Complex magnetism of small clusters on surfaces An approach from first principles Phivos Mavropoulos IFF, Forschungszentrum Jülich Collaboration: S. Lounis, H. Höhler, R. Zeller, S. Blügel, P.H. Dederichs (FZ Jülich) J. Kroha (Universität Bonn) V. Popescu, H. Ebert (LMU München) N. Papanikolaou (NCRS “Demokritos”, Athens)

Appetizer: Adatoms and small clusters transition from atomic to bulk behaviour Spin moments: 4d & 5d on Ag(001), shape & size dependence Wildberger et al, PRL 75, 509 (1995)

Appetizer: Adatoms and small clusters transition from atomic to bulk behaviour I. Cabria et al., PRB 65, (2002) Spin and orbital moments: 3d & 4d on Ag(001) Orbital moments of Co clusters on Pt P. Gambardella et al., Science 300, 1130 (2003)

Ingredients for the study of clusters Magnetic clusters on surfaces Surface electronic structure Real-space embedding method Charge and spin density Non-collinear magnetism Transport properties (STM) Static and dynamic correlations Spin and Orbital moments Lattice relaxations

Ingredients for the study of clusters Magnetic clusters on surfaces Surface electronic structure Real-space embedding method Charge and spin density Non-collinear magnetism Static correlations Dynamic correlations Spin and Orbital moments Transport properties (STM) ? competing interactions

Ingredients for the study of clusters Magnetic clusters on surfaces Surface electronic structure Real-space embedding method Charge and spin density Non-collinear magnetism Static correlations Dynamic correlations Spin and Orbital moments Transport properties (STM)

Calculations from first principles Density-functional theory –Maps the many-electron problem to effective mean-field problem. –Accurate for ground-state electronic & magnetic properties in bulk, surfaces, interfaces, defects. –Successful for transition metals. –No adjustable parameters. –Designed for ground state, but gives reasonable excitation spectrum in many cases. Green-function method of Korringa, Kohn and Rostoker (KKR) –Multiple-scattering approach. –Reciprocal and real-space method. –Suitable for impurities & clusters, no supercell needed.

KKR Green-function method Green function G connected to G 0 of a reference system via Dyson eq.: KKR Representation of Green function: Method suitable for: Bulk calculations, Interfaces, Surfaces Impurity clusters on surfaces Magnetism in clusters (non-collinear) Disordered systems (CPA) Electronic transport: STM etc. Accurate calculation of: Charges & magn. moments Total energies Forces on atoms Lattice relaxations P.H. Dederichs and R. Zeller, Jülich

Adatoms: FM vs. AF Atoms on Fe(001) and on Fe/Cu(001) Stepanyuk et al, PRB 61, 2356 (2000) Alexander-Anderson model

Adatoms on ferromagnetic surfaces Nonas et al., PRB 57, 84 (1998) Stepanyuk et al, PRB 61, 2356 (2000) 3d and 4d adatoms on Fe (001) 3d on Fe antiferro ferro Early transition elements align antiferromagnetically Late transition elements align ferromagnetically Interpretation via Alexander-Anderson model 3d adatoms on Ni (001)

Fe clusters on Ni(001) Motivation: recent experimental results (Lau et al, PRL 89, (2002)) Trend: spin moment as function of: Cluster size Coordination of Fe Result: linear behavior Similar on Ni(111) and Cu

Fe clusters on Ni(111), Cu(001), Cu(111)

Comparison: Fe clusters vs. Co clusters

Non-collinear magnetism ? Driving mechanism: magnetic frustration Example: Mn dimer on Ni(001) Collinear result (frustrated): Mn-Ni: ferro, Mn-Mn: antiferro Competing interactions Non-collinear result θ=72.5º E.g.: Trimer on (111) of paramagnetic metal Dimer/trimer on ferromagnetic surface

Dimers on Ni(001)-collinear vs. noncollinear Cr or Mn first neighbours are AF coupled. → Candidates for frustration Second & third neighbours are always FM coupled to each other (coupling with substrate prevails). Cr dimer on Ni(001) Noncollinear:  (Cr)=94.2 ,  Collinear result: Frustrated state

Non-collinear dimers and trimers Fit to Heisenberg model: how good is it? J is fit by collinear total energy calculations of ferro- and antiferro allignment

Example: Mn trimer on Ni(001) Side view Top view Plan: bigger clusters, include relaxations, relate to XMCD. Mn-Ni: ferro, Mn-Mn: antiferro Mn 1,Mn 3Mn 2Ni 1,2,3,5Ni 4Ni 6,7Ni 8 θ(degrees) φ(degrees)

Fe clusters on W(001): c2×2 Antiferromagnetic order (Collaboration with P. Ferriani and S. Heinze. Recent experiment: Kubetzka et al.) Antiferro Ferro Antiferro c2×2

Dynamical correlations: Kondo behaviour Approach based on the theory of Logan [Logan et al., J. Phys: C.M. 10, 2673 (1998)] UHF spin-polarised solution of Anderson model Impurity spin fluctuations within the RPA Construct Self-energy New Green function: Kondo peak emerges at Fermi level Self-consistency to satisfy Friedel sum rule

Dynamical correlations: Kondo behaviour Outlook: Extend the theory to LDA Impurity Green function from KKR Describe Kondo behaviour of impurities in bulk and on surfaces The Logan approximation captures low and high-energy characteristics: Kondo-peak Scaling behaviour Correction to Hubbard bands LDA GF → new GF: G (Kondo) = G (LDA) + G (LDA) Σ G (Kondo) Scaling with U Scaling with 1/N

Conclusion: Realistic, material-specific description Magnetic clusters on surfaces Surface electronic structure + lattice relaxations Real-space embedding method Charge and spin density Spin and Orbital moments Static correlations Dynamic correlations Non-collinear magnetism Transport properties (STM) OK OK (LDA+U) OK On the way Mavropoulos et al., PRB(2004) (to be published)

Non-collinear Green function method GF for spin up & spin down becomes a matrix in spin space Density for spin up & spin down becomes density matrix

STM results Papanikolaou et al, PRB 62, (2000) Caculations with Tersoff-Hamann model

Dynamical correlations: Kondo behaviour Outlook: Extend the theory to LDA Impurity Green function from KKR Describe Kondo behaviour of impurities in bulk and on surfaces The Logan approximation captures low and high-energy characteristics: Kondo-peak Scaling behaviour Correction to Hubbard bands LDA GF → new GF: G (Kondo) = G (LDA) + G (LDA) Σ G (Kondo)