We used a mobility edge transport model and solved the drift-diffusion equation to characterize the space-charge-limited current of a rubrene single-crystal hole-only diode. The current-voltage characteristics suggest that current is injection-limited at high voltage when holes are injected from the bottom contact (reverse bias). In contrast, the low-voltage regime shows that the current is higher when holes are injected from the bottom contact as compared to hole injection from the top contact (forward bias), which does not exhibit injection-limited current in the measured voltage range. This behavior is attributed to an asymmetric distribution of trap states in the semiconductor, specifically, a distribution of traps located near the top contact. Accounting for a localized trap distribution near the contact allows us to reproduce the temperature-dependent current-voltage characteristics in forward and reverse bias simultaneously, i.e., with a single set of model parameters. We estimated that the local trap distribution contains 1.19×1011 cm -2 states and decays as exp(-x/32.3nm) away from the semiconductor-contact interface. The local trap distribution near one contact mainly affects injection from the same contact, hence breaking the symmetry in the charge transport. The model also provides information of the band mobility, energy barrier at the contacts, and bulk trap distribution with their corresponding confidence intervals. © 2012 American Physical Society.