The nanowire geometry offers significant advantages for exploiting the potential of III-Sb materials. Strain due to lattice mismatch is efficiently accommodated, and carrier confinement effects can be utilized in tunneling and quantum devices for which the III-Sb materials are of particular interest. It has however proven difficult to grow thin (below a few tens of nanometers), epitaxial III-Sb nanowires, as commonly no growth is observed below some critical diameter. Here we explore the processes limiting the diameter of III-Sb nanowires in a model system, in order to develop procedures to control this effect. The InAs-GaSb heterostructure system was chosen due to its great potential for tunneling devices in future low-power electronics. We find that with increasing growth temperature or precursor partial pressures, the critical diameter for GaSb growth on InAs decreases. To explain this trend we propose a model where the Gibbs-Thomson effect limits the Sb supersaturation in the catalyst particle. This understanding enabled us to further reduce the nanowire diameter down to 32 nm for GaSb grown on 21 nm InAs stems. Finally, we show that growth conditions must be carefully optimized for these small diameters, since radial growth increases for increased precursor partial pressures beyond the critical values required for nucleation. © 2013 American Chemical Society.