Green Functions Approach to Graphene Nanostructures

Due to their fascinating optoelectronic properties, finite graphene nanostructures are expected to find use in a number of technological applications, ranging from field effect transistors and solar cells to quantum computers and even biomedical treatments. However, despite their small size, these systems can contain several hundred to thousands of electrons, unfortunately, making their theoretical modeling a major challenge. The main drawback of the established theoretical formalism, the nonequilibrium Green functions approach, is its high numerical effort, which scales cubically with the number of required time steps. Therefore, performing time-dependent simulations of the nonequilibrium dynamics of excited finite graphene nanostructures is not feasible, which makes further improvements urgently necessary.

Such a feat was achieved by the author and coworkers during the work on this thesis by developing the G1–G2 scheme. It constitutes the first formulation of the nonequilibrium Green functions approach with linearly-scaling numerical effort with respect to the propagation time. Because of the great importance of this discovery, this thesis addresses two main topics. The first is the aforementioned theoretical framework in general and its application to finite graphene nanostructures. The focus is on special topologically protected states that can occur in these systems. Moreover, the G1–G2 scheme is used to study the ultrafast response of various graphene nanostructures to an external laser pulse. The second aspect includes a detailed discussion of the G1–G2 scheme. Many questions that have arisen in previous publications on the subject are answered. The central insight is that the derivation of the G1–G2 scheme holds many more advantages, besides the obvious numerical ones. These findings can contribute decisively to the further development of approximation methods in many-particle theory.


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