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Computation in the past decades has been driven by deterministic computers based on classical deterministic bits. Recently, alternative computing paradigms and domain-based computing like quantum computing and probabilistic computing have gained traction. While quantum computers based on q-bits utilize quantum effects to advance computation, probabilistic computers based on probabilistic (p-)bits are naturally suited to solve problems that require large amount of random numbers utilized in Monte Carlo and Markov Chain Monte Carlo algorithms. These Monte Carlo techniques are used to solve important problems in the fields of optimization, numerical integration or sampling from probability distributions. However, to efficiently implement Monte Carlo algorithms the generation of random numbers is crucial. In this paper, we present and benchmark a probabilistic coprocessor based on p-bits that are naturally suited to solve these problems. We present multiple examples and project that a nanomagnetic implementation of our probabilistic coprocessor can outperform classical CPU and GPU implementations by multiple orders of magnitude.

This paper draws attention to a hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state. The basic constituents are stochastic nanomagnets which switch randomly between the $\pm 1$ Ising states and can be monitored continuously with standard electronics. Their mutual interactions can be short or long range, and their strengths can be reconfigured as needed to solve specific problems and to anneal the system at room temperature. The natural laws of statistical mechanics guide the network of stochastic nanomagnets at GHz speeds through the collective states with an emphasis on the low energy states that represent optimal solutions. As proof-of-concept, we present simulation results for standard NP-complete examples including a 16-city traveling salesman problem using experimentally benchmarked models for spin-transfer torque driven stochastic nanomagnets., Comment: 12 pages, 3 figures

This paper analyzes write errors in spin torque switching due to thermal fluctuations in a system with Perpendicular Magnetic Anisotropy (PMA). Prior analytical and numerical methods are summarized, a physics based Fokker-Planck Equation (FPE) chosen for its computational efficiency and broad applicability to all switching regimes. The relation between write error slope and material parameters is discussed in detail to enable better device engineering and optimization. Finally a 2D FPE tool is demonstrated that extends the applicability of FPE to write error in non PMA systems with built-in asymmetry., Comment: 7 pages, 8 figures

Belief networks represent a powerful approach to problems involving probabilistic inference, but much of the work in this area is software based utilizing standard deterministic hardware based on the transistor which provides the gain and directionality needed to interconnect billions of them into useful networks. This paper proposes a transistor like device that could provide an analogous building block for probabilistic networks. We present two proof-of-concept examples of belief networks, one reciprocal and one non-reciprocal, implemented using the proposed device which is simulated using experimentally benchmarked models., Comment: Keywords: stochastic, sigmoid, phase transition, spin glass, frustration, reduced frustration, Ising model, Bayesian network, Boltzmann machine. 23 pages, 9 figures

The possible use of spin and magnets in place of charge and capacitors to store and process information is well known. Magnetic tunnel junctions are being widely investigated and developed for magnetic random access memories. These are two terminal devices that change their resistance based on switchable magnetization of magnetic materials. They utilize the interaction between electron spin and magnets to read information from the magnets and write onto them. Such advances in memory devices could also translate into a new class of logic devices that offer the advantage of nonvolatile and reconfigurable information processing over transistors. Logic devices having a transistor-like gain and directionality could be used to build integrated circuits without the need for transistor-based amplifiers and clocks at every stage. We review device characteristics and basic logic gates that compute with spins and magnets from the mesoscopic to the atomic scale, as well as materials, integration, and fabrication challenges and methods., Comment: 18 pages, 5 figures. Key Words: Magnetic Memory, Gain, Directionality, Non-Volatile Spin Logic, Reconfigurable Logic, Logic-in-Memory, Memory-in-Logic, Spintronics, Magnetic Tunnel Junction, Spin Transfer Torque, Magneto-Resistance, Perpendicular Magnetic Anisotropy, Hysteresis, Spin-Polarized Scanning Tunneling Microscopy, RKKY Interaction, MRS Bulletin, Volume 39, August 2014

Nanomagnets driven by spin currents provide a natural implementation for a neuron and a synapse: currents allow convenient summation of multiple inputs, while the magnet provides the threshold function. The objective of this paper is to explore the possibility of a hardware neural network (HNN) implementation using a spin switch (SS) as its basic building block. SS is a recently proposed device based on established technology with a transistor-like gain and input-output isolation. This allows neural networks to be constructed with purely passive interconnections without intervening clocks or amplifiers. The weights for the neural network are conveniently adjusted through analog voltages that can be stored in a non-volatile manner in an underlying CMOS layer using a floating gate low dropout voltage regulator. The operation of a multi-layer SS neural network designed for character recognition is demonstrated using a standard simulation model based on coupled Landau-Lifshitz-Gilbert (LLG) equations, one for each magnet in the network., Comment: 7 pages, 9 figures and 1 table

Progress in the last two decades has effectively integrated spintronics and nanomagnetics into a single field, creating a new class of spin-based devices that are now being used both to Read (R) information from magnets and to Write (W) information onto magnets. Many other new phenomena are being investigated for nano-electronic memory as described in Part II of this book. It seems natural to ask whether these advances in memory devices could also translate into a new class of logic devices. What makes logic devices different from memory is the need for one device to drive another and this calls for gain, directionality and input-output isolation as exemplified by the transistor. With this in mind we will try to present our perspective on how W and R devices in general, spintronic or otherwise, could be integrated into transistor-like switches that can be interconnected to build complex circuits without external amplifiers or clocks. We will argue that the most common switch used to implement digital logic based on complementary metal oxide semiconductor (CMOS) transistors can be viewed as an integrated W-R unit having an input-output asymmetry that give it gain and directionality. Such a viewpoint is not intended to provide any insight into the operation of CMOS switches, but rather as an aid to understanding how W and R units based on spins and magnets can be combined to build transistor-like switches. Next we will discuss the standard W and R units used for magnetic memory devices and present one way to integrate them into a single unit with the input electrically isolated from the output. But we argue that this integrated W-R unit would not provide the key property of gain. We will then show that the recently discovered giant spin Hall effect could be used to construct a W-R unit with gain and suggest other possibilities for spin switches with gain., Comment: 27 pages. To appear in Emerging Nanoelectronic Devices, Editors: An Chen, James Hutchby, Victor Zhirnov and George Bourianoff, John Wiley & Sons (to be published)

Multiple beyond-CMOS information processing devices are presently under active research and require methods of benchmarking them. A new approach for calculating the performance metric, energy-delay product, of such devices is proposed. The approach involves estimating the device properties of resistance and switching charge, rather than dynamic evolution characteristics, such as switching energy and time. Application of this approach to a wide class of charge-based and non-charge based devices is discussed. The approach suggests pathways for improving the performance of beyond-CMOS devices and a new realistic limit for energy-delay product in terms of the Planks constant., Comment: 7 pages 2 figures. Submitted to IEEE Transactions on Nanotechnology

The significant experimental advances of the last few decades in dealing with the interaction of spin currents and nanomagnets, at the device level, has allowed envisioning a broad class of devices that propose to implement information processing using spin currents and nanomagnets. To analyze such spin-magnet logic circuits, in general, we have developed a coupled spin-transport/ magnetization-dynamics simulation framework that could be broadly applicable to various classes of spin-valve/ spin-torque devices. Indeed, the primary purpose of this chapter is to describe in detail, the overall approach we have developed to include a description of spin transport coupled with magnetization dynamics and to show how it was benchmarked against available data on experiments. We address non-collinear spin-transport in Section-2 using a lumped "4-component spin-circuit formalism" that describes the interaction of non-collinear magnets (required for modeling spin torque), by computing 4-component currents and voltages at every node of a "circuit". For modeling the magnetization dynamics, we use the standard Landau-Lifshitz-Gilbert (LLG) equation with the Slonczewski and the field-like terms included for spin torque. Section-3 describes how this LLG model is coupled with the spin transport model to analyze spin-torque experiments and spin-magnet circuits in general. We include MATLAB codes in the Appendix to facilitate a "hands-on" understanding of our model and hope it will enable interested readers to conveniently analyze their own experiments, develop a deeper insight into spin-magnet circuits or come up with their own creative designs., Comment: To appear in Handbook of Spintronics, eds. D Awschalom, J Nitta & Y Xu, Springer (2013). 52 pages and 20 figures. Comments and suggestions are welcome. Code typos corrected in (v3)

We show that the established physics of spin valves together with the recently discovered giant spin-Hall effect could be used to construct Read and Write units that can be integrated into a single spin switch with input-output isolation, gain and fan-out similar to CMOS inverters, but with the information stored in nanomagnets making it non-volatile. Such spin switches could be interconnected, with no external amplification, just with passive circuit elements, to perform logic operations. Moreover, since the digitization and storage occurs naturally in the magnets, the voltages can be used to implement analog weighting for non-Boolean logic., Comment: 10 pages. 5 figures in the main article. 4 figures in the appendix. Also available at http://apl.aip.org/resource/1/applab/v101/i25/p252411_s1?bypassSSO=1