Ab Initio Calculation of Phase Boundaries in Iron Along the bcc-fcc Transformation Path and Magnetism of Iron Overlayers

A detailed theoretical study of magnetic behavior of iron along the bcc-fcc (Bain’s) transformation paths at various atomic volumes, using both the local spin-density approximation (LSDA) and the generalized gradient approximation (GGA), is presented. The total energies are calculated by the spin-polarized fullpotential linearized augmented plane waves method and are displayed in contour plots as functions of tetragonal distortionc/aand volume; borderlines between various magnetic phases are shown. Stability of tetragonal magnetic phases of γ-Fe is discussed. The topology of phase boundaries between the ferromagnetic and antiferromagnetic phases is somewhat similar in LSDA and GGA; however, the LSDA fails to reproduce correctly the ferromagnetic bcc ground state and yields the ferromagnetic and antiferromagnetic tetragonal states at a too low volume. The calculated phase boundaries are used to predict the lattice parameters and magnetic states of iron overlayers on various (001) substrates. Disciplines Atomic, Molecular and Optical Physics | Engineering | Materials Science and Engineering | Physics | Semiconductor and Optical Materials This journal article is available at ScholarlyCommons: http://repository.upenn.edu/mse_papers/234 Ab initio calculation of phase boundaries in iron along the bcc-fcc transformation path and magnetism of iron overlayers M. Friák, M. Šob,* and V. Vitek Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic Department of Solid State Physics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic Department of Materials Science and Engineering, University of Pennsylvania, 3231 Walnut Street, Philadelphia, Pennsylvania 19104-6272 ~Received 17 August 2000; published 10 January 2001! A detailed theoretical study of magnetic behavior of iron along the bcc-fcc ~Bain’s! transformation paths at various atomic volumes, using both the local spin-density approximation ~LSDA! and the generalized gradient approximation ~GGA!, is presented. The total energies are calculated by the spin-polarized full-potential linearized augmented plane waves method and are displayed in contour plots as functions of tetragonal distortion c/a and volume; borderlines between various magnetic phases are shown. Stability of tetragonal magnetic phases of g-Fe is discussed. The topology of phase boundaries between the ferromagnetic and antiferromagnetic phases is somewhat similar in LSDA and GGA; however, the LSDA fails to reproduce correctly the ferromagnetic bcc ground state and yields the ferromagnetic and antiferromagnetic tetragonal states at a too low volume. The calculated phase boundaries are used to predict the lattice parameters and magnetic states of iron overlayers on various ~001! substrates. DOI: 10.1103/PhysRevB.63.052405 PACS number~s!: 75.50.Bb, 71.15.Nc, 71.20.Be Iron exists in both bcc and fcc modifications and has many magnetic phases, especially in thin films. In particular, fcc iron films exhibit a large variety of structural and magnetic properties that depend delicately on the iron layer thickness and preparation conditions. The close competition between different magnetic states has also been confirmed by first-principles electronic structure calculations. In the case of Fe, the local spin-density approximation ~LSDA! predicts a nonmagnetic close-packed ground state instead of the ferromagnetic bcc phase found in nature. It turns out that inclusion of nonlocal effects through the generalized gradient approximation ~GGA! overcomes this problem and stabilizes the bcc ferromagnetic state. This has also been verified in a number of recent studies ~see, e.g., Refs. 7–10!. Recently, Qiu, Marcus, and Ma found that the behavior of the total energy of iron along the tetragonal deformation path at the constant volume of 11.53 Å is quite similar within both LSDA and GGA. As we show in the present paper, the relaxation of volume brings a somewhat different picture of the energetics of iron and, in this way, the similarity mentioned in the above paper is rather a happy coincidence. Very recently, Spišák and Hafner included also the double-layer antiferromagnetic state (↑↑↓↓ . . . ) into their considerations of energetics of g-iron multilayers on Cu~001! and found that this magnetic ordering is energetically favored at the lattice constant equal to the equilibrium lattice constant of Cu. The purpose of this contribution is to perform the calculation of the total energy of iron along the bcc-fcc tetragonal ~Bain’s! transformation path at various volumes, to identify the stable and metastable phases, and to find the phase boundaries between various iron modifications. This path is significant for energetics of ultrathin films at the ~001! substrates, as pseudomorphic epitaxy on a ~001! surface of a cubic metal usually results in a strained tetragonal structure of the film. We include ferromagnetic ~FM!, nonmagnetic ~NM!, and both single-layer ~AFM1 – ↑↓ . . . ) and doublelayer ~AFMD! antiferromagnetic states. The calculated total energies are used to predict lattice parameters and the type of magnetic ordering of iron layers at various ~001! substrates. Craievich et al. have shown that some energy extrema on constant-volume transformation paths are dictated by the symmetry. Namely, most of the structures encountered along the transformation paths between some higher-symmetry structures, say between bcc and fcc at the Bain’s path, have a symmetry that is lower than cubic. At those points of the transformation path where the symmetry of the structure is higher the derivative of the total energy with respect to the parameter describing the path must be zero. These are the so-called symmetry-dictated extrema. However, other extrema may occur that are not dictated by symmetry and reflect properties of the specific material. Configurations corresponding to energy minima at the transformation paths represent stable or metastable structures and may mimic atomic arrangements that could be encountered when investigating thin films and extended defects such as interfaces and dislocations. We start with the bcc structure and consider it as a tetragonal one with the c/a ratio equal to 1. Subsequently, we perform a tetragonal deformation ~uniaxial deformation along the @001# axis!, i.e., we change the c/a ratio and the structure becomes tetragonal. However, at c/a5A2, we arrive at the fcc structure, which has again cubic symmetry. The points c/a51 and c/a5A2 correspond to the only highsymmetry structures along the tetragonal deformation path and, therefore, symmetry-dictated extrema of the total energy may be expected here. Let us note here that many papers define the c/a such that the fcc structure is considered a tetragonal one with (c/a)*51; then (c/a)*5(c/a)/A2 and the bcc structure corresponds to (c/a)*5A2/2. We calculate the total energy of NM, FM, AFM1, and AFMD iron along the tetragonal deformation paths keeping PHYSICAL REVIEW B, VOLUME 63, 052405 0163-1829/2001/63~5!/052405~4!/$15.00 ©2001 The American Physical Society 63 052405-1 the atomic volume constant; the region of atomic volumes studied extends from V/Vexp50.84 until V/Vexp51.05. For the total-energy calculation, we utilize the full-potential linearized augmented plane waves code described in detail in Ref. 17. The calculations are performed using both the GGA ~Ref. 18! and the LSDA. The muffin-tin radius of iron atoms of 1.90 au is kept constant for all calculations, the product of the number of k-points and number of nonequivalent atoms in the basis is equal to 6000, and the product of the muffin-tin radius and the maximum reciprocal space vector, RMTkmax, is equal to 10. The maximum l value for the waves inside the atomic spheres, lmax , and the largest reciprocal vector G in the charge Fourier expansion, Gmax , is set to 12 and 15, respectively. Figures 1~a! and 1~b! display the variation of total energies of iron along the tetragonal deformation path for the experimental lattice volume of the FM bcc iron of 11.72 Å calculated using the GGA and LSDA, respectively. The NM and FM states exhibit energy extrema at c/a51 and c/a 5A2 corresponding to higher-symmetry structures ~bcc and fcc!. However, the AFM1 iron keeps its cubic symmetry only for c/a51, i.e., for the bcc structure. At c/a5A2, the atoms occupy the fcc lattice positions, but as the atoms with spins up and down are not equivalent, the resulting symmetry is tetragonal, and no higher-symmetry structure occurs here. Therefore, no symmetry-dictated extremum of total energy at c/a5A2 is to be expected. And, indeed, the total-energy curves of AFM1 states exhibit, in general, a nonzero derivative at c/a5A2 ~Figs. 1 and 2!; this was also found by Qiu, Marcus, and Ma. The minima of the AFM1 curves are not dictated by symmetry. As to the AFMD iron, it is never cubic and no symmetry-dictated extrema occur along the tetragonal deformation path. It may be seen from Figs. 1~a! and 1~b! that the behavior of energies calculated within the GGA and LSDA for the experimental lattice volume is quite alike, in accordance with the findings of Qiu, Marcus, and Ma, who used a slightly different atomic volume of 11.53 Å. In both cases, the FM bcc iron has the lowest energy. In the region from c/a '1.4 till c/a'1.7, the AFMD ordering is most favorable. The shape of the energy curves corresponding to the same type of magnetic order is remarkably similar, but their mutual shift is somewhat different in the GGA and LSDA. The situation is completely changed at sufficiently lower atomic volumes. Figures 2~a! and 2~b! show the total energies of iron for the atomic volume of 10.21 Å. In the GGA, the energy of the FM bcc state is still the lowest one, in contrast to the LSDA case where AFM1 states lie distinctly lower than the FM bcc state. For 1.4&c/a&1.8, the AFM1 states are most favorable in both cases. Here the shape of the FIG. 1. Variations of total energies of iron along the constantvolume bcc-fcc transformation path for experimental volume Vexp 511.72 Å calculated within the GGA ~a! and LSDA ~b! relative to the equilibrium energy E0 of FM bcc iron. FIG. 2. The same as Fig. 1, but for the atomic volume V 510.21 Å. BRIEF REPORTS PHYSICAL REVIEW B 63 052405