Electrode-passive layer-ferroelectric-electrode sandwich structure. The thickness of the passive layer is , that of the ferroelectric is .
(a) Hysteresis loop of a ferroelectric, and the calculated loops of the sandwich for different values of . (b) The calculated field amplitude dependence of the coercive field for the sandwich structure (ferroelectric-thin passive layer) with , compared to that of the ferroelectric part.
Inverse tilt of the polarization loop at , , vs inverse thickness of the film, , for a set of PZT sol-gel films. The data seem to validate the passive layer model prediction of a linear dependence of the two quantities—cf. Eq. (2.6). After Ref. 5.
(a) Predicted dependence of the coercive field on maximum polarization and on film thickness according to the insulating passive layer with threshold conduction model; and (b) experimental confirmation of the prediction for films with Pt electrodes. After Ref. 10.
Schematic distribution of polarization across a film of thickness for the out-of-plane (a) and in-plane (b) cases. , where —cf. Eqs. (2.27) and (2.30). Here, is the dielectric constant of the material, its background dielectric constant, is the correlation length, and is the “bulk value” of polarization.
(a) Film thickness dependences of the inverse zero-bias capacitance density of film based plate capacitors at different Ti content (Ref. 25). (b) The inverse of the zero-bias capacitance density of film based plate capacitors as a function of film thickness at different temperatures (Ref. 26).
Dense domain pattern in a ferroelectric capacitor system with passive nearby-electrode layers. is the period of the domain pattern, is the width of the positive (negative) domains, is the electric potential, and is the spontaneous polarization from Landau-Devonshire theory.
The function appearing in Eq. (2.41), for three different values of the passive layer permittivity . When the passive layer thickness is much greater than the domain pattern period , the dense pattern approximation (2.40) reduces to the simple in-series capacitor formula (2.14). After Ref. 33.
Effect of domain spacing on the offset of the film thickness dependence of the normalized inverse capacitance, . The dashed line corresponds to the simple in-series capacitor formula, Eq. (2.14), the dotted lines to the dense pattern approximation, Eq. (2.40), and the continuous lines to the exact formula, Eq. (2.39). After Ref. 33.
The graph shows how the offset of the dependence of is affected by the distribution of the dielectric response of the ferroelectric system between the intrinsic and extrinsic contributions, for a dense domain pattern . As in Fig. 9, the dashed line corresponds to the in-series capacitor formula, Eq. (2.14), while the continuous line to the dense pattern approximation, Eq. (2.39). After Ref. 34.
Time dependence of the voltage offset , in the case of weak compensation of the depolarizing field by carrier transport across the passive layer, Eq. (2.66).
Distribution of the built-in electric field in a film with depleted charge carriers, in the case of partial depletion and of full depletion . The shaded areas correspond to the depleted regions of the film.
The model devised in Refs. 58 and 59 to explain the voltage offset of ferroelectric loops in terms of the nearby-electrode poling effect of the strain gradient. In the convention adopted herein, the top electrode is grounded and the voltage is applied to the bottom electrode.
Nucleus of a reverse domain in a “down-polarized” ferroelectric material which has its polar axis in the vertical direction and normal to the electrode. An external field in the “up” direction is applied to the system.
The effect of parameter on the domain energy (normalized to the Landauer activation energy ). The nucleation barrier and critical size (i.e., the point of maximum ) both decrease with increasing values.
Nucleation barrier (normalized to the Landauer value ) as a function of the anisotropy factor , the external field , and the ratio .
The method used for the characterization of internal field effects in ferroelectric capacitor systems. The capacitor is subjected to a sequence of positive ( and ) and negative ( and ) voltage pulses, and for each pulse the switching charge is measured. The difference between and gives information about the sign of the internal bias field.
(a) The method used by Abe et al. (Ref. 81) to evaluate the built-in internal bias in thin films by comparing the value of the maximum switching current for each sense of switching. (b) The loop for the same film. Good agreement is observed between the voltage offsets determined from the measurements and from the loop. After Ref. 81.
Imprint in PLZT film capacitors: (a) difference between the switching charge of the imprinted and the nonimprinted capacitor, , as a function of exposition time for different exposition temperatures; (b) switching charge as a function of applied voltage; and (c) calculated voltage offset as a function of exposition time and exposition temperature. After Ref. 35.
Thickness dependence of the voltage offset measured in (a) films with built-in internal bias (after Ref. 58) and (b) PZT films with photoinduced imprint (after Ref. 77).
(a) The effect of ratio on the voltage offset evaluated from the piezoelectric and polarization loop measurements (after Ref. 85), and (b) the effect of annealing at on the voltage offset for different values of the ratio (after Ref. 84).
Effect on the built-in internal field of the oxygen pressure at which a PZT capacitor with oxide electrodes was maintained during cooling. The voltage offset of the loop increases with decreasing oxygen pressure. After Ref. 87.
Temperature dependence of the parameters and appearing in the analytical theory for imprint due to nearby-electrode trapping—cf. Eqs. (2.66), (2.70), and (2.71). These parameters have been determined from the fit of the time dependence of the voltage offset, shown in Fig. 19(c).
Thickness of a dielectric layer with , which affects the dielectric permittivity of the film identically to the effects listed in the first column. —film thickness, —thickness of the layer, —correlation radius, and —permittivity and background permittivity of the ferroelectric, —extrapolation length for the polarization boundary conditions, —Thomas-Fermi screening length, —coefficient of the dielectric nonlinearity, —space charge density in the depletion layer, and —depletion layer width.
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