3.1.3 Restrictions of Analytical Approaches

The charge-pumping techniques, described in the previous section, rely on analytical expressions which are approximate, but which enable efficient processing of experimental data in practice. In order to see what we can benefit from a rigorous numerical model of the charge-pumping experiment, we will discuss the general restrictions which are common to most of these analytical approaches:

1. An instantaneous response of the minority and the majority carriers to the variations of terminal biases has been assumed in the analytical models. A retarded response of the minority carriers has been found experimentally in the small-rectangular-pulse method in long channel devices [494] and in the large-signal techniques for very short off-switching times, when the so-called geometric current component is produced (noted as effect 2 in Section 3.1). Unlike for the minorities, the instantaneous response of the majorities is probably always fulfilled in bulk MOSFETs, because the dielectric relaxation time in the bulk (with specific conductivity ) is much shorter than the rise and fall times of the gate pulses. For example: assuming doping and , it follows for Si . The shortest and , applied in the charge pumping in practice, are always longer than . The retarded response is the topic of an extensive study in Section 3.4.
2. To account for the influence of the interface trapped charge on the surface potential (under transient conditions), the so-called charge-potential feedback effect [149], a system of integro-differential equations must be solved. This approach does not permit any analytical solution in the general case.
3. An exact determination of the energy interval which contributes to the charge-pumping current is difficult in the general case. The local charge-pumping threshold voltage and the charge-pumping flat-band potential depend on the actual trapped charge in transient conditions, local dopant concentration and the characteristic time intervals involved in the definition of these voltages (introduced in Section 3.3).
4. A general analytical model which accounts for the capture in the both strong and weak inversion and/or strong and weak accumulation cannot be given in a closed form. The only model in the literature [49] consists of a system of implicit algebraic equations.
5. Additional two-dimensional effects which make any analytical approach hard to deal with:
   • The charge-pumping threshold voltage and the charge-pumping flat-band potential vary along the interface, particularly in the junctions, due to lateral variation in the dopant concentration (Figure 3.30). Moreover, after nonuniform degradation (due to hot-carrier stress), high densities of the localized interface states and fixed oxide charges can produce very large shifts of local threshold and flat-band voltages. As a consequence of the previous effects, the emission times and, therefore, the energy interval which contributes to the charge-pumping current, are position dependent in some charge-pumping techniques. Additionally, due to a large local shift of the potential, a part of the interface can be screened, thereby it becomes unavailable for the free-carrier flow [196]. A combined effect of the high local densities of both, interface traps and fixed oxide charges frequently takes place in practice. It is difficult to interpret the combined effect, even qualitatively in some cases.
     Speaking in general, when reducing the channel length the charge-pumping current becomes more and more influenced by two-dimensional effects within the source and drain junctions. Because the analytical approaches are invalid in these regions, the interpretation of the experimental charge-pumping characteristics becomes more qualitative [196][194].
   • The coupling between the front and the back interface in thin-film SOI devices [357] and the spatial nonuniformity observed in those devices [116].
   • The lateral nonuniformity produced by the lateral current flow, as described in [49]. In my opinion, this effect could only have an influence (if any) in those methods which are sensitive to the minority-carrier capture.
6. As a rule, a very simple physical description of the interface traps is assumed in analytical charge-pumping models. A typical engineering model used for traps in MOS devices are the Shockley-Read-Hall (SRH) equations with constant capture cross-sections for electrons and holes (Section 3.2). It is believed, however, that a strong energy-dependence holds for the interface traps near the band edges [331]. This dependence has been included in the models [397][357][49] and experimentally confirmed by the charge pumping in [397]. The model for dispersion of the capture cross-sections proposed by Preier [376] has been included in some works [52][51] also. Three reasons are, probably, responsible for such a state of the art in the charge pumping modeling:
   1) inclusion of complex trap models makes an analytical approach difficult,
   2) the present charge-pumping techniques cannot resolve the fine properties of interface traps. Accounting for the physically more accurate models does not seem to be necessary. The situation, however, changes when very small MOS devices which only contain a few active traps are studied by using the charge pumping.
   3) due to the complex physics of interface traps and their different characteristics observed in experiments (a short review is given in [411][241]), no simple and unique (a priori) model of the SiO-Si interface has been established so far.
   As a matter of fact, unlike for the bulk trap, the energy-position of an interface trap with respect to the band edges can change with the surface electric field, if the trap is located within the oxide and not exactly at the interface ([381]). Due to the changing trap energy with the surface field, the emission-time constant becomes gate-bias dependent, while the capture-time constant can deviate from the typical inverse proportionality to the surface carrier concentration. The last effect is determined by a complex local selfconsistent interaction between the charge in the trap and the charge in the channel (screening of the trap by the channel-charge included). It is modeled analytically in [411] and numerically in [410]. A temperature activation with the large activation barriers for the electron emission and capture is reported in [411][251][155], whereas a negligible activation for the hole capture and emission is found in [241]. The activation depends strongly on the type of trap. These observations should be taken into account while interpreting the charge-pumping current at different temperatures ([97]). A barrier-lowering for the Coulomb potential induced by the trap located at the interface, due to the Frenkel-Poole effect, is analyzed in e.g. [388]. The influence of the surface field can be formally modeled either by a field-dependent capture cross-section at a given energy level , with , or as a shift of the trap energy level relative to the band edges with constant .
   Note that up to now there is no evidence that the SRH balance equations cannot be applied to the interface states. Still questionable is the nature of the capture and the emission transitions, namely the microscopic model for .

   A finite time for the capture event is neglected in the Shockley-Read-Hall theory [409][179] (clearly pointed out in [409]). An extension of the SRH equations to include this effect is proposed in [99]. An inclusion of quantum effects in direction perpendicular to the interface is done in [432][431]. The authors have formulated an extension of the Shockley-Read-Hall theory and have experimentally determined the electron capture cross-section, both consistent with the 2D quantum model of the inverted MOS interface [12]. Moreover, they demonstrated a case where quantum effects take place thereby influencing the charge-pumping current!

7. The analytical models, of course, do not contain any information on parasitic effects. The charge-pumping techniques proposed in the literature have not been confirmed by a rigorous numerical model, so far. In other words, it is presently unknown, whether our understanding of the processes is correct, or whether some non-ideal effects take place in these techniques.

An one-dimensional transient model of the MOS structure selfconsistently including trap dynamics can account for the restrictions discussed under 2, 3, 4 and 6. Such a rigorous numerical model is presented for the MOS capacitors by T.Collins and J.Churchill in [87] (see also e.g. [230][149]) and for MOSFETs by G.Ghibaudo and N.Saks in [144] where an instantaneous response of both, the minority and majority carriers to the gate-bias variations is assumed. Using this numerical model the large-signal charge-pumping methods have been studied in [395][145][144][11].

A two-dimensional (2D) transient model including dynamics of bulk traps is presented by J.McMacken and S.Chamberlain in [305], although it is applied to amorphous-Si thin-film transistors and by H.Yano et al. in [519], who employed it to simulate the deep-trap effects in time domain in compound devices. According to my knowledge, the only 2D transient model which accounts for the interface-state dynamics, proposed to simulate the charge pumping in MOSFETs is given by F.Hofmann and W.Hänsch in [205].

The approach presented in this work is based on the selfconsistent numerical solution of the time-dependent basic semiconductor equations including the trap-dynamics equations, assuming arbitrary voltage pulses at the terminals. Arbitrary interface and bulk trap distributions in energy and position space can be specified. After the simulation (lasting sufficient pulse periods) the charge-pumping current is calculated as the DC value of the electron and the hole components of the terminal currents, just as in experiments where the DC component of the terminal currents is measured. Using this rigorous approach the present charge-pumping techniques are critically evaluated from a different point of view.

In Section 3.2 the physical model and its implementation in MINIMOS ([415][160]) are presented. Analytical modeling of the charge pumping is considered in Section 3.3. The geometric component and other non-ideal effects in the charge-pumping experiment are investigated in Section 3.4. The charge-pumping characteristics for virgin and electrically stressed devices are calculated and discussed in Section 3.5 and Appendix D. A study of the degraded region in MOSFETs after hot-carrier stress is the subject of Section 3.5. Finally, the results are summarized and some further potential applications of the model are suggested in Section 3.6.