Triple-Point Fermions in Ferroelectric GeTe

Juraj Krempaský, Laurent Nicolaï, Martin Gmitra, Houke Chen, Mauro Fanciulli, Eduardo B. Guedes, Marco Caputo, Milan Radović, Valentine V. Volobuev, Ondřej Caha, Gunther Springholz, Jan Minár, and J. Hugo Dil

Phys. Rev. Lett. 126, 206403 – Published 17 May 2021

Abstract

Ferroelectric $α$ -GeTe is unveiled to exhibit an intriguing multiple nontrivial topology of the electronic band structure due to the existence of triple-point and type-II Weyl fermions, which goes well beyond the giant Rashba spin splitting controlled by external fields as previously reported. Using spin- and angle-resolved photoemission spectroscopy combined with ab initio density functional theory, the unique spin texture around the triple point caused by the crossing of one spin-degenerate and two spin-split bands along the ferroelectric crystal axis is derived. This consistently reveals spin winding numbers that are coupled with time-reversal symmetry and Lorentz invariance, which are found to be equal for both triple-point pairs in the Brillouin zone. The rich manifold of effects opens up promising perspectives for studying nontrivial phenomena and multicomponent fermions in condensed matter systems.

Received 13 November 2020
Revised 16 February 2021
Accepted 8 April 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.206403

Physics Subject Headings (PhySH)

Electronic structure Ferroelectricity Topological phases of matter

Topological materials

Angle-resolved photoemission spectroscopy Density functional calculations Spin-resolved photoemission spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Juraj Krempaský ^1,†, Laurent Nicolaï ², Martin Gmitra ^3,*, Houke Chen ⁴, Mauro Fanciulli ⁵, Eduardo B. Guedes ¹, Marco Caputo ¹, Milan Radović¹, Valentine V. Volobuev ^6,7, Ondřej Caha⁸, Gunther Springholz ⁹, Jan Minár², and J. Hugo Dil ^1,10

¹Photon Science Division, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
²New Technologies-Research Center, University of West Bohemia, 301 00 Plzeň 3, Czech Republic
³Institute of Physics, P. J. Šafárik University in Košice, Park Angelinum 9, 040 01 Košice, Slovakia
⁴Department of Physics, Tsinghua University, Beijing 100084, China
⁵Laboratoire de Physique des Matériaux et Surfaces, CY Cergy Paris Université, 95031 Cergy-Pontoise, France
⁶International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
⁷National Technical University “KhPI”, Kyrpychova Street 2, 61002 Kharkiv, Ukraine
⁸Department of Condensed Matter Physics, Masaryk University, Kotlářská 267/2, 61137 Brno, Czech Republic
⁹Institut für Halbleiter-und Festkörperphysik, Johannes Kepler Universität, A-4040 Linz, Austria
¹⁰Institut de Physique, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

^*Corresponding author. martin.gmitra@upjs.sk
^†Corresponding author. juraj.krempasky@psi.ch

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 126, Iss. 20 — 21 May 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Bulk BZ and surface BZ of rhombohedral $α$ -GeTe with $⟨ 111 ⟩$ ferroelectric axis along the $k_{z}$ direction. (b) BSF initial state calculations along $k_{x}$ for 16 selected $k_{z}$ values denoted in (a). (c) Band structure along $k_{z}$ with the position of ${TP}_{-}^{σ}$ and ${TP}_{+}^{σ}$ indicated by orange arrows.
Reuse & Permissions
Figure 2
(a) Calculated band structure along the $U^{'} - Z - U$ mirror plane, with TP, WP II, and DP. (b) Formation of a TP pair ( ${TP}_{-}^{-}$ , ${TP}_{-}^{+}$ ) by spin-degenerate band $Λ_{6}$ and two spin-split bands $Λ_{4, 5}$ . (c)–(e) Close-up of band structure along $k_{y}$ at three different values of $k_{z}$ across the TP pair. Three types of touching points are indicated with black, green, and magenta markers, and black arrows indicate the $k_{z}$ dispersion of the latter. (f) Band structure around the TP under 10 T magnetic field along the $z$ direction. The TP splits into a pair of Weyl points (WPs). (g) Band structure for rocksalt $β$ -GeTe above the ferroelectric phase transition. The triple fermions merges into a single DP. Inset: band structure along $k_{x}$ .
Reuse & Permissions
Figure 3
Top: high-resolution ARPES data for various photon energies around the TP (with corresponding reduced $k_{z}$ values with respect to $Γ$ ). Bottom: corresponding curvature maps showing the TP formation at 30.5 eV photon energy inside the orange frame. The arrows indicate surface states ${SS}_{2}$ , SR, and B.
Reuse & Permissions
Figure 4
(a) ARPES band maps along $\bar{M Γ M}$ measured at the COPHEE setup with $h ν = 58$ and 89 eV. Orange bullets indicate ${TP}_{-}^{σ}$ and ${TP}_{+}^{σ}$ . MDC $A$ and MDC $B$ indicate where data in Fig. 5 are obtained. (b) Constant energy map (CEM) measured at $h ν = 89$ and $- 0.4 eV$ binding energy at COPHEE. Marker $A$ indicates the $k_{x, y}$ locus used for SARPES EDC, next to the bulk bands denoted with dashed red and blue lines. (c),(d) SARPES EDC cut $A$ total intensity and related $P_{x, y, z}$ spin polarizations with corresponding $B_{1, 2, 3, 4}$ bulk band fitting indicated in red and blue and surface-derived bands in gray. (e) 3D TP cartoon illustrating EDC cut $A$ with corresponding in-plane spin textures, showing spin vectors projected mainly along $P_{x}$ with a characteristic $(+ - - +)$ pattern around the TP.
Reuse & Permissions
Figure 5
(a) Tabulated spin textures for ${TP}_{-}^{σ}$ and ${TP}_{+}^{σ}$ measured along the MDC $A$ (MDC $B$ ) cuts 250 meV above (below) the TP, as denoted in Fig. 4. (b),(c) Corresponding SARPES intensities and 3D spin polarization fitting for ${TP}_{-}^{σ}$ and ${TP}_{+}^{σ}$ , respectively. Red and blue curves indicate the $B_{1, 2, 3, 4}$ peak fitting, and surface-derived bands are in gray. (d) Analogous spin textures measured by rotating the sample by 180°; the frames in green and magenta indicate two inequivalent directions denoted in Fig. 1. (e) BSF constant energy maps; the black arrow indicates the Weyl-II fermion, and the dashed line its energy dispersion and hybridization with TP bands. (f) Close-ups of spin-resolved CEMs at $- 1.55$ and $- 1.6 eV$ , respectively, showing a change in spin winding number from $\pm 1$ to $\pm 2$ around the WP II point inclusion with a conspicuous “spin swirl” denoted by the dashed circle.
Reuse & Permissions

Physical Review Letters