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Towards Magnon Valleytronics

Valleytronics is a pioneering technological field relying on the fermionic valley degree of freedom to achieve novel electronic functionality. Topological valley-projected electronic states confined to domain walls in bilayer graphene were extensively studied in view of their potentials in valleytronics. Here, we study the magnonic version of domain wall states in ferromagnetic bilayers characterized by Dzyaloshinskii-Moriya interaction and uniform layer-dependent electrostatic doping. In particular, we explore topological confinement of magnon modes on layer stacking domain walls separating two semi-infinite ferromagnetic regions with AB and BA stacking. We predict that layer-dependent electrostatic doping induces a pair of valley-projected chiral boundary modes that admit topological interpretation. We conclude that magnon valley index can constitute an additional degree of freedom for 2D magnonic applications.

The magnon spectrum in 2D honeycomb ferromagnets mimic the electronic band structure of graphene [11], with Dirac cones at the corners of the hexagonal Brillouin zone (BZ). Similar to graphene, the Dirac spectrum of 2D magnons in honeycomb ferromagnets is characterized by 2 non-equivalent valleys and ′ = − . The energy degeneracy at valleys is protected by inversion and time reversal symmetries. Next-nearest-neighbor Dzyaloshinskii-Moriya interaction (DMI), present in several 2D ferromagnets, breaks the inversion symmetry and opens topological gaps at ± . In bilayer ferromagnets, the valley gap can also be induced by layerdependent electrostatic doping [23]. A recent study [24] predicts that inversion symmetry breaking by electrostatic doping can generate topological magnon valley currents in quasi-infinite ferromagnetic bilayers (FBL). The valley degree of freedom in electronic systems is proved very promising for quantum technologies, leading to the novel technological field of valleytronics [25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In particular, topological confinement of electronic states on domain walls of bilayer graphene (BLG) has been proposed to realize novel nano-electronic devices, like valley filters and valves. Applying a perpendicular electric field in BLG breaks inversion symmetry and consequently opens a valley band gap [39][40][41][42][43] that admits a topological interpretation [26,44]. Whilst the integral of the Berry curvature over the full BZ is zero, the Berry integral within a single valley is nonzero resulting in a valley-projected topological index [26,27,[46][47][48][49]. Novel domain walls can then be formed in BLG by a sign reversal of the perpendicular electric field. Analysis of the sub-gap electronic states near ± valleys demonstrates the existence of two topologically protected chiral modes confined to the electric-field domain wall (EFW). Domain walls can also be formed in BLG by interlayer stacking reversal, where the layer stacking switches from AB to BA configurations. In the presence of a uniform interlayer electric field, layer stacking domain walls (LSW) also support topological chiral modes, analogous to their EFW counterparts. The technological interest in the valley degree of freedom was extended to 2D photonic [50] and phononic [51] crystals. Magnon valleytronics, however, did not yet receive the deserved attention.
In this work, we study magnon topological confinement in LSW embedded in a 2D ferromagnetic bilayer (FBL). The theoretical study is based on a four-band continuum model, developed near ± valleys, including weak DMI and uniform layer-dependent electrostatic doping (ED). The four-band continuum Hamiltonian of FBLs does not have a fermionic counterpart, and is hence a candidate for new physics not present in 2D electron systems. For zero DMI and uniform ED, we find that the LSW confines topologically protected chiral 1D magnon modes analogous to their fermionic counterpart. In addition, we observe flat energy magnon modes localized on the LSW that are absent in fermionic systems. Introducing weak DMI results in a topologically compensated LSW, and the intragap magnon spectrum becomes completely gapped. These prediction of 1D magnon modes on LSW shall constitute an important step towards magnon valleytronics.
and are the momenta along and directions. As stated previously, due to the presence of 0 and on the diagonal, ℋ ± does not have an electronic analogue.
For an infinite BA-stacked FBL, the equivalent Hamiltonian reads We next consider a 2D bilayer ferromagnet with an LSW (at = 0) separating two semi-infinite regions, with AB and BA stacking in the regions > 0 and < 0 respectively. The corresponding four-components magnon wavefunctions in the regions ≷ 0 are denoted ± respectively, with ± = { ± , = 1, … ,4}. The translational symmetry in the system is broken (preserved) along the − direction ( − direction). The behavior of ± is hence dictated by the functional form − , where = + denotes the generalized phase factor in the − direction [15,18,20,21,26,27]. For intragap modes confined to the LSW, is real and the wavefunction decays exponentially away from the LSW.
Reinstating the momenta as differential operators in ℋ ± (equivalently ℋ ± ), the wave equation  This development allows us to determine the magnon propagating and confined modes in the FBL system, composed of local AB and BA stacking regions separated by a domain wall. Figure 1 presents numerical results near the valley, for selected electrostatic doping potential and negligible DMI. The shaded regions represent the continuous bulk magnon spectrum which differs significantly from the electronic structure of BLG. The red dispersion curves correspond to the magnon modes confined to the LSW. The bulk spectrum profile allows the presence of a boundary flat energy mode, absent in BLG with embedded LSW or EFW. More important, the numerical results in figure 1 demonstrate the existence of pairs of valley intragap chiral magnon modes, confined to the LSW, and propagating in opposite velocities at the ± valleys. We can readily draw a topological interpretation for these modes, in analogy with their fermionic counterparts [27]. Magnon valley Berry curvatures, Hall conductivity and Chern numbers are indeed well defined in FBLs with layer-dependent electrostatic doping [24], and the presence of the magnon chiral modes is associated with the change in the valley Chern number sign across the LSW [27]. In figure 2, we investigate the effect of a weak DMI on the LSW confined modes. In the presence of DMI, the LSW is topologically compensated and the LSW spectrum near ± valleys is completely gapped. This is analogous to the compensated domain wall in BLG separating two regions with reversed interlayer electric field and layer stacking [27]. Discussion-This work constitutes an important step towards magnon valleytronics utilizing magnon modes confined to LSW in FBL. We demonstrate that LSW in FBL with negligible DMI and uniform ED binds a pair of gapless valley-projected chiral magnon modes. Similar to their fermionic counterparts, the predicted chiral magnon modes admit a topological interpretation in term of the variation of the magnon valley index across the LSW. The magnon valley index hence serve as a good quantum number offering an additional degree of freedom promising for magnonic applications. Topologically confined 1D magnon modes on domain walls might have significant advantages over edge modes in nanoribbons, as their realization might be less challenging. Since the chiral valley modes propagate in opposite directions at the ± valleys, LSW can serve to realize magnon valley filters and valves [26], indispensable for magnonic circuitry. Given their topological protection, damping of these modes should be negligible. With the rapid advancement in the experimental techniques to fabricate and characterize 2D magnetic materials, experimental testing of our predictions shall be feasible.