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Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

1.3 Time Dependent Dielectric Breakdown

Silicon dioxide, the native oxide of Si, has been traditionally used as dielectric in Si based devices due to its large bandgap and its high breakdown field. However, application of stress, such as high electric field, for long stress times can cause the dielectric to loose its insulating property. Moreover, the continuous down-scaling of transistors causes high electric fields across the dielectric which further increase the risk of failure. Dielectric breakdown can be intrinsic or extrinsic. Intrinsic breakdown occurs due to the structure of the dielectric material like crystal defects, dielectric thickness, etc. Extrinsic breakdown, on the other hand, can be caused by introduction of defects during the technological steps to manufacture the device.

Time dependent dielectric breakdown (TDDB) is observed in devices which use silicon dioxide or high-K materials. TDDB refers to the transition of the insulator from its insulating phase to a conductive phase with time when a constant electric field, smaller than the breakdown field, is applied. While the complete molecular picture is still speculative, the phenomenon is often attributed to the polar nature of the dielectric [92]. In \( \mathrm {SiO_{2}} \), local electric fields cause distortion in the silicon-oxygen bond leading to the possible creation of an oxygen vacancy (dangling bond) via bond breakage or coordination breakage.

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Figure 1.8: The hole-catalyzed, field-enhanced bond breakage process. A hole-capture event weakens the bond which can then be broken thermally or by a high electric field, thus, creating a defect. From [93] .

The coordination breakage refers to displacement of the Si ion from the fourfold coordination to a threefold coordination in the tetrahedral \( \mathrm {SiO_{2}} \). The coordination breakage requires much less energy than bond breakage and is considered the primary mechanism behind TDDB [93]. TDDB stress leads to generation of traps which create a percolation path, as shown in Figure 1.6, which triggers the breakdown of the insulator [94]. Time dependent dielectric breakdown can only be described statistically as nominally identical devices subjected to the same stress break down at different times. This is also the case for BTI and HCD in scaled devices. Thus, the failure probability has to be described statistically. This description allows to extract the device lifetime [95].

Breakdown in oxides thicker than 5 nm occurs abruptly after a certain electric field. Thus, a large jump in the voltage/current vs. stress time curve is observed. This effect is known as hard breakdown. In thinner oxides, on the other hand, soft breakdown occurs where the change in voltage/current with stress time is not very abrupt and the magnitude of change is many orders lower than in hard breakdown [43, 96].

Various TDDB models which consider field-induced degradation, current-induced degradation or a combination of both have been suggested [93, 97, 98, 99]. The field dependent models suggest the logarithm of time to breakdown to be proportional to the oxide electric field. One of these models is the thermochemical \( E \)-model. Within this model, bond breakage is triggered by the interaction of the Si-O bond, which is strained due to the local field (see Figure 1.7), with the lattice [97]. Thus, the bond breakage is considered to be a phonon driven process. The time to breakdown is given as \( t_{\mathrm {BD}}=A_{0}\exp \left [\gamma E_{\mathrm {ox}}\right ]\exp \left [E_{\mathrm {a}}/k_{\mathrm {B}}T\right ] \). However, the \( E \)-model was unable to explain the polarity dependence of TDDB [100].

The current driven model is the so-called \( 1/E \) model (the logarithm of time to breakdown is inversely proportional to the oxide electric field) where the damage is associated with the current flowing through the dielectric [93]. According to the \( 1/E \) model, the electrons tunneling through the dielectric from the cathode to the anode excite an electron from the valence band to the conduction, and leave a hole behind [101, 102]. On gaining energy, the holes can tunnel back into the oxide and create oxide traps. Thus, this model is also called Anode Hole Injection model (AHI). Emission of carriers through SiO (math image) is due to Fowler-Nordheim (FN) tunneling which requires high fields (>6 MV/cm) [103]. The \( 1/E \) dependence of time to breakdown is given as \( t_{\mathrm {BD}}=\tau _{0}\exp \left [G/E_{\mathrm {ox}}\right ] \). If deep traps exist, trap assisted tunneling can lead to pre FN conduction. Since FN conduction is temperature independent, the current model was unable to explain the temperature dependence of TDDB [104]. Moreover, according to this model, breakdown occurs when a critical hole fluence is reached which has no physical explanation [105].

Other models for specific devices, such as the power-law voltage model [106, 107, 108] for thin dielectrics with transport in the ballistic regime and the exponential model [109, 110] for low-k dielectrics, have also been proposed. It has been suggested that TDDB can be better explained by a combination of field and current driven approaches [42, 111]. This combination suggests that the field strains the Si-O bonds which can then be excited by the carriers constituting the current flow. Figure 1.8 shows one of these scenarios where the bond breakage process is hole-catalyzed and enhanced by the field. Thus, the activation energy is reduced and the bond can be easily broken by phonon interactions. The combination of both field- and current-based models seems the most practical solution for modeling TDDB.