# Dissipation properties of anomalous Hall effect: Intrinsic vs extrinsic magnetic materials

V Desbuis, D Lacour, M Hehn, S Geiskopf, L Michez, J Rial, Vincent Baltz,

J-E Wegrowe

► To cite this version:

V Desbuis, D Lacour, M Hehn, S Geiskopf, L Michez, et al.. Dissipation properties of anomalous Hall effect: Intrinsic vs extrinsic magnetic materials. Journal of Applied Physics, 2025, 138, pp.233906. 10.1063/5.0281311. hal-05430373

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Dissipation properties of anomalous Hall effect: intrinsic vs. extrinsic magnetic materials

V. Desbuis,¹ D. Lacour,² M. Hehn,² S. Geiskopf,² L. Michez,³ J. Rial,⁴ V. Baltz,⁴ and J.-E. Wegrowe¹, a)

¹) Laboratoire des Solides Irradiés, CEA/DRF/IRAMIS, UMR CNRS 7642, Institut Polytechnique de Paris, Ecole Polytechnique, F-91120 Palaiseau, France

²) Universit de Lorraine, CNRS, IJL, F-54000 Nancy, France

³) Aix-Marseille University, CNRS, CINaM, Marseille France

⁴) Universit Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-SPINTEC, Grenoble, France

(Dated: 27 November 2025)

The power efficiency of Hall devices is still an open question. How is it possible to extract power from a current produced by the Hall effect? In what extent it is related to the nature of the effective magnetic field produced in the material? In order to answer these questions, a comparative study of anomalous-Hall current injection and anisotropic current injection (through planar Hall effect) are studied in Hall devices contacted to a lateral load circuit. Hall currents are injected into the load circuit from three different kinds of magnetic Hall bars: Mn₅Si₃ altermagnet, Co₇₅Gd₂₅ ferrimagnet, and Ni₈₀Fe₂₀ ferromagnet. The current, the voltage and the power are measured as a function of the load resistance and the Hall angle. It is observed that the power dissipated for the three kinds of materials follow the same law as a function of load resistance and Hall angle, at the leading order in the Hall angle. Since the anomalous Hall effect in the altermagnetic Hall-bar is due to the intrinsic topological structure (i.e. due to the presence of a Berry phase in the reciprocal space), these observations suggest that the dissipative properties of anomalous Hall effect are dominated by the injection of electric charges accumulated at the edges, instead of the very mechanism responsible for it.

PACS numbers: 72.25.Mk, 85.75.-d

<footnote>a) Electronic mail: jean-eric.wegrowe@polytechnique.edu</footnote>
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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# I. INTRODUCTION

Is it possible to light a lamp - or feed a load circuit - with using the current produced by the Hall effect? What is the power that can be extracted from the Hall current? The question is not as simple as it appears, especially if the Hall current is produced by the anomalous Hall effect (AHE)¹,². Despite the fundamental character of the problem, this basic question is also of great practical interest for Hall sensors or Hall actuators in the context of low consumption electronic devices. The goal of this work is consequently to measure and to understand the power dissipated by an electric device composed of a Hall cross of different materials connected to a load circuit.

The Hall current is often assumed dissipationless because it is normal to the driving field, or in other terms because it is produced by a curl-force³ (in relation to gauge invariance and geometrical phase), or again in other words, because it is due to time-reversal symmetry breaking at the microscopic scales in relation with antireciprocal Onsager relations⁴. In the case of electric transport in magnetic systems, both the anomalous Hall effect (AHE), and the planar Hall effect⁵,⁶,⁸,⁹ (PHE) (or transverse magnetoresistance) can be measured. The former effect is described by an antisymmetric conductivity matrix, while the latter is described by a symmetric conductivity matrix¹⁰.

Since seventy years¹¹, the study of AHE has been the object of deep and fundamental theoretical developments¹²⁻¹⁵, especially since the description of the effective magnetic field generated in the reciprocal space by the Berry curvature¹⁶⁻¹⁸, following the works of Aharonov and Stern¹⁹, or Haldane²⁰. In the last years, the studies about AHE focused on the different possible origins of the effect, pointing-out the qualitative difference between *extrinsic* and *intrinsic* mechanisms²¹. The *extrinsic* mechanisms for AHE being related to spin-dependent scattering (typically skew scattering and side-jump scattering), while the *intrinsic* mechanisms are related to non-dissipative topological phase (i.e. Berry curvature, although we note that spin-orbit coupling is always needed to generate AHE). The last property is today intensively studied in new topological materials like Heusler alloys²², topological semimetals²³, or altermagnetic materials²⁴⁻²⁶. In this context, the experimental characterizations are based on the comparison between the longitudinal conductivity and the Hall conductivity, measured as a function of the temperature, of the crystal structure and composition, or of a controlled density of impurities.
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Yet, as shown in a recent experimental work$^{27}$, the electric power carried by the anomalous Hall-current can be measured directly from the Joule power dissipated in a load circuit, which is connected between the two lateral edges of the Hall bar: see Fig.1. This protocol allows an independent characterization of the AHE to be performed (the protocol has also been applied to anomalous Nernst effect$^{28}$). The power dissipated on the load resistance has been shown to be approximately proportional to the square of the Hall angle, and depends on the load resistance in agreement with theoretical predictions based on non-equilibrium thermodynamics$^{10,29}$. However, the observations about Co$_{75}$Gd$_{25}$ ferrimagnetic Hall-device reported in reference$^{27}$ were non-conclusive from the point of view of the role played by the Berry phase, because extrinsic scattering mechanisms could dominate in Co$_{75}$Gd$_{25}$, even at low temperature.

The aim of the present report is to extend these observations to various magnetic materials, including the altermagnet Mn$_5$Si$_3$ (thin crystalline layer) that generates *intrinsic* AHE (i.e. related to its Berry curvature)$^{9,30}$. More precisely, the measurements performed on the altermagnetic material are compared to that performed on Co$_{75}$Gd$_{25}$ ferrimagnetic and Ni$_{80}$Fe$_{20}$ ferromagnetic Hall bars of identical geometry and measured under the same experimental protocol. The Ni$_{80}$Fe$_{20}$ (thin polycrystalline layer) plays here the role of *yardstick* for the conventional dissipation, as it has a strong anisotropic magnetoresistance - hence a strong PHE- together with negligible AHE. The extension of the comparative study to PHE is also highly instructive in altermagnets (as shown in the case of MnTe$^{8,31}$).

The observations show that the profiles of the power dissipation as a function of both the load resistance $R_{load}$ and the Hall angle $\Theta$ are approximately superimposed (at the leading order in the Hall angle $\Theta$) for the three materials. The maximum power dissipated is approximately proportional to $\Theta^2$, and obeys the maximum power transfer theorem (maximum for the resistance matching $R = R_{load}$). These observations are in agreement with the phenomenological theory based on dissipative transport developed in reference$^{10}$ that describes the lateral current injection$^{10,29,32-34}$.

## II. PHENOMENOLOGICAL MODEL

The systems studied are Hall bars composed of a thin-film (defining the plane {$x, y$}) that are contacted to an electric generator along the longitudinal direction $x$. The usual Hall
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 1. : (a) Diagram of the Hall bar contacted to a load resistance. An electric generator imposes a constant current through the longitudinal bar (x direction). A transverse anomalous-Hall current is generated inside the materials (Mn$_{5}$Si$_{3}$, GdCo, and NiFe), and injected through the lateral electrodes (y direction). Angles $\theta$ and $\phi$ give the direction for the magnetization $\vec{m}$ (or for the magnetic order parameter for the alternate magnet). (b) Optical picture of a Hall cross with the definition of width $2\ell$ and length L.</figcaption></figure>

measurements consists in measuring the voltage on a voltmeter placed transversely between the two edges of the Hall bar, along the direction $y$ (open circuit configuration), while applying an external magnetic field. In magnetic materials, the magnetic field is replaced by the magnetization, which direction is defined by the radial angle $\theta$ and the azimuthal angle $\phi$ (see Fig.1). In antiferromagnets or alternate magnets, the magnetization is replaced by the Néel vector.

The protocol used here consists in closing the circuit by placing a load resistance $R_{load}$ connecting the two edges of the Hall bar, and measuring the current and the Joule power dissipated in this resistance: see Fig.1. This configuration allows the dissipative properties of the Hall-current (generated by both the AHE and the PHE) to be measured directly. The corresponding theoretical analysis has been performed in recent works$^{10,29}$, and the phenomenological description is summarized below.

The local electric transport properties of a magnetic material is described by the transport equation$^{5-7}$:

$$ \vec{\mathcal{E}} = \rho \vec{J} + \Delta\rho (\vec{m} \cdot \vec{J}) \vec{m} + \rho_H \vec{m} \times \vec{J} \quad (1) $$

where $\vec{\mathcal{E}}$ is the electric field, $\vec{J}$ the electric current density, and $\vec{m}$ is the unit vector which gives the direction of the magnetization (assumed uniform), $\rho$ is the resistivity of the isotropic
material, $\Delta\rho$ is the anisotropic resistivity, and $\rho_H$ is the anomalous Hall resistivity, that couples the two directions of the space in the plane of the Hall bar. If the magnetization is not totally saturated, an analysis of the magnetization states is necessary$^{27}$. Note that the energy of the carriers also contains the contribution due to the electric charge accumulation and screening, so that the effective electric field $\vec{\mathcal{E}}$ should be generalized with the gradient of the chemical potential$^{35}$.

In a first step, we assume homogeneous and isotropic materials. In addition, we assume that the electric current flowing in the $xy$-plane only. The Ohm's law Eq.(1) can then be written $\vec{\mathcal{E}} = \hat{\rho}\vec{J}$. The resistivity matrix $\hat{\rho}$ can be reduced to the $2 \times 2$ matrix in an arbitrary orthonormal 2D frame:

$$ \hat{\rho} = \rho \begin{pmatrix} 1 + \frac{\Delta\rho}{\rho}m_x^2 & \Theta_{pl} + \Theta_{an} \\ \Theta_{pl} - \Theta_{an} & 1 + \frac{\Delta\rho}{\rho}m_y^2 \end{pmatrix} \qquad (2) $$

where the diagonal term $\frac{\Delta\rho}{\rho}m_x^2 \equiv \frac{\Delta\rho}{\rho}sin^2(\varphi)sin^2(\theta)$ is the anisotropic magnetoresistance (AMR) that is measured longitudinally, along the $x$ axis. The expression $\Theta_{pl} \pm \Theta_{an}$ for the cross-coefficient of the conductivity matrix comes from the fact that, after removing the diagonal, any matrix is a sum of a symmetric matrix and an antisymmetric matrix. This decomposition is hence universal and is valid whatever the mechanisms responsible for it.

Note that the AMR is of the order of one percent in usual materials, so it is small with respect to unity ($\frac{\Delta\rho}{\rho} \ll 1$), and we will not consider the AMR contribution in the following. More interesting for the present study are the two Hall angles$^{36}$, defined as (see details in reference$^{10}$):

$$ \Theta_{pl} = \frac{\Delta\rho}{2\rho} \sin^2(\theta) \sin(2\varphi) \quad \text{and} \quad \Theta_{an} = -\frac{\rho_H}{\rho} \cos(\theta) \qquad (3) $$

where the index $pl$ stands for PHE and the index $an$ stands for AHE. Note that the material is characterized by the maximum values $\Theta_{max}^{pl} = \frac{\Delta\rho}{2\rho}$ reached for $\theta = \pi/2$, $\varphi = \pi/4$, and $\Theta_{max}^{an} = \frac{\rho_H}{\rho}$ reached for $\theta = 0$.

In the case of an open circuit ($R_{load} \to \infty$) the transverse current density $J_y$ vanishes$^{32}$ and the Hall voltage measured between the two lateral edges is, after integration of $\mathcal{E}_y$ over the length of the Hall bar:

$$ V_{xy}^{0}(\theta, \varphi) = 2\ell \rho I_0 (\Theta_{pl}(\theta, \varphi) + \Theta_{an}(\theta)) \qquad (4) $$
where $V_{xy}^0 = V^0(y = -\ell) - V^0(y = \ell)$ and the superscript $V^0$ stands for the open-circuit condition. From the experimental point of view, the relations Eqs.(3) allows the two effects, PHE and AHE, to be characterized in a univocal way while measuring both the voltage and the current in the load circuit. This expression of the Hall voltage justify the use of the term PHE, i.e. the reference to Hall despite the fact that the effect is just the transverse anisotropic magnetoresistance. We are aware that the name planar Hall angle could be problematic, and it was criticized since the first publication$^5$. The criticism is justified: the PHE is fundamentally different from Hall effects because it conserves the time reversal symmetry at the microscopic scale, while AHE breaks this symmetry. This striking microscopic property is precisely reflected at the macroscopic scale by the Onsager reciprocity relations of the first$^{37}$ kind (PHE), and of the second$^4$ kind (AHE) through the dichotomy between the symmetrical part (PHE) and the antisymmetrical part (AHE) of the conductivity matrix, as shown in Eq.(3). However, as will be shown in the following, this fundamental difference is not reflected in a striking way at the level of the Hall-current measured in a load circuit.

Due to the accumulation of electric charges at the edges, and the resulting non-equilibrium screening effect, the problem is not entirely defined by the transport equation Eq.(1) and the continuity equation, that are local. For this reason, a specific variational approach has been developed$^{10}$ in order to define the stationary states. The method is based on the minimization of the power dissipated in the system under the constrains imposed by the generator and the circuit's configuration. The power density takes the usual form: $p_J = \vec{J} \cdot \vec{\mathcal{E}}$, and the total power is obtained after integrating $p_J$ over the whole system, including the load circuit. The resulting Hall-voltage at the load terminals is given by the expressions:

$$V_{xy}^{pl} = 2\ell\rho J_x^0 \frac{1}{1+\alpha} \frac{\Theta_{pl}}{1-\Theta_{pl}^2} \quad \text{and} \quad V_{xy}^{an} = 2\ell\rho J_x^0 \frac{1}{1+\alpha} \frac{\Theta_{an}}{1+\Theta_{an}^2} \quad (5)$$

where the parameter $\alpha = R/R_{load}$ is the ratio of the resistance of the transverse part of the Hall bar over the load resistance $R_{load}$, and $J_x^0$ is the current injected by the generator at $\Theta = 0$. Beside the specific behavior of $\Theta(\theta, \phi)$ shown in Eq.(3), the only difference between AHE and PHE is contained in the sign $\pm$ that appears in the expression $1/(1 \pm \Theta^2)$.

On the other hand, the Joule power $P = V_{xy}I_{xy}$ dissipated at the load terminals by the
anomalous or planar Hall-currents injection reads:

$$P(\Theta_{pl}, \alpha) = P_0 \frac{\alpha}{(1+\alpha)^2} \frac{\Theta_{pl}^2}{1-\Theta_{pl}^2}, \quad \text{and} \quad P(\Theta_{an}, \alpha) = P_0 \frac{\alpha}{(1+\alpha)^2} \frac{\Theta_{an}^2}{1-\Theta_{an}^2}, \quad (6)$$

where $P_0$ is the power injected into the Hall bar for open circuit (or for $\Theta = 0$). As expected, the PHE dissipates more than the AHE.

The objective of the two next sections is to confirm experimentally the validity and the “universality” of the expressions Eq. (5) and Eq. (6) for the three specific materials.

### III. TRANSVERSE VOLTAGE FOR OPEN CIRCUIT CONFIGURATION

The characterization of the voltage $V_{xy}^0(\theta, \phi)$ for the open-circuit configuration is of fundamental importance in the present study, because we are exclusively interested in the currents generated by AHE or PHE injected into the lateral circuit. This means that two main artifacts has to be corrected that are (i) the thermoelectric contributions and (ii) the effect of the misalignment ($\approx 50$nm) of the two lateral contacts of the Hall cross (see Supplementary material).

The ferromagnetic material Ni$_{80}$Fe$_{20}$ (permalloy) has been chosen because of its strong PHE and negligible AHE ($\Delta\rho \gg \rho_H$)$^{38}$, while the ferrimagnetic alloy Co$_{75}$Gd$_{25}$ has been chosen because of its strong AHE and negligible PHE ($\rho_H \gg \Delta\rho$). The longitudinal current injected is $I_{xx} = 0.43$ mA for CoGd layer (longitudinal voltage 0.148 V) and 1 mA for NiFe (longitudinal voltage 25 mV). Both materials are measured at room temperature. In contrast, the altermagnet Mn$_{5}$Si$_{3}$ has been chosen because the AHE is generated by an intrinsic mechanism$^{30}$, due to the presence of a Berry curvature in the reciprocal space (together with spin-orbit coupling). The longitudinal current injected is $I_{xx} = 0.5$ mA (longitudinal voltage 0.22 V). The experiments on Mn$_{5}$Si$_{3}$ are performed at 70K, temperature at which the material exhibits the altermagnetic phase with the presence of spontaneous AHE and vanishing net magnetization$^{9}$. The out-of-plane signal $V_{xy}(\theta)$ (for open circuit configuration) is due to AHE, and a detailed description of the sample can be found in reference$^{30}$ (See also Supplementary material).

The experimental protocol for the voltage measurements $V_{xy}(\theta, \phi)$ at the open-circuit configuration is the following. If the magnetization is saturated by the external field $\vec{H}_{app}$, the direction of the magnetization $\vec{m}$ coincides with the direction of the vector $\vec{H}_{app}$. In that
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case, the magnetization state is well known, so that $\theta = \theta_{H_{app}}$ and $\varphi = \varphi_{H_{app}}$, with $\theta_{H_{app}}$ (resp. $\varphi_{H_{app}}$) being the radial (resp. azimuthal) angle of the applied field $\vec{H}_{app}$. Thus, the contributions of AHE and PHE to the voltage $V_{xy}$ can then be directly deduced from Eqs. (4). This is the case for the measurements of Ni$_{80}$Fe$_{20}$ due to the magnetic field in the plane. The red line accounts for the $PHE \propto \sin(2\varphi)$ profile. If the magnetization is not fully aligned with the applied field, a correction should be applied, from micromagnetic considerations.

<figure><img src="image_4.png" /><figcaption>FIG. 2. Left: Planar Hall voltages in Ni$_{80}$Fe$_{20}$ versus in-the plane azimuthal angle $\varphi_{Happ}$ of the applied field $H_{app}$ at $H_{app} = 1.5$ T. The red curve is the expected profile $\propto \sin(2\varphi)$ for saturated magnetization. Right: Anomalous Hall voltages in GdCo versus out-of-plane radial angle $\theta_{Happ}$ under the applied field $H_{app} = 1.5$ T. The magnetization of Co$_{75}$Gd$_{25}$ is not aligned with the applied field at 1.2 T. The red line is the result of the calculation of the equilibrium states of the magnetization for each orientation $\theta_{Happ}$ of the applied field (see reference$^{27}$).</figcaption></figure>
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<figure><img src="image_5.png" /><figcaption>FIG. 3. Hall voltages in Mn₅Si₃ versus radial angle θHapp for an applied field Happ = 1.2T (white circles), measured at T=70K. The blue arrows indicate the hysteresis. The current is injected along the [2\bar{1}10] direction³⁰. The red curve (Model) is the AHE with the cos(θ) behavior.</figcaption></figure>

Indeed, for Co₇₅Gd₂₅, we see that the voltage does not strictly follow the AHE profile proportional to cos(θHapp). However, after having corrected the effect of non-saturation, the calculated profile (see Eq.(3) and Eq.(4)), coincides with the measurements (as shown in Fig.2), thus validating the expected AHE ∝ cos(θ) behavior. The relation between θ and the angle of the applied field θHapp has been calculated numerically, as described in reference²⁷. In the case of the altermagnet Mn₅Si₃, the problem is more subtle because there is no real saturation (the magnetization is nearly zero but the order parameter depends on the magnetic field), and hysteresis loops can be seen in Fig.3 for θHapp = ±30. The deviation from the standard AHE (red curve in the figure) can be attributed to the role of the magnetocrystaline anisotropy of the crystal, but may also be due to the contribution of specific transport properties of the altermagnetic order parameter. However, as for the Co₇₅Gd₂₅ sample, the fact that the magnetization state is not well known is not an obstacle for the application of our measurement protocol (see below).

In order to compare the three different materials, the measured voltage has to be normal-
ized by the maximum value of the anomalous or planar Hall angles. These maximum values
are reached for Θ<sup>GdCo</sup><sub>max</sub> ≈ 8.10<sup>-3</sup> for Co<sub>75</sub>Gd<sub>25</sub><sup>27,38</sup>, Θ<sup>NiFe</sup><sub>max</sub> ≈ 9.6·10<sup>-3</sup> for Ni<sub>80</sub>Fe<sub>20</sub> for the
permalloy Hall bar, and $\Theta_{max}^{Mn_5Si_3} \approx 7.6.10^{-4}$ for the altermagnet. Note that for all three materials, the voltage $V_{xy}(\theta_{H_{app}}, \varphi_{H_{app}})$ plotted in Fig.2 and Fig.3 are centered at zero. However, voltage offsets have been systematically been removed (see *Supplementary Materials*).

## IV. LOAD CIRCUIT AND POWER EFFICIENCY

As shown in Fig.(4) below, when a load resistance $R_{load}$ is contacted to the lateral edges of the Hall bar - anything else being equal - the corresponding Hall voltage $V_{xy}$ decreases from $V_{xy}^0$ down to zero linearly with $R_{load}$ (tuned from 1Ω up to 12kΩ), while keeping the same angular profile as for the open circuit $V_{xy}^0(\theta, \varphi, H_{app})$. More precisely, the profiles of the voltage $V_{xy}(R_{load})$ follow the relation Eq.(5). In parallel, the measurement of the electric current $I_{xy}$ at the edges of $R_{load}$ shows a linear increase, while also keeping the same angular profile as the Voltage $V_{xy}^0$. This behavior corresponds to Ohm's law $I_{xy} = V_{xy}/R_{load}$ for the load resistance. These qualitative observations suggest that the electric charges accumulated at the edges of the Hall bar due to AHE or PHE are injected into the load resistance. The quadripole depicted in Fig.1 becomes a standard *Hall bar* when $R_{load}$ tends to infinite (open circuit: maximum charge accumulation), and it is equivalent to the standard *Corbino disk*$^{38,39}$ when $R_{load}$ tends to zero (zero charge accumulation).

Consequently, the Hall current $I_{xy}$ flowing into the lateral circuit dissipates the *Joule power* $P = V_{xy} \cdot I_{xy}$ inside the load resistance. The voltage, the current, and the power measured on the load resistance are plotted in Fig.(4) (raw data) as a function of the magnetic field for GdCo and Mn$_5$Si$_3$, and as a function of the in-plane angle $\varphi = \varphi_{H_{app}}$ (under a saturation field) for NiFe in the last column of Fig.(4). Note that the reason why this measurement on NiFe is not presented as a function of the applied field is that the coercitive field is too small in the plane of the sample. Furthermore, for practical reason, the power $P_0$ injected from the generator is not the same for the three samples (see Fig.5 below for normalized data).
<figure><img src="image_6.png" /><figcaption>FIG. 4. Effect of the load resistance, for the profiles of the Voltage $V_{xy}$ (first column), the current $I_{xy}$ (second column), and the power $P = V_{xy} \cdot I_{xy}$ (third column) dissipated in the load resistance as a function of the applied magnetic field for GdCo (first ligne) and Mn$_5$Si$_3$ (second ligne), and as a function of the in-plane angle for NiFe (third ligne).</figcaption></figure>

Furthermore, the profiles of the power as a function of the inverse of the load resistance $1/R_{load}$, and as a function of the Hall angle $\Theta$ are plotted in Fig. 5 for the three materials (NiFe, GdCo and Mn$_5$Si$_3$), after being normalized by the power $P_0 = V_{xx} I_{xx}$ (constant for a given sample) injected by the generator in the open circuit configuration. Furthermore, the load resistance $R_{load}$ is normalized with the resistance $R$ of the material, corresponding to the region defined by the cross of the Hall device (see Fig. 1-b)), through which the lateral Hall-current is flowing (as studied in reference$^{27}$). The parameter $R$ can be measured with injecting a known current along the transverse direction in the Hall cross. The values obtained are $R^{GdCo} = 270 \ \Omega$ for Co$_75$Gd$_{25}$, $R^{NiFe} = 135 \ \Omega$ for permalloy, and $R^{Mn_5Si_3} = 2800 \ \Omega$ for the alternmagnet.

The left plot of Fig. 5 shows the dependence of the normalized dissipated power $P/(P_0\Theta_{max}^2)$ as a function of $\Theta/\Theta_{max}$, and the right plot of Fig. 5 shows the normalized dissipated power
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as a function of the ratio $R/R_{load}$. The first important observation is that the power is of the order of $\Theta_{max}^2$, i.e., it is a factor $10^{-4}$ to $10^{-7}$ of the power injected from the generator. From this point of view, PHE and AHE (intrinsic and extrinsic) are indeed dissipative, but the resulting dissipation is small.

The second important observation is that the normalized power profiles can be superimposed at the leading order of $\Theta$ as seen on Fig. 5. The correction that allows PHE to be differentiated from AHE is contained in the factor $1/(1 \pm \Theta^2)$ in Eq. (5). Considering that Hall angles for all materials are below $10^{-2}$, the factor between the two extreme cases AHE $(1/(1+\Theta^2))$ and PHE $(1/(1-\Theta^2))$ does not exceed $10^{-4}$, compared to 1. This correction can only be seen with a resolution of the next to leading order in $\Theta$. In the present work (which is a first comparative study based on this experimental protocol), the next to leading order resolution has not been reached, and the discussion will then focus on the similarity of the three profiles at the leading order for all three materials.

<figure><img src="image_7.png" /><figcaption>FIG. 5. Normalized dissipated power in the load resistance for the three materials as a function of Normalized Hall angle (left) and Normalized load resistance (right). Continuous black line are deduced from the model discussed in the text. The error bars on the left indicate the uncertainty due to the hysteresis shown in Fig. 3.</figcaption></figure>

A third observation can be made on Fig. 5: the normalized power profiles can also be superimposed as a function of $R/R_{load}$ and all measured points follow the curve given by Eq. (6). The maximum of the normalized power equals 0.25 and is reached for the same value $R/R_{load} = 1$ for all three materials. The results shown on Fig. 5 demonstrates the “universality” of the power dissipation of Hall currents for the three specific materials studied. More precisely,
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the dissipated power do not depend on the mechanisms involved, nor on the materials used. These very properties of the different mechanism for AHE and anisotropic magnetoresistance are contained in the two parameters $\Theta_{max}$ and the resistivity of the material through the resistance $R$.

V. CONCLUSION

We have studied the effect of the transverse current injection and the corresponding dissipation into a load circuit, due to PHE and AHE in three different materials. The materials have been chosen in order to compare Hall-like currents generated by very different mechanisms. Indeed, the Co₇₅Gd₂₅ ferrimagnetic material shows a typical AHE due to extrinsic effect, while the altermagnet Mn₅Si₃ shows spontaneous AHE, the amplitude of which is due to intrinsic properties related to Berry curvature in the reciprocal space (when spin-orbit coupling still triggers the effect). On the other hand, the Ni₈₀Fe₂₀ material shows a typical PHE effect (with negligible AHE) due to s-d scattering.

The first observation shows that the amplitude of the power is of the order of the square of the Hall angle $\Theta^2$ in all cases. This is a small quantity in general, varying from $10^{-4}$ to $10^{-7}$ fraction of the power $P_0$ injected from the generator in our case (some exceptions of large Hall angles have, however, been reported, e.g. measured in topological semimetals$^{40}$). The second observation- presented in Eq. (6) - shows that the curves of the normalized power are superimposed for the three materials, at the leading order in $\Theta$.

This behavior is predicted in a phenomenological model based on the Onsager reciprocity relations and the second law of thermodynamics$^{10,29}$. Both the model and the experiments show that the nature of the microscopic mechanisms responsible for the different kinds of Hall effects (intrinsic topological properties, extrinsic scattering, etc.) do not play a crucial role for the phenomenological electric properties of the Hall devices. The main physical picture, at the macroscopic scale, is the accumulation of the electric charges at the edges that are injected into the load, as shown in the measurements plotted in Fig(4).

This study is being pursued in order to reach the next to leading order in the Hall angle $\Theta$. More importantly, the resistance matching property shown in the present work should find an application.

Beyond, from a more fundamental viewpoint, the characteristic proper to altermagnets$^{31,41}$
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should also be observed through the power dissipated in the load circuit at the next to leading order. An appropriate description would then be necessary, that goes beyond the basic phenomenological model proposed here.

## VI. SUPPLEMENTARY MATERIAL

This supplementary materials contains four sections, that are presented as complement to the main text. The four sections are the following:

* Section I ("Sample preparation") presents the basic information regarding the fabrication of the three samples Co<sub>75</sub>Gd<sub>25</sub>, Ni<sub>80</sub>Fe<sub>20</sub>, and Mn<sub>5</sub>Si<sub>3</sub>.

* Section II ("Measurement of transverse currents") describes the configuration of the circuit.

* Section III ("Thermoelectric and geometric offsets") gives a short description of the spurious effects, that are contained in the part of the signal that does not depend on the applied magnetic field.

* Section III ("Correction due to the geometry of the contacts") shows the small deviation of the current lines from the uniform longitudinal current density $J_x^0$ due to the presence of the lateral contacts.

## VII. DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

The groups of the CINaM and SPINTEC is supported by French national research agency (ANR-22-EXSP-0007) and the Deutsche Forschungsgemeinschaft (DFG), Germany (Project HEXAS - Grant No. ANR-24-CE92-0038-02). The group of IJL is supported by the inter-disciplinary project LUE MAT-PULSE, part of the French PIA project Lorraine Université d'Excellence reference ANR-15-IDEX-04-LUE and by the ANR through the France 2030 government grants PEPR SPIN SPINMAT ANR-22-EXSP-0007.
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