Topological thermal Hall effect and nondissipative transport of the nanometric skyrmion lattice in Gd₂PdSi₃

Abstract

We report the first observation of a topological thermal Hall effect (tTHE) of electrons moving through a magnetic skyrmion lattice (SkL) of short period, where the skyrmion diameter is only a few nanometers. In the hexagonal intermetallic Gd₂PdSi₃, we observe a characteristic anomaly in the magnetic field dependence of the thermal Hall conductivity ${\kappa }_{xy}$, which scales well with the anomaly in the electrical Hall conductivity ${\sigma }_{xy}$ and is closely related to the topological winding number of the SkL ground state. The relative magnitude of entropy and charge currents, defined as the Lorenz ratio $\sim {\kappa }_{xy}/({\sigma }_{xy}T)$, is consistent with a nondissipative, or intrinsic, mechanism for the topological Hall effect. We stress that the Berry phase in momentum space ($\mathit{k}$-space) causes such a nondissipative Hall transport, independent of the carrier relaxation time.

Significance Statement.

The transverse thermal gradient induced by longitudinal heat current, thermal Hall effect, is precisely measured in a material with a spin vortex lattice of nanometric modulation period. The Hall effect is called topological, as it is related to the quantum geometric curvature of spin texture, and is anticipated to be free of energy loss. In this report, the scaling between thermal and electrical Hall conductivities confirms the energy-loss-less nature of the topological Hall electronic current. This provides an experimental clarification of the nondissipative nature of the topological quantum current for future energy efficient electronics.

Introduction

A key trend in contemporary research on solids is the detection of a fictitious Lorentz force, or emergent magnetic field, through the measurement of heat carried by fundamental excitations. The transverse deflection of heat current density ${\mathit{J}}^Q$ due to this Lorentz force is picked up in experiments as a thermal Hall effect $J_y^Q={\kappa }_{xy}({\mathrm{\partial }}_{x}T)$, where $(-{\mathrm{\partial }}_{x}T)$ is the temperature gradient applied along a sample. For lattice vibrations (phonons), thermal Hall conductivity ${\kappa }_{xy}$ can be caused by extrinsic skew scattering from magnetic or charged defects (1, 2), or by intrinsic quantum phenomena related to the coupling of spin waves (magnons) and phonons (3, 4). Both sources of phononic ${\kappa }_{xy}$ explored so far can be traced back to relativistic spin-orbit coupling (SOC). Spin waves in long-range ordered magnets create ${\kappa }_{xy}$ due to SOC or due to the noncoplanar twisting of magnetic spins in space, termed scalar spin chirality (5). If the magnetic unit cell is large enough, and if the magnetization can be considered as a continuous field $\mathit{n}$ in space, which is normalized to unit length, there is a contribution to ${\kappa }_{xy}$ directly proportional to the topological charge

$$n_{\mathrm{sk}}=\frac{1}{4\pi }\int \mathit{n}\cdot (\frac{\mathrm{\partial }\mathit{n}}{\mathrm{\partial }x}\times \frac{\mathrm{\partial }\mathit{n}}{\mathrm{\partial }y})\mathrm{d}x\mathrm{d}y.$$ (1)

For example, magnonic ${\kappa }_{xy}\propto n_{\mathrm{sk}}$ has been observed in skyrmion-hosting insulators recently (6, 7).

In this work, we focus on the ${\kappa }_{xy}$ of electrons due to $n_{\mathrm{sk}}$. Figure 1A illustrates the anomalous ${\kappa }_{xy}$ in metallic ferromagnets, which is attributed to SOC and has been thoroughly studied (8–10). However, the observation of an electronic ${\kappa }_{xy}$ for a magnetic skyrmion lattice (SkL), where ${\kappa }_{xy}$ is directly related to $n_{\mathrm{sk}}$ as illustrated in Fig. 1B, has remained an open challenge in the field. We report the topological thermal Hall effect (tTHE) of electrons under the influence of $n_{\mathrm{sk}}$ in the SkL of Gd${ }_2$PdSi${ }_3$. Our analysis of energy dissipation based on the Lorenz ratio of entropy and charge conductivities confirms that ${\kappa }_{xy}$ is not affected by inelastic scattering, consistent with an intrinsic and nondissipative mechanism for the topological Hall effect in Gd${ }_2$PdSi${ }_3$ (11). We discuss the general relevance of this electronic tTHE as a testing bed for adiabatic spin transport and the theory of spin-Berry phases in solids.

Fig. 1.

Topological thermal Hall effect (tTHE) of skyrmions in Gd${ }_2$PdSi${ }_3$. A, B) Anomalous thermal Hall effect and tTHE (schematic) have opposite signs in Gd${ }_2$PdSi${ }_3$; the former is too small to be easily visible in the data. The itinerant conduction electron is represented in blue/red large sphere with fat arrow, and the blue circles around individual atoms in (A) represent the orbital angular momentum that is essential for the former effect. C) Schematic crystal structure of Gd${ }_2$PdSi${ }_3$. D) Setup for thermal Hall effect measurement. The plate-shape sample (red/blue plate) connects heater (top) and heat bath (light blue plate); thermometers detect longitudinal ($\mathrm{\Delta }{T}_x$) and transverse ($\mathrm{\Delta }{T}_y$) temperature gradients. The magnetic field (B) is applied perpendicular to the sample plane (along the c-axis). E) Magnetic field dependence of thermal (${\kappa }_{xy}$, symbols) and electrical (${\sigma }_{xy}$, dashed lines) Hall effects in pristine and lightly electron-doped Gd${ }_2$PdSi${ }_3$ (samples A and B, respectively). The ${\sigma }_{xy}$ scale (right) is stretched to match the ${\kappa }_{xy}$.

Results

Topological thermal Hall effect and Lorenz ratio

Figure 1C shows the hexagonal AlB${ }_2$-type structure of Gd${ }_2$PdSi${ }_3$ (12, 13). The crystal is built of magnetic layers comprising a triangular lattice of Gd${ }^{3+}$ atoms, sandwiched by nonmagnetic honeycomb layers of Pd/Si. The highly symmetric crystal structure and weak magnetic anisotropy of Gd${ }^{3+}$ ions allow the formation of a SkL with nanometric magnetic vortices of 2–3 nm arrayed in the hexagonal basal plane (ab-plane) of Gd${ }_2$PdSi${ }_3$. Here, electronic states have significant Gd-$5d$ character, resulting in a strong exchange coupling between local magnetic moments and the spin of conducting states on the order of several 100 meV (14–16).

Figure 1D illustrates the setup for the thermal Hall effect measurement. A heat current ${\mathit{J}}^Q$ is applied to the plate-shape sample along the negative x-direction. The magnetic field $\mathit{B}$ is perpendicular to the sample plane (along the z-direction). Note that the z-direction in Fig. 1D corresponds to the c-axis of the hexagonal structure (Fig. 1C). See also Materials and methods section and Supplementary material for more detail.

Figure 1E shows representative results of tTHE (${\kappa }_{xy}$) and electrical Hall effect (${\sigma }_{xy}$) around 11 K for two single crystals of Gd${ }_2$PdSi${ }_3$: sample A (pristine) and sample B ($5\mathrm{\%}$ Ag substitution for Pd, slightly electron-doped). Here, the electrical conductivity tensor is defined as $J_i={\sigma }_{ij}{E}_j$, where $\mathit{J}$ and $\mathit{E}$ are the electric current density and the electric field, respectively. The general shape of ${\sigma }_{xy}$ and ${\kappa }_{xy}$ is closely comparable, with a flat-top profile suggestive of the calculated behavior of $n_{\mathrm{sk}}(B)$ for a SkL state induced by magnetic field (17, 18). We also note that ${\kappa }_{xy}$ and ${\sigma }_{xy}$ have the same sign throughout the experimentally accessible range of temperature and magnetic field.

To set the stage for a more quantitative analysis of these results, we discuss the temperature dependence of ${\sigma }_{xx}$ and ${\kappa }_{xx}$ in Fig. 2A. Analyzing the thermal transport data from the viewpoint of the generalized Lorenz ratio,

Fig. 2.

Temperature and magnetic field dependence of tTHE. A) Temperature (T) dependence of ${\kappa }_{xx}/T$, ${\sigma }_{xx}{\mathcal{L}}_0$, where ${\mathcal{L}}_0=({\pi }^2/3)(k_B/e{)}^2$ is the Lorenz number. Blue circles (solid lines) and red triangles (dashed lines) indicate ${\kappa }_{xx}/T$ (${\sigma }_{xx}/{\mathcal{L}}_0$) of samples A and B of Gd${ }_2$PdSi${ }_3$, respectively. B) T dependence of ${\kappa }_{xx}/{\sigma }_{xx}T$ in units of ${\mathcal{L}}_0$. Blue solid and red dashed lines correspond to samples A and B, respectively. The vertical black dashed lines in (A) and (B) indicate the transition temperature $T_{\mathrm{N}}$ of each sample. C–J) Magnetic field dependence of ${\kappa }_{xy}/T$ (blue circles and red triangles) and ${\sigma }_{xy}{\mathcal{L}}_0$ (fat black dashed lines) in sample A (C–F) and sample B (G–J) of Gd${ }_2$PdSi${ }_3$ at various $T<T_{\mathrm{N}}$.

$${\mathcal{L}}_{ij}=\frac{{\kappa }_{ij}}{{\sigma }_{ij}T},$$ (2)

we start by focusing on ${\mathcal{L}}_{xx}$. Note that the thermal conductivity is considered to have phonon and electron contributions: ${\kappa }_{xx}={\kappa }_{xx}^{\mathrm{ph}}+{\kappa }_{xx}^{\mathrm{el}}$. When ${\kappa }_{xx}^{\mathrm{el}}\gg {\kappa }_{xx}^{\mathrm{ph}}$, ${\mathcal{L}}_{xx}$ is expected to approach the Lorenz number ${\mathcal{L}}_0=({\pi }^2/3)(k_{\mathrm{B}}/e{)}^2$ at low T (19); we observe ${\mathcal{L}}_{xx}>15\times {\mathcal{L}}_0$ in the accessible temperature range (Fig. 2B) and infer that ${\kappa }_{xx}^{\mathrm{ph}}$ cannot be ignored in the heat transport of Gd${ }_2$PdSi${ }_3$. Moreover, magnon heat flow seems to be negligible, as there is no clear anomaly or enhancement in ${\kappa }_{xx}(T)$ below $T_{\mathrm{N}}\sim 20$ K (see also Fig. S6A).

Further attempting to isolate the electronic term, we move on to ${\mathcal{L}}_{xy}$. In conductors, the electrons’ contribution to the thermal Hall conductivity is expected to dominate: ${\kappa }_{xy}\approx {\kappa }_{xy}^{\mathrm{el}}$, and ${\mathcal{L}}_{xy}$ should also approach ${\mathcal{L}}_0$ as $T\to 0$ (20). Figure 2C–J overplots ${\kappa }_{xy}/T$ and ${\sigma }_{xy}{\mathcal{L}}_0$, which are the constituent parts of ${\mathcal{L}}_{xy}/{\mathcal{L}}_0$. Over the entire temperature range, the curves track each other well for both A and B, especially in the low magnetic field region of the SkL state. In contrast, there is a significant temperature-dependent separation between ${\kappa }_{xy}/T$ and ${\sigma }_{xy}{\mathcal{L}}_0$ in the high magnetic field region. Encouragingly, these results suggest that a discussion using ${\mathcal{L}}_{xy}$ may be applicable for the tTHE in Gd${ }_2$PdSi${ }_3$, indicating that the topological Hall transport in the SkL state is distinct from the normal Hall transport, as discussed in the following.

Nondissipative transport

We separate the thermal Hall conductivity into two contributions: (i) the normal Hall channel, from orbital motion in the applied external field B and (ii) the topological Hall channel, arising from the topological winding number $n_{\mathrm{sk}}$ of the SkL; viz. ${\kappa }_{xy}={\kappa }_{xy}^{\mathrm{N}}+{\kappa }_{xy}^{\mathrm{T}}$. The ${\kappa }_{xy}^{\mathrm{T}}$ is responsible for the enhancement of ${\kappa }_{xy}$ within the SkL state, while ${\kappa }_{xy}^{\mathrm{N}}$ dominates at high-B. Likewise, ${\sigma }_{xy}$ is written as a sum of two terms, since the anomalous Hall effect from the net magnetization and SOC is very weak in Gd${ }_2$PdSi${ }_3$ (18). Accordingly, for each sample, Fig. 3A and B shows the T dependence of ${\kappa }_{xy}/T$ and ${\sigma }_{xy}{\mathcal{L}}_0$ at their peak value in the SkL phase, representing the topological component. Moreover, the figure also shows ${\kappa }_{xy}/T$ and ${\sigma }_{xy}{\mathcal{L}}_0$ at $B=8$ T, corresponding to the normal (thermal) Hall effect. Thus, the high field and peak values are identified with ${\kappa }_{xy}^{\mathrm{N}}$ (${\sigma }_{xy}^{\mathrm{N}}$) and ${\kappa }_{xy}^{\mathrm{T}}$ (${\sigma }_{xy}^{\mathrm{T}}$), respectively. We further define a Lorenz ratio ${\mathcal{L}}^{\mathrm{T}}$ for the tTHE by setting ${\sigma }_{ij}={\sigma }_{xy}^{\mathrm{T}}$ and ${\kappa }_{ij}={\kappa }_{xy}^{\mathrm{T}}$ in Eq. 2 and plot it as a function of T in Fig. 3C: ${\mathcal{L}}^{\mathrm{T}}$ is found to be close to ${\mathcal{L}}_0$ at low temperatures. From now on, we focus on ${\mathcal{L}}^{\mathrm{T}}$ in the SkL state; the Lorenz ratio for the normal Hall effect, ${\mathcal{L}}^{\mathrm{N}}$, is analyzed in the Supplementary material, where a stronger temperature dependence is observed, with a deviation from ${\mathcal{L}}_0$.

Fig. 3.

Lorenz ratio for the tTHE in Gd${ }_2$PdSi${ }_3$. A, B) For samples A and B, we show the peak values of tTHE (${\kappa }_{xy}/T$) and electrical topological Hall effect (${\sigma }_{xy}{\mathcal{L}}_0$) in the SkL phase (positive, red and blue) and the high-field values at $B=8$ T (negative, green and black). C) T-dependence of the topological Hall Lorenz ratio (${\mathcal{L}}_{xy}^{\mathrm{T}}/{\mathcal{L}}_0$) in sample A (blue circles) and sample B (red triangles). An outlier is shown in gray color. Inset of (C): elastic and inelastic electronic scattering processes for charge current $\mathit{J}$ and entropy current ${\mathit{J}}_Q$ (see text). Solid and open circles represent occupied (filled) and empty states (holes), respectively. D) Parameter space of Matsui et al. (21), where J, $v_F$, τ, ${\lambda }_{sk}$, and a are the coupling strength between itinerant and local moments, Fermi velocity, carrier relaxation time, and sizes of magnetic and crystallographic unit cells, respectively. Points are placed for sample A (blue circle) and sample B (red triangle). E) Logarithmic plot of electrical Hall conductivity ${\sigma }_{xy}$ vs. electrical conductivity ${\sigma }_{xx}$ for materials with a Hall effect of magnetic origin, following Refs. (22, 23). We add corresponding symbols for the topological Hall effect of Gd${ }_2$PdSi${ }_3$, where the change with T (red and blue arrows) relates to a T-dependent change of the ordered moment—not a change of ${\sigma }_{xx}$. The solid symbols mark the values at base-T for sample A and B, which are most relevant to the scaling analysis. Dashed lines: impurity scattering, intrinsic, and extrinsic regimes of the anomalous Hall effect.

What is the physical importance of ${\mathcal{L}}_{xy}^{\mathrm{T}}\sim {\mathcal{L}}_0$? The insets in Fig. 3C show schematic illustrations of charge and entropy flow in momentum ($\mathit{k}$-) space. In principle, electrons carry both charge $-e$ and entropy $k_{\mathrm{B}}T$ at the same rate as they move in a crystal. The derivation of ${\mathcal{L}}_{ij}={\mathcal{L}}_0$ at low T rests on the assumption that there is a fixed relation between loss of momentum and kinetic energy in the relevant scattering processes that relax the (charge or heat) current (19). This is generally true for elastic scattering but certain inelastic scattering processes—from above to below the Fermi energy $E_{\mathrm{F}}$—dramatically change the kinetic energy of a moving electron, while they hardly change its momentum. Assuming that electrons dominate both heat and charge currents, inelastic scattering causes ${\mathcal{L}}_{ij}<{\mathcal{L}}_0$.

Hall effect from Berry phase in momentum space

Quantum mechanical wave propagation of elementary excitations in solids can create nondissipative charge or entropy currents, as modeled by the Berry phase. These currents are by definition insensitive to inelastic scattering processes (energy loss). In the nondissipative case, ${\mathcal{L}}_{xy}^{\mathrm{T}}/{\mathcal{L}}_0=1$ is realized even when electrons are involved in inelastic scattering from phonons, spin waves, or other excitations in solids. Under the influence of relativistic SOC, the ${\sigma }_{xy}$ and ${\kappa }_{xy}$ in electron transport of ferromagnets (24), and of noncollinear antiferromagnets (25, 26) have been shown to be independent of the carrier relaxation time τ, including inelastic scattering. The topological charge $n_{\mathrm{sk}}$ provides one route to realize ${\sigma }_{xy}$ and ${\kappa }_{xy}$ without the need for SOC (23). In the limit where the magnetic skyrmions are much larger than a crystallographic unit cell, electrons move under the influence of a Berry phase in real space ($\mathit{r}$-space approximation) (27–30). Here, ${\sigma }_{xy}^{\mathrm{T}}\propto n_{\mathrm{sk}}$ depends on τ and a deviation from ${\mathcal{L}}_{xy}^{\mathrm{T}}/{\mathcal{L}}_0=1$ is expected (23, 27, 29). Figure 1B in fact illustrates electron flow in this approximation, where wavepackets of Bloch electrons move through a twisted magnetic background, adiabatically adjusting their spin to the local quantization axis. When the skyrmion is small enough, Berry phases should be considered in $\mathit{k}$-space and ${\sigma }_{xy}$ is predicted to be independent of τ; this is the limit of clean crystals, where the mean-free path is comparable to or larger than the size of a skyrmion (21, 31, 32). Our analysis of the Lorenz ratio shows ${\mathcal{L}}_{xy}^{\mathrm{T}}/{\mathcal{L}}_0\sim 1$, in both our samples of Gd${ }_2$PdSi${ }_3$, which indicates that there is no significant effect of inelastic scattering processes on ${\kappa }_{xy}^{\mathrm{T}}$. Thus, the $\mathit{k}$-space limit for the topological Hall current is appropriate in Gd${ }_2$PdSi${ }_3$, rather than the $\mathit{r}$-space approximation.

Matsui et al. (21) discuss the crossover between $\mathit{r}$-space approximation and the $\mathit{k}$-space limit from the viewpoint of a simple one-band model. We adopt the essence of their analysis in Fig. 3D, where the right-hand side corresponds to the limit of large skyrmion size (${\lambda }_{\mathrm{sk}}$), while on the vertical axis, the upper side corresponds to the disordered limit with shorter mean-free-path. Here, clean materials with small skyrmions are classified into the $\mathit{k}$-space regime with adiabatic coupling of conduction electron spin and local moment. Consistent with the analysis of the Lorenz ratio ${\mathcal{L}}_{xy}^{\mathrm{T}}$, our samples A and B of Gd${ }_2$PdSi${ }_3$ are estimated to be in the $\mathit{k}$-space regime based on ${\lambda }_{\mathrm{sk}}=2.5$ nm, Fermi velocity $v_{\mathrm{F}}=5.3\times {10}^5\mathrm{m}{\mathrm{s}}^{-1}$, exchange coupling strength $J=0.2$ eV, and relaxation time $\tau =44$ fs (sample A) and 18 fs (sample B) from Refs. (16, 33).

Another perspective on the intrinsic or nondissipative $\mathit{k}$-space regime is provided by the universal scaling law of ${\sigma }_{xy}$ versus ${\sigma }_{xx}$ for the Hall effect, as proposed by Onoda et al. (22, 23). Figure 3E emphasizes the moderately dirty, intrinsic regime: $3\times {10}^3<{\sigma }_{xx}<5\times {10}^5{\mathrm{\Omega }}^{-1}$cm${ }^{-1}$ in good metals. Here, the Hall current is directly related to the Berry phase of conduction electrons in $\mathit{k}$-space, with only minor contamination by extrinsic effects, such as skew scattering. Samples A and B of Gd${ }_2$PdSi${ }_3$ are located at around ${10}^4$  ${\mathrm{\Omega }}^{-1}$ cm${ }^{-1}$, within the intrinsic regime. Note that the carrier density of $\approx {10}^{22}$ cm${ }^{-3}$ in Gd${ }_2$PdSi${ }_3$ (16, 34) is comparable to typical metals, such as elemental Fe (35). This scaling plot is designed in relation to defect (quenched impurity) scattering, and thus ${\sigma }_{xx}$ on the horizontal axis essentially relates to the relaxation time τ.

Discussion

In conclusion, we report the first example of an electronic tTHE in a SkL host. In Gd${ }_2$PdSi${ }_3$, the entropy and charge currents of electrons passing through the nanometric SkL are directly proportional to each other, establishing ${\mathcal{L}}_{xy}^{\mathrm{T}}\sim {\mathcal{L}}_0$ for the tTHE. On the other hand, the Lorenz ratio for the normal Hall conductivity from conventional orbital motion of conduction electrons, ${\mathcal{L}}_{xy}^{\mathrm{N}}$, shows stronger temperature dependence, possibly due to a contributions from the phonon thermal Hall effect (36, 37) and from inelastic magnetic scattering (38–40) (see Supplementary material for detailed discussion). This contrast between ${\mathcal{L}}_{xy}^{\mathrm{T}}$ and ${\mathcal{L}}_{xy}^{\mathrm{N}}$ further confirms that inelastic scattering processes do not contribute to the relaxation of the topological Hall current in Gd${ }_2$PdSi${ }_3$. Our finding supports the intrinsic ($\mathit{k}$-space limit) scenario for the tTHE from electron transport through a nanometric SkL (11), rather than the $\mathit{r}$-space approximation that was developed for large-scale spin superstructures in MnSi and related SkL hosts.

Materials and methods

Single crystals of Gd${ }_2$PdSi${ }_3$ are obtained by the high-vacuum optical floating zoning method (16). A plate-shape sample is cut from the crystal and set up to connect between the heat bath and a metal film heater, subject to a heat current ${\mathit{J}}^Q$ along the negative x-direction as in Fig. 1D. The longitudinal ($\mathrm{\Delta }{T}_x$) and transverse ($\mathrm{\Delta }{T}_y$) temperature differences are measured with semiconducting chip thermometers and converted to thermal conductivity ${\kappa }_{xx}$ and thermal Hall conductivity ${\kappa }_{xy}$ (see Supplementary material for more detail). For measurements of the electrical conductivity ${\sigma }_{xx}$ and the Hall conductivity ${\sigma }_{xy}$, we use the same sample plate with the same contacts, wires, etc., as those for thermal conductivity measurements. We additionally calibrate the absolute values of ${\kappa }_{xx}$ and ${\sigma }_{xx}$ using zero-field measurements on bar-shape samples.

Supplementary Material

Supplementary material is available at PNAS Nexus online.

Funding

P.M. and M.H. are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Transregio TRR 360 - 492547816. Y.Ta. is supported by the RIKEN TRIP initiative (Many-body Electron Systems and Advanced General Intelligence for Science Program). This work was supported by JSPS KAKENHI grant nos. JP22K20348, JP23H05431, JP23K13057, JP24H01607, JP24H01604, and JP25K17336. It was also supported by the Japan Science and Technology Agency via JST CREST grant nos. JPMJCR1874 and JPMJCR20T1 (Japan), JST FOREST grant no. JPMJFR2238 (Japan), and JST Adopting Sustainable Partnerships for Innovative Research Ecosystem (ASPIRE) grant no. JPMJAP2426. The authors are grateful for support from the Murata Science Foundation, the Yamada Science Foundation, the Hattori Hokokai Foundation, the Iketani Science and Technology Foundation, the Mazda Foundation, the Casio Science Promotion Foundation, the Takayanagi Foundation, and the Yashima Environment Technology Foundation.

Author Contributions

M.H. conceived and supervised the project. D.Y., P.M., and R.Y. performed thermal and electrical transport measurements, data analysis/visualization, and calculation. A.K. grew single crystals. D.Y., P.M., and M.H. wrote the original draft. Y.Ta. and Y.To. reviewed and edited the draft. All authors discussed the results and commented on the manuscript.

Data Availability

All data needed to evaluate the conclusions in the article are present in the article and/or Supplementary material.