Resolving spin, valley, and moir\'e quasi-angular momentum of interlayer excitons in WSe2/WS2 heterostructures

Moir\'e superlattices provide a powerful way to engineer properties of electrons and excitons in two-dimensional van der Waals heterostructures. The moir\'e effect can be especially strong for interlayer excitons, where electrons and holes reside in different layers and can be addressed separately. In particular, it was recently proposed that the moir\'e superlattice potential not only localizes interlayer exciton states at different superlattice positions, but also hosts an emerging moir\'e quasi-angular momentum (QAM) that periodically switches the optical selection rules for interlayer excitons at different moir\'e sites. Here we report the observation of multiple interlayer exciton states coexisting in a WSe2/WS2 moir\'e superlattice and unambiguously determine their spin, valley, and moir\'e QAM through novel resonant optical pump-probe spectroscopy and photoluminescence excitation spectroscopy. We demonstrate that interlayer excitons localized at different moir\'e sites can exhibit opposite optical selection rules due to the spatially-varying moir\'e QAM. Our observation reveals new opportunities to engineer interlayer exciton states and valley physics with moir\'e superlattices for optoelectronic and valleytronic applications

Liouville theorems for harmonic maps

We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space ( R m , g 0 ) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at ∞.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46575/1/222_2005_Article_BF02100594.pd

Three problems concerning the elliptic equations and systems

The first result is a counterexample of the following problem: Suppose u:M\sp{\rm m}{\to} N\sp{\rm n} is a harmonic map from Riemannian manifold M\sp{\rm m} to Riemannian manifold N\sp{\rm n}, and $\Omega$ $\subset$ M\sp{\rm m} is an open set. If rank(du) $\leq$ k on $\Omega$, can we conclude that rank(du) $\leq$ k on M\sp{\rm m}? It is known that the answer is affirmative for k = 0,1. For k $\geq$ 2, J. Kazdan and I construct a metric g on R\sp3, a harmonic map u: (R\sp3,g) $\to$ (R\sp3,g\sb{\rm o}), where g\sb{\rm o} is the Euclidean metric on R\sp3, such that rank(du) = 3 on some open set in R\sp3, while rank(du) = 2 on some other open set in R\sp3. The next result is a unique continuation property on the boundary for solutions of an elliptic equation: Let $\Omega$ be a smooth domain in R\sp{\rm n}, x\sb{\rm o}\in\partial\Omega, and u a harmonic function in $\Omega$. If there are constants a,b $\u3e$ 0, such that $\vert$u(x)$\vert$ $\leq$ a exp$\{-$b/$\vert$x $-$ x\sb{\rm o}\vert\} for x $\in\Omega$, and $\vert$x $-$ x\sb{\rm o}\vert small, then u = 0 in $\Omega$. Also if n = 2, the same conclusion holds for the solutions of a general second order linear elliptic equation. The last result is the nonexistence of non-parametric minimal submanifolds with a given boundary in a Riemannian manifold. Let \Omega\sp{\rm n} be a bounded domain, $\varphi$:$\partial\Omega\ \to$ R\sp{\rm k} be a map, is there a map u:$\Omega$ $\to$ R\sp{\rm k}, such that u = $\varphi$ on $\partial\Omega$ and the graph V = $\{$(x,u(x))$\vert$x $\in$ $\Omega\}$ is a minimal submanifold in the Riemannian manifold $\{$R\sp{\rm n+k},ds\sp2) ? Lawson and Osserman proved that if \Omega\sp{\rm n} = B\sp{\rm n} is the unit ball, $\varphi$:S\sp{\rm n} $\to$ S\sp{\rm k} is homotopically nontrivial, then there is a t\sb{\rm o} = t\sb{\rm o} (n,k,$\varphi$), so that there is no u:B\sp{\rm n} $\to$ R\sp{\rm k} with the properties that u = t$\varphi$ on $\partial\Omega$, t $\u3e$ t\sb{\rm o} and the graph is a minimal submanifold in R\sp{\rm n+k} with the Euclidean metric. We generalized this result to certain classes of domains, boundary maps and the metrics on R\sp{\rm n+k}. In particular, we can conclude the nonexistence result in the hyperbolic space

Growth rate and existence of solutions to Dirichlet problems for prescribed mean curvature equations on unbounded domains

We prove growth rate estimates and existence of solutions to Dirichlet problems for prescribed mean curvature equation on unbounded domains inside the complement of a cone or a parabola like region in $mathbb{R}^n$ ($ngeq 2$). The existence results are proved using a modified Perron's method by which a subsolution is a solution to the minimal surface equation, while the role played by a supersolution is replaced by estimates on the uniform $C^{0}$ bounds on the liftings of subfunctions on compact sets

DIRICHLET PROBLEMS FOR SEMILINEAR ELLIPTIC EQUATIONS WITH A FAST GROWTH COEFFICIENT ON UNBOUNDED DOMAINS

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations that are similar either to the singular Emden-Fowler equation or a sublinear elliptic equation. The result obtained can be applied to equations with coefficients of the nonlinear term growing exponentially. The proof is based on the super and sub-solution method. A super solution itself is constructed by solving a quasilinear elliptic equation via a modified Perron’s method

Rate of convergence for solutions to Dirichlet problems of quasilinear equations

We obtain rates of convergence for solutions to Dirichlet problems of quasilinear elliptic (possibly degenerate) equations in slab-like domains. The rates found depend on the convergence of the boundary data and of the coefficients of the operator. These results are obtained by constructing appropriate barrier functions based on the structure of the operator and on the convergence of the boundary data

Existence of positive solutions for Dirichlet problems of some singular elliptic equations

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations similar to the singular Emden-Fowler equation. The proof is based on a super and sub-solution method. A super solution is constructed by Perron's method together with a family of auxiliary functions