  We study the size-dependent exciton fine structure in monolayer black
phosphorus quantum dots (BPQDs) deposited on different substrates (isolated, Si
and SiO$_2$) using a combination of tight-binding method to calculate the
single-particle states, and the configuration interaction formalism to
determine the excitonic spectrum. We demonstrate that the substrate plays a
dramatic role on the excitonic gaps and excitonic spectrum of the QDs. For
reasonably high dielectric constants ($\varepsilon_{sub} \sim \varepsilon_{Si}
= 11.7 \varepsilon_0$), the excitonic gap can be described by a single power
law $E_X(R) = E_X^{(bulk)} + C/R^{\gamma}$. For low dielectric constants
$\varepsilon_{sub} \leq \varepsilon_{SiO_2} = 3.9 \varepsilon_0$, the size
dependence of the excitonic gaps requires the sum of two power laws $E_X(R) =
E_g^{(bulk)} + A/ R^{n} - B/R^{m}$ to describe both strong and weak quantum
confinement regimes, where $A$, $B$, $C$, $\gamma$, $n$, and $m$ are
substrate-dependent parameters. We also predict that the exciton lifetimes
exhibit a strong temperature dependence, ranging between 2-8 ns (Si substrate)
and 3-11 ns (SiO$_2$ substrate) for QDs up 10 nm in size.
