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High‐performance silicon heterojunction (SHJ) solar cells use carrier‐selective contact structures based on hydrogentated amorphous Si (a‐Si:H) to maximize collection of photogenerated carriers. The high open circuit voltages observed experimentally in SHJ cells require that the carrier‐selective contacts provide selectivity and passivation. However, a microscopic understanding of the dynamics of carrier transport through the a‐Si layer is currently lacking. In this paper, we explicitly simulate the transport of holes across the a‐Si:H(i) layer using a novel kinetic Monte Carlo approach. The hole‐selective contact structure investigated in this paper uses p‐type doped a‐Si:H(p) and intrinsic a‐Si:H(i) on an n‐type crystalline silicon wafer, where the selectivity is provided by the a‐Si:H(p) and the passivation is provided by the a‐Si:H(i). However, in addition to the passivation provided by the a‐Si:H(i), this layer also creates a potential barrier to the collection of photogenerated holes. There have been experimental studies in the literature that have suggested that multi‐phonon processes are the main transport mechanism that assists in the transport of holes across the intrinsic a‐Si:H barrier. Simulations presented here show that multi‐phonon injection of holes into the a‐Si:H(i) layer is the rate limiting step for transport across the a‐Si:H(i) layer. Our results indicate that multi‐phonon transport is strongly dependent on the electric field at the a‐Si:H(i)/c‐Si heterointerface as well. Transport simulations presented in this paper are consistent with experimental findings that multi‐phonon processes limit transport across the a‐Si:H(i) layer and are responsible for photocurrent suppression at the a‐Si:H(i)/c‐Si heterointerface when these processes are slower than the associated incident hole flux due to photo‐excitation.