Spatially resolved mapping of disorder type and distribution in random systems using artificial neural network recognition

The spatial variability of the polarization dynamics in thin film ferroelectric capacitors was probed by recognition analysis of spatially resolved spectroscopic data. Switching spectroscopy piezoresponse force microscopy (SSPFM) was used to measure local hysteresis loops and map them on a two dimensional (2D) random-bond, random-field Ising model. A neural-network based recognition approach was utilized to analyze the hysteresis loops and their spatial variability. Strong variability is observed in the polarization dynamics around macroscopic cracks because of the modified local-elastic and electric-boundary conditions, with the most pronounced effect on the length scale of ∼100 nm away from the crack. The recognition approach developed here is universal and can potentially be applied for arbitrary macroscopic and spatially resolved data, including temperature- and field-dependent hysteresis, I-V curve mapping, electron microscopy electron energy loss spectroscopy (EELS) imaging, and many others.

Figure 1

(a) Surface topography (arrow indicates the propagating crack), (b) PFM image collected at 250 kHz using a probing amplitude of 1V, (c) resonance frequency, and (d) Q-factor images of the polycrystalline capacitor surface. Histograms of (e) resonance frequency and (f) PFM amplitude obtained across the surface.

Figure 2

Data acquisition and processing in band excitation switching spectroscopy PFM. (a) At a single spatial location, the switching signal is formed by the triangular waveform modulated by rectangular pulses. The on-state pulse induces switching, while the electromechanical response is detected in the off-state in a broad frequency range centered at the tip-surface resonance using a band-excitation waveform. At each pixel, the measured amplitude response represents a 2D (b) amplitude and (c) cantilever response phase versus frequency and time spectrogram. The evolution of amplitude and phase with applied DC waveform is clearly seen. (d) The signal in the 2D spectrogram is (i) integrated over the frequency axis or (ii) fitted by a simple harmonic oscillator (SHO) model to yield the single-point hysteresis loops.

Figure 3

(a) Switchable polarization map and (b) hysteresis loops at selected locations across the capacitor surface from the same region as Fig. 1.

Figure 4

Switching parameters of the selected region on the surface. Shown are (a) positive coercive field, (b) negative coercive field, (c) imprint, (d) positive remanent response, (e) negative remanent response, (f) switching polarization, (g) work of switching, (h) positive nucleation bias, and (i) negative nucleation bias, as extracted from the experimental spatially resolved hysteresis loops array.

Figure 5

(a) 2D random-bond, random-field Ising model describing grain-grain interactions $J_{\mathit{ij}}$ in the random field of defects $h_i$. (b) Hysteresis loops for $J$ $=$ 0, δh $=$ 0, and δJ $=$ 1, 2, 3, 4 calculated for continuous field sweep.

Figure 6

Schematics of recognition analysis of experimental data utilizing principal-component-analysis based decorrelation and neural-network recognition.

Figure 7

Schematics of the normalization process applied to experimental data. The loops are shifted in vertical and lateral directions and normalized to unity. Thus extracted parameters are stored as imprint, Im; width, W; height, H; and vertical offset, Off.

Figure 8

(a)–(f) Loading maps for first six PCA components, (g) normalized eigenvectors for first three PCA components, and (h) normalized eigenvectors for next three PCA components. (i) Plot of eigenvalue versus principal-component number (scree plot) suggesting that PCA components above 10 do not have significant correlation and hence do not provide information about the adopted model.

Figure 9

Recognition analysis of the S-PFM data set (90 × 90 spatial pixels, 64 bias points per loop). Top row illustrates normalization parameter maps, including (a) vertical offset, (b) imprint, (c) loop height. The middle row illustrates recognition parameter maps, including (d) $J$, (e) δJ, and (f) $h$. Histograms for the reconstructed disorder parameters are (a) $J$, (b) δJ, and (c) $h$.

Figure 10

Spatial correlation functions for (a) normalization parameters and (b) reconstructed disorder parameters for the $J$-δJ-$h$ family. The distance is normalized to image size.

Figure 11

Reconstructed maps for (a) $J$, (b) δJ, and (c) $h$ for the $J$-δJ-$h$ family. (d) Average experimental and average of closest simulated loops for the area 1 and (e) area 2 selected in map (a).

Figure 12

Reconstructed parameters (a) $J$, (b) δJ, (c) average experimental and closest simulated loops in the selected area for $J$-δJ family, (d) $J$, (e) $h$, (f) average experimental and closest simulated loops in the selected area for $J$-$h$ family, (g) δJ, (h) $h$, and (i) average experimental and closest simulated loops in the selected area for δJ-$h$ family.

Figure 13

(a) Quality map obtained by comparing the fitting errors (difference between reconstructed and normalized experimental families) from reconstructed $J$-δJ, $J$-$h$, and $J$-δJ-$h$ families (b) Minimum-error map obtained by comparing the errors from reconstructed families and the best fit between the families is demarcated at each pixel in (a).