Alexander Schwarz:

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3 physics department prison football stadium FC St. Pauli AFM: Abteilung Fördernde Mitglieder (Department of Supporting Members)

4 Magnetic Sensitive Force Microscopy Alexander Schwarz, Universität Hamburg, Jungiusstr. 11, Hamburg, Germany Magnetism Basics (Part I & II) atomistic approach (origin of magnetism, magnetic exchange mechanisms) phenomenological approach (hysteresis, domains, domain walls) Magnetic Force Microscopy (MFM; Part I) tip preparation and tip properties separation of forces examples (thin films, superconductors, ) Magnetic Exchange Force Microscopy (MExFM; Part II) tip preparation and tip properties separation of forces examples (spectroscopy, switching, dissipation ) Review: Magnetic sensitive force microscopy : A. Schwarz & R. Wiesendanger, Nanotoday 3, 28 (2008).

5 Magnetism-Based Applications compass medical MRI door bell AMR angle sensor (1 mm 2 ) electro motor loud speaker hard drive with GMR head

6 Technological Impact of Magnetic Sensitive Force Microscopy MFM image continuous media the envisaged ultimate limit of magnetic data storage: MFM image patterned media bits from single atoms with spin up and spin down investigation/analysis requires nanonmeter down to atomic resolution

7 Forces in Force Microscopy Electromagnetic Forces F ts electrostatic force magnetostatic force van der Waals force 100 nm 10 nm 0 nn steps (topography) MFM (z > 10 nm) chemical force magnetic exchange force repulsive forces 1 nm -1 nn -0.3 nm AFM/MExFM atomic resolution z magnitude of forces: magnetic exchange 0.1 nn (d < 0.5 nm) magnetostatic 1 pn (d > 10 nm)

8 Part I Magnetic Force Microscopy (MFM)

9 Magnetic Order

10 Dipolar Interactions and Magnetic Ordering in Solids macroscopic observation: dipoles µ E = 0 µ 1 µ 3 2 µ 1 r µ 2 r 3 2 4πr r ( ) ( ) assumptions: atoms in solids = interacting atomic dipoles; interatomic distance 300 pm; dipole moment of atoms 1 µ B E = µ µ 0 B 1,6 10 J 30 μev 3 4πr 0.3 K Curie temperature of ferromagnets (Fe, Ni, Co) > 300 K (> 20 mev)! magnetic ordering is a true quantum effect and rare in pure elements superconductivity occurs much more often, although not at RT!

11 Two Interacting Electrons with Spin QM: spin operator (S x, S y, S z ) S 2 χ = s (s+1) ħ 2 χ = ¾ ħ 2 χ ; s = ½ S z χ = m s ħ χ = ± ħ/2 χ ; m s = -s,, +s = ± ½ ( m s = 1 ) µ S = -g µ B S/ħ ; g 2 (Stern-Gerlach) spin states χ i χ 1+ = χ 1- = χ 2+ = > > χ 2- = > > Dirac Bra-Ket Notation 2 interacting electrons with spin s i = ½ s = s 1 ± s 2 = 0, 1 QM: product states of indistinguishable quantum particles

12 Two Interacting Electrons with Spin spin states χ i χ 1+ = χ 1- = χ 2+ = > > χ 2- = > > 2 electrons each with s = ½ & m s = ±½ 2 electron state χ(s,m) s m > singlett antismmetric 1 0 > 1-1 triplett symmetric s = s 1 - s 2,, s 1 + s 2 = s 1 ± s 2 = 0,1 and m = -s,, +s = -1,0,+1 ( m = 1) no spin dependent term! Coulomb Interaction V(r) + Pauli Principle & Indistinguishability Alignment of Spins Magnetic Order

13 Orbital Momentum and Spin-Orbit Coupling electron spin S : intrinsic degree of freedom of the electron with associated magnetic moment µ S orbital momentum L : movement of electrons around the nucleus generates a magnetic moment µ L µ S and µ L are usually expressed in units of µ B = eħ/2m e spin-orbit coupling : H so = λ L S (λ atomic number Z strong for heavy elements) spin-orbit coupling is the origin of the magneto-crystalline anisotropy energy (MAE) in many solids µ L is quenched and µ = µ L + µ S is dominated by µ S! many electron systems: total angular momentum J LS-coupling for small Z with J = L+S with L = Σl i and S = Σs i JJ-coupling for large Z with J = Σj i with j i = l i + s i Note: LS-coupling is not spin-orbit coupling, but a way of adding up momenta in Quantum Mechanics

14 Hamilton Operator of an Electron H = -ħ/2m + V p (r) + λ L S + g l µ B BL + g s µ B BS + V cf (r) energy scales: kinetic energy > 1 ev potential energy > 1 ev crystal field 10 mev spin-orbit interaction 1 mev Zeeman interaction 1 mev geometry dependent crystal field splitting V cf (r): tetrahedral coordination octahedral coordination

15 Magnetic Exchange: Heisenberg Model in multi-electron systems: Heisenberg spin operator H : J is the magnetic coupling constant (not the total angular momentum) J > 0 ferromagnetic order J < 0 antiferromagnetic order skalar product: no magnetic interaction for perpendicular spins (more details in Part II)

16 Ferro- and Antiferromagnetism Bethe-Slater Curve simplified picture to explain ferro- (fm) and antiferromagnetic (afm) order in solids based on direct magnetic exchange: for small separation ferromagnetic spins are not allowed to share the same orbital (Pauli exclusion principle)!

17 Zoo of Magnetic Order in Nature ferromagnet antiferromagnet ferrimagnet and more (Skyrmions) canted antiferromagnet spiraling spin structure apart from direct magnetic exchange, other mechanisms can become important as well: biquadratic, Dzyaloshinskii-Moriya,

18 Indirect Coupling via Super Exchange mediation of the magnetic exchange interactions between isolated localized spins via bridging atoms (note: many similarities to magnetism in molecules!) super exchange: prototypical in transition metal monoxides like NiO with afm order TM d-state oxygen p-state TM d-state afm order fm order

19 Magnetic Domains and Domain Walls

20 Magnetization Curves and Hysteresis magnetization M parallel to external magnetic field H is favored (Zeeman energy) (for single atoms: E Z = ±µb) M R M M S Barkhausen jumps magnetic domains! H saturation magnetization M S remanent magnetization M R coercive field H C paramagnets: no hysteresis!

21 The Origins of Magnetic Domains in Ferromagnets magnetic exchange energy: atomic spins parallel short range, strong stray field energy: atomic spins antiparallel long range, weak domain formation (energy minimization) large stray field small stray field

22 The Origins of Magnetic Domains in Ferromagnets magnetic exchange energy: atomic spins parallel short range, strong stray field energy: atomic spins antiparallel long range, weak external domain magnetic formation field H (energy (additional minimization) Zeeman energy) large stray field small stray field

23 The Origins of Magnetic Domains in Ferromagnets magnetic exchange energy: atomic spins parallel short range, strong stray field energy: atomic spins antiparallel long range, weak external domain magnetic formation field H (energy (additional minimization) Zeeman energy) large stray field small stray field

24 Domain Walls if there are domains with oppositely oriented magnetizations, there must be domain walls as well due to the strong and short-ranged magnetic exchange energy, domain walls cannot be atomically sharp 180 Bloch wall: out-of-plane rotation 180 Neel wall: in-plane rotation flux closure domains (180 & 90 walls) 100 nm K: uniaxial anisotropy (preferred magnetization direction) A: exchange stiffness (related to exchange energy)

25 Magnetic Anisotropy Energies exchange energy E ex stray field energy E sf Zeeman energy E Z magnetocrystalline anisotropy energy (MAE) E mae (spin orbit coupling) shape anisotropy energy E shape surface/interface anisotropy energy E surf/interf (symmetry breaking) magnetic state is determined by the total magnetic energy: E tot = E ex + E sf + E Z + E mae + E shape + E surf/interf + uniaxial anisotropy energy E = K u V sin 2 θ (simplest case!) K u : energy density along the easy axis, θ : angle relative to easy axis, V : volume

26 Summary: Magnetism atomistic origin: spin and angular momentum alignment of spins: Coulomb interaction and Pauli exclusion principle of indistinguishable particles theoretical description with Heisenberg modell direct and indirect magnetic exchange mechanisms (super exchange, double exchange, RKYY, ) ferromagnetism, antiferromagnetism, ferrimagnetism, but also complex non-collinear spin structures, origin of magnetic domains: interplay between short range magnetic exchange interaction and long range magnetostatic forces domain structure also influenced by shape, interfaces and external field domain walls: complex magnetic structures between domains

27 Magnetic Force Microscopy (MFM) demonstrated 1987 on a hard disk, just one year after AFM Magnetic imaging by force microscopy with 1000 Å resolution. Y. Martin, Y. and H. K. Wickramasinghe, Appl. Phys. Lett. 50, 1455 (1987).

28 Forces in Force Microscopy Electromagnetic Forces F ts electrostatic force magnetostatic force van der Waals force 100 nm 10 nm 0 nn steps (topography) MFM (z > 10 nm) chemical force magnetic exchange force repulsive forces 1 nm -1 nn -0.3 nm AFM/MExFM atomic resolution z magnitude of forces: magnetic exchange 0.1 nn (d < 0.5 nm) magnetostatic 1 pn (d > 10 nm)

29 Force Sensor Si 1-7 µm µm pyramidal tip magnetic coating µm Cantilever resolution of near field techniques: effective tip size and tip-sample distance metal wire (bulk) tips: possible, but usually not so sharp; complex domain structure at tip apex (closure domains )

30 CA-FM-AFM Set-Up Deflection Sensor Amplitude Detector Frequency Demodulator Sample XYZ- Scanner A (x,y) A-Feedback Cantilever Shaker Tip A-Set-Point a exc (x,y): Dissipation constant height z (x,y): Topography f (x,y) z-feedback f -Set-Point

31 Magnetostatic Tip-Sample Interaction F ts int S T T S Sample tip stray field interacts with sample magnetization or, equivalently, vice versa = E = µ ts 0 m H ( t s ) Tip z F z = µ 0 m J: Magnetic Polarization (= Magnetization M) H: Stray Field IMPORTANT: it is impossible to unambiguously infer the sample magnetization by sensing the stray field additional information is required! dipole approximation for tip: m t = (m x, m y, m z ) magnitudes of force: 1 pn soft cantilevers (1-5 N/m) force gradients: 10-5 N/m decay length (depends on domain size): λ 100 nm oscillation amplitude: A 10 nm x 2 H 2 z x + m y 2 H z y 2 + m z 2 H 2 z in general all three components of the sample stray field contribute to the magnetic signal! if λ >> A f F/ z z

32 Optimized Properties of Thin Film Tips Shape Anisotropy and Film Thickness large magnetic signal large tip stray field (thick film) high spatial resolution localized tip stray field tip tip stray field should not alter the magnetic structure of the sample small stray field sample stray field should not switch magnetization direction in the tip hard magnetic tip shape anisotropy easy axis follows tip shape H H x y H z F = + + z µ 0 m x m m 2 y 2 z 2 z z z z out-of-plane sensitive tip 2 Hz x Fz = µ 0 mz x 2 z z potentially complex domain pattern at tip apex difficult image interpretation (true also for bulk magnetic tips: closure domains!) single domain tip with well defined easy axis easier image interpretation (only H z!) h

33 Bad Tip - Good Tip CoPt multilayer sample with out-of-plane anisotropy 500 nm tip with in-plane component 500 nm pure out-of-plane tip

34 Separation of Forces Van-der-Waals Force lift mode: disadvantage: long scan times, large magnetostatic interaction during topography scan 1st scan: topography z feedback on 2nd scan: true constant height (z-feedback off) using 1st scan data plane-subtraction-mode: obtain tilt from one topography scan (z-feedback on) better: obtain scan from two perpendicular topography scan lines (z-feedback on) even better: apply large bias to obtain tilt from topography scan (z-feedback on ) far away from surface 1st: compensate tilt between tip and sample h 20 nm 2nd: constant height (h > 10 nm) with z- feedback off (tilt remains constant over time!)

35 Separation of Forces: Electrostatic Forces +3.4 V +2.4 V +1.4 V +0.4 V topography (1 µm 2 ) plane-subtraction-mode: (non-magnetic tip) particularly important on inhomogeneous surfaces if signals are weak not every contrast visible with the tip far away from the surface is of magnetic origin a real proof is needed!

36 Contrast Formation and Interpretation Thin Films: Out-Of-Plane vs. In-Plane Anisotropy thickness dependent reorientation transition of Co on Au(111) below 4 ML: out-of-plane magnetization; surface/interface anisotropy dominates out-of-plane tip images magnetic domains above 4 ML: out-of-plane magnetization; shape anisotropy dominates out-of-plane tip images domain walls 180 Bloch wall with out-of-plane component!

37 Contrast Formation and Interpretation Magnetic Nanodots: In-Plane vs. Out-Of-Plane Anisotropy patterned media: polycrystalline cobalt nanodots on silicon diameter: 100 nm, height: 40 nm single domain; shape anisotropy favors in-plane magnetization diameter: 70 nm, height: 100 nm single domain; shape anisotropy favors out-of-plane magnetization dots appear as dipoles A. Fernandez et al., IEEE Trans. Mag. 32, 4472 (1996).

38 Field Dependent Experiments Magnetization Reversal Behavior of Thin Film Tips 30 mt manganite (La 0.7 Sr 0.3 MnO 3 ) on LaAlO 3 sample exhibits interface (stress) induced out-of-plane anisotropy 50 mt 20 mt area scan line scan (slow axis disabled) 0 mt contrast change along whole scan line indicates switching of the tip magnetization contrast inversion indicate complete reversal by 108 : well behaved single domain tip!

39 Hysteresis Loop on LSMO on LAO A. Schwarz et al., PRL 92, (2004) & M. Liebmann et al., PRB (2005): EPAPS

40 Visualization of Single Barkhausen Events 275 mt 270 mt difference images A-B: revealing Barkhausen events merging images A + C: position of Barkhausen events A A - B 1 µm B A + C 1 µm identification of nucleation of new domains and growth of old domains due to domain wall propagation N P C 1 µm D 1 µm resolution 10 nm depends on scan height and localization of the stray field emanating from the tip A. Schwarz et al., PRL 92, (2004)

41 Summary: MFM senses stray field above the surface long range magnetostatic interaction separation from other long range forces (vdw, electrostatic) lift mode or plane-subtraction mode compensation of CPD resolution limit about 10 nm tip-sample distance localization of the stray field emanating from the tip usually only qualitative, but not quantitative (e.g., not possible to obtain magnetization like with SEMPA)