Optical properties of transition metal dichalcogenide bilayers

Two-dimensional transition metal dichalcogenide (TMD) semiconductors exhibit unique optoelectronic properties. Rational assembly of individual monolayers into homobilayers or heterostructures provides versatile means for realizations of a wide range of novel quantum materials with tunable optical and transport phenomena based on unique spin-valley degrees of freedom and strong electron correlations. Monolayer crystals synthesized by chemical vapor deposition (CVD) serve as elementary building blocks of layered van der Waals systems with distinct properties determined by layer number, composition and orientation. In the first part of this work, bilayers of WSe2 were obtained with CVD synthesis in two contrasting crystal structures and studied with cryogenic optical spectroscopy. Control of CVD growth parameters resulted in high-quality large-area homobilayer stacks with contamination-free interfaces and strictly parallel and antiparallel alignment. Using complementary optical spectroscopy techniques at room and cryogenic temperatures, and employing theoretical calculations, we identified distinct signatures of momentum-direct excitons in optical absorption and momentum-indirect excitons in the photoluminescence of WSe2 homobilayers with parallel and antiparallel crystallographic orientation. The study not only relates optical properties of homobilayers to crystal symmetry and stacking, it also highlights the role of crystal structure for the formation of hybrid interlayer exciton states. In the second part, we report studies of MoSe2-WSe2 heterostructures built as stacks of separately grown monolayers. For such nearly-commensurate heterostacks, theory predicts atomic reconstruction of the rigid moiré lattice into periodic domains of nanoscale patterns. In finite-size samples of heterostacks, however, we found effects of lattice reconstruction on mesoscopic length scales, with direct consequences for local optical properties dictated by excitons. Using extensive optical spectroscopy studies, correlative secondary electron imaging of reconstructed domains, and theoretical modeling, we identified the coexistence of nanoscale quantum arrays, quantum wires, and extended domains of only one atomic registry in the same sample as the main source of the diverse spectral features reported for excitons in MoSe2-WSe2 heterostacks. This notion of mesoscopic reconstruction provides a unifying perspective on exciton phenomena in heterobilayers with and without moiré effects. Finally, the last part of the work presents the results on MoSe2-WSe2 heterostacks obtained as vertically stacked triangular crystals from CVD-synthesis. With combined studies by cryogenic optical spectroscopy and high-resolution transmission electron microscopy, we identified extended moiré-free domains of energetically favored Hhh and RMh registries in H- and R-type stacks, in addition to grain boundaries of alternative registries and moiré-type cores. Remarkably, optical spectroscopy suggests that the RMh registry hosts the majority of exciton population with in-plane dipolar emission, as opposed to the predominance of RXh exciton features with out-of-plane luminescence in exfoliation-stacked MoSe2-WSe2 heterostructures. Overall, the work provides a unifying perspective on diverse signatures of excitons in semiconductor homobilayers and heterobilayers in parallel and antiparallel alignment. The results contribute to improved understanding of exciton phenomena that dictate the optical properties of layered semiconductor van der Waals systems

Accurate gradient computations at interfaces using finite element methods

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}. For 2D interface problems, the idea is to introduce a small tube near the interface and introduce the gradient as part of unknowns, which is similar to a mixed finite element method, except only at the interface. Thus the computational cost is just slightly higher than the standard finite element method. We present rigorous one dimensional analysis, which show second order convergence order for both of the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface

Towards a Multimodal Charging Network: Joint Planning of Charging Stations and Battery Swapping Stations for Electrified Ride-Hailing Fleets

This paper considers a multimodal charging network in which charging stations and battery swapping stations are built in tandem to support the electrified ride-hailing fleet in a synergistic manner. Our central thesis is predicated on the observation that charging stations are cost-effective, making them ideal for scaling up electric vehicles in ride-hailing fleets in the beginning, while battery swapping stations offer quick turnaround and can be deployed in tandem with charging stations to improve fleet utilization and reduce operational costs for the ride-hailing platform. To fulfill this vision, we consider a ride-hailing platform that expands the multimodal charging network with a multi-stage investment budget and operates a ride-hailing fleet to maximize its profit. A multi-stage network expansion model is proposed to characterize the coupled planning and operational decisions, which captures demand elasticity, passenger waiting time, charging and swapping waiting times, as well as their dependence on fleet status and charging infrastructure. The overall problem is formulated as a nonconvex program. Instead of pursuing the globally optimal solution, we establish a theoretical upper bound through relaxation, reformulation, and decomposition so that the global optimality of the derived solution to the nonconvex problem is verifiable. In the case study for Manhattan, we find that the two facilities complement each other and play different roles during the expansion of charging infrastructure: at the early stage, the platform always prioritizes building charging stations to electrify the fleet, after which it initiates the deployment of swapping stations to enhance fleet utilization. Compared to the charging-only case, ..

Sharp estimates, uniqueness and nondegeneracy of positive solutions of the Lane-Emden system in planar domains

We study the Lane-Emden system $$\begin{cases} -\Delta u=v^p,\quad u>0,\quad\text{in}~\Omega, -\Delta v=u^q,\quad v>0,\quad\text{in}~\Omega, u=v=0,\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as $p,q\to+\infty$ and $|q-p|\leq \Lambda$. In this paper, we obtain sharp estimates of such multi-bubble solutions, including sharp convergence rates of local maxima and scaling parameters, and accurate approximations of solutions. As an application of these sharp estimates, we show that when $\Omega$ is convex, then the solution of this system is unique and nondegenerate for large $p, q$.Comment: 63 pages. This is a revised version of arXiv:2205.15055v1. We fix a gap in the previous version and add some details of the proo