Switching Characteristics of Electrically Connected Stochastically Actuated Magnetic Tunnel Junction Nanopillars

Abstract: We investigate the stochastic dynamics of nanoscale perpendicular magnetic tunnel junctions (pMTJs) and the correlations that arise when they are electrically coupled. Individual junctions exhibit thermally activated spin-transfer torque switching with transition probabilities that are well described by a Poisson process. When two junctions are connected in parallel, circuit-mediated redistribution of voltages that occurs in real time as the junction resistances change leads to correlated switching behavior. A minimal stochastic model based on single-junction statistical switching properties and Kirchhoff's laws captures the coupled switching probabilities, while a Markov-chain formalism describes nonequilibrium steady states under multi-pulse driving. Further, these circuit-mediated interactions can be mapped onto the parameters of an Ising Hamiltonian, providing an interpretation in terms of effective spin-spin interactions. Our results demonstrate how simple electrical connections can generate Ising-like couplings and tunable stochastic dynamics in nanoscale magnets.

• The paper evaluates the stochastic switching characteristics of electrically connected pMTJs, finding that simple electrical interconnections could produce correlations observed as effective ferromagnetic- or antiferromagnetic-like Ising couplings.
• The authors use a minimal model combining Poisson statistics and Kirchhoff's laws to accurately mimic the coupled dynamics of the junctions, demonstrating that multi-pulse driving creates programmable nonequilibrium steady states.
• A key implication is that the output probability distributions of the coupled pair can be programmed through pulse sequence design, allowing for controllable stochastic interactions relevant to Ising machines and probabilistic computing systems.

Overview

This paper reports an experimental and theoretical study of correlated stochastic switching in two electrically connected perpendicular magnetic tunnel junctions (pMTJs). The central finding is that simple electrical interconnection—two 40 nm pMTJs wired in parallel through a series resistor—produces measurable correlations in thermally activated, spin-transfer-torque-driven switching that can be described as effective ferromagnetic- or antiferromagnetic-like Ising couplings. The authors show that a minimal model combining single-junction Poisson switching statistics with Kirchhoff's laws reproduces the coupled dynamics, that multi-pulse driving is captured by a Markov-chain formalism yielding programmable nonequilibrium steady states, and that these steady-state distributions map onto parameters $(J, h_1, h_2)$ of an Ising Hamiltonian.

Single-junction characterization

The experiments are performed at room temperature ($T = 295$ K) on two circular pMTJs (Devices A and B) of 40 nm diameter, each connected in series with $R_0 = 2\,\text{k}\Omega$. Switching probabilities are measured with a reset–write–read pulse sequence (200 µs reset, 200 µs variable-amplitude write, 500 µs read), repeated $N = 10{,}000$ times per amplitude. The switching probability follows the macrospin thermally activated form

$$P(V,t) = 1 - \exp[-\Gamma(V)t], \qquad \Gamma(V) = \frac{1}{t_0}\exp\left[-\xi\left(1 - \frac{V}{V_0}\right)\right],$$

with attempt time fixed at $t_0 = 1$ ns and fitted barriers $\xi \approx 29$–$45$ and critical voltages $|V_0| \approx 0.14$–$0.24$ V for both switching directions of both devices. An appendix verifies the Poisson-process assumption directly by varying pulse duration at fixed voltage, obtaining fits with $T = 295$0–$T = 295$1. This validation matters because the entire modeling framework rests on treating each junction as an inhomogeneous Poisson process with voltage-dependent rate.

Correlated switching in parallel-connected junctions

When Devices A and B are connected in parallel, four resistance levels distinguish all four joint states (00, 01, 10, 11), since the devices have slightly different resistances. The coupled system exhibits polarity-dependent correlations:

• Ferromagnetic-like behavior: initialized in 00 and driven at $T = 295$2 V, the system ends predominantly in symmetric configurations, with $T = 295$3 and $T = 295$4. When one junction switches from low to high resistance, the voltage across the other increases, raising its switching probability.
• Antiferromagnetic-like behavior: initialized in 11 and driven at $T = 295$5 V, the system ends predominantly in asymmetric configurations, with $T = 295$6 and $T = 295$7, because a high-to-low transition lowers the voltage across the partner junction.

These correlations arise purely from real-time voltage redistribution—there is no direct magnetic interaction between the junctions. Using a two-step waveform exploiting the antiferromagnetic-like regime, the authors measure the complete voltage-dependent transition matrix from all four initial states, with $T = 295$8 repetitions per point.

Predictive modeling

The simulation discretizes the write pulse into time steps $T = 295$9, computes instantaneous junction voltages from the current state via Kirchhoff's laws, and updates each junction stochastically using the independently measured single-junction rates $R_0 = 2\,\text{k}\Omega$0. This model, requiring no fitted coupled-system parameters, captures the overall shape, symmetry, and trends of the full experimental transition matrices, though not exact quantitative agreement. The authors attribute residual discrepancies to simplifications such as neglecting the bias dependence of junction resistance in the antiparallel state; they present this as satisfactory given the model's minimalism rather than as a precise quantitative theory.

Extending to pulse trains, the long pulse duration relative to relaxation allows each pulse to be treated as independent, so the state evolution under a sequence $R_0 = 2\,\text{k}\Omega$1 is a product of measured transition matrices $R_0 = 2\,\text{k}\Omega$2. Repeated application drives the system to a stationary nonequilibrium distribution given by the unit-eigenvalue eigenvector of $R_0 = 2\,\text{k}\Omega$3. Experiments with alternating two-pulse trains ($R_0 = 2\,\text{k}\Omega$4 cycles) confirm good agreement between measured and predicted steady-state distributions. A key implication is that the output probability distribution of the coupled pair is programmable purely through pulse-sequence design, without hardware modification.

Mapping to Ising parameters

Identifying the parallel state with $R_0 = 2\,\text{k}\Omega$5 and antiparallel with $R_0 = 2\,\text{k}\Omega$6, and assuming a Boltzmann form $R_0 = 2\,\text{k}\Omega$7 (with $R_0 = 2\,\text{k}\Omega$8), the authors derive a linear transformation from $R_0 = 2\,\text{k}\Omega$9 to effective parameters $N = 10{,}000$0. Applying this mapping to steady states reachable by two- and three-pulse trains shows that tailored sequences access a variety of effective couplings and local fields.

Two structural constraints are worth noting. First, the mapping is not surjective: for $N = 10{,}000$1 only same-sign local fields are accessible, while for $N = 10{,}000$2 only opposite-sign fields appear. Second, because Devices A and B are not identical, the accessible distributions lie on a gently curved surface in probability space rather than the flat plane $N = 10{,}000$3 that identical devices would occupy. The authors leave open whether alternative mappings or additional circuit elements (resistors, nonlinear components) could expand the accessible parameter space.

Limitations and open questions

Several limitations are acknowledged within the paper. The predictive model does not exactly reproduce experiment, and identified refinements—more accurate single-junction fitting functions and bias-dependent resistance—are not implemented. The Markov treatment assumes inter-pulse relaxation, valid for the 200 µs pulses used here but untested at shorter durations where sequential pulses would correlate. The Boltzmann assumption underlying the Ising mapping is adopted as a convention rather than derived from the driven, nonequilibrium nature of the system, so the extracted $N = 10{,}000$4 should be read as effective descriptors of the steady-state distribution, not equilibrium thermodynamic couplings. Finally, all results concern a single pair of devices; scaling to larger networks of coupled junctions, where the characterization burden motivates the modeling framework, remains to be demonstrated.

Conclusion

This work establishes that Kirchhoff-law-mediated voltage redistribution between parallel pMTJs generates tunable, polarity-dependent correlated switching that mimics ferro- and antiferromagnetic Ising interactions. The combination of validated Poisson single-junction statistics, a parameter-free circuit-level stochastic model, and a Markov-chain description of pulse-train dynamics provides both physical insight into nonequilibrium steady states in nanoscale magnets and a practical route to electrically programmable stochastic interactions relevant to Ising-machine and probabilistic-computing architectures.