Magnetic field-induced non-trivial electronic topology in Fe3GeTe2

The anomalous Hall, Nernst and thermal Hall coefficients of Fe$_{3-x}$GeTe$_2$ display several features upon cooling, like a reversal in the Nernst signal below $T = 50$ K pointing to a topological transition (TT) associated to the development of magnetic spin textures. Since the anomalous transport variables are related to the Berry curvature, a possible TT might imply deviations from the Wiedemann-Franz (WF) law. However, the anomalous Hall and thermal Hall coefficients of Fe$_{3-x}$GeTe$_2$ are found, within our experimental accuracy, to satisfy the WF law for magnetic-fields $\mu_0H$ applied along its inter-layer direction. Surprisingly, large anomalous transport coefficients are also observed for $\mu_0H$ applied along the planar \emph{a}-axis as well as along the gradient of the chemical potential, a configuration that should not lead to their observation due to the absence of Lorentz force. However, as $\mu_0H$ $\|$ \emph{a}-axis is increased, magnetization and neutron scattering indicate just the progressive canting of the magnetic moments towards the planes followed by their saturation. These anomalous planar quantities are found to not scale with the component of the planar magnetization ($M_{\|}$), showing instead a sharp decrease beyond $\sim \mu_0 H_{\|} = $ 4 T which is the field required to align the magnetic moments along $\mu_0 H_{\|}$. We argue that locally chiral spin structures, such as skyrmions, and possibly skyrmion tubes, lead to a field dependent spin-chirality and hence to a novel type of topological anomalous transport. Locally chiral spin-structures are captured by our Monte-Carlo simulations incorporating small Dzyaloshinskii-Moriya and biquadratic exchange interactions.


I. INTRODUCTION
The current interest in topologically non-trivial compounds is in the promise of observing novel magneto-opto-electronic phenomena with potential technological relevance, hitherto unobserved in conventional materials. Examples include the generation of photocurrents with circularly polarized light 1,2 , the observation of a Hall like signal in the absence of broken time-reversal symmetry in Weyl semimetals 3 , or the observation of large anomalous Hall and Nernst-effects at room temperature in a non-collinear antiferromagnet displaying magnetic Weyl fermions [4][5][6] . For instance, the large anomalous Nernst effect observed at room temperature in magnetic Weyl semimetals has been proposed as an effective alternative to thermoelectric energy conversion 5,7 In this context, Fe 3−x GeTe 2 is a layered, van der Waals like ferromagnet displaying a simple collinear spin arrangement with the magnetic moments oriented along the out-ofthe plane direction 8,9 . However, this simple magnetic order is claimed to trigger complex phenomena such as i) the development of a Kondo lattice below a coherence temperature of ≈ 150 K 10 , ii) electric field tuning of its Curie temperature T c = (220 ± 10) K up to room temperature 11 , iii) skyrmions 12-14 , iv) and very large anomalous Hall and Nernst coefficients claimed to result from its non-trivial electronic topology 15,16 .
Fe 3−x GeTe 2 displays a comparatively high Curie temperatures T c , relative to the magnetic ordering temperature of other two-dimensional magnetic systems, that is ranging from 150 to 220 K depending on the Fe occupancy 8,9,17,18 . Fe 3−x GeTe 2 can be understood as containing van der Waals (vdW) bonded Fe 3−x GeTe 2 slabs or, as discussed in Ref. 19 , as a scaffold with a lattice akin to that of the transition metal dichalcogenides but stuffed with Fe atoms. Its structure leads to two inequivalent Fe sites, Fe 3+ I and Fe 2+ II , within the Fe 3−x GeTe 2 slab 8, 11 . Partially filled Fe-d orbitals dominate the band structure around the Fermi level producing itinerant ferromagnetism in bulk Fe 3−x GeTe 2 20 . As a result of the reduced crystallographic symmetry inherent to its layered structure (space group 194; P 6 3 /mmc), bulk Fe 3−x GeTe 2 exhibits a strong magneto-crystalline anisotropy 21 . The observation of skyrmions 12-14 and spin spirals on its surface 22 can only be reconciled with a sizeable Dzyaloshinskii-Moriya interaction, whose origin is discussed in Ref. 23 , although inelastic neutron scattering would suggest negligible inter-layer exchange interactions but a prominent role for the single ion anisotropy 24 .
Fe 3−x GeTe 2 is claimed to correspond to a rare example of a ferromagnetic topological nodal line semimetallic system for which electronic correlations are claimed to be relevant 15,25 . In particular, the existence of a gapped Dirac nodal line, and its remanent Berry curvature, would explain the very large anomalous Hall and Nernst coefficients in Fe 3−x GeTe 2 with concomitantly large anomalous Hall and Nernst angles 15,16 . Therefore, it is an ideal system to explore the relation between the off-diagonal anomalous variables associated to charge (Hall-effect) and heat/entropy (thermal Hall-effect) transport since this is a relatively unexplored subject from both the theoretical and experimental perspectives 26 .
Although the anomalous transport variables were found to satisfy the Wiedemann-Franz (WF) law in few compounds 27 , Mn 3 Ge is claimed to violate it due to a mismatch in thermal and electrical summations of the Berry curvature over the Fermi surface 26 . At first glance, Fe 3−x GeTe 2 offers a relatively simple magnetic system to test its general validity for the anomalous transport variables since previous assertions in favor of its violation (see, e.g. 28,29 ) were claimed to result from artifacts 26,27 .
This compound has a potential for spintronics applications through the electrical control of its magnetic domains and skyrmions. Neutron scattering and magnetization measurements reveal a ground state with magnetic moments pointing collinearly along the c-axis 9,30 as confirmed by us. This collinear spin arrangement leads to stripe like magnetic domains according to in-situ Lorentz transmission electron microscopy 12, 13,31 . Remarkably, application of a magnetic field along the c-axis, induces the formation of magnetic bubbles or magnetic skyrmions as those domains with spins pointing against the field are suppressed by it 12, 13 . Skyrmions could result from the Dzyaloshinskii-Moriya interaction since the inequivalent Fe sites form a lattice that lacks inversion symmetry 23 . The size of these domains are susceptible to manipulation via pulses of electrical current which apparently can also induce skyrmion bubbles 13 . As Fe 3−x GeTe 2 still exhibits robust ferromagnetism with a strong perpendicular anisotropy even when mechanically exfoliated down to the monolayer limit 32 , it has potential for 2D spintronics applications.
In this manuscript, we report on novel anomalous Hall, Nernst and thermal Hall effects in Fe 3−x GeTe 2 having clear topological origin associated to magnetic field-induced spin textures. Typically, the anomalous transport variables are observed in solids characterized by broken time-reversal symmetry (e.g. ferromagnets) result from the spin-orbit coupling and are conceptually treated in terms of the Berry phase 33 . For magnetic fields µ 0 H along the inter-planar direction (c-axis) and electrical currents flowing a long a planar direction (a-axis), we confirm the observation of a very large anomalous Hall conductivity σ xy accompanied by a concomitantly large anomalous Nernst coefficient S A xy and a very large anomalous thermal Hall signal κ A xy . In contrast to Ref. 26, we find that the anomalous variables in Fe 3−x GeTe 2 do satisfy the Wiedemann-Franz law over the range of temperatures measured, i.e. 2 to 225 K. However, the diagonal as well as the off-diagonal components of the thermal transport variables reveal anomalies around ≈ 150 K, as well as around 50 K which corresponds to the onset of a change in the sign of the Nernst signal upon cooling. Concomitant anomalies are seen in the magnetization but not in the heat capacity or neutron scattering data, suggesting the possibility of a very small canting of the magnetic moments with respect to the inter-planar direction. Surprisingly, in Fe 3−x GeTe 2 we observe a novel type of anomalous transport variables even when the external magnetic field is aligned along the gradient of the chemical potential (or along any planar direction), e.g. implying a Hall-like signal in absence of Lorentz force. For this geometry, ρ A xy , S A xy , κ A xy all increase as the in-plane field µ 0 H increases, peaking around µ 0 H 4 T and then saturating beyond µ 0 H 6 T, the field where the magnetization M is also observed to saturate. The size of the effect is comparable among samples having distinct geometries (e.g. different inter-layer thicknesses t). Therefore, we conclude that a field dependent chiral spin texture leads to a finite fieldinduced spin chirality that affects the Berry phase of the charge carriers leading to hitherto unreported anomalous Hall, Nernst and thermal Hall effects having a topological origin.
Monte-Carlo simulations that incorporate small Dzyaloshinskii-Moriya and biquadratic exchange interactions yield spiral spin textures for the domain walls separating the serpentine domains with the emergence of skyrmions, and possibly also skyrmion tubes, upon application of an external magnetic field. An asymmetric anomalous planar Hall-effect is also seen in the paramagnetic state suggesting that even in the absence of magnetic order Fe 3−x GeTe 2 bears non-trivial electronic topology.

A. Basic properties
The onset of ferromagnetic order in a Fe 3.05 GeTe 2 single-crystal at T c ≈ 210 K is observed as a sharp increase of the magnetization M per Fe atom as a function of the temperature T (Fig. 1a). A mild anomaly is seen in M (T ) for T s between 150 K and 170 K, a range of temperatures claimed to coincide with the onset of a Kondo lattice 10 in this d -electron system. Below 100 K, one observes a pronounced deviation between traces collected under zero-field (blue) and field cooled (magenta) conditions due to the development of striped or labyrinthine domains 12,13,31 with their opposite magnetic moments oriented along the c-axis.
The same panel plots the µ Fe extracted from the scattered intensity by the (100) magnetic Bragg peak (associated with the out of the plane magnetic moment) as a function of T . As seen, µ FE decreases continuously as T increases hence not providing evidence for additional magnetic phase-transitions; the Bragg peak at (002), which is sensitive to in-plane moment, remains constant through T c indicating the absence of an in-plane moment at zero-field.
Neutron scattering supports previous reports indicating a collinear ferromagnetic ground state with moments along the c-axis 9 although it leads to skyrmions 12,14,31 and chiral spin spirals at its surface 22 . A more detailed neutron scattering study exploring a broader region in k -space is required to reconcile these contrasting observations. The derivative of M as a function of T (Fig. 1b) highlights the magnetization anomalies, corresponding to the middle point T c as well as subtler anomalies around 150 K and 50 K. In SI, we provide M (T ) for a second sample, along with its derivative as a function of T revealing clearer anomalies at T c = 204 K (Fig. S1), and also at T = 156 K and 78 K. The sharpness and exact locations in temperature of both low-temperature anomalies are sample dependent, and contrast with the heat capacity data (Fig. S2) that reveals a single anomaly at T c = 215 K in good agreement with the change in slope seen in the resistivity ρ xx as a function of T (Fig.   1c). In our opinion, a plausible scenario for the anomalies in M (T < T c ) are subsequent small variations in the tilt angle of the moments with respect to the out of the plane caxis. Broad anomalies centered around these temperatures are also seen in the thermal transport, namely in the thermopower S xx (T ) and thermal conductivity κ xx (T ) measured show a couple of anomalies below T c implying that these are intrinsic to the material and not due to Fe deficiencies.
We can further understand the behaviour of M (T ) via atomistic simulations 34-36 using the following spin Hamiltonian: where i, j represent the atoms index, α, β = x, y, z, J αβ ij represents the exchange tensor that includes the anti-symmetric exchange − the Dzyaloshinskii-Moryia interaction (DMI), K ij the biquadratic exchange interaction 34 , D i the uniaxial anisotropy, which for Fe 3−x GeTe 2 is orientated out of plane (e = (0, 0, 1)) and B the external magnetic field applied during the field cooling (see additional details in SI). It has been previously reported that higher-order exchange interactions are fundamental in the description of 2D magnetic layers [34][35][36] . We also noticed that Fe 3−x GeTe 2 develops substantial biquadratic interactions in its magnetic properties. We estimated a magnitude of K ij within the range of 1.0 to 1.5 meV, which follows those previously calculated for several vdW materials 34 . The inclusion of K ij provides the best agreement between atomistic simulations and measurements as can be observed in  spectively. The data was collected at several temperatures indicating that i) the c-axis is the easy magnetization axis, and ii) the saturation moment µ ≈1.2 µ B per Fe atom matches the zero-field ordered moment extracted from neutron scattering as T → 0 K. However the extracted magnitude of µ is smaller than that (≈3 µ B ) measured from the Curie-Weiss susceptibility and neutron scattering data above T c 9 . We observed that for fields along the ab−plane, the integration of the neutron scattering Bragg peak (whose intensity is given by both the lattice and magnetic order scattering) reveals a continuous decrease in the out-of-plane magnetic moment (Fig. 2c) in favor of the planar one (Fig. 2d). This is consistent with the planar magnetization data as a function of µ 0 H which increases continuously with field. Our analysis suggests that the Fe moments originally aligned along the c−axis cant progressively towards the planar direction as the field increases. This variation in spin orientation might contribute to generate non-coplanar spin textures between the three Fe moments within the formula unit (e.g. spin chirality) leading to hitherto unreported transport properties.
B. Anomalous transport variables for magnetic fields along the inter-planar direction and currents along a planar direction Usually the anomalous transport properties of magnetic compounds are measured in a configuration implying thermal/electrical currents oriented perpendicularly to the magnetic field which is dictated by the texture of the Berry curvature at and below the Fermi level 33 .
For fields aligned along the inter-planar direction 16 , the anomalous Hall resistivity ρ A xy (Fig.  3a) scales with the magnetization M and is given by ρ xy = λM ρ n xx with n ≈ 2, where λ represents the strength of the spin-orbit coupling, M the magnetization, and ρ xx the longitudinal resistivity. For the many crystals measured in this work, ρ A xy (T ) consistently tends to saturate at a value within 10−12 µΩ cm. We obtained a Hall angle θ H = σ A xy /σ xx 0.04 at T = 25 K (Figs. 3a and 3b) roughly smaller than that previously reported (≈0.07) 16 .
Nevertheless, this value of θ H is consistent with the Fe deficiency δ, which can cause Hall angles in the range of 0.04 to 0.085 15 . Moreover, ρ A xy and σ A xy decrease slightly upon cooling below T = 50 K (Fig. 3b) consistently with previous works 15, 16 . The Nernst signal, where a transverse electric field E y is generated by a thermal gradient ∇T x under an external field where µ 0 H z is the magnetic field along the interlayer direction, was collected on the same crystal using the same electrodes (Fig. 3c) and reveals a maximum in the neighbourhood of 150 K followed by a change in sign below 50 K.
At 150 K one obtains a Nernst angle θ N = S A xy /S xx 0.073 which is slightly smaller than 0.09 earlier reported 16 .
The Nernst signal results from a combination of terms in the thermoelectric α αβ and charge conductivity σ αβ (α, β = x, y, z) tensors: where the transverse thermoelectric conductivity is given by: and Ω z is the z component of the momentum integrated Berry curvature, and the only term that can lead to a change in the sign of S xy in Eq. 2 is the transverse thermoelectric conductivity term α A xy . To expose this point, we use S A xy (T ) (Fig. 3d) and Eq. 2 to calculate α A xy (T ) via: σ xx (T ) = ρ −1 xx (T ) (Fig. 1c), α xx (T ) (Fig. 1e), and σ A xy (T ). It results that α A xy (T ) (Fig. 3d) follows the overall behavior of S A xy (T ) also changing its sign below T ≈30 K. Equation 3 implies that this change in the sign of α A xy (T ) ought to result from a sharp reconfiguration of the Berry curvature at the Fermi level occurring below ≈50 K. A topological transition either associated with a new magnetic texture or an electronic Lifshitz transition likely coupled to the magnetic order is feasible to occur. In contrast, the change in the sign of α xy observed at T c is associated to a change in the entropy density.
The final anomalous transport variable, that is, the thermal Hall or Righi-Leduc con-

C. Wiedemann-Franz law
Following the anomalies observed in κ xy , it is pertinent to ask if the observed anomalous transport quantities in Fe 3−x GeTe 2 would satisfy the Wiedemann-Franz (WF) law, i.e. κ xy /T σ xy L 0 where L 0 is the Lorentz number. This would imply that the same carriers would transport heat and charge. Such issue was recently addressed for both Mn 3 Sn 27 and Mn 3 Ge 26 compounds that are claimed to display Weyl nodes relatively close to their Fermi level 6 . The former is claimed to satisfy the WF-law whereas the latter shows a pronounced deviation approaching room temperature that is not attributed to inelastic scattering 26 .
The aforementioned anomalies that are quite marked in κ xy /T (Fig. 4a) contrast with the smooth evolution of σ A xy = −ρ A xy /(ρ 2 xy + ρ 2 xx ) ( Fig. 4b) upon decreasing T . Although both quantities were measured on the same crystal, which precludes the errors inherent to the measurement of its geometrical factors, there is an intrinsic experimental error within these measurements as illustrated at T = 250 K (Fig. 4c). Within the conservative error bars estimated by us, the anomalous quantities in Fe 3−x GeTe 2 seem to satisfy the WF-law.
D. Anomalous transport variables for magnetic fields along the gradient of the chemical potential Subsequently, we discuss the most intriguing aspects of the anomalous transport properties of Fe 3−x GeTe 2 . When the magnetic field is oriented along the gradient of chemical potential (parallel to either the applied current or thermal gradient), we observe anomalous planar Hall and anomalous planar heat transport variables that are truly antisymmetric as a function of field orientation (see Fig. 5). This observation is not to be confused with the conventional planar Hall-effect discussed for example, in the context of Weyl semimetals 37 , that is a measure of the anisotropy of the magnetoresistivity as a function of field orientation and therefore is an even signal of the magnetic field. In contrast, our observations are an odd function of magnetic field.
At lower T s the maximum value of ρ A xy , that ranges from ≈ 3 µΩ cm and ≈ 8 µΩ cm is sample dependent, indicating that sample quality, the exact slight deficiency in the occupancy of Fe, and errors in the precise determination of t play a role on the extracted numbers.
We performed detailed checks for experimental artifacts, more specifically whether the anti-symmetric signal could be caused by slight misalignments of the magnetic field (see, Fig. 5, S4 and S5 in SI), but in the process we became aware of Ref. 38 showing similar experimental results. Ref. 38 suggests that this unexpected observation would result from an internal gauge field resulting from a complex spin texture associated either with the formation of skyrmions 12,14,31 or a complex non-coplanar spin texture 22 . We observed this anomalous planar Hall-effect in more than 6 samples of different geometries and thicknesses (see, Figs.5, S4 and S5), despite differences in sample quality and Fe deficiency, suggesting that this effect indeed has a topological origin. Concerning this anomalous Hall signal for currents along the magnetic-field (Figs. 5, S4 and S5), one of our initial concerns was inhomogenous current distribution with a fraction of it flowing along the inter-planar direction. Although this might create a Hall-like signal for currents originally expected to flow along a planar direction aligned along the field, it would not explain the observation of this anomalous Hall signal for currents flowing along a planar direction but oriented perpendicularly to the field.
We also measured the anomalous variables of Fe 3−x GeTe 2 for fields applied along the gradient of the chemical potential through pulsed methods. For these measurements, we used an electrical current (or thermal heat) pulse method to minimize sample self-heating within the vacuum cell. Although ρ A xy for j µ 0 H collected through this experimental method is subjected to a poorer signal to noise ratio (Fig. 6a), it yields essentially the previously discussed behavior (Fig. 5). For instance, one sees that ρ A xy at the lowest T s displays a maximum in the neighborhood of µ 0 H ≈ 4 T (Fig. 6b). This behavior is followed by the Nernst signal S A xy (µ 0 H j Q ) collected on the same sample ( Fig. 6c) which also shows a maximum around the same field value. Although not seen in this data-set, due to limitations in the signal to noise ratio, we also observed a change in the sign of S A xy upon cooling below T = 50 K in another crystal using the constant heat gradient method and a different experimental set-up (Fig. S6).
, with the maxima seen in both thermal transport variables as a function of µ 0 H disappearing as T is lowered.
This contrasts with ρ A xy (µ 0 H j) whose maxima increase as T is lowered down to ≈ 50 K, from which point ρ A xy (µ 0 H j) decreases slightly. We understand the shape of the anomalous planar variables (Figs. 5 and 6) as resulting from a magnetic-field induced canted and non-coplanar spin texture that leads to a fielddependent spin chirality χ ijk that acts as an effective gauge field and reaches a maximum value in the neighborhood of µ 0 H 4.5 T. Remarkably, as the moments align along the field thus suppressing χ ijk (µ 0 H), ρ A xy remains finite instead of reaching zero. We speculate that this results from the intrinsic topological nature of the electronic band structure of To support this assertion, we point to the observation of an anomalous planar Hall signal already in the paramagnetic state, or for 220 K < T ≤ 300 K (Figs. 7, S4 and S5). In contrast to what is seen in the ferromagnetic state, where ρ A xy shows a maximum somewhere between µ 0 H = 4 T and 5 T (or where the planar magnetization begins to saturate), this Hall like signal initially increases linearly with field but tends to saturate as µ 0 H increases beyond 6 T. This implies, within the gauge field scenario proposed by Ref. 38 , that Fe 3−x GeTe 2 would already display a topological non-trivial character in its paramagnetic state. Given the absence of magnetic order, its origin would have to rely on its electronic band structure due, for example, to the existence of a Dirac nodal line as proposed by Ref. 15 , although the electronic structure calculations in Ref. 15 explicitly take into account the FM order. Therefore, Fe 3−x GeTe 2 might provide an unique example of a compound displaying coexisting mechanisms affecting the texture of its Berry phase for fields applied along a planar direction.

E. Non-trivial topological spin textures
We can shed some light on the unique magnetic features observed on the experimental results using atomistic spin dynamics [34][35][36] considering the spin Hamiltonian in Eq.1. The system is thermally equilibrated above the Curie temperature (T = 300 K) and then linearly cooled for 2 ns to a temperature of T = 0 K, followed by a 1 ns relaxation at zero-temperature ( Figure 9). Following the results of M (T ) and the importance of higher-order interactions ( Fig. 1a) in the magnetic properties of Fe 3 GeTe 2 , a biquadratic nearest-neighbours exchange of K ij = 1 meV has been used. We also included an in-plane DMI value of 10% of the Heisenberg exchange implemented following the symmetry presented in Laref et al. 23 . In the absence of an applied field, the ground-state is a stripe domain phase (Fig. 9a,c) with a topological number Q 39 (see section III in Methods for details) that indicates a trivial topology (Q = 0.05) or no topological protection. Once a magnetic field is applied perpendicularly to the surface Néel skyrmions are observed (Fig. 9b,d) resulting in a topological number Q = −10.63 in agreement with the amount observed on the top view (Fig. 9b). We also noticed that at each unit cell of monolayer Fe 3 GeTe 2 composing the bulk, there is the for-mation of three skyrmions resulting of the two inequivalent Fe atoms. That is, one skyrmion per Fe atomic layer disposed parallel to each other in the crystal structure (Fig. 9d). In such sandwich arrangement, substantial interactions occur between the skyrmions along the interlayer direction which aligned them roughly at the same position but at different heights.
The skyrmion diameter D sk can be extracted numerically from the computations (see section III in Methods for details) resulting in an average value of D sk = 2.424 nm. This skyrmion size can be directly related to the amount of DMI included in our simulations which can be additionally tuned at different magnitudes 40 . We however used a more qualitative approach to show the existence of non-trivial spin textures into the system.
It is worth mentioning that the formation of skyrmions on Fe 3−x GeTe 2 not only depends on DMI, but also on the biquadratic exchange K ij (Eq. 1) as shown in Figure 10. As long as K ij = 0 no skymions are observed under zero (Fig. 10a) or finite magnitudes of B (Fig.   10d). The scenario changes substantially as a finite value of K ij is included into the system with the stabilisation of skyrmions throughout the layer once the magnetic field is switched on (Fig. 10e-f). The different values of K ij induce variations on the size of the stripe domains as well as on the skyrmions and their topological number. At B = 0 T, the lengthscale of the stripe domains increases with the increment of K ij (Fig. 10b-c) become broader and more extended over the entire crystal. At B = 2 T, the skyrmions decrease in diameter and number which suggest a critical interplay between magnetic field and higher-order exchange processes. On that, the biquadratic exchange favors a perpendicular alignment of the spins in order to minimise the total energy which interacts with the in-plane DMI allowing the spins to precess around some perimeter. The higher the value of K ij the smaller the precession of the spins as they will be pointing along the same direction (e.g. out-of-plane). This implicates shorter radius and consequently less possibility of skyrmion nucleation.

III. CONCLUDING REMARKS
The Fe 3−x GeTe 2 transport data discussed here can only be understood in terms of a gauge field intrinsic to the electronic band structure of Fe 3−x GeTe 2 that acts as an effective "band magnetic field" that bends the electronic orbits to yield, for example, an anomalous Hall-like signal in the paramagnetic state in the absence of interaction between the electrical current and the external magnetic field. At lower temperatures within the ferromagnetic state, this anomalous and antisymmetric planar Hall signal is accompanied by anomalous planar Nernst and Righi-Leduc effects, which to the best of our knowledge have yet to be reported. We propose that the local Dzyaloshinskii-Moriya interaction, claimed to stabilize skyrmions 12-14 and chiral spin spirals at its surface 22 , favors a magnetic-field dependent canted and non-coplanar spin structure that scatters the charge carriers affecting their Berry phase and leading to the novel effects observed by us. To support this point, we used neutron diffraction to measure magnetic order with fields along the planar direction (Fig. 8). For fields along the planar direction, the area of the diffracted neutron peak corresponding to the in-plane moment increases sharply in the neighborhood of µ 0 H ≈ 4 K (Fig. 8a) which is consistent with a metamagnetic transition leading to the pronounced dips observed in the anomalous planar transport quantities (Figs. 5 and 6). At T = 50 K, and for fields along the a-axis, neutrons reveal a very pronounced hysteresis between field increasing and decreasing sweeps (Figs. 8b and 8c), pointing to either a temperature-dependent change in the magnetic anisotropy, a metamagnetic transition, or the emergence of another magnetic order. The elucidation of its origin will require a small angle neutron scattering study.
However, these observations, combined with a temperature dependent magnetic anisotropy found in our neutron study (Fig. S7), suggest changes in the magnetic domain structures and their textures, possibly leading to local regions with chiral spin order 22,41 instead of a simple collinear one, or even the possibility of a transition between both states as function of T . For example, as discussed in Ref. 41 , conical spin spirals would naturally lead to a planar topological Hall-effect (due to spin-chirality), as exposed here. We re-emphasize that heat capacity measurements do not support the notion of thermodynamic phase-transitions, as one would expect for electronic, magnetic, or structural transitions, albeit these could be subtle requiring further investigations.
Regardless of the precise field-induced spin texture, we have found that it also leads to planar topological Nernst and thermal Hall-effects that have yet to be reported or dis- Notice that for tilted magnetic fields one tends to observe the development of a pronounced peak in ρ A xy (Fig. S8) requiring fields in excess of µ 0 H = 10 T for its suppression. This peak is likely to correlate with the development of the aforementioned complex textures, e.g. skyrmions, skyrmion tubes, and spin spirals 41 , as implied by our Monte Carlo simulations, but become suppressed at higher magnetic fields where the Hall effect becomes dominated by the anomalous contribution. Additional neutron studies will be required to elucidate this behavior. However, we propose that deviations from spin collinearity as seen in the surface of this compound 22 , are likely to be responsible for the seemingly topological phase transition seen by the Nernst signal in the neighborhood of T = 50 K. Anomalies associated with this transition are seen in all other transport variables with the exception of the resistivity. These subtle transitions are not detected in the heat capacity either.
Finally, the Nernst effect is often used as a probe for topological excitations in quantum materials providing a means to convert heat into electricity 6,7 . Our study reveals a new way to detect, or expose such excitations as well as a new geometry for heat conversion that expands the horizon of thermoelectric technology. Notice, that the Curie temperature of

Neutron diffraction measurements
Neutron diffraction measurements were performed on the BT-4 triple axis spectrometer at the NIST Center for Neutron Research. The neutron energy was fixed at 14.7 meV (λ = 2.359Å), with pyrolytic graphite filters placed between the sample and analyzer to remove higher-order contamination. The sample was sealed in a helium environment and placed in a closed-cycle refrigerator to vary the temperature between 4 and 300 K. Temperaturedependent measurements were performed without a magnetic field; field-dependent measurements were performed with a magnetic field between 0 and 7 T applied within the ab-plane, approximately perpendicular to the (100)-axis.

Thermal transport measurements
Thermal conductivity and the thermal Hall-effect were measured using a one-heater threethermometer method. Additional electrical contacts allowed us to measure four-probe resistivity, Hall-effect, Seebeck and Nernst effects simultaneously. For the thermal transport measurements a heat pulse was applied in order to generate a longitudinal thermal gradient corresponding to a ≈ 3 % of the sample base temperature. After applying the heat pulse, the temperature of all three thermometers were monitored until they reached a stable condition

Magnetization and heat capacity measurements
Magnetization measurements under fields up to µ 0 H = 7 T were performed in a commercial superconducting quantum interference device magnetometer (Quantum Design -SQUID).
Heat capacity measurements through the thermal relaxation method were collected in a Quantum Design-PPMS.

Atomistic spin dynamics
The magnetization as a function of temperature were calculated by using Monte Carlo method as described in Ref. [34][35][36]

Topological number calculation
To calculate the skyrmion or topological number we follow the procedure shown in Ref. 39 .
The hexagonal lattice is split into nearest-neighbour triangles of spins. The topological charge associated with the system region defined by spins S 1 − S 4 will be given by the summation between two neighbour triangles (ω i 1 , ω i 2 ) as shown in Fig. S9 in the SI, where simulations, as extracted from Deng et al. 11 . The DMI value has been set to 10% of the exchange, with the symmetry as given in Ref. 23 .
the spherical area of each triangle is 47 : The total topological number will be given by the summation over all triangles in the system: The negative sign of the topological number is given by the orientation of the core of the spin structures. The topological number has been calculated per each individual atomic layer, then the average between the three atomic layers in the Fe 3−x GeTe 2 system has been performed.

Skyrmion radius calculation
In order to calculate the skyrmion radius, the profile of the magnetisation has been fitted to 40,48 : where x 0 is the position of skyrmion, R its radius and w the width of the domain wall. The radial direction the magnetisation profile provides a 360 o Néel domain wall 40,48 . Once the temperature is cooled in Fe 3 GeTe 2 , we find multiple skyrmions throughout the layers. In order to calculate numerically the average radius of the skyrmions, we determine the region along the x-axis where S z is minimum and fit the profile for y = constant via Eq. 6. The average skyrmion diameter D sk = 2R obtained for the first layer of skyrmions that consists of 11 skyrmions shown in panel c), Fig. 9 gave a value of D sk = 2.424 nm.

SUPPLEMENTARY MATERIAL
See the Supplementary Material for single-crystal x-ray refinement parameters from a

DISCLAIMER
The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology.

Competing interests
The authors declare no competing interests.

IV. DATA AVAILABILITY
Data are available from the corresponding authors upon request.     Integrated area of the Bragg peaks for both increasing (blue) and decreasing (orange) magnetic fields for a, the (002) peak at T = 3 K, b, the (002) peak at T = 50 K, and c, the (100) peak at T = 50 K. Increasing and decreasing field sweeps were taken at subsequently increasing temperatures, alternating between the (100) and (002) peaks at each field point.