Size and temperature effects on dielectric breakdown of PLZT film capacitors

The paper introduces a model of dielectric breakdown strength. The model integrated thermal breakdown and defect models, the relationship between the electric field of ferroelectric films and dimensional parameters and operating temperature ( T ). The thickness effect is thermal breakdown in origin, whereas the area effect is statistical in nature. This model is verified with experimental results of the lead lanthanum zirconate titanate (PLZT) films of various d (0.8 – 3 m), A (0.0020 – 25 mm 2 ) tested under a range of T (300 – 400 K) with satisfying fitting results. Also learned is a relationship that the recoverable electric energy density is directly proportional to the square of breakdown electric field. This relationship is found viable in predicting the electric energy density in terms of variables of d , A , and T for the PLZT films. model on the correlation between E b and other key materials factors, such as dielectric thickness, area, and temperature. The validity of the model was tested by measurements of E b of PLZT films.


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There has been a great interest in the dielectric breakdown (DBD) of the ferroelectric films for their important applications in passive devices, field emission transistors, and ferroelectric random-access memory (FeRAM). In these devices, DBD strength determines the degree of the integration and the magnitude of energy loss [1][2][3]. In particular, high performance energy storage capacitors require large DBD electric field (E b ) and relative permittivity ( r ). The recoverable electric energy density (U reco ) is calculated using the following equation [4] 0 0 b E r reco EdE U (1) where 0 is the permittivity of vacuum. Early works on the mechanisms of DBD in high permittivity dielectrics (like ferroelectrics) found that the failure was not intrinsic in nature. That is, electronic breakdown featuring bond breaking and oxygen removal was not observed [5][6][7][8]. In this fashion, thermal breakdowns (TBD), such as dc TBD, impulse TBD, and avalanche breakdown, always dominate prior to intrinsic DBD. High-permittivity ferroelectric materials are characterized by their wide band-gap and electrical-to-thermal conductivity ratio ( / Therefore, TBD is a commonly observed behavior in these materials including semiconductors and insulators [6]. Key factors dominating the breakdown strength (E b ) of ferroelectric devices extrinsically include dielectric/electrode material properties (composition, microstructure), device structure (size and shape), and operational conditions (voltage ramp rate, temperature) [5,6,[9][10][11][12][13][14][15][16][17] Previous research suggested the dc TBD is the main mechanism, responsible for failure in highr dielectrics [5,6,16], although defect density was not considered. In these models, the dc electric field ramp rate is ≫1 s and the relationship between T and E b is as follows [6,15].
It has also been reported that E b is inversely proportional to thickness (d) which is mainly determined by the thermal distribution across the dielectric thickness and the bulk/surface thermal conductivity of the material [17][18][19]. Below a critical dielectric thickness, a power relationship between E b and d has been established: [5,9,11] where c is a constant, E a is the activation energy, and k is Boltzmann constant. The exponent has a theoretical value of 0.5 [11,17,18] while experimental values have been observed to vary between 0 and 1 depending on the processing conditions, thickness, etc. [5,7,9].
Although the TBD model in Eq. 3 provides the relationship between d and E b of the ferroelectric films, the electrode area is not included. According to the thermal models [5,6,16,19], heat generated during TBD can be effectively removed from the surface areas of the dielectric, leading to an increase in E b . This is in contrast to the experimentally observed decrease in E b with increasing electrode area. However, this discrepancy can be explained by considering the probability of finding a more defective column with increase in area [12]. In practice, due to the defective nature of the capacitors, a number of samples are measured to statistically determine the reliable E b values. For example, one widely adopted method to determine average Defect models based on probability start with a small volume of defect randomly distributed in the bulk of the material. If the ferroelectric thin film is divided into a large number of small cubes with an edge length of a representing the smallest defect size, the column (consisting of a stack of the cubes vertical to the surface) with the most defects can form a conducting path for the breakdown (Fig. 1a). This model is known as the weakest phase breakdown model [5,12,21,22]. In case of a larger electrode area, the probability of finding a conductive path under the electrode increases, leading to a reduction in E b . As mentioned previously, the typical defective failure analysis is usually based on testing samples by varying several parameters (T, d, and A) while keeping one of which a constant. In Weibull analysis, the cumulative probability (F) corresponding to E b is expressed as follows [20], where F is the sample cumulative distributive parameter and k is the fitting parameter.
Using Eq. 4 and Eq. 5, we get E A of a finite electrode area is smaller than that of a unit column since k is typically greater than that of an increasing failure rate [23]. As shown in Fig. 1b where c, , k are the fitting parameters, E a is the activation energy, and k is the Boltzmann constant. Eq. 7 describes the interdependence of three variables, d, A, and T, that determine E A of the ferroelectric films. For any of the two variables being constant, the equation is reduced to Eq.
3 and 6, which has been widely discussed [5,9,11,15].  The effects of thickness, area, and temperature on E A are fitted using individual models based on Eq. 3 and 6 and represented as solid red lines in Fig. 3a c. As can be seen in the figure, the individual equations effectively describe the dependence of E A on all three individual parameters with correlation coefficients between 0.88-0.99. It is observed that the of PLZT films follows the TBD model based on d and T similar to that observed by others [10,11,13]. The exponent ' ' calculated in Fig 3a, is ~0.393, which is close to that of reported for PZT [6,12]. The model can also be used for device optimization in specific applications. For example, the aspect ratio (diameter/thickness) of the thin film is essential to capacitors for energy storage application. For some energy storage properties such as U reco , the corresponding polarization hysteresis loops were measured with the maximum electric field at E A .
As shown in Fig. 4, the PLZT films exhibit polarization hysteresis loops with low coercive electric field (E c ) and high saturation polarization (P s ), typical of the relaxors which are prime candidates for energy storage applications [25,26]. Based on Eq. 1, high U reco is expected for materials with large E b , for a given permittivity. Similar to breakdown strength, U reco is also found to be strongly dependent on area and less so on temperature. Greater U reco is observed for samples with higher breakdown strengths as shown These fitting parameters of k, U are approximately twice of those obtained in Fig 3, indicating that U reco is proportional to the square of E A . This implies that the polarization linearly increases as a function of the applied electric field at fields much greater than E c . This square relationship agrees well with the fact that the PLZT is characteristically relaxor with nearly reversible P -E loops (E c ~0.02 MV/cm).
In conclusion, a unified equation has been developed on the relationship between the breakdown strength and related parameters such as the film thickness, electrode area, and temperature. The model has been found to agree well with the experimental results of the PLZT thin films in terms of all variables. Theoretical fits to the experimental data indicate the interdependence of the processing parameters. Polarization hysteresis loops of PLZT films are also obtained for predicting U reco based on the developed model. A unique relationship of U reco ∝ E A 2 is found, that is particularly useful in predicting the electric energy density in terms of variables d, A, and T.