  For decades, nonlinear optics has been used to control the frequency and
propagation of light in unique ways enabling a wide range of applications such
as ultrafast lasing, sub-wavelength imaging, and novel sensing methods. Through
this, a key thread of research in the field has always been the development of
new and improved nonlinear materials to empower these applications. Recently,
epsilon-near-zero (ENZ) materials have emerged as a potential platform to
enhanced nonlinear interactions, bolstered in large part due to the extreme
refractive index tuning ({\Delta}n~ 0.1 - 1) of sub-micron thick films that has
been demonstrated in literature. Despite this experimental success, the theory
has lagged and is needed to guide future experimental efforts. Here, we
construct a theoretical framework for the intensity-dependent refractive index
of the most popular ENZ materials, heavily doped semiconductors. We demonstrate
that the nonlinearity when excited below bandgap, is due to the modification of
the effective mass of the electron sea which produces a shift in the plasma
frequency. We discuss trends and trade-offs in the optimization of excitation
conditions and material choice (such material loss, band structure, and index
dispersion), and provide a figure of merit through which the performance of
future materials may be evaluated. By illuminating the framework of the
nonlinearity, we hope to propel future applications in this growing field.
