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We consider a neutral exciton localized on a model p-i-n junction defined ina two-dimensional crystal: MoSe$_2$ and phosphorene, using a variationalapproach to the effective mass Hamiltonian. The non-homogeneous electric fieldat the junction prevents the separation of the center of mass. The variationalsolution provides the exciton density in the real space and accounts for thekinetic energy due to the exciton localization.  For low values of the potential step across the junction, the excitonoccupies an area which is much larger than the nominal range of the junctionand the energy remains essentially insensitive to the value of the step.Localization of the exciton within the junction area is accompanied by theappearance of the dipole moment induced by the local electric field. The dipolemoment becomes a linear function of the potential step only when the step issufficiently large. In consequence, the energy dependence on the step value isnon-parabolic. We demonstrate that the exciton gets localized not exactly atthe center of the junction but on the side which is more energeticallyfavourable for the heavier carrier: electron or hole.

Exciton localization on p-i-n junctions in two-dimensional crystals