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Developing permanent magnets with fewer critical elements requires understanding hysteresis effects and coercivity through visualizing magnetization reversal. Here, we numerically investigate the effect of the geometry of nanoscale ferromagnetic inclusions in a paramagnetic/non-magnetic matrix to understand the key factors that maximize the magnetic energy product of such nanocomposite systems. Specifically, we have considered a matrix of 3 micron x 3 micron x40 nanometer dimension, which is a sufficiently large volume, two-dimensional representation considering that the ferromagnetic inclusions thickness is less than 3.33% of the lateral dimensions simulated. Using this approach that is representative of bulk behavior while being computationally tractable for simulation, we systematically studied the effect of the thickness of ferromagnetic strips, the separation between the ferromagnetic strips due to the nonmagnetic matrix material, and the length of these ferromagnetic strips on magnetic coercivity and remanence by simulating the hysteresis loop plots for each geometry. Furthermore, we study the underlying micromagnetic mechanism for magnetic reversal to understand the factors that could help attain the maximum magnetic energy densities for ferromagnetic nanocomposite systems in a paramagnetic/non-magnetic material matrix. In this study, we have used material parameters of an exemplary Alnico alloy system, a rare-earth-free, thermally stable nanocomposite, which could potentially replace high-strength NdFeB magnets in applications that don't require large energy products. This can stimulate further experimental work on the fabrication and large-scale manufacturing of RE-free PMs with such nanocomposite systems.

Selective control of individual spin qubits is needed for scalable quantum computing based on spin states. Achieving high-fidelity in both single and two-qubit gates, essential components of universal quantum computers, necessitates highly localized control fields. These fields must be capable of addressing specific spin qubits while minimizing gate errors and cross-talk in adjacent qubits. Overcoming the challenge of generating a localized radio-frequency magnetic field, in the absence of elementary magnetic monopoles, we introduce a technique that combines divergent and convergent nanoscale magnetic skyrmions. This approach produces a precise control field that manipulates spin qubits with high fidelity. We propose the use of 2D skyrmions, which are 2D analogues of 3D hedgehog structures. The latter are emergent magnetic monopoles, but difficult to fabricate. The 2D skyrmions, on the other hand, can be fabricated using standard semiconductor foundry processes. Our comparative analysis of the density matrix evolution and gate fidelities in scenarios involving proximal skyrmions and nanomagnets indicates potential gate fidelities surpassing 99.95% for \pi/2-gates and 99.90% for \pi-gates. Notably, the skyrmion configuration generates a significantly lower field on neighboring spin qubits, i.e. 15 times smaller field on a neighboring qubit compared to nanomagnets that produces the same field at the controlled qubit, making it a more suitable candidate for scalable quantum control architectures by reducing disturbances in adjacent qubits.

In the autoencoder based anomaly detection paradigm, implementing the autoencoder in edge devices capable of learning in real-time is exceedingly challenging due to limited hardware, energy, and computational resources. We show that these limitations can be addressed by designing an autoencoder with low-resolution non-volatile memory-based synapses and employing an effective quantized neural network learning algorithm. We propose a ferromagnetic racetrack with engineered notches hosting a magnetic domain wall (DW) as the autoencoder synapses, where limited state (5-state) synaptic weights are manipulated by spin orbit torque (SOT) current pulses. The performance of anomaly detection of the proposed autoencoder model is evaluated on the NSL-KDD dataset. Limited resolution and DW device stochasticity aware training of the autoencoder is performed, which yields comparable anomaly detection performance to the autoencoder having floating-point precision weights. While the limited number of quantized states and the inherent stochastic nature of DW synaptic weights in nanoscale devices are known to negatively impact the performance, our hardware-aware training algorithm is shown to leverage these imperfect device characteristics to generate an improvement in anomaly detection accuracy (90.98%) compared to accuracy obtained with floating-point trained weights. Furthermore, our DW-based approach demonstrates a remarkable reduction of at least three orders of magnitude in weight updates during training compared to the floating-point approach, implying substantial energy savings for our method. This work could stimulate the development of extremely energy efficient non-volatile multi-state synapse-based processors that can perform real-time training and inference on the edge with unsupervised data.

Multistate memory systems have the ability to store and process more data in the same physical space as binary memory systems, making them a potential alternative to existing binary memory systems. In the past, it has been demonstrated that voltage-controlled magnetic anisotropy (VCMA) based writing is highly energy-efficient compared to other writing methods used in non-volatile nano-magnetic binary memory systems. In this study, we introduce a new, VCMA-based and skyrmion-mediated non-volatile ternary memory system using a perpendicular magnetic tunnel junction (p-MTJ) in the presence of room temperature thermal perturbation. We have also shown that ternary states -1, 0, +1 can be implemented with three magnetoresistance values obtained from a p-MTJ corresponding to ferromagnetic up, down, and skyrmion state, with 99% switching probability in the presence of room temperature thermal noise in an energy-efficient way, requiring ~3 fJ energy on an average for each switching operation. Additionally, we show that our proposed ternary memory demonstrates an improvement in area and energy by at least 2X and ~60X respectively, compared to state-of-the-art spin-transfer torque (STT)-based non-volatile magnetic multistate memories. Furthermore, these three states can be potentially utilized for energy-efficient, high-density in-memory quantized deep neural network implementation.

Physical Reservoir Computing (PRC) is an unconventional computing paradigm, which exploits nonlinear dynamics of reservoir blocks to perform recognition and classification tasks. Here we show with simulations that patterned thin films hosting several skyrmions, particularly one, two, four and nine skyrmions, can implement energy efficient reservoir computing. This reservoir computing (RC) block is based on nonlinear breathing dynamics of skyrmions, which are coupled to each other through dipole interaction and spin waves, in response to a voltage generated strain. This nonlinear and coupled magnetization dynamics is exploited to perform temporal pattern recognition. Two performance metrics, namely Short-Term Memory (STM) and Parity Check (PC) capacity are studied to demonstrate the potential of such skyrmion based PRC in addition to showing it can classify sine and square waves with 100% accuracy. Furthermore, our study demonstrates that nonlinear magnetization dynamics and interaction through spin waves and dipole coupling have a strong influence on STM and PC capacity, thus explaining the role of physical interactions in a dynamical system on its ability to perform Reservoir Computing (RC).

We demonstrate using micromagnetic simulations that a nanomagnet array excited by Surface Acoustic Waves (SAWs) can work as a reservoir that can classify sine and square waves with high accuracy. To evaluate memory effect and computing capability, we study the Short-Term Memory (STM) and Parity Check (PC) capacities respectively. The simulated nanomagnet array has an input nanomagnet that is excited with focused SAW and coupled to several nanomagnets, seven of which serve as output nanomagnets. The SAW has a carrier frequency of 4 GHz whose amplitude is modulated to provide different inputs of sine and square waves of 100 MHz frequency. The responses of the selected output nanomagnets are processed by reading the envelope of their magnetization state, which is used to train the output weights using regression method (e.g. Moore-Penrose pseudoinverse operation). For classification, a random sequence of 100 square and sine wave samples are used, of which 80 % are used for training, and the rest of the samples used for testing. We achieve 100 % training accuracy and 100 % testing accuracy for different combination of nanomagnets as outputs. Further, the STM and PC is calculated to be ~ 5.5 bits and ~ 5.3 bits respectively, which is indicative of the proposed acoustically driven nanomagnet oscillator array being well suited for physical reservoir computing applications. Finally, the ability to use high frequency (4GHz, wavelength ~1 micron) SAW makes the device scalable to small dimensions, while the ability to modulate the envelope at lower frequency (100 MHz) adds flexibility to encode different signals beyond the sine and square waves demonstrated here.

We demonstrate that extremely low resolution quantized (nominally 5-state) synapses with large stochastic variations in Domain Wall (DW) position can be both energy efficient and achieve reasonably high testing accuracies compared to Deep Neural Networks (DNNs) of similar sizes using floating precision synaptic weights. Specifically, voltage controlled DW devices demonstrate stochastic behavior as modeled rigorously with micromagnetic simulations and can only encode limited states; however, they can be extremely energy efficient during both training and inference. We show that by implementing suitable modifications to the learning algorithms, we can address the stochastic behavior as well as mitigate the effect of their low-resolution to achieve high testing accuracies. In this study, we propose both in-situ and ex-situ training algorithms, based on modification of the algorithm proposed by Hubara et al. [1] which works well with quantization of synaptic weights. We train several 5-layer DNNs on MNIST dataset using 2-, 3- and 5-state DW device as synapse. For in-situ training, a separate high precision memory unit is adopted to preserve and accumulate the weight gradients, which are then quantized to program the low precision DW devices. Moreover, a sizeable noise tolerance margin is used during the training to address the intrinsic programming noise. For ex-situ training, a precursor DNN is first trained based on the characterized DW device model and a noise tolerance margin, which is similar to the in-situ training. Remarkably, for in-situ inference the energy dissipation to program the devices is only 13 pJ per inference given that the training is performed over the entire MNIST dataset for 10 epochs.

The desire to perform information processing, computation, communication, signal generation and related tasks, while dissipating as little energy as possible, has inspired many ideas and paradigms. One of the most powerful among them is the notion of using magnetostrictive nanomagnets as the primitive units of the hardware platforms and manipulating their magnetizations with electrically generated static or time varying mechanical strain to elicit myriad functionalities. This approach has two advantages. First, information can be retained in the devices after powering off since the nanomagnets are non-volatile unlike charge-based devices such as transistors. Second, the energy expended to perform a given task is exceptionally low since it takes very little energy to alter magnetization states with strain. This field is now known as "straintronics", in analogy with electronics, spintronics, valleytronics, etc. We review the recent advances and trends in straintronics, including digital information processing (logic), information storage (memory), domain wall devices operated with strain, control of skyrmions with strain, non-Boolean computing and machine learning with straintronics, signal generation (microwave sources) and communication (ultra-miniaturized acoustic and electromagnetic antennas) implemented with strained nanomagnets, hybrid straintronics-magnonics, and interaction between phonons and magnons in straintronic systems. We identify key challenges and opportunities, and lay out pathways to advance this field to the point where it might become a mainstream technology for energy-efficient systems.

We show that a surface acoustic wave (SAW) applied across the terminals of a magnetic tunnel junction (MTJ) decreases both the (time-averaged) parallel and antiparallel resistances of the MTJ, with the latter decreasing much more than the former. This results in a decrease of the tunneling magnetoresistance (TMR) ratio. The coercivities of the free and fixed layer of the MTJ, however, are not affected significantly, suggesting that the SAW does not cause large-angle magnetization rotation in the magnetic layers through the inverse magnetostriction (Villari) effect at the power levels used. This study sheds light on the dynamical behavior of an MTJ under periodic compressive and tensile strain.

Implementation of skyrmion based energy efficient and high-density data storage devices requires aggressive scaling of skyrmion size. Ferrimagnetic materials are considered to be a suitable platform for this purpose due to their low saturation magnetization (i.e. smaller stray field). However, we show by performing rigorous micromagnetic simulation that such scaling of skyrmion size by lowering saturation magnetization while applicable in infinite films or where the skyrmion size is very small compared to the film's lateral dimension, does not hold in confined geometries. We also found in confined geometries, where skyrmion occupies the whole volume of a nanodot, high saturation magnetization helps form stable skyrmions. Specifically, such skyrmions can be formed in 20 nm lateral dimension nanodots with high saturation magnetization (1.6-1.71 MA/m) and moderate DMI (3 mJ/m2). This result could stimulate experiments on implementation of highly dense skyrmion devices. In particular, we show that Voltage Controlled Magnetic Anisotropy (VCMA) based switching mediated by an intermediate skyrmion state can be achieved in the soft layer of a ferromagnetic p-MTJ of lateral dimensions 20 nm with sub 1fJ/bit energy in the presence of room temperature thermal noise with reasonable DMI ~3 mJ/m2.

We present a systematic numerical modeling investigation of magnetization dynamics and thermal magnetic moment fluctuations of single magnetic domain nanoparticles in a configuration applicable to enhancing inductive magnetic resonance detection signal to noise ratio (SNR). Previous proposals for oriented anisotropic single magnetic domain nanoparticle amplification of magnetic flux in MRI coil focused only on the coil pick-up voltage signal enhancement. Here we extend the analysis to the numerical evaluation of the SNR by modeling the inherent thermal magnetic noise introduced into the detection coil by the insertion of such anisotropic nanoparticle-filled coil core. We utilize the Landau-Lifshitz-Gilbert equation under the Stoner-Wohlfarth single magnetic domain (macrospin) assumption to simulate the magnetization dynamics in such nanoparticles due to AC drive field as well as thermal noise. These simulations are used to evaluate the nanoparticle configurations and shape effects on enhancing SNR. Finally, we explore the effect of narrow band filtering of the broadband magnetic moment thermal fluctuation noise on the SNR. Our results provide the impetus for relatively simple modifications to existing MRI systems for achieving enhanced detection SNR in scanners with modest polarizing magnetic fields.

We propose energy efficient strain control of domain wall (DW) in a perpendicularly magnetized nanoscale racetrack on a piezoelectric substrate that can implement multi state synapse to be utilized in neuromorphic computing platforms. In conjunction with SOT from to a current flowing in the heavy metal layer, strain is generated by applying a voltage across the piezoelectric. Such a strain is mechanically transferred to the racetrack and modulates the Perpendicular Magnetic Anisotropy (PMA). When different voltages are applied (i.e. different strains are generated), it can translate the DW to different distances for the same current which implements different synaptic weights. We have shown using micromagnetic simulations that 5-state and 3-state synapse can be implemented in a racetrack that is modeled with natural edge roughness and room temperature thermal noise. Such strain-controlled synapse has an energy consumption of few fJs and could thus be very attractive to implement energy-efficient quantized neural networks, which has been shown recently to achieve near equivalent classification accuracy to the full-precision neural networks.

We have investigated Surface Acoustic Wave (SAW) induced ferromagnetic resonance (FMR) assisted Spin Transfer Torque (STT) switching of perpendicular MTJ (p-MTJ) with inhomogeneities using micromagnetic simulations that include the effect of thermal noise. With suitable frequency excitation, the SAW can induce ferromagnetic resonance in magnetostrictive materials, and the magnetization can precesses in a cone with high deflection from the perpendicular direction. With incorporation of inhomogeneity via lateral anisotropy variation as well as room temperature thermal noise, the magnetization precession in different gains can be significantly incoherent. Interestingly, the precession in different grains are found to be in phase, even though the precession amplitude (angle of deflection from the perpendicular direction) vary across grains of different anisotropy. Nevertheless, the high mean deflection angle can complement the STT switching by reducing the STT current significantly; even though the applied stress induced change in anisotropy is much lower than the total anisotropy barrier. This work indicates that SAW assisted switching can improve energy efficiency while being scalable to very small dimensions, which is technologically important for STT-RAM and elucidates the physical mechanism for the potential robustness of this paradigm in realistic scenarios with thermal noise and material inhomogeneity

One method of creating and annihilating skyrmions in confined geometries is to use Voltage-Controlled Magnetic Anisotropy (VCMA) [1, 2, 3]. Previous study shows that robust voltage controlled ferromagnetic reversal from up to down state in the soft layer of a perpendicular Magnetic Tunnel Junction (p-MTJ) can be achieved by creating and subsequently annihilating an intermediate skyrmion state [4] in the presence of room temperature thermal noise and anisotropy variation across grains [4]. However, when scaling to 20 nm, thermal noise can annihilate the skyrmions, for example, by randomly moving the core towards the boundary of the nanostructure. In this work, we study three p-MTJs of different dimensions, particularly lateral dimensions of 100nm, 50nm and 20 nm and investigate the change in switching behavior as the dimension is decreased. Particularly, our focus is to investigate to what extent skyrmion mediated switching scheme can be employed below ~50nm lateral dimensions in the presence of thermal perturbation.

An energy-efficient voltage controlled domain wall device for implementing an artificial neuron and synapse is analyzed using micromagnetic modeling in the presence of room temperature thermal noise. By controlling the domain wall motion utilizing spin transfer or spin orbit torques in association with voltage generated strain control of perpendicular magnetic anisotropy in the presence of Dzyaloshinskii-Moriya interaction, different positions of the domain wall are realized in the free layer of a magnetic tunnel junction to program different synaptic weights. The feasibility of scaling of such devices is assessed in the presence of thermal perturbations that compromise controllability. Additionally, an artificial neuron can be realized by combining this DW device with a CMOS buffer. This provides a possible pathway to realize energy efficient voltage controlled nanomagnetic deep neural networks that can learn in real time.

We report the possibility of achieving an order of magnitude reduction in the energy dissipation needed to write bits in perpendicular magnetic tunnel junctions (p-MTJs) by simulating the magnetization dynamics under a combination of resonant surface acoustic waves (r-SAW) and spin-transfer-torque (STT). The magnetization dynamics were simulated using the Landau-Lifshitz-Gilbert equation under macrospin assumption with the inclusion of thermal noise. The resonant magnetization dynamics in the magnetostrictive nanomagnet build over few 10s of cycles of SAW application that drives the magnetization to precess in a cone with a deflection of ~45 degrees from the perpendicular direction. This reduces the STT current density required to switch the magnetization direction without increasing the STT application time or degrading the switching probability in the presence of room temperature thermal noise. This could lead to a pathway to achieve energy efficient switching of spin transfer torque random access memory (STTRAM) whose lateral dimensions can be scaled aggressively despite using materials with low magnetostriction by employing resonant excitation.

In this work, we utilize voltage controlled magnetic anisotropy (VCMA) to manipulate magnetic skyrmions that are fixed in space. Memory devices based on this strategy can potentially be of smaller footprint and better energy efficiency than current-controlled motion-based skyrmionic devices. To demonstrate VCMA induced manipulation of skyrmions, we fabricate antiferromagnet/ferromagnet/oxide heterostructure films where skyrmions can be stabilized without any external magnetic field due to the presence of exchange bias. These isolated skyrmions were annihilated by applying a voltage pulse that increased PMA. On the other hand, decreasing PMA promoted formation of more skyrmions. Furthermore, skyrmions can be created from chiral domains by increasing PMA of the system. To corroborate our experimental observations, we performed micromagnetic simulation. The proposed method could potentially lead to novel skyrmion-based memory devices.

In the brain, the membrane potential of many neurons oscillates in a subthreshold damped fashion and fire when excited by an input frequency that nearly equals their eigen frequency. In this work, we investigate theoretically the artificial implementation of such "resonate-and-fire" neurons by utilizing the magnetization dynamics of a fixed magnetic skyrmion in the free layer of a magnetic tunnel junction (MTJ). To realize firing of this nanomagnetic implementation of an artificial neuron, we propose to employ voltage control of magnetic anisotropy or voltage generated strain as an input (spike or sinusoidal) signal, which modulates the perpendicular magnetic anisotropy (PMA). This results in continual expansion and shrinking (i.e. breathing) of a skyrmion core that mimics the subthreshold oscillation. Any subsequent input pulse having an interval close to the breathing period or a sinusoidal input close to the eigen frequency drives the magnetization dynamics of the fixed skyrmion in a resonant manner. The time varying electrical resistance of the MTJ layer due to this resonant oscillation of the skyrmion core is used to drive a Complementary Metal Oxide Semiconductor (CMOS) buffer circuit, which produces spike outputs. By rigorous micromagnetic simulation, we investigate the interspike timing dependence and response to different excitatory and inhibitory incoming input pulses. Finally, we show that such resonate and fire neurons have potential application in coupled nanomagnetic oscillator based associative memory arrays.

We theoretically study the effect of a material defect (material void) on switching errors associated with magneto-elastic switching of magnetization in elliptical magnetostrictive nanomagnets having in-plane magnetic anisotropy. We find that the error probability increases significantly in the presence of the defect, indicating that magneto-elastic switching is particularly vulnerable to material imperfections. Curiously, there is a critical stress value that gives the lowest error probability in both defect-free and defective nanomagnets. The critical stress is much higher in defective nanomagnets than in defect-free ones. Since it is more difficult to generate the critical stress in small nanomagnets than in large nanomagnets (having the same energy barrier for thermal stability), it would be a challenge to downscale magneto-elastically switched nanomagnets in memory and other applications where reliable switching is required. This is likely to be further exacerbated by the presence of defects.

We studied the depth dependent magnetization profile of the magnetostrictive Co thin film layer in a PMN-PT (011)/Ta/Co/Ta structure under both zero and nonzero applied electric field using polarized neutron reflectometry. Application of electric field across the PMN-PT substrate generates a strain, which rotates the magnetization of the Co layer consistent with the Villari effect. At low magnetic fields (near remanence and coercive field conditions), we find that the depth dependent magnetization profile is non-uniform, under both zero and nonzero applied electric fields. These variations are attributable to the depth dependent strain profile in the Co film, as determined by finite element analysis simulations.

Recent work [1] demonstrated high coercivity and magnetic moment in cobalt carbide nanoparticle assemblies and explained the high coercivity from first principles in terms of the high magnetocrystalline anisotropy of the cobalt carbide nanoparticles. In this work, we comprehensively model the interaction between the nanoparticles comprising the assembly and systematically understand the effect of particle size, distribution of the orientations of the nanoparticles' magnetocrystalline anisotropy axis with respect to the applied magnetic field, and dipole coupling between nanoparticles on the temperature dependent magnetic behavior of the nanoparticle assembly. We show that magnetocrystalline anisotropy alone is not enough to explain the large hysteresis over the 50K-400K temperature range and suggest that defects and inhomogeneties that pin the magnetization could also play a significant role on this temperature dependent magnetic behavior.

We propose a two terminal nanomagnetic memory element based on magnetization reversal of a perpendicularly magnetized nanomagnet employing a unipolar voltage pulse that modifies the perpendicular anisotropy of the system. Our work demonstrates that the presence of Dzyaloshinskii-Moriya Interaction (DMI) can create alternative route for magnetization reversal that obviates the need for utilizing precessional magnetization dynamics as well as a bias magnetic field that are employed in traditional voltage control of magnetic anisotropy (VCMA) based switching of perpendicular magnetization. We show with extensive micromagnetic simulation, in the presence of thermal noise, that the proposed skyrmion mediated VCMA switching mechanism is robust at room temperature leading to extremely low error switching while also being potentially 1-2 orders of magnitude more energy efficient than state of the art spin transfer torque (STT) based switching.

Rotating the magnetization of a shape anisotropic magnetostrictive nanomagnet with voltage-generated stress/strain dissipates much less energy than most other magnetization rotation schemes, but its application to writing bits in non-volatile magnetic memory has been hindered by the fundamental inability of stress/strain to rotate magnetization by full 180 degrees. Normally, stress/strain can rotate the magnetization of a shape anisotropic elliptical nanomagnet by only up to 90 degrees, resulting in incomplete magnetization reversal. Recently, we predicted that applying uniaxial stress sequentially along two different axes that are not collinear with the major or minor axis of the elliptical nanomagnet will rotate the magnetization by full 180 degrees. Here, we demonstrate this complete 180 degree rotation in elliptical Co-nanomagnets (fabricated on a piezoelectric substrate) at room temperature. The two stresses are generated by sequentially applying voltages to two pairs of shorted electrodes placed on the substrate such that the line joining the centers of the electrodes in one pair intersects the major axis of a nanomagnet at ~+30 degrees and the line joining the centers of the electrodes in the other pair intersects at ~ -30 degrees. A finite element analysis has been performed to determine the stress distribution underneath the nanomagnets when one or both pairs of electrodes are activated, and this has been approximately incorporated into a micromagnetic simulation of magnetization dynamics to confirm that the generated stress can produce the observed magnetization rotations. This result portends an extremely energy-efficient non-volatile "straintronic" memory technology predicated on writing bits in nanomagnets with electrically generated stress.

Hardware based image processing offers speed and convenience not found in software-centric approaches. Here, we show theoretically that a two-dimensional periodic array of dipole-coupled elliptical nanomagnets, delineated on a piezoelectric substrate, can act as a dynamical system for specific image processing functions. Each nanomagnet has two stable magnetization states that encode pixel color (black or white). An image containing black and white pixels is first converted to voltage states and then mapped into the magnetization states of a nanomagnet array with magneto-tunneling junctions (MTJs). The same MTJs are employed to read out the processed pixel colors later. Dipole interaction between the nanomagnets implements specific image processing tasks such as noise reduction and edge enhancement detection. These functions are triggered by applying a global strain to the nanomagnets with a voltage dropped across the piezoelectric substrate. An image containing an arbitrary number of black and white pixels can be processed in few nanoseconds with very low energy cost.

Recent work [1,2] suggests that ferromagnetic reversal with spin transfer torque (STT) requires more current in a system in the presence of DMI than switching a typical ferromagnet of the same dimensions and perpendicular magnetic anisotropy (PMA). However, DMI promotes stabilization of skyrmions and we report that when the perpendicular anisotropy is modulated (reduced) for both the skyrmion and ferromagnet, it takes much smaller current to reverse the fixed skyrmion than to reverse the ferromagnet in the same time, or the skyrmion reverses much faster than the ferromagnet at similar levels of current. We show with rigorous micromagnetic simulations and energy portraits that the skyrmion switching proceeds along a different path at very low PMA which results in a significant reduction in the spin current required or time required for reversal. This can have potential for memory application where a relatively simple modification of the standard STT-RAM to include a heavy metal adjacent to the soft magnetic layer and with appropriate design of the tunnel barrier can lead to energy efficient and fast magnetic memory device based on the reversal of fixed skyrmions.

We have observed a super-giant (~10,000,000%) negative magnetoresistance at 39 mT field in Cu nanowires contacted with Au contact pads. In these nanowires, potential barriers form at the two Cu/Au interfaces because of Cu oxidation that results in an ultrathin copper oxide layer forming between Cu and Au. Current flows when electrons tunnel through, and/or thermionically emit over, these barriers. A magnetic field applied transverse to the direction of current flow along the wire deflects electrons toward one edge of the wire because of the Lorentz force, causing electron accumulation at that edge and depletion at the other. This lowers the potential barrier at the accumulated edge and raises it at the depleted edge, causing a super-giant magnetoresistance at room temperature.

It has recently been possible to synthesize ordered assemblies composed of magnetic superatomic clusters Ni9Te6(PEt3)8 separated by C60 and study their magnetic behavior. We have carried out theoretical studies on model systems consisting of magnetic superatoms separated by non-magnetic species to examine the evolution in magnetic response as the nature of the magnetic superatom (directions of spin quantization), the strength of isotropic and anisotropic interactions, the magnetic anisotropy energy, and the size of the assembly are varied. We have examined square planar configurations consisting 16, 24 and 48 sites with 8, 12 and 24 magnetic superatoms respectively. The magnetic atoms are allowed 2 or 5 orientations. The model Hamiltonian includes isotropic exchange interactions with second nearest neighbor ferromagnetic and nearest neighbor antiferromagnetic couplings and anisotropic Dzyaloshinskii-Moriya interactions. It is shown that the inclusion of Dzyaloshinskii-Moriya interaction that cause spin canting is necessary to get qualitative response as observed in experiments.

We report nanomagnetic switching with Acoustic Waves (AW) launched from interdigitated electrodes that modulate the stress anisotropy of elliptical cobalt nanoscale magnetostrictive magnets (340 nm x 270 nm x 12 nm) delineated on 128 degree Y-cut lithium niobate. The dipole-coupled nanomagnet pairs are in a single-domain state and are initially magnetized along the major axis of the ellipse, with their magnetizations parallel to each other. The magnetizations of nanomagnets having lower shape anisotropy are reversed upon acoustic wave propagation. Thereafter, the magnetization of these nanomagnets remains in the reversed state and demonstrate non-volatility. This executes a 'NOT' operation. This proof of acoustic wave induced magnetic state reversal in dipole-coupled nanomagnets implementing a 'NOT' gate operation could potentially lead to the development of extremely energy-efficient nanomagnetic logic. Furthermore, fabrication complexity is reduced immensely due to the absence of individual contacts to the nanomagnets, leading to lower energy dissipation

Using micromagnetic simulations we demonstrate core reversal of a fixed magnetic skyrmion by modulating the perpendicular magnetic anisotropy of a nanomagnet with an electric field. We can switch reversibly between two skyrmion states and two ferromagnetic states, i.e. skyrmion states with the magnetization of the core pointing down/up and periphery pointing up/down, and ferromagnetic states with magnetization pointing up/down, by sequential increase and decrease of the perpendicular magnetic anisotropy. The switching between these states is explained by the fact that the spin texture corresponding to each of these stable states minimizes the sum of the magnetic anisotropy, demagnetization, Dzyaloshinskii-Moriya interaction (DMI) and exchange energies. This could lead to the possibility of energy efficient nanomagnetic memory and logic devices implemented with fixed skyrmions without using a magnetic field and without moving skyrmions with a current.

Nanomagnetic logic has emerged as a potential replacement for traditional CMOS-based logic because of superior energy-efficiency. One implementation of nanomagnetic logic employs shape-anisotropic (e.g. elliptical) ferromagnets (with two stable magnetization orientations) as binary switches that rely on dipole-dipole interaction to communicate binary information. Normally, circular nanomagnets are incompatible with this approach since they lack distinct stable in-plane magnetization orientations to encode bits. However, circular magnetoelastic nanomagnets can be made bi-stable with a voltage induced anisotropic strain, which provides two significant advantages for nanomagnetic logic applications. First, the shape anisotropy energy barrier is eliminated which reduces the amount of energy to reorient the dipole. Second, the in-plane size can be reduced (~20nm) which was previously impossible due to thermal stability issues. In circular magnetoelastic nanomagnets, a voltage induced strain stabilizes the magnetization even at this size overcoming the thermal stability issue. In this paper, we analytically demonstrate a binary logic wire implemented with an array of circular nanomagnets that are clocked with voltage-induced strain applied by an underlying piezoelectric substrate. This leads to an energy-efficient logic paradigm orders of magnitude superior to existing CMOS-based logic that is scalable to dimensions substantially smaller than those for existing nanomagnetic logic approaches. The analytical approach is validated with experimental measurements conducted on dipole coupled Ni nanodots fabricated on a PMN-PT sample.

Strain-mediated voltage control of magnetization in piezoelectric/ferromagnetic systems is a promising mechanism to implement energy-efficient spintronic memory devices. Here, we demonstrate giant voltage manipulation of MgO magnetic tunnel junctions (MTJ) on a Pb(Mg1/3Nb2/3)0.7Ti0.3O3 (PMN-PT) piezoelectric substrate with (001) orientation. It is found that the magnetic easy axis, switching field, and the tunnel magnetoresistance (TMR) of the MTJ can be efficiently controlled by strain from the underlying piezoelectric layer upon the application of a gate voltage. Repeatable voltage controlled MTJ toggling between high/low-resistance states is demonstrated. More importantly, instead of relying on the intrinsic anisotropy of the piezoelectric substrate to generate the required strain, we utilize anisotropic strain produced using local gating scheme, which is scalable and amenable to practical memory applications. Additionally, the adoption of crystalline MgO-based MTJ on piezoelectric layer lends itself to high TMR in the strain-mediated MRAM devices.

We report manipulation of the magnetic states of elliptical cobalt magnetostrictive nanomagnets (of nominal dimensions ~ 340 nm x 270 nm x 12 nm) delineated on bulk 128\deg Y-cut lithium niobate with Surface Acoustic Waves (SAWs) launched from interdigitated electrodes. Isolated nanomagnets that are initially magnetized to a single domain state with magnetization pointing along the major axis of the ellipse are driven into a vortex state by surface acoustic waves that modulate the stress anisotropy of these nanomagnets. The nanomagnets remain in the vortex state until they are reset by a strong magnetic field to the initial single domain state, making the vortex state non-volatile. This phenomenon is modeled and explained using a micromagnetic framework and could lead to the development of extremely energy efficient magnetization switching methodologies.

Micromagnetic studies of the magnetization change in magnetostrictive nanomagnets subjected to stress are performed for nanomagnets of different sizes. The interplay between demagnetization, exchange and stress anisotropy energies is used to explain the rich physics of size-dependent magnetization dynamics induced by modulating stress anisotropy in planar nanomagnets. These studies have important implications for strain mediated ultralow energy magnetization change in nanomagnets and its application in energy-efficient nanomagnetic computing systems.

An energy based stochastic model for temperature dependent anhysteretic magnetization curves of ferromagnetic materials is proposed and bench marked against experimental data. This is based on the calculation of macroscopic magnetic properties by performing an energy weighted average over all possible orientations of the magnetization vector. Most prior approaches that employ this method are unable to independently account for the effect of both inhomogeneity and temperature in performing the averaging necessary to model experimental data. Here we propose a way to account for both effects simultaneously and benchmark the model against experimental data from ~5K to ~300K for two different materials in both annealed (fewer inhomogeneities) and deformed (more inhomogeneities) samples. This demonstrates that the independent accounting for the effect of both inhomogeneity and temperature is necessary to correctly model temperature dependent magnetization behavior.

Spintronic devices usually rely on long spin relaxation times and/or lengths for optimum performance. Therefore, the ability to modulate these quantities with an external agent offers unique possibilities. The dominant spin relaxation mechanism in most technologically important semiconductors is the D'yakonov-Perel' (DP) mechanism which vanishes if the spin carriers (electrons) are confined to a single conduction subband in a quantum wire grown in certain crystallographic directions, or polycrystalline quantum wires. Here, we report modulating the DP spin relaxation rate (and hence the spin relaxation length) in self assembled 50-nm diameter InSb nanowires with infrared light at room temperature. In the dark, almost all the electrons in the nanowires are in the lowest conduction subband at room temperature, resulting in near-complete absence of DP relaxation. This allows observation of spin-sensitive effects in the magnetoresistance. Under infrared illumination, electrons are photoexcited to higher subbands and the DP spin relaxation mechanism is revived, leading to a three-fold decrease in the spin relaxation length. Consequently, the spin sensitive effects are no longer observable under illumination. This phenomenon may have applications in spintronic room-temperature infrared photodetection.

Reversible straintronic switching of a nanomagnet's magnetization between two stable or metastable states promises ultra-energy-efficient non-volatile memory. Here, we report strain-induced magnetization switching in ~300 nm sized FeGa nanomagnets delineated on a piezoelectric PMN-PT substrate. Voltage of one polarity applied across the substrate generates compressive strain in a nanomagnet and switches its magnetization to one state, while voltage of the opposite polarity generates tensile strain and switches the magnetization back to the original state, resulting in "non-toggle" switching. The two states can encode the two binary bits, and, using the right voltage polarity, one can write either bit deterministically.

We report observation of a "non-volatile" converse magneto-electric effect in elliptical FeGa nanomagnets delineated on a piezoelectric PMN-PT substrate. The nanomagnets are initially magnetized with a magnetic field directed along their nominal major axis and subsequent application of an electric field across the substrate generates strain in the substrate, which is partially transferred to the nanomagnets and rotates the magnetizations of some of them to metastable orientations. There they remain after the field is removed, resulting in "non-volatility". In isolated nanomagnets, the angular separation between the initial and final magnetization directions is < 90 deg, but in a dipole-coupled pair consisting of one "hard" and one "soft" nanomagnet, the soft nanomagnet's magnetization is rotated by > 90 deg from the initial orientation because of the dipole influence of the hard nanomagnet. These effects can be utilized for ultra-energy-efficient logic and memory devices.

In artificial neural networks, neurons are usually implemented with highly dissipative CMOS-based operational amplifiers. A more energy-efficient implementation is a 'spin-neuron' realized with a magneto-tunneling junction (MTJ) that is switched with a spin-polarized current (representing weighted sum of input currents) that either delivers a spin transfer torque or induces domain wall motion in the soft layer of the MTJ. Here, we propose and analyze a different type of spin-neuron in which the soft layer of the MTJ is switched with mechanical strain generated by a voltage (representing weighted sum of input voltages) and term it straintronic spin-neuron. It dissipates orders of magnitude less energy in threshold operations than the traditional current-driven spin neuron at 0 K temperature and may even be faster. We have also studied the room-temperature firing behaviors of both types of spin neurons and find that thermal noise degrades the performance of both types, but the current-driven type is degraded much more than the straintronic type if both are optimized for maximum energy-efficiency. On the other hand, if both are designed to have the same level of thermal degradation, then the current-driven version will dissipate orders of magnitude more energy than the straintronic version. Thus, the straintronic spin neuron is superior to current-driven spin neurons.

We propose a reconfigurable bit comparator implemented with a nanowire spin valve whose two contacts are magnetostrictive with bistable magnetization. Reference and input bits are "written" into the magnetization states of the two contacts with electrically generated strain and the spin-valve's resistance is lowered if they match. Multiple comparators can be interfaced in parallel with a magneto-tunneling junction to determine if an N-bit input stream matches an N-bit reference stream bit by bit. The system is robust against thermal noise at room temperature and a 16-bit comparator can operate at roughly 416 MHz while dissipating at most 420 aJ per cycle.

This is a reply to a comment on our work recently posted in arXiv. To our knowledge, this comment has not been published anywhere else. We show that the points raised in the comment are invalid

The magnetic properties of two important passive magnetic shielding materials (A4K and Amumetal) for accelerator applications, subjected to various processing and heat treatment conditions are studied comprehensively over a wide range of temperatures: from Cryogenic to room temperature. We analyze the effect of processing on the extent of degradation of the magnetic properties of both materials and investigate the possibility of restoring these properties by re-annealing.

This is a comment on two recent arXiv postings.

A long-standing goal of computer technology is to process and store digital information with the same device in order to implement new architectures. One way to accomplish this is to use nanomagnetic `non-volatile' logic gates that can perform Boolean operations and then store the output data in the magnetization states of nanomagnets, thereby doubling as both logic and memory. Unfortunately, many proposed nanomagnetic gates do not possess the seven essential characteristics of a Boolean logic gate: concatenability, non-linearity, isolation between input and output, gain, universal logic implementation, scalability and error resilience. More importantly, their energy-delay products and error-rates vastly exceed that of conventional transistor-based logic gates, which is a drawback. Here, we propose a non-volatile voltage-controlled nanomagnetic logic gate that possesses all the necessary characteristics of a logic gate and whose energy-delay product is ~2 orders of magnitude less than that of other nano-magnetic (non-volatile) logic gates and ~1 order of magnitude less than that of (volatile) CMOS-based logic gates. The error-resilience is also superior to that of other known nanomagnetic gates.

Strain-clocked dipole-coupled nanomagnetic logic is an energy-efficient Boolean logic paradigm whose progress has been stymied by its propensity for high error rates. In an effort to mitigate this problem, we have studied the effect of nanomagnet geometry on error rates, focusing on elliptical and cylindrical geometries. We had previously reported that the out-of-plane excursion of the magnetization vector during switching creates a precessional torque that is responsible for high switching error probability in elliptical nanomagnet geometries. The absence of this torque in cylindrical magnets portends lower error rates, but our simulations show that the error rate actually does not improve significantly compared to elliptical magnets while the switching becomes unacceptably slow. Here, we show that dipole coupled nanomagnetic logic can offer relatively high reliability (switching error probability < 10^-8), moderate clock speed (~ 100 MHz) and 2-3 orders of magnitude energy saving compared to CMOS devices, provided the shape anisotropy energy barrier of the magnet is increased to at least ~5.5 eV to allow engineering a stronger dipole coupling between neighboring nanomagnets.

Rotating the magnetization of a magnetostrictive nanomagnet with electrically generated mechanical strain dissipates miniscule amount of energy compared to any other rotation method and would have been the ideal method to write bits in nonvolatile magnetic memory, except strain cannot ordinarily rotate the magnetization of magnet by more than 90 degrees and "flip" it. Here, we describe a scheme to achieve complete 180 degree rotation of the magnetization of a nanomagnet with strain that will enable writing of binary bits in non-volatile magnetic memory implemented with magneto-tunneling junctions whose soft layers are two-phase magnetostrictive/piezoelectric multiferroics. At room temperature, this writing method results in: (1) energy dissipation < 6200 kT per bit, (2) write error probability < 10^-6, (3) write time of ~ 1 ns, and (4) low read error. This could potentially lead to a new genre of non-volatile memory that is extremely reliable, fast and, at the same time, ultra-energy-efficient.

Dipole-coupled nanomagnetic logic (NML), where nanomagnets with bistable magnetization states act as binary switches and information is transferred between them via dipole coupling and Bennett clocking, is a potential replacement for conventional transistor logic since magnets dissipate less energy than transistors when they switch in response to the clock. However, dipole-coupled NML is much more error-prone than transistor logic because thermal noise can easily disrupt magnetization dynamics. Here, we study a particularly energy-efficient version of dipole-coupled NML known as straintronic multiferroic logic (SML) where magnets are clocked/switched with electrically generated mechanical strain. By appropriately shaping the voltage pulse that generates strain, the error rate in SML can be reduced to tolerable limits. In this paper, we describe the error probabilities associated with various stress pulse shapes and discuss the trade-off between error rate and switching speed in SML.

Nanomagnetic implementations of Boolean logic [1,2] have garnered attention because of their non-volatility and the potential for unprecedented energy-efficiency. Unfortunately, the large dissipative losses that take place when nanomagnets are switched with a magnetic field [3], or spin-transfer-torque [4] inhibit the promised energy-efficiency. Recently, there have been experimental reports of utilizing the Spin Hall effect for switching magnets [5-7], and theoretical proposals for strain induced switching of single-domain magnetostrictive nanomagnets [8-12], that might reduce the dissipative losses significantly. Here, we 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 unidirectional bit information propagation in dipole-coupled nanomagnets chains. This portends ultra-low-energy logic processors and mobile electronics that may be able to operate solely by harvesting energy from the environment without ever requiring a battery.

We propose an improved scheme for low-power writing of binary bits in non-volatile (multiferroic) magnetic memory with electrically generated mechanical stress. Compared to an earlier idea [Tiercelin, et al., J. Appl. Phys., 109, 07D726 (2011)], our scheme improves distinguishability between the stored bits when the latter are read with magneto-tunneling junctions. More importantly, the write energy dissipation and write error rate are reduced significantly if the writing speed is kept the same. Such a scheme could be one of the most energy-efficient approaches to writing bits in magnetic non-volatile memory.

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 (LLG) equation for a single domain macro-spin state. Different factors were found to affect the switching probability in the presence of thermal noise at room temperature: (i) dipole coupling strength, (ii) stress levels, and (iii) 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.

The potential energy profile of a binary switch is a symmetric double well. Switching between the wells without energy dissipation requires time-modulating the potential barrier separating them and tilting the profile towards the desired well at the precise juncture when the barrier disappears. This 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. Here, we present a third strategy that requires a time modulated barrier but no timing synchronization. It is therefore fault-tolerant in the absence of thermal noise and yet it dissipates arbitrarily small energy since an arbitrarily small tilt is required for slow and adiabatic switching. This case is exemplified with stress induced switching of a shape-anisotropic single-domain nanomagnet dipole coupled to a neighbor. We also show by examining various energy profiles and the corresponding probability distributions that when thermal noise is present, the minimum energy dissipated to switch in this scheme is 2kTln(1/p) [p = switching error probability].

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 $\sim$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 $\sim$100% probability in less than 1 ns while dissipating less than 2 aJ at \it 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 is ramped up or down.

The general methodology of binary switching requires tilting of potential landscape along the desired direction of switching. The tilt generates a torque along the direction of switching and the degree of tilt should be sufficient enough to beat thermal agitations with a tolerable error probability. However, we show here that such tilt is not necessary. Considering the full three-dimensional motion, we point out that the built-in dynamics can facilitate switching without requiring any asymmetry in potential landscape even in the presence of thermal noise. With experimentally feasible parameters, we theoretically demonstrate such intriguing possibility in electric field-induced magnetization switching of a magnetostrictive nanomagnet.

A circular magnetic disk with biaxial magnetocrystalline anisotropy has four stable magnetization states which can be used to encode a pixel's shade in a black/gray/white image. By solving the Landau-Lifshitz- Gilbert equation, we show that if moderate noise deflects the magnetization slightly from a stable state, it always returns to the original state, thereby automatically de-noising the corrupted image. The same system can compare a noisy input image with a stored image and make a matching decision using magneto-tunneling junctions. These tasks are executed at ultrahigh speeds (~2 ns for a 512\times512 pixel image).

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 2-phase (magnetostrictive/piezoelectric) multiferroic elements that are clocked with electrostatic potentials of ~50 mV applied to the piezoelectric layer generating 10 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 ~ 1000 kT in the gate and ~3000 kT in the 12-magnet array comprising two input and two output wires for fan-in and fan-out. This makes it respectively 3 and 5 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 3 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. Keywords: Nanomagnetic logic, multiferroics, straintronics and spintronics, Landau-Lifshitz-Gilbert equation.

A common ploy to reduce the switching current and energy dissipation in spin-transfer-torque driven magnetization switching of shape-anisotropic single-domain nanomagnets is to employ magnets with low saturation magnetization $M_s$ and high shape-anisotropy. The high shape-anisotropy compensates for low $M_s$ to keep the static switching error rate constant. However, this ploy increases the switching delay, its variance in the presence of thermal noise, and the dynamic switching error rate. Using the stochastic Landau-Lifshitz-Gilbert equation with a random torque emulating thermal noise, we show that pumping some excess spin-polarized current into the nanomagnet during switching will keep the mean switching delay and its variance constant as we reduce $M_s$, while still reducing the energy dissipation significantly.

We predict the existence of a new metastable magnetization state in a single-domain nanomagnet with uniaxial shape anisotropy. It emerges when a spin-polarized current, delivering a spin-transfer-torque, is injected into the nanomagnet. It can trap the magnetization vector and prevent spin-transfer-torque from switching the magnetization from one stable state along the easy axis to the other. Above a certain threshold current, the metastable state no longer appears. This has important technological consequences for spin-transfer-torque based magnetic memory and logic systems.

Nanomagnets with biaxial magnetocrystalline anisotropy have four stable magnetization orientations that can encode 4-state logic bits (00), (01), (11) and (10). Recently, a 4-state NOR gate derived from three such nanomagnets, interacting via dipole interaction, was proposed. Here, we devise a Bennett clocking scheme to propagate 4-state logic bits unidirectionally between such gates. The nanomagnets are assumed to be made of 2-phase strain-coupled magnetostrictive/piezoelectric multiferroic elements, such as nickel and lead zirconate titanate (PZT). A small voltage of 200 mV applied across the piezoelectric layer can generate enough mechanical stress in the magnetostrictive layer to rotate its magnetization away from one of the four stable orientations and implement Bennett clocking. We show that a particular sequence of positive and negative voltages will propagate 4-state logic bits unidirectionally down a chain of such multiferroic nanomagnets for logic flow.

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.

The authors show that the magnetization of a 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 $\sim$45 $kT$ at room temperature for a switching delay of $\sim$100 ns and $\sim$70 $kT$ for a switching delay of $\sim$10 ns, if the energy barrier separating the two stable magnetization directions is $\sim$32 $kT$. Such devices can be powered by harvesting energy exclusively from the environment without the need for a battery.

The authors show how to implement a 4-state universal logic gate (NOR) using three strain-coupled magnetostrictive-piezoelectric multiferroic nanomagnets (e.g. Ni/PZT) with biaxial magnetocrystalline anisotropy. Two of the nanomagnets encode the 2-state input bits in their magnetization orientations and the third nanomagnet produces the output bit via dipole interaction with the input nanomagnets. A voltage pulse alternating between -0.2 V and +0.2 V is applied to the PZT layer of the third nanomagnet and generates alternating tensile and compressive stress in its Ni layer to produce the output bit, while dissipating ~ 57,000 kT (0.24 fJ) of energy per gate operation.

In magnetic memory and logic devices, a magnet's magnetization is usually flipped with a spin polarized current delivering a spin transfer torque (STT). This mode of switching consumes too much energy and considerable energy saving can accrue from using a multiferroic nanomagnet switched with a combination of STT and mechanical stress generated with a voltage (VGS). The VGS mode consumes less energy than STT, but cannot rotate magnetization by more than 90?, so that a combination of the two modes is needed for energy-efficient switching.

It has been recently shown that multiferroic logic - where logic bits are encoded in the magnetization orientation of a nanoscale magnetostrictive layer elastically coupled to a piezoelectric layer - can be Bennett clocked with small electrostatic potentials of few tens of mV applied to the piezoelectric layer. The potential generates stress in the magnetostrictive layer and rotates its magnetization by a large angle to carry out Bennett clocking. This method of clocking is far more energy-efficient than using spin transfer torque. In order to assess if such a clocking scheme can be also reasonably fast, we have studied the magnetization dynamics of a multiferroic logic array with nearest neighbor dipole coupling using the Landau-Lifshitz-Gilbert (LLG) equation. We find that switching delays of ~ 3 ns (clock rates of 0.33 GHz) can be achieved with proper design provided we clock non-adiabatically and dissipate ~48,000 kT (at room temperature) of energy per clock cycle per bit flip in the clocking circuit. This dissipation far exceeds the energy barrier separating the two logic states, which we assumed to be 32 kT to yield a bit error probability of . Had we used spin transfer torque to switch with the same ~ 3 ns delay, the energy dissipation would have been much larger (~ $6 \times 106$ kT). This shows that spin transfer torque, widely used in magnetic random access memory, is an inefficient way to switch a magnet, and multiferroic logic clocked with voltage-induced stress is a superior nanomagnetic logic scheme.

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.

We propose a spintronic strain sensor capable of sensing strain with a sensitivity of 1E-13/sqrtHz at room temperature with an active sensing area of 1 cmE2 and power dissipation of 1 watt. This device measures strain by monitoring the change in the spin-polarized current in a parallel array of free standing nanowire spin valves when the array is subjected to compressive or tensile stress along the wires' length. The change in the current is linearly proportional to the strain, which makes the sensor relatively distortion-free. Such a sensor can be fabricated using a variety of techniques involving nanolithography, self assembly and epitaxial growth.