A Sc2C2@C88 cluster based ultra-compact multi-level probabilistic bit for matrix multiplication

Information units are progressively approaching the fundamental physical limits of the integration density, including in terms of extremely small sizes, multistates and probabilistic traversal. However, simultaneously encompassing all of these characteristics in a unit remains elusive. Here, via real-time in situ electrical monitoring, we clearly observed stochastic alterations of multiple conductance states in Sc2C2@C88. The true random bit sequence generated exhibited an autocorrelation function whose confidence interval fell within \pm 0.02, demonstrating high-quality randomness. The alterations of multiple conductance states are controllable, that is, whose probability distributions could traverse from 0 to 1, enabling us to factorize 551 into its prime factors. Furthermore, we proposed a matrix-chain multiplication scheme and experimentally verified the multiplication of two 4 \times 4 state-transition matrices with a small maximum error < 0.05. Combined with theoretical calculations, the stochastic but controllable multistates are probably attributed to the rich energy landscape, which could be stepwise changed by the electric field. Our findings reveal extremely small multi-level probabilistic bit for matrix multiplication, which pave the way for ultracompact intelligent electronic devices.

💡 Research Summary

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The authors present an ultra‑compact probabilistic bit (p‑bit) based on a single Sc₂C₂@C₈₈ endohedral fullerene cluster, achieving the simultaneous realization of three coveted attributes for next‑generation information units: sub‑nanometer size, multistate capability, and controllable stochastic behavior. Devices were fabricated by defining a 50 nm hour‑glass nanowire via electron‑beam lithography and creating a nanogap through feedback‑controlled electromigration break‑junction (FCEBJ) at 1.8 K. After depositing Sc₂C₂@C₈₈ molecules, low‑temperature transport measurements revealed clear Coulomb blockade with a charging energy exceeding 75 meV, confirming that the measured signal originates from the encapsulated cluster rather than metallic residues.

When a bias voltage is applied, the source‑drain current randomly switches among three to four discrete conductance levels. The number of observable states and their relative probabilities depend systematically on the bias voltage: at 140 mV four states appear, while at 180 mV only three remain. Statistical analysis of the current‑time traces shows that the probability of a “high‑conductance” (assigned binary 1) versus “low‑conductance” (binary 0) can be tuned continuously from 0 to 1, following a sigmoid curve. Autocorrelation analysis of the derived 0‑1 sequences yields confidence intervals within ±0.02, indicating high‑quality true randomness comparable to, or better than, previously reported atomic‑scale stochastic devices.

The controllable stochasticity is exploited in two proof‑of‑concept applications. First, a physical implementation of a p‑bit‑based integer factorization algorithm successfully factors 551 (19 × 29). The algorithm operates in a closed‑loop: a voltage array is fed sequentially into the device, the instantaneous conductance state is read as a binary output, and a LabVIEW routine updates the next voltage based on a cost function that drives the probability distribution toward the target binary vector. Repeating the process 20 times always converges to the correct factor pair, demonstrating that the device can act as a physical random number generator with programmable bias‑dependent probabilities.

Second, the authors demonstrate high‑precision matrix multiplication using the intrinsic Markovian state‑transition behavior of the device. By measuring the transition probabilities between conductance states at two distinct bias voltages (140 mV and 160 mV), they construct two 4 × 4 stochastic matrices. Alternating the bias every second and recording the initial and final states yields a composite transition matrix that matches the mathematical product of the two individual matrices. The experimentally obtained matrix elements deviate from the calculated values by less than 0.05 (maximum) and 0.03 (average), confirming that a single cluster can embody both matrix elements and the multiplication operation without external weighting circuitry.

The underlying mechanism is attributed to a rich energy landscape of the Sc₂C₂@C₈₈ system, where the electric field reshapes the potential surface of the encapsulated Sc₂C₂ unit, enabling stepwise changes among multiple metastable configurations. Density‑functional calculations support this picture, showing that different field strengths stabilize distinct charge distributions and thus distinct conductance levels.

Compared with conventional memristor cross‑bar arrays, which require many devices to store matrix weights and rely on Kirchhoff’s laws for multiplication, the Sc₂C₂@C₈₈ p‑bit accomplishes both storage and computation within a single nanoscale element. The authors propose a practical roadmap: by characterizing a library of bias‑dependent transition matrices (T₁, T₂, …, Tₙ), a user could program complex linear algebra operations simply by sequencing voltage levels, effectively “programming” the hardware through its physical tuning parameters.

In summary, this work demonstrates that an endohedral fullerene cluster can serve as an ultra‑compact, multistate, probabilistic information unit capable of generating high‑quality random bits, performing integer factorization, and executing matrix multiplication with low error. The findings open a pathway toward densely integrated, physics‑driven computing architectures that blend storage, randomness, and arithmetic in a single molecular‑scale device. Future challenges include achieving room‑temperature operation, scaling to larger arrays, and integrating with conventional CMOS control circuitry to realize practical probabilistic and neuromorphic processors.