Domain wall energy calculation

Domain wall energy and thickness for a 180° wall, calculated from the minimum in exchange and anisotropy contributions (Landau Lifshitz, 1935) can be expressed respectively as ... 

To estimate the temperature evolution of the entire domain structure we express its energy at T = 0 (i.e. at f = —1) in the Kittel approximation, assuming a kink-like structure of the domain wall and a flat profile of a polarization inside domains. The calculations are similar to those for the domain structure of the uniaxial ferroelectric film surrounded by paraelectric dead layers and embedded into short circuited capacitor [27]. The free energy is superimposed from the domain walls energy and electrostatic contributions ... 

Fp, as observed. At lower T, non-linearity in the free energy functional is expected to favor sharp domain walls [21], The transition to the monodomain ferroelectric phase F1 has been predicted by treating the ferroelectric as a semiconductor, in which carriers can be created by the field effect [16]. This transition can be calculated to occur at Tc - T = 325 K for d = 10 nm [4], in qualitative agreement with our observations. The rich phase diagram we have observed in this simple system makes it an excellent test case for more quantitative development of these concepts. 

The decrease in mechanical strain energy resulting from the formation of domains is counterbalanced by the increase in wall energy as the domains develop. At equilibrium the total energy will be a minimum. The strain energy s per unit of volume occupied by 90° domains has been calculated to be... 

A qualitative change occurs only at about 1.056 monolayers, where it is no more possible to unambiguously identify commensurate stripes [182]. There, the situation can be described as commensurate modulations in an otherwise uniform uniaxial incommensurate structure. Following an earlier suggestion [362], test calculations where the lateral substrate energy is increased by a factor of two compared to the original potential [326] were performed at several coverages. The results are qualitatively similar to what is shown in Fig. 40, only the boundaries between the domain walls seem to... 

Several studies have been made by Vanderbilt et al. [17] on Ihe ab initio calculation of energy of the domain wall having a narrow width of Ihe... 

---