  We address a fundamental problem for the advance of phononics: the search of
a feasible thermal diode. We establish sufficient conditions for the existence
of thermal rectification in general graded materials. By starting from simple
assumptions satisfied by the usual anharmonic models that describe heat
conduction in solids, we derive an expression for the rectification. The
analytical formula shows how to increase the rectification, and the conditions
to avoid its decay with the system size, a problem present in the recurrent
model of diodes given by the sequential coupling of two or three different
parts. Moreover, for these graded systems, we show that the regimes of
non-decaying rectification and of normal conductivity do not overlap. Our
results indicate the graded systems as optimal materials for a thermal diode,
the basic component of several devices of phononics.
