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General Carrier Transport Picture for Organic Semiconductors: A Nearly Exact TD-MPS Study on the Nonlocal Electron-Phonon Couplings

Weitang Li, Jiajun Ren, Zhigang Shuai

Published Dec 17, 2020• 0 citations
OpenAlex

Abstract

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 ($V$) and the nonlocal EPC ($\Delta V$). 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.

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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.
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.
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).
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.
Organic/polymeric materials are of emerging importance for thermoelectric conversion. The soft nature of these materials implies strong electron-phonon coupling, often leading to carrier localization. This poses great challenges for the conventional Boltzmann transport description based on relaxation time approximation and band structure calculations. In this work, combining Kubo formula with finite-temperature time-dependent density matrix renormalization group (FT-TD-DMRG) in the grand canonical ensemble, we developed a nearly exact algorithm to calculate the thermoelectric power factor PF=α^2 σ, where α is the Seebeck coefficient and σ is electrical conductivity, and apply the algorithm to organic materials modeled by Holstein Hamiltonian with electron-phonon coupling. Our algorithm can provide a unified description covering the weak coupling limit described by bandlike Boltzmann transport to the strong coupling hopping limit.
The Holstein model is a benchmark model of systems with electron-phonon interaction. However, its electrical transport properties are not yet fully understood. In this work, we performed numerically exact calculations of the imaginary-time current-current correlation function of the Holstein Hamiltonian for a broad range of model parameters. These calculations were performed using a path-integral-based quantum Monte Carlo method. We compared these results with the results obtained under the assumption of conventional band transport, small polaron hopping, and polaron band transport. From this comparison, we identified the regions in parameter space where each of these transport regimes is valid. In some cases when an imaginary-time comparison of current-current correlation functions could not give conclusive results, we complemented it with real-time comparisons or a comparison with the literature data on numerically exact dc mobilities. Overall, we found that the parameter space is almost completely covered by the three mentioned transport regimes.
In this thesis, I investigate the transport and dynamical properties of electron-phonon coupled systems using numerical methods. I focus on the one-dimensional Holstein model, which incorporates both polaron and charge density wave (CDW) physics, and use matrix-product-state-based algorithms. In the first part, I present work where we combine purification with local basis optimization to study the finite-temperature properties of the model. This gives access to polaron spectral functions, polaron and bipolaron optical conductivity, and the energy-transport coefficient at finite filling. The second part focuses on the Holstein model coupled to two tight-binding leads. There, we investigate how current is transported through the structure and how CDW states break down. In the last part, we analyze the spread of an electron in a lattice and the decay of CDWs. We aim to understand if a full quantum mechanical treatment of the phonons is necessary or if trajectory-based algorithms are applicable.