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Copyright copy 2012 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofNanoscience and Nanotechnology

Vol 12 7516ndash7519 2012

Noncollinear Magnetism in Ultrathin Films withStrong Spin-Orbit Coupling from Ab Initio

D Boumlttcher12lowast A Ernst1 and J Henk11Max-Planck-Institut fuumlr Mikrostrukturphysik Weinberg 2 D-06120 Halle Saale Germany

2Institut fuumlr Physik Martin-Luther-Universitaumlt Halle-Wittenberg D-06120 Halle Saale Germany

Fe and Pt are paradigms for ferromagnetism and strong spin-orbit coupling respectively Theircombinationmdashin an ultrathin Fe film on a Pt(111) substratemdashis thus expected to modify the mag-netic structures We report on a theoretical investigation of a monolayer of Fe on Pt(111) usinga generalized Heisenberg model that includes the complete spin interaction matrices Iij computedfrom first principles We find a noncollinear periodic configuration that is strongly determined by theDzyaloshinskii-Moriya interaction Taking into account a magnetic field to mimic recent experimentsthis noncollinear structure solves the disagreement between the experimental magnetization andthe average magnetization for a ferromagnetic system The critical temperature decreases from670 K to 590 K due to spin-orbit coupling

Keywords Dzyaloshinskii-Moryia Interaction Ultrathin Films Magnetic Ground StateProperties

1 MOTIVATION

The properties of ultrathin magnetic films depend stronglyon the substrate1 as has been shown experimentallyby spin-polarized scanning tunneling microscopy andmagneto-optical Kerr spectroscopy In particular spin-orbitcoupling (SOC) induces a symmetry breaking in mag-netic systems which eg shows up as magnetocrys-talline anisotropy Together with the exchange interactionthe magnetic anisotropy manifests itself in the forma-tion of domain walls In non-centrosymmetric systems theDzyaloshinskii-Moriya (DM) interaction results in non-collinear magnetic structures preferably at surfaces andin ultrathin films2 The magnetic properties of a mono-layer Fe on Pt(111) are still not understood completelyOn one hand Moulas et al3 showed that by applying anexternal magnetic field of 5 T the magnetization is barely12 B the saturation field is roughly estimated to 10 TOn the other hand first-principles electronic-structure cal-culations using a KorringandashKohnndashRostoker Greenrsquos func-tion method3 or the Vienna Ab-Initio Simulation Package4

support ferromagnetic order an Fe magnetic moment ofabout 30 B and an induced Pt moment of 025 B Weregard these contradictory results as an indication for anoncollinear magnetic structure in FePt(111) which maybe driven by the strong SOC in Pt Noncollinear mag-netism due to SOC has been shown for FePt-alloy clusters

lowastAuthor to whom correspondence should be addressed

deposited on Pt(111)56 In this paper we report on a first-principles investigation of a monolayer Fe on Pt(111) thatincludes SOC Using a generalized Heisenberg model wefocus on the question on how spin-orbit coupling mani-fests itself in the magnetic ground-state properties and themagnetic structure of FePt(111)

2 OUTLINE OF THETHEORETICAL APPROACH

21 First-Principles Calculations

To investigate the magnetic properties of a monolayerFe on Pt(111) we performed first-principles electronic-structure calculations using a relativistic KorringandashKohnndashRostoker Greenrsquos function method78 Solving the Diracequation for a spin-polarized system spin-orbit couplingand magnetism are treated on equal footing Wavefunc-tions and scattering matrices have been calculated up to anorbital momentum of lmax = 3 The site-dependent poten-tials are described in the atomic sphere approximation Thefilm-substrate system is taken as translationally invariantparallel to the layers but semi-infinite perpendicular to thelayers We adopt the interlayer spacing derived by Hardratet al1 who found an inward relaxation of the monolayerfcc Fe by 127 of the Pt bulk interlayer distance (latticeconstant a= 281 Aring Fig 1)From the ab initio calculations we derived the com-

plete spin interaction matrix of a generalized Heisenberg

7516 J Nanosci Nanotechnol 2012 Vol 12 No 9 1533-48802012127516004 doi101166jnn20126555

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IP 19210869177Wed 26 Sep 2012 091823

RESEARCH

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Boumlttcher et al Noncollinear Magnetism in Ultrathin Films with Strong Spin-Orbit Coupling from Ab Initio

Fig 1 Schematic view of a monolayer Fe on Pt(111) Only the firsttwo layers of Pt (gray spheres) are important for the spin structure in theFe film (red spheres)

model7 within the framework of the magnetic forcetheorem9ndash14

22 Spin Hamiltonian and Interaction Matrix

The Heisenberg model

H =minussumi =j

Jijmi middotmj (1)

where mi and mj are classical normalized magneticmoments and Jij is their exchange constant is a standardmodel for magnetic materials Since the Jij are isotropicand symmetric it does not account for spin-orbit couplingeffects A straightforward generalization is given by

H =minussumij

miIijmj (2)

where Iij is the full 3times 3 interaction matrix (Iij = ITij )Decomposing Iij as

Iij equiv Jij1+Sij +Aij (3)

one recovers the Heisenberg exchange Jij = 13trIij Theanisotropic and traceless parts Sij and Aij split into a sym-metric and an antisymmetric contribution Sij = 12 middotIij + ITij minus Jij1 and Aij = 12Iij minus ITij This impliesthat only the Jij force the system into a collinear stateThe anisotropic antisymmetric components include theDzyaloshinskii-Moriya vectors

Dij equiv

12

sum

Iij = x y z (4)

that describe the noncollinearity between two magneticmoments15 they are closely linked to the symmetry of thesystemThe complete Hamiltonian then reads

H = minussumi =j

Jijmi middotmj + Dij middot mitimesmj +miSijmj

minus sumi

miIiimiminussumij

miQijmj +Hext (5)

The fourth term correlates to the uniaxial anisotropyof the system whereas the fifth term accounts for the

dipolendashdipole interaction (shape anisotropy) The dipolarmatrix Qij is given by

Q13ij = 0

8

3rij r13ijminusr2ij

13

r5ij rij =riminusrj 13=xyz (6)

where ri is the location of the atom i Eventually an exter-nal magnetic field is described by the Zeeman term in HextThe magnetic properties of the systems in thermal equilib-rium are obtained by a standard Monte-Carlo method1617

using the Metropolis algorithm

3 RESULTS AND DISCUSSION

31 Spin-Resolved Electronic Properties

The electronic-structure calculations yield that the major-ity bands in the Fe layer are almost completely filledunlike the minority bands that show a sharp maximum atthe Fermi level (Fig 2) As a consequence of the reduceddimensionality of the film the Fe magnetic moment(303 B) is increased with respect to Fe bulk (226 B)and agrees with that given by Hardrat et al1 (310 B)Hybridization of Fe and Pt electronic states results ininduced Pt magnetic moments that decrease rapidly towardthe bulk (first Pt-layer 026 B second Pt-layer 001 BFig 2) This is a first hint on that the Pt substratewith its large spin-orbit coupling can indeed have aprofound influence on the magnetic structure in the Feadlayer

Fig 2 Spin-resolved electronic structure of a monolayer Fe on Pt(111)The density of states (DOS) is given for the Fe layer (top) and the firstPt layer (bottom) The strong exchange splitting in Fe (303 B) and thehybridization of Fe with Pt electronic states leads to a sizable inducedmoment in the first Pt layer (026 B)

J Nanosci Nanotechnol 12 7516ndash7519 2012 7517

Delivered by Ingenta toMPI f Mikrostrukturphysik

IP 19210869177Wed 26 Sep 2012 091823

RESEARCH

ARTIC

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Noncollinear Magnetism in Ultrathin Films with Strong Spin-Orbit Coupling from Ab Initio Boumlttcher et al

32 Magnetic Configuration of aMonolayer Fe on Pt(111)

In the Monte Carlo simulations for the magnetic groundstate we consider the first three layers (Fe layer and thetwo subsequent Pt layers) because only these show relevantmagnetic moments and spin interactionsFirst we discuss the effects of the individual contribu-

tions in the spin Hamiltonian on the magnetic configura-tions The nearest-neighbor Heisenberg exchange constantexhibits ferromagnetic behavior (J1NN gt 0) the magne-tocrystalline anisotropy is in-plane (parallel to the surface)Hence without the DM contribution we obtain an in-planecollinear ground state with an average atomic moment of159 B in clear contrast to experiment (10 B fromRef [3])Inclusion of the symmetric anisotropic part Sij in the

Hamiltonian maintains this spin configuration in generalIn contrast to the DM part this term tends to tilt pairs ofmagnetic moments in the same direction Hence it can beinterpreted as an additional contribution to the magneto-crystalline anisotropy that slightly changes the easy axisfor each pair of moments As a consequence the localmagnetic moments deviate more strongly from the generaleasy axis than without Sij even at very low temperaturesThese fluctuations of the easy axes can be viewed as abroadening of the global energy minimum that also givesrise to fluctuations of the transition region width wThe Dzyaloshinskii-Moriya contribution turns out to be

significant only for nearest neighbor sites as was deducedby successively reducing the interaction range in the MCcalculations it tends to tilt magnetic moments mutuallyand is strongly linked to the symmetry of the system(Fig 3) It produces as an outcome of our calculationsa periodic noncollinear configuration which shows up inboth the Fe layer and the subsurface Pt layers (Fig 4 notshown for Pt) this corroborates the significant coupling ofFe and Pt found in the density of states (Fig 2) Sincethe magnetic structure contains in-plane and perpendicularcomponents it reminds at a combination of (very narrow)Bloch and Neacuteel walls

Fig 3 Dzyaloshinskii-Moriya interaction in a monolayer Fe onPt(111) Arrows depict the DM vectors Dij between two magneticmoments i (central site fixed) and j in the Fe adlayer Because of thethree-fold symmetry DM contributions from an in-plane ring structurewith antisymmetric interactions with respect to site i

Fig 4 Noncollinear magnetism in a monolayer Fe on Pt(111) Becauseof strong spin-orbit coupling effects the local magnetic moments(arrows) form a periodic noncollinear magnetic structure The momentaveraged over one magnetic unit cell vanishes but the moment averagedover the length w of the transition range is about 10 B

An impression of the interplay of the individual contribu-tions to the Hamiltonian is provided by ratios J KSD Weobtain for the FendashFe interaction 751916 in contrast to theFendashPt interaction 8112 While in both cases the Heisen-berg exchange dominates the ratio J D is almost identicalfor FendashFe and FendashPt (about 81) giving further support tothe importance of the SOC and hybridization The mag-netic lattice constant can be estimated from domain-walltheory within a continuum model Without DM interactionthe width w of the transition region is given by

w =radic

A

K(7)

where A and K are the exchange density and the averageanisotropy respectively With DM contribution an ana-lytical solution for w has been achieved only for DMvectors aligned along the z-direction enlarging the transi-tion region w From the calculated parameters we obtainw = 38 nm which is slightly smaller than half of the mag-netic lattice constant (Fig 4) Taking into account the DMcontribution w is increased to 67 nm A closer analysisof the MC simulations yields that this increase can indeedbe attributed to the DM contributionsAveraging over the magnetic unit cell gives a vanishing

net magnetization Restricting the average to the transi-tion region as indicated by the double arrow in Figure 4produces a net moment of about 10 B

The perpendicular components of the local moments arevery well described by an arithmetic function Further thesymmetric anisotropic interaction leads to tiny fluctuationsof the magnetic lattice constant these are attributed tothe anisotropic symmetric contributions to the exchangematricesWe now focus on the magnetic phase transition Since a

noncollinear configuration is not well characterized by itsaverage magnetization we deduce the critical temperaturefrom the nearest-neighbor spin correlation function

S = 1N

sumi

1Ni

sumj

mi middotmj (8)

Here N is the number of sites in the sample and Ni thenumber of nearest neighbor atoms of site i An alternative

7518 J Nanosci Nanotechnol 12 7516ndash7519 2012

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IP 19210869177Wed 26 Sep 2012 091823

RESEARCH

ARTIC

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Boumlttcher et al Noncollinear Magnetism in Ultrathin Films with Strong Spin-Orbit Coupling from Ab Initio

measure is the spinndashspin correlation function (SSCF)

sdr= mr middotmr+drr (9)

of magnetic moments at a distance dr The SSCF ofFePt(111) behaves like cosdrw at very low temper-atures For increasing thermal fluctuations its amplitudedecreases and eventually vanishes at the Curie tempera-ture TC From this general behavior we derive Curie tem-peratures of 670 K without DM interaction With DMinteraction the Curie temperature is reduced to 590 Kwhich is in good agreement with 06 of TC for bulk Fe(1043 K) as found by Rausch and Nolting18

Eventually we consider the dependency of the aver-age magnetization on an external magnetic field perpen-dicular to the surface We recall that a magnetization of12 B has been found experimentally at 5 T even at10 T the sample was not driven into saturation3 To mimicthese experiments we performed MC simulations with themagnetic-field term (up to 10 T) As consequence of thestrong SOC the local magnetic moments maintain sizablein-plane components (cf Fig 4 for zero field) but are tiltedout-of-plane with increasing field strength For a field of5 T we obtain a net magnetization of 13 B Also at 10 Tthe local magnetic moments are not completely rotatedout-of-plane indicating incomplete saturation Since thesefindings agree nicely with experiment3 we conclude thatthe theoretically predicted noncollinear structure solves theaforementioned puzzle

4 CONCLUSION

Ultrathin films with strong spin-orbit coupling show non-collinear spin structures as is demonstrated for a mono-layer Fe on Pt(111) Using Monte-Carlo calculations fora generalized Heisenberg model in which spin-orbit con-tributions are taken into account and whose parametersare obtained from first-principles calculations we finda periodic arrangement of toroidal structures This non-collinear structure solves the discrepancy of the exper-imental magnetization and the theoretical magnetizationfor a ferromagnetic configuration Our findings call fornew experiments in order to verify the predicted magneticstructure

We expect a strong impact of the spin-orbit interactionon time-dependent phenomena as is currently investi-gated within an atomistic approach based on the stochas-tic LandaundashLifshitzndashGilbert equation First calculations forsystems perturbed by a magnetic field pulse show long-term excitations which may excite magnons

Acknowledgments D Boumlttcher is member of the Inter-national Max Planck School for Science and Technologyof Nanostructures This work is supported by the Sonder-forschungsbereich 762 lsquoFunctionality of Oxide Interfacesrsquo

References and Notes

1 B Hardrat A Al-Zubi P Ferriani S Bluumlgel G Bihlmayer andS Heinze Phys Rev B 79 094411 (2009)

2 K von Bergmann S Heinze M Bode G Bihlmayer S Bluumlgeland R Wiesendanger New J Phys 9 396 (2007)

3 G Moulas A Lehnert S Rusponi J Zabloudil C Etz S OuaziM Etzkorn P Bencok P Gambardella P Weinberger and H BrunePhys Rev B 78 214424 (2008)

4 A Lehnert S Dennler P Blonski S Rusponi M EtzkornG Moulas P Bencok P Gambardella H Brune and J HafnerPhys Rev B 82 094409 (2010)

5 A Enders R Skomski and J Honolka J Phys Condens Matt22 433001 (2010)

6 S Mankovsky S Bornemann J Minaacuter S Polesya H Ebert J BStaunton and A I Lichtenstein Phys Rev B 80 014422 (2009)

7 J Zabloudil R Hammerling L Szunyogh and P Weinberger (eds)Electron Scattering in Solid Matter Springer Berlin (2005)

8 D Boumlttcher A Ernst and J Henk J Phys Condens Matt 23 29(2011)

9 A I Liechtenstein M I Katsnelson V P Antropov and V AGubanov J Magn Magn Mater 67 65 (1987)

10 P Strange H Ebert J B Staunton and B L Gyorffy J PhysCondens Matt 1 3947 (1989)

11 G H O Daalderop P J Kelly and M F H Schuurmans PhysRev B 41 11919 (1990)

12 X Wang D-S Wang R Wu and A J Freeman J Magn MagnMater 159 337 (1996)

13 M D Stiles S V Halilov R A Hyman and A Zangwill PhysRev B 64 104430 (2001)

14 S S Dhesi G van der Laan E Dudzik and A B Shick PhysRev Lett 87 067201 (2001)

15 T Moriya Phys Rev 120 1 (1960)16 N Metropolis A W Rosenbluth M N Rosenbluth and E Teller

J Chem Phys 21 1087 (1953)17 K Binder (ed) Monte Carlo Methods in Statistical Physics

Springer Berlin (1979)18 R Rausch and W Nolting J Phys Condens Matt 21 376002

(2009)

Received 29 March 2011 Accepted 29 October 2011

J Nanosci Nanotechnol 12 7516ndash7519 2012 7519

Delivered by Ingenta toMPI f Mikrostrukturphysik

IP 19210869177Wed 26 Sep 2012 091823

RESEARCH

ARTIC

LE

Noncollinear Magnetism in Ultrathin Films with Strong Spin-Orbit Coupling from Ab Initio Boumlttcher et al

32 Magnetic Configuration of aMonolayer Fe on Pt(111)

In the Monte Carlo simulations for the magnetic groundstate we consider the first three layers (Fe layer and thetwo subsequent Pt layers) because only these show relevantmagnetic moments and spin interactionsFirst we discuss the effects of the individual contribu-

tions in the spin Hamiltonian on the magnetic configura-tions The nearest-neighbor Heisenberg exchange constantexhibits ferromagnetic behavior (J1NN gt 0) the magne-tocrystalline anisotropy is in-plane (parallel to the surface)Hence without the DM contribution we obtain an in-planecollinear ground state with an average atomic moment of159 B in clear contrast to experiment (10 B fromRef [3])Inclusion of the symmetric anisotropic part Sij in the

Hamiltonian maintains this spin configuration in generalIn contrast to the DM part this term tends to tilt pairs ofmagnetic moments in the same direction Hence it can beinterpreted as an additional contribution to the magneto-crystalline anisotropy that slightly changes the easy axisfor each pair of moments As a consequence the localmagnetic moments deviate more strongly from the generaleasy axis than without Sij even at very low temperaturesThese fluctuations of the easy axes can be viewed as abroadening of the global energy minimum that also givesrise to fluctuations of the transition region width wThe Dzyaloshinskii-Moriya contribution turns out to be

significant only for nearest neighbor sites as was deducedby successively reducing the interaction range in the MCcalculations it tends to tilt magnetic moments mutuallyand is strongly linked to the symmetry of the system(Fig 3) It produces as an outcome of our calculationsa periodic noncollinear configuration which shows up inboth the Fe layer and the subsurface Pt layers (Fig 4 notshown for Pt) this corroborates the significant coupling ofFe and Pt found in the density of states (Fig 2) Sincethe magnetic structure contains in-plane and perpendicularcomponents it reminds at a combination of (very narrow)Bloch and Neacuteel walls

Fig 3 Dzyaloshinskii-Moriya interaction in a monolayer Fe onPt(111) Arrows depict the DM vectors Dij between two magneticmoments i (central site fixed) and j in the Fe adlayer Because of thethree-fold symmetry DM contributions from an in-plane ring structurewith antisymmetric interactions with respect to site i

Fig 4 Noncollinear magnetism in a monolayer Fe on Pt(111) Becauseof strong spin-orbit coupling effects the local magnetic moments(arrows) form a periodic noncollinear magnetic structure The momentaveraged over one magnetic unit cell vanishes but the moment averagedover the length w of the transition range is about 10 B

An impression of the interplay of the individual contribu-tions to the Hamiltonian is provided by ratios J KSD Weobtain for the FendashFe interaction 751916 in contrast to theFendashPt interaction 8112 While in both cases the Heisen-berg exchange dominates the ratio J D is almost identicalfor FendashFe and FendashPt (about 81) giving further support tothe importance of the SOC and hybridization The mag-netic lattice constant can be estimated from domain-walltheory within a continuum model Without DM interactionthe width w of the transition region is given by

w =radic

A

K(7)

where A and K are the exchange density and the averageanisotropy respectively With DM contribution an ana-lytical solution for w has been achieved only for DMvectors aligned along the z-direction enlarging the transi-tion region w From the calculated parameters we obtainw = 38 nm which is slightly smaller than half of the mag-netic lattice constant (Fig 4) Taking into account the DMcontribution w is increased to 67 nm A closer analysisof the MC simulations yields that this increase can indeedbe attributed to the DM contributionsAveraging over the magnetic unit cell gives a vanishing

net magnetization Restricting the average to the transi-tion region as indicated by the double arrow in Figure 4produces a net moment of about 10 B

The perpendicular components of the local moments arevery well described by an arithmetic function Further thesymmetric anisotropic interaction leads to tiny fluctuationsof the magnetic lattice constant these are attributed tothe anisotropic symmetric contributions to the exchangematricesWe now focus on the magnetic phase transition Since a

noncollinear configuration is not well characterized by itsaverage magnetization we deduce the critical temperaturefrom the nearest-neighbor spin correlation function

S = 1N

sumi

1Ni

sumj

mi middotmj (8)

Here N is the number of sites in the sample and Ni thenumber of nearest neighbor atoms of site i An alternative

7518 J Nanosci Nanotechnol 12 7516ndash7519 2012

Delivered by Ingenta toMPI f Mikrostrukturphysik

IP 19210869177Wed 26 Sep 2012 091823

RESEARCH

ARTIC

LE

Boumlttcher et al Noncollinear Magnetism in Ultrathin Films with Strong Spin-Orbit Coupling from Ab Initio

measure is the spinndashspin correlation function (SSCF)

sdr= mr middotmr+drr (9)

of magnetic moments at a distance dr The SSCF ofFePt(111) behaves like cosdrw at very low temper-atures For increasing thermal fluctuations its amplitudedecreases and eventually vanishes at the Curie tempera-ture TC From this general behavior we derive Curie tem-peratures of 670 K without DM interaction With DMinteraction the Curie temperature is reduced to 590 Kwhich is in good agreement with 06 of TC for bulk Fe(1043 K) as found by Rausch and Nolting18

Eventually we consider the dependency of the aver-age magnetization on an external magnetic field perpen-dicular to the surface We recall that a magnetization of12 B has been found experimentally at 5 T even at10 T the sample was not driven into saturation3 To mimicthese experiments we performed MC simulations with themagnetic-field term (up to 10 T) As consequence of thestrong SOC the local magnetic moments maintain sizablein-plane components (cf Fig 4 for zero field) but are tiltedout-of-plane with increasing field strength For a field of5 T we obtain a net magnetization of 13 B Also at 10 Tthe local magnetic moments are not completely rotatedout-of-plane indicating incomplete saturation Since thesefindings agree nicely with experiment3 we conclude thatthe theoretically predicted noncollinear structure solves theaforementioned puzzle

4 CONCLUSION

Ultrathin films with strong spin-orbit coupling show non-collinear spin structures as is demonstrated for a mono-layer Fe on Pt(111) Using Monte-Carlo calculations fora generalized Heisenberg model in which spin-orbit con-tributions are taken into account and whose parametersare obtained from first-principles calculations we finda periodic arrangement of toroidal structures This non-collinear structure solves the discrepancy of the exper-imental magnetization and the theoretical magnetizationfor a ferromagnetic configuration Our findings call fornew experiments in order to verify the predicted magneticstructure

We expect a strong impact of the spin-orbit interactionon time-dependent phenomena as is currently investi-gated within an atomistic approach based on the stochas-tic LandaundashLifshitzndashGilbert equation First calculations forsystems perturbed by a magnetic field pulse show long-term excitations which may excite magnons

Acknowledgments D Boumlttcher is member of the Inter-national Max Planck School for Science and Technologyof Nanostructures This work is supported by the Sonder-forschungsbereich 762 lsquoFunctionality of Oxide Interfacesrsquo

References and Notes

1 B Hardrat A Al-Zubi P Ferriani S Bluumlgel G Bihlmayer andS Heinze Phys Rev B 79 094411 (2009)

2 K von Bergmann S Heinze M Bode G Bihlmayer S Bluumlgeland R Wiesendanger New J Phys 9 396 (2007)

3 G Moulas A Lehnert S Rusponi J Zabloudil C Etz S OuaziM Etzkorn P Bencok P Gambardella P Weinberger and H BrunePhys Rev B 78 214424 (2008)

4 A Lehnert S Dennler P Blonski S Rusponi M EtzkornG Moulas P Bencok P Gambardella H Brune and J HafnerPhys Rev B 82 094409 (2010)

5 A Enders R Skomski and J Honolka J Phys Condens Matt22 433001 (2010)

6 S Mankovsky S Bornemann J Minaacuter S Polesya H Ebert J BStaunton and A I Lichtenstein Phys Rev B 80 014422 (2009)

7 J Zabloudil R Hammerling L Szunyogh and P Weinberger (eds)Electron Scattering in Solid Matter Springer Berlin (2005)

8 D Boumlttcher A Ernst and J Henk J Phys Condens Matt 23 29(2011)

9 A I Liechtenstein M I Katsnelson V P Antropov and V AGubanov J Magn Magn Mater 67 65 (1987)

10 P Strange H Ebert J B Staunton and B L Gyorffy J PhysCondens Matt 1 3947 (1989)

11 G H O Daalderop P J Kelly and M F H Schuurmans PhysRev B 41 11919 (1990)

12 X Wang D-S Wang R Wu and A J Freeman J Magn MagnMater 159 337 (1996)

13 M D Stiles S V Halilov R A Hyman and A Zangwill PhysRev B 64 104430 (2001)

14 S S Dhesi G van der Laan E Dudzik and A B Shick PhysRev Lett 87 067201 (2001)

15 T Moriya Phys Rev 120 1 (1960)16 N Metropolis A W Rosenbluth M N Rosenbluth and E Teller

J Chem Phys 21 1087 (1953)17 K Binder (ed) Monte Carlo Methods in Statistical Physics

Springer Berlin (1979)18 R Rausch and W Nolting J Phys Condens Matt 21 376002

(2009)

Received 29 March 2011 Accepted 29 October 2011

J Nanosci Nanotechnol 12 7516ndash7519 2012 7519

Delivered by Ingenta toMPI f Mikrostrukturphysik

IP 19210869177Wed 26 Sep 2012 091823

RESEARCH

ARTIC

LE

Boumlttcher et al Noncollinear Magnetism in Ultrathin Films with Strong Spin-Orbit Coupling from Ab Initio

measure is the spinndashspin correlation function (SSCF)

sdr= mr middotmr+drr (9)

of magnetic moments at a distance dr The SSCF ofFePt(111) behaves like cosdrw at very low temper-atures For increasing thermal fluctuations its amplitudedecreases and eventually vanishes at the Curie tempera-ture TC From this general behavior we derive Curie tem-peratures of 670 K without DM interaction With DMinteraction the Curie temperature is reduced to 590 Kwhich is in good agreement with 06 of TC for bulk Fe(1043 K) as found by Rausch and Nolting18

Eventually we consider the dependency of the aver-age magnetization on an external magnetic field perpen-dicular to the surface We recall that a magnetization of12 B has been found experimentally at 5 T even at10 T the sample was not driven into saturation3 To mimicthese experiments we performed MC simulations with themagnetic-field term (up to 10 T) As consequence of thestrong SOC the local magnetic moments maintain sizablein-plane components (cf Fig 4 for zero field) but are tiltedout-of-plane with increasing field strength For a field of5 T we obtain a net magnetization of 13 B Also at 10 Tthe local magnetic moments are not completely rotatedout-of-plane indicating incomplete saturation Since thesefindings agree nicely with experiment3 we conclude thatthe theoretically predicted noncollinear structure solves theaforementioned puzzle

4 CONCLUSION

Ultrathin films with strong spin-orbit coupling show non-collinear spin structures as is demonstrated for a mono-layer Fe on Pt(111) Using Monte-Carlo calculations fora generalized Heisenberg model in which spin-orbit con-tributions are taken into account and whose parametersare obtained from first-principles calculations we finda periodic arrangement of toroidal structures This non-collinear structure solves the discrepancy of the exper-imental magnetization and the theoretical magnetizationfor a ferromagnetic configuration Our findings call fornew experiments in order to verify the predicted magneticstructure

We expect a strong impact of the spin-orbit interactionon time-dependent phenomena as is currently investi-gated within an atomistic approach based on the stochas-tic LandaundashLifshitzndashGilbert equation First calculations forsystems perturbed by a magnetic field pulse show long-term excitations which may excite magnons

Acknowledgments D Boumlttcher is member of the Inter-national Max Planck School for Science and Technologyof Nanostructures This work is supported by the Sonder-forschungsbereich 762 lsquoFunctionality of Oxide Interfacesrsquo

References and Notes

1 B Hardrat A Al-Zubi P Ferriani S Bluumlgel G Bihlmayer andS Heinze Phys Rev B 79 094411 (2009)

2 K von Bergmann S Heinze M Bode G Bihlmayer S Bluumlgeland R Wiesendanger New J Phys 9 396 (2007)

3 G Moulas A Lehnert S Rusponi J Zabloudil C Etz S OuaziM Etzkorn P Bencok P Gambardella P Weinberger and H BrunePhys Rev B 78 214424 (2008)

4 A Lehnert S Dennler P Blonski S Rusponi M EtzkornG Moulas P Bencok P Gambardella H Brune and J HafnerPhys Rev B 82 094409 (2010)

5 A Enders R Skomski and J Honolka J Phys Condens Matt22 433001 (2010)

6 S Mankovsky S Bornemann J Minaacuter S Polesya H Ebert J BStaunton and A I Lichtenstein Phys Rev B 80 014422 (2009)

7 J Zabloudil R Hammerling L Szunyogh and P Weinberger (eds)Electron Scattering in Solid Matter Springer Berlin (2005)

8 D Boumlttcher A Ernst and J Henk J Phys Condens Matt 23 29(2011)

9 A I Liechtenstein M I Katsnelson V P Antropov and V AGubanov J Magn Magn Mater 67 65 (1987)

10 P Strange H Ebert J B Staunton and B L Gyorffy J PhysCondens Matt 1 3947 (1989)

11 G H O Daalderop P J Kelly and M F H Schuurmans PhysRev B 41 11919 (1990)

12 X Wang D-S Wang R Wu and A J Freeman J Magn MagnMater 159 337 (1996)

13 M D Stiles S V Halilov R A Hyman and A Zangwill PhysRev B 64 104430 (2001)

14 S S Dhesi G van der Laan E Dudzik and A B Shick PhysRev Lett 87 067201 (2001)

15 T Moriya Phys Rev 120 1 (1960)16 N Metropolis A W Rosenbluth M N Rosenbluth and E Teller

J Chem Phys 21 1087 (1953)17 K Binder (ed) Monte Carlo Methods in Statistical Physics

Springer Berlin (1979)18 R Rausch and W Nolting J Phys Condens Matt 21 376002

(2009)

Received 29 March 2011 Accepted 29 October 2011

J Nanosci Nanotechnol 12 7516ndash7519 2012 7519