The anomalous-Hall current injection is studied in a Hall device contacted to a lateral load circuit. This anomalous-Hall current is generated inside a Co 75 Gd 25 ferrimagnetic Hall bar and injected into a lateral load circuit contacted at the edges. The current, the voltage, and the power are measured as a function of the magnetization states, the load resistance R l, and the temperature. It is shown that (1) the resistance associated with the anomalous-Hall current flowing inside the Hall bar is that of the portion of the ferrimagnet located between the lateral contacts, (2) the role of the non-uniformity of the current due to the lateral contacts is small, (3) the maximum power efficiency of the current injection into the load circuit corresponds to the condition of the resistance matching of the two sub-circuits, and (4) this maximum power efficiency is of the order of the square of the anomalous-Hall angle. These observations are in agreement with recent predictions based on a non-equilibrium variational approach.

The search for low power consumption electronic devices is one of the main motivations for the development of spintronics. Indeed, the spins attached to the charge carriers allow a direct manipulation of the magnetization states. Accordingly, the power used is that of the spin degrees-of-freedom and not directly the electric power. Yet the transport of the spins attached to the charge carriers follows the thermodynamic rules that determine Joule dissipation. In the case of Hall effect (HE), anomalous Hall effect (AHE),1–7 or spin-Hall effect (SHE),8 the Joule dissipation is minimized due to the presence of an effective magnetic field that breaks the time-invariance symmetry at the microscopic scale.9 The effect of the effective magnetic field on the electric carriers is then taken into account by a typical supplementary Hall-like term in Ohm’s law10,11 [see Eq. (1)].

It is well-known that the force associated with a magnetic field—typically the Lorentz force for the HE—cannot produce mechanical work in vacuum. More generally, due to the Onsager reciprocity relations, it is often assumed that the Hall current produced by Hall-like effects is dissipationless.12–17 However, it is not necessarily the case: typically, the power associated with the Hall voltage in a perfect Hall bar is indeed null, but the power associated with a Corbino disk (all other things being equivalent) dissipates.11,18 The Hall bar contacted to a lateral circuit presents an intermediate state between these two limiting cases of zero load resistance (Corbino disk) and infinite load resistance (perfect Hall bar).

The injection of spin current produced by SHE or AHE has attracted much attention in the context of spin–orbit torque (SOT) effects or charge-to-spin conversion mechanisms. Indeed, SOT allows a ferromagnetic layer to be reversed by the injection of spin-polarized current from an adjacent non-magnetic Hall bar (for the SHE) or from an adjacent magnetic Hall bar (for the AHE).19–23 The SOT device is hence related to the Hall bar contacted to a lateral load circuit as described above (the lateral layer would then play the role of the lateral resistance). However, in the literature, the parameter used so far for the evaluation of the power efficiency is the Joule dissipation J s w 2 ρ that is generated by the switching current density J s w injected from the generator (or ρ / θ s o t 2 as shown in Fig. 8 of the review Roadmap of Spin–Orbit Torques24), without taking into account the load resistance of the adjacent ferromagnetic layer.

The goal of the present report is to study the anomalous Hall current that flows inside the Hall bar, and the conditions that govern the injection of this current into a load circuit. In the experimental protocol proposed here (see Fig. 1), the load circuit is devoted to measure the amount of Joule power injected from the ferrimagnetic layer due to the AHE, regardless of the spin properties.25 

It is shown that the transverse resistance experienced by the anomalous-Hall current flowing from one edge to the other edge inside the sample is defined by the resistivity ρ of the ferrimagnet and the section defined by the lateral electrodes. The role of the non-homogeneity of the current lines due to the electrodes is not significant [it is characterized by the pink zone in Fig. 3(b)]. Furthermore, the power efficiency is measured to be of the order of the square of the anomalous Hall angle θ A H 2 10 5 and a sharp maximum is observed that corresponds to the resistance matching between the magnetic layer and the load circuit. Note that these observations are not trivial because the system does not obey Kirchhoff’s laws and cannot be described by a simple lumped-resistor circuit.26 These observations are in agreement with recent predictions based on a non-equilibrium variational approach.27–29 Beside, these results show that—like for direct spin injection into semiconductors30,31—the impedance matching could also be a crucial issue for transverse spin-current injection into the adjacent layers of a Hall bar for SOT.

The sample is a Co 75 Gd 25 layer of thickness t = 30 nm, sputtered on a glass substrate and the buffer layer. The choice of the ferrimagnet Co 75 Gd 25 is motivated by its high amplitude of A H E, its negligible planar Hall effect (i.e., its negligible anisotropic magnetoresistance), and its negligible magnetocrystalline anisotropy.11 The magnetic layer is sandwiched between 5 nm thick Ta buffer and 3 nm thick Pt cap. As shown in Fig. 1, the length of the Hall bar is L x x = 400 μ m and the width W x x = 200 μ m. The magnetic properties and the transport properties of the thin layers have been previously studied (see the supplementary material). The magnetization is uniform for the quasi-static states, which are the only states under consideration in this study. The out-of-plane shape anisotropy corresponds to a field of about 0.8 T defined by the magnetization at saturation. The structure of CoGd is amorphous and there is no texture. The study of the magnetization states is presented in the supplementary material.

At room temperature, the longitudinal voltage is Δ V = 0.148 V, the longitudinal current flowing through the CoGd is 0.43 mA, and the resistance of the layer is R C o G d 344 Ω. The anomalous Hall voltage per Tesla is V x y 0 = 0.886 mV T 1 (the index 0 sands for the open lateral circuit). The anomalous Hall angle per Tesla θ A H = V x y 0 Δ V = 6 10 3 T 1 is deduced, leading to the value of the anomalous resistivity per Tesla of about ρ A H 1.02 10 2 μ Ω m T 1.

Gold contact pads are formed thanks to a standard two-step UV lithographic process [yellow pads shown in Fig. 1(b)]. As shown in Fig. 1, the pads at the extremity along the x axis of the Hall bar allow the Co 75 Gd 25 layer to be contacted to the electric generator, and two opposite lateral pads at the edges ( L x y = 450 μm and W x y = 200 μ m) define the lateral terminals for the load resistances R l, range between 1 Ω and 100 k Ω (decade resistance box). Voltmeters and ampermeter allow the lateral voltage V x y and the lateral current I x y to be measured while injecting a DC longitudinal current (see Fig. 1). An external magnetic field H a p p varying between ±1.5 T is applied at an angle Φ Happ defined with respect to the direction of the injected current I x x.

Figure 2 shows the lateral voltage V x y (a), the lateral current I x y (b), and the power P = V x y I x y (c) measured on the load circuit and plotted as a function of the amplitude of the magnetic field at Φ Happ = 90 ° (applied normal to the layer) at room temperature. The load resistance R l is used as a parameter, depicted in the color code shown in the right (green for 1  Ω up to red for 8000  Ω).

Figure 2(d) shows the profiles of the anomalous Hall current as a function of the angle Φ Happ in a load resistance of R l = 270 Ω [close to the maximum of the power in the profile of Fig. 3(b)], for three different amplitudes of the applied field. At H a p p = 1.5 T (at saturation), the magnetization direction m follows approximately the applied field. For H a p p < 1 T, the angle of the magnetization and that of the magnetic field are significantly different due to the shape anisotropy.

The continuous lines in Fig. 2(d) correspond to the angular profile of the AHE: V x y ( H a p p ) m ( H a p p ) . e z, where m is the magnetization direction and e z gives the direction normal to the plane of the Hall bar. The contribution of the Planar Hall effect (PHE) of the C o G d layer is about two orders of magnitude below the AHE and can be neglected. The important point is that the observed profiles show a typical signature of the AHE. It is worth pointing out that the analysis about AHE in Fig. 2(d) is performed on the measurement of the anomalous-Hall current injected in the lateral circuit and not on the voltage of the open circuit: such studies have not been reported in the literature so far (to the best of our knowledge).

The calculation of m ( H a p p ) and the details of the magnetic simulation are presented in the supplementary material and in Ref. 11. The excellent agreement between the fit and the data in Fig. 2(d)—together with the specificity of AHE profiles—confirms that the current measured in the load circuit is due uniquely to the anomalous Hall current generated by the AHE of the ferrimagnetic layer.

As can be seen in Fig. 2(a), when a load resistance R l is contacted to the edges, the amplitude of the anomalous Hall voltage V x y is a monotonously increasing function of R l. On contrary, the anomalous Hall current injected into the lateral circuit is a monotonous decreasing function of R l. This is compatible with the intuitive interpretation that the electric charges accumulated at the edges of the Hall bar—that is a consequence of the AHE—are extracted and injected into the load circuit in proportion of the load resistance.

Note that the sign of the anomalous-Hall current is inverted when the direction of the magnetization is changed. This is a consequence of the change of the sign of the accumulated charges when the magnetization is rotated from up to down direction (this sign is equally inverted when the direction of the longitudinal current is reversed). The profile of power shown in Fig. 2(d) is well-characterized by a quadratic shape interrupted by the two horizontal lines corresponding to the up and down saturation states of the magnetization.32 In Fig. 3(a), the voltage variation (in Volt per Tesla) is plotted as a function of the load resistance R l. On the other hand, Fig. 3(b) shows the profile of the Joule power dissipated in the load resistance. The profile of the power is no longer a monotonous function of the load resistance and a sharp maximum appears for a well-defined resistance. The two typical profiles shown in Figs. 3(a) and 3(b) are analyzed in Sec. III.

The transverse Hall current is zero for the open circuit. In contrast, if the Hall bar is in contact with the load circuit, an anomalous-Hall current is generated that injects the charge carriers accumulated at the edges into the load circuit. The anomalous-Hall voltage decreases accordingly, as seen in Fig. 2(a). At stationary regime, the out-of-equilibrium balance between the charge accumulation and current injection into the load circuit is determined by the principle of minimum power dissipation under the constraints imposed on the system.33 This scenario is investigated in Ref. 28 for the usual Hall effect in a perfect Hall bar (assuming the invariance by translation along the longitudinal direction x). The main results are summarized below.

The generalized Ohm’s law relates the electric current density J = { J x , J y , J z } to the electrochemical potential μ. The gradient μ of the electrochemical potential is used instead of the electric field in order to include the effect of the electric screening.28 The Hall device is defined in the plane { x , y }, in which x is the longitudinal direction defined by the current injection from the generator, y is the transverse direction, and z is the direction perpendicular to the plane of the layer. Ohm’s law then reads
J = σ ( μ θ A H e z × μ ) ,
(1)
where × denotes the cross product, = { x , y , z } is the gradient, and σ = 1 / ( ρ ( 1 + θ A H 2 ) ) is the conductivity of the C o G d ferrimagnetic layer, while θ A H is the anomalous Hall angle introduced above. The total electric power dissipated in the device is given by P = J . μ d v (integrated over the volume of the device). After performing the functional minimization of the electric power P under the two constraints of global charge conservation and global current conservation, the Hall voltage at the edges is given by the expression (for constant temperature and small charge accumulation)28 
V x y ( R l ) = V x y 0 1 1 + ρ α R l ,
(2)
where ρ is the resistivity of the CoGd layer and the unknown geometrical parameter α is such that the ratio ρ / α defines the resistance R of the active region of the ferrimagnetic Hall cross. The fit of the data in Fig. 3(a) with the adjustable parameter α = 0.93 × 10 8 m is given by the red curve. This coefficient α ( t ) / W x x defines the geometry of the lateral current injection through the active part of the sample, as shown in Fig. 1(a). Indeed, the anomalous-Hall current is injected through the effective section . t where t = 3 × 10 8 m is the thickness of the sample, is the size of the lateral pad in contact to the Hall bar, and W x x = 3 . It is worth pointing out that the transverse resistance R experienced by the anomalous Hall current while flowing from one edge to the other inside the sample is defined by the resistivity ρ of the ferrimagnet and the section located between lateral electrodes. As pointed out in the introduction, this is not a trivial result because the Hall device cannot be described by a simple lumped-resistor circuit.17,26
On the other hand, the expression of the power dissipated in the load circuit reads28 
P ( R l ) = P 0 θ A H 2 ρ α R l ( 1 + ρ α R l ) 2 ,
(3)
where P 0 64 μ W is the input power injected from the electric generator into the CoGd Hall bar. This value takes into account the correction due to the current flowing through the buffer layer and through the cap layer. The profile of the power P ( R l ) [Eq. (3)] is calculated and plotted in Fig. 3(b) (continuous line), together with the measured power (squares).

As can be seen in Fig. 3, the measurements are qualitatively in agreement with the model. A sharp maximum is measured for a load resistance equal to the resistance R l = R = ρ / α. The pink zone between the calculated curve and the experimental points shows a global shift between the measurements and the theory. It is surprising to see that the calculated profile is below the measured profile, since the model is based on an optimized ideal Hall bar (invariant by translation), for which the maximum efficiency is expected. We ascribe the shift to the under-estimation of the measured anomalous Hall angle θ A H, which is due to the inhomogeneity of the current lines near the lateral contacts. This counter-intuitive effect has been discussed in the context of spin-Hall measurements34 and is studied in the supplementary material. After performing this correction, the calculated curve can be superimposed to the experimental points.

This observation hence validates qualitatively and quantitatively the prediction derived in Ref. 28 and confirms the general assumption that the anomalous-Hall current (like the Hall current) is very sober in terms of power consumption (of the order of θ A H 2 times the power injected in the Hall bar), but invalidate the claim that it is dissipationless for a perfect Hall bar.

Before concluding it is important to point out that—as shown in Fig. 4—the same measurements performed at different temperatures, from T = 30 K to T = 295 K, do not change the profiles [described by Eqs. (2) and (3)]. There is no qualitative change in the dissipation regime for the temperature range under consideration. In our context, the temperature dependence of the power is defined by the temperature dependence of the parameters θ A H and ρ in Eq. (3), whatever the mechanism responsible for the presence of the effective magnetic field: either spin–orbit coupling or Berry curvature (as discussed in Refs. 1–6).

In conclusion, the anomalous-Hall current flowing inside a ferrimagnetic GdCo layer and the electric power generated by this current into a load circuit have been studied as a function of the magnetization states Φ Happ, the load resistance R l, and the temperature T. The experimental protocol allows the physical properties of the transverse anomalous Hall current to be characterized and quantified.

The observations show that the Hall current behaves as a usual current inside the ferrimagnetic layer (resistance R = ρ α), and the maximum electric-power dissipated in the load circuit is indeed very small, of the order of the square of the anomalous-Hall angle θ A H 2. Furthermore, the profile shows a sharp maximum corresponding to the resistance matching. The matching occurs when the resistance of the load circuit R l equals the resistance R of the active part for AHE, defined by the geometrical parameter α t / W x x. This result is not trivial, because the system cannot be described by Kirchhoff’s laws (i.e., by a lumped-resistor circuit).17,26

Finally, it is important to point out that the presence of a sharp peak in the profile of the power injected in the load circuit could have important consequences for the optimization of SOT. Indeed, the power carried by the spins in a spin current is supposed to be controlled by the Joule power carried by the electric charges. The resistance R l is suspected to play a crucial role in the efficiency of the spin injection, which could be drastically reduced in the case of impedance mismatch. This study suggests that for the optimization of SOT devices, the ensemble of writing processes could be separated into an upstream anomalous-Hall transducer that transforms a part of the electric power of the generator into the Hall current [Eq. (3)], and a downstream charge-to-spin converter that uses the spin polarization of the current in order to switch a magnetic layer located at a nanoscopic distance from the Hall bar.

The supplementary material contains four sections that complement the main text. The corresponding information is not necessary for the understanding of the message delivered in the present report, but it presents the magnetic and electric properties of the samples (Secs. I–III) and discusses some basic questions related to the context of the study (Sec. IV). Section I (Magnetic characterization and simulation) presents and discusses the magnetic characterization of the GdCo layer, together with the corresponding magneto-transport properties: anisotropic magnetoresistance (AME), planar Hall effect (PHE), and anomalous Hall effect (AHE). The characterization allows the values of the parameters used in the main text to be defined. Section II (Resistance and resistivity) establishes the resistance of the C o G d layer from the total longitudinal resistance that includes the P t cap layer and the T a buffer layer. Section III (Correction of θ A H due to the real geometry) discusses the deviation of the current lines from the uniform longitudinal current density J x 0 due to the presence of the lateral contacts. This estimation explains the deviation (pink zone) observed in Fig. 3(a). Section IV (Overlooked properties about dissipation in Hall and Anomalous Hall devices) presents some important and basic questions about dissipation in Hall devices, in the spirit of the present study. In principle, these general results should be known at the basic level, but they are surprisingly overlooked in the textbooks about solid-state physics, semiconductor physics, or monographs about Hall devices.

The authors have no conflicts to disclose.

D. Lacour: Conceptualization (equal); Data curation (lead); Formal analysis (equal). M. Hehn: Conceptualization (supporting); Data curation (equal); Formal analysis (equal). Min Xu: Data curation (equal). J.-E. Wegrowe: Conceptualization (lead); Data curation (supporting); Formal analysis (equal); Writing – original draft (lead).

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

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