## The Hubbard dimmer: a case study crossing the bridge between DFT and strongly correlated systems

##### Other title:

El dimero de Hubbard: un caso de estudio que enlaza DFT y los sistemas fuertemente correlacionados

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Física de la materia condensada

Física del estado sólido

Química física

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The use of Density Functional Theory (DFT) has become a standard procedure in the recent years for the simulation of nanostructures. However it is well known that DFT suffers from severe limitations when covering the description of strongly correlated systems for which those approximations are not good. Strongly correlated systems can be described by means of tight-binding models. Such models are much simpler than the more realistic real-space models as they depend on few parameters and in some cases can be solved algebraically. The first three chapters of this thesis are devoted to the use of the DFT machinery to the description of such lattice models in order to assess the true limitations when DFT is applied to strongly correlated systems. The fourth and fifth chapters are oriented to the direct application of DFT to the study of two different and interesting problems: the search of molecular magnets with large magnetic anisotropy energy (MAE) and the study of molecular junctions using graphene leads. In the first chapter we derive the exact functional and the exchange-correlation energies that let us building the Kohn-Sham (KS) potentials of the single and double site Anderson-Hubbard models in order to find out if the exact DFT gives the correct description of such systems. We find that this description is in general incomplete. The second chapter presents an analysis of the double-site asymmetric Hubbard model with 2 electrons. The properties of its ground and excited states are discussed. We then use the Mean-Field (MF) and the Bethe-ansatz LDA (BALDA) approximations to assess their performance as a function of the external potential-interactions ratio. We then apply the concepts of DFT to this model, establishing its KS description and the corresponding adiabatic connection. A parametrized functional is also devised using a method that allows improving the accuracy of any approximated functional whenever the relation between the kinetic and the potential correlation terms is known. The third chapter covers the DFT estimation of the fundamental gap in strongly correlated systems. We study the asymmetric Hubbard dimer. We find the exact quasi-particle excitation spectrum, including the ionization potential and electron affinity, which let us derive the real gap. We set a method to calculate the KS potentials for non-integer particle numbers as a necessary step to find the KS Green's function. We prove that the Koopmans' and the Janak's theorems are obeyed. We find that the KS spectrum is in general incorrect, except for the HOMO by definition, when strong correlations are dominant. The KS gap is always smaller than the true gap with interactions and we find the role of the derivative discontinuity on this. In the fourth chapter we focus on the practical use of DFT and we test several organometallic molecules based with the objective of finding stable molecular magnets at room temperatures. The current technology uses thin-sheets for manufacturing magnetic bits for data storage. Different alternatives such as the use of single molecular magnets or clusters of transition metals to reduce the physical size of a bit. Heavy metal dimers may show huge magnetic anisotropies provided by a strong spin-orbit coupling (SOC). We analyze the magnetic properties of several organometallic molecules based on transition metals to determine if such huge anisotropies survive when the metal is encapsulated into the organic molecule. The fifth chapter analyzes the use of graphene in the development of nanoelectronic circuits based on molecular junctions. The graphene has unique properties excellently matching the requirements for ideal electrodes. However the use of graphene nanoelectronics is difficulty because the shape of the graphene edges can change arbitrarily the coupling and switching functionality of the molecules attached. The aim of this chapter is to investigate the electronics and transport properties of different edge morphologies of graphene-wedge single-molecule junctions. It is shown that some graphene-wedge BDT and BPD junctions show negative differential resistance (NDR), spin-filtering (SP) behavior and Fano resonances.

The use of Density Functional Theory (DFT) has become a standard procedure in the recent years for the simulation of nanostructures. However it is well known that DFT suffers from severe limitations when covering the description of strongly correlated systems for which those approximations are not good. Strongly correlated systems can be described by means of tight-binding models. Such models are much simpler than the more realistic real-space models as they depend on few parameters and in some cases can be solved algebraically. The first three chapters of this thesis are devoted to the use of the DFT machinery to the description of such lattice models in order to assess the true limitations when DFT is applied to strongly correlated systems. The fourth and fifth chapters are oriented to the direct application of DFT to the study of two different and interesting problems: the search of molecular magnets with large magnetic anisotropy energy (MAE) and the study of molecular junctions using graphene leads. In the first chapter we derive the exact functional and the exchange-correlation energies that let us building the Kohn-Sham (KS) potentials of the single and double site Anderson-Hubbard models in order to find out if the exact DFT gives the correct description of such systems. We find that this description is in general incomplete. The second chapter presents an analysis of the double-site asymmetric Hubbard model with 2 electrons. The properties of its ground and excited states are discussed. We then use the Mean-Field (MF) and the Bethe-ansatz LDA (BALDA) approximations to assess their performance as a function of the external potential-interactions ratio. We then apply the concepts of DFT to this model, establishing its KS description and the corresponding adiabatic connection. A parametrized functional is also devised using a method that allows improving the accuracy of any approximated functional whenever the relation between the kinetic and the potential correlation terms is known. The third chapter covers the DFT estimation of the fundamental gap in strongly correlated systems. We study the asymmetric Hubbard dimer. We find the exact quasi-particle excitation spectrum, including the ionization potential and electron affinity, which let us derive the real gap. We set a method to calculate the KS potentials for non-integer particle numbers as a necessary step to find the KS Green's function. We prove that the Koopmans' and the Janak's theorems are obeyed. We find that the KS spectrum is in general incorrect, except for the HOMO by definition, when strong correlations are dominant. The KS gap is always smaller than the true gap with interactions and we find the role of the derivative discontinuity on this. In the fourth chapter we focus on the practical use of DFT and we test several organometallic molecules based with the objective of finding stable molecular magnets at room temperatures. The current technology uses thin-sheets for manufacturing magnetic bits for data storage. Different alternatives such as the use of single molecular magnets or clusters of transition metals to reduce the physical size of a bit. Heavy metal dimers may show huge magnetic anisotropies provided by a strong spin-orbit coupling (SOC). We analyze the magnetic properties of several organometallic molecules based on transition metals to determine if such huge anisotropies survive when the metal is encapsulated into the organic molecule. The fifth chapter analyzes the use of graphene in the development of nanoelectronic circuits based on molecular junctions. The graphene has unique properties excellently matching the requirements for ideal electrodes. However the use of graphene nanoelectronics is difficulty because the shape of the graphene edges can change arbitrarily the coupling and switching functionality of the molecules attached. The aim of this chapter is to investigate the electronics and transport properties of different edge morphologies of graphene-wedge single-molecule junctions. It is shown that some graphene-wedge BDT and BPD junctions show negative differential resistance (NDR), spin-filtering (SP) behavior and Fano resonances.

##### Local Notes:

DT(SE) 2015-037

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