Quantized topological Hall effect in skyrmion crystal

Abstract

We theoretically study the quantized anomalous Hall effect (QAHE) in skyrmion crystal (SkX) without external magnetic field. The emergent magnetic field in SkX could be gigantic as much as ∼ 4000T when its lattice constant is ∼ 1nm. The band structure is not flat but has a finite gap in the low electron-density regime. We also study the conditions to realize the QAHE for the skyrmion size, carrier density, disorder strength and temperature. Comparing the SkX and the system under the corresponding uniform magnetic field, the former is more fragile against the temperature compared with the latter since the gap is reduced by a factor of ∼ 1/5, while they are almost equally robust against the disorder. Therefore, it is expected that the QAHE of SkX system is realized even with strong disorder at room temperature when the electron density of the order of one per a skyrmion.Magnetic skyrmion is a topological spin texture in ferromagnets 1 . After the early theoretical proposals in magnets 2-4 , the study of magnetic skyrmion is growing rapidly since it was discovered experimentally [5][6][7] . The periodic array of skyrmions, i.e., a skyrmion crystal (SkX), is realized at interfaces 7 or in bulk chiral magnets such as B20 compounds 5,6 . An emergent magnetic field, generated in the background skyrmion spin texture. Namely, a skyrmion has one magnetic flux Φ 0 = h/e acting on the conduction electrons coupled to it. When the skyrmions form a periodic lattice, i.e., a SkX, the emergent magnetic field reaches ∼ 4000T assuming the uniform averaged flux for the skyrmion size and the lattice constant of SkX of the order of ∼ 1nm. The effective magnetic field is proportional to λ −2 , where λ is the skyrmion radius. Since the size of the skyrmion is 3nm for MnGe 11 , 18nm for MnSi 9 , and 70nm for FeGe 13 , the corresponding emergent magnetic field is ∼ 1100T, 28T, and 1T, respectively.This emergent magnetic field leads to the Hall effect 8 . Most of the studies focus on the Hall effect in metallic systems with large electron density 9-12 . This so-called topological Hall conductivity σ xy is usually small compared with the longitudinal conductivity σ xx , i.e., the Hall angle σ xy /σ xx is typically of the order of 10 −2 at most 11 . Up to now, we regard skyrmions as the source of the real space emergent magnetic field. When the size of the skyrmion becomes comparable to the mean free path, it is expected that the crossover from the real to momentum space Berry curvature occurs. One can regard the case of pyrochlore ferromagnet as the limit of large emergent magnetic field, where the spin chirality is defined for each unit cell of tetrahedron 14 . In this case, there is no Landau Level (LL) formation, but the band structure is formed by taking into account the solid angle of the spin, and the intrinsic anomalous Hall effect appears whose conductance is given by the integral of the Berry curvature in momentum space 15 . A more drastic example is the quantized anomalous Hall effect (QAHE) in magnetic topological insulator (TI)...

Quantum anomalous Hall effect in graphene coupled to skyrmions

“…During the final stages of the completion of this manuscript, we became aware of a work predicting a quantum anomalous phase for electrons on a square lattice coupled to skyrmions [26] in the strong-coupling limit (J t). …”

“…The length of the magnetization field M 0 is assumed to be the same for all sites. Unlike most of the previous work [26], we focus on the weak-coupling limit J t, adequate for proximity induced magnetism. Therefore, the exchange field acts as a perturbation on the Dirac spectrum.…”

Topological Hall and spin Hall effects in disordered skyrmionic textures

“…These sizes correspond to emergent magnetic fields ranging from 1 to 4000 T. The topological properties of skyrmions ensure that the total flux d 2 rB z (r) over a sample equals N 0 with N the quantized topological winding number or Chern number and 0 = h/e the quantum of flux. Recently, the discretized topological Hall effect has been observed [34] in constricted geometry and the emergence of quantum AHE in a skyrmion crystal has been theoretically explored [35]. An intriguing topological spin Hall effect (TSHE) has been obtained numerically in a single skyrmion [36] and vortex [37].…”

Insights into the orbital magnetism of noncollinear magnetic systems

“…16,17 The link between the magnetic structure and orbital electronic properties was explored for other kinds of magnetic systems before, [18][19][20][21][22][23] with renewed interest since the experimental discovery of skyrmions. [24][25][26][27][28][29][30] Inspired by the investigations of nanosized skyrmions in the PdFe/Ir(111) system, 31,32 we uncovered another manifestation of their topological nature: a new kind of orbital magnetism. In Ref.…”