Write Error Rate of Spin-Transfer-Torque Random Access Memory Including Micromagnetic Effects Using Rare Event Enhancement

Abstract

Spin-transfer-torque random access memory (STT-RAM) is a promising candidate for the next-generation of random-access-memory due to improved scalability, read-write speeds and endurance. However, the write pulse duration must be long enough to ensure a low write error rate (WER), the probability that a bit will remain unswitched after the write pulse is turned off, in the presence of stochastic thermal effects. WERs on the scale of 10 −9 or lower are desired. Within a macrospin approximation, WERs can be calculated analytically using the Fokker-Planck method to this point and beyond. However, dynamic micromagnetic effects within the bit can affect and lead to faster switching. Such micromagnetic effects can be addressed via numerical solution of the stochastic Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation. However, determining WERs approaching 10 −9 would require well over 10 9 such independent simulations, which is infeasible. In this work, we explore calculation of WER using "rare event enhancement" (REE), an approach that has been used for Monte Carlo simulation of other systems where rare events nevertheless remain important. Using a prototype REE approach tailored to the STT-RAM switching physics, we demonstrate reliable calculation of a WER to 10 −9 with sets of only approximately 10 3 ongoing stochastic LLGS simulations, and the apparent ability to go further.Index Terms-spin-transfer-torque, write error rate, micromagnetic, rare event enhancement.

Machine Learning for Statistical Modeling

“…As is evident, probability distributions obtained from micromagnetic simulations can be very different from ones obtained from macrospin simulations. Average switching times obtained from micromagnetic simulations are always found to be lower than that obtained from macrospin simulations, in agreement with previous studies [33,34]. The probability distribution function (PDF) of t sw resulting from the thermal fluctuation has been described previously in terms of a skew-normal PDF, where the PDF is characterized by the mean, standard deviation, and skewness [30][31][32] or Pearson IV form [34].…”

“…Even within the macrospin approximation, attempts have been made to reduce the computational load by modeling variation of parameters in a multi-dimensional space in terms of linear analytical equations that describe the stochastic magnetization dynamics [8]. However, it has been shown that a macrospin approximation cannot always capture the complicated physics arising from incoherent switching [33], such as sub-volume excitation [37,38], anomalous branching of write error variation with increasing write voltage or current [21], and effects coming from higher-order spin wave modes [9]. Thus, full micromagnetic simulations may be needed for a more accurate study.…”

Overcoming thermal noise in non-volatile spin wave logic

“…Since an error-rate calculation using brute force micromagnetic simulation can be computationally exhaustive, we have restricted simulations to only 1000 trials that gave an error rate of <10 −3 . The recently developed “rare-event enhancement” (REE) technique for micromagnetics 58 cannot be trivially applied to our fast picosecond magnetization switching dynamics. Hence, to capture the extreme tails of error-rate, we use a less computationally intensive equivalent single domain stochastic LLG simulation for the SW detector 46 and the “rare-event enhancement” (REE) technique to reach an error-rate of less than 10 −9 .…”

Low Energy Barrier Nanomagnet Design for Binary Stochastic Neurons: Design Challenges for Real Nanomagnets With Fabrication Defects

“…In logic, errors propagate and consequently the switching error probability of a switch should be less than 10 -15 for logic applications [14]. Memory is more forgiving since errors do not propagate, but in order to avoid repetitive write/read/re-write operations, the error probability still needs to be below 10 -9 [15]. Such low error probabilities may be out of reach for voltage-controlled binary nanomagnetic switches, and maybe a tall order for even currentcontrolled (more dissipative) ones if we wish to expend no more than ~1 fJ of write energy per bit.…”