Hardware emulation of stochastic p-bits for invertible logic

The common feature of nearly all logic and memory devices is that they make use of stable units to represent 0's and 1's. A completely different paradigm is based on three-terminal stochastic units which could be called "p-bits", where the output is a random telegraphic signal continuously fluctuating between 0 and 1 with a tunable mean. p-bits can be interconnected to receive weighted contributions from others in a network, and these weighted contributions can be chosen to not only solve problems of optimization and inference but also to implement precise Boolean functions in an inverted mode. This inverted operation of Boolean gates is particularly striking: They provide inputs consistent to a given output along with unique outputs to a given set of inputs. The existing demonstrations of accurate invertible logic are intriguing, but will these striking properties observed in computer simulations carry over to hardware implementations? This paper uses individual micro controllers to emulate p-bits, and we present results for a 4-bit ripple carry adder with 48 p-bits and a 4-bit multiplier with 46 p-bits working in inverted mode as a factorizer. Our results constitute a first step towards implementing p-bits with nano devices, like stochastic Magnetic Tunnel Junctions

Modular Approach to Spintronics

There has been enormous progress in the last two decades, effectively combining spintronics and magnetics into a powerful force that is shaping the field of memory devices. New materials and phenomena continue to be discovered at an impressive rate, providing an ever-increasing set of building blocks that could be exploited in designing transistor-like functional devices of the future. The objective of this thesis is to provide a quantitative foundation for this building block approach, so that new discoveries can be integrated into functional device concepts, quickly analyzed and critically evaluated. Through careful benchmarking against available theory and experiments we establish a set of elemental modules representing diverse materials and phenomena. These elemental modules can be integrated seamlessly to model composite devices involving both spintronic and nanomagnetic phenomena, even when subtle quantum mechanical properties of spin are involved. We envision the library of modules to evolve both by incorporating new modules and by improving existing modules as the field progresses. The primary contribution of this thesis is to establish the ground rules or protocols for a modular approach that can build a lasting bridge between materials scientists and circuit designers in the field of spintronics and nanomagnetics

Modular Approach to Spintronics.

There has been enormous progress in the last two decades, effectively combining spintronics and magnetics into a powerful force that is shaping the field of memory devices. New materials and phenomena continue to be discovered at an impressive rate, providing an ever-increasing set of building blocks that could be exploited in designing transistor-like functional devices of the future. The objective of this paper is to provide a quantitative foundation for this building block approach, so that new discoveries can be integrated into functional device concepts, quickly analyzed and critically evaluated. Through careful benchmarking against available theory and experiment we establish a set of elemental modules representing diverse materials and phenomena. These elemental modules can be integrated seamlessly to model composite devices involving both spintronic and nanomagnetic phenomena. We envision the library of modules to evolve both by incorporating new modules and by improving existing modules as the field progresses. The primary contribution of this paper is to establish the ground rules or protocols for a modular approach that can build a lasting bridge between materials scientists and circuit designers in the field of spintronics and nanomagnetics

Stochastic p-Bits for Invertible Logic

Conventional semiconductor-based logic and nanomagnet-based memory devices are built out of stable, deterministic units such as standard metal-oxide semiconductor transistors, or nanomagnets with energy barriers in excess of ≈40–60  kT. In this paper, we show that unstable, stochastic units, which we call “p-bits,” can be interconnected to create robust correlations that implement precise Boolean functions with impressive accuracy, comparable to standard digital circuits. At the same time, they are invertible, a unique property that is absent in standard digital circuits. When operated in the direct mode, the input is clamped, and the network provides the correct output. In the inverted mode, the output is clamped, and the network fluctuates among all possible inputs that are consistent with that output. First, we present a detailed implementation of an invertible gate to bring out the key role of a single three-terminal transistorlike building block to enable the construction of correlated p-bit networks. The results for this specific, CMOS-assisted nanomagnet-based hardware implementation agree well with those from a universal model for p-bits, showing that p-bits need not be magnet based: any three-terminal tunable random bit generator should be suitable. We present a general algorithm for designing a Boltzmann machine (BM) with a symmetric connection matrix [J] (J_{ij}=J_{ji}) that implements a given truth table with p-bits. The [J] matrices are relatively sparse with a few unique weights for convenient hardware implementation. We then show how BM full adders can be interconnected in a partially directed manner (J_{ij}≠J_{ji}) to implement large logic operations such as 32-bit binary addition. Hundreds of stochastic p-bits get precisely correlated such that the correct answer out of 2^{33} (≈8×10^{9}) possibilities can be extracted by looking at the statistical mode or majority vote of a number of time samples. With perfect directivity (J_{ji}=0) a small number of samples is enough, while for less directed connections more samples are needed, but even in the former case logical invertibility is largely preserved. This combination of digital accuracy and logical invertibility is enabled by the hybrid design that uses bidirectional BM units to construct circuits with partially directed interunit connections. We establish this key result with extensive examples including a 4-bit multiplier which in inverted mode functions as a factorizer

Spin Funneling for Enhanced Spin Injection into Ferromagnets.

It is well-established that high spin-orbit coupling (SOC) materials convert a charge current density into a spin current density which can be used to switch a magnet efficiently and there is increasing interest in identifying materials with large spin Hall angle for lower switching current. Using experimentally benchmarked models, we show that composite structures can be designed using existing spin Hall materials such that the effective spin Hall angle is larger by an order of magnitude. The basic idea is to funnel spins from a large area of spin Hall material into a small area of ferromagnet using a normal metal with large spin diffusion length and low resistivity like Cu or Al. We show that this approach is increasingly effective as magnets get smaller. We avoid unwanted charge current shunting by the low resistive NM layer utilizing the newly discovered phenomenon of pure spin conduction in ferromagnetic insulators via magnon diffusion. We provide a spin circuit model for magnon diffusion in FMI that is benchmarked against recent experiments and theory

Intrinsic optimization using stochastic nanomagnets.

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 ±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

Intrinsic optimization using stochastic nanomagnets.

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 ±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