^{1,a)}, I. N. Krivorotov

^{2}, E. N. Bankowski

^{3}, T. J. Meitzler

^{3}, V. S. Tiberkevich

^{4}and A. N. Slavin

^{4}

### Abstract

We studied the operation of a dual-free-layer (DFL) spin-torque nano-oscillator (STNO) and demonstrated that in a practically interesting regime when the magnetizations of the two free layers (FLs) precess in opposite directions along large-angle out-of-plane trajectories, thus doubling the generation frequency, the operation of the DFL STNO is strongly hysteretic as a function of a bias dc current. The stable magnetization dynamics starts at a rather large magnitude of the bias dc current density when the bias current is increased, but the regime of stable counter-precession of the FLs persists till rather low magnitudes of the bias dc current density when the bias current is decreased. This hysteresis is caused by the dipolar coupling between the FLs, and is especially pronounced for small distances between the FLs and the small magnetic damping in them. The discovered hysteretic behavior of the DFL STNO implies the possibility of application of a strong initial pulse of the bias current (greater than the upper threshold of the stable dynamics) and subsequent reduction of the bias current to a working point ( ) corresponding to the required output frequency . The obtained results are important for the practical development of DFL STNOs with optimized operation characteristics.

This work was supported in part by the contract from the U.S. Army TARDEC, RDECOM, by the Grant Nos. ECCS-1001815, ECCS-1002358, DMR-1015175, and DMR-0748810 from the National Science Foundation of the USA, by the CNFD Grant from the “Semiconductor Research Corporation,” by the Nanoelectronics Research Initiative through the Western Institute of Nanoelectronics, by the DARPA MTO/MESO Grant N66001-11-1-4114, and by the Grant No. UU34/008 from the State Fund for Fundamental Research of Ukraine.

I. INTRODUCTION

II. MODEL

A. Schematic and fundamentals of the DFL STNO

B. A simplified model of a DFL STNO

C. The equations for the magnetization dynamics in a DFL STNO

D. Parameters of the DFL STNO

III. RESULTS AND DISCUSSION

A. Hysteresis in the dependence of the generated frequency on the bias current

B. Threshold bias current densities of the DFL STNO

C. Stable magnetization dynamics in a DFL STNO for bias current densities below the upper threshold

IV. SUMMARY

### Key Topics

- Current density
- 39.0
- Magnetization dynamics
- 25.0
- Magnetic anisotropy
- 13.0
- Torque
- 13.0
- Magnetic fields
- 11.0

## Figures

(a) Layout of the symmetric DFL STNO consisting of four magnetic layers: two inner FLs of the thickness L and two outer pinned layers with perpendicular polarization separated by the GMR spacer of the thickness d. (b) The torques acting on the magnetization vectors of the FLs in the DFL STNO driven by the transverse bias dc current of the density J dc. The conservative torque (T P ) caused by the effective magnetic field B eff induces magnetization precession in the FLs around the normal to the STNO layers. The non-conservative spin-torques (T S ) caused by the perpendicular polarizers adjacent to each of the FLs tend to increase the precession angle, while the non-conservative torques (T D ) caused by the Gilbert damping in the FLs tend to reduce it.

(a) Layout of the symmetric DFL STNO consisting of four magnetic layers: two inner FLs of the thickness L and two outer pinned layers with perpendicular polarization separated by the GMR spacer of the thickness d. (b) The torques acting on the magnetization vectors of the FLs in the DFL STNO driven by the transverse bias dc current of the density J dc. The conservative torque (T P ) caused by the effective magnetic field B eff induces magnetization precession in the FLs around the normal to the STNO layers. The non-conservative spin-torques (T S ) caused by the perpendicular polarizers adjacent to each of the FLs tend to increase the precession angle, while the non-conservative torques (T D ) caused by the Gilbert damping in the FLs tend to reduce it.

(a) Numerically calculated dependences of the precession frequencies in the top and bottom FLs in a DFL STNO on the density J dc of the bias dc current, demonstrating strong hysteresis. Dynamics starts at when the bias current increases (red squares and cyan circles) and stops at when the current decreases (orange hollow squares and blue crosses). The system can be in the IPS or in the out-of-plane precessional (OPP) state. (b) Numerically calculated dependence of the frequency of an output signal f in a DFL STNO on the bias direct current density J dc. The generated frequency is equal to the sum of the precession frequencies in the FLs, . Yellow star marks a possible working point, which could be reached by using the procedure described in Sec. III C . The curves are calculated for α = 0.08, . All the other parameters of the DFL STNO are presented in Sec. II D .

(a) Numerically calculated dependences of the precession frequencies in the top and bottom FLs in a DFL STNO on the density J dc of the bias dc current, demonstrating strong hysteresis. Dynamics starts at when the bias current increases (red squares and cyan circles) and stops at when the current decreases (orange hollow squares and blue crosses). The system can be in the IPS or in the out-of-plane precessional (OPP) state. (b) Numerically calculated dependence of the frequency of an output signal f in a DFL STNO on the bias direct current density J dc. The generated frequency is equal to the sum of the precession frequencies in the FLs, . Yellow star marks a possible working point, which could be reached by using the procedure described in Sec. III C . The curves are calculated for α = 0.08, . All the other parameters of the DFL STNO are presented in Sec. II D .

Numerically calculated dependences of the current density thresholds (blue circles and line) and (red squares and line) on the different parameters of the DFL STNO. (a) The dependence of and on the Gilbert damping parameter α. (b) The dependence of and on the magnitude of the in-plane anisotropy field B A. (c) The dependence of and on the distance between the FLs d. Green stars and dashed line show the dependence of the cross-demagnetization coefficient ρ on d, ρ(d). All the curves are calculated for α = 0.08, B A = 5 mT (if other values of parameters are not mentioned specifically). All the other parameters of the DFL STNO are presented in Sec. II D .

Numerically calculated dependences of the current density thresholds (blue circles and line) and (red squares and line) on the different parameters of the DFL STNO. (a) The dependence of and on the Gilbert damping parameter α. (b) The dependence of and on the magnitude of the in-plane anisotropy field B A. (c) The dependence of and on the distance between the FLs d. Green stars and dashed line show the dependence of the cross-demagnetization coefficient ρ on d, ρ(d). All the curves are calculated for α = 0.08, B A = 5 mT (if other values of parameters are not mentioned specifically). All the other parameters of the DFL STNO are presented in Sec. II D .

Numerically calculated time profiles of the in-plane ( , blue curve) and out-of-plane ( , red curve) magnetization components in the top FL of the DFL STNO for different regimes of time-dependent current biasing (b, d, f). The DFL STNO is biased by a dc current density having a constant and a pulsed components and its time profile is shown by the green curve with green shading (a, c, e). Dashed horizontal lines show the levels corresponding to the higher (orange lines) and lower (violet lines) current thresholds . (a) The constant current density below the higher threshold induces only small perturbations of the IPS magnetization state (b). (c) The current density containing a constant part and a pulse of the duration 0.5 ns with the amplitude (still below the higher threshold) induces some transient dynamics, and the IPS magnetization state remains stable (d). (e) The current density containing a constant part (above the lower threshold) and a strong pulse of the duration T pulse = 0.5 ns with the amplitude (exceeding the higher threshold) induces a stable magnetization precession (f). The curves are calculated for α = 0.08, B A = 5 mT. All other parameters of the DFL STNO are presented in Sec. II D .

Numerically calculated time profiles of the in-plane ( , blue curve) and out-of-plane ( , red curve) magnetization components in the top FL of the DFL STNO for different regimes of time-dependent current biasing (b, d, f). The DFL STNO is biased by a dc current density having a constant and a pulsed components and its time profile is shown by the green curve with green shading (a, c, e). Dashed horizontal lines show the levels corresponding to the higher (orange lines) and lower (violet lines) current thresholds . (a) The constant current density below the higher threshold induces only small perturbations of the IPS magnetization state (b). (c) The current density containing a constant part and a pulse of the duration 0.5 ns with the amplitude (still below the higher threshold) induces some transient dynamics, and the IPS magnetization state remains stable (d). (e) The current density containing a constant part (above the lower threshold) and a strong pulse of the duration T pulse = 0.5 ns with the amplitude (exceeding the higher threshold) induces a stable magnetization precession (f). The curves are calculated for α = 0.08, B A = 5 mT. All other parameters of the DFL STNO are presented in Sec. II D .

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