Simulation of current-driven domain structure evolution by ...



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Micromagnetic simulation of electric field-modulation on precession dynamics of spin torque nano-oscillator

Congpeng Zhao1, Xingqiao Ma1,*, Houbing Huang1, Zhuhong Liu1, Hasnain Mehdi Jafri1, Jianjun Wang2, Xueyun Wang3 and Long-Qing Chen2

1Department of Physics, University of Science and Technology Beijing, Beijing, 100083, China

2Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

3School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

*Corresponding author: xqma@sas.ustb.

Uniform precession discussion

This is an additional discussion for uniform precession region (See region C of Fig. 2a in main text) under zero bias field. In this simulation, nano-rectangular of free layer will induce inhomogeneous dipole interaction between local spins. To investigate oscillation behavior of the device with the nano-rectangular, trajectories of local spin are projected on sphere surface under[pic]. As illustrated in Fig. S1a, we label grid-point trajectory in blue line, corresponding effective field in red line and average magnetization trajectory in green line. Trajectory of grid-point possesses an unstable precession axis which can be directed by total effective field (Fig. S1c). Effective field trajectory further exhibits the formation of loop for precession axis (Fig. S1d). On the other hand, trajectory of average magnetization is relatively simple with small amplitude (Fig. S1b). The corresponding behaviors of oscillation for average and grid-point trajectories are shown in Fig. S1e and f, respectively. The[pic]and[pic]in Fig. S1f show extra peak value compared to that in Fig.S1e. Above results reveal various oscillations among grid points. Despite such oscillation discrepancy, synchronization of all local spins produces a persistent trajectory (average magnetization trajectory) due to adequate current density (produces enough torque to local spin for overcoming damping) and micromagnetic interactions, resulting in a single frequency spectrum (Fig. 2f-h or Fig. 3c).

[pic]

Figure S1. (a) Self-oscillating magnetization trajectories projected on sphere surface at current density value: [pic]. (b) Average magnetization trajectory of total simulation grids (green line in a). (c) Local spin (brown arrow in a) trajectory of selected grid-point (blue line in a). (d) Corresponding effective field of local spin (red line in a). The spin motions are in following order: A→B→C→D→E→A (c and d). (e) Average oscillation vs. time. (f) Grid-point oscillation vs. time.

Electric field induced non-uniform or uniform precession

Figure 2a in main text shows that there is a critical current density (a boundary between region B and C) due to inhomogeneous dipole interaction (strayfield) between local spins, which means large stray field in central area of film and smaller in peripheral area. Under such non-uniform stray field, there is more anisotropy-induced damping in peripheral area than that in central area. Generally, for region C, current-induced torque is large enough to overcome damping and maintain oscillations in peripheral area and subsequently, local spins at all sites are synchronized under micromagnetic interaction to produce same frequency (green spectrum in Fig. S24b). This process is called uniform precession (Fig. S2b,e). In the uniform mode, phase difference among local spins is related to spin position but is irrelavent to precession time. However, for unsynchronized precession in region B, relatively weak current provides adequate torques for central area but it may supply inadequate torques in peripheral area which fails to overcome damping. In this case, phase difference among local spins will become time-dependent, which is responsible for the propagation of spin waves (energy dispersion).

In addition to above current-induced torque, the boundary between uniform (region C) or non-uniform (region B) precession also depends on damping torque. Electric field (E) can also be used to modulate the boundary because E affects anisotropy (actually, the anisotropy-induced damping). Fig S2a-c describes the details of E effects on precession modes. Spatial magnetization dynamic behavior by fixing current value [pic] revealed average magnetization precession trajectories under different E values ( -0.3, 0, 0.3 V/nm) (Fig. S2a) and their corresponding spectra obtained from Fast Fourier Transform (Fig. S2b). Trajectories for [pic] (Fig. S2a) show more persistent behaviors than that for E= -0.3, because of decreasing IPMA. To clarify the uniform and non-uniform modes, normalized magnetization ([pic]) distribution under E=0, 0.3, -0.3 is illustrated by three snapshots at 24.78 ns, 25.46 ns, 26.15 ns (Fig. S2c), where non-uniform [pic] at [pic]V/nm can be distinguished from another two E values ([pic]V/nm) for distribution asymmetry. Note that the direction of magnetization is changed from red color (out-of-plane) to blue (in-plane direction). Asymmetric configuration with larger IPMA (E < 0) also indicates an asynchronized motion of all local spins during oscillation, which is equivalent to spin wave propagation. Corresponding spectrum (Fig. S2b) also reveals discrepancy between non-uniform (in red) and uniform precession (in blue and green).

[pic]

Figure S2. (a) Trajectories of average magnetization modulated by different E value -0.3, 0, 0.3 V/nm at current density value of[pic]. (b) Frequency spectrum corresponding to three different E value in a. (c) Snapshots of out-of-plane magnetization([pic]).

Confirmation of simulation by Larmor model theoretical calculation

A theoretical calculation was perfromed using Larmor precession model to confirm our simulation, electric field explaination and rate of frequency modulation. Larmor model [pic] describes magnetic moments precessing around the external magnetic field. Furthermore, any micromagnetic interaction could be equivalent to an effective magnetic field, its angular frequency of magnetic moment can be considered as:

[pic] (S1)

where angular frequency ω relates to output frequency f as:[pic], while effective field strength acting on magnetic momentums contain magnetocrystalline anisotropy field ([pic]), demagnetization field ([pic]), coupling field (between pinned-layer and free-layer caused by spin polarized current [pic]) and exchange field ([pic]). To estimate the scale of frequency shift induced by electric field, single spin precessing around the field is considered for simplicity and, therefore, the effect of [pic] term is neglected. High order terms of [pic] is also neglected because [pic](see Equation (3)). Total effective field [pic]with electric bias can be rewritten in Cartesian coordinate system (x, y, z) as:

[pic] (S2)

Where, J and d are coupling field varables for current density and thickness of free-layer, respectively, [pic]represents scalar function of MTJ and θ means angle between [pic] and [pic]. The demagnetization shape factor in z direction, [pic], is considered as 0.95 ([pic]). Thus, frequency modulation scale per 0.1 V/nm electric field can be estimated as [pic]. The value of [pic] was calculated to be approximately [pic] with initial stable local spin [pic]. We define value of frequency change per 0.1 V/nm electric field as [pic], which was calculated to be [pic]. The results show that relative modulation value of frequency [pic] is 3.7%. Scale of [pic] is 1~102 MHz (estimated roughly by 3.7% of f scale 0.1~10 GHz), which concurs with our simulation result (31.2MHz).

Spectrum calculation

Frequency spectrum was obtained by computing Fast Fourier Transformation of resistance-time traces for each computational cell with sampling interval 43.02 picosecond. [pic][S1] represents total normalized output power estimated by following fomular:

[pic] (S3)

Here, k represents total number of simulation cells. Current passing through each cells is described by [pic], where S denotes cross-section area of nano-pillar. Resistance dependent parameters, [pic] and [pic] are defined as: [pic], [pic] , where [pic] and [pic] denote anti-parallel and parallel state resistance of the frame. Fourier Transformation of output power [pic] illustrated in spectrum is expressed by:

[pic] (S4)

where, ω is angular frequency described by [pic].

Reference

S1 B. Georges, et al. Origin of the spectral linewidth in nonlinear spin-transfer oscillators based on MgO tunnel junctions 80 060404 (2009)

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