Abstract We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all solvable Einstein metrics of dimension seven or less with codimension-one nilradical. For each linearly stable metric, dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.

BiFeO 3 (BFO) is one of the most widely studied magneto-electric multiferroics. The magneto-electric coupling in BiFeO 3, which allows for the control of the ferroelectric and magnetic domain structures via applied electric fields, can be used to incorporate BiFeO 3 into novel spintronics devices and sensors. Before BiFeO 3 can be integrated into such devices, however, a better understanding of the dynamics of ferroelectric switching, particularly in the vicinity of extended defects, is needed. We use in situ transmission electron microscopy (TEM) to investigate the response of ferroelectric domains within BiFeO 3 thin films to applied electric fields at high temporal and spatial resolution. This technique is well suited to imaging the observed intermediate ferroelectric switching regimes, which occur on a time- and length-scale that are too fine to study via conventional scanning-probe techniques. Additionally, the spatial resolution of transmission electron microscopy allows for the direct study of the dynamics of domain nucleation and propagation in the presence of structural defects. In this article, we show how this high resolution technique captures transient ferroelectric structures forming during biasing, and how defects can both pin domains and act as a nucleation source. The observation of continuing domain coalescence over a range of times qualitatively agrees with the nucleation-limited-switching model proposed by Tagantsev We demonstrate that our in situ transmission electron microscopy technique is well-suited to studying the dynamics of ferroelectric domains in BiFeO 3 and other ferroelectric materials. These biasing experiments provide a real-time view of the complex dynamics of domain switching and complement scanning-probe techniques. © 2012 American Institute of Physics.

The switching of domains in ferroelectric and multiferroic materials plays a central role in their application to next-generation computer systems, sensing applications, and memory storage. A detailed understanding of the response to electric fields and the switching behavior in the presence of complex domain structures and extrinsic effects (e.g., defects and dislocations) is crucial for the design of improved ferroelectrics. In this work, in situ transmission electron microscopy is coupled with atomistic molecular dynamics simulations to explore the response of 71° ferroelastic domain walls in BiFeO3 with various orientations under applied electric-field excitation. We observe that 71° domain walls can have intrinsically asymmetric responses to opposing biases. In particular, when the electric field has a component normal to the domain wall, forward and backward domain-wall velocities can be dramatically different for equal and opposite fields. Additionally, the presence of defects and dislocations can strongly affect the local switching behaviors through pinning or nucleation of the domain walls. These results offer insight for controlled ferroelastic domain manipulation via electric-field engineering.

We use in situ transmission electron microscopy to directly observe, at high temporal and spatial resolution, the interaction of ferroelectric domains and dislocation networks within BiFeO3 thin films. The experimental observations are compared with a phase field model constructed to simulate the dynamics of domains in the presence of dislocations and their resulting strain fields. We demonstrate that a global network of misfit dislocations at the film-substrate interface can act as nucleation sites and slow down domain propagation in the vicinity of the dislocations. Networks of individual threading dislocations emanating from the film-electrode interface play a more dramatic role in pinning domain motion. These dislocations may be responsible for the domain behavior in ferroelectric thin-film devices deviating from conventional Kolmogorov-Avrami-Ishibashi dynamics toward a Nucleation Limited Switching model.