Anomalous and planar Righi-Leduc effects in ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ ferromagnets

In this paper, we report experimental evidence of anomalous and planar Righi-Leduc effects on $\mathrm{NiFe}$. The Righi-Leduc effect is the thermal analog of the Hall effect, in which the electric current is replaced by the heat current and the electric field by the temperature gradient. When the material is ferromagnetic, it is well known that there are two other contributions to the Hall voltage which depend on the orientation of the magnetization. These two extra contributions are called the anomalous Hall effect when the magnetization is out of the plane of the sample and the planar Hall effect when the magnetization is in the plane of the sample. In the same way, an anomalous and a planar Righi-Leduc effects are shown to appear when a transverse temperature gradient is generated by a heat current.

Figure 1

Schematic views of a typical NiFe device for (a) the anomalous and planar Righi-Leduc effect. The electrode and the heater have a width of $400 \mu \mathrm{m}$ and are spaced 5 mm apart. (b) Measurements of the anomalous and planar Hall effect and anisotropic magnetoresistance. The magnetization vector $\vec{m}$ is defined by the angles $\theta$ and $\phi$. Similarly, ${\theta }_H$ and ${\phi }_H$ define the orientation of the external magnetic field orientation.

Figure 2

(a) Magnetoresistance as a function of the amplitude of the external perpendicular field at $\phi =0$ for (a) $\theta =5^{\circ },\theta ={23}^{\circ },$ and $\theta ={50}^{\circ }$. The points correspond to the measured data and the line is the fit calculated from the minimization of the magnetic energy $F$. (b) Magnetoresistance as a function of the out-of-plane angle ${\theta }_H$ for a weak external magnetic field $H=0.2T$ at ${\phi }_H=\phi =0^{\circ }$ (upper curve) and ${\phi }_H=\phi ={90}^{\circ }$ (lower curve).

Figure 3

Anomalous Hall measurement with Cu (5 nm) / Pt (10 nm) as a function of (a) the amplitude of the magnetic field $H$ for different out-of-plane angles and (b) the radial angle ${\theta }_H$ for an applied field of $H=0.2$ T. The lines correspond to the calculation of the magnetic states.

Figure 4

Time dependance of the transverse voltage for a 50-nm Pt electrode. The modulation frequency is $2f=\omega /\pi =0.04\text{Hz}$. The external field is 1 T and is applied along ${\theta }_H=0$, which is perpendicular to the sample plane. We define $\mathrm{\Delta }{V}_y=V_y^{2\omega }-V_0^{2\omega }$ where $V_0^{2\omega }$ is the $2\omega$ voltage measured without external magnetic field.

Figure 5

(a) Measurement of the voltage $V_y^{2\omega }$ as a function of the heat power injected for different values of the out-of-plane applied field. (b) Magnetothermal response $\delta \left(\mathrm{\Delta }{V}_y\right)=\mathrm{\Delta }{V}_y(\theta =0^{\circ })-\mathrm{\Delta }{V}_y(\theta ={180}^{\circ })$ as a function of the power injected.

Figure 6

(a) Thermoelectric voltage at 0 field, $V_0$, and (b) magnetovoltaic signal $\delta \left(\mathrm{\Delta }{V}_y\right)$ (as defined in the caption of Fig. 5) as a function of the square root of the freqency $\sqrt{2f}$. The squares are the experimental data and the gray line is the exponential relaxation with the adjustable thermal diffusivities $\kappa$ (a) and K (b) that are compared to the known values.

Figure 7

Thermoelectric voltage generated by a contact wire of Al on (a) NiFe (Py), (b) Cu, and (c) Pt, (d) Bi as a function of time. The temperature difference of $\mathrm{\Delta }T=15.6^{\mathrm{o}}$ is imposed at $t=0s$, then the metallic electrode thermalizes to a steady state temperature gradient.

Figure 8

Transverse voltages $\mathrm{\Delta }{V}_y=$ as a function of the direction of the $1T$ magnetic field $H$. (a) In-plane configuration for which the polar angle ${\theta }_H$ is fixed and equal to ${90}^{\circ }$, the azimuthal angle ${\phi }_H$ is varied $(◂)$; (b) out-of-plane configuration for which the azimuthal angle ${\phi }_H$ is fixed to ${90}^{\circ }$ and the polar angle is varied $\left(▴\right)$.

Figure 9

Transverse voltage $\mathrm{\Delta }{V}_y$ on a Cu (5 nm)/Pt (10 nm) electrode as a function of (a) the amplitude of the magnetic field $H$ for three out-of-plane angles and (b) the radial angle ${\theta }_H$ for an applied field of $H=0.2$ T. The gray line corresponds to the calculation of Eq. (6).

Figure 10

Transverse voltage $\mathrm{\Delta }{V}_y$ for a 20-nm Cu probe electrode as a function of (a) the in-plan angle ${\phi }_H$ and (b) the out-of-plane angle ${\theta }_H$.

Figure 11

Transverse voltage $\mathrm{\Delta }{V}_y$ vs ${\phi }_H$ for in-the-plan configuration with (a) $\text{Pt}$, (b) $\text{Cu,}$ and (c) $\text{Bi}$ electrodes. The signal with Bi electrode is 10 times higher than that with Pt and Cu electrodes.

Figure 12

(a) Maximum out-of-plane voltage response $\delta \left(\mathrm{\Delta }{V}_y\right)$ to the thermal excitation $J_x^Q$ vs thickness of the $\text{Pt}$ electrode. (b) Electrical counterpart of $\delta \left(\mathrm{\Delta }{V}_y\right)$, i.e., the voltage response to the electric excitation $J_x^e$ vs thickness of the $Pt$ electrode (anomalous Hall effect). The gray line presents the expected dependence taking into account only the shunt effect.