  We study the thermodynamics and critical behavior of su($m|n$) supersymmetric
spin chains of Haldane-Shastry type with a chemical potential term. We obtain a
closed-form expression for the partition function and deduce a description of
the spectrum in terms of the supersymmetric version of Haldane's motifs, which
we apply to obtain an analytic expression for the free energy per site in the
thermodynamic limit. By studying the low-temperature behavior of the free
energy, we characterize the critical behavior of the chains with $1\le
m,n\le2$, determining the critical regions and the corresponding central
charge. We also show that in the su($2|1$), su($1|2$) and su($2|2$) chains the
bosonic or fermionic densities can undergo first-order (discontinuous) phase
transitions at $T=0$, in contrast with the previously studied su(2) case.
