DTP-Backbone Polymer with TPA for p-Bit Devices

• DTP-backbone polymer is a conjugated polymer functionalized with TPA groups that exhibits stochastic memristive switching via redox-induced dynamics.
• It employs a logistic transfer function to convert continuous resistance fluctuations into discrete binary outputs, forming probabilistic bits.
• Discrete Shannon entropy and dielectric spectroscopy quantify its tunable randomness, enabling energy-efficient thermodynamic computing.

A dithieno[3,2-b:2',3'-d]pyrrole (DTP)-backbone polymer functionalized with pendant triphenylamine (TPA) groups defines a class of conjugated polymers suited for use as active elements in organic memristive devices. These polymers exhibit distinctive resistance fluctuations that arise from thermally driven and field-induced molecular dynamics. This characteristic makes them well-positioned to serve as stochastic circuit components—specifically, probabilistic bits (p-bits)—within energy-efficient thermodynamic computing hardware. In such architectures, the stochastic electrical behavior rooted in redox activity and conformational variability is harnessed and quantized at the circuit level, providing a chemically tunable entropy source.

1. DTP-Backbone Polymer Architecture in Memristive Probabilistic Bits

The polymer investigated, denoted as pTPAC₆DTP, employs a conjugated DTP backbone decorated with TPA groups to create a memristive device whose resistance (Rₘₑₘ) is inherently non-deterministic under bias due to molecular scale variability. Under external electric field, the polymer does not transition between binary states in a deterministic fashion; rather, it continuously samples a range of resistance states. These transitions are a consequence of redox-driven changes and conformational rearrangements in both the backbone and side chains.

In typical circuit applications, the polymer memristor is configured in series with a reference resistor (Rₛ) to form a voltage divider, generating an intermediate voltage (Vₘᵥd):

• Vₘᵥd is then compared to a reference voltage (V_ref) within a comparator circuit.
• When Vₘᵥd exceeds V_ref, the output is logic "1"; otherwise, logic "0." These operations convert time-dependent analog resistance fluctuations into discrete binary outputs, which form the functional core of a p-bit.

2. Statistical Mapping: Logistic Transfer Function of Bias to Probability

The relation between input bias and probabilistic output is mathematically described by a logistic transfer function:

$$P_1 = \frac{1}{1 + \exp\left[-k(V_{\text{offset}} - V_{\circ})\right]}$$

where:

• $V_{\text{offset}}$: external bias voltage offset,
• $V_\circ$: the threshold at which the output crosses 50% probability,
• $k$: steepness factor of the sigmoid.

At low supply voltages, minimal noise yields a sharp logistic curve, while at higher voltages, increased stochasticity results in a broader transition range between logic states. This function is characteristic of stochastic binary neurons and dictates how the analog-to-digital conversion threshold responds to physical fluctuations.

3. Quantification of Stochasticity: Discrete Shannon Entropy

Stochastic device behavior is quantitatively assessed by constructing histograms from ensembles of pulsed I–V measurements, parsed into logarithmically spaced current bins. The discrete Shannon entropy (H₍disc₎) is calculated as:

$H_{\text{disc}} = -\sum_i P(i|V) \log_2 P(i|V)$

and the effective number of equiprobable states:

$N_{\text{eff}} = 2^{H_{\text{disc}}}$

Maximal entropy values indicate conditions under which the memristor explores a large number of current states, signaling optimal probabilistic operation. Peaks in $H_{\text{disc}}$ directly coincide with biasing regimes that maximize memristor voltage variability, linking device-level entropy to underlying material fluctuations.

4. Functional Role of Pendant Triphenylamine (TPA) Groups

TPA pendants are essential for dynamic stochasticity due to their molecular flexibility and redox responsiveness. These groups can reorient and rearrange under applied bias, modulating the local conductive landscape by opening or closing alternative transport pathways within the polymer matrix. Consequently, these frequent and reversible changes manifest as rapid alterations in resistance, which—amplified by the comparator—result in random switching between binary outputs.

The TPA units thus act both as transport modulators and as intrinsic noise sources, with their behavior directly influencing the device's ability to act as a chemically tunable entropy source.

5. Dielectric Relaxation Dynamics and Frontier Orbital Alignment

Dielectric spectroscopy reveals multiple relaxation modes within the polymer:

• A low-temperature process ($E_a \sim 0.44$ eV) related to localized motion, such as ether linkage rotations or phenyl flips in TPA.
• A higher-temperature process ($E_a \sim 1.04$ eV) linked to more cooperative segmental or backbone rearrangements.

Energy-resolved electrochemical impedance spectroscopy (ER–EIS) in conjunction with density functional theory (DFT) calculations demonstrates the alignment of DTP, TPA, and ITO electrode frontier orbitals within a polymer transport gap ($\sim 2.8$ eV). In the neutral state, DTP backbone orbitals dominate; post-oxidation, TPA-related orbitals activate, enabling bifurcated conduction (via DTP- or TPA-localized channels). When these dual conduction pathways approach degeneracy near specific bias conditions, stochastic switching and entropy output are maximized.

6. Implications for Chemically Tunable Entropy and Thermodynamic Computing

The device’s experimentally validated attributes—the bias-tunable logistic response, entropy quantification, dynamic TPA fluctuations, and bifurcation in transport networks—establish the DTP–TPA polymer membrane as a customizable entropy source. Modifying molecular structure or applied bias allows for the engineering of desired stochasticity. This property supports energy-efficient thermodynamic computing regimes where fluctuations are a resource, not a limitation.

Probabilistic bits based on these polymers are suitable for hardware that naturally executes probabilistic inference or optimization, typically energy-intensive tasks in deterministic frameworks. Their ability to leverage molecular-scale uncertainty offers scalability and customization potential in low-power, high-density computational platforms.

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In summary, the DTP-backbone polymer with pendant TPA groups enables stochastic memristive behavior via redox-induced molecular dynamics. This stochasticity underlies the device’s probabilistic bit output, following a logistic transfer function, and is quantified through Shannon entropy analysis. Dielectric and electronic structure analysis affirms the active role of TPA pendants and the existence of dual conductive channels, enabling chemically tunable entropy essential for thermodynamic computing architectures (Foulger et al., 22 Sep 2025).