Theory of anisotropic magnetoresistance in atomic-sized ferromagnetic metal contacts

Recent experiments in ferromagnetic atomic-sized contacts have shown that the anisotropic magnetoresistance (AMR) is greatly enhanced and has an asymmetric angular dependence as compared with that of bulk samples. The origin of these effects is still under debate. In this work, we present a theoretical analysis of the AMR in atomic contacts of the $3d$ ferromagnetic materials. Our results strongly suggest that the anomalous AMR stems from the reduced symmetry of the atomic contact geometries. We also present calculations supporting the idea that the pronounced voltage and temperature dependence in some experiments can be attributed to impurities near the constrictions.

Figure 1

(Color online) (a) Ideal Ni one-atom contact in fcc [111] direction with atoms on lattice positions. Green (gray) atoms are those in the atomic constriction, yellow (light gray) ones are part of the surfaces used to model the leads. [(b) and (c)] Channel decomposition and the total linear conductance as a function of $\theta$ for different angles $\varphi$. The relative conductance is defined as $G_{\text{rel}}=G\left(\theta ,\varphi \right)/{⟨G\left(\theta ,\varphi \right)⟩}_{\theta }-1$. (c) Channel decomposition of (b). [(d)–(f)] Same as (a)–(c) but with the contact distorted by randomly shifting the red (dark gray) atoms by up to 5% of the nearest-neighbor distance.

Figure 2

(Color online) Contact evolution of a Ni junction grown in the fcc [001] direction as obtained from the simulations of Ref. 15. (a) Spin projected $G_{\uparrow ,\downarrow }$ and total conductance in the absence of SOC and total conductance averaged over $\theta ,\varphi$ in the presence of SOC (indicated with arrows). Vertical lines correspond to the contact geometries in (b). Inset: relative AMR amplitude $\Delta G/{⟨G⟩}_{\theta }=\left[G_{\text{max},\theta }\left(\varphi \right)-G_{\text{min},\theta }\left(\varphi \right)\right]/{⟨G\left(\theta ,\varphi \right)⟩}_{\theta }$ vs inverse averaged conductance. (c) Conductance vs $\theta$ for the geometries in (b) and with $\varphi$ in steps of $\pi /6$. (d) Relative AMR $\left[G\left(\theta ,\varphi \right)/{⟨G\left(\theta ,\varphi \right)⟩}_{\theta ,\varphi }-1\right]$ in % on a “Bloch sphere.” (e) Same as (b)–(d) for a thick regular geometry with 324 atoms with the relative AMR values multiplied by 10 for visibility.

Figure 3

(Color online) (a) $\tau \left(\epsilon ,V=0\right)$ for the contact of Fig. 1a with $\theta =0$,$\pi /2$. (b) Sketch of the chain model, (c) corresponding transmission for $\theta =0$, $\pi /2$ in the absence of impurities, (d) $\tau \left(\epsilon ,V=0\right)$ with an impurity $N=751$ sites from the scattering region, and (e) relative nonlinear conductance $G_{\text{rel}}\left(\theta ,V\right)=G\left(\theta ,V\right)/{⟨G\left(\theta ,V\right)⟩}_{\theta }-1$. (f) Relative linear conductance vs $\theta$ and (g) voltage dependence of nonlinear conductance at $\theta =\pi /2$ for indicated temperatures. Results at 0 K (blue line) in (f) and (g) correspond to the dashed lines in (e).