In this work, we propose a spin nano-oscillator (SNO) device where information is encoded in the phase (time-shift) of the output oscillations. The spin current required to set up the oscillations in the device is generated through spin pumping from an input nanomagnet that is precessing at RF frequencies. We discuss the operation of the SNO device, in which either the in-plane (IP) or out-of-plane (OOP) magnetization oscillations are utilized toward implementing ultra-low-power circuits. Using physical models of the nanomagnet dynamics and the spin transport through non-magnetic channels, we quantify the reliability of the SNO device using a “scaling ratio”. Material requirements for the nanomagnet and the channel to ensure correct logic functionality are identified using the scaling ratio metric. SNO devices consume (2-5)× lower energy compared to CMOS devices and other spin-based devices with similar device sizes and material parameters. The analytical models presented in this work can be used to optimize the performance and scaling of SNO devices in comparison to CMOS devices at ultra-scaled technology nodes.

## I. INTRODUCTION

Spintronics technology exploits the interaction between electric currents and the magnetic order parameter in conducting magnetic nanostructures through the spin-transfer-torque (STT) effect.^{1} In majority of the proposed spin-based devices, the STT effect is used to switch the magnetization vector of a nanomagnet, thereby switching between the stored logical values of “1” and “0”.^{2,3} The STT effect can also be used to excite steady-state magnetization oscillations at RF frequencies when the negative damping due to the STT effect balances the intrinsic damping of the nanomagnet.^{4}

In this paper, we study the dynamics of spin nano-oscillators (SNOs) that can be used as substrates for Boolean and non-Boolean architectures.^{5–8} The proposed SNO device is forced into oscillation by a pure spin current generated through the spin-pumping mechanism. Spin-pumping mechanism is the reciprocal effect of the STT effect as governed by the Onsager reciprocity relationships that relate the thermodynamic forces and currents in the linear response matrix.^{9} The precessing input nanomagnet pumps spin current into the non-magnetic channel that communicates information encoded in the phase of oscillations between the transceiver nanomagnets. Electrical current flow is required only in the input stage to start the oscillations. This implies significant energy savings in circuits with a higher logic depth, i.e. circuits with series-connected SNO gates.

To quantify the performance of the SNO device, we estimate the metrics of both in-plane (IP) and out-of-plane (OOP) magnetization oscillations of a thin film nanomagnet with biaxial anisotropy. Specifically, we analyze the spin current requirement and the frequency of both IP and OOP oscillations. Using the analytical models of magnetization oscillations combined with the spin pumping description, we derive a scaling ratio that allows us to relate the material- and device-level metrics to the circuit-level performance, i.e. logic reliability and energy dissipation. We also compare the performance of the SNO device against other spin-based devices and CMOS devices at ultra-scaled technology nodes.

## II. DEVICE OPERATION

The four-step “copy” function implemented using the proposed SNO device is shown in Fig. 1. To initiate the operation, a nanomagnet precessing at RF frequencies pumps a pure spin current into the non-magnetic channel of the SNO device. As no electric field is present across the non-magnetic channel, a pure spin diffusion current carrying the phase-encoded information flows through the non-magnetic channel. Upon reaching the receiver, it transfers the information to the output stage by forcing oscillations in the magnetization of the output nanomagnet. Each logic stage consists of two nanomagnets that are connected in a push-pull manner with their magnetizations oriented opposite to each other. In the push-pull configuration, only one of the two nanomagnets exhibits oscillations ensuring that the copy function will be correctly implemented for complementary inputs. This configuration is reminiscent of complementary MOS logic in which only one of the two n-type or p-type transistors is turned on in the steady state. To initiate magnetization oscillations in the input stage of a multi-stage network of SNO devices, electrical current will be required. However, this contains the Joule heating to the input stage and leads to significant energy savings at the circuit level for complex logic. The necessary condition to achieve the correct functionality of the device is that the spin current reaching the receiver nanomagnet must meet the spin current requirements to sustain magnetization oscillations in the output nanomagnet.

### A. Spin pumping

In non-magnet/ferromagnet heterostructures, spin accumulation in the non-magnetic material can exert a torque on the ferromagnet known as the spin-transfer torque or STT. The reciprocal process of STT is known as spin pumping in which, a precessing magnet induces a spin current in the adjacent non-magnetic material.^{10} The pumped spin current also creates a spin accumulation in the non-magnetic channel, which induces a spin backflow into the nanomagnet. The ac or the oscillating transverse component of the spin current vanishes inside most of the normal metal when $\u210fDN/dN2<<\u210f\omega $.^{11} Here, $\u210f$ is the reduced Planck’s constant, *D*_{N} and *d*_{N} are the spin diffusion coefficient and the length of the non-magnetic channel, and $\omega $ is the frequency of the nanomagnet that is pumping spin current. By accounting for spin backflow and in the limit of vanishing transverse oscillating spin accumulation, the time-averaged spin current in the non-magnetic channel is given as^{12}

In the above equations, *e* is the elementary charge, *L*_{sN} and $\nu DOS$ are the spin-relaxation length and the one-spin density of states, respectively, in the non-magnetic channel, $Re(g\u2191\u2193)$ is the (dimensionless) real spin mixing conductance, *A*_{cross} is the cross-sectional area of the device, and $\theta $ is the precession cone angle of magnetization vector and the effective magnetic field. The reduction factor $\eta z$ is related to the spin back-flow and causes the net spin current in the channel to reduce. Within the channel, the spin current decays with position according to $js(x)=\beta (x)jspump$, where the factor $\beta (x)$ is given as

The solution of *j*_{s}(*x*) is obtained by solving the spin diffusion equation in the channel and assuming that the spin accumulation vanishes at the boundary *x* = *d*_{N}, while the spin current at the boundary is finite. The term $\beta (x)$ in Eq. 2 increases with an increase in the ratio (*L*_{sN}/*d*_{N}). On the contrary, the factor $\eta z$ in Eq. 1b reduces with an increase in the ratio (*L*_{sN}/*D*_{N}), which reduces the net current *j*_{s}(*x*). Therefore, the spin current reaching the receiver nanomagnet will vary non-monotonically with *L*_{sN} for a fixed value of *d*_{N}.

### B. Spin oscillator

We consider a monodomain nanomagnet with biaxial magnetic anisotropy, where the anisotropy energies are given as $K=(1/2)\mu 0MSHKV$ (easy-axis) and $KM=\mu 0MS2V$ (hard-axis). Here, $\mu 0=4\pi 10\u22127$ H/m is the free-space permeability, *M*_{S} is the saturation magnetization in A/m, *H*_{K} is the Stoner-Wohlfarth field in A/m, and *V* is the volume of the nanomagnet. The dynamical evolution of monodomain $m^$ under the STT effect is accurately described through the stochastic Landau Lifshitz Gilbert Slonczewski (LLGS) equation, which is given as^{13}

where $H\u2192eff$ is the sum of the intrinsic anisotropy field and the thermal field acting on the nanomagnet, $\alpha $ is the Gilbert damping constant and is set equal to 0.01 for all simulations in this paper, $Is,\u22a5$ is the perpendicular component of the spin current, $Ns=2MSV/\gamma \u210f$. Here, $\gamma $ is the gyromagnetic ratio. Thermal field is modeled as a three-dimensional Wiener process with statistical properties discussed by Brown.^{14,15} The IP and OOP trajectories that are typically excited in the monodomain due to the STT effect are shown in Fig. 2a. The characteristics that differentiate the two oscillation trajectories are: (i) the current required to initiate and sustain oscillations and (ii) the frequency (or time period) of oscillations.^{16} The spin current for which magnetization oscillations are observed is shown as the shaded yellow region in Figs. 2(b,c) as a function of the ratio of hard- and easy-axis anisotropies, denoted as *D* = *M*_{S}/*H*_{K} in the figures. While OOP trajectories only exist for $D>0.2$, IP trajectories exist for all values of *D* although their susceptibility to noise increases below $D<5.09$. In addition, the frequency of OOP oscillations exceeds that of IP oscillations for larger values of *D*. A higher oscillation frequency increases the device reliability by increasing the spin current pumped into the non-magnetic channel of the device.

## III. DESIGN AND MATERIAL REQUIREMENTS

To ensure the correct logic functionality of the proposed device, the magnitude of the spin current reaching the output nanomagnet must lie between the minimum and maximum spin current to sustain oscillations in the nanomagnet as shown in Fig. 2(b,c). To characterize the reliability of operation, we compute two scaling ratios *r*_{min} = *j*_{s}(*x* = *d*_{N})/*j*_{min} and *r*_{max} = *j*_{s}(*x* = *d*_{N})/*j*_{max}. Here, *j*_{min} and *j*_{max} are the threshold and maximum spin current densities, respectively, to sustain IP or OOP oscillations of the magnetization vector. For correct device operation, we impose the conditions: $rmin\u22651$ and $rmax\u22641$. The scaling ratios for both IP and OOP oscillations are given in Table I.

Oscillation
. | r
. _{min} | r
. _{max} |
---|---|---|

In-plane (IP) | $\kappa (x)8\pi 1+1/D1+D/2$ | $\kappa (x)161D$ ($D>5.09$) |

as above | r_{min} ($D<5.09$) | |

Out-of-plane (OOP) ($D>0.2$) | $\kappa (x)16$ | $\kappa (x)8\pi 1+DD+1/2$ |

$\kappa (x)=\u210f\gamma \mu 02\xi kBT\eta zg0\u2191\u2193sin2(\theta )AcrossdF\beta (x)$ |

Oscillation
. | r
. _{min} | r
. _{max} |
---|---|---|

In-plane (IP) | $\kappa (x)8\pi 1+1/D1+D/2$ | $\kappa (x)161D$ ($D>5.09$) |

as above | r_{min} ($D<5.09$) | |

Out-of-plane (OOP) ($D>0.2$) | $\kappa (x)16$ | $\kappa (x)8\pi 1+DD+1/2$ |

$\kappa (x)=\u210f\gamma \mu 02\xi kBT\eta zg0\u2191\u2193sin2(\theta )AcrossdF\beta (x)$ |

In the above set of equations, $g0\u2191\u2193=Re(g\u2191\u2193)/Across$, and *d*_{F} denotes the thickness of the nanomagnet. In writing the above equations, we fix the energy barrier of the macrospin as $\Delta U=(1/2)\mu 0MsHKAcrossdF=\xi kBT$ ($\xi >>1)$. All scaling factors increase with a reduction in the nanomagnet thickness and an increase in the cross-sectional area of the device. However, increasing the cross-sectional area will require a comprehensive treatment of multi-domain effects^{19} in nanomagnets and will be pursued in follow-up work. In the case of IP oscillations for $D<5.09$, the condition for correct logic functionality becomes *r*_{min} = *r*_{max} = 1. The values of *L*_{sN} and *D*_{N} for which both these conditions are simultaneously satisfied are plotted in Fig. 3a. Engineering non-magnetic channels for which the specific combination of *D*_{N} and *L*_{sN} values is obtained will be prohibitive for the SNO devices utilizing IP oscillations. Interestingly, $rminOOP$ is independent of the ratio *D* implying that by proper selection of material parameters, OOP trajectories can be excited for larger values of *D*, easing the constraints on nanomagnet design. In Figs. 3(b,c), we map the region of correct functionality of the SNO device utilizing OOP oscillations. The dark blue regions correspond to $rmin<1$ and, therefore, will lead to incorrect functionality. In contrast to IP oscillations, OOP oscillations are more robust in that they provide a broad range of values of *L*_{sN} and *D*_{N} where the device can operate correctly. The range of material properties, which lead to reliable device functionality with OOP oscillations, increases with an increase in the value of *D*. This is especially advantageous for thin film nanomagnets that have a high demagnetization field perpendicular to the nanomagnet plane.

### A. Comparison with CMOS logic

A comparison of the energy dissipation of spin-based and CMOS devices is shown in Fig. 4. For a fair comparison, we keep the size, the energy barrier, and the material properties of nanomagnets same across all spin-based devices. First, the amount of threshold spin current needed for each device is calculated based on the device geometry, dimensions, and material properties.^{15} The amount of input spin current is taken to be twice the threshold current for each spin device. The impact of thermal-noise-induced variability on the performance of spin-based devices is also included through comprehensive Monte Carlo simulations to obtain the probability density function of magnetization switching.^{15} The data for high performance (HP) and low-power (LP) CMOS devices is taken from the 2013 edition of the International Technology Roadmap for Semiconductors (ITRS) for the technology year of 2021. The energy dissipation of *N*-SNOs connected in parallel is given as

where $\nu i$ is the multiple of the time period, $\tau per,i$, of the SNO (required for injection locking to an external microwave signal), *R*_{i} is the interface resistance in the path of current flow, $\eta i$ is the efficiency of spin injection, and *I*_{osc,spin,i} is the spin current required for the *i*^{th} SNO to achieve stable limit cycles. For series connected SNO devices, *N* = 1 in Eq. 4, since energy is consumed only in the input stage and is independent of the logic depth so long as correct logic functionality is ensured. Spin-based devices, such as the all-spin logic (ASL) and the the giant spin Hall logic (GSHL), rely on complete magnetization reversal to switch between logic states and, therefore, consume more energy than CMOS and SNO devices. SNOs compare favorably against CMOS devices in terms of energy dissipation. The SNO device utilizing OOP oscillations is an order of magnitude more energy efficient compared to SNO devices utilizing IP oscillations for $D>1$.

## IV. SUMMARY

Spin nano-oscillators (SNOs) driven by pure spin current exhibit either in-plane (IP) or out-of-plane (OOP) magnetization oscillations, which can be used to implement energy-efficient phase-based logics. In this paper, we focus on the material requirements of the non-magnetic channel and the nanomagnet to ensure correct logic functionality of the SNO device. While IP oscillations incur less area of implementation, they are more susceptible to noise and require precise values of material parameters, i.e. spin-relaxation length, diffusion coefficient, and the magnetic anisotropies. On the other hand, OOP-based SNOs offer a broader range of material options to choose from and can be used reliably for information encoding. The energy efficiency of SNO devices, particularly using OOP oscillations, is at least (2-5)× better than that of CMOS devices for ultra-scaled technology nodes.