Here, Pi is the polarization along direction i, 𝜀 𝑘𝑙 is the strain tensor, 𝑥 𝑗 is a coordinate along the direction 𝑗, and 𝜇 𝑖𝑗𝑘𝑙 is the flexoelectric tensor. Given the flat, flexible nature of the Mn3O4 in FNGs, certain types of strain gradients are expected to be more relevant. Furthermore, we are former is an inhomogeneous shear strain along an in-plane direction while the latter is an in-plane uniaxial strain that varies along the out-of-plane direction. These inhomogeneous strains are chosen because they are common in a distorted sheet, such as the FNG cell during operation. While there are density functional perturbation methods for calculating flexoelectric coefficients, we elected for a simpler strategy involving calculations on a supercell with a manual, periodic strain imposed (see details in SI section 2, Theoretical calculations). We refer to the periodic strain modes corresponding to the calculation of 𝜇 3𝑘𝑘3 and 𝜇 33𝑘𝑘 as the shear and bending modes, respectively. Figures and show abstractions of the shear and bending modes, respectively (also Figure ). These abstractions are expressed more concretely in Figure shows the four flexoelectric coefficients calculated for three geometries of Mn3O4: monolayer, bilayer, and bulk. The magnitudes of several of these coefficients exceed 100 pC/m; this value is significantly larger than common 2D semiconducting materials, such as transition metal dichalcogenides, which only reach up to 10 pC/m in MoS2. Furthermore, we find an additional contribution to the flexoelectric coefficient of about 100 pC/m from finite surface effects for the bending modes. Details can be found in SI Sections 2 and 7. In monolayer and bilayer Mn3O4, the shear modes contribute negligibly to the flexoelectric response due to cancelations of positive and negative responses caused by anomalous positive Born effective charges in undercoordinated surface O atoms (see Figure ). For the bending mode in the monolayer structure, the anomalous Born effective charges do not lead to a cancellation and instead cause a sign reversal compared to the bulk structure. In contrast, both modes contribute significantly to flexoelectricity in bulk Mn3O4.