Topological phase transitions and Berry-phase hysteresis in exchange-coupled nanomagnets

Topological phase in magnetic materials yields a quantized contribution to the Hall effect known as the topological Hall effect, which is often caused by skyrmions, with each skyrmion creating a magnetic flux quantum $\pm h/e$. The control and understanding of topological properties in nanostructured materials is the subject of immense interest for both fundamental science and technological applications, especially in spintronics. In this work, the electron-transport properties and spin structure of exchange-coupled cobalt nanoparticles with an average particle size of 13.7 nm are studied experimentally and theoretically. Magnetic and Hall-effect measurements identify topological phase transitions in the exchange-coupled cobalt nanoparticles and were used to discover a qualitatively new type of hysteresis in the topological Hall effect—namely, Berry-phase hysteresis. Micromagnetic simulations reveal the origin of the topological Hall effect—namely, the chiral domains, with domain-wall chirality quantified by an integer skyrmion number. These spin structures are different from the skyrmions formed due to Dzyaloshinskii–Moriya interactions in B20 crystals and multilayered thin films, and caused by cooperative magnetization reversal in the exchange-coupled cobalt nanoparticles. An analytical model is developed to explain the underlying physics of Berry-phase hysteresis, which is strikingly different from the iconic magnetic hysteresis and constitutes one aspect of 21st-century reshaping of our view on nature at the borderline of physics, chemistry, mathematics, and materials science.

Figure 1

Phase transitions: (a) Curie transition (magnetic Landau transition), (b) magnetic hysteresis, (c) Lifshitz transition in metal, and (d–i) topological phase transitions in a magnetic thin film with perpendicular anisotropy. In (c), the gray areas denote the $k$-space region occupied by electrons at the Fermi level. In (d–i), red and blue regions indicate positive (↑) and negative (↓) magnetizations with respect to the film plane. Topological phase transitions are characterized by topological numbers $Q$. The topological protection in the micromagnetic case is experimentally established—for example, through the “blowing” of skyrmions [14, 15, 16].

Figure 2

Magnetic hysteresis, Berry-phase hysteresis, and spin structure: (a) experiment, (b) simulation, and (c) simulated spin structure in a field of $-0.7\mathrm{T}$. In (b), $Q$ is the number of skyrmions per unit area ($240\mathrm{nm}\times 240\mathrm{nm}\times 60\mathrm{nm}$), and $m$ is the normalized magnetization, $M_z/M_s$. The origin of the topological Hall effect due to spin texture—namely, the noncoplanar spin structure—is visible in (c).

Figure 3

Real structure and temperature effects: (a) experimental magnetization and susceptibility, (b) simulated magnetization and susceptibility, (c) the real-structure origin of the Berry-phase hysteresis, and (d) Berry-phase hysteresis at 10 K and 300 K. The susceptibility peaks in (a) and (b) reflect Barkhausen jumps and are strongly real-structure dependent.

Figure 4

MFM image and magnetization reversal: MFM image of magnetization reversal in a region with an even surface (blue region). The magnetization reversal starts randomly making domains at a field of 0.06 T, and these domains expand as the field is increased, and finally, coalescence of the domain occurs at a high field. Note that not only do we have topological Hall effect contributions due to magnetic domains, but also we have topological Hall effect contributions due to chiral spin inhomogeneity, and imaging of these chiral spins texture is difficult [47, 60].

Figure 5

Analytical modeling of Berry-phase hysteresis: (a) most general cubic plot of skyrmion number $Q$ as a function of magnetization, (b) experimental $Q-M$ plot, (c) and (d) topological phase transition in a simple circular-domain model, (e) and (f) topologically equivalent version of the same model, and (g) Berry-phase hysteresis loop for the model of (c) and (d). The transitions from (c) to (d) and from (e) to (f) are triggered by a magnetic field increase and are accompanied by an incremental magnetization increase only. In (g), the transition occurs at a fairly high value of $M_z=\pm 0.814M_s$. At $M_z=\pm M_s, Q$ jumps to zero, because the residual domains are annihilated at saturation. The light-blue and light-red areas in (g) are a duality effect caused by the triangular skyrmion lattice assumed in (a) and (b). As in other parts of this work, $m$ is the normalized magnetization ($m=M_z/M_s$).