Micromagnetic instabilities in spin-transfer switching of perpendicular magnetic tunnel junctions

Micromagnetic instabilities and nonuniform magnetization states play a significant role in spin-transfer induced switching of nanometer scale magnetic elements. Here we model domain wall mediated switching dynamics in perpendicularly magnetized magnetic tunnel junction nanopillars. We show that domain wall surface tension always leads to magnetization oscillations and instabilities associated with the disk shape of the junction. A collective coordinate model is developed that captures aspects of these instabilities and illustrates their physical origin. Model results are compared to those of micromagnetic simulations. The switching dynamics are found to be very sensitive to the domain wall position and phase, which characterizes the angle of the magnetization in the disk plane. This sensitivity is reduced in the presence of spin torques, and the spin current needed to displace a domain wall can be far less than the threshold current for switching from a uniformly magnetized state. A prediction of this model is conductance oscillations of increasing frequency during the switching process.

Figure 1

Time evolution of the spatially averaged normalized z-axis disk magnetization during a nucleation process with $j/j_c=1.23$ and $d/d_c=1.95$ ($r_1=15$ nm). The other material parameters can be found in Appendix pp1-s5. Snapshots show the disk magnetization at different times indicated in the curve by red dots. The inset is a schematic of the tunnel junction nanopillar.

Figure 2

(a) Schematic of a reversed domain in a thin disk of radius $r_1$. The domain boundary–the red line–is characterized by its position from the center of the disk $q$ and the angle of the magnetization from the boundary normal $\varphi$. The curvature of the DW is represented by radius $r_2$. (b) Energy of the DW as a function of position in the disk normalized to its energy ${\mathcal{U}}_0$ at $q=0$, its energy at the center of the disk.

Figure 3

Domain wall dynamics in the ($q,\varphi$) model starting at $t=0$ in a Bloch configuration ($\varphi =\pi /2$) with $q/r_1=0.06$. The upper panel shows $q$ and the lower panel shows the phase, restricted to $[0,2\pi ]$, as a function of time. There is no spin-polarized current or applied magnetic field.

Figure 4

Domain wall dynamics in the ($q,\varphi$) model starting from $q/r_1=0.06$ and different initial DW phases: Néel ($\varphi =0$) and Bloch ($\varphi =\pi /2$) and an intermediate phase, $\varphi =\pi /4$. Again, the applied field and spin current are zero.

Figure 5

Relaxation phase diagram showing the final magnetization state as a function of the initial conditions for the analytic model. Red represents a final state with magnetization up. Blue represents magnetization down.

Figure 6

Time evolution of $m_z$ for the same initial DW position but different initial phases $\varphi$. We simulate a CoFeB disk with radius 15 nm and thickness 2.3 nm with $q/r_1=0.06$ nm, with no spin current or applied magnetic field. Starting from both Néel ($\varphi =0$) and a Bloch DW ($\varphi =\pi /2$) the disk relaxes to down state. Whereas, for an initial phase of $\varphi =\pi /4$, the magnetization relaxes into an up state for the same initial DW position. The bottom panel shows the time evolution of the Bloch DW's phase, which oscillates with an increasing frequency as the domain is expelled to the edge of the disk. In the relaxation panels, starting from a Néel DW ($\varphi =0$), the disk relaxes to down state. Whereas in the panels labeled switch the initial phase is $\varphi =\pi /4$ and relaxation is to an up state for the same initial DW position. White arrows indicate the direction of the DW displacement.

Figure 7

(a) Relaxation phase diagram as a function of $q$ and $\varphi$ with no spin current or field applied. We simulate a disk of radius 15 nm for different initial domain sizes and different initial DW phases. The color scale indicates the normalized magnetization of the final state, where blue represents relaxation to a down domain and red that the magnetization switches. The blue, orange and green dots ($q/r_1=0.06$ and $\varphi =0,\pi /4,$ and $\pi /2$) show the parameters used in Fig. 6. (b) Relaxation phase diagram for the $(q,\varphi )$ model. An asymmetry can be observed between positive and negative angles.

Figure 8

(a) Relaxation phase diagram as a function of $q$ and $\varphi$ for a spin current $a_I/a_c=0.05$, i.e., 5% of the threshold current for switching starting from a uniform magnetization state. We simulate a 15 nm radius disk for different initial DW positions and phases. The color scale is the same as in Fig. 7. (b) Relaxation phase diagram for the parameters $q$ and $\varphi$ calculated with the $(q,\varphi )$ model for $a_I/a_c=0.07$.

Figure 9

Switching current as a function of the initial position of the DW for three different initial phases. Each point indicates the minimum current required for magnetization switching.

Figure 10

Relaxation phase diagram showing the final magnetization state as a function of the initial conditions for the analytic model in the presence of an applied field. (a) Field in the $-z$ direction and (b) field in the $+z$ direction. In both cases the applied field magnitude is half the coercive field $H=H_c/2$ and the spin current is zero. Red represents a final state with magnetization up, and blue magnetization down.

Figure 11

Switching current as a function of the initial position of the DW for $\varphi =0$ (Bloch DW) computed in micromagnetics. Each point indicates the minimum current required for magnetization switching.