Anomalous Nernst effect in the noncollinear antiferromagnet Mn₅Si₃

Abstract

Investigating the off-diagonal components of the conductivity and thermoelectric tensor of materials hosting complex antiferromagnetic structures has become a viable method to reveal the effects of topology and chirality on the electronic transport in these systems. In this respect, Mn₅Si₃ is an interesting metallic compound that exhibits several antiferromagnetic phases below 100 K with different collinear and noncollinear arrangements of Mn magnetic moments determined from neutron scattering. Previous electronic transport measurements have shown that the transitions between the various phases give rise to large changes of the anomalous Hall effect. Here, we report measurements of the anomalous Nernst effect of Mn₅Si₃ single crystals that also show clear transitions between the different magnetic phases. In the noncollinear phase, we observe an unusual sign change of the zero-field Nernst signal with a concomitant decrease of the Hall signal and a gradual reduction of the remanent magnetization. Furthermore, a symmetry analysis of the proposed magnetic structures shows that both effects should actually vanish. These results indicate a symmetry-breaking modification of the magnetic state with a rearrangement of the magnetic moments at low temperatures, thus questioning the previously reported models for the noncollinear magnetic structure obtained from neutron scattering.

Introduction

Antiferromagnetic materials have attained a renaissance in condensed-matter research due to technical advantages like low stray-fields and ultrafast switching compared to ferromagnets, leading to the development of a new field coined antiferromagnetic spintronics^1,2. A particular class of materials is antiferromagnets in which the magnetic moments of atoms are ordered in a noncollinear fashion. They often exhibit a nonzero Berry curvature leading to an emergent electromagnetic response, which can be harnessed for practical purposes^3,4,5,6. In noncollinear antiferromagnets and spin liquids, a nonzero Berry curvature gives rise to an unusually large anomalous Hall effect (AHE)^7,8,9,10,11. Like in ferromagnets, the intrinsic part of the AHE is obtained by integration of the Berry curvature of occupied electronic bands over the entire Brillouin zone^12,13. Similar to the AHE, its thermoelectric counterpart, the anomalous Nernst effect (ANE), generates a voltage transverse to the heat flow and magnetization. This transverse thermopower provides a measure of the Berry curvature only at the Fermi energy E_F^12,13. It is therefore not a priori clear that a large AHE guarantees a large ANE.

The AHE and ANE have been the subject of intense research and development in recent years and hold great promise for practical applications in the fields of spintronics and thermoelectronics. Both effects can be extraordinary large in chiral antiferromagnets like Mn₃Sn^7,10,14 and Mn₃Ge^8,9,11,15,16 despite their tiny magnetization. Here, the enhanced Berry-phase curvature is associated with the existence of Weyl points near the Fermi level where the Berry curvature diverges. In addition to the thermoelectric ANE, a sizeable thermal Hall effect (Righi-Leduc effect) generated by the Berry curvature has been reported for these two materials^10,11.

The intermetallic compound Mn₅Si₃ is a noncollinear antiferromagnet that has gained attention due to unusual thermodynamic and electronic transport phenomena^{17,18,19,20,21,22}. Mn₅Si₃ has a hexagonal crystal structure (space group P6₃/mcm) with two inequivalent Mn lattice sites Mn₁ and Mn₂ at room temperature and undergoes two structural phase transitions toward orthorhombic symmetry below 100 K.

Figure 1 shows the magnetic phase diagram of Mn₅Si₃ in a magnetic field H oriented along the crystallographic c axis. Various methods including neutron scattering have confirmed the existence of an antiferromagnetic phase AF2 between the Néel temperatures T_N1 = 60 K and T_N2 = 100 K with zero Mn₁ moments and a collinear arrangement of two-thirds of Mn₂ moments with opposite orientations on adjacent sites of the orthorhombic structure [Fig. 2a]^{17,21,23,24,25}.

Fig. 1: Magnetic phases of Mn₅Si₃.

Magnetic phase diagram and magnetic moment obtained from individual magnetization curves, see Fig. 3a. AF1 and AF1' exhibit noncollinear antiferromagnetic structures while AF2 is a collinear antiferromagnetic phase.

Fig. 2: Magnetic structures of Mn₅Si₃.

Crystal structures of the low-temperature (T < 100 K) antiferromagnetic phases in the orthorhombic a-b plane indicated by the black outlined rectangle (b > a). Mn and Si atoms are shown in magenta and blue color, respectively. Red arrows indicate different Mn magnetic moments. a Collinear AF2 phase observed for 60 K  < T < 100 K^17,24,29,56. b Noncoplanar AF1 phase without inversion symmetry proposed in ref. ²⁶ for T < 60 K⁵⁶. c Illustration of the AF1 phase proposed in ref. ²⁸ for T < 60 K, also noncoplanar and noncentrosymmetric.

In the antiferromagnetic AF1 phase below T_N1 = 60 K, a highly noncollinear and noncoplanar arrangement of magnetic moments is observed where the Mn₁ atoms also acquire a magnetic moment in addition to the two thirds of Mn₂ moments, although the details of the size and orientation of the Mn moments are still under dispute^26,27,28, see Fig. 2b, c. The noncollinearity results from the combined effects of magnetic frustration and anisotropy.

Inelastic neutron-scattering measurements of the low-energy dynamics reveal a single-anisotropy energy gap in the AF1 phase and a double gap in the AF2 phase arising from a bi-axial anisotropy^28,29. Concomitantly performed density-functional theory calculations indicate that in the AF1 phase the crystallographic b axis is the easy axis, whereas in the AF2 phase the b and c axes are the first and second preferred magnetic easy axes, respectively, and a is the hard axis³⁰.

As soon as the magnetic structure changes from collinear AF2 to noncollinear AF1 below T_N1 = 60 K, a strong AHE is observed^18,19,20. In addition, the existence of a magnetic-field induced intermediate phase AF1’ has been inferred from neutron scattering and electronic transport measurements^20,21,25. Finally, at high magnetic fields, a collinear phase with similar properties to the zero-field AF2 phase is stabilized^18,20,21,28. We mention that an additional phase has been reported to exist in single crystals grown out of Cu flux which was attributed to strain or stress introduced by Cu inclusions from the crystal growth³¹. It is these different magnetic moment configurations of noncollinear and collinear order achieved at different magnetic fields and temperatures, that make the Mn₅Si₃ compound so interesting for investigations of the effect of topology and complex magnetic order on the magnetotransport properties.

For the noncollinear AF1 phase, previous first-principle calculations assumed a Heisenberg Hamiltonian that takes into account magnetic exchange and biaxial anisotropy²⁹. Through modeling of the low-temperature spin-wave spectrum obtained by inelastinc neutron scattering at 10 K, two additional exchange constants representing the Mn₁-Mn₁ and Mn₁-Mn₂ interactions have been proposed by Biniskos et al.²⁸, resulting in a ground-state spin arrangement in the AF1 phase that is different from the previous AF1 phase reported by Brown et al.²⁶. Here, in zero magnetic fields the Mn₁ atoms carry a magnetic moment oriented along the orthorhombic b axis while the two-thirds of Mn₂ atoms carry a magnetic moment with a significant component along the c axis, respectively (Fig. 2c). Furthermore, the field-induced phase transition between AF1 and AF1’ was simulated by including a Zeeman term in the Heisenberg Hamiltonian²⁸. In this model, it is assumed that the Mn₁ moments are longitudinally susceptible, i.e., their size is affected by an external field. Starting from the proposed ground state (Fig. 2c) and increasing the magnetic field applied along the c axis, a “spin flop” phase with coplanar Mn₂ moments in the a-b plane is succeeded by an AF1’-like phase, where the Mn₁ moments are also predominantly orientated in the a-b plane, but start to have a component along the c axis. At even higher fields, a transition to a field-induced phase is proposed, with collinear and antiparallel oriented Mn₂ moments along the b axis and nonvanishing Mn₁ moments aligned parallel to the direction of the magnetic field. All models for the magnetic structure reported so far suggest that the field-induced phase at high magnetic fields has similar properties as the zero-field AF2 phase, although it is not clear whether the Mn₁ moment eventually collapses or not^25,28.

In addition to single crystals, thin epitaxially strained Mn₅Si₃ layers have been grown that exhibit collinear order up to higher temperatures and a spontaneous AHE due to the breaking of time-reversal symmetry by an unconventional staggered spin-momentum interaction³². In the ideal case considered by theory, the zero net magnetization allows for a non-relativistic electronic structure with altermagnetic collinear spin polarization in momentum space^33,34. This gives rise to anomalous Hall and Nernst effects experimentally observed in thin epitaxial Mn₅Si₃ films with a small net magnetization of 20 - 60 mμ_B/f.u.³⁵. Recently, spin-orbit torque switching of the Néel vector has been demonstrated for strained Mn₅Si₃ films and a strong enhancement of the ANE upon Mn doping of a Mn₅Si₃ film has been attributed to the shift of the Fermi level^36,37.

The observation of a nonzero AHE and its strong changes at the magnetic phase boundaries, the link between the AHE and ANE via the Berry-phase concept, and the hitherto reports of both effects observed in other noncollinear antiferromagnets, motivated this study of the ANE in Mn₅Si₃, where we focus on the noncollinear magnetic phase at low temperatures.

Results

Magnetization and anomalous Hall effect

Figure 3a shows a typical magnetization curve of Mn₅Si₃ at T = 35 K with the magnetic field applied parallel to the crystallographic c axis. The two jumps of the magnetization ΔM(H_c1) and ΔM(H_c2) at magnetic fields H_c1 and H_c2, respectively, are attributed to the AF1/AF1’ and AF1’/AF2 phase transitions. Before sweeping the magnetic field, the sample has a magnetization much lower than the remanent magnetization M_R(0), see left inset Fig. 3a. This indicates that the antiferromagnetic structure is not fully compensated in the magnetically pristine state and exhibits weak moments that allow the domain configuration to be changed by applying a magnetic field and generate a net magnetization along the field direction. The hysteresis around zero field is not observed when the magnetic field is oriented perpendicularly to the c axis for all temperatures below 60 K¹⁹. The heights of the jumps are plotted in Fig. 3b together with the remanent magnetization M_R(0) vs. temperature. A remarkable detail is the decrease of M_R(0) when cooling below 20 K while ΔM(H_c1) and ΔM(H_c2) continuously increase with decreasing temperature below T_N1 = 60 K. Moreover, the absolute value of M_R(0) below 20 K depends on the maximum applied magnetic field, i.e.  ± 2 T or  ± 5 T, see Fig. 3b, indicating either a reformation of antiferromagnetic domains with opposite Néel vectors in this temperature range³² or a modification of the local magnetic structure. Magnetic polarization at fields larger than 5 T was not possible without entering the AF1’ phase.

Fig. 3: Magnetization and anomalous Hall effect of Mn₅Si₃.

a Magnetization M(H) at T = 35 K. Insets show a close up of M(H) near zero field and around the transition at H_c1. b Remanent field M_R(0) and magnetization jumps ΔM(H_c1) and ΔM(H_c2) obtained from M(H) vs temperature. Closed (open) circles indicate data obtained after applying a magnetic field of  ± 5 T ( ± 2 T) before measuring M_R(0). The accuracy with which the values from M(H) loops could be determined is indicated by error bars. c–f Anomalous Hall effect for various temperatures. g Anomalous Hall conductivity σ_xy(H) at T = 25 K. Open symbols represent data in the virgin magnetic state before sweeping the field from zero to +8 T. h Temperature dependence of σ_xy(0) and σ_xy(H_c1). Inset shows the resistivity ρ_xx(T) with indicated Néel temperatures T_N1 and T_N2 at the magnetic phase transitions AF1/AF1' and AF1'/AF2, respectively. The magnetic field was always oriented along the crystallographic c axis.

The phase transitions are clearly observed in the AHE of Mn₅Si₃, see Fig. 3c–f. At 50 K, below T_N1, the AHE strongly changes around zero field and at magnetic fields H_c1 and H_c2. In the collinear AF2 phase the AHE vanishes²⁰. The absolute value of the transverse resistivity ρ_yx(0) in the remanent state at zero field, i.e., after completing a full magnetic field cycle, increases with decreasing temperature down to 25 K and then decreases and even vanishes at T = 5 K while M_R(0) remains nonzero.

Figure 3g shows the Hall conductivity ${\sigma }_{{{\rm{xy}}}}={\rho }_{{{\rm{yx}}}}/({\rho }_{{{\rm{xx}}}}^{2}+{\rho }_{{{\rm{yx}}}}^{2})$, where ρ_xx is the longitudinal resistivity, see inset Fig. 3h. In the magnetically virgin state before sweeping the magnetic field, σ_xy is very small compared to the remanent state σ_xy(0) observed after completion of a field sweep to high fields, similar to the magnetization M(H), Fig. 3a. A strong difference between the temperature dependences of ρ_yx(0) was already reported earlier when the sample was cooled down in zero field from above T_N1 and subsequently polarized in a magnetic field of  ± 3 T²⁰.

σ_xy is affected by extrinsic contributions arising from skew scattering or side-jump mechanism and by the intrinsic Berry-phase contribution³. In the dirty and ultraclean regimes of electron scattering the dependence of σ_xy on the longitudinal conductivity σ_xx can be described by a power law ${\sigma }_{{{\rm{xy}}}}\propto {\sigma }_{{{\rm{xx}}}}^{\alpha }$ with α > 0. In the intermediate regime of moderate conductvity 3 × 10³ − 5 × 10⁵ Ω⁻¹cm⁻¹ the AHE is predominated by the intrinsic Berry-phase contribution where σ_xy is independent of σ_xx^38,39. In the present case, the single crystals with σ_xy = 150 Ω⁻¹cm⁻¹ and σ_xx = 7400 Ω⁻¹cm⁻¹ fall just within this intermediate regime as discussed earlier^20,40, similar to the noncollinear antiferromagnets Mn₃Ge and Mn₃Sn⁴¹. From ρ_xx = 135 μΩcm, a Fermi velocity v_F = 2 × 10⁶ m/s⁴², and an electron density n = 1 × 10²² cm⁻³²⁰ we estimate an electron mean free-path of 5.2 nm, larger than several lattice parameters, supporting the classification of this material as a moderately conductive metal.

The absolute values of the remanent σ_xy(0) and σ_xy(H_c1) plotted in Fig. 3h gradually increase between T_N1 = 60 K and  ≈ 25 K but then drop to very low values for T ≤ 20 K. We emphasize that σ_xy(0) completely vanishes for T ≤ 5 K while σ_xy(H_c1) remains at a low value of 17 Ω⁻¹cm⁻¹ at T = 2 K. The decrease of the AHE toward low temperatures is unusual for a well established magnetic order but could arise from domain reformation around zero field mentioned above. However, the fact that both contributions to the AHE, σ_xy(0) and σ_xy(H_c1), decrease toward zero temperature while the magnetization ΔM(H_c1) remains constant, demonstrates that a different mechanism might be at hand. For antiferromagnetic Mn₃Sn, a sharp drop of σ_zx and σ_yz in connection with a strong increase of $\left\vert {\sigma }_{{{\rm{xy}}}}\right\vert$ was observed in the low temperature spin-glass phase below 50 K, where a large ferromagnetic compound evolves due to spin canting^7,41. This suggests that in Mn₅Si₃ the magnetic moments possibly rearrange at low temperatures as previously inferred from analyzing the evolution of spin-wave energies, revealing another field induced phase-transition below the AF1/AF1’ phase boundary²⁸.

Anomalous Nernst effect

In the following we focus on the ANE obtained on the same Mn₅Si₃ single crystal. Theoretically, the ANE generates an electric field E = Q_Sμ₀M × ( − ∇ T) perpendicular to the directions of the magnetization M and temperature gradient  − ∇ T, where Q_S is the anomalous Nernst coefficient and μ₀ is the magnetic permeability of free space. Figure 4c shows a cartoon of the ANE setup. For the present configuration with H_z parallel to the crystallographic c axis, $\left\vert {{\bf{M}}}\right\vert ={M}_{{{\rm{z}}}}$, and with a temperature gradient along x, this simplifies to E_y = − S_yx ∇ T_x, with S_yx= Q_S μ₀M_z.

Fig. 4: Characterization of Mn₅Si₃ single crystals.

Laue diffraction patterns of a Mn₅Si₃ single crystal oriented with the x-ray beam along the a [001] and b [210] directions of the hexagonal structure at room temperature. At T < 100 K, the [100], [210], and [001] directions correspond to the a, b, and c axes, respectively, of the orthorhombic phase. c Temperature difference ΔT = T_h − T_c between the hot and the cold end of the sample vs heater power for different average temperatures T₀. Solid lines indicate a linear behavior. Inset shows a cartoon of the experimental set up and the orientation of the sample with crystallographic axes a, b, and c of low-temperature orthorhombic phase with respect to the sample holder.

The Nernst signal S_yx is experimentally measured by

$${S}_{{{\rm{yx}}}}=\frac{\Delta {V}_{{{\rm{y}}}}\,{l}_{{{\rm{x}}}}}{\Delta {T}_{{{\rm{x}}}}\,{l}_{{{\rm{y}}}}}$$

(1)

where we made use of the fact that E_y = − ΔV_y/l_y and  ∇ T_x = ΔT_x/l_x.

Figure 5a–d shows S_yx vs magnetic field H applied parallel to the c direction and heat flow along the b direction for different temperatures below T_N1. At 50 K, the transitions between the different magnetic phases are observed as clear changes of S_yx with a hysteresis, very similar to the behavior of the AHE [Fig. 3c–f].

Fig. 5: Anomalous Nernst effect of Mn₅Si₃.

a–d S_yx(H) for various temperatures with the magnetic field H applied along the crystallographic c axis. e S_yx(0) and S_yx(H_c1) vs temperature T. Solid symbols indicate data taken from individual S_yx(H) sweeps, see panels a–d. Open circles were calculated from the difference between S_yx(H) measured in two consecutive temperature sweeps at positive or negative remanence. The accuracy with which the values from S_yx(H) loops could be determined is indicated by error bars. Dashed line indicates a linear behavior toward zero temperature. f Temperature dependence of the Seebeck coefficient S_xx in zero field. Inset shows S_xx(H) at T = 55 K. g Transverse Peltier coefficients α_yx(0) and α_yx(H_c1) vs temperature T. Grey curve shows the ordinary Hall constant R₀ of Mn₅Si₃ thin films¹⁸.

S_yx is zero in the collinear AF2 phase for temperatures T > 60 K, see Fig. 6, concomitant with a vanishing AHE and magnetization. Again, after reaching a maximum the magnitude of S_yx decreases with decreasing temperature below 25 K, similar to the behavior of the AHE, cf. Fig. 3c–f. In addition, the coercive fields of M(H), ρ_yx, and S_yx around zero magnetic field are of similar size and continuously decrease from 15 K to 60 K as previously observed for ρ_yx(T)²⁰. This indicates that the effects of the magnetic field on ρ_yx and S_yx are of similar origin.

Fig. 6: Orientation dependence of the anomalous Nernst effect.

S_yx(H) of another Mn₅Si₃ single crystal for magnetic fields H and heat flows Q applied along different crystallographic directions a, b, or c. a–c H along c, Q along the orthorhombic a direction over a length of 0.52 mm. d, e H applied along the orthorhombic a direction, Q along the orthorhombic b direction over a length of 1.2 mm. f H along the orthorhombic b direction, Q along the orthorhombic a direction over a length of 0.52 mm.

A remarkable difference is the sign change of the remanent S_yx(0) below 25 K [see Fig. 5b, c] before it vanishes at lower temperatures. This behavior is also observed on another single crystal for H applied parallel to the c direction and heat flow along the a direction but not for H applied parallel to the a or b directions where no AHE appeared at zero field²⁰, see Fig. 6.

The temperature dependence is seen more clearly in plots of S_yx(0, T) and S_yx(H_c1, T), Fig. 5e. Whereas σ_xy(0) (in phase AF1) and σ_xy(H_c1) (in phase AF1’) have opposite signs but are of the same magnitude [Fig. 3h], S_yx(0) also has the opposite sign of S_yx(H_c1) but only reaches half of its value. Both effects, AHE and ANE, do not scale with the change of the magnetization at zero field and at H_c1, confirming that they arise from topology and from the intrinsic Berry-curvature effect like other antiferromagnets with a nontrivial magnetic structure.

S_yx(0) rapidly changes sign by cooling to below 20 K, strongly deviating from an approximately linear temperature dependence observed for S_yx(H_c1). Both coefficients vanish toward zero temperature due to Nernst’s theorem³⁸. The sign change of S_yx(0, T) occurs at the same temperature where we observe a decrease of M_R(0), in contrast to ΔM(H_c1) which appears more like a magnetically saturated state at low temperatures, see Fig. 3b. While the vanishing σ_xy(0) and reduced M_R(0) could be considered as indications for the reformation of antiferromagnetic domains, this cannot explain the reappearance of a positive S_yx(0) below 20 K supporting the idea of a weak modification of the magnetic structure at low temperatures.

The transverse thermoconductivity, i.e., the transverse Peltier coefficient, can be calculated in zero magnetic field by

$${\alpha }_{{{\rm{yx}}}}={S}_{{{\rm{yx}}}}{\sigma }_{{{\rm{xx}}}}+{S}_{{{\rm{xx}}}}{\sigma }_{{{\rm{yx}}}}$$

(2)

Here, S_xx = ΔV_x/ΔT_x is the Seebeck coefficient measured in the same setup and plotted in Fig. 5f for different temperatures using σ_yx = − σ_xy.

An interesting detail is the magnetic field behavior of S_xx(H) in the inset of Fig. 5f. At magnetic fields below H_c2 the Seebeck coefficient is almost independent of H but precipituously drops at H_c2 when crossing the phase boundary between the field-induced noncollinear AF1’ phase and the high-field collinear AF2-like phase. This extraordinary sharp change of the magneto-Seebeck effect with magnetic field is observed only at H_c2 where the magnetic order changes from noncollinear to collinear. It is correlated with the strong change of the magnetoresistance ρ_xx(H) by  ≈ 10 % at H_c2 and T = 45 K²⁰. This is expected from the Mott relation between the thermo-electric and electronic conductivity

$${\alpha }_{{{\rm{xx}}}}={\sigma }_{{{\rm{xx}}}}{S}_{{{\rm{xx}}}}=\frac{{\pi }^{2}{k}_{{{\rm{B}}}}^{2}}{3e}T{\left(\frac{\partial {\sigma }_{{{\rm{xx}}}}}{\partial E}\right)}_{\mu }.$$

(3)

where k_B is the Boltzmann constant, e the electron charge, and μ the electrochemical potential. It has been shown previously that the Mott relation between σ_xx and α_xx can also be applied for the transverse coefficient α_yx^12,43,44

$${\alpha }_{{{\rm{yx}}}}=-{\alpha }_{{{\rm{xy}}}}=-\frac{{\pi }^{2}{k}_{{{\rm{B}}}}^{2}}{3e}T{\left(\frac{\partial {\sigma }_{{{\rm{xy}}}}}{\partial E}\right)}_{\mu }.$$

(4)

The transverse coefficient α_yx [Fig. 5g] obtained from Eq. (2) essentially shows the same behavior as S_yx(0) [Fig. 5e]. Since $\left\vert {S}_{{{\rm{xx}}}}{\sigma }_{{{\rm{xy}}}}\right\vert \, \ll \, \left\vert {S}_{{{\rm{yx}}}}{\sigma }_{{{\rm{xx}}}}\right\vert$, α_yx is dominated by S_yxσ_xx (Eq. (2)). Hence, the ANE mostly arises from the transverse heat flow (90 %) and the sign change of α_yx occurs due to the sign change of S_yx.

For Mn₅Si₃, maximum values $\left\vert {S}_{{{\rm{yx}}}}(0)\right\vert$ = 0.05 μVK⁻¹ and $\left\vert {S}_{{{\rm{yx}}}}({H}_{{{\rm{c1}}}})\right\vert$ = 0.11 μVK⁻¹ are reached at magnetizations M_R(0) = 0.028 μ_B/f.u. and M_R(H_c1) = 0.17 μ_B/f.u., respectively. Note that the values of $\left\vert {S}_{{{\rm{yx}}}}(0)\right\vert$ and $\left\vert {S}_{{{\rm{yx}}}}({H}_{{{\rm{c1}}}})\right\vert$ should be considered as lower limits due to the thermal resistances between the sample holder and the single crystal which give rise to a lower applied thermal gradient in the crystal than the measured ΔT_x = T_h − T_c. Hence, an even larger $\left\vert {S}_{{{\rm{yx}}}}\right\vert$ derived from the low magnetization is expected. Similar values $\left\vert {S}_{{{\rm{yx}}}}\right\vert =0.6\,\mu {{{\rm{VK}}}}^{-1}$ and μ₀M = 1 mT (corresponding to M = 0.01 μ_B/f.u.) have been reported for chiral Mn₃Sn¹⁴. Compared to ferromagnetic metals, $\left\vert {S}_{{{\rm{yx}}}}\right\vert$ of Mn₅Si₃ is strongly enhanced outside the broad range for which $\left\vert {S}_{{{\rm{yx}}}}\right\vert \propto M$ is observed, in close vicinity to values of the chiral antiferromagnets Mn₃Sn^14,41 and Mn₃Ge^15,41.

In various topological magnets, the ratio $\left\vert {\alpha }_{{{\rm{xy}}}}/{\sigma }_{{{\rm{xy}}}}\right\vert$ increases with temperature from 1 μV/K and often approaches k_B/e = 86 μV/K at room temperature or above^10,11,45 and thus follows a universal scaling⁴⁶, with the exception of UCo_0.8Ru_0.2Al which exhibits a colossal ANE⁴⁷. In the present case, maximum values of $\left\vert {\alpha }_{{{\rm{yx}}}}(0)/{\sigma }_{{{\rm{xy}}}}(0)\right\vert =3.5\,\mu$V/K and $\left\vert {\alpha }_{{{\rm{yx}}}}({{{\rm{H}}}}_{{{\rm{c}}}1})/{\sigma }_{{{\rm{xy}}}}({{{\rm{H}}}}_{{{\rm{c}}}1})\right\vert =5.8\,\mu$V/K are obtained for T = 40 K, which are only somewhat smaller when compared to values of other topological materials at this low temperature^10,11,45. The similar ratios obtained in the AF1 and AF1’ phases of Mn₅Si₃ support the correlation between the two properties α_yx and σ_xy that are predominated by intrinsic effects of the Berry curvature in both magnetic phases.

If the Berry curvature Ω_n,z(k) is known, the intrinsic contributions to α_yx and σ_xy can be obtained from refs. ^12,13,41

$${\alpha }_{{{\rm{yx}}}}=-{\alpha }_{{{\rm{xy}}}}=-\frac{e}{T\hslash }{\sum}_{n}\int\frac{d{{\boldsymbol{k}}}}{{(2\pi )}^{3}}{\Omega }_{n,z}({{\boldsymbol{k}}}){g}_{n,{{\boldsymbol{k}}}}$$

(5)

${g}_{n,{{\boldsymbol{k}}}}=({E}_{n,{{\boldsymbol{k}}}}-\mu ){f}_{n,{{\boldsymbol{k}}}}+{k}_{{{\rm{B}}}}T\,{{\rm{\ln }}}\left[1+\exp \left(-\frac{{E}_{n,{{\boldsymbol{k}}}}-\mu }{{k}_{{{\rm{B}}}}T}\right)\right]$

$${\sigma }_{{{\rm{xy}}}}=-\frac{{e}^{2}}{\hslash }{\sum}_{n}\int\frac{d{{\boldsymbol{k}}}}{{(2\pi )}^{3}}{\Omega }_{n,z}({{\boldsymbol{k}}}){f}_{n,{{\boldsymbol{k}}}}$$

(6)

where k = (k_x, k_y, k_z) is the three-dimensional crystal momentum, E_n,k is the energy of the band of index n, and f_n,k is the Fermi-Dirac distribution function.

Finally, we argue that it is unlikely that the magnetocrystalline anisotropy is the origin of the anisotropic behaviour of both thermal effects. In the AF1 phase, nonzero AHE and ANE values are observed at zero field only for H oriented parallel to the c axis but not along the a or b axes, see Fig. 6 and ref. ¹⁹, where the latter is the magnetic easy axis in the AF1 phase³⁰.

Discussion

Temperature dependence of the ANE

The ANE represented by S_yx or α_yx is a Fermi surface property (Eq. (5)). A sign change of these parameters at 20 K inevitably means that they reach zero around 20 K. Hence, from Eq. (4) it follows that ${(\frac{\partial {\sigma }_{{{\rm{xy}}}}}{\partial E})}_{\mu }$ = 0, i.e., σ_xy(E) has a maximum or minimum at the Fermi surface which is shifting through the Fermi level with increasing temperature from below 20 K to above 20 K. A change of the Fermi surface at low temperatures can be also inferred from the variation of the ordinary Hall effect of Mn₅Si₃ thin films¹⁸. In that case, the ordinary Hall effect ${\rho }_{{{\rm{yx}}}}^{0}={R}_{0}{\mu }_{0}H$ could be separated from the measured signal, revealing a change of the ordinary Hall constant R₀ or the effective charge carrier density n_H with temperature and a maximum (minimum) of R₀ (n_H) around 20 K, see Fig. 5g. At this point, ab initio calculations of the electronic bandstructure and σ_xy are necessary for a more quantitative analysis of the AHE and ANE following the Berry-phase concept by application of Eqs. (5) and (6). Such calculations involve a knowledge of the correct magnetic configuration in the ground state of bulk Mn₅Si₃, which has not been determined so far.

A sign change of the ANE with temperature was reported earlier for the ferromagnetic semiconductor Ga_1−xMn_xAs and was attributed to the scattering-independent nature of the intrinsic AHE following a behavior ${\rho }_{{{\rm{yx}}}}\propto {\rho }_{{{\rm{xx}}}}^{2}$⁴⁴. The sign change of S_yx was found to be correlated with a sharp maximum in S_xx thus compensating this contribution to the transverse coefficient α_yx which did not change sign. In contrast, for Mn₅Si₃ the sign change occurs for S_yx and α_yx but not for S_xx. An analysis of the temperature dependence of α_xy based on the power law ${\rho }_{{{\rm{yx}}}}^{{{\rm{AHE}}}}=\lambda {M}_{z}{\rho }_{{{\rm{xx}}}}^{n}$⁴⁴ is not possible because λ of Mn₅Si₃ is strongly temperature dependent¹⁸ in contrast to itinerant ferromagnets where it is usually temperature independent at low temperatures.

In the pyrochlore molybdate R₂Mo₂O₇ (R = Nd, Sm) with noncoplanar spin structure the variations of S_xy and α_xy were attributed to a contribution from the spin chirality of the system which was considered to be responsible for an enhanced AHE and a positive ANE at low temperatures⁴⁸. A strong magnetic field reduces the amplitude of the spin chirality by aligning the Mn moments along the magnetic field direction leading to a reduced σ_xy and α_xy. It is interesting to note that this is observed only for R = Nd, Sm with non-coplanar spin structure and not for collinearly ordered Gd₂Mo₂O₇. In the present case, it is therefore conceivable that a similar behavior occurs due to a small change of the spin structure by moment reorientation and decrease of chirality/noncollinearity at small magnetic fields.

We only mention that for the Weyl semimetal Co₃Sn₂S₂ different behaviors of α_yx have been reported^45,49,50. In these cases, however, S_xy did not change its sign with the temperature.

Below T_N1 = 60 K, α_yx(H_c1) first increases with decreasing temperature together with a concomitant increase of ΔM(H_c1), eventually saturates and then decreases towards low temperatures. This is similar to the behavior of an itinerant ferromagnet and due to the dominant T-linear term in Eq. (4) after saturation of M³⁸. In contrast, in the AF1 phase below H_c1 the ANE strongly deviates from this behavior at temperatures below  ≈ 20 K. We speculate that the reduction of M_R(0) and the simultaneous sign changes of α_yx(H_c1) and S_yx(0) with temperature could arise from the different compensation of opposed Mn moments. In this respect it bears some similarity to the magnetization behavior of garnets or rare-earth transition-metal ferrimagnets below and above the compensation point.

Symmetry analysis

To gain further insights into the low-temperature behavior of Mn₅Si₃, we analyze the symmetries of the anomalous Hall and Nernst effects under a small external magnetic field. The Onsager relations establish strong symmetry requirements for each response tensor. The (thermo-)electric responses measured in our experiments are approximately odd under the reversion of a small external magnetic field (Figs. 3 and 5). Up to second order, the only odd-in-field transverse electric (thermo-electric) responses are the intrinsic and quadratic anomalous Hall (Nernst) effects⁵¹. These response tensors are even under space inversion ${{\mathcal{P}}}$ and odd under the combined time reversal and lattice translation symmetry ${{\mathcal{T}}}{{\boldsymbol{t}}}$^51,52. According to Neumann’s principle, the transport coefficients must also be invariant under all the symmetries of the material. In the next paragraph, we combine the implications of the Onsager relations with Neumann’s principle to investigate the symmetry constraints of the AHE and ANE in the antiferromagnetic phases AF1 and AF2 of Mn₅Si₃ (Fig. 2).

Consider the collinear AF2 phase shown in Fig. 2a. The system is invariant under both ${{\mathcal{P}}}$ and ${{\mathcal{T}}}{{\boldsymbol{t}}}$. Because the response tensors are also invariant under ${{\mathcal{P}}}$, this symmetry operation does not impose any constraint. On the other hand, the ${{\mathcal{T}}}{{\boldsymbol{t}}}$ invariance of the system imposes that the response coefficients be even under time reversal. Since the Onsager relations require the response tensors to be odd under ${{\mathcal{T}}}$, the intrinsic and quadratic anomalous Hall/Nernst effects are prohibited in the AF2 phase, in agreement with our experimental results for 60 K  < T < 100 K [Figs. 3h and 5e, g]^18,19,20. Although a neutron-diffraction study on polycrystalline Mn₅Si₃ in the AF2 phase suggested a spin canting of up to 8^∘ toward the b axis for the spins still confined in the a-b plane due to Dzyaloshinskii-Moriya interaction, this would still preserve ${{\mathcal{P}}}$ and ${{\mathcal{T}}}{{\boldsymbol{t}}}$ symmetry with vanishing Hall and Nernst effects^17,29.

As mentioned above, the magnetic configurations of Mn₅Si₃ for T < 60 K are still under discussion²⁸. The AF1 phases shown in Fig. 2b and c break ${{\mathcal{P}}}$ but (as well as other proposals in the literature) are invariant under ${{\mathcal{T}}}{{\boldsymbol{t}}}$, which allows for a p-wave non-relativistic spin splitting of the energy bands⁵³. However, even though the bands are not spin-degenerate, the ${{\mathcal{T}}}{{\boldsymbol{t}}}$ symmetry prohibits the anomalous responses. We conclude that, due to ${{\mathcal{T}}}{{\boldsymbol{t}}}$ invariance, none of the proposed AF1 phases can explain our experimental signals. This suggests a slight tilting of the magnetic state for T < 60 K, which breaks the combined ${{\mathcal{T}}}{{\boldsymbol{t}}}$ symmetry.

The question is whether this could be caused by a residual strain in the single crystal similar to epitaxially strain prevailing in Mn₅Si₃ films^32,35,36. Strain caused by Cu impurities was presumably responsible for generating additional magnetic phases in Mn₅Si₃ single crystals³¹. Uniaxial residual stress was reported to cause a change of the antiferromagnetic structure of polycrystalline Mn₃Sn from non-collinear and coplanar to non-coplanar⁵⁴. In the present case, where the different phases observed in the electronic transport measurements agree with the results from neutron-diffraction experiments on different single crystals, it is unlikely that the residual stress in the crystal only evolves along the c axis to generate a nonzero AHE and ANE. The samples were not clamped in the ANE device with pressure [Fig. 4c] but only glued to the Cu banks of which the hot end was always freely suspended. ANE measurements shown in Figs. 5 and 6 with the magnetic field along the c axis were done on different samples with c oriented perpendicular to the plane or in the plane of the sample plate, respectively. Moreover, the strong variation of the ANE and the sign change in a small range of temperature around 20 K is even more unlikely as being due to strain. Thus, we believe that residual strain cannot explain the appearance of a finite AHE and ANE in contrast to what is expected from the symmetry analysis.

Possible deviations from the stoichiometry must always be considered. It has been previously reported that large displacements of Mn atoms from their high-symmetry positions lower the structural symmetry in Mn₃SnN or Mn₃GaN resulting in spin canting and allowing the observation of non-vanishing AHE⁵⁵. We cannot completely rule out such effects but emphasize that an x-ray analysis of a pulverized piece from the ingot sample yielded lattice parameters that deviate less than 0.1 % from recently published data of a polycrystal^17,20.

We conclude that the reduction in the magnitude of the anomalous Hall conductivity and sign change of the anomalous Nernst conductivity below T≤20 K hints to either a new phase or a weak rearrangement of the spins due to magnetic frustration or magnetic anisotropies, as pointed out by N. Biniskos et al.²⁸.

Conclusion

The anomalous Nernst effect of Mn₅Si₃ single crystals displays characteristic variations with magnetic field and temperature in agreement with the magnetic-field induced transitions between different antiferromagnetic phases. The anomalous Hall effect, and Nernst effect are sizeable in the the remanent state, i.e., in absence of a magnetic field. Detailed analysis of the low-temperature behavior below 25 K shows that the anomalous Nernst effect, anomalous Hall conductivity, and magnetization in low magnetic fields exhibit an unusual temperature dependence hinting a subtle modification of the magnetic structure in the AF1 phase. Furthermore, the experimentally observed Hall and Nernst effects in the noncollinear AF1 phase are in contrast to a symmetry analysis of the proposed magnetic AF1 structures, according to which these effects should disappear. These results should be taken into account in a refined model of the magnetic structure. While first ab initio calculations of the electronic band structure and Berry curvature have been performed for the collinear magnetic phase observed in thin films^32,33, the behavior of the AHE and ANE in the noncollinear AF1 phase of Mn₅Si₃ at low temperatures is not yet fully understood and requires further investigation.

Methods

Mn₅Si₃ single crystals were obtained by a combined Bridgman and flux-growth technique and were characterized by X-ray diffraction as described earlier²⁰. The single crystals have been polished and oriented by Laue diffraction, Fig. 4a, b. Magnetization data were obtained in a superconducting-quantum-interference device (SQUID) equipped with a 5 T superconducting magnet and in a vibrating sample magnetometer (VSM) with a 12 T magnet. Measurements of the AHE were performed in a physical-property measurement system (PPMS) as described in ref. ²⁰ with 50-μm Pt wires attached to the crystal in an appropriate fashion with conductive silver-epoxy. The ANE was obtained on the same Mn₅Si₃ single crystal of thickness t = 0.6 mm and lengths l_x = 1.2 mm and l_y = 1.4 mm along z, x, and y, respectively. The crystal was mounted between two Cu clamps electrically isolated by 20-μm Kapton foil and a resistor R_H was attached on one clamp serving as a heater generating a temperature difference ΔT_x = T_h − T_c between the hot and cold side, inset Fig. 4c. Both temperatures were determined by using calibrated resistive thermometers. The power of the heater was always adjusted to keep ΔT_x below 10% of the average temperature T₀ = (T_h + T_c)/2. The Nernst voltage ΔV_y was measured along the y direction with a nanovoltmeter. The crystal was mounted with the crystallographic a, b, and c axes oriented parallel to the y, x, and z directions, respectively, with the heat flow Q along the b axis and the magnetic field oriented along the c axis, Fig. 4c.