exotic quasiparticles having mass in one direction yet being massless in the perpendicular direction: Semi-Dirac Fermions in a Topological Metal
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Source: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.14.041057
Semi-Dirac Fermions in a Topological Metal
Yinming Shao
<https://journals.aps.org/search/field/author/Yinming%20Shao><https://orcid.org/0000-0002-2891-0028>^1,2,*
,Seongphill Moon
<https://journals.aps.org/search/field/author/Seongphill%20Moon>^3,4
,A. N. Rudenko
<https://journals.aps.org/search/field/author/A%20N%20Rudenko><https://orcid.org/0000-0003-4313-3690>^5
,Jie Wang
<https://journals.aps.org/search/field/author/Jie%20Wang><https://orcid.org/0000-0002-3750-9340>^6,7,8
,Jonah Herzog-Arbeitman
<https://journals.aps.org/search/field/author/Jonah%20HerzogArbeitman><https://orcid.org/0000-0003-4993-3381>^9
,Mykhaylo Ozerov
<https://journals.aps.org/search/field/author/Mykhaylo%20Ozerov><https://orcid.org/0000-0002-5470-1158>^4
,David Graf
<https://journals.aps.org/search/field/author/David%20Graf>^4 ,Zhiyuan
Sun
<https://journals.aps.org/search/field/author/Zhiyuan%20Sun><https://orcid.org/0000-0002-0342-6248>^7
,Raquel Queiroz
<https://journals.aps.org/search/field/author/Raquel%20Queiroz>^1 /et al./
Phys. Rev. X*14*, 041057 –*Published 5 December, 2024*
DOI:https://doi.org/10.1103/PhysRevX.14.041057
<https://doi.org/10.1103/PhysRevX.14.041057>
Abstract
Topological semimetals with massless Dirac and Weyl fermions represent
the forefront of quantum materials research. In two dimensions, a
peculiar class of fermions that are massless in one direction and
massive in the perpendicular direction was predicted 16 years ago. These
highly exotic quasiparticles—the semi-Dirac fermions—ignited intense
theoretical and experimental interest but remain undetected. Using
magneto-optical spectroscopy, we demonstrate the defining feature of
semi-Dirac fermions—𝐵2/3scaling of Landau levels—in a prototypical
nodal-line metal ZrSiS. In topological metals, including ZrSiS, nodal
lines extend the band degeneracies from isolated points to lines, loops,
or even chains in the momentum space. With/ab initio/calculations and
theoretical modeling, we pinpoint the observed semi-Dirac spectrum to
the crossing points of nodal lines in ZrSiS. Crossing nodal lines
exhibit a continuum absorption spectrum but with singularities that
scale as𝐵2/3at the crossing. Our work sheds light on the hidden
quasiparticles emerging from the intricate topology of crossing nodal
lines and highlights the potential to explore quantum geometry with
linear optical responses.
Popular Summary
Among the emerging quasiparticles, semi-Dirac fermions stand out. In 2D
systems, these exotic quasiparticles are thought to have mass in one
direction yet be massless in the perpendicular direction. These peculiar
fermions have so far evaded detection in solid materials. Here, we
present experimental evidence of the defining feature of semi-Dirac
fermions in samples of the metal ZrSiS.
In a typical metal, the presence of an external magnetic field induces
cyclotron motion of electrons, and the ensuing cyclotron energy scales
linearly with the strength of the magnetic field. In graphene, the
presence of massless Dirac fermions leads to a cyclotron energy that
scales with the square root of the field. Semi-Dirac fermions are
predicted to have a different response: Their cyclotron energy scales
with the two-thirds power of the field. Initial proposals to realize
semi-Dirac fermions require stretching graphene until its two Dirac
points—features in the energy band structure that signify the presence
of Dirac fermions—merge in momentum space. However, monolayer graphene
breaks down before reaching the desired strain level.
In ZrSiS, the continuous extension of Dirac points in momentum space,
known as a nodal line, presents exciting opportunities to realize
semi-Dirac fermions. Through high-precision magnetoinfrared
spectroscopy, we observe compelling evidence of the two-thirds power-law
behavior of the cyclotron energy with magnetic field. Combining these
findings with calculations and modeling, we identify semi-Dirac fermions
at special crossings of nodal lines in ZrSiS. Those crossings facilitate
the merging of Dirac points without the need for the unrealistic large
strain required in graphene.
Our work sheds light on the hidden quasiparticles emerging from the
intricate topology and geometry of crossing nodal lines.
Article Text
I. INTRODUCTION
Conventional 2D fermions are described by parabolic energy (𝐸) momentum
(𝐤) dispersion𝐸(𝐤)=ℏ2𝑘2/(2𝑚)with effective mass𝑚. In contrast,
Dirac and Weyl fermions[1,2]have linear dispersion𝐸𝐷(𝐤)=ℏ𝑣𝐹𝑘and
are massless. The striking manifestations of massless Dirac fermions are
revealed through the anomalous half-integer quantum Hall effect[3,4],
Klein tunneling[5,6], and√𝐵scaling of Landau levels (LLs) with magnetic
fields (𝐵)[7–9]in graphene. All these effects are observed in graphene,
where the characteristic√𝐵scaling provides a litmus test for Dirac
quasiparticles.
Semi-Dirac fermions[10–12], with
dispersion𝐸SD(𝐤)=±√(ℏ𝑣𝑘1)2+[ℏ2𝑘22/(2𝑚)]2being linear in one
momentum direction (𝑘1) and quadratic in the orthogonal direction
(𝑘2), have been proposed to appear in materials where multiple Dirac
points merge[10,13]into a semi-Dirac point. Despite intense theoretical
and experimental interests[14–22], semi-Dirac fermions remain
undetected. Strained graphene may be a candidate system to host
semi-Dirac quasiparticles. However, the required uniaxial strain level
is unrealistically large[10,23,24]. Black phosphorus is proposed as
another candidate semi-Dirac fermions system upon strong doping[25]. Yet
the precise semi-Dirac dispersion in black phosphorus has not been
established either experimentally[26]or theoretically[27]. Thus far, the
semi-Dirac dispersion𝐸SDhas been experimentally explored only in
synthetic platforms including honeycomb lattices of ultracold atoms[28],
photonic resonators[29,30], and polaritons systems[31]. Identifying the
fermionic counterpart is crucial to realize the diverse
topological[13,17]and correlated[18,32]phases predicted for semi-Dirac
fermions, but remains challenging in 2D systems. A defining feature of
semi-Dirac fermions is the unique𝐵2/3dependence[10,12]of inter-LL
transitions [Fig. 1(a)]. Here we report on the first observation of this
characteristic𝐵2/3power law in a topological metal, ZrSiS, through LL
spectroscopy.
FIG. 1.
Semi-Dirac fermions at nodal-line crossing points in ZrSiS.
(a) Cyclotron energy (ℏ𝜔𝑐) as a function of magnetic field (𝐵) for
conventional fermions (black line), Dirac fermions (orange line), and
semi-Dirac fermions (purple line). Insets show three-dimensional plots
of their band structures (gray surfaces) overlaid with Landau levels
(red and blue contour lines). (b) The lattice structure of ZrSiS,
showing the square lattice of Si atoms (blue) and the Zr (brown)-S
(yellow) layers above and below. (c) /Ab initio/calculation of the Fermi
surface and nodal-line structure of ZrSiS. Only the𝑘𝑥>0part of the
Fermi surface is shown for better visualization of the nodal-line
structures (gray lines). Black spheres indicate the crossing points
(CPs) of multiple nodal lines. Purple shaded planes
at𝑘𝑥=±𝑘CP1=±0.1971(2𝜋/𝑎)cross the CP1 formed by nodal lines
at𝑘𝑧=0and𝑘𝑥=±𝑘𝑦. The circular purple arrow illustrates the
cyclotron motion around one of the CP1 for magnetic field (green arrow)
applied along𝑘𝑥(𝑎axis of the crystal). Calculated band structure (see
the Appendix and Supplemental Material Sec. VII[33]) near CP1
at𝑘𝑥=𝑘CP1plane shows quadratic dispersion along𝑘𝑧(d) and linear
dispersion along𝑘𝑦(e), characteristic of semi-Dirac fermions.
Semiclassically, a magnetic field induces cyclotron motion and the area
of the cyclotron orbit at energy𝐸is𝑆(𝐸)∝𝐸3/2√𝑚/𝑣for the
semi-Dirac dispersion𝐸SD[10]. Following the Onsager
quantization[34]𝑆(𝐸)=2𝜋(𝑛+𝛾)𝑒𝐵/ℏ, the
characteristic𝐵2/3scaling of LLs is obtained:𝐸𝑛∝[(𝑛+𝛾)𝐵]2/3,
where𝑛is the LL index and𝛾is the phase factor(0≤𝛾<1). The 2D
semi-Dirac spectrum can also arise as singularity points of a continuum
absorption spectrum of a 3D material. For example, the LL spectrum of a
nodal ring𝐸NR=±√[(𝑘2𝑥+𝑘2𝑦−𝑘20)/2𝑚]2+𝑣2𝑧𝑘2𝑧with field in
the𝑥-𝑦plane exhibits a continuum absorption with a lower edge scaling
as𝐵2/3, arising from the semi-Dirac structure (see the Appendix).
The prototypical nodal-line semimetal ZrSiS[35–37][Fig. 1(b)] hosts two
planar nodal squares linked by vertical nodal-lines[38,39], forming a
chainlike structure[40–44][gray lines in Fig. 1(c)] in momentum space.
The low-energy physics of the ZrSiS family of nodal metals is further
enriched by the Fermi energy variations along the Dirac nodal
lines[45–47], reflected by the coexisting electron (blue) and hole (red)
pockets [Fig. 1(c)]./Ab initio/calculations and theoretical modeling
show that the observed semi-Dirac fermions originate from the crossing
points (CPs) of the nodal lines in ZrSiS [black dots in Fig. 1(c)]. Near
the CP, the band structure at𝑘𝑥=𝑘CPshows quadratic [Fig. 1(d)] and
linear dispersion [Fig. 1(e)] along𝑘𝑧and𝑘𝑦, respectively. Under
magnetic field oriented along the𝑎axis (𝐵∥𝑘𝑥), the cyclotron motion
of electrons becomes quantized in the(𝑘𝑦,𝑘𝑧)plane and reflects the
semi-Dirac fermions through the unique LL scaling. The
anticipated𝐵2/3power law in ZrSiS is robust against material
complexities and can be readily identified in infrared magneto-optics
experiments.
II. EXPERIMENTAL RESULTS
We now proceed to the magnetoreflectance spectra𝑅(𝜔,𝐵)normalized by
the zero-field data𝑅(𝜔,0T)for ZrSiS with in-plane magnetic fields up
to 17.5 T (see the Appendix), shown in Fig. 2(a). The most prominent
features are a series of dips in the reflectance spectra hardening with
increasing field (gray dashed lines). For a highly metallic system like
ZrSiS, the infrared reflectance approaches unity and, therefore, dips
in𝑅(𝐵)/𝑅(0)correspond to absorption𝐴(𝜔)=1−𝑅(𝜔)[45,47]. We
attributed these absorption features to interband LL transitions from
massive Dirac fermions (𝐸±𝑛=√2𝑒ℏ|𝑛|𝐵¯𝑣2+Δ2), which exhibit
notable departures from the linear-in-𝐵scaling expected for fermions in
parabolic bands [Fig. 1(a)]. Here,Δis half of the spin-orbit coupling
(SOC) gap[47–50]and we find2Δ≈28 meV, in excellent agreement with
previous lower-field studies[45,51]and calculations[35]. Surprisingly,
above a critical field𝐵𝑐≈7 T, weaker subgap features [red and purple
dots in Fig. 2(a)] appear at around100 cm−1and harden with increasing
field. To better visualize these subgap structures, we report the second
derivative𝑑2𝑅/𝑑𝐵2analysis in Fig. 2(b). The local minima of the
second derivative coincide with the dips in𝑅(𝐵)/𝑅(0)(see
Supplemental Material Sec. III[33]), which we identify as the LL
transition energies in all analyses.
FIG. 2.
Landau level (LL) spectroscopy of ZrSiS with in-plane magnetic fields.
(a) Magnetoreflectance𝑅(𝜔,𝐵)normalized by zero-field
reflectance𝑅(𝜔,0T)for sample S1. Gray dashed lines mark the
positions of the series of interband LL transitions across the
spin-orbit coupling gap (2Δ≈28 meV). The red-shaded region indicates a
potential phonon feature[52–54]that is not dispersing with increasing
field. Above a critical field of𝐵𝑐≈7 T, additional subgap transitions
(red and purple dots) emerge and harden with increasing field. The inset
is a schematic of the experimental configuration with near-normal
incident and unpolarized light while the magnetic field is applied
in-plane (𝐵⊥𝑐,𝐵∥𝑎, Voigt geometry). (b) Second
derivative𝑑2𝑅/𝑑𝐵2data of sample S1 overlaid with model fitting
(gray dashed lines) of the LL transitions across the gapped Dirac cone.
The top schematic shows a gapped Dirac cone with gap2Δand the first LL
transitionsLL−0(−1)→LL1(+0)(gray arrows). Orange lines are the guides
for the subgap LL transitions [red dots in (a)] that follow
approximately√𝐵scaling, characteristic of a Dirac-like fermion (bottom
schematic). The subgap features near the purple-shaded region follow a
distinct𝐵2/3power law and originate from semi-Dirac fermions.
In Fig. 2(b), we show the𝑑2𝑅/𝑑𝐵2spectra for ZrSiS obtained with
in-plane magnetic fields up to 17.5 T. The gray dashed lines denote the
model calculation of interband LL transitions across the gapped Dirac
cone[48,55]:𝐸𝑇=√2𝑒ℏ|𝑛|𝐵¯𝑣2+Δ2+√2𝑒ℏ(|𝑛|+1)𝐵¯𝑣2+Δ2,
where𝑛is the LL index and¯𝑣is the averaged Fermi velocity. The
intraband LL transitions[56], if present, would follow a field
dependence distinct from the observed subgap features (see Supplemental
Material Figs. S8 and S25). Importantly, beyond the two series of subgap
transitions labeled as√𝐵and𝐵2/3(red and purple dots, respectively),
two additional dispersive features are apparent above150 cm−1. As
indicated by the thick orange lines, the dispersions of these latter
features also follow approximately the√𝐵scaling.
The dipole selection rule𝛿|𝑛|=±1[7,9]for Dirac fermions dictates that
the energy ratios of the lowest three interband LL transitions
are1:1+√2:√2+√3. Using a single averaged velocity of 0.82 eV Å, the
three branches of the transitions can be approximated by the lowest
three interband LL transitions from massless Dirac-like fermions (orange
lines) and are labeledLL0→1,LL−1→2, andLL−2→3. Detailed analysis on the
multiple peak splitting ofLL0→1below shows evidence of a small gap and
spin splitting due to Zeeman effect. On the other hand, the remaining
subgap features [near purple shaded region in Fig. 2(b)] follow
sublinear𝐵dependence that is distinct from√𝐵. We will confirm next
that these peculiar LL transitions’ field dependence scales precisely
as𝐵2/3, a fingerprint of semi-Dirac fermions in ZrSiS.
To quantify the power-law scaling of the subgap features, we extract the
transition energies from Fig. 2(b)(see Supplemental Material Sec. III
and Figs. S7 and S8[33]) and plot them against√𝐵and𝐵2/3in Figs.
3(a)and3(b), respectively. Figure 3(a)shows the experimentally
determined LL transition energies (red dots) for the three groups of
transitions labeledLL0→1,LL−1→2,LL−2→3in Fig. 2(b). Remarkably, all
these LL transitions can be understood as originating from a Dirac
fermion with a small gap2Δ′=2.4 meVand Zeeman-split LLs. The resulting
model calculations (orange lines) show good agreement with the data (see
Supplemental Material Figs. S24 and S35). ForLL−2→3, variation
of¯𝑣𝐷shows a logarithmic reduction with increasing𝐵field
[¯𝑣𝐷∝−ln(𝐵); see Fig. S27 in Supplemental Material]. Alternatively,
theLL−1→2andLL−2→3transitions can arise from the cyclotron resonance of
another gapped Dirac cone[56](Fig. S27 of Supplemental Material), and
the exact origin of these transitions awaits future studies.
FIG. 3.
andpower-law behaviors of Landau levels in ZrSiS. (a) Subgap transition
energies (red dots) in Fig. 2are plotted as a function of. Orange lines
represent the fitting based on Dirac-like fermions with an averaged
Fermi velocity, a small gap, and Zeemanfactor. Solid and dashed lines
represent the spin-conserving and spin-flip transition, respectively.
Orange shaded areas indicate the uncertainties infor the() and()
transitions. (b) Higher-energy subgap transitions in Fig. 2(purple dots)
are plotted as a function of, following the exact power-law behavior
expected for semi-Dirac fermions (green dashed lines). The bottom inset
shows the calculated Fermi surface of ZrSiS. The shaded purple plane
indicates theplane, which cuts through CP1 (black dot at). The top inset
shows the calculated band structureversus,for CP1 at the plane, with
linear dispersion alongand quadratic dispersion along, characteristic of
semi-Dirac dispersion.
As we alluded to previously, a series of subgap features displays
thescaling that is characteristic of semi-Dirac fermions [Fig. 3(b)].
Fine splitting of LL transitions is also apparent and all the split
peaks agree with the predicted power-law behavior (green dashed lines)
for semi-Dirac fermions. These latter features are reminiscent of the
spin and valley splitting of Landau levels in Dirac fermions[57,58]and
we discuss several possible scenarios for peak splitting in the
Supplemental Material Sec. IV and Figs. S10–S13[33]. Because of the
nonanalytical nature of the LLs of semi-Dirac fermions[10], the
selection rules have only been explored numerically for type-I
semi-Dirac fermions. Nevertheless, we discuss several possibilities for
the absence of additional high-order semi-Dirac LL transitions in ZrSiS
in the Supplemental Material Sec. IX and Fig. S28.
Importantly, the semi-Dirac fermions in ZrSiS are confirmed both
from/ab initio/calculations and from theoretical modeling of the
crossing points of nodal lines. As shown in the bottom inset of Fig.
3(b), there are two nonequivalent CPs in ZrSiS, labeled as CP1 (at) and
CP2 (at). The observedscaling LL transition is dominated by CP1 since
the energy of the CP is very close to the Fermi level, while the energy
of CP2 is about 0.1 eV below the Fermi level (see Fig. S19[33]). The
calculated semi-Dirac bands near the CP1 [top inset of Fig. 3(b)] are
also asymmetric in, in contrast to the usual type-I semi-Dirac
dispersion[10]:. We demonstrate below that a unique semi-Dirac fermion
that originates from the merging of three Dirac points[13]is realized
near CP1 in ZrSiS, distinct from the merging of two Dirac
points[10,14]realized in a single nodal ring (see Supplemental Material
Video 1 [33]).
III. THEORY AND CALCULATION
We now turn to the theoretical interpretation of the results. The
complex 3D nodal-line cage obtained in density functional theory (DFT)
reveals 8 CPs each in theandplane. Building on the 2D Hamiltonian for
square net motifs[59], we add the 3D hoppings () and obtain a minimal
four-band (not including spin) tight-binding model which reproduces all
nodal lines and CPs, as shown in Fig. 4(see Supplemental Material Sec. V
for details[33]):
(1)
where,represent the sublattice and,orbital degrees of freedom,
respectively. Theandare the in-plane and out-of-plane lattice constants,
respectively. Despite the complexity of the band structure, this model
is analytically solvable, and yields closed form expressions for the
nodal lines in terms of a small number of physical parameters. This
model not only captures the global features of the nodal cage over the
entire Brillouin zone [Fig. 4(b)], but also allows us to obtainmodels at
each CP derived directly from the microscopic hoppings. In contrast to
the intersection of two straight nodal lines[41]or two nodal rings[44],
the CP1 in ZrSiS is composed of a straight nodal line and a curved nodal
line [see Figs. 1(c)and4], which can be described by a minimum two-band
model for the CP:
(2)
whereis the identity matrix,are the Pauli matrices,is the effective mass
along thedirection,is the Fermi velocity alongdirection,controls the
electron-hole asymmetry (tilt of the nodal line along),controls the
dispersion along(parallel to the in-plane nodal line), andis half of the
SOC gap. As shown in Fig. 5(a), the spectrum of Eq. (2)describes the
crossing of a straight in-plane nodal line (NL1, bounded by
theandplanes) and a curved vertical nodal line (NL2, bounded by
theandplanes). The Fermi surface (FS) of the model [Fig. 5(a)] captures
the main features of the FS in ZrSiS near the CP2 [bottom inset of Fig.
3(b)]. In particular, the crescent-shaped contour refers to the purple
dashed line in Fig. 5(a)at the CP (purple dashed line) is consistent
with band structure calculations of ZrSiS and further corroborated with
quantum oscillation measurements (Supplemental Material Figs. S5 and
S6[33]). Under the external magnetic field along, the 2D plane normal to
the field at negativewill cross NL1 once and cross NL2 twice, forming
three isolated Dirac points (D1 and). As the 2D plane moves to the
right, the three Dirac points get closer and merge at(purple shaded
plane), realizing a type-II semi-Dirac fermion[13]:([Fig. 5(b)].
FIG. 4.
Nodal-line structure of ZrSiS. The 3D nodal-line structure of ZrSiS
calculated using DFT (a) and the tight-binding model (b). The
tight-binding model Eq. (1)(Supplemental Material Sec. V[33]) captures
faithfully the complex structure of nodal-line crossings in DFT. Gray
dashed lines connect the high symmetry points at. Blue spheres indicate
the location of the four symmetry-related crossing points at(CP1). The
crossing of curved vertical nodal line and straight horizontal nodal
line at CP1 hosts semi-Dirac fermions. (c) Expanding the tight-binding
model (b) near CP1 and retaining the leading order terms inleads to the
two-band continuum model Eq. (2). The CP1 is formed by two nodal lines:
the vertical nodal line defined bywith finite curvature and the
horizontal nodal line onwith negligible curvature.
FIG. 5.
Semi-Dirac fermions and quantum geometry at the crossing point of two
nodal lines. (a) Fermi surface (blue) of the two-band model Eq. (2). The
orange shaded plane atcrosses the nodal line NL1 atand crosses NL2 at.
Purple shaded plane atcuts through the CP (black sphere) of NL1 and NL2
at the origin. (b) Band structureversus,at, showing a semi-Dirac point
(black sphere) as a result of the merging of three Dirac points atand.
Purple dashed lines in (a) and (b) show the crescent-shaped Fermi
surface contour of semi-Dirac fermions. Purple shaded planes represent
the Fermi level for CP1 () and CP2 (). (c) Calculated LL spectrum based
on model parameters for CP1 at. The right-hand panel indicates the
corresponding density of states (DOS) of the LLs. Purple dashed line
represents the Fermi level of CP1. Purple dots label the extremal points
in the LLs and purple arrow indicates the lowest momentum-conserving
transition (); see Supplemental Material Sec. VIII for details[33].
(d),(e) Calculated momentum space distribution of the Fubini-Study
metricat a Dirac point [(d)] and at the semi-Dirac point [(e)]. Near the
semi-Dirac point,is nonzero and shows a stronger divergence than the
Dirac case.
Atthe spectrum becomes a single Dirac again. The unique semi-Dirac
fermion described byis predicted to exhibit nontrivial Berry phase and
finite Chern number when a gap opens[13]. This is in stark contrast with
the zero Berry phase for semi-Dirac fermions formed by merging an even
number of Dirac points (modulo). Crossing nodal lines in Eq.
(2)therefore offer a new platform for studying the rich phenomena of
merging Dirac points[28,60](see Supplemental Material Video 2[33]),
where topology and correlation effects intertwine.
We have calculated the Landau level spectrum of Eq. (2)in the presence
of an in-plane field directed alongas a function of, and the results are
shown in Fig. 5(c). Sinceis a good quantum number for, at eachthe LL
spectrum is a series of discrete levels, with level spacing determined
by the projection of the constant energy contours onto the plane
perpendicular to. The LLs exhibit extremal points nearand lead to peaks
in the density of states (DOS), indicated by purple dots in Fig. 5(c).
Optical transitions (vertical purple arrow) between these LL
singularities are observed experimentally in LL spectroscopy. We remark
again that only the lowest-order momentum-conserving LL transitions ()
have been observed in this work (Supplemental Material Sec. IX[33]).
Higher-order LL transitions are in general weaker and may be forbidden
by selection rules (see, e.g., Ref. [61]for type-I semi-Dirac fermions).
We further demonstrate thescaling of the LLs at the CP using both
semiclassical quantization below and full LL calculation with SOC
(Supplemental Material Sec. VIII). Apart from enhanced DOS, we remark
that the semi-Dirac fermions realized by Eq. (2)also show stronger
divergence in the quantum metriccompared to Dirac fermions [see Fig.
5(d)]. In Fig. 5(d), we compared the calculated-space distribution
ofat(semi-Dirac) and(Dirac) based on Eq. (2)(Supplemental Material
Sec. VI). We remark that the effects of quantum geometry in solids have
been discussed mostly theoretically in terms of nonlinear optical
response[62,63], but recent[64]and classic work[65,66]have connected the
linear response to the integrated Fubini-Study metric. Our experimental
observation of semi-Dirac fermions highlights the potential to explore
quantum geometry from novel quasiparticles using linear optical and
magneto-optical response.
Before concluding, we comment on the robustness of the observedscaling
of Landau levels from crossing nodal lines in ZrSiS. There are several
different ways that two nodal lines cross in momentum space[41,44]. We
illustrate three of the lowest orders crossing in Fig. 6, where the two
nodal lines are Fig. 6(a)both straight[41](), Fig. 6(b)both
parabolic[44], or Fig. 6(c)consisting of one straight and one parabolic
nodal line, whereis half of the SOC gap. Under an external magnetic
field along, the cyclotron motions of electrons will be quantized in the
2D plane normal to the magnetic field direction (constantplanes). The
band structures corresponding to the three models in Figs. 6(a)–6(c)at
the planeare shown in Figs. 6(d)–6(f), respectively. The cyclotron
motions for charge carriers from the three different nodal-line crossing
points can now be assessed by their distinct Fermi surface cross
sections at the Fermi level [black dashed lines in Figs. 6(d)–6(f)].
Importantly, the band structure at the CP from[Fig. 6(f)] is
electron-hole asymmetric (see also Fig. S30[33]), disctinct fromand.
FIG. 6.
Nodal-line crossing point models. Three different two-band models for
the crossing point of two nodal lines at. (a) Fermi surface (FS) near
the crossing of two straight nodal lines, described by the
Hamiltonian[41]. (b) FS near the crossing of two parabolic nodal lines,
described by the Hamiltonian[44]. (c) FS near the crossing of one
straight and one parabolic nodal line, described by the Hamiltonian,
whereis half of the spin-orbit coupling (SOC) gap. The band structures
atfor the three different crossing points are shown in (d)–(f) for,,
and, respectively. (g) The FS contour atfor, showing an open orbit for
magnetic field. (h) The area of the closed orbitatforincreases linearly
with energy. (i) increases asatforand remainsfor energies higher than
the SOC gap. Here,,,,. Note thatisasymmetric and exhibits open orbits
for hole doping (Fig. S30[33]).
We can evaluate the Landau level scaling from a semiclassical approach
for these three different crossing points. First, it is evident that the
crossing of two straight nodal lines () will leads to an open orbit at
the CP [Fig. 6(g)], and therefore will not give rise to quantized energy
levels. For the crossing of two parabolic nodal lines (), the area of
the cyclotron orbitincreases linearly with energy[Fig. 6(h)]. According
to the Onsager quantization relation[34],, whereis the Landau level
index andis the phase factor, the LL scaling will be linear in. In
contrast, as derived in the Appendix, the cyclotron orbit area increases
as[Fig. 6(i)] for the CP of a straight and parabolic nodal line (),
which results in an LL scaling that is strictly. This semiclassical
analysis is consistent with full quantum mechanical calculations for the
CP of a straight and parabolic nodal lines, which also exhibits theLL
scaling (see Supplemental Material Sec. VIII and Fig. S23[33]).
The generic nature of our nodal-line crossing point model and analysis
(Figs. 4–6) guarantees the robustness of the experimental observable
even in real and complex materials such as ZrSiS. In Fig. 7, we show the
power-law fitting of one of the Landau level transitions from semi-Dirac
fermions (purple dots) with functionand the correspondingconfidence
interval (purple shaded region). The data points are consistent with
thescaling and statistically different from(conventional massive
fermions),(three-quarter Dirac fermions[67]), and(Dirac fermions).
FIG. 7.
Comparison of the power law of Landau level transitions for different
fermions in a log-log scale plot. Power-law fitting (purple dashed line)
of the interband Landau level transitions (purple dots) associated with
the semi-Dirac fermions in ZrSiS. Purple-shaded area indicates
theconfidence interval. Orange and black lines show the power-law
scaling of LLs for Dirac () and massive fermions (), respectively.
Using Landau level spectroscopy, we uncovered the semi-Dirac fermions
inside bulk ZrSiS under in-plane magnetic fields. Our results
demonstrate novel magnetic field effect in ZrSiS not explored in
previous experiments utilizing out-of-plane magnetic fields[68–70]. In
contrast to the conventional expectation of 2D electrons at the surface
or interface of a layered material, the observed semi-Dirac fermions
reside within planes perpendicular to the atomic layers of ZrSiS and
originate from the vicinity of points where nodal lines cross. The
crossing point of nodal lines in ZrSiS offers a unique and generic
platform for realizing semi-Dirac fermions through the merging of three
Dirac points. Our findings advance the understanding of exotic 2D
electrons in natural bulk crystals, establish the existence of novel
quasiparticles associated with crossing nodal lines in momentum
space[40,44], and open new directions in exploring quantum geometry and
topological effects in metals.
Notes
The data files for the materials growth recipe and the characterization
data are available at[71].
ACKNOWLEDGMENTS
Magneto-optical spectroscopy of ZrSiS is supported by National Science
Foundation, Division of Materials Research 2210186. Research on the
electrodynamics of semi-Dirac quasiparticles, including the theoretical
analysis of Landau levels spectra at Columbia, is supported as part of
Programmable Quantum Materials, an Energy Frontier Research Center
funded by the U.S. Department of Energy (DOE), Office of Science, Basic
Energy Sciences (BES), under Award No. DE-SC0019443. D. N. B. is Moore
Investigator in Quantum Materials EPIQS GBMF9455. Support for crystal
growth and characterization at Penn State was provided by the National
Science Foundation through the Penn State 2D Crystal
Consortium-Materials Innovation Platform (2DCC-MIP) under NSF
Cooperative Agreement No. NSF-DMR 2039351. The National High Magnetic
Field Laboratory is supported by the National Science Foundation through
NSF/DMR-1644779, NSF/DMR-2128556, and the State of Florida. The work of
A. N. R. and M. I. K. was supported by the European Union’s Horizon 2020
research and innovation program under European Research Council Synergy
Grant No. 854843 “FASTCORR.” The work of M. I. K. was further supported
by the Dutch Research Council (NWO) via the “TOPCORE” consortium.
J. H. A. gratefully acknowledges support from a Hertz Fellowship. The
Flatiron Institute is a division of the Simons Foundation.
Appendices
APPENDIX: METHODS
1. Voigt magneto-optical spectroscopy and the absence of surface
states
High-field magneto-optical measurements were performed atunder Voigt
geometry (and B) at the National High Magnetic Field Laboratory. A
Bruker Vertex 80V FTIR spectrometer combined with a 17.5 T
superconducting magnet was utilized to record the reflectance spectra of
ZrSiS at zero and high magnetic field. Infrared beam from a Globar lamp
was focused on the (001) surface of ZrSiS crystal. The typical size of
ZrSiS crystals used in our measurements isand the spot size of the
infrared beam is smaller than the lateral size of the sample.
We remark that there are various surface states observed by
angular-resolved photoemission measurements[35,37,72,73]from the (001)
surface of the ZrSiS family of topological semimetals. These surface
states are strictly confined to the top and bottom surfaces of ZrSiS and
do not contribute to the out-of-plane orbital motion when the magnetic
field is applied in-plane. Therefore, we conclude that the previously
reported surface states from (001) surface of ZrSiS cannot explain the
nearly massless Dirac fermions observed in our Voigt geometry
magneto-optical data.
2. Density functional theory calculation
DFT calculations were carried out using the plane-wave pseudopotential
method as implemented in thequantumespressosimulation package[74,75].
Norm-conserving pseudopotentials[76]were used in conjunction with the
local density approximation for exchange-correlation potential. An
energy cutoff of 70 Ry for the plane waves and a convergence threshold
ofwere used for the self-consistent solution of the Kohn-Sham equations.
The Brillouin zone was sampled by aMonkhorst-Pack[77]-point mesh.
Lattice constants were relaxed, resulting inand. The atomic structure
within the unit cell was relaxed until the residual forces were less than.
To ensure the numerical accuracy of the Fermi surface calculations and
related properties, an interpolation scheme based on the maximally
localizedwannierfunctions (MLWF)[78]was used. For this purpose, we used
thewannier90 code[79]to construct an extended tight-binding Hamiltonian
for ZrSiS in the MLWF basis, which included theandstates for Si and S as
well as theandstates for Zr. The interpolated band energies ensure a
correct description of the DFT band structure within the range of at
leastrelative to the Fermi energy. Thewannier-interpolated band energies
were calculated on a-point mesh, and were further used to calculate the
de Haas–van Alphen (dHvA) frequencies using the supercell-space extremal
area finder (skeaf) code[80].
3. Quantum oscillations in ZrSiS with torque magnetometry
measurements
The torque magnetometry measurements up to 14 T were performed using a
piezoresistive cantilever in a superconducting magnet equipped with a
variable temperature insert. A single crystal of ZrSiS is fixed to the
end of a 0.30 mm cantilever arm with vacuum grease. A jet of helium-4
gas from the inlet of the variable temperature insert onto the cryostat
kept the samples at a constant temperature of 1.6 K during the
measurements. There are two resistive elements on the cantilever, one of
which is located at the base of the arm and experiences strain with a
change in the sample magnetization. The second resistive element is not
affected by the torque but mimics the temperature and magnetic field
dependence of the first. These are combined with two more resistors at
room temperature to form a Wheatstone bridge that can be balanced at low
temperatures before changing the magnetic field. A small current is
applied across the bridge circuit and the measured voltage records the
changing torquecreated by the dHvA effect.
4. Semi-Dirac fermions in the nodal-ring model
To illustrate the semi-Dirac fermions in the nodal-ring model, we
consider the nodal-ring Hamiltonian[81,82],
(A1)
whereis the Fermi velocity along,is the identity matrix, and,are the
Pauli matrices for orbitals. The corresponding energy spectrum is
(A2)
and a Dirac nodal ring atwith radiuscan be identified. This nodal ring
marks a protected crossing of two bands along a ring in momentum space,
and at any point on the ring the spectrum is Dirac. Remarkably, at, the
projection of constant energy contours onto theplane yields contours of
the semi-Dirac form[14,82,83]that describes a massive fermion along()
and massless Dirac fermion along(see Supplemental Material Video 1[33]).
Under the in-plane magnetic field, the area of the cyclotron orbit at
energyis, which leads to thescaling of LLs[10,82]. Here, we briefly
describe the solution of the Hamiltonian in an in-plane magnetic
fielddirected alongthat reveals the main features of the absorption
spectrum. Choosing the Landau gaugeand rescaling the coordinatewith
magnetic lengthand defining field scaleand rescaled field, we havewith.
From this form ofone can see that for,is minimized at a
particularexpanding around the twovalues that give Dirac spectra, while
whenwe have a semi-Dirac equation. We have solved the corresponding
Schrödinger equation numerically by observing that at each,can be
diagonalized trivially yielding two second-order differential equations
that we solve by discretization. The current operator can then be
obtained as a differential operator and applied to the solutions.
Integration of the resulting absorption spectrum overgives the continuum
absorption, with singularities at the upper edge scaling asand lower
edge as, consistent with previous results[82]. While the nodal-ring
model is not directly relevant to the interpretation of our experimental
results, it does display the generic features of an absorption continuum
with upper and lower edges scaling differently with, and with the
characteristic Dirac and semi-Dirac behaviors.
5. Semiclassical quantization of the crossing nodal-line model
To obtain the Landau level scaling of the semi-Dirac fermion at the CP
of nodal lines, we consider the following Hamiltonian:
(A3)
which has an extremal Fermi surface atfor. This model also describes the
crossing of one straight and one curved nodal line and is related to the
model Eq. (2)by a 45-deg in-plane rotation. The corresponding eigenvalue
is given by
(A4)
For a magnetic fieldapplied along, the eigenvalue spectrum has extremal
values at. Without loss of generality, we consider the Fermi surface of
the electron pockets () and choose,, and. The cross section of the
electron Fermi surface at constant energyis then determined by
(A5)
which is bounded by two parabolic curves:
𝑘+𝑦=𝐸𝑣−𝑡−12𝑚𝑣𝑘2𝑧and𝑘−𝑦=−𝐸𝑣+𝑡+12𝑚𝑣𝑘2𝑧.
(A6)
These two parabolic curves intersect at𝑘2𝑧=𝐸/𝑡≡𝑎2, forming a
crescent-shaped closed contour for electron orbits [see inset of Fig.
6(i)]. If𝑘2𝑧>𝐸/𝑡, the hole FS is obtained
with𝐸=𝑡𝑘2𝑧−(𝑘2𝑧/2𝑚)−𝑣𝑘𝑦. The area of the electron FS
contour can be obtained by evaluating the difference between the areas
of the parabolic segments of𝑘−𝑦and𝑘+𝑦:
𝑆(𝐸)=𝑆−(𝐸)−𝑆+(𝐸)=∫𝑎−𝑎(𝑘+𝑦−𝑘−𝑦)𝑑𝑘𝑧=∫𝑎−𝑎2𝐸/𝑣(1−𝑘2𝑧𝑎2)𝑑𝑘𝑧=83𝐸3/2√𝑡𝑣.
(A7)
Following Onsager’s quantization relation𝑆(𝐸)=2𝜋(𝑛+𝛾)𝑒𝐵/ℏ,
where𝑛is the Landau level index and𝛾is the phase factor, we obtain
the𝐵2/3scaling of LLs:𝐸𝑛∝(𝑛+𝛾)2/3𝐵2/3.
In the presence of finite SOC, the degeneracy at the semi-Dirac point is
lifted but the characteristic band dispersions and LL scaling recover at
energies higher than the SOC gap. We illustrate the case with SOC using
the following Hamiltonian based on Eq. (A3):
𝐻′3=𝑡𝑘2𝑧𝜏0+(𝑘2𝑧2𝑚−𝑣𝑘𝑦)𝜏𝑧+𝑘𝑥𝑘𝑧𝜏𝑥+Δ𝜏𝑦,
(A8)
where2Δis the size of the SOC gap. The area of the electron FS contour
with SOC becomes
𝑆SOC(𝐸)=83√𝐸+Δ√𝑡𝑣×(𝑒𝐸[1−2Δ𝐸+Δ]𝐸−𝑒𝐾[1−2Δ𝐸+Δ]Δ),
(A9)
where𝑒𝐸[𝑥]and𝑒𝐾[𝑥]denote the complete elliptic integral of the
second kind and first kind, respectively. Equation (A9)can be solved
numerically and the resulting𝑆SOC(𝐸)is shown in Fig. 6(i), where
the𝑆SO(𝐸)∝𝐸3/2behavior is recovered at energies higher than around2Δ.
Supplemental Material
Supplementary Material files contain crystal characterizations of ZrSiS,
quantum oscillation measurements and band structure calculations.
Details of the tight-binding model and continuum model for the crossing
nodal-lines are provided and compared with DFT calculations.
Calculations of quantum geometry and Landau level spectra based on the
continuum model are provided.
Two Supplementary Videos are also provided, comparing the semi-Dirac
fermions formed by merging of two Dirac points and three Dirac points.
/
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Source: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.14.041057
Semi-Dirac Fermions in a Topological Metal
Yinming Shao
<https://journals.aps.org/search/field/author/Yinming%20Shao><https://orcid.org/0000-0002-2891-0028>^1,2,*
,Seongphill Moon
<https://journals.aps.org/search/field/author/Seongphill%20Moon>^3,4
,A. N. Rudenko
<https://journals.aps.org/search/field/author/A%20N%20Rudenko><https://orcid.org/0000-0003-4313-3690>^5
,Jie Wang
<https://journals.aps.org/search/field/author/Jie%20Wang><https://orcid.org/0000-0002-3750-9340>^6,7,8
,Jonah Herzog-Arbeitman
<https://journals.aps.org/search/field/author/Jonah%20HerzogArbeitman><https://orcid.org/0000-0003-4993-3381>^9
,Mykhaylo Ozerov
<https://journals.aps.org/search/field/author/Mykhaylo%20Ozerov><https://orcid.org/0000-0002-5470-1158>^4
,David Graf
<https://journals.aps.org/search/field/author/David%20Graf>^4 ,Zhiyuan
Sun
<https://journals.aps.org/search/field/author/Zhiyuan%20Sun><https://orcid.org/0000-0002-0342-6248>^7
,Raquel Queiroz
<https://journals.aps.org/search/field/author/Raquel%20Queiroz>^1 /et al./
Phys. Rev. X*14*, 041057 –*Published 5 December, 2024*
DOI:https://doi.org/10.1103/PhysRevX.14.041057
<https://doi.org/10.1103/PhysRevX.14.041057>
Abstract
Topological semimetals with massless Dirac and Weyl fermions represent
the forefront of quantum materials research. In two dimensions, a
peculiar class of fermions that are massless in one direction and
massive in the perpendicular direction was predicted 16 years ago. These
highly exotic quasiparticles—the semi-Dirac fermions—ignited intense
theoretical and experimental interest but remain undetected. Using
magneto-optical spectroscopy, we demonstrate the defining feature of
semi-Dirac fermions—𝐵2/3scaling of Landau levels—in a prototypical
nodal-line metal ZrSiS. In topological metals, including ZrSiS, nodal
lines extend the band degeneracies from isolated points to lines, loops,
or even chains in the momentum space. With/ab initio/calculations and
theoretical modeling, we pinpoint the observed semi-Dirac spectrum to
the crossing points of nodal lines in ZrSiS. Crossing nodal lines
exhibit a continuum absorption spectrum but with singularities that
scale as𝐵2/3at the crossing. Our work sheds light on the hidden
quasiparticles emerging from the intricate topology of crossing nodal
lines and highlights the potential to explore quantum geometry with
linear optical responses.
Popular Summary
Among the emerging quasiparticles, semi-Dirac fermions stand out. In 2D
systems, these exotic quasiparticles are thought to have mass in one
direction yet be massless in the perpendicular direction. These peculiar
fermions have so far evaded detection in solid materials. Here, we
present experimental evidence of the defining feature of semi-Dirac
fermions in samples of the metal ZrSiS.
In a typical metal, the presence of an external magnetic field induces
cyclotron motion of electrons, and the ensuing cyclotron energy scales
linearly with the strength of the magnetic field. In graphene, the
presence of massless Dirac fermions leads to a cyclotron energy that
scales with the square root of the field. Semi-Dirac fermions are
predicted to have a different response: Their cyclotron energy scales
with the two-thirds power of the field. Initial proposals to realize
semi-Dirac fermions require stretching graphene until its two Dirac
points—features in the energy band structure that signify the presence
of Dirac fermions—merge in momentum space. However, monolayer graphene
breaks down before reaching the desired strain level.
In ZrSiS, the continuous extension of Dirac points in momentum space,
known as a nodal line, presents exciting opportunities to realize
semi-Dirac fermions. Through high-precision magnetoinfrared
spectroscopy, we observe compelling evidence of the two-thirds power-law
behavior of the cyclotron energy with magnetic field. Combining these
findings with calculations and modeling, we identify semi-Dirac fermions
at special crossings of nodal lines in ZrSiS. Those crossings facilitate
the merging of Dirac points without the need for the unrealistic large
strain required in graphene.
Our work sheds light on the hidden quasiparticles emerging from the
intricate topology and geometry of crossing nodal lines.
Article Text
I. INTRODUCTION
Conventional 2D fermions are described by parabolic energy (𝐸) momentum
(𝐤) dispersion𝐸(𝐤)=ℏ2𝑘2/(2𝑚)with effective mass𝑚. In contrast,
Dirac and Weyl fermions[1,2]have linear dispersion𝐸𝐷(𝐤)=ℏ𝑣𝐹𝑘and
are massless. The striking manifestations of massless Dirac fermions are
revealed through the anomalous half-integer quantum Hall effect[3,4],
Klein tunneling[5,6], and√𝐵scaling of Landau levels (LLs) with magnetic
fields (𝐵)[7–9]in graphene. All these effects are observed in graphene,
where the characteristic√𝐵scaling provides a litmus test for Dirac
quasiparticles.
Semi-Dirac fermions[10–12], with
dispersion𝐸SD(𝐤)=±√(ℏ𝑣𝑘1)2+[ℏ2𝑘22/(2𝑚)]2being linear in one
momentum direction (𝑘1) and quadratic in the orthogonal direction
(𝑘2), have been proposed to appear in materials where multiple Dirac
points merge[10,13]into a semi-Dirac point. Despite intense theoretical
and experimental interests[14–22], semi-Dirac fermions remain
undetected. Strained graphene may be a candidate system to host
semi-Dirac quasiparticles. However, the required uniaxial strain level
is unrealistically large[10,23,24]. Black phosphorus is proposed as
another candidate semi-Dirac fermions system upon strong doping[25]. Yet
the precise semi-Dirac dispersion in black phosphorus has not been
established either experimentally[26]or theoretically[27]. Thus far, the
semi-Dirac dispersion𝐸SDhas been experimentally explored only in
synthetic platforms including honeycomb lattices of ultracold atoms[28],
photonic resonators[29,30], and polaritons systems[31]. Identifying the
fermionic counterpart is crucial to realize the diverse
topological[13,17]and correlated[18,32]phases predicted for semi-Dirac
fermions, but remains challenging in 2D systems. A defining feature of
semi-Dirac fermions is the unique𝐵2/3dependence[10,12]of inter-LL
transitions [Fig. 1(a)]. Here we report on the first observation of this
characteristic𝐵2/3power law in a topological metal, ZrSiS, through LL
spectroscopy.
FIG. 1.
Semi-Dirac fermions at nodal-line crossing points in ZrSiS.
(a) Cyclotron energy (ℏ𝜔𝑐) as a function of magnetic field (𝐵) for
conventional fermions (black line), Dirac fermions (orange line), and
semi-Dirac fermions (purple line). Insets show three-dimensional plots
of their band structures (gray surfaces) overlaid with Landau levels
(red and blue contour lines). (b) The lattice structure of ZrSiS,
showing the square lattice of Si atoms (blue) and the Zr (brown)-S
(yellow) layers above and below. (c) /Ab initio/calculation of the Fermi
surface and nodal-line structure of ZrSiS. Only the𝑘𝑥>0part of the
Fermi surface is shown for better visualization of the nodal-line
structures (gray lines). Black spheres indicate the crossing points
(CPs) of multiple nodal lines. Purple shaded planes
at𝑘𝑥=±𝑘CP1=±0.1971(2𝜋/𝑎)cross the CP1 formed by nodal lines
at𝑘𝑧=0and𝑘𝑥=±𝑘𝑦. The circular purple arrow illustrates the
cyclotron motion around one of the CP1 for magnetic field (green arrow)
applied along𝑘𝑥(𝑎axis of the crystal). Calculated band structure (see
the Appendix and Supplemental Material Sec. VII[33]) near CP1
at𝑘𝑥=𝑘CP1plane shows quadratic dispersion along𝑘𝑧(d) and linear
dispersion along𝑘𝑦(e), characteristic of semi-Dirac fermions.
Semiclassically, a magnetic field induces cyclotron motion and the area
of the cyclotron orbit at energy𝐸is𝑆(𝐸)∝𝐸3/2√𝑚/𝑣for the
semi-Dirac dispersion𝐸SD[10]. Following the Onsager
quantization[34]𝑆(𝐸)=2𝜋(𝑛+𝛾)𝑒𝐵/ℏ, the
characteristic𝐵2/3scaling of LLs is obtained:𝐸𝑛∝[(𝑛+𝛾)𝐵]2/3,
where𝑛is the LL index and𝛾is the phase factor(0≤𝛾<1). The 2D
semi-Dirac spectrum can also arise as singularity points of a continuum
absorption spectrum of a 3D material. For example, the LL spectrum of a
nodal ring𝐸NR=±√[(𝑘2𝑥+𝑘2𝑦−𝑘20)/2𝑚]2+𝑣2𝑧𝑘2𝑧with field in
the𝑥-𝑦plane exhibits a continuum absorption with a lower edge scaling
as𝐵2/3, arising from the semi-Dirac structure (see the Appendix).
The prototypical nodal-line semimetal ZrSiS[35–37][Fig. 1(b)] hosts two
planar nodal squares linked by vertical nodal-lines[38,39], forming a
chainlike structure[40–44][gray lines in Fig. 1(c)] in momentum space.
The low-energy physics of the ZrSiS family of nodal metals is further
enriched by the Fermi energy variations along the Dirac nodal
lines[45–47], reflected by the coexisting electron (blue) and hole (red)
pockets [Fig. 1(c)]./Ab initio/calculations and theoretical modeling
show that the observed semi-Dirac fermions originate from the crossing
points (CPs) of the nodal lines in ZrSiS [black dots in Fig. 1(c)]. Near
the CP, the band structure at𝑘𝑥=𝑘CPshows quadratic [Fig. 1(d)] and
linear dispersion [Fig. 1(e)] along𝑘𝑧and𝑘𝑦, respectively. Under
magnetic field oriented along the𝑎axis (𝐵∥𝑘𝑥), the cyclotron motion
of electrons becomes quantized in the(𝑘𝑦,𝑘𝑧)plane and reflects the
semi-Dirac fermions through the unique LL scaling. The
anticipated𝐵2/3power law in ZrSiS is robust against material
complexities and can be readily identified in infrared magneto-optics
experiments.
II. EXPERIMENTAL RESULTS
We now proceed to the magnetoreflectance spectra𝑅(𝜔,𝐵)normalized by
the zero-field data𝑅(𝜔,0T)for ZrSiS with in-plane magnetic fields up
to 17.5 T (see the Appendix), shown in Fig. 2(a). The most prominent
features are a series of dips in the reflectance spectra hardening with
increasing field (gray dashed lines). For a highly metallic system like
ZrSiS, the infrared reflectance approaches unity and, therefore, dips
in𝑅(𝐵)/𝑅(0)correspond to absorption𝐴(𝜔)=1−𝑅(𝜔)[45,47]. We
attributed these absorption features to interband LL transitions from
massive Dirac fermions (𝐸±𝑛=√2𝑒ℏ|𝑛|𝐵¯𝑣2+Δ2), which exhibit
notable departures from the linear-in-𝐵scaling expected for fermions in
parabolic bands [Fig. 1(a)]. Here,Δis half of the spin-orbit coupling
(SOC) gap[47–50]and we find2Δ≈28 meV, in excellent agreement with
previous lower-field studies[45,51]and calculations[35]. Surprisingly,
above a critical field𝐵𝑐≈7 T, weaker subgap features [red and purple
dots in Fig. 2(a)] appear at around100 cm−1and harden with increasing
field. To better visualize these subgap structures, we report the second
derivative𝑑2𝑅/𝑑𝐵2analysis in Fig. 2(b). The local minima of the
second derivative coincide with the dips in𝑅(𝐵)/𝑅(0)(see
Supplemental Material Sec. III[33]), which we identify as the LL
transition energies in all analyses.
FIG. 2.
Landau level (LL) spectroscopy of ZrSiS with in-plane magnetic fields.
(a) Magnetoreflectance𝑅(𝜔,𝐵)normalized by zero-field
reflectance𝑅(𝜔,0T)for sample S1. Gray dashed lines mark the
positions of the series of interband LL transitions across the
spin-orbit coupling gap (2Δ≈28 meV). The red-shaded region indicates a
potential phonon feature[52–54]that is not dispersing with increasing
field. Above a critical field of𝐵𝑐≈7 T, additional subgap transitions
(red and purple dots) emerge and harden with increasing field. The inset
is a schematic of the experimental configuration with near-normal
incident and unpolarized light while the magnetic field is applied
in-plane (𝐵⊥𝑐,𝐵∥𝑎, Voigt geometry). (b) Second
derivative𝑑2𝑅/𝑑𝐵2data of sample S1 overlaid with model fitting
(gray dashed lines) of the LL transitions across the gapped Dirac cone.
The top schematic shows a gapped Dirac cone with gap2Δand the first LL
transitionsLL−0(−1)→LL1(+0)(gray arrows). Orange lines are the guides
for the subgap LL transitions [red dots in (a)] that follow
approximately√𝐵scaling, characteristic of a Dirac-like fermion (bottom
schematic). The subgap features near the purple-shaded region follow a
distinct𝐵2/3power law and originate from semi-Dirac fermions.
In Fig. 2(b), we show the𝑑2𝑅/𝑑𝐵2spectra for ZrSiS obtained with
in-plane magnetic fields up to 17.5 T. The gray dashed lines denote the
model calculation of interband LL transitions across the gapped Dirac
cone[48,55]:𝐸𝑇=√2𝑒ℏ|𝑛|𝐵¯𝑣2+Δ2+√2𝑒ℏ(|𝑛|+1)𝐵¯𝑣2+Δ2,
where𝑛is the LL index and¯𝑣is the averaged Fermi velocity. The
intraband LL transitions[56], if present, would follow a field
dependence distinct from the observed subgap features (see Supplemental
Material Figs. S8 and S25). Importantly, beyond the two series of subgap
transitions labeled as√𝐵and𝐵2/3(red and purple dots, respectively),
two additional dispersive features are apparent above150 cm−1. As
indicated by the thick orange lines, the dispersions of these latter
features also follow approximately the√𝐵scaling.
The dipole selection rule𝛿|𝑛|=±1[7,9]for Dirac fermions dictates that
the energy ratios of the lowest three interband LL transitions
are1:1+√2:√2+√3. Using a single averaged velocity of 0.82 eV Å, the
three branches of the transitions can be approximated by the lowest
three interband LL transitions from massless Dirac-like fermions (orange
lines) and are labeledLL0→1,LL−1→2, andLL−2→3. Detailed analysis on the
multiple peak splitting ofLL0→1below shows evidence of a small gap and
spin splitting due to Zeeman effect. On the other hand, the remaining
subgap features [near purple shaded region in Fig. 2(b)] follow
sublinear𝐵dependence that is distinct from√𝐵. We will confirm next
that these peculiar LL transitions’ field dependence scales precisely
as𝐵2/3, a fingerprint of semi-Dirac fermions in ZrSiS.
To quantify the power-law scaling of the subgap features, we extract the
transition energies from Fig. 2(b)(see Supplemental Material Sec. III
and Figs. S7 and S8[33]) and plot them against√𝐵and𝐵2/3in Figs.
3(a)and3(b), respectively. Figure 3(a)shows the experimentally
determined LL transition energies (red dots) for the three groups of
transitions labeledLL0→1,LL−1→2,LL−2→3in Fig. 2(b). Remarkably, all
these LL transitions can be understood as originating from a Dirac
fermion with a small gap2Δ′=2.4 meVand Zeeman-split LLs. The resulting
model calculations (orange lines) show good agreement with the data (see
Supplemental Material Figs. S24 and S35). ForLL−2→3, variation
of¯𝑣𝐷shows a logarithmic reduction with increasing𝐵field
[¯𝑣𝐷∝−ln(𝐵); see Fig. S27 in Supplemental Material]. Alternatively,
theLL−1→2andLL−2→3transitions can arise from the cyclotron resonance of
another gapped Dirac cone[56](Fig. S27 of Supplemental Material), and
the exact origin of these transitions awaits future studies.
FIG. 3.
andpower-law behaviors of Landau levels in ZrSiS. (a) Subgap transition
energies (red dots) in Fig. 2are plotted as a function of. Orange lines
represent the fitting based on Dirac-like fermions with an averaged
Fermi velocity, a small gap, and Zeemanfactor. Solid and dashed lines
represent the spin-conserving and spin-flip transition, respectively.
Orange shaded areas indicate the uncertainties infor the() and()
transitions. (b) Higher-energy subgap transitions in Fig. 2(purple dots)
are plotted as a function of, following the exact power-law behavior
expected for semi-Dirac fermions (green dashed lines). The bottom inset
shows the calculated Fermi surface of ZrSiS. The shaded purple plane
indicates theplane, which cuts through CP1 (black dot at). The top inset
shows the calculated band structureversus,for CP1 at the plane, with
linear dispersion alongand quadratic dispersion along, characteristic of
semi-Dirac dispersion.
As we alluded to previously, a series of subgap features displays
thescaling that is characteristic of semi-Dirac fermions [Fig. 3(b)].
Fine splitting of LL transitions is also apparent and all the split
peaks agree with the predicted power-law behavior (green dashed lines)
for semi-Dirac fermions. These latter features are reminiscent of the
spin and valley splitting of Landau levels in Dirac fermions[57,58]and
we discuss several possible scenarios for peak splitting in the
Supplemental Material Sec. IV and Figs. S10–S13[33]. Because of the
nonanalytical nature of the LLs of semi-Dirac fermions[10], the
selection rules have only been explored numerically for type-I
semi-Dirac fermions. Nevertheless, we discuss several possibilities for
the absence of additional high-order semi-Dirac LL transitions in ZrSiS
in the Supplemental Material Sec. IX and Fig. S28.
Importantly, the semi-Dirac fermions in ZrSiS are confirmed both
from/ab initio/calculations and from theoretical modeling of the
crossing points of nodal lines. As shown in the bottom inset of Fig.
3(b), there are two nonequivalent CPs in ZrSiS, labeled as CP1 (at) and
CP2 (at). The observedscaling LL transition is dominated by CP1 since
the energy of the CP is very close to the Fermi level, while the energy
of CP2 is about 0.1 eV below the Fermi level (see Fig. S19[33]). The
calculated semi-Dirac bands near the CP1 [top inset of Fig. 3(b)] are
also asymmetric in, in contrast to the usual type-I semi-Dirac
dispersion[10]:. We demonstrate below that a unique semi-Dirac fermion
that originates from the merging of three Dirac points[13]is realized
near CP1 in ZrSiS, distinct from the merging of two Dirac
points[10,14]realized in a single nodal ring (see Supplemental Material
Video 1 [33]).
III. THEORY AND CALCULATION
We now turn to the theoretical interpretation of the results. The
complex 3D nodal-line cage obtained in density functional theory (DFT)
reveals 8 CPs each in theandplane. Building on the 2D Hamiltonian for
square net motifs[59], we add the 3D hoppings () and obtain a minimal
four-band (not including spin) tight-binding model which reproduces all
nodal lines and CPs, as shown in Fig. 4(see Supplemental Material Sec. V
for details[33]):
(1)
where,represent the sublattice and,orbital degrees of freedom,
respectively. Theandare the in-plane and out-of-plane lattice constants,
respectively. Despite the complexity of the band structure, this model
is analytically solvable, and yields closed form expressions for the
nodal lines in terms of a small number of physical parameters. This
model not only captures the global features of the nodal cage over the
entire Brillouin zone [Fig. 4(b)], but also allows us to obtainmodels at
each CP derived directly from the microscopic hoppings. In contrast to
the intersection of two straight nodal lines[41]or two nodal rings[44],
the CP1 in ZrSiS is composed of a straight nodal line and a curved nodal
line [see Figs. 1(c)and4], which can be described by a minimum two-band
model for the CP:
(2)
whereis the identity matrix,are the Pauli matrices,is the effective mass
along thedirection,is the Fermi velocity alongdirection,controls the
electron-hole asymmetry (tilt of the nodal line along),controls the
dispersion along(parallel to the in-plane nodal line), andis half of the
SOC gap. As shown in Fig. 5(a), the spectrum of Eq. (2)describes the
crossing of a straight in-plane nodal line (NL1, bounded by
theandplanes) and a curved vertical nodal line (NL2, bounded by
theandplanes). The Fermi surface (FS) of the model [Fig. 5(a)] captures
the main features of the FS in ZrSiS near the CP2 [bottom inset of Fig.
3(b)]. In particular, the crescent-shaped contour refers to the purple
dashed line in Fig. 5(a)at the CP (purple dashed line) is consistent
with band structure calculations of ZrSiS and further corroborated with
quantum oscillation measurements (Supplemental Material Figs. S5 and
S6[33]). Under the external magnetic field along, the 2D plane normal to
the field at negativewill cross NL1 once and cross NL2 twice, forming
three isolated Dirac points (D1 and). As the 2D plane moves to the
right, the three Dirac points get closer and merge at(purple shaded
plane), realizing a type-II semi-Dirac fermion[13]:([Fig. 5(b)].
FIG. 4.
Nodal-line structure of ZrSiS. The 3D nodal-line structure of ZrSiS
calculated using DFT (a) and the tight-binding model (b). The
tight-binding model Eq. (1)(Supplemental Material Sec. V[33]) captures
faithfully the complex structure of nodal-line crossings in DFT. Gray
dashed lines connect the high symmetry points at. Blue spheres indicate
the location of the four symmetry-related crossing points at(CP1). The
crossing of curved vertical nodal line and straight horizontal nodal
line at CP1 hosts semi-Dirac fermions. (c) Expanding the tight-binding
model (b) near CP1 and retaining the leading order terms inleads to the
two-band continuum model Eq. (2). The CP1 is formed by two nodal lines:
the vertical nodal line defined bywith finite curvature and the
horizontal nodal line onwith negligible curvature.
FIG. 5.
Semi-Dirac fermions and quantum geometry at the crossing point of two
nodal lines. (a) Fermi surface (blue) of the two-band model Eq. (2). The
orange shaded plane atcrosses the nodal line NL1 atand crosses NL2 at.
Purple shaded plane atcuts through the CP (black sphere) of NL1 and NL2
at the origin. (b) Band structureversus,at, showing a semi-Dirac point
(black sphere) as a result of the merging of three Dirac points atand.
Purple dashed lines in (a) and (b) show the crescent-shaped Fermi
surface contour of semi-Dirac fermions. Purple shaded planes represent
the Fermi level for CP1 () and CP2 (). (c) Calculated LL spectrum based
on model parameters for CP1 at. The right-hand panel indicates the
corresponding density of states (DOS) of the LLs. Purple dashed line
represents the Fermi level of CP1. Purple dots label the extremal points
in the LLs and purple arrow indicates the lowest momentum-conserving
transition (); see Supplemental Material Sec. VIII for details[33].
(d),(e) Calculated momentum space distribution of the Fubini-Study
metricat a Dirac point [(d)] and at the semi-Dirac point [(e)]. Near the
semi-Dirac point,is nonzero and shows a stronger divergence than the
Dirac case.
Atthe spectrum becomes a single Dirac again. The unique semi-Dirac
fermion described byis predicted to exhibit nontrivial Berry phase and
finite Chern number when a gap opens[13]. This is in stark contrast with
the zero Berry phase for semi-Dirac fermions formed by merging an even
number of Dirac points (modulo). Crossing nodal lines in Eq.
(2)therefore offer a new platform for studying the rich phenomena of
merging Dirac points[28,60](see Supplemental Material Video 2[33]),
where topology and correlation effects intertwine.
We have calculated the Landau level spectrum of Eq. (2)in the presence
of an in-plane field directed alongas a function of, and the results are
shown in Fig. 5(c). Sinceis a good quantum number for, at eachthe LL
spectrum is a series of discrete levels, with level spacing determined
by the projection of the constant energy contours onto the plane
perpendicular to. The LLs exhibit extremal points nearand lead to peaks
in the density of states (DOS), indicated by purple dots in Fig. 5(c).
Optical transitions (vertical purple arrow) between these LL
singularities are observed experimentally in LL spectroscopy. We remark
again that only the lowest-order momentum-conserving LL transitions ()
have been observed in this work (Supplemental Material Sec. IX[33]).
Higher-order LL transitions are in general weaker and may be forbidden
by selection rules (see, e.g., Ref. [61]for type-I semi-Dirac fermions).
We further demonstrate thescaling of the LLs at the CP using both
semiclassical quantization below and full LL calculation with SOC
(Supplemental Material Sec. VIII). Apart from enhanced DOS, we remark
that the semi-Dirac fermions realized by Eq. (2)also show stronger
divergence in the quantum metriccompared to Dirac fermions [see Fig.
5(d)]. In Fig. 5(d), we compared the calculated-space distribution
ofat(semi-Dirac) and(Dirac) based on Eq. (2)(Supplemental Material
Sec. VI). We remark that the effects of quantum geometry in solids have
been discussed mostly theoretically in terms of nonlinear optical
response[62,63], but recent[64]and classic work[65,66]have connected the
linear response to the integrated Fubini-Study metric. Our experimental
observation of semi-Dirac fermions highlights the potential to explore
quantum geometry from novel quasiparticles using linear optical and
magneto-optical response.
Before concluding, we comment on the robustness of the observedscaling
of Landau levels from crossing nodal lines in ZrSiS. There are several
different ways that two nodal lines cross in momentum space[41,44]. We
illustrate three of the lowest orders crossing in Fig. 6, where the two
nodal lines are Fig. 6(a)both straight[41](), Fig. 6(b)both
parabolic[44], or Fig. 6(c)consisting of one straight and one parabolic
nodal line, whereis half of the SOC gap. Under an external magnetic
field along, the cyclotron motions of electrons will be quantized in the
2D plane normal to the magnetic field direction (constantplanes). The
band structures corresponding to the three models in Figs. 6(a)–6(c)at
the planeare shown in Figs. 6(d)–6(f), respectively. The cyclotron
motions for charge carriers from the three different nodal-line crossing
points can now be assessed by their distinct Fermi surface cross
sections at the Fermi level [black dashed lines in Figs. 6(d)–6(f)].
Importantly, the band structure at the CP from[Fig. 6(f)] is
electron-hole asymmetric (see also Fig. S30[33]), disctinct fromand.
FIG. 6.
Nodal-line crossing point models. Three different two-band models for
the crossing point of two nodal lines at. (a) Fermi surface (FS) near
the crossing of two straight nodal lines, described by the
Hamiltonian[41]. (b) FS near the crossing of two parabolic nodal lines,
described by the Hamiltonian[44]. (c) FS near the crossing of one
straight and one parabolic nodal line, described by the Hamiltonian,
whereis half of the spin-orbit coupling (SOC) gap. The band structures
atfor the three different crossing points are shown in (d)–(f) for,,
and, respectively. (g) The FS contour atfor, showing an open orbit for
magnetic field. (h) The area of the closed orbitatforincreases linearly
with energy. (i) increases asatforand remainsfor energies higher than
the SOC gap. Here,,,,. Note thatisasymmetric and exhibits open orbits
for hole doping (Fig. S30[33]).
We can evaluate the Landau level scaling from a semiclassical approach
for these three different crossing points. First, it is evident that the
crossing of two straight nodal lines () will leads to an open orbit at
the CP [Fig. 6(g)], and therefore will not give rise to quantized energy
levels. For the crossing of two parabolic nodal lines (), the area of
the cyclotron orbitincreases linearly with energy[Fig. 6(h)]. According
to the Onsager quantization relation[34],, whereis the Landau level
index andis the phase factor, the LL scaling will be linear in. In
contrast, as derived in the Appendix, the cyclotron orbit area increases
as[Fig. 6(i)] for the CP of a straight and parabolic nodal line (),
which results in an LL scaling that is strictly. This semiclassical
analysis is consistent with full quantum mechanical calculations for the
CP of a straight and parabolic nodal lines, which also exhibits theLL
scaling (see Supplemental Material Sec. VIII and Fig. S23[33]).
The generic nature of our nodal-line crossing point model and analysis
(Figs. 4–6) guarantees the robustness of the experimental observable
even in real and complex materials such as ZrSiS. In Fig. 7, we show the
power-law fitting of one of the Landau level transitions from semi-Dirac
fermions (purple dots) with functionand the correspondingconfidence
interval (purple shaded region). The data points are consistent with
thescaling and statistically different from(conventional massive
fermions),(three-quarter Dirac fermions[67]), and(Dirac fermions).
FIG. 7.
Comparison of the power law of Landau level transitions for different
fermions in a log-log scale plot. Power-law fitting (purple dashed line)
of the interband Landau level transitions (purple dots) associated with
the semi-Dirac fermions in ZrSiS. Purple-shaded area indicates
theconfidence interval. Orange and black lines show the power-law
scaling of LLs for Dirac () and massive fermions (), respectively.
Using Landau level spectroscopy, we uncovered the semi-Dirac fermions
inside bulk ZrSiS under in-plane magnetic fields. Our results
demonstrate novel magnetic field effect in ZrSiS not explored in
previous experiments utilizing out-of-plane magnetic fields[68–70]. In
contrast to the conventional expectation of 2D electrons at the surface
or interface of a layered material, the observed semi-Dirac fermions
reside within planes perpendicular to the atomic layers of ZrSiS and
originate from the vicinity of points where nodal lines cross. The
crossing point of nodal lines in ZrSiS offers a unique and generic
platform for realizing semi-Dirac fermions through the merging of three
Dirac points. Our findings advance the understanding of exotic 2D
electrons in natural bulk crystals, establish the existence of novel
quasiparticles associated with crossing nodal lines in momentum
space[40,44], and open new directions in exploring quantum geometry and
topological effects in metals.
Notes
The data files for the materials growth recipe and the characterization
data are available at[71].
ACKNOWLEDGMENTS
Magneto-optical spectroscopy of ZrSiS is supported by National Science
Foundation, Division of Materials Research 2210186. Research on the
electrodynamics of semi-Dirac quasiparticles, including the theoretical
analysis of Landau levels spectra at Columbia, is supported as part of
Programmable Quantum Materials, an Energy Frontier Research Center
funded by the U.S. Department of Energy (DOE), Office of Science, Basic
Energy Sciences (BES), under Award No. DE-SC0019443. D. N. B. is Moore
Investigator in Quantum Materials EPIQS GBMF9455. Support for crystal
growth and characterization at Penn State was provided by the National
Science Foundation through the Penn State 2D Crystal
Consortium-Materials Innovation Platform (2DCC-MIP) under NSF
Cooperative Agreement No. NSF-DMR 2039351. The National High Magnetic
Field Laboratory is supported by the National Science Foundation through
NSF/DMR-1644779, NSF/DMR-2128556, and the State of Florida. The work of
A. N. R. and M. I. K. was supported by the European Union’s Horizon 2020
research and innovation program under European Research Council Synergy
Grant No. 854843 “FASTCORR.” The work of M. I. K. was further supported
by the Dutch Research Council (NWO) via the “TOPCORE” consortium.
J. H. A. gratefully acknowledges support from a Hertz Fellowship. The
Flatiron Institute is a division of the Simons Foundation.
Appendices
APPENDIX: METHODS
1. Voigt magneto-optical spectroscopy and the absence of surface
states
High-field magneto-optical measurements were performed atunder Voigt
geometry (and B) at the National High Magnetic Field Laboratory. A
Bruker Vertex 80V FTIR spectrometer combined with a 17.5 T
superconducting magnet was utilized to record the reflectance spectra of
ZrSiS at zero and high magnetic field. Infrared beam from a Globar lamp
was focused on the (001) surface of ZrSiS crystal. The typical size of
ZrSiS crystals used in our measurements isand the spot size of the
infrared beam is smaller than the lateral size of the sample.
We remark that there are various surface states observed by
angular-resolved photoemission measurements[35,37,72,73]from the (001)
surface of the ZrSiS family of topological semimetals. These surface
states are strictly confined to the top and bottom surfaces of ZrSiS and
do not contribute to the out-of-plane orbital motion when the magnetic
field is applied in-plane. Therefore, we conclude that the previously
reported surface states from (001) surface of ZrSiS cannot explain the
nearly massless Dirac fermions observed in our Voigt geometry
magneto-optical data.
2. Density functional theory calculation
DFT calculations were carried out using the plane-wave pseudopotential
method as implemented in thequantumespressosimulation package[74,75].
Norm-conserving pseudopotentials[76]were used in conjunction with the
local density approximation for exchange-correlation potential. An
energy cutoff of 70 Ry for the plane waves and a convergence threshold
ofwere used for the self-consistent solution of the Kohn-Sham equations.
The Brillouin zone was sampled by aMonkhorst-Pack[77]-point mesh.
Lattice constants were relaxed, resulting inand. The atomic structure
within the unit cell was relaxed until the residual forces were less than.
To ensure the numerical accuracy of the Fermi surface calculations and
related properties, an interpolation scheme based on the maximally
localizedwannierfunctions (MLWF)[78]was used. For this purpose, we used
thewannier90 code[79]to construct an extended tight-binding Hamiltonian
for ZrSiS in the MLWF basis, which included theandstates for Si and S as
well as theandstates for Zr. The interpolated band energies ensure a
correct description of the DFT band structure within the range of at
leastrelative to the Fermi energy. Thewannier-interpolated band energies
were calculated on a-point mesh, and were further used to calculate the
de Haas–van Alphen (dHvA) frequencies using the supercell-space extremal
area finder (skeaf) code[80].
3. Quantum oscillations in ZrSiS with torque magnetometry
measurements
The torque magnetometry measurements up to 14 T were performed using a
piezoresistive cantilever in a superconducting magnet equipped with a
variable temperature insert. A single crystal of ZrSiS is fixed to the
end of a 0.30 mm cantilever arm with vacuum grease. A jet of helium-4
gas from the inlet of the variable temperature insert onto the cryostat
kept the samples at a constant temperature of 1.6 K during the
measurements. There are two resistive elements on the cantilever, one of
which is located at the base of the arm and experiences strain with a
change in the sample magnetization. The second resistive element is not
affected by the torque but mimics the temperature and magnetic field
dependence of the first. These are combined with two more resistors at
room temperature to form a Wheatstone bridge that can be balanced at low
temperatures before changing the magnetic field. A small current is
applied across the bridge circuit and the measured voltage records the
changing torquecreated by the dHvA effect.
4. Semi-Dirac fermions in the nodal-ring model
To illustrate the semi-Dirac fermions in the nodal-ring model, we
consider the nodal-ring Hamiltonian[81,82],
(A1)
whereis the Fermi velocity along,is the identity matrix, and,are the
Pauli matrices for orbitals. The corresponding energy spectrum is
(A2)
and a Dirac nodal ring atwith radiuscan be identified. This nodal ring
marks a protected crossing of two bands along a ring in momentum space,
and at any point on the ring the spectrum is Dirac. Remarkably, at, the
projection of constant energy contours onto theplane yields contours of
the semi-Dirac form[14,82,83]that describes a massive fermion along()
and massless Dirac fermion along(see Supplemental Material Video 1[33]).
Under the in-plane magnetic field, the area of the cyclotron orbit at
energyis, which leads to thescaling of LLs[10,82]. Here, we briefly
describe the solution of the Hamiltonian in an in-plane magnetic
fielddirected alongthat reveals the main features of the absorption
spectrum. Choosing the Landau gaugeand rescaling the coordinatewith
magnetic lengthand defining field scaleand rescaled field, we havewith.
From this form ofone can see that for,is minimized at a
particularexpanding around the twovalues that give Dirac spectra, while
whenwe have a semi-Dirac equation. We have solved the corresponding
Schrödinger equation numerically by observing that at each,can be
diagonalized trivially yielding two second-order differential equations
that we solve by discretization. The current operator can then be
obtained as a differential operator and applied to the solutions.
Integration of the resulting absorption spectrum overgives the continuum
absorption, with singularities at the upper edge scaling asand lower
edge as, consistent with previous results[82]. While the nodal-ring
model is not directly relevant to the interpretation of our experimental
results, it does display the generic features of an absorption continuum
with upper and lower edges scaling differently with, and with the
characteristic Dirac and semi-Dirac behaviors.
5. Semiclassical quantization of the crossing nodal-line model
To obtain the Landau level scaling of the semi-Dirac fermion at the CP
of nodal lines, we consider the following Hamiltonian:
(A3)
which has an extremal Fermi surface atfor. This model also describes the
crossing of one straight and one curved nodal line and is related to the
model Eq. (2)by a 45-deg in-plane rotation. The corresponding eigenvalue
is given by
(A4)
For a magnetic fieldapplied along, the eigenvalue spectrum has extremal
values at. Without loss of generality, we consider the Fermi surface of
the electron pockets () and choose,, and. The cross section of the
electron Fermi surface at constant energyis then determined by
(A5)
which is bounded by two parabolic curves:
𝑘+𝑦=𝐸𝑣−𝑡−12𝑚𝑣𝑘2𝑧and𝑘−𝑦=−𝐸𝑣+𝑡+12𝑚𝑣𝑘2𝑧.
(A6)
These two parabolic curves intersect at𝑘2𝑧=𝐸/𝑡≡𝑎2, forming a
crescent-shaped closed contour for electron orbits [see inset of Fig.
6(i)]. If𝑘2𝑧>𝐸/𝑡, the hole FS is obtained
with𝐸=𝑡𝑘2𝑧−(𝑘2𝑧/2𝑚)−𝑣𝑘𝑦. The area of the electron FS
contour can be obtained by evaluating the difference between the areas
of the parabolic segments of𝑘−𝑦and𝑘+𝑦:
𝑆(𝐸)=𝑆−(𝐸)−𝑆+(𝐸)=∫𝑎−𝑎(𝑘+𝑦−𝑘−𝑦)𝑑𝑘𝑧=∫𝑎−𝑎2𝐸/𝑣(1−𝑘2𝑧𝑎2)𝑑𝑘𝑧=83𝐸3/2√𝑡𝑣.
(A7)
Following Onsager’s quantization relation𝑆(𝐸)=2𝜋(𝑛+𝛾)𝑒𝐵/ℏ,
where𝑛is the Landau level index and𝛾is the phase factor, we obtain
the𝐵2/3scaling of LLs:𝐸𝑛∝(𝑛+𝛾)2/3𝐵2/3.
In the presence of finite SOC, the degeneracy at the semi-Dirac point is
lifted but the characteristic band dispersions and LL scaling recover at
energies higher than the SOC gap. We illustrate the case with SOC using
the following Hamiltonian based on Eq. (A3):
𝐻′3=𝑡𝑘2𝑧𝜏0+(𝑘2𝑧2𝑚−𝑣𝑘𝑦)𝜏𝑧+𝑘𝑥𝑘𝑧𝜏𝑥+Δ𝜏𝑦,
(A8)
where2Δis the size of the SOC gap. The area of the electron FS contour
with SOC becomes
𝑆SOC(𝐸)=83√𝐸+Δ√𝑡𝑣×(𝑒𝐸[1−2Δ𝐸+Δ]𝐸−𝑒𝐾[1−2Δ𝐸+Δ]Δ),
(A9)
where𝑒𝐸[𝑥]and𝑒𝐾[𝑥]denote the complete elliptic integral of the
second kind and first kind, respectively. Equation (A9)can be solved
numerically and the resulting𝑆SOC(𝐸)is shown in Fig. 6(i), where
the𝑆SO(𝐸)∝𝐸3/2behavior is recovered at energies higher than around2Δ.
Supplemental Material
Supplementary Material files contain crystal characterizations of ZrSiS,
quantum oscillation measurements and band structure calculations.
Details of the tight-binding model and continuum model for the crossing
nodal-lines are provided and compared with DFT calculations.
Calculations of quantum geometry and Landau level spectra based on the
continuum model are provided.
Two Supplementary Videos are also provided, comparing the semi-Dirac
fermions formed by merging of two Dirac points and three Dirac points.
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