Thickness dependence of anomalous Hall and Nernst effects in Ni-Fe thin films

We systematically investigate the Ni-Fe layer thickness ($t$) dependence of the anomalous Hall effect (AHE) and anomalous Nernst effect (ANE). The AHE and ANE show different behavior in the $t$ dependence; the sign of the anomalous Hall resistivity changes around $t=9\mathrm{nm}$, whereas the anomalous Nernst coefficient ($S_{\mathrm{ANE}}$) keeps almost constant regardless of $t$, namely, no sign change of $S_{\mathrm{ANE}}$ with $t$. We analyze $S_{\mathrm{ANE}}$ and separate it into the contribution coming from the transverse thermoelectric conductivity (${\alpha }_{xy}$) and the AHE contribution coming from the Seebeck effect. The detailed analysis for $S_{\mathrm{ANE}}$ concludes that the AHE contribution is negligibly small and ${\alpha }_{xy}$, which is related to the anomalous Hall conductivity (${\sigma }_{xy}$) via the Mott relation, is also independent of $t$. To gain insight into the difference in the $t$ dependence of ${\sigma }_{xy}$ and ${\alpha }_{xy}$, we clarify the origin of sign reversal in the AHE. The sign reversal in the AHE is attributed to the competing contribution of the different sources of the AHE: the extrinsic and intrinsic mechanisms. The present experimental finding of the difference between the $t$ dependence on ${\sigma }_{xy}$ and ${\alpha }_{xy}$ suggests that the relative degree between the extrinsic and intrinsic processes for the AHE is quite different from that for the ANE in the case of Ni-Fe.

Figure 1

Schematic illustrations of (a) the permalloy (Py) wedged film, (b) microfabricated pattern of Hall bars aligned in a wedged film for the electrical transport measurements, and (c) rectangular-shaped sample for the anomalous Ettingshausen effect (AEE) measurement using lock-in thermography (LIT). Schematic illustrations of (d) the Hall device structure and measurement setup and (e) the sample for the anomalous Nernst effect (ANE) and Seebeck effect (SE) measurements. $\mathbf{H}, {\mathbf{J}}_{\mathrm{c}}$, and $\nabla T$ denote the magnetic field, charge current, and temperature gradient, respectively.

Figure 2

Thickness $t$ dependence of (a) the longitudinal resistivity ${\rho }_{xx}$ and (b) the anomalous Hall resistivity ${\rho }_{yx}$ in the Py thin film measured at 300 K. The inset of (b) shows the magnetic field $H$ dependence of the transverse resistance $R_{yx}$ for $t=12\mathrm{nm}$.

Figure 3

(a) Longitudinal thermoelectric voltage $V_x$ as a function of temperature difference $\mathrm{\Delta }T$ for the 5-, 8-, 10-, and 12-nm-thick Py thin film measured at ∼ 300 K. (b) Thickness $t$ dependence of the Seebeck coefficient $S_{\mathrm{SE}}$ measured at ∼ 300 K. Blue and gray plots denote $S_{\mathrm{SE}}$ measured for the microfabricated devices and the unpatterned blanket films, respectively. (c) Transverse thermoelectric voltage $V_y$ as a function of magnetic field $H$ for the 12-nm-thick Py thin film measured at ∼ 300 K. The applied temperature gradient $\nabla T$ was set at $1.5\mathrm{K}\mathrm{m}{\mathrm{m}}^{-1}$. The inset of (c) shows the $\nabla T$ dependence of the anomalous Nernst effect (ANE)-induced transverse electric field $E_{\mathrm{ANE}}$ for the 12-nm-thick Py thin film. (d) $t$ dependence of the anomalous Nernst coefficient $S_{\mathrm{ANE}}$ measured at ∼ 300 K.

Figure 4

(a) Lock-in amplitude $A_{\mathrm{AEE}}$ and (b) phase ${\varphi }_{\mathrm{AEE}}$ images of the anomalous Ettingshausen effect (AEE)-induced temperature modulation. These images were taken for the wedged Py layer with the rectangular shape, in which the thickness $t$ is continuously varied from 2 nm (left side) to 12 nm (right side). The black (white) dashed lines in (a) [(b)] denote the edges of the rectangular-shaped Py thin film. (c) $t$ dependence of $A_{\mathrm{AEE}}$ and the anomalous Nernst coefficient $S_{\mathrm{ANE}}$. The numerical values of $A_{\mathrm{AEE}}$ are obtained by averaging the $y$-directional profile around the upper edge in (a), where the averaging width was defined as a full width at half maximum.

Figure 5

Thickness $t$ dependence of (a) $S_{\mathrm{I}}$ and $S_{\mathrm{II}}$ terms of the anomalous Nernst effect, which are denoted by orange circles and gray diamonds, respectively, and (b) the transverse thermoelectric conductivity ${\alpha }_{xy}$. The inset of (a) shows the $t$ dependence of $S_{\mathrm{II}}$ term.

Figure 6

Temperature $T$ dependence of (a) the longitudinal resistivity ${\rho }_{xx}$, (b) the anomalous Hall resistivity ${\rho }_{yx}$ for the various thicknesses, and (c) ${\rho }_{yx}$ with $t=8.2$, 9.1, and 9.9 nm.

Figure 7

(a) Anomalous Hall conductivity ${\sigma }_{xy}$ as a function of residual longitudinal conductivity ${\sigma }_{xx0}$ measured at 5 K. The blue line is the linear fit for $t \ge$ 3.1 nm. (b) ${\sigma }_{xy}-\alpha {\sigma }_{xx0}^{-1}{\sigma }_{xx}^2$ as a function of ${\sigma }_{xx}$ for the Py thin films with $t=3.1$, 6.5, 9.1, and 12.0 nm. The solid lines are the fitted curves using Eq. (4). (c) Thickness $t$ dependence of scaling parameters ${\beta }_0$, γ, and ${\beta }_1$.

Figure 8

Thickness $t$ dependence of the anomalous Hall resistivity ${\rho }_{yx}, {\rho }_{yx}^{\mathrm{ss}}$, and ${\rho }_{yx}^{\mathrm{sj}+\mathrm{int}}$ at 5 K. The inset shows the $t$ dependence of the absolute values of ${\rho }_{yx}^{\mathrm{ss}}$ and ${\rho }_{yx}^{\mathrm{sj}+\mathrm{int}}$.

Figure 9

Measurement setup for the Seebeck effect and anomalous Nernst effect. $\mathbf{H}$ and $\nabla T$ denote the magnetic field and temperature gradient, respectively.

Figure 10

Thickness $t$ dependence of the anomalous Hall resistivity ${\rho }_{yx}, {\rho }_{yx}^{\mathrm{ss}}, {\rho }_{yx}^{\mathrm{sj}+\mathrm{int},{\beta }_0}, {\rho }_{yx}^{\mathrm{sj}+\mathrm{int},\gamma }$, and ${\rho }_{yx}^{\mathrm{sj}+\mathrm{int},{\beta }_1}$ at 300 K. The inset shows $t$ dependence of the absolute values of ${\rho }_{yx}^{\mathrm{ss}}, {\rho }_{yx}^{\mathrm{sj}+\mathrm{int},{\beta }_0}, {\rho }_{yx}^{\mathrm{sj}+\mathrm{int},\gamma }$, and ${\rho }_{yx}^{\mathrm{sj}+\mathrm{int},{\beta }_1}$.