Observation of gapless nodal-line states in NdSbTe
Sabin Regmi1, Robert Smith1, Anup Pradhan Sakhya1, Milo Sprague1, Mazharul
Islam Mondal1, Iftakhar Bin Elius1, Nathan Valadez1 Andrzej Ptok2, Dariusz
Kaczorowski3, Madhab Neupane*1

1Department of Physics, University of Central Florida, Orlando, Florida 32816,
USA
2Institute of Nuclear Physics, Polish Academy of Sciences,
W. E. Radzikowskiego 152, PL-31342 Kraków, Poland
3Institute of Low Temperature and Structure Research,
Polish Academy of Sciences, Okólna 2, PL-50-422 Wrocław, Poland

∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

Lanthanide ($Ln$)-based systems in the $\mathrm{ZrSiS}$-type nodal-line
semimetals have been subjects of research investigations as grounds for
studying the interplay of topology with possible magnetic ordering and
electronic correlations that may originate from the presence of $Ln~{}4f$
electrons. In this study, we carried out a thorough study of a
$Ln\mathrm{SbTe}$ system - $\mathrm{NdSbTe}$ \- by using angle-resolved
photoemission spectroscopy along with first-principles calculations and
thermodynamic measurements. We experimentally detect the presence of multiple
gapless nodal-line states, which is well supported by first-principles
calculations. A dispersive and an almost non-dispersive nodal-line exist along
the bulk $\mathrm{X}-\mathrm{R}$ direction. Another nodal-line is present well
below the Fermi level across the $\mathrm{\Gamma}-\mathrm{M}$ direction, which
is formed by bands with high Fermi velocity that seem to be sensitive to light
polarization. Our study provides an insight into the electronic structure of
an alternative $Ln\mathrm{SbTe}$ material system that will aid towards
understanding the connection of $Ln$ elements with topological electronic
structure in these systems.

I. INTRODUCTION

Experimental discovery of topological semimetal (TSM) phases has unleashed a
flurry of research investigations in semimetallic quantum materials. The
family of TSMs has ever since expanded to Dirac semimetals [1, 2, 3], Weyl
semimetals [1, 4, 5, 6, 7], nodal-line semimetals [8, 9, 10, 11] and beyond
[12, 13, 14, 15]. Different from the point degeneracy at the topological band
crossings between the bulk valence and conduction bands in other classes of
TSM, the nodal-line semimetals feature band crossing that extends along a line
or a loop [8]. Among the nodal-line semimetals, $\mathrm{ZrSiS}$ and similar
$\mathrm{111}$ materials have been the most studied ones because of their
ability to host multiple fermionic states including the nodal-line fermion and
non-symmorphic fermion originating from the square net of Group-IV elements
[10, 11, 16, 17, 18, 19, 20, 21, 22, 23]. In addition to the topological
fermions, these materials may also exhibit exotic properties including flat
optical conductivity, unconventional magnetotransport, and unconventional mass
enhancement [26, 24, 25, 27, 28, 29].

Lanthanide ($Ln$)-based $Ln\mathrm{SbTe}$ systems belonging to the
$\mathrm{ZrSiS}$-type family of materials have recently become attractive to
researchers because of (a) Sb-square net, less studied compared to Si-square
net materials, (b) possible magnetic ordering from the $Ln$ magnetic moments,
and (c) possible correlation effects brought by the $4f$ electrons of $Ln$
elements. Despite these attractions, the studies on the electronic structure
of these $Ln\mathrm{SbTe}$ systems have been limited. One of the earlier works
on one of such systems - antiferromagnetic (AFM) $\mathrm{GdSbTe}$ with Neel
temperature of about $\mathrm{12~{}K}$ [30] \- revealed a nodal-line state as
well as an AFM Dirac state protected by the combination of broken time-
reversal symmetry and rotoinversion symmetry [31]. $\mathrm{CeSbTe}$ has been
shown to be a potential ground for several topological features [32] and also
a more symmetric non-symmorphic Dirac state due to higher spin-orbit coupling
(SOC) than $\mathrm{ZrSiS}$ [33]. While $\mathrm{HoSbTe}$ [34, 35] and
$\mathrm{DySbTe}$ [36] exhibit SOC gap along all high-symmetry directions
leading to weak topological insulating state [36, 37], $\mathrm{LaSbTe}$
features gapless nodal-line that persists even with SOC included [38].
Previous work on $\mathrm{SmSbTe}$ revealed multiple Dirac nodes that
correspond to gapless nodal-line in this material system [39]. Presence of
Dirac nodal-lines in this system was also reported in another study, which
also demonstrated the possibility of Kondo effects and electronic correlation
enhancements [40]. These results demonstrate that the electronic structure of
$Ln\mathrm{SbTe}$ depends on the choice of $Ln$ element, however, a
comprehensive understanding of how the electronic structure and topology in
this family of materials evolve demands more electronic structure studies of
$Ln\mathrm{SbTe}$ systems with different $Ln$ elements. Out of such systems,
$\mathrm{NdSbTe}$ has been reported to feature metamagnetic transitions and
potentially a coexisting Kondo localization [41], which has been reported in
$\mathrm{CeSbTe}$ as well [42, 43, 44]. Although transport and magnetic
measurements have been performed in $\mathrm{NdSbTe}$ [41, 45], the
experimental measurement of the electronic structure is yet to be reported.

Figure 1: Bulk properties of $\mathrm{NdSbTe}$. (a) Nonsymmorphic tetragonal
crystal structure of $\mathrm{NdSbTe}$. (b) Temperature dependence of the
inverse magnetic susceptibility measured in a magnetic field of
$\mathrm{0.5~{}T}$ applied along the tetragonal axis. Solid red line
represents the Curie-Weiss fit. Top left inset: low-temperature magnetic
susceptibility data. Bottom right inset: magnetic field variation of the
magnetization taken at $\mathrm{1.72~{}K}$ with increasing (solid symbols) and
decreasing (open symbols) field strength. Dashed line emphasizes a linear
behavior below the metamagnetic transition. (c) Temperature variation of the
specific heat. Inset: low-temperature data plotted as the specific heat over
temperature ratio versus temperature. (d) Bulk Brillouin zone and its
projection onto $\mathrm{(001)}$ surface Brillouin zone. High symmetry points
are marked. (e) Left: Calculated bulk bands along various high-symmetry
directions without considering the spin-orbit coupling. Right: Calculated bulk
bands with spin-orbit coupling taken into account.

In this paper, we report a detailed study of $\mathrm{NdSbTe}$ through high-
resolution angle-resolved photoemission spectroscopy (ARPES) measurements
complemented by first-principles calculations and thermodynamic
characterizations. Thermodynamic measurements show that $\mathrm{NdSbTe}$ is
AFM with a transition temperature of around $\mathrm{2~{}K}$. Our experimental
data in the paramagnetic phase shows the presence of gapless nodal-lines along
multiple high-symmetry directions in agreement with the predictions from the
first-principles calculations. The bands associated with the nodal-line around
the $\mathrm{\Gamma}$ point across the $\mathrm{\Gamma}-\mathrm{M}$ direction
are observed to be sensitive to light polarization. Couple of nodal-lines are
observed along the $\mathrm{X}-\mathrm{R}$ direction, with one in the vicinity
of the Fermi level and the other residing around $\mathrm{800~{}meV}$ below
it. Overall, this study reveals the electronic structure and topological
states in $\mathrm{NdSbTe}$, for the first time, and presents a valuable
platform towards having an understanding of the interplay of topology and $Ln$
elements in the $Ln\mathrm{SbTe}$ family of quantum materials.

Figure 2: Fermi surface and constant energy contours. (a) ARPES measured Fermi
surface (first panel) and constant energy contours at various binding energies
as noted on top of each plots. (b) Respective Fermi surface and energy
contours obtained from first-principles calculations.

II. METHODS

High-quality single crystals of $\mathrm{NdSbTe}$ were synthesized using
chemical vapor transport technique and characterized using energy dispersive
X-ray spectroscopy and single crystal X-ray diffraction. Thermodynamic (heat
capacity, magnetic susceptibility, magnetization) measurements were performed
using commercial equipment. ARPES experiments were carried out at the Stanford
synchrotron radiation lightsource endstation 5-2 at a temperature of
$18~{}\mathrm{K}$. The energy resolution of the ARPES analyzer was set better
than 20 meV. Density-functional theory [46, 47] based first-principles
calculations were carried out using the Vienna ab initio simulation package
[48, 49, 50] on the basis of projector augmented wave potential [51]. For more
details, see Section 1 in the supplemental material (SM) [52].

III. RESULTS

A. Bulk properties of $\mathrm{NdSbTe}$
The crystal structure of $\mathrm{NdSbTe}$ is presented in Fig. 1(a), where
$\mathrm{Nd-Te}$ atomic layers sandwich the square nets formed by the
$\mathrm{Sb}$ atoms. The structure is tetragonal in the nonsymmorphic space
group number 129 ($P4/nmm$). Figure 1(b) depicts the temperature dependence of
the inverse magnetic susceptibility of $\mathrm{NdSbTe}$ measured in a
magnetic field applied along the crystallographic c-axis. Above
$\mathrm{50~{}K}$, $\chi^{-1}(T)$ can be approximated by the Curie-Weiss (CW)
law (note the solid straight line) with the effective magnetic moment
$\mu\mathrm{{}_{eff}=3.57(1)~{}\mu_{B}}$ and the paramagnetic Curie
temperature $\theta\mathrm{{}_{p}=-17.4(2)~{}K}$. The value of
$\mu\mathrm{{}_{eff}}$ is close to the theoretical Russell-Saunders prediction
for trivalent $\mathrm{Nd}$ ion ($\mathrm{3.62~{}\mu_{B}}$). The negative sign
of $\theta\mathrm{{}_{p}}$ signals AFM exchange interactions. Below
$\mathrm{50~{}K}$, $\chi^{-1}(T)$ deviates from the CW behavior, likely due to
the crystalline electric field (CEF) effect. As displayed in the top left
inset of Fig. 1(b), $\chi(T)$ forms a maximum at $T\mathrm{{}_{N}=2.7~{}K}$
that signals the long-range AFM ordering. Another indication of the AFM
character of the magnetically ordered state is the behavior of the
magnetization measured at $T\mathrm{=1.72~{}K}$ as a function of magnetic
field. As can be inferred from the lower inset to Fig. 1(b), $\sigma(H)$ is
initially linear and exhibits a metamagnetic-like transition near
$\mathrm{2~{}T}$. In stronger fields, the magnetization shows a faint convex
curvature and remains far from saturation. In a magnetic field of
$\mathrm{5~{}T}$, the magnetization attains a value of
$\mathrm{14.2(1)~{}emu/g}$ that corresponds to the magnetic moment of
$\mathrm{1~{}\mu_{B}}$ that is much smaller than the theoretical prediction
for $\mathrm{Nd^{3+}}$ ion ($\mathrm{3.27~{}\mu_{B}}$). This reduction results
mostly from the CEF interaction and single-ion anisotropy. The $\sigma(H)$
variation exhibits no hysteresis effect, in concert with the AFM nature of the
magnetic ground state of $\mathrm{NdSbTe}$. Figure 1(c) presents the
temperature dependence of the specific heat of $\mathrm{NdSbTe}$. Close to
room temperature, $C(T)$ attains a value of about $\mathrm{74~{}J/(mol~{}K)}$
that is nearly equal to the Dulong-Petit limit
($\mathrm{74.8~{}J/(mol~{}K)}$). With decreasing temperature, the specific
heat decreases in an usual manner. The AFM phase transition manifests itself
as a distinct mean-field-like anomaly in $C(T)$. As clearly seen in the inset
to Fig. 1(c), the latter feature exhibits an extended tail above
$T\mathrm{{}_{N}}$ that can be attributed to short-range exchange
interactions. It is worth noting that overall the thermodynamic data collected
for the single crystals of $\mathrm{NdSbTe}$ used in our ARPES study is very
similar to the results obtained before by other research groups [41, 45].

Figure 3:  Surface state along
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$. (a)-(b)
Dispersion maps along
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$ measured
with linear horizontal and linear vertical polarized light with energy
$\mathrm{60~{}eV}$, respectively. (c) Calculated surface spectrum along
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$.

The lattice parameters of $\mathrm{NdSbTe}$ were theoretically optimized, and
the so-obtained values $a=b=\mathrm{4.371~{}\AA}$, and
$c=\mathrm{9.457~{}\AA}$ are close to those reported before
($a=b=\mathrm{4.312~{}\AA}$, and $c=\mathrm{9.371~{}\AA}$ [53]). The Wyckoff
positions occupied by $\mathrm{Nd}$, $\mathrm{Sb}$, and $\mathrm{Te}$ atoms
are respectively: $2c$ ($1/4$,$1/4$,$0.2769$), $2a$ ($3/4$,$1/4$,$0$), and
$2c$ ($1/4$,$1/4$,$0.6279$). In Fig. 1(d), we present the three-dimensional
bulk Brillouin zone and its projection onto the two-dimensional (001) surface
Brillouin zone. The high-symmetry points are marked on both the bulk and
surface Brillouin zones. The left and the right panels in Fig. 1(e) show the
calculated bulk bands along different high-symmetry directions without and
with the consideration of SOC, respectively. Dirac nodal-lines are realized
along the $\mathrm{X}-\mathrm{R}$ and $\mathrm{M}-\mathrm{A}$ directions (due
to $P4/nmm$ symmetry). Within $\mathrm{\pm 1~{}eV}$ of the Fermi level, the
band structure without SOC shows Dirac nodes at the X and R points that
correspond to nodal-lines along the $\mathrm{X}-\mathrm{R}$ direction. Dirac
nodes are also present along $\mathrm{\Gamma}-\mathrm{X}$ and
$\mathrm{Z}-\mathrm{R}$ (shown by black arrows in Fig. 1(e)). However, when
SOC is considered, these Dirac nodes along the $\mathrm{\Gamma}-\mathrm{X}$
and $\mathrm{Z}-\mathrm{R}$ directions gap out. These connect to the gapped
nodal-line along $\mathrm{\Gamma}-\mathrm{M}$ (or $\mathrm{Z}-\mathrm{A}$)
forming the diamond-like structure (see SM Fig. S2 for the exact bulk gap in
the diamond-like structure [52]). The Dirac nodes at the $\mathrm{X}$ and
$\mathrm{R}$ points remain gapless leading to gapless nodal-lines along
$\mathrm{X}-\mathrm{R}$, even with SOC considered (also see Fig. S2 and
Section 2 in the SM [52]). Specifically, the nodal-line near the Fermi level
(enclosed in red rectangle) is dispersive with Dirac node below the Fermi
level at the $\mathrm{X}$ point, which moves upwards and above the Fermi level
on reaching the $\mathrm{R}$ point. The other nodal line that lies slightly
above $\mathrm{-1~{}eV}$ (enclosed in green rectangle) shows very little
dispersion along the $\mathrm{X}-\mathrm{R}$ direction.

B. ARPES measured energy contours and surface state at the
$\mathrm{\overline{\textbf{X}}}$ point.
Figures 2-5 present the results of the ARPES measurements on the
$\mathrm{NdSbTe}$ single crystal. Note that the light source is linear
horizontal (LH) polarized unless specified on top of the figure. The ARPES
measured Fermi surface (FS) in Fig. 2(a) (first panel) is similar to the
$\mathrm{ZrSiS}$-type materials with a diamond shaped pocket centered at the
$\mathrm{\overline{\Gamma}}$ point. The diamond pocket has double sheets, out
of which the inner sheet is strongly visible and the outer one has very faint
intensity. The double sheet nature of the FS is well reproduced in the
calculated FS (Fig. 2(b), first panel). The sheet separation in the momentum
space reduces with increasing binding energy and finally turns into a single
sheet at higher values of binding energies as seen in the constant energy
contours at $\mathrm{300~{}meV}$ binding energy, where a new pocket emerges at
the center of the Brillouin zone. Several bulk pockets appear at higher
binding energies, which can be clearly visualized from the constant energy
contour plots at binding energies of $\mathrm{350~{}meV}$,
$\mathrm{500~{}meV}$, and $\mathrm{700~{}meV}$ (see section 2 in the SM [52]).
The calculated energy contours well reproduce the experimental ones. The
exception of the intense circular pocket at higher binding energy can be
explained from the polarization dependence, which can be seen from the
polarization dependent maps presented in section 3 of the SM [52].

Figure 4: Band dispersion along
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$.
(a)-(b) Dispersion maps along the
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$
direction measured at photon energies of $\mathrm{90~{}eV}$ and
$\mathrm{60~{}eV}$, respectively. Left panel in (b) shows the dispersion map
measured with linear vertical polarization. (c) Dispersion map at
$\mathrm{60~{}eV}$ photon energy with both linear horizontal and liner
vertical data added. Right panel shows the second derivative plot of the
dispersion map in the left panel. (d) Calculated surface band spectrum along
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$.

In order to understand the underlying electronic structure of
$\mathrm{NdSbTe}$ at low temperatures, we analyze the dispersion maps along
various high-symmetry directions in Figs. 3-5. We begin with the dispersion
maps along the
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$, which is
presented in Fig. 3. A gapped Dirac-like state is observed at the
$\mathrm{\overline{X}}$ point. The dispersion of this state does not seem to
depend on the choice of photon energy (see section 5 in the SM [52] for more
photon energy measurements), which implies that this state is surface
originated. The surface state co-exists with bulk states (see slab calculation
in the SM section 4 [52]). As seen in Fig. 3(a), the surface state is strong
in measurement with LH polarized photon source and strongly suppressed in
measurement with linear vertical (LV) polarized light source (Fig. 3(b)). The
calculated surface spectrum is presented in Fig. 3(c), which fairly reproduces
the experimental observation of the gapped surface state at the
$\mathrm{\overline{X}}$ point.

Figure 5: Band dispersion along
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$. (a)
Dispersion map along the
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
direction measured with different photon energy of $\mathrm{90~{}eV}$. (b)
Photon energy dependent dispersion maps along
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$ with
the value of photon energy noted on top of each plot. (c)
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$ band
structure measured with $\mathrm{60~{}eV}$ linear vertical polarized photon
source. (d) Calculated surface projected band structure along the
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
direction.

C. Observation of gapless nodal-line states
Next, we move on to the experimental electronic band dispersion along the
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$
direction. Figure 4(a) represents a dispersion map measured with photon energy
of $\mathrm{90~{}eV}$ along this direction. A single band can be seen crossing
the Fermi level. Similar band dispersion is observed in measurement carried
out with $\mathrm{60~{}eV}$ photon energy (Fig. 4(b), left panel). This band
has almost linear dispersion up to $\mathrm{\sim 0.8~{}eV}$ below the Fermi
level and is slightly converging towards the $\mathrm{\overline{\Gamma}}$
point. The Fermi velocity is very large ($\mathrm{11.43\pm 0.89~{}eV\AA}$ as
obtained from the slope of the linear fit of peaks in the momentum
distribution curves within $\mathrm{-360~{}meV}$ binding energy, see Fig. 4(b)
left panel) compared to the $\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$
bands in $\mathrm{ZrSiS}$ [54]. The observation of a double sheet FS means
that there should be two bands crossing the Fermi level along this direction.
Strong suppression of outer sheet in the FS map and the observation of single
band extending upto the Fermi level along
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$ in
the measurements with LH polarization pointed to the possibility of
polarization dependence of the outer band. We, therefore, performed
measurement with LV polarized light (Fig. 4(b), right panel), where we again
observed a single band crossing the Fermi level. This band is also almost
linear above $\mathrm{~{}-500~{}meV}$. Interestingly, unlike the band observed
in LH polarization, this band seems to slightly diverge away from the
$\mathrm{\overline{\Gamma}}$ point. In Fig. 4(c), we present the dispersion
map along
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$ with
the LH and LV polarization matrices added. We can clearly observe two bands
crossing the Fermi level, out of which the inner one is sensitive to LH
polarization and the outer one is sensitive to LV polarization (also see
section 3 in the SM [52]). This explains the double sheet nature of the
diamond pocket in the FS. These two bands cross each other at around
$\mathrm{300~{}meV}$ below the Fermi level and disperse almost linearly across
this crossing point giving an impression of a gapless nodal-line along
$\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$ direction. This can be
clearly visualized in the second derivative plot presented in Fig. 4(c) (right
panel), where the crossing between the bands is shown by two red-colored
dashed lines. The calculated band structure along this direction is presented
in Fig. 4(d), which clearly shows this crossing feature. Despite the
experimental observation of the gapless nodal-line in experiments, one
important thing to note is that the bulk calculation shows gapped nature along
this direction (Fig. 1(d)) and the calculation in Fig. 4(d) is integrated for
all values of $k_{z}$. The $k_{z}$ plane dependent bulk bands presented in
section 6 of the SM [52] also shows a gapped nature for any particular value
of $k_{z}$, which is undetectable in ARPES data. Such a nodal-line with
undetectable gap within the experimental resolution was reported in a similar
$Ln\mathrm{SbTe}$-type material [40]. The ARPES measurement in the vacuum
ultraviolet region has poor $k_{z}$ resolution and hence the data observed in
the experiments probably covers a certain range of $k_{z}$ value rather than a
certain $k_{z}$ plane. The other possibility could be because of the
discrepancy between experimental data and theoretical calculations that is
usual for metallic/semimetallic systems.

In Fig. 5, we present the dispersion maps along the
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
direction. The measured dispersion map along this direction with a photon
energy of $\mathrm{90~{}eV}$ (Fig. 5(a)) shows a linearly dispersing band
(green dashed line) crossing the Fermi level around the
$\mathrm{\overline{X}}$ point. In addition, another linear band touching is
present at the $\mathrm{\overline{X}}$ point about $\mathrm{760~{}meV}$ below
the Fermi level. In Figs. 5(b-f),
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$ band
dispersion measured with different photon energies are presented. The linear
dispersion in the vicinity of the Fermi level seems to be strongly photon
energy dependent. This photon energy dependence is indicative of the bulk
origination of the bands associated with the linear dispersion. The crossing
point of the linear dispersion lies well below the Fermi level at a photon
energy of $\mathrm{56~{}eV}$ ($k_{z}\sim\mathrm{0}$, Fig. 5(f)) and it shifts
upwards and above the Fermi level on increasing the photon energy up to
$\mathrm{70~{}eV}$($k_{z}\sim\mathrm{\pi}$, Fig. 5(b)). This agrees well with
the theoretical prediction of the dispersive and gapless nodal-line along the
$\mathrm{X}-\mathrm{R}$ direction. The presence of gapless nodal-line in the
vicinity of the Fermi level has been recently reported in similar compounds
$\mathrm{SmSbTe}$ [39] and $\mathrm{LaSbTe}$ [38]. The band-touching feature
around $\mathrm{760~{}meV}$, on the other hand, seems to have very little
change with photon energy variation, in agreement with the almost
dispersionless and gapless nodal-line along $\mathrm{X}-\mathrm{R}$ direction
as seen in the bulk-band calculations. This feature is also observed to be
sensitive to LH polarization and is strongly suppressed when the incident
photon is LV polarized (see Fig. 5(g)). The observation of the Dirac nodes
along the
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
direction is well reproduced in the calculated surface electronic structure
shown in Fig. 5(h) (also see section 6 in the SM [52]).

III. CONCLUSION

In summary, through high-resolution ARPES measurements supported by
theoretical calculations, we studied the detailed low-temperature electronic
structure in $Ln$ based $\mathrm{ZrSiS}$-type material $\mathrm{NdSbTe}$. The
thermodynamic measurements indicated an AFM transition in this material with
Neel temperature, $T_{N}\mathrm{\sim 2~{}K}$. Our ARPES results in the
paramagnetic phase detect three gapless nodal lines in this material system -
two along bulk $\mathrm{X}-\mathrm{R}$ direction and one around the
$\mathrm{\Gamma}$ point formed by steep bands with high Fermi velocity that
constitute the diamond shape. Although theory predicts that the nodal-line
around the $\mathrm{\Gamma}$ point is gapped, experimental results show
gapless feature. The theoretical calculations well reproduce the experimental
results. This work presents $\mathrm{NdSbTe}$ as a multiple nodal-line fermion
system and provides a new platform to understand the evolution of topology and
electronic structure among the $Ln\mathrm{SbTe}$ family of materials.

ACKNOWLEDGMENTS

M.N. acknowledges the support from the National Science Foundation under
CAREER award DMR-1847962 and the Air Force Office of Scientific Research MURI
Grant No. FA9550-20-1-0322. D.K. was supported by the National Science Centre
(Poland) under research grant 2021/41/B/ST3/01141. A.P. acknowledges the
support by National Science Centre (NCN, Poland) under Projects No.
2021/43/B/ST3/02166 and also appreciates the funding in the frame of
scholarships of the Minister of Science and Higher Education (Poland) for
outstanding young scientists (2019 edition, No. 818/STYP/14/2019). The use of
Stanford Synchrotron Radiation Lightsource (SSRL) in SLAC National Accelerator
Laboratory is supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. We thank
Makoto Hashimoto and Donghui Lu for the beamline assistance at SSRL endstation
5-2.

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Supplemental Material

1\. METHODS
1.1 Experimental techniques
Crystal Structure and Sample Characterization

Synthesis of single-crystalline NdSbTe was carried out in two steps. First, a
stoichiometric mixture of the elemental constituents (purity:
$\mathrm{Nd~{}–~{}99.9~{}wt.\%}$, $\mathrm{Sb~{}–~{}99.999~{}wt.\%}$,
$\mathrm{Te~{}–~{}99.999~{}wt.\%}$) was placed in an evacuated quartz ampoule
and slowly heated up to $\mathrm{1000~{}\degree C}$ in a resistance furnace,
dwelled at this temperature for 7 days, and then cooled quickly to room
temperature. The so-obtained polycrystalline material was utilized as a charge
in single crystals growth performed by chemical vapor transport method with
iodine as a transport agent. The evacuated quartz tube was placed in a two-
zone furnace with the precursor hold at $\mathrm{950~{}\degree C}$ and the
crystal growth taking place in a colder end of the tube maintained at
$\mathrm{850~{}\degree C}$. After 20 days, black shiny crystals with platelet
shape and dimensions up to $\mathrm{1\times 1\times 0.2~{}mm^{3}}$ were
obtained (see Fig. S1 inset). They were found stable against air and moisture
for few days only.

Chemical composition and phase homogeneity of the prepared crystals were
determined by energy-dispersive X-ray (EDX) analysis performed using a FEI
scanning electron microscope equipped with an EDAX Genesis XM4 spectrometer.
The EDX results proved single-phase character of the specimens and indicated
the stoichiometry close to $\mathrm{1:1:1}$ (see Fig. S1).

Figure S1: Exemplary result of the microprobe analysis of the chemical
composition of one of the $\mathrm{NdSbTe}$ crystals used in the physical
properties measurements. Inset shows the single crystals of $\mathrm{NdSbTe}$
grown by chemical vapor transport method.

Crystal symmetry of the obtained single crystals was examined on an Oxford
Diffraction X’calibur four-circle single-crystal X-ray diffractometer equipped
with a CCD Atlas detector. The experiment confirmed a tetragonal unit cell of
the $\mathrm{ZrSiS}$-type with the lattice parameters close to those reported
in the literature [1, 2]. Crystallinity and crystallographic orientation of
the crystals selected for physical measurements were determined by means of
Laue X-ray backscattering technique implemented in a LAUE-COS (Proto) system.

Thermodynamic properties Measurements

Magnetic properties were investigated in the temperature range
$\mathrm{1.72-400~{}K}$ and in magnetic fields up to $\mathrm{5~{}T}$ using a
Quantum Design MPMS-5 superconducting quantum interference device (SQUID)
magnetometer. Heat capacity was measured in the temperature interval
$\mathrm{0.37-300~{}K}$ employing the relaxation technique and
$\mathrm{two-\tau}$ model implemented in a Quantum Design PPMS-9 platform
equipped with a $\mathrm{{}^{3}He}$ refrigerator.

Spectroscopic Characterization

Spectroscopic characterization of the electronic band structure using ARPES
was performed at the Stanford Synchrotron Radiation Lightsource endstation 5-2
at a temperature of $\mathrm{18~{}K}$. The energy resolution of the
experimental set up was set better than $\mathrm{0.02~{}eV}$. The samples
mounted on copper posts were loaded and then cleaved inside the ARPES chamber
maintained at ultra-high vacuum better than $\mathrm{10^{-10}~{}Torr}$ before
carrying out the measurements.

1.2 Computational details

The density-functional theory (DFT) [3, 4] based first-principles calculations
were performed using projector augmented-wave (PAW) potentials [5] implemented
in the Vienna Ab initio simulation package VASP [6, 7, 8]. Calculations were
made within the generalized gradient approximation (GGA) in the Perdew, Burke,
and Ernzerhof (PBE) parameterization [9]. The energy cutoff for the plane-wave
expansion was set to $\mathrm{350~{}eV}$. The structural parameters (lattice
constants and atomic positions) were theoretically optimized because of the
lack of experimental data about $z_{\mathrm{Nd}}$ and $z_{\mathrm{Te}}$
values. The optimization was performed in the conventional unit cell using the
$15\times 15\times 7$ k-point grid in the Monkhorst–Pack scheme [10]. The
convergence was checked for these values. The excellent agreement of the
lattice constants suggests a realistic value of the atom position. The atomic
position in other compounds with similar structure e.g.,
$\mathrm{Sm}\mathrm{Sb}_{0.93}\mathrm{Te}_{1.07}$, $z_{\mathrm{Sm}}=0.7244$
(or $0.2756$) and $z_{\mathrm{Te}}=0.3734$ (or $0.6266$) [11], for
$\mathrm{HoSbTe}$, $z_{\mathrm{Ho}}=0.2762$ and $z_{\mathrm{Te}}=0.3761$ (or
$0.6239$) [12], for $\mathrm{TbSbTe}$ $z_{\mathrm{Tb}}=0.2758$ and
$z_{\mathrm{Te}}=0.3756$ (or $0.6244$) [12], is close to our theoretical
result, i.e., $z_{\mathrm{Nd}}=0.2769$ and $z_{\mathrm{Te}}=0.6279$. In the
optimization procedure, we assumed the AFM order, while the $4f$ states of Nd
were treated within the DFT+$U$ scheme introduced by Liechtenstein et al. [13]
(with $\mathrm{U=4.0~{}eV}$ and $\mathrm{J=0.5~{}eV)}$. As a break of the
optimization loop, we took the condition with an energy difference of
$\mathrm{10^{-6}~{}eV}$ and $\mathrm{10^{-8}~{}eV}$ for ionic and electronic
degrees of freedom.

The electronic band structure was also evaluated within Quantum ESPRESSO [14,
15, 16] with PAW GGA PBE pseudopotential included in PsLibrary [17]. Exact
results of the DFT calculations (with $\mathrm{Nd}$ $4f$ states as core
states) were used to find the tight binding model in the basis of the
maximally localized Wannier orbitals [18, 19, 20]. It was performed using the
Wannier90 software [21, 22, 23]. In our calculations, we used the $8\times
8\times 6$ full ${\bm{k}}$-point DFT calculation, starting from the $d$
orbitals of $\mathrm{Nd}$, and $p$ orbitals for $\mathrm{Sb}$ and
$\mathrm{Te}$. The wannierization procedure (i.e., procedure of the tight
binding model finding) is based on the exact DFT band structure. Within this
procedure, storage of each wavefunction for all ${\bm{k}}$-points are
necessary. Next, the Bloch wavefunction from DFT are used in the construction
of the “rotation” and overlap matrices. More technical details can be found in
the documentation of wannier90 or Ref. [18]. Finally, the found $22$-orbital
tight binding model of the $\mathrm{NdSbTe}$ was used for the investigation of
the surface Green’s function for semi-infinite system [70] using WannierTools
[25] software. Within this calculation, the tight binding models of the bulk
system is used to perform the properties of the system with “edge” (e.g.
surface). More detailed theoretical description can be found in Ref. [24]. In
order to match the experimental results, the Fermi level was shifted upwards
by $\mathrm{200~{}meV}$ in the calculations.

Figure S2: Calculated bulk bands. (a) Bulk-band structure in the presence of
spin-orbit coupling. (b-c) Gapped state formed by the bands 2 and 3 in (a) in
the diamond-like structure for $k_{z}=0$ (b) and $k_{z}=\pi$ (c).

2\. CALCULATED BULK BANDS INCLUDING SPIN-ORBIT COUPLING

In S2(a), we reproduce the bulk band calculation including spin-orbit coupling
presented in main text Fig. 1(e) with four bands labeled 1 – 4 in different
colors. The bands 3 and 4 produce crossings at the $\mathrm{X}$ and
$\mathrm{R}$ points (enclosed in red circles). These crossings produce a
nodal-line along $\mathrm{X}-\mathrm{R}$ in the vicinity of the Fermi level
(enclosed in red colored rectangle). The crossing starts well below the Fermi
level at the X point ($k_{z}=0$) and shifts upward and above the Fermi level
on reaching the R point ($k_{z}=\pi$) giving rise to a nodal-line that is
dispersive in energy axis. The bands 1 and 2 also produce crossings at the
$\mathrm{X}$ and $\mathrm{R}$ points (enclosed in green circles), forming
another nodal-line $\mathrm{X}-\mathrm{R}$ (enclosed in green rectangle). The
bands 2 and 3 are responsible for the diamond shaped pockets seen in the FS
and energy contours. These bands form a gapped state (enclosed in black
circles) along $\mathrm{\Gamma}-\mathrm{X}$ and $\mathrm{\Gamma}-\mathrm{M}$
($k_{z}=0$) as well as $\mathrm{Z}-\mathrm{R}$ and $\mathrm{Z}-\mathrm{A}$
($k_{z}=\pi$). The exact bulk gap in the diamond-like structure formed by
bands 2 (upper) and 3(lower) can be seen in Figs. S2(b) and S2(c),
respectively.

3\. EXPERIMENTAL AND THEORETICAL ENERGY CONTOURS

Figure S3: FS and energy contours measured using a photon source of
$\mathrm{90~{}eV}$. (a) FS and (b)-(f) energy contours at binding energy noted
on top. The left panel in each plot represents the ARPES measured energy
counter and the right panel represents the calculated energy contour.

Figure S3 presents the ARPES measured energy contours (left panels) and
corresponding calculated energy contours (right panels) at the Fermi level and
various binding energies as noted on top of each experimental plot. The
measurements are done using $\mathrm{70~{}eV}$ photon energy. A typical
$\mathrm{ZrSiS}$-type FS containing a $\mathrm{\Gamma}$ point centered diamond
pocket is observed. The diamond pocket has double sheets (the inner one is
intense and the outer one is faint because of polarization, see section 3
below), which is well reproduced in first-principles calculation. Double
sheets merge towards each other and around $\mathrm{-300~{}meV}$ constant
energy contour evolving into a single diamond pocket, where an extra inner
pocket emerges. Several bulk pockets appear at higher binding energies, which
can be clearly visualized from the constant energy contour plots at binding
energies of $\mathrm{350~{}meV}$, $\mathrm{500~{}meV}$, and
$\mathrm{700~{}meV}$. The experimental energy contours show fine agreement
with the calculated ones. The exception of the intense circular pocket at
higher binding energy can be explained from the polarization dependence (see
section 4 below).

4\. FS AND ENERGY CONTOURS MEASURED WITH $\mathrm{60~{}eV}$ PHOTON ENERGY

Figure S4 shows the ARPES measured energy contours at the FS and various
binding energies by using LH (left panels) and LV (right panels) polarized
photon source of energy $\mathrm{60~{}eV}$. The FS contains a diamond-shaped
pocket centered at the $\mathrm{\overline{\Gamma}}$ point. This pocket has a
double sheet nature, where the sheets are sensitive to polarization of the
incident photon. While the inner sheet is strong in LH polarized light, the
outer sheet becomes prominent in LV polarized light. The sheets merge into a
single-sheet on going down to higher binding energies creating a Dirac-like
crossing obtained along the
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$
direction. Several bulk pockets emerge at higher binding energies, some of
which show strong polarization dependence. For example, the hole pocket that
appers at the center of surface Brillouin zone at a binding energy of
$\mathrm{-500~{}meV}$ is more sensitive to LV polarization and the pockets
emerging at the corner of the surface Brillouin zone at a binding energy of
$\mathrm{-700meV}$ shows strong sensitivity to LH polarization.

Figure S4: FS and energy contours measured using a photon source of
$\mathrm{60~{}eV}$. The left panels are measured using LH polarization and the
right panels are measured using LV polarization. Corresponding binding
energies are noted on top for each energy contours.

5\. SLAB CALCULATIONS

In Fig. S5, we present the slab calculations along the
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$,
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$, and
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
directions. The grey bands represent the surface projected bulk bands, red
bands are surface bands for $\mathrm{Sb}$ square net termination (termination
in the experimental data), and the blue bands are the surface states with
$\mathrm{NdTe}$ plane termination. The comparison with our experimental data
shows that the termination of the sample is on the $\mathrm{GdTe}$ plane. The
STEM images for (001) surface of $\mathrm{GdSbTe}$ was presented Ref. [26],
which shows that the $\mathrm{NdTe}$ termination is preferred. It can suggest
general features of this type of compounds and termination on the
$R\mathrm{Te}$ ($R$ = rare earth) plane.

Figure S5: Slab calculations along the
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$,
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$, and
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
directions. Color of the line correspond to the surface termination as
labeled.

6\. BAND STRUCTURE ALONG
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$ AT
DIFFERENT PHOTON ENERGIES

In Fig. S6, we present the ARPES measured electronic band dispersion along the
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$ direction
measured at different photon energies as noted on top of each plot. The gapped
Dirac-like feature seems to have a little or no change in dispersion with
photon energy variation implying the associated bands are coming from the
surface. The surface bands co-exist with the bulk bands as seen from the slab
calculation in Fig. S5 (blue bands).

Figure S6: ARPES measured band dispersion along
$\mathrm{\overline{M}}-\mathrm{\overline{X}}-\mathrm{\overline{M}}$ measured
at a photon energy of (a) $\mathrm{90~{}eV}$, (b) $\mathrm{80~{}eV}$, (c)
$\mathrm{70~{}eV}$, and (d) $\mathrm{60~{}eV}$.

7\. EXPERIMENTAL AND CALCULATED ELECTRONIC STRUCTURE ALONG
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$ AND
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$

In Fig. S7, we present the comparison of the experimental and calculated band
structure along the
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$ and
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$
directions. The experimental band dispersion in Fig. S7(a) is obtained from
the addition of matrices of LH polarized and LV polarized data. A gapless
Dirac crossing can be observed around $\mathrm{300~{}meV}$ below the Fermi
level. This feature is well resolved in calculated band structure. The
calculated band structure is the projection from all $k_{z}$ values and a gap
is not resolved, although the bulk band calculations (main text Fig. 1)
predict a gap along $\mathrm{\Gamma}-\mathrm{M}$. A small gap can be observed
in the bulk bands obtained from slab calculations. The disagreement with the
experimental results, where we observe gapless Dirac node, comes from the poor
$k_{z}$ resolution in the vacuum ultra-violet ARPES. The experimental band
dispersion along
$\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$ shows
a linearly dispersing hole band crossing the Fermi level. This band likely
forms the Dirac node above the Fermi level, corresponding to the dispersive
nodal-line along the $\mathrm{X}-\mathrm{R}$ direction. Below
$\mathrm{800~{}meV}$ binding energy, another linear crossing exists between an
electron band and a W-shaped band. These observations are well reproduced in
calculated band calculations. The dispersive nature of the nodal-line in the
vicinity of Fermi level and a minimal dispersion of the nodal-line observed
below $\mathrm{\sim 800~{}meV}$ is well supported by the slab calculated bulk
band dispersion.

Figure S7: ARPES measured band dispersion (left panel), surface projected band
structure calculation (middle panel), and $k_{z}depended$ bulk bands
calculation (right panel) along (a)
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$ and
(b) $\mathrm{\overline{X}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{X}}$.

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