Topological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers
TTopological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with TunableChern Numbers
Gyungchoon Go, Se Kwon Kim, ∗ and Kyung-Jin Lee
1, 3, † Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA KU-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 bulkbands of hybrid excitations, which are referred to as magnon-polarons, are analytically shown to be topologicallynontrivial, possessing finite Chern numbers. We also show that the Chern numbers of magnon-polaron bandsand the number of band-crossing lines can be manipulated by an external magnetic field. For experiments, wepropose to use the thermal Hall conductivity as a probe of the finite Berry curvatures of magnon-polarons. Ourresults 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 quantizedHall effect without Landau levels [1], intrinsic topologicalproperties of electronic bands have emerged as a central themein condensed matter physics. The band topology can be char-acterized by emergent vector potential and associated magneticfield defined in momentum space for electron wavefunctions,called Berry phase and Berry curvature, respectively [2]. TheBerry curvature is responsible for various phenomena on elec-tron transport such as anomalous Hall effect [3, 4] and spinHall effect [5–7]. In addition, nontrivial topology of bulkbands gives rise to chiral or helical edge states according tothe bulk-boundary correspondence [8].Recently, research on the effects of Berry curvature on trans-port properties, which was initiated for electron systems orig-inally, has expanded to transport of collective excitations invarious systems. In particular, magnetic insulators, whichgather great attention in spintronics due to their utility forJoule-heat-free devices [9], have been investigated for nontriv-ial Berry phase effects on their collective excitations [10–15]:spin waves (magnons) and lattice vibrations (phonons). Pre-vious studies exclusively considering either only magnons oronly phonons showed that they can have the topological bandsof their own, thereby exhibiting either the magnon Hall ef-fect in chiral magnetic systems [10–14] or the phonon Halleffect [15] when the Raman spin-phonon coupling is present.Interestingly, the hybridized excitation of magnons andphonons, called a magnetoelastic wave [16] or magnon-polaron [17], is able to exhibit the Berry curvature and thusnontrivial topology due to magnon-phonon interaction [18–20], even though each of magnon system and phonon systemhas a trivial topology. In noncollinear antiferromagnets, thestrain-induced change (called striction) of the exchange in-teraction is able to generate the nontrivial topology in themagnon-phonon hybrid system [18]. In ferromagnets, whichare of main focus in this work, nontrivial topology of magnon-polarons is obtained by accounting for long-range dipolar in-teraction [19]. In addition, in ferromagnets with broken mirrorsymmetry, the striction of Dzyaloshinskii-Moriya (DM) inter-action leads to topological magnon-polaron bands [20].In this Letter, we theoretically investigate the topological
C=0C=2C=1 (a) (b) J ( m e V ) H (meV) eff FIG. 1: (a) The schmematic illustration of the magnon and phononsystem. The ground state of the magnetization is given by the uniformspin state along the z axis (red arrow). (b) The Chern number ofour magnon-phonon hybrid system. H eff represents the effectivemagnetic field including the anisotropy field and the external magneticfield, H eff = K z S + B . Here we use the parameters S = 3 / , (cid:126) ω = 10 meV, and Mc = 5 × eV. aspects of the magnon-phonon hybrid excitation in a simpletwo-dimensional (2D) square-lattice ferromagnet with perpen-dicular magnetic anisotropy [see Fig. 1(a) for the illustrationof the system]. Several distinguishing features of our modelare as follows. Our model is optimized for atomically thinmagnetic crystals, i.e. , 2D magnets. The recent discovery ofmagnetism in 2D van der Waals materials opens huge oppor-tunities for investigating unexplored rich physics and futurespintronic devices in reduced dimensions [21–31]. Becausewe consider 2D model, we ignore the non-local dipolar inter-action, which is not a precondition for a finite Berry curvaturein 2D magnets. Moreover, the Berry curvature we find doesnot require a special spin asymmetry such as the DM interac-tion nor a special lattice symmetry: Our 2D model descriptionis applicable for general thin film ferromagnets. Therefore,we show in this work that even without such long-range dipo-lar interaction, DM interaction, or special lattice symmetry,the nontrivial topology of magnon-phonon hybrid can emergeby taking account of the well-known magnetoelastic interac-tion driven by Kittel [32]. As the Kittel’s magnetoelasticinteraction originates from the magnetic anisotropy, which isubiquitous in ferromagnetic thin film structures [33], our re-sult does not rely on specific preconditions but quite generic. a r X i v : . [ c ond - m a t . s t r- e l ] J u l Furthermore, we show that the topological structures of themagnon-polaron bands can be manipulated by effective mag-netic fields via topological phase transition. We uncover theorigin of the nontrivial topological bands by mapping ourmodel to the well-known two-band model for topological in-sulators [7], where the Chern numbers are read by counting thenumber of topological textures called skyrmions of a certainvector in momentum space. At the end of this Letter, we pro-pose the thermal Hall conductivity as an experimental probefor our theory.
Model—
Our model system is a 2D ferromagnet on a squarelattice described by the Hamiltonian H = H mag + H ph + H mp , (1)where the magnetic Hamiltonian is given by H mag = − J (cid:88) (cid:104) i,j (cid:105) S i · S j − K z (cid:88) i S i,z − B (cid:88) i S i,z , (2)where J > is the ferromagnetic Heisenberg exchange inter-action, K z > is the perpendicular easy-axis anisotropy, and B is the external magnetic field applied along the easy axis.Throughout the paper, we focus on the cases where a groundstate is the uniform spin state along the z axis: S i = ˆ z . Thephonon system accounting for the elastic degree of freedom ofthe lattice is described by the following Hamiltonian: H ph = (cid:88) i p i M + 12 (cid:88) i,j,α,β u αi Φ α,βi,j u βj , (3)where u i is the displacement vector of the i th ion from its equi-librium position, p i is the conjugate momentum vector, M isthe ion mass, and Φ α,βi,j is a force constant matrix. The mag-netoelastic coupling is modeled by the following Hamiltonianterm [32, 34]: H mp = κ (cid:88) i (cid:88) e i ( S i · e i ) (cid:0) u zi − u zi + e i (cid:1) , (4)where κ is the strength of the magnon-phonon interaction and e i ’s are the nearest neighbor vectors. Equation (4) describesthe magnetoelastic coupling as a leading order in the magnonamplitude, where the in-plane components of the displacementvector do not appear.We note here that our model Hamiltonian does not includethe dipolar interaction and the DM interaction, distinct fromthe model considered in Refs. [19] and [20]. Because theabove-mentioned interactions are absent in our model, neitherferromagnetic system nor elastic system exhibits the thermalHall effect when they are not coupled. In other words, they areinvariant under the combined action of time-reversal ( T ) andspin rotation by ◦ around an in-plane axis ( C ) [20]. It is themagnetoelastic coupling term H mp that breaks the combinedsymmetry T C and thus can give rise to the thermal Hall effectas will be shown below.
Magnon-phonon hybrid excitations—
We first diagonalizethe magnetic Hamiltonian H mag and the phonon Hamiltonian H ph separately, and then obtain the magnon-phonon hybridexcitations, which are called magnon-polarons, by taking ac-count of the coupling term H mp .The magnetic Hamiltonian is solved by performing theHolstein-Primakoff transformation S xi ≈ ( √ S/ a i + a † i ) , S yi ≈ ( √ S/ i )( a i − a † i ) , S zi = S − a † i a i , where a i and a † i are the annihilation and the creation operators ofa magnon at site i . By taking the Fourier transformation, a i = (cid:80) k e i k · R i a k / √ N , where N is the number of sites inthe system, we diagonalize the magnetic Hamiltonian in themomentum space: H mag = (cid:88) k (cid:126) ω m ( k ) a † k a k , (5)where the magnon dispersion is given by ω m ( k ) =[2 JS (2 − cos k x − cos k y ) + K z S + B ] / (cid:126) .For the elastic Hamiltonian H ph , it is also convenient todescribe in the momentum space: H ph = (cid:88) k (cid:20) p z − k p z k M + 12 u z − k Φ( k ) u z k (cid:21) , (6)where only nearest-neighbor elastic interactions are main-tained as dominant terms and the momentum-dependent springconstant is Φ( k ) = M ω (4 − k x − k y ) , where thecharacteristic vibration frequency ω corresponds to the elas-tic interaction between two nearest-neighbor ions. To obtainthe quantized excitations of the phonon system, we introducethe phonon annihilation operator b k and the creation operator b † k in such a way that u z k = (cid:115) (cid:126) M ω p ( k ) (cid:32) b k + b †− k √ (cid:33) , (7) p z k = (cid:113) (cid:126) M ω p ( k ) (cid:32) b − k − b † k √ i (cid:33) , (8)where the phonon dispersion is given by ω p ( k ) = ω (cid:112) − k x − k y . This leads to the following di-agonalized phonon Hamiltonian: H ph = (cid:88) k (cid:126) ω p ( k ) (cid:18) b † k b k + 12 (cid:19) . (9)In terms of the magnon and phonon operators introducedabove, the magnetoelastic coupling term is recast into the fol-lowing form in the momentum space: H mp = H mp1 + H mp2 ,where H mp1 = ˜ κ (cid:88) k (cid:104) a † k b k ( − i sin k x + sin k y ) (cid:105) + h.c. , (10) H mp2 = ˜ κ (cid:88) k (cid:104) a †− k b † k ( i sin k x − sin k y ) (cid:105) + h.c. , (11) Γ X M Γ E ( k ) ( m e V ) (b) Γ X M Γ E ( k ) ( m e V ) (a) (d) −π π k x −ππ k y (c) −π π k x −ππ k y Γ(Ω) d z FIG. 2: The band structure and its topology for | C | = 1 case. Theband structure for κ = 0 (a) and κ = 10 meV/Å (b). The red dashedline represents the band-crossing points. (c) Berry curvatures of theupper band in log-scale Γ(Ω z ) = sign(Ω z )log(1 + | Ω z | ) for κ = 10 meV/Å (d) Schematic illustration of ˆd ( k ) for κ = 10 meV/Å. Thein-plane components ( ˆ d x , ˆ d y ) are shown in red arrows. with ˜ κ = κ (cid:112) (cid:126) S/ ( M ω k ) . Note that H mp1 conserves the totalparticle number, whereas H mp2 does not. Because of H mp2 ,the total Hamiltonian takes the Bogoliubov-de-Gennes (BdG)form.The band structure of magnon-phonon hybrid system is ob-tained by solving the Heisenberg equations with the aboveresults [Eqs. (1)-(10)] (see the supplementary information forthe detailed calculation and the schematic illustration of theband structure [35]). Without magnon-phonon interaction,there are two positive branches consisting of a magnon bandand a phonon band. The two bands cross at k points satisfying ω m ( k ) = ω p ( k ) . Different from the conventional Dirac sys-tem, there are innumerable band-crossing points which forma closed line. These band-crossing lines are removed by themagnon-phonon interaction ∝ κ , which induces the nontrivialtopological property of the bands, characterized by the Berrycurvatures. In the BdG Hamiltonian, the Berry curvature isgiven by [18, 20, 36] Ω n ( k ) = ∇ × A n ( k ) , (12)where A n = i (cid:104) ψ n, k |J ∇ k | ψ n, k (cid:105) and ψ n, k are the n -th eigen-states (see supplementary information for details). The topo-logical property of the whole system is determined by theChern number of bands, which is the integral of the Berry cur-vature over the Brillouin zone [37]. In Fig. 1(b), we show theChern number of our bosonic system with nonzero magnon-phonon interaction κ . In our system, the Chern number can beone of three integers (0, 1, and 2) depending on the effectivemagnetic field H eff = K z S + B and exchange interaction J .This is one of our central results: The magnon-polaron bandsin a 2D simple square-lattice ferromagnet are topologically Γ X M Γ E ( k ) ( m e V ) (a) (d) Γ X M Γ E ( k ) ( m e V ) (b) −π π k x −ππ k y (c) −π π k x −ππ k y Γ(Ω) d z FIG. 3: The band structure and its topology for | C | = 2 case. Theband structure for κ = 0 (a) and κ = 10 meV/Å (b). The red dashedline represents the band-crossing points. (c) Berry curvatures of theupper band in log-scale Γ(Ω z ) = sign(Ω z )log(1 + | Ω z | ) for κ = 10 meV/Å. (d) Schematic illustration of ˆd ( k ) for κ = 10 meV/Å. Thein-plane components ( ˆ d x , ˆ d y ) are shown in red arrows. nontrivial even in the absence of dipolar or DM interactionand their topological property can be controlled by the effec-tive magnetic field. Origin of the topological property—
The origin of thenontrivial magnon-polaron bands obtained above can beunderstood through the mapping our system to the well-known model for two-dimensional topological insulators suchas HgTe [7, 37]. Considering H mp as a weak perturba-tion with unperturbed Hamiltonian with well-defined ener-gies of magnons and phonons, the effect of particle-number-nonconserving component H mp2 on the band structure is muchsmaller than that of particle-number-conserving part H mp1 .Neglecting H mp2 , the total Hamiltonian is simplified into asingle-particle two-band Hamiltonian H ≈ (cid:88) k (cid:0) a † k b † k (cid:1) H k (cid:18) a k b k (cid:19) , (13)where H k = (cid:18) (cid:126) ω m ( k ) ˜ κ (sin k y − i sin k x )˜ κ (sin k y + i sin k x ) (cid:126) ω p ( k ) (cid:19) . (14)In terms of the Pauli matrices σ = ( σ x , σ y , σ z ) , we writeEq. (14) in a more compact form H k = (cid:126) ω m ( k ) + ω p ( k )] I × + d ( k ) · σ , (15)where d ( k ) = (cid:18) ˜ κ sin k y , ˜ κ sin k x , (cid:126) ω m ( k ) − ω p ( k )) (cid:19) . (16)The band structure for the above Hamiltonian is given by E ± ( k ) = (cid:126) ω m ( k ) + ω p ( k )] ± | d ( k ) | . (17)In terms of d vectors, the Berry curvature is written explicitlyas Ω z ± ( k ) = ∓ ˆd ( k ) · (cid:32) ∂ ˆd ( k ) ∂k x × ∂ ˆd ( k ) ∂k y (cid:33) . (18)The corresponding expression for the Chern number is givenby [37–39] C ± = 12 π (cid:90) dk x dk y Ω z ± ( k ) , (19)which is the skyrmion number of the d vector [37], countinghow many times ˆd wraps the unit sphere in the Brillouin zone.From Eq. (14), we read that the magnon band and phononband cross at k points satisfying ω m ( k ) = ω p ( k ) without themagnon-phonon interaction. These band crossing points areopened by the magnon-phonon interaction ∝ κ and the finiteBerry curvatures are induced near the gap opening region.After integrating the Berry curvatures over the Brillouin zone,we obtain C ± = 0 , ± or ± . The two-band model has almostidentical band structures and Berry curvatures to those of fullHamiltonian, where H mp2 is additionally considered (see thesupplemental information).In Fig. 2 and Fig. 3, we show that the bulk band structuresand their topological properties for | C | = 1 and | C | = 2 ,respectively. For calculation, we use the parameters of themonolayer ferromagnet CrI in Ref. [20, 22, 25, 40] ( J = 2 . meV, K z = 1 . meV, S = 3 / , and M c = 5 × eV). The force constant between the nearest-neighbor phononis assumed as (cid:126) ω = 10 meV. The external magnetic field B = − . meV is chosen for Fig. 2 and B = 0 . meV ischosen for Fig. 3. In Fig. 2(a), we find a band-crossing line(red dashed line) which is removed by the magnon-phononinteraction [Fig. 2(b)]. In this case, a dominant contributionof the Berry curvature comes from vicinity of the Γ -point[Fig. 2(c)]. An intuitive way to verify the topological nature ofthe system is the number of skyrmions of the unit vector ˆd ( k ) .In Fig. 2(d), we find that there is a skyrmion at the Γ -pointcorresponding to | C | = 1 . By changing the sign of externalmagnetic field B , we can modify the band structure with twoband-crossing lines [Fig. 3(a)]. In this case, the dominantcontribution of the Berry curvature comes from vicinity of the Γ - and M -points [Fig. 3(c)]. In terms of ˆd ( k ) , we find thatone skyrmion is located at Γ -point and the other skyrmion isat M -point corresponding to | C | = 2 [Fig. 3(d)]. Thermal Hall effect—
The finite Berry curvatures ofmagnon-phonon hybrid excitations give rise to the intrinsicthermal Hall effect as shown below. The semiclassical equa-tions of motion for the wave packet of magnon-phonon hybridare given by [2, 41] ˙ r n = 1 (cid:126) ∂E n ( k ) ∂ k − ˙ k × Ω n ( k ) , (cid:126) ˙ k = −∇ U ( r ) , (20)
20 K30 K40 K10 K (a) κ ( W / K ) − xy H (meV) eff H (meV) eff (b) C h e r n nu m b e r FIG. 4: (a) Dependence of thermal Hall conductivity on the effectivemagnetic field H eff and with different temperatures T using the pa-rameters in the main text. (b) Dependence of Chern number on theeffective magnetic field H eff using the parameters in the main text. where U ( r ) is the potential acting on the wave packet which canbe regarded as a confining potential of the bosonic excitation.Near the edge of sample, the gradient of the confining potentialproduces the anomalous velocity, ∇ U ( r ) × Ω n ( k ) . In equilib-rium, the edge current circulates along the whole edge and netmagnon current is zero along any in-plane direction. However,if the temperature varies spatially, the circulating current doesnot cancel, which causes the thermal Hall effect [12].The Berry-curvature-induced thermal Hall conductivity isgiven by [12, 13] κ xy = − k B T (cid:126) V (cid:88) n, k c ( ρ n, k )Ω zn ( k ) , (21)where c ( ρ ) = (1 + ρ ) ln [(1 + ρ ) /ρ ] − ln ρ − ( − ρ ) , ρ n, k = [ e ( E n ( k )) /k B T − ] − is the Bose-Einstein distributionfunction with a zero chemical potential, k B is the Boltzmannconstant, T is the temperature, and Li ( z ) is the polylogarithmfunction. In Fig. 4(a), we show the dependence of thermal Hallconductivity on the effective magnetic field H eff at differenttemperatures T . For small H eff , the thermal Hall conductivityincreases with increasing H eff . However, for large H eff , itdecreases with increasing H eff . This behavior of the thermalHall conductivity can be understood through the Chern numberof magnon-polaron bands depicted in Fig. 4(b). The absolutevalue of the Chern number is 1 for small H eff , then it jumpsup to 2 for a certain value of H eff and vanishes for large H eff . Discussions—
In this paper, we investigate the topology ofthe magnon-polaron bands in a simple 2D ferromagnet with-out long-range dipolar interaction and DM interaction. In ourmodel, the topological structure can be controlled by the effec-tive magnetic field which changes the number of band-crossinglines. Using a perturbation approach, we develop a two-bandmodel Hamilitonian which provides an intuitive understand-ing of the topological structure of the model. In the two-bandmodel, the nontrivial topology of the magnon-polaron bandsare reflected in the skyrmion number of d ( k ) in momentumspace. As an experimental demonstration, we propose that thethermal Hall conductivity arises from the non-trivial topologyof the magnon-polaron bands. The thermal Hall conductivitydepends on the effective magnetic field which can be manip-ulated by the external magnetic field or voltage-induced mag-netic anisotropy change [42]. 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