Modeling of Dispersive Transport in the Context of Negative Bias Temperature Instability Tibor Grasser ∗ , Wolfgang G¨ os ∗ , and Ben Kaczer ◦ ∗ Christian Doppler Laboratory for TCAD at the Institute for Microelectronics, Gußhausstraße 27–29, A-1040 Wien, Austria ◦ IMEC, Kapeldreef 75, B-3001 Leuven, Belgium Abstract— Negative bias temperature instability (NBTI) is one of the most serious reliability concerns for highly scaled pMOSFETs. It is most commonly interpreted by some form of reaction-diffusion (RD) model, which assumes that some hydrogen species is released from previously passivated interface defects, which then diffuses into the oxide. It has been argued, however, that hydrogen motion in the oxide is trap-controlled, resulting in dispersive transport behavior. This defect- controlled transport modifies the characteristic exponent in the power-law that describes the threshold-voltage shift. However, previously published models are contradictory and both an increase and a decrease in the power-law exponent have been reported. We clarify this discrepancy by identifying the boundary condition which couples the transport equations to the electro-chemical reaction at the interface as the crucial component of the physically-based description. I. I NTRODUCTION Amongst the various reliability issues in modern CMOS technol- ogy, negative bias temperature instability (NBTI) has been identified as one of the most serious concerns for highly scaled pMOSFETs [1–4]. Recently, a lot of effort has been put into refining the classic reaction-diffusion theory [2, 5, 6] originally proposed by Jeppson and Svensson nearly thirty years ago [7, 8]. The RD model assumes that Si–H bonds at the interface are broken at higher temperatures and electric fields, causing the released hydrogen species to diffuse into the oxide. Analytic solutions of the RD model can been shown to follow a power-law [2] ΔV th (t) ∝ ΔNit (t)= A(T,Eox) t n , (1) where the change in the threshold voltage is often assumed to be proportional to the change in the silicon dangling bond density Nit = [Si • ] at the interface. As diffusing species H2 is often assumed because in the RD framework it results in a characteristic time exponent of 1/6 for the threshold shift, consistent with recent no- delay measurements [9], while H 0 and H + result in n =1/4 and n =1/2, respectively [5]. During the last couple of years a variety of alternative explanations for NBTI have been put forward [3, 10–13]. In particular it has been argued that transport of the hydrogen species inside the oxide is dis- persive [2, 10–12], consistent with hydrogen diffusion measurements and available models for irradiation damage. Interestingly, in these models the slope depends on a temperature-dependent dispersion parameter. Also, it is possible to incorporate technology-dependent behavior into the model by adjusting the dispersion parameter. In addition, these models brought H + back into the game, which had originally been dismissed due to the 1/2 slope resulting from the RD model. One feature common to trap-controlled dispersive NBTI models is that they predict that dispersive diffusion reduces the slope compared to their conventional counterparts [10–12]. However, in contrast to that it was observed that inclusion of traps into a standard RD model increases the slope [5]. In addition, our own simulations showed [14] that a straight-forward application of the multiple-trapping transport Et Et ρ(Et) Ec E d (t) Emin DOS g(E t ) Deep Traps Shallow Traps Fig. 1. Schematic illustration of dispersive transport. Particles in the conduction band fall into the traps and are re-emitted into the conduction band. Re-emission is more likely for shallow traps. The time-dependent demarcation energy separates shallow from deep traps. With time, the demarcation energy becomes more negative, until the bottom of the trap distribution is reached and equilibrium is obtained. As a result, the motion of the particle packet slows down with time. Note how the individual trap levels, which microscopically correspond to the different bonding energies of hydrogen in an amorphous material, are approximated by a macroscopic density-of-states. model [15, 16], which is the basis for many dispersive transport models [17], increases the slope, also in contradiction to published reaction-dispersive-diffusion models [2, 10–12]. A detailed analysis reveals that the boundary condition at the Si/SiO2 interface is the main reason for this discrepancy. It is shown that the choice of the boundary condition is essential for the overall behavior of the dispersive system. II. DISPERSIVE TRANSPORT In contrast to drift-diffusion transport, dispersive transport is trap- controlled. This implies that most particles are trapped, that is, bonded. Depending on the bonding energy (the distance to the ’con- duction band’) hydrogen can be easily released from shallow traps but have large release times from deep traps. This is schematically illustrated in Fig. 1. Reaction-dispersive-diffusion models proposed so far have relied on simplified transport models developed either for the time evolution of an initial hydrogen profile after an irradiation pulse [11, 12] or on a phenomenological time-dependent diffusivity as observed in hydrogen diffusion and annealing experiments [18]. The applicability of these simplified equations to the problem at hand has not been rigorously assessed and, as we will show in the following, contains some pitfalls: First, during NBTI stress, one has to deal with a continuous influx of particles which has to be properly accounted for in the boundary condition of the model equations. Care has to be taken to inject the particles into the mobile state only, an issue often not obvious in simplified equation sets. Second, the reverse rate of the depassivating reaction depends on the concentration of available hydrogen at the interface. Here, one might have to consider 2006 IIRW FINAL REPORT 1-4244-0297-2/06/$20.00 ©2006 IEEE 5