Flexocoupling impact on the kinetics of polarization reversal

The impact of flexoelectric coupling on polarization reversal and space-charge variation in thin films of ferroelectric semiconductors has been studied theoretically. The relaxation-type Landau-Khalatnikov equation together with the Poisson equation and the theory of elasticity equations have been used to calculate in a self-consistent way the spatial-temporal development of ferroelectric polarization, electric potential, space charge, elastic stresses and strains. The analysis of the obtained results reveals a moderate increase in the flexocoupling influence on the polarization, elastic strain, electric potential, and space-charge development with a decrease in the ferroelectric film thickness. In contrast, the dependence of polarization switching time on the applied electric field is remarkably affected by the flexocoupling strength. The polarization reversal process consists typically of two stages; the first stage has no characteristic time, whereas the second one exhibits a switching time strongly dependent on the applied electric field.

Figure 1

Schematics of the single-domain ferroelectric-semiconductor film with electrodes in the flat capacitor geometry. Polarization direction is shown by the arrow.

Figure 2

(a) and (b) Spatial distributions of polarization, as well as (c) and (d) electric potential calculated for the films of 100-nm ((a) and (c)) and 10-nm ((b) and (d)) thicknesses. Different curves are calculated with ($\mathrm{FA}=1$) and without ($\mathrm{FA}=0$) flexocoupling, as well as with ($U=1$ or $U=0.1\mathrm{V}$) and without $(U=0)$ applied electric voltage. Material parameters are listed in Table C1 in Appendix C of Ref. [55].

Figure 3

(a) Donor and (b) electron concentrations for the film with a thickness of 100 nm with (blue solid curve) and without (red dashed curve) the flexoelectric effect [compare with Figs. 2 and 2]. The voltage applied to the film is 1 V. Material parameters are listed in Table C1 in Appendix C of Ref. [55].

Figure 4

(a) Donor and (b) electron concentrations for the film with a thickness of 10 nm with (blue solid curve) and without (red dashed curve) the flexoelectric effect [compare with Figs. 2 and 2]. The voltage applied to the film is 0.1 V. Material parameters are listed in Table C1 in Appendix C of Ref. [55].

Figure 5

(a) Donor and (b) electron concentration distributions in the 100-nm film without an applied electric field, $U=0$ [compare with Fig. 2]. Material parameters are listed in Table C1 in Appendix C of Ref. [55]. The blue solid line represents the distribution calculated with an account of the flexoelectric effect, and the red dashed line represents the distribution without the flexoelectric effect.

Figure 6

(a) and (b) Single-domain polarization reversal kinetics together with (c) and (d) correspondent displacement currents calculated for the 100-nm films with ((a) and (c)) and without ((b) and (d)) the flexoelectric effect. The applied electric field is measured in megavolts per meter and increases from right to left in all plots. Parameters used in the modeling are listed in Table C1 in Appendix C of Ref. [55].

Figure 7

(a) Dependence of the polarization switching time τ on the strength of the applied electric field. The solid line denotes the dependences calculated with the flexocoupling ($E_{\mathrm{cr}}^{\mathrm{fl}}=1.50\mathrm{MV}/\mathrm{m}$), and the dashed line shows the one calculated without it ($E_{\mathrm{cr}}^{\mathrm{nofl}}=1.63\mathrm{MV}/\mathrm{m}$). (b) The dashed-dotted line shows the asymptotic power behavior of both dependences on the logarithmic scale. Film thickness is 100 nm. Material parameters are listed in Table C1 in Appendix C of Ref. [55].

Figure 8

Switching time dependence on the relative FA coefficient under the applied electric fields of about 1.631 MV/m (curve 1), 1.633 MV/m (curve 2), and 1.637 MV/m (curve 3). Material parameters are listed in Table C1 in Appendix C of Ref. [55]. $\mathrm{FA}=1$ corresponds to the flexocoefficient of $2.46\times {10}^{-11}{\mathrm{m}}^3/\mathrm{C}$, and the other FA values are the corresponding parts of this value. The inset: thickness dependence of the critical field in films with (solid lines) and without (dashed lines) the flexoelectric effect.

Figure 9

(a) Single-domain polarization reversal kinetics and (b) correspondent displacement current calculated for the 100-nm film under the applied electric field of 1.631 MV/m. Each curve corresponds to different values of the flexocoupling coefficient defined as $\mathrm{FA}\times F_{33}$, where $F_{33}$ is the flexoelectric tensor component (see Table C1 in Appendix C of Ref. [55]) and FA is a relative amplitude coefficient that increases from 0 (the right curve in the plot) to 1 (the left curve) adopting 40 equidistant values. The switching time of the system is maximal at $\mathrm{FA}=0$ (no flexoeffect) and minimal at $\mathrm{FA}=1$ (full flexoeffect present).