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Journal of Chemical Theory and Computation

Computational Method for Evaluating the Thermoelectric Power Factor for Organic Materials Modeled by the Holstein Model: A Time-Dependent Density Matrix Renormalization Group Formalism

Yufei Ge, Weitang Li, Jiajun Ren, Zhigang Shuai

Published Sep 29, 2022• 15 citations
Publisher

Abstract

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.

DOI
10.1021/acs.jctc.2c00651

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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.
Heavy doping in organic materials induces complex many-body interactions characterized by the coexistence of strong electron correlation and electron-phonon coupling, arising from their soft structures and low dielectric constants. Simulating charge transport and thermopower in this regime poses a significant computational challenge. In this work, we propose a finite-temperature time-dependent tree tensor network state (TTNS) method to calculate the conductivity and Seebeck coefficient of organic materials modeled by the Holstein-Hubbard Hamiltonian. Our TTNS ansatz features a tailored “bus-branch” structure, where the “bus” captures the electronic entanglement, and the “branches” encode the local electron-phonon correlations. Benchmarks against exact diagonalization (ED) on small clusters demonstrate that this network structure achieves numerically exact results with smaller bond dimensions than the Matrix Product States (MPS). Furthermore, the “branch” topology efficiently isolates the local electron-phonon entanglement, thereby avoiding the globally large bond dimensions required by MPS. These algorithmic advantages enable us to constrain the relative error of transport properties to within 10% across a wide range of parameter regimes. To validate the capability of this framework, we study the complex interplay between Hubbard repulsion and phonon-mediated attraction, successfully illustrating the roles of spin degrees of freedom and effective interactions in thermodynamics and charge transport, and reproducing the correlation-induced anomalous sign change of the Seebeck coefficient. Our work establishes the “bus-branch” TTNS as a powerful and efficient tool for exploring quantum dynamics in strongly correlated electron-phonon systems.
Quantum transport in molecular junctions has attracted great attention. The charge motion in a molecular junction can cause geometric deformation, leading to strong electron phonon coupling, which was often overlooked. We have formulated a nearly exact method to assess the time-dependent current and occupation number in the molecular junction modeled by the electron-phonon coupled bridge state using the time-dependent density matrix renormalization group (TD-DMRG) method. The oscillation period and amplitude of the current are found to be dependent on the electron phonon coupling strength and energy level alignment with the electrodes. In an attempt to better understand these phenomena, we have devised a new approximation that explains the bistability phenomenon and the behavior of steady currents in the strong electron-phonon coupling regime. Comparisons have been made with the multilayer-multiconfiguration time-dependent Hartree (ML-MCTDH) method and the analytical result in the purely electronic limit. Furthermore, we explore the entropy of different orderings, extending to the electron phonon model problems. Regarding finite temperature, the thermal Bogoliubov transformation of both fermions and bosons is used and compared with imaginary time evolution results.
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.
A comprehensive picture of polaron and bipolaron physics is essential to understand the optical absorption spectrum in many materials with electron-phonon interactions. In particular, the finite-temperature properties are of interest since they play an important role in many experiments. Here, we combine the parallel two-site time-dependent variational principle algorithm (p2TDVP) with local basis optimization (LBO) and purification to calculate time-dependent current-current correlation functions. From this information, we extract the optical conductivity for the Holstein polaron and bipolaron with dispersive phonons at finite temperatures. For the polaron in the weak and intermediate electron-phonon coupling regimes, we analyze the influence of phonon dispersion relations on the spectra. For strong electron-phonon coupling, the known result of an asymmetric Gaussian is reproduced for a flat phonon band. For a finite phonon bandwidth, the center of the Gaussian is either shifted to larger or smaller frequencies, depending on the sign of the phonon hopping. We illustrate that this can be well understood by considering the Born-Oppenheimer surfaces. Similar behavior is seen for the bipolaron for strong coupling. For the bipolaron with weak and intermediate coupling strengths and a flat phonon band, we obtain two very different spectra. The latter also has a temperature-dependent resonance at a frequency below the phonon frequency.
The motion of polarons, which serve as charge carriers in conjugated polymers, is of fundamental importance for understanding transport properties of organic optoelectronic devices. We investigate the dynamics of a charged polaron in the presence of both electron-phonon and electron-electron interactions under the influence of an external electric field, which is modeled by the one-dimensional tight-binding Su-Schrieffer-Heeger (SSH) model supplemented with a Hubbard on-site repulsion term. For this many-body dynamical evolution problem, we develop an adaptive time-dependent density matrix renormalization group (ttt-DMRG) method in combination with a Newtonian equation of motion for atomic displacements. Our results show that the velocity of the polaron is suppressed by the on-site Coulomb interaction UUU. The polaron moves with a supersonic velocity, about four times the sound velocity at the small UUU limit, and approaches the sound velocity at the large UUU limit. Furthermore, the dependence of the polaron velocity and the polaron effective mass on the lattice structures are discussed.
Recent advances in experimental techniques have enabled precise characterization of fundamental nonadiabatic and even electronic dynamics in molecules, on ultrafast time scales reaching femtosecond to attosecond resolution. However, accurate theoretical simulation of these ultrafast chemical dynamics processes in large systems remains challenging, largely due to the overwhelming number of degrees of freedom (DoFs) and the pronounced many-body correlations. In recent years, by leveraging efficient decomposition schemes for high-dimensional wave function and operator tensors, the time-dependent density matrix renormalization group (TD-DMRG) has emerged as a powerful and accurate quantum dynamics method for simulating nonadiabatic and electron dynamics in large chemical systems, in conjunction with realistic electron/exciton-vibration/phonon models or ab initio quantum chemistry many-electron Hamiltonians. This review outlines the fundamentals of TD-DMRG for chemical dynamics, covering matrix product state/operator (MPS/MPO) frameworks and algorithms from ground-state calculations to time evolution. We discuss thermal/environmental effects and compare TD-DMRG with other tensor network methods such as multiconfiguration time-dependent Hartree (MCTDH) and multilayer MCTDH (ML-MCTDH). Demonstrated applications include simulations of pyrazine absorption, singlet fission in rubrene crystal, and charge migration in chloroacetylene cation. These show TD-DMRG’s capability for modeling complex ultrafast processes from femtoseconds to attoseconds with controlled accuracy.