A high-speed single sideband generator using a magnetic tunnel junction spin torque nano-oscillator

An important property of spin-torque nano-oscillators (STNOs) is their ability to produce a frequency modulated (FM) signal, which is very critical for communication applications. We here demonstrate a novel single sideband (SSB) modulation phenomenon using a magnetic tunnel junction (MTJ)-based STNO, which saves transmission bandwidth and in principle should minimize attenuation for wireless communication. Experimentally, lower single sidebands (LSSBs) have been successfully demonstrated over a wide range of modulation frequency, f m = 150 MHz-1 GHz. The observed LSSBs are determined by the intrinsic properties of the device, which can be modeled well by a nonlinear frequency and amplitude modulation formulation and reproduced in macrospin simulations. Moreover, our macrospin simulation results show that the range of modulation current and modulation frequency for generating SSBs can be controlled by the field-like torque and biasing conditions.


Onset of LSSB
The onset modulation frequency, f onset for observing lower single side band (LSSB) generation in STNOs can be predicted from the frequency difference ( f max -f 0 ). Here, f max is the resonance frequency at low bias current or the maximum operating frequency of the STNO and f 0 is the current dependent STNO frequency as defined in the main text. Figure 1 (a) shows the experimentally measured f onset and calculated frequency difference ( f max -f 0 ), while Fig. 1 (b) shows corresponding results from macrospin simulations. The agreement shows that f onset at any operating dc current can be obtained from the frequency difference, ( f max − f 0 ).

Upper single side band generation
We have achieved upper single sideband (USSB) modulation for an experimental condition of H app = 200 Oe and ϕ = 260 • . At this condition, the STNO frequency shows a blue shift with dc bias current. The free running STNO frequency and power at H app = 200 Oe and ϕ = 260 • are shown in Fig. 2(a) and (b), respectively. The blue shift of STNO frequency with bias current we observe here is likely due to magnetization precession in synthetic antiferromagnetic (SyF) layer 1 , as this behavior was not reproduced in our macrospin simulations of the free layer. Figure 2(c) is an example of USSB spectra at I dc = 4.4 mA, I m = 1.2 mA and f m = 700 MHz. As expected from the NFAM formulation, USSB is produced in this case due to the blue shift of STNO frequency with bias current and strong amplitude non-linearity. However, the free-running linewidth at this bias condition is relatively high, which makes the modulation experiment difficult due to the overlap of the carrier and sidebands.

Determination of frequency and amplitude modulation sensitivity coefficients from freerunning properties
In the NFAM formulation, 2, 3 the instantaneous frequency is assumed to depend nonlinearly on the modulating signal: where m(t) is the modulating signal and the coefficients k i represent the i-th order frequency modulation sensitivity. Similarly, the output amplitude A c is given by: where λ i is the i-th order amplitude sensitivity coefficient. The coefficients k i and λ i describe the nonlinear current dependence of f i and A c of the free-running STNO. We use sine wave modulation, m(t) = I m sin(2π f m t), where I m is the amplitude and f m is the frequency of modulation.   Table 1 and 2, respectively. Table 2. Amplitude modulation sensitivity coefficients from the polynomial fits of the amplitude of the free-running STO.

Macrospin simulation of STNO frequency vs. bias current with varying field-like torque
Macrospin simulations were performed by solving the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation 4,5 : Here,m is the normalized magnetic moment, γ is the gyromagnetic ratio of the electron, J is the spin polarized current density, e is the electronic charge and φ is the angle between the free and fixed layers. The saturation magnetization of the free layer M s and the Gilbert damping constant α were taken to be 10 6 A/m and 0.022, respectively. The thickness of the free layer, t fl = 3.5 nm. The magnetization of the fixed layer,ê p , is taken alongx axis with a polarization efficiency, P = 0.65. The value of the field-like torque is changed by varying the ratio between the field like torque and the spin transfer torque, b f to check the tunability of the free-running STNO frequency ( f ) vs. dc bias current (I dc ). H eff is the effective field acting on the STNO, which includes a contribution from the applied external field ( H ext ), the demagnetizing field ( H demag ), interlayer exchange coupling ( H IEC ) and the thermal field ( H therm ). H app is applied in the plane of the sample having onlyx andŷ components. H demag is defined as: where, N x , N y and N z are the demagnetization factors used to characterize the film geometry. The values of demagnetization factors have been approximated for the case of a thin circular disk where the thickness is much lower than the diameter 6 . The calculated values of N x , N y and N z are 0.01125, 0.01125 and 0.9775, respectively.
An interlayer exchange coupling ( H IEC ) of ∼ 117 Oe is included in the net effective field. The thermal field, H therm is defined according to Brown's approximation 7, 8 by adding a random fluctuating field whose components in different directions satisfy the following criteria:  Using such a criteria leads to a Boltzmann distribution of energies at equilibrium condition. Here, the variables i and j refer to components of the time-varying thermal field in different directions. k B is the Boltzmann constant, T is the temperature, V is the volume of the free layer and µ 0 the magnetic permeability of free space. Figure 4(a) shows the f vs. I dc data for varying b f in the range of 0 to 0.5. For the case of b f = 0, we found weak tunability of the STNO frequency with the bias current. However, as b f increases, the current tunability becomes stronger. We did not see strong variation in the power with the increase in field-like torque [ Fig. 4(b)], which is expected, as the field-like torque affects only the resonance frequency of the STNO 9-11 . The weak tunability for low values of b f is the reason for observing LSSB in Fig. 5 of the main paper.