《纺织复合材料》课程参考文献（Computational Materials Science，From Basic Principles to Material Properties）08 Magnetism, Structure and Interactions at the Atomic Scale

8 Magnetism,Structure and Interactions at the Atomic Scale V.S.Stepanyuk!and W.Hergert2 1 Max Planck Institute of Microstructure Physics,Weinberg 2,06120 Halle, Germany 2 Martin-Luther-University Halle-Wittenberg,Department of Physics, Von-Seckendorff-Platz 1,06120 Halle,Germany Abstract.An efficient scheme is developed to study magnetism and structure as well as interaction between supported particles on the atomic scale.Starting by ab initio calculations of the electronic structure in the framework of density func- tional theory,interaction potentials for molecular dynamics simulations of metallic nanostructures supported on metallic surfaces are carefully optimized. The two methods are shortly explained.Examples for the application of the methods are given.Mainly electronic and structural properties of Co nanostructures on Cu(001)and Cu(111)surfaces are investigated. 8.1 Introduction The essence of nanoscience and technology is the ability to understand and manipulate matter at the atomic level.Structures behave differently when their dimensions are reduced to dimensions between 1 and 100 nm. Such structures show novel physical and chemical properties,due to their nanoscopic size. In the frontier field of nanomagnetism,understanding of the relationship between magnetism and structure plays a central role.During the past few years experimental investigations of metallic nanostructures in the initial stage of heteroepitaxial growth revealed a lot of information which asks for a consistent theoretical explanation.Some important effects experimentally observed recently are: Surface alloying is found also for metals immiscible in bulk form (i.e.Co onCu(001)).[8.1,8.2 Burrowing of Co clusters into Au,Cu and Ag surfaces has been observed. [8.3.8.4 It was observed,that the motion of adatoms on top of islands is not the same as on a fat surface.[8.5 Fast island decay in homoepitaxial growth was observed by Giesen et al. [8.6-8.9] By using STM(scanning tunnelling microscope)adsorbate manipulation techniques,it is possible to construct atomic-scale structures on metal sur- faces and to study artificially confined quantum systems.[8.10] V.S.Stepanyuk and W.Hergert,Magnetism,Structure and Interactions at the Atomic Scale, Lect.Notes Phys.642,159-176(2004) http://www.springerlink.com/ C Springer-Verlag Berlin Heidelberg 2004

8 Magnetism, Structure and Interactions at the Atomic Scale V.S. Stepanyuk1 and W. Hergert2 1 Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany 2 Martin-Luther-University Halle-Wittenberg, Department of Physics, Von-Seckendorff-Platz 1, 06120 Halle, Germany Abstract. An efficient scheme is developed to study magnetism and structure as well as interaction between supported particles on the atomic scale. Starting by ab initio calculations of the electronic structure in the framework of density func￾tional theory, interaction potentials for molecular dynamics simulations of metallic nanostructures supported on metallic surfaces are carefully optimized. The two methods are shortly explained. Examples for the application of the methods are given. Mainly electronic and structural properties of Co nanostructures on Cu(001) and Cu(111) surfaces are investigated. 8.1 Introduction The essence of nanoscience and technology is the ability to understand and manipulate matter at the atomic level. Structures behave differently when their dimensions are reduced to dimensions between 1 and 100 nm. Such structures show novel physical and chemical properties, due to their nanoscopic size. In the frontier field of nanomagnetism, understanding of the relationship between magnetism and structure plays a central role. During the past few years experimental investigations of metallic nanostructures in the initial stage of heteroepitaxial growth revealed a lot of information which asks for a consistent theoretical explanation. Some important effects experimentally observed recently are: – Surface alloying is found also for metals immiscible in bulk form (i.e. Co on Cu(001) ). [8.1, 8.2] – Burrowing of Co clusters into Au, Cu and Ag surfaces has been observed. [8.3, 8.4] – It was observed, that the motion of adatoms on top of islands is not the same as on a flat surface. [8.5] – Fast island decay in homoepitaxial growth was observed by Giesen et al. [8.6–8.9] – By using STM (scanning tunnelling microscope) adsorbate manipulation techniques, it is possible to construct atomic-scale structures on metal sur￾faces and to study artificially confined quantum systems. [8.10] V.S. Stepanyuk and W. Hergert, Magnetism, Structure and Interactions at the Atomic Scale, Lect. Notes Phys. 642, 159–176 (2004) http://www.springerlink.com/
c Springer-Verlag Berlin Heidelberg 2004

160 V.S.Stepanyuk and W.Hergert To discuss all the effects from theoretical point of view,to get a deep under- standing of the underlying physics,it is absolutely necessary to investigate the real structure of the system as well as the electronic and magnetic struc- ture of the nanosystems,because these aspects are strongly interconnected on the atomic scale. Our combination of the Korringa-Kohn-Rostoker (KKR)Green's func- tion(GF)method with a molecular dynamics(MD)scheme allows us to study the effects mentioned above in detail. We will discuss the methods briefly.The magnetic properties of metallic nanostructures are discussed.We start from an ideal lattice structure and take into account step by step imperfections,mixing and relaxations.The effect of quantum interference and the implications for long-range interactions and self-organization are discussed next.Finally,we introduce the new concept of mesoscopic misfit and discuss the consequences for strain fields,adatom motion and island decay. 8.2 Theoretical Methods 8.2.1 Calculation of Electronic Structure Our calculations are based on the density functional theory and multiple- scattering approach using the Korringa-Kohn-Rostoker Green's function method for low-dimensional systems [8.11].The basic idea of the method is a hierarchical scheme for the construction of the Green's function of adatoms on a metal surface by means of successive applications of Dyson's equation. We treat the surface as an infinite two-dimensional perturbation of the bulk. fcc(001) Fig.8.1.Structure to calculate the surface Green's function for the (001)surface of the fcc-structure (blue -decoupled half-crystals,brown-vacuum layers). For the construction of the ideal surface the nuclear charges of several monolayers are removed,thus creating two half crystals being practically

160 V.S. Stepanyuk and W. Hergert To discuss all the effects from theoretical point of view, to get a deep under￾standing of the underlying physics, it is absolutely necessary to investigate the real structure of the system as well as the electronic and magnetic struc￾ture of the nanosystems, because these aspects are strongly interconnected on the atomic scale. Our combination of the Korringa-Kohn-Rostoker (KKR) Green’s func￾tion(GF) method with a molecular dynamics (MD) scheme allows us to study the effects mentioned above in detail. We will discuss the methods briefly. The magnetic properties of metallic nanostructures are discussed. We start from an ideal lattice structure and take into account step by step imperfections, mixing and relaxations. The effect of quantum interference and the implications for long-range interactions and self-organization are discussed next. Finally, we introduce the new concept of mesoscopic misfit and discuss the consequences for strain fields, adatom motion and island decay. 8.2 Theoretical Methods 8.2.1 Calculation of Electronic Structure Our calculations are based on the density functional theory and multiple￾scattering approach using the Korringa-Kohn-Rostoker Green’s function method for low-dimensional systems [8.11]. The basic idea of the method is a hierarchical scheme for the construction of the Green’s function of adatoms on a metal surface by means of successive applications of Dyson’s equation. We treat the surface as an infinite two-dimensional perturbation of the bulk. Fig. 8.1. Structure to calculate the surface Green’s function for the (001) surface of the fcc-structure (blue -decoupled half-crystals, brown - vacuum layers). For the construction of the ideal surface the nuclear charges of several monolayers are removed, thus creating two half crystals being practically

8 Magnetism,Structure and Interactions at the Atomic Scale 161 uncoupled.Taking into account the 2D periodicity of the ideal surface,we calculate the structural Green's function by solving a Dyson equation self- consistently: ()(+)(E)G EX8.1) j"L Here G is the structural Green's function of the bulk in a k-layer representa- tion(j,j-layer indices).The k wave vector belongs to the 2D Brillouin zone. △t,(E)is the perturbation of the t matrix to angular momentum L=(亿，m) in the j-th layer. The consideration of adsorbate atoms on the surface destroys the trans- lation symmetry.Therefore the Green's function of the adsorbate adatom on the surface has to be calculated in a real space formulation.The structural Green's function of the ideal surface in real space representation is then used as the reference Green's function for the calculation of the adatom-surface system from an algebraic Dyson equation: G(E)=Gt(E)+∑G2(E)4t%(E)G院(E, (8.2) n"Eu where G(E)is the energy-dependent structural Green's function matrix onn and GLL(E)the corresponding matrix for the ideal surface,serving as a reference system.At(E)describes the difference in the scattering properties at site n induced by the existence of the adsorbate atom. Exchange and correlation effects are included in the local density approx- imation.The full charge density and the full potential approximation can be used in the calculations.Details of the method and several of its applications can be found elsewhere [8.11]. 8.2.2 Molecular Dynamics Simulations In the last years we developed a method which connects the ab initio elec- tronic structure calculations with large scale molecular dynamics simulations. Our approach is based on fitting of the interaction parameters of potentials for molecular dynamic simulations to accurate first-principle calculations of selected cluster-substrate properties,bulk properties and forces acting on adatoms of the system under investigation.8.12 To describe metallic clusters on noble metal substrates,many body poten- tials in the second moment tight-binding approximation are used.[8.13,8.14] The cohesive energy Ecoh is the sum of the band energy EB and the repulsive part ER

8 Magnetism, Structure and Interactions at the Atomic Scale 161 uncoupled. Taking into account the 2D periodicity of the ideal surface, we calculate the structural Green’s function by solving a Dyson equation self￾consistently: Gjj
LL
(k , E) = G˚jj
LL
(k , E) +  j

L

G˚jj

LL

(k , E)∆tj

L

(E)Gj

j
L

L
(k , E)(8.1) . Here G is the structural Green’s function of the bulk in a ˚ k -layer representa￾tion (j, j - layer indices). The k wave vector belongs to the 2D Brillouin zone. ∆tj L(E) is the perturbation of the t matrix to angular momentum L = (l, m) in the j-th layer. The consideration of adsorbate atoms on the surface destroys the trans￾lation symmetry. Therefore the Green’s function of the adsorbate adatom on the surface has to be calculated in a real space formulation. The structural Green’s function of the ideal surface in real space representation is then used as the reference Green’s function for the calculation of the adatom-surface system from an algebraic Dyson equation: Gnn
LL
(E) = G˚nn
LL
(E) +  n

L

G˚nn

LL

(E)∆tn

L

(E)Gn

n
L

L
(E), (8.2) where Gnn
LL
(E) is the energy-dependent structural Green’s function matrix and G˚nn

LL

(E) the corresponding matrix for the ideal surface, serving as a reference system. ∆tn L(E) describes the difference in the scattering properties at site n induced by the existence of the adsorbate atom. Exchange and correlation effects are included in the local density approx￾imation. The full charge density and the full potential approximation can be used in the calculations. Details of the method and several of its applications can be found elsewhere [8.11]. 8.2.2 Molecular Dynamics Simulations In the last years we developed a method which connects the ab initio elec￾tronic structure calculations with large scale molecular dynamics simulations. Our approach is based on fitting of the interaction parameters of potentials for molecular dynamic simulations to accurate first-principle calculations of selected cluster-substrate properties, bulk properties and forces acting on adatoms of the system under investigation. [8.12] To describe metallic clusters on noble metal substrates, many body poten￾tials in the second moment tight-binding approximation are used. [8.13, 8.14] The cohesive energy Ecoh is the sum of the band energy EB and the repulsive part ER

162 V.S.Stepanyuk and W.Hergert Eoh=∑(E路+R) (8.3) 1/2 -2aa/-l (8.4) E=∑(（Aier/r88-1)+4A8a)ep(-pas(r/n6-10 (8.5) whererij represents the distance between the atoms i and j,and ro is the first-neighbour distance in the a,B lattice structure,while it is just an adjustable parameter in the case of the cross interaction.is an effective hopping integral and depends on the material and gos and pas describe the dependence of the interaction strength on the relative interatomic distance. Table 8.1.Data used for the fitting of the potential together with the values calculated with the optimized potential.(cohesive energy Ec,bulk modulus B,elas- tic constants C from Cleri et al.[8.13],first and second neighbour interaction energies Efgc,Eco from Hoshino et al.8.15]solution energy Eon Cu from Drittler etal 816]and binding energies of small CoclustersE)E E2x2island oncu(oo)are calculated using the KKR Green's function method. quantity data fitted value Cu aCu 3.615A 3.614A (fcc) e 3.544eV 3.545eV 1.42 Mbar 1.42 Mbar C11 1.76 Mbar 1.76 Mbar C12 1.25 Mbar 1.25 Mbar C44 0.82 Mbar 0.82 Mbar Co aC 2.507A 2.515A Ee 4.386eV 4.395eV B 1.948 Mbar 1.989 Mbar C1 3.195 Mbar3.337 Mbar C12 1.661 Mbar 1.426 Mbar C13 1.021 Mbar 1.178 Mbar C33 3.736 Mbar 3.665 Mbar C44 0.824 Mbar 0.646 Mbar ●o-Cu Eso in Cu 0.4eV 0.38eV -0.12eV -0.18eV 0.03eV -0.05eV ECu(001) -1.04eV -1.04eV ECn Cu -0.26eV -0.35eV -2.06eV -1.96eV -3.84eV -3.86eV

162 V.S. Stepanyuk and W. Hergert Ecoh =  i Ei B + Ei R  (8.3) Ei B = −   j ξ2 αβ exp(−2qαβ(rij/rαβ 0 − 1))   1/2 (8.4) Ei R =  j  A1 αβ(rij/rαβ 0 − 1)) + A0 αβ exp(−pαβ(rij/rαβ 0 − 1)) (8.5) where rij represents the distance between the atoms i and j, and r αβ 0 is the first-neighbour distance in the α, β lattice structure, while it is just an adjustable parameter in the case of the cross interaction. ξ is an effective hopping integral and depends on the material and qαβ and pαβ describe the dependence of the interaction strength on the relative interatomic distance. Table 8.1. Data used for the fitting of the potential together with the values calculated with the optimized potential. (cohesive energy Ec, bulk modulus B, elas￾tic constants Cij from Cleri et al. [8.13], first and second neighbour interaction energies ECo-Co 1,b , ECo-Co 2,b from Hoshino et al. [8.15] solution energy ECo in Cu S from Drittler et al. [8.16] and binding energies of small Co clusters ECo-Co 1,on Cu(001), ECo-Co 1,in Cu, Etrimer on Cu(100), E2×2island on Cu(100) are calculated using the KKR Green’s function method. quantity data fitted value Cu aCu 3.615 ˚A 3.614 ˚A (fcc) Ec 3.544 eV 3.545 eV B 1.42 Mbar 1.42 Mbar C11 1.76 Mbar 1.76 Mbar C12 1.25 Mbar 1.25 Mbar C44 0.82 Mbar 0.82 Mbar Co aCo 2.507 ˚A 2.515 ˚A Ec 4.386 eV 4.395 eV B 1.948 Mbar 1.989 Mbar C11 3.195 Mbar 3.337 Mbar C12 1.661 Mbar 1.426 Mbar C13 1.021 Mbar 1.178 Mbar C33 3.736 Mbar 3.665 Mbar C44 0.824 Mbar 0.646 Mbar Co-Cu ECo in Cu S 0.4 eV 0.38 eV ECo-Co 1,b -0.12 eV -0.18 eV ECo-Co 2,b 0.03 eV -0.05 eV ECo-Co 1,on Cu(001) -1.04 eV -1.04 eV ECo-Co 1,in Cu -0.26 eV -0.35 eV Etrimer on Cu(100) -2.06 eV -1.96 eV E2×2 island on Cu(100) -3.84 eV -3.86 eV

8 Magnetism,Structure and Interactions at the Atomic Scale 163 We will explain the method for the system Co/Cu(001).Co and Cu are not miscible in bulk form.Therefore the determination of the cross interac- tion is a problem.A careful fitting to accurate first-principles calculations of selected cluster substrate properties solves the problem.The result is a manageable and inexpensive scheme able to account for structural relaxation and including implicitly magnetic effects,crucial for a realistic determination of interatomic interactions in systems having a magnetic nature.After de- termination of the Cu-Cu parameters,which are fitted to experimental data only 8.14,the Co-Co and Co-Cu parameters are optimized simultaneously by including in the fit the results of first-principles KKR calculations.To this purpose,we have taken the solution energy of a single Co impurity in bulk Cu ESo in Cu [8.16],energies of interaction of two Co impurities in Cu bulk 815]EoEand binding energies of small supported Co clusters oCu(100)E2x2 island on Cu(001)-E)E(terrace position),E on Cu(100) The set of data used to define the potential and the corresponding values cal- culated by means of the optimized potential are given in Table 8.1.The bulk and surface properties are well reproduced. The method,discussed so far has been further improved.We are able to calculate forces on atoms above the surface on the ab initio level.The forces are also included in the fitting procedure.This gives a further improvement of the potentials used in the MD simulations.It should be mentioned that our method allows also to use only ab initio bulk properties from KKR cal- culations.Therefore,we can construct ab initio based many-body potentials. 8.3 Magnetic Properties of Nanostructures on Metallic Surfaces Using the KKR Green's function method we have studied the properties of 3d,4d and 5d adatoms on Ag(001),Pd(001)and Pt(001)systematically. 8.17,8.18 One central point of investigation was the study of imperfect nanostructures.We have investigated the influence of Ag impurities on the magnetism on small Rh and Ru clusters on the Ag(001)surface.[8.19]The change of the magnetic moments could be explained in the framework of a tight-binding model.Nevertheless it was observed that the magnetism of Rh nanostructures shows some unusual effects.8.20]An anomalous increase in the magnetic moments of Rh adatoms on the Ag(001)surface with decreasing interatomic distance between atoms was observed,whereas for dimers of other transition metals the opposite behaviour is found. In this chapter we will discuss some selected results for the real,electronic and magnetic structure of metal nanostructures on noble metal surfaces.We will concentrate our discussion on one special system:Co nanostructures on Cu surfaces.Although a special system is investigated general conclusions can be drawn

8 Magnetism, Structure and Interactions at the Atomic Scale 163 We will explain the method for the system Co/Cu(001). Co and Cu are not miscible in bulk form. Therefore the determination of the cross interac￾tion is a problem. A careful fitting to accurate first-principles calculations of selected cluster substrate properties solves the problem. The result is a manageable and inexpensive scheme able to account for structural relaxation and including implicitly magnetic effects, crucial for a realistic determination of interatomic interactions in systems having a magnetic nature. After de￾termination of the Cu-Cu parameters, which are fitted to experimental data only [8.14], the Co-Co and Co-Cu parameters are optimized simultaneously by including in the fit the results of first-principles KKR calculations. To this purpose, we have taken the solution energy of a single Co impurity in bulk Cu ECo in Cu S [8.16], energies of interaction of two Co impurities in Cu bulk [8.15] ECo-Co 1,b , ECo-Co 2,b and binding energies of small supported Co clusters on Cu(001) - ECo-Co 1,on Cu(001), ECo-Co 1,in Cu (terrace position), Etrimer on Cu(100), E2×2 island on Cu(100). The set of data used to define the potential and the corresponding values cal￾culated by means of the optimized potential are given in Table 8.1. The bulk and surface properties are well reproduced. The method, discussed so far has been further improved. We are able to calculate forces on atoms above the surface on the ab initio level. The forces are also included in the fitting procedure. This gives a further improvement of the potentials used in the MD simulations. It should be mentioned that our method allows also to use only ab initio bulk properties from KKR cal￾culations. Therefore, we can construct ab initio based many-body potentials. 8.3 Magnetic Properties of Nanostructures on Metallic Surfaces Using the KKR Green’s function method we have studied the properties of 3d, 4d and 5d adatoms on Ag(001), Pd(001) and Pt(001) systematically. [8.17, 8.18] One central point of investigation was the study of imperfect nanostructures. We have investigated the influence of Ag impurities on the magnetism on small Rh and Ru clusters on the Ag(001) surface. [8.19] The change of the magnetic moments could be explained in the framework of a tight-binding model. Nevertheless it was observed that the magnetism of Rh nanostructures shows some unusual effects. [8.20] An anomalous increase in the magnetic moments of Rh adatoms on the Ag(001) surface with decreasing interatomic distance between atoms was observed, whereas for dimers of other transition metals the opposite behaviour is found. In this chapter we will discuss some selected results for the real, electronic and magnetic structure of metal nanostructures on noble metal surfaces. We will concentrate our discussion on one special system: Co nanostructures on Cu surfaces. Although a special system is investigated general conclusions can be drawn

164 V.S.Stepanyuk and W.Hergert 8.3.1 Metamagnetic States of 3d Nanostructures on the Cu(001)Surface The existence of different magnetic states like high spin ferromagnetic (HSF),low spin ferromagnetic (LSF)and antiferromagnetic (AF)states is well known for bulk systems. A theoretical investigation of Zhou et al.8.21]shows that up to five different magnetic states are found for y Fe.(LSF,HSF,AF,and two ferri- magnetic states).Different theoretical investigations have shown,that energy differences between the magnetic states can be of the order of 1 meV.In such a case magnetic fluctuations can be excited by temperature changes or exter- nal fields.Magneto-volume effects play also an important role in the theory of the Invar effect.[8.22] Lee and Callaway [8.23]have studied the electronic and magnetic proper- ties of free V and Cr clusters.They found that for some atomic spacings as many as four or five magnetic states exists for a Vo or Cro cluster.The typical low and high spin moments are 0.33 uB and 2.78 uB for the Vo cluster. We have calculated the electronic and magnetic properties of small 3d transition metal clusters on the Cu(001)surfaces.Dimers,trimers and tetramers,as given in Fig.8.2,are investigated.All atoms occupy ideal lat- tice sites.No relaxation at the surface is taken into account.[8.24,8.25] While larger clusters might show a non-collinear structure of the magnetic moments,such a situation is not likely for the clusters studied here.Dimers and tetramers have only one non-equivalent site in the paramagnetic state. [010 ① E E M3 M Q● M4 [100] Fig.8.2.Metallic nanostructures (Dimer,trimer and tetramer)on the fcc(001) surface

164 V.S. Stepanyuk and W. Hergert 8.3.1 Metamagnetic States of 3d Nanostructures on the Cu(001)Surface The existence of different magnetic states like high spin ferromagnetic (HSF), low spin ferromagnetic (LSF) and antiferromagnetic (AF) states is well known for bulk systems. A theoretical investigation of Zhou et al. [8.21] shows that up to five different magnetic states are found for γ Fe. (LSF, HSF, AF, and two ferri￾magnetic states). Different theoretical investigations have shown, that energy differences between the magnetic states can be of the order of 1 meV. In such a case magnetic fluctuations can be excited by temperature changes or exter￾nal fields. Magneto-volume effects play also an important role in the theory of the Invar effect. [8.22] Lee and Callaway [8.23] have studied the electronic and magnetic proper￾ties of free V and Cr clusters. They found that for some atomic spacings as many as four or five magnetic states exists for a V9 or Cr9 cluster. The typical low and high spin moments are 0.33 µB and 2.78 µB for the V9 cluster. We have calculated the electronic and magnetic properties of small 3d transition metal clusters on the Cu(001) surfaces. Dimers, trimers and tetramers, as given in Fig. 8.2, are investigated. All atoms occupy ideal lat￾tice sites. No relaxation at the surface is taken into account. [8.24, 8.25] While larger clusters might show a non-collinear structure of the magnetic moments, such a situation is not likely for the clusters studied here. Dimers and tetramers have only one non-equivalent site in the paramagnetic state. Fig. 8.2. Metallic nanostructures (Dimer, trimer and tetramer) on the fcc(001) surface

8 Magnetism,Structure and Interactions at the Atomic Scale 165 The trimers have two non-equivalent sites(C-center,E,E'-edge positions). Ferromagnetic states of the trimers,either low spin(LSF)or high spin(HSF) states,have parallel moments at the sites C and E,E,but the moments have different sign at C and E,E'in the antiferromagnetic state(AF).The atoms at the edge positions (E,E')have the same moment (MEME)for LSF, HSF and AF states.Another possible magnetic state,which is compatible with the chemical symmetry of the system is an antisymmetric (AS)one. The magnetic moment at the central atom of the trimer is zero and the moments at the edge positions have different sign (ME=-ME). We concentrate our discussion on the multiplicity of magnetic states to V and Mn.For the single V adatom only a high spin state with a moment of 3.0 uB is obtained.For the V2 dimer we find both a ferromagnetic and an antiferromagnetic state with moments of 2.85 and 2.58 uB respectively.The antiferromagnetic state has the lowest energy being about 0.2 eV/atom lower than the ferromagnetic one. The magnetic moments for all the different magnetic states of the V and Mn trimers are summarized in Table 8.2.All the magnetic states have a lower total energy than the paramagnetic state.The AF state is the ground state of the V trimer.The energy difference between the AF and LSF state in V is about 8 meV/atom.The LSF state is more stable than the HSF state.The ground state of the Mn trimer is also the antiferromagnetic state. The energy difference between the ground state and the HSF state is only 2 meV/atom.This energy difference corresponds to a temperature difference of 25 K.A transition between the two states caused by temperature changes or an external field leads to a change of the total moment of the Mn trimer of 7.8 uB.Such a strong change of the total moment,controlled by an external parameter opens a new field for an experimental proof of the theoretical results. Table 8.2.Magnetic moments (in uB)for the atoms of the trimers V3 and Mng onCu(001). V3 Mns state Me Mc ME ME Mc ME' HSF2.852.582.854.033.834.03 LSF 2.631.412.634.040.014.04 AF 2.63-2.022.633.99-3.883.99 AS 2.620.00-2.623.980.00-3.98 We have shown,that metamagnetic behaviour exists in supported clus- ters.It is shown,that the energy differences between different magnetic states can be small,which can lead to a change of the magnetic state of the cluster by an external parameter.The energy differences between different magnetic

8 Magnetism, Structure and Interactions at the Atomic Scale 165 The trimers have two non-equivalent sites (C - center, E, E - edge positions). Ferromagnetic states of the trimers, either low spin (LSF) or high spin (HSF) states, have parallel moments at the sites C and E, E , but the moments have different sign at C and E, E in the antiferromagnetic state (AF). The atoms at the edge positions (E, E ) have the same moment (ME = ME
) for LSF, HSF and AF states. Another possible magnetic state, which is compatible with the chemical symmetry of the system is an antisymmetric (AS) one. The magnetic moment at the central atom of the trimer is zero and the moments at the edge positions have different sign (ME = −ME
). We concentrate our discussion on the multiplicity of magnetic states to V and Mn. For the single V adatom only a high spin state with a moment of 3.0 µB is obtained. For the V2 dimer we find both a ferromagnetic and an antiferromagnetic state with moments of 2.85 and 2.58 µB respectively. The antiferromagnetic state has the lowest energy being about 0.2 eV/atom lower than the ferromagnetic one. The magnetic moments for all the different magnetic states of the V and Mn trimers are summarized in Table 8.2. All the magnetic states have a lower total energy than the paramagnetic state. The AF state is the ground state of the V trimer. The energy difference between the AF and LSF state in V is about 8 meV/atom. The LSF state is more stable than the HSF state. The ground state of the Mn trimer is also the antiferromagnetic state. The energy difference between the ground state and the HSF state is only 2 meV/atom. This energy difference corresponds to a temperature difference of 25 K. A transition between the two states caused by temperature changes or an external field leads to a change of the total moment of the Mn trimer of 7.8 µB. Such a strong change of the total moment, controlled by an external parameter opens a new field for an experimental proof of the theoretical results. Table 8.2. Magnetic moments (in µB) for the atoms of the trimers V3 and Mn3 on Cu(001). V3 Mn3 state ME MC ME
ME MC ME
HSF 2.85 2.58 2.85 4.03 3.83 4.03 LSF 2.63 1.41 2.63 4.04 0.01 4.04 AF 2.63 -2.02 2.63 3.99 -3.88 3.99 AS 2.62 0.00 -2.62 3.98 0.00 -3.98 We have shown, that metamagnetic behaviour exists in supported clus￾ters. It is shown, that the energy differences between different magnetic states can be small, which can lead to a change of the magnetic state of the cluster by an external parameter. The energy differences between different magnetic

166 V.S.Stepanyuk and W.Hergert states will strongly depend on the cluster size.Therefore such ab inito calcu- lations can help to select interesting systems for experimental investigations. 8.3.2 Mixed Co-Cu Clusters on Cu(001) The magnetic properties of Co nanostructures on Cu substrate can be strongly influenced by Cu atoms.For example,Cu coverages as small as three hundredths of a monolayer drastically affect the magnetization of Co films.[8.26 Experiments and theoretical studies demonstrated that magneti- zation of mixed clusters of Co and Cu depends on the relative concentration of Co and Cu in a nonobvious way.Quenching of ferromagnetism in Co clus- ters embedded in copper was reported.8.27]Calculations by means of our MD method showed that surface alloying is energetically favourable in the case of Co/Cu(001)and mixed Co-Cu clusters are formed in the early stages of heteroepitaxy.Recent experiments 8.2 suggest that mixed Co-Cu clusters indeed exist. Cu Cu -0.004 Co 1.680 0.018 1.660 1.830 0.011 Fig.8.3.Spin polarization of Co-Cu mixed clusters on Cu(001).Magnetic moments in Bohr magnetons are given for all inequivalent site. We have studied all possible mixed configurations in 3x 3-atoms islands on Cu(001)surfaces.[8.28]We observe a small induced moment at the Cu atoms in the island and a decrease of the moments at the Co atoms in comparison with the 3x 3 Co-island.A stronger reduction of the Co moments is achieved, if the Cog cluster is surrounded by a Cu brim and capped by a Cu cluster.A reduction of 14 is obtained for the average moment of the Cog cluster.This effect should have a strong influence on the properties of the Co-Cu interface in the early stages of growth.Coating of Co clusters with Cu atoms has been found recently in experiments.8.4 8.3.3 Effect of Atomic Relaxations on Magnetic Properties of Adatoms and Small Clusters Possible technological applications of supported magnetic clusters are con- nected with the magnetic anisotropy energy (MAE),which determines the

166 V.S. Stepanyuk and W. Hergert states will strongly depend on the cluster size. Therefore such ab inito calcu￾lations can help to select interesting systems for experimental investigations. 8.3.2 Mixed Co-Cu Clusters on Cu(001) The magnetic properties of Co nanostructures on Cu substrate can be strongly influenced by Cu atoms. For example, Cu coverages as small as three hundredths of a monolayer drastically affect the magnetization of Co films. [8.26] Experiments and theoretical studies demonstrated that magneti￾zation of mixed clusters of Co and Cu depends on the relative concentration of Co and Cu in a nonobvious way. Quenching of ferromagnetism in Co clus￾ters embedded in copper was reported. [8.27] Calculations by means of our MD method showed that surface alloying is energetically favourable in the case of Co/Cu(001) and mixed Co-Cu clusters are formed in the early stages of heteroepitaxy. Recent experiments [8.2] suggest that mixed Co-Cu clusters indeed exist. Fig. 8.3. Spin polarization of Co-Cu mixed clusters on Cu(001). Magnetic moments in Bohr magnetons are given for all inequivalent site. We have studied all possible mixed configurations in 3×3-atoms islands on Cu(001) surfaces. [8.28] We observe a small induced moment at the Cu atoms in the island and a decrease of the moments at the Co atoms in comparison with the 3×3 Co-island. A stronger reduction of the Co moments is achieved, if the Co9 cluster is surrounded by a Cu brim and capped by a Cu cluster. A reduction of 14 % is obtained for the average moment of the Co9 cluster. This effect should have a strong influence on the properties of the Co-Cu interface in the early stages of growth. Coating of Co clusters with Cu atoms has been found recently in experiments. [8.4] 8.3.3 Effect of Atomic Relaxations on Magnetic Properties of Adatoms and Small Clusters Possible technological applications of supported magnetic clusters are con￾nected with the magnetic anisotropy energy (MAE), which determines the

8 Magnetism,Structure and Interactions at the Atomic Scale 167 orientation of the magnetization of the cluster with respect to the surface. Large MAE barriers can stabilize the magnetization direction in the cluster and a stable magnetic bit can be made.Ab initio calculations have predicted very large MAE and orbital moments for 3d,5d adatoms and 3d clusters on Ag(001).[8.29-8.31] The interplay between magnetism and atomic structure is one of the cen- tral issues in physics of new magnetic nanostructures.Performing ab initio and tight-binding calculations we demonstrate the effect of atomic relaxations on the magnetic properties of Co adatoms and Co clusters on the Cu(001) surface.[8.32]We address this problem by calculating magnetic properties of the Co adatom and the Cog cluster on the Cu(100)surface.First,we calcu- late the effect of relaxations on the spin moment of the Co atom using the KKR Green's function method.(cf.Fig.8.4) 1.90 1.85 ￥°1.80 relaxed position 1.75 1.70 04402000204●n Energy (Ry) 0 2 4 6 8101214161820 Vertical displacement (% Fig.8.4.The dependence of the spin magnetic moment of the Co adatom on the distance from the Cu substrate.The relaxed position of the adatom is indicated:0% correspond to Co at a Cu interlayer separation above the Cu surface.The magnetic moments per atom are given in Bohr magnetons.Inset:ab initio (thick line)and TB(thin line)results for the d component of the local density of states (LDOS)of the Co adatom for unrelaxed position. We employ the tight binding electronic Hamiltonian with parameters cho- sen to fit the KKR local densities of electronic states and the local magnetic moments of Co overlayers and small clusters on Cu(100)to calculate the MAE.To evaluate the MAE the intraatomic spin-orbit coupling is presented by the operator L.s where is the spin-orbit coupling parameter. The results for the orbital moments and the MAE of the Co adatoms in the unrelaxed and relaxed positions are presented in Table 8.3.Calculation of

8 Magnetism, Structure and Interactions at the Atomic Scale 167 orientation of the magnetization of the cluster with respect to the surface. Large MAE barriers can stabilize the magnetization direction in the cluster and a stable magnetic bit can be made. Ab initio calculations have predicted very large MAE and orbital moments for 3d, 5d adatoms and 3d clusters on Ag(001). [8.29–8.31] The interplay between magnetism and atomic structure is one of the cen￾tral issues in physics of new magnetic nanostructures. Performing ab initio and tight-binding calculations we demonstrate the effect of atomic relaxations on the magnetic properties of Co adatoms and Co clusters on the Cu(001) surface. [8.32] We address this problem by calculating magnetic properties of the Co adatom and the Co9 cluster on the Cu(100) surface. First, we calcu￾late the effect of relaxations on the spin moment of the Co atom using the KKR Green’s function method. (cf. Fig. 8.4) Fig. 8.4. The dependence of the spin magnetic moment of the Co adatom on the distance from the Cu substrate. The relaxed position of the adatom is indicated: 0% correspond to Co at a Cu interlayer separation above the Cu surface. The magnetic moments per atom are given in Bohr magnetons. Inset: ab initio (thick line) and TB (thin line) results for the d component of the local density of states (LDOS) of the Co adatom for unrelaxed position. We employ the tight binding electronic Hamiltonian with parameters cho￾sen to fit the KKR local densities of electronic states and the local magnetic moments of Co overlayers and small clusters on Cu(100) to calculate the MAE. To evaluate the MAE the intraatomic spin-orbit coupling is presented by the operator ξL · s where ξ is the spin-orbit coupling parameter. The results for the orbital moments and the MAE of the Co adatoms in the unrelaxed and relaxed positions are presented in Table 8.3. Calculation of

168 V.S.Stepanyuk and W.Hergert Table 8.3.Magnetic orbital moments and magnetic anisotropy energy of a single Co adatom on the Cu(100)surface:L2 and L are the orbital moments for mag- netization along the normal Z and in-plane X direction;the electronic part of the mangetic anisotropy energy AE(meV)is presented. Unrelaxed geometry Relaxed geometry L四 1.06 0.77 1.04 0.80 AE 1.70 -0.37 the MAE reveal that for the unrelaxed position above thethe surface,out-of plane magnetization for the Co adatom is more stable.The relaxation of the vertical position of the adatom by 14%shortens the first nearest neighbour Co-Cu separation from 2.56 to 2.39 A and has a drastic effect on MAE.We find that the relaxation of the Co adatom leads to in-plane magnetization. Ideal geometry Relaxed geometry 1.757 Y -1.708 1.757 1.699 -1.829 1,783 X Atom 兕 叹 △E(X,Z) 以 以 △EX,Z) 1 0.11 0.25 -1.48 0.09 0.18 -1.13 2 0.17 0.24 -1.35 0.13 0.19 -1.05 2* 0.17 0.27 -0.87 0.13 0.25 -1.26 3 0.21 0.43 -2.44 0.18 0.39 -2.60 Atom 吗 LXY △EX+Y,Z) 喝 L △E(X+Y,Z 1 0.11 0.25 -1.47 0.09 0.21 -1.06 2 0.17 0.25 -1.20 0.13 0.20 -1.15 3 0.21 0.40 -2.17 0.18 0.37 -2.38 3* 0.21 0.40 -2.30 0.18 0.35 -2.71 Fig.8.5.Magnetic properties of Cog cluster on the Cu(100)surface in unrelaxed and relaxed geometries;spin magnetic moments in Bohr magnetons are shown for each atom in the cluster;orbital magnetic moments and electronic part of the MAE are presented in the table for the normal 2,in-plane X and X+Y directions of the magnetization.For the unrelaxed cluster the the average MAE is-1.74 meV/Co atom for X-direction;for X+Y direction these values are-1.69 meV/Co atom. For the relaxed cluster,the above energies are-1.79 meV/Co atom for X direction and -1.76 meV/Co atom for X+Y direction

168 V.S. Stepanyuk and W. Hergert Table 8.3. Magnetic orbital moments and magnetic anisotropy energy of a single Co adatom on the Cu(100) surface:Lm Z and Lm X are the orbital moments for mag￾netization along the normal Z and in-plane X direction; the electronic part of the mangetic anisotropy energy ∆E (meV) is presented. Unrelaxed geometry Relaxed geometry Lm Z 1.06 0.77 Lm X 1.04 0.80 ∆E 1.70 -0.37 the MAE reveal that for the unrelaxed position above the the surface, out-of plane magnetization for the Co adatom is more stable. The relaxation of the vertical position of the adatom by 14% shortens the first nearest neighbour Co-Cu separation from 2.56 to 2.39 ˚A and has a drastic effect on MAE. We find that the relaxation of the Co adatom leads to in-plane magnetization. Fig. 8.5. Magnetic properties of Co9 cluster on the Cu(100) surface in unrelaxed and relaxed geometries; spin magnetic moments in Bohr magnetons are shown for each atom in the cluster; orbital magnetic moments and electronic part of the MAE are presented in the table for the normal Z, in-plane X and X +Y directions of the magnetization. For the unrelaxed cluster the the average MAE is −1.74 meV/Co atom for X-direction; for X + Y direction these values are −1.69 meV/Co atom. For the relaxed cluster, the above energies are −1.79 meV/Co atom for X direction and −1.76 meV/Co atom for X + Y direction