J 12 J 23 J 34. Driving forces in the nano-magnetism world. Intra-atomic exchange, electron correlation effects: Inter-atomic exchange: MAGNETIC ORDER

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1 Driving forces in the nano-magnetism world Intra-atomic exchange, electron correlation effects: LOCAL (ATOMIC) MAGNETIC MOMENTS m d or f electrons Inter-atomic exchange: MAGNETIC ORDER H exc J S S i j ij i j J 1 J 3 J 34 Hund s rules Spin-Orbit Coupling: MAGNETOCRYSTALLINE ANISOTROPY: K Dipolar Interaction: SHAPE ANISOTROPY r m m 1 H s.o. LS s i l i H dip m 1 m ( m1 )( ) 3 r m r 3 5 r r

2 The He atom H p 1 Ze 1 e H i He H 0 i1 m 4 0 i1 ri 4 0 r r1 ee r r 1 He atom Ground state -> the two electrons occupy the 1s orbital -> the spatial part of the wavefunction is symmetric (electrons have identical quantum numbers nlm=100) -> the spin part must be antisymmetric (electrons are fermions) 1 ground sym (r 1,r ) antisym (s 1,s ) [ 100 (r 1 ) 100 (r ) 100 (r 1) 100 (r )][ ] From the previous Hamiltonian we can calculate the energy of the ground state. Because H He does not contain spin terms, the spin part of the wavefunction only needs to satisfy the antisymmetric condition Excited state -> one electron occupies the 1s orbital (nlm=100); the second electron is in an excited state nlm 1) the spatial part of the wavefunction is symmetric and the spin part is antisymmetric S sym (r 1,r ) antisym (s 1,s ) [ 100 (r 1 ) nlm (r ) nlm (r 1) 100 (r )][ ] Singlet S=0 exci 1 ) the spatial part of the wavefunction is symmetric and the spin part is antisymmetric T 1 exci antisym (r 1,r ) sym (s 1,s ) [ 100 (r 1) nlm (r ) nlm (r 1) 100 (r )] sym (s 1,s ) ; sym (s 1,s ) ; ; ( ) Triplet S=1

3 The exchange interaction The singlet and triplet wavefunctions applied to H 0 give the same energy; the H e-e contribution can be calculated with the perturbation theory e I (r ) (r ) dr dr nlm r1 r e J (r ) (r ) (r ) (r )dr dr * * nlm 100 nlm 4 0 r1 r 1 1 E I J S ee E I J I is the Coulomb integral -> electrostatic repulsion between the electrons ((r) -> r electron density J is the exchange integral -> energy associated with a change of quantum states between the electrons T ee Exchange interaction: -> Coulomb repulsion between electrons -> total anti-symmetric wave function (Pauli exclusion principle) J is positive -> triplet ground state S=1 -> parallel spins J is negative -> singlet ground state S=0 -> antiparallel spins Note: we get the exchange energy from an Hamiltonian not including spin terms!!!!!

4 The H molecule r 1 r R Atom1 Atom Atom1 Atom

5 The H molecule

6 Inter-atomic exchange

7 The H molecule Ground state depends on the relative strength of J 1 (always > 0) in respect to -4l (I 1 +I +K 1 ) J 1 > 4l (I 1 +I +K 1 ) -> ground state is magnetic (triplet) J 1 < 4l (I 1 +I +K 1 ) -> ground state is non magnetic (singlet) Ex: H molecule -> singlet ground state The molecule magnetic ground state depends on the interaction between electrons belonging to two different atoms

8 The Heisenberg Hamiltonian The Heisenberg Hamiltonian is an effective Hamiltonian extending to larger atoms the electronelectron interaction seen in the He atom H H H H Heisenberg 0 Coulomb exchange H J s s ; N exchange ij i j i j e J (r ) (r ) (r ) (r )dr dr 4 r r * * ij i i j j i j j i i j 0 i j The Heisenberg Hamiltonian is used to describe: -> coupling of individual spins located on the same atom (intra-atomic exchange) -> coupling of atomic spin moments on different atoms (inter-atomic exchange) [Stö06]

9 Magnetic coupling between atoms: inter-atomic exchange Generalization to N-atoms system: Coulomb interaction + Pauli s principle The spins of the electrons are correlated i.e., there is a magnetic splitting in the energy spectrum of electrons in systems of atoms with open el. shells, true for systems of any size, doesn t tell what type of magnetic coupling The energy spectrum is represented by a model system of pairwise interacting spins H J S S i j ij i j correct to nd order in the overlap orbitals, cannot be proven rigorously Heisenberg model There are many possible exchange-interaction Hamiltonians H J z S S J S S S S z z x x y y i j i j i j ij ij Anisotropic Heisenberg model and XY model (J z =0) H J z i j SS z i z j Ising model

10 Magnetic coupling between atoms: FM and AFM coupling H S S J J ij i j i j J 0 0 ferromagnetic coupling antiferromagnetic coupling (10-50 mev in Fe, Co, Ni) ( mev in Mn, Cr; 0 mev in NiO superexchange) J < 0, depending on formula unit, system dimensions, topology: fully compensated AFM ferrimagnetism (uncompensated AFM) frustrated AFM Cr(100) surface

11 Exchange energy All spins in the grain must be ferromagnetically aligned exchange energy J coupling spins H exc J S S i j ij i j H exc << kt Coupling is destroyed and the net magnetic moment is zero Domain formation -> magnetic moment is strongly reduced Gain in the magnetostatic energy at the expenses of the exchange energy

12 Weiss-Heisenberg model Which is the origin of the aligned atomic spin moments in a ferromagnetic domain (spontaneous magnetization)? Weiss -> molecular field H W (or mean field theory) of unknown origin Heisenberg -> molecular field originating from the interaction of the atomic spin moment with the spin sea of all the other atoms Ei sijijs j mih W; HW Jijs j g j B j Example: single spin s=1/ Two energy levels: +/- B H with an occupation probability given by exp(+/- B H/kT) x x e e BH M(H,T) M M Ng B( ) N x x x x Btanh(x); x e e e e k T B Sea of spin s=1/ x B(H M(T)) kt B M(T) -> molecular field Curie temperature: temperature at which the spontaneous magnetization (H=0) goes to zero M(T) ->0 when tanh(x) -> 0 when x-> 0 i.e. tanh(x) = x ( M(T )) N M(0) M(T ) N x N T B c B c B B C B kbtc kb kb The magnetization goes to zero when the thermal energy equals the energy of a single spin in the molecular field (H W =gs B M0

13 Curie and Neel temperatures For a general spin value s and moment m=-g B s s J T ; J J NJ 0 C 0 ij 01 3kB j H W s J m 0 Neel temperature: is the equivalent to the Curie temperature for an antiferromagnet A few Neel temperatures NiO T N = 55 K CoO T N = 90 K FeO T N = 98 K oxigen metal

14 Curie temperature size dependence T C depends on the atomic coordination s J T ; J J NJ 0 C 0 ij 01 3kB j S. Bornemann et al. Phase Transitions 78, 701 (005) J ij as a function of distance T c nanostructures < T c bulk The number of magnetic neighbours (N) is reduced in a thin film interface: N i bulk-like: N b t layers N zb interface: N i b i t = t b + t i t N t N N N b b i i b i t t t 1/t dependence

15 Curie temperature size dependence Phys. Rev. B 1, 35 (1970) T ( ) T ( t) t, =1 C C Experiments and finite-size scaling model: TC( ) T ( t) t T ( ) C C t0 C. Domb, J. Phys. A 1, 196 (1973); Y. Li et al. Phys. Rev. Lett. 68, 108 (199); etc. ' ', 11.6 This model accounts for the decrease of T C with t down to a critical thickness t 0 4 monolayers. t is a continuous parameter -> in the ultra thin limit t becomes discrete (number of atomic layers) Ultra thin limit: linear decrease TC( ) TC( t) T ( ) R. Zhang et al. Phys. Rev. Lett. 86, 665 (001) C A ( n 1) / N 0 N o is the spin-spin coupling range (typically a few atomic sites) A takes into account the observation that for n=1 ferromagnetism can exist

16 Paramagnet the magnetic moments are randomly oriented due to thermal fluctuations Curie temperature and band structure T>T C T<T C Ferromagnet unlike the moments in a paramagnet, these moments will remain parallel even when a magnetic field is not applied Antiferromagnet Adjacent magnetic moments from the magnetic ions align antiparallel to each other without an applied field. T>T C T<T C Spin resolved inverse photoemission spectroscopy for 3d bands as a function of temperature for Ni(110) T=T C -> spin transition from minority and majority states are equal -> D=0 [Stö06] E - E F

17 Spin resolved inverse photoemission - Monochromatic low-energy spin polarized electron beam is incident on a single crystal surface. -Upon entering the crystal the electron momentum in vacuum k and inside the crystal k need to match. - The incident electron (with E > E V ) occupies excited states and then decays in lower unoccupied bands with the emission of a photon (dipolar transition conserves momentum k and spin s) [Stö06] Energy conservation (assuming free electron like electrons) F =E V E F -> work function k m ' E V k m Momentum conservation (assuming free electron like electrons) G -> reciprocal space vector p e = (me) 1/ -> ~ for 1 ev electron p hv = E/c -> ~ for 1 ev photon k k G ' // // k (k ) me ' V First Brillouin zone G=0

18 Angle resolved photoemission spectroscopy (ARPES) k // Surface states in metals: free like electrons k k k me //

19 Spin resolved inverse photoemission Electrons that are photoemitted from a sample by ultraviolet radiation are energy and angle selected by an electrostatic analyzer and detected in two orthogonal Mott polarimeters. In an electrostatic beam deflection system the spin direction is conserved and polarimeter I measures the polarization components Py and Pz, while polarimeter II measures Px and Pz. The beam is switched between the two in order to allow quasi-simultaneous data collection. In the figure, the polarimeter system is shown rotated by 90 for graphical clarity, i.e. in reality the z axis is directed straight to the left and parallel to the electron lens of the spectrometer J. Osterwalder Lect. Notes Phys. 697, 95 (006)

20 Spin detector (polarimeter) Mott spin detector Exchange-scattering spin detector Reflection spin asymmetry A from the surface of a Fe(100) crystal. I I A I I minority majority [Stö06] D. Tillmann Z Phys. B 77, 1 (1989) Electron scattering from a thin film of atoms with large Z (typically Au). In the rest system of the electron the atomic core is moving and then generating an inhomogeneous magnetic field -> A force F=m grad(h) deflects the electron depending on the electron spin The unoccupied bands above the vacuum level are the relevant bands for the incident polarized electrons. Asymmetry peak at 10 ev where the majority spins have available states at the upper edge of the gap. majority spins are less reflected compared to the minority ones

21 Bias exchange a) In an external field and above the Néel temperature T N, antiferromagnetic spins located at the interface with the ferromagnet are aligned, like the ferromagnet, with the external field. b) Once cooled below T N, these interface spins keep their orientation and appear "pinned" because they are tightly locked to the spin lattice in the bulk of the antiferromagnet, which is not sensitive to external fields. Consequently these pinned spins produce a constant magnetic field at the interface that causes the hysteresis loop of the ferromagnet to shift. H c c) This intuitive picture overestimates the magnitude of the loop shift by orders of magnitude. H b / H c W / ev / 50 mev 30 When T < T N you need several tens of tesla to reverse the AFM coupled spins ( pinning ) H c H b J. Nogués et al., J. Magn. Magn. Mater. 19, 03 (1999)

22 Bias exchange: real case only a small fraction (5%) of interfacial spins is actually pinned, and these cause the horizontal hysteresis loop shifts. H. Ohldag et al., Phys. Rev. Lett. 91, (003)

23 AFM order at the atomic scale 1 monolayer Mn/W(110): Ground state W tip SP-STM 1x1 unit cell c x magnetic unit cell Fe coated W tip Sensibility limited to the sample surface S. Heinze et al., Science 88, 1805 (000)

24 AFM domains and bias exchange XMLD XMCD PEEM F. Nolting et al. Nature 405, 767 (000) Photoemission Electron microscopy (PEEM) utilizes electron emission to generate image contrast. The excitation is produced by synchrotron radiation or X-ray sources. PEEM collects the emitted secondary electrons generated in the electron cascade that follows the creation of the primary core hole in the absorption process. PEEM is a surface sensitive technique because the emitted electrons originate from a very shallow layer

25 GMR: giant magneto resistance DR/R(%) Pinned FM NM spacer (metallic or insulating ) Free FM Electric current DR/ R R R R

26 GMR: giant magneto resistance M (arb. un.) DR/R (%) Electric current H E Pinned layer Free layer J. Appl. Phys. 69, 4774 (1991) AFM Pinned FM Free FM

27 Role of the NM spacer The atom spins are coupled together In non magnetic materials J = 0 Inter-atomic exchange: MAGNETIC ORDER H exc J S S i j ij i j J 1 J 3 J 34 Domain wall between two pinned ferromagnetic materials with opposite orientation of the magnetization Without the spacer two (negative) scenarios depending on the strength of the exchange force in respect to the pinning force: a) The free layer magnetization always aligns parallel to the magnetization of the pinned layer b) A domain wall forms which produces the spin current depolarization J. Appl. Phys. 69, 4774 (1991)

28 GMR applications Magnetic random access memory (MRAM) Ferromagnetic (FM) - nonmagnetic (NM) - ferromagnetic (FM) junction Science 8, 1660 (1998); Nat. mater. 6, 813 (007)

29 GMR applications Reading-writing head in HDD Reading: the bit stray field defines the magnetization direction of the free layer

30 Magnetic order in 1D: atomic chains Finite system (N localized moments): ground state: E=E 0 = -J(N-1) M tot 0 lowest excited state: E=E 0 + J ( N-1 states) M tot 0 change in free energy: DF= DU-TS) = J - kt ln(n-1) If DF<0 the excited state is more favourable than the ground state-> no ferromagnetism for N-1 > exp(j/kt) or T C = J/(k ln(n-1)) Infinite chain -> T C = 0 i.e. no magnetism (M tot 0) at any temperature 0

31 Magnetic order in D NxN localized Ising moments [Bel89] exact solution (due to Onsager) : long-range magnetic order prevails for J kt ln 1 A D Ising system is less sensitive to thermal fluctuations with respect to the 1D case. A finite Curie temperature exists, given by J Tc k ln 1 Mermin & Wagner, PRL 1966; Bruno, PRL 001. In general (Infinite systems) Heisenberg model -> no FM state in 1D and D at T > 0 1 Hheisen Jijsi s j si H i, j Ising model -> no FM state in 1D at T > 0 1 H J s s H s z z z ising i j i i, j i i Ising, Z. Phys. 195; Bander & Mills, PRB 1989

32 The 1D case: exchange+mae Finite chain Inter-atomic exchange energy for Co -> J 15-0 mev T C = J/k ln(n-1) ->ferromagnetism only for N < 50 atoms at T = 50 K Ising model + anisotropy-> FM state in 1D at T > 0 in an infinite system 1 1 H J s s s K s z z z easy H i ising i j i i, j i i Spin blocks oriented along the easy axis by the external field needs to overcome the MAE barrier to reverse the orientation K The MAE stabilizes the direction of the spin blocks -> M tot 0 when the external field is put back to 0

33 Atomic chains: growth Ni/Pt(997) 1 x alloy T = 00 K Phys. Rev. B 61, 54 (000); Surf. Sci. 449, 93 (000); Surf. Sci. 475, L9 (001)

34 Atomic chains: Ag/Pt(997) z (Å) T = 340 K, Q Ag = 0.04 ML Q Ag = 0.13 ML Preferential nucleation at the lower step edges: 1D Ag islands x (Å) Ag Coalescence of 1D islands: continuous atomic chains P. Gambardella et al., Phys. Rev. B 61, 54 (000).

35 Mono-atomic chains: Co/Pt(997) 10 nm nm deposition template: stepped platinum surface detail of the Pt steps array of parallel cobalt monatomic chains diffusion of Co atoms at the step edges results in self-assembled monatomic chains P. Gambardella et al., Nature 416, 301 (00). Phys. Rev. B 61, 54 (000); Surf. Sci. 449, 93 (000)

36 Mono-atomic chains: Co/Pt(997) From STM N = atoms/chain T C = J/(k ln(n-1)) = 50 K? (N=80) E(, 0, ) μ B K cos ( easy μ) m L m S FM coupled atoms: N = 15 short-range FM K =.0 mev/atom M n cos M z sat cos exp [ E(,, )/ k T] dd exp [ E(,, )/ k T] dd 0 0 Metastable FM state stabilized by the MAE long-range FM m s =.07 B LDA calculations (S. Blügel, FZ Jülich) m L = / B XMCD measurements P. Gambardella et al., Nature 416, 301 (00).

37 From 1D atomic chains to a D layer: theoretical prediction «One-dimensional finite and infinite chains show Magnetic Anisotropy Energies which are an order of magnitude larger than in two-dimensional thin films. The Magnetic Anisotropy Energy of multichains oscillates as a function of chain width and depends strongly on the transversal structure» Phys. Rev. Lett. 81, 08 (1998) 1-atom chain -atoms chain 3-atoms chain

38 Remanent Magnetization (a. u.) From 1D atomic chains to a D layer: easy axis Phys. Rev. Lett. 93, (004) F 0 0 Q

39 From 1D atomic chains to a D layer: MAE P. Gambardella et al. Phys. Rev. Lett. 93, (004). K =.1 mev/atom K = 0.34 mev/atom K = 0.47 mev/atom K = 0.13 mev/atom m L = 0.68 B m L =0.37 B m L =0.33 B m L =0.31 B

40 From 1D atomic chains to a D layer: orbital moment m L 0.68 B 0.37 B 0.33 B 0.31 B Orbital moment decreases with increasing chain thickness

41 From 1D atomic chains to a D layer: MAE Dipolar magnetic field produced by an array of uniformly magnetized chains of length N and width w N = inf Dipolar magnetic energy: E dip = mb dip << K with m = 3.8 B E dip = 0. mev/(tesla atom) For w = 3 B dip < mev/atom K = 0.4 mev/atom A. Vindigni et al., Appl. Phys. A 8, 385 (006).

42 Dipolar interaction Long range interaction between magnetic moments H dip m 1 m ( m1 )( ) 3 r m r r 3 5 m 1 r r m m 1 and m can be the magnetic moments of two atoms in a particle or the moments of two particles In out-of-plane configuration the dipolar interaction is reduced

43 Magnetic domains The magnetic configurations are determined by the competition, at a local scale, of four different energies: Zeeman, exchange, magnetocrystalline anisotropy, and dipolar coupling. exchange, magnetocrystalline energy dipolar energy -> short range -> long range Structure of a domain wall between two ferromagnetic domains with opposite orientation of the local magnetization (180 wall) SP-STM of 1.3 monolayers Fe / stepped W(110) M. Bode, Rep. Progr. Phys. 66, 53 (003)

44 Nanoparticle magnetization state: staid state Magnetic phase diagram for ultrathin films with perpendicular anisotropy (l ex = nm) R. Skomski et al. Phys.Rev. B 58, 33 (1998) A. Vaterlaus et al. J. Magn. Magn. Mater. 7-76, 1137 (004) magnetic domain pattern of perpendicularly magnetized ultra-thin Fe particles grown on Cu(0 0 1) 4 m m 1 m Magnetic phase diagram for ultrathin particles with in-plane anisotropy (Fe/W(001)) magnetic domain pattern of in-plane magnetized ultra-thin Fe particles grown on W(0 0 1) SP-STM Calculated vortex R. Skomski et al. Phys.Rev. Lett. 91, 1701 (003)

45 Magnetization (a.u.) Demagnetizing field: shape anisotropy Pushes the magnetization M along the longer side of the nanostructure: Cylinder -> M // axis Disk -> M // disk surface Co/Pt(111) Orientation and shape of Co magnetic domains out-of-plane in-plane Co wedge Pt substrate thickness (ML) 0 m X-ray photoemission electron microscopy, SIM Swiss Light Source