On the Onsager–Casimir reciprocal relations in a tilted Weyl semimetal

Onsager's reciprocity relations state that in thermodynamic systems out of equilibrium, the linear response coefficient, linking a thermodynamic driving force and the resultant flow, is a symmetric tensor. That is, ${ {\mathcal L} }_{ij}={ {\mathcal L} }_{ji}$. In the presence of time-reversal symmetry breaking fields, it becomes the Onsager–Casimir reciprocal relations, ${ {\mathcal L} }_{ij}({\mathcal B})={ {\mathcal L} }_{ji}(- {\mathcal B})$, where ${\mathcal B}$ denotes all time-reversal symmetry breaking fields. These relations, rooted in the principle of microscopic reversibility, are a fundamental symmetry of nonequilibrium statistical systems. When multiple coupled pairs of flows and thermodynamic forces are involved, the relations can offer insights into the connection between different response coefficients that seem, at first glance, unrelated. For instance, in his seminal papers in 1931, Onsager derived the relations for several coupled irreversible processes and related different coefficients to one another.^[1,2] The reciprocal relations have also played an important role on understanding various spin-related transport.^[3]

Recently, an unusual chirality-dependent Hall effect and magnetoresistance that are antisymmetric in both magnetic field and magnetization were predicted to occur in tilted topological Weyl semimetals.^[4–10] It was suggested that such an antisymmetry violates the Onsager–Casimir reciprocal relations.^[8] Soon, the antisymmetric Hall and magnetoresistance were experimentally observed in a magnetic Weyl semimetal, Co₃Sn₂S₂.^[11] The phenomenon is originated from the Berry curvature of energy bands and involves several effects. In particular, the longitudinal linear magnetoresistance results from the Berry curvature correction to the phase space volume, while the Hall effect consists of contributions from the chiral chemical potential and the anomalous velocity. Here, we study the chirality-dependent Hall effect in Co₃Sn₂S₂, with a focus on the validity of the Onsager–Casimir reciprocal relations. Two methods were employed and it was found that the reciprocal relations hold for this Hall effect. The relations connect the effect of the chiral chemical potential to that of the anomalous velocity. Despite only one pair of flow and force at work, our results reveal an intriguing connection between two effects, which deepens our understanding of Berry-curvature-related electrical transport.

Co₃Sn₂S₂ single crystals were grown by the chemical vapor transport method^[11] and polished to a bar shape for the transport measurements. We utilized silver paste to fabricate the electrical contacts. Before applying the paste, the crystal was treated with Ar plasma to improve the contact. The device is shown in the inset of Fig. 1(a) and the x axis in the diagram is along the crystal [100] direction. The electrical transport measurements were performed using a low frequency lock-in method. A typical current of 2 mA–3 mA was employed. The angular dependence experiment was carried out using a motorized rotator stage with a resolution of 0.02°. Our samples exhibit transport properties similar to previous reports.^[11–14] The temperature dependence of the longitudinal resistivity ρ_yy is shown in Fig. 1(a), where a ferromagnetic phase transition manifests as a clear kink at the Curie temperature T_c ∼ 175 K. Below T_c, the anomalous Hall effect is observed, as shown in Fig. 1(b). The Hall resistivity ρ_xy jumps at the coercive field B_c and shows a linear dependence on the magnetic field above B_c, from which we obtain an ordinary Hall coefficient R_H = 0.51 μΩ⋅cm/T and a corresponding carrier density of 1.2 × 10²¹ cm⁻³.

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Co₃Sn₂S₂ is a magnetic Weyl semimetal.^[14–16] The interplay of band topology and magnetism gives rise to a number of unusual phenomena, attracting intensive research interest.^[12–20] A very recent experiment has uncovered a chirality-dependent Hall effect that is antisymmetric in both the in-plane magnetic field and the magnetization.^[11] It is believed to be associated with the titled Weyl cones. The chirality-dependent Hall effect is observed in our samples. When the magnetic field is rotated from the z axis to the x axis, as depicted in the inset of Fig. 1(c), the in-plane field generates an extra Hall resistivity proportional to B_x . Therefore, the transverse signals at ±θ diverge, shown in Fig. 1(c) (θ = ±70° for instance). The chirality-dependent Hall effect ${\rho }_{xy}^{\chi }$ can be obtained by data antisymmetrizing as ${\rho }_{xy}^{\chi }(\theta)=[{{\rho }^{\prime}_{xy}}(+\theta)-{{\rho }^{\prime}_{xy}}(-\theta)]/2$, where ${{\rho }^{\prime}_{xy}}={\rho }_{xy}-{\rho }_{xy}^{{\rm{AHE}}}$. Considering that the amplitude and orientation of the magnetization remain almost unchanged at 50 K due to a strong magnetocrystalline anisotropy,^[17,18] a constant anomalous Hall resistivity ${\rho }_{xy}^{{\rm{AHE}}}$ is subtracted. The obtained ${\rho }_{xy}^{\chi }$ for positive and negative magnetization at θ = ±70° are plotted in Fig. 1(d), which is consistent with the previously reported results of ${\rho }_{xy}^{\chi }\propto {B}_{x}{M}_{z}$.^[11]

The chirality-dependent Hall effect satisfies ρ_xy (B,M) = ρ_xy (–B,–M), which is consistent with the Onsager–Casimir reciprocal relations.^[11] However, to prove that the reciprocal relations hold, one needs to verify ρ_xy (B,M) = ρ_yx (–B,–M). One method is to attest the reciprocity theorem

$$\begin{eqnarray}&&{R}_{12,34}(B,M)={R}_{34,12}(-B,-M),\end{eqnarray} \tag{ 1 }$$

where the two pairs of indices denote the current leads and voltage leads, respectively.^[31,32] Equation (1) can be derived from the Onsager–Casimir reciprocal relations. Figure 2 shows the Hall effect measured under two configurations. In the first configuration, the current is applied through contacts 1 and 2, and the voltage between 3 and 4 is measured to obtain R_12,34, as depicted in the bottom inset of Fig. 2(b). In the second configuration, the current and voltage leads are swapped. The current is applied through contacts 3 and 4, and the voltage between 1 and 2 is measured to obtain R_34,12, as depicted in the bottom inset of Fig. 2(c). Measurements were performed at several angles. Data are shown in Fig. 2(a). In all cases, R_34,12(–B,–M) follows R_12,34(B,M) exactly. Consequently, the extracted chirality-dependent Hall effect satisfies the reciprocity theorem, ${R}_{12,34}^{\chi }(B,M)={R}_{34,12}^{\chi }(-B,-M)$, as shown in Figs. 2(b) and 2(c). We further measured the magnetic field angular dependence of two resistances, shown in Figs. 2(a) and 2(b). The magnetic field was rotated in the xz plane. The chirality-dependent Hall resistivity can be extracted as R^χ (θ) = [R(+θ) – R(–θ)]/2 and is presented in Figs. 3(c) and 3(d). The angular dependences of ${R}_{12,34}^{\chi }(B,M)$ and ${R}_{34,12}^{\chi }(-B,-M)$ show almost the same behavior, further supporting the validity of Eq. (1).

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However, equation (1) is a global symmetry. In principle, a whole series of four-probe measurements must be carried out and all results must comply with Eq. (1) in order to prove the validity of the Onsager–Casimir relations.^[32,33] We now attempt another method, i.e., direct verification of the local symmetry of ρ_xy (B,M) = ρ_yx (–B,–M). Note that there is a practical disadvantage in this method, as it is difficult to prepare Hall bars along two directions on the same crystal. In Fig. 2, R_12,34(B,M), corresponding to ρ_xy (B,M), was measured. We had to prepare a Hall bar with another sample to measure ρ_yx (–B,–M). Since no two samples can have exactly the same resistivity, only a semi-quantitative comparison will be made.

Figure 4(a) displays the measured ρ_yx (–B) at θ = 80°, 0°, –80° for sample #2. Above B_c, ρ_yx exhibits a linear B-dependence, from which a carrier density of 8.4 × 10²⁰ cm⁻³ is obtained. It is slightly smaller than the carrier density of sample #1. The resistivity at zero field is about 125 μΩ⋅cm, compared to 65 μΩ⋅cm of sample #1. With increasing magnetic field or angle θ, ρ_yx (–B) at ± θ diverge. This behavior qualitatively agrees with that observed in sample #1. The chirality-dependent Hall resistivity ${\rho }_{yx}^{\chi }(-B)$ can be extracted by symmetrizing, depicted in Fig. 4(b). It is antisymmetric in both magnetic field and magnetization, mimicking ρ_xy (B,M), which confirms the local Onsager–Casimir reciprocal relations. The amplitude of ${\rho }_{yx}^{\chi }$ for sample #2 is nearly twice as much as that for sample #1, which we believe can be attributed to the difference of a factor of 2 in the resistivity.

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To reveal the implication of the reciprocal relations on the chirality-dependent Hall effect, it is necessary to identify various effects of the Berry curvature involved in the phenomenon. Start from the equations of motion for band electrons. The velocity can be expressed as

$$\begin{eqnarray}&&\dot{{\boldsymbol{r}}}=D({\boldsymbol{B}},{{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}})\left[{{\boldsymbol{v}}}_{{\boldsymbol{k}}}+\displaystyle \frac{e}{\hslash }{\boldsymbol{E}}\times {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}+\displaystyle \frac{e}{\hslash }({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){\boldsymbol{B}}\right],\end{eqnarray} \tag{ 2 }$$

where ${{\boldsymbol{v}}}_{{\boldsymbol{k}}}=\displaystyle \frac{1}{\hslash }\displaystyle \frac{\partial \epsilon }{\partial {\boldsymbol{k}}}$ is the band group velocity, ${{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}\propto \chi \displaystyle \frac{{\boldsymbol{k}}}{{k}^{3}}$ is the Berry curvature, and $D({\boldsymbol{B}},{{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}})={[1+\displaystyle \frac{e}{\hslash }({\boldsymbol{B}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}})]}^{-1}$ is the modification of the phase space volume.^[34,35] e, ℏ, E , and B are the elementary charge, the reduced Planck constant, electric field, and magnetic field, respectively. χ = ±1 is the chirality of Weyl cones. $\displaystyle \frac{e}{\hslash }({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){\boldsymbol{B}}$ is the anomalous velocity related to the Berry curvature. One can calculate the nonequilibrium distribution function by solving the Boltzmann equation under a relaxation time approximation and arrive at^[7,8]

$$\begin{eqnarray}\begin{array}{lll}{f}_{{\boldsymbol{k}}} & = & {f}_{0}+\left[eD\tau {\boldsymbol{E}}\cdot {{\boldsymbol{v}}}_{{\boldsymbol{k}}}+\displaystyle \frac{{e}^{2}}{\hslash }D\tau ({\boldsymbol{B}}\cdot {\boldsymbol{E}})({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}})\right]\\ & & \times \left(-\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{{\boldsymbol{k}}}}\right),\end{array}\end{eqnarray} \tag{ 3 }$$

neglecting higher-order corrections. Here, f₀ is the equilibrium distribution function, and τ is the mean free time. The first term in the square bracket is the shifted distribution induced by the electric field, while the second term is the chiral chemical potential, resulting from pumping of electrons between Weyl cones of opposite chiralities. The current can be calculated by integration of the velocity

$$\begin{eqnarray}&&{\boldsymbol{j}}=-e\displaystyle \int \displaystyle \frac{{\rm{d}}{\boldsymbol{k}}}{{(2\pi)}^{3}}{D}^{-1}\dot{{\boldsymbol{r}}}{f}_{{\boldsymbol{k}}}.\end{eqnarray} \tag{ 4 }$$

When calculating the integral, it should be born in mind that Weyl cones appear in pairs. There are three contributions that are linear in magnetic field and electric field, which are non-zero only when Weyl cones are tilted. One is the product of the first term of Eq. (2) and the second term in Eq. (3). It generally contributes to the diagonal current and hence is not of our interest. We write down the other two contributions as

$$\begin{eqnarray}\begin{array}{lll}{\boldsymbol{j}} & = & -\displaystyle \frac{{e}^{3}\tau }{\hslash }\displaystyle \int \displaystyle \frac{{\rm{d}}{\boldsymbol{k}}}{{(2\pi)}^{3}}D[({\boldsymbol{E}}\cdot {{\boldsymbol{v}}}_{{\boldsymbol{k}}})({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){\boldsymbol{B}}\\ & & +({\boldsymbol{B}}\cdot {\boldsymbol{E}})({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){{\boldsymbol{v}}}_{{\boldsymbol{k}}}]\times \left(-\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{{\boldsymbol{k}}}}\right).\end{array}\end{eqnarray} \tag{ 5 }$$

The first current is the contribution of the anomalous velocity from the shifted distribution, denoted as j _A. The second current is the contribution of the velocity from the chiral chemical potential, denoted as j _C.

For simplicity, let us consider the particular configurations in our measurements. As depicted in Fig. 5, the tilt of Weyl cones is along the y axis,^[11] while the in-plane magnetic field is along the x axis. Various terms in the integrand are sketched to provide an intuitive estimation on the corresponding currents. For ρ_xy (Since the Hall resistivity is much smaller than the longitudinal resistivity, ${\rho }_{xy}=-{\sigma }_{xy}/({\sigma }_{xx}^{2}+{\sigma }_{xy}^{2})\propto {\sigma }_{xy}$. Thus, σ_xy calculated by Eq. (5) can be directly compared with ρ_xy obtained by experiments.), the electric field is in the y direction. j _C vanishes because B ⋅ E = 0. j _A is parallel to B (transverse). Although the anomalous velocity $\displaystyle \frac{e}{\hslash }({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){\boldsymbol{B}}$ are opposite in Weyl cones of opposite chiralities, the shifted distributions are not symmetric in the presence of a tilt, yielding a finite current. For ρ_yx , the electric field is in the x direction. j _A is in the x direction and affects the longitudinal conductivity. The transverse component of current comes from j _C. It is worth pointing out that for Weyl cones without a tilt, when considering the effect of the chiral chemical potential, the Fermi level lies horizontally and only shifts in energy. In contrast, for tilted Weyl cones, the Fermi level not only shifts in energy, but also tilts along the y direction, as sketched in Fig. 5(d). When Weyl cones are not tilted, j _C vanishes because v_k is odd in k within a Weyl cone. A tilt breaks the inversion symmetry of v_k within a Weyl cone and the balance between two cones, giving rise to a finite current. The current is along the direction of the tilt (the y axis), as sketched in Fig. 5(f).

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Our experiments indicate that the Onsager–Casimir reciprocal relations hold for the chirality-dependent Hall effect, ρ_xy (B,M) = ρ_yx (–B,–M). Based on the above analysis on the origin of ρ_xy and ρ_yx , it can be therefore concluded that j _A(B,M) = j _C(–B,–M) in our experiment configurations. Although these two currents are manifests of different effects, our results suggest that they are intricately connected.

Having verified the reciprocal relations under a particular electric and magnetic field configuration and pointed out its implication, we generalize the conclusion to all configurations. Substituting Eqs. (2) and (3) into Eq. (4) and dropping the term for the anomalous Hall effect, one can express the conductivity tensor as

$$\begin{eqnarray}\begin{array}{lll}{\sigma }_{ij} & = & -{e}^{2}\tau \displaystyle \int \displaystyle \frac{{\rm{d}}{\boldsymbol{k}}}{{(2\pi)}^{3}}D\left[{v}_{i}+\displaystyle \frac{e}{\hslash }({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){B}_{i}\right]\\ & & \times \left[{v}_{j}+\displaystyle \frac{e}{\hslash }({{\boldsymbol{v}}}_{{\boldsymbol{k}}}\cdot {{\boldsymbol{\varOmega }}}_{{\boldsymbol{k}}}){B}_{j}\right]\times \left(-\displaystyle \frac{\partial {f}_{0}}{\partial {\epsilon }_{{\boldsymbol{k}}}}\right),\end{array}\end{eqnarray} \tag{ 6 }$$

where v_i denotes the i-th component of v_k . It was incorrectly concluded that equation (6) violates the reciprocal relations.^[8] This misunderstanding stems from the implicit dependence of the anomalous velocity on the magnetization. In a time-reversal-symmetry-broken Weyl semimetal, reversing the magnetization leads to a reversal of the chiralities of Weyl nodes.^[36] Thus, the sign of Ω_k depends on the sign of the magnetization. Taking the magnetization into consideration, equation (6) indeed satisfies the Onsager–Casimir reciprocal relations, σ_ij (B,M) = σ_ji (–B,–M). When Ω = 0, equation (6) is reduced to the conventional formula. The effect of the Berry curvature is to introduce a correction (the anomalous velocity) to the velocity, besides the change of the phase space volume.

Acknowledgements