Low offset frequency $1/f$ flicker noise in spin-torque vortex oscillators

Low-frequency noise close to the carrier remains little explored in spin-torque nano-oscillators. However, it is crucial to investigate as it limits the oscillator's frequency stability. This work addresses the low offset frequency flicker noise of a spin-torque vortex oscillator in the regime of large-amplitude steady oscillations. We first phenomenologically expand the nonlinear auto-oscillator theory, aiming to reveal the properties of this noise. We then present a thorough experimental study of the oscillator's $1/f$ flicker noise and discuss the results based on the theoretical predictions. Thereby, we connect the oscillator's nonlinear dynamics with the concept of flicker noise and furthermore refer to the influence of a standard $1/f$ noise description based on the Hooge formula, taking into account the nonconstant magnetic oscillation volume, which contributes to the magnetoresistance.

Figure 1

(a) Schematics of forces acting on the magnetic vortex core: gyroforce (green arrow), confinement force (yellow), spin transfer (purple), and damping force (blue). (b) Noise schematics of vortex motion: amplitude and phase noise, ${\delta }_s$ and ${\delta }_{\varphi }$, respectively.

Figure 2

Characteristic oscillation parameters, such as the oscillation frequency $f_c$, output power $p_0$, nonlinear parameter $\nu$, linewidth $\mathrm{\Delta }{f}_0$, and damping rate $f_p$, as a function of the applied current $I_{dc}$ at ${\mu }_0{H}_{\perp }=500\mathrm{mT}$.

Figure 3

PSD of phase (PN) and amplitude (AN) noise, ${\mu }_0{H}_{\perp }=500\mathrm{mT}$, $I=4.6\mathrm{mA}$, as well as theoretical curves based on Eqs. (2) and (3) and described in the text. The inset shows an analytical representation [Eqs. (2) and (3), but with parameters that are different from the measurement] of the higher offset frequency noise due to thermal effects. The phase noise of a linear oscillator ($\nu =0$, dashed line) is shown, along with its increase due to nonlinearity.

Figure 4

Evolution with applied current of the prefactors ${\alpha }_{\exp }$ from the ${\alpha }_{\exp }/f^{\beta }$ fits of (i) the experimental phase noise (PN) data (red points), (ii) the pure flicker PN (difference between black and violet curves in Fig. 3), represented by green points, and (iii) the amplitude noise (AN) data (blue points).

Figure 5

(a) Experimental active magnetic surface $A$ and (b) Hooge proportionality $P_{dc}/A$ vs applied dc current $I_{dc}$.

Figure 6

Calculated pure flicker noise prefactors evaluated through Eqs. (2) and (3) with measured experimental magnitudes without ${\alpha }_H$.