{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/244667"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/244667","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Modeling and Numerics for Two Partial Differential Equation Systems Arising From Nanoscale Physics","abstract":"The mathematical field of analysis of partial differential equations has undergone rapid changes in recent years. Consistent improvement of mathematical computation allows more and more questions to be addressed in the form of numerical simulations. At the same time, novel materials arising from advances in physics and material sciences are creating new problems which must be addressed. This thesis focuses on two such materials, organic semiconductors and graphene. In Chapter 2 we derive a generalized model for organic photovoltaic devices - solar cells based on organic semiconductors. After selecting an appropriate self-consistent model, we derive asymptotic estimates for the operation of the device in several regimes. We use these estimates to partially justify several assumptions used in the formulation of simplified models which have already been discussed in the literature. Furthermore, we simulate such devices using a 2D hybrid discontinuous Galerkin finite element scheme and compare the numerical results to our asymptotics. We then discuss the potential applicability of the simulations to real-devices by identifying which regimes correctly model physical device behavior and discussing the limitations of the model choice. Finally, several perspectives are given on proving existence and uniqueness of the model. In Chapter 3 we derive a convergent second-order finite difference numerical scheme for simulation of the 2D Dirac equation. We demonstrate the convergence of the numerical scheme with several examples for which explicit solutions are known and consider how errors appear in the simulations. We furthermore extend the Dirac system with Poisson's equation (in 3D) for the electrical potential and investigate the application to electrons in graphene. In particular, we show that the numerical scheme captures several interesting physical effects which have been predicted to appear in graphene.","abstract_html":"The mathematical field of analysis of partial differential equations has undergone rapid changes in recent years. Consistent improvement of mathematical computation allows more and more questions to be addressed in the form of numerical simulations. At the same time, novel materials arising from advances in physics and material sciences are creating new problems which must be addressed. This thesis focuses on two such materials, organic semiconductors and graphene. In Chapter 2 we derive a generalized model for organic photovoltaic devices - solar cells based on organic semiconductors. After selecting an appropriate self-consistent model, we derive asymptotic estimates for the operation of the device in several regimes. We use these estimates to partially justify several assumptions used in the formulation of simplified models which have already been discussed in the literature. Furthermore, we simulate such devices using a 2D hybrid discontinuous Galerkin finite element scheme and compare the numerical results to our asymptotics. We then discuss the potential applicability of the simulations to real-devices by identifying which regimes correctly model physical device behavior and discussing the limitations of the model choice. Finally, several perspectives are given on proving existence and uniqueness of the model. In Chapter 3 we derive a convergent second-order finite difference numerical scheme for simulation of the 2D Dirac equation. We demonstrate the convergence of the numerical scheme with several examples for which explicit solutions are known and consider how errors appear in the simulations. We furthermore extend the Dirac system with Poisson's equation (in 3D) for the electrical potential and investigate the application to electrons in graphene. 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Consistent improvement of mathematical computation allows more and more questions to be addressed in the form of numerical simulations. At the same time, novel materials arising from advances in physics and material sciences are creating new problems which must be addressed. This thesis focuses on two such materials, organic semiconductors and graphene. In Chapter 2 we derive a generalized model for organic photovoltaic devices - solar cells based on organic semiconductors. After selecting an appropriate self-consistent model, we derive asymptotic estimates for the operation of the device in several regimes. We use these estimates to partially justify several assumptions used in the formulation of simplified models which have already been discussed in the literature. Furthermore, we simulate such devices using a 2D hybrid discontinuous Galerkin finite element scheme and compare the numerical results to our asymptotics. We then discuss the potential applicability of the simulations to real-devices by identifying which regimes correctly model physical device behavior and discussing the limitations of the model choice. Finally, several perspectives are given on proving existence and uniqueness of the model. In Chapter 3 we derive a convergent second-order finite difference numerical scheme for simulation of the 2D Dirac equation. We demonstrate the convergence of the numerical scheme with several examples for which explicit solutions are known and consider how errors appear in the simulations. We furthermore extend the Dirac system with Poisson's equation (in 3D) for the electrical potential and investigate the application to electrons in graphene. 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Finally, several perspectives are given on proving existence and uniqueness of the model. In Chapter 3 we derive a convergent second-order finite difference numerical scheme for simulation of the 2D Dirac equation. We demonstrate the convergence of the numerical scheme with several examples for which explicit solutions are known and consider how errors appear in the simulations. We furthermore extend the Dirac system with Poisson's equation (in 3D) for the electrical potential and investigate the application to electrons in graphene. 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