Micromagnetic simulations of spin-torque driven magnetization dynamics with spatially resolved spin transport and magnetization texture

We present a simple and fast method to simulate spin-torque driven magnetization dynamics in nanopillar spin-valve structures. The approach is based on the coupling between a spin transport code based on random matrix theory and a micromagnetics finite-elements software. In this way the spatial dependence of both spin transport and magnetization dynamics is properly taken into account. Our results are compared with experiments. The excitation of the spin-wave modes, including the threshold current for steady-state magnetization precession and the nonlinear frequency shift of the modes are reproduced correctly. The giant magneto resistance effect and the magnetization switching also agree with experiment. The similarities with recently described spin-caloritronics devices are also discussed.

Figure 1

(a) Sketch of the nanopillar under study, with the arrows representing the magnetization vectors. In the steady state, the spin-transfer torque $\mathit{M}\times \mathit{S}$ compensates the damping torque $\mathit{M}\times \mathit{d}$ and the local magnetization vector ${\mathit{M}}_s$ precesses regularly at the Larmor frequency along a circular orbit. (b) The SW modes in each disk are Bessel functions $J_{\ell m}$, with $\left(\ell \right)$ $\left(m\right)$ the azimuthal (radial) index. Modes with $\ell =0$ correspond to a uniform in-phase precession of the magnetization, while modes with $\ell =1$ correspond to a nonuniform precession, with the magnetization vector rotating along the azimuthal direction. (c) In the case where two disks are coupled through the dipolar interaction, the SW $J_{\ell m}^{B/A}$ modes separate into binding (B) and antibinding (A), which correspond, respectively, to an antiphase precession (occurring mainly in the thick layer) and in-phase precession (occurring mainly in the thin layer).

Figure 2

SW power spectrum for the $\ell =0$ (a) and $\ell =1$ (b) modes, obtained from the volume-averaged magnetization. The $B$ (respectively, $A)$ modes corresponds to antiphase (respectively, in phase) precession, whose amplitude is larger in the thick (respectively, thin) layer.

Figure 3

(a) Scattering matrix relating incoming and outgoing modes ${\psi }_n$ in region $n=0,1$. The $+$ and $-$ sign represent, respectively, right and left propagating modes. (b) Illustration of the “hat” matrix approach. We define the regions 0, 1, and 2, respectively, on the left, middle, and right of the two samples $S_a$ and $S_b$. In each region the four vectors ${\mathit{P}}_{i\pm \sigma }$ represents the probability to find right $(+)$ propagating and left $(-)$ propagating electrons. (c) The CRMT equations allow one to calculate the hat matrices as a function of the length $L$ of the sample, using the sum law for hat matrices.

Figure 4

Illustration of the nanopillar simulated with CRMT. Electrons enter from the left and are spin-polarized along the direction of $M_b$. When they enter the thin layer, they change their polarization along the direction of $M_a$. Because of the conservation of angular momentum, the transverse component of the spin-polarisation is transferred to the thin and thick layers as a spin torque.

Figure 5

Calculation of the angular dependence of spin torque (a) and resistance (b) obtained keeping ${\mathit{m}}_b$ fixed and rotating ${\mathit{m}}_a$. Magnetoresistance hysteresis curve of the nanopillar for an in-plane magnetic field. The dark (respectively, light) symbols indicate the magnetic field being ramped up (respectively, down).

Figure 6

$z$ component of the spin current along the $z$ axis of the nanopillar. Black circles correspond to the collinear configuration, while other symbols correspond to different rotation angles of the magnetization of the thin layer in the $x\text{−}y$ plane.

Figure 7

Spin polarization for $\uparrow$ (a) and $\downarrow$ (b) electrons.

Figure 8

Schematic of the method adopted to couple CRMT to Nmag, with the nanopillar viewed from profile and the current flowing along the $\mathit{z}$ axis. (a) One considers the sites lying on the inner surfaces of disks $a$ and $b$ that face each other. Then one selects a site of surface 1 (thin layer, red dots) and looks for its nearest neighbor among the sites of surface 2 (thick layer, blue dots). (b) To each couple of such sites, one associates a column $k$, which corresponds to a CRMT system, with magnetization ${\mathit{m}}_1^k$ and ${\mathit{m}}_2^k$. The system is then represented as an assembly of CRMT columns connected in parallel. From the magnetic configuration in each column $k$, STT and conductance are computed. (c) Illustration of the system divided into CRMT pillars, view from top.

Figure 9

Time evolution of $M_z$ for different values of the dc current. (a) Subcritical regime, where both magnetization vectors are aligned. (b) Slightly higher than critical current, where the thin layer precesses and the thick layer stand still. (c) High current, with reversal of thin layer and coupled precession.

Figure 10

Resistance difference $\mathrm{\Delta }R$ between parallel and antiparallel configuration as a function of the dc current $I_{\text{dc}}$, calculated for different values of the applied field $H_{\mathrm{ext}}$.

Figure 11

SW power spectrum of the $\ell =0$ (a) and $\ell =+1$ (b) modes, computed for different values of the dc current. The yellow squares (respectively, circles) indicate the positions of the $B$ (respectively, $A)$ modes.

Figure 12

Line widths (a) and frequencies (b) of the modes $A_{00}$ and $A_{10}$ as a function of the dc current near the critical threshold. The symbols are numerical calculations, while dashed lines are linear fits.

Figure 13

Magnetization current from thick to thin layer as a function of dc current $I_{\text{dc}}$, computed for different values of the applied field.