Tracing Dirac points of topological surface states by ferromagnetic resonance

Laura Pietanesi, Magdalena Marganska, Thomas Mayer, Michael Barth, Lin Chen, Ji Zou, Adrian Weindl, Alexander Liebig, Rebeca Díaz-Pardo, Dhavala Suri, Florian Schmid, Franz J. Gießibl, Klaus Richter, Yaroslav Tserkovnyak, Matthias Kronseder, and Christian H. Back

Phys. Rev. B 109, 064424 – Published 26 February 2024

Abstract

Ferromagnetic resonance is used to reveal features of the buried electronic band structure at interfaces between ferromagnetic metals and topological insulators. By monitoring the evolution of magnetic damping, the application of this method to a hybrid structure consisting of a ferromagnetic layer and a 3D topological insulator reveals a clear fingerprint of the Dirac point and exhibits additional features of the interfacial band structure not otherwise observable. The underlying spin-pumping mechanism is discussed in the framework of dissipation of angular momentum by topological surface states (TSSs). Tuning of the Fermi level within the TSS was verified both by varying the stoichiometry of the topological insulator layer and by electrostatic backgating and the damping values obtained in both cases show a remarkable agreement. The high-energy resolution of this method additionally allows us to resolve the energetic shift of the local Dirac points generated by local variations of the electrostatic potential. Calculations based on the chiral tunneling process naturally occurring in TSSs agree well with the experimental results.

Received 12 April 2023
Revised 3 November 2023
Accepted 16 January 2024

DOI:https://doi.org/10.1103/PhysRevB.109.064424

Physics Subject Headings (PhySH)

Spin pumping Topological materials

Topological insulators

Ferromagnetic resonance Green's function methods Molecular beam epitaxy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Laura Pietanesi^1,*, Magdalena Marganska ², Thomas Mayer², Michael Barth², Lin Chen ¹, Ji Zou³, Adrian Weindl ², Alexander Liebig ², Rebeca Díaz-Pardo ¹, Dhavala Suri ^1,4, Florian Schmid², Franz J. Gießibl ², Klaus Richter ², Yaroslav Tserkovnyak³, Matthias Kronseder ^2,*, and Christian H. Back ^1,5,†

¹School of Natural Sciences, Department of Physics, Technical University of Munich, 85748 Garching, Germany
²Institute for Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany
³Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
⁴Center for Nanoscience and Engineering, Indian Institute of Science, Bengaluru 560 012, India
⁵Center for Quantum Engineering (ZQE), Technical University Munich, 85748 Garching, Germany

^*These authors contributed equally to this work.
^†christian.back@tum.de

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Issue

Vol. 109, Iss. 6 — 1 February 2024

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Images

Figure 1
(a) Schematic of the spin-pumping FMR setup comprising a coplanar-waveguide (CPW) to externally drive a ferromagnetic system while an external magnetic field $H_{ext}$ is applied to the system. The temperature of the CPW and sample, which lies face down on the CPW, can be varied between 8–300K. Application of an electric field between the backside and the Py layer in the sample stack ${SrTiO}_{3}$ (111)/(1)BS/(10)BSTS/Py/ ${AlO}_{x}$ is used to gate the system leading to a change of the position of the Fermi level of BSTS. (b) Linewidth dependence on frequency for Py (black curve) at 9 K and (1)BS/(10)BSTS/Py with $x = 0.67$ at 10 K and 0 V (red curve) as well as 9 V backgate (blue curve) voltages. The inset shows a typical FMR absorption curve and the corresponding fit (derivative of a Lorentzian line).
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Figure 2
(a) Schematics of the band diagrams along the growth direction of the heterostructure of BS/BSTS with a Py ferromagnet on top grown on the substrate ${SrTiO}_{3}$ . At $V_{BG} \sim 0$ , the intentional internal band bending sets the Fermi energy in the top-TSS close to the Dirac point. Application of BG voltages distorts the conduction and valence bands changing also the energetic alignment of the top-Dirac cone. (b) Gilbert damping versus applied backgate voltage for a sample with (1)BS/(10)BSTS/Py with $x = 0.67$ measured at 10 K. The damping for STO/Py is subtracted: $Δ α = α_{TI / FM} - α_{FM}$ . (c) Simplified sketch of the band structure considering bulk states and the Dirac cone at the top surface; the Fermi level position for different Sb concentrations is shown on the left and a sketch of the density of states for TSS and bulk at $T = 0$ is shown on the right. (d) Schematic representation of different contributions of the bulk, $σ_{x x, b}$ (green line) and of the TSSs, $σ_{x x, TSS}$ (yellow line) to the longitudinal conductivity $σ_{x x}$ , resulting in a spin-pumping efficiency $ξ_{SP}$ (red dashed-dotted line).
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Figure 3
To enable a direct comparison, a part of the data presented already in Fig. 2 is replotted in (a) for the sample with $x \sim 0.67$ measured at 10 K. At $V_{BG} \sim$ −5V, the Fermi energy is at the Dirac point. (b) $Δ α$ dependence on Sb concentration measured at 10 K at $V_{BG} = 0$ . The Fermi energy shifts towards the conduction band with increasing $V_{BG}$ or decreasing $x$ as shown in the sketch of the Dirac cone between the panels. The measurements shown in (a) have been recorded on the same sample as shown in (b) at $x = 0.67$ with 12 months separation.
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Figure 4
(a) High-resolution scan of $Δ α$ around $V_{BG} \sim 0$ . Both measurements (red and black dots) have been performed on the same sample with (1)BS/(10)BSTS/Py and $x = 0.7$ , measured at $T = 10 K$ . (b) Band alignment for a spatially extended potential barrier in a single TSS. (c) Corrugated potential energy landscape with local n-p junctions for $E_{F}$ between the Dirac points [purple dashed line in (b)]. A right moving electron with spin up which crosses the n-p boundary has to become a right moving hole with spin up (b), undergoing the chiral tunneling and reflection process as shown in (d). In chiral systems, some incidence angles lead to enhanced transmission [(e), left], for a nonchiral zero-gap semiconducting system transmission is exponentially suppressed (right). (f) Analytically calculated quantum mechanical transmission for three systems (single TSS, double TSS, and zero-gap semiconductor) with a potential barrier [as in (b)] of height $V = 60$ meV and width $D = 60$ nm, weighted by the DOS. The chiral systems, single and double TSS, show two dips at the Dirac point energies with a pronounced maximum between them. Removing the chiral nature of the system results in zero transmission for energies at which n-p or p-n junctions occur, as in this case the transmission is mediated by evanescent waves. (g) Numerical transport calculations for a single potential step in the same systems as in (f) at $T = 10$ K reveal the same qualitative features as the quantum mechanical approach in (f). The results in (f) and (g) are rescaled for visibility (rescaling factor in brackets in the legend; see also the Supplemental Material).
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