  We study bi-segment thermal rectifiers whose forward heat fluxes are greater
than reverse counterparts. Presently, a shortcoming of thermal rectifiers is
that the rectification ratio, namely the forward flux divided by the reverse
flux, remains too small for practical applications. In this study, we have
managed to discover and theoretically derive the ultimate limit of such ratios,
which are validated by numerical simulations, experiments, and micro-scale
Hamiltonian-oscillator analyses. For rectifiers whose thermal conductivities
are linear with the temperature, this limit is simply a numerical value of 3.
For those whose conductivities are nonlinear with temperatures, the maxima
equal $\kappa_{max}/\kappa_{min}$, where the two extremes denote values of the
solid segment materials that can be possibly found or fabricated within a
reasonable temperature range on earth. Recommendations for manufacturing
high-ratio rectifiers are also given with examples.
