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Home , Topological order

Topological - Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers

Gyungchoon Go,1 Se Kwon Kim,2, ∗ and Kyung-Jin Lee1, 3, † 1Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 2Department of 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-, are analytically shown to be topologically nontrivial, possessing finite Chern numbers. We also show that the Chern numbers of magnon- 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 . The band can be char- 3.5 C=2 (meV)

acterized by emergent vector potential and associated magnetic J field defined in momentum space for wavefunctions, C=1 called Berry 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 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, ω = 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 () and lattice vibrations (). 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 , 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 . 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 by√ performing the Sx ≈ ( 2S/2)(a + a†) number of topological textures called skyrmions of a certain Holstein-Primakoff√ transformation i i i , vector in momentum space. At the end of this Letter, we pro- y † z † Si ≈ ( 2S/2i)(ai − ai ), Si = S − ai ai, where ai pose the thermal Hall conductivity as an experimental probe † and ai are the annihilation and the creation operators of for our theory. a magnon at site i.√ 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 † Hmag = ~ωm(k)akak, (5) where the magnetic Hamiltonian is given by k

X Kz X 2 X where the magnon dispersion is given by ωm(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 = zˆ. 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) = Mω0 (4 − 2 cos kx − 2 cos ky), where the X p2 1 X characteristic vibration frequency ω0 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- † 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 † ! z ~ bk + b−k netoelastic coupling is modeled by the following Hamiltonian uk = √ , (7) term [32, 34]: Mωp(k) 2 ! q † X X z z  z b−k − bk Hmp = κ (Si · ei) u − u , (4) pk = ~Mωp(k) √ , (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 ωp(k) = p ei’s are the nearest neighbor vectors. Equation (4) describes ω0 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 = ω (k) b† 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. In other words, they are where invariant under the combined action of time-reversal (T ) and X h i H =κ ˜ a† b (−i sin k + sin k ) + h.c. , (10) spin rotation by 180◦ around an in-plane axis (C) [20]. It is the mp1 k k x y k magnetoelastic coupling term Hmp that breaks the combined X h † † i symmetry TC and thus can give rise to the thermal Hall effect Hmp2 =κ ˜ a−kbk (i sin kx − sin ky) + h.c. , (11) as will be shown below. k 3

30 30 30 30 (a) (b) (a) (b) 20 20 20 20

10 10 10 10 E ( k ) (meV) E ( k ) (meV) E ( k ) (meV) E ( k ) (meV)

0 0 0 0 Γ X M Γ Γ X M Γ Γ X M Γ Γ X M Γ

Γ(Ω) dz Γ(Ω) dz (c) π (d) π (c) π (d) π

0 ky 0 ky ky 0 ky 0

−π −π −π −π −π π −π 0 π 0 −π 0 π −π 0 π kx kx kx kx

FIG. 2: The band structure and its topology for |C| = 1 case. The FIG. 3: The band structure and its topology for |C| = 2 case. The band structure for κ = 0 (a) and κ = 10 meV/Å (b). The red dashed band structure for κ = 0 (a) and κ = 10 meV/Å (b). The red dashed line represents the band-crossing points. (c) Berry curvatures of the line represents the band-crossing points. (c) Berry curvatures of the upper band in log-scale Γ(Ωz) = sign(Ωz)log(1 + |Ωz|) for κ = 10 upper band in log-scale Γ(Ωz) = sign(Ωz)log(1 + |Ωz|) for κ = 10 meV/Å (d) Schematic illustration of dˆ(k) for κ = 10 meV/Å. The meV/Å. (d) Schematic illustration of dˆ(k) for κ = 10 meV/Å. The ˆ ˆ in-plane components (dˆx, dˆy) are shown in red arrows. in-plane components (dx, dy) are shown in red arrows.

p with κ˜ = κ ~S/(Mωk). Note that Hmp1 conserves the total nontrivial even in the absence of dipolar or DM interaction particle number, whereas Hmp2 does not. Because of Hmp2, and their topological property can be controlled by the effec- the total Hamiltonian takes the Bogoliubov-de-Gennes (BdG) tive magnetic field. form. Origin of the topological property— The origin of the The band structure of magnon-phonon hybrid system is ob- nontrivial magnon-polaron bands obtained above can be tained by solving the Heisenberg equations with the above understood through the mapping our system to the well- results [Eqs. (1)-(10)] (see the supplementary information for known model for two-dimensional topological insulators such the detailed calculation and the schematic illustration of the as HgTe [7, 37]. Considering Hmp as a weak perturba- band structure [35]). Without magnon-phonon interaction, tion with unperturbed Hamiltonian with well-defined ener- there are two positive branches consisting of a magnon band gies of magnons and phonons, the effect of particle-number- and a phonon band. The two bands cross at k points satisfying nonconserving component Hmp2 on the band structure is much ωm(k) = ωp(k). Different from the conventional Dirac sys- smaller than that of particle-number-conserving part Hmp1. tem, there are innumerable band-crossing points which form Neglecting Hmp2, the total Hamiltonian is simplified into a a closed line. These band-crossing lines are removed by the single-particle two-band Hamiltonian magnon-phonon interaction ∝ κ, which induces the nontrivial   topological property of the bands, characterized by the Berry X † †  ak H ≈ ak bk Hk , (13) bk curvatures. In the BdG Hamiltonian, the Berry curvature is k given by [18, 20, 36] where Ωn(k) = ∇ × An(k) , (12)   ~ωm(k)κ ˜(sin ky − i sin kx) Hk = . (14) where An = ihψn,k|J ∇k|ψn,ki and ψn,k are the n-th eigen- κ˜(sin ky + i sin kx) ~ωp(k) states (see supplementary information for details). The topo- logical property of the whole system is determined by the In terms of the Pauli matrices σ = (σx, σy, σz), we write Chern number of bands, which is the integral of the Berry cur- Eq. (14) in a more compact form vature over the Brillouin zone [37]. In Fig. 1(b), we show the ~ Chern number of our bosonic system with nonzero magnon- Hk = [ωm(k) + ωp(k)] I2×2 + d(k) · σ, (15) phonon interaction κ. In our system, the Chern number can be 2 one of three integers (0, 1, and 2) depending on the effective where magnetic field Heff = KzS + B and exchange interaction J.   ~ This is one of our central results: The magnon-polaron bands d(k) = κ˜ sin ky, κ˜ sin kx, (ωm(k) − ωp(k)) . (16) in a 2D simple square-lattice ferromagnet are topologically 2 4

The band structure for the above Hamiltonian is given by 6 2 (a) 40 K ~ 4 E±(k) = [ωm(k) + ωp(k)] ± |d(k)|. (17) 2 −12 30 K 1 2 20 K xy

In terms of d vectors, the Berry curvature is written explicitly W/K) κ (10 10 K Chern number (b) as 0 0 0.5 5.5 10.5 0.5 5.5 10.5 ! ˆ ˆ H (meV) H (meV) z 1 ˆ ∂d(k) ∂d(k) eff eff Ω±(k) = ∓ d(k) · × . (18) 2 ∂kx ∂ky FIG. 4: (a) Dependence of thermal Hall conductivity on the effective The corresponding expression for the Chern number is given magnetic field Heff and with different temperatures T using the pa- by [37–39] rameters in the main text. (b) Dependence of Chern number on the effective magnetic field Heff using the parameters in the main text. 1 Z C = dk dk Ωz (k) , (19) ± 2π x y ± where U(r) is the potential acting on the wave packet which can which is the skyrmion number of the d vector [37], counting ˆ be regarded as a confining potential of the bosonic excitation. how many times d wraps the unit sphere in the Brillouin zone. Near the edge of sample, the gradient of the confining potential From Eq. (14), we read that the magnon band and phonon produces the anomalous velocity, ∇U(r)×Ωn(k). In equilib- band cross at k points satisfying ωm(k) = ωp(k) without the rium, the edge current circulates along the whole edge and net magnon-phonon interaction. These band crossing points are magnon current is zero along any in-plane direction. However, opened by the magnon-phonon interaction ∝ κ and the finite if the temperature varies spatially, the circulating current does Berry curvatures are induced near the gap opening region. not cancel, which causes the thermal Hall effect [12]. After integrating the Berry curvatures over the Brillouin zone, The Berry-curvature-induced thermal Hall conductivity is we obtain C± = 0, ±1 or ±2. The two-band model has almost given by [12, 13] identical band structures and Berry curvatures to those of full Hamiltonian, where Hmp2 is additionally considered (see the 2 xy kBT X z supplemental information). κ = − c2(ρn,k)Ωn(k), (21) ~V In Fig. 2 and Fig. 3, we show that the bulk band structures n,k and their topological properties for |C| = 1 and |C| = 2, 2 2 respectively. For calculation, we use the parameters of the where c2(ρ) = (1 + ρ) ln [(1 + ρ)/ρ] − ln ρ − 2Li2(−ρ), (En(k))/kB T −1 −1 ρn,k = [e ] is the Bose-Einstein distribution monolayer ferromagnet CrI3 in Ref. [20, 22, 25, 40] (J = 2.2 2 10 function with a zero chemical potential, kB is the Boltzmann meV, Kz = 1.36 meV, S = 3/2, and Mc = 5 × 10 eV). The force constant between the nearest-neighbor phonon constant, T is the temperature, and Li2(z) is the polylogarithm function. In Fig. 4(a), we show the dependence of thermal Hall is assumed as ~ω0 = 10 meV. The external magnetic field B = −0.1 meV is chosen for Fig. 2 and B = 0.1 meV is conductivity on the effective magnetic field Heff at different chosen for Fig. 3. In Fig. 2(a), we find a band-crossing line temperatures T . For small Heff , the thermal Hall conductivity (red dashed line) which is removed by the magnon-phonon increases with increasing Heff . However, for large Heff , it interaction [Fig. 2(b)]. In this case, a dominant contribution decreases with increasing Heff . This behavior of the thermal of the Berry curvature comes from vicinity of the Γ-point Hall conductivity can be understood through the Chern number [Fig. 2(c)]. An intuitive way to verify the topological nature of of magnon-polaron bands depicted in Fig. 4(b). The absolute the system is the number of skyrmions of the unit vector dˆ(k). value of the Chern number is 1 for small Heff , then it jumps In Fig. 2(d), we find that there is a skyrmion at the Γ-point up to 2 for a certain value of Heff and vanishes for large Heff . corresponding to |C| = 1. By changing the sign of external Discussions— In this paper, we investigate the topology of magnetic field B, we can modify the band structure with two the magnon-polaron bands in a simple 2D ferromagnet with- band-crossing lines [Fig. 3(a)]. In this case, the dominant out long-range dipolar interaction and DM interaction. In our contribution of the Berry curvature comes from vicinity of the model, the topological structure can be controlled by the effec- Γ- and M-points [Fig. 3(c)]. In terms of dˆ(k), we find that tive magnetic field which changes the number of band-crossing one skyrmion is located at Γ-point and the other skyrmion is lines. Using a perturbation approach, we develop a two-band at M-point corresponding to |C| = 2 [Fig. 3(d)]. model Hamilitonian which provides an intuitive understand- Thermal Hall effect— The finite Berry curvatures of ing of the topological structure of the model. In the two-band magnon-phonon hybrid excitations give rise to the intrinsic model, the nontrivial topology of the magnon-polaron bands thermal Hall effect as shown below. The semiclassical equa- are reflected in the skyrmion number of d(k) in momentum tions of motion for the wave packet of magnon-phonon hybrid space. As an experimental demonstration, we propose that the are given by [2, 41] thermal Hall conductivity arises from the non-trivial topology of the magnon-polaron bands. The thermal Hall conductivity 1 ∂En(k) ˙ ˙ depends on the effective magnetic field which can be manip- r˙ n = − k × Ωn(k), ~k = −∇U(r), (20) ~ ∂k ulated by the external magnetic field or voltage-induced mag- 5 netic anisotropy change [42]. 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