Electron-phonon interactions (EPIs) are ubiquitous in condensed matter physics and materials science. They are crucial for understanding numerous phenomena, including conventional superconductivity, charge-transport and, most pertinent for this thesis, polaron formation. A polaron is a charge carrier (electron or hole) dressed with a “cloud” of phonons. The polaronic quasiparticle may have vastly different ground- and excited-state properties from that of the bare, constituent charge carrier.
While polarons are well-studied and largely understood in canonical model Hamiltonians, recent advances have made it possible to study more complex, fully ab initio systems. Here, the numerically exact methods which are available for some model systems become much more challenging to apply, so accurate approximate methods are a necessity. In this dissertation, we present several advancements in approximate but accurate methods for different polaronic problems and polaron observables.
With respect to polaron dynamics, we focus on low-scaling methods to produce wave vector-dependent single-particle spectral function. We present a thorough study of the accuracy of the second- and fourth-order cumulant expansions (CE) of the electronic Green’s function by comparing them against numerically-exact reference data for the one-dimensional Holstein model. We find that the second-order CE is accurate at zero electronic-momentum across a wide range of temperatures, while for non-zero electronic momenta, the CE is only accurate at high-temperatures. The fourth-order cumulant expansion improves on the dynamics at short times and can improve the spectra; however, it can also introduce non-physical divergences and negative spectral weight. The second-order cumulant expansion is thus a useful tool for determining spectral functions in some instances. However, increasing the order of the CE introduces pathologies that may persist at arbitrarily high-order.
As an alternate approach to improving the CE, we introduce a new self-consistent cumulant expansion (SC-CE) which remedies many of the deficits of the CE. We compare the results for this new approximation against those from the second-order cumulant expansion as well as reference data for the one-dimensional Holstein model. Unlike the CE, the SC-CE can produce accurate spectra across the entire Brillouin-zone, and captures non-perturbative features excellently. The trade-off for this increased accuracy is the introduction of some degree of negative spectral-weight and the potential for rapid divergences in time in some instances. We find that these problems can be minimized, but not completely eliminated in the thermodynamic limit and in more realistic cases where phonon dispersion exists. We also demonstrate how the SC-CE fits into the greater scheme of Green’s function methods which approximate the self-energy non-diagrammatically as has recently been proposed by Pandey and Littlewood, and we note the potential applications of the SC-CE both in ab initio polaron problems and in general many-body problems.
We finally consider a new method to determine the ground-state structure of the polaron in ab initio materials, a topic which has only recently appeared in the literature. We present a new all-coupling variational method based on the Nagy-Markoš variational ansatz for the Fröhlich model. The ansatz is a projected unitary transform which naturally interpolates between the weak-coupling (Lee-Low-Pines) ansatz and the strong-coupling adiabatic ansatz by modulating the momentum conservation of the electron-phonon scattering processes. We demonstrate our ab initio Nagy-Markoš ansatz on the Holstein model and the Fröhlich model, and show that it always improves upon the better of the weak or strong coupling result. We consider the ab initio case of lithium fluoride (LiF), and find that the ansatz provides accurate polaron binding energies for both the hole-polaron and the electron-polaron which are classical cases of small and large polarons, respectively. We note how our flexible variational ansatz is an ideal starting point for perturbative energy corrections and cumulant Green’s function methods.
Future developments and applications of the efficient methodologies presented in this dissertation may enable quantitative calculations of polarons in large-intermediately coupled ab initio systems, such as the lead-halide perovskites and other systems where it has hitherto been difficult to fully understand the effects of the electron-phonon interactions.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/fbhn-mc12 |
| Date | January 2023 |
| Creators | Robinson, Paul Joseph Pagano |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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