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Spectral function analysis on the Holstein polaron problem : extraction of the self-energy and coupling strength, and their implications for angle resolved photoemission spectroscopy

C. N. Veenstra

Published Jan 1, 2011• 0 citations
Publisher

Abstract

This thesis first reviews and examines the Angle Resolved Photoemission Spectroscopy (ARPES) experiment. It is shown that the spectral density function, familiar from the Green's function method of studying correlated systems, can be directly measured. A model spectral function with a nontrivial self-energy is then used to test an improvement to a recently arrived method[15,17] to analyze ARPES data. This new method relies on self-consistency between the real and imaginary parts of the self-energy (as measured through the Kramers-Kronig transform) to overcome the requirement of knowing the bare electronic structure. Through this, the method extracts both the complex self-energy and the bare electronic structure from the spectral function. The method described here is an improvement on this idea as previously implemented in that a strict form for the bare band dispersion (previously considered linear or quadratic) is never assumed. Although the method here utilized a polynomial of arbitrary degree, it could be trivially expanded to use any other functional form so long as both the value and first derivative are known analytically as a function of the fitting parameters. Using Mona Berciu's first order momentum average (MA) approximation[2] as implemented by Glen Goodvin[8] the spectral function as well as momentum independent self-energy were calculated for the Holstein polaron for a wide range of parameters. It was found that self-consistent spectral function analysis was highly successful at extracting the self energy and bare electronic dispersion from the spectral function over a consistent subset of these parameters. For studies outside this range of parameters the more traditional ARPES analysis method of measuring renormalization is also examined.

DOI
10.14288/1.0084909

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This thesis first reviews and examines the Angle Resolved Photoemission Spectroscopy (ARPES) experiment. It is shown that the spectral density function, familiar from the Green's function method of studying correlated systems, can be directly measured. A model spectral function with a nontrivial self-energy is then used to test an improvement to a recently arrived method[15,17] to analyze ARPES data. This new method relies on self-consistency between the real and imaginary parts of the self-energy (as measured through the Kramers-Kronig transform) to overcome the requirement of knowing the bare electronic structure. Through this, the method extracts both the complex self-energy and the bare electronic structure from the spectral function. The method described here is an improvement on this idea as previously implemented in that a strict form for the bare band dispersion (previously considered linear or quadratic) is never assumed. Although the method here utilized a polynomial of arbitrary degree, it could be trivially expanded to use any other functional form so long as both the value and first derivative are known analytically as a function of the fitting parameters. Using Mona Berciu's first order momentum average (MA) approximation[2] as implemented by Glen Goodvin[8] the spectral function as well as momentum independent self-energy were calculated for the Holstein polaron for a wide range of parameters. It was found that self-consistent spectral function analysis was highly successful at extracting the self energy and bare electronic dispersion from the spectral function over a consistent subset of these parameters. For studies outside this range of parameters the more traditional ARPES analysis method of measuring renormalization is also examined.
Using an efficient variational exact diagonalization method, we computed the electron removal spectral function within the framework of the Holstein-Hubbard model containing two electrons with opposite spins coupled to dispersive quantum optical phonons. We focus on how the interplay between the electron-phonon coupling and on-site Coulomb repulsion shapes the spectral function, a quantity directly relevant to angle-resolved photoemission spectroscopy (ARPES). Tuning the strengths of the electron-phonon coupling and the Hubbard interaction allows us to examine the evolution of the spectral properties of the system as it crosses over from a bound bipolaron to separate polarons. We find that phonon dispersion plays an important role: It substantially redistributes the spectral weight across the low- and the high-energy features and controls the emergence and the structure of distinct multiphonon continua. In regimes of strong electron-phonon coupling and low Coulomb repulsion, the spectral function exhibits a broad distribution reflecting bound bipolaronic states. Increasing the Coulomb repulsion weakens bipolaron binding, shifts the primary quasiparticle peak closer to polaronic behavior and confines the spectral weight near the Brillouin-zone center. Our results provide a framework for interpreting ARPES measurements in systems where phonon–mediated attraction competes with direct Coulomb repulsion.
In this work we present a highly efficient and accurate analytical approximation for the Green's function of a polaron: the momentum average (MA) approximation. It is obtained by summing all of the self-energy diagrams, but with each diagram averaged over the momenta of its free propagators. The result becomes exact for both zero bandwidth and for zero electron-phonon coupling, and is accurate everywhere in the parameter space. The approximation is first used to investigate the Holstein model. A detailed analysis of its accuracy is provided through diagrammatics and spectral weight sum rules. It is shown that the resulting Green's function satisfies exactly the first six spectral weight sum rules, and all higher order sum rules are satisfied with great accuracy. Comparison with numerical data also confirms this accuracy. We then show how to improve the MA approximation by systematically improving the accuracy of the self-energy diagrams in such a way that they can still all be summed efficiently. This allows us to fix some of the problems of the zeroth-order MA approximation. The quantitative agreement with numerical data is also improved. Next, we generalize the MA approximation to study the properties of models with momentum-dependent electron-phonon coupling, and then show that further improvements can be obtained based on variational considerations, using the 1D breathing-mode Hamiltonian as a specific example. For example, by using this variational MA, we obtain ground state energies within at most 0.3% error of the numerical data. Finally, we study the effects of a nearby surface on the spectral weight of a Holstein polaron. The broken translational symmetry is accounted for without any additional approximations, and the resulting inhomogeneous MA approximation continues to be accurate for all coupling strengths. We show that the surface changes properties significantly, with bulk values being recovered only very far away from it. We find that the electron-phonon coupling gives rise to an additional surface potential which is responsible for binding surface states even when they are not normally expected. These results demonstrate that interpretation in terms of bulk properties of spectroscopic data sensitive only to a few surface layers is not straightforward.
An analytical approach to the one-dimensional spinless Holstein model is proposed, which is valid at finite charge-carrier concentrations. Spectral functions of charge carriers are computed on the basis of self-energy calculations. A generalization of the Lang–Firsov canonical transformation method is shown to provide an interpolation scheme between the extreme weak-coupling and strong-coupling cases. The transformation depends on a variationally determined parameter that characterizes the charge distribution across the polaron volume. The relation between the spectral functions of the polaron and electron, the latter corresponding to the photoemission spectrum, is derived. Particular attention is paid to the distinction between the coherent and incoherent parts of the spectra, and their evolution as a function of band filling and model parameters. Results are discussed and compared with recent numerical calculations for the many-polaron problem.
We calculate the spectral properties of the one-dimensional Holstein and breathing polarons using the self-consistent Born approximation. The Holstein model electron-phonon coupling is momentum independent while the breathing coupling increases monotonically with the phonon momentum. We find that for a linear or tight binding electron dispersion: (i) for the same value of the dimensionless coupling the quasiparticle renormalization at small momentum in the breathing polaron is much smaller, (ii) the quasiparticle renormalization at small momentum in the breathing polaron increases with phonon frequency unlike in the Holstein model where it decreases, and (iii) in the Holstein model the quasiparticle dispersion displays a kink and a small gap at an excitation energy equal to the phonon frequency ${\ensuremath{\omega}}_{0}$ while in the breathing model it displays two gaps, one at excitation energy ${\ensuremath{\omega}}_{0}$ and another one at $2{\ensuremath{\omega}}_{0}$. These differences have two reasons: first, the momentum of the relevant scattered phonons increases with increasing polaron momentum and second, the breathing bare coupling is an increasing function of the phonon momentum. These result in an effective electron-phonon coupling for the breathing model which is an increasing function of the total polaron momentum, such that the small momentum polaron is in the weak coupling regime while the large momentum one is in the strong coupling regime. However, the first reason does not hold if the free electron dispersion has low energy states separated by large momentum, as in a higher dimensional system, for example, in this situation the difference between the two models becomes less significant.
We explore the ground state and thermodynamic properties of the polaron coupled to quantum dispersive optical phonons in one spatial dimension. Calculations are performed using the finite--temperature Lanczos method augmented by a highly efficient construction of the variational Hilbert space. We focus on the electron removal spectral function as relevant for the angle-resolved photoemission experiments. We show that photoemission spectroscopy can be used to measure the phonon dispersion relation in a dilute system of polarons. The spectral weight of observed phonon bands is proportional to the phonon contribution to the wavefunction at finite phonon momentum. In addition, we demonstrate that when removing an electron from a polaron ground state, the polaron band does not appear in the spectral function. The latter becomes observable only at elevated temperatures.
In many cases the standard perturbation approach appears to be too simple to describe precisely the angle-resolved photoemission spectrum of a strongly correlated electron system. In particular, to describe the momentum asymmetry observed in the photoemission spectra of high-Tc{T}_{c}Tc​ cuprates, a phenomenological approach based on an extremely correlated Fermi-liquid model has been recently introduced. Here we analyze the general structure of the Green's function of quasiparticles in strongly correlated electron systems and stress that it is defined not only by the self-energy of Hubbard quasiparticles but also by a strength operator. The latter leads to an additional odd momentum contribution to the spectral function and alone can explain the observed asymmetry. So, the asymmetry of the angle-resolved photoemission spectra can be a measure of the strength of electron correlations in materials.