­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Magnetic anisotropy of transition­metal dimers
and isolated adatoms on non­magnetic substrates
P. Błoński, J. Hafner Fakultät für Physik and Center for Computational Materials Science, Universität Wien ­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Outline
➢
Cluster geometry
➢
Motivation
➢
Computational method ➢
Trends in binding energies, geometries, magnetic moments – collinear and non­collinear calculations ➢
Summary and conclusions
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Cluster geometry
CLUSTERS – finite aggregates of atoms and molecules bound by forces which may be metallic, covalent, ionic, hydrogen­bonded or van­der­Waals in characters and can contain from a few to tens of thousand of atoms
MOLECULES usually have a well defined composition and structure Dozens of isomers even for a small cluster (Ar13) Davis et al. J. Chem. Phys. 86 (1987) 6456
Type
Nature of bonding
Binding energy
Ionic
Examples
(NaCl)n, Nan Fn­1
ionic bonds
2 – 4 eV
Covalent
C60, Sn
covalent bonding
1 – 4 eV
Simple and noble metals
Nan, Aln , Agn
metallic bonds
0.5 – 3 eV
Transition metal
van der Waals
Nin, Pdn, Ptn
metallo­covalent
rare gas clusters dispersion Magnetic anisotropy of small transition­metal clusters
0.5 – 3 eV
< 0.3 eV
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Cluster geometry
isolated adatom
dumbbell
tetrahedron
equilateral triangle ➢
trigonal bipyramid
cluster geometry optimization – without any symmetry constraints
structural distortion may be driven by different mechanism:
➢
●
a genuine instability of the assumed structure under the action of interatomic forces ●
a Jahn­Teller mechanism if partially occupied eigenstates exist ●
magneto­structural effects
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Spin­orbit constant
Fe Co Ni
Ru Rh Pd
Os Ir Pt
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Motivation
➢
Unique catalytic properties: ultra­fine dispersed Pd clusters supported on Al – superior CO oxidation catalysts than single crystals of Pd; enhanced activity for reduction for nitric oxide by carbon monoxide; higher activity and better selectivity in hydrogenation process
➢
Unusual magnetic properties: non­magnetic in the bulk materials – magnetic when dimensionality is reduced ➢
Possibility to assume non­crystallographic arrangements (icosahedral or other) – an important factor distinguishing clusters from other low­
dimensional structures Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Motivation
➢
Unique catalytic properties: ultra­fine dispersed Pd clusters supported on Al – superior CO oxidation catalysts than single crystals of Pd; enhanced activity for reduction for nitric oxide by carbon monoxide; higher activity and better selectivity in hydrogenation process
➢
Unusual magnetic properties: non­magnetic in the bulk materials – magnetic when dimensionality is reduced ➢
Possibility to assume non­crystallographic arrangements (icosahedral or other) – an important factor distinguishing clusters from other low­
dimensional structures A football (soccer ball) with full icosahedral symmetry. This commonplace object is not, however, a regular icosahedron; it is a spherical truncated icosahedron.
Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups.
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Motivation
➢
Unique catalytic properties: ultra­fine dispersed Pd clusters supported on Al – superior CO oxidation catalysts than single crystals of Pd; enhanced activity for reduction for nitric oxide by carbon monoxide; higher activity and better selectivity in hydrogenation process
➢
Unusual magnetic properties: non­magnetic in the bulk materials – magnetic when dimensionality is reduced ➢
Possibility to assume non­crystallographic arrangements (icosahedral or other) – an important factor distinguishing clusters from other low­
dimensional structures A football (soccer ball) with full icosahedral symmetry. This object is not, however, a regular icosahedron; it is a spherical truncated icosahedron.
Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups.
Non­crystallographic symmetry, together with variation in bond lengths and coordination – important changes in the electronic properties – significant consequence for the magnetic and chemical properties Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Motivation
➢
Magnetic properties:in present state­of­the­art hard disk drives hundreds of single­domain particles hold one bit of information – an important line of research has centered on molecular magnets1 ­ very small transition metal clusters embedded on appropriate matrix might re­establish the limit for magnetic storage2,3 ➢
quantity that is of interest is the magnetic anisotropy representing the barrier for magnetic moments re­orientation and arises from the spin­orbit interaction
➢
the calculation of the magnetic anisotropy is a problem of fundamental importance. Magnetic anisotropy, magneto­optical spectra, magnetic dichroism and other important properties are caused by spin­orbit coupling.
1
S. Maekawa and T. Shinjo, Spin Dependent Transport in Magnetic Structures, Taylor & Francis, London, 2002.
P. Błoński and J. Hafner, PRB, submitted (2009).
3
T. O. Strandberg, et al., Nature Materials 6, 648 (2007); Phys. Rev. B 77, 174416.
2
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Computational details DFT­based code (VASP)
➢
Exc – spin­polarized GGA (PW91)
➢
➢
Electron­ion interaction – PAW – full­potential all­electron projector augmented wave method ➢
Hybrid functionals mixing density­functional and Hartree­Fock exchange (PBE0 and HSE03 functionals)
➢
DFT+U method (an orbital­dependent on­site Coulomb repulsion U)
Ecut = 250  700 eV
➢
➢
Scalar relativistic and fully relativistic calculations including spin­
orbit coupling (allowing for non­collinearity of spin and orbital moments)
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Computational details Cluster calc.: 10 x 10 x 10 Å3 cubic supercell ➢
➢
Adatom adsorption: slabs consisting of 5 to 10 atomic layers with 5 x 5 surface unit cell (up to 251 atoms) and a vacuum space of 16 Å ➢
Stringent relaxation criterion, 10­7 eV ­ essential
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Computational details Cluster calc.: 10 x 10 x 10 Å3 cubic supercell ➢
➢
Adatom adsorption: slabs consisting of 5 to 10 atomic layers with 5 x 5 surface unit cell (up to 251 atoms) and a vacuum space of 16 Å ➢
Stringent relaxation criterion, 10­7 eV ­ essential
All moments set along global z axis, x axis, [1 1 1 ] direction Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Computational details Cluster calc.: 10 x 10 x 10 Å3 cubic supercell ➢
➢
Adatom adsorption: slabs consisting of 5 to 10 atomic layers with 5 x 5 surface unit cell (up to 251 atoms) and a vacuum space of 16 Å ➢
Stringent relaxation criterion, 10­7 eV ­ essential
All moments set along global z axis, x axis, [1 1 1 ] direction in­plane
out of plane
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Co2
scalar relat.
inc. SO coupl.
z
S
2.00
J
x
1.96
Rh2
scalar relat.
S
2.34
1.96
2.00
Ir2
inc. SO coupl.
J
2.00
2.21
scalar relat.
S
2.84
2.21
2.00
inc. SO coupl.
J
2.00
2.21
2.92
2.22
2.00
2.34
2.84
2.92
S,L = 0.0o
S,L = 0.0o
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
3.90 0.78 3.90 0.32 7.1
4.09 2.00 4.14 0.33 50

32
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
Axial
Perp.
µS
µL
µS
µL MAE

3.86 1.82 3.80 0.5 47.3 136
3.98 2.12 3.93 0.63 104
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
Magnetic anisotropy of small transition­metal clusters
Axial
Perp.
µS
µL
µS
µL MAE

3.88 1.96 3.42 0.94 69.8 413
4.00 1.34 4.10 1.24 100
L. Fernandez­Seivane and J. Ferrer, PRL 99, 183401 (2007)
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Co2
scalar relat.
inc. SO coupl.
z
S
2.00
J
x
1.96
Rh2
scalar relat.
S
2.34
1.96
2.00
Ir2
J
2.00
2.21
scalar relat.
inc. SO coupl.
S
2.84
2.21
2.00
inc. SO coupl.
J
2.00
2.21
2.92
2.22
2.00
2.34
2.84
2.92
S,L = 0.0o
S,L = 0.0o
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
3.90 0.78 3.90 0.32 7.1
4.09 2.00 4.14 0.33 50

32
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
Axial
Perp.
µS
µL
µS
µL MAE

3.86 1.82 3.80 0.5 47.3 136
3.98 2.12 3.93 0.63 104
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
Axial
Perp.
µS
µL
µS
µL MAE

3.88 1.96 3.42 0.94 69.8 413
4.00 1.34 4.10 1.24 100
L. Fernandez­Seivane and J. Ferrer, PRL 99, 183401 (2007)
➢
the easy magnetic axis oriented along the dimer axis
➢
scalar­relativistic calculations ­ ground state with S=2 ➢
spin­orbit coupling (SOC) leaves the interatomic distance unchanged for Co2 and Rh2, only for the Ir2 the bond­length increases by 0.01 Å ➢
SOC mixes different spin­eigenstates ­ a slight reduction of the magnetic spin­
moment ­ more pronounced with increasing strength of SOC
➢
strong SOC induces an anisotropy of the spin­moment
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Co2
scalar relat.
inc. SO coupl.
z
S
2.00
Rh2
J
x
S
2.34
1.96
1.96
2.00
inc. SO coupl.
J
scalar relat.
S
2.00
2.84
2.21
2.21
2.00
inc. SO coupl.
J
2.00
2.21
2.92
2.22
2.00
2.34
2.84
2.92
S,L = 0.0o
S,L = 0.0o
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
3.90 0.78 3.90 0.32 7.1
4.09 2.00 4.14 0.33 50

32
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
➢
scalar relat.
Ir2
Axial
Perp.
µS
µL
µS
µL MAE

3.86 1.82 3.80 0.5 47.3 136
3.98 2.12 3.93 0.63 104
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
Axial
Perp.
µS
µL
µS
µL MAE

3.88 1.96 3.42 0.94 69.8 413
4.00 1.34 4.10 1.24 100
L. Fernandez­Seivane and J. Ferrer, PRL 99, 183401 (2007)
Bruno's1 expression: MAE   µS ∆µL
MAE should be about 12 times as large for Rh2 than for Co2 ­ we find an increase by a factor of 7 ➢
Ir2 ­ a substantial anisotropy of the spin moment ­ Bruno's formula cannot be expected to be valid ➢
Van der Laan's2 approximate expression accounts for the spin anisotropy in the limit of a weak SOC ­ not the case for Ir2 1
P. Bruno, Phys. Rev. B 39, 865 (1989).
G. van der Laan, J. Phys.: Condens. Matter 10, 3239 (1998).
2
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Co2
Rh2
Magnetic anisotropy of small transition­metal clusters
Ir2
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Co2
➢
Rh2
Ir2
HOMO ­ a doubly degenerate antibonding d* minority state occupied by one electron only
➢
For d* the SOC splitting is 0.94 (Ir2), 0.41 (Rh2) and 0.26 eV (Co2)
➢
Lowering of the d* states ­ dominant effect determining the MAE
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Ni2
scalar relat.
inc. SO coupl.
z
S
1.00
Pd2
J
x
S
1.28
2.08
scalar relat.
2.09
1.00
2.48
1.00
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
1.00
2.33
2.31
2.38
1.00
1.17
Axial
Perp.
µS
µL
µS
µL MAE
1.96 0.02 1.98 0.36 2.3
1.94 0.86 1.98 0.53 5.0
inc. SO coupl.
J
2.49
S,L = 0.0
2.31
o

404
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE

1.88 2.74 1.34 0.80 46.3 742
1.90 2.40 1.65 1.20 220
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
L. Fernandez­Seivane and J. Ferrer, PRL 99, 183401 (2007)
L = ~1.00 B if d > 2.5 Å
d = 2.35 Å for perp. magnet.
1.18
2.09
1.18
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
1.98 0.12 1.98 0.38 5.9
7.6
scalar relat.
S
1.17
1.00
S,L = 0.0o

101
inc. SO coupl.
J
1.28
Axial
Perp.
µS
µL
µS
µL MAE
1.98 0.58 1.96 0.30 6.5
1.99 0.88 1.99 0.45 11
Pt2

101
T. O. Strandberget al., PRB 77, 174416 (2008)
∆E = 27 meV
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Ni2
scalar relat.
inc. SO coupl.
z
S
1.00
Pd2
J
x
scalar relat.
S
1.28
2.08
J
1.00

101
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
1.00
2.31
2.33
2.38
1.00
1.17
Axial
Perp.
µS
µL
µS
µL MAE
1.96 0.02 1.98 0.36 2.3
1.94 0.86 1.98 0.53 5.0
inc. SO coupl.
J
2.49
S,L = 0.0
S,L = 0.0o
scalar relat.
S
1.17
1.00
1.28
Axial
Perp.
µS
µL
µS
µL MAE
1.98 0.58 1.96 0.30 6.5
1.99 0.88 1.99 0.45 11
inc. SO coupl.
2.48
2.09
1.00
Pt2
2.31
o

404
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE

1.88 2.74 1.34 0.80 46.3 742
1.90 2.40 1.65 1.20 220
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
L. Fernandez­Seivane and J. Ferrer, PRL 99, 183401 (2007)
L = ~1.00 B if d > 2.5 Å
d = 2.35 Å for perp. magnet.
1.18
2.09
➢
1.18
length and almost equal spin moments, but widely S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
1.98 0.12 1.98 0.38 5.9
7.6

101
different orbital moments ➢
Pd2 ­ discontinuous change of the orbital magnetic moment if the dimer bond length is increased beyond T. O. Strandberget al., PRB 77, 174416 (2008)
∆E = 27 meV
Ni2 ­ two locally stable solutions with equal bond 2.50 Å
➢
Pt2 ­ SOC influences the bond length
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Ni2
Pd2
Magnetic anisotropy of small transition­metal clusters
Pt2
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Ni2
➢
➢
Pd2
Pt2
ground state ­ triplet (S=1) state, but the Kohn­Sham eigenvalue spectra differ already at the scalar­relativistic level ­ different electronic configurations of the free atoms in their ground state
all doubly degenerate dimer eigenstates ­ either fully occupied by two electrons or empty ­ MAE relatively low unless SOC leads to a re­ordering and a change of occupation of the levels close to the Fermi energy.
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Fe2
scalar relat.
inc. SO coupl.
z
S
3.00
J
x
1.98
Ru2
scalar relat.
S
3.08
1.98
3.00
2.07
2.00
2.09
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
Magnetic anisotropy of small transition­metal clusters
Perp.
µS
µL MAE

3.94 0.24 36.5 334
inc. SO coupl.
J
2.00
2.09
2.07
S,L = 0.0o
Axial
µS
µL
3.98 0
scalar relat.
S
2.09
2.00
S,L = 0.0o

24
inc. SO coupl.
J
3.08
Axial
Perp.
µS
µL
µS
µL MAE
5.84 0.32 5.84 0.16 0.3
5.84 1.74
6.00 1.89 6.00 0.19 32
Os2
2.92
2.10
2.00
2.92
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE

3.74 ­0.8 3.48 0.62 28.8 885
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Fe2
scalar relat.
inc. SO coupl.
z
S
3.00
Ru2
J
x
scalar relat.
S
3.08
1.98
1.98
3.00
2.07
2.00
scalar relat.
S
2.09
2.00
S,L = 0.0o

24
inc. SO coupl.
J
3.08
Axial
Perp.
µS
µL
µS
µL MAE
5.84 0.32 5.84 0.16 0.3
5.84 1.74
6.00 1.89 6.00 0.19 32
Os2
J
2.00
2.92
2.09
2.07
2.09
Perp.
µS
µL MAE

3.94 0.24 36.5 334
2.10
2.00
2.92
S,L = 0.0o
S,L = 0.0o
Axial
µS
µL
3.98 0
inc. SO coupl.
Axial
Perp.
µS
µL
µS
µL MAE

3.74 ­0.8 3.48 0.62 28.8 885
D. Fritsch et al., J. Comp. Chem. 29, 2210 (2008)
➢
Fe2 ­ the result for the orbital magnetic moment depends on the initialization – the high orbital moment solution higher in energy by 86 meV
➢
Ru2 ­ zero orbital magnetic moment for a magnetization along the dimer axis (as for Pd2)
➢
Os2 – spin anisotropy ­ larger than the orbital anisotropy
Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Fe2
Ru2
Magnetic anisotropy of small transition­metal clusters
Os2
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Fe2
➢
➢
➢
➢
Ru2
Os2
the electronic ground state configurations are s2d6 for Fe and Os, sd7 for Ru
Fe2 – SOC coupling splits for a parallel magnetic axis the degenerate d HOMO, but as the center of gravity is up­shifted, the energy gain is very modest Ru2 – the change in the sum of the one­electron energies is ­40 meV ­ calculated perpendicular MAE = ­35.5 meV
Os2 – the axial anisotropy (MAE = 29 meV) results from the splitting of the doubly degenerate d* HOMO occupied by only one electron Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results – post­DFT calculations Ni2
Pd2
Hybrid­Functional calc.
scalar relat.
S
J
1.00
2.26
Hybrid­Functional calc.
inc. SO coupl.
z
scalar relat.
S
1.24
x
2.26
1.00
inc. SO coupl.
2.56
scalar relat.
S
1.24
0.00
1.24
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
1.96 0.02 1.98 0.50 0.3

343
inc. SO coupl.
J
1.00
2.50
0.00
S,L = 0.0o

101
Hybrid­Functional calc.
J
1.24
Axial
Perp.
µS
µL
µS
µL MAE
1.98 0.50 1.96 0.30 3.7
Pt2
2.31
2.31
2.43
1.00
2.31
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE

1.88 3.02 1.28 1.06 30.5 743
d = 2.42 Å for perp. magnet.
DFT+U (3 eV)
DFT+U (3 eV)
Axial
Perp.
µS
µL
µS
µL MAE

2.00 0.42 2.00 0.36 17.0 114

384
Axial
Perp.
µS
µL
µS
µL MAE

1.90 2.90 1.48 0.88 19.5 644
1.18
1.21
2.08
1.21
Axial
Perp.
µS
µL
µS
µL MAE
1.94 0.02 1.98 0.40 2.1
DFT+U (3 eV)
2.07
1.18
2.38
1.18
S,L = 0.0o
Axial
Perp.
µS
µL
µS
µL MAE
1.98 0.02 2.00 0.36 6.2

117
Magnetic anisotropy of small transition­metal clusters
1.18
DFT+U (2 eV)
Axial
Perp.
µS
µL
µS
µL MAE

1.90 2.82 1.40 0.84 13.3 708
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Pt2 – Hybrid Functional Pt2 – GGAU (U = 2 eV)
Magnetic anisotropy of small transition­metal clusters
Pt2 – GGAU (U = 5 eV)
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results Pt2 – Hybrid Functional Pt2 – GGAU (U = 2 eV)
➢
➢
➢
Pt2 – GGAU (U = 5 eV)
the main effect of the admixture of a fraction of Hartree­Fock exchange is to increase the exchange splitting of the partially occupied eigenstates
the effect of the Hubbard­type on­site potential U is again to increase the exchange splitting of partially occupied eigenstates At a modest value of U=2 eV – enhanced spin and orbital moments, an increased exchange splitting of the d* states and an increased SOC splitting Magnetic anisotropy of small transition­metal clusters
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Summary ➢
SOC has almost no influence on the dimer bond­length
➢
the mixing of different spin­eigenstates – the spin moment is no longer an integer multiple of µB when SOC is included ➢
SOC induces a slight reduction of the spin moment by 0.02 to 0.25 µB, increasing with the strength of the SOC ➢
for the heavier elements SOC induces an anisotropy of the spin moments
➢
the determination of the relativistic ground state – hampered by the existence of multiple local minima with different orbital moments
➢
mixing orbital­dependent Hartree­Fock with DFT exchange – an increased bond­length for Ni and Pt dimers, spin moments ­ unaffected, orbital moments slightly increased for Pd and Pt ­ this hardly affects the MAE
➢
the effect of U is to increase the exchange splitting of the partially occupied d­states. For Pt2 (U = 5 eV) ­ a change in sign of the MAE ­ unrealistic eigenvalue spectrum
Magnetic anisotropy of small transition­metal clusters
Results ­ triangles z
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Ni
Pd
Pt
All moments set along z axis
All moments set along z axis
All moments set along z axis or [1 1 1]
equilateral triangle
x
S
isosceles triangle
S,L = 3o
0.67
2.20
0.67
3
2
2.20
µS
µL
µJ
0.67
1.07
0.91
0.91
0.67
2.21
|µ|
x
y
z
­0.02 0.00 2.19 2.19
0.00 0.00 0.58 0.58
­0.04 0.00 2.77 2.77
All moments set along [1 1 1] direction
isoscales triangle
|µ|
x
y
z
1.43 0.00 1.66 2.19
0.37 0.00 0.44 0.58
1.80 0.00 2.10 2.77
0.92
equilateral triangle
S,L = 0o
0.67
2.52
µS
µL
µJ
S,L = 0o
2.50
0.84
0.84
0.67
2.51
|µ|
x
y
z
0.00 0.00 1.66 1.66
0.00 0.00 0.94 0.94
0.00 0.00 2.60 2.60
All moments set along [1 1 1] direction
isoscales triangle
|µ|
x
y
z
1.44 0.00 0.83 1.66
0.79 0.00 0.32 0.85
2.23 0.00 1.15 2.51
0.67
2.50
S,L = 25o
S,L 0o
µS
µL
µJ
isosceles triangle
0.67
2.53
S,L = 48o
S,L 0o
µS
µL
µJ
isosceles triangle
J
1
0.67
equilateral triangle
µS
µL
µJ
1.40
0.82
0.82
2.51
S,L = 46o
|µ|
x
y
z
0.00 0.00 1.51 1.51
0.00 0.00 1.53 1.53
0.00 0.00 3.04 3.04
S,L 0 o
All moments set along [1 1 1] direction
isoscales triangle
µS
µL
µJ
|µ|
x
y
z
0.62 0.00 1.34 1.49
0.46 0.00 0.99 1.09
1.08 0.00 2.33 2.57
∆Ez­[1 1 1] < 0.2 meV
∆Ez­[1 1 1] < 0.1 meV
∆Ez­[1 1 1] < 0.1 meV
All moments set along y axis
equilateral triangle, d = 2.21 Å
All moments set along y axis
equilateral triangle, d = 2.53 Å
All moments set along y axis
isoscales triangle
µS
µL
µJ
|µ|
x
y
z
0.00 2.19 0.00 2.19
0.00 1.80 0.00 1.80
0.00 3.99 0.00 3.99
∆Ez­y = 7.0 meV
µS
µL
µJ
|µ|
x
y
z
0.00 1.74 0.00 1.74
0.00 0.75 0.00 0.75
0.00 2.49 0.00 2.49
∆Ez­y = 2.9 meV
Magnetic anisotropy of small transition­metal clusters
µS
µL
µJ
|µ|
x
y
z
0.00 ­0.20 0.00 0.20
0.00 ­0.05 0.00 0.05
0.00 ­0.25 0.00 0.25
∆Ez­y = 13.2 meV
Results ­ triangles z
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Ni
Pd
Pt
All moments set along z axis
All moments set along z axis
All moments set along z axis or [1 1 1]
equilateral triangle
x
S
isosceles triangle
S,L = 3o
0.67
2.20
0.67
3
2
2.20
µS
µL
µJ
0.67
1.07
equilateral triangle
0.91
0.91
2.21
S,L = 48
s
isosceles triangle
0.67
2.52
o
|µ|
x
y
z
­0.02 0.00 2.19 2.19
0.00 0.00 0.58 0.58
­0.04 0.00 2.77 2.77
All moments set along [1 1 1] direction
isoscales triangle
|µ|
x
y
z
1.43 0.00 1.66 2.19
0.37 0.00 0.44 0.58
1.80 0.00 2.10 2.77
0.67
µS
µL
µJ
0.84
2.50
0.84
0.67
S,L = 25
o
|µ|
x
y
z
0.00 0.00 1.66 1.66
0.00 0.00 0.94 0.94
0.00 0.00 2.60 2.60
S,L 0o
All moments set along [1 1 1] direction
isoscales triangle
µS
µL
µJ
S,L = 0o
0.67
0.92
l
S,L 0o
µS
µL
µJ
isosceles triangle
S,L = 0o
J
1
0.67
equilateral triangle
|µ|
x
y
z
1.44 0.00 0.83 1.66
0.79 0.00 0.32 0.85
2.23 0.00 1.15 2.51
0.67
2.50
µS
µL
µJ
1.40
0.82
0.82
2.51
S,L = 46o
|µ|
x
y
z
0.00 0.00 1.51 1.51
0.00 0.00 1.53 1.53
0.00 0.00 3.04 3.04
S,L 0 o
All moments set along [1 1 1] direction
isoscales triangle
µS
µL
µJ
|µ|
x
y
z
0.62 0.00 1.34 1.49
0.46 0.00 0.99 1.09
1.08 0.00 2.33 2.57
∆Ez­[1 1 1] < 0.2 meV
∆Ez­[1 1 1] < 0.1 meV
∆Ez­[1 1 1] < 0.1 meV
All moments set along y axis
equilateral triangle, d = 2.21 Å
All moments set along y axis
equilateral triangle, d = 2.53 Å
All moments set along y axis
isoscales triangle
µS
µL
µJ
|µ|
x
y
z
0.00 2.19 0.00 2.19
0.00 1.80 0.00 1.80
0.00 3.99 0.00 3.99
∆Ez­y = 7.0 meV
µS
µL
µJ
|µ|
x
y
z
0.00 1.74 0.00 1.74
0.00 0.75 0.00 0.75
0.00 2.49 0.00 2.49
∆Ez­y = 2.9 meV
Magnetic anisotropy of small transition­metal clusters
µS
µL
µJ
|µ|
x
y
z
0.00 ­0.20 0.00 0.20
0.00 ­0.05 0.00 0.05
0.00 ­0.25 0.00 0.25
∆Ez­y = 13.2 meV
Results ­ tetrahedron z
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Ni
Pd
Pt
All moments set along z axis
All moments set along z axis
All moments set along z axis and [1 1 1]
tetrahedron – ­ broken symmetry, S4
tetrahedron – ­ d = 2.61 Å
x
S
2.20
1.00
1.00
1.00
1.00
2.20
J
0.95
2.20
2.20
0.95
0.50
0.50
0.58
0.58
0.25
0.25
0.52
0.52
0.50
0.50
0.58
0.58
0.75
0.75
0.52
0.52
0.95
0.95
S,L = 18o
|µ|
x
y
z
0.00 0.02 3.44 3.44
0.00 0.00 0.36 0.36
0.00 0.02 3.80 3.80
S,L 0
o
All moments set along [1 1 1] direction
broken symetry, S4
µS
µL
µJ
S,L = 24o
|µ|
x
y
z
2.00 2.00 1.96 3.44
0.20 0.20 0.20 0.35
2.20 2.20 2.16 3.79
µS
µL
µJ
|µ|
x
y
z
0.00 0.00 1.84 1.84
0.00 0.00 0.48 0.48
0.00 0.00 2.32 2.32
S,L 0
o
All moments set along [1 1 1] direction
tetrahedron
µS
µL
µJ
|µ|
x
y
z
1.08 1.02 1.06 1.82
0.29 0.29 0.29 0.50
1.37 1.31 1.35 2.32
∆Ez­[1 1 1] = 0.5 meV
∆Ez­[1 1 1] = 1.1 meV
All moments set along x axis
broken symetry, S4
All moments set along x axis
tetrahedron
µS
µL
µJ
tetrahedron – ­ d = 2.59 Å
2.69
Remaining = 2.32 Å
µS
µL
µJ
tetrahedron – ­ broken symmetry, C2
|µ|
x
y
z
3.44 0.00 0.06 3.44
0.32 0.00 0.00 0.32
3.76 0.00 0.06 3.76
∆Ez­x = 1.9 meV
µS
µL
µJ
|µ|
x
y
z
1.84 0.00 0.00 1.84
0.48 0.00 0.00 0.48
2.32 0.00 0.00 2.32
∆Ez­x = 0.01 meV
Magnetic anisotropy of small transition­metal clusters
S,L = 0o
Remaining = 2.58 Å
x
S1
S2
S3
S4
L1
L2
L3
L4
0.21
y
z
|m|
0.21 ­0.21 0.36
­0.21 ­0.21 ­0.21 0.36
0.21 ­0.21 0.21
­0.21 0.21
0.09
0.21
0.36
0.36
0.09 ­0.09 0.16
­0.09 ­0.09 ­0.09 0.16
0.09 ­0.09 0.09
­0.09 0.09
0.09
0.16
0.16
All moments set along x axis
broken symetry, S4
µS
µL
µJ
|µ|
x
y
z
2.72 0.00 0.00 2.72
0.96 0.00 0.00 0.96
3.68 0.00 0.00 3.68
∆Ez­x = 24.6 meV
Results – trigonal bipyramid
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Ni
Pd
Pt
All moments set along [1 1 1] direction
All moments set along z axis
All moments set along z or x axis, trigonal bipyramid –D3h
S
J
0.72
2.33
0.85
0.85
2.25
0.72
0.76
2.33
0.91
S,L = 8o
S,L = 6o
0.91
|µ|
x
y
z
2.48 2.86 0.48 3.82
0.29 0.32 0.05 0.43
2.77 3.18 0.53 4.25
All moments set along x axis
D3h
|µ|
x
y
z
3.82 0.00 0.00 3.82
0.45 0.00 0.00 0.45
4.27 0.00 0.00 4.27
2.58
0.35
2.91
0.47
S,L = 8o
S,L 8o
µS
µL
µJ
0.35
µS
µL
µJ
0.50
S,L = 7o
0.45
0.45
2.89
0.50
S,L = 4o
|µ|
x
y
z
0.66 1.69 ­0.10 1.82
0.21 0.49 0.04 0.53
0.87 2.18 ­0.06 2.35
S,L 4o
All moments set along x axis
D3h
µS
µL
µJ
|µ|
x
y
z
1.79 ­0.05 0.05 1.79
0.56 0.00 0.00 0.56
2.35 ­0.05 0.05 2.35
∆E[1 1 1]­x = 0.5 meV
∆Ez­x = 8.1 meV
All moments set along Z axis
D3h
All moments set along [1 1 1] direction
D3h
µS
µL
µJ
|µ|
x
y
z
0.00 0.00 3.82 3.82
0.00 0.00 0.50 0.50
0.00 0.00 4.32 4.32
∆Ez­x = 1.4 meV
µS
µL
µJ
|µ|
x
y
z
0.34 1.79 0.00 1.82
0.10 0.52 0.00 0.53
0.44 2.31 0.00 2.35
∆E[1 1 1]­x = 4.6 meV
Magnetic anisotropy of small transition­metal clusters
trigonal bipyramid – D3h
S,L = 4o
0.47
2.56
2.26
0.76
z
x
µS
µL
µJ
trigonal bipyramid – D3h
2.62
2.62
0.47
S,L = 12o
0.42
S,L = 3o
0.35
0.35
2.60
0.47
µS
µL
µJ
0.95
0.95
2.60
0.42
S,L = 12o
|µ|
x
y
z
2.76 ­0.26 0.00 2.77
0.93 ­0.07 0.00 0.93
3.69 ­0.33 0.00 3.70
S,L 8 o
All moments set along [1 1 1] direction
D3h
µS
µL
µJ
|µ|
x
y
z
1.81 2.10 0.04 2.77
0.62 0.71 0.00 0.94
2.43 2.81 0.04 3.71
∆E[1 1 1]­x = 0.2 meV
Results – magnetic moment (per atom), average bond length, binding energy
Properties of small transition metal clusters – scalar­relativistic vs fully relativistic calc.
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Co and Fe on Pt(111)
Scalar relat.
∆E(fcc ­ hcp) = 5 meV ∆d Fe­Pt = 0.1 Å
Including SOC
Perpendicular
µL
µtot
µS
µS
MFe 3.306
3.282
0.104
3.386
MPt
2.377
2.153
0.402
Meff
5.683
5.435
MCo 2.190
MPt
Meff
∆E(hcp ­ fcc) = 28 meV ∆d Co­Pt = 0.7 Å
Adatoms on non­magnetic substrates
In­plane
µL
µtot
∆E [meV]
µS
2.05
3.283
0.108
3.391
2.555
2.172
0.518
2.690
0.506
5.941
5.455
0.626
6.081
2.155
0.126
2.281
2.155
0.088
2.243
4.826
3.319
0.684
4.003
3.325
0.829
4.154
7.016
5.474
0.810
6.284
5.480
0.917
6.397
1.19
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Co and Fe on Pt(111)
Scalar relat.
MFe Perpendicular
µL
µtot
µS
µS
3.306
3.282
3.395
3.366
0.104
0.628
0.343
3.386
4.023
3.709
2.153
0.402
1
∆E(fcc ­ hcp) = 5 meV ∆d Fe­Pt = 0.1 Å
Including SOC
SKKR
LSDA – ideal geometry
MPt
2.377
In­plane
µL
µ tot
∆E [meV]
µS
2.05
5.31
2.35
3.283
3.514
3.366
0.108
0.266
0.234
3.391
3.780
2.555
2.172
0.518
2.690
5.455
0.626
6.081
2.155
1.973
2.151
0.088
0.483
0.350
2.243
2.456
2.501
Meff
5.683
5.435
0.506
5.941
MCo 2.190
2.155
2.153
2.152
0.126
0.726
0.631
2.281
2.879
2.783
3.319
0.684
4.003
3.325
0.829
4.154
5.474
0.810
6.284
5.480
0.917
6.397
1
SKKR
LSDA – ideal geometry
MPt
4.826
Meff
7.016
∆E(hcp ­ fcc) = 28 meV ∆d Co­Pt = 0.7 Å
1
C. Etz et al. PRB 77, 184425 (2008).
Adatoms on non­magnetic substrates
1.19
5.02
6.41
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Co and Fe on Pt(111)
Scalar relat.
MFe µS
3.306
3.282
3.395
3.366
0.104
0.628
0.343
3.386
4.023
3.709
2.153
0.402
SKKR
LSDA – ideal geometry
MPt
2.377
In­plane
µL
µtot
∆E [meV]
µS
2.05
5.31
2.35
3.283
3.514
3.366
0.108
0.266
0.234
3.391
3.780
2.555
2.172
0.518
2.690
5.455
0.626
6.081
2.155
1.973
2.151
0.088
0.483
0.350
2.243
2.456
2.501
Meff
5.683
5.435
0.506
5.941
MCo 2.190
2.155
2.153
2.152
0.126
0.726
0.631
2.281
2.879
2.783
3.319
1.131
0.684
0.194
4.003
1.325
3.325
1.258
0.829
0.279
4.154
1.537
5.474
0.810
6.284
5.480
0.917
6.397
1
SKKR
LSDA – ideal geometry
MPt
4.826
LSDA – ideal geometry
Meff
7.016
●
∆E(hcp ­ fcc) = 28 meV ∆d Co­Pt = 0.7 Å
Perpendicular
µL
µtot
µS
1
∆E(fcc ­ hcp) = 5 meV ∆d Fe­Pt = 0.1 Å
Including SOC
1.19
5.02
6.41
2Experiment – Fe/Pt – out of plane orientation, L/Seff = 0.18±0.05
Experiment – Co/Pt – out of plane orientation, L/Seff = 0.61±0.05, µPt = 1.8±0.7
●
1
2
C. Etz et al. PRB 77, 184425 (2008).
A. Lehnert, PhD Thesis, Ecole Polytechnique Fédérale de Lausanne (2009).
Adatoms on non­magnetic substrates
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Fe on Pt(111)
the lifting of degeneracies of the states (dyz and dxz) far away from the Fermi surface (energy window of –4 ÷ –2 eV)
Adatoms on non­magnetic substrates
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Fe on Pt(111)
the lifting of degeneracies of the states (dyz and dxz) far away from the Fermi surface (energy window of –4 ÷ –2 eV)
Adatoms on non­magnetic substrates
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Fe and Co on Rh and Pd (111)
Scalar relat.
Including SOC
Perpendicular
µL
µ tot
µS
µS
3.205
3.204
0.083
3.287
MRh
0.191
0.957
­0.014
Meff
3.396
4.161
M
2.071
MRh
In­plane
µL
µtot
∆E [meV]
µS
0.39
3.203
0.090
3.293
0.943
0.992
0.028
1.020
0.069
4.230
4.195
0.118
4.313
2.155
0.143
2.210
2.064
0.155
2.219
0.814
1.386
0.015
1.401
1.431
0.057
1.488
Meff
2.885
3.453
0.158
3.611
3.495
0.212
3.707
M
3.381
3.375
0.088
3.463
3.376
0.084
3.460
MPd
0.811
0.866
0.054
0.920
0.886
0.085
0.971
Meff
4.192
4.241
0.142
4.383
4.262
0.169
4.431
MCohcp­hollow 2.249
2.245
0.220
2.465
2.244
0.188
2.432
MPd
0.253
­0.282
­0.039
­0.321
­0.250
­0.046
­0.296
Meff
2.502
1.963
0.181
2.144
1.994
0.142
2.136
M
hcp­hollow
Fe
hcp­hollow
Co
fcc­hollow
Fe
Adatoms on non­magnetic substrates
­
0.09
1.61
∆E [meV]
­
0.19
­
­
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Fe and Co on Rh and Pd (111)
Scalar relat.
Including SOC
Perpendicular
µL
µ tot
µS
µS
3.205
3.204
0.083
3.287
MRh
0.191
0.957
­0.014
Meff
3.396
4.161
M
2.071
MRh
In­plane
µL
µtot
∆E [meV]
µS
0.39
3.203
0.090
3.293
0.943
0.992
0.028
1.020
0.069
4.230
4.195
0.118
4.313
2.155
0.143
2.210
2.064
0.155
2.219
0.814
1.386
0.015
1.401
1.431
0.057
1.488
Meff
2.885
3.453
0.158
3.611
3.495
0.212
3.707
M
3.381
3.375
0.088
3.463
3.376
0.084
3.460
MPd
0.811
0.866
0.054
0.920
0.886
0.085
0.971
Meff
4.192
4.241
0.142
4.383
4.262
0.169
4.431
MCohcp­hollow 2.249
2.245
0.220
2.465
2.244
0.188
2.432
MPd
0.253
­0.282
­0.039
­0.321
­0.250
­0.046
­0.296
Meff
2.502
1.963
0.181
2.144
1.994
0.142
2.136
M
hcp­hollow
Fe
hcp­hollow
Co
fcc­hollow
Fe
­
0.09
1.61
∆E [meV]
­
0.19
­
­
●
Experiment – Fe/Rh – out of plane orientation, L/Seff = 0.15±0.05
●
Experiment – Co/Rh – in­plane orientation, L/Seff = 0.57±0.05, µRh = 2.9±0.1, MAE = 0.5 meV ●
Experiment – Fe/Pd – out of plane orientation, L/Seff = 0.12±0.05
●
Experiment – Co/Pd – out of plane orientation, L/Seff = 0.70±0.05, MAE = 3 meV
Adatoms on non­magnetic substrates
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Results: Fe and Co on Rh and Pd (111)
Scalar relat.
Including SOC
Perpendicular
µL
µ tot
µS
µS
3.205
3.204
0.083
3.287
MRh
0.191
0.957
­0.014
Meff
3.396
4.161
M
2.071
MRh
In­plane
µL
µtot
∆E [meV]
µS
0.39
3.203
0.090
3.293
0.943
0.992
0.028
1.020
0.069
4.230
4.195
0.118
4.313
2.155
0.143
2.210
2.064
0.155
2.219
0.814
1.386
0.015
1.401
1.431
0.057
1.488
Meff
2.885
3.453
0.158
3.611
3.495
0.212
3.707
M
3.381
3.375
0.088
3.463
3.376
0.084
3.460
MPd
0.811
0.866
0.054
0.920
0.886
0.085
0.971
Meff
4.192
4.241
0.142
4.383
4.262
0.169
4.431
M
2.249
2.245
0.220
2.465
2.244
0.188
2.432
MPd
0.253
­0.282
­0.039
­0.321
­0.250
­0.046
­0.296
Meff
2.502
1.963
0.181
2.144
1.994
0.142
2.136
M
3.388
3.377
0.080
3.457
3.382
0.078
3.460
MPd
­0.528
9.341
0.994
10.335
5.532
0.611
6.143
Meff
2.860
12.718
1.074
13.792
8.914
0.718
9.632
M
hcp­hollow
Fe
hcp­hollow
Co
fcc­hollow
Fe
hcp­hollow
Co
hcp­hollow
Fe
­
0.09
1.61
­
∆E [meV]
­
0.19
­
­
39
●
Experiment – Fe/Rh – out of plane orientation, L/Seff = 0.15±0.05
●
Experiment – Co/Rh – in­plane orientation, L/Seff = 0.57±0.05, µRh = 2.9±0.1, MAE = 0.5 meV ●
Experiment – Fe/Pd – out of plane orientation, L/Seff = 0.12±0.05
●
Experiment – Co/Pd – out of plane orientation, L/Seff = 0.70±0.05, MAE = 3 meV
Adatoms on non­magnetic substrates
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
➢
Slab thickness – nonmagnetic ground state of the bare surfaces
➢
➢
Magneto­structural effect – relaxations and height of the adatom above the surface
➢
Exc – LDA or GGA? ➢
Dependence of the magnetic anisotropy on the adsorption sites
➢
...
Adatoms on non­magnetic substrates
­k Workshop on Magnetism in Complex Systems, Wien, 16­19.04.2009
Thank you for your attention!
Adatoms on non­magnetic substrates