Fig. 2: Strain-gradients and lateral piezoresponses at the domain walls.

a Schematic of the shear strain-gradients at the macro-domain walls and their piezoresponses. A Cartesian coordinate system (\(x^{\prime} y^{\prime} z^{\prime}\)) is defined so that the \(y^{\prime}\)-axis is parallel to the walls. b Schematic of the tip orientations for the PFM measurements with the surface image. c Real part signal images of the in-plane PFM for four different cantilever orientations. The color scale represents the direction and magnitude of the in-plane PFM signal as indicated by the central double arrow per panel. Horizontal scale bars in b, c represent 200 nm. d Line profiles of the real part signals along the lines (starting at the circles) in c. While a meaningful signal of the piezoresponse is not observed in the interior domains, indicated by the pale green and cyan rectangles, due to the centrosymmetric behavior of WO₃, the ferroelastic domain walls between the A and B domains reveal largely enhanced piezoresponses. Black, blue, magenta, and green colors represent the cantilever orientation angles. 1 mV is estimated to ~30 pm. e Angle-resolved lateral PFM maps for the white dashed boxes in c. Black arrows represent the in-plane piezoresponse vectors, which are comprehensively obtained by sinusoidal fittings of the real part signals. The color scale represents the magnitude of the arrow. Horizontal scale bars represent 20 nm.