Mathematical modelling of disorder in organic solar cells

马特将研究数学方法，以考虑太阳能电池无序的影响。特别强调的是有机太阳能电池以及含有有机成分的混合太阳能电池。与硅电池等替代品相比，这些电池具有至关重要的优势，因为它们可以直接从冷却的溶液中产生。这种生产过程意味着无序比其他太阳能电池发挥更大的作用，而这个项目的目的将是系统地处理由这种无序引起的影响。现有的太阳能电池的数学方法包括利用微分方程对载流子的经典运动进行建模。这些方程描述了导电电子和空穴浓度的变化率，由扩散、漂移以及电荷载流子的产生和重组所控制。这些方程也可以适用于激子（由电子和空穴形成的束缚态）。有机太阳能电池的另一个模型是高斯无序模型。这个模型比微分方程更全面，并且明确地考虑了量子效应和统计力学。在该模型中，载流子的可能状态被定位在随机分布的位置。这些状态的能量遵循高斯分布，位置之间的跳跃率是根据统计力学的思想选择的。最后，太阳能电池可以用多粒子量子力学的完整机制来描述，第二个量子化哈密顿量考虑了太阳能电池中载流子的运动、与光的相互作用等。在这个项目中需要遵循两个方向：1。开发了一个改进版本的漂移扩散方程，它结合了高斯无序模型的影响，使用了连续统极限。2. 在有机太阳能电池的第二量化模型中实现无序平均。这将利用凝聚态理论中用于执行无序平均的方法，例如非线性sigma模型。这些研究都是基于初步结果，包括导师的工作和Matt的硕士论文，在那里他考虑了基于Hubbard模型和Jaynes-Cummings模型的太阳能电池的第二量化模型。这些主题之间的平衡将根据项目的进展进行调整。这项工作将允许将数学物理中开发的先进方法转移到光伏发电，这主要是由实验工作和更多的现象学方法主导。它将有助于清洁能源的发展，具有明显的环境影响。它连接到EPSRC的数学物理（数学科学主题）、太阳能技术（能源主题）和凝聚态物质：电子结构（物理科学主题）的研究领域。

Matt will work on mathematical methods to take into account the effects of disorder in solar cells. A particular emphasis will be on organic solar cells as well as hybrid solar cells containing organic components. These cells have crucial advantages over alternatives such as silicon as they can be produced directly from a solution that is cooled down. This production process means that disorder plays a larger role than in other solar cells, and the aim of this project will be systematically treat effects arising from this disorder. Existing mathematical approaches to solar cells involve modelling the classical motion of charge carriers using differential equations. These equations describe the rate rate of change for the concentration of conducting electrons and holes as governed by diffusion, drift, and well as generation and recombination of charge carriers. The equations can also be adapted to include excitons (bound states formed out of electrons and holes). A further model for organic solar cells is the Gaussian disorder model. This model is more comprehensive than the differential equations, and explicitly takes account of quantum effects as well as statistical mechanics. In this model, the possible states of the charge carriers are localised at randomly distributed sites. The energies of these states follow a Gaussian distribution, and jump rates between sites are chosen in line with ideas from Statistical Mechanics. Finally, the solar cells could be described using the full machinery of many-particle quantum mechanics, with a second quantised Hamiltonian taking into account the motion of charge carriers in the solar cell, interaction with light, etc. Two directions to be followed in this project are 1. The development an improved version of the drift diffusion equations that incorporates effects from the Gaussian disorder model, using a continuum limit. 2. Implementing disorder averages in the second quantised model of organic solar cells. This will make use of methods used to perform disorder averages in condensed matter theory, such as nonlinear sigma models. These lines of research are based on preliminary results, including work of the supervisor and in Matt's MSc dissertation, where he considered second quantised models for solar cells based on e.g. the Hubbard model and the Jaynes-Cummings model. The balance between these topics will be adapted to the progress of the project. This work will allow for a transfer of advanced methods developed in Mathematical Physics to photovoltaics, which is mostly dominated by experimental work and more phenomenological approaches. It will contribute to the development of clean energy, with an obvious environmental impact. It is connected to the EPSRC research areas of Mathematical Physics (in the Mathematical Sciences theme), Solar Technology (in the Energy theme) and Condensed Matter: Electronic Structure (in the Physical Sciences theme).