Converting topological insulators into topological metals within the tetradymite family

K.-W. Chen, N. Aryal, J. Dai, D. Graf, S. Zhang, S. Das, P. Le Fèvre, F. Bertran, R. Yukawa, K. Horiba, H. Kumigashira, E. Frantzeskakis, F. Fortuna, L. Balicas, A. F. Santander-Syro, E. Manousakis, and R. E. Baumbach

Phys. Rev. B 97, 165112 – Published 9 April 2018

Abstract

We report the electronic band structures and concomitant Fermi surfaces for a family of exfoliable tetradymite compounds with the formula $T_{2} C h_{2} P n$ , obtained as a modification to the well-known topological insulator binaries ${Bi}_{2}$ ( ${Se, Te)}_{3}$ by replacing one chalcogen ( $C h$ ) with a pnictogen ( $P n$ ) and Bi with the tetravalent transition metals $T =$ Ti, Zr, or Hf. This imbalances the electron count and results in layered metals characterized by relatively high carrier mobilities and bulk two-dimensional Fermi surfaces whose topography is well-described by first-principles calculations. Intriguingly, slab electronic structure calculations predict Dirac-like surface states. In contrast to ${Bi}_{2} {Se}_{3}$ , where the surface Dirac bands are at the $Γ$ point, for ( ${Zr, Hf)}_{2} {Te}_{2}$ (P,As) there are Dirac cones of strong topological character around both the $\bar{Γ}$ and $\bar{M}$ points, which are above and below the Fermi energy, respectively. For ${Ti}_{2} {Te}_{2} P$ , the surface state is predicted to exist only around the $\bar{M}$ point. In agreement with these predictions, the surface states that are located below the Fermi energy are observed by angle-resolved photoemission spectroscopy measurements, revealing that they coexist with the bulk metallic state. Thus this family of materials provides a foundation upon which to develop novel phenomena that exploit both the bulk and surface states (e.g., topological superconductivity).

Received 21 November 2017
Revised 8 February 2018

DOI:https://doi.org/10.1103/PhysRevB.97.165112

Physics Subject Headings (PhySH)

Electronic structure Fermi surface Topological materials de Haas-van Alphen effect

Density functional calculations Quantum oscillation techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

K.-W. Chen^1,2, N. Aryal^1,2, J. Dai³, D. Graf¹, S. Zhang^1,2, S. Das^1,2, P. Le Fèvre⁴, F. Bertran⁴, R. Yukawa⁵, K. Horiba⁵, H. Kumigashira⁵, E. Frantzeskakis³, F. Fortuna³, L. Balicas^1,2, A. F. Santander-Syro³, E. Manousakis^1,2, and R. E. Baumbach^1,2

¹National High Magnetic Field Laboratory, Florida State University, Florida, USA
²Department of Physics, Florida State University, Florida, USA
³CSNSM, Universié Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 91405 Orsay Cedex, France
⁴Synchrotron SOLEIL, L'Orme des Merisiers, Saint-Aubin-BP48, 91192 Gif-sur-Yvette, France
⁵Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba 305-0801, Japan

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Vol. 97, Iss. 16 — 15 April 2018

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Figure 1
Summary of quantum oscillation results for ${Zr}_{2} {Te}_{2} P$ . (a) $τ$ ( $μ_{0} H$ ) at various angles $θ$ , where low (main panel) and high (inset) frequency dHvA oscillations are seen. $θ$ is defined as the angle between the crystallographic $c$ axis and $μ_{0} H$ , where $θ = 0^{\circ}$ and $90^{\circ}$ correspond to fields parallel to the $c$ and the $a$ axes, respectively. (b) Taken from Ref. [35]: background subtracted magnetization $Δ M$ vs $μ_{0} H$ showing the low frequency de Haas-van Alphen (dHvA) oscillations for different temperatures. (Inset) Fast Fourier transforms (FFT) for $Δ M$ at various $T s$ showing the low frequency $α$ and $β$ pockets. (c) Background subtracted $Δ τ$ ( $μ_{0} H$ ) emphasizing the high frequency dHvA oscillations. Another frequency, $γ$ is observed for the ${Hf}_{2} {Te}_{2} P$ compound, see Fig. S1(f) [36]. (d) Amplitudes of the low and high frequency peaks observed in the FFT spectra as a function of $T$ . Solid lines are fits to the Lifshitz-Kosevich formula from which the effective masses $m^{*}$ of the charge carriers for different branches are obtained. (e) Fast Fourier transforms of the dHvA signal collected at angles $- 40^{\circ} \leq θ \leq 90^{\circ}$ as functions of the cyclotron frequency $F$ . The red dashed line is a fit to the low frequency orbit $β$ , see main text. (f) Fast Fourier transforms of the dHvA signal collected at angles $0^{\circ} \leq θ \leq 32^{\circ}$ as functions of the cyclotron frequency $F$ . The red dashed line is a fit to the high frequency orbit $δ$ , see main text.
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Figure 2
Fermi surface sheets calculated from density functional theory (DFT): (a) calculated full Fermi surface for the ${Zr}_{2} {Te}_{2} P$ . (b) The projected Fermi surface on the $k_{x} − k_{y}$ plane as measured by ARPES. (c) The projected Fermi surface on the $k_{x} − k_{z}$ plane as measured by ARPES. (d), (e), (f), and (g) are, respectively, the ${Zr}_{2} {Te}_{2} P$ , ${Hf}_{2} {Te}_{2} P$ , ${Zr}_{2} {Te}_{2} As$ , and ${Ti}_{2} {Te}_{2} P$ hole pockets around the $Γ$ point.
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Figure 3
Band structure of ${Zr}_{2} {Te}_{2} P$ in the rhombohedral unit cell along the TRIMs (a) without the inclusion of the spin-orbit coupling (SOC) and (b) with the inclusion of SOC. The colors used indicate the orbital character of the bands: red, blue, and green colors corresponds to Zr $d$ , Te $p$ , and P $p orbitals$ , respectively. The superposition of bulk bands (shown as Gray ribbons) with bands obtained from a slab of five quintuple layers (shown as blue lines) depicting the presence of a Dirac-like surface state at the $\bar{Γ}$ point is shown for ${Zr}_{2} {Te}_{2} P$ in (c). (d) shows the absence of such a surface state for ${Ti}_{2} {Te}_{2} P$ at the $\bar{Γ}$ point.
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Figure 4
Comparison between ARPES measurements and band-structure calculations for cuts along the $K − Γ − K$ direction. (a) ARPES determined band dispersion and (b) calculated DFT bands for ${Zr}_{2} {Te}_{2} P$ , respectively. (c) ARPES and (d) DFT for ${Ti}_{2} {Te}_{2} P$ , respectively. Fermi energy $E_{F}$ is indicated by the horizontal black line. Blue solid lines are bands obtained from a five-layer slab calculation and therefore represent surface states and grey ribbons represent the bulk bands.
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Figure 5
Comparison between ARPES measurements and band structure calculations for cuts along the $K − M − K$ direction. (a) ARPES determined band dispersion and (b) calculated DFT bands for ${Zr}_{2} {Te}_{2} P$ , respectively. (c) ARPES and (d) DFT for ${Ti}_{2} {Te}_{2} P$ , respectively. Fermi energy $E_{F}$ is indicated by horizontal line. Solid blue lines depict bands obtained from a five-layer slab calculation and represent surface states and grey ribbons represent the bulk bands.
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