Granular vortex spin-torque nano oscillator for reservoir computing

In this paper, we investigate the granularity in the free layer of the magnetic tunnel junctions (MTJ) and its potential to function as a reservoir for reservoir computing where grains act as oscillatory neurons while the device is in the vortex state. The input of the reservoir is applied in the form of a magnetic field which can pin the vortex core into different grains of the device in the magnetic vortex state. The oscillation frequency and MTJ resistance vary across different grains in a non-linear fashion making them great candidates to be served as the reservoir's outputs for classification objectives. Hence, we propose an experimentally validated area-efficient single granular vortex spin-torque nano oscillator (GV-STNO) device in which pinning sites work as random reservoirs that can emulate neuronal functions. We harness the nonlinear oscillation frequency and resistance exhibited by the vortex core granular pinning of the GV-STNO reservoir computing system to demonstrate waveform classification.


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The grains present in the free layer of the magnetic tunnel junction and their effect can be easily monitored through the energy landscape.In the main text, Fig. 2 shows how different input frequencies (equal, less, and greater than natural frequency) change the energy landscape due to the interplay between the several energy components exchange, demagnetization, and Zeeman.Fig. S1 shows that there is one local energy minimum for the natural frequency of the magnetic vortex core, fgyro irrespective of the presence or absence of grains [1].The vortex core is translated to 100 nm distance from the center.
The nonlinearity observed for the MTJ resistance experimentally (refer to Fig. 3 (c)) indicates the presence of a granular effect in the system otherwise the relation is supposed to be linear [2].Moreover, the randomness in the nonlinear characteristics of the resistance in the vortex state for different devices with varying diameters is also observed and presented in Fig. S2.Consequently, the experiment provides proof of the concept that reservoir computing can be a good candidate for future applications provided by granular effects present in the system.Fig. 4 of the main text wherein the amplitude of the applied Magnetic field is varied to investigate the nonlinear feature of reservoir computing.The output frequency and MTJ resistance for different amplitudes for sine and square wave inputs are presented here in Table 1.Moreover, the MTJ resistance is calculated as RMTJ = ((my×MR) + RDC) for which the maximum value of my is considered.This performance metric is useful for reading out through a CMOS circuitry.For the waveform classification application, we have considered four different combinations of sine and square waveform inputs-Sine-Sine, Sine-Square, Square-Sine, and Square-Square as presented in Fig. 5 of the main text.Table 2 presents the frequency range of pinned vortex core (fout) and oscillating resistance (ΔR) (which gives rise to MTJ resistance) for four combinational signals.
For Fig. 7 in the main text, we present the frequency detection through the spin diode effect.A front-end CMOS circuit is also proposed [2] that generates a DC voltage if and only if the frequency range of the input signal is closed to the oscillation frequency of the nano-oscillator, which mimics a down-converter mixer followed by a low-pass filter with an ultra-high-quality factor or an ideal in-phase only locking amplifier.This suppresses all out-of-band noise and allows the output signal to be detected.To prove this claim, a noisy and pure sinusoidal signal has been applied to the system.Similarly, if we want to detect the granular frequency, then we might need a filter to choose the specific band of granular output frequency while discarding the input frequency.For instance, we performed FFT for input sine and square wave signals with frequencies 25 and 50 MHz, respectively.As shown in Figs.S3 and S4, if input frequency (with high power though) will be filtered, then on amplifying the power of granular frequency we can detect the grain frequency.
FFT signals and their description for sine and square wave inputs for without (wo) and with (w) grain systems are shown hereby: Fig. S3: FFT of output signal for given input frequency as 25 MHz for sine and square waveforms wherein the with grain system shows more than one output power peak due to fout obtained at different pinning sites Sine input: The power at input frequency (25 MHz), without grain is -25.8 dBm whereas for with grains -24.2 dBm.Moreover, there is no other peak frequency for without grains system (maybe a slight peak at gyrotropic frequency) whereas we can see some peaks for with grains system (at 75 MHz, 125 MHz, 225 MHz, and 473 MHz with power -42 dBm, -46.2 dBm, -52 dBm, and -53 dBm, respectively).

Square input:
The power at input frequency (25MHz), without grain is -23.68 dBm whereas for with grains it is -23.3 dBm.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas we can see some peaks for with grains system (at 74.8 MHz, 100 MHz, and 124 MHz with power -29.3 dBm, -31.9 dBm, and -31.2 dBm, respectively).Moreover, for the square wave input, we see some harmonics in the without grains system too but those signals are too weak which could be of the order of thermal noise floor.Fig. S5: FFT of output signal for given input frequency as 100 MHz for sine and square waveforms wherein the without and with grain system only show one high output peak power at the input frequency which is the gyrotropic frequency.

Sine input:
The power for resonant frequency (at 100 MHz) is -12.5 dBm and -11.4 dBm, for without grains and with grains systems, respectively.

Square input:
The power for resonant frequency (at 100 MHz) is -11.5 dBm and -10.3 dBm, for without grains and with grains systems, respectively.
Fig. S6: FFT of output signal for given input frequency as 200 MHz for sine and square waveforms wherein the with grain system shows more than one output power peak due to fout obtained at different pinning sites Sine input: The power at input frequency (200 MHz), without grain is -45.8 dBm whereas for with grains there is no frequency peak at input signal.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas can see some peaks for with grains system (at 50 MHz, 80 MHz, and 109 MHz with power -41.1 dBm, -29.8 dBm, and -32.3 dBm, respectively).

Square input:
The power at input frequency (200 MHz), without grain is -26.9 dBm whereas for with grains there is no frequency peak at input signal.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas we can see some peaks for with grains system (at 59.7 MHz, 84.7 MHz, 100 MHz, 115 MHz, 125 MHz, 139.5 MHz, 598 MHz, and 1 GHz with power -36 dBm, -27.7 dBm, -34.6 dBm, -31.8 dBm, -34.4 dBm, -34.6 dBm, -47.6 dBm, and -51 dBm, respectively).Interestingly, we also see some high-frequency peaks in the without grains system, however, the signals are too weak to be detected by CMOS readout circuit whereas the system with grains provides more fout peaks with detectable output power.Sine input: The power at input frequency (400 MHz), without grain is -43 dBm whereas for with grains it is -41.9 dBm.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas we can see some peaks for with grains system (at 80 MHz, 100 MHz, 140 MHz, 160 MHz, and 219.5 MHz with power -31.8 dBm, -28.6 dBm, -33 dBm, -34.1 dBm, and -43.6 dBm, respectively).

Square input:
The power at input frequency (400 MHz), without grain is -43 dBm whereas for with grains it is -41.9 dBm.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas we can see some peaks for with grains system (at 40 MHz, 60 MHz, 90 MHz, 139.7 MHz, 169.6 MHz, 219 MHz, 400 MHz, 528.8 MHz, 1.2 GHz, and 2 GHz with power -35.2 dBm, -30.4 dBm, -28 dBm, -34 dBm, -44 dBm, -45.9 dBm, -42.2 dBm, -50.3 dBm, -47.6 dBm, and -50 dBm respectively).We can also see some high-frequency peaks in the without grain system which might be due to the harmonics.
The oscillation power can be evaluated using the following expression [4]: (∆) 2 . (  +  ) 2 (1 where ∆R is the AC resistance which is the product of my with magnetoresistance (MR) and RDC is the average of P and AP resistances (here, 100 Ω and 200 Ω, respectively).Further, the load resistance, RL, is 50 Ω which is also used for impedance matching.Iread is the applied current to read out the vortex state of the nanodisk.The readability in some cases is low, especially for lower amplitude of the applied fields, however, it is much higher when compared to skyrmions [5].The magnetization, my, for skyrmions, is predicted to be less than 0.1 through the simulation as reported in [5] and [6], however, using GV-STNO, a higher and easily detectable outcome is achieved.

Fig. S1 .
Fig. S1.Total energy density landscape of GV-STNO plotted against the circular replacement of the vortex core position on the FL surface considering (a) no granular effect and (b) with granular effect.

Fig. S2 .
Fig. S2.For different samples, the MTJ resistance in the vortex state shows randomness when it comes to the nonlinear behavior caused by the grains.

Fig. S7 :
Fig.S7: FFT of output signal for given input frequency as 400 MHz for sine and square waveforms wherein the with grain system shows more than one output power peak due to fout obtained at different pinning sites

Table 1 :
Oscillation frequency and MTJ resistance for sine and square wave inputs with different amplitude and input frequencies

Table 2 :
Oscillation frequency and AC component of resistance for different combinations of sine and square wave inputs with different FFT of output signal for given input frequency as 50 MHz for sine and square waveforms wherein the with grain system shows more than one output power peak due to fout obtained at different pinning sites Sine input: The power at input frequency (50MHz), without grain is -24.4 dBm whereas for with grains -21.2 dBm.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas we can see some peaks for with grains system (at 55 MHz, 81.8 MHz, and 100 MHz with power -37.3 dBm, -38.1 dBm, and -37.6 dBm, respectively).
input: The power at input frequency (50MHz), without grain is -22.2 dBm whereas for with grains -21.3 dBm.Moreover, there is no other peak frequency for the without grains system (except at gyrotropic frequency) whereas we can see some peaks for with grains system (at 83.8 MHz, 95 MHz, 104 MHz, and 149 MHz with power -38.7, -29.6, -29.2, and -37.6 dBm, respectively).