Thermodynamics of high-T C materials in the rotating antiferromagnetism theory Mohamed Azzouz Department of Physics and Astronomy, Laurentian University, Ramsey Lake Road, Sudbury, Ontario, Canada P3E 2C6 ~Received 20 June 2003; published 21 November 2003! In high-temperature superconductors the pseudogap energy affects the electronic specific-heat data for dop- ing densities smaller than the optimal point. Within the rotating antiferromagnetism theory I calculated the superconducting and rotating antiferromagnetic parameters, then got the phase diagram. Also, the specific-heat coefficient and entropy are evaluated as a function of temperature. In addition, the doping dependence of the condensation energy, specific-heat anomaly, and entropy at T C are calculated. Experimental data are analyzed by focusing on the trends in the doping and temperature dependence. As far as the above quantities are concerned this theory yields very good agreement with experiment, and can therefore be said to be applicable to high-temperature superconductors. DOI: 10.1103/PhysRevB.68.174523 PACS number~s!: 74.25.Bt, 74.25.Ha I. INTRODUCTION The rotating antiferromagnetism theory ~RAFT! I have recently proposed 1 addresses the issue of the pseudogap ~PG! phenomenon 2 and the doping dependence of the electronic structure in high-temperature superconductors 3 ~HTSC’s!. RAFT is based on spin antiferromagnetism ~AF! and is therefore fundamentally different from the density d-wave ~DDW! theory 4 which is based on orbital AF. Furthermore, rotational symmetry is not broken in RAFT because the ro- tating order parameter characterizing rotating antiferromag- netism ~RAF! is a magnetization that has a finite average magnitude but a random phase angle. Also, very recently I proposed to treat the CuO 2 layers of HTSC’s as open sys- tems in contact with a particle reservoir, which is formed by the atoms doped between these layers. To leading order, one can model the low-energy physics of HTSC’s by considering a two-dimensional ~2D! lattice of electrons in contact with a thermodynamics electrons reservoir. As a consequence, the charge-carrier density on the CuO 2 layers is temperature dependent. 5 This T dependence should not be surprising in the light of the strong T dependence of the Hall coefficient. 6–8 In the present work I tested the applicability of RAFT and the above ideas at finite temperature by calcu- lating several thermodynamic functions and comparing them with their experimental data. I calculated the temperature dependence of RAFT’s order parameters, the electronic en- tropy S el ( T ) and the specific-heat C el ( T ) or coefficient g ( T ) 5C el ( T )/ T . I also investigated the doping dependence of the condensation energy U 0 , the entropy at T C , S ( T C ), and the jump in g ( T ) at the superconducting transition tem- perature T C . One of the important questions that I also addressed in this work is whether there exists a true phase transition be- low which the PG appears or not. Experimentally, no specific-heat anomaly due to the PG has been reported so far. In this paper, I propose two possible scenarios to explain the absence of experimental evidence for such an anomaly. In the first scenario, the anomaly in g ( T ) due to the PG goes unnoticed because it is significantly smaller than the one due to superconductivity ~SC! especially in the neighborhood of the optimal point. In the second scenario, the PG temperature T * stays much higher than the superconducting transition temperature T C as doping increases away form half filling even though the PG order parameter itself is significantly reduced and vanishes near the optimal point. According to both scenarios, to observe the anomaly at T * one should not be looking for a l -shape ~or peak! anomaly but rather for a steplike one in the specific-heat coefficient. In the second scenario, this anomaly occurs at temperatures much higher than T C even near optimal doping. Furthermore, I show that RAFT yields specific-heat results that are consistent with the conclusion of Loram and co-workers. 9 These authors con- cluded that the simplest interpretation of the specific-heat data can be done in terms of carriers that are fermions with a very low Fermi temperature, which is of the order of 10 3 K. RAFT has been found to agree well with ground-state ex- perimental data when I indeed assumed that the electronic hopping energies are small. 1 This paper is organized as follows. In Sec. II, RAFT is briefly reviewed, and the idea of the CuO 2 layers behaving as open systems is explained. For the results presented in this paper, two Hamiltonian parameters sets are used to carry on the numerical calculations. The analysis of the temperature dependence of the rotating and superconducting order pa- rameters is presented in Sec. III. In the latter, the temperature dependence of doping is analyzed as well, and used to dis- cuss the T dependence of the Hall coefficient in HTSC’s. Furthermore, I focus on the doping dependence of the super- conducting transition temperature T C and PG temperature T * in Sec. IV, where the phase diagrams are calculated for both Hamiltonian parameters sets. The specific-heat coeffi- cient and entropy results are presented in Sec. V. The doping dependence of superfluidity is studied in Sec. VI. Finally, the summary and conclusions are given in Sec. VII. II. DESCRIPTION OF THE METHOD RAFT is a theory that describes HTSC’s in terms of two competing order parameters. One of these is proportional to the superconducting gap amplitude, and is called D 0 , and the second one is the RAF parameter labeled Q. The latter is the amplitude of a rotating magnetization, which I have pro- posed in order to model the PG behavior in the cuprates. PHYSICAL REVIEW B 68, 174523 ~2003! 0163-1829/2003/68~17!/174523~12!/$20.00 ©2003 The American Physical Society 68 174523-1