Giant Anomalous Hall Effect due to Double-Degenerate Quasiflat Bands

We propose a novel approach to achieve a giant anomalous Hall effect (AHE) in materials with flat bands (FBs). FBs are accompanied by small electronic bandwidths, which consequently increases the momentum separation ($\mathcal{K}$) within pair of Weyl points and, thus, the integrated Berry curvature. Starting from a simple model with a single pair of Weyl nodes, we demonstrated the increase of $\mathcal{K}$ and the AHE by decreasing the bandwidth. It is further expanded to a realistic pyrochlore lattice model with characteristic double-degenerated FBs, where we discovered a giant AHE while maximizing the $\mathcal{K}$ with nearly vanishing band dispersion of FBs. We identify that such a model system can be realized and modulated through strain engineering in both pyrochlore and spinel compounds based on first-principles calculations, validating our theoretical model and providing a feasible platform for experimental exploration.

Figure 1

Band structure and anomalous Hall conductivity of generic WSMs. (a) Typical band structure of a magnetic WSM derived from Eq. (1). We set an exchange field along the $x$ direction and the following parameters: $\beta =2\mathrm{eV}$, $t_x=t_y=t_z=1\mathrm{eV}$, and $a=1\AA$. (b) Weyl node separation as a function of $t_x$ for several $\beta$ values. (c) Berry curvature (${\mathrm{\Omega }}^x$) distribution at the $k_x-k_y$ plane with the inset showing a linear scan along the dashed line. (d) Calculated AHC based on Eq. (3) (symbols) and Weyl node separation (${\sigma }_{xy}=(e^2\mathcal{K}/4{\pi }^2\hslash )$, solid curves) as a function of the exchange splitting strength $\beta$ for different $t_x$ values. The horizontal line highlights the maximum value $-2\pi$ for which ${\sigma }_{yz}=e^2/ha$.

Figure 2

Band structure of the pyrochlore lattice. (a) Crystal structure of the pyrochlore lattice with four atoms ($A–D$) in one unit cell. $t$ and $t^{\prime }$ indicate the NN and NNN hoppings, respectively. (b) High symmetry $k$ path in the first Brillouin zone. (c) Band structure of ideal pyrochlore lattice without considering NNN interaction and SOC effect. (d) Band structure with NNN interaction $t^{\prime }=0.2t$. The triple degenerate point (TP) and nodal lines (NL) are highlighted by red ellipses. (e) Band structure considering SOC ($\lambda =-0.2t$) and broken TRS (${\lambda }_z=5t$) with different Weyl points (WPs) highlighted.

Figure 3

Evolution of AHC with the change of NNN hopping strength $t^{\prime }$. (a) The left and right panels show band structure and AHC, respectively, of the pyrochlore model with zero $t^{\prime }$. (b) The same as (a) for evolution of the band structure and AHC with the change of NNN hopping strength $t^{\prime }$, respectively. AHC decreases with the increase of the bandwidth due to the increase with the $t^{\prime }$, as indicated by the colored arrow. (c) Evolution of $\mathcal{K}$ for $\mathrm{WP}({\mathrm{\Gamma }}_1)$ and $\mathrm{WP}(X)$ and their corresponding AHC, $P^{C,A}$ with different $t^{\prime }$. (d) Energy evolution of $\mathrm{WP}({\mathrm{\Gamma }}_1,X,{\mathrm{\Gamma }}_2)$ and $L$ point in comparison with evolution of peaks $P^{A,B,C}$ and $D^1$ with different $t^{\prime }$.

Figure 4

Giant AHE in pyrochlore and spinel compounds. (a) Crystal structure of the pyrochlore compounds with the $A$ cations forming the 3D kagome lattice. (b) Band structure of hole-doped ${\mathrm{Sn}}_2{\mathrm{Nb}}_2{\mathrm{O}}_7$. (c) DFT-calculated AHC. (d)–(f) The same as (a)–(c) for the spinel compounds ${\mathrm{MgV}}_2{\mathrm{O}}_4$, where cations $B$ form the 3D kagome lattice.