Observation of the anomalous Nernst effect in
alternmagnetic candidate Mn5Si3

Antonin Badura, Warlley Campos, Venkata Bharadwaj, Ismaïla Kounta, Lisa

Michez, Matthieu Petit, Javier Rial, Miina Leiviskä, Vincent Baltz, Filip

Krizek, et al.

► To cite this version:

Antonin Badura, Warlley Campos, Venkata Bharadwaj, Ismaïla Kounta, Lisa Michez, et al. Observation of the anomalous Nernst effect in altermagnetic candidate Mn5Si3. 2024. hal-04523482

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# Observation of the anomalous Nernst effect in altermagnetic candidate Mn5Si3

Antonin Badura, Warlley Campos, Venkata Bharadwaj, Ismaïla Kounta, Lisa Michez, Matthieu Petit, Javier Rial, Miina Leiviskä, Vincent Baltz, Filip Krizek, et al.

► To cite this version:

Antonin Badura, Warlley Campos, Venkata Bharadwaj, Ismaïla Kounta, Lisa Michez, et al. Observation of the anomalous Nernst effect in altermagnetic candidate Mn5Si3. Nature Communications, 2025, 16, pp.7111. 10.1038/s41467-025-62331-7. hal-04523482v2

HAL Id: hal-04523482

https://hal.science/hal-04523482v2

Submitted on 4 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
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<figure><img src="image_1.png" /></figure>

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Observation of the anomalous Nernst effect in altermagnetic candidate Mn₅Si₃

Antonín Badura,¹,²,* Warlley H. Campos,³ Venkata K. Bharadwaj,³ Ismaïla Kounta,⁴ Lisa

Michez,⁴ Matthieu Petit,⁴ Javier Rial,⁵ Miina Leiviskä,⁵ Vincent Baltz,⁵ Filip Krizek,¹ Dominik

Krieger,¹ Jan Zemen,⁶ Sjoerd Telkamp,⁷ Sebastian Sailler,⁸ Michaela Lammel,⁸ Rodrigo

Jaeschke Ubiergo,³ Anna Birk Hellenes,³ Rafael González-Hernández,⁹,³ Jairo Sinova,³,¹⁰

Tomáš Jungwirth,¹,¹¹ Sebastian T. B. Goennenwein,⁸ Libor Šmejkal,³,¹ and Helena Reichlova¹

¹Institute of Physics, Czech Academy of Sciences, Prague, Czechia

²Faculty of Mathematics and Physics, Charles University, Prague, Czechia

³Institute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany

⁴Aix Marseille Univ, CNRS, CINAM, AMUTECH, Marseille, France

⁵Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-SPINTEC, Grenoble, France

⁶Faculty of Electrical Engineering, Czech Technical University, Prague, Czechia

⁷Solid State Physics Laboratory, ETH, Zürich, Switzerland

⁸Department of Physics, University of Konstanz, Konstanz, Germany

⁹Grupo de Investigación en Física Aplicada, Departamento de Física, Universidad del Norte, Barranquilla, Colombia

¹⁰Department of Physics, Texas A&M University, College Station, Texas, USA

¹¹School of Physics and Astronomy, University of Nottingham, Nottingham, United Kingdom

The anomalous Nernst effect generates transverse voltage to the applied thermal gradient in magnetically ordered systems. The effect was previously considered excluded in compensated magnetic materials with collinear ordering. However, in the recently identified class of compensated magnetic materials, dubbed altermagnets, time-reversal symmetry breaking in the electronic band structure makes the presence of the anomalous Nernst effect possible despite the collinear spin arrangement. In this work, we investigate epitaxial Mn₅Si₃ thin films known to be an altermagnetic candidate. We show that the material manifests a sizable anomalous Nernst coefficient despite the small net magnetization of the films. The measured magnitudes of the anomalous Nernst coefficient reach a scale of microVolts per Kelvin. We support our magneto-thermoelectric measurements by density-functional theory calculations of the material's spin-split electronic structure, which allows for the finite Berry curvature in the reciprocal space. Furthermore, we present our calculations of the intrinsic Berry-curvature Nernst conductivity, which agree with our experimental observations.

The anomalous Nernst effect (ANE, see Fig. 1a) was traditionally considered only in ferromagnetic materials. However, recent studies have shown that it can also be observed in non-collinear antiferromagnets, such as Mn₃Sn [1] and Mn₃NiN [2]. In these materials, the ANE can arise from a finite Berry curvature in the momentum space and is allowed by the symmetry of the atomic positions and non-collinear spins of the frustrated magnetic lattice [3–6]. This leads to the presence of a sizable ANE despite the vanishing net magnetization. The potential to enhance the ANE response has renewed interest in this phenomenon both in ferromagnets [7, 8] and antiferromagnets [6, 9], with possibilities for its application in heat harvesting elements or heat flux sensors. Additionally, the presence of the ANE in a material enables the use of novel magnetic imaging microscopy techniques that rely on thermal gradients [10–13]. However, the complex spin arrangement of non-collinear antiferromagnets does not support long spin coherence, and the high ANE magnitude is often observed in noncollinear systems with heavy elements. In the broad family of compensated magnets with a robust unfrustrated collinear antiparallel ordering, the ANE was thought to be absent. In this combined experimental and theoretical work, we demonstrate

that in the recently identified class of altermagnets [14, 15], the spontaneous anomalous Nernst effect can be observed. Our calculations, consistent with experimental findings, show the spontaneous ANE in altermagnet candidate Mn₅Si₃ thin films, which contain light and abundant elements. Our results suggest a new direction for observing and utilizing the ANE within the broad family of collinear compensated magnets.

In the paramagnetic state, Mn₅Si₃ crystallizes in a hexagonal unit cell with the space group P6₃/mcm. There are two non-equivalent sites of manganese atoms in its unit cell: four Mn atoms (Mn1) occupy the Wyckoff position 4d, whereas the remaining six are at the Wyckoff position 6g (Mn2) [16, 17]. Bulk Mn₅Si₃ undergoes two successive first-order transitions into antiferromagnetic phases as established by neutron-diffraction experiments [17–19]: Below 100 K, the crystal symmetry is reduced to orthorhombic accompanied by a collinear antiferromagnetic configuration with a symmetry combining time-reversal and half unit-cell translation. The second transition occurs at 60 K when the magnetic order becomes highly noncollinear and noncoplanar [19].

Due to epitaxial strain at the interface with the substrate, the in-plane lattice parameters of our epitaxial Mn₅Si₃ films exhibit no structural transitions, as evidenced by X-ray diffraction measurements. Consequently, the unit cell remains hexagonal in the tempera-

<footnote>* badura@fzu.cz</footnote>
<figure><img src="image_2.png" /></figure>

# Observation of the anomalous Nernst effect in altermagnetic candidate Mn₅Si₃

Received: 6 June 2025

Accepted: 18 July 2025

Published online: 02 August 2025

<figure><img src="image_3.png" /></figure>

Check for updates

Antonín Badura Ⓔ<fnref n="1" />,²✉, Warlley H. Campos Ⓔ³,⁴, Venkata K. Bharadwaj⁴, Ismaïla Kounta⁵, Lisa Michez⁵, Matthieu Petit Ⓔ⁵, Javier Rial⁶, Miina Leiviskä<fnref n="1" />, Vincent Baltz Ⓔ⁶, Filip Krizek Ⓔ<fnref n="1" />, Dominik Kriegner Ⓔ<fnref n="1" />, Jakub Železný<fnref n="1" />, Jan Zemen Ⓔ⁷, Sjoerd Telkamp Ⓔ⁸, Sebastian Sailler Ⓔ⁹, Michaela Lammel Ⓔ⁹, Rodrigo Jaeschke-Ubiergo Ⓔ⁴, Anna Birk Hellenes Ⓔ⁴, Rafael González-Hernández⁴,¹⁰, Jairo Sinova Ⓔ⁴,¹¹, Tomáš Jungwirth Ⓔ<fnref n="1" />,¹², Sebastian T. B. Goennenwein Ⓔ⁹, Libor Šmejkal Ⓔ<fnref n="1" />,³,⁴,¹³✉ & Helena Reichlova Ⓔ<fnref n="1" />✉

The anomalous Nernst effect generates a voltage transverse to an applied thermal gradient in some magnetically ordered systems. While the effect was considered excluded in compensated magnetic materials with collinear ordering, in the recently identified symmetry-class of altermagnets, the anomalous Nernst effect is possible despite the compensated collinear spin arrangement. In this work, we show that epitaxial Mn₅Si₃ thin films grown on Si manifest an anomalous Nernst effect with a finite spontaneous signal at zero magnetic field despite the vanishing spontaneous magnetization. We attribute this to the previously theoretically predicted and experimentally corroborated altermagnetism of epitaxial Mn₅Si₃ thin films grown on Si. The observed spontaneous anomalous Nernst coefficient reaches the value of 0.26 µV/K with the corresponding spontaneous Nernst conductivity of 0.22 A/(K·m). To complement our measurements, we perform density-functional theory calculations of the momentum-resolved anomalous Nernst conductivity, highlighting the contributions of altermagnetic pseudonodal surfaces and ladder transitions to the Berry curvature. Our results illustrate the value of unconventional d-wave wave altermagnets composed of abundant and non-toxic light elements for thermo-electrics and spin-caloritronics.

The anomalous Nernst effect (ANE, see Fig. 1a) was traditionally considered only in ferromagnetic materials. However, recent studies have shown that it can also be observed in non-collinear antiferromagnets, such as Mn₃Sn<fnref n="1" /> and Mn₃NiN². In these materials, the ANE can arise from

a finite Berry curvature in the momentum space and is allowed by the symmetry of the atomic positions and non-collinear spins of the frustrated magnetic lattice³⁻⁶. This leads to the presence of a sizable ANE despite the vanishing net magnetization. The potential to enhance the

<footnote n="1">¹Institute of Physics, Czech Academy of Sciences, Prague, Czechia. ²Faculty of Mathematics and Physics, Charles University, Prague, Czechia. ³Max Planck Institute for the Physics of Complex Systems, Dresden, Germany. ⁴Institute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany. ⁵Aix Marseille University, CNRS, CINAM, AMUTECH, Marseille, France. ⁶University. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-SPINTEC, Grenoble, France. ⁷Faculty of Electrical Engineering, Czech Technical University in Prague, Prague 6, Czechia. ⁸Solid State Physics Laboratory, ETH, Zürich, Switzerland. ⁹Department of Physics, University of Konstanz, Konstanz, Germany. ¹⁰Departamento de Física, Grupo de Investigación en Física Aplicada, Universidad del Norte, Barranquilla, Colombia. ¹¹Department of Physics, Texas A&M University, College Station, TX, USA. ¹²School of Physics and Astronomy, University of Nottingham, Nottingham, UK. ¹³Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany. ✉e-mail: badura@fzu.cz; lsmejkal@pks.mg.de; reichlh@fzu.cz</footnote>
a

<figure><img src="image_1.png" /></figure>

b

<figure><img src="image_2.png" /></figure>

c

<figure><img src="image_3.png" /></figure>

d

<figure><img src="image_4.png" /></figure>

FIG. 1. The anomalous Nernst effect in Mn₅Si₃. **a** Schematic illustration of the experiment: A longitudinal temperature gradient ∇T induces a spontaneous transverse voltage V_y. **b** A cross-section of a typical sample, i.e. substrate / MnSi buffer layer / Mn₅Si₃ layer, captured by scanning transmission electron microscopy, including crystal structure models in the left panel. **c** Transverse Nernst signal S_{yx} as a function of applied magnetic field for a sample temperature of 216 K. The figure also includes the field dependence of the anomalous Hall conductivity for comparison. **d** *Ab initio* calculations of the band structure of Mn₅Si₃ (the left panel) including the calculated anomalous Nernst conductivities at 58 K and 216 K (the right panel).

ture range of 10–300 K [20, 21]. The expected collinear
magnetic order between ≈ 80 K and ≈ 240 K is con-
sistent with the density-functional theory calculations
and magnetotransport experiments [20]. This particu-
lar crystal and magnetic structure leads to the breaking
of translation and inversion symmetries while preserving
a rotation symmetry connecting sublattices with oppo-
site spins. These non-relativistic spin symmetries cate-
gorize the material as altermagnetic [15]. Experimental
evidence further supports the existence of the altermag-
netic phase below the ordering temperature of ≈ 240 K in
these films by the measured anomalous Hall effect (AHE)
[20, 22] despite a negligible net remanent magnetization.
The AHE response in Mn₅Si₃ epilayers is spontaneous,
i.e., it is sizable even in the absence of the magnetic field.

It was ascribed to the altermagnetic mechanism of the
time-reversal symmetry breaking in the electronic struc-
ture. The non-relativistic spin-split electronic structure
is anisotropic in the momentum space with the d-wave
symmetry [20]. Lowering the symmetry by the relativis-
tic spin-orbit coupling then allows for a non-zero Berry
curvature in the momentum space. Correspondingly, the
AHE can be detected in measurements of the (0001)-
oriented Mn₅Si₃ films, except for the case of the Néel
vector aligned with the [0001]-axis for which the inte-
gral of the Berry curvature vanishes by symmetry, or for
the Néel vector within the (0001)-plane, (2110)-plane or
(0110)-plane for which the Hall vector, if allowed, is con-
strained by symmetry to the (0001)-plane of the thin
film. The ANE, linked to the AHE by the Mott rela-
<figure><img src="image_4.png" /><figcaption>b</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>c</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>d</figcaption></figure>

<figure><img src="image_7.png" /></figure>

**Fig. 1** | The anomalous Nernst effect in Mn₅Si₃. a Schematic illustration of the experiment: A longitudinal temperature gradient ∇T induces a spontaneous transverse voltage V_y. b A cross-section of a typical sample, i.e., substrate/MnSi buffer layer/Mn₅Si₃ layer, captured by scanning transmission electron microscopy, including crystal structure models in the left panel. c Transverse Nernst signal S_yx as

a function of applied magnetic field for a sample temperature of 216 K. The figure also includes the field dependence of the anomalous Hall conductivity for comparison. d Ab initio calculations of the band structure of Mn₅Si₃ (the left panel) including the calculated anomalous Nernst conductivities at 58 K and 216 K (the right panel).

ANE response has renewed interest in this phenomenon both in ferromagnets<sup>7,8</sup> and antiferromagnets<sup>6,9</sup>, with possibilities for its application in heat harvesting elements or heat flux sensors. Furthermore, the presence of ANE in a material enables the use of novel magnetic imaging microscopy techniques that rely on thermal gradients<sup>10–13</sup>. However, the complex spin arrangement of non-collinear antiferromagnets does not support long spin coherence, and the high ANE magnitude is often observed in non-collinear systems with heavy elements. In the broad family of compensated magnets with an unfrustrated collinear antiparallel ordering, the ANE was thought to be absent. This understanding should be revised because of the recently identified class of altermagnets, which combine collinear compensated magnetic order with anisotropic spin splitting in their electronic structure<sup>14,15</sup>. Epitaxial thin films of Mn₅Si₃ have recently been recognized as a candidate representative of this class<sup>16,17</sup>.

In the paramagnetic state, Mn₅Si₃ crystallizes in a hexagonal unit cell with the space group *P*6₃/*mcm*. There are two non-equivalent sites of manganese atoms in its unit cell: four Mn atoms (Mn1) occupy the Wyckoff position 4d, while the remaining six are at the Wyckoff position 6g (Mn2)<sup>18,19</sup>. Upon the onset of magnetic order the behavior of bulk and thin films differ. While bulk Mn₅Si₃ undergoes two successive first-order transitions into antiferromagnetic phases, epitaxial strain causes an altermagnetic order in thin films<sup>16,20–22</sup>.

For bulk Mn₅Si₃, neutron-diffraction experiments<sup>19,23,24</sup> establish that below 100 K, the crystal symmetry is reduced to orthorhombic accompanied by a collinear antiferromagnetic configuration with a symmetry combining time-reversal and half unit-cell translation, which forbids the ANE<sup>25</sup>. The second transition occurs at 60 K when the magnetic order becomes highly noncollinear and noncoplanar<sup>24</sup>. The presence of the ANE in this low-temperature phase is under discussion<sup>25</sup>.

Due to epitaxial strain, epitaxial Mn₅Si₃ thin films exhibit no structural transitions, as evidenced by X-ray diffraction measurements. Consequently, the unit cell remains hexagonal in the temperature range of 10–300 K<sup>16,17</sup>. This particular crystal structure in combination with a compensated collinear order between (90 ± 30) K and (240 ± 30) K<sup>16,20–22</sup> leads to the breaking of translation and inversion symmetries while preserving a rotation symmetry connecting sublattices with opposite spins. These non-relativistic spin symmetries categorize the material as altermagnetic<sup>15</sup>. Although direct measurements of spin structure in thin films of compensated magnets are notoriously difficult, indirect experimental evidence supports the existence of the altermagnetic phase in these films by the measured anomalous Hall effect (AHE)<sup>16,22</sup> despite a negligible net remanent magnetization. The proposed magnetic structure of Mn₅Si₃ films is consistent with the previous experimental studies<sup>16,20,22</sup>, including a recent work demonstrating that this structure is a key ingredient in
tion, obeys analogous symmetry requirements, and its
existence is thus allowed.

ANOMALOUS NERNST EFFECT IN Mn₅Si₃
FILMS

For our experiments, we use 20nm Mn₅Si₃(0001) films grown by molecular-beam epitaxy on intrinsic Si(111) substrates. Fig. 1b shows a section of our Mn₅Si₃ epi-layer along its [0001] direction as captured by scanning transmission electron microscopy (STEM). The STEM image displays the high quality of our film and its inter-face, as well as the buffer layer of MnSi, which forms at the interface after annealing. See Methods and Ref. [21] for more details about the layer growth and character-ization. Our samples exhibit the spontaneous AHE, as shown in Fig. 1c (black line).

To measure the ANE in our samples, we generate an in-plane temperature gradient along the [2\bar{1}10] direction by a macroscopic heater as illustrated in Fig. 2a: The sample is supported by a plastic block with a resistive heater on one side and by a brass block on the other side [23]. The different thermal conductivities of the plastic and brass blocks enhance the temperature gradient generated by the heater. To detect the thermovoltage generated in the Mn₅Si₃ layer, we lithographically defined Mn₅Si₃ strip structures, as shown in the detail of our design in Extended Data Fig. E1. We detect the thermovoltage *V*<sub>y</sub> transverse to the gradient direction as a function of the magnetic field applied along the z-axis (film's c-direction) as illustrated in Fig. 1a. An example of such a field-dependent transverse thermopower *S*<sub>yx</sub> is shown in Fig. 1c for a sample average temperature of 216 K. *S*<sub>yx</sub> shows clear hysteretic behaviour and saturation, a signature of the finite ANE in our samples. The saturation field and coercivity of *S*<sub>yx</sub> match the field dependence of the transverse conductivity *σ*<sub>yx</sub> shown in the same panel. The experimental manifestation of ANE is supported by our *ab initio* calculations of the anomalous Nernst conductivity *α*<sub>yx</sub>. The energy dependence of *α*<sub>yx</sub> for temperature 58 K and 216 K is shown in Fig. 1d next to the spin-resolved electronic band structure of alternative Mn₅Si₃. It reaches the value of 0.25 A/(K·m) at the Fermi energy for 58 K.

A crucial step in quantifying the Nernst coefficient $S_{xy}$ is to determine the temperature gradient in the sample precisely. To do so, we measure the temperature at two places on the sample by deposited platinum strips. The precise spatial temperature distribution is simulated by the finite-element method in COMSOL MULTIPHYSICS [24], where we used realistic geometry of our experimental setup and adjusted the surface heat transfer to fit the measured sample temperatures. The result is shown as a colour map in Fig. 2a for the average sample temperature of 121 K. An example of the temperature gradient profile determined by the simulation is shown in Extended Data Fig. E1: The temperature gradient is linear in the

central patterned part of the sample and deviates from
the linearity at the edges. The magnitude of the simu-
lated thermal gradient along the x-axis varied between
0.04 K/mm at 58 K and 0.93 K/mm at 216 K.

Fig. 2b presents $S_{yx}$ measured at four different sample temperatures between 58 K and 216 K. We take into account only the odd-in-field part of the $S_{yx}$ field dependence to remove the effect of the temperature gradient not being perfectly perpendicular to the lithographic bars. The figure also features the field dependence of the transverse conductivity $\sigma_{yx}$ measured on the same sample at the same temperatures. Both $S_{yx}$ and $\sigma_{yx}$ are presented without the linear-in-field contribution of the ordinary Nernst effect and the ordinary Hall effect, respectively. The magnitude of the saturated $S_{yx}$ is strongly temperature dependent: It changes from $-(3 \pm 2) \mu\text{V/K}$ at 58 K to $-(0.04 \pm 0.01) \mu\text{V/K}$ at 216 K. In contrast, the saturated $\sigma_{yx}$ only slightly decreases from $(10.6 \pm 0.1) \text{ S/cm}$ to $(5.91 \pm 0.02) \text{ S/cm}$. We also measured the field dependence of the Seebeck coefficient $S_{xx}$. This allows us to determine the spontaneous Nernst conductivity $\alpha_{yx}$, which varies from $\alpha_{yx} = (0.11 \pm 0.08) \text{ A/(K·m)}$ at 58 K to $\alpha_{yx} = (0.015 \pm 0.005) \text{ A/(K·m)}$ at 216 K.

Our Mn₅Si₃ samples show a sizable ANE despite their
small magnetization as illustrated in Fig. 2c: The fig-
ure shows the magnetic-field dependence of the out-
of-plane component of the magnetization measured by
SQUID magnetometry at different temperatures. The
main figure corresponds to the spontaneous magnetiza-
tion of our Mn₅Si₃ after subtracting the saturating con-
tribution of the substrate and the sample holder. The
raw data of the same measurements are given in the
inset. layer. We present the data without the domi-
nating diamagnetic signal of the silicon substrate. The
data show a clear hysteretic behaviour with the satura-
tion value of (45 ± 20) mμ<sub>B</sub>/f.u. at 58 K, which decreases
to (10 ± 8) mμ<sub>B</sub>/f.u. at 216 K. The field dependence of
the magnetization suggests that the Néel vector reversal
is accompanied by small canting of the magnetic sub-
lattice moments. The observed spontaneous moment of
(20 ± 10) mμ<sub>B</sub>/f.u. at 58 K is of the same order as the
value 40 mμ<sub>B</sub>/f.u. obtained from our *ab initio* calcula-
tions.

To quantitatively understand the origin of the observed ANE, we performed first-principles calculations of the Nernst conductivity $\alpha_{yx}$. The electronic band structure (left panel in Fig. 1d) shows a large anisotropic spin-splitting, the hallmark of altermagnets, together with a spin-mixing induced by spin-orbit coupling. This combination allows for a finite anomalous Hall conductivity in Mn₅Si₃ [20] and for a finite ANE. In the right panel of Fig. 1d, we show the calculated anomalous Nernst conductivity $\alpha_{yx}$ for different temperatures as a function of the chemical potential. The Nernst conductivity calculations yield 0.25 A/(K·m) for 58 K and 0.12 A/(K·m) for 216 K, which agrees with the order of magnitude determined by the experiment.

$S_{yx}$ shows a complex field-dependence compared to $\sigma_{yx}$
explaining the characteristic anisotropy of angular-dependent mag-
neto-transport²¹.

The AHE response in Mn₅Si₃ epilayers is spontaneous, i.e., it is
sizable even in the absence of the magnetic field. It corroborates the
alternmagnetic mechanism of the time-reversal symmetry breaking in
the electronic structure. The non-relativistic spin-split electronic
structure is anisotropic in the momentum space with the d-wave
symmetry¹⁶. Lowering the symmetry by the relativistic spin-orbit
coupling then allows for a non-zero Berry curvature in the momen-
tum space. Correspondingly, the AHE can be detected in measure-
ments of the (0001)-oriented Mn₅Si₃ films, except for the case of the
Néel vector aligned with the [0001]-axis for which the integral of the
Berry curvature vanishes by symmetry, or for the Néel vector within
the (0001)-plane, (2\110)-plane or (0\110)-plane for which the Hall
vector, if allowed, is constrained by symmetry to the (0001)-plane of
the thin film¹⁶.

In this combined experimental and theoretical work, we demon-
strate that the spontaneous anomalous Nernst effect can also be
observed in altermagnetic materials. Our calculations, in agreement
with experimental results, reveal a spontaneous ANE in the alter-
magnetic candidate Mn₅Si₃ thin films, composed of light and earth-
abundant elements. Our results suggest a new direction for observing
and utilizing the ANE within the broad family of altermagnets, which
also includes semiconductors that are favorable conduction types for
thermoelectric.

Results

For our experiments, we use 20 nm Mn₅Si₃(0001) films grown by
molecular-beam epitaxy on intrinsic Si(111) substrates. Figure 1b shows
a section of our Mn₅Si₃ epilayer along its [0001] direction as captured
by scanning transmission electron microscopy (STEM). The STEM
image displays the high quality of our film and its interface, as well as
the 5-nm buffer layer of MnSi, which forms at the interface after
annealing. The hexagonal crystal symmetry in magnetically ordered
phase, obeying the altermagnetic symmetry requirements was con-
firmed by the X-ray diffraction measurements of lattice parameters¹⁶.
Measurement of the temperature-dependent resistivity (see Supple-
mentary Fig. S1a) shows characteristic features reported in Mn₅Si₃
epitaxial films¹⁶. Our samples exhibit the spontaneous AHE, as shown in
Fig. 1c (black line). The spontaneous AHE is present in a wide tem-
perature range as illustrated in Supplementary Figs. S1b and S1d with a
characteristic feature below 120 K which was ascribed to the topolo-
gical Hall effect arising from the non-collinear magnetic order¹⁶,²⁶. See
Methods and ref. 17 for more details about the layer growth and
characterization.

To measure the ANE in our samples, we generate an in-plane
temperature gradient along the [2\110] direction by a macroscopic
heater as illustrated in Fig. 2a: The sample is supported by a plastic
block with a resistive heater on one side and by a brass block on the
other side²⁷. The different thermal conductivities of the plastic and
brass blocks enhance the temperature gradient generated by the
heater. To detect the thermovoltage generated in the Mn₅Si₃ layer, we
lithographically defined Mn₅Si₃ strip structures, as shown in the detail
of our design in Supplementary Fig. S2a. We detect the thermovoltage
V<sub>y</sub> transverse to the gradient direction as a function of the magnetic
field applied along the z-axis (film's c-direction) as illustrated in Fig. 1a.
An example of such a field-dependent transverse thermopower S<sub>yx</sub> is
shown in Fig. 1c for a sample average temperature of 216 K. S<sub>yx</sub> shows
clear hysteretic behavior and saturation, a signature of the finite ANE in
our samples. The saturation field and coercivity of S<sub>yx</sub> match the field
dependence of the transverse conductivity σ<sub>yx</sub> shown in the same
panel. The experimental manifestation of ANE is supported by our ab
initio calculations of the anomalous Nernst conductivity α<sub>yx</sub>. The
energy dependence of α<sub>yx</sub> for temperature 58 K and 216 K is shown in
Fig. 1d next to the spin-resolved electronic band structure of

alternmagnetic Mn₅Si₃. It reaches the value of 0.25 A/(K·m) at the Fermi
energy for 58 K.

A crucial step in quantifying the Nernst coefficient S<sub>xy</sub> is to
determine the temperature gradient in the sample precisely. To do so,
we measure the temperature at two places on the sample by deposited
platinum strips. The precise spatial temperature distribution is simu-
lated by the finite-element method in COMSOL MULTIPHYSICS²⁸,
where we used realistic geometry of our experimental setup and
adjusted the surface heat transfer to fit the measured sample tem-
peratures. The result is shown as a color map in Fig. 2a for the average
sample temperature of 121 K. An example of the temperature gradient
profile determined by the simulation is shown in Supplementary
Fig. S2a: The temperature gradient is linear in the central patterned
part of the sample and deviates from the linearity at the edges. The
magnitude of the simulated thermal gradient along the x-axis varied
between 0.04 K/mm at 58 K and 0.93 K/mm at 216 K.

Figure 2b presents S<sub>yx</sub> measured at five different sample tem-
peratures between 58 K and 176 K, complementing the 216 K mea-
surement in Fig. 1c. We take into account only the odd-in-field part of
the S<sub>yx</sub> field dependence to remove the effect of the temperature
gradient not being perfectly perpendicular to the lithographic bars.
The figure also features the field dependence of the transverse con-
ductivity σ<sub>yx</sub> measured on the same sample at the same temperatures.
Both S<sub>yx</sub> and σ<sub>yx</sub> are presented without the linear-in-field contribution
of the ordinary Nernst effect and the ordinary Hall effect, respectively.
The magnitudes of the saturated and zero-field (spontaneous) S<sub>yx</sub> are
summarized in Supplementary Fig. S4a. Both quantities are strongly
temperature dependent: Saturated S<sub>yx</sub> drops from -(2.8 ± 1.1) μV/K at
58 K to -(0.03 ± 0.01) μV/K at 216 K, and spontaneous S<sub>yx</sub> changes
from -(0.26 ± 0.10) μV/K at 58 K to -(0.03 ± 0.01) μV/K at 216 K. In
contrast, the saturated σ<sub>yx</sub> only slightly decreases from (10.6 ± 0.1) S/
cm to (5.91 ± 0.02) S/cm. We also measured the field dependence of
the Seebeck coefficient S<sub>xx</sub>. This allows us to determine the sponta-
neous Nernst conductivity α<sub>yx</sub>, which varies from |α<sub>yx</sub>| = (0.22 ± 0.11) A/
(K · m) at 58 K to |α<sub>yx</sub>| = (0.04 ± 0.01) A/(K · m) at 216 K.

Our Mn₅Si₃ samples show a sizable ANE despite their small mag-
netization as illustrated in Fig. 2c: The figure shows the magnetic-field
dependence of the out-of-plane component of the magnetization
measured by SQUID magnetometry at different temperatures. The
data corresponds to the spontaneous magnetization of our Mn₅Si₃
after subtracting the saturating contribution of the substrate and the
sample holder. The raw data of the same measurements are given in
Fig. 2d. We present the data without the dominating diamagnetic
signal of the silicon substrate. The data show a clear hysteretic beha-
vior with the saturation value of (45 ± 20) mµ<sub>B</sub>/f.u. at 58 K, which
decreases to (10 ± 8) mµ<sub>B</sub>/f.u. at 216 K. The field dependence of the
magnetization suggests that the Néel vector reversal is accompanied
by small canting of the magnetic sublattice moments. We present an
extended dataset of the magnetization curves in Extended Data Fig. 1c
which highlights the vanishing hysteresis at 300 K. The observed
spontaneous moment of (20 ± 10) mµ<sub>B</sub>/f.u. at 58 K is of the same order
as the value 40 mµ<sub>B</sub>/f.u. obtained from our ab initio calculations.

To quantitatively understand the origin of the observed ANE, we
performed first-principles calculations of the Nernst conductivity α<sub>yx</sub>.
The electronic band structure (left panel in Fig. 1d) shows a large ani-
rotropic spin-splitting, the hallmark of altermagnets, together with a
spin-mixing induced by spin-orbit coupling. In Fig. 3a, we show the
relativistic Fermi surface for the k<sub>z</sub> = 0 plane (for the non-relativistic
Fermi surface, see Supplementary Fig. S9a). The avoided band cross-
ings along the Γ - K and Γ - M paths are induced by spin-orbit coupling
(SOC) as illustrated in Fig. 3b for -K - Γ - K, showing the momentum-
dependent band spin-splitting.

We have studied the momentum dependence of the Berry cur-
vature and momentum-resolved anomalous Nernst conductivity
(ANC) in altermagnetic Mn₅Si₃ from first principles (see Methods). We
<figure><img src="image_5.png" /></figure>

FIG. 2. Temperature dependence of the anomalous Nernst effect in Mn₅Si₃. **a** Detailed image of the experimental setup: The longitudinal temperature gradient in the sample is induced by heating up the sample with a platinum resistive heater. The colour map of the sample temperature for a particular setting was calculated by a finite-element simulation. **b** Magnetic-field dependence of the anomalous Nernst signal *S*<sub>yx</sub> for four average sample temperatures. Each figure shows the anomalous Hall conductivity measured at the same temperature for comparison. **c** Dependence of the detected out-of-plane magnetization on the applied magnetic field for different sample temperatures. The main figure features the spontaneous magnetization of our Mn₅Si₃ layers after subtracting a saturating contribution from the substrate, whereas the raw data are given in the inset. In the presented data, the dominating diamagnetic signal of the silicon substrate has been subtracted.

as illustrated in Fig. 2b: The anomalous Nernst and Hall
dependencies are nearly identical above 170 K, while the
low-temperature *S*<sub>yx</sub> have pronounced sigmoid charac-
ter. This leads us to the hypothesis that the *S*<sub>yx</sub> sig-
nal is composed of at least two competing contributions
with different temperature dependencies. To disentan-
gle them, we analytically describe the low-temperature
sigmoid character of the *S*<sub>yx</sub> curves by an error func-
tion. When the sigmoid contribution is subtracted, only
a spontaneous part of *S*<sub>yx</sub> prevails with the identical field
dependence as σ<sub>yx</sub>. The analysis is robust in the whole
temperature range, always giving spontaneous *S*<sub>yx</sub> pro-
portional to σ<sub>yx</sub>. This analysis strategy is illustrated
in Extended Data Fig. E2. The sigmoid contribution
vanishes above 170 K, as quantitatively shown in Ex-
tended Data Fig. E2c. We attribute the spontaneous

component of S<sub>yx</sub> to be a direct consequence of the al-
termagnetic spin-split electronic structure of our Mn<sub>5</sub>Si<sub>3</sub>
epilayers, analogous to our previous reports of the AHE
in the material [20]. The interpretation of the sigmoid
contribution is less clear. It cannot be attributed to the
Brillouin magnetism which would not be expected up to
100 K. Because it is present only in the lower temperature
range it may be related to the complex low-temperature
magnetic phase of Mn<sub>5</sub>Si<sub>3</sub> [17–19] where the topological
Hall effect was reported [20, 25]. However, a systematic
study of the low-temperature sigmoid contribution is be-
yond the scope of this manuscript.

The separation of the spontaneous and sigmoid S<sub>yx</sub>
contributions enables us to quantify the temperature de-
pendence of the S<sub>yx</sub> coercive field and compare it with
σ<sub>yx</sub> as shown in Fig. 3b. Both signals show the same
<figure><img src="image_8.png" /><figcaption>Fig. 2 | Temperature dependence of the anomalous Nernst effect in Mn₅Si₃.</figcaption></figure>

a Detailed image of the experimental setup: The longitudinal temperature gradient in the sample is induced by heating up the sample with a platinum resistive heater. The color map of the sample temperature for a particular setting was calculated by a finite-element simulation. b Magnetic-field dependence of the anomalous Nernst signal $S_{yx}$ for five average sample temperatures. Each figure shows the anomalous Hall conductivity measured at the same temperature for comparison.

c Dependence of the detected out-of-plane magnetization on the applied magnetic field for different sample temperatures. The figure features the spontaneous magnetization of our Mn₅Si₃ layers after subtracting a saturating contribution from the substrate. In the presented data, the dominating diamagnetic signal of the silicon substrate has been subtracted. d The raw data of magnetization in panel c, including a saturating contribution from the substrate.

have identified that the leading characteristic contributions to the effects can be best illustrated on the $k_z = 0$ plane shown in Fig. 3a, c, d. Contributions to the Berry curvature were analyzed in an earlier theoretical study of the d-wave altermagnetic phase of RuO₂. Prominent contributions were found to originate from Weyl points at crossings between bands of the same spin, pseudonodal surfaces whose spin degeneracy is protected by the altermagnetic spin symmetries in the absence of SOC and is lifted by SOC, and ladder transitions from regions where a very small gap between spin-up and spin-down bands exists without SOC²⁹. We emphasize that the presence of spin-flip pseudonodal surfaces distinguishes altermagnets from other magnetic systems, such as ferromagnets and non-collinear antiferromagnetic. As illustrated in Fig. 3c, our calculations show that the main contribution to the Berry curvature in altermagnetic Mn₅Si₃ comes from the pseudonodal surfaces and ladder transitions between altermagnetic split bands. (Note that Weyl points are found to be absent near the Fermi energy in altermagnetic Mn₅Si₃). In Fig. 3d, we show the corresponding momentum-resolved ANC for the

temperature of 58 K with highlighted pseudonodal surfaces (green arrows) and ladder transitions (purple arrows). The ANC-momentum-space distributions calculated at 121 K, 176 K, and 216 K are shown in Supplementary Fig. S9b–d. The reciprocal space dependence of the ANC is computed via Eq. (4), without performing the integration over the Brillouin zone (in Eq. (3)).

In the right panel of Fig. 1d, we show the integrated anomalous Nernst conductivity $\alpha_{yx}$ values for different temperatures as a function of the chemical potential. The Nernst conductivity calculations yield 0.25 A/(K·m) for 58 K and 0.12 A/(K·m) for 216 K. The temperature dependencies of calculated and measured $\alpha_{yx}$ are compared in the Supplementary Fig. S4b. Both dependencies decrease with increasing temperature. The low-temperature experimental values agree with the calculated ones within the indicated uncertainty. The high-temperature values differ by a factor of two. This discrepancy might be caused by the slight shift of the expected position of the Fermi energy (see Supplementary Fig. S8) or by the possible phononic contributions which were not taken into account in the calculations.
<figure><img src="image_6.png" /><figcaption>FIG. 3. Decomposition of the anomalous Nernst signal. a The anomalous Nernst signal measured at 121 K decomposed into its spontaneous and sigmoid contributions. The anomalous Hall signal is also plotted for comparison. b Temperature dependence of the coercive field of the spontaneous ANE contribution together with the respective dependence for the AHE.</figcaption></figure>

increasing temperature dependence of coercivity. In a ferromagnet, the common expectation is that the coercive (reorientation) field decreases with increasing temperature, usually because of the decreasing magnetocrystalline anisotropy [26, 27]. The situation in compensated magnetic materials is, however, more complex: for example, the observed increase of the reorientation (spin-flop) field in collinear MnF₂ was ascribed to its temperature-dependent and anisotropic response to the applied magnetic field [28]. In our Mn₅Si₃ thin films, however, we do not observe any reorientation transition in the reachable magnetic field (see Fig. 2c). Instead, we expect that the hysteresis in the Néel vector reversal originates from the competition of the magnetocrystalline anisotropy and the Dzyaloshinskii-Moriya interaction. These effects have generally different temperature dependencies, which can result in an increasing coercivity with temperature. The temperature dependence of the magnetocrystalline anisotropy may be affected by the complex temperature dependence of the lattice parameters [20].

## DISCUSSION

We have demonstrated a significant and spontaneous ANE in an altermagnetic candidate composed of light elements. Fig. 4 presents a comparison of the ANE coefficients observed in various magnetic compounds depending on their magnetization per formula unit, including ferromagnets (hollow circles), non-collinear antiferromagnets (full circles), and our altermagnetic Mn₅Si₃ (crossed circle). Our material showcases ANE values comparable to those of other materials, despite having vanishing magnetization, collinear magnetic order, and being constituted of light elements.

Although we observe finite but small saturation magnetization of 45 mμ<sub>B</sub>/f.u., let us emphasise that it is not the prime source of AHE and ANE as shown by the first-principles calculations. Instead, the AHE and ANE are dominated by the strong altermagnetic breaking of the time-reversal symmetry in the electronic structure in the absence of a net magnetization, combined with the relativistic spin-orbit coupling. An absence of a correlation between the atomic mass and the ANE is highlighted in Fig. 4 where the symbols' size linearly decreases with the highest atomic number of the compound (the largest symbols represent the lightest elements). This observation highlights the importance of not limiting the search for high ANE performance materials solely to heavy elements. Additionally, we note that, unlike many materials listed in Fig. 4, our Mn₅Si₃ is in thin film form, which often reduces the ANE magnitude. Therefore, we anticipate that higher ANE values could be discovered in altermagnetic bulks.

## CONCLUSION

In summary, we provide experimental evidence that altermagnetic materials can host a spontaneous ANE despite their collinear magnetic order and vanishing net magnetization, and without the need of heavy elements. We demonstrate a sizable anomalous Nernst coefficient in the altermagnetic candidate Mn₅Si₃ in the form of epitaxial thin layers. The effect reaches $S_{yx} = -(3 \pm 2) \mu\text{V/K}$ and spontaneous $\alpha_{yx} = (0.11 \pm 0.08) \text{ A/(K} \cdot \text{m)}$ at 58 K, and drops to $S_{yx} = -(0.04 \pm 0.01) \mu\text{V/K}$ and spontaneous $\alpha_{yx} = (0.015 \pm 0.005) \text{ A/(K} \cdot \text{m)}$ at 216 K. The Nernst effect measurements are supported by our first-principles calculations of the Mn₅Si₃ electronic band structure and the Nernst conductivity, which yield $\alpha_{yx} = 0.25 \text{ A/(K} \cdot \text{m)}$ at 58 K.

By experimentally demonstrating the ANE in the collinear altermagnets, we introduce a new category of compensated magnetic materials for exploring the ANE. This expands the field beyond the more complex non-collinear magnetic structures or materials with heavy elements, marking a significant broadening of the scope for the ANE research. The search for materials with high and robust ANE magnitude is one of the key challenges
<figure><img src="image_9.png" /></figure>

**Fig. 3** | Relativistic Fermi surface with momentum-resolved Berry curvature and anomalous Nernst conductivity in the $k_z = 0$ plane. a Spin-resolved relativistic Fermi surface with gray solid lines outlining the first Brillouin zone. The color map shows the spin texture of the Fermi surface, with positive (negative) $\langle S_z \rangle$ values shown in red (blue). b SOC-split electronic band structure along the high-symmetry path –K–Γ–K. c z-component of the Berry curvature distribution with the color map showing positive (negative) values in red (blue). The black solid lines show the

relativistic Fermi surface, and the green dotted line highlights the high-symmetry path used in (b). d Distribution of the momentum-resolved anomalous Nernst conductivity for the temperature 58 K, with positive (negative) values in red (blue). The green arrows show the contributions coming from the SOC-split pseudo-nodal surfaces and the purple arrows indicate contributions coming from the ladder transitions between altermagnetic-split energy bands.

$S_{yx}$ shows a complex field-dependence compared to $\sigma_{yx}$ as illustrated in Fig. 2b: The anomalous Nernst and Hall dependencies are nearly identical above 140 K, while the low-temperature $S_{yx}$ have pronounced sigmoid character. This leads us to the hypothesis that the $S_{yx}$ signal is composed of at least two competing contributions with different temperature dependencies. To disentangle them, we analytically describe the low-temperature sigmoid character of the $S_{yx}$ curves by an error function. When the sigmoid contribution is subtracted, only a spontaneous part of $S_{yx}$ prevails with the identical field dependence as $\sigma_{yx}$. The analysis is robust in the whole temperature range, always giving spontaneous $S_{yx}$ proportional to $\sigma_{yx}$. This analysis strategy is illustrated in Supplementary Fig. S3. The sigmoid contribution vanishes above 170 K, as quantitatively shown in Supplementary Fig. S4a. We attribute the spontaneous component of $S_{yx}$ to be a direct

consequence of the altermagnetic spin-split electronic structure of our Mn₅Si₃ epilayers, analogous to our previous reports of the AHE in the material¹⁶. The interpretation of the sigmoid contribution is less clear. It cannot be attributed to the Brillouin magnetism which would not be expected up to 100 K. Because it is present only in the lower temperature range it may be related to the complex low-temperature magnetic phase of Mn₅Si₃¹⁹,²³,²⁴ where the topological Hall effect was reported¹⁶,²⁶. However, a systematic study of the low-temperature phase of Mn₅Si₃³⁰ including the sigmoid contribution to the ANE and comparing the low-temperature thin-film and bulk phases²⁵ is an important future task.

The separation of the spontaneous and sigmoid $S_{yx}$ contributions enables us to quantify the temperature dependence of the $S_{yx}$ coercive field and compare it with $\sigma_{yx}$ as shown in Fig. 4b. Both signals show the
<figure><img src="image_7.png" /><figcaption>FIG. 4. Comparison of the anomalous Nernst effect measured in epitaxial Mn₅Si₃ and in other materials [1, 2, 7, 9, 23, 29-37]. The ANE magnitude *S*<sub>xy</sub> is captured as a function of materials' magnetization. The symbol size linearly decreases with the highest atomic number of materials' formula. Antiferromagnetic materials are indicated by full circles, while ferromagnets are denoted by hollow circles and the altermagnet by a crossed circle. Materials showing the spontaneous Nernst effect are labelled in bold font.</figcaption></figure>

for fully exploiting their application potential.

## METHODS

a. *Sample preparation* The Mn₅Si₃ thin films were grown by molecular beam epitaxy (MBE) in an ultra-high vacuum system with a base pressure better than 2·10⁻⁷ Pa. Elemental Mn and Si were flux matched in a 5:3 ratio using a quartz microbalance and co-deposited on the intrinsic Si(111) substrate (*R* > 10000 Ω·cm) heated to 170°C. The films were subsequently annealed up to ≈ 300°C, the temperature for which a clear (√3 × √3)*R*30° reconstruction is observed on the RHEED pattern. Ex situ X-ray diffraction (XRD) using a high brilliance rotating anode showed a good crystalline quality of the Mn₅Si₃(0001) thin films whose growth was promoted by the formation of a thin MnSi(111). A more detailed description of the growth process can be found in [21].

For the measurement of the anomalous Nernst effect, we used optical lithography and argon ion beam etching to pattern our Mn₅Si₃ layers into two distinct lithographic designs illustrated in the Extended Data Fig. E1. The platinum thermometers of ≈ 50 nm thickness were deposited by magnetron sputtering.

b. *Magnetometry measurements* Magnetic characterization of our Mn₅Si₃ thin films was carried out using a SQUID MPMS XL 5.0 magnetometer. The as-grown sample was cleaned with acetone, alcohol, and DI water, before measurement and mounted with two dots of

glue (X60) on a sample holder made of two quartz capillaries. Field-dependent magnetization was measured at different temperatures for magnetic field strengths ranging from ±4 T applied along the film normal. The signal is dominated by the diamagnetism of the silicon substrate, which accounts for 96% of the signal at 4 T. After eliminating the linear-in-field diamagnetic contribution, the magnetic response contains two components: a non-hysteretic part linked to the substrate and sample holder and a hysteretic part that we attribute to the Mn₅Si₃ film, due to a canting of the moments of the two sublattices.

c. *High-resolution scanning transmission electron microscopy imaging and lamellae preparation* The cross-sectional lamellae of selected samples were prepared by Focused Ion Beam (FIB) (Helios 5 UX from Thermo Fisher Scientific) using AutoTEM 5 software (Thermo Scientific, the Netherlands) at ScopeM, ETH Zurich. A protective carbon layer was deposited on the selected region of interest first by an electron beam (2 KV, 13 nA) and subsequently by an ion beam (30 kV, 1.2 nA). The chuck milling and lamellae thinning were done at 30 kV with FIB current from 9 nA to 90 pA. Finally, the lamellae were polished at 5 KV (17 pA) and finished at 2 kV (12 pA). The expected thickness was below 50 nm. The samples were kept in vacuum between the FIB and TEM measurements (except for loading and unloading).

The STEM images were acquired with a JEOL ARM200F cold field emission gun scanning transmission electron microscope located at the Binnig and Rohrer Nanotechnology Center Noise-free laboratories at IBM
same increasing temperature dependence of coercivity. In a ferro-magnet, the common expectation is that the coercive (reorientation) field decreases with increasing temperature, usually because of the decreasing magnetocrystalline anisotropy³¹,³². The situation in compensated magnetic materials is, however, more complex: for example,

<figure><img src="image_10.png" /><figcaption>**Fig. 4** | Decomposition of the anomalous Nernst signal. a The anomalous Nernst signal measured at 121 K decomposed into its spontaneous and sigmoid contributions. The anomalous Hall signal is also plotted for comparison. b Temperature dependence of the coercive field of the spontaneous ANE contribution together with the respective dependence for the AHE.</figcaption></figure>

the observed increase of the reorientation (spin-flop) field in collinear MnF₂ was ascribed to its temperature-dependent and anisotropic response to the applied magnetic field³³. In our Mn₅Si₃ thin films, however, we do not observe any reorientation transition in the reachable magnetic field (see Fig. 2c). Instead, we expect that the hysteresis in the Néel vector reversal originates from the competition of the magnetocrystalline anisotropy and the Dzyaloshinskii-Moriya interaction. These effects have generally different temperature dependencies, which can result in an increasing coercivity with temperature. The temperature dependence of the magnetocrystalline anisotropy may be affected by the complex temperature dependence of the lattice parameters¹⁶.

Discussion

We have demonstrated a significant and spontaneous ANE in an altermagnetic candidate composed of light elements. Fig. 5 presents a comparison of the ANE coefficients observed in various magnetic compounds depending on their magnetization per formula unit, including ferromagnets (hollow circles), non-collinear antiferromagnets (full circles), and our altermagnetic Mn₅Si₃ (crossed circle). Materials showing spontaneous ANE are labeled by an asterisk. Our material showcases ANE values comparable to those of other materials, despite having vanishing magnetization, collinear magnetic order, and being constituted of light elements.

Unlike ferromagnets or non-collinear antiferromagnets, altermagnets are characterized by the presence of symmetry pseudonodal surfaces in their electronic bands. This unique feature can contribute to the Berry curvature and enhance the ANC as demonstrated by our calculations in Mn₅Si₃.

The maximum anomalous Nernst conductivity value $|\alpha_{yx}| = (0.22 \pm 0.11) A/(K \cdot m)$ we determine for Mn₅Si₃ at 58 K is comparable to $\alpha_{yx}$ reported for noncollinear antiferromagnets, such as Mn₃ Sn with 0.30 A/(K·m) at 150 K¹, though it is smaller than the values for YMnBi₂ (10 A/(K·m) at 75 K)⁹ and Co₂MnGa with 2.0 A/(K·m) at 300 K³⁴.

Unlike non-collinear magnets, the collinear spin arrangement of altermagnets allows for the generation of spin currents with well-defined spin-quantization axis or long spin coherence³⁵. Consequently, altermagnets are highly advantageous for multi-functional spintronic and spincaloritronics devices. In such devices, a single layer can, for

<figure><img src="image_11.png" /><figcaption>**Fig. 5** | Comparison of the anomalous Nernst effect measured in epitaxial Mn₅Si₃ and in other materials¹,²,⁷,⁹,²⁷,⁵⁴⁻⁶². The ANE magnitude $S_{xy}$ is captured as a function of materials' magnetization. The symbol size linearly decreases with the highest atomic number of materials' formula. Antiferromagnetic</figcaption></figure>

materials are indicated by full circles, while ferromagnets are denoted by hollow circles and the altermagnet by a crossed circle. Materials showing the spontaneous Nernst effect are labeled by an asterisk.
Research Europe. The images were acquired at 200 kV and were further processed using FIJI - Fiji is just ImageJ freeware [38]. The FFT pattern of the images was masked by a circular aperture. After masking, a constant threshold was subtracted before performing an inverse FFT of the spectra to remove the background between the high-intensity spots corresponding to Bragg peaks. In selected images, the contrast was normalized by the contrast normalization plugin for better visualisation of lighter elements.

d. Density-functional theory calculations We calculated the electronic structure of Mn₅Si₃ (space group P6₃/mcm, No. 193) in the pseudo-potential DFT code Vienna Ab initio Simulation Package (VASP) [39] within Perdew-Burke-Ernzerhof (PBE) + SOC generalized gradient approximation (GGA) [40-42]. The lattice constants for the hexagonal unit cell containing two formula units (f.u.) are a = b = 6.901 Å and c = 4.795 Å – consistent with the experimental values reported in Refs. [20, 21]. We switched off symmetrization and converged the self-consistent calculations on a 9 × 9 × 12 k-point grid using an energy cutoff of 520 eV, energy convergence criterion of 0.5 × 10⁻⁸ eV and the tetrahedron smearing method with Blöchl corrections and smearing parameter 0.1 eV. The converged ground state has Fermi energy 9.945 eV and a total magnetization of 40 mμB/f.u., where μB is the Bohr magneton. We plotted the spin-resolved electronic band structure (left panel in Fig. 1d) using the PyProcar Python library for electronic structure pre/post-processing [43].

To calculate the anomalous Nernst conductivity (ANC; right panel in Fig. 1d) from our DFT results, we initially obtained the Wannier functions of Mn₅Si₃ using the Wannier90 code [44]. Then, we incorporated these Wannier functions on the WannierBerri python library [45] using a 26 × 26 × 30 k-point grid and a Fermi-Dirac smoother with smearing parameter 0.86 meV to evaluate the quantities reliant on the Berry curvature [46]

$$ \Omega_{yx}^{n}(\mathbf{k}) = -i \left[ \frac{\partial}{\partial k_x} \left\langle u_{n\mathbf{k}} \left| \frac{\partial}{\partial k_y} \right| u_{n\mathbf{k}} \right\rangle - \frac{\partial}{\partial k_y} \left\langle u_{n\mathbf{k}} \left| \frac{\partial}{\partial k_x} \right| u_{n\mathbf{k}} \right\rangle \right], \quad (1) $$

where $\mathbf{k} = (k_x, k_y, k_z)$ is the crystal momentum and $|u_{n\mathbf{k}}\rangle$ is the periodic part of the Bloch state with band index n. Finally, we obtained the ANC for temperature T = 1 K via the Mott relation valid in the low-temperature limit [47]

$$ \alpha_{yx}(\mu) \approx \frac{\pi^2 k_B^2 T}{3 e} \sum_n \int_{BZ} \frac{d\mathbf{k}}{(2\pi)^3} \Omega_{yx}^n(\mathbf{k}) \frac{\partial f(\varepsilon_{n\mathbf{k}}, \mu)}{\partial \varepsilon_{n\mathbf{k}}}, \quad (2) $$

where e is the electron elementary charge, μ is the chemical potential, εn_k are the energy eigenvalues and f(ε, μ) = {1 + exp[(ε - μ)/k_B T]}⁻¹ is the Fermi-Dirac distribution with k_B the Boltzmann constant. The integration runs over the whole Brillouin zone (BZ).

For the ANC at higher temperatures, we first obtained the anomalous Hall conductivity (AHC) at T = 0 K [46]

$$ \sigma_{yx}^{T=0}(\varepsilon) = -\frac{e^2}{\hbar} \sum_n \int_{BZ} \frac{d\mathbf{k}}{(2\pi)^3} \Omega_{yx}^n(\mathbf{k}) \Theta(\varepsilon - \varepsilon_{n\mathbf{k}}), \quad (3) $$

where Θ(x) is the Heaviside function. Then, we used our AHC results - consistent with the calculations previously reported in Ref. [20] - to obtain the ANC via the Mott relation [47-51]

$$ \alpha_{yx}(\mu) = -\frac{1}{e} \int d\varepsilon \frac{\partial f(\varepsilon, \mu)}{\partial \mu} \sigma_{yx}^{T=0}(\varepsilon) \frac{\varepsilon - \mu}{T}. \quad (4) $$

At low temperatures, Eq. (4) reduces to αyx ≈ π² k_B² T / 3 e dσyx / dε, which we used to verify our results for T = 1 K obtained from Eq. (2).

We test our Nernst conductivity calculations on iron. For iron at 300 K, we obtain αxy = -0.31 A/(K · m) which is in very good agreement with the experimental observation of αxy ≈ -0.5 A/(K · m) [52].

e. Magnetotransport measurements The transport experiments were conducted in an Oxford Instruments Integra AC cryostat equipped with a 3D superconducting magnet. The temperature gradient in the sample was generated by an external heater as illustrated in Fig. 2a. As a heater, we used the platinum resistor Pt2000. The applied heating power ranged from 0.2 to 2 W. The sample temperature was detected by two on-chip platinum strips located on the opposite ends of the sample with respect to the applied in-plane temperature gradient. We calibrated the resistance of the platinum thermometers by a slow cool-down of the sample. The transverse Nernst voltage was detected by Keithley Nanovoltmeters 2182A. The Nernst coefficient Syx was determined from the measured voltage V_y and the simulated temperature gradient x-component (∇T)_x as: Syx = V_y/(l · (∇T)_x) where l is the length of the transverse Mn₅Si₃ contact. The main contribution to the error of the Syx is the determination of the temperature gradient, which we determine with the absolute error of 0.02 K/mm.

To evaluate the magnetic field dependence of Syx, we only consider its odd-in-field component and subtract the linear-in-field contribution of the ordinary Nernst effect. An example of the data processing, including the raw data, is shown in Extended Data Fig. E2a. We suppose that the strong even-in-field contribution corresponds to the magneto-Seebeck effect arising due to the misalignment of contact bonds.

We determined the Nernst conductivity αyx via the relation

$$ \alpha_{yx} = \sigma_{xx} \cdot S_{yx} + \sigma_{yx} \cdot S_{xx}, \quad (5) $$

where Sxx is the Seebeck coefficient and σxx is longitudinal conductivity [51]. The latter two quantities were measured at a separate sample with a lithographically patterned Hall bar (see Extended Data Fig. E1b). Sxx was determined in the same experimental geometry as
instance, enable both the coherent generation or propagation of spin currents and the efficient spontaneous ANE.

Although we observe finite but small saturation magnetization of 45 mμ<sub>B</sub>/f.u., let us emphasize that it is not the prime source of AHE and ANE as shown by the first-principles calculations. Instead, the AHE and ANE are dominated by the strong altermagnetic breaking of the time-reversal symmetry in the electronic structure in the absence of a net magnetization, combined with the relativistic spin-orbit coupling. An absence of a correlation between the atomic mass and the ANE is highlighted in Fig. 5 where the symbols' size linearly decreases with the highest atomic number of the compound (the largest symbols represent the lightest elements). This observation highlights the importance of not limiting the search for high ANE performance materials solely to heavy elements. Additionally, we note that, unlike many materials listed in Fig. 5, our Mn<sub>5</sub>Si<sub>3</sub> is in thin film form, which often reduces the ANE magnitude. Therefore, we anticipate that higher ANE values could be discovered in altermagnetic bulks.

In summary, we provide experimental evidence that alter-
magnetic materials can host a spontaneous ANE despite their colli-
near magnetic order and vanishing net magnetization, and without
the need of heavy elements. We demonstrate a sizable anomalous
Nernst coefficient in the altermagnetic candidate Mn<sub>5</sub>Si<sub>3</sub> in the form
of epitaxial thin layers. At 58 K, the effect shows spontaneous
S<sub>yx</sub> = -(0.26 ± 0.10) µV/K, saturated S<sub>yx</sub> = -(2.8 ± 1.1) µV/K, and spon-
taneous |α<sub>yx</sub>| = (0.22 ± 0.11) A/(K · m). Both the spontaneous and
saturated S<sub>yx</sub> drop to S<sub>yx</sub> = -(0.03 ± 0.01) µV/K at 216 K with the
spontaneous |α<sub>yx</sub>| = (0.04 ± 0.01) A/(K · m). The Nernst effect mea-
surements are supported by our first-principles calculations of the
Mn<sub>5</sub>Si<sub>3</sub> electronic band structure and the Nernst conductivity, which
yield α<sub>yx</sub> = 0.25 A/(K · m) at 58 K. Our momentum-resolved ANC
calculation in Mn<sub>5</sub>Si<sub>3</sub> reveals contributions from altermagnetic
pseudonodal surfaces and ladder transitions to the Berry curvature,
while suggesting the absence of Weyl points near the Fermi level.

By demonstrating the ANE in an altermagnet, we introduce a new
category of compensated magnetic materials for exploring the ANE.
This expands the field beyond the more complex non-collinear mag-
netic structures or materials with heavy elements, marking a significant
broadening of the scope for the ANE research. The search for materials
with high and robust ANE magnitude is one of the key challenges for
fully exploiting their application potential. Furthermore, it will be
important to explore the universal validity of the Mott rule in this class
of materials. Although the ANE and the AHE typically share the same
symmetry, the Mott rule is not a fundamental principle and can be
violated³⁶. Moreover, in materials where the AHE and the ANE have the
common Berry-curvature origin, the ANE can exhibit some features
distinct from the AHE³⁷ and its magnitude can be influenced by various
means including spin fluctuations at transition temperature³⁸.

**Methods**

Sample preparation

The Mn₅Si₃ thin films were grown by molecular beam epitaxy (MBE) in an ultra-high vacuum system with a base pressure better than 2 × 10⁻⁷ Pa. Elemental Mn and Si were flux matched in a 5: 3 ratio using a quartz microbalance and co-deposited on the intrinsic Si(111) substrate (R > 10,000 Ω · cm) heated to 170 °C. The films were subsequently annealed up to ≈300 °C, the temperature for which a clear (√3 × √3)R 30° reconstruction is observed on the RHEED pattern. Ex situ X-ray diffraction (XRD) using a high brilliance rotating anode showed a good crystalline quality of the Mn₅Si₃(0001) thin films whose growth was promoted by the formation of a thin MnSi(111). A more detailed description of the growth process can be found in ref. 17.

For the measurement of the anomalous Nernst effect, we used
optical lithography and argon ion beam etching to pattern our Mn₅Si₃
layers into two distinct lithographic designs illustrated in the

Supplementary Fig. S2. The platinum thermometers of ≈50 nm thickness were deposited by magnetron sputtering.

Magnetometry measurements

Magnetic characterization of our Mn₅Si₃ thin films was carried out using a SQUID MPMS XL 5.0 magnetometer. The as-grown sample was cleaned with acetone, alcohol, and DI water, before measurement and mounted with two dots of glue (X60) on a sample holder made of two quartz capillaries. Field-dependent magnetization was measured at different temperatures for magnetic field strengths ranging from ±4 T applied along the film normal. The signal is dominated by the diamagnetism of the silicon substrate, which accounts for 96% of the signal at 4 T. After eliminating the linear-in-field diamagnetic contribution, the magnetic response contains two components: a non-hysteretic part linked to the substrate and sample holder and a hysteretic part that we attribute to the Mn₅Si₃ film, due to a canting of the moments of the two sublattices.

High-resolution scanning transmission electron microscopy
imaging and lamellae preparation

The cross-sectional lamellae of selected samples were prepared by
Focused Ion Beam (FIB) (Helios 5 UX from Thermo Fisher Scientific)
using AutoTEM 5 software (Thermo Scientific, the Netherlands) at
ScopeM, ETH Zurich. A protective carbon layer was deposited on the
selected region of interest first by an electron beam (2 KV, 13 nA) and
subsequently by an ion beam (30 kV, 1.2 nA). The chuck milling and
lamellae thinning were done at 30 kV with FIB current from 9 nA to
90 pA. Finally, the lamellae were polished at 5 KV (17 pA) and finished
at 2 kV (12 pA). The expected thickness was below 50 nm. The samples
were kept in vacuum between the FIB and TEM measurements (except
for loading and unloading).

The STEM images were acquired with a JEOL ARM200F cold field emission gun scanning transmission electron microscope located at the Binnig and Rohrer Nanotechnology Center Noise-free laboratories at IBM Research Europe. The images were acquired at 200 kV and were further processed using FIJI - Fiji is just ImageJ freeware³⁹. The FFT pattern of the images was masked by a circular aperture. After masking, a constant threshold was subtracted before performing an inverse FFT of the spectra to remove the background between the high-intensity spots corresponding to Bragg peaks. In selected images, the contrast was normalized by the contrast normalization plugin for better visualization of lighter elements.

Density-functional theory calculations

We calculated the electronic structure of Mn₅Si₃ (space group P6₃/ mcm, No. 193) in the pseudo-potential DFT code Vienna Ab initio Simulation Package (VASP)⁴⁰ within Perdew-Burke-Ernzerhof (PBE) + SOC generalized gradient approximation (GGA)⁴¹⁻⁴³. The lattice constants for the hexagonal unit cell containing two formula units (f.u.) are a = b = 6.901 Å and c = 4.795 Å – consistent with the experimental values reported in refs. 16,17. We switched off symmetrization and converged the self-consistent calculations on a 9 × 9 × 12 k-point grid using an energy cutoff of 520 eV, energy convergence criterion of 0.5 × 10⁻⁸ eV and the tetrahedron smearing method with Blöchl corrections and smearing parameter 0.1 eV. The converged ground state has Fermi energy 9.945 eV and a total magnetization of 40 mμ<sub>B</sub>/f.u., where μ<sub>B</sub> is the Bohr magneton. We plotted the spin-resolved electronic band structure (left panel in Fig. 1d) using the PyProcar Python library for electronic structure pre/post-processing⁴⁴.

The Néel vector was defined along the direction [111] and remains
virtually unchanged after the self-consistent electronic minimization.
The magnetic Mn atoms have magnetic moments of ≈2.52 μ<sub>B</sub> along the
direction of the Néel vector. The net magnetization is ≈0.0409 μ<sub>B</sub>/f.u.
in the direction defined in the spherical coordinate system by θ = 121°
$S_{xy}$, i.e. in the elevated configuration illustrated in Fig. 2a. The measured field-dependencies of $S_{xx}$ and $\sigma_{xx}$ are shown in Extended Data Fig. E3a. We present the $\alpha_{yx}$ field dependencies at different temperatures in Extended Data Fig. E3b.

f. Finite-element simulations of temperature distribution We simulated the distribution of the temperature in our sample by the finite-element method (FEM) in COMSOL MULTIPHYSICS [24]. We used realistic simulation geometry illustrated in Fig. 2a, which precisely reflects our experimental setup, including the chip carrier, the acrylonitrile styrene acrylate (ASA) and brass blocks, the platinum heater (in an alumina ceramic package), as well as the sample itself. The bonding wires were also taken into account. The simulation was performed using material-specific physical parameters as illustrated in Tab. I showing the used thermal conductivity and heat capacity values. For certain materials, we used temperature-dependent material properties.

<table><caption>TABLE I. Values of heat capacity at constant pressure $C_p$ and thermal conductivity $\kappa$ used for the finite-element modelling of the temperature gradient. For values marked by an asterisk, we used temperature-dependent $C_p$ or $\kappa$.</caption><thead><tr><th>material</th><th>C<sub>p</sub> (J kg<sup>-1</sup>K<sup>-1</sup>)</th><th>&kappa; (W K<sup>-1</sup>m<sup>-1</sup>)</th></tr></thead><tbody><tr><td>silicon</td><td>*[53]</td><td>*[54]</td></tr><tr><td>brass</td><td>380</td><td>40</td></tr><tr><td>ASA</td><td>1300</td><td>0.17</td></tr><tr><td>GE varnish</td><td>983</td><td>*[55]</td></tr><tr><td>aluminium</td><td>900</td><td>238</td></tr><tr><td>alumina ceramics</td><td>730</td><td>23</td></tr></tbody></table>

Our FEM model includes the heat equation with the

mixed boundary condition of the constant temperature at the interfaces in contact with the chip carrier and constant heat flux elsewhere. The heat transfer coefficient is a fitting parameter of our model that we adjust to reach the temperatures experimentally observed on the on-chip platinum thermometers. The coefficient is temperature-dependent and increases from 26 W/(m²·K) at 58 K to 89 W/(m²·K) at 216 K. The sample heating is simulated via the Joule heating by the charge current in the platinum heater.

As illustrated in the Extended Data Fig. 1a, the simulated temperature gradient is linear in the measurement area of the sample, where it reaches $-\partial T/\partial x = 0.04$ K/mm for the sample temperature of 58 K, 0.34 K/mm for 121 K, 0.74 K/mm for 176 K, 0.93 K/mm 216 K. Outside the measurement area, the temperature gradient is not monotonous and gains a finite out-of-plane component.

ACKNOWLEDGEMENTS

The authors acknowledge the support of Peng Zheng from ScopeM ETH Zurich and the support of Marielyne Sousa (IBM) via the Binning and Rohrer Nanotechnology Center (BRNC). The study was supported by Charles University, project GA UK No. 266723, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project-IDs 445976410 and 490730630), by the French National Research Agency (ANR) (Project MATHEEIAS - Grant No. ANR-20-CE92-0049-01 and Project PEPR SPIN - Grant No. ANR-22-EXSP-0007), and by the Grant Agency of the Czech Republic Grant No. 22-17899K and the Max Planck Dioscuri Program. D.K. acknowledges the support from the Czech Academy of Sciences (project No. LQ100102201).

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and $\phi = 244^{\circ}$. The magnitude of the net magnetization is then $\approx 1.6\%$ of the Mn atoms.

To calculate the anomalous Nernst conductivity (ANC; right panel in Fig. 1d) from our DFT results, we initially obtained the Wannier functions of Mn$_5$Si$_3$ using the Wannier90 code$^{45}$. Then, we incorporated these Wannier functions on the WannierBerri python library$^{46}$ using a 26 × 26 × 30 k-point grid and a Fermi-Dirac smoother with smearing parameter 0.86 meV to evaluate the quantities reliant on the Berry curvature$^{47}$

$$ \Omega_{yx}^{n}(\mathbf{k}) = -i \left[ \frac{\partial}{\partial k_x} \langle u_{nk} | \frac{\partial}{\partial k_y} | u_{nk} \rangle - \frac{\partial}{\partial k_y} \langle u_{nk} | \frac{\partial}{\partial k_x} | u_{nk} \rangle \right], \quad (1) $$

where $\mathbf{k} = (k_x, k_y, k_z)$ is the crystal momentum and $|u_{nk}|$ is the periodic part of the Bloch state with band index $n$. Finally, we obtained the ANC for temperature $T=1$ K via the Mott relation valid in the low-temperature limit$^{48}$

$$ \alpha_{yx}(\mu) \approx \frac{\pi^2 k_B^2 T}{3 e} \sum_n \int_{BZ} \frac{d\mathbf{k}}{(2\pi)^3} \Omega_{yx}^n(\mathbf{k}) \frac{\partial f(\varepsilon, \mu)}{\partial \varepsilon_{nk}}, \quad (2) $$

where $e$ is the electron elementary charge, $\mu$ is the chemical potential, $\varepsilon_{nk}$ are the energy eigenvalues and $f(\varepsilon, \mu) = \{1 + \exp[(\varepsilon - \mu)/k_B T]\}^{-1}$ is the Fermi-Dirac distribution with $k_B$ the Boltzmann constant. The integration runs over the whole Brillouin zone (BZ).

For the ANC at higher temperatures, we first obtained the anomalous Hall conductivity (AHC) at $T=0$ K$^{47}$

$$ \sigma_{yx}^{T=0}(\varepsilon) = -\frac{e^2}{\hbar} \sum_n \int_{BZ} \frac{d\mathbf{k}}{(2\pi)^3} \Omega_{yx}^n(\mathbf{k}) \Theta(\varepsilon - \varepsilon_{nk}), \quad (3) $$

where $\Theta(x)$ is the Heaviside function. Then, we used our AHC results - consistent with the calculations previously reported in ref. 16 - to obtain the ANC via the Mott relation$^{29,48-51}$

$$ \alpha_{yx}(\mu) = -\frac{1}{e} \int d\varepsilon \frac{\partial f(\varepsilon, \mu)}{\partial \mu} \sigma_{yx}^{T=0}(\varepsilon) \frac{\varepsilon - \mu}{T}. \quad (4) $$

At low temperatures, Eq. (4) reduces to $\alpha_{yx} \approx \frac{n^2 k_B^2 T}{3 e} \frac{d\sigma_{yx}}{d\varepsilon}$, which we used to verify our results for $T=1$ K obtained from Eq. (2).

We test our Nernst conductivity calculations on iron. For iron at 300 K, we obtain $\alpha_{xy} = -0.31$ A/(K·m) which is in very good agreement with the experimental observation of $\alpha_{xy} \approx -0.5$ A/(K·m)$^{52}$.

## Magnetotransport measurements

The transport experiments were conducted in an Oxford Instruments Integra AC cryostat equipped with a 3D superconducting magnet. The temperature gradient in the sample was generated by an external heater as illustrated in Fig. 2a and photographed in Supplementary Fig. S2d. As a heater, we used the platinum resistor Pt2000. The applied heating power ranged from 0.2 to 2 W. The sample temperature was detected by two on-chip platinum strips located on the opposite ends of the sample with respect to the applied in-plane temperature gradient. We calibrated the resistance of the platinum thermometers by a slow cool-down of the sample.

The transverse Nernst voltage was detected by Keithley Nanovoltmeters 2182A. The Nernst coefficient $S_{yx}$ was determined from the measured voltage $V_y$ and the simulated temperature gradient $x$-component $(\nabla T)_x$ as: $S_{yx} = E_y/(\nabla T)_x = V_y/(l \cdot (\nabla T)_x)$ where $l$ is the length of the transverse Mn$_5$Si$_3$ contact. The main contribution to the error of the $S_{yx}$ is the determination of the temperature gradient, which we determine with the absolute error of 0.02 K/mm. As shown in Supplementary Fig. S5 for 121 K, we evaluated $S_{yx}$ at multiple devices on the chip with different widths of the contact. For each device, we used the value of

the $(\nabla T)_x$ determined from the finite-element simulations as illustrated in Supplementary Fig. S2a for 121 K. The mean value of saturated $S_{yx}$ at 121 K for the three presented devices is $-0.646$ µV/K with the standard deviation of 0.067 µV/K which agrees with the value presented in Fig. 2b and Supplementary Fig. S4a of $(-0.679 \pm 0.097)$ µV/K.

To evaluate the magnetic field dependence of $S_{yx}$, we treat the data as illustrated in Supplementary Fig. S3a: We only consider its odd-in-field component to eliminate the effect of the temperature gradient not being perfectly perpendicular to the lithographic bars. We suppose that the strong even-in-field contribution corresponds to the magnetoo-Seebeck effect arising due to the misalignment of contact bonds. Furthermore, we subtract the linear-in-field contribution of the ordinary Nernst effect. The slope of the ordinary Nernst effect reaches $-2.44$ µV/(K·T) at 58 K and increases to $-0.12$ µV/(K·T) at 216 K. For the further analysis of anomalous $S_{yx}$, we separate the signal into its spontaneous and sigmoid contributions. We describe the sigmoid contribution by an error function $f(\mu_0 H_2)$:

$$ f(\mu_0 H_z) = \frac{2A}{\pi} \int_{0}^{\mu_0 H_z} e^{-\frac{t^2}{B}} dt. \quad (5) $$

where A and B are parameters, describing the magnitude and saturation field of the sigmoid contribution, respectively. Note that our choice of the error function over other sigmoid functions (e.g., hyperbolic tangent) is arbitrary. The spontaneous contribution to $S_{yx}$ is the difference between anomalous $S_{xy}$ and $f(\mu_0 H_z)$.

We determined the magnetic field dependence of the Nernst conductivity $\alpha_{yx}$ via the relation

$$ \alpha_{yx} = \sigma_{xx} \cdot S_{yx} + \sigma_{yx} \cdot S_{xx}, \quad (6) $$

where $S_{xx}$ is the Seebeck coefficient and $\sigma_{xx}$ is longitudinal conductivity$^{51}$. In Eq. (6), we use the $S_{yx}$ and $\sigma_{yx}$ values from Fig. 2b, i.e., odd-in-field component of raw transverse thermoelectric signal and conductivity without the linear-in-field contributions, which is not influenced by the buffer layer or the substrate. The values of $S_{xx}$ and $\sigma_{xx}$ were measured at a separate sample with a lithographically patterned Hall bar (see Supplementary Fig. S2b). $S_{xx}$ was determined in the same experimental geometry as $S_{xy}$, i.e., in the elevated configuration illustrated in Fig. 2a, via the formula $S_{xx} = E_x/(\nabla T)_x = V_x/(d \cdot (\nabla T)_x)$ (d is the length of the longitudinal contact). The measured field-dependencies of $S_{xx}$ and $\sigma_{xx}$ are shown in Supplementary Fig. S6a. We present the $\alpha_{yx}$ field dependencies at different temperatures in Supplementary Fig. S6b. The values of spontaneous $\alpha_{yx}$ correspond to $\alpha_{yx}$ at zero magnetic field.

When determining $\alpha_{yx}$ from Eq. (6), the resulting value may be influenced by the MnSi buffer and the substrate contributions to the measured $S_{xx}$ and $\sigma_{xx}$. We conducted a dedicated finite-element simulation to describe the $S_{xx}$ signal generated by a Mn$_5$Si$_3$/MnSi stack on a silicon substrate. The left panel of the Supplementary Fig. S7 shows a distribution of the temperature gradient in such a stack simulated by COMSOL when keeping the opposite sides of the sample at 214 K and 216 K, respectively. The figure shows that the silicon substrate fully dictates the temperature distribution in the device despite the different thermal conductivities of Mn$_5$Si$_3$ and MnSi compared to silicon. The right panel presents the distribution of the Seebeck signal, showing that the large Seebeck coefficient of the substrate does not influence the signal detected at the device. This is due to the very small electric conductivity of the silicon substrate. The Seebeck signal at the device is homogeneous, and slightly influenced by the presence of the MnSi buffer – the difference to the pure Mn$_5$Si$_3$ value is approx. 26%. As a consequence, we use the as-measured $S_{xx}$ values in Eq. (6) and include this error of $S_{xx}$ to the error bar of $\alpha_{yx}$. The finite-element simulations at other temperatures show the same trend keeping the error below 26%. Since the electrical conductivity of the
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MnSi is similar to Mn₅Si₃ (e.g., 6500 S/cm for Mn₅Si₃ and 4500 S/cm for MnSi at 216 K) and due to the small thickness of the MnSi buffer layer, we use the as-measured values of σₓₓ to calculate αᵧᵧ.

Finite-element simulations of temperature distribution

We simulated the distribution of the temperature in our sample by the finite-element method (FEM) in COMSOL MULTIPHYSICS²⁸. We used realistic simulation geometry illustrated in Fig. 2a, which precisely reflects our experimental setup, including the chip carrier, the acrylonitrile styrene acrylate (ASA) and brass blocks, the platinum heater (in an alumina ceramic package), as well as the sample itself. The bonding wires were also taken into account. The simulation was performed using material-specific physical parameters as illustrated in Supplementary Table S1 showing the used thermal conductivity and heat capacity values. For certain materials, we used temperature-dependent material properties.

Our FEM model includes the heat equation with the mixed boundary condition of the constant temperature at the interfaces in contact with the chip carrier and constant heat flux elsewhere. The heat transfer coefficient is a fitting parameter of our model that we adjust to reach the temperatures experimentally observed on the on-chip platinum thermometers. The coefficient is temperature-dependent and increases from 26 W/(m²·K) at 58 K to 89 W/(m²·K) at 216 K. The sample heating is simulated via the Joule heating by the charge current in the platinum heater. The numerical mesh used in the simulation is visualized in the Supplementary Fig. S2c.

As illustrated in the Supplementary Fig. S2a, the simulated temperature gradient is linear in the measurement area of the sample, where it reaches $-\partial T/\partial x = 0.04$ K/mm for the sample temperature of 58 K, 0.34 K/mm for 121 K, 0.74 K/mm for 176 K, 0.93 K/mm 216 K. Outside the measurement area, the temperature gradient is not monotonous and gains a finite out-of-plane component.

Data availability

All data supporting the findings of this study are included within the main text or Supplementary Information. Any additional requests for information can be directed to the corresponding author. Source data for figures in the main text are available via Zenodo data repository⁵³.

Code availability

In the manuscript, we used the Vienna Ab initio Simulation Package (VASP) for our pseudo-potential DFT calculations. The package can be purchased from https://www.vasp.at/. We also used the Wannier90 code available at https://wannier.org/ and the WannierBerri python library (accessible in the repository https://github.com/wannier-berri). All input files, post-processing procedures and scripts used to evaluate experimental and theoretical data are available from the authors on request.

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## Acknowledgements

We would like to thank Pascal Manuel for fruitful discussions about the magnetic structure of Mn₅Si₃. The authors acknowledge the support of Peng Zheng from ScopeM ETH Zurich and the support of Marilyne Sousa (IBM) via the Binning and Rohrer Nanotechnology Center (BRNC). The study was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project-IDs 445976410 and 490730630), by the French National Research Agency (ANR) (Project MATHEEIAS – Grant No. ANR-20-CE92-0049-01 and Project PEPR SPIN – Grant No. ANR-22-EXSP-0007), and by the Czech Science Foundation Grant No. 22-17899K and Dioscuri Program LV23025 funded by MPG and MEYS, TERAFIT CZ.02.01.01/00/22 008/0004594. A.B. was supported by Charles University, project GA UK No. 266723. D.K. acknowledges the support from the Czech Academy of Sciences (project No. LQ100102201). A.B.H. acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) grant no. TRR 173 268565370 (project A03). T.J. acknowledges support by the ERC Advanced Grant No. 101095925. We acknowledge CzechNanoLab Research Infrastructure supported by MEYS CR (LM2023051). W.H.C. and L.Š. acknowledge support from the ERC Starting Grant No. 101165122.

## Author contributions

The transport experiments were performed by A.B., S.S., and M.La. I.K., L.M., and M.P. grew the thin-film samples. M.Le., J.R., and V.B. performed the magnetometry measurements. F.K. and S.T. conducted the TEM characterization. The DFT calculations were performed by W.H.C., V.K.B., R.J.U., A.B.H., R.G.H., J.S., and L.Š. A.B., D.K., S.T.B.G., J.Ž., and H.R. interpreted the experimental transport data. A.B. and J.Z. did the finite-element calculations of temperature gradient. The manuscript was written by A.B., H.R., and T.J. with contributions from all the authors.

## Competing interests

The authors declare no competing interests.

## Additional information

**Supplementary information** The online version contains supplementary material available at [https://doi.org/10.1038/s41467-025-62331-7](https://doi.org/10.1038/s41467-025-62331-7).

**Correspondence** and requests for materials should be addressed to Antonín Badura.

**Peer review information** *Nature Communications* thanks the anonymous, reviewers for their contribution to the peer review of this work. A peer review file is available.

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