Conditions for quantized anisotropic magnetoresistance

Conditions for the appearance of quantized anisotropic magnetoresistance (QAMR) in ferromagnetic conductors have been investigated from the viewpoint of the transmission channels calculated by method of the nonequilibrium Green's function–density functional theory. We demonstrate that the spin-orbital interaction is a precondition and the transverse size is a crucial condition for QAMR to emerge. Furthermore, we show the evolution of QAMR from a stepwise shape (corresponding to small transverse sizes) to a classical smooth cosine shape (corresponding to large transverse sizes). In addition, we prove the strong temperature dependence of QAMR, so a low temperature is necessary. Our research reconciles the different experiments and enlightens experimenters on QAMR.

Figure 1

The real-space (a) and the reciprocal-space (b) unit cells of the iron bcc structure. In (a), the system extends infinitely in the axes of $x$, $y$, and $z$. The current direction (red arrow) is along the $z$ axis, and the magnetization direction (blue arrow) changes in the $x-y$ plane. The angle between the current and the magnetization is represented by θ. In (b), the green area denotes the transverse BZ, and the color spots represent the $(k_x,k_y)$ points at which the transmissions are calculated.

Figure 2

The $k$-resolved transmission in the transverse BZ in the absence (a) and in the presence (c) of SOI for $\theta =0$ (left color plots) and $\theta =\pi /2$ (right color plots). The values are painted with colors. (b) The angle dependence of conductance in the absence of SOI. (d) The angle dependence in the presence of SOI. The red line in (d) is the fitting curve based on the Döring equation.

Figure 3

Angle dependence of the conductance at three representative $(k_x,k_y)$ points: (a) $(k_x,k_y)=(0,0.33)$, (c) $(k_x,k_y)=(0.54,0.03)$, and (e) $(k_x,k_y)=(0.1,0)$. Their corresponding band structures are plotted in (b), (d), and (f), respectively. The left and the right band plots represent $\theta =0$ and $\theta =\pi /2$, respectively. The Fermi energy is set to zero and indicated by green lines. The red circles are used to stress the important differences.

Figure 4

The relative changes of the conductance for conductors having transverse cells $N_x\times N_y$: (a) from $1\times 1$ to $9\times 9$ and (b) from $10\times 10$ to $100\times 100$. In (a), the relative changes with $5\times 5$ and $7\times 7$ are zero as with $1\times 1$.

Figure 5

The relative changes of the conductance for (a) $N_x\times N_y=3\times 3$ and (b) $N_x\times N_y=100\times 100$ at different temperatures.