We investigate the scattering of an electron by phonons in a small structure between two one-dimensional tight-binding leads. This model mimics the quantum electron transport through atomic wires or molecular junctions coupled to metallic leads. The electron-phonon-coupled structure is represented by the Holstein model. We observe permanent energy transfer from the electron to the phonon system (dissipation), transient self-trapping of the electron in the electron-phonon-coupled structure (due to polaron formation and multiple reflections at the structure edges), and transmission resonances that depend strongly on the strength of the electron-phonon coupling and the adiabaticity ratio. A recently developed TEBD algorithm, optimized for bosonic degrees of freedom, is used to simulate the quantum dynamics of a wave packet launched against the electron-phonon-coupled structure. Exact results are calculated for a single electron-phonon site using scattering theory and analytical approximations are obtained for limiting cases.
We investigate the scattering of an electron by phonons in a small structure between two one-dimensional tight-binding leads. This model mimics the quantum electron transport through atomic wires or molecular junctions coupled to metallic leads. The electron-phonon-coupled structure is represented by the Holstein model. We observe permanent energy transfer from the electron to the phonon system (dissipation), transient self-trapping of the electron in the electron-phonon-coupled structure (due to polaron formation and multiple reflections at the structure edges), and transmission resonances that depend strongly on the strength of the electron-phonon coupling and the adiabaticity ratio. A recently developed TEBD algorithm, optimized for bosonic degrees of freedom, is used to simulate the quantum dynamics of a wave packet launched against the electron-phonon-coupled structure. Exact results are calculated for a single electron-phonon site using scattering theory and analytical approximations are obtained for limiting cases.
In this dissertation a newly developed numerical method is presented, which is optimized for the time evolution of one-dimensional lattice systems with large local Hilbert spaces. This method extends the time-evolving block-decimation (TEBD) to include a local basis optimization (LBO), which has already been successfully combined with ground state methods. The algorithm is based on matrix product states (MPS), which can represent the quantum state of a one-dimensional chain in most cases with a number of parameters, that is not exponentially increasing with the chain length. The LBO causes a reduction of the simulation times that is linear in the bond dimension of the MPS. To demonstrate the advantages of this method, we apply the TEBD-LBO to electron-phonon (e-p) systems. In this thesis these are described by the Holstein model, which goes beyond semi-classical approximations and covers the full quantum statistics of the phonons. The understanding of the nonequilibrium dynamics of charge carriers coupled to lattice vibrations is of great importance for research areas like transport through quasi one-dimensional conductors, photo-generated phase transitions and time-resolved spectroscopy. First, the energy transfer from a highly excited electron to the phononic degrees of freedom on a small chain is studied. In the various parameter regimes different types of relaxation occur. In any case, after a certain time the system reaches a state, where on average no energy is exchanged between the electron and phonons. This can either mean a constant kinetic energy or oscillations with a constant amplitude and frequency. Next, long perfectly conducting leads without coupling to phonons were attached on both sides of the small chain. In the left lead an electron with density distribution in the shape of a Gaussian wave packet is injected with momentum towards the e-p coupled structure in the middle. This structure acts as an impurity in an otherwise perfectly conducting chain. The investigation shows resonance effects in the transmission and reflection at this impurity. Further, the electron can transfer a part of its energy permanently to the phonons, which results in a reduction of the velocity. Finally, two mechanisms are presented that lead to self-trapping of the electron on the e-p coupled structure.
We study the quantum dynamics of small polaron formation and polaron transport through finite quantum structures in the framework of the one-dimensional Holstein model with site-dependent potentials and interactions. Combining Lanczos diagonalization with Chebyshev moment expansion of the time evolution operator, we determine how different initial states, representing stationary ground states or injected wave packets, after an electron-phonon interaction quench, develop in real space and time. Thereby, the full quantum nature and dynamics of electrons and phonons is preserved. We find that the decay out of the initial state sensitively depends on the energy and momentum of the incoming particle, the electron-phonon coupling strength, and the phonon frequency, whereupon bound polaron-phonon excited states may emerge in the strong-coupling regime. The tunneling of a Holstein polaron through a quantum wall or dot is generally accompanied by strong phonon number fluctuations due to phonon emission and reabsorption processes.
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.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
Employing a largely unbiased variational exact diagonalization technique, we analyze the consequences of longer-ranged electron hopping and electron-phonon interaction on polaron formation in one dimension. Having at our disposal the accurate ground-state energy and wave function, we calculate and discuss various physical quantities, such as the renormalized band structure, effective mass, wave-function renormalization factor, phonon dressing, and Drude weight, characterizing the properties of the polaronic quasiparticle. We demonstrate that the electron-phonon coupling affects the relative strength of the nearest neighbor (NN) and next nearest neighbor (NNN) hopping processes in a dynamic way. Most notably we observe that the minimum of the polaron band, occurring at a finite momentum for large negative ratio between NN and NNN transfer, jumps to zero momentum as the electron-phonon coupling exceeds a critical one, thereby causing a rather sharp polaron transition in the one-dimensional extended Holstein model. The signatures of this transition are seen in the effective mass and polaron mobility, and therefore should be easily detectable by transport measurements.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.