Abstract
Absolute values of surface energy and surface stress of solids are hardly accessible by experiment. Here, we investigate the temperature dependence of both parameters for the (001) and (110) surface facets of bodycentered cubic Fe from firstprinciples modeling taking into account vibrational, electronic, and magnetic degrees of freedom. The monotonic decrease of the surface energies of both facets with increasing temperature is mostly due to lattice vibrations and magnetic disorder. The surface stresses exhibit nonmonotonic behaviors resulting in a strongly temperature dependent excess surface stress and surface stress anisotropy.
Introduction
Introduced by Gibbs, the surface free energy (γ) and the surface stress (σ), are the two essential macroscopic parameters that characterize the thermodynamic properties of crystalline surfaces^{1,2}. Today, there is growing appreciation that both parameters play an important role in understanding various surface phenomena, e.g., faceting, roughening, segregation, surface reconstruction, crystal growth, adsorption, bottomup selforganization, and surface melting^{1,2,3,4,5,6}. Surface stress also bears enormous potential for prospective molecular sensing and actuation devices exploiting its sensitivity to changes of chemical bonding and morphology in the surface region^{7,8}.
Although Langmiur^{9} pointed out already a century ago that rising thermal agitation of atoms in solids would lead to a reduced work of creation of surface (the surface energy), its temperature dependence and that of the surface stress are still largely unknown in the absence of direct measurements. Today, we face the situation that the polar dependence of γ and σ is accessible experimentally—the anisotropy of the surface energy is encoded in the equilibrium shape of 3D crystals—, but the determination of their absolute values often lacks reliability and accuracy hampered by experimental difficulties^{1,2,3,10,11}. Available absolute surface energies were in most cases either obtained at temperatures slightly below the melting point (T_{m}) where plastic flow occurs or were derived from the surface tension in the liquid phase (γ_{LV})^{12,13,14}. Absolute surface stress has proved elusive for direct measurements so far, merely few data for noble metals were derived from surface stress induced lattice strain in nanometersized particles^{11}.
In view of the experimental difficulties, the determination of surface energy and surface stress from theory has become a convenient alternative route^{15,16,17,18}, although modeling has been essentially limited to the ground state (T = 0 K) hitherto. Semiempirical approaches^{12,19} estimated configurational and vibrational contributions to the surface excess entropy for an “average” highindex surface facet to access γ(T). These estimations rely, however, on dependable surfacetension data and carry errors of unknown magnitude at lowtemperatures due to the uncertainty in the extrapolation procedure based on certain assumptions for the temperature dependence, e.g., a linear dependence resembling the behavior of the surface tension (Eötvös rule)^{19}.
Various thermodynamic integration (TI) methods in combination with offlattice Monte Carlo or molecular dynamics (MD) simulations have been employed to compute the vibrational contribution to the surface parameters, both potentialbased^{20,21,22} and by means of ab inito simulations^{23}. Unfortunately, the applicability of most TI variants is limited to solid systems that do not undergo an allotropic or magnetic phase transition in the temperature interval of interest.
The phase stability and mechanical properties of Fe are strongly influenced by magnetism and temperature^{24,25,26}. Magnetic long range order and magnetic correlations in Fe are responsible for the stability of the bodycentered cubic (bcc) (α) phase up to 1189 K significantly above the Curie temperature (T_{C} = 1043 K). The facecentered cubic (fcc) phase is stable in the temperature interval 1189 K–1662 K, but iron eventually melts from the reoccurring bcc (δ) phase at 1807 K. Surface magnetism in Fe was shown to have a pronounced impact on the magnitudes of surface energy and surface stress in the ground state^{17,27}.
This paper brings forwards an densityfunctional theory (DFT) description of γ(T) and σ(T) for the two most stable surface facets of bcc Fe [(001) and (110)]^{18,28}. Free energies [F = F(T, V)] were computed for a surface subsystem and for a bulk subsystem, from which the surface energies, γ = A^{−1}(F^{surf} − F^{bulk}), and the surface stresses, σ = (2A)^{−1}∂(F^{surf} − F^{bulk})/∂ε_{ε = 0}, were obtained^{2}. Here, ε specifies an isotropic and elastic inplane deformation of the surface with unit area A and σ represents half the trace of the surface stress tensor.
Theoretical modeling
We rely on the adiabatic approximation shown to be applicable to Fe^{29,30} and model the individual vibrational, electronic, and magnetic excitations to the free energy, F = E_{0} + F_{vib} + F_{el} + F_{mag}. E_{0} is the total energy of the ferromagnetic ground state. The vibrational free energy contribution (F_{vib}) and the bulk thermal expansion were derived within the quasiharmonic approximation (QHA)^{31} in the ferromagnetically ordered state using a gradientcorrected exchangecorrelation parameterisation (PBE)^{32}. The forceconstant matrix was obtained within the framework of densityfunctional perturbation theory^{31,33} as implemented in the projectoraugmented wave method (VASP)^{34} and employing Phonopy^{35} to compute the phonon density of states (DOS) and F_{vib}. The electronic (F_{el}) and magnetic (F_{mag} = E_{mag} − TS_{mag}) contributions to the free energy were obtained with the exact muffintin orbitals (EMTO) method and the full chargedensity technique^{36,37,38} using PBE in conjunction with the coherentpotential approximation (CPA)^{39,40}. F_{el} takes into account electronic excitations due to smearing of the FermiDirac distribution^{41,42}. The magnetic disorder effect on the total energy (E_{mag}) was approximated by the partially disorderedlocal moment (PDLM) model described within the framework of CPA^{43,44}. The PDLM model connects the ordered ferromagnetic phase with the disordered paramagnetic (DLM) phase without taking into account short range order^{45}. Accordingly, the gradual loss of magnetic longrange order, characterised by the staggered magnetisation (m, 1 ≥ m ≥ 0), is captured by a gradual concentration change (x, 0 ≤ x ≤ 0.5) of a random binary alloy with antiparallel moment orientation, , where x = 0 and x = 0.5 correspond to the ferromagnetic phase (m = 1) and the paramagnetic (DLM) phase (m = 0), respectively. To describe the loss of magnetisation as a function of temperature, m was mapped to T via an analytic representation (m = [1 − 0.35τ^{3/2} − 0.65τ^{4}]^{1/3}, τ = T/T_{C})^{46} of the experimental magnetisation curve of Fe^{47}. Magnetic entropy (S_{mag}) within the PDLM model was considered by the meanfield expression, S_{mag}/k_{B} = 2x ln(μ + 1), where k_{B} and μ denote the Boltzmann constant and the local magnetic moment in units of Bohr magneton, respectively. We notice that the measured magnetic entropy of paramagnetic bcc iron is in agreement with this meanfield estimate^{48}. Thermal lattice expansion was incorporated into the magnetic and electronic contributions via the lattice parameters obtained from the QHA for F_{vib}.
For the surface energy γ full explicit phononic contributions and thermal expansion were considered. For the surface stress σ, we included the thermal expansion effect only. That is because previous investigations found that the dominating phononic contribution to the surface stress (of atomically ordered surfaces) arises from thermal expansion rather than from a temperatureinduced change of the vibrational amplitude, even for open surface facets^{21}. The computational details can be found in the Method’s section.
Results and Discussion
Figure 1 shows the computed temperature dependence of surface free energy and surface stress for the (001) and (110) surface facets of bcc Fe. Rising temperature from 0 K to T_{m} leads to a monotonically decreasing surface energy for both faces, i.e., by 28% and 29% for (110) and (001), respectively. The closepacked (110) surface facet maintains the lower surface energy in the entire stability range of the bcc phase, but the anisotropy γ_{(110)}/γ_{(001)} slightly decreases with increasing temperature.
The surface stress of the (001) and (110) facets exhibits a nonmonotonic dependence on T (Fig. 1). While σ_{(001)} is significantly lower than σ_{(110)} in the fullyordered magnetic state at 0 K, they approach each other with increasing temperature mainly due to a significant increase of the surface stress of the (001) facet. The surface stresses of the two facets are nearly of the same magnitude in the paramagnetic phase.
Figure 2 shows how the individual magnetic, electronic, and vibrational contributions add up to the surface free energy change as a function of temperature for the (110) surface facet. Accordingly, lattice vibrations reduce the surface energy most significantly followed by magnetic disorder. The magnetic disorder contribution to γ below T_{C} originates mainly from the continuous loss of magnetic long range order, while the magnetic entropy determines the magnetic contribution above T_{C}. The overall small electronic disorder effect becomes significant only at temperatures higher than several hundreds of Kelvin. We note that the relative magnitudes of the individual contributions to γ are similar for the (001) surface facet. The absolute vibrational (magnetic) contribution to the surface free energy is, however, larger (smaller) for the open (001) facet than for the closepacked (110) facet. The different contributions due to phonons can be understood by comparing the phonon DOS for the two surface atomic layers with the bulk one as shown in the inset of Fig. 2 (DOSs computed for the equilibrium geometry). The reduced coordination number at both surface leads to an enhanced DOS at low phonon frequencies and a lower Debye temperature compared to the bulk counterpart. This softening of the vibrational modes is more pronounced for the (001) surface of Fe than for the (110) one^{49} resulting in a larger excess entropy at the more open surface facet.
In contrast to the monotonic trends seen for γ and the weakly temperature dependent σ_{(110)}, σ_{(001)} increases significantly between 0 K and T_{C} (Fig. 1) resulting in a strongly temperaturedependent surface stress anisotropy (σ_{(110)}/σ_{(001)}). We analyzed this striking behavior by considering the magnetic contribution (σ_{mag}) to the surface stress, where σ_{mag} is defined as the difference between the nonmagnetic value of the surface stress (no spinpolarization considered) and the surface stress from Fig. 1 (spinpolarization considered). The geometry of the lattice was kept fixed in this additional study. According to Punkkinen et al.’s^{17} analysis for magnetic metals, σ_{mag} was found to be proportional to the surface moment enhancement (Δμ^{2}), viz., , where . The surface magnetic moment (μ_{surf}) of Fe is larger than the bulk one (μ_{bulk}) leading to a compressive σ_{mag}, i.e., σ_{mag} tends to expand the lattice and to reduce the total tensile surface stress^{17}. Here, we introduced the facetdependent positive proportionality constant to reflect the influence of atomic coordination and surface electronic structure on σ_{mag}. Δμ^{2} for the open (001) facet was obtained by adding up the contributions from both the surface layer and the first subsurface layer, since μ differed significantly from μ_{bulk} in these two layers.
The proportionality between σ_{mag} and Δμ^{2} shown in Fig. 3 is excellent for the (110) surface facet and describes somewhat less satisfactorily the data of the (001) facet. σ_{mag} is nearly constant above T_{C} due to the small influence of thermal expansion on σ_{mag} and Δμ^{2} in the DLM state. We find that a temperature increase from 0 K to T_{m} has a more significant effect on Δμ^{2} for the (001) facet than for the (110) facet. This is plausible since the magnetic moments in the surface region of the densely packed (110) facet are smaller and converge more rapidly towards the bulk value compared to the situation at the open (001) facet^{18}. We also observe that indicating that the enhanced surface magnetic moment for the (110) surface facet generates a larger surface stress than in the case of (001). Taking into account that the unit area per surface atom for (110) is a factor of smaller than for (001), the magnetic surface stress per atom for (001) is still ≃43% smaller than that for (110). A possible explanation lies in the lower coordination number, and hence the smaller dbandwidth, for an Fe atom situated at the (001) surface compared to when it is located at the (110) surface. Since is inversely proportional to the height of the singleparticle DOS^{17,50}, should roughly scale with the width of the dband. Thus, the difference in the surface atom coordination number confirms our finding .
Available experimental data for γ of δ Fe allows for a comparison with the present ab initio surface energies. Price reported γ = 1.95 J/m^{2} ± 10% in the temperature range 1673 K – T_{m}^{14}. Our prediction for γ slightly below T_{m} lies within the uncertainty of the experiment, see Fig. 1. The jump of γ across the liquidsolid phase transition may further be used to estimate γ(T_{m}) from γ_{LV} adding to it the latent heat of fusion per surface area, viz., γ(T_{m}) ~ ΔH_{f}/A + γ_{LV}^{51}. Using available values for Fe, ΔH_{f} = 13.81 kJ/mol^{52} and γ_{LV} = 1.78 ± 0.08 J/m^{2} ^{13}, we arrive at 2.04 ± 0.08 J/m^{2} for the (001) facet and at 2.15 ± 0.08 J/m^{2} for the (110) facet. These values are somewhat larger than our predicted surface energies at T_{m}, γ_{(110)} = 1.76 J/m^{2} and γ_{(001)} = 1.79 J/m^{2}. A possible reason for the underestimation of γ near T_{m} could be that the vibrational contribution to γ for paramagnetic Fe is somewhat overestimated by the present theory. An estimation of the influence of the magnetic state of Fe on the vibrational contributions to the surface energy can be found in the Supplementary Information online. For the (110) surface facet, we found that the surface vibrational excess is nearly identical in the two magnetic phases. This indicates that the here reported vibrational contribution to γ_{(110)} is not expected to be significantly altered by the magnetic state of Fe. For the (001) surface facet, we estimated that the surface vibrational excess to γ in the paramagnetic state is ≃40% smaller than in the ferromagnetic state. This corresponds to an underestimation of γ_{(001)} by 0.1 J/m^{2} at T_{m}. The updated estimate for γ_{(001)}(T_{m}), accounting for the reduced vibration excess in the paramagnetic state of Fe, is 1.89 J/m^{2} in closer agreement with the experimental data from Price^{14} and the above value derived from γ_{LV}.
Semiempirical estimates of γ(0 K) for an average surface facet, derived from experimental surfacetension data and estimated configurational and vibrational contributions, are available for comparison: γ(0 K) = 2.42 J/m^{2} from ref. 12 based on a nonlinear extrapolation scheme, and γ(0 K) = 2.48 J/m^{2} from ref. 19 assuming that γ is a linear function of T. Magnetic disorder was not considered in either approach, although the decomposition of γ(T) (Fig. 2) suggests that the magnetic part should be taken into account for a more accurate low temperature estimate. We quantified the magnetic part for an average surface facet on the basis of the presently calculated magnetic contribution to the surface energy. To this end, the magnetic contributions to γ_{(001)} and γ_{(110)} in the range 0 K to T_{m} were weighted by their fractional areal contributions to the equilibrium shape of a 3D crystallite according to the Wulff construction^{10} (other crystal faces possessing larger surface energies were not considered, as they contribute only by small fractions to the equilibrium shape^{18}). Using the present 0 K values, γ_{(110)} = 2.46 J/m^{2} and γ_{(001)} = 2.53 J/m^{2}, the fractions of {001} facets and of {110} facets to the equilibrium crystal shape are 0.23 and 0.77, respectively. This gives an average magnetic contribution J/m^{2}, which when added to the aforementioned estimated figures results in 2.56 J/m^{2} (with the value from ref. 12) and 2.62 J/m^{2} (ref. 19) respectively. The former estimate is in very good agreement with the present values.
A possible measure to indicate surface reconstruction is based on a bulk continuum elastic model, viz., 0.1–0.2, where the upper limit corresponds to more corrugated surface facets^{1,53}. Accordingly, the larger is the excess surface stress (γ − σ), or the smaller is the shear modulus (G) [the length of the Burger’s vector (b) is nearly constant], the higher is the tendency to reconstruct. The predicted trends for γ and σ (Fig. 1) indicate that the excess surface stress and hence the propensity to reconstruct would decrease with rising temperature for both surface facets. However, the observed strong softening of the shear elastic constants with increasing temperature^{24,25} works against the excess surface stress. Expressing G through the Hill average^{54} of the experimental singlecrystalline elastic constants from refs 24,25 and evaluating κ for α Fe in the ferromagnetic and paramagnetic phases, we found that κ decreases as T increases. The largest κ values (at 0 K) are 0.09 and 0.02 for the (001) facet and the (110) facet, respectively, indicating stable surfaces. This prediction is in line with observations, namely, no surface reconstruction was observed for the (110) and (001) facets of Fe in the temperature range 300–500 K^{55}.
To conclude, temperature has a pronounced effect on the surface energy and the surface stress of bcc Fe. The surface energies of the (001) and (110) facets decrease nonlinearly by approximately 30% from 0 K up to the melting point with slight reduction of their anisotropy. This reduction of the work of separation is primarily due to lattice vibrations and magnetic disorder. There is good agreement between the present surface energies and the 0 K values from Tyson et al.’s and de Boer et al.’s databases^{12,19} especially if the latter values are corrected for the missing magnetic term. The surface stresses are nonmonotonic functions of temperature exhibiting a strongly temperature dependent surface stress anisotropy. Surface stress originating from the thermal magnetic disorder at the (001) surface is more strongly affected by temperature than the one at (110) and is mainly responsible for the pronounced increase of the stress at (001) below the Curie temperature. We predict that the propensity for surface reconstruction at (001) and (110) lowers as temperature increases.
Today, the theoretical description of metal surfaces at elevated temperature is a very challenging task. In order to make such a study feasible in the case of complex systems, one has to adopt several simplifications and approximations. Despite of that, the present theoretical results turn out to follow closely the previous semiempirical estimates, demonstrating the predictive power of our ab initio tools in the case of hightemperature surface properties. These finding offers a solid platform for extending the existing theoretical methodology to study the surface properties of technological alloys.
Methods
Convergence of all numerical parameters was carefully checked. Bulk and surface reference systems were of identical size to ensure numerical error cancellation. F_{vib} for the surface subsystem was obtained from 32–34 Å thick slabs separated by 10–14 Å vacuum. The relaxation at 0 K included all interlayer distances perpendicular to the surfaces until the residual forces on each layer were smaller than 0.05 meV/Å. To compute the force constant matrix (VASP), we used a plane wave cutoff of 500 eV and a kpoint mesh equivalent to a 24 × 24 × 2(1) MonkhorstPack mesh of the primitive bulk (surface) reference system. The phonon properties were sampled on a 3 × 3 × 1 mesh. All derived vibrational quantities that involve an integration of the phonon density of states (i.e., free energies) are converged to within ±2% against a denser 4 × 4 × 1 sampling mesh. We checked the temperaturedependent surface interlayer relaxation of Fe within the QHA according to ref. 56. We found that relaxation at the (110) facet, determined at 1043 K and at 1805 K, change γ and σ by <2% and by <4%, respectively. Since this relaxation effect represents only a small correction to both γ and σ at significantly higher computational cost, finite temperature surface interlayer relaxation was not included in the present F_{vib}, i.e., the relaxed T = 0 K interatomic distances were rigidly rescaled according to the bulk thermal expansion.
The magnetic and electronic disorder effects were modelled by slabs with a converged thicknesses of 13 atomic layers and 9 atomic layers for the (001) surface and the (110) surface, respectively, separated by ~10 Å vacuum. Surface energies and surface stresses calculated with EMTO were sampled on a 15 × 15 × 2(1) kpoint mesh in the case of the bulk (surface) reference system.
Additional Information
How to cite this article: Schönecker, S. et al. Thermal surface free energy and stress of iron. Sci. Rep. 5, 14860; doi: 10.1038/srep14860 (2015).
References
 1.
Ibach, H. The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures. Surf. Sci. Rep. 29, 193 (1997).
 2.
Müller, P. & Saúl, A. Elastic effects on surfaces physics. Surf. Sci. Rep. 54, 157 (2004).
 3.
Métois, J. J., Saúl, A. & Müller, P. Measuring the surface stress polar dependence. Nat. Mat. 4, 238 (2005).
 4.
Bach, C. E., Giesen, M., Ibach, H. & Einstein, T. L. Stress relief in reconstruction. Phys. Rev. Lett. 78, 4225–4228 (1997).
 5.
Ibach, H. The relation between the straindependence of the heat of adsorption and the coverage dependence of the adsorbate induced surface stress. Surf. Sci. 556, 71–77 (2004).
 6.
Olivier, S., Saúl, A. & Tréglia, G. Relation between surface stress and (1×2) reconstruction for (110) fcc transition metal surfaces. Appl. Surf. Sci. 212–213, 866–871 (2003).
 7.
Biener, J. et al. Surfacechemistrydriven actuation in nanoporous gold. Nat. Mat. 8, 47 (2008).
 8.
Ndieyira, J. W. et al. Surfacestress sensors for rapid and ultrasensitive detection of active free drugs in human serum. Nat. Nanotechnol. 9, 225–232 (2014).
 9.
Langmuir, I. The constitution and fundamental properties of solids and liquids. J. Am. Chem. Soc. 38, 2221–2295 (1916).
 10.
Wulff, G. Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Krystallflächen. Z. Kristall. Mineral. 34, 449–530 (1901).
 11.
Sander, D. Surface stress: implications and measurements. Curr. Opin. Solid St. M. 7, 51–57 (2003).
 12.
Tyson, W. R. & Miller, W. A. Surface free energies of solid metals: estimation from liquid surface tension measurements. Surf. Sci. 62, 267 (1977).
 13.
Allen, B. C. The surface tension of liquid transition metals at their melting points. Trans. Metal. Soc. AIME 227, 1175 (1963).
 14.
Price, A. T., Holl, H. A. & Greenough, A. P. The surface energy and self diffusion coefficient of solid iron above 1350 °C. Acta Metall. 12, 49 (1964).
 15.
Vitos, L., Ruban, A., Skriver, H. L. & Kollár, J. The surface energy of metals. Surf. Sci. 411, 186–202 (1998).
 16.
Needs, R. J. Calculations of the surface stress tensor at aluminum (111) and (110) surfaces. Phys. Rev. Lett. 58, 53 (1987).
 17.
Punkkinen, M. P. J., Kwon, S. K., Kollár, J., Johansson, B. & Vitos, L. Compressive surface stress in magnetic transition metals. Phys. Rev. Lett. 106, 057202 (2011).
 18.
Błoński, P. & Kiejna, A. Structural, electronic, and magnetic properties of bcc iron surfaces. Surf. Sci. 601, 123–133 (2007).
 19.
de Boer, F. R., Boom, R., Mattens, W. C. M., Miedema, A. R. & Niessen, A. K. Cohesion in metals: Transition Metal Alloys (NorthHolland, Amsterdam, 1988).
 20.
Hansen, U., Vogl, P. & Fiorentini, V. Quasiharmonic versus exact surface free energies of Al: A systematic study employing a classical interatomic potential. Phys. Rev. B 60, 5055–5064 (1999). URL http://link.aps.org/doi/10.1103/PhysRevB.60.5055.
 21.
Frolov, T. & Mishin, Y. Temperature dependence of the surface free energy and surface stress: An atomistic calculation for Cu(110). Phys. Rev. B 79, 045430 (2009).
 22.
Grochola, G., Russo, S. P., Snook, I. K. & Yarovsky, I. On simulation methods to compute surface and interfacial free energies of disordered solids. J. Chem. Phys. 116, 8547 (2002).
 23.
Fox, H., Horsfield, A. P. & Gillan, M. J. Density functional calculations of surface free energies. J. Chem. Phys. 124, 134709 (2006).
 24.
Rayne, J. A. & Chandrasekhar, B. S. Elastic constants of iron from 4.2 to 300 K. Phys. Rev. 122, 1714–1716 (1961). URL http://link.aps.org/doi/10.1103/PhysRev.122.1714.
 25.
Dever, D. J. Temperature dependence of the elastic constants in αiron single crystals: relationship to spin order and diffusion anomalies. J. Appl. Phys. 43, 3293 (1972).
 26.
Zhang, H. et al. Anomalous elastic hardening in FeCo alloys at high temperature. Phys. Rev. B 89, 184107 (2014). URL http://link.aps.org/doi/10.1103/PhysRevB.89.184107.
 27.
Aldén, M., Skriver, H. L., Mirbt, S. & Johansson, B. Calculated surfaceenergy anomaly in the 3d metals. Phys. Rev. Lett. 69, 2296 (1992).
 28.
Punkkinen, M. P. J. et al. Surface properties of 3d transition metals. Philos. Mag. 91, 3627 (2011).
 29.
Körmann, F. et al. Free energy of bcc iron: Integrated ab initio derivation of vibrational, electronic, and magnetic contributions. Phys. Rev. B 78, 033102 (2008).
 30.
Sabiryanov, R. F. & Jaswal, S. S. Magnons and magnonphonon interactions in iron. Phys. Rev. Lett. 83, 2062–2064 (1999). URL http://link.aps.org/doi/10.1103/PhysRevLett.83.2062.
 31.
Baroni, S., Giannozzi, P. & Isaev, E. Densityfunctional perturbation theory for quasiharmonic calculations. Rev. Mineral. Geochem. 71, 39–57 (2010). URL http://rimg.geoscienceworld.org/content/71/1/39.short.
 32.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
 33.
Baroni, S., De Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal properties from densityfunctional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001).
 34.
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
 35.
Togo, A., Oba, F. & Tanaka, I. Firstprinciples calculations of the ferroelastic transition between rutiletype and CaCl_{2}type SiO_{2} at high pressures. Phys. Rev. B 78, 134106 (2008).
 36.
Andersen, O. K., Jepsen, O. & Krier, G. Lectures on Methods of Electronic Structure Calculations, 63 (World Scientific, Singapore, 1994).
 37.
Vitos, L. Totalenergy method based on the exact muffintin orbitals method. Phys. Rev. B 64, 014107 (2001).
 38.
Vitos, L., Skriver, H. L., Johansson, B. & Kollár, J. Application of the exact muffintin orbitals theory: the spherical cell approximation. Comp. Mat. Sci 18, 24 (2000).
 39.
Györffy, B. L. Coherentpotential approximation for a nonoverlappingmuffintinpotential model of random substitutional alloys. Phys. Rev. B 5, 2382 (1972).
 40.
Vitos, L., Abrikosov, I. A. & Johansson, B. Anisotropic lattice distortions in random alloys from firstprinciples theory. Phys. Rev. Lett 87, 156401 (2001).
 41.
Mermin, N. D. Thermal properties of the inhomogeneous electron gas. Phys. Rev. 137, A1441–A1443 (1965). URL http://link.aps.org/doi/10.1103/PhysRev.137.A1441.
 42.
Wildberger, K., Lang, P., Zeller, R. & Dederichs, P. H. FermiDirac distribution in ab initio Green’sfunction calculations. Phys. Rev. B 52, 11502 (1995).
 43.
Khmelevskyi, S., Turek, I. & Mohn, P. Large negative magnetic contribution to the thermal expansion in ironplatinum alloys: Quantitative theory of the Invar effect. Phys. Rev. Lett. 91, 037201 (2003). URL http://link.aps.org/doi/10.1103/PhysRevLett.91.037201.
 44.
Ruban, A. V., Korzhavyi, P. A. & Johansson, B. Firstprinciples theory of magnetically driven anomalous ordering in bcc FeCr alloys. Phys. Rev. B 77, 094436 (2008).
 45.
Staunton, J., Györffy, B. L., Pindor, A. J., Stocks, G. M. & Winter, H. The “disordered local moment” picture of itinerant magnetism at finite temperatures. J. Magn. Magn. Mater. 45, 15–22 (1984).
 46.
Kuz’min, M. D. Shape of temperature dependence of spontaneous magnetization of ferromagnets: Quantitative analysis. Phys. Rev. Lett. 94, 107204 (2005).
 47.
Crangle, J. & Goodman, G. M. The magnetization of pure iron and nickel. Proc. Roy. Soc. Lond. A 321, 477–491 (1971).
 48.
Grimvall, G. Polymorphism in metals II. electronic and magnetic free energy. Phys. Scripta 12, 173 (1975). URL http://stacks.iop.org/14024896/12/i=3/a=011.
 49.
Łażewski, J., Korecki, J. & Parlinski, K. Phonons of (100) and (110) iron surfaces from firstprinciples calculations. Phys. Rev. B 75, 054303 (2007). URL http://link.aps.org/doi/10.1103/PhysRevB.75.054303.
 50.
Andersen, O. K., Madsen, J., Poulsen, U. K., Jepsen, O. & Kollár, J. Magnetic ground state properties of transition metals. Physica 86–88B, 249–256 (1977).
 51.
Bondi, A. The spreading of liquid metals on solid surfaces. Surface chemistry of highenergy substances. Chem. Rev. 52, 417–458 (1953).
 52.
Haynes, W. M. (ed.) CRC Handbook of Chemistry and Physics (CRC Press/Taylor and Francis, Boca Raton, USA, 2014), 95 (internet version 2015) edn.
 53.
Cammarata, R. C. Continuum model for surface reconstruction in (111) and (100) oriented surfaces of fcc metals. Surf. Sci. 279, 341–348 (1992).
 54.
Hill, R. The elastic behaviour of a crystalline aggregate. P. Phys. Soc. Lond. A 65, 349 (1952). URL http://stacks.iop.org/03701298/65/i=5/a=307.
 55.
Fasolino, A., Selloni, A. & Shkrebtii, A. 2.2 Surface reconstruction and relaxation. In Chiarotti, G. (ed.) LandoltBörnsteinGroup III Condensed Matter: Numerical Data and Functional Relationships in Science and Technology, vol. 24a (SpringerVerlag, Berlin, 1993).
 56.
Xie, J., de Gironcoli, S., Baroni, S. & Scheffler, M. Temperaturedependent surface relaxations of Ag(111). Phys. Rev. B 59, 970–974 (1999). URL http://link.aps.org/doi/10.1103/PhysRevB.59.970.
Acknowledgements
The Swedish Research Council, the Swedish Foundation for Strategic Research, the China Scholarship Council, the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (NRF2014R1A2A1A12067579), and the Hungarian Scientific Research Fund (research projects OTKA 84078 and 109570) are acknowledged for financial support. The computations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre in Linköping.
Author information
Affiliations
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE10044, Sweden
 Stephan Schönecker
 , Xiaoqing Li
 , Börje Johansson
 & Levente Vitos
Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE75120, Uppsala, Sweden
 Börje Johansson
 & Levente Vitos
Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang 790784, Korea
 Se Kyun Kwon
Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, Budapest H1525, P.O. Box 49, Hungary
 Levente Vitos
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Contributions
S.S. and X.L. performed the calculations. S.S. analysed the results. S.S. and L.V. wrote the manuscript. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Stephan Schönecker.
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Further reading

1.
Scientific Reports (2016)
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