Snowball Earth is a hypothesized state in the deep past of Earth in which the ocean was completely or nearly completely covered by sea ice, resulting from a runaway ice-albedo feedback. Here, we address how the treatment of sea-ice thermodynamics affects the initiation of a Snowball Earth in the global climate model ICON-A run in an idealized slab-ocean aquaplanet setup. Specifically, we study the impact of vertical resolution and brine pockets of ice by comparing the 3-layer Winton and a 0-layer Semtner scheme, and we investigate the impact of limiting ice thickness to 5 m. The internal heat storage of ice is increased by higher vertical resolution and brine pockets, which weakens surface melting and increases global albedo by allowing snow and ice to persist longer into the summer season. The internal heat storage weakens the melt-ratchet effect, as is confirmed with offline simulations with the two ice schemes. The result is a substantially easier Snowball Earth initiation and an increase in the critical CO2 for Snowball initiation by 50%. Limiting ice thickness impedes Snowball initiation as the removal of excess ice leads to an artificial heat source. Yet, the impact is minor and critical CO2 is decreased by 5% only. The results show that while the sea-ice thickness limit plays only a minor role, the internal heat storage of ice represents an important factor for Snowball initiation and needs to be taken into account when modeling Snowball Earth initiation.