Simulation of the Variability in Next-Generation Microelectronic Capacitors with Polycrystalline Dielectrics
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Simulation of the Variability in Next-Generation Microelectronic Capacitors with Polycrystalline Dielectrics Jesse L. Cousins and David E. Kotecki Department of Electrical and Computer Engineering, University of Maine 5708 Barrows Hall Orono, ME 04469-5708 ABSTRACT Monte Carlo simulations of capacitors with polycrystalline (Bax, Sr1-x)TiO3 (BST) dielectrics were performed. The variation in capacitors due to the polycrystalline microstructure of the dielectric was investigated, as well as the effects of varying the distribution of crystal sizes. When a lognormal probability density function (pdf) was used to approximate the crystal area pdf and the average number of crystals per capacitor was near 100, it was found that the minimum capacitance value was nearly independent of the standard deviation of crystal area distribution. Both the mean and maximum capacitance values were found to increase as the width of the standard deviation increased. INTRODUCTION As the minimum feature size of integrated circuits continues to shrink, it is becoming difficult to scale down the area of on-chip capacitors while leaving the value of capacitance unchanged. The minimum thickness of a dielectric film is limited, so a material with a high dielectric constant is desired. Paraelectric dielectrics such as BST or SrTiO3 (STO) have higher permittivities than currently used dielectrics such as SiO2 or Si3N4. These polycrystalline dielectrics are a major change from currently used amorphous dielectrics since the microstructure could introduce variations in capacitance that are not present with amorphous dielectrics. Many characterizations of these films have been performed [1, 2]. The dielectric constant of these films has been found to increase with increasing crystal area. No one has investigated how variations in the dielectric constant with crystal area will affect capacitors made using these films. In the present study, published data on BST crystal area and dielectric constant were used to create a model for a BST thin film. The standard deviation of the crystal area pdf was varied while the average crystal area was held constant. These distribution functions were then used in a Monte Carlo simulation to generate constant area capacitors. The resulting capacitor variations were then analyzed. SIMULATION BST Crystal Size Distribution and Capacitance To model capacitors with polycrystalline dielectrics, it was necessary to generate a probability density function for the crystal area, as well as equations for how capacitance varied with crystal area. Cross-sectional TEM images of BST thin films [1] were manually measured to generate an array of crystal sizes. This array was sorted and cumulatively summed to generate a
C7.24.1
cumulative density function of crystal area (1), where FX(x) is the cumulative density function and x is the crystal area in nm2: FX ( x) = −(4.0 × 10 −9 ) x 4 + (6.1 × 10 −6 ) x 3 − (2.9 × 10 −3 ) x 2 + 0.41x + 10
(1)
The derivative of this function is the crystal area pdf. In order to study the effects of va
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