Structures and velocities of noisy ferroelectric domain walls

Ferroelectric domain wall motion is fundamental to the switching properties of ferroelectric devices and is influenced by a wide range of factors including spatial disorder within the material and thermal noise. We build a Landau-Ginzburg-Devonshire (LGD) model of ${180}^{\circ }$ ferroelectric domain wall motion that explicitly takes into account the presence of both spatial disorder and thermal noise. We demonstrate both creep flow and linear flow regimes of the domain wall dynamics by solving the LGD equations in a Galilean frame moving with the wall velocity $v$. Thermal noise plays a key role in the wall depinning process at small fields $E$. We study the scaling of the velocity $v$ with the applied DC electric field $E$ and show that noise and spatial disorder strongly affect domain wall velocities. We also show that domain walls develop a local, metastable paraelectric region and widen significantly in the presence of thermal noise in materials with “multiwell” potentials, representative of ferroelectrics at temperatures $T$ just below a first-order phase transition (Curie) temperature $T_c$. These calculations point to the potential of noise and disorder to become control factors for the switching properties of ferroelectric materials, for example for advancement of microelectronic applications.

Figure 1

(a) and (b) Piezoresponse force microscopy images of domain walls in representative ferroelectric materials, ${\mathrm{CuInP}}_2{\mathrm{S}}_6$ (a) and ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ (b). The color indicates the response of the material to a locally applied AC electric field (via an AFM tip). We show our numerically simulated domain wall under an applied field without noise in (c), with quenched spatial noise in (d), and with thermal and quenched spatial noise in (e). The black dashed line indicates the initial position of the wall, which moves in response to an applied DC electric field when the field (in combination with the thermal noise) is large enough to overcome the pinning due to the quenched spatial disorder, as in the case of (c) and (e). In (d), the domain wall is pinned by the quenched spatial disorder. The color bars in (c)–(e) indicate the value of the polarization $P$ in C/$\mathrm{m}{}^2$.

Figure 2

(a) The LGD potential used in this study at various temperatures $T$. Note the development of the metastable paraelectric state (with polarization $P=0$) as we increase $T$ to right below the Curie temperature $T_c$ (red curve). This metastable state has profound consequences for the profile of domain walls, shown at various temperatures (without noise) in (b), where the vertical dotted line indicates the center of the domain wall. As $T$ approaches $T_c$ from below, the domain wall widens and eventually develops a plateau at $P=0$ (red line). These shapes were derived using the numerical solution to the LDG equation and are in perfect agreement with the analytic predictions (see the Appendix).

Figure 3

We show successive snapshots of a simulated wall (in the static laboratory frame) at an initial time $t=0$ (a) and a final time $t=\tau =75\mathrm{ms}$ (75 000 time steps) (b) under an applied field $E=1050\mathrm{V}/\mathrm{cm}$. Note that the domain wall will leave the simulation window at later times. Using the co-moving frame, shown in (c) and (d), it is possible to simulate a moving domain wall over a much longer period of time by adjusting the velocity $v$ [see Eqs. (6) and (7)]. The distances are measured in lattice spacings (taken to be 0.4 nm).

Figure 4

Polarization distribution of a region of either positive (red) or negative (blue) polarization evolved over a fixed system size and time period with thermal noise given by noise power $N$. The mean polarization plus or minus 3 standard deviations is plotted as the red/blue lines with shading in between, and the black lines show histograms of the polarization at certain $N$ values. (a) The distributions at 300 K, and (b) the distributions at just below the Curie temperature $T=392.275\mathrm{K}<T_c$. The system size is $500\times 500$ lattice spacings and polarization distributions are taken from regions of size $200\times 500$ lattice spacings in both the positive and negative polarization regions. The distributions are taken over a 500 time step period after evolving the system for an initial 10 000 time steps.

Figure 5

(a) Simulated domain wall shapes for various values of the thermal noise magnitude $N$ at a fixed temperature $T=300$ K. Note that the $N=0$ case is known analytically and matches our numerical solution. The inset shows the domain wall width $\xi$ as a function of $N$, which increases with increasing $N$. (b) Simulated domain wall shapes for different noise magnitudes for domain walls just below the Curie temperature $T=392.275\mathrm{K}<T_c$. The inset plots the size $\ell$ of the paraelectric ($P=0$) region as a function of the noise magnitude $N$ for this material just below $T_c$. The paraelectric region broadens with increasing $N$.

Figure 6

(a) Domain wall velocity as a function of electric field for systems with spatial disorder $\chi =0.05$ for various temperatures approaching $T_c$ (without thermal noise). The dotted lines show the analytic result for the domain wall velocity for each temperature (for $\chi =0$). In the inset we plot the depinning field magnitude $E_{\text{dp}}$ at a function of $T$. We see that $E_{\text{dp}}$ decreases to zero as $T\to T_c$. (b) Domain wall velocity for the $T=300$ K case for various noise magnitudes $N$ and disorder magnitudes $\chi$ indicated in the legend. Bars denote 1 standard error of the fluctuating velocity as measured in the co-moving frame. The system size is $50\times 50$ lattice spacings and the motion is measured over $1.5$ million time steps. The dotted line is the analytic result for the noiseless wall motion.

Figure 7

Domain wall speed as a function of applied field $E$ for both the double-well (DW) and triple-well (TW) potentials [see Eq. (11)], with a spatial quenched disorder with magnitude $\chi =0.2$ [see Eq. (6)]. Bars denote the standard error of the velocity measurement in the co-moving frame. Note that for $N=0$ (circles), both walls are pinned at small fields $E$, with the TW domain wall depinning at a much smaller $E_{\text{dp}}$. When we add some thermal noise with magnitude $N=0.0112$ (squares), the thermal noise can completely depin the TW domain wall, while the DW domain wall remains pinned at small applied fields $E$. The system size is $50\times 500$ lattice spacings and the motion was measured over $6\times {10}^6$ time steps. The black dotted lines indicate the analytic results for the velocities (without noise and disorder) in the DW and TW cases, with the orange TW data lying along the TW prediction.