Sophomore (Fall 2016 - Spring 2017)

Freshman (Fall 2015 - Spring 2016)

Abstract: The stochastic variational method is applied to excitonic formations within semiconducting transition metal dichalcogenides using a correlated Gaussian basis. The energy and structure of two- to six-particle systems are investigated along with their dependence on the effective screening length of the two-dimensional Keldysh potential and the electron-hole effective mass ratio. Excited state biexcitons are shown to be bound, with binding energies of the L=0 state showing good agreement with experimental measurements of biexciton binding energies. Ground and newly discussed excited state exciton-trions are predicted to be bound and their structures are investigated.

Abstract: The Stochastic Variational Method (SVM) is used to show that the effective mass model correctly estimates the binding energies of excitons and trions but fails to predict the experimental binding energy of the biexciton. Using high-accuracy variational calculations, it is demonstrated that the biexciton binding energy in transition metal dichalcogenides is smaller than the trion binding energy, contradicting experimental findings. It is also shown that the biexciton has bound excited states and that the binding energy of the $L=0$ excited state is in very good agreement with experimental data. This excited state corresponds to a hole attached to a negative trion and may be a possible resolution of the discrepancy between theory and experiment.

Abstract: Recently, experimental measurements and theoretical modeling have been in a disagreement concerning the binding energy of biexctions in transition metal dichalcogenides. While theory predicts a smaller binding energy (∼20 meV) that is, as logically expected, lower than that of the trion, experiment finds values much larger (∼60 meV), actually exceeding those for the trion. In this work, we show that there exists an excited state of the biexciton which yields binding energies that match well with experimental findings and thus gives a plausible explanation for the apparent discrepancy. Furthermore, it is shown that the electron-hole correlation functions of the ground state biexciton and trion are remarkably similar, possibly explaining why a distinct signature of ground state biexcitons would not have been noticed experimentally.

Pre-college (Spring 2015 and prior)

Abstract: Traditionally, linear multistep methods (LMMs) for the numerical solution of initial value problems, such as Adams methods and backward differentiation formulas, have been derived through the use of polynomial interpolation and collocation through continuous schemes. While these methods can be implemented in modern computer algebra systems, they require the use of highly expensive operations such as symbolic matrix inversion. This imposes a severe limit on the complexity of LMMs that can be derived. In this presentation, we present a generalized algorithm for deriving LMMs based upon Taylor series expansion. By our approach, we show that the derivation of a LMM containing $k + 1$ terms is reducible to the numerical solution of a $k \times k$ linear system, allowing for the efficient derivation of methods including hundreds or thousands of terms. Furthermore, we show that this algorithm is trivially generalizable to methods including arbitrarily many off-grid points, and that it can be generalized to create LMMs for directly solving initial value problems of arbitrarily high order, with the inclusion of all intermediate derivative terms. Specific methods are stated and tested numerically on well-known problems given in the literature.