Quantitative Evaluation of Hardware Binary Stochastic Neurons

In the era of internet of things (IoT), combinatorial optimization problems are ubiquitous Yamaoka et al. (2015). In fact, most of the real-problems that quantum computers are aiming to solve can be formulated as combinatorial optimization problems.From directing traffic flow Neukart et al. (2017), to routing interconnections in integrated circuit design Barahona et al. (1988); Cook et al. (2018), to making financial decisions Rosenberg et al. (2016), drug discoveries Sakaguchi et al. (2016), etc. - all involve solving a form of combinatorial optimization problems. The demand for solving these problems faster and more efficiently is ever-increasing. But such problems typically fall into the category of NP-hard or NP-complete class in complexity theory Barahona (1982), with no known polynomial time solution, making them notoriously difficult to solve in digital computers using traditional computing methods. This has given rise to a new paradigm in computing, namely Ising computing. Ising computing maps combinatorial optimization problems to an Ising model, and solves it by searching for the ground state of the system described by Lucas (2014); Sutton et al. (2017):

where, $m$ denotes the Ising spin, $J$ is the coupling co-efficient, $h$ is the external bias, and $I_{0}$ is the annealing parameter which is proportional to the inverse of temperature. In the machine learning field, the same underlying principle is used for Boltzmann Machines with the annealing parameter being 1. The binary stochastic neurons (BSNs) BSN of stochastic neural networks are well suited to function as a ‘spin’ is such systems, described mathematically by:

where $r_{i}$ is a random number between $\rm+1$ and $\rm-1$, and $I_{i}=-\partial E/\partial m_{i}$ is the input to the neuron.

Given the importance of optimization problems, a lot of research has gone into developing algorithms and identifying appropriate hardware for Ising computing. Various approaches including quantum computers based on quantum annealing (QA) or adiabatic quantum optimization (AQC) implemented with superconducting circuits Johnson et al. (2011), coherent Ising machines (CIMs) implemented with laser pulses McMahon et al. (2016), phase-change oscillators Dutta et al. (2020), or CMOS oscillators Goto, Tatsumura, and Dixon (2019); Wang and Roychowdhury (2019); Ahmed, Chiu, and Kim (2020); Chou et al. (2019) and digital annealers based on simulated annealing (SA) Kirkpatrick, Gelatt, and Vecchi (1983) implemented with digital circuits Baity-Jesi et al. (2014); Yamaoka et al. (2015); Takemoto et al. (2019); Aramon et al. (2019); Yamamoto et al. (2020); Patel et al. (2020); Patel, Canoza, and Salahuddin (2020) are being explored.

In this paper we comprehensively evaluate and characterize a stochastic magnetic tunnel junction (sMTJ) based realization of the Ising spin (eqn.2) where random numbers are generated using the natural physics of low barrier nanomagnets Camsari, Salahuddin, and Datta (2017) in a compact design. A network of these BSN units can be coupled with a memristive crossbar array Xia et al. (2016); Cai et al. (2019a); Bayat et al. (2018) to perform the synaptic operation as shown in Fig. 1 can drastically improve the area requirements and accelerate computation speed of Ising Machines. We evaluate the performance of the BSN device in terms of its energy and delay metrics and connect these to the problem and substrate-independent metric of flips per second that the probabilistic system makes Sutton et al. (2019).

Our evaluation of 1MTJ-3T BSN design considers different types of low-barrier nanomagnet realizations of MTJs. As the MTJ essentially functions as a two-terminal stochastic resistor (SR), we first take a general 3T-1SR design approach, classifying necessary and sufficient conditions for achieving the BSN operation for different types of SRs in Section II. We relate these conditions to the different sMTJ realizations in Section III. We report the timescale of operation, power and energy for each case based on benchmarked SPICE simulations of the BSN hardware consisting of spintronic elements from a modular circuit framework Torunbalci et al. (2018) coupled to 14 nm FinFET PTM models pre , and provide analytical results for relevant quantities in Section IV. Lastly, we use these device performance metrics to project onto hardware performance figures of merit such as flips per second that a probabilistic sampler makes. Our projections indicate orders of magnitude improvement potential over current digital implementations.

Binary stochastic neurons (BSNs) are well suited to function as a ‘spin’ in Ising machines for solving combinatorial optimization problems BSN ; Hassan et al. (2019). A compact and efficient hardware realization of the BSN leveraging the natural physics of stochastic nanomagnets can be made by using unstable magnetic tunnel junctions (MTJs) Daniels et al. (2020); Parks et al. (2018); Grollier et al. (2020); Abeed and Bandyopadhyay (2019); Drobitch and Bandyopadhyay (2019) as shown in Fig. 1.

The compact design of BSN based on low-barrier magnet (LBM) stochastic MTJs (sMTJs) was first proposed in 2017 Camsari, Salahuddin, and Datta (2017). Using magnet and circuit physics to analyze the performance, it was reported that using an LBM in a circular disk geometry with energy barriers below $\rm k_{B}T$ as the free layer of an MTJ results in sub-ns response times requiring only $\sim$ a few fJ of energy per random bit Hassan et al. (2019). The proposed design and the performance analysis considers a very specific type of sMTJ which had circular in-plane magnetic anisotropy (IMA) whose fluctuations are undisturbed by the current in the circuit for typical current drive conditions. However, in 2019, a version of the BSN design that was implemented in hardware to solve an 8-bit factorization problem Borders et al. (2019), consisted of an sMTJ with perpendicular anisotropy (PMA) and a barrier of a few $\rm k_{B}T$ as its free layer. Unlike the circular in-plane design, the PMA design relied on its resistance being tunable by the spin-transfer-torque effect in order to achieve the BSN operation. This has called for an extension of our initial analysis presented in Hassan et al. (2019) which we systematically perform in this paper.

As the MTJs in the BSN circuit effectively act as a fluctuating resistor, $R$ Parks et al. (2020) and the design principle is independent of this realization, for establishing the fundamental design rules we approach it from a general perspective and we hope these design rules stimulate discussion in the realization of different stochastic resistors that use different mechanisms Cheemalavagu et al. (2005); Shukla et al. (2014); Kumar, Strachan, and Williams (2017); Stampfer et al. (2018); Cai et al. (2019b); Camsari et al. (2020).

A magnetic-tunnel-junction (MTJ) whose free layer is a low-barrier magnet (LBM) could serve as a physical realization of fluctuating resistors. Depending on the nature and characteristics of the LBM magnetization fluctuations, we can get different types of R. Our previous analysis Hassan et al. (2019) was restricted to one type of LBM, the circular IMA with barrier $\rm<k_{B}T$, in this section we extend it to include all possible LBMs.

A general description of the energy associated with a magnet is given by Hassan et al. (2019):

where, $\rm H_{kp}=2K_{s}/t-4\pi M_{s}$ is the perpendicular anisotropy field along the x-axis, $\rm K_{s}$ is the surface anisotropy density, $\rm H_{ki}$ is the in-plane anisotropy along z-axis, $\rm H_{ext}$ is the external field, $\rm M_{s}$ is the saturation magnetization and $\rm\Omega=\pi(D/2)^{2}t$ is the volume of the magnet. By adjusting the thickness or the shape of the magnet, the magnetic anisotropy of the magnet can be scaled to behave like a low-barrier magnet Debashis et al. (2018); Hassan et al. (2019).Second order magnetic anisotropy effect and in-plane components of demagnetization fields have not been considered here and left for future investigation since the macroscopic models without it seems to be reasonably consistent with recent experimental results involving low barrier magnets Debashis et al. (2016); Safranski et al. (2020); Parks et al. (2020); Zhang et al. (2021). We use the stochastic LLG module from our spintronics library nanohub.org to simulate the LBM dynamics in HSPICE using its transient noise function. This model has been carefully benchmarked against general Fokker-Planck based methods Torunbalci et al. (2018).

Fig. 8 shows the fluctuation dynamics, the magnetization distribution, and the auto-correlation time ($\tau_{CORR}$) for low barrier magnets. Magnetization fluctuations translate into resistance fluctuations in MTJ, and we see that magnets with barrier $\rm<k_{B}T$ act like continuous resistances, while slightly higher barrier magnets, which have a more defined two states, give telegraphic fluctuations, and in both cases IMA magnets fluctuate orders of magnitude faster than their PMA counterparts due to a novel mechanism where the demagnetization field plays a central role Pufall et al. (2004); Safranski et al. (2020); Hassan et al. (2019); Kaiser et al. (2019); Faria, Camsari, and Datta (2017).

An important thing to note here is the current tunability in presence of an external field which can arise, for example, due to the fixed, stable layer that acts as a reference to the free layer in the MTJ. In the case of high-barrier magnets, the spin-current induced magnetic switching hysteresis loop just shifts in case of PMA magnets depending on the direction of field, but for IMA magnets the shape of the hysteresis and magnet dynamics is changed Sun (2000). The large demagnetizing field present perpendicular to the magnetization plane in IMA magnets causes the magnetization to precess around it when spin-current is applied in the opposite direction to the external field. The same is observed in low-barrier magnets as shown in Fig. 10. The larger the external field the more pronounced the effect is. The uniform precessional motion kicks in at high-field, when the current is close to the biasing current or higher applied in the opposite direction to the field. Very recently, this has been observed experimentally for low fields Safranski et al. (2020). While this is an undesired effect in case of our BSN operation, this can be useful in context to oscillator based networks Romera et al. (2018).

This has important implications in terms of acting as a fluctuating resistance in a BSN circuit. IMA magnets with external fields (i.e. uncompensated dipolar fields in MTJ Jenkins et al. (2019)) greater than its pinning field is not suited to function as a tunable or non-tunable resistor. IMA magnets with continuous magnetization coupled to a transistor with small saturation current ($\rm tens~of~\mu A$) compared to the biasing current of IMA ($\rm hundreds~of~\mu A$) can work as non-tunable resistors, and as experimental observations in ref. Safranski et al. (2020) suggest, it can withstand small (compared to its pinning field) stray fields.

PMA magnet MTJs with their small biasing current ($\sim$ few to few tens of $\mu A$) when coupled to typical transistors act as tunable resistors in BSN circuit. In this case the external bias field is actually preferred, since this enables positive $\rm I_{50}$ current Borders et al. (2019).

So, if we coupled an MTJ with a 14 nm FinFET ($\rm V_{DD}=0.8$ and $\rm I_{Dsat}=15\mu A$) pre , the table in Fig. 9 summarizes the resistance mapping and the associated parameters.

In the final section we compare the physical performance of these different sMTJs in a BSN.

Fig. 11 shows that the response time, $\tau_{N}$ for non-tunable resistor is independent of the fluctuation time of the resistance, it is rather proportional to the RC delay of the circuit. While for the tunable cases, the response time is related to the characteristic timescales of the resistor. But the time to give new numbers or flip rate $\tau_{C}$ at $\rm V_{IN}=0$ is entirely resistance fluctuation time dependent for all cases ($\tau_{C}\approx\tau_{CORR}$). So for the tunable case, the two said timescales of operation are likely to be similar as they are governed by the magnet fluctuation characteristics while for the non-tunable case, the response time which is RC dependent has the potential to be very short compared to the magnet dependent correlation time. For most applications this difference may not be of importance but for some applications where the network is directed, like Bayesian inference having two different timescales seems to be a requisite Faria et al. (2020).

Energy: As there are two timescales associated with the BSN operation, we can define two energy as well. The energy to produce first random number after the input changes, $\rm E_{N}\sim\tau_{N}\langle P\rangle$ and the energy required to produce new random numbers at a given input state, $E_{C}=\tau_{C}\langle P\rangle$. Fig. 12(a) shows an energy delay plot indicating the ranges for each type of MTJs. When describing the performance of a hardware BSN, we generally refer to the correlation time $\tau_{C}$ for delay and $E_{C}$ for the energy. The individual energy-delay numbers can be used to project performance parameters for processors built with them.

Hardware Projections: Typically the performance of an Ising hardware is measured in terms of time and energy it takes to solve a specific problem. Time to solution depends not only on the physical hardware performance but also on the algorithm that is being implemented. Here, we emphasize measuring the hardware performance in terms of a purely hardware metric flips per second (fps) Sutton et al. (2019); Isakov et al. (2015); Baity-Jesi et al. (2014), which refers to the maximum number of spin configurations the hardware can cycle through per second. It depends on the number of spins in the system (N) and the time it takes for a spin to flip ($\rm\tau$), $f=N/\tau$.

For the digital annealers the spin update time is usually determined by its clock period ($\rm\tau_{clk}$) which ranges typically in tens of ns range. To ensure fidelity simultaneous updates of connected spins needs to be avoided Aarts, Aarts, and Lenstra (2003) forcing digital annealers that operate on clock edge to update spins sequentially. So in a network where all spins are connected effectively only one spin can update per clock cycle Aramon et al. (2019). But it need not be if some spins are unconnected (i.e. nearest neighbor Yamaoka et al. (2015); Baity-Jesi et al. (2014), or king-graph Takemoto et al. (2019) connection, or if spins are parallelized by implementing special algorithms Yamamoto et al. (2020); Patel et al. (2020); Patel, Canoza, and Salahuddin (2020). Based on the reported total spin number and clock speeds of digital annealing hardware today which have about $\rm\sim 10K$ neurons that can update per $\rm\sim 10ns$ clock period, we derive an estimation of their performance at $\rm f\sim 10^{4}/10^{-8}=10^{12}$ flips per second Yamaoka et al. (2015); Sutton et al. (2019) as shown in Fig. 13.

Compared to digital annealers the Ising spin hardware we presented in this work can work autonomously, i.e, without a synchronizing clock or a sequencer Sutton et al. (2019); Faria et al. (2020); Kaiser et al. (2020). In this mode, the speeds are governed by neuron ($\tau_{neu}$) and synapse ($\tau_{syn}$) time only, and to ensure fidelity and avoid simultaneous updates of connected BSNs the synapse needs to update faster than the the neuron ($\rm\tau_{syn}<\tau_{neu}$). Sutton et. al.Sutton et al. (2019) defines a metric $\rm s=\tau_{syn}/\tau_{neu}$ and showed that to ensure the fidelity of operations $\rm s$ needs to be less than 1. The exact requirements are problem and architecture dependent. Memristive crossbar arrays paired with a fast summing amplifier synapse could operate very efficiently at as low as few tens of ps speeds Xia et al. (2016); Cai et al. (2019c); Huang et al. (2015); Bayat et al. (2018); Hu et al. (2018); Cai et al. (2019a).

The digital annealers mimic the Ising spin using a combination of random-number generators (LFSR, Xoshiro, etc.), look-up-tables (LUT) and comparators. The random number generator (RNG) unit is one of the most are expensive elements in the design Gyoten, Hiromoto, and Sato (2018). Even in the most optimized design, the RNG unit take up $\rm\sim 11\%$ of the total logic gate area Yamamoto et al. (2020). The 3T-1MTJ design offers drastic reduction in the area footprint, promising massive scalability leveraging existing 1T-1MTJ Magnetic RAM technology that already has 1Gbit integrated cells Aggarwal et al. (2019); eve (2019).

Fig. 12(b) projects fps number considering $\rm\tau\equiv\tau_{neu}\approx\tau_{CORR}$ for different no of spins, N. An MTJ realization with circular IMA, with $\sim$ ns timescale can offer almost two orders of magnitude speedup with $\rm<10k$ neurons. If spins are implemented in Gbit densities all stochastic implementations seem to outperform the CMOS implementations. For such systems the upper bound for N is ultimately determined either by area or by power budget of the chip. Note that the fps number does not reflect the connectivity of the spins or the algorithm implemented by the hardware. It also does not indicate the solution accuracy obtainable for specific problems Zhang et al. (2020). What we highlight here is that using the natural physics of the MTJ we can design a very compact realization of eq. 2 compared to current state of the art CMOS implementations, and despite being a magnetic circuit, low barrier magnet implementations even offer an overall speed up due to their fast fluctuation rates.

In this paper, we presented a comprehensive evaluation of naturally stochastic magnetic building blocks for implementing probabilistic algorithms compactly and efficiently. We generalized the proposed 1MTJ-3T design to a 1SR-3T design and presented necessary design rules for BSN operation that we hope will stimulate further interest in finding stochastic resistance (1SR) with suitable properties. We extended the physical performance analysis of the 1MTJ-3T BSN design to include unstable MTJ’s with different low-barrier-magnets as free layers. They are evaluated as physical realizations of the general stochastic resistor (SR) with respect to 14 nm FinFET transistors. IMA magnets with barrier $\leq k_{B}T$ proved to be the best option, low-barrier PMA can function as current-tunable resistors as well. While careful optimization of the fixed layer to cancel the stray fields in IMA MTJ is preferred, PMA can benefit from the presence of stray fields (can be a source of the $\rm I_{50}$). The most challenging set of working conditions are set for telegraphic IMA magnets, even if they are highly optimized and no stray fields are present in the circuit, they need to be coupled with high current transistors due to their high pinning currents, because if paired with low current transistors like 14 nm FinFET results in a staircase-like functional behavior which does not work as a p-bit as we discussed.

These BSNs are an integral part of Ising machines which are often referred to as annealing processors. Using 1MTJ-3T BSN could speed up the operation of these processors by orders of magnitude. Another important application space for these BSN is stochastic neural networks Kaiser et al. (2020); Nasrin et al. (2019); Schuman et al. (2017); Hinton (2002). In fact, binary stochastic neurons are desired for deep learning networks, but are typically avoided because it is harder to generate random bits in CMOS hardware Courbariaux et al. (2016). Use of this compact neuron that relies on MTJs natural physics to provide stochastic binarization could accelerate computation in custom hardware Tsai et al. (2017); Park et al. (2015) by faster evaluation of BSN function Hassan et al. (2019) and also encourage algorithmic advancement using BSN.