Geometrical Hall effect and momentum-space Berry curvature from spin-reversed band pairs

When nanometric, noncoplanar spin textures with scalar spin chirality (SSC) are coupled to itinerant electrons, they endow the quasiparticle wave functions with a gauge field, termed Berry curvature, in a way that bears analogy to relativistic spin-orbit coupling (SOC). The resulting deflection of moving charge carriers is termed the geometrical (or topological) Hall effect. Previous experimental studies modeled this signal as a real-space motion of wave packets under the influence of a quantum-mechanical phase. In contrast, we here compare the modification of Bloch waves themselves and of their energy dispersion due to SOC and SSC. Using the canted pyrochlore ferromagnet ${\mathrm{Nd}}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ as a model compound, our transport experiments and first-principles calculations show that SOC impartially mixes electronic bands with equal or opposite spin, while SSC is much more effective for opposite-spin band pairs.

Figure 1

Smooth transition of the molybdenum sublattice in (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x{)}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ (green arrows) from (a) and (b) canted ferromagnet (C-FM) at low field to (c) field-aligned ferromagnet (FA-FM). (d) Magnetization $M$ as a function of increasing temperature $T$ at $x=0.00$ after preparation in a field-cooled (FC) or high-field-cooled (HFC) state. Ferromagnetic ordering on the molybdenum sublattice at $T_C$ is marked by a dashed line. (e) and (f) High-field magnetization and Hall conductivity measurements. The anisotropy of transport and magnetic properties is exemplified using two directions of the magnetic field $\mathbf{H}$. In (e) and (f), curves for increasing and decreasing magnetic field are marked by solid and dashed lines, while white and orange shaded regions in (e) and (f) indicate the C-FM and FA-FM regimes, respectively.

Figure 2

Separation of Hall conductivities in ${\mathrm{Nd}}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ at $T=2\mathrm{K}$ and $\mathbf{H}//〈001〉$. We show the anomalous Hall conductivity ${\sigma }_{xy}^{\mathrm{A}}$ (violet curves) after subtraction of the field-linear normal Hall signal and the geometrical Hall signal ${\sigma }_{xy}^{\mathrm{G}}$ generated by SSC (green curves). The Hall signal originating from spin-orbit coupling ${\sigma }_{xy}^{\mathrm{KL}}$ is marked by a violet shaded box in each plot.

Figure 3

(a) Low-field Hall conductivity ${\sigma }_{xy}^{0.5\mathrm{T}}$ in (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x{)}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$, a proxy for the anomalous Hall signal ${\sigma }_{xy}^{\mathrm{A}}={\sigma }_{xy}^{\mathrm{G}}+{\sigma }_{xy}^{\mathrm{KL}}$. (b)–(e) Measured and calculated SOC-driven Hall conductivity ${\sigma }_{xy}^{\mathrm{KL}}$ as well as SSC-driven geometrical Hall conductivity ${\sigma }_{xy}^{\mathrm{G}}$. In (d) and (e), a second $y$ axis indicates the result of the calculation (${\sigma }_{xy}^{\mathrm{KL},\mathrm{int}.}$ and ${\sigma }_{xy}^{\mathrm{G},\mathrm{int}}$, see text) before applying the universal correction for finite defect concentration. Moreover, contributions to ${\sigma }_{xy}^{\mathrm{KL}}$ and ${\sigma }_{xy}^{\mathrm{G}}$ from equal-spin and opposite-spin Berry curvatures (${\mathrm{\Omega }}_{\mathbf{k}}^{\mathrm{same}}$ and ${\mathrm{\Omega }}_{\mathbf{k}}^{\mathrm{diff}.}$) are displayed separately. Thick magenta and green lines in (a) and (b) are guides to the eye. Error bars in (a)–(c) are calculated from the estimated sample geometry error ($\sim \pm 17\%$). In (c), the thick purple line indicates the ${(1-x)}^3$ power law expected from dilution of rare-earth magnetic moments (see text).

Figure 4

(a)–(d) Contour maps of Berry curvature $-{\mathrm{\Omega }}_{\mathbf{k}}^z$ in the $k_z=0$ plane, where same-spin and opposite-spin contributions, as defined in the text, are shown separately in two columns. The top row is for the collinear ferromagnetic state with spin-orbit coupling (SOC); the bottom row is for the canted ferromagnetic (C-FM) state without SOC. Black curves indicate Fermi surface cross sections. For comparison, cross-sectional lines of the minority (FM-$\downarrow$) and majority (FM-$\uparrow$) Fermi surfaces in the collinear state, without SOC, are also shown. The direction of the $\mathrm{\Gamma }\text{−}X$ line in reciprocal space is indicated by a dashed line. (e) and (f) Along $\mathrm{\Gamma }\text{−}X$, anticrossing points of electronic bands are present both in the ferromagnetic state with SOC and in the C-FM state without SOC. In magenta and green, ferromagnetic bands (no SOC) are plotted for comparison. Some band-anticrossing points can be directly associated with features in the $-{\mathrm{\Omega }}_{\mathbf{k}}^z$ maps (highlighted by shaded circles). Dashed lines mark the position of the Fermi energy ${\varepsilon }_F$.