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Influence of chemical composition and magnetic effects on the elastic properties of fcc Fe–Mn alloys
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Acta Materialia 59 (2011) 1493–1501 www.elsevier.com/locate/actamat
Influence of chemical composition and magnetic effects on the elastic properties of fcc Fe–Mn alloys T. Gebhardt a,⇑, D. Music a, M. Ekholm b, I.A. Abrikosov b, J. von Appen c, R. Dronskowski c, D. Wagner d, J. Mayer d, J.M. Schneider a b
a Materials Chemistry, RWTH Aachen University, D-52056 Aachen, Germany Department of Physics, Chemistry and Biology (IFM), Linko¨ping University, SE-58183 Linko¨ping, Sweden c Chair of Solid-State and Quantum Chemistry, RWTH Aachen University, D-52056 Aachen, Germany d Central Facility for Electron Microscopy, RWTH Aachen University, D-52056 Aachen, Germany
Received 19 July 2010; received in revised form 29 October 2010; accepted 4 November 2010 Available online 10 December 2010
Abstract The influence of the Mn content on the elastic properties of face centered cubic Fe–Mn alloys was studied using the combinatorial approach. Fe–Mn thin films with a graded chemical composition were synthesized. Nanoindentation experiments were carried out to investigate the elastic properties as a function of the Mn content. As the Mn content increases from 23 to 39 at.%, the average bulk modulus varies from 143 to 105 GPa. Ab initio calculations served to probe the impact of magnetic effects on the elastic properties. The magnetic state description with disordered local moments yields the best agreement with the experimental results, whereas with non-magnetic and antiferromagnetic configurations the bulk modulus is overestimated. The strong impact of the magnetic configuration may be understood based on the differences in the chemical bonding and the magnetovolume effect. It is suggested that, owing to minute energy differences of competing antiferromagnetic configurations, a mixture of these with a “notional magnetic disorder” is present, which is in fact well described by the disordered local moments model. These results show that the combinatorial thin film synthesis with subsequent nanoindentation is an appropriate tool for investigating the elastic properties of Fe–Mn alloys systematically as a function of the chemical composition, to validate theoretical models. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Iron alloys; Sputtering; Elastic properties; Ab initio calculations; Nanoindentation
1. Introduction Fe–Mn alloys exhibit fascinating properties for a wide range of applications [1–4]. These properties are mainly related to the face centered cubic (fcc) ? hexagonal close packed (hcp) martensitic transformation [1,2] as well as the magnetism and magnetic transformations present in these alloys [3,5]. Regarding the martensitic transformation, Enami et al. [1] reported the occurrence of the socalled shape-memory effect (SME) in binary Fe–Mn alloys. The SME strongly depends on the alloy composition, ⇑ Corresponding author. Tel.: +49 241 80 25984; fax: +49 241 80 22295.
E-mail address: [email protected] (T. Gebhardt).
where additions of Si result in a nearly complete SME [6,7]. High-Mn steels feature high strength and exceptional plasticity owing to twin formation (twinning induced plasticity (TWIP effect)) or via multiple martensitic transformations (transformation induced plasticity (TRIP effect)) under mechanical load [2,8,9], therefore providing great potential for structural components in automotive engineering [10]. The stacking fault energy (SFE) of the austenite governs the deformation behavior. The chemical composition exhibits a strong impact on the SFE [9]. With respect to magnetism, the compositionally disordered fcc Fe–Mn alloys are antiferromagnetically ordered [5], with several inconsistent proposals for the antiferromagnetic (AFM) ground state [11–14]. Furthermore, fcc Fe–Mn
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.11.013
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alloys feature the so-called Invar and anti-Invar effects in the AFM and paramagnetic (PM) state, respectively, caused by magnetovolume instabilities [3], resulting in anomalous magnetization, thermal expansion, heat capacity and elastic properties [15]. The Invar and, even more pronounced, the anti-Invar effect strongly depend on the chemical composition [3]. Recently, Hermawan et al. [4] proposed the use of Fe–Mn alloys as a new class of metallic degradable biomaterials. The chemical composition has a strong impact on the aforementioned effects and properties. A previous study investigated the influence of the Mn content on the elastic properties of fcc Fe–Mn random alloys with non-magnetic (NM) and PM configurations using ab initio calculations [16]. However, a systematic experimental study of the dependence of the elastic properties of Fe–Mn alloys on the Mn content has not been reported. The present work systematically investigates the elastic properties of these alloys by combinatorial thin film synthesis and subsequent nanoindentation experiments. Furthermore, the impact of magnetic effects on the elastic properties is probed using ab initio calculations. It is the purpose of the present paper to contribute towards understanding the correlation between structure, chemical composition, magnetic configuration and elastic properties. Within a Mn content range from 22 to 40 at.% phase pure fcc Fe–Mn thin films with a graded chemical composition were synthesized. As the Mn content increases from 23 to 39 at.%, the average bulk modulus decreases from 143[±34] to 105[±12] GPa. The experimental results were compared with ab initio data, whereas describing the magnetic state of fcc Fe–Mn random alloys with disordered local moments (DLM) yields the best agreement. NM and AFM configurations overestimate the bulk modulus. The strong impact of the magnetic state on the elastic properties can be understood, on the one hand, based on the differences in chemical bonding and, on the other hand, based on the magnetovolume effect, which softens the lattice and which also leads to the observed decrease in bulk modulus with increasing Mn content. The fact that the DLM and not the AFM ground state description of the magnetic configuration results in the best agreement between measured and calculated elastic properties might be due to minute energy differences between competing AFM configurations and therefore a “notional magnetic disorder” at room temperature (RT). This configuration is well described by the DLM model. This description of the magnetic configuration may be useful for the determination of the elastic properties of other Fe-based alloys with AFM ground states at elevated temperature approaching the magnetic transition temperature.
4.0 W cm 2. The deposition geometry results in the formation of a lateral gradient in chemical composition, since the deposited flux from each plasma source is not uniformly distributed over the whole substrate surface area. This constitutes the so-called combinatorial thin film synthesis (Fig. 1). Depositions took place in an Ar atmosphere at 0.8 Pa. Sapphire (0 0 0 1) wafers with a diameter of 50 mm were used as substrates. The substrate, fixed on a 1 mm Cu plate to homogenize the temperature, was heated to 400 °C. After a deposition time of 120 min, which resulted in a film thickness of 1.7 lm, the samples were cooled down in the load-lock chamber. The substrate-totarget distance was 10 cm, with an inclination angle of the cathodes of 19°. For phase identification, grazing incidence X-ray diffraction measurements were carried out in a Bruker AXS D8 Discover General Area Diffraction Detector System equipped for micro-diffraction with a Cu Ka source. A 13 1 grid with a lateral spacing of 3.5 mm along the Mn gradient was analyzed on each sample. The primary beam was collimated using a pin hole with a diameter of 0.5 mm at an incidence angle of 15°, resulting in an irradiated surface ellipse with a = 1.24 mm and b = 0.32 mm. Energy dispersive X-ray analysis with an EDAX Genesis2000 detector was used to map the chemical composition along the 13 1 grid. Morphological investigations were performed by scanning electron microscopy (SEM) using a JEOL JSM-6480 microscope. The elastic properties of the graded Fe–Mn thin films were investigated at RT using a depth-sensing nanoindenter (Hysitron TriboIndentere) equipped with a Berkovich indenter tip. The Oliver–Pharr method was used to deter-
2. Experimental methods Fe–Mn films were grown in an ultra-high-vacuum chamber using DC-magnetron sputtering from an elemental Fe target and a FeMn compound target (Fe/Mn-ratio equal to 1) applying power densities of 1.5–2.5 and 3.0–
Fig. 1. Deposition setup for the combinatorial Fe–Mn alloy thin film synthesis. The color gradient schematically shows the evolving gradient in chemical composition. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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mine the reduced elastic modulus from the unloading curves [17]. A fused quartz specimen served as the reference material for calibration of the tip area function. A series of nine indentations with loads of 1.5 mN was performed at each grid position selected for further investigation, resulting in contact depths <10% of the film thickness. The bulk modulus values used in this work were calculated from the reduced elastic modulus values using the Poisson’s ratio obtained from ab initio calculations [16]. The deformation behavior of the Fe–Mn films was studied by means of a combination of nanoindentation and cross-sectional transmission electron microscopy (TEM) techniques. A single indent was carried out with a sharp cube-corner tip at 7 mN at a medium Mn concentration. The cube-corner tip was used to keep the plastic deformation to a more confined volume. For preparation of the TEM specimen, a FEI Strata FIB 205 workstation was used. The TEM sample was examined in a JEOL JEM 2000 FX TEM instrument operating at an acceleration voltage of 200 kV. 3. Theoretical methods The theoretical study was carried out based on density functional theory [18], as implemented in the exact muffin-tin orbitals (EMTO) formalism [19,20], the Vienna ab initio simulation package (VASP) [21,22], the OpenMX code [23] and the linear muffin-tin orbital (LMTO) theory [24–26]. The EMTO, VASP and OpenMX codes were used to calculate the elastic properties with different magnetic configurations. Chemical bonding analysis was carried out by the LMTO method. The EMTO code, based on Green’s function [27] and full charge density techniques [20], served to calculate the elastic properties of fcc Fe–Mn random alloys with AFM and ferromagnetic (FM) configurations. The AFM alloys were described with a 1Q, 2Q and 3Q spin structure. The local density approximation was used to calculate self-consistent charge density, the total energy was evaluated using the generalized gradient approximation [28], and the ion core states were kept frozen. The chemical disorder, reported by Endoh and Ishikawa [5], was treated with the coherent potential approximation [29,30]. The k-space sampling of the irreducible part of the Brillouin zone was done using a 21 21 21 grid for the AFM and a 13 13 13 grid for the FM alloys, respectively. The total energy convergence criterion was 10 7 Ry. The bulk moduli of the investigated Fe–Mn alloys were obtained by fitting the energy vs volume data with the Birch–Murnaghan equation of state [31]. The VASP code was used to treat chemically disordered 3Q configurations with 25.0 and 37.5 at.% Mn. The projector augmented wave potentials [32] with the generalized gradient approximation [33] were employed. The following parameters were applied: convergence criterion for the total energy of 0.01 meV; Blo¨chl corrections [34] for the total energy cut-off of 300 eV; and integration in the Brillouin zone according to Monkhorst–Pack [35] with
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5 5 5 k-points. Following the special quasirandom structure (SQS) approach [36], 2 2 2 supercells containing 32 atoms were created with random positions of Fe and Mn, using the locally self-consistent Green’s function software package [37,38]. In the software, the Warren–Cowley SRO parameter [39] was employed to quantify randomness, which was close to zero within seven coordination shells. The OpenMX code, based on basis functions in the form of linear combination of localized pseudoatomic orbitals [40] was used to treat chemically disordered non-collinear FM configurations with 25.0 and 37.5 at.% Mn. The electronic potentials were fully relativistic pseudopotentials with partial core corrections [41,42], and the generalized gradient approximation was applied [28]. The basis functions were generated by a confinement scheme [40,43] and specified as follows: Fe5.0-s1p2d1 and Mn7.0-s3p3d2. The first symbol designates the chemical name, followed by the cut-off radius (in Bohr radius) in the confinement scheme, and the last set of symbols defines primitive orbitals applied. The energy cut-off (150 Ry) and k-point grid (1 1 1) within the real space grid technique [44] were adjusted to reach an accuracy of 10 6 H atom 1. As described above for the 3Q VASP calculations, 2 2 2 supercells containing 32 atoms were created with SQS random positions of Fe and Mn. The calculations based on the LMTO method for NM Fe–Mn alloys and structures with 1Q configuration with 25.0 at.% Mn were performed in its tight-binding representation [45]. The TB-LMTO-ASA 4.7 program code [46] with the atomic-spheres approximation was used. All calculations were checked for convergence of total energies and magnetic moments with respect to the number of kpoints. The chemical bonding situation was investigated using Crystal Orbital Hamilton Population (COHP) analysis [47] with a real-space partitioning technique of the band-structure energy (sum of the one-electron eigenvalues). A COHP curve is an energy resolved plot for each bond. Integrated up to the Fermi level, the so-called ICOHP values measure the sum of all bonding and anti-bonding interactions and thus serve as an indication of the bond strength. 4. Results and discussion Two Fe–Mn combinatorial thin films were synthesized to obtain the desired Mn content range from 20 to 40 at.%. Applying power densities of 1.5 and 4.0 W cm 2 at the Fe and FeMn target, respectively, resulted in a film with a compositional gradient from 32 to 40 at.% Mn. The crystallographic structure of this film is shown in Fig. 2. Within the whole Mn content range, the sample consists of phase pure fcc Fe–Mn. A compositional gradient from 20 to 32 at.% Mn was obtained by applying power densities of 2.5 and 3.0 W cm 2 at the Fe and FeMn target, respectively. The crystallographic structure of this film is shown in Fig. 3.
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Fig. 2. Phase evolution as a function of the Mn content for 32–40 at.% Mn. The intensity is displayed with a logarithmic scale.
Fig. 3. Phase evolution as a function of the Mn content for 20–32 at.% Mn. The intensity is displayed with a logarithmic scale.
The film consists of phase pure fcc Fe–Mn for Mn contents above 23 at.%. At lower Mn contents, the film is X-ray amorphous. Thus, phase pure fcc Fe–Mn was synthesized within a Mn content range from 23 to 40 at.%. Both films are textured, which is often observed in magnetron sputtered thin films. Owing to an inhomogeneous temperature distribution at the substrate during deposition, both films exhibit a pronounced surface roughness at the edges of the wafer. Therefore these areas were excluded from further characterization of the elastic properties by nanoindentation. A bright-field TEM image of a Fe–Mn thin film with 34 at.% Mn after being indented with a cube-corner tip is displayed in Fig. 4. No evidence for the formation of pores or cavities can be seen in this image, suggesting the growth of dense Fe–Mn. Furthermore, it shows that, at this
composition, the deformation features underneath and in the vicinity of the indented spot are primarily manifested by dislocation activities (positions 1, 2 and 3). Sub-grain boundaries (position 4) and twin formation can be seen (position 5). Selected area electron diffraction verified that regions with low (position 6) as well as with high dislocation densities (positions 1, 2 and 3) are composed of phase pure fcc Fe–Mn. Thus, no martensitic phase transformation was induced with a cube-corner tip at a load of 7 mN at this composition. The elastic properties of the Fe–Mn films were measured along the Mn concentration gradient by nanoindentation. The resulting bulk modulus values (white and black pentagons, respectively) are plotted in Fig. 5. Furthermore, calculated bulk moduli for magnetically ordered (AFM, 1blue triangles; FM, green circles) and disordered (red squares) fcc Fe–Mn random alloys as well as bulk data from Lenkkeri and Levoska [48] (open star) and Cankurtaran et al. [49] (filled star) are presented. The magnetic state of the magnetically disordered fcc Fe–Mn random alloys is described using the DLM model [50]. Details of these calculations can be found elsewhere [16]. The experimentally obtained bulk modulus values were calculated from the measured reduced elastic modulus values using a linear dependence of Poisson’s ratio on the Mn content, obtained from ab initio calculations of the elastic constants of DLM fcc Fe–Mn alloys [16] assuming crystallographic isotropy and using the Voigt–Reuss–Hill method [51] to determine the shear modulus (white pentagons) as well as using a mean Poisson’s ratio of 0.25 (black pentagons). The Poisson’s ratios calculated using the linear dependence are consistent with the data from Lenkkeri and Levoska [48] ( 6.7% deviation) and Cankurtaran et al. [49] (+1.3% deviation). Taking the dependence of Poisson’s ratio on the Mn content into account, the experimentally determined bulk modulus decreases from 143[±34] GPa at 23 at.% Mn to 105[±12] GPa at 39 at.% Mn. The nanoindentation data are in good agreement with the bulk data as well as with the DLM solutions of fcc Fe–Mn random alloys. The decreasing trend with increasing Mn content is depicted very well. This decrease in the bulk modulus can be understood based on the magnetovolume effect [16]. It is reasonable to assume that the elastic properties are affected in addition to the composition by the deposition geometry employed here, as well as by the presence of residual stresses. Generally, the elastic modulus of thin films is primarily dependent on the thin film density [52]. Lintymer et al. [53] investigated the modification of mechanical properties of sputter-deposited chromium thin films by glancing angle deposition using nanoindentation. Up to a vapor incidence angle of a = 30°, no significant change in Young’s modulus could be determined. When a was further increased, a noticeable decrease in Young’s
1 For interpretation of color in Figs. 5 and 8, the reader is referred to the web version of this article.
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Fig. 4. Bright-field TEM image of Fe–Mn thin film with 34 at.% Mn subjected to an indentation load of 7 mN with a cube-corner tip. The diffraction patterns show that areas 1–3 are austenitic.
Fig. 5. Bulk moduli as obtained using nanoindentation (white and black pentagons) as well as using ab initio calculations (AFM, blue triangles; FM, green circles; DLM, Music et al. [16], red squares). Results are compared with experimental data from Lenkkeri and Levoska [48] (open star) and Cankurtaran et al. [49] (filled star).
modulus was observed, which was attributed to the evolution of shadowing-induced porosity. TEM investigations revealed the growth of dense Fe–Mn thin films. Furthermore, and most importantly, the effect of elastic anisotropy
on the indentation modulus according to Nix [54] and Kooi et al. [55] is not large, even for highly anisotropic materials. Based on the good agreement between the present nanoindentation results and published bulk data [48,49], a minor influence of residual stresses is inferred, as well as the deposition geometry on the elastic properties reported here. The nanoindentation data were measured at RT. Within the regarded composition range, fcc Fe–Mn random alloys have previously been shown to exhibit AFM order at RT [5]. Hence the DLM solution does not represent the ground state. However, with respect to the elastic properties, the description of the magnetic state as completely disordered within the DLM model yields the best agreement with experiment, whereas the AFM 3Q ordered configuration results in bulk modulus values 10% and 30% larger than the experimental ones for 25.0 and 37.5 at.% Mn, respectively. Furthermore, Music et al. [16] observed that the bulk modulus values for NM structures are a factor of 2 higher compared with the DLM state. To explore the influence of the magnetic configuration on the elastic properties of Fe–Mn random alloys, the chemical bonding of magnetically ordered and NM structures at ambient and high pressures was analyzed. The ground-state volume of all considered magnetic structures is 9% larger than for NM ones, which is mirrored in
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the atomic (on-site) and bonding (off-site) energies of the band-structure energy partitioning. Fig. 6a–d shows the total COHP of NM and AFM states at ambient and high pressures (10 GPa). In the NM ambient-pressure state, the interactions are perfectly bonding up to 0.5 eV (Fig. 6a), but at the Fermi level (eF) they become antibonding. This electronic instability may be lifted by spin polarization. Consequently, the AFM phase (Fig. 6c) has
essentially lost these anti-bonding states owing to the splitting of the spin up/down channels. The latter is not visible from the total COHP, since there is an equal number of up and down spins in the AFM state, which has a zero net magnetic moment. To visualize the underlying spin polarization, the chemical bonding of a single, ferromagnetically coupled nextnearest Fe–Fe bond within the AFM cell is considered
Fig. 6. (a–d) Total COHP of the NM and AFM states at their ground-state volume and the volume at 10 GPa. (e–f) COHP of a ferromagnetically coupled single bond within the AFM structure at the same pressures. COHP values to the right/left indicate bonding/anti-bonding interactions. In addition, the ICOHP values are given for each total COHP.
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(Fig. 6e). As in any magnetic system, the up/down spins lower/raise the total energy owing to a less/more effective shielding of the core [56]. Consequently, the down spin density distribution (black) is more diffuse and dominates the chemical bonding; this is the reason why magnetic structures adopt larger volumes. At ambient pressure, the up/down splitting is 1.7 eV. At 10 GPa, however, the COHP (Fig. 5f) has become more disperse [57] and the up/down splitting is much smaller, namely 0.9 eV, so that one already observes anti-bonding interactions for the up spins (red) at eF. This is all a consequence of the partial suppression of spin polarization, since the formerly more diffuse down spins are compressed into a smaller volume. For the NM state, the COHP at ambient and high pressures (Fig. 6a and b) appear more similar as a consequence of the rigidness of the electronic structure. Apparently, two things do change when pressure is exerted. The COHP becomes more disperse, and the number of anti-bonding states below eF increases slightly. The missing degree of electronic freedom—no spin polarization allowed—forces the electrons directly into these anti-bonding states, and the resistance against this unfavorable effect might be looked upon as the “covalent” reason for the larger bulk modulus. However, the growing bonding interactions below eF misleadingly suggest that the structures are stabilized by pressure. Indeed, ICOHP values as shown in Fig. 6a–d indicate stronger bonding for the NM type at ambient pressure and a remarkable increase in stability for both phases at elevated pressure. This argument holds for the Fe–Fe covalent bonding because, upon applying pressure, the orbitals overlap, and thus the bonding interaction increases. Nonetheless, these bonding interactions are only part of the band-structure energy, namely the “off-site” entries of its real-space partitioning. In addition, there are the “on-site” (atomic-like) entries, and these strongly destabilize the structures when pressure is exerted, because the tails of the next-neighbors valence orbitals enter the core-like regions. For illustration, Fig. 7 shows the course of these on-site ICOHP for the NM and AFM structures as a function of the pressure. As stated above, the off-site ICOHP (chemical bonding part) is stabilized by 2.4 (NM) and 5.5 eV (AFM) upon increasing the pressure from ambient to 10 GPa. However, the destabilization due to the off-site interactions (atomic-like part) amounts to 10 (NM) and 17 eV (AFM), so that off- and on-site contributions add up to a less favorable total energy, as expected. It is even more important that at all pressures the on-site ICOHP is more unfavorable for the NM structure. Because the structure is denser, the orbitals are more compressed, and the electron density is higher. This leads not only to larger electron–electron repulsion, but also to larger kinetic energy, and one might coin it the “atomic” reason for the larger bulk modulus. The enormous resistance against compression for the NM structure due to the “covalent” and the “atomic” reason is clearly reflected by the much smaller volume decrease of only 11% and, trivially, by the increased bulk modulus, which is about twice as large.
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Fig. 7. On-site ICOHPs for the NM (black circles) and AFM 1Q (white circles) states as a function of pressure.
To continue with the discussion of the elastic properties’ dependence on magnetic ordering, from Fig. 5 it can be seen that all magnetically ordered fcc Fe–Mn configurations (AFM and FM) exhibit a larger bulk modulus than the configuration with both chemical and magnetic disorder (DLM). In Fig. 8, the average local magnetic moments of Fe (blue squares) and Mn (red circles) for the DLM (filled symbols) and AFM (1Q, horizontally half filled symbols; 2Q, vertically half filled symbols; 3Q, open symbols) configurations as functions of the lattice parameter for the Fe–Mn random alloy with a Mn content of 25.0 at.% as well as the equilibrium lattice parameter for each magnetic configuration are shown. The amplitude of local magnetic moments for the DLM configuration is much more sensitive to changes in volume than for AFM configurations. A strong dependence of local magnetic moments on volume is known to soften the lattice [58]. As the volume changes during uniform compression for the determination of the bulk modulus, the softening is reflected by a
Fig. 8. Local magnetic moment vs Wigner–Seitz radius for DLM and AFM (1Q, 2Q, and 3Q) Fe0.75Mn0.25 configuration.
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lower bulk modulus of Fe–Mn alloys with DLM configuration compared with magnetically ordered alloys. Thus the difference in bulk modulus emerges as a direct result of the so-called magnetovolume effect [58]. All the AFM configurations considered overestimate the experimentally determined bulk moduli, whereas the DLM configuration corresponding to the magnetically disordered state above the Ne´el temperature yields good agreement with experiment. This observation might be explained based on minute differences in total energy at 0 K between the different AFM configurations. For both investigated alloys with 25.0 and 37.5 at.% Mn, differences in total energy of 1.3 and 1.8 mRy atom 1 were calculated with the EMTO code, which is consistent with the literature [14]. Therefore, it is conceivable that, at RT and depending on the Mn content, a mixture of AFM configurations is present, resulting in a “notional magnetic disorder”, and, thus, with respect to the determination of elastic properties, the magnetic configuration of fcc Fe–Mn random alloys at RT is better described using the DLM model. The description of the magnetic configuration with the DLM model may be useful for determining the elastic properties of other Fe-based alloys with AFM ground states at elevated temperatures approaching the magnetic transition temperature. 5. Conclusions The effect of the Mn content on the elastic properties of fcc Fe–Mn alloys was investigated using the so-called combinatorial thin film approach. Fe–Mn thin films with a graded chemical composition were synthesized using DCmagnetron sputtering from two plasma sources on sapphire (0001) substrates at a substrate temperature of 400°. The elastic properties were measured by nanoindentation experiments as a function of the Mn content. With an increase in Mn content from 23 to 39 at.%, the average bulk modulus decreases from 143 to 105 GPa. The impact of the magnetic state on the elastic properties was probed with ab initio calculations. Describing the magnetic state of fcc Fe–Mn alloys with the DLM model yields the best agreement with the experimental results, whereas NM and AFM configurations overestimate the bulk modulus. The strong impact of the magnetic state on the elastic properties can be understood based on differences in the chemical bonding and the magnetovolume effect. Bonding in the NM configuration is stiffer than in the AFM configuration owing to stronger electron–electron atomic-like interactions and less pronounced anti-bonding increase upon compression. The DLM configuration exhibits a lower bulk modulus than the magnetically ordered configuration, owing to a larger magnetovolume effect which softens the lattice. It is suggested that, owing to minute energy differences between competing AFM configurations, a mixture of these configurations may be present at RT. This magnetic configuration is well described by the DLM model. These results indicate that the magnetic con-
figuration described by DLM may be useful for the determination of the elastic properties of other Fe-based alloys with AFM ground states. Acknowledgements The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center (SFB) 761 “Steel – ab initio”. The results calculated with the EMTO code were obtained using implementations from Andrei Ruban and Levente Vitos. ME and IAA are grateful to the Go¨ran Gustafsson Foundation for Research in Natural Sciences and Medicine for financial support. RD and JvA would like to thank the Research Center Ju¨lich for letting them use their computational resources. TG wishes to thank Tetsuya Takahashi for the design of Fig. 1. References [1] Enami K, Nagasawa A, Nenno S. Scripta Metall 1975:941–8. [2] Frommeyer G, Brux U, Neumann P. ISIJ Int. 2003:438–46. [3] Schneider T, Acet M, Rellinghaus B, Wassermann EF, Pepperhoff W. Phys Rev B: Condens Matter 1995:8917–21. [4] Hermawan H, Alamdari H, Mantovani D, Dube D. Powder Metall 2008:38–45. [5] Endoh Y, Ishikawa Y. J Phys Soc Jpn 1971:1614–27. [6] Sato A, Chishima E, Soma K, Mori T. Acta Metall 1982:1177–83. [7] Gu Q, Vanhumbeeck J, Delaey L. J Phys IV 1994:135–44. [8] Grassel O, Kruger L, Frommeyer G, Meyer LW. J Plast 2000:1391–409. [9] Brux U, Frommeyer G, Grassel O, Meyer LW, Weise A. Steel Res 2002:294–8. [10] Grassel O, Frommeyer G. Stahl Eisen 2002:65–9. [11] Bisanti P, Mazzone G, Sacchetti F. J Phys F: Met Phys 1987:1425–35. [12] Schulthess TC, Butler WH, Stocks GM, Maat S, Mankey GJ. J Appl Phys 1999:4842–4. [13] Spisak D, Hafner J. Phys Rev B: Condens Matter 2000:11569–75. [14] Stocks GM, Shelton WA, Schulthess TC, Ujfalussy B, Butler WH, Canning A. J Appl Phys 2002:7355–7. [15] Kawald U, Mitze O, Bach H, Pelzl J, Saunders GA. J Phys Condens Matter 1994:9697–706. [16] Music D, Takahashi T, Vitos L, Asker C, Abrikosov IA, Schneider JM. Appl Phys Lett 2007:191904. [17] Oliver WC, Pharr GM. J Mater Res 1992:1564–83. [18] Hohenberg P, Kohn W. Phys Rev B: Condens Matter 1964:864–71. [19] Andersen OK, Jespen O, Krier G. Lectures on methods of electronic structure calculations. Singapore: World Scientific; 1994. [20] Vitos L. Phys Rev B: Condens Matter 2001:014107. [21] Kresse G, Hafner J. J Phys Rev B: Condens Matter 1993:13115–8. [22] Kresse G, Hafner J. J Phys Rev B: Condens Matter 1994:14251–69. [23] Ozaki T, Kino H. Phys Rev B: Condens Matter 2005:8. [24] Andersen OK. Phys Rev B: Condens Matter 1975:3060–83. [25] Skriver HL. The LMTO method. Berlin: Springer; 1984. [26] Andersen OK. The electronic structure of complex systems. In: Phariseau P, Temmerman WM, editors. The electronic structure of complex systems. New York: Plenum; 1984. [27] Vitos L, Skriver HL, Johansson B, Kollar J. J Comput Mater Sci 2000:24–38. [28] Perdew JP, Burke K, Ernzerhof M. Phys Rev Lett 1996:3865–8. [29] Soven P. Phys Rev 1967:809–13. [30] Vitos L, Abrikosov IA, Johansson B. Phys Rev Lett 2001:156401. [31] Birch F. Phys Rev 1947:809. [32] Kresse G, Joubert D. Phys Rev B: Condens Matter 1999:1758–75.
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Perdew JP, Wang Y. Phys Rev B: Condens Matter 1992:13244–9. Blochl PE. Phys Rev B: Condens Matter 1994:17953–79. Monkhorst HJ, Pack JD. Phys Rev B: Condens Matter 1976:5188–92. Zunger A, Wei SH, Ferreira LG, Bernard JE. Phys Rev Lett 1990:353–6. Abrikosov IA, Niklasson AMN, Simak SI, Johansson B, Ruban AV, Skriver HL. Phys Rev Lett 1996:4203–6. Abrikosov IA, Simak SI, Johansson B, Ruban AV, Skriver HL. Phys Rev B: Condens Matter 1997:9319–34. Cowley JM. J Appl Phys 1950:24–9. Ozaki T. Phys Rev B: Condens Matter 2003:5. Troullier N, Martins JL. Phys Rev B: Condens Matter 1991:1993–2006. Blochl PE. Phys Rev B: Condens Matter 1990:5414–6. Ozaki T, Kino H. Phys Rev B: Condens Matter 2004:19. Soler JM, Artacho E, Gale JD, Garcia A, Junquera J, Ordejon P, et al. J Phys Condens Matter 2002:2745–79. Andersen OK, Jepsen O. Phys Rev Lett 1984:2571–4. Krier G, Jepsen O, Burkhardt A, Andersen OK. The TB-LMTO-ASA program. Stuttgart: Max-Planck-Institut fu¨r Festko¨rperforschung.
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[47] Dronskowski R, Blochl PE. J Phys Chem 1993:8617–24. [48] Lenkkeri JT, Levoska J. Philos Mag A 1983:749–58. [49] Cankurtaran M, Saunders GA, Ray P, Wang Q, Kawald U, Pelzl J, et al. Phys Rev B: Condens Matter 1993:3161–70. [50] Staunton J, Gyorffy BL, Pindor AJ, Stocks GM, Winter H. J Magn Magn Mater 1984:15–22. [51] Hill R. Proc Phys Soc London, Sec A 1952:349–55. [52] Warren OL, Wyrobek TJ. Meas Sci Technol 2005:100–10. [53] Lintymer J, Gavoille J, Martin N, Takadoum J. Surf Coat Technol 2003:316–23. [54] Nix WD. Mater Sci Eng A 1997:37–44. [55] Kooi BJ, Poppen RJ, Carvalho NJM, De Hosson JTM, Barsoum MW. Acta Mater 2003:2859–72. [56] Dronskowski R. Itinerant ferromagnetism and antiferromagnetism from a chemical bonding perspective. In: Kramer B, editor. Adv Solid State Phys. Berlin Heidelberg: Springer; 2002. [57] Drickamer HG, Frank CW. Electronic transitions and the high pressure chemistry and physics of solids. London: Chapman & Hall; 1973. [58] van Schilfgaarde M, Abrikosov IA, Johansson B. Nature 1999:46–9.
Acta Materialia 59 (2011) 1493–1501 www.elsevier.com/locate/actamat
Influence of chemical composition and magnetic effects on the elastic properties of fcc Fe–Mn alloys T. Gebhardt a,⇑, D. Music a, M. Ekholm b, I.A. Abrikosov b, J. von Appen c, R. Dronskowski c, D. Wagner d, J. Mayer d, J.M. Schneider a b
a Materials Chemistry, RWTH Aachen University, D-52056 Aachen, Germany Department of Physics, Chemistry and Biology (IFM), Linko¨ping University, SE-58183 Linko¨ping, Sweden c Chair of Solid-State and Quantum Chemistry, RWTH Aachen University, D-52056 Aachen, Germany d Central Facility for Electron Microscopy, RWTH Aachen University, D-52056 Aachen, Germany
Received 19 July 2010; received in revised form 29 October 2010; accepted 4 November 2010 Available online 10 December 2010
Abstract The influence of the Mn content on the elastic properties of face centered cubic Fe–Mn alloys was studied using the combinatorial approach. Fe–Mn thin films with a graded chemical composition were synthesized. Nanoindentation experiments were carried out to investigate the elastic properties as a function of the Mn content. As the Mn content increases from 23 to 39 at.%, the average bulk modulus varies from 143 to 105 GPa. Ab initio calculations served to probe the impact of magnetic effects on the elastic properties. The magnetic state description with disordered local moments yields the best agreement with the experimental results, whereas with non-magnetic and antiferromagnetic configurations the bulk modulus is overestimated. The strong impact of the magnetic configuration may be understood based on the differences in the chemical bonding and the magnetovolume effect. It is suggested that, owing to minute energy differences of competing antiferromagnetic configurations, a mixture of these with a “notional magnetic disorder” is present, which is in fact well described by the disordered local moments model. These results show that the combinatorial thin film synthesis with subsequent nanoindentation is an appropriate tool for investigating the elastic properties of Fe–Mn alloys systematically as a function of the chemical composition, to validate theoretical models. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Iron alloys; Sputtering; Elastic properties; Ab initio calculations; Nanoindentation
1. Introduction Fe–Mn alloys exhibit fascinating properties for a wide range of applications [1–4]. These properties are mainly related to the face centered cubic (fcc) ? hexagonal close packed (hcp) martensitic transformation [1,2] as well as the magnetism and magnetic transformations present in these alloys [3,5]. Regarding the martensitic transformation, Enami et al. [1] reported the occurrence of the socalled shape-memory effect (SME) in binary Fe–Mn alloys. The SME strongly depends on the alloy composition, ⇑ Corresponding author. Tel.: +49 241 80 25984; fax: +49 241 80 22295.
E-mail address: [email protected] (T. Gebhardt).
where additions of Si result in a nearly complete SME [6,7]. High-Mn steels feature high strength and exceptional plasticity owing to twin formation (twinning induced plasticity (TWIP effect)) or via multiple martensitic transformations (transformation induced plasticity (TRIP effect)) under mechanical load [2,8,9], therefore providing great potential for structural components in automotive engineering [10]. The stacking fault energy (SFE) of the austenite governs the deformation behavior. The chemical composition exhibits a strong impact on the SFE [9]. With respect to magnetism, the compositionally disordered fcc Fe–Mn alloys are antiferromagnetically ordered [5], with several inconsistent proposals for the antiferromagnetic (AFM) ground state [11–14]. Furthermore, fcc Fe–Mn
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.11.013
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alloys feature the so-called Invar and anti-Invar effects in the AFM and paramagnetic (PM) state, respectively, caused by magnetovolume instabilities [3], resulting in anomalous magnetization, thermal expansion, heat capacity and elastic properties [15]. The Invar and, even more pronounced, the anti-Invar effect strongly depend on the chemical composition [3]. Recently, Hermawan et al. [4] proposed the use of Fe–Mn alloys as a new class of metallic degradable biomaterials. The chemical composition has a strong impact on the aforementioned effects and properties. A previous study investigated the influence of the Mn content on the elastic properties of fcc Fe–Mn random alloys with non-magnetic (NM) and PM configurations using ab initio calculations [16]. However, a systematic experimental study of the dependence of the elastic properties of Fe–Mn alloys on the Mn content has not been reported. The present work systematically investigates the elastic properties of these alloys by combinatorial thin film synthesis and subsequent nanoindentation experiments. Furthermore, the impact of magnetic effects on the elastic properties is probed using ab initio calculations. It is the purpose of the present paper to contribute towards understanding the correlation between structure, chemical composition, magnetic configuration and elastic properties. Within a Mn content range from 22 to 40 at.% phase pure fcc Fe–Mn thin films with a graded chemical composition were synthesized. As the Mn content increases from 23 to 39 at.%, the average bulk modulus decreases from 143[±34] to 105[±12] GPa. The experimental results were compared with ab initio data, whereas describing the magnetic state of fcc Fe–Mn random alloys with disordered local moments (DLM) yields the best agreement. NM and AFM configurations overestimate the bulk modulus. The strong impact of the magnetic state on the elastic properties can be understood, on the one hand, based on the differences in chemical bonding and, on the other hand, based on the magnetovolume effect, which softens the lattice and which also leads to the observed decrease in bulk modulus with increasing Mn content. The fact that the DLM and not the AFM ground state description of the magnetic configuration results in the best agreement between measured and calculated elastic properties might be due to minute energy differences between competing AFM configurations and therefore a “notional magnetic disorder” at room temperature (RT). This configuration is well described by the DLM model. This description of the magnetic configuration may be useful for the determination of the elastic properties of other Fe-based alloys with AFM ground states at elevated temperature approaching the magnetic transition temperature.
4.0 W cm 2. The deposition geometry results in the formation of a lateral gradient in chemical composition, since the deposited flux from each plasma source is not uniformly distributed over the whole substrate surface area. This constitutes the so-called combinatorial thin film synthesis (Fig. 1). Depositions took place in an Ar atmosphere at 0.8 Pa. Sapphire (0 0 0 1) wafers with a diameter of 50 mm were used as substrates. The substrate, fixed on a 1 mm Cu plate to homogenize the temperature, was heated to 400 °C. After a deposition time of 120 min, which resulted in a film thickness of 1.7 lm, the samples were cooled down in the load-lock chamber. The substrate-totarget distance was 10 cm, with an inclination angle of the cathodes of 19°. For phase identification, grazing incidence X-ray diffraction measurements were carried out in a Bruker AXS D8 Discover General Area Diffraction Detector System equipped for micro-diffraction with a Cu Ka source. A 13 1 grid with a lateral spacing of 3.5 mm along the Mn gradient was analyzed on each sample. The primary beam was collimated using a pin hole with a diameter of 0.5 mm at an incidence angle of 15°, resulting in an irradiated surface ellipse with a = 1.24 mm and b = 0.32 mm. Energy dispersive X-ray analysis with an EDAX Genesis2000 detector was used to map the chemical composition along the 13 1 grid. Morphological investigations were performed by scanning electron microscopy (SEM) using a JEOL JSM-6480 microscope. The elastic properties of the graded Fe–Mn thin films were investigated at RT using a depth-sensing nanoindenter (Hysitron TriboIndentere) equipped with a Berkovich indenter tip. The Oliver–Pharr method was used to deter-
2. Experimental methods Fe–Mn films were grown in an ultra-high-vacuum chamber using DC-magnetron sputtering from an elemental Fe target and a FeMn compound target (Fe/Mn-ratio equal to 1) applying power densities of 1.5–2.5 and 3.0–
Fig. 1. Deposition setup for the combinatorial Fe–Mn alloy thin film synthesis. The color gradient schematically shows the evolving gradient in chemical composition. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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mine the reduced elastic modulus from the unloading curves [17]. A fused quartz specimen served as the reference material for calibration of the tip area function. A series of nine indentations with loads of 1.5 mN was performed at each grid position selected for further investigation, resulting in contact depths <10% of the film thickness. The bulk modulus values used in this work were calculated from the reduced elastic modulus values using the Poisson’s ratio obtained from ab initio calculations [16]. The deformation behavior of the Fe–Mn films was studied by means of a combination of nanoindentation and cross-sectional transmission electron microscopy (TEM) techniques. A single indent was carried out with a sharp cube-corner tip at 7 mN at a medium Mn concentration. The cube-corner tip was used to keep the plastic deformation to a more confined volume. For preparation of the TEM specimen, a FEI Strata FIB 205 workstation was used. The TEM sample was examined in a JEOL JEM 2000 FX TEM instrument operating at an acceleration voltage of 200 kV. 3. Theoretical methods The theoretical study was carried out based on density functional theory [18], as implemented in the exact muffin-tin orbitals (EMTO) formalism [19,20], the Vienna ab initio simulation package (VASP) [21,22], the OpenMX code [23] and the linear muffin-tin orbital (LMTO) theory [24–26]. The EMTO, VASP and OpenMX codes were used to calculate the elastic properties with different magnetic configurations. Chemical bonding analysis was carried out by the LMTO method. The EMTO code, based on Green’s function [27] and full charge density techniques [20], served to calculate the elastic properties of fcc Fe–Mn random alloys with AFM and ferromagnetic (FM) configurations. The AFM alloys were described with a 1Q, 2Q and 3Q spin structure. The local density approximation was used to calculate self-consistent charge density, the total energy was evaluated using the generalized gradient approximation [28], and the ion core states were kept frozen. The chemical disorder, reported by Endoh and Ishikawa [5], was treated with the coherent potential approximation [29,30]. The k-space sampling of the irreducible part of the Brillouin zone was done using a 21 21 21 grid for the AFM and a 13 13 13 grid for the FM alloys, respectively. The total energy convergence criterion was 10 7 Ry. The bulk moduli of the investigated Fe–Mn alloys were obtained by fitting the energy vs volume data with the Birch–Murnaghan equation of state [31]. The VASP code was used to treat chemically disordered 3Q configurations with 25.0 and 37.5 at.% Mn. The projector augmented wave potentials [32] with the generalized gradient approximation [33] were employed. The following parameters were applied: convergence criterion for the total energy of 0.01 meV; Blo¨chl corrections [34] for the total energy cut-off of 300 eV; and integration in the Brillouin zone according to Monkhorst–Pack [35] with
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5 5 5 k-points. Following the special quasirandom structure (SQS) approach [36], 2 2 2 supercells containing 32 atoms were created with random positions of Fe and Mn, using the locally self-consistent Green’s function software package [37,38]. In the software, the Warren–Cowley SRO parameter [39] was employed to quantify randomness, which was close to zero within seven coordination shells. The OpenMX code, based on basis functions in the form of linear combination of localized pseudoatomic orbitals [40] was used to treat chemically disordered non-collinear FM configurations with 25.0 and 37.5 at.% Mn. The electronic potentials were fully relativistic pseudopotentials with partial core corrections [41,42], and the generalized gradient approximation was applied [28]. The basis functions were generated by a confinement scheme [40,43] and specified as follows: Fe5.0-s1p2d1 and Mn7.0-s3p3d2. The first symbol designates the chemical name, followed by the cut-off radius (in Bohr radius) in the confinement scheme, and the last set of symbols defines primitive orbitals applied. The energy cut-off (150 Ry) and k-point grid (1 1 1) within the real space grid technique [44] were adjusted to reach an accuracy of 10 6 H atom 1. As described above for the 3Q VASP calculations, 2 2 2 supercells containing 32 atoms were created with SQS random positions of Fe and Mn. The calculations based on the LMTO method for NM Fe–Mn alloys and structures with 1Q configuration with 25.0 at.% Mn were performed in its tight-binding representation [45]. The TB-LMTO-ASA 4.7 program code [46] with the atomic-spheres approximation was used. All calculations were checked for convergence of total energies and magnetic moments with respect to the number of kpoints. The chemical bonding situation was investigated using Crystal Orbital Hamilton Population (COHP) analysis [47] with a real-space partitioning technique of the band-structure energy (sum of the one-electron eigenvalues). A COHP curve is an energy resolved plot for each bond. Integrated up to the Fermi level, the so-called ICOHP values measure the sum of all bonding and anti-bonding interactions and thus serve as an indication of the bond strength. 4. Results and discussion Two Fe–Mn combinatorial thin films were synthesized to obtain the desired Mn content range from 20 to 40 at.%. Applying power densities of 1.5 and 4.0 W cm 2 at the Fe and FeMn target, respectively, resulted in a film with a compositional gradient from 32 to 40 at.% Mn. The crystallographic structure of this film is shown in Fig. 2. Within the whole Mn content range, the sample consists of phase pure fcc Fe–Mn. A compositional gradient from 20 to 32 at.% Mn was obtained by applying power densities of 2.5 and 3.0 W cm 2 at the Fe and FeMn target, respectively. The crystallographic structure of this film is shown in Fig. 3.
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Fig. 2. Phase evolution as a function of the Mn content for 32–40 at.% Mn. The intensity is displayed with a logarithmic scale.
Fig. 3. Phase evolution as a function of the Mn content for 20–32 at.% Mn. The intensity is displayed with a logarithmic scale.
The film consists of phase pure fcc Fe–Mn for Mn contents above 23 at.%. At lower Mn contents, the film is X-ray amorphous. Thus, phase pure fcc Fe–Mn was synthesized within a Mn content range from 23 to 40 at.%. Both films are textured, which is often observed in magnetron sputtered thin films. Owing to an inhomogeneous temperature distribution at the substrate during deposition, both films exhibit a pronounced surface roughness at the edges of the wafer. Therefore these areas were excluded from further characterization of the elastic properties by nanoindentation. A bright-field TEM image of a Fe–Mn thin film with 34 at.% Mn after being indented with a cube-corner tip is displayed in Fig. 4. No evidence for the formation of pores or cavities can be seen in this image, suggesting the growth of dense Fe–Mn. Furthermore, it shows that, at this
composition, the deformation features underneath and in the vicinity of the indented spot are primarily manifested by dislocation activities (positions 1, 2 and 3). Sub-grain boundaries (position 4) and twin formation can be seen (position 5). Selected area electron diffraction verified that regions with low (position 6) as well as with high dislocation densities (positions 1, 2 and 3) are composed of phase pure fcc Fe–Mn. Thus, no martensitic phase transformation was induced with a cube-corner tip at a load of 7 mN at this composition. The elastic properties of the Fe–Mn films were measured along the Mn concentration gradient by nanoindentation. The resulting bulk modulus values (white and black pentagons, respectively) are plotted in Fig. 5. Furthermore, calculated bulk moduli for magnetically ordered (AFM, 1blue triangles; FM, green circles) and disordered (red squares) fcc Fe–Mn random alloys as well as bulk data from Lenkkeri and Levoska [48] (open star) and Cankurtaran et al. [49] (filled star) are presented. The magnetic state of the magnetically disordered fcc Fe–Mn random alloys is described using the DLM model [50]. Details of these calculations can be found elsewhere [16]. The experimentally obtained bulk modulus values were calculated from the measured reduced elastic modulus values using a linear dependence of Poisson’s ratio on the Mn content, obtained from ab initio calculations of the elastic constants of DLM fcc Fe–Mn alloys [16] assuming crystallographic isotropy and using the Voigt–Reuss–Hill method [51] to determine the shear modulus (white pentagons) as well as using a mean Poisson’s ratio of 0.25 (black pentagons). The Poisson’s ratios calculated using the linear dependence are consistent with the data from Lenkkeri and Levoska [48] ( 6.7% deviation) and Cankurtaran et al. [49] (+1.3% deviation). Taking the dependence of Poisson’s ratio on the Mn content into account, the experimentally determined bulk modulus decreases from 143[±34] GPa at 23 at.% Mn to 105[±12] GPa at 39 at.% Mn. The nanoindentation data are in good agreement with the bulk data as well as with the DLM solutions of fcc Fe–Mn random alloys. The decreasing trend with increasing Mn content is depicted very well. This decrease in the bulk modulus can be understood based on the magnetovolume effect [16]. It is reasonable to assume that the elastic properties are affected in addition to the composition by the deposition geometry employed here, as well as by the presence of residual stresses. Generally, the elastic modulus of thin films is primarily dependent on the thin film density [52]. Lintymer et al. [53] investigated the modification of mechanical properties of sputter-deposited chromium thin films by glancing angle deposition using nanoindentation. Up to a vapor incidence angle of a = 30°, no significant change in Young’s modulus could be determined. When a was further increased, a noticeable decrease in Young’s
1 For interpretation of color in Figs. 5 and 8, the reader is referred to the web version of this article.
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Fig. 4. Bright-field TEM image of Fe–Mn thin film with 34 at.% Mn subjected to an indentation load of 7 mN with a cube-corner tip. The diffraction patterns show that areas 1–3 are austenitic.
Fig. 5. Bulk moduli as obtained using nanoindentation (white and black pentagons) as well as using ab initio calculations (AFM, blue triangles; FM, green circles; DLM, Music et al. [16], red squares). Results are compared with experimental data from Lenkkeri and Levoska [48] (open star) and Cankurtaran et al. [49] (filled star).
modulus was observed, which was attributed to the evolution of shadowing-induced porosity. TEM investigations revealed the growth of dense Fe–Mn thin films. Furthermore, and most importantly, the effect of elastic anisotropy
on the indentation modulus according to Nix [54] and Kooi et al. [55] is not large, even for highly anisotropic materials. Based on the good agreement between the present nanoindentation results and published bulk data [48,49], a minor influence of residual stresses is inferred, as well as the deposition geometry on the elastic properties reported here. The nanoindentation data were measured at RT. Within the regarded composition range, fcc Fe–Mn random alloys have previously been shown to exhibit AFM order at RT [5]. Hence the DLM solution does not represent the ground state. However, with respect to the elastic properties, the description of the magnetic state as completely disordered within the DLM model yields the best agreement with experiment, whereas the AFM 3Q ordered configuration results in bulk modulus values 10% and 30% larger than the experimental ones for 25.0 and 37.5 at.% Mn, respectively. Furthermore, Music et al. [16] observed that the bulk modulus values for NM structures are a factor of 2 higher compared with the DLM state. To explore the influence of the magnetic configuration on the elastic properties of Fe–Mn random alloys, the chemical bonding of magnetically ordered and NM structures at ambient and high pressures was analyzed. The ground-state volume of all considered magnetic structures is 9% larger than for NM ones, which is mirrored in
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the atomic (on-site) and bonding (off-site) energies of the band-structure energy partitioning. Fig. 6a–d shows the total COHP of NM and AFM states at ambient and high pressures (10 GPa). In the NM ambient-pressure state, the interactions are perfectly bonding up to 0.5 eV (Fig. 6a), but at the Fermi level (eF) they become antibonding. This electronic instability may be lifted by spin polarization. Consequently, the AFM phase (Fig. 6c) has
essentially lost these anti-bonding states owing to the splitting of the spin up/down channels. The latter is not visible from the total COHP, since there is an equal number of up and down spins in the AFM state, which has a zero net magnetic moment. To visualize the underlying spin polarization, the chemical bonding of a single, ferromagnetically coupled nextnearest Fe–Fe bond within the AFM cell is considered
Fig. 6. (a–d) Total COHP of the NM and AFM states at their ground-state volume and the volume at 10 GPa. (e–f) COHP of a ferromagnetically coupled single bond within the AFM structure at the same pressures. COHP values to the right/left indicate bonding/anti-bonding interactions. In addition, the ICOHP values are given for each total COHP.
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(Fig. 6e). As in any magnetic system, the up/down spins lower/raise the total energy owing to a less/more effective shielding of the core [56]. Consequently, the down spin density distribution (black) is more diffuse and dominates the chemical bonding; this is the reason why magnetic structures adopt larger volumes. At ambient pressure, the up/down splitting is 1.7 eV. At 10 GPa, however, the COHP (Fig. 5f) has become more disperse [57] and the up/down splitting is much smaller, namely 0.9 eV, so that one already observes anti-bonding interactions for the up spins (red) at eF. This is all a consequence of the partial suppression of spin polarization, since the formerly more diffuse down spins are compressed into a smaller volume. For the NM state, the COHP at ambient and high pressures (Fig. 6a and b) appear more similar as a consequence of the rigidness of the electronic structure. Apparently, two things do change when pressure is exerted. The COHP becomes more disperse, and the number of anti-bonding states below eF increases slightly. The missing degree of electronic freedom—no spin polarization allowed—forces the electrons directly into these anti-bonding states, and the resistance against this unfavorable effect might be looked upon as the “covalent” reason for the larger bulk modulus. However, the growing bonding interactions below eF misleadingly suggest that the structures are stabilized by pressure. Indeed, ICOHP values as shown in Fig. 6a–d indicate stronger bonding for the NM type at ambient pressure and a remarkable increase in stability for both phases at elevated pressure. This argument holds for the Fe–Fe covalent bonding because, upon applying pressure, the orbitals overlap, and thus the bonding interaction increases. Nonetheless, these bonding interactions are only part of the band-structure energy, namely the “off-site” entries of its real-space partitioning. In addition, there are the “on-site” (atomic-like) entries, and these strongly destabilize the structures when pressure is exerted, because the tails of the next-neighbors valence orbitals enter the core-like regions. For illustration, Fig. 7 shows the course of these on-site ICOHP for the NM and AFM structures as a function of the pressure. As stated above, the off-site ICOHP (chemical bonding part) is stabilized by 2.4 (NM) and 5.5 eV (AFM) upon increasing the pressure from ambient to 10 GPa. However, the destabilization due to the off-site interactions (atomic-like part) amounts to 10 (NM) and 17 eV (AFM), so that off- and on-site contributions add up to a less favorable total energy, as expected. It is even more important that at all pressures the on-site ICOHP is more unfavorable for the NM structure. Because the structure is denser, the orbitals are more compressed, and the electron density is higher. This leads not only to larger electron–electron repulsion, but also to larger kinetic energy, and one might coin it the “atomic” reason for the larger bulk modulus. The enormous resistance against compression for the NM structure due to the “covalent” and the “atomic” reason is clearly reflected by the much smaller volume decrease of only 11% and, trivially, by the increased bulk modulus, which is about twice as large.
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Fig. 7. On-site ICOHPs for the NM (black circles) and AFM 1Q (white circles) states as a function of pressure.
To continue with the discussion of the elastic properties’ dependence on magnetic ordering, from Fig. 5 it can be seen that all magnetically ordered fcc Fe–Mn configurations (AFM and FM) exhibit a larger bulk modulus than the configuration with both chemical and magnetic disorder (DLM). In Fig. 8, the average local magnetic moments of Fe (blue squares) and Mn (red circles) for the DLM (filled symbols) and AFM (1Q, horizontally half filled symbols; 2Q, vertically half filled symbols; 3Q, open symbols) configurations as functions of the lattice parameter for the Fe–Mn random alloy with a Mn content of 25.0 at.% as well as the equilibrium lattice parameter for each magnetic configuration are shown. The amplitude of local magnetic moments for the DLM configuration is much more sensitive to changes in volume than for AFM configurations. A strong dependence of local magnetic moments on volume is known to soften the lattice [58]. As the volume changes during uniform compression for the determination of the bulk modulus, the softening is reflected by a
Fig. 8. Local magnetic moment vs Wigner–Seitz radius for DLM and AFM (1Q, 2Q, and 3Q) Fe0.75Mn0.25 configuration.
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lower bulk modulus of Fe–Mn alloys with DLM configuration compared with magnetically ordered alloys. Thus the difference in bulk modulus emerges as a direct result of the so-called magnetovolume effect [58]. All the AFM configurations considered overestimate the experimentally determined bulk moduli, whereas the DLM configuration corresponding to the magnetically disordered state above the Ne´el temperature yields good agreement with experiment. This observation might be explained based on minute differences in total energy at 0 K between the different AFM configurations. For both investigated alloys with 25.0 and 37.5 at.% Mn, differences in total energy of 1.3 and 1.8 mRy atom 1 were calculated with the EMTO code, which is consistent with the literature [14]. Therefore, it is conceivable that, at RT and depending on the Mn content, a mixture of AFM configurations is present, resulting in a “notional magnetic disorder”, and, thus, with respect to the determination of elastic properties, the magnetic configuration of fcc Fe–Mn random alloys at RT is better described using the DLM model. The description of the magnetic configuration with the DLM model may be useful for determining the elastic properties of other Fe-based alloys with AFM ground states at elevated temperatures approaching the magnetic transition temperature. 5. Conclusions The effect of the Mn content on the elastic properties of fcc Fe–Mn alloys was investigated using the so-called combinatorial thin film approach. Fe–Mn thin films with a graded chemical composition were synthesized using DCmagnetron sputtering from two plasma sources on sapphire (0001) substrates at a substrate temperature of 400°. The elastic properties were measured by nanoindentation experiments as a function of the Mn content. With an increase in Mn content from 23 to 39 at.%, the average bulk modulus decreases from 143 to 105 GPa. The impact of the magnetic state on the elastic properties was probed with ab initio calculations. Describing the magnetic state of fcc Fe–Mn alloys with the DLM model yields the best agreement with the experimental results, whereas NM and AFM configurations overestimate the bulk modulus. The strong impact of the magnetic state on the elastic properties can be understood based on differences in the chemical bonding and the magnetovolume effect. Bonding in the NM configuration is stiffer than in the AFM configuration owing to stronger electron–electron atomic-like interactions and less pronounced anti-bonding increase upon compression. The DLM configuration exhibits a lower bulk modulus than the magnetically ordered configuration, owing to a larger magnetovolume effect which softens the lattice. It is suggested that, owing to minute energy differences between competing AFM configurations, a mixture of these configurations may be present at RT. This magnetic configuration is well described by the DLM model. These results indicate that the magnetic con-
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