Dualities in theoretical physics are important tools to gain information on a physical model from its proposed duality with another typically more accessible physical theory. One such duality is mirror symmetry, which is a duality theory stemming from string theory. The mathematical implications of this duality are manifold. In the proposed project we are interested in relating these mathematical implications with ideas coming from number theory and representation theory. Namely, we propose to find patterns in the character tables of some finite matrix groups which explain this mirror symmetry from the perspective of Langlands duality. This latter is a vast program in modern number theory which in a special case implies Fermat's Last Theorem by the work of Andrew Wiles. In this proposal we are connecting via the study of the character tables of finite matrix groups, these two seemingly far dualities: mirror symmetry in string theory, and Langlands duality in number theory.

With the global population projected to reach 12 billion by 2050 coupled with moderate economic growth, and despite increased energy efficiency, the total global energy consumption is estimated to more than double to ~28 TW (1 billion billion watts) from the current use of ~13 TW. To cap or reduce carbon dioxide levels, most of this additional energy must come from carbon-free sources, the largest of which is solar energy (100,000 TW). However, solar energy has to be converted into a useable form at reasonably low cost. Indeed, in the UK, recent increases in renewable energy generation have mostly relied upon increased use of wind power due to the relatively high cost of solar power. One of the most promising approaches to reducing the cost of solar power, is to use small-area high efficiency cells with light concentrated on them by low cost, large-area plastic lenses. The highest efficiency solar cells to date consist of three junction III-V semiconductor devices containing both arsenides and phosphides. The failure of the band gaps of these materials to match the wavelength range of the solar spectrum limits the maximum efficiency obtainable.The proposed work will develop nitride materials for future demonstration of full spectrum super-high efficiency photovoltaics. Nitride-based solar cells with concentrator technologies promise to deliver significant advances in efficiency and reductions in cost over the current state-of-the-art. This potential is a result of the newly discovered narrow band gap of indium nitride (InN), making the band gaps of the ternary alloys indium gallium nitride and indium aluminium nitride span the entire solar spectrum (0.6 to 3.4 eV for InGaN and 0.6 to 6.2 eV for InAlN). Solar cells made from these material systems are predicted to attain the maximum theoretical efficiency of a double-junction cell of 50%. This is almost twice as efficient as the current generation of triple-junction solar cell devices. Nitride based cells with three or more junctions could achieve efficiencies approaching 60%. However, research on these materials is not very advanced and the material quality is still being optimised. The epitaxial growth of InGaN and InAlN continues to be developed, with improvements being made by the project partners who will provide samples for the proposed work. In this project, a comprehensive programme of structural, optical and electrical characterisation will be undertaken to optimise these alloys for application in nitride-based photovoltaic devices. In parallel with these activities, experiments will also be undertaken on III-nitride structures to achieve reproducible n- and p-type doping, to develop tunnel junctions, to determine the role of defects in photovoltaic performance, and to optimise metal contacts and transparent conducting oxide solar cell windows. Solar cell modelling will be performed using material parameters determined from the experiments to produce optimized designs for high-efficiency nitride solar cells and to investigate new integrated optical/electrical solar cell designs which circumvent traditional current and lattice matching constraints. The proposed programme will ultimately allow the UK's exceptionally high expertise in the broad area of nitrides to be extended to include indium-rich nitride alloys for low cost, low carbon energy generation.

With the global population projected to reach 12 billion by 2050 coupled with moderate economic growth, and despite increased energy efficiency, the total global energy consumption is estimated to more than double to ~28 TW (1 billion billion watts) from the current use of ~13 TW. To cap or reduce carbon dioxide levels, most of this additional energy must come from carbon-free sources, the largest of which is solar energy (100,000 TW). However, solar energy has to be converted into a useable form at reasonably low cost. Indeed, in the UK, recent increases in renewable energy generation have mostly relied upon increased use of wind power due to the relatively high cost of solar power. One of the most promising approaches to reducing the cost of solar power, is to use small-area high efficiency cells with light concentrated on them by low cost, large-area plastic lenses. The highest efficiency solar cells to date consist of three junction III-V semiconductor devices containing both arsenides and phosphides. The failure of the band gaps of these materials to match the wavelength range of the solar spectrum limits the maximum efficiency obtainable.The proposed work will develop nitride materials for future demonstration of full spectrum super-high efficiency photovoltaics. Nitride-based solar cells with concentrator technologies promise to deliver significant advances in efficiency and reductions in cost over the current state-of-the-art. This potential is a result of the newly discovered narrow band gap of indium nitride (InN), making the band gaps of the ternary alloys indium gallium nitride and indium aluminium nitride span the entire solar spectrum (0.6 to 3.4 eV for InGaN and 0.6 to 6.2 eV for InAlN). Solar cells made from these material systems are predicted to attain the maximum theoretical efficiency of a double-junction cell of 50%. This is almost twice as efficient as the current generation of triple-junction solar cell devices. Nitride based cells with three or more junctions could achieve efficiencies approaching 60%. However, research on these materials is not very advanced and the material quality is still being optimised. The epitaxial growth of InGaN and InAlN continues to be developed, with improvements being made by the project partners who will provide samples for the proposed work. In this project, a comprehensive programme of structural, optical and electrical characterisation will be undertaken to optimise these alloys for application in nitride-based photovoltaic devices. In parallel with these activities, experiments will also be undertaken on III-nitride structures to achieve reproducible n- and p-type doping, to develop tunnel junctions, to determine the role of defects in photovoltaic performance, and to optimise metal contacts and transparent conducting oxide solar cell windows. Solar cell modelling will be performed using material parameters determined from the experiments to produce optimized designs for high-efficiency nitride solar cells and to investigate new integrated optical/electrical solar cell designs which circumvent traditional current and lattice matching constraints. The proposed programme will ultimately allow the UK's exceptionally high expertise in the broad area of nitrides to be extended to include indium-rich nitride alloys for low cost, low carbon energy generation.