Fig. 1: Concept of nonlinear vortex Nernst effect.

a A schematic illustration of a nonlinear vortex Nernst effect. E, ΔT, and B denote an electric field, a temperature difference, and an external magnetic field, respectively. The bottom surface of a superconductor is in contact with a magnetic insulator. E due to the vortex Nernst effect is nonlinear with respect to ∆T applied along the y direction. The direction of E is perpendicular to the temperature gradient and the magnetic field. b The cross-sectional area (the x-y plane) of the superconductor/magnetic-insulator bilayer system. Vortex strings are depicted by gray dots. c The B dependence of the resistivity ρ of a type-II superconductor MoGe film on a ferrimagnetic insulator Y₃Fe₅O₁₂ (YIG) at T = 4 K. The green shaded area represents the magnetic field region of the vortex liquid phase, where vortex strings can move freely. The gray (red) shaded area shows the field region of the vortex solid (normal conductor) phase. d The B-T phase diagram of the MoGe. The green, gray, and red shaded areas represent the vortex liquid, vortex solid, and normal conductor phases, respectively. \({B}_{{{{{\rm{c}}}}}2}\) (black dots) and B_m (black triangles) are the upper critical field and the vortex-solid melting field, respectively, determined by using the conditions⁶ \(\rho ({B}_{{{{{\rm{c}}}}}2})=0.95{\rho }_{{{{{\rm{N}}}}}}\) and \(\rho ({B}_{{{{{\rm{m}}}}}})={10}^{-3}{\rho }_{{{{{\rm{N}}}}}}\), where \({\rho }_{{{{{\rm{N}}}}}}\) is the resistivity of the MoGe in the normal conducting state at B = 9 T.