# Signatures of van Hove singularities in the anisotropic in-plane optical
conductivity of the topological semimetal Nb3SiTe6

J. Ebad-Allah  Experimentalphysik II, Institute for Physics, University of
Augsburg, D-86135 Augsburg, Germany  Department of Physics, Tanta University,
31527 Tanta, Egypt  A. A. Tsirlin  Felix Bloch Institute for Solid-State
Physics, Leipzig University, 04103 Leipzig, Germany  Experimentalphysik VI,
Center for Electronic Correlations and Magnetism, Institute for Physics,
University of Augsburg, D-86135 Augsburg, Germany  Y. L. Zhu  2D Crystal
Consortium, Materials Research Institute, Pennsylvania State University,
University Park, PA 16802, USA  Department of Physics, Pennsylvania State
University, University Park, Pennsylvania 16802, USA  Z. Q. Mao  2D Crystal
Consortium, Materials Research Institute, Pennsylvania State University,
University Park, PA 16802, USA  Department of Physics, Pennsylvania State
University, University Park, Pennsylvania 16802, USA  C. A. Kuntscher
<EMAIL_ADDRESS>Experimentalphysik II, Institute
for Physics, University of Augsburg, D-86135 Augsburg, Germany

###### Abstract

We present a temperature-dependent infrared spectroscopy study on the layered
topological semimetal Nb3SiTe6 combined with density-functional theory (DFT)
calculations of the electronic band structure and optical conductivity. Our
results reveal an anisotropic behavior of the in-plane ($ac$-plane) optical
conductivity, with three pronounced excitations located at around 0.15, 0.28,
and 0.41 eV for the polarization of the incident radiation along the $c$ axis.
These excitations are well reproduced in the theoretical spectra. Based on the
ab initio results, the excitations around 0.15 eV and 0.28 eV are interpreted
as fingerprints of van Hove singularities in the electronic band structure and
compared to the findings for other topological semimetals.

##  I Introduction

Novel layered Dirac materials hosting nontrivial band crossings in the
vicinity of the Fermi level (EF) attract considerable attention in condensed-
matter research due to their unusual physical properties and phenomena such as
anisotropic electron transport Hu  _et al._ (2015), chiral anomaly Son and
Spivak (2013), nodal chains Wang  _et al._ (2017), hourglass dispersions Li
_et al._ (2018), drumhead-like states Yu  _et al._ (2017), van Hove
singularities Chen  _et al._ (2015); Biswas  _et al._ (2020); van Heumen  _et
al._ (2019); Kim  _et al._ (2018); Neupane  _et al._ (2015); Havener  _et al._
(2014), and surface superconductivity Heikkilä  _et al._ (2011). The layered
ternary telluride compounds $\textit{M}_{3}$SiTe6 (M= Nb and Ta) are one class
of these materials, where the band structure calculations predicted several
nontrivial band features near EF Li  _et al._ (2018); Sato  _et al._ (2018);
Naveed  _et al._ (2020). Nb3SiTe6 is a van-der-Waals layered material with the
crystal structure very similar to that of MoS2 Lee  _et al._ (2010), and can
be thinned down to atomically thin 2D crystals Hu  _et al._ (2015); Zhu  _et
al._ (2020). Bulk Nb3SiTe6 has an orthorhombic symmetry with the space group
$Pnma$ Ohno (1999). The layers stack via van-der-Waals forces, forming bundles
of sandwich layers with the order Te-(Nb,Si)-Te. Each Te-(Nb,Si)-Te layer is
composed of face- and edge-sharing NbTe6 prisms with Si ions inserted into
interstitial sites among these prisms, as illustrated in Fig. 1(a). We also
depict in Fig. 1(b) the first Brillouin zone of bulk Nb3SiTe6 with the high-
symmetry points.

In the absence of the spin-orbit coupling (SOC), the electronic band structure
of $\textit{M}_{3}$SiTe6 contains (i) a nodal loop related to the linear-band-
crossing points along the $\Gamma-Y$ and $\Gamma-Z$ paths, and (ii) a fourfold
nodal line formed along the $S-R$ path Li  _et al._ (2018); Sato  _et al._
(2018); Naveed  _et al._ (2020). Adding SOC leads to several new features, for
instance: (i) gapping the nodal loop around the $\Gamma$ point, (ii) fourfold
degeneracy of each band along the paths $U-X$, $R-U$, and $Z-S$, and (iii) the
emergence of an hourglass Dirac loop along the $S-R$ path instead of the nodal
line as well as along $S-X$. The degeneracies of features (ii) and (iii)
result from the nonsymmorphic space group symmetry. Furthermore, the charge
carrier mobility along the a direction was predicted to be much higher due to
the Dirac dispersion along this direction, leading to a strong anisotropy in
the electronic properties.

Temperature-dependent resistivity measurements on a bulk sample of Nb3SiTe6
showed typical metallic behavior with an anisotropy along in-plane and out-of-
plane directions, related to the specific bonding state of Nb ions. In
particular, according to the projected band structure and density of states
the conduction bands crossing the Fermi level are mainly derived from Nb $4d$
orbitals Hu  _et al._ (2015). Furthermore, an in-plane anisotropy is also
expected for the $\textit{M}_{3}$SiTe6 compounds due to the large difference
between the in-plane lattice parameters Ohno (1999). Consistently, an angle-
resolved photoemission spectroscopy study reported a strong anisotropy of the
Fermi surface of Ta3SiTe6 Sato  _et al._ (2018). In Nb3SiTe6, both hole and
electron pockets exist, hence in the bulk the presence of free charge carriers
with different scattering rates is expected. The hole-type charge carriers are
suggested to prevail in the transport properties of thin flakes An  _et al._
(2018), while their density will be affected by the deficiency of Te atoms
Pang  _et al._ (2020).

Despite Nb3SiTe6 revealing very interesting electronic properties, its optical
conductivity has not been studied yet. In this paper we investigate the
temperature-dependent in-plane ($ac$-plane) optical conductivity of bulk
Nb3SiTe6 single crystal, obtained by frequency-dependent reflectivity
measurements for the polarization directions E$\|a$ and E$\|c$. The optical
conductivity shows an anisotropic behaviour between the two in-plane
polarization directions. Several pronounced interband excitations are observed
in the optical conductivity along the $c$ axis, which sharpen as the
temperature decreases. Based on density-functional theory (DFT) calculations
we relate these excitation peaks to specific transitions between electronic
bands and propose that they are related to van Hove singularities in the
electronic band structure.

Figure 1: (a) Crystal structure of Nb3SiTe6. (b) First Brillouin zone of
Nb3SiTe6 with the notation of the high-symmetry points. Figure 2: Temperature-
dependent reflectivity spectra of Nb3SiTe6 for the polarization directions (a)
E$\|a$ and (b) E$\|c$. Inset of (a): Comparison between the reflectivity
spectra at 6 K for the two polarization directions over a broad frequency
range. (c) and (d) Temperature-dependent optical conductivity $\sigma_{1}$
spectra of Nb3SiTe6 for E$\|a$ and E$\|c$, respectively. Inset of (c):
Comparison between the $\sigma_{1}$ spectra at 6 K for the two polarization
directions in the entire measured range. Inset of (d): Difference spectra
$\Delta\sigma_{1}$ calculated according to
$\Delta\sigma_{1}(\omega,T)=\sigma_{1}(\omega,T)-\sigma_{1}(\omega,295~{}K)$.
The gray area marks the full-width-at-half-maximum of the L1 peak at 295 K.

##  II SAMPLE PREPARATION AND EXPERIMENTAL DETAILS

Single crystals of Nb3SiTe6 were grown using chemical vapor transport with a
mixture of Nb, Si, ant Te at a molar ratio of 3:1:6. During synthesis, the
temperature of the hot and cold ends of the double-zone tube furnace was set
at 950∘C and 850∘C, respectively Hu  _et al._ (2015); An  _et al._ (2018).

The temperature-dependent reflectivity measurements at ambient pressure were
performed between 295 and 6 K in the frequency range from 0.025 to 2.48 eV
(200 to 20000 cm-1). Measurements have been conducted on a single crystal on a
freshly cleaved E$\|ac$ surface. A silver layer was evaporated onto half of
the sample surface to serve as a reference, and the obtained spectra have been
corrected with the mirror reflectivity later on. The sample was mounted on a
cold-finger microcryostat. The measurements were carried out for the in-plane
polarization directions E$\|a$ and E$\|c$ using an infrared microscope (Bruker
Hyperion), equipped with a 15$\times$ Cassegrain objective, coupled to a
Bruker Vertex 80v FT-IR spectrometer.

The Kramers-Kronig (KK) relations were applied to transform the reflectivity
spectra into the complex optical conductivity
$\sigma(\omega)=\sigma_{1}(\omega)+i\sigma_{2}(\omega)$ and the complex
dielectric function
$\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$. The
extrapolation of the reflectivity data were done in a manner similar to our
previous publications Ebad-Allah  _et al._ (2019); Köpf  _et al._ (2020);
Ebad-Allah  _et al._ (2021). To this end, the reflectivity was extrapolated to
low frequencies based on a Drude-Lorentz fit, while for the high-frequency
extrapolation we used the x-ray atomic scattering functions Tanner (2015). To
obtain the contributions to the optical conductivity, the reflectivity and
optical conductivity spectra were simultaneously fitted with the Drude-Lorentz
model.

Band structure of Nb3SiTe6 was calculated in the Wien2K code Blaha  _et al._ ;
Blaha  _et al._ (2020) using the Perdew-Burke-Ernzerhof (PBE) type of the
exchange-correlation potential Perdew  _et al._ (1996). Lattice parameters and
atomic positions from Ref. Li  _et al._ (1992) were employed without further
optimization. The corresponding notation of the high-symmetry points is shown
in Fig. 1(b). Charge density was converged on the $8\times 4\times 4$
$k$-mesh. Consequently, optical conductivity was calculated with the internal
routines of Wien2K Ambrosch-Draxl and Sofo (2006) on the dense $24\times
12\times 12$ mesh.

##  III RESULTS AND DISCUSSION

The temperature-dependent reflectivity spectra of Nb3SiTe6 for the in-plane
polarization directions E$\|a$ and E$\|c$ are shown in Fig. 2 (a) and (b),
respectively, and in Fig. S1 in the Supplemental Material Sup . The plasma
edge in the reflectivity and the increase of the low-frequency reflectivity
level during cooling down, which almost approaches unity at 6 K for both
polarization directions, reveals the metallic nature of the compound
consistent with transport measurements Hu  _et al._ (2015); Sato  _et al._
(2018); An  _et al._ (2018); Pang  _et al._ (2020). The plasma edge in the
reflectivity depends not only on the temperature but also on the polarization
direction as illustrated in the inset of Fig 2 (a). Figures 2 (c) and (d)
display the temperature-dependent real part of the optical conductivity
$\sigma_{1}$ for both polarization directions, as derived from the
reflectivity spectra through KK relations. Corresponding plots on a lin-log
scale can be found in Fig. S1 in the Supplemental Material Sup . For both in-
plane polarization directions, the $\sigma_{1}$ spectrum shows intraband
excitations at low frequencies described by Drude contributions, which become
sharper during cooling down due to reduced scattering.

For the further analysis and discussion of other excitations (besides the
intraband transitions) we divide the measured energy range into two regions:
(i) the low-energy region (energies between 0.08 eV and 0.7 eV) and (ii) the
high-energy region (energies from 0.7 eV up to 2.25 eV). In the low-energy
region, the $\sigma_{1}$ spectrum along both axes shows a drop at around 0.1
eV followed by several polarization-dependent interband excitations at
energies below 0.7 eV. Interestingly, most of the observed excitations in this
energy range hardly shift with decreasing temperature but only sharpen. This
temperature evolution of the low-energy excitations can be explained by a
simple approach taking the temperature dependence of the Fermi-Dirac
distribution function into account, causing a sharpening of the low-energy
interband transitions with cooling. This approach was recently demonstrated
for the Weyl semimetal TaP Yaresko and Pronin (2021). In the high-energy
region, $\sigma_{1}$ exhibits a monotonic increase with increasing frequency
overlaid with various excitations, whose positions depend on the polarization
direction. The overall changes in the optical conductivity in this energy
region during cooling down are modest for both $a$ and $c$ axis.

Figure 3: Real part of the optical conductivity $\sigma_{1}$ of Nb3SiTe6 at 6
K for (a) E$\|a$ and (b) E$\|c$ together with the total Drude-Lorentz fit and
the various fitting contributions (D: Drude term, L: Lorentz term). Figure 4:
Temperature-dependent effective plasma frequency of Nb3SiTe6 for the
polarization directions E$\|a$ ($\omega_{p,a}$) and E$\|c$ ($\omega_{p,c}$).
Figure 5: (a) Comparison between the experimental and DFT interband
conductivity $\sigma_{1,interband}$ for energies up to 2.2 eV for the
polarization directions E$\|a$ and E$\|c$ at 6 K. (b) Calculated band
structure of Nb3SiTe6 with SOC. (c) Temperature-dependent experimental
interband conductivity $\sigma_{1,interband}$ below 1.0 eV for the
polarization directions E$\|c$ compared with the DFT interband conductivity
$\sigma_{zz}$. The colored peaks are the fitting contributions for L1, L2, L3
using the Lorentz model. (d) Contributions of different band combinations to
the optical conductivity $\sigma_{zz}$. (e) The experimental
$\sigma_{1,interband}$ at 6 K for the polarization E$\|a$ compared with the
DFT interband conductivity $\sigma_{xx}$. (f) Contributions of different band
combinations to the optical conductivity $\sigma_{xx}$.

Focusing on the experimental spectra at the lowest studied temperature 6 K,
one observes that the optical response contains several pronounced
contributions for E$\|c$ compared to E$\|a$, as illustrated in the insets of
Figs. 2(a) and (c). In order to distinguish the optical contributions of each
axis, we fitted the $\sigma_{1}$ spectra (and simultaneously the reflectivity
spectra) using the Drude-Lorentz model. As an example, we display in the two
panels of Fig. 3 the fitting of the $\sigma_{1}$ spectrum together with the
fitting contributions for E$\|a$ and E$\|c$ at 6 K (see also Figs. S2 and S3
in the Supplemental Material Sup ). From the fitting we obtained the following
contributions for the polarization E$\|c$: (i) Below 0.7 eV, the $\sigma_{1}$
spectrum consists of two Drude contributions related to the itinerant charge
carriers Pang  _et al._ (2020). The presence of different types of free
carriers is reasonable according to the reported electronic band structure, as
already mentioned in the Introduction. In fact, two Drude components had to be
included in the model for the E$\|c$ polarization direction to obtain a
reasonable fit quality. Most interestingly, the E$\|c$ $\sigma_{1}$ spectrum
shows a sharp peak-like excitation (L1) at around 0.15 eV, followed by two
less sharp but still pronounced excitations at around 0.28 and 0.41 eV (L2 and
L3, respectively), and a shoulder (L4) at around 0.49 eV. The main interband
contributions (L1 – L3) to the E$\|c$ optical conductivity are also
highlighted in Fig. 5(c). (ii) Above 0.7 eV, another three high-energy
excitations overlay the monotonic increase, and are positioned at
approximately 0.93 eV, 1.5 eV, and 2.0 eV [see Fig. 3(b)].

For the polarization direction E$\|a$ [see Fig. 3(a)], one Drude term was
sufficient for obtaining a good fit quality. However, for consistency reasons,
we included two Drude contributions for this polariziation direction as well,
like for the E$\|c$ optical spectrum, accounting for different types of
carriers. The observed interband excitations below 0.7 eV are smeared out
compared to E$\|c$. In addition, most of the observed excitations below and
above 0.7 eV are located at slightly different energies, namely at around 0.12
eV, 0.29 eV, 0.41 eV, and 0.52 eV in the low-energy region, and at around 0.9
eV, 1.5 eV, and 2.2 eV in the high-energy region [see Fig. 3(a) for the
fitting contributions]. These differences between the observed excitations for
the two directions give a clear evidence for the in-plane anisotropy of
Nb3SiTe6, consistent with the in-plane anisotropy of the crystal structure,
which is further illustrated by the large difference between the in-plane
lattice parameters $a$ and $c$ ($a$=6.353 ${\AA}$, $b$=11.507 ${\AA}$,
$c$=13.938 ${\AA}$ according to Ref. Li  _et al._ (1992)). An in-plane
anisotropy was also observed in the Fermi surface of the sister compound
Ta3SiTe6 Sato  _et al._ (2018).

The in-plane anisotropy is also revealed by the effective plasma frequency
$\omega_{p}$, which was calculated from the plasma frequencies $\omega_{p1}$
and $\omega_{p2}$ of the two Drude contributions according to
$\omega_{p}=\sqrt{\omega_{p1}^{2}+\omega_{p2}^{2}}$. The temperature-dependent
plasma frequency for the two measured polarization directions is shown in Fig.
4. For ${\bf E}\|a$ $\omega_{p}$ decreases as the temperature decreases,
namely from $\omega_{p,300K}$ = 1.18 eV to $\omega_{p,6K}$ = 1.09 eV. Such a
temperature dependence is expected for a semimetal, with less free carriers at
low temperatures Crassee  _et al._ (2018). Along the $c$ axis, $\omega_{p}$
exhibits an unusual behaviour, where it slightly increases from
$\omega_{p,300K}$ = 1.05 eV to $\omega_{p,6K}$ = 1.11 eV upon cooling (see
Fig. 4). Generally, a direct relation should be expected between the
electronic kinetic energy and the Drude spectral weight, which is directly
proportional to $\omega_{p}^{2}$. An increase in $\omega_{p}$ with cooling
thus indicates an increase in electronic kinetic energy, which can be due to
reduced electron correlation effects, i.e., reduced effective electronic mass
Xu  _et al._ (2020); Shao  _et al._ (2020). Another possible reason could be a
temperature-induced shift of the Fermi level, as was recently demonstrated for
the semimetal ZrTe5 Xu  _et al._ (2018). However, a Fermi level shift should
affect the plasma frequency for both polarization directions in the same
manner, which is inconsistent with our results.

The temperature-dependent $\sigma_{1}$ spectrum for E$\|c$ suggests a
considerable redistribution of the spectral weight from low to high
frequencies upon cooling [see Fig. 2(d)]. The difference spectra
$\Delta\sigma_{1}$, defined as
$\Delta\sigma_{1}(\omega,T)=\sigma_{1}(\omega,T)-\sigma_{1}(\omega,295K)$,
illustrate the temperature-induced reshuffling of the spectral weight among
various contributions. Such kind of analysis is in particular interesting for
the ${\bf E}\|c$ optical conductivity spectrum with the most pronounced
excitations. We observe that the spectral weight redistribution mainly occurs
between the Drude contributions and the mid-infrared excitations [see inset of
Fig. 2 (d)]. At first sight, it seems that some Drude spectral weight is
transferred to the L1 excitation during cooling down. However, when taking
into account the full-width-at-half-maximum of the L1 peak at room
temperature, as indicated by the grey area, it is clear that the spectral
weight of the L1 peak is decreased in certain energy ranges during cooling.
For better illustration, we plot in Fig. S4 in the Supplemental Material Sup
the difference spectra $\Delta\sigma_{1}$ together with the temperature-
dependent L1 Lorentz peak. From our detailed fitting analysis of the
experimental data we find that the Drude spectral weight is slightly
increasing with decreasing temperature (increasing plasma frequency
$\omega_{p}$ for E$\|c$, see Fig. 4), whereas the spectral weight of the L1
peak (as well as the L2 peak) is decreasing, as will be discussed in more
detail later. The transfer of the spectral weight indicates the reconstruction
of the electronic energy bands over the energy range below 0.7 eV.

For the interpretation of the observed excitations in the experimental optical
conductivity spectra, we carried out DFT electronic band structure
calculations and calculations of the optical conductivity. Hereby, SOC was
taken into account. The calculated electronic band structure, as depicted in
Fig. 5(b), is in agreement with earlier results Li  _et al._ (2018). Namely,
we observe the gapping of the nodal loop around $\Gamma$ point, the appearance
of an hourglass Dirac loop along the $S-R$ and $S-X$ paths, and the degeneracy
of the bands along several paths. The electronic band structure contains
several nontrivial bands near EF, mainly originating from Nb $4d$ orbitals Hu
_et al._ (2015), as evidenced by the partial density of states depicted in
Fig. 6 (b). For the further discussion, we group adjacent electronic bands
into pairs, labeled A, B, C, and D, with the C and D bands lying in direct
vicinity of $E_{F}$. The theoretical plasma frequencies from the DFT
calculations amount to 0.138 eV for E$\|a$, 0.699 eV for E$\|b$, and 0.943 eV
for E$\|c$. Accordingly, the one for E$\|c$ matches the experimental value,
while the one for E$\|a$ is lower compared to the experimental result.

Figure 5 (a) displays a comparison between the calculated interband
conductivity $\sigma_{xx}$ and $\sigma_{zz}$ together with the corresponding
experimental results $\sigma_{1,interband}$ at 6 K for the polarization
directions E$\|a$ and E$\|c$, respectively, over a broad energy range. As DFT
provides only the interband contribution, we subtracted the Drude terms from
the experimental $\sigma_{1}$ spectra and show $\sigma_{1,interband}$.
Obviously, the overall experimental interband conductivity spectra for both
polarization directions are well reproduced by theory, especially in the low-
energy region below 1 eV, which will be the main focus in the following. The
somewhat less favorable agreement above 1 eV is probably caused by
inaccuracies in the description of the excited states within DFT. It is
important to note that the agreement between theory and experiment is in
particular obvious for the E$\|c$ polarization direction, since the
theoretical $\sigma_{zz}$ spectrum also shows a low-energy three-peak profile
with peak energies 0.15, 0.26, and 0.38 eV, corresponding to the L1, L2, and
L3 peaks at positions 0.15, 0.28, and 0.41 eV in the measured $\sigma_{1}$
spectrum.

The calculated optical conductivity reveals a tiny peak below 0.1 eV for both
polarization directions, which is absent in the experimental spectra and most
likely hidden behind the Drude contribution. This tiny peak might be due to
transitions within the hourglass Dirac loop very close to EF. One furthermore
observes that the experimental E$\|c$ $\sigma_{1,interband}$ below the first
sharp L1 excitation is approximately linear in frequency and temperature
independent, although the L1 peak itself shows a significant temperature
dependence in its intensity. A similar behavior was recently observed for the
nodal-line semimetal BaNiS2 Santos-Cottin  _et al._ (2021). Here, temperature-
dependent peaks in the theoretical optical conductivity spectra, located at
the high-energy limit of a temperature-independent isosbestic linear-in-
frequency conductivity, were associated with van Hove singularities (VHS’s).
These VHS’s are related to saddle points of the electronic bands which result
from the connections between Dirac cones in the reciprocal space. Although the
sharp peaks were not fully developed in the experimental optical data of
BaNiS2, it was suggested that such sharp VHS’s in $\sigma_{1}$ are signatures
of open Dirac nodal lines Santos-Cottin  _et al._ (2021). In case of BaNiS2 a
pronounced transfer of the spectral weight from the Drude contributions to the
sharp peaks via the temperature-independent isosbestic line was observed
during cooling down, in contrast to our findings. It is important to note that
the appearance of a VHS is not a generic feature of open Dirac nodal lines.
For example, for dispersive open Dirac nodal lines the linear-in-frequency
conductivity may end with a flat spectral response without any noticeable
peaks Ahn  _et al._ (2017); Shao  _et al._ (2019).

Figure 6: (a) Calculated electronic band structure of Nb3SiTe6. (b) Total
density of states together with the partial density of states for Nb $4d$, Si
$3p$, and Te $5p$. The two vertical arrows mark the possible electronic
transitions, which could explain the two low-frequency absorption peaks L1 and
L2 in the E$\|c$ optical conductivity.

The presence of Dirac nodal lines or loops with corresponding two-dimensional
Dirac electrons is expected to cause characteristic fingerprints in the
optical response, namely a frequency-independent interband optical
conductivity related to the transitions within the Dirac cones Carbotte
(2017); Mukherjee and Carbotte (2017); Ahn  _et al._ (2017). It was
furthermore shown that energy-dispersive nodal lines (in contrast to flat
nodal lines) can cause a linear-in-frequency behavior in the optical
conductivity, similar to semimetals with separate Dirac nodal points Shao  _et
al._ (2019). As mentioned in the Introduction, the electronic band structure
of Nb3SiTe6 contains several nodal lines and loops in the vicinity of EF.
However, these nontrivial features are neither revealed in the experimental
nor in the theoretical interband conductivity spectra [see Fig. 5(a)].

Based on the DFT calculations we can decompose the interband optical
conductivity into contributions of different band combinations. In Figs. 5(d)
and (f) we display the contributions of the interband transitions C-D, B-C,
B-D, A-C, and A-D to the theoretical conductivity spectra $\sigma_{zz}$ and
$\sigma_{xx}$, respectively. In the following, we will focus on understanding
the origin of these excitations and relate them to the observed excitation
features in $\sigma_{1,interband}$.

By using the Drude-Lorentz fit, we managed to separate each contribution to
the experimental $\sigma_{1}$ spectrum and to follow the respective
temperature dependence. A direct comparison between the fitting components L1,
L2, L3, and L4 in the experimental $\sigma_{1,interband}$ spectra and the C-D,
B-C, B-D, A-C and A-D excitations in the theoretical spectra reveals the
following: (i) For E$\|c$, the sharpest peak L1 originates from transitions
between the C-D bands [Figs. 5 (c) and (d)]. The L2 peak is due to transitions
between the B-C bands, while the L3 peak results from transitions between A-C
and B-D bands. The shoulder [Lorentz contribution L4, see Fig. 3(b)] above the
L3 peak can be associated with A-D transitions. However, based on the results
of the DFT calculations we could not locate the positions in momentum space,
where the transitions take place. (ii) For E$\|a$, the lowest-energy peak can
be attributed to C-D transitions, like for E$\|c$ [see Figs. 5 (e) and (f)].
However, the interpretation of the higher-energy contributions to E$\|a$
$\sigma_{1,interband}$ is less straightforward, since the C-D transitions
contribute spectral weight over a rather broad frequency range, namely up to
0.6 eV, hence overlaying the contributions of other band combinations B-C, B-D
etc.

Figure 7: Temperature-dependent (a) oscillator strength and (b) width
($\gamma$) of L1, L2, and L3 excitations for the polarization E$\|c$, obtained
from Drude-Lorentz fittings of $\sigma_{1}$.

Further insight into the origin of these spectral features can be obtained
from the electronic density of states (DOS) shown in Fig. 6(b). There are two
DOS peaks located below the Fermi level and manifest VHS’s. Another VHS is
located right above EF. The accumulation of electronic states around these
energies and the transitions between them [see arrows in Fig. 6(b)] could
cause absorption peaks at energies 0.15 and 0.25 eV, which match the energy
positions of the first two low-energy peaks in $\sigma_{zz}$, and hence could
serve as an explanation for the L1 and L2 peaks in the experimental
$\sigma_{1}$ spectrum [see Fig. 5(c)]. On the other hand, the fact that no DOS
peak accompanies the higher-energy $\sigma_{zz}$ peaks (and thus L3 and L4)
suggests a different origin of these features.

As we mentioned above, the L1, L2, and L3 excitations get sharper on
decreasing temperature, while their energy positions are almost frequency-
independent. Thus, by comparing the temperature dependence of the scattering
rate (width) and oscillator strength of these three excitations we can gain
insight whether these excitations have the same origin or not. In Figs. 7 (a)
and (b) we display the temperature-dependent scattering rate and oscillator
strength, as extracted from the Drude-Lorentz fittings. The L3 excitation
shows a different behaviour compared to L1 and L2: whereas its width is hardly
affected (slight narrowing) during cooling down, its oscillator strength
monotonically increases with decreasing temperature and saturates below 100 K.
This suggests that the L3 excitation has a different origin than the L1 and L2
excitations. Interestingly, both L1 and L2 excitations display a very similar
temperature behaviour: below 250 K their width as well as the oscillator
strength monotonically decrease down to 100 K. The reduction in the width can
be related to the reduced occupation of the bands D that lie immediately above
EF and become less populated when temperature decreases. Below 100 K, both the
scattering rate and oscillator strength for the L1 and L2 excitations are
approximately constant. Since the L1 and L2 excitations show a similar
temperature dependence, we conclude that they have a similar origin,
contrasted with L3. This analysis underpins our assignment of L1 and L2 to the
VHS features of the band structure.

The VHS interpretation of the L1 and L2 excitations is further supported by
the low-dimensional character of the material, which leads to an increasing
probability for the existence of VHS’s in the electronic band structure, like
in the layered Dirac semimetal ZrTe5 Chen  _et al._ (2015). Additionally, the
observed transfer of the spectral weight upon cooling, combined with the
temperature-independent behaviour of $\sigma_{1,interband}$ at energies below
L1, also suggest that these excitations are due to VHS’s related to the nodal
lines. As already mentioned above, a recent optical conductivity study of the
nodal-line semimetal BaNiS2 Santos-Cottin  _et al._ (2021) showed spectral
weight transfer from low to high energies via a temperature-independent
isosbestic line, ending at a VHS, which was proposed to result from the
connections between Dirac cones in the reciprocal space. Pronounced features
in the optical conductivity spectra related to VHS’s have also been observed
in other Dirac materials, such as ZrTe5 Chen  _et al._ (2015); Martino  _et
al._ (2019), PbTe2 van Heumen  _et al._ (2019), $T_{d}$-MoTe2 Santos-Cottin
_et al._ (2020), and TaAs Santos-Cottin  _et al._ (2022), and in the kagome
metals Fe3Sn2 Biswas  _et al._ (2020), Co3Sn2S2 Xu  _et al._ (2020), and
AV3Sb5 with A = K, Rb, Cs Uykur  _et al._ (2021, 2022); Wenzel  _et al._
(2022). These VHS’s may have different origin. In ZrTe5, the quasi-divergent
peak observed in the optical conductivity was interpreted in terms of a VHS in
the joint density of states. In AV3Sb5, they are due to band saddle points,
while in Fe3Sn2 and Co3Sn2S2 the observed sharp absorption peaks in the low-
energy optical conductivity were attributed to the flat bands. In the case of
the Dirac semimetal PbTe2 the decrease and the collapse of the scattering rate
of the low-energy charge carriers (i.e., the reduction in phase space for
scattering) was interpreted as an experimental evidence for a VHS close to the
Fermi level van Heumen  _et al._ (2019).

##  IV CONCLUSION

In summary, we have performed a polarization-dependent in-plane infrared
spectroscopy study of Nb3SiTe6 at low temperature, combined with DFT
calculations of the electronic band structure and optical conductivity. The
comparative experimental and theoretical study revealed a similar profile of
the interband optical conductivity for both in-plane polarization directions,
namely, several peaks due to interband transitions followed by a monotonic
increase overlaid with various high-energy excitations. We found that the
interband excitations along the $c$ axis lead to pronounced and sharp peaks,
in contrast to the less pronounced excitations for E$\|a$, indicating an in-
plane anisotropic behavior of Nb3SiTe6. Based on calculations of the band
structure and optical conductivity, we assign the pronounced peaks at around
0.15 and 0.28 eV in the E$\|c$ optical conductivity to van Hove singularities
in the electronic density of states.

###### Acknowledgements.

C.A.K. acknowledges financial support by the Deutsche Forschungsgemeinschaft
(DFG), Germany, through Grant No. KU 1432/15-1. Z.Q.M. acknowledges financial
support by the US Department of Energy under grant DE-SC0019068.

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