Logistic Modeling of Progressive Breakdown in Ultrathin Gate Oxides E. Miranda * , L. Bandiera, A. Cester and A. Paccagnella Dipartimento di Ingegneria dell’ Informazione, Università degli Studi di Padova, via Gradenigo 6B, 35131 Padova, Italy. * Present address: Facultad de Ingeniería, Universidad de Buenos Aires, Paseo Colón 850, (1063) Buenos Aires, Argentina. (emirand@fi.uba.ar) Abstract: The sigmoidal behavior exhibited by the current-time characteristics of constant voltage stressed MOS capacitors with ultrathin oxides is ascribed to a self- constrained increase of the leakage sites population that assist the conduction process between the electrodes. To analytically describe this dynamical process we consider a classical model of population growth theories such as the Verhulst differential equation. The role played by the background tunneling current in the detection of the breakdown event is also discussed. 1. Introduction Recently, it has been reported that the degradation dynamics of ultrathin (sub-2 nm) gate oxides in metal- oxide-semiconductor (MOS) devices subjected to electrical stress exhibits different features than those observed in thicker oxides [1,2]. The occurrence of distinctive breakdown modes in these latter oxides, such as the so-called soft and hard breakdown, not only is easily recognizable by abrupt changes in the magnitude of the leakage current, but also by the shape of the post- breakdown current-voltage (I-V) characteristic [3]. On the contrary, thinner oxides may not exhibit a sudden change of conducting properties but a gradual increase of the leakage current, which has led to name this conduction mode as progressive breakdown (PBD) [1]. In addition, when the oxide thickness is around or below 2 nm, the magnitude of the tunneling current flowing through the undamaged area of the device is comparable with the excess leakage current itself. This fact was shown to set a detection window that can mask the true onset of the failure event [4]. This is a major concern from the reliability viewpoint, because this would mean that an oxide might be locally damaged without showing previous anomalous external signs. Moreover, combined electrical and optical experiments have demonstrated that the size of the breakdown spot evolves with the stress time, thus becoming potentially dangerous for a particular application [1,5]. To our knowledge, modeling of PBD dynamics has not been analytically faced until the recent work of Hosoi et al. [6]. These authors have accomplished to fit the excess leakage current by means of an iterative scheme that incorporates the redistribution of the potential drops along the device as the damaged area increases. This involves an empirical power-law model for the current- time ( I-t ) characteristic of the form I=at b , where a and b are constants, together with the point contact resistance of the breakdown path across the oxide layer. Instead, we consider a systemic approach in which the damaged zone of the device is formed by an ensemble of many leakage sites, whose population evolves over time according to a self-constrained growth dynamics [7]. We will show that a suitable mathematical device arising from population growth theories with limited resources, such as the Verhulst differential equation, is able to account for consistently the behavior exhibited by our p- and n-type substrate samples. Fig. 1. Typical I-t characteristics measured on p- and n-type substrate samples during a CVS at ±3.75 V. The arrows indicate the breakdown event detection considered as the onset of the logistic growth. 0 2 4 6 8 10 1.0x10 -4 2.0x10 -4 3.0x10 -4 6.0x10 -4 7.0x10 -4 8.0x10 -4 t=4.1 seg t=680 mseg P-type, V g =-3.75 V N-type, V g =+3.75V Current [A] time [s]