AB INITIO STUDY OF IRON AND Cr/Fe(1) H. C. HERPER,E.HOFFMANNandP.ENTEL Theoretical Low-Temperature Physics, Gerhard Mercator University, 78 Duisburg, Germany (Received...) Abstract In the present paper interfacial mixing of a thin chromium overlayer on bcc iron is studied. The calculations are performed in the framework of a pseudo-potential technique using the generalized gradient approximation for the exchange-correlation functional. Although Fe and Cr do not alloy in the bulk system at low temperatures, strong intermixing effects have been observed with Auger spectroscopy, if Cr is epitaxially grown on bcc Fe(1). Besides these structural effects we discuss the magnetic structure of the interface. It can be shown that the results are in good agreement with the experimental findings of Pfandzelter (Pfandzelter, Igel and Winter, 1996), provided we allow for lattice relaxation. Additionally, the structural and magnetic properties of iron investigated by the pseudo-potential method are compared to our former full potential results. Keywords: Magnetic multilayers, Mixing effects, Density-functional theory 1. INTRODUCTION Hetero-epitaxial growth processes of metals on metals, oxides or semiconductors are a great technologically challenge, because the occurrence of islands and interdiffusion often suppresses the layer-by-layer growth. Today, there are still many open questions concerning the interface structure and the penetration depth of the surface atoms, because Auger spectroscopy and other surface sensitive methods do not give detailed information about the structure of more than two layers. Therefore, they give no evidence of the alloying effects at deeper layers. A metal on metal system of special interest Corresponding author, Tel: +9-3-379-356, fax: +9-3-379-3665, email: heike@thp.uni-duisburg.de 1
is chromium on bcc iron. Here the effects of interface alloying are discussed for a Cr monolayer on bcc Fe(1) and are compared to the results from Auger spectroscopy (Pfandzelter, Igel and Winter, 1996). Measurements of bulk Fe-Cr samples have shown that a miscibility gap exists, which stretches nearly over all concentrations (Kubaschewski, 198). Only two stable phases have been found. A fcc phase on the iron rich side and a σ-phase around 5 at.% Cr exist. Both phases vanish with decreasing temperature. However, intermixing effects have been observed, if Cr is deposited on an Fe substrate. The magnetic properties also exhibit interesting features. An odd number of Cr layers is expected to favor ferromagnetic (FM) coupling, but instead of that antiferromagnetic (AFM) spin structures are observed (Unguris, Celotta and Pierce, 199), due to alloying effects at the interface. In addition, it is known that Cr itself has an incommensurate spin density wave (Fawcett, 1988), but if it is deposited on a surface the magnetic behavior changes, and collinear spin structures are favored. The magnetic and structural properties of the Cr-Fe interface are investigated within an ab initio pseudo-potential method. First, we check the accuracy of the pseudo potentials, because it is well known that the treatment of the magnetovolume effects of iron is complex. The present results for bulk iron are compared to the experimental findings, as well as to our former full potential results (Herper, Hoffmann and Entel, 1999).. BULK PROPERTIES The magnetic and structural properties of iron are investigated by using two different pseudo potentials. The calculations are performed using the VASP code (Kresse and Hafner, 1996; Kresse and Furtmüller, 1996) and the generalized gradient approximation (GGA) (Perdew and Wang, 199). Both methods employ ultra soft Vanderbilt pseudo potentials, but the description of the 3p semi-core electrons is different. In the first potential (PS1) the 3p orbitals are part of the pseudo core and possible dispersion effects are neglected. In the second a pseudo potential (PS) is used, which does not include the semi-core states. In this case the 3p states are treated selfconsistently. The results for the magnetic and structural properties of iron are summarized in Fig. 1. The choice of the pseudo potential has no significant influence on the properties of bcc iron, which can be seen from Table 1. Even for the simpler PS1 potential there are only small deviations from the experimental values and from the full potential results. The calculated bulk modulus is about 5.8% smaller than the experimental value, whereas the volume is slightly overestimated. These deviations decrease if the PS potential is used. This has also been found by Moroni and coworkers (Moroni, Kresse, Hafner and Furthmüller, 1997). Nevertheless, both pseudo potentials are at least sufficient for the description of the ground state properties of bcc iron. This does not hold for fcc iron, which is known to show magneto-volume
3 3 M( B ) 1 1 PS1 (s 3d) PS (3p 3d s) fcc AFM-I fcc AFM-I E tot (mry/atom) 16 8 fcc NM 16 8 fcc NM fcc FM fcc FM bcc FM bcc FM 6 7 8 88 V/atom (a.u.) 6 7 8 88 V/atom (a.u.) Figure 1: Calculated phase diagram of iron. Two different pseudo potentials are used. The PS1 (left panel) includes the 3p electrons and in the PS (right panel) the 3p electrons are treated as valence states. The FM bcc phase is marked through dotted lines. Full lines indicate fcc states. instabilities leading to the anti-invar effect. In a previous work we have shown that the anti-invar effect can be well understood from full-potential calculations (Herper, Hoffmann and Entel, 1999). However, the PS1 fails for γ-fe, because fcc Fe would have a FM ground state, which is obviously not the case. The AFM state with a smaller volume has a higher energy and can only be occupied at higher temperatures. This situation would be expected for a typical Invar system like Fe 65 Ni 35, but not for an anti-invar system. If the PS is used instead of PS1 the AFM state is more stable than all other fcc states, which is in accordance with the experimental results. This was not clear in a previous work (Moroni, Kresse, Hafner and Furthmüller, 1997) where the AFM and FM high-spin states of γ-fe are nearly degenerated. The difference can be accounted for on the basis that we use a higher plane wave cut-off, therefore, a larger number of plane waves, which enhances the accuracy. In the present work the self-consistent treatment of the 3p orbitals 3
PS1 PS FLAPW EXPT. bcc Fe (FM) V 79.33 78.36 77. 78.9 B 16 166 17 17 bcc Cr (AFM) V 8. 8.3 81.1 B 13 173 16 Table 1: Calculated bulk moduli B (in GPa) and ground state volumes V in (a.u.) for FM bcc Fe. The pseudo-potential calculations are performed by using the VASP program (Kresse and Hafner, 1996; Kresse and Furtmüller, 1996). The full potential results are taken from Herper et al. (1999). All calculations are done within the GGA (Perdew and Wang, 199). Experimental data for iron are from Acet et al. (199). The Cr data are taken from Donohue (198); Guillermet and Grimvall (1989). is obviously an improvement, but the description of the two FM states is still problematic. In contrast to what is expected from the full potential calculations, the sequence of the high-spin and low-spin state is wrong, because the low-spin state has a higher energy. Summarizing we can state that the PS obviously works better for fcc Fe, but unfortunately it is rather time consuming and offers no real advantage over the full potential. As mentioned above, the ground state properties of bcc iron come out sufficiently well for both pseudo potentials. Therefore, the PS1 can be used for the investigations of a Cr overlayer on bcc Fe(1). In the following part we discuss several aspects of the Fe-Cr bulk system in order to make sure that the interfacial effects discussed later on are no artefacts of the bulk properties. Therefore, bcc Fe 1 x Cr x compounds with x varying in steps of 5% are examined in view of their miscibility and magnetic properties. All compounds prefer more or less an antiparallel alignment of the Fe and Cr spins, see Fig. ). On the average the iron moment is enhanced compared to pure bcc iron as long as the compounds contain less than 5 at.% Fe. The Cr moment breaks down for FeCr, but with increasing Cr content it increases again. Therefore, the net moment of the unit cell decreases from pure Fe to FeCr 3. In addition, the total energies of the compounds have been used to calculate the mixing energy e m at T =K E m = E Fe Cr [(1 x)e Fe + xe Cr ] (1) with E Fe and E Cr being the total energies of FM bcc iron and AFM bcc chromium. The mixing energy provides useful information about ordering and disordering trends in the system. The mixing energies for the Fe-Cr system are always positive, see Fig.. This means that no cubic Fe-Cr compounds exist at T = K. Structures other than cubic have not been
M( B /atom) 3 1-1 Fe 1-x Cr x - 6 8 1 x (at.% Cr) Mixing energy (mry/atom) 16 Fe 1-x Cr x 1 8 6 8 1 x (at.% Cr) Figure : Distribution of the average magnetic moments versus the chromium amount in bcc Fe 1 x Cr x compounds (left panel). Open symbols correspond to the magnetic moment of the Fe atoms and filled squares mark the Cr moments. The full circles indicate the net moment per unit cell. The mixing energy for bcc Fe 1 x Cr x compounds versus the Cr amount is shown in the right panel. investigated. The observed decomposition agrees qualitatively with the experimental findings (Kubaschewski, 198; Pepperhoff and Acet, ). Experimentally the decomposition starts below 8 K. Therefore, calculated absolute values are by a factor of 3 higher than the measured values (Pepperhoff and Acet, ), but nevertheless, these simple calculations reveal the right chemical trend. 3. Cr ON BCC Fe(1) It has been already mentioned in the beginning that no bulk Fe-Cr alloys exist at low temperatures. Therefore, one could suspect that Cr atoms deposited on a bcc Fe substrate remain on the surface, which obviously is not the case. Numerous experiments (Pfandzelter, Igel and Winter, 1996; Davies, Stroscio, Pierce and Celotta, 1996) show that the Cr atoms penetrate into the substrate. Some theoretical work has also been done in this field (Turek, Weinberger, Freyss, Stoeffler et al., 1998; Freyss, Stoeffler and Dreyssé, 1997). However, there are still a number of questions concerning the penetration depth, interface alloying as well as the magnetic behavior. In order to study the mixing effects at the Fe-Cr interface and the related magnetic effects, we have performed supercell calculations for one Cr monolayer on a bcc Fe(1) substrate. All calculations are done with the VASP program employing the 5
Cr vacuum Fe vacuum 7 ML 11 ML Figure 3: Periodic supercells for 1 ML Cr on bcc Fe(1). The present systems consist of 9 ML Fe/ ML Cr and 5 ML Fe/ ML Cr, whereas the second Cr layer is added for symmetry reasons. The atoms are covered with 11 ML and 3 ML of vacuum, respectively. GGA. We use the PS1 potential, which is discussed in Section. We have used bcc supercells in slab geometry with one monolayer (ML) Cr on an Fe substrate. For symmetry reasons a second monolayer has been added to the bottom of the cell, which can be seen in Fig. 3. In order to study the influence of the system size and to rule out side effects from the periodic structure, we use two different types of super cells. They consist of 7 and 11 layers of atoms covered with 3 and 11 layers of vacuum, respectively. The initial lattice constant is chosen as a = 5.1a.u., which corresponds to the calculated value for bcc FM iron (Table 1). Starting from a perfect Cr monolayer on the Fe substrate for the two different unit cells, we change the Cr amount on the surface. In order to examine the interface structure, some of the Cr atoms are interchanged with the Fe atoms from the first layer below the surface, until the Cr layer is completely covered with an Fe monolayer. Thereby, we change the Cr amount in steps of 5 at.%. It should be mentioned that all investigated structures are ordered surface compounds, and no disorder effects are taken into account. All calculations have been performed twice using the 7 ML (+3 ML vacuum) and the larger 11 ML (+ 11 ML vacuum) supercells. The results show no significant differences for the total energy or the magnetic moments. There are only small deviations of about 1.5 mry in the absolute values of the total energy, and the magnetic moments are identical for both systems. Only the size of the relaxation effects seems to depend on the size of the unit cell, which will be discussed further below. However, the results for the concentration dependence show a good agreement with the experimental findings (Pfandzelter, Igel and Winter, 1996), see Fig.. The multilayer structure 6
Energy (mry/interface) 16 8 7 ML, ideal positions 7 ML, relaxed 11 ML, ideal positions 11 ML, relaxed 1 ML Cr on Fe(1) -8 6 8 1 x (at.% Cr in top layer) Figure : Total energy per interface versus the Cr amount of the top layer. Both results, without lattice relaxation (dotted lines, open symbols) and after the relaxation (full lines, filled symbols) are given. is most stable, if half of the Cr atoms are located in the 1st Fe layer below the surface. This is connected with an energy gain of about.8 mry per interface relative to the energy of the ideal Cr monolayer (1 at.% Cr on the surface). The gain is somewhat smaller for the smaller supercell, but it shows the same trend. Further penetration of Cr atoms into the 1st Fe layer leads to a strong increase of the total energy. Therefore, such configurations are less stable and cannot be expected at low temperatures. This corresponds well to the experimental results from Auger spectroscopy. After growing a single monolayer Cr on a bcc Fe substrate the remaining Cr amount on the surface is 5 at.%. The rest of the Cr atoms move into the nd layer. Deeper layers solely consist of Fe atoms (Pfandzelter, Igel and Winter, 1996). This is in nearly perfect agreement with the present results, whereas the Cr atoms are not allowed to interchange with the Fe atoms from deeper layers. It should be emphasized that the agreement with the experimental findings is connected with the allowance of lattice relaxation. If the atoms are kept fixed on the positions of the ideal lattice no interface mixing would occur and most of the Cr atoms would remain on the surface (Fig. ). The configuration with 5 at.% Cr on the surface would then have mry higher energy relative to the energy of the perfect Cr overlayer. In the following we have investigated lattice relaxation effects in detail for the considered surface compounds. The layer-resolved results for the relative movements a/a from the ideal positions are given in Fig. 5. The presented 7
M( B /atom) M( B /atom) M( B /atom) M( B /atom) M( B /atom) - Layer 1% Cr - - 75% Cr - - 5% Cr - - 5% Cr - - - % Cr Layer a/a (%) a/a (%) a/a (%) a/a (%) a/a (%) 1 5-5 Layer 1% Cr -1 1 75% Cr 5-5 -1 5% Cr 1-1 1 5% Cr 5-5 -1 1 % Cr 5-5 -1 Cr,7ML Fe,7ML Cr,11ML Fe, 11 ML Layer Figure 5: Magnetic moment (left column) and relative lattice relaxation (right column) versus the number of the layer. The numbers in the boxes express the relative Cr content on the surface. 8
values are averages over all Fe or Cr atoms per layer. As expected, the relaxation effects for the deeper layers are rather small, except for the totally covered Cr layer ( at.% on the surface). In the latter case the relative lattice relaxations are of about %. This confirms out concentration dependent total energy results, because this state has a much higher energy and tries to lower it by strong movements of all atoms, compare Fig.. In all other cases only the first three layers show considerable lattice relaxation effects. Especially large relaxation effects are observed for the surface atoms of the system with 5 at.% Cr on top. The Cr atoms move out of the multilayer, whereas the Fe atoms move inwards. This means that the Cr atoms on the surface and in the nd layer try to repel each other, which supports the experimental findings that in Cr surface alloys nearest neighbor occupation is suppressed (Davies, Stroscio, Pierce and Celotta, 1996). In view of this large repulsion it can be understood that the calculations without lattice relaxation do not give the correct structure. In this case the ideal Cr monolayer will be preferred, because this structure shows the smallest relaxation effects, see Fig. 5. The interchange of Fe and Cr atoms is also connected with strong changes of the magnetic structure. It should be mentioned that the Fe moments of the deeper layers are slightly higher than the bulk value of. µ B,whichmay be an artifact of the periodic structure. The chromium atoms on the surface are always antiparallel aligned to the Cr atoms in the nd layer, whereas the Fe atoms are ferromagnetically ordered, which is expected for this lattice constant (Fig. 1). In accordance with the experimental findings we observe an increase of the surface magnetic moments as compared to the bulk moments. The magnetic moments for the Cr atoms amount to 3. 3.3 µ B depending on the concentration of Cr on the surface. These values are in agreement with the results from in situ measurements (Turtur and Bayreuther, 199). In accordance with photoemission experiments (Hillebrecht, Roth, Jungblut, Kisker et al., 199), we observe an antiparallel alignment of the Cr monolayer on the Fe substrate. This holds even if the Cr monolayer is completely covered with a monolayer Fe (Fig. 5). If the relative Cr content in the nd layer becomes larger than 5 at.%, the Cr moments in this layer align antiparallel to the bulk magnetic moments, whereas the Cr atoms on the surface accept a parallel alignment to the magnetic moments of the Fe atoms. The AFM coupling of the Cr atoms seems to be stronger than the coupling between the Fe and Cr atoms. The structures with incomplete Cr layers can be understood as a simple model for non-layer-by-layer growth. This means that a new Cr layer, which starts growing on an incomplete Cr layer, is always antiparallel aligned to the Cr atoms in the former layer. One should keep in mind that the total number of atoms corresponds to one monolayer. It has to be investigated, whether this behavior changes with increasing the number of Cr layers. Finally, it should be mentioned that the influence of the lattice relaxation on the magnetic structure is less important. Apart from the absolute value of the moments, the same magnetic structures have already been found when the atoms are kept fixed. 9
. SUMMARY In the first part we have studied the phase diagram of iron within a pseudopotential method using two different pseudo potentials. It has been shown that the semi-core states of Fe have to be calculated self-consistently in order to describe the magneto-volume effects of fcc Fe. For bcc Fe, and therefore for the present multilayer calculations, it is sufficient to treat the 3p states as core states. Secondly, we have discussed the intermixing effects as well as the magnetic structure of one monolayer Cr on bcc Fe(1) using the same method. The penetration of the Cr atoms in the substrate has been changed in steps of 5 at.% Cr. From our results we can conclude that 5% of the Cr atoms penetrate into the nd layer. This is connected with large lattice relaxation effects. Therefore, this is not observed if the atoms are kept fixed. In this case all Cr atoms would remain on the surface. References Acet, M., H. Zähres, E. F.Wassermann and W. Pepperhoff (199). Hightemperature moment-volume instability and anti-invar of γ-fe. Phys. Rev. B, 9, 61. Davies, A., J. A. Stroscio, D. T. Pierce and R. J. Celotta (1996). Atomic-scale observations of alloying at the Cr-Fe(1) interface. Phys. Rev. Lett., 76, 175. Donohue, J. (198). The Structures of the Elements. R. E. Krieger, Malabar, Florida. Fawcett, E. (1988). Spin-density-wave antiferromagnetism in chromium. Rev. Mod. Phys., 6, 9. Freyss, M., D. Stoeffler and H. Dreyssé (1997). Interfacial alloying and interfacial coupling in Cr/Fe(1). Phys. Rev. B, 56, 67. Guillermet, A. F. and G. Grimvall (1989). Homology of interatomic forces and debye temperatures in transition metals. Phys. Rev. B,, 151. Herper, H. C., E. Hoffmann and P. Entel (1999). Ab initio full-potential study of the structural and magnetic phase stability of iron. Phys. Rev. B, 6, 3839. Hillebrecht, F. U., C. Roth, R. Jungblut, E. Kisker et al. (199). Antiferromagnetic coupling of a Cr overlayer to Fe(1). Europhys. Lett., 19, 711. Kresse, G. and J. Furtmüller (1996). Efficiency of ab initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6, 15. Kresse, G. and J. Hafner (1996). Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements. Comput. Mater. Sci., 6, 15. Kubaschewski, O. (198). Iron-Binary Phase Diagrams. Springer, New York. 1
Moroni, E. G., G. Kresse, J. Hafner and J. Furthmüller (1997). Ultrasoft pseudopotentials applied to magnetic Fe,Co, and Ni: From atoms to solids. Phys. Rev. B, 56, 1569. Pepperhoff, W. and M. Acet (). Konstitution und Magnetismus des Eisens und seiner Legierungen. Springer, Berlin. Perdew, J. P. and Y. Wang (199). Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B, 5, 13. Pfandzelter, R., T. Igel and H. Winter (1996). Intermixing during growth on Fe(1) studied by proton- and electron-induced Auger-electron spectroscopy. Phys. Rev. B, 5, 96. Turek, I., P. Weinberger, M. Freyss, D. Stoeffler et al. (1998). Cr-Fe surface alloy on Fe substrate: CPA-TB-LMTO and semi-empirical tb calculations. Phil. Mag. B, 78, 637. Turtur, C. and G. Bayreuther (199). Magnetic moments in ultrathin Cr films on Fe(1). Phys. Rev. Lett., 7, 1557. Unguris, J., R. J. Celotta and D. T. Pierce (199). Magnetism in Cr thin films on Fe(1). Phys. Rev. Lett., 69, 115. 11