calculations. Our calculations have shown that the $E_{\rm F}$ lies near
the top of a narrow band in the DOS. This sensitivity of $T_{\rm c}$ most
likely originates from changes in the DOS at $E_{F}$ due to small changes in
$E_{F}$ either by pressure effects (as evidenced by difference in unit cell
sizes) or due to a difference in the electron densities (by minute variation
in the stoichiometry) in the samples. A slight change in $E_{F}$ will lead to
large changes in the DOS at $E_{F}$ because the $E_{F}$ is located on top of a
narrow band in the band structure.

Figure 5: (Color online) Resistivity versus temperature at zero field for two
LaRh3B2 samples. Inset shows the variation in the superconducting transition
temperature.

Our phonon calculations support the motif of weak-coupling phonon-mediated
superconductivity, and also indicate the reason for the absence of a
correlation-induced CDW instability for LaRh3B2. The phonon dispersion for
LaRh3B2 are shown in Figs. 6 a, b. The phonon dispersion does not exhibit any
imaginary frequency mode; this is a sign of dynamical stability, confirming
the absence of experimental signatures of charge density wave states. Low-
frequency phonon modes mostly originate from La atoms, while intermediate
frequencies are essentially due to Rh atoms; the manifold of low-dispersing
bands around 120 cm-1 is then due to the kagome network. Finally, B atoms
contribute to the high-frequency modes. We computed the electron-phonon
coupling to be $\lambda_{\mathrm{e-ph}}\approx 0.55-0.65$. This suggests that
LaRh3B2 is a weak-to-moderate coupling superconductor. The McMillan formula
was then used to estimate the superconducting critical temperature $T_{c}$
McMillan ; Carbotte :

$T_{c}=\frac{\omega_{log}}{1.2}e^{\bigl{[}\frac{-1.04(1+\lambda)}{\lambda(1-0.62\mu^{*})-\mu^{*}}\bigr{]}}$
(1)

with $\omega_{log}$ being related to the Eliashberg function:

$\omega_{log}=e^{\bigl{[}\frac{2}{\lambda}\int{\frac{d\omega}{\omega}\alpha^{2}F(\omega)\log{\omega}}\bigr{]}}$
(2)

While the Coulomb pseudo-potential $\mu^{*}$ lies in the typical range [0.1 -
0.2], we obtain values for $T_{c}$ in fair agreement with the experimental
results. Specifically, $T_{c}\approx 2.6$ K is obtained for $\mu^{*}=0.17$
(Fig. 6 d).

Figure 6: a) Phonon dispersion along high-symmetry lines; b) Corresponding
density of states; c) Eliashberg function; d) Superconducting critical
temperature $T_{c}$ as a function of the Coulomb pseudo-potential $\mu^{*}$.

##  V Summary and Discussion:

We report on the electronic structure, phonon spectrum and physical properties
of a kagome lattice superconductor LaRh3B2. The structure of LaRh3B2 is built
up of kagome planes of Rh stacked along the $c$-axis with La-B planes
separating the kagome planes. The electronic structure contains all features
expected for a 2D kagome lattice including a flat band and Dirac bands and van
Hove singularities at various positions in the Brillouin zone and in
particular near $E_{F}$. This is qualitatively consistent with the band-
structure observed for the $A$V3Sb5 kagome metals. In contrast with the
$A$V3Sb5 materials, however, we did not observe signatures of strong
electronic correlations in LaRh3B2. The van Hove singularities in the
electronic band structure are situated further away from $E_{F}$ than for
$A$V3Sb5, which is a fermiological reason why one would not expect charge
ordering or density wave like instabilities as reported for $A$V3Sb5.

The superconductivity in LaRh3B2 seems conventional, and there is no
experimental evidence for charge density wave instabilities as reported for
the $A$V3Sb5 materials. The majority contribution to the DOS at $E_{F}$ in
LaRh3B2 derives from Rh $4$d bands. This suggests that the role of electronic
correlations is weakened in LaRh3B2 compared to the family of $A$V3Sb5 kagome
metals, since the Rh $4d$ orbitals are less strongly-correlated than the V
$3d$ orbitals. Interestingly, the computed $\lambda_{\mathrm{e-ph}}$ for
LaRh3B2 is in good agreement with experimentally reported values of
$\lambda_{\mathrm{e-ph}}$ for the CsV3Sb5 compound Zhong  which, together with
the phonon dispersions, suggests a similarity in the principal phonon sector
between LaRh3B2 and $A$V3Sb5. It suggests that a central difference between
LaRh3B2 and the by now more established kagome metals must be found in terms
of differing electronic correlations, which are lower in strength for LaRh3B2.
This may explain the absence of CDW in the LaRh3B2 kagome metal, and points
towards phonon-mediated $s$-wave superconductivity. Given the large number of
materials in the $RT_{3}$B2 and $RT_{3}$Si2 families, their possibility of
Fermiology-tuning around $E_{F}$ presents an exciting direction for future
work.

_Acknowledgments.–_ We thank the X-ray facility at IISER Mohali. JS
acknowledges UGC-CSIR India for a fellowship. The phonon-DFT work was
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through Project-ID 258499086-SFB 1170 and by the Würzburg-Dresden
Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat
Project-ID 390858490-EXC 2147. The research leading to these results has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie Grant Agreement No.
897276. The authors acknowledge the Gauss Centre for Supercomputing e.V. for
providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz
Supercomputing Centre.

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