Free energy of bcc iron: Integrated ab initio derivation of vibrational, electronic, and magnetic contributions F. Körmann, 1, * A. Dick, 1 B. Grabowski, 1 B. Hallstedt, 2 T. Hickel, 1 and J. Neugebauer 1 1 Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf D-40237, Germany 2 Materials Chemistry, RWTH Aachen University, Aachen, Germany Received 10 March 2008; published 3 July 2008 We present ab initio derived thermodynamic properties of ferromagnetic bcc iron up to the bcc-fcc phase- transition temperature 1200 K, including vibrational, electronic, and magnetic contributions. The quasihar- monic approximation and finite-temperature density-functional theory are employed to account for vibrational and electronic excitations. The magnetic contribution is derived from the solution of the quantum Heisenberg model within many-body theory using the mean-field and random-phase approximation. The calculated ther- modynamic properties show an excellent agreement with available experimental data and reveal the necessity to consider all three types of excitations. DOI: 10.1103/PhysRevB.78.033102 PACS numbers: 71.15.-m, 63.20.-e, 75.40.Cx, 75.50.Bb A key quantity in fully characterizing the thermodynamic properties of materials is the Helmholtz free energy of their individual structural and magnetic phases. Well established simulation tools, e.g., CALPHAD Refs. 1 and 2, use empiri- cal interpolation formulas and experimental input to describe the temperature dependence of this energy. However, experi- mental input data, e.g., for novel alloy systems or metastable structures, are not always available. Therefore, there is a strong interest in incorporating ab initio results into these simulation tools. 3 Based on density-functional theory DFT, substantial progress has been made in determining ab initio free energies for elementary nonmagnetic materials. 4,5 Here, lattice vibrations and in particular for transition metals electronic excitations yield the dominant contributions to the entropy. Despite its great importance for many practical ap- plications e.g., steel, magnetic actuators, the ab initio treat- ment of the magnetic contribution is discussed to a much less extent. However, only if all relevant excitation processes can be described with a total accuracy of a few meV within an integrated approach, a reliable ab initio prediction of thermo- dynamic properties in these materials can be expected. We have chosen iron as the most prominent magnetic material in order to develop such an integrated computational treatment and to test whether the accuracy of common DFT functionals is sufficient to achieve this goal. We will concentrate our investigation on the treatment of magnetic properties. In the past, several methods e.g., a multiband Hubbard model, 6 spin-fluctuation theory, 7,8 the Heisenberg model 9 solved within the Monte Carlo method, 10–13 or many-body theory 11,14,15  have been used for calculating quantitative temperature-dependent magnetic properties from ab initio input. However, most of them fo- cused on the calculation of the magnetization and the Curie temperature. Much less attention is paid to the magnetic free energy, although it is well established that magnetic excita- tions are of crucial importance for the structural phase sta- bility in iron. 16,17 An existing approach, based on the single- band spin-fluctuation theory within the mean-field approximation, gives a good qualitative agreement with ex- periment but fails to provide a reasonable quantitative de- scription of the magnetic free energy. 17 In this Brief Report, we, therefore, propose an accurate method for calculating the magnetic free energy using ab initio input and many-body theory for the Heisenberg model. We show that the magnetic contribution is essential for the description of thermody- namic properties and, if properly taken into account, allows for a quantitative description over the entire temperature range relevant for the considered phase. For the present purpose, the Helmholtz free-energy sur- face FT , V, as a function of the temperature T and the crys- tal volume V, is considered in the adiabatic approximation FT, V = F vib T, V + F el T, V + F mag T, V , 1 that treats the vibrational F vib , the electronic F el including the energy of the static lattice, and the magnetic F mag con- tribution separately. For iron, the adiabatic approximation is well justified since the underlying mechanisms, i.e., phonon, electron, and magnon excitations, reside on different time scales. 18 The validity of the adiabatic approximation is fur- ther supported by recent explicit ab initio calculations of the magnon-phonon coupling in iron. 19 To calculate F vib , F el , and F mag , we employ the VASP Ref. 20 package using the projector augmented wave method 21 within the generalized gradient approximation Perdew- Burke-Ernzerhof parametrization 22 . 23 Anharmonic contribu- tions are expected to be small a few meV and will, there- fore, not be considered in this study. 24 The contributions of F vib and F el are calculated within the quasiharmonic approxi- mation and finite-temperature DFT, 25 respectively, as dis- cussed in detail in Ref. 5. As the first step, we have, there- fore, calculated the phonon dispersion of ferromagnetic bcc iron, which serves as an input to the partition function needed for computing F vib . The comparison with the experi- mental phonon spectrum shows a good agreement Fig. 1. 29 The resulting combined vibronic and electronic free energy F vib + F el  is shown in Fig. 2 black line and compared to CALPHAD data. Clearly, the deviation between ab initio F vib + F el  and CALPHAD data rapidly increases with tempera- ture. For instance, at 1200 K, which corresponds to the ex- perimental bcc to fcc transition temperature, the difference is 45 meV. An even more sensitive physical quantity PHYSICAL REVIEW B 78, 033102 2008 1098-0121/2008/783/0331024 ©2008 The American Physical Society 033102-1