Supriyo Bandyopadhyay

Hybrid spintronics and straintronics: A magnetic technology for ultra low energy computing and signal processing

The authors show that the magnetization of a 2-phase magnetostrictive/piezoelectric multiferroic single-domain shape-anisotropic nanomagnet can be switched with very small voltages that generate strain in the magnetostrictive layer. This can be the basis of ultralow power computing and signal processing. With appropriate material choice, the energy dissipated per switching event can be reduced to ∼45 kT at room temperature for a switching delay of ∼100 ns and ∼70 kT for a switching delay of ∼10 ns, if the energy barrier separating the two stable magnetization directions is ∼32 kT. Such devices can be powered by harvesting energy exclusively from the environment without the need for a battery.

Bennett clocking of nanomagnetic logic using multiferroic single-domain nanomagnets

The authors show that it is possible to rotate the magnetization of a multiferroic (strain-coupled two-layer magnetostrictive-piezoelectric) nanomagnet by a large angle with a small electrostatic potential. This can implement Bennett clocking in nanomagnetic logic arrays resulting in unidirectional propagation of logic bits from one stage to another. This method of Bennett clocking is superior to using spin-transfer torque or local magnetic fields for magnetization rotation. For realistic parameters, it is shown that a potential of ~ 0.2 V applied to a multiferroic nanomagnet can rotate its magnetization by nearly 900 to implement Bennett clocking.The authors show that it is possible to rotate the magnetization of a multiferroic (strain-coupled two-layer magnetostrictive-piezoelectric) nanomagnet by a large angle with a small electrostatic potential. This can implement Bennett clocking [Int. J. Theor. Phys. 21, 905 (1982)] in nanomagnetic logic arrays resulting in unidirectional propagation of logic bits from one stage to another. This method is potentially more energy efficient than using spin-transfer torque for magnetization rotation. For realistic parameters, it is shown that a potential of ∼0.2 V applied to a multiferroic nanomagnet can rotate magnetization by nearly 90° to implement Bennett clocking.

Energy dissipation and switching delay in stress-induced switching of multiferroic nanomagnets in the presence of thermal fluctuations

Switching the magnetization of a shape-anisotropic 2-phase multiferroic nanomagnet with voltage-generated stress is known to dissipate very little energy (<1 aJ for a switching time of ∼0.5 ns) at 0 K temperature. Here, we show by solving the stochastic Landau-Lifshitz-Gilbert equation that switching can be carried out with ∼100% probability in less than 1 ns while dissipating less than 1.5 aJ at room temperature. This makes nanomagnetic logic and memory systems, predicated on stress-induced magnetic reversal, one of the most energy-efficient computing hardware extant. We also study the dependence of energy dissipation, switching delay, and the critical stress needed to switch, on the rate at which stress on the nanomagnet is ramped up or down.

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.

Switching dynamics of a magnetostrictive single-domain nanomagnet subjected to stress

The temporal evolution of the magnetization vector of a single-domain magnetostrictive nanomagnet, subjected to in-plane stress, is studied by solving the Landau-Lifshitz-Gilbert equation. The stress is ramped up linearly in time and the switching delay, which is the time it takes for the magnetization to flip, is computed as a function of the ramp rate. For high levels of stress, the delay exhibits a non-monotonic dependence on the ramp rate, indicating that there is an {\it optimum} ramp rate to achieve the shortest delay. For constant ramp rate, the delay initially decreases with increasing stress but then saturates showing that the trade-off between the delay and the stress (or the energy dissipated in switching) becomes less and less favorable with increasing stress. All of these features are due to a complex interplay between the in-plane and out-of-plane dynamics of the magnetization vector induced by stress.

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.

Conductance modulation of spin interferometers

We study the conductance modulation of gate controlled electron spin interferometers (also known as spin field effect transistors) based on the Rashba spin--orbit coupling effect. It is found that the modulation is dominated by Ramsauer (or Fabry-Perot) type transmission resonances rather than the Rashba effect in typical structures. These transmission resonances are due to reflections at the interferometers contacts caused by large interface potential barriers and effective mass mismatch between the contact material and the semiconductor. They are particularly strong in quasi-one-dimensional structures which, in fact, are preferred for spin interferometers because of the energy independence of the spin precession angle. Thus, unless particular care is taken to eliminate Ramsauer resonances by proper contact engineering, any observed conductance modulation of spin interferometers may not have its origin in the Rashba effect.

Phase coherent quantum mechanical spin transport in a weakly disordered quasi one-dimensional channel

A transfer matrix technique is used to model phase-coherent spin transport in the weakly disordered quasi-one-dimensional channel of a gate-controlled electron spin interferometer [S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990)]. The model includes the effects of an axial magnetic field in the channel of the interferometer (caused by the ferromagnetic contacts), a Rashba spin-orbit interaction, and elastic (nonmagnetic) impurity scattering. We show that in the presence of an axial magnetic field, nonmagnetic impurities can cause spin relaxation in a manner similar to the Elliott-Yafet mechanism. The amplitudes and phases of the conductance oscillations of the interferometer and the degree of spin-conductance polarization are found to be quite sensitive to the height of the interface barrier at the contact, as well as the strength, locations, and nature (attractive or repulsive) of just a few elastic nonmagnetic impurities in the channel. This can seriously hinder practical applications of spin interferometers.

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.

Four-state nanomagnetic logic using multiferroics

The authors theoretically demonstrate the implementation of a low-power 4-state universal logic gate (NOR) using a linear array of three dipole-coupled magnetostrictive-piezoelectric multiferroic nanomagnets (e.g. Ni/PZT) with biaxial magnetocrystalline anisotropy. The two peripheral nanomagnets in the array encode the 4-state input bits in their magnetization orientations and the central nanomagnets magnetization orientation represents the output bit. Numerical simulations are performed to confirm that the 4-state output bit is the Boolean NOR function of the two 4-state inputs bits when the array reaches its ground state. A voltage pulse alternating between ?0.2 and +0.2?V, applied to the piezoelectric layer of the central nanomagnet, generates alternating tensile and compressive stress in its magnetostrictive layer. This drives the array to the correct ground state where dipole interaction between the magnets ensures that the output is the NOR function of the input. For the system considered, the gate operation is executed while dissipating only ~33?000?kT (0.138?fJ) of energy.