Temperature-dependence of magnetism of free Fe clusters

Temperature-dependence of magnetism of free Fe clusters O. Šipr 1, S. Bornemann 2, J. Minár 2, S. Polesya 2, H. Ebert 2 1 Institute of Physics, Academy of Sciences CR, Prague, Czech Republic 2 Universität München, Department Chemie, München, Germany p.1/19

Outline Introducing finite-temperature magnetism (a bit of theory) p.2/19

Outline Introducing finite-temperature magnetism (a bit of theory) Exchange coupling in clusters p.2/19

Outline Introducing finite-temperature magnetism (a bit of theory) Exchange coupling in clusters Dependence of mean magnetization of clusters on temperature and on cluster size p.2/19

Outline Introducing finite-temperature magnetism (a bit of theory) Exchange coupling in clusters Dependence of mean magnetization of clusters on temperature and on cluster size Critical ( Curie ) temperature T c of clusters p.2/19

Motivation Magnetism is about ordering of magnetic moments p.3/19

Motivation Magnetism is about ordering of magnetic moments Rule of thumb for studying different geometries: follow the coordination number! p.3/19

Motivation Magnetism is about ordering of magnetic moments Rule of thumb for studying different geometries: follow the coordination number! Lowering the coordination number increase of magnetic moments enhanced magnetization decrease of coupling reduced magnetization When comparing surfaces with bulk: competition between enhancement of magnetic moments and reduction of their coupling p.3/19

Bulk, surfaces, clusters,... Clusters contain a large portion of surface atoms p.4/19

Bulk, surfaces, clusters,... Clusters contain a large portion of surface atoms However, their properties cannot be expressed as a mere linear combination of surface and bulk contributions p.4/19

Bulk, surfaces, clusters,... Clusters contain a large portion of surface atoms However, their properties cannot be expressed as a mere linear combination of surface and bulk contributions Finite cluster dimensions and associated quantum size effects are significant factors p.4/19

Systems to be studied Free spherical Fe clusters, geometry taken as if cut from a bcc Fe crystal Cluster sizes: between 9 atoms (1 coordination shell) and 89 atoms (7 coordinations shells) shells atoms radius [a.u.] Fe cluster 27 atoms view in Z direction 3rd shell 2nd shell 1st shell center 1 9 4.70 2 15 5.42 3 27 7.67 4 51 8.99 5 59 9.39 6 65 10.85 7 89 11.82 p.5/19

Calculations for T =0 Ab-initio within LDA framework (material specific) Scalar-relativistic real-space spin-polarized multiple-scattering formalism Atomic sphere approximation (ASA) Using SPRKKR code http://olymp.cup.uni-muenchen.de/ak/ebert/sprkkr p.6/19

Calculations for T 0 For localized moments, finite-temperature magnetism can be described by a classical Heisenberg hamiltonian H eff = i j J ij e i e j e i e j p.7/19

Mapping DFT onto Heisenberg Coupling constants J ij can be obtained from ground-state electronic properties: J ij = 1 EF 4π Im de Tr [ (t 1 i t 1 i ) τ ij (t 1 j t 1 j ) τ ji ] [Liechtenstein et al. JMMM 67, 65 (1987)] p.8/19

From J ij to M(T ) Mean magnetization M(T ) of a system described by a classical Heisenberg hamiltonian is M(T ) = k M k exp( E k k exp( E k k B T ) k B T ) M k is the magnetization of the system for a particular configuration k of the directions of spins E k is the energy of configuration k Practical evaluation: Monte Carlo method with the importance sampling Metropolis algorithm p.9/19

Fe clusters at T =0 mag. moment per atom [ B ] 3.2 3.0 2.8 2.6 2.4 2.2 Average mag. moments bulk 2.0 0 20 40 60 80 100 cluster size (number of atoms) Average magnetic moment of clusters oscillates with cluster size Further reading: O. Šipr et al. Phys. Rev. B 70, 174423 (2004) p.10/19

Fe clusters at T =0 mag. moment per atom [ B ] 3.2 3.0 2.8 2.6 2.4 2.2 Average mag. moments bulk 2.0 0 20 40 60 80 100 cluster size (number of atoms) mag. moment at site [ B ] 3.2 3.0 2.8 2.6 2.4 2.2 2.0 89-atom cluster 0 2 4 6 8 10 12 distance from center [a.u.] Average magnetic moment of clusters oscillates with cluster size Local magnetic moments increase when going from the center outwards in an oscillatory way Further reading: O. Šipr et al. Phys. Rev. B 70, 174423 (2004) p.10/19

J ij in bulk and in clusters 50 Exchange coupling between atoms 40 coupling parameter J ij [mev] 30 20 10 bulk crystal bulk 0-10 4 6 8 10 12 distance between atoms i and j [a.u.] Atom i is fixed, atom j scans coordination shells around i p.11/19

J ij in bulk and in clusters 50 Exchange coupling between atoms 40 coupling parameter J ij [mev] 30 20 10 bulk crystal 89-atom cluster, central atom bulk central 0 central -10 4 6 8 10 12 distance between atoms i and j [a.u.] Atom i is fixed, atom j scans coordination shells around i p.11/19

J ij in bulk and in clusters 50 Exchange coupling between atoms 40 coupling parameter J ij [mev] 30 20 10 bulk outermost crystal 89-atom cluster, central atom 89-atom cluster, outermost atom bulk central 0 central -10 4 6 8 10 12 distance between atoms i and j [a.u.] outermost Atom i is fixed, atom j scans coordination shells around i Oscillatory decay of J ij with distance J ij for a given distance differs a lot between clusters and crystals p.11/19

Site-dependence of j J ij Total strength with which one spin (at site i) is held in its direction: Energy needed to flip the spin of atom i while keeping all the remaining spins collinear: J i = j i J ij p.12/19

Site-dependence of j J ij sum of all coupling parameters J i = j J ij [mev] 300 200 100 Total coupling of a particular atom 15 cluster of 15 atoms 0 2 4 6 8 10 12 distance of atom i from the center of the cluster [a.u.] p.12/19

Site-dependence of j J ij sum of all coupling parameters J i = j J ij [mev] 300 200 100 51 Total coupling of a particular atom 15 cluster of 51 atoms cluster of 15 atoms 0 2 4 6 8 10 12 distance of atom i from the center of the cluster [a.u.] p.12/19

Site-dependence of j J ij sum of all coupling parameters J i = j J ij [mev] 300 200 100 51 Total coupling of a particular atom 15 cluster of 89 atoms cluster of 51 atoms cluster of 15 atoms 0 2 4 6 8 10 12 distance of atom i from the center of the cluster [a.u.] 89 p.12/19

Site-dependence of j J ij sum of all coupling parameters J i = j J ij [mev] 300 200 100 51 Total coupling of a particular atom 15 bulk crystal cluster of 89 atoms cluster of 51 atoms cluster of 15 atoms 0 2 4 6 8 10 12 distance of atom i from the center of the cluster [a.u.] 89 Mild differences in J ij translate into large differences in j J ij No systematics in cluster size, no systematics in the position of atom within a cluster p.12/19

Decay of magnetization with T 3.5 3.0 Temperature-dependence of magnetization mean magnetization M(T) [ B ] 2.5 2.0 1.5 1.0 0.5 bulk cluster of 89 atoms crystal 89 [Bulk M(T ) curve was extrapolated to calculated T C ] 0.0 0 200 400 600 800 1000 1200 temperature T [K] M(T ) curves are more shallow in clusters than in bulk p.13/19

Decay of magnetization with T 3.5 3.0 Temperature-dependence of magnetization mean magnetization M(T) [ B ] 2.5 2.0 1.5 1.0 0.5 bulk cluster of 59 atoms cluster of 89 atoms crystal 89 59 [Bulk M(T ) curve was extrapolated to calculated T C ] 0.0 0 200 400 600 800 1000 1200 temperature T [K] M(T ) curves are more shallow in clusters than in bulk Small clusters more shallow M(T ) curves than large clusters p.13/19

Decay of magnetization with T 3.5 3.0 Temperature-dependence of magnetization mean magnetization M(T) [ B ] 2.5 2.0 1.5 1.0 0.5 bulk cluster of 27 atoms cluster of 59 atoms cluster of 89 atoms crystal 27 89 59 [Bulk M(T ) curve was extrapolated to calculated T C ] 0.0 0 200 400 600 800 1000 1200 temperature T [K] M(T ) curves are more shallow in clusters than in bulk Small clusters more shallow M(T ) curves than large clusters p.13/19

Decay of magnetization with T 3.5 3.0 Temperature-dependence of magnetization mean magnetization M(T) [ B ] 2.5 2.0 1.5 1.0 0.5 bulk cluster of 15 atoms cluster of 27 atoms cluster of 59 atoms cluster of 89 atoms crystal 15 27 89 59 [Bulk M(T ) curve was extrapolated to calculated T C ] 0.0 0 200 400 600 800 1000 1200 temperature T [K] M(T ) curves are more shallow in clusters than in bulk Small clusters more shallow M(T ) curves than large clusters p.13/19

Experimental and theoretical M(T ) 3.5 mean magnetization M(T) [ B ] 3.0 2.5 2.0 1.5 1.0 0.5 50-60 82-92 M(T) by experiment and by theory 50-60 atoms, experiment 82-92 atoms, experiment 59 atoms, theory 89 atoms, theory 89 59 Experiment from Billas et al. PRL 71, 4067 (1993) 0.0 0 200 400 600 800 1000 1200 temperature T [K] Caution: Each experimental curve corresponds to a range of cluster sizes (it represents an average over several M(T ) curves which may differ quite a lot one from another) p.14/19

M as function of cluster size mean magnetization M(T) [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Magnetization as function of cluster size 0 K 0.0 0 20 40 60 80 100 cluster size (number of atoms) p.15/19

M as function of cluster size mean magnetization M(T) [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Magnetization as function of cluster size 0 K 90 K 290 K 0.0 0 20 40 60 80 100 cluster size (number of atoms) Dependence of M on cluster size does not really vary with T for low ( experimental ) temperatures p.15/19

M as function of cluster size mean magnetization M(T) [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Magnetization as function of cluster size 0 K 90 K 290 K 610 K 810 K 1130 K 0.0 0 20 40 60 80 100 cluster size (number of atoms) Dependence of M on cluster size does not really vary with T for low ( experimental ) temperatures For large T, magnetization of large clusters is significantly reduced p.15/19

Dependence of T c on cluster size 3.5 3.0 M(T) and its derivative mean magnetization M(T) [ B ] 2.5 2.0 1.5 1.0 0.5 59 atoms, theory 89 atoms, theory 89 59 0.0 0 200 400 600 800 1000 1200 temperature T [K] Critical temperature T c defined as the inflection point of M(T ) curves (no phase transition for finite systems) p.16/19

Dependence of T c on cluster size 1200 1000 bulk critical temperature T c [K] 800 600 400 200 clusters crystal clusters, calculated with proper J ij mean magnetization M(T) [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 59 atoms, theory 89 atoms, theory M(T) and its derivative 0.0 0 200 400 600 800 1000 1200 temperature T [K] 89 59 0 0 50 100 150 200 cluster size (number of atoms) Critical temperature T c defined as the inflection point of M(T ) curves (no phase transition for finite systems) T c oscillates with cluster size p.16/19

Dependence of T c on cluster size 1200 1000 bulk critical temperature T c [K] 800 600 400 200 proper J ij bulk-like J ij crystal clusters, calculated with proper J ij clusters, calculated with bulk-like J ij mean magnetization M(T) [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 59 atoms, theory 89 atoms, theory M(T) and its derivative 0.0 0 200 400 600 800 1000 1200 temperature T [K] 89 59 0 0 50 100 150 200 cluster size (number of atoms) Critical temperature T c defined as the inflection point of M(T ) curves (no phase transition for finite systems) T c oscillates with cluster size Proper cluster-adjusted J ij have to be taken into account p.16/19

Shell-resolved magnetization Expectations: outer shells have smaller coordination numbers than inner shells M in outer shells should decay more quickly with T than M of inner shells p.17/19

Shell-resolved magnetization 1.0 normalized projection of magnetization 0.8 0.6 0.4 0.2 Shell-resolved projection of M(T) for a 89-atom cluster 1 st shell 7 th shell whole cluster Projecting M of a shell onto the direction of the total M of the 89-atom cluster (Projections are normalized to T =0) 0.0 0 200 400 600 800 1000 1200 temperature T [K] p.17/19

Shell-resolved magnetization 1.0 normalized projection of magnetization 0.8 0.6 0.4 0.2 Shell-resolved projection of M(T) for a 89-atom cluster 1 st shell 2 nd shell 7 th shell whole cluster Projecting M of a shell onto the direction of the total M of the 89-atom cluster (Projections are normalized to T =0) 0.0 0 200 400 600 800 1000 1200 temperature T [K] Not monotonous in order of shells p.17/19

Shell-resolved magnetization 1.0 normalized projection of magnetization 0.8 0.6 0.4 0.2 Shell-resolved projection of M(T) for a 89-atom cluster 1 st shell 2 nd shell 3 rd shell 7 th shell whole cluster Projecting M of a shell onto the direction of the total M of the 89-atom cluster (Projections are normalized to T =0) 0.0 0 200 400 600 800 1000 1200 temperature T [K] Not monotonous in order of shells Although M of outer shells usually decays faster than M of inner shells, no systematics can be found. p.17/19

Magnetic profile for T 0 projection of shell-resolved magnetization [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 4 6 8 10 12 distance from the center of the cluster [a.u.] 0 K Cluster of 89 atoms (7 coordination shells) p.18/19

Magnetic profile for T 0 projection of shell-resolved magnetization [ B ] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Magnetic profile of 89-atom cluster depending on temperature 0.0 4 6 8 10 12 distance from the center of the cluster [a.u.] 0 K 90 K 290 K 610 K 810 K 1130 K Cluster of 89 atoms (7 coordination shells) Magnetic profile gets more flat if temperature increases p.18/19

Summary Exchange coupling constants J ij in clusters differ from the bulk, with no obvious systematics p.19/19

Summary Exchange coupling constants J ij in clusters differ from the bulk, with no obvious systematics Magnetization M(T ) curves are more shallow for small clusters than for large clusters p.19/19

Summary Exchange coupling constants J ij in clusters differ from the bulk, with no obvious systematics Magnetization M(T ) curves are more shallow for small clusters than for large clusters For large T, magnetization of large clusters is significantly reduced Critical temperature T c oscillates with cluster size (Normalized) M of the outer shells usually decreases with T more quickly than M of inner shells p.19/19