# Presentation: 920396

## Domain walls are to Ferroelectricity

**Domain walls**** are to Ferroelectricity**
**what ****Dislocations ****are to Mechanical **
**properties of Materials**

**Volkmar Dierolf, Lehigh Univeristy, DMR 0349632, 0602986**

- Polarization switching is central to *ferroelectrics*, and their applications ranging from piezoelectric actuators to high speed electro-optic devices. However a fundamental question remains open: **Why are coercive fields in ferroelectrics much lower than theory?**   The PIs have shown using phase-field modeling that the presence of ferroelectric domain walls with even 1-2nm width lowers the coercive field dramatically by many orders of magnitude.  Using piezoelectric force microscopy, they have shown that unlike ideal wall width expectations of a few unit cells, **wall widths (2***********o*****) can be surprisingly up to 60nm in real materials such as lithium niobate (LiNbO****3****).**  The predicted coercive fields for these walls are close to experiments (horizontal lines, top right figure). Domain walls are, in this sense, equivalent to dislocations with finite strain fields, which dramatically lower the mechanical deformation stresses in real materials.
- (In review *Phys. Rev. Lett.* (2007).)

- Phase Field Modeling

- exp

- Piezoelectric force
- microscopy

**Notes:**
This work addresses a fundamental problem in all ferroelectrics, namely, why coercive fields in ferroelectrics are much lower than that predicted by Landau theory.

Topological defects in materials play a critical role in understanding their real-world physical behavior.  For example, dislocations explain the low deformation stresses required to overcome the otherwise large intrinsic Peierls potential barrier predicted for a perfect lattice. Additionally, the threshold stress to move a dislocation is inversely proportional to the spatial extent of the local stress field around a dislocation. Similarly, the coercive field to move a magnetic domain wall decreases exponentially as the wall width increases.  Thus, one might expect similar trends in ferroelectrics, which contain two or more switchable states of built-in electrical polarization under the application of an electric field. While these ideas have been theoretically proposed in the ferroelectrics literature, the inverse relationship between a ferroelectric wall width and wall coercive field is not intuitive, besides being experimentally challenging to verify.  This is because, unlike magnetic walls, a ferroelectric domain wall is theoretically predicted to possess an intrinsic width on a unit-cell level (<0.5nm). Here we show that antiparallel domain walls can become diffuse on tens of nanometers scale in real ferroelectrics.  We also show that even a broadening of the wall on 1-2nm scale can dramatically lower the domain wall coercive field by orders of magnitude, bringing them in agreement with experimental coercive fields.  These two facts, taken together, imply that the motion of slightly diffuse ferroelectric domain walls can be a mechanism to explain low experimental coercive fields in ferroelectrics. Since antiparallel domain walls are symmetry allowed in all ferroelectrics, such a mechanism can be general.

We focus on two well-known trigonal (3m) ferroelectrics, namely, lithium niobate (LiNbO3) and lithium tantalate (LiTaO3). There are only two domain states, with polarization Ps along +c or –c crystallographic 3-fold rotation direction. 

Figure 1 shows the possible consequence of nanometer scale widths on domain wall coercive field required to move a single domain wall by one unit cell, a=0.515nm. Phase-field simulation method was used in these numerical calculations.[i] The complete analytical Ginzburg-Landau-Devonshire (GLD) total free energy for the prototype paraelectric phase for these materials is given in Ref [[ii]].  A single domain wall with a polarization profile is defined, where x is the coordinate normal to the wall. The temporal and spatial evolution of the polarization vector field is obtained by solving the time-dependent GLD equations using the semi-implicit Fourier spectral method15. We approximate the lattice friction felt by a moving wall by discretizing the continuum G-L equation using a grid size that is equal to the lattice spacing, a, in the crystal. A variable electric field is applied in the z (or c) directions till the wall moved by a. 
From Fig. 1 (top right), in LiNbO3, the threshold electric field decreases drastically, a change of two orders of magnitude (from 787 kV/cm to 1.95 kV/cm), with an increase in domain wall width from 2o~0.76nm to 2o~2.1nm.  For larger domain wall widths, another order of magnitude decrease is observed, reaching a value of ~ 0.3 kV/cm for a wall width of 2o~103nm.  Similar results are seen for LiTaO3, but with lower coercive fields. To put these results in context, the theoretical coercive field in LiNbO3 to simply hop between the two Landau-potential wells (without a domain wall and ignoring elastic and electrostrictive energy terms) can be calculated to be 5420 kV/cm. On the other hand, the lowest reported experimental coercive field for LiNbO3 to-date is ~2 kV/cm.  The corresponding domain wall width from Fig.  1, if that were assumed as the dominant mechanism, would be ~2nm.  Similarly, for LiTaO3, the lowest reported experimental coercive field is ~0.55kV/cm, which would correspond to a wall width of ~2.7nm.  Thus, even a small broadening on 2-3nm scale can dramatically lower coercive fields in these materials.

In further support of the wall broadening, Figure 2 (bottom right) presents the piezoelectric force microscopy (PFM) response across a domain wall. PFM detects the surface displacements Ui related to piezoelectric strain ij=dkij Ek induced by applying an oscillating electric field Ek to the tip in contact with the sample surface. Since piezoelectric coefficient dijk=ijklPs, where Ps is the spontaneous polarization, and ijkl is the electrostrictive coefficient of the material which is invariant across the wall, measurement of the piezoresponse across a wall is expected to provide direct information on the primary order parameter, Ps across a wall. Until recently, the reported experimental PFM wall widths were limited by a lack of theoretical modeling of the instrumental resolution limits. Frequency independent vertical PFM signal Uz as a function of domain wall normal coordinate, x is shown in Figure 2. Also shown are two theoretical PFM profiles calculated using Finite Element Method (FEM) and analytical theory.  The FEM simulation, performed using ANSYS, includes the complete dielectric, piezoelectric and elastic tensors of the material, in addition to the precise description of the tip shape and tip sample contact.  The end of the tip was realistically defined as a disk of radius r, which was determined from scanning electron microscopy images of the same tip used in the specific PFM scan.  The conical part of the tip was defined as silicon coated with metal, with a 25 micron length and 15 degree half- angle.  The tip was in true contact with the sample.  In addition to the above, a diffuse wall was defined in terms of piezoelectric dijk = dbulk tanh(x/o), 2o=0-60nm, in analytical theory. (Diffuse wall was not included in FEM). Figure 2 shows that an excellent fit is obtained only when 2o =60nm.  This analysis was repeated for several tip radii below 50nm and for several domain walls, and 2o was found to vary from 20-60nm. 
It is thus realistic to think in terms of a mechanism of domain walls with nanometric widths possessing low domain wall coercive fields. Since antiparallel domain walls are symmetry allowed in all ferroelectrics, and since defect-domain wall interactions are widely reported in ferroelectrics literature, we believe that this can serve as a general mechanism to explain low coercive fields in all ferroelectrics.

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- Volkmar Dierolf, Lehigh University, DMR 0349632, 602986