Topological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers Gyungchoon Go,1 Se Kwon Kim,2, ∗ and Kyung-Jin Lee1, 3, y 1Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA 3KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea We theoretically investigate magnon-phonon hybrid excitations in two-dimensional ferromagnets. The bulk bands of hybrid excitations, which are referred to as magnon-polarons, are analytically shown to be topologically nontrivial, possessing finite Chern numbers. We also show that the Chern numbers of magnon-polaron bands and the number of band-crossing lines can be manipulated by an external magnetic field. For experiments, we propose to use the thermal Hall conductivity as a probe of the finite Berry curvatures of magnon-polarons. Our results show that a simple ferromagnet on a square lattice supports topologically nontrivial magnon-polarons, generalizing topological excitations in conventional magnetic systems. Introduction— Since Haldane’s prediction of the quantized 6.5 Hall effect without Landau levels [1], intrinsic topological (a) (b) properties of electronic bands have emerged as a central theme C=0 in condensed matter physics. The band topology can be char- 3.5 C=2 (meV) acterized by emergent vector potential and associated magnetic J field defined in momentum space for electron wavefunctions, C=1 called Berry phase and Berry curvature, respectively [2]. The 0.5 Berry curvature is responsible for various phenomena on elec- 0.5 5.5 10.5 H (meV) tron transport such as anomalous Hall effect [3, 4] and spin eff Hall effect [5–7]. In addition, nontrivial topology of bulk FIG. 1: (a) The schmematic illustration of the magnon and phonon bands gives rise to chiral or helical edge states according to system. The ground state of the magnetization is given by the uniform the bulk-boundary correspondence [8]. spin state along the z axis (red arrow). (b) The Chern number of Recently, research on the effects of Berry curvature on trans- our magnon-phonon hybrid system. Heff represents the effective port properties, which was initiated for electron systems orig- magnetic field including the anisotropy field and the external magnetic inally, has expanded to transport of collective excitations in field, Heff = KzS + B. Here we use the parameters S = 3=2, ! = 10 meV, and Mc2 = 5 × 1010 eV. various systems. In particular, magnetic insulators, which ~ 0 gather great attention in spintronics due to their utility for Joule-heat-free devices [9], have been investigated for nontriv- aspects of the magnon-phonon hybrid excitation in a simple ial Berry phase effects on their collective excitations [10–15]: two-dimensional (2D) square-lattice ferromagnet with perpen- spin waves (magnons) and lattice vibrations (phonons). Pre- dicular magnetic anisotropy [see Fig. 1(a) for the illustration vious studies exclusively considering either only magnons or of the system]. Several distinguishing features of our model only phonons showed that they can have the topological bands are as follows. Our model is optimized for atomically thin of their own, thereby exhibiting either the magnon Hall ef- magnetic crystals, i.e., 2D magnets. The recent discovery of fect in chiral magnetic systems [10–14] or the phonon Hall magnetism in 2D van der Waals materials opens huge oppor- effect [15] when the Raman spin-phonon coupling is present. tunities for investigating unexplored rich physics and future Interestingly, the hybridized excitation of magnons and spintronic devices in reduced dimensions [21–31]. Because phonons, called a magnetoelastic wave [16] or magnon- we consider 2D model, we ignore the non-local dipolar inter- polaron [17], is able to exhibit the Berry curvature and thus action, which is not a precondition for a finite Berry curvature nontrivial topology due to magnon-phonon interaction [18– in 2D magnets. Moreover, the Berry curvature we find does 20], even though each of magnon system and phonon system arXiv:1907.02224v1 [cond-mat.str-el] 4 Jul 2019 not require a special spin asymmetry such as the DM interac- has a trivial topology. In noncollinear antiferromagnets, the tion nor a special lattice symmetry: Our 2D model description strain-induced change (called striction) of the exchange in- is applicable for general thin film ferromagnets. Therefore, teraction is able to generate the nontrivial topology in the we show in this work that even without such long-range dipo- magnon-phonon hybrid system [18]. In ferromagnets, which lar interaction, DM interaction, or special lattice symmetry, are of main focus in this work, nontrivial topology of magnon- the nontrivial topology of magnon-phonon hybrid can emerge polarons is obtained by accounting for long-range dipolar in- by taking account of the well-known magnetoelastic interac- teraction [19]. In addition, in ferromagnets with broken mirror tion driven by Kittel [32]. As the Kittel’s magnetoelastic symmetry, the striction of Dzyaloshinskii-Moriya (DM) inter- interaction originates from the magnetic anisotropy, which is action leads to topological magnon-polaron bands [20]. ubiquitous in ferromagnetic thin film structures [33], our re- In this Letter, we theoretically investigate the topological sult does not rely on specific preconditions but quite generic. 2 Furthermore, we show that the topological structures of the Magnon-phonon hybrid excitations— We first diagonalize magnon-polaron bands can be manipulated by effective mag- the magnetic Hamiltonian Hmag and the phonon Hamiltonian netic fields via topological phase transition. We uncover the Hph separately, and then obtain the magnon-phonon hybrid origin of the nontrivial topological bands by mapping our excitations, which are called magnon-polarons, by taking ac- model to the well-known two-band model for topological in- count of the coupling term Hmp. sulators [7], where the Chern numbers are read by counting the The magnetic Hamiltonian is solved byp performing the Sx ≈ ( 2S=2)(a + ay) number of topological textures called skyrmions of a certain Holstein-Primakoffp transformation i i i , vector in momentum space. At the end of this Letter, we pro- y y z y Si ≈ ( 2S=2i)(ai − ai ), Si = S − ai ai, where ai pose the thermal Hall conductivity as an experimental probe y and ai are the annihilation and the creation operators of for our theory. a magnon at site i.p By taking the Fourier transformation, Model— Our model system is a 2D ferromagnet on a square P ik·Ri ai = k e ak= N, where N is the number of sites in lattice described by the Hamiltonian the system, we diagonalize the magnetic Hamiltonian in the momentum space: H = Hmag + Hph + Hmp ; (1) X y Hmag = ~!m(k)akak; (5) where the magnetic Hamiltonian is given by k X Kz X 2 X where the magnon dispersion is given by !m(k) = Hmag = −J Si · Sj − Si;z − B Si;z; (2) 2 [2JS (2 − cos kx − cos ky) + KzS + B]= . hi;ji i i ~ For the elastic Hamiltonian Hph, it is also convenient to where J > 0 is the ferromagnetic Heisenberg exchange inter- describe in the momentum space: action, Kz > 0 is the perpendicular easy-axis anisotropy, and X pz pz 1 B is the external magnetic field applied along the easy axis. H = −k k + uz Φ(k)uz ; (6) ph 2M 2 −k k Throughout the paper, we focus on the cases where a ground k state is the uniform spin state along the z axis: Si = z^. The phonon system accounting for the elastic degree of freedom of where only nearest-neighbor elastic interactions are main- the lattice is described by the following Hamiltonian: tained as dominant terms and the momentum-dependent spring 2 constant is Φ(k) = M!0 (4 − 2 cos kx − 2 cos ky), where the X p2 1 X characteristic vibration frequency !0 corresponds to the elas- H = i + uαΦα,βuβ; (3) ph 2M 2 i i;j j tic interaction between two nearest-neighbor ions. To obtain i i;j,α,β the quantized excitations of the phonon system, we introduce the phonon annihilation operator bk and the creation operator where ui is the displacement vector of the ith ion from its equi- y bk in such a way that librium position, pi is the conjugate momentum vector, M is α,β the ion mass, and Φi;j is a force constant matrix. The mag- s y ! z ~ bk + b−k netoelastic coupling is modeled by the following Hamiltonian uk = p ; (7) term [32, 34]: M!p(k) 2 ! q y X X z z z b−k − bk Hmp = κ (Si · ei) u − u ; (4) pk = ~M!p(k) p ; (8) i i+ei 2i i ei where κ is the strength of the magnon-phonon interaction and where the phonon dispersion is given by !p(k) = p ei’s are the nearest neighbor vectors. Equation (4) describes !0 4 − 2 cos kx − 2 cos ky. This leads to the following di- the magnetoelastic coupling as a leading order in the magnon agonalized phonon Hamiltonian: amplitude, where the in-plane components of the displacement X 1 vector do not appear. H = ! (k) by b + : (9) ph ~ p k k 2 We note here that our model Hamiltonian does not include k the dipolar interaction and the DM interaction, distinct from the model considered in Refs. [19] and [20]. Because the In terms of the magnon and phonon operators introduced above-mentioned interactions are absent in our model, neither above, the magnetoelastic coupling term is recast into the fol- ferromagnetic system nor elastic system exhibits the thermal lowing form in the momentum space: Hmp = Hmp1 + Hmp2, Hall effect when they are not coupled.