Figure 1

(a) (Color online) Schematic of ferroelectric $180°$-domain-wall boundary curved by the strong localized electric field of the biased probe in contact with the sample surface. (b) Wall curvature at the sample surface in quasicontinuous media approximation (solid curves and color scale). Dashed rectangle corresponds to the schematic of activation field calculations used by Miller and Weinreich (Ref. 1) for rigid polarization model where the distance $a$ is equal to the lattice constant.Reuse & Permissions

Figure 2

(Color online) (a) and (b) Bias dependence of $P_V$ on applied bias $V$ for different domain-wall initial position $x_0$ (labels near the curves in nanometer). Effective distance $d=5\text{nm}$ and material parameters for ${\text{LiNbO}}_3$ are ${\epsilon }_{11}=84$, $\alpha =-2\times {10}^9$, $\eta ={10}^{-9}$ in SI units (i.e., $L_{\perp }=0.5\text{nm}$), and $P_S=0.75\text{C}/{\text{m}}^2$. Dotted curves is linear approximation $P_V=V$ that works satisfactorily up to 5 V for the chosen material parameters. (c) and (d) Polarization below the probe apex $P_3\left(0\right)$ for different $x_0$ values (labels near the curves in nanometer).Reuse & Permissions

Figure 3

(Color online) Equilibrium surface profile of domain wall affected by biased probe. Material parameters for ${\text{LiNbO}}_3$ are ${\epsilon }_{11}=84$, $\alpha =-2\times {10}^9$, $\eta ={10}^{-9}$ in SI units (i.e., $L_{\perp }=0.5\text{nm}$), and $P_S=0.75\text{C}/{\text{m}}^2$. Effective distance $d=5\text{nm}$ and initial domain-wall position $x_0=1\text{nm}$ for plots (a) and (b), while $x_0=2.5\text{nm}$ for plots (c) and (d). Different curves correspond to different voltages applied to the probe. Positive $V=1,2,5,10\text{V}$ for plots (a) and (c) and negative $V=-1,-2,-5,-10\text{V}$ for plots (b) and (d).Reuse & Permissions

Figure 4

(Color online) Equilibrium surface profile of domain-wall boundary affected by biased probe (for $x_0=0,2,3\text{nm}$) and probe-induced domain formation (for $x_0=5,7.5\text{nm}$). Effective distance $d=5\text{nm}$, applied bias $V=5\text{V}$, and ${\text{LiNbO}}_3$ material parameters are listed in Fig. 3.Reuse & Permissions

Figure 5

(Color online) (a) Dependence of probe-induced domain formation coercive biases $V_c^{+}$ (dashed curves) and $V_c^{-}$ (solid curves) vs the distance $x_0$ between the domain wall and the probe apex. Separation $d$ between the effective charge modeling the probe field and sample surface is fixed as 5, 10 and 15 nm (right labels with arrows). In the region where hysteresis is absent, dotted curves represent the bias $V_I$ at inflection point of polarization dependence on bias. Polarization $P_3$ bias dependence in the regions I (no hysteresis loop, inflection point bias $V_I$ increases with distance $x_0$ increase as shown by arrow cross section with solid, dashed, and dotted curves), II (no hysteresis loop, inflection point bias $V_I$ slightly decreases with distance $x_0$ increase as shown by arrow cross section with solid, dashed, and dotted curves), and III (hysteresis loop appears and becomes symmetric with distance $x_0$ increase as shown by solid, dashed, and dotted curves) is schematically shown below at insets I–III. (b) Coercive biases $V_c^{+}$ (dashed curves) and $V_c^{-}$ (solid curves) dependence vs the effective distance $d$ for the wall-probe distance $x_0$ fixed as 10, 20 and 50 nm (labels near the curves). Parameters of ${\text{LiNbO}}_3$ are the same as in Fig. 3.Reuse & Permissions

Figure 6

(Color online) Dependence of probe-induced domain formation hysteresis loop width $\Delta V_c=\left(V_c^{+}-V_c^{-}\right)/2$ in linear-log (a) and linear (b) scales as the function of the distance $x_0$ between the domain wall and the probe apex for different effective charge-surface distances $d=5$, 10 and 30 nm (labels near the curves). (c) The inflection point bias $V_I$ (dotted curves) and hysteresis loop imprint bias $V_I=\left(V_c^{+}+V_c^{-}\right)/2$ (solid curves) vs the distance $x_0$ for the same $d=5$, 10 and 30 nm (labels near the curves). (d) Absolute values of threshold biases $V_{\text{th}}^{+}$ (solid curves) and $V_{\text{th}}^{-}$ (dashed curves) required to move the domain-wall boundary by overcoming the effect of the lattice-constant discreteness as the function of the distance $x_0$ for $d=5$, 10 and 30 nm (labels near the curves). Parameters of ${\text{LiNbO}}_3$ are the same as in Fig. 3.Reuse & Permissions

Figure 7

(Color online) Phase diagram in coordinates $\left\{x_0,d\right\}$. Boundaries between the regions of different bias dependence of $P_3$. Activationless domain-wall bending, domain nucleation far from the wall (hysteresis), and intermediate regimes are shown schematically. Material parameters of ${\text{LiNbO}}_3$ are the same as in Fig. 5.Reuse & Permissions

Figure 8

(Color online) (a), (b), and (c) Free-energy excess $\Delta G$ related to the polarization redistribution caused by the probe electric field vs the polarization amplitude $P_V$ (in volts) at different applied bias $V=0$ (a), 2V (b), and $-2\text{V}$ (c). (d) Polarization orientation barrier $E_b$ of nascent domain existing in the range of hysteresis (dashed curves) and the energy $E_S$ of metastable saddle point (dotted curves) vs the distance $x_0$ between the domain wall and probe apex for fixed charge-surface separation $d=5$, 10, 15 nm (marked near the curves) and $V=0$. Material parameters of ${\text{LiNbO}}_3$ are the same as in Fig. 5.Reuse & Permissions