Quantum thermodynamics of the magnetic quantum phase-transition in P-doped graphene under biaxial strain*

We explore the quantum-thermodynamic effects in a phosphorous (P)-doped graphene biaxial strain. We obtain thermodynamic quantities such as the electronic entropy and specific heat as a function of temperature and strain within a Fermi-Dirac statistical model. At zero temperature, a magnetic moment of 1.0 µ_B is induced by the P atom in the system, which can be tuned by the strain control parameter. From zero to about 5% of strain, the system exhibits a magnetic phase, transforming it into a paramagnetic phase for strain >5%. When the temperature is different from zero, we find entropy changes presenting sign changes within both a magnetic regime with magnetic moment of 1.0 µ_B (0-1.5% strain) and the paramagnetic regime, while in the intermediate regime with magnetic moment < 1.0 µ_B (~1.5-5% strain), the entropy changes are negative. Notably, the electronic specific heat for P-doped graphene increases in the magnetic phase up to 5% strain and abruptly drops in the paramagnetic regime at 5.5% strain, showing a magnetic quantum phase transition at low temperatures.