Abstract
The Wiedemann-Franz (WF) law has been tested in numerous solids, but the extent of its relevance to the anomalous transverse transport and the topological nature of the wave function, remains an open question. Here, we present a study of anomalous transverse response in the noncollinear antiferromagnet Mn3Ge extended from room temperature down to sub-kelvin temperature and find that the anomalous Lorenz ratio remains close to the Sommerfeld value up to 100 K but not above. The finite-temperature violation of the WF correlation is caused by a mismatch between the thermal and electrical summations of the Berry curvature and not by inelastic scattering. This interpretation is backed by our theoretical calculations, which reveals a competition between the temperature and the Berry curvature distribution. The data accuracy is supported by verifying the anomalous Bridgman relation. The anomalous Lorenz ratio is thus an extremely sensitive probe of the Berry spectrum of a solid.
INTRODUCTION
The Berry curvature of electrons can give rise to the anomalous Hall effect (AHE) (1, 2). This happens if the host solid lacks time-reversal symmetry, which impedes cancellation after integration over the whole Fermi surface. Explored less frequently (3–5), the thermoelectric and thermal counterparts of the AHE (the anomalous Nernst and anomalous Righi-Leduc effects) also arise by the same fictitious magnetic field (6–8). How do the magnitudes of these anomalous off-diagonal coefficients correlate with each other? Do the established correlations between the ordinary transport coefficients hold? Satisfactory answers to these questions are still missing. A semiclassical formulation of AHE (9) is laborious because the concept of Berry connection [or the “anomalous velocity” (10)] is based on off-diagonal matrix elements linking adjacent Bloch functions and not wave packets of semiclassical transport theory (9). This makes any intuitive picture of how Berry curvature combined to a longitudinal thermal gradient can produce a transverse electric field (an anomalous Nernst response) (7, 11) or a transverse thermal gradient (an anomalous thermal Hall response) (12, 13) even more challenging.
Here, we present a study of correlations between the anomalous off-diagonal transport coefficients of a magnetic solid, with a focus on the relation between anomalous electrical,
We track
We find that over a wide temperature range (0.5 K < T < 100 K),
Following theoretical propositions (15, 16), a large AHE was found in Mn3X (X= Sn and Ge) family of noncollinear antiferromagnets (17–19) below a high Néel temperature (20–22). These newcomers to the emerging field of antiferromagnetic spintronics (23) present a distinct profile of the Hall resistivity in which the extraction of the anomalous Hall conductivity becomes straightforward. An anomalous thermoelectric (Nernst) (24–26) and Righi-Leduc (25), counterparts of AHE, were also observed in Mn3Sn. In the case of Mn3Sn, the triangular order is destroyed at finite temperature (17, 24, 25). This is not the case for Mn3Ge where the fate of these signals can be followed down to sub-kelvin temperatures.
RESULTS
Basic properties
The room temperature field dependence of the three transport properties in Mn3Ge is shown in Fig. 1. Like in Mn3Sn, a hysteretic jump is triggered at a well-defined magnetic field, marking the nucleation of domains of opposite polarity induced by magnetic field (27). The large jump, the small magnetic field required for inverting polarity, and the weakness of the ordinary Hall response lead to step-like profiles contrasting with other topological solids exhibiting AHE (28–30). A step-like profile of anomalous transverse response [for other varieties (31, 32)] makes the extraction of the anomalous component straightforward. The panels of Fig. 1 show the measured Hall resistivity, Nernst signal, and thermal Hall resistivity, which were used to extract electric, thermoelectric, and thermal Hall conductivities.
As shown in the three left panels, they link to four vectors, which are charge density current (
Figure 2 presents a number of basic properties of the system under study. The spin texture (17, 20, 21) is shown in Fig. 2A. This magnetic order is stabilized thanks to the combination of Heisenberg and Dzyaloshinskii-Moriya interactions (33). As seen in Fig. 2B, which shows the magnetization, it emerges below TN = 370 K. The small residual ferromagnetism has been attributed to the residual magnetic moment of octupole clusters of Mn atoms (34) in this pseudo-Kagomé lattice.
(A) A sketch of the magnetic texture of Mn3Ge, showing the orientation of spins of Mn atoms. Red and blue represent two adjacent planes. (B) Temperature dependence of the magnetization with Néel temperature visible at 370 K. emu, electromagnetic unit. (C) Temperature dependence of resistivity along two orientations. (D) The Seebeck coefficient, S, as a function of temperature. (E) Low-temperature specific heat, C/T, as a function of T2. Extrapolation to T = 0 yields γ = 24.3 mJ mol−1 K−2. (F) Plot of the absolute value of S/T versus γ for a number of correlated metals including Mn3X and MnSi (37, 38).
A carrier density of n = 3.1 × 1022 cm−3 is extracted from the magnitude of the ordinary Hall number (Supplementary Materials) in agreement with a previous report (19). The electrical resistivity shows little variation with temperature (Fig. 2C), and its magnitude of 150 μΩ·cm implies a mean free path as short as 0.9 nm, compatible with the fact that Mn3X crystals are not stoichiometric (19, 21). In our crystals, we found the Mn:Ge ratio ranges from 3.32:1 to 3.35:1 (Supplementary Materials). Since one-tenth of Ge sites are occupied by Mn atoms, the average distance between these defects is
The Seebeck coefficient (Fig. 2D) has a nonmonotonic temperature dependence with a peak around 60 K, a sign change above 200 K, and a large low-temperature slope indicative of electronic correlations. The T-linear electronic specific heat (Fig. 2E) is as large as γ = 24.3 mJ mol−1 K−2, 30 times larger than copper and 5 times larger than iron (35). Assuming a single spherical Fermi surface corresponding to the known carrier density, such a γ implies an effective mass as large as m* = 14.5 me, which should not be taken literally given that the system is multiband. The slope of the Seebeck coefficient at low temperature (S/T ≃ − 0.2 μV K−2) correlates with γ, yielding
The anomalous transverse WF law
For each temperature, we measured
Our main finding is presented in Fig. 3. Below 100 K, the anomalous Lorenz ratio,
Temperature dependence of the anomalous Hall conductivity
The Bridgman relation
Several previous reports of the violation of WF law have been refuted afterward. One may therefore wonder whether our data can be validated by independent criteria. The answer is affirmative. Their validity is supported by the verification of the Kelvin relation (for normal longitudinal transport coefficients) and the Bridgman relation (for anomalous transverse coefficients). According to the thermodynamics of irreversible processes, these relations should remain valid irrespective of microscopic details.
The same data (namely, the electric field and the thermal gradient produced by imposing a heat current) and the same setup were used for both thermal and thermoelectric studies. Therefore, the validity of Kelvin and Bridgman relations is a guarantee of the validity of the collected thermal data.
To check the Bridgman relation, we directly measured both the Nernst (Fig. 4A) and the Ettingshausen (Fig. 4B) effects. The former is the transverse electric field generated by a longitudinal thermal gradient,
(A) The transverse electric field created by a finite longitudinal temperature gradient as a function of magnetic field (the Nernst effect). (B) The transverse thermal gradient produced by a finite longitudinal charge current (the Ettingshausen effect) at the same temperature. Insets show experimental configurations. (C) The temperature dependence of the anomalous Nernst (
As seen in Fig. 4C, the two sides of the equation remain close to each other in the whole temperature range. The Bridgman relation, derivable by a thermodynamic argument (39), is based on Onsager reciprocity (14). Its experimental validity has been confirmed in semiconductors (40) and in superconductors hosting mobile vortices (41). While there is a previous report on simultaneous measurements of anomalous Nernst and Ettingshausen coefficients (42), the present study is the first experimental confirmation of the validity of Bridgman relation in the context of topological transverse response. We also verified the Kelvin relation linking the Seebeck and Peltier coefficients (Supplementary Materials).
The temperature dependence of the anomalous transverse thermoelectric conductivity,
DISCUSSION
Origin of the finite-temperature violation
Having discussed the thermoelectric response, let us now turn back to the thermal transport. The zero-temperature validity of the WF law implies that the transverse flow of charge and entropy caused by Berry curvature conforms to a ratio of
To identify the origin of the observed drop in the anomalous transverse Lorenz number, let us begin by recalling what is known about ordinary transport and Lorenz number. The WF law ceases to be valid in the presence of inelastic scattering. This is because small-angle inelastic collisions decay the momentum flow less efficiently than the energy flow (45). In the semiclassic picture of electronic transport, charge and heat conductivity are set by the mean free path and the Fermi radius averaged over the whole Fermi surface with different pondering factors (45, 46, 47)
This expression has been obtained for both Boltzmann (45) and Landauer (46) formalisms. It has been shown that n = 0 for charge transport, n = 1 for thermoelectric transport, and n = 2 for thermal transport (see Fig. 5A).
(A) The pondering functions for electric (∂f(ϵk)/∂ϵk), thermoelectric ([(ϵk − μ)/kBT]∂f(ϵk)/∂ϵk), and thermal ([(ϵk − μ)/kBT]2∂f(ϵk)/∂ϵk) transport (49, 51, 52). (B) A representation of how a mismatch between thermal and electrical summations of the Berry curvature can occur if the two pondering functions do not average the overall contribution of the Berry curvature sources and sinks. Disorder makes these functions step-like in k-space. Note four distinct length scales in the k-space relevant to these summations: the inverse of the thermal wavelength (Λ), the inverse of the mean free path (𝓁), the typical size of a Berry sink (or source) (s), and its distance (d) from the Fermi surface.
One can see that in the presence of energy-dependent scattering, the difference between the electric and the thermal pondering functions can give rise to a difference between thermal and electrical conductivities. Electron-phonon scattering is expected to give rise to electric and thermal resistivities with different exponents in their temperature dependence [ρ ∝ T5 and T/k ∝ T3] (45, 48). This is because of the abundance of small wave vector phonons at low temperatures. In the case of electron-electron scattering, the electric and the thermal resistivities are both expected to be quadratic in temperature, with mismatch in the size of the prefactor (48). On the other hand, for electron-impurity scattering, the WF law is expected to hold.
With all this in mind, let us examine the case of Mn3X metals. The magnitude of the ordinary Hall coefficient yields a carrier density of 2 (3) ×1022 cm−3 in Mn3Sn (25) (Mn3Ge) (Supplementary Materials; 19). Combined with the magnitude of resistivity, this implies a mean free path in Mn3Sn (Mn3Ge), which is 0.7 (0.55) nm at 300 K and 0.9 (0.7) nm at 4 K. Now, these samples are not stoichiometric (Supplementary Materials; 19), and there is always an excess of Mn content, pointing that a fraction of Sn or Ge sites are occupied by Mn atoms. The measured value of Mn excess (Supplementary Materials; 19) yields an average distance between such sites of the order of 1 nm. This is almost identical to our estimation of the mean free path and leads us to conclude that the dominant scattering mechanism in both Mn3Sn and Mn3Ge is scattering off these antisite defects. There is little room for inelastic scattering in our context of investigation and one should look for an alternative route toward the violation of the WF law.
The pondering functions plotted in Fig. 5A remain relevant to the anomalous transverse transport. They imply that states probed by each transport coefficient are centered at a given location in the k-space. In this context, the summation of the Berry curvature over the Fermi sheets with these pondering factors can potentially generate a mismatch between
We note that our picture of the violation of the anomalous WF law shares a formal similarity with the well-established route toward violation of the ordinary WF law in a degenerate Fermi liquid (where T < TF). In the latter case, it is caused by a sufficiently large energy dependence in the scattering time near the Fermi energy. In our interpretation, it is caused by a sufficiently large energy dependence in the Berry curvature near the Fermi energy. The driving source of violation is totally different, but in both cases, the departure is caused by the fact that thermal and electric conductivities do not average identically near the Fermi level.
Theoretical calculations of the anomalous Lorenz ratio
The anomalous Hall conductivity
Here,
Here, we summarize general rules about the anomalous WF law. At zero temperature,
The total Berry curvature
By shifting μ slightly above, we can reproduce the general trend of T dependence of
The theoretical zero-temperature Berry curvature
We note that the theoretical violation for Mn3Ge is smaller than the experimental one. Let us recall that electronic correlations are neglected in the density functional theory used to calculate the band structure of the system and are absent in Eqs. 5 to 8. In addition, calculations that were performed for an ideal clean system may cause quantitative discrepancies when comparing to the dirty metal in the experiment. Given these deviations, our theory qualitatively demonstrates the different Berry spectrum in Mn3Sn and Mn3Ge, which leads to different behaviors of
Different behaviors between two materials originate in their different Berry curvature
CONCLUDING REMARKS
To summarize, we measured counterparts of the AHE associated with the flow of entropy and found that the WF law linking the magnitude of the thermal and electrical Hall effects is valid at zero temperature, but a finite deviation emerges above 100 K. We show that the mean free path of carriers is short and comparable to the distance between Mn antisite defects. As a consequence, the dominant scattering is elastic. We propose that the deviation from the WF law is caused by a mismatch in thermal and electrical summations of Berry curvature over the Fermi surface. Our theoretical calculations support this interpretation by showing that the Berry spectrums in the two systems are not identical. The Bridgman relation, which links anomalous Nernst and Ettingshausen coefficients, is satisfied over the whole temperature range. Last, we observed that the room temperature
MATERIALS AND METHODS
Sample preparation and transport measurement
Single crystals of Mn3Ge were grown from polycrystalline samples using Bridgman-Stockbarger technique. The raw materials, Mn (99.99% purity) and Ge (99.999% purity), were weighed and mixed in an argon glove box with a molar ratio of 3.3:1, loaded in an alumina crucible and then sealed in a vacuum quartz ampule. The mixture was heated up to 1050°C, remained for 2 hours to ensure homogeneity of melt, and then was cooled slowly down to 800°C to obtain polycrystalline samples. The polycrystalline Mn3Ge were ground, loaded in an alumina crucible, and sealed into another vacuum quartz ampule. The growth temperature was controlled at 980° and 800°C for high-temperature and low-temperature end, respectively. Last, to obtain high-temperature hexagonal phase, the quartz ampule was quenched with water. The single crystals were cut by a wire saw into typical dimensions of 0.3 mm by 1.5 mm by 2 mm for transport measurements. The stoichiometry was found to be Mn3.08Ge0.92 (Mn:Ge = 3.32 to 3.35:1) using energy-dispersive X-ray spectroscopy. This is close to the ratio of the raw materials and comparable to previous reports (19).
Longitudinal and Hall resistivity were measured by the standard four-probe method using a current source (Keithley 6221) with a dc nanovoltmeter (Keithley 2182A) in a commercial measurement system [Physical Property Measurement System (PPMS), Quantum Design]. The thermal conductivity and thermal Hall effect were performed using a heater and two pairs of thermocouples in the PPMS in a high-vacuum environment (25). For temperatures below 4.2 K, the measurements were performed in a dilution refrigerator inserted in a 14-T superconducting magnet using a one heater–three thermometers setup, allowing the measurement of longitudinal and transverse transport coefficients with the same contacts.
Theoretical calculations
The band structure was calculated with the density functional theory in the framework of the generalized-gradient approximation (52). The Bloch wave functions were projected to atomic orbital–like Wannier functions (53). On the basis of the Wannier-projected tight-binding Hamiltonian, we calculated the Berry curvature and the anomalous Hall conductivity in the clean limit. More details can be found in (54). As shown in Fig. 6 (A and C), the
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/17/eaaz3522/DC1
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REFERENCES AND NOTES
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