Modeling the Motion of Ferroelectric Domain Walls with the Classical Stefan Problem

P. V. Yudin, M. Yu. Hrebtov, A. Dejneka, and L. J. McGilly

Phys. Rev. Applied 13, 014006 – Published 6 January 2020

Abstract

With advances in nanotechnology, ferroelectric switching by individual domain walls (DWs) has become a subject of broad interest. Conventional models consider DW motion in a fixed homogeneous or inhomogeneous electric field. However, it is clear that the electric field commonly evolves in time due to the redistribution of bound charges and screening free charges on the ferroelectric surface, particularly due to surface conductance. Taking this effect into account remains a serious challenge. Here we propose a simple concept to describe simultaneously the evolution of the electric field and the DW motion in a ferroelectric sample. The approach is based on a full analogy between charge transport during ferroelectric switching and heat transport in a moving melting front: the classical Stefan problem. The analogy helps in the establishment of control of DW motion in thin films. Experimentally, DWs are displaced by voltage pulses under electrodes with a limited conductivity and detected using piezoresponse force microscopy. In the frame of a combined theoretical and experimental approach, DW-friction mechanisms and the impact of the DW energy on its shape are discussed. The study facilitates the development of nanoelectronic devices.

Received 1 August 2019
Revised 14 August 2019

DOI:https://doi.org/10.1103/PhysRevApplied.13.014006

Physics Subject Headings (PhySH)

Domain walls Electrical properties Ferroelectricity Phase transitions Transport phenomena

Ferroelectrics Functional materials

Numerical techniques

General PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

P. V. Yudin^1,2,*, M. Yu. Hrebtov¹, A. Dejneka², and L. J. McGilly³

¹Kutateladze institute of Thermophysics, Siberian Branch of Russian Academy of Science, Lavrent’eva av. 1, Novosibirsk, Russia
²Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic
³Department of Physics, Columbia University, New York, New York 10027, USA

^*yudin@fzu.cz

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Vol. 13, Iss. 1 — January 2020

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Images

Figure 1
Experimental setup for investigation of domain-wall motion and the analogy with the classical Stefan problem. (a) 3D topography image and schematic showing an EBID $Pt$ line electrode deposited on the ferroelectric thin film and contacted by a conducting probe, which is biased relative to the SRO lower electrode. (b) Cross-section view of the experimental setup showing partial switching of the capacitor structure. The polarization is shown by red arrows and free charges by pluses and minuses, whilst bound charges are circled. (c) Schematic of the analogous experiment for melting of a fusible metallic rod, where heat is supplied by a soldering iron in contact with its edge. (d) Heat conduction in the rod, which is described by the one-dimensional classical Stefan problem. The meanings of the symbols are given in Table 1.
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Figure 2
DW motion under linear electrodes. (a) Series of PFM phase images from the initial switched state (top) and the DW position after pulses of the durations shown on the left-hand side. (b) Calculated voltage distribution according to the Stefan model for a moving DW at 1 ms after voltage application (blue), 10 ms (orange), and 100 ms (red) (c) DW displacement as a function of switching time for the system shown in (a).
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Figure 3
Growth of a circular domain. (a) Schematic drawing of the experimental configuration; screening charges and their motion accompanying the growth of the domain are shown in the magnified image. (b) PFM phase images for three selected pulse durations of 1, 50, and 350 ms (left to right). (c) Calculated voltage distribution according to the 2D axisymmetric one-phase Stefan model at 5 ms after voltage application (blue), 50 ms (orange), and 500 ms (red). (d) Time dependence of domain radius: experimental averaged radius (rhombs) and theoretical prediction, Eqs. (12)–(14) (solid line). The inset shows the results of a numerical simulation of domain growth in ${Li Ta O}_{3}$ [29] [curve 30 V in Fig. 6(a) in Ref. 29], our fit by Eqs. (12)–(14), and a fit by the theory of Ref. [20] [Eq. (7) therein].
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Figure 4
DW propagation under a linear electrode with three wider sections. (a) 3D topography image of the top $Pt$ electrode used. (b) Comparison of the simulated (dry-friction model) and PFM-detected DW positions and shapes for several snapshots in time. (c) Mean velocity as a function of displacement: experiment (dots), and two theoretical fits corresponding to the dry-friction model (dashed blue line) and a model with both the dry- and viscous-friction mechanisms (solid red line). (d) Voltage distributions along the central line of the electrode for several instances.
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Figure 5
Impact of the intrinsic energy of a DW on its shape. Simulations with zero DW energy (top row) and with an energy of 0.41 J/m² (middle row), and the experiment (bottom row). The moments of time correspond to entrance into the first wider section (left column), a short time after (middle column), and propagation in the narrow part (right column).
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