Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-modules $\phi: E\to E\otimes \Omega^1_X$ (the Higgs field) such that $$\phi\wedge \phi=0: E\to E\otimes \Omega^2_X.$$

If we replace the holomorphic vector bundle $E$ by a coherent sheaf $\mathcal{E}$, then we get a Higgs sheaf on $X$. See for example On Gieseker stability for Higgs sheaves.

Since the bounded derived category of coherent sheaves has been intensively studied in algebraic geometry, I wonder whether the bounded derived category of Higgs sheaves (or related concept) has ever been defined or studied in literatures.