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Growth and properties of small Co islands on a strained Pt surface
Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
Growth and properties of small Co islands on a strained Pt surface R.F. Sabiryanov*, K. Cho, M.I. Larsson1, W.D. Nix, B.M. Clemens Stanford University, Stanford, USA
Abstract The effect of strain on the surface diffusion of Co on a Pt surface is studied using first-principles density functional theory. We employed the projector augmented wave method in the 7-layer slab geometry. Co tends to occupy FCC sites except for surfaces under large tensile strain where HCP sites are energetically favorable. We find that energy barrier for single atom diffusion increases linearly with strain. The total energy of adatoms decreases with increase of strain. The magnetization of the Co adatom is around 2 mB. Pt atoms are strongly polarized making the total magnetic moment per Co atom at FCC sites 3.6 mB. The magnetic moment of the system is sensitive to the adatom position and the strain. For the saddle point, for example, the magnetization changes from 3.1 mB per Co atom for an unstrained lattice to 3.8 mB for a 3% compressed lattice. The results suggest that both FCC and HCP islands formation is possible on the strained Pt(1 1 1) surface. Exchange coupling between adatoms is analyzed using Green’s function method. There is a strong exchange interaction between Co moments in the small clusters on the surface, suggesting that they should behave like giant moments. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface diffusion; Surface strain; Diffusion barrier; Small clusters; Dimer; Trimer; Cobalt; Platinum; Magnetic moments; Exchange interactions
Diffusion of magnetic atoms on surfaces of 4d-metals such as Pt and Pd is very different from the diffusion of adatoms of nonmagnetic elements. It is very important to understand how the growth occurs on these surfaces because it is a key to understand the way to control growth of magnetic nanostructures. Recently, several groups have presented an elegant way of controlling nanostructure growth using elastic properties of interfaces [1–3]. The idea is to utilize the misfit between the lattice constants of the substrate and that of the thin film. The stress due to the misfit is relaxed by creation of dislocations. Due to the repulsive interaction between dislocations a regular array of buried dislocations can be achieved [2]. The array of buried dislocations produces elastic strains at the surface of the thin film. The *Corresponding author. Tel.: 1-650-723-6328. E-mail address: [email protected] (R.F. Sabiryanov). 1 On leave from Department of Physics, Karlstad University, Sweden.
diffusion of the adatoms is known to depend strongly on the strain of the surface [4,5]. As a result small clusters of an adsorbing element can nucleate preferentially in the areas where the surface experiences tensile or compressive strain. In the present paper, we study the diffusion of Co adatoms on the Pt(1 1 1) surface as function of strain, the interaction between Co adatoms and the formation of small Co islands and their magnetic properties. We show that the magnetism of Pt plays an important role in the diffusion of Co adatoms on strained surfaces as well as magnetic properties of small Co clusters. Pt and Pd have large induced magnetic moments due to the impurity atom. A single impurity of Fe in Pd has total moment of about 10 mB [6–8]. Actual local magnetic moment of Fe is smaller, while several shells of Pd have induced magnetic moments [6–8]. We find similar behavior at the surface. Pt has the FCC structure. In the (1 1 1) plane, the FCC structure can be viewed as a layered system where each layer has a
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 0 9 1 - 0
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R.F. Sabiryanov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
triangular lattice and the period is three layers. It is conventional to say that FCC lattice has the ABC stacking (HCP structure has AB stacking of the same layers). There are three symmetric locations for an adatom. HCP and FCC sites have the same coordination with the surface layer of Pt. However, the HCP site has a Pt neighbors in the second layer, while the FCC site does not. We find that for the lattice at small strain the FCC site is most stable. The diffusion of an individual adatoms from one FCC site to another occurs by means of a hopping process via the saddle point and metastable HCP sites (see Fig. 1). For a given geometry it is the bridge position between two Pt sites at the surface. We employ the projector augmented-wave (PAW) method [9]. The detail of the calculations can be found in Ref. [3]. All the structures were relaxed to find the optimized geometries and the energy of the system at the FCC, HCP and saddle point sites. The interactions between Co atoms were analyzed by considering a Co dimer and trimer. We find that the Pt (1 1 1) surface has a small (0.8%) outward relaxation, while the second layer has a very small inward one. Uniform in-plane biaxial strain has been applied simulating the experimental conditions. There is a strong out-of-plane relaxation with a Poisson’s ratio of about 0.4 (experimental bulk value is 0.38). The surface layer has slightly larger Poisson’s ratio. We find from the densities of states that Pt states for the surface layer are more localized, its bandwidth for d-states is smaller, and states near the Fermi energy are shifted slightly towards lower energies, while at the bottom of the valence band they are shifted towards higher energies. With no strain applied a single Co adatom at an FCC ( above the surface, at site resides at the distance of 1.79 A
( and at the saddle an HCP site this distance is 1.774 A ( The underlying nearest-neighbor Pt sites point 1.823 A. sink a little under the surface plane. When tensile strain is applied the distance between adatom and surface decreases as shown in the Fig. 2. The decrease is larger for FCC and HCP sites and less for the bridge position. With large compressive strain applied, adatoms again getting closer to the surface due to the increased polarization of Pt, and magnetic interactions between Pt and Co increase. For the saddle point, for example, the magnetization changes from 3.1 mB for an unstrained lattice atom to 3.8 mB for 3% compressed lattice. The total energy of the slab, calculated as a function of strain, decreases non-linearly with increasing strain. The barrier heights, presented in Fig. 3, show linear increase with strain. Most of the previous calculations of surface diffusion processes on metal surfaces predict linear variation of both energies and barrier height. Dobbs et al. [10] explained these observations in terms of linear elastic theory. The cohesive energy of an adatom is proportional to the strain with coefficient equal to the change of the intrinsic surface stress, Dtij ; in the presence of an adatom compared to the clean surface: ð1Þ
DE ¼ ADtij eij :
This theory assumes that the excess elastic modulus does not change with the presence of an adatom. However, in case of a magnetic adatom it is not always true. Pt sites have a large cumulative magnetic moment of 1.65 mB per Co atom. The magnetically polarized atoms of Pt have additional contributions to their interatomic interaction of magnetic origin. With strain applied the induced magnetic polarization of Pt changes in a non-linear fashion. For tensile strains polarization does not change much, while at compressive strain it
1.84
Distance, A
1.80
1.76
fcc 1.72
hcp saddle 1.68 -0.04
Fig. 1. Geometry of (1 1 1) surface. Arrow shows diffusion path from FCC site to HCP site. The energy profile for the path is presented on the right side.
-0.02
0.00
0.02
0.04
Strain Fig. 2. Distance between Co adatom and the surface as a function of strain.
R.F. Sabiryanov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
1.93 mB for trimer, and 1.85 mB for the monolayer. The magnetic moment of bulk Co is 1.63 mB. We employ the tight-binding linear-muffin-tin-orbitals Green’s function method [11] to study the exchange interactions between Co atoms of the small Co clusters. The infinitesimal angle formalism for exchange coupling has the Heisenberg form in the lowest order [11,12]:
Barrier energy, eV
0.4
0.3
EB2 (hcp)
DE ¼ Jij y2 ¼ 0.2
-0.04
E B1(fcc)
-0.02
367
0.00
0.02
0.04
Strain Fig. 3. Barrier height of hopping diffusion processes from FCC to HCP site (Ebarrier =Esaddle Evalley ) as a function of strain.
increases dramatically. Because the magnetic contribution to the elastic interaction changes in the same manner, the variation for adatom energy as function of strain is non-linear. In addition, calculations for nonmagnetic atoms like Pt or Ag show no variation in energy difference between FCC and HCP positions. In the case of a Co adatom on a Pt(1 1 1) HCP site, it becomes more energetically favorable than the FCC position at 2% tensile strain. This is again due to the magnetic interaction between the Co atom and Pt atoms. HCP site has additional neighbors in subsurface layer and interaction with this atom seems to reduce the energy. As a result both HCP and FCC islands can be formed. With increase of the Co coverage, Co adatoms start interacting with each other and form small islands. Two Co atoms form a strongly bound dimer. Fully relaxed geometry for a Co dimer with both atoms located at FCC sites exhibit considerably shorter distance between the Co atoms than between Pt atoms in the surface. The ( vs. the Pt–Pt distance 2.762 A ( Co–Co distance is 2.206 A or about 20% lower. This forces adatoms to shift from exact FCC positions more in the direction of the bridge site (not exactly although). However, this shift does not result in lifting up the dimer from the surface. The distance of the Co dimer from the surface is 1.742 vs. ( for the single adatom. 1.79 A There is a strong attraction between Co atoms in trimers as well. The distance between Co atoms in a ( and trimer is slightly larger than in a dimer, 2.297 A, trimer atoms are shifted from FCC positions in the direction of the underlying Pt site. This results in lifting ( The spin the trimer from the surface to 1.813 A. magnetic moment on Co atoms decreases from about 2 mB for a single Co adatom, to 1.96 mB for a dimer,
1 Im p
Z
EF
N
m 2 1 k dEDt1 i T0ij Dtj T0ji y ;
ð2Þ
m;k is the where Jij are exchange coupling parameters, T0ij scattering path operator in site representation and 1 1 Dt1 i ¼ tim tik is the difference of inverse single-site scattering matrices, y is the rotation P angle. The on-site (total) exchange parameter J0 ¼ ja0 Joj can be calculated analytically as well [11,12]. The calculated exchange parameters, Jij ; show that Co has strong exchange coupling with nearest neighbors. For dimer the exchange coupling parameter between Co atoms is 2.14 mRy, while for trimer it is 1.97 mRy. The results with coupling between nearest-neighbor Co atoms in the monolayer gives 1.47 mRy and in the bulk HCP Co the value is 1.1 mRy. Comparing these results it can be seen that individual Co pair interactions increase with decrease in coordination. However, the onsite exchange parameter decreases due to the reduction of coordination. For example, most contribution to J0 ¼ 12:9 mRy of the Co bulk comes from the first shell of neighbors (12 first neighbors J01 ¼ 1:1 mRy, while 6 second neighbors J02 ¼ 0:09 mRy). The exchange coupling between Co moments is very strong and we conclude that small clusters will behave like giant moments. Roughly half of this giant moment will be due to the Co atoms and half due to the induced Pt moment. In conclusion, we performed the analysis of surface diffusion of magnetic Co adatoms on the strained Pt(1 1 1) surface and studied elastic and magnetic properties of small Co clusters. We show that magnetism plays an important role in the diffusion process due to the induced magnetic polarization of the Pt atoms. The total spin moment induced on the Pt atoms near the Co adatom is similar to the moment of the Co atom itself. This induced polarization is not a linear function of applied strain. As a result the magnetic contribution to the elastic energy is not a linear function of the strain contrary to the diffusion of nonmagnetic impurities. Co adatoms have strong attractive interaction with nearestneighbor Co or Pt adatoms. The exchange coupling is very strong between Co neighbors in small clusters (larger than in the bulk) and it suggests that these clusters will behave as giant moment.
This work is supported by the NSF Grant EEC0085569. DFT calculations are performed on the
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R.F. Sabiryanov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
NERSC facility and supported by Computational Materials Science Network. References [1] T.R. Mattson, H. Matiu, Appl. Phys. Lett. 75 (1999) 926. [2] K. Bromann, M. Giovanni, H. Brune, K. Kern, Eur. Phys. J. D 9 (1999) 2528. [3] R. Sabiryanov, M. Larsson, K. Cho, W. Nix, B. Clemens, submitted for publication. [4] E. Penev, P. Kratzer, M. Scheffler, Phys. Rev. B 64 (2001) 085401. [5] C. Ratsch, A.P. Seitonen, M. Scheffler, Phys. Rev. B 55 (1997) 6750.
[6] A. Oswald, R. Zeller, P.H. Dederichs, Phys. Rev. Lett. 56 (1991) 1419. [7] J.F. van Acker, W. Speier, R. Zeller, Phys. Rev. B 43 (1991) 9558. [8] P. Mohn, K. Schwarz, J. Phys.: Condens. Matter 5 (1993) 5099. [9] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [10] H.T. Dobbs, A. Zangwill, D.D. Vvedensky, in: M. Tringides (Ed.), Surface Diffusion: Atomistic and Collective Processes, Plenum Press, New York, 1997, p. 263. [11] R.F. Sabiryanov, S.K. Bose, O.N. Mryasov, Phys. Rev. B 51 (1995) 8958. [12] A. Liechtenstein, V. Antropov, M. Katsnelson, V. Gubanov, J. Magn. Magn. Mater. 67 (1987) 65.
Growth and properties of small Co islands on a strained Pt surface R.F. Sabiryanov*, K. Cho, M.I. Larsson1, W.D. Nix, B.M. Clemens Stanford University, Stanford, USA
Abstract The effect of strain on the surface diffusion of Co on a Pt surface is studied using first-principles density functional theory. We employed the projector augmented wave method in the 7-layer slab geometry. Co tends to occupy FCC sites except for surfaces under large tensile strain where HCP sites are energetically favorable. We find that energy barrier for single atom diffusion increases linearly with strain. The total energy of adatoms decreases with increase of strain. The magnetization of the Co adatom is around 2 mB. Pt atoms are strongly polarized making the total magnetic moment per Co atom at FCC sites 3.6 mB. The magnetic moment of the system is sensitive to the adatom position and the strain. For the saddle point, for example, the magnetization changes from 3.1 mB per Co atom for an unstrained lattice to 3.8 mB for a 3% compressed lattice. The results suggest that both FCC and HCP islands formation is possible on the strained Pt(1 1 1) surface. Exchange coupling between adatoms is analyzed using Green’s function method. There is a strong exchange interaction between Co moments in the small clusters on the surface, suggesting that they should behave like giant moments. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface diffusion; Surface strain; Diffusion barrier; Small clusters; Dimer; Trimer; Cobalt; Platinum; Magnetic moments; Exchange interactions
Diffusion of magnetic atoms on surfaces of 4d-metals such as Pt and Pd is very different from the diffusion of adatoms of nonmagnetic elements. It is very important to understand how the growth occurs on these surfaces because it is a key to understand the way to control growth of magnetic nanostructures. Recently, several groups have presented an elegant way of controlling nanostructure growth using elastic properties of interfaces [1–3]. The idea is to utilize the misfit between the lattice constants of the substrate and that of the thin film. The stress due to the misfit is relaxed by creation of dislocations. Due to the repulsive interaction between dislocations a regular array of buried dislocations can be achieved [2]. The array of buried dislocations produces elastic strains at the surface of the thin film. The *Corresponding author. Tel.: 1-650-723-6328. E-mail address: [email protected] (R.F. Sabiryanov). 1 On leave from Department of Physics, Karlstad University, Sweden.
diffusion of the adatoms is known to depend strongly on the strain of the surface [4,5]. As a result small clusters of an adsorbing element can nucleate preferentially in the areas where the surface experiences tensile or compressive strain. In the present paper, we study the diffusion of Co adatoms on the Pt(1 1 1) surface as function of strain, the interaction between Co adatoms and the formation of small Co islands and their magnetic properties. We show that the magnetism of Pt plays an important role in the diffusion of Co adatoms on strained surfaces as well as magnetic properties of small Co clusters. Pt and Pd have large induced magnetic moments due to the impurity atom. A single impurity of Fe in Pd has total moment of about 10 mB [6–8]. Actual local magnetic moment of Fe is smaller, while several shells of Pd have induced magnetic moments [6–8]. We find similar behavior at the surface. Pt has the FCC structure. In the (1 1 1) plane, the FCC structure can be viewed as a layered system where each layer has a
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 0 9 1 - 0
366
R.F. Sabiryanov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
triangular lattice and the period is three layers. It is conventional to say that FCC lattice has the ABC stacking (HCP structure has AB stacking of the same layers). There are three symmetric locations for an adatom. HCP and FCC sites have the same coordination with the surface layer of Pt. However, the HCP site has a Pt neighbors in the second layer, while the FCC site does not. We find that for the lattice at small strain the FCC site is most stable. The diffusion of an individual adatoms from one FCC site to another occurs by means of a hopping process via the saddle point and metastable HCP sites (see Fig. 1). For a given geometry it is the bridge position between two Pt sites at the surface. We employ the projector augmented-wave (PAW) method [9]. The detail of the calculations can be found in Ref. [3]. All the structures were relaxed to find the optimized geometries and the energy of the system at the FCC, HCP and saddle point sites. The interactions between Co atoms were analyzed by considering a Co dimer and trimer. We find that the Pt (1 1 1) surface has a small (0.8%) outward relaxation, while the second layer has a very small inward one. Uniform in-plane biaxial strain has been applied simulating the experimental conditions. There is a strong out-of-plane relaxation with a Poisson’s ratio of about 0.4 (experimental bulk value is 0.38). The surface layer has slightly larger Poisson’s ratio. We find from the densities of states that Pt states for the surface layer are more localized, its bandwidth for d-states is smaller, and states near the Fermi energy are shifted slightly towards lower energies, while at the bottom of the valence band they are shifted towards higher energies. With no strain applied a single Co adatom at an FCC ( above the surface, at site resides at the distance of 1.79 A
( and at the saddle an HCP site this distance is 1.774 A ( The underlying nearest-neighbor Pt sites point 1.823 A. sink a little under the surface plane. When tensile strain is applied the distance between adatom and surface decreases as shown in the Fig. 2. The decrease is larger for FCC and HCP sites and less for the bridge position. With large compressive strain applied, adatoms again getting closer to the surface due to the increased polarization of Pt, and magnetic interactions between Pt and Co increase. For the saddle point, for example, the magnetization changes from 3.1 mB for an unstrained lattice atom to 3.8 mB for 3% compressed lattice. The total energy of the slab, calculated as a function of strain, decreases non-linearly with increasing strain. The barrier heights, presented in Fig. 3, show linear increase with strain. Most of the previous calculations of surface diffusion processes on metal surfaces predict linear variation of both energies and barrier height. Dobbs et al. [10] explained these observations in terms of linear elastic theory. The cohesive energy of an adatom is proportional to the strain with coefficient equal to the change of the intrinsic surface stress, Dtij ; in the presence of an adatom compared to the clean surface: ð1Þ
DE ¼ ADtij eij :
This theory assumes that the excess elastic modulus does not change with the presence of an adatom. However, in case of a magnetic adatom it is not always true. Pt sites have a large cumulative magnetic moment of 1.65 mB per Co atom. The magnetically polarized atoms of Pt have additional contributions to their interatomic interaction of magnetic origin. With strain applied the induced magnetic polarization of Pt changes in a non-linear fashion. For tensile strains polarization does not change much, while at compressive strain it
1.84
Distance, A
1.80
1.76
fcc 1.72
hcp saddle 1.68 -0.04
Fig. 1. Geometry of (1 1 1) surface. Arrow shows diffusion path from FCC site to HCP site. The energy profile for the path is presented on the right side.
-0.02
0.00
0.02
0.04
Strain Fig. 2. Distance between Co adatom and the surface as a function of strain.
R.F. Sabiryanov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
1.93 mB for trimer, and 1.85 mB for the monolayer. The magnetic moment of bulk Co is 1.63 mB. We employ the tight-binding linear-muffin-tin-orbitals Green’s function method [11] to study the exchange interactions between Co atoms of the small Co clusters. The infinitesimal angle formalism for exchange coupling has the Heisenberg form in the lowest order [11,12]:
Barrier energy, eV
0.4
0.3
EB2 (hcp)
DE ¼ Jij y2 ¼ 0.2
-0.04
E B1(fcc)
-0.02
367
0.00
0.02
0.04
Strain Fig. 3. Barrier height of hopping diffusion processes from FCC to HCP site (Ebarrier =Esaddle Evalley ) as a function of strain.
increases dramatically. Because the magnetic contribution to the elastic interaction changes in the same manner, the variation for adatom energy as function of strain is non-linear. In addition, calculations for nonmagnetic atoms like Pt or Ag show no variation in energy difference between FCC and HCP positions. In the case of a Co adatom on a Pt(1 1 1) HCP site, it becomes more energetically favorable than the FCC position at 2% tensile strain. This is again due to the magnetic interaction between the Co atom and Pt atoms. HCP site has additional neighbors in subsurface layer and interaction with this atom seems to reduce the energy. As a result both HCP and FCC islands can be formed. With increase of the Co coverage, Co adatoms start interacting with each other and form small islands. Two Co atoms form a strongly bound dimer. Fully relaxed geometry for a Co dimer with both atoms located at FCC sites exhibit considerably shorter distance between the Co atoms than between Pt atoms in the surface. The ( vs. the Pt–Pt distance 2.762 A ( Co–Co distance is 2.206 A or about 20% lower. This forces adatoms to shift from exact FCC positions more in the direction of the bridge site (not exactly although). However, this shift does not result in lifting up the dimer from the surface. The distance of the Co dimer from the surface is 1.742 vs. ( for the single adatom. 1.79 A There is a strong attraction between Co atoms in trimers as well. The distance between Co atoms in a ( and trimer is slightly larger than in a dimer, 2.297 A, trimer atoms are shifted from FCC positions in the direction of the underlying Pt site. This results in lifting ( The spin the trimer from the surface to 1.813 A. magnetic moment on Co atoms decreases from about 2 mB for a single Co adatom, to 1.96 mB for a dimer,
1 Im p
Z
EF
N
m 2 1 k dEDt1 i T0ij Dtj T0ji y ;
ð2Þ
m;k is the where Jij are exchange coupling parameters, T0ij scattering path operator in site representation and 1 1 Dt1 i ¼ tim tik is the difference of inverse single-site scattering matrices, y is the rotation P angle. The on-site (total) exchange parameter J0 ¼ ja0 Joj can be calculated analytically as well [11,12]. The calculated exchange parameters, Jij ; show that Co has strong exchange coupling with nearest neighbors. For dimer the exchange coupling parameter between Co atoms is 2.14 mRy, while for trimer it is 1.97 mRy. The results with coupling between nearest-neighbor Co atoms in the monolayer gives 1.47 mRy and in the bulk HCP Co the value is 1.1 mRy. Comparing these results it can be seen that individual Co pair interactions increase with decrease in coordination. However, the onsite exchange parameter decreases due to the reduction of coordination. For example, most contribution to J0 ¼ 12:9 mRy of the Co bulk comes from the first shell of neighbors (12 first neighbors J01 ¼ 1:1 mRy, while 6 second neighbors J02 ¼ 0:09 mRy). The exchange coupling between Co moments is very strong and we conclude that small clusters will behave like giant moments. Roughly half of this giant moment will be due to the Co atoms and half due to the induced Pt moment. In conclusion, we performed the analysis of surface diffusion of magnetic Co adatoms on the strained Pt(1 1 1) surface and studied elastic and magnetic properties of small Co clusters. We show that magnetism plays an important role in the diffusion process due to the induced magnetic polarization of the Pt atoms. The total spin moment induced on the Pt atoms near the Co adatom is similar to the moment of the Co atom itself. This induced polarization is not a linear function of applied strain. As a result the magnetic contribution to the elastic energy is not a linear function of the strain contrary to the diffusion of nonmagnetic impurities. Co adatoms have strong attractive interaction with nearestneighbor Co or Pt adatoms. The exchange coupling is very strong between Co neighbors in small clusters (larger than in the bulk) and it suggests that these clusters will behave as giant moment.
This work is supported by the NSF Grant EEC0085569. DFT calculations are performed on the
368
R.F. Sabiryanov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 365–368
NERSC facility and supported by Computational Materials Science Network. References [1] T.R. Mattson, H. Matiu, Appl. Phys. Lett. 75 (1999) 926. [2] K. Bromann, M. Giovanni, H. Brune, K. Kern, Eur. Phys. J. D 9 (1999) 2528. [3] R. Sabiryanov, M. Larsson, K. Cho, W. Nix, B. Clemens, submitted for publication. [4] E. Penev, P. Kratzer, M. Scheffler, Phys. Rev. B 64 (2001) 085401. [5] C. Ratsch, A.P. Seitonen, M. Scheffler, Phys. Rev. B 55 (1997) 6750.
[6] A. Oswald, R. Zeller, P.H. Dederichs, Phys. Rev. Lett. 56 (1991) 1419. [7] J.F. van Acker, W. Speier, R. Zeller, Phys. Rev. B 43 (1991) 9558. [8] P. Mohn, K. Schwarz, J. Phys.: Condens. Matter 5 (1993) 5099. [9] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [10] H.T. Dobbs, A. Zangwill, D.D. Vvedensky, in: M. Tringides (Ed.), Surface Diffusion: Atomistic and Collective Processes, Plenum Press, New York, 1997, p. 263. [11] R.F. Sabiryanov, S.K. Bose, O.N. Mryasov, Phys. Rev. B 51 (1995) 8958. [12] A. Liechtenstein, V. Antropov, M. Katsnelson, V. Gubanov, J. Magn. Magn. Mater. 67 (1987) 65.