Nonlinear Hall Effect in Metal-Organic Frameworks

Abstract

We propose metal-organic frameworks (MOFs) as a versatile platform for realizing the nonlinear Hall effect. We develop an analytical down-folding scheme that maps a broad class of MOFs onto a universal effective low-energy model. As representative examples, we analyze two $C_3$-symmetric frameworks: Cu-dicyanoanthracene and triphenyl-metal monolayer, demonstrating how their low-energy bands can be efficiently captured by a star-lattice geometry. First-principles calculations corroborate this mapping and show that both Fermi levels lie close to symmetry-protected Dirac points. Spin-orbit coupling or inversion-symmetry breaking gaps these Dirac cones, generating Berry-curvature hotspots near the Fermi level. Supported with symmetry-based indicators, these MOFs thus suggest themselves for strain and substrate engineering as well as doping to achieve a finite nonlinear Hall response. We formulate a synthesis-oriented strategy that implements the Dirac gap directly within the framework architecture without externally applied strain. Our results establish MOFs as a broadly designable platform for engineering Berry-curvature physics and nonlinear Hall transport.

Figures (7)

• Figure 1: From the Cu--DCA framework to the effective star lattice. (a) Relaxed geometry of the Cu--DCA metal--organic framework obtained from first-principles calculations. Inset: because Cu orbitals contribute negligibly near the Fermi level, the three N atoms surrounding a Cu site can be represented as an effective triangle after the first decimation step. (b) Orbital-projected band structure of Cu--DCA showing that the states near the Fermi level are predominantly derived from the $p_z$ orbitals of C and N atoms. (c) Second decimation step for the linker segment connecting neighboring triangles. Sites removed in the downfolding are marked in red, and the renormalized hoppings between the terminal sites $L$ and $R$ are generated recursively. (d) Effective star lattice with a six-site unit cell spanned by lattice vectors $\vec{a}_1$ and $\vec{a}_2$. Blue and red triangles carry intra-triangle hoppings $t_{\triangle}$ and $t_{\bigtriangledown}$, respectively; inter-triangle bonds are described by $t$ (and by $t_\perp$ when uniaxial anisotropy is introduced). Green arrows denote the intrinsic Kane--Mele-type SOC $\pm i\lambda_{\rm SO}$.
• Figure 2: Berry curvature and its dipolar component in the star lattice. (a) Berry-curvature-projected band structure for the onsite-modulated model with $\delta_1=-0.08\,t$, $\delta_2=0.08\,t$, and $t_{\triangle}=t_{\bigtriangledown}=0.5\,t$, shown for two SOC strengths. Increasing $\lambda_{\rm SO}$ drives a band inversion near $K$ and reverses the Berry-curvature hot spots at the avoided crossings. (b) Corresponding Berry-curvature dipole $D^{xz}$ as a function of Fermi level for representative SOC values in the presence of a small hopping anisotropy $t_\perp\neq t$ (here taken to model a $2\%$ tensile strain). The strong enhancement and sign reversal near the critical point reflect the topological transition.
• Figure 3: Dipolar component of Berry curvature in metal--organic frameworks. (a) DFT band structure of Cu--DCA showing two energetically separated kagome-like band manifolds, with the Fermi level located close to the Dirac point of the upper kagome-like band. Inset: replacing one Cu atom by Zn provides one route to inversion-symmetry breaking. (b) Phase diagram for the dipolar component obtained from DFT-consistent hopping parameters ($t_{\triangle}=t_{\bigtriangledown}=-0.11$ eV, $t=0.81$ eV, and $t_\perp=-0.78$ eV) in the presence of SOC, inversion-symmetry breaking, and $2\%$ uniaxial strain. The dipolar component switches sign across the topological transition. (c) Proposed synthesis-based route for generating the required symmetry lowering intrinsically, by modifying one of the three equivalent linker directions. (d) Generalization to a triphenyl--metal MOF (Pb shown here), which can likewise be downfolded to a star lattice. Because the two Pb sites lie on different planes, a transverse electric field can break inversion symmetry. (e) DFT band structure of the pristine triphenyl--Pb framework, showing the characteristic star-lattice-like band structure.
• Figure S1.1: Representative band structures of the pristine star lattice. Band structures along $\Gamma\rightarrow M\rightarrow K\rightarrow\Gamma$ for $t_{\triangle}=t_{\bigtriangledown}=0.5\,t$, $t_{\triangle}=t_{\bigtriangledown}=\frac{2}{3}t$, and $t_{\triangle}=t_{\bigtriangledown}=t$. Throughout the calculations, we set $t=1$ eV.
• Figure S2.1: Wannier charge centers and topological phase transition. Upper panels (a,b): WCC evolution for the onsite-modulated model with $\delta_1=-0.08\,t$, $\delta_2=0.08\,t$, and $t_{\triangle}=t_{\bigtriangledown}=0.5\,t$. Lower panels (c,d): WCC evolution for the breathing star lattice with $t_{\triangle}=0.45\,t$ and $t_{\bigtriangledown}=0.5\,t$. In both cases, increasing $\lambda_{\rm SO}$ drives a $\mathds{Z}_2$ transition. An odd (even) number of crossings of a reference line within half of the Brillouin zone identifies the nontrivial (trivial) phase.