Topological Nernst Effect of the Two-Dimensional Skyrmion Lattice

Max Hirschberger, Leonie Spitz, Takuya Nomoto, Takashi Kurumaji, Shang Gao, Jan Masell, Taro Nakajima, Akiko Kikkawa, Yuichi Yamasaki, Hajime Sagayama, Hironori Nakao, Yasujiro Taguchi, Ryotaro Arita, Taka-hisa Arima, and Yoshinori Tokura

Phys. Rev. Lett. 125, 076602 – Published 12 August 2020

Abstract

The topological Hall effect (THE) and its thermoelectric counterpart, the topological Nernst effect (TNE), are hallmarks of the skyrmion lattice phase (SkL). We observed the giant TNE of the SkL in centrosymmetric ${Gd}_{2} {PdSi}_{3}$ , comparable in magnitude to the largest anomalous Nernst signals in ferromagnets. Significant enhancement (suppression) of the THE occurs when doping electrons (holes) to ${Gd}_{2} {PdSi}_{3}$ . On the electron-doped side, the topological Hall conductivity approaches the characteristic threshold $\sim 1000 {(Ω おめが cm)}^{- 1}$ for the intrinsic regime. We use the filling-controlled samples to confirm Mott’s relation between TNE and THE and discuss the importance of Gd-5d orbitals for transport in this compound.

Received 8 October 2019
Revised 16 May 2020
Accepted 10 July 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.076602

Physics Subject Headings (PhySH)

Magnetism RKKY interaction Skyrmions Topological materials

Helimagnets Thermoelectrics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Max Hirschberger ^1,2,*, Leonie Spitz ^1,†, Takuya Nomoto², Takashi Kurumaji ^1,‡, Shang Gao^1,§, Jan Masell ¹, Taro Nakajima^1,∥, Akiko Kikkawa¹, Yuichi Yamasaki^3,4, Hajime Sagayama⁵, Hironori Nakao⁵, Yasujiro Taguchi ¹, Ryotaro Arita², Taka-hisa Arima^1,6, and Yoshinori Tokura ^1,2,7

¹RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
²Department of Applied Physics and Quantum-Phase Electronics Center, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
³Research and Services Division of Materials Data and Integrated System (MaDIS), National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan
⁴PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan
⁵Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan
⁶Department of Advanced Materials Science, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
⁷Tokyo College, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan

^*hirschberger@ap.t.u-tokyo.ac.jp
^†Present address: Physik-Department, Technical University of Munich, Garching 85748, Germany.
^‡Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
^§Present address: Materials Science & Technology Division and Neutron Science Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA.
^∥Present address: Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan.

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Vol. 125, Iss. 7 — 14 August 2020

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Images

Figure 1
(a) Hexagonal structure (basic ${AlB}_{2}$ -type) of ${Gd}_{2} {PdSi}_{3}$ . Inset: high-density SkL state with nanometer-sized interskyrmion distance $d$ . (b) Magnetic phase diagram for $B ∥ c$ and decreasing field $\partial B / \partial t < 0$ , as adapted from Ref. [61]. Labels indicate the IC-1 ground state, the skyrmion lattice phase, the IC-2 fanlike state, and the paramagnetic regime (PM). (c) Hall resistivity, (e) thermopower, (f) Nernst effect, and (g) Nernst conductivity. For the latter three, entropy factor $\sim T$ was removed. (d) Magnetic susceptibility ${χ かい}_{DC} (B)$ at the lowest $T = 2 K$ . Curves in panels (c),(e)–(g) were shifted by vertical offsets of $2 μ みゅー Ω おめが cm$ , $15 nV K^{- 2}$ , $10 nV K^{- 2}$ , and $0.01 {AK}^{- 2} m^{- 1}$ , respectively. The zero level for each curve is indicated by a colored triangle at the right side of the panel. Solid arrows mark the direction of the field ramp. In (e), red, green, and purple twinned lines also indicate zero levels for $T = 7.6$ , 9.7, and 14.8 K.
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Figure 2
Comparison of large Nernst conductivity ${α あるふぁ}_{x y} / T$ driven by magnetic order for various materials reported in the literature. Grey shading indicates anomalous Nernst response (ANE, proportional to the net magnetization $M$ ), while red shading is reserved for the topological Nernst effect (TNE, proportional to the scalar spin chirality).
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Figure 3
(a) Néel temperature $T_{N}$ of ${Gd}_{2} ({Pd}_{1 - x} M_{x}) {Si}_{3}$ from magnetic susceptibility. $M = Rh$ , Ag corresponds to $z < 0$ ( $z > 0$ ), respectively (see text). Inset: sample geometry for resonant elastic x-ray diffraction (REXS) at the Gd- $L_{2}$ edge in reflection geometry. (b) Magnetic wave number $q$ , determined from REXS. The data point for pure ${Gd}_{2} {PdSi}_{3}$ (grey disk) is reproduced from Ref. [23]. Inset: line scans of REXS intensity along $(2 - h, 1, 0)$ in reciprocal space for two samples, after the subtraction of a constant background. The beam energy was set to $E = 7.935 keV$ and the sample temperature was $T = 4 K$ (5 K) for the Ag-doped (Rh-doped) crystal. Scale of $y$ axis (not shown) is intensity normalized by monitor counts (arbitrary units). Statistical errors of detector counts are indicated. (c) Longitudinal resistivity as well as extremal (d) topological Hall resistivity and (f) topological Hall conductivity. Systematic (sample shape) errors are indicated. In (e), raw data of Hall conductivity as a function of magnetic field. Inset of (d), illustration of the real-space Berry-phase mechanism for the THE and TNE [6, 12]. Red lines in (b),(d) are guides to the eye, while the green line in (e) was calculated from the Nernst conductivity (see text). Vertical dashed lines mark $z = 0$ .
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Figure 4
Density functional theory calculations, neglecting Gd-4f, in the spin-polarized state of pure ${Gd}_{2} {PdSi}_{3}$ . (a) Partial density of states (P-DOS) $g_{P} (ϵ)$ ; solid and dashed lines represent spin-up and spin-down bands, respectively. (b),(c) Total DOS $g (ϵ)$ for ${Gd}_{2} {PdSi}_{3}$ and two doped derivatives. The vertical fat lines are guides to the eye, marking the shift of prominent features in $g (ϵ)$ with band filling.
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