  In this work, trions in two-dimensional (2D) space are studied by variational
method with trial wavefunctions being constructed by linear combinations of 2D
slater-type orbitals (STOs). Via this method, trion energy levels and
wavefunctions can be calculated efficiently with fairly good accuracy. We first
apply this method to study trion energy levels in a 2D hydrogen-like system
with respect to a wide range of mass ratios and screening lengths. We find that
the ground-state trion is bound for the whole parameter range, and an
excited-state trion with antisymmetric permutation of electrons with finite
angular momentum is bound for large electron-hole mass ratios or long screening
lengths. The binding energies of ground-state trions calculated by the present
method agree well with those calculated by more sophisticated but
computationally-demanding methods. We then calculate trion states in various
monolayer transition metal dichalcogenides (TMDCs) by using this method with
the inclusion of electron-hole exchange (EHX) interaction. For TMDCs, we found
that the effect of EHX can be significant in determining the trion binding
energy and the possible existence of stable excited-state trions.
