Mohammad Salehi Fashami

Magnetization dynamics, Bennett clocking and associated energy dissipation in multiferroic logic

It has been recently shown that the magnetization of a multiferroic nanomagnet, consisting of a magnetostrictive layer elastically coupled to a piezoelectric layer, can be rotated by a large angle if a tiny voltage of a few tens of millivolts is applied to the piezoelectric layer. The potential generates stress in the magnetostrictive layer and rotates its magnetization by ~90° to implement Bennett clocking in nanomagnetic logic chains. Because of the small voltage needed, this clocking method is far more energy efficient than those that would employ spin transfer torque or magnetic fields to rotate the magnetization. In order to assess if such a clocking scheme can also be reasonably fast, we have studied the magnetization dynamics of a multiferroic logic chain with nearest-neighbor dipole coupling using the Landau-Lifshitz-Gilbert (LLG) equation. We find that clock rates of 2.5 GHz are feasible while still maintaining the exceptionally high energy efficiency. For this clock rate, the energy dissipated per clock cycle per bit flip is ~52,000 kT at room temperature in the clocking circuit for properly designed nanomagnets. Had we used spin transfer torque to clock at the same rate, the energy dissipated per clock cycle per bit flip would have been ~4 x 10⁸ kT, while with current transistor technology we would have expended ~10⁶ kT. For slower clock rates of 1 GHz, stress-based clocking will dissipate only ~200 kT of energy per clock cycle per bit flip, while spin transfer torque would dissipate about 10⁸ kT. This shows that multiferroic nanomagnetic logic, clocked with voltage-generated stress, can emerge as a very attractive technique for computing and signal processing since it can be several orders of magnitude more energy efficient than current technologies.

Magnetization dynamics, throughput and energy dissipation in a universal multiferroic nanomagnetic logic gate with fan-in and fan-out

The switching dynamics of a multiferroic nanomagnetic NAND gate with fan-in/fan-out is simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation while neglecting thermal fluctuation effects. The gate and logic wires are implemented with dipole-coupled two-phase (magnetostrictive/piezoelectric) multiferroic elements that are clocked with electrostatic potentials of ~50 mV applied to the piezoelectric layer generating 10.1 MPa stress in the magnetostrictive layers for switching. We show that a pipeline bit throughput rate of ~0.5 GHz is achievable with proper magnet layout and sinusoidal four-phase clocking. The gate operation is completed in 2 ns with a latency of 4 ns. The total (internal + external) energy dissipated for a single gate operation at this throughput rate is found to be only ~500 kT in the gate and ~1250 kT in the 12-magnet array comprising two input and two output wires for fan-in and fan-out. This makes it respectively three and five orders of magnitude more energy-efficient than complementary-metal-oxide-semiconductor-transistor (CMOS)-based and spin-transfer-torque-driven nanomagnet-based NAND gates. Finally, we show that the dissipation in the external clocking circuit can always be reduced asymptotically to zero using increasingly slow adiabatic clocking, such as by designing the RC time constant to be three orders of magnitude smaller than the clocking period. However, the internal dissipation in the device must remain and cannot be eliminated if we want to perform fault-tolerant classical computing.

Experimental Clocking of Nanomagnets with Strain for Ultralow Power Boolean Logic

Nanomagnetic implementations of Boolean logic have attracted attention because of their nonvolatility and the potential for unprecedented overall energy-efficiency. Unfortunately, the large dissipative losses that occur when nanomagnets are switched with a magnetic field or spin-transfer-torque severely compromise the energy-efficiency. Recently, there have been experimental reports of utilizing the Spin Hall effect for switching magnets, and theoretical proposals for strain induced switching of single-domain magnetostrictive nanomagnets, that might reduce the dissipative losses significantly. Here, we experimentally demonstrate, for the first time that strain-induced switching of single-domain magnetostrictive nanomagnets of lateral dimensions ∼200 nm fabricated on a piezoelectric substrate can implement a nanomagnetic Boolean NOT gate and steer bit information unidirectionally in dipole-coupled nanomagnet chains. On the basis of the experimental results with bulk PMN-PT substrates, we estimate that the energy dissipation for logic operations in a reasonably scaled system using thin films will be a mere ∼1 aJ/bit.

Switching of Dipole Coupled Multiferroic Nanomagnets in the Presence of Thermal Noise: Reliability of Nanomagnetic Logic

The stress-induced switching behavior of a multiferroic nanomagnet, dipole coupled to a hard nanomagnet, is numerically studied by solving the stochastic Landau-Lifshitz-Gilbert equation for a single-domain macrospin state. Different factors were found to affect the switching probability in the presence of thermal noise at room temperature: 1) dipole coupling strength, 2) stress levels, and 3) stress withdrawal rates (ramp rates). We report that the thermal broadening of the magnetization distribution causes large switching error rates. This could render nanomagnetic logic schemes that rely on dipole coupling to perform Boolean logic operations impractical whether they are clocked by stress or field or other means.

Energy dissipation and error probability in fault-tolerant binary switching

The potential energy profile of an ideal binary switch is a symmetric double well. Switching between the wells without energy dissipation requires time-modulating the height of the potential barrier separating the wells and tilting the profile towards the desired well at the precise juncture when the barrier disappears. This, however, demands perfect timing synchronization and is therefore fault-intolerant even in the absence of noise. A fault-tolerant strategy that requires no time modulation of the barrier (and hence no timing synchronization) switches by tilting the profile by an amount at least equal to the barrier height and dissipates at least that amount of energy in abrupt switching. Here, we present a third strategy that requires a time modulated barrier but no timing synchronization. It is therefore fault-tolerant, error-free in the absence of thermal noise, and yet it dissipates arbitrarily small energy in a noise-free environment since an arbitrarily small tilt is required for slow switching. This case is exemplified with stress induced switching of a shape-anisotropic single-domain soft nanomagnet dipole-coupled to a hard magnet. When thermal noise is present, we show analytically that the minimum energy dissipated to switch in this scheme is ~2kTln(1/p) [p = switching error probability].

Corrigendum: Energy dissipation and error probability in fault-tolerant binary switching

Scientific Reports 3: Article number: 320410.1038/srep03204; published online: November132015; updated: February222016 In the original paper our stochastic simulations had a systematic error that lead to an incorrect error probability in Figure 4 (b). The corrected version of this figure is Fig. 1 of this corrigendum. The actual error probability turns out to be smaller than what we had reported earlier. However, the new plots convey the same physics and overall message; hence, the key message of this paper and our conclusions remain unchanged. Figure 1 Energy dissipated vs. the natural logarithm of the inverse dynamic switching error probability at various temperatures. We plot the simulation results for the switching error probability vs. dipole energy for the 4.2, 77 and 300 K cases up to the point where the dipole coupling (tilt in profile) is still low enough that the error probability exceeds ~10−6 and 10 million simulations suffice to capture the statistics accurately. (NOTE: Errors of 10−5 and 10−6 approximately correspond to 11.5 and 13.8 respectively on the y-axis). The leveling off of switching error probability with increasing dipole energy at 300 K is still observed, albeit at a much smaller switching error probability value. The error probabilities up to which we perform simulations are too high for the leveling off behavior to set in at 4.2 and 77 K.