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.
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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 spintronics1,2. 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 purposes3,4,5,6. 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 zone12,13. 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 EF12,13. 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 Mn3Sn7,10,14 and Mn3Ge8,9,11,15,16 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 materials10,11.
The intermetallic compound Mn5Si3 is a noncollinear antiferromagnet that has gained attention due to unusual thermodynamic and electronic transport phenomena17,18,19,20,21,22. 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 TN1 = 60 K and TN2 = 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.
In the antiferromagnetic AF1 phase below TN1 = 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 dispute26,27,28, 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 anisotropy28,29. 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 TN1 = 60 K, a strong AHE is observed18,19,20. In addition, the existence of a magnetic-field induced intermediate phase AF1’ has been inferred from neutron scattering and electronic transport measurements20,21,25. Finally, at high magnetic fields, a collinear phase with similar properties to the zero-field AF2 phase is stabilized18,20,21,28. 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 not25,28.
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 space33,34. This gives rise to anomalous Hall and Nernst effects experimentally observed in thin epitaxial Mn5Si3 films with a small net magnetization of 20 - 60 mμB/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 level36,37.
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(Hc1) and ΔM(Hc2) at magnetic fields Hc1 and Hc2, 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 MR(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 MR(0) vs. temperature. A remarkable detail is the decrease of MR(0) when cooling below 20 K while ΔM(Hc1) and ΔM(Hc2) continuously increase with decreasing temperature below TN1 = 60 K. Moreover, the absolute value of MR(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.
The phase transitions are clearly observed in the AHE of Mn5Si3, see Fig. 3c–f. At 50 K, below TN1, the AHE strongly changes around zero field and at magnetic fields Hc1 and Hc2. In the collinear AF2 phase the AHE vanishes20. The absolute value of the transverse resistivity ρyx(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 MR(0) remains nonzero.
Figure 3g shows the Hall conductivity \({\sigma }_{{{\rm{xy}}}}={\rho }_{{{\rm{yx}}}}/({\rho }_{{{\rm{xx}}}}^{2}+{\rho }_{{{\rm{yx}}}}^{2})\), where ρxx is the longitudinal resistivity, see inset Fig. 3h. In the magnetically virgin state before sweeping the magnetic field, σxy is very small compared to the remanent state σxy(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 ρyx(0) was already reported earlier when the sample was cooled down in zero field from above TN1 and subsequently polarized in a magnetic field of ± 3 T20.
σxy 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 σxy on the longitudinal conductivity σxx 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 σxy is independent of σxx38,39. In the present case, the single crystals with σxy = 150 Ω−1cm−1 and σxx = 7400 Ω−1cm−1 fall just within this intermediate regime as discussed earlier20,40, similar to the noncollinear antiferromagnets Mn3Ge and Mn3Sn41. From ρxx = 135 μΩcm, a Fermi velocity vF = 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 σxy(0) and σxy(Hc1) plotted in Fig. 3h gradually increase between TN1 = 60 K and ≈ 25 K but then drop to very low values for T ≤ 20 K. We emphasize that σxy(0) completely vanishes for T ≤ 5 K while σxy(Hc1) 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, σxy(0) and σxy(Hc1), decrease toward zero temperature while the magnetization ΔM(Hc1) remains constant, demonstrates that a different mechanism might be at hand. For antiferromagnetic Mn3Sn, a sharp drop of σzx and σyz 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 canting7,41. 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 = QSμ0M × ( − ∇ T) perpendicular to the directions of the magnetization M and temperature gradient − ∇ T, where QS 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 Hz 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 Ey = − Syx ∇ Tx, with Syx= QS μ0Mz.
The Nernst signal Syx is experimentally measured by
where we made use of the fact that Ey = − ΔVy/ly and ∇ Tx = ΔTx/lx.
Figure 5a–d shows Syx vs magnetic field H applied parallel to the c direction and heat flow along the b direction for different temperatures below TN1. At 50 K, the transitions between the different magnetic phases are observed as clear changes of Syx with a hysteresis, very similar to the behavior of the AHE [Fig. 3c–f].
Syx 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 Syx 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), ρyx, and Syx around zero magnetic field are of similar size and continuously decrease from 15 K to 60 K as previously observed for ρyx(T)20. This indicates that the effects of the magnetic field on ρyx and Syx are of similar origin.
A remarkable difference is the sign change of the remanent Syx(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 Syx(0, T) and Syx(Hc1, T), Fig. 5e. Whereas σxy(0) (in phase AF1) and σxy(Hc1) (in phase AF1’) have opposite signs but are of the same magnitude [Fig. 3h], Syx(0) also has the opposite sign of Syx(Hc1) 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 Hc1, confirming that they arise from topology and from the intrinsic Berry-curvature effect like other antiferromagnets with a nontrivial magnetic structure.
Syx(0) rapidly changes sign by cooling to below 20 K, strongly deviating from an approximately linear temperature dependence observed for Syx(Hc1). Both coefficients vanish toward zero temperature due to Nernst’s theorem38. The sign change of Syx(0, T) occurs at the same temperature where we observe a decrease of MR(0), in contrast to ΔM(Hc1) which appears more like a magnetically saturated state at low temperatures, see Fig. 3b. While the vanishing σxy(0) and reduced MR(0) could be considered as indications for the reformation of antiferromagnetic domains, this cannot explain the reappearance of a positive Syx(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
Here, Sxx = ΔVx/ΔTx is the Seebeck coefficient measured in the same setup and plotted in Fig. 5f for different temperatures using σyx = − σxy.
An interesting detail is the magnetic field behavior of Sxx(H) in the inset of Fig. 5f. At magnetic fields below Hc2 the Seebeck coefficient is almost independent of H but precipituously drops at Hc2 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 Hc2 where the magnetic order changes from noncollinear to collinear. It is correlated with the strong change of the magnetoresistance ρxx(H) by ≈ 10 % at Hc2 and T = 45 K20. This is expected from the Mott relation between the thermo-electric and electronic conductivity
where kB is the Boltzmann constant, e the electron charge, and μ the electrochemical potential. It has been shown previously that the Mott relation between σxx and αxx can also be applied for the transverse coefficient αyx12,43,44
The transverse coefficient αyx [Fig. 5g] obtained from Eq. (2) essentially shows the same behavior as Syx(0) [Fig. 5e]. Since \(\left\vert {S}_{{{\rm{xx}}}}{\sigma }_{{{\rm{xy}}}}\right\vert \, \ll \, \left\vert {S}_{{{\rm{yx}}}}{\sigma }_{{{\rm{xx}}}}\right\vert\), αyx is dominated by Syxσxx (Eq. (2)). Hence, the ANE mostly arises from the transverse heat flow (90 %) and the sign change of αyx occurs due to the sign change of Syx.
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 MR(0) = 0.028 μB/f.u. and MR(Hc1) = 0.17 μB/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 ΔTx = Th − Tc. 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 μB/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 Mn3Sn14,41 and Mn3Ge15,41.
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 kB/e = 86 μV/K at room temperature or above10,11,45 and thus follows a universal scaling46, with the exception of UCo0.8Ru0.2Al 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 temperature10,11,45. The similar ratios obtained in the AF1 and AF1’ phases of Mn5Si3 support the correlation between the two properties αyx and σxy that are predominated by intrinsic effects of the Berry curvature in both magnetic phases.
If the Berry curvature Ωn,z(k) is known, the intrinsic contributions to αyx and σxy can be obtained from refs. 12,13,41
\({g}_{n,{{\boldsymbol{k}}}}=({E}_{n,{{\boldsymbol{k}}}}-\mu ){f}_{n,{{\boldsymbol{k}}}}+{k}_{{{\rm{B}}}}T\,{{\rm{\ln }}}\left[1+\exp \left(-\frac{{E}_{n,{{\boldsymbol{k}}}}-\mu }{{k}_{{{\rm{B}}}}T}\right)\right]\)
where k = (kx, ky, kz) is the three-dimensional crystal momentum, En,k is the energy of the band of index n, and fn,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 Syx or αyx 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., σxy(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 nH with temperature and a maximum (minimum) of R0 (nH) around 20 K, see Fig. 5g. At this point, ab initio calculations of the electronic bandstructure and σxy 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 Ga1−xMnxAs 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 Syx was found to be correlated with a sharp maximum in Sxx thus compensating this contribution to the transverse coefficient αyx which did not change sign. In contrast, for Mn5Si3 the sign change occurs for Syx and αyx but not for Sxx. An analysis of the temperature dependence of αxy 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 Sxy and αxy 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 σxy and αxy. 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 αyx have been reported45,49,50. In these cases, however, Sxy did not change its sign with the temperature.
Below TN1 = 60 K, αyx(Hc1) first increases with decreasing temperature together with a concomitant increase of ΔM(Hc1), 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 Hc1 the ANE strongly deviates from this behavior at temperatures below ≈ 20 K. We speculate that the reduction of MR(0) and the simultaneous sign changes of αyx(Hc1) and Syx(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 effects17,29.
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 films32,35,36. 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 polycrystal17,20.
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 films32,33, 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 lx = 1.2 mm and ly = 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 RH was attached on one clamp serving as a heater generating a temperature difference ΔTx = Th − Tc 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 ΔTx below 10% of the average temperature T0 = (Th + Tc)/2. The Nernst voltage ΔVy 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.
Data availability
The data that support the findings of this study are available in the KITopen repository with the identifier https://doi.org/10.35097/XqLocxHPRfSlZYQh.
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Acknowledgements
We thank N. Biniskos for helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC TRR 288 - 422213477 “ElastoQMat” (Projects A08, A09). We acknowledge support by the KIT-Publication Fund of the Karlsruhe Institute of Technology.
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C.S. conceived the experiments and carried out the electronic transport measurements. G.F. measured the magnetization. W.H.C., A.B.H., L. Š., and J. S. contributed to the theory and symmetry analysis. M. M. performed the Laue measurements. T. Wolf grew the single crystals. C.S., W.H.C., M.M. and W.W. analyzed the data and wrote the manuscript.
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Communications Materials thanks Mingquan He, Cheng Song and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: Xiaokang Li and Aldo Isidori. A peer review file is available.
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Sürgers, C., Fischer, G., Campos, W.H. et al. Anomalous Nernst effect in the noncollinear antiferromagnet Mn5Si3. Commun Mater 5, 176 (2024). https://doi.org/10.1038/s43246-024-00617-x
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DOI: https://doi.org/10.1038/s43246-024-00617-x
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