🔬 applied physics

Switching Characteristics of Electrically Connected Stochastically Actuated Magnetic Tunnel Junction Nanopillars

This paper demonstrates that electrically coupling stochastically actuated magnetic tunnel junctions in parallel induces correlated switching behavior through real-time circuit-mediated voltage redistribution, which can be modeled via Kirchhoff's laws and mapped onto an Ising Hamiltonian to create tunable effective spin-spin interactions.

Original authors: Dairong Chen, Ahmed Sidi El Valli, Jonathan Z. Sun, Flaviano Morone, Dries Sels, Andrew D. Kent

Published 2026-02-04

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CC BY 4.0

Original authors: Dairong Chen, Ahmed Sidi El Valli, Jonathan Z. Sun, Flaviano Morone, Dries Sels, Andrew D. Kent

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have two tiny, magical switches called Magnetic Tunnel Junctions (MTJs). Think of these not as simple on/off light switches, but as fidgety coins that are constantly trying to flip between heads (state 0) and tails (state 1).

In this paper, the researchers are playing with these "coins" to see what happens when they are connected together by a wire.

The Setup: Two Fidgety Coins

First, the scientists looked at a single coin. They found that if you give it a little electrical "nudge" (a voltage pulse), it might flip. The bigger the nudge, the more likely it is to flip. This flipping isn't perfectly predictable; it's a bit like rolling a die. Sometimes it flips, sometimes it doesn't. The researchers mapped out exactly how likely the coin is to flip based on how hard they push it. They described this behavior using a math concept called a Poisson process, which is just a fancy way of saying "random events happening at a specific rate."

The Experiment: Wiring Them Together

Next, they took two of these coins (Device A and Device B) and wired them in parallel, sharing a single power source and a resistor (like a shared water pipe with a narrow section).

Here is the magic: The coins start talking to each other through the electricity.

Because they share the same circuit, when one coin flips, it changes the electrical pressure (voltage) for the other coin.

Scenario 1 (The "Team Players"): When the researchers applied a negative voltage, they noticed something interesting. If Coin A flipped from heads to tails, the electrical pressure on Coin B increased, making it more likely to flip too. The two coins tended to end up in the same state (both heads or both tails). The researchers call this ferromagnetic-like coupling. It's like two friends who finish each other's sentences; if one jumps, the other jumps too.
Scenario 2 (The "Opposites"): When they applied a positive voltage, the effect reversed. If Coin A flipped, the electrical pressure on Coin B dropped, making it less likely to flip. The coins tended to end up in opposite states (one heads, one tails). The researchers call this antiferromagnetic-like coupling. It's like two rivals; if one jumps, the other stays put.

Crucially, the coins aren't touching magnetically. They are just "talking" through the wires. The circuit itself creates this relationship.

The Prediction: A Game of Chance

The researchers built a computer model to predict this behavior. They didn't need to know the complex physics inside the coins; they just used the "flip rules" they learned from testing the single coins and applied basic circuit laws (Kirchhoff's laws).

The Result: The computer model successfully predicted the real-world behavior. It showed that you can simulate how two connected coins will behave just by knowing how a single coin behaves and how the wires connect them.

The Advanced Move: The Pulse Train

The researchers didn't just stop at one push. They tried pushing the coins with a sequence of different pulses (a "pulse train").

They treated the system like a board game where the state of the coins changes with every roll of the dice (every pulse).
By using a mathematical tool called a Markov Chain, they could predict exactly what the final distribution of heads and tails would be after a long sequence of pushes.
The Takeaway: By simply changing the pattern of electrical pushes, they could "program" the system to settle into any specific mix of states they wanted, without ever changing the physical wires or the coins themselves.

The Big Picture: An "Ising Machine"

Finally, the researchers connected this to a famous concept in physics called the Ising Model.

Imagine a grid of magnets that want to align with their neighbors. This is a classic problem in physics used to solve complex puzzles.
The researchers showed that their two connected coins act exactly like two interacting magnets in this model.
By adjusting the electrical pulses, they could tune the "strength" of the connection between the coins. They could make them act like strong friends (ferromagnetic) or strong rivals (antiferromagnetic).

Summary

In simple terms, this paper shows that you can create complex, "smart" interactions between tiny magnetic switches just by wiring them together and controlling the electricity. You don't need to build complex magnetic structures; the circuit itself does the work. This proves that simple electrical connections can create tunable, random behaviors that mimic the interactions of physical magnets, offering a new way to build computers that solve problems using probability and randomness.

Technical Summary: Switching Characteristics of Electrically Connected Stochastically Actuated Magnetic Tunnel Junction Nanopillars

Problem Statement
While magnetic tunnel junctions (MTJs) are established as promising building blocks for probabilistic computing (p-bits) and Ising machines, most existing architectures treat junctions as individually driven entities. In these systems, the coupling terms ( $J_{ij}$ ) required to realize an Ising Hamiltonian are typically embedded via external computation rather than arising from physical interactions between the devices. There is a lack of understanding regarding the emergent dynamics of electrically connected junctions, specifically how circuit-mediated voltage redistribution influences the stochastic switching of multiple MTJs and whether such interactions can be predicted from single-junction properties.

Methodology
The authors experimentally and theoretically investigate two perpendicular magnetic tunnel junctions (pMTJs) connected in parallel through a shared series resistor ( $R_0 = 2\text{ k}\Omega$ ). The study operates at room temperature ( $295\text{ K}$ ) using 40 nm diameter devices.

Experimental Characterization:
- Single-Junction: The stochastic switching behavior of individual pMTJs (Device A and Device B) was characterized under voltage pulses. Transition probabilities were measured as a function of pulse amplitude and duration, fitting the data to a thermally activated macrospin model described by a Poisson process.
- Coupled Junctions: The two devices were connected in parallel. The system's response was measured using a reset–write–read pulse sequence. The authors mapped the transition probabilities between all four possible states (00, 01, 10, 11) as a function of the applied write voltage. Special waveforms were employed to isolate transition probabilities from asymmetric states (01 and 10).
Modeling Framework:
- Poisson Process Model: A minimal stochastic model was developed where the switching of each junction is treated as a Poisson process with a voltage-dependent rate $\Gamma(V)$ . The coupling is incorporated via Kirchhoff's laws: as one junction switches, its resistance changes, instantly redistributing the voltage across the parallel network and altering the switching probability of the other junction.
- Markov Chain Formalism: To analyze multi-pulse driving, the system was modeled as a discrete-time Markov process. Transition matrices $T(V)$ were constructed from experimental data to predict the evolution of the system under arbitrary pulse sequences and to determine non-equilibrium steady-state distributions.
Mapping to Ising Hamiltonian:
- The steady-state probability distributions obtained from the Markov model were mapped onto the parameters of an effective Ising Hamiltonian ( $H = -\sum J_{ij}s_i s_j - \sum h_i s_i$ ), treating the low-resistance state as spin $-1$ and the high-resistance state as spin $+1$ .

Key Results

Emergent Correlated Switching: The experimentally connected pMTJs exhibit strong correlations driven purely by circuit-mediated voltage redistribution.
- Ferromagnetic-like Coupling: When initialized in the low-resistance state (00) and driven with negative voltage, the system preferentially transitions to symmetric states (00 or 11). This occurs because a switch to a high-resistance state increases the voltage across the remaining junction, promoting a simultaneous switch.
- Antiferromagnetic-like Coupling: When initialized in the high-resistance state (11) and driven with positive voltage, the system preferentially transitions to asymmetric states (01 or 10). Here, a switch to a low-resistance state decreases the voltage across the other junction, suppressing its switching probability.
Model Accuracy: The minimal Poisson-based model, using only single-junction parameters and Kirchhoff's laws, successfully reproduces the overall shape, symmetry, and trends of the coupled switching probabilities observed in experiments.
Steady-State Control: The Markov-chain formalism accurately predicts the non-equilibrium steady-state distributions generated by arbitrary pulse trains. By tailoring pulse sequences, the authors demonstrated the ability to tune the system's output distribution.
Ising Parameter Mapping: The study successfully mapped the experimentally accessible probability distributions to effective coupling strengths ( $J$ ) and local fields ( $h$ ). The results show that simple electrical connections can generate tunable effective spin-spin interactions, although the accessible parameter space is constrained by the specific asymmetry of the devices and the nature of the circuit-mediated coupling.

Significance and Claims
The paper claims to demonstrate that simple electrical interconnections can mediate effective couplings between magnetically stable pMTJs without direct magnetic interaction. The significance lies in:

Physical Realization of Coupling: Providing a physical mechanism for generating Ising-like couplings ( $J_{ij}$ ) through circuit topology rather than external computation.
Predictive Modeling: Establishing that the complex dynamics of coupled stochastic systems can be captured by a minimal model based on single-junction statistics and circuit laws.
Tunability: Showing that the nature of the interaction (ferromagnetic vs. antiferromagnetic) and the steady-state distribution can be controlled purely through electrical pulse sequences.
Platform for Nonequilibrium Physics: Offering a platform to investigate correlated stochastic dynamics and nonequilibrium statistical mechanics in circuit-mediated nanoscale magnetic systems.

The authors position this work as a foundational step toward scalable, reconfigurable stochastic computing architectures where interactions are programmable via electrical control.