An electrically reconfigurable logic gate intrinsically enabled by spin-orbit materials

The spin degree of freedom in magnetic devices has been discussed widely for computing, since it could significantly reduce energy dissipation, might enable beyond Von Neumann computing, and could have applications in quantum computing. For spin-based computing to become widespread, however, energy efficient logic gates comprising as few devices as possible are required. Considerable recent progress has been reported in this area. However, proposals for spin-based logic either require ancillary charge-based devices and circuits in each individual gate or adopt principals underlying charge-based computing by employing ancillary spin-based devices, which largely negates possible advantages. Here, we show that spin-orbit materials possess an intrinsic basis for the execution of logic operations. We present a spin-orbit logic gate that performs a universal logic operation utilizing the minimum possible number of devices, that is, the essential devices required for representing the logic operands. Also, whereas the previous proposals for spin-based logic require extra devices in each individual gate to provide reconfigurability, the proposed gate is ‘electrically’ reconfigurable at run-time simply by setting the amplitude of the clock pulse applied to the gate. We demonstrate, analytically and numerically with experimentally benchmarked models, that the gate performs logic operations and simultaneously stores the result, realizing the ‘stateful’ spin-based logic scalable to ultralow energy dissipation.

S1. Room temperature modeling of the Spin-Orbit Perpendicular-Anisotropy (SOPE) gate The magnetic dynamics under the effect of the current induced spin-orbit torques at room temperature is governed by the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) 1 equation, where m is a unit vector along the magnetization, α is the Gilbert damping factor causing relaxation of the magnetization to its equilibrium state, and γ = gµ B / is the gyromagnetic ratio, where µ B is the Bohr magneton and g denotes the g-factor. At any instant of time, m makes an angle of ϑ withẑ, while the plane of m andẑ makes an angle ϕ withx.
The spin-orbit torque comprises a damping-like torque and a field-like torque, where ζ and ζ ⊥ denote the efficiency of the current in producing the damping-like and the field-like components, respectively. Since the field-like torque may be negligible in comparison with the damping-like torque 2,3,4 , the auxiliary effect of the field-like spinorbit torque is skipped here. M s denotes the saturation magnetization and t F denotes the thickness of the nanomagnet placed on the spin-orbit channel. H ef f represents the effective field experienced by the magnetization and is expressed as Here H k denotes the perpendicular magnetic anisotropy field, where K u is the magnetic anisotropy and M s is the saturation magnetization. H dp is the dipole field exerted by the reference layer. H d denotes the demagnetization field, where N i , N j , and N z are demagnetizing factors and N i + N j + N z = 1. Here,î andĵ represent the unit vectors along the length and width of the nanomagnet, respectively, Angle Θ denotes the tilt angle enclosed by the length of the nanomagnet and current flow as illustrated in Fig. 1(a).
A nonzero temperature introduces thermal fluctuations to the magnetization, which is modeled by the Langevin random field H L = (H L,x , H L,y , H L,z ). Each component of H L follows a zero-mean Gaussian random process whose standard deviation is a function of Here v F is the volume of the nanomagnet, T denotes the temperature, and ∆t is the duration of the constant effective thermal fluctuation field. As the current flows within the channel, the temperature increases through the Joule heating effect, and is proportional to the square of the current where I is the amplitude of the current, T 0 is the temperature at zero current (room temperature), and k is the Joule heating parameter which relates the temperature to the current.
Joule heating decreases M s and K u as 2,4,6 where M s0 and K u0 are, respectively, the saturation magnetization and magnetic anisotropy at temperature T 0 . Coefficients β and η represent the change in, respectively, M s and K u as the temperature changes by T − T 0 . Substituting (8) in (9), the saturation magnetization and magnetic anisotropy are proportional to the square of the current amplitude as, respectively,

S2. Channel current control for logic operations
To perform a logic operation, a clock pulse is applied to CLK P , thereby producing a current that flows into the channel underneath Q. Depending on the bit stored in P (stable magnetization state of P ), the magnetic tunnel junction comprising nanomagnets R P and P (MTJ P ) exhibits a low or a high resistance due to the magnetoresistance effect, where TMR P denotes the tunneling magnetoresistance ratio of MTJ P and R p 1 (R p 0 ) is the resistance of MTJ P when p = 1 (p = 0). Accordingly, the channel current density is Here V CLK P is the clock pulse amplitude, R c is the interconnect resistance, and R ef f = R s ||R b ||R m , where R s , R b , and R m denote the resistance of the spin-orbit layer, resistance of the buffer layer, and shunt resistance of nanomagnet Q. The interconnect is made of a low resistivity metal, such as copper, thus R c is negligible compared to R p 0(1) + R ef f . Accordingly, from (11) and (12), we have Therefore, to switch the channel current density between J p 0 and J p 1 , TMR P should satisfy For the gate with the switching probability diagram illustrated in Fig. 1, that J p 0 lies within the J n region (J+ region) when V CLK P is set such that J p 1 lies within the J− region (J n region). Hence, by setting R p 0 = R ef f , TMR P = 1.5 (150%) is sufficient to switch the channel current density between J p 1 ∈ J− and J p 0 ∈ J n or between J p 1 ∈ J n and J p 0 ∈ J+.

S3. Channel current control for state transferring in cascaded gates
Taking the same steps as in (11) to (15), and noting that J q 1 is larger than J q 0 , the TMR Q 1 should satisfy the following constraint so that the channel current density can be switched between J q 0 and J q 1 .
where R 1 is the resistance of MTJ Q 1 when q 1 = 1 and R ef f = R s ||R b ||R m is the channel resistance seen from Q 1 . Here, R s , R b , and R m denote the resistance of the spin-orbit layer, resistance of the buffering layer, and shunt resistance of nanomagnet P 2 .

S4. Energy dissipation
The average energy dissipated by performing a NOR or NAND operation using the SOPE gate is where I 0 and I 1 denote the amplitude of the current produced by applying a clock pulse with a duration τ p to CLK P when p is, respectively, 0 and 1. By setting TMR P = 1.5, from (13) and (14) we have Here, I base is the channel current required to perform the operation when ζ = 1, and is a constant that captures the effect of finite shunt resistance of nanomagnet Q, where ρ m , ρ s , and ρ b (d m , d s , and d b ) demote the resistivity (thickness) of the ferromagnetic layer, spin-orbit layer, and buffer layer, respectively. As illustrated in Fig.   S1, the energy dissipation per operation can range from a few aJ to a few fJ.