Phase-field Modeling of Flexoelectric Effect in Perovskite Ferroelectrics

Open Access
Graduate Program:
Materials Science and Engineering
Doctor of Philosophy
Document Type:
Date of Defense:
September 02, 2014
Committee Members:
  • Long Qing Chen, Dissertation Advisor
  • Long Qing Chen, Committee Chair
  • Venkatraman Gopalan, Committee Member
  • Clive A Randall, Committee Member
  • Susan E Trolier Mckinstry, Committee Member
  • Qiang Du, Committee Member
  • James Chen, Special Member
  • Flexoelectric effect
  • ferroelectrics
  • phase-field method
Ferroelectrics are multifunctional materials that have many applications in devices such as actuators, sensors, memory storage, microelectromechanical systems (MEMS), and others. The multifunctionality derives from the couplings among internal order parameters, such as ferroelectric polarization and spontaneous strain, and external thermodynamic variables, such as temperature, stress, and electric field. Although the thermodynamics of such couplings has been well established, the coupling among order parameters and their gradients is much less well-understood. The main goal of the dissertation is to fundamentally understand the role of flexoelectric effect, the coupling between polarization and the gradient of strain, in the ferroelectricity of a crystal, e.g., domain structures, polarization distributions across domain walls, and domain switching. There are sufficient experimental evidences showing that the flexoelectric effect, which is small and generally ignored in macroscopic systems, may become significant or even dominant in nanostructures, particularly in ferroelectric materials which exhibit strong dielectric properties. However, there are few well-developed theories, especially mesoscale models, available to explain or to further predict the flexoelectric effect-induced phenomenon in ferroelectrics. In this dissertation, a phase-field model of ferroelectric domains incorporating flexoelectric contribution is developed. The flexoelectric effect in ferroelectric single crystals is investigated using the developed model. The present phase-field model of ferroelectric domain structures and switching is extended to include the flexoelectric effect by adding the flexoelectric coupling terms to the total free energy. The semi-implicit Fourier-spectral method is modified to solve the time-dependent Ginzburg-Landau equation with anisotropic gradient energy coefficients. Then by calculating the domain wall profiles of BaTiO3 and SrTiO3 and comparing with other numerical solutions, this model is validated. In order to incorporate the interactions of different soft modes or order parameters, many new terms, including biquadratic coupling, trilinear coupling and flexoelectric coupling, are introduced to the classic Landau-Ginzburg-Devonshire (LGD) theory of ferroelectrics. However, it is still unclear how each of these terms modifies the domain wall structures and its thermodynamic stabilities. As a matter of fact, it is also uncertain whether the continuum LGD theory can be used to study domain walls, which may be as thin as one unit cell. From phase-field simulations, it was found that polar domain walls of incipient ferroelectric CaTiO3 are likely to be induced by flexoelectric coupling. The polarization component parallel to the twin walls shows an even distribution; while the polarization normal to the walls has an odd distribution. The peak value of the induced polarization is on the magnitude of 1 μC/cm2. The simulation results of domain wall structures agree very well with existing experimental observations and other computational predictions. Thus, the developed phase-field model, which is based on LGD theory, can be used to study the nano scale domain structures at least qualitatively. The pure ferroelectric (180°) domain walls of tetragonal ferroelectrics have long been believed to be Ising type and charge neutral. However, recent theoretical studies show that the wall exhibits additional Bloch-like or Néel-like features. Using the phase-field simulations, it is demonstrated that the wall has Ising-Bloch-Néel characters. The new features are strongly anisotropic and are entirely due to the flexoelectric effect. From thermodynamic analysis, it is shown that the polarization at the domain wall is determined by the competition between the flexoelectric field and the depolarization field. The flexoelectric effect-induced polarization shows an odd distribution at the domain walls with the peak values on the magnitude of 0.1 μC/cm2. Flexoelectricity can be regarded as a mechanical analogue to the electric field, which can modify the free energy profile of a ferroelectric material asymmetrically. Therefore, the flexoelectric effect can be used to switch ferroelectric domains if large enough. By approximating its geometry as a spherical indenter, an AFM tip is shown to induce a strong flexoelectric effect to an ultrathin BaTiO3 film (~5 nm) under a 1000 nN load. The radius of contact area is 10 nm. The flexoelectric field is on the magnitude of 107 V/m, which is well above the coercive field and thus can induce 180-degree switching. The stresses under the AFM tip, on the magnitude of several GPa, are also larger than the coercive stresses which can induce nucleation of new domains. Due to the strong compressive biaxial strain exerted by SrTiO3 substrate, only flexoelectricity induced (180-dgree) domain remains after unloading. This process which can induce 180-dgree switching, is more similar to electric field-induced switching rather than the conventional mechanical switching via piezoelectricity. From the simulations, it is also shown that this type of mechanical switching is only possible in nanoscale films with the upper bound for the film thickness on the order of ~25 nm.