Discovery of topological chiral crystals with helicoid arc states

TL;DR

This paper reports the discovery of topological chiral crystals in the RhSi family, exhibiting unique helicoid Fermi arc surface states and topological charges, advancing the understanding of topological quantum materials.

Contribution

First observation of topological quantum properties in chiral RhSi crystals, revealing helicoid Fermi arcs and high topological charges linked to bulk chiral fermions.

Findings

Helicoid Fermi arcs stretch across the entire Brillouin zone.

Surface states exhibit a highly unusual helicoid structure.

Topological charges of ±2 are experimentally characterized.

Abstract

The quantum behaviour of electrons in materials lays the foundation for modern electronic and information technology. Quantum materials with novel electronic and optical properties have been proposed as the next frontier, but much remains to be discovered to actualize the promise. Here we report the first observation of topological quantum properties of chiral crystals in the RhSi family. We demonsrate that this material hosts novel phase of matter exhibiting nearly ideal topological surface properties that emerge as a consequence of the crystals' structural chirality or handedness. We also demonstrate that the electrons on the surface of this crystal show a highly unusual helicoid structure that spirals around two high-symmetry momenta signalling its topological electronic chirality. Such helicoid Fermi arcs on the surface experimentally characterize the topological charges of $\pm{2}$, which arise from the bulk chiral fermions. The existence of bulk high-fold degenerate fermions are guaranteed by the crystal symmetries, however, in order to determine the topological charge in the chiral crystals it is essential to identify and study the helical arc states. Remarkably, these topological conductors we discovered exhibit helical Fermi arcs which are of length $\pi$, stretching across the entire Brillouin zone and orders of magnitude larger than those found in all known Weyl semimetals. Our results demonstrate novel electronic topological state of matter on a structurally chiral crystal featuring helicoid Fermi arc surface states. The exotic electronic chiral fermion state realised in these materials can be used to detect a quantised photogalvanic optical response or the chiral magnetic effect and its optical version in future devices as described by G. Chang \textit{et.al.,} `Topological quantum properties of chiral crystals' Nature Mat. 17, 978-985 (2018).

Tables

Table 1. Extended Data Table 1: Single crystal crystallographic data for CoSi at 100(2) and 300(2)K.

CoSi	T = 100(2) K	T = 300(2) K
Scan	1	2
F.W. (g/mol)	87.02	87.02
Space Group; $Z$	$P{2}_{1}3$ (No.198); 4	$P{2}_{1}3$ (No.198); 4
$a\left(\text{Å}\right)$	4.433(4)	4.4245(16)
$V({\text{Å}}^3$)	87.1(2)	86.61(9)
Absorption Correction	Numerical	Numerical
Extinction Coefficient	0.39(9)	0.5(2)
$\theta$ range (deg)	19.301- 31.714	18.740- 31.781
No. Reflections; Rint	165; 0.0222	167;0.0232
No. Independent Reflections	79	77
No. Parameters	9	9
$R_1$; $w{R}_2$ (all I)	0.0223; 0.0561	0.0533; 0.1209
Goodness of fit	1.123	1.142
Diffraction peak and hole (e${}_{}{}^{-}/{\text{Å}}^3$)	0.637; -0.736	1.155; -1.895

Full text

Discovery of topological chiral crystals with helicoid arc states

Daniel S. Sanchez

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Ilya Belopolski

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Tyler A. Cochran

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Xitong Xu

International Center for Quantum Materials, School of Physics, Peking University, China

  

Jia-Xin Yin

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Guoqing Chang

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Weiwei Xie

Department of Chemistry, Louisiana State University, Baton Rouge, LA, 70803, USA

  

Kaustuv Manna

Max Planck Institute for Chemical Physics of Solids, Dresden, D-01187, Germany

  

Vicky Süß

Max Planck Institute for Chemical Physics of Solids, Dresden, D-01187, Germany

  

Cheng-Yi Huang

Institute of Physics, Academia Sinica, Taipei 11529, Taiwan

  

Nasser Alidoust

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

Rigetti Computing, Berkeley, California 94720, USA

  

Daniel Multer

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

  

Songtian S. Zhang

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Nana Shumiya

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Xirui Wang

International Center for Quantum Materials, School of Physics, Peking University, China

  

Guang-Qiang Wang

International Center for Quantum Materials, School of Physics, Peking University, China

  

Tay-Rong Chang

Department of Physics, National Cheng Kung University, Tainan 701, Taiwan

  

Claudia Felser

Max Planck Institute for Chemical Physics of Solids, Dresden, D-01187, Germany

  

Su-Yang Xu

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  

Shuang Jia

International Center for Quantum Materials, School of Physics, Peking University, China

Collaborative Innovation Center of Quantum Matter, Beijing,100871, China

CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Science, Beijing 100190, China

  

Hsin Lin

Institute of Physics, Academia Sinica, Taipei 11529, Taiwan

  

M. Zahid Hasan*†000†*Corresponding author (email): mzhasan@princeton.edu

Submitted on August 10, 2018; References updated.

Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

The quantum behaviour of electrons in materials lays the foundation for modern electronic and information technology natnews ; kmoore ; revHK ; revQZ ; jia ; rev4 ; rev6 ; rev6a ; rev7 ; KramersWeyl ; News . Quantum materials with novel electronic and optical properties have been proposed as the next frontier, but much remains to be discovered to actualize the promise. Here we report the first observation of topological quantum properties of chiral crystals KramersWeyl ; News in the RhSi family. We demonsrate that this material hosts novel phase of matter exhibiting nearly ideal topological surface properties that emerge as a consequence of the crystals’ structural chirality or handedness KramersWeyl ; News . We also demonstrate that the electrons on the surface of this crystal show a highly unusual helicoid structure that spirals around two high-symmetry momenta signalling its topological electronic chirality. Such helicoid Fermi arcs on the surface experimentally characterize the topological charges of $\pm{2}$, which arise from the bulk chiral fermions. The existence of bulk high-fold degenerate fermions are guaranteed by the crystal symmetries, however, in order to determine the topological charge in the chiral crystals it is essential to identify and study the helical arc states. Remarkably, these topological conductors we discovered exhibit helical Fermi arcs which are of length $\pi$, stretching across the entire Brillouin zone and orders of magnitude larger than those found in all known Weyl semimetals rev4 ; rev6 ; jia ; rev6a . Our results demonstrate novel electronic topological state of matter on a structurally chiral crystal featuring helicoid Fermi arc surface states. The exotic electronic chiral fermion state realised in these materials can be used to detect a quantised photogalvanic optical response or the chiral magnetic effect and its optical version in future devices as described by G. Chang et.al., “Topological quantum properties of chiral crystals” Nature Mat. 17, 978-985 (2018) KramersWeyl .

The discovery of topological insulators has inspired the search for a wide variety of topological conductors natnews ; kmoore ; revHK ; revQZ ; rev4 ; rev6 ; jia ; rev7 ; NielsenNinomiya1 ; filling_constraint1 ; rev6a ; Topo.gapless.phase ; Wan ; HgCr2Se4 ; News ; Weyl-Multilayer ; unconventionalWeyl ; RhSi ; CoSi ; TaAs1 ; TaAs2 ; ARPES-TaAs1 ; KramersWeyl ; Ben1 ; Manes ; ARPES-TaAs2 ; Nobel . One example of topological conductor is the Weyl semimetal (WSM), featuring emergent Weyl fermions as low-energy excitations of electrons. These Weyl fermions are associated with topological chiral charges (Chern numbers) that locate at two-fold degenerate band crossings natnews ; kmoore ; rev4 ; rev6 ; rev6a ; rev7 ; NielsenNinomiya1 ; Topo.gapless.phase ; Wan ; Weyl-Multilayer in momentum space. In general, such emergent topological chiral fermions may appear in a variety of types including not only two-fold degenerate Weyl fermions rev4 ; rev7 ; rev6 ; rev6a ; NielsenNinomiya1 ; Topo.gapless.phase ; Wan ; Weyl-Multilayer ; HgCr2Se4 , Kramers-Weyl fermions KramersWeyl ; News or higher-fold fermions unconventionalWeyl ; RhSi ; CoSi . Recently, a few non-centrosymmetric crystals were identified where a band inversion gives rise to a WSM state TaAs1 ; TaAs2 ; ARPES-TaAs1 ; ARPES-TaAs2 ; rev4 ; rev7 ; rev6 . However, all these materials suffer from several drawbacks: a large number of Weyl fermions, Weyl fermions close to each other in momentum space, and short Fermi arcs which are much (orders of magnitude) less topologically robust. In order to thoroughly explore and utilise the robust and unusual quantum phenomena induced by chiral fermions in optics or magneto-transport, topological conductors with near-ideal electronic properties or novel types of topological conductors are needed natnews ; kmoore .

A different approach toward searching for ideal topological conductors is to examine crystalline symmetries, which can also lead to topological band crossings revHK ; revQZ ; rev6 . For instance, it has been shown that non-symmorphic symmetries can guarantee the existence of band crossings for certain electron fillings unconventionalWeyl ; filling_constraint1 ; Ben1 . As another example, we might consider structurally chiral crystals whose lattice possess no inversion, mirror and roto-inversion symmetries KramersWeyl . Structurally chiral crystals are expected to host a variety of topological band crossings which are guaranteed to be pinned to time-reversal invariant momenta (TRIMs) KramersWeyl ; Manes . Moreover, structurally chiral topological crystals naturally give rise to a quantised circular photogalvanic current, the chiral magnetic effect and other novel transport and optics effects forbidden in known topological conductors, such as TaAs rev6 ; TaAs1 ; TaAs2 ; ARPES-TaAs1 ; ARPES-TaAs2 .

Incorporating these paradigms into a broader search, we have studied various candidate nonmagnetic and magnetic conductors, such as pyrochlore iridates, MoTe2, WTe2, Mn3Ge, Ge3Sn, Mn3Sn, Na3Bi, GdPtBi, the LuPtSb family, HgCr2Se4, LaPtBi, Co3Sn2S2, Fe3Sn2 and the RhSi family, with advanced spectroscopic techniques. Many of the materials exhibit either large co-existing trivial bulk Fermi surfaces or surface reconstruction masking topological states. And as often is the case with surface-sensitive techniques, the experimentally realised surface potential associated with a cleaved crystal may or may not allow these unusual electronic states to be observed. Thus, despite the new search paradigms, the discovery of topological materials that are suitable for spectroscopic experiments has remained a significant challenge. Of all the materials we explored, we observed that the $X$Si ($X=$ Co, Rh) family of chiral crystals comes close to the experimental realization of the sought-after ideal topological conductor. Additionally it is a novel type of topological conductor beyond Weyl semimetals. Here we report high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements in combination with state-of-the-art ab initio calculations to demonstrate novel topological chiral properties in CoSi and RhSi. These chiral crystals approach ideal topological conductors because of their large Fermi arcs which is related to the fact that they host the minimum non-zero number of chiral fermions—topological properties which we experimentally visualise for the first time.

The $X$Si ($X=$ Co, Rh) family of materials crystallises in a structurally chiral cubic lattice with space group $P2_{1}3$, No. 198 (Fig. 1a). We confirmed the chiral crystal structure of our CoSi samples by single crystal X-ray diffraction (XRD; Fig. 1b), with associated 3D Fourier map (Fig. 1c). We found a Flack factor of $\sim 91\%$, which indicates that our samples are predominantly of a single structural chirality. Ab initio electronic bulk band structure calculations predict that both chiral crystals exhibit a 3-fold degeneracy at $\Gamma$ near the Fermi level, $E_{\textrm{F}}$ (Fig. 1d). This degeneracy is described by a low-energy Hamiltonian which exhibits a 3-fold fermion associated with Chern number $+2$ RhSi ; CoSi . We refer to this Chern number as a chiral charge, a usage of the term “chiral” which is distinct from the notion of structural chirality defined above and which also motivates our use of the term “chiral fermion” to describe these topological band crossings. The $R$ point hosts a 4-fold degeneracy corresponding to a 4-fold fermion with Chern number $-2$. These two higher-fold chiral fermions are pinned to opposite TRIMs and are consequently constrained to be maximally separated in momentum space (Fig. 1e), suggesting that CoSi might provide a near-ideal platform for accessing topological phenomena using a variety of techniques. The hole pocket at $M$ is topologically trivial at the band relevant for low-energy physics, but it is well-separated in momentum space from the $\Gamma$ and $R$ topological crossings. It is not expected to affect topological transport, such as the chiral anomaly (note that $\sigma_{xx}$ is not topological transport). The two higher-fold chiral fermions lead to a net Chern number of zero in the entire bulk Brillouin zone (BZ), as expected from theoretical considerations NielsenNinomiya1 . CoSi and RhSi also satisfy a key criterion for an ideal topological conductor, namely that they have only two chiral fermions in the bulk BZ, the minimum non-zero number allowed.

The two chiral fermions remain topologically non-trivial over a wide energy range. In particular, CoSi maintains constant-energy surfaces with non-zero Chern number over an energy window of 0.85 eV, while for RhSi this window is 1.3 eV (Fig. 1d). This prediction suggests that $X$Si satisfies another criterion for an ideal topological conductor—a large topologically non-trivial energy window. An ab initio calculated Fermi surface shows that the projection of the higher-fold chiral fermions to the (001) surface results in a hole (electron) pocket at $\bar{\Gamma}$ ($\bar{M}$) with Chern number $+2$ ($-2$; Fig. 1f). As a result, we expect that $X$Si hosts Fermi arcs of length $\pi$ spanning the entire surface BZ, again suggesting that these materials may realise a near-ideal topological conductor.

Using low-photon-energy ARPES, we experimentally study the (001) surface of CoSi and RhSi to reveal their surface electronic structure. For CoSi, the measured constant-energy contours show the following dominant features: two concentric contours around the $\bar{\Gamma}$ point, a faint contour at the $\bar{X}$ point, and long winding states extending along the $\bar{M}-\bar{\Gamma}-\bar{M}$ direction (Fig. 2a). Both the $\bar{\Gamma}$ and $\bar{X}$ pockets show a hole-like behaviour (Extended Data Fig. 8). The measured surface electronic structure for RhSi shows similar features (Fig. 2b; Extended Data Fig. 1). Using only our spectra, we first sketch the key features of the experimental Fermi surface for CoSi (Fig. 2c). Then, to better understand the k-space trajectory of the long winding states in CoSi, we study Lorentzian fits to the momentum distribution curves (MDCs) of the ARPES spectrum. We plot the Lorentzian peak positions as the extracted band dispersion (Fig. 2d, e) and we find that the long winding states extend from the center of the BZ to the $\bar{M}$ pocket (Fig. 2f). To better understand the nature of these states, we perform an ARPES photon energy dependence and we find that the long winding states do not disperse as we vary the photon energy, suggesting that they are surface states (Extended Data Fig. 6). Moreover, we observe an overall agreement between the ARPES data and the ab initio calculated Fermi surface, where topological Fermi arcs connect the $\bar{\Gamma}$ and $\bar{M}$ pockets (Fig. 1f). Taken together, these results suggest that the long winding states observed in ARPES may be topological Fermi arcs.

Grounded in the framework of topological band theory, the bulk-boundary correspondence of chiral fermions makes it possible for ARPES (spectroscopic) measurements to determine the Chern numbers of a crystal by probing the surface state dispersion (Fig. 3a; Methods). Such spectroscopic methods to determine Chern numbers have become well-accepted in the field Nobel . Using such approach and its spectroscopic analogs, we provide two spectral signatures of Fermi arcs in CoSi. We first look at the dispersion of the candidate Fermi arcs along a pair of energy-momentum cuts on opposite sides of the $\bar{\Gamma}$ pocket, taken at fixed $+k_{x}$ (Cut I) and $-k_{x}$ (Cut II; Fig. 2f). In Cut I, we observe two right-moving chiral edge modes (Fig. 3b,c). Since the cut passes through two BZs (Fig. 2f), we associate one right-moving mode with each BZ. Next, we fit Lorentzian peaks to the MDCs and we find that the extracted dispersion again suggests two chiral edge modes (Fig. 3d). Along Cut II, we observe two left-moving chiral edge modes (Fig. 3e,f). Consequently, one chiral edge mode is observed for each measured surface BZ on Cuts I and II, but with opposite Fermi velocity direction. In this way, our ARPES spectra suggest that the number of chiral edge modes $n$ changes by $+2$ when the $k$-slice is swept from Cut I to Cut II. This again suggests that the long winding states are topological Fermi arcs. Moreover, these ARPES results imply that projected topological charge with net Chern number $+2$ lives near $\bar{\Gamma}$.

Next we search for other Chern numbers encoded by the surface state band structure. We study an ARPES energy-momentum cut on a loop $\mathcal{P}$ enclosing $\bar{M}$ (Fig. 4a, inset). Again following the bulk-boundary correspondence, we aim to extract the Chern number of chiral fermions projecting on $\bar{M}$. The cut $\mathcal{P}$ shows two right-moving chiral edge modes dispersing towards $E_{\textrm{F}}$ (Fig. 4a,b), suggesting a Chern number $-2$ on the associated bulk manifold. Furthermore, the ab initio calculated surface spectral weight along $\mathcal{P}$ is consistent with our experimental results (Fig. 4c). Our ARPES spectra on Cut I, Cut II and $\mathcal{P}$ suggest that CoSi hosts a projected chiral charge of $+2$ at $\bar{\Gamma}$ with its partner chiral charge of $-2$ projecting on $\bar{M}$. This again provides evidence that the long winding states are a pair of topological Fermi arcs which traverse the surface BZ on a diagonal, connecting the $\bar{\Gamma}$ and $\bar{M}$ pockets. Our ARPES spectra on RhSi also provide evidence for gigantic topological Fermi arcs following a similar analysis (Extended Data Fig. 1).

To further explore the topological properties of CoSi, we examine in greater detail the structure of the Fermi arcs near $\bar{M}$. We consider the dispersion on $\mathcal{P}$ (plotted as a magenta loop in Fig. 4d, inset) and we also extract a dispersion from Lorentzian fitting on a second, tighter circle (black loop; Extended Data Fig. 7). We observe that as we decrease the binding energy (approach $E_{\textrm{F}}$), the extracted dispersion spirals in a clockwise fashion on both loops, suggesting that as a given k point traverses the loop, the energy of the state does not return to its initial value after a full cycle. Such a non-trivial electronic dispersion directly signals a projected chiral charge at $\bar{M}$ (Fig. 4d). In fact, the extracted dispersion is characteristic of the helicoid structure of topological Fermi arcs as they wind around a chiral fermion (Fig. 4e), suggesting that CoSi provides a rare example of a non-compact surface in nature HelicodalFermiArcs ; KramersWeyl .

To further understand these experimental results, we consider the ab initio calculated spectral weight for the (001) surface and we observe a pair of Fermi arcs winding around the $\bar{\Gamma}$ and $\bar{M}$ pockets in a counterclockwise and clockwise manner, respectively, with decreasing binding energy (approaching $E_{\textrm{F}}$; Fig. 4f). The clockwise winding around $\bar{M}$ is consistent with our observation by ARPES of a $-2$ projected chiral charge. Moreover, from our ab initio calculations, we predict that the $-2$ charge projecting to $\bar{M}$ arises from a 4-fold chiral fermion at the bulk $R$ point (Fig. 1d). The $+2$ chiral charge which we associate with $\bar{\Gamma}$ from ARPES (Fig. 3) is further consistent with the 3-fold chiral fermion predicted at the bulk $\Gamma$ point. By fully accounting for the predicted topological charges in experiment, our ARPES results suggest the demonstration of a topological chiral crystal in CoSi. We can similarly account for the predicted topological charges in RhSi from our ARPES data (Extended Data Fig. 1, 2).

The surface state dispersions in our ARPES spectra, taken together with the topological bulk-boundary correspondence established in theory Wan ; arcDetect1 , demonstrate that CoSi is a topological chiral crystal. This experimental result is further consistent with the numerical result determined from first-principles calculations of the surface state dispersions and topological invariants. Unlike previously-reported WSMs, the Fermi arcs which we observe in CoSi and RhSi stretch diagonally across the entire (001) surface Brillouin zone, from $\bar{\Gamma}$ to $\bar{M}$. In fact, the Fermi arcs in $X$Si are longer than those found in TaAs by a factor of thirty. Our surface band structure measurements also demonstrate two well-separated Fermi pockets carrying Chern number $\pm{2}$. Lastly, we observe for the first time in an electronic material the helicoid structure of topological Fermi arcs, offering an example of a non-compact surface HelicodalFermiArcs ; KramersWeyl . Our results suggest that CoSi and RhSi are excellent candidates for studying topological phenomena distinct to chiral fermions, using a variety of techniques rev4 ; rev7 ; rev6 .

Crucial for applications, the topologically non-trivial energy window in CoSi is an order of magnitude larger than that in TaAs ARPES-TaAs1 ; ARPES-TaAs2 , rendering its quantum properties robust against changes in surface chemical potential and disorder. Moreover, the energy offset between the higher-fold chiral fermions at $\Gamma$ and $R$ is predicted to be $\sim 225$ meV. Such an energy offset is essential for inducing the chiral magnetic effect and its optical analog chiralmagenticeffect1 and the quantised photogalvanic effect (optical) chiral_photogalvanic . When coupled to a compatible superconductor, CoSi is a compelling platform for studying the superconducting pairing of Fermi surfaces with non-zero Chern numbers, which may be promising for realizing a new type of topological superconducting phase recently proposed by Li and Haldane Haldane which can be probed with STM-based spectroscopy. CoSi further opens the door to exploring other exotic quantum phenomena when combined with the isochemical material FeSi. Fe1-xCoxSi may simultaneously host k-space topological defects (chiral fermions) and real-space topological defects (skyrmions) and their interplay which can also be probed by STM/STS. Through our observation of a helicoid surface state and its topological properties, our results suggest the discovery of the first structurally chiral crystal that is also topological. In this way, our work provides a much-needed new and next-generation platform for further study and search for novel types of topological conductors.