few picoseconds range resulting in exciton annihilation. The capture rates exhibit linear as well as quadratic dependence on the exciton density. The quadratic dependence of the exciton annihilation rate on the exciton density is generally considered to be an exclusive characteristic of exciton-exciton annihilation processes via interband Auger scattering. Although the discussion in this paper focuses on monolayer MoS$_{2}$, the analysis and the results presented here are expected to be relevant to all 2D-TMDs, and are expected to be useful in designing metal dichalcogenide optoelectronic devices as well as in helping to understand and interpret experimental data [@Shi13; @Lagarde14; @Korn11; @Wangb14].

Theoretical Model
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![Two basic Auger processes for the capture of an electron (a) or a hole (b) of an exciton by a defect state are depicted [@Landsberg92; @Landsberg80].[]{data-label="fig:auger_1"}](./auger_1){width=".3\textwidth"}

Introduction
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The two basic Auger processes for the capture of an electron (a) or a hole (b) of an exciton by a defect state are depicted in Fig.\[fig:auger\_1\]. Proper partitioning of the Hamiltonian is important in order to compute the rates of these processes. We discuss the terms in the Hamiltonian describing various processes below.

The Non-interacting Hamiltonian {#sec:nonintH}
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The crystal structure of a monolayer of group-VI dichalcogenides 