Fig. 1: Diverse polar SkLs in PTO thin films.

a Three-dimensional OP polarization distribution of T-SkL in the film, simulated at the condition of E_[001] = –1.25 MV/cm and f₁₂ = 5 V, f₁₁ = f₄₄ = 0 V. The configurations of individual polar skyrmion bubble at the top, middle, and bottom (001) planes of the film are schematically shown on the right. b Polarization distributions of T-SkL formed near the top plane of the film at the condition of E_[001] = –1.25 MV/cm and f₁₁ = 3 V, f₁₂ = f₄₄ = 0 V. c–g Schematics of different polar SkL patterns predicted in PTO thin films. All the polarization vectors are normalized, and the color represents Pz, sharing the color bar in b. f_ij are the flexocoupling coefficients and can be found in Methods, and E_[001] is the out-of-plane electric field.