Figure 1

(Color online) Through the spin-transfer torque, an unpolarized electrical current flowing perpendicular to a thin ferromagnetic layer can enhance or suppress spin waves. Electrons backscattered from the ferromagnet at point 1 have their spin predominantly polarized antiparallel to magnetization direction $\mathbf{m}\left(1\right)$. These electrons exert a torque on the ferromagnet’s magnetization $\mathbf{m}\left(2\right)$ if they reach the ferromagnet a second time at point 2, the direction of the torque being to enhance an existing spin-wave [i.e., to increase any preexisting difference between $\mathbf{m}\left(1\right)$ and $\mathbf{m}\left(2\right)$]. When electrons transmitted through the ferromagnet reach the ferromagnet a second time at point 3, they exert a torque that suppresses an existing spin wave. If source and drain contacts are not symmetric, there is a net torque on the ferromagnet, which enhances or suppresses the spin wave, depending on the current direction.Reuse & Permissions

Figure 2

(Color online) Spin will accumulate in normal metals on both sides of a ferromagnetic layer with uniform magnetization if an unpolarized current is passed through the ferromagnet (top left). A large-amplitude spin wave in the ferromagnet reduces the amount of spin polarization in the normal-metal regions adjacent to the ferromagnet and lowers the total resistance of the device (bottom left). This is shown schematically in the circuit diagrams (right). The top two circuit diagrams show the resistances seen by majority and minority electrons when the magnetization is spatially uniform, the short and long resistor symbols referring to minority and majority resistances, respectively. The ferromagnet with a large-amplitude spin wave can be seen as a parallel configuration of ferromagnets with opposite magnetization directions. The bottom two circuit diagrams show the resistances seen by two spin directions in this case. The net resistance is lower in the presence of a large-amplitude spin wave.Reuse & Permissions

Figure 3

(Color online) Schematic picture of the normal-metal-ferromagnet-normal-metal junction considered in our calculations. The ferromagnetic layer (F) is connected to source and drain reservoirs though normal metal spacers (N). We consider the maximally asymmetric case with only one spacer of length $L⪢l_{\mathrm{sf}}$.Reuse & Permissions

Figure 4

(Color online) Schematic drawing of the model solved numerically. The continuous magnet is replaced by $N$ magnets (left), each coupled to a normal-metal wire (right). The wires are coupled via transverse diffusion (shown schematically as solid lines); the magnets are coupled via the exchange interaction (shown schematically as dashed lines).Reuse & Permissions

Figure 5

(Color online) The main panel shows the magnetization component $m_3\left(1\right)$ of the first magnet, as a function of the applied current. The solid line is obtained from the perturbation theory result (41), while the dashed line is a guide to the eye. In a large magnetic field, the motion is circular. An example is shown in the inset, where $j=1.5j_c$.Reuse & Permissions

Figure 6

(Color online) Resistance of the ferromagnetic layer, as a function of the applied current (crosses). The solid line is obtained from the perturbation theory result (44), while the dashed line is a guide to the eye.Reuse & Permissions

Figure 7

(Color online) Typical elliptical trajectory for one of the discrete nanomagnets $\mathbf{m}\left(n\right)$ for weak easy axis and strong easy plane anisotropy with $j_c<j<2j_c$ (left panel). The upper and lower right panels show the corresponding Poincaré sections for $j=1.2j_c$ and $1.5j_c$, respectively. This regime agrees with the perturbative calculation of Sec. 2, where the lowest energy spin-wave mode is excited and increasing the current only changes the amplitude of elliptical oscillation.Reuse & Permissions

Figure 8

(Color online) First manifestations of further dynamical instabilities in the range $2j_c<j<2.5j_c$. The upper right panel shows a Poincaré section for $j=2.2j_c$, where the motion is no longer symmetric about the easy axis. The lower right panel shows the motion for $j=2.4j_c$, where the motion is trapped between the $\pm {\widehat{e}}_3$ easy axes direction. The left panel shows what this motion looks like on the unit sphere.Reuse & Permissions

Figure 9

(Color online) Poincaré sections for the magnetization direction of one of the magnets at $j=2.5j_c$ (left) and $j=3.2j_c$ (right).Reuse & Permissions

Figure 10

(Color online) The upper panel shows the time trace of resistance where the spin-wave instability causes a decrease in the observed resistance. The lower left plot shows how the amplitude and period of the resistance oscillation change with the driving current, while the lower right panel shows the decrease of dc resistance.Reuse & Permissions