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The Journal of Chemical Physics

Carrier mobility in Holstein–Peierls models of organic materials: A tensor-train HEOM approach

H. Takahashi, Raffaele Borrelli

Published Nov 17, 2025• 2 citations
Publisher

Abstract

We present a comprehensive theoretical study of charge transport in organic materials on the basis of the one-dimensional dispersive Holstein-Peierls model. The hierarchical equations of motion in tensor-train format (HEOM-TT) method is employed to accurately describe both local Holstein-type and nonlocal Peierls-type electron-phonon couplings in the presence of low-frequency dispersive phonons. Linear response formulation of carrier mobility is obtained by employing the generating functional method. We demonstrate the capability of the method by calculating current correlation functions, diffusivities, and mobilities for various transport regimes. Our results provide new insights into the mechanisms underlying charge transport in organic semiconductors and establish HEOM-TT as a powerful tool for studying complex vibronic quantum dynamics and transport properties in condensed matter systems.

DOI
10.1063/5.0300292

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We present a comprehensive theoretical study of charge transport in organic materials on the basis of the one-dimensional dispersive Holstein-Peierls model. The hierarchical equations of motion in tensor-train format (HEOM-TT) method is employed to accurately describe both local Holstein-type and nonlocal Peierls-type electron-phonon couplings in the presence of low-frequency dispersive phonons. Linear response formulation of carrier mobility is obtained by employing the generating functional method. We demonstrate the capability of the method by calculating current correlation functions, diffusivities, and mobilities for various transport regimes. Our results provide new insights into the mechanisms underlying charge transport in organic semiconductors and establish HEOM-TT as a powerful tool for studying complex vibronic quantum dynamics and transport properties in condensed matter systems.
Hierarchical Equations of Motion (HEOM) in the Tensor-Train (TT) representation is applied to study the charge-transfer dynamics in organic semiconductors (OSCs). The theoretical formulation as well as the basic computational aspects of HEOM-TT are discussed in detail. Charge transfer in OSCs is modeled using dissipative polaronic models that incorporate the effects of both high- and low-frequency molecular vibrations, and it is simulated in a fully quantum and nonperturbative manner, which has not been studied intensively. The capability of treating complex electron-vibrational systems is examined by analyzing and comparing the numerical behavior of the time-dependent variational approach and the time-Alternating Minimal Energy methods and by calculating the current autocorrelation function and diffusivity across various models. Our results indicate that the HEOM-TT framework offers a robust tool for the detailed analysis of complex polaronic systems, suggesting its potential for broader applications.
Studying charge transport in models with nonlocal carrier-phonon interaction is difficult because it requires finite-temperature real-time correlation functions of mixed carrier-phonon operators. Focusing on models with discrete undamped phonon modes, we show that such correlation functions can be retrieved from the hierarchical equations of motion (HEOM), although phonons have been integrated out. Our procedure relies on the explicit expression of HEOM auxiliaries in terms of phonon creation and annihilation operators. It reveals that the auxiliaries describe multiphonon-assisted carrier transitions induced by genuine many-phonon correlations, from which lower-order correlations are subtracted according to the finite-temperature Wick's theorem. Applying the procedure to our recently developed momentum-space HEOM method featuring a specific hierarchy closing, we compute the numerically exact dynamical mobility of a carrier within the one-dimensional Peierls model. The carrier mobility at moderate temperatures decreases with increasing interaction, whereas high temperatures see the opposite trend, reflecting the prevalence of the phonon-assisted current over the purely electronic band current. The pronounced finite-size effects and HEOM instabilities delimit the range of applicability of our approach to moderate interactions, moderate to high temperatures, and not too fast phonons. Importantly, this range comprises the values relevant for charge transport in crystalline organic semiconductors, and we present and discuss the corresponding numerically exact results in a companion paper (arXiv:2501.05055).
Evaluation of the charge transport property of organic semiconductors requires exact quantum dynamics simulation of large systems. We present a numerically nearly exact approach to investigate carrier transport dynamics in organic semiconductors by extending the non-Markovian stochastic Schrödinger equation with complex frequency modes to a forward-backward scheme and by solving it using the matrix product state (MPS) approach. By utilizing the forward-backward formalism for noise generation, the bath correlation function can be effectively treated as a temperature-independent imaginary part, enabling a more accurate decomposition with fewer complex frequency modes. Using this approach, we study the carrier transport and mobility in the one-dimensional Peierls model, where the nonlocal electron-phonon interaction is taken into account. The reliability of this approach was validated by comparing carrier diffusion motion with those obtained from the hierarchical equations of motion method across various parameter regimes of the phonon bath. The efficiency was demonstrated by the modest virtual bond dimensions of MPS and the low scaling of the computational time with the system size.
The role of nonlocal electron-phonon couplings (EPC) in organic semiconductors has the been center of interest recently. Several irreconcilable scenarios coexist for the description of the nonlocal EPC, such as phonon-assisted (PA) transport, transient localization (TL), and band-like (BL) transport. In this Letter, we present a nearly exact numerical study for the carrier mobility of the one-dimensional Holstein-Peierls model using finite temperature time-dependent matrix product states approach. We are able to locate the PA, TL and BL regimes as a function of the electronic coupling (VVV) and the nonlocal EPC (ΔV\Delta VΔV). The distinct transport behaviors with respect to EPC are analyzed by carrier mobility, mean free path, optical conductivity and one-particle spectral function. We have also identified an intermediate where none of the established pictures applies, and the generally perceived hopping regime is found to be at a very limited end in the proposed general paradigm.
Numerically "exact" methods addressing the dynamics of coupled electron-phonon systems have been intensively developed. Nevertheless, the corresponding results for the electron mobility μdc are scarce, even for the one-dimensional (1d) Holstein model. Building on our recent progress on single-particle properties, here we develop the momentum-space hierarchical equations of motion (HEOM) method to evaluate real-time two-particle correlation functions of the 1d Holstein model at a finite temperature. We compute numerically "exact" dynamics of the current-current correlation function up to real times sufficiently long to capture the electron's diffusive motion and provide reliable results for μdc in a wide range of model parameters. In contrast to the smooth ballistic-to-diffusive crossover in the weak-coupling regime, we observe a temporally limited slow-down of the electron on intermediate time scales already in the intermediate-coupling regime, which translates to a finite-frequency peak in the optical response. Our momentum-space formulation lowers the numerical effort with respect to existing HEOM-method implementations, while we remove the numerical instabilities inherent to the undamped-mode HEOM by devising an appropriate hierarchy closing scheme. Still, our HEOM remains unstable at too low temperatures, for too strong electron-phonon coupling, and for too fast phonons.
We present a Hierarchical Equations of Motion (HEOM) approach in the Matrix Product State (MPS) formalism to simulate carrier transport in molecular aggregates described by an electron-phonon Hamiltonian with bosonic dissipation. Transport properties are evaluated through time-dependent population analysis and mobility calculations. The method's validity is rigorously established through benchmarking against conventional HEOM. Comparative analysis with Thermo-Field Dynamics combined with MPS (TFD + MPS) reveals fundamental similarities and differences in their effective Hamiltonians and demonstrates the superior accuracy and computational efficiency of our HEOM + MPS framework. For single-electron systems, we introduce state-vector space configurations that enhance performance beyond traditional Fock space approaches. Results confirm that our method provides a robust, nearly exact, and efficient numerical quantum dynamic approach for carrier transport in dissipative bosonic environments.