  In this work, we obtained the black hole solutions in the expansion
f(R)-gravitation (R is not regarded as a constant here), and studied their
thermodynamics, especially the phase transition and criticality in anti-de
Sitter Behavior (AdS) expands the phase space. We obtained accurate Banados,
Teitelboim and Zanelli (BTZ) corresponding solutions in dilaton f(R)-gravity,
which is the basis of our work. We also obtained the exact form of the f(R)
model for some solutions. In thermodynamic analysis, we calculate thermodynamic
quantities such as temperature and entropy of these solutions and compare them
with the corresponding quantities of BTZ. After that, we study the Ruppeiner
geometry, such as the f(R)-gravity of the BTZ black hole, and analyze its phase
transition process Ruppeiner geometry.
