Recent Progress In Spin Wave Spintronics

Transcription

1 Recent Progress In Spin Wave Spintronics Ryuichi Shindou International Center for Quantum Materials (ICQM), Peking University

2 Collaborators and Acknowledge Jun-ichiro Ohe (Toho Univ.) Micromagnetic calculations and discussions Shuichi Murakami (Tokyo Institute of Tech.) Ryo Matsumoto (Tokyo Institute of Tech.) Eiji Saitoh (Tohoku Univ.) Discussions and theoretical suggestions Jing Shi (UC Riverside, ICQM at PKU ) Z. Q. Qiu (UC Berkley, ICQM at PKU) Experimental suggestions

3 Content of this Tutorial talk (former half) Magnetostatic (MS) spin wave (Theory and Experiment) Magnetostatics in ferromagnetic thin film Forward, Backward volume modes, and Damon-Eshbach chiral surface mode Experimental method for detecting MS waves Quasi-particle (MS waves) Bose-Einstein Condensation Reference [1] H. Ibach and H. Luth, Sec. 8 in `Solid State Physics (Springer) [2] A. G. Gurevich and G. A. Melkov, `Magnetization Oscillations, and Waves'. (CRC Press, 1996, Boca Raton) [3] S. O. Demokritov et al., Nature (London) 443, 430 (2006).

4 Content of this Tutorial talk (latter half) Magnetostatic wave analog of Integer Quantum Hall physics (Theory) Topological Chern integer for MS spin-wave bands Properties of the integer; sum rules, topological chiral spin-wave edge modes Realistic model-calculations; chiral spin-wave bands in ferromagnetic thin film models Justification via micromagnetic simulations Reference R. Shindou et.al. Phys. Rev. B 87, (2013) R. Shindou et. al., Phys. Rev. B 87, (2013) R. Shindou and J-i. Ohe, Phys. Rev. B 89, (2014)

5 Magnetostatic spin wave Spin wave : collective propagation of magnetic moments in magnets Magnetostatic spin wave : driven by magnetic dipole-dipole interaction Landau-Lifshitz equation Exchange-interaction field dipolar field Maxwell equation (magnetostatic approximation) The dipolar field is given by magnetization itself a closed EOM for M.

6 Magnetostatic spin wave Spin wave : collective propagation of magnetic moments in magnets Magnetostatic spin wave : driven by magnetic dipole-dipole interaction Landau-Lifshitz equation (1/λ) 2 Exchange-interaction field dipolar field Wavelength of spin waves (λ) >> exchange-interaction length Dipole regime Dipolar field >> exchange-interaction field Spin wave is mainly driven by magnetic dipole-dipole intertaction. exchange regime Dipole regime um~sub-um GHz~subGHz c.f. typical Exch.-interaction length = several nm (iron) ~ 10nm (YIG)

7 Ferromagnetic thin film D (width) L (Length) In the dipolar regime, spin-wave dispersions depend on system shape. Ferromagnetic thin film (μm thickness) Magnetostatic energy MW eq. with MS approximation Where the magnetic charge is

8 Ferromagnetic thin film D (width) L (Length) +Ms (per area) -Ms (per area) L D -Ms +Ms Where the magnetic charge is

9 Ferromagnetic thin film +Ms (per area) Demagnetization field = -Ms -Ms (per area) Two parallel (infinitely large) plates with charges -Ms +Ms An external field which sets off the demagnetization field can makes the System magnetized along z-direction. H ext > Ms H ext < Ms ; perpendicularly magnetized ; has in-plane magnetized Where the magnetic charge is

10 MS wave in Ferromagnetic thin film In-plane magnetized case Backward volume mode Damon-Eshbach (DE) chiral surface mode Damon-Eshbach, J. Phys. Chem. Solids, 19, 308, (1961) Out-of-plane magnetized case forward volume mode Damon-Van De Vaart, J. Appl. Phys, 36, 3453, (1965)

11 MS wave in Ferromagnetic thin film Backward volume mode Wave vector k is parallel to in-plane magnetization Group velocity dω/dk is negative backward propagation of spin-wave wavepacket Why backward?

12 Why backward?

13 Why backward? Antiferromagnetic coupling larger k has a smaller excitation (MS) energy Smaller k has a larger MS energy To be more precise, the argument holds true for thin wire or those thin films whose one of the length is smaller than the other

14 In the short wavelength limit. When k becomes comparable to the exchange interaction length, the MS wave will be transformed into the usual spin wave which is driven by short-range quantum-mechanical exchange Interaction. Exchange regime

15 MS wave in Ferromagnetic thin film forward volume mode Wave vector k is transverse to out-of-plane magnetization Group velocity dω/dk is positive forward propagation of SW

16 MS wave in Ferromagnetic thin film forward volume mode Wave vector k is transverse to out-of-plane magnetization Group velocity dω/dk is positive forward propagation of SW Why forward? When H ext Uniformly in-plane polarized When the external field Hext is just above the demagnetization field Ms (Hext Ms), the ferromagnetic moments should start to lay down within the plane uniformly with an assistant of smaller energy (Hext - Ms). The MSW resonance frequency at k=0 is lower than those of k 0.

17 MS wave in Ferromagnetic thin film DE chiral surface mode Wave vector k is transverse to in-plane magnetization Localized at the surface (top surface and bottom surface) propagate transverse to ferromagnetic moment and along the surface Anti-clockwise with the ferromagnetic moment set to be inward Localization length

18 Why chiral surface state? Holstein-Primakov (HP) boson field z : Normal hopping term of HP boson field y 2 Pairing terms (bb, b*b*) acquires a phase which depends on the relative spatial angle between position 1 and 2 1 Θ 12 x Under the time-reversal operations Broken `time-reversal symmetry

19 Why chiral surface state? where Damon-Eshbach chiral surface spin-wave mode c.f. Two-dimensional electron gas under strong magnetic field : gapless chiral (Dirac) edge modes Aharonov-Bohm (AB) phase due to external gauge field Chiral p-wave superconductors : chiral Majorana edge modes chiral p-wave pairing term due to effective attractive interactions

20 Experiments on MS waves How to measure these MS waves? Standard methods for detecting spin-waves : inelastic neutron scattering (NS) It is hard for NS to get access to small wavevector regime (dipolar regimes) Brillouin light scattering ω k,sw,k sw Magneto-optical coupling ε I,q I ε S,q s Incident light Momentum and energy conservation Scattered light (transmitted light) By measuring scattering (transmitted) light, k and ω k can be experimentally determined. Demokritov, Hillebrands Slavin Physics Report (2001)

21 Microwave antenna: microwave AC current introduced in `in-put antenna excites spin waves, which can be detected in `out-put antenna spatially separated from the `input. Serga et.al., Journal of Physics D (2010)

22 Infrared Camera : magnetic energy are dissipated into lattice system measuring heat associated with phonon YIG (Yttrium iron garnet; Y 3 Fe 5 O 12 ) : ferromagnetic insulators Contrary to ferromagnetic metals (Fe, Co, Ni ), magnetic energies are dissipated only into lattice system with very long life-time An et.al. Nature Materials (2013)

23 Experiments on MS waves Quasi-particle Bose Einstein Condensation MS waves (Magnon,HP boson) : quasi-particle, not intrinsic boson particle In other words..... At equilibrium, the total number of magnon is determined by Temperature (T), volumes (V), and other external variables (A, B, ). Total number of magnon (N) is not independent from temperature, volumes and other external variables. Free energy F(T, V, A, B,... ) but not F(T, N, V, A, B, ) Conjugate chemical potential μ is note defined (or μ=0). All the magnetostatic (MS) magnon (backward, forward, DE surface modes) have finite (positive) resonance frequencies. No Bose-Einstein Condensation of MS magnons is expected at equilibrium.

24 Experiments on MS waves Quasi-particle Bose-Einstein Condensation MS waves (Magnon,HP boson) : quasi-particle, not intrinsic boson particle In YIG (Yttrium iron garnet; Y 3 Fe 5 O 12 ).... Demokritov et.al. Nature (2006) MS magnons have very long life time (~ 1μs) No ``spin bath other than phonon (lattice) systems Typical phonon-magnon interaction energy scale is 1μs, While that of magnon-magnon interaction is much shorter than this in time (10ns) Magnon-magnon interaction conserves total number of magnons. Quasi-equilibrium with non-zero chemical potentials by pumping magnon quasi-equilibrium BEC by pumping magnons

25 Quasi-particle (MS magnon) BEC The set-up for MS magnon BEC To interferometer H From laser Frequency (log-plot) Backward volume mode Wavevector (log-plot) YIG film with in-plane magnetization MS backward volume mode Incident light (from laser) applied onto YIG film from above and Send reflected lights to an interferometer intensity of BLS YIG film Microstrip resonator Demokritov et.al. Nature (2006)

26 Quasi-particle (MS magnon) BEC The set-up for MS magnon BEC To interferometer H From laser Frequency (log-plot) ν p (magnon) 2ν p (from resonator) Wavevector (log-plot) YIG film with in-plane magnetization MS backward volume mode ν p (magnon) Incident light (from laser) applied onto YIG film from above and Send reflected lights to an interferometer intensity of BLS YIG film Microstrip resonator Magnon pumping Demokritov et.al. Nature (2006) Microstrip resonator beneath the YIG film. Send microwave with (2ν p and zero wavevector) split into two magnons (ν p and finite wavectors with opposite signs)

27 Quasi-particle (MS magnon) BEC Without Magnon Pumping (thermal equilibrium MS magnons) Circle points : measured BLS intensity Solid line : D(ν) n B (ν) (theory fitting) Dotted line : D(ν) (theory) ν m ν m Demokritov et.al. Nature (2006)

28 Quasi-particle (MS magnon) BEC With Magnon Pumping Starting two magnons with ν p are going to be relaxed into other lower frequency magnons via magnon-magnon interaction This relaxation time is on the order of 50ns ~200ns Since the interaction preserve the number of magnon, the total number of magnons and total energy of magnons are preserved during this relaxation process Demokritov et.al. Nature (2006) Demidov et.al. Phys. Rev. Lett (2007) After sufficient long time (>1μs), the magnon energy will be dissipated into lattice (phonon) system For t>1μs, total number and energy of magnons are no longer conserved. From t = 50ns ~ 1μs, the magnon system reaches an quasi-equilibrium state, where the total number and energy of magnons are conserved. with ~10-19 cm -3

29 Quasi-particle (MS magnon) BEC With Magnon Pumping (t > 50ns) Demokritov et.al. Nature (2006) Quasi-equilibrium with non-zero chemical potential for magnon 200ns 300ns Both are well-fitted by D(ν) n B (ν) Fitting parameter chemical potential (μ) Number of pumped magnons are larger in τ=300ns than in τ=200ns Chemical potential increases when pumping time (τ) increases. At τ=300ns, μ reaches the lowest MS magnon resonance frequency (ν m ).

30 Quasi-particle (MS magnon) BEC Demokritov et.al. Nature (2006) After μ reaches ν m (t > 300ns); when further pumping magnons, Quasi-particle (MS waves) BEC 400ns 500ns deviates from D(ν) n B (ν) (μ=ν m ; green dotted lines) Well-fitted by D(ν) (n B (ν) + A δ(ν-ν m )) (μ=ν m ; blue solid lines) Higher frequency resolution experiment confirms that the BEC peak width is 50MHz (10-4 cm reciprocal lattice unit) C.f. Thermal magnons are distributed over BZ up to k T 0.5 reciprocal lattice unit

31 Summary so far (former half) Magnetostatic (MS) spin wave (theory and Experiment) Magnetostatics in ferromagnetic thin film Forward, Backward volume modes, and Damon-Eshbach chiral surface mode Experimental method for detecting MS waves Bose-Einstein Condensation of MS spin waves by microwave pumpings (Experiment) Reference [1] H. Ibach and H. Luth, Sec. 8 in `Solid State Physics (Springer) [2] A. G. Gurevich and G. A. Melkov, `Magnetization Oscillations, and Waves'. (CRC Press, 1996, Boca Raton) [2] S. O. Demokritov et al., Nature (London) 443, 430 (2006).

32 Content of the latter half Magnetostatic wave analog of Integer Quantum Hall physics Topological Chern integer for MS spin-wave bands Properties of the integer; sum rules, topological chiral spin-wave edge modes Realistic model-calculations; chiral spin-wave bands in ferromagnetic thin film models Justification via micromagnetic simulations Reference R. Shindou et.al. Phys. Rev. B 87, (2013) R. Shindou et. al., Phys. Rev. B 87, (2013) R. Shindou and J-i. Ohe, Phys. Rev. B 89, (2014)

33 Experiments on MS waves periodically modulated magnetic materials Permalloy (Ni 80 Fe 20 ) Scanning electron microscope Polarized microscope YIG Brillouin light Scattering (BLS) Two-dimensional Lithography technique in semiconductors engineering enables us to makes an periodic array of holes in ferromagnetic thin film. `multiple-band character. One-dimensional Gulyaev et.al. JETP letters (2003) Adeyeye et.al. J. Phys. D (2008) Wang et.al. App. Phys. Letters (2009) Tacchi et.al. Phys. Rev. Lett. (2011)

34 Our Proposal = MS spin-wave analog of integer quantum Hall state 2-d ferromagnetic insulators Zeeman field periodic structuring Spin-wave volume(bulk)-mode band C 3 = -1 ω Shindou et.al. PRB (13,13,14) External frequency C 2 = -2 MS spin-wave analog of Integer quantum Hall state normally magnetized `2-d magnetic materials with periodic structurings magnetostatic spin-wave (boson) multiple band character Bloch w.f. for each band 1 st Chern integer for each band ω=0 chiral edge mode for spin-wave Spin-wave Volume-mode band k C 1 = 1 Number of chiral edge modes within a gap := sum of the Chern integers over the bands below the gap chiral edge modes for spin-wave free from static backward scatterings

35 magnetic superlattice structure Landau-Lifshitz equation Maxwell equation (magnetostatic approx.) FM material ext Minimize the magnetostatic energy E MS classical spin configuration M 0 ext Transverse Landau-Lifshitz moments equation around is linearized the classical w.r.t. spin m ± configuration: m 2 real-valued fields : Holstein-Primakoff (HP) boson field Hermite matrix

36 magnetic superlattice structure Spin-wave Hamiltonian (quadratic boson Hamiltonian) HP boson field Because H 2 2 has a particle-particle pairing term (# of the particle is non-conserved) Due to the spin-orbit locking nature of magnetic dipole-dipole interaction, there is no U(1) rotation symmetry in the spin-space

37 Topological Chern number from quadratic boson Hamiltonian BdG (Bogoliubov-de-Gennes)-type Hamiltonian where 2N 2N Hermite matrix k : crystal momentum N: # (degree of freedom within a unit cell of the magnetic superlattice) A bosonic BdG Hamiltonian is diagonalized in terms of para-unitary transformation T k Commutation relation of boson field Orthogonality and Completeness of (new) bosonic fields Because this satisfies Projection operator filtering out the j-th bosonic and

38 Topological Chern number from quadratic boson Hamiltonian Projection operator filtering out the j-th bosonic (First) Chern number for the j-th bosonic band Avron et.al. PRL (83) Gauge field (connection) TKNN Integer Thouless et.al. PRL (82) Kohmoto, Annal of Physics (85)

39 Topological Chern number from quadratic boson Hamiltonian A sum rule for the integer... (When summed over particle bands and hole bands) 1 1 How about a sum only over all particle bands? a sum can be non-zero?

40 Topological Chern number from quadratic boson Hamiltonian How about a sum only over all particle bands? a sum can be non-zero? ω=0 particle bands +1 Could we have a topological boson (spin-wave) mode which is gapless? Hole bands -1 A sum only over all particle bands and that over all hole bands are required to be zero respectively. No gapless topological boson (spin-wave) mode

41 Topological Chern number from quadratic boson Hamiltonian All the bosonic excitations are required to be fully gapped, because of the stability of the classical solution..... : Paraunitarily equivalent to positive definite diagonal matrix Or Sylvester s law of inertia : unitarily equivalent to positive definite diagonal matrix

42 Topological Chern number from quadratic boson Hamiltonian Interpolation to the trivial limit.. : unitarily equivalent to positive definite diagonal matrix

43 Topological Chern number from quadratic boson Hamiltonian Interpolation to the trivial limit.. : paraunitarily equivalent to positive definite diagonal matrix λ=0 λ=1 ω=0 particle bands hole bands Particle bands and Hole bands are disconnected from Each other during the interporation from λ=0 to λ=1. ω=0 particle bands hole bands

44 Topological Chern number from quadratic boson Hamiltonian The sum of the integer over all particle bands are invariant during the interpolation.

45 Topological Chern number from quadratic boson Hamiltonian The sum of the integer over all particle bands are invariant during the interpolation. ω chiral spin-wave edge mode Ch 3 = -3 Su-Schrieffer Heeger, PRL (79), Niemi-Semenoff, PR (86),... The number and the direction of the chiral edge modes within a given band gap is determined by the sum of the integer over those particle bands below the gap. Ch 2 = 2 Ch 1 = 1 k

46 magnetic superlattice structure Landau-Lifshitz equation Maxwell equation (magnetostatic approx.) FM material ext Minimize the magnetostatic energy E MS classical spin configuration M 0 ext Landau-Lifshitz equation is linearized w.r.t. m ± : Holstein-Primakoff (HP) boson field

47 without external magnetic field.. =0 Vortex configuration minimizes MS energy. Moment lies within the x-y plane: `stray-field-free configuration: =0 Moment is tangential along the boundary, while being divergence-free within the body no magnetic charge

48 spin excitations within a single ring... ``Atomic orbitals for ``tight-binding models at zero field... Moment is almost tangential along the ring Spin excitations along the ring becomes like the so-called backward volume mode in ferromagnetic thin film or thin wire. ferromagnetic thin film or wire Negative slope Damon-Eshbach (1961),.... Arias-Mills (2001),... From Damon-Eshbach JPCS 19, 308 (1961) Group velocity ω/ k is antiparallel to the vector k ``backward volume mode

49 spin excitations within a single ring ``Atomic orbitals for ``tight-binding models near zero field Atomic orbitals with higher angular momenta (n J ) come in the low-frequency side of those with lower n J (as far as the dipole regime is concerned) Resonance frequency ω atomic orbitals with low n J Dipole regime atomic orbitals with higher n J exchange regime Atomic orbitals with higher n J have many nodes along the rings.... The inter-ring transfer integrals between orbitals with higher n J become very small, due to the cancellation b.t.w. the opposite phases. Volume mode bands in the low frequency regime becomes less dispersive and featureless. Chern integers for them = 0 angular momentum n J

50 spin excitations within a single ring ``Atomic orbitals for ``tight-binding models Near the saturation field (H s )... Moments are fully polarized above H s, while start to acquire a finite in-plane component below H s H ext > H s The atomic orbital with zero angular momentum (n J =0) becomes gapless at H ext =H s ω Dipole regime atomic orbitals with higher n J atomic orbitals with low n J n J H ext < H s Volume mode bands in the low frequency regime becomes more dispersive. chance to have non-zero Chern integers.

51 -4π/L -2π/L 0 2π/L 4π/L spin excitations within a single ring ω atomic orbitals with low n J Dipole regime ``Atomic orbitals for ``tight-binding models Near the saturation field (H s )... Four-fold rotational anisotropy (e.g. depolarization fields coming from neighboring rings) leads to the mixing among n J, n J ±2π/L*4, n J ±2π/L*8,.... n J (L: length of the ring) All the atomic orbitals within a ring are classified only into four angular momenta; n J =0, ±2π/L, 4π/L. Mod 4

52 spin excitations within a single ring ω atomic orbitals with low n J Dipole regime ``Atomic orbitals for ``tight-binding models Near the saturation field (H s )... Four-fold rotational anisotropy (e.g. depolarization fields coming from neighboring rings) leads to the mixing among n J, n J ±2π/L*4, n J ±2π/L*8,.... n J (L: length of the ring) All the atomic orbitals within a ring are classified into four angular momenta; n J =0, ±2π/L, 4π/L. +i Symmetry of `atomic orbitals -1 x +1 n J = 0 s-wave like orbital n J = 2π/L p + -wave (p x +ip y ) orbital n J = -2π/L p - -wave (p x -ip y ) orbital -4π/L D x2-y2-2π/l P - 0 s 2π/L P + 4π/L D x2-y2 -i n J = 4π/L d x2-y2 -wave orbital

53 Nearest-Neighbor Tight Binding s-p + model s p + p - d x2-y2 Hopping between nearest neighbor ``Atomic orbitals +1 S-wave orbital +1 Δ d x2-y2 p - p + `atomic-orbital levels s -it sp i S-wave orbital i +1 p + -wave orbital +t sp

54 Nearest-Neighbor Tight Binding s-p + model Topological number which tells us about each of these two spin-wave bands is topological or not where Hamiltonian in momentum space 3-component vector 1 st Chern integer (topological number) 2 by 2 Pauli matrix normalized vector which counts how many times the vector wraps the unit sphere, when the momentum k wraps the 2-dim momentum space.

55 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four `phases 2-bands (S-P + ) NN tight-binding model on -lattice (i) 2-dim k-space (torus) a unit sphere Δ=-4(t ss +t pp ) Δ=0 Δ=4(t ss +t pp ) t ss : NN transfer between s-orbitals t pp : NN transfer between p-orbitals Δ =ε s - ε P+ Bernevig-Hughes-Zhang, Science (2006), Fu-Kane PRB (2007),.... Δ k Hamiltonian in momentum space h(k)

56 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four phases (-π,π) k y (π,π) (i) Γ-point k x a unit sphere h 3 <0 (-π,-π) (π,-π) h(k) Hamiltonian in momentum space

57 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four phases (-π,π) k y (π,π) (i) k x a unit sphere h 3 <0 (-π,-π) (π,-π) h(k) h(k) rotate within a southern hemisphere. Hamiltonian in momentum space

58 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four phases k y (π,π) (i) a unit sphere Γ k x h(k) h 3 =0 h(k) rotate along the equator. Hamiltonian in momentum space

59 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four phases (-π,π) k y (π,π) (i) k x h(k) a unit sphere h 3 >0 (-π,-π) (π,-π) h(k) rotate within a northern hemisphere. Hamiltonian in momentum space

60 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer (i) Which distinguishes the four phases a unit sphere h(k) The wrapping (winding) number is +1. : `topological phase h(k) also reaches the north pole. Hamiltonian in momentum space (-π,π) (-π,-π) k y Γ-point (π,π) h 3 >0 k x (π,-π)

61 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four phases (-π,π) k y (π,π) (ii) Γ-point k x a unit sphere h(k) h 3 >0 (-π,-π) (π,-π) h(k) is on the north pole. Hamiltonian in momentum space

62 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer Which distinguishes the four phases (-π,π) k y (π,π) (ii) k x a unit sphere h(k) h 3 >0 (-π,-π) (π,-π) h(k) always rotates within the northern hemisphere Hamiltonian in momentum space

63 Nearest-Neighbor Tight Binding s-p + model 1 st Chern integer (ii) Which distinguishes the four phases : `non-topological phase (-π,π) k y Γ-point (π,π) k x a unit sphere h(k) h 3 <0 The wrapping (winding) number is 0. h(k) cannot lap the unit sphere. Hamiltonian in momentum space (-π,-π) (π,-π)

64 Nearest-Neighbor Tight Binding s-p + model s p + p - d x2-y2 ``Atomic orbitals for ``tight-binding models Near the saturation field (H s )... 2-bands (S-P + ) NN tight-binding model on -lattice Bernevig-Hughes-Zhang, Science (2006), Fu-Kane PRB (2007),.... Δ d x2-y2 p - p + `atomic-orbital levels s Ch 2 = 0 Ch 1 = 0 Ch 2 = +1 Ch 1 = -1 Ch 2 = -1 Ch 1 = +1 Ch 2 = 0 Ch 1 = 0 Δ=-4(t ss +t pp ) Δ=0 Δ=4(t ss +t pp ) t ss : NN transfer between s-orbitals t pp : NN transfer between p-orbitals Δ = ε s - ε p+ Δ

65 2-bands NN tight-binding model on -lattice Near the saturation field (H s )... Ch 1 = +1 Ch 1 = -1 Ch 1 = +1 Ch 1 = -1 H ext = 0.94*H s A similar effective model is valid for the other model. Minor details Sometimes, coupling between 2 nd lowest band and 3 rd or 4 th bands further transfers Ch 2 =+1 into Ch 3 =+1 or Ch 4 =+1

66 Above the saturation field... All the spins are fully polarized along the field spins at the four corner regions feel stronger demagnetization field. spins at the four corner regions are much softer than those in other regions. Low frequency spin waves are composed of those spins at these corner regions. where Linearized EOM for magnons demagnetization field part Exchange process part

67 Above the saturation field... To construct a proper tight-binding model... We first include `exchange process between two nearest neigboring spins, which is due to Magnetic dipole-dipole interaction... where Linearized EOM for magnons demagnetization field part Exchange process part

68 Above the saturation field... To construct a proper tight-binding model... We first include `exchange process between two nearest neighboring spins, which is due to Magnetic dipole-dipole interaction... Bonding orbital and anti-bonding orbital on nearest neighbor x-link and y-link antibonding bonding Atomic energy for bonding orbital comes lower than that for antibonding orbital

69 Above the saturation field... To construct a proper tight-binding model... We first include `exchange process between two nearest neighboring spins, which is due to Magnetic dipole-dipole interaction... Bonding orbital and anti-bonding orbital on nearest neighbor x-link and y-link antibonding We finally include `exchange process between these atomic orbitals defined on different links bonding

70 Above the saturation field... Two bands out of two Anti-bonding orbitals Two bands out of two bonding orbitals Transfer between Nearest neighboring links Lowest two bands comprises two massless Dirac cones at two inequivalent symmetric k-points (X-points); graphene like

71 Above the saturation field... Two bands out of two Anti-bonding orbitals Ch 1 = +1 Ch 1 = -1 Two bands out of two bonding orbitals Transfer between Nearest neighboring links Lowest two bands comprises two massless Dirac cones at two inequivalent symmetric k-points (X-points); graphene like Transfer between Next Nearest neighboring links two massless Dirac cones acquire finite gap, such that the lowet two bands acquire a finite Chern integers (±1). Topological chiral edge modes between these lowest two bands.

72 Above the saturation field... Two bands out of two Anti-bonding orbitals Ch 1 = +1 Ch 1 = -1 Two bands out of two bonding orbitals Transfer between Nearest neighboring links Lowest two bands comprises two massless Dirac cones at two inequivalent symmetric k-points (X-points) Transfer between Next Nearest neighboring links two massless Dirac cones acquire finite gap, such that the lowet two bands acquire a finite Chern integers (±1). Topological chiral edge modes between these lowest two bands.

73 Micromagnetic Simulations Landau-Lifshitz-Gilbert (LLG) equation : static external magnetic field (H z ) H eff : spatially-local pulse field in the transverse direction (1ps) : long-ranged dipolar field α : Gilbert damping term (of YIG) After calculating the time-evolution of the system.... Open boundary conditions 1um X 1um (100 x 100) YIG (5nm thickness) ΔT = 50 ps, n = 1024 : Spatial-resolved frequency power spectrum ``local density of state for a given ω : integrated frequency power spectrum

74 Integrated frequency power spectrum in the lowest resonating ω region... (near saturation field) Pulse field at edge X X Pulse field at center Location of the initial pulse field

75 Spatial-resolved frequency power spectrum Ch 1 = +1 Ch 1 = -1 Chiral edge mode (anticlockwise) lower bulk band Bulk does not reponse in this frequency regime Bulk band gap Upper bulk band

76 Applications to other systems Above the saturation field... Spins in thinner regions feel stronger demagnetization fields than those spins in thicker region. H Soft magnons are localized in thinner regions. A system with gutters shares a similar low-frequency spin-wave bands with those with ring model above the saturation field.

77 Applications to other systems Above the saturation field... Spins in thinner regions feel stronger demagnetization fields than those spins in thicker region. H Soft magnons are localized in thinner regions. A system with gutters shares a similar low-frequency spin-wave bands with those with ring model above the saturation field. PMA (perpendicularly magnetic anisotropy) materials Due to the surface magnetism, ultrathin magnetic film (e.g. several monolayers Fe) often realizes PMA. When the thickness increases, the bulk magnetism overcomes the surface magnetism, so that the spins lie down within the plane, minimizing the shape anisotropy. Qiu et.al. PRL 70, ( 93) `Critical thickness for spin reorientation transition. M

78 Applications to other systems Above the saturation field... Spins in thinner regions feel stronger demagnetization fields than those spins in thicker region. H Soft magnons are localized in thinner regions. A system with gutters shares a similar low-frequency spin-wave bands with those with ring model above the saturation field. PMA (perpendicularly magnetic anisotropy) materials Near (but below) the critical thickness... Soft magnons are localized near the thicker regions, because the spins there nearly acquires transverse components A PMA system with chambers shares a similar low-frequency spin-wave bands with those with ring model above the saturation field. M

79 Summary of the latter half Magnetostatic spin-wave analog of integer quantum Hall states chiral spin-wave edge modes in dipolar regime Chiral edge mode is robust against elastic scatterings Fault-Tolerant spin-wave devices Spin-wave `Fabry-Perot interferometer Halperin, PRB (`82) PS1, PS2 Kruglyak et.al. (`10)

80 Summary of this Tutorial talk Magnetostatic (MS) spin wave (Theory and Experiment) Forward, Backward volume modes, and Damon-Eshbach chiral surface mode Bose-Einstein Condensation of MS waves (Experiment) MS waves analog of integer Quantum Hall states (Theory) Thank you for your attention