Recent studies have demonstrated that in modern deca-nanometer devices the variability due to random discrete dopants (RDD) and oxide defects has become critical in the context of reliability [100, 101, 102, 103]. For instance, discrete charges in the oxide or at the interface lead to the occurrence of potentially huge ΔVth shifts due to the non-uniform current flow in the channel (Figure 5.10). RDD needs to be considered when investigating the ‘step heights’ in the drain current and threshold voltage observed in field effect devices under bias temperature or hot carrier stress. Most studies have considered either fixed positive/negative charges or described charge capture into a fixed number of defects by standard SRH trapping kinetics [104], where the goal is to explain the statistical distribution of ΔVth shifts in an ensemble of devices (cf. Figure 5.9).
In this section the mere electrostatic effects of random discrete charges (traps) and random dopants on the drain current are discussed. A discussion on the actual number of available defects in a time-dependent manner and their activation process itself is delayed to Chapter 6.
| Figure 5.9: | The cumulative distribution functions (CDF) of ΔVth over gate overdrive for 500 different microscopic p-channel MOSFETs due to RDD only (no traps). Clearly visible is the large deviation in maximum step-heights of the CDFs over gate overdrive. This is due to the emergence of new percolation paths with higher gate overdrive. |
A
C
B
D
| Figure 5.10: | The percolation path in a selected microscopic MOSFET due to random discrete dopants and equi-potential surfaces. In sub-figure A there are no oxide traps and a single percolation path dominates. Upon the the formation of a fully charged trap right above the dominant percolation path, the device is switched off and the dominant percolation path vanishes (sub-figure B) leading to a huge ΔVth. When an oxide trap is being charged next to the percolation path (sub-figure C), the trap has a negligibly small influence on the current flow and causes only a small ΔVth. Sub-figure D shows the formation of six fully charged traps perpendicular to the current flow. In such a case the device becomes much harder to switch on. |
The effect of a single charged oxide defect on the drain current in a MOSFET can be explained without considering the time dependence of its occupancy and has been studied in depth using a first-order quantum corrected drift diffusion model [103]. Depending on the spatial location of the trap the resulting ΔVth, that is its ‘step height’, varies. In the course of this thesis, together with the author of [90], this was investigated for planar devices.
In order to obtain the spatial dependence of the ΔVth for a single defect for a single microscopic device (constant dopant configuration), first a discrete random dopant configuration is determined (cf. Section 5.1.1) for the planar MOSFET under investigation. Then for each position of a single discrete elementary oxide charge in a predefined grid a single device simulation is carried out to determine the Id - Vgcurve. At last these Id - Vgcurves are compared to the Id - Vgcurve for the device without any oxide charges. In Figure 5.11 the ΔVth maps for a planar p-channel MOS, with a channel doping of ND = 1018cm-3, a channel length and width of 35nm and an effective oxide thickness (EOT) of 1nm silicon-dioxide are shown.
| Figure 5.11: | Spatial distribution of ΔVth in 35nm×35nm p-channel MOSFET, with a channel doping of ND = 1018cm-3. To generate these maps 1500 simulations of Id - Vgcurves with Vds = 1mV have been carried out, where in each simulation the elementary point charge has been moved along the channel surface and was located directly at the interface. |
The ΔVth maps are a practical tool to show the trap position dependence of the ΔVth caused by a single elementary charge, however they are not suitable to compare simulation and measurement. For this the cumulative distribution function, obtained by considering a statistical representative ensemble of devices, of the individual step heights is used. In order to obtain the cumulative distribution of the step-heights of a statistically representative sample of microscopic devices has to be simulated. In Figure 5.12 the cumulative distribution function of the ‘step heights’ for a single trap in a p-channel MOSFET is plotted. From the plot the dependence of the step-height maximum on gate bias and channel doping is clearly visible. Figure 5.13 finally shows that in order to describe the occurrence of large steps in Vth a microscopic description of the doping is necessary by comparing the CDFs for RDD and continuous doping.
| Figure 5.12: | The CDF of the ΔVth in a 35nm×35nm p-channel MOSFET due to a single trap for two distinct channel dopings. It can be seen that a higher channel doping results in bigger ΔVth steps for a single charge at the interface. This is due to the increasing number of dopants in the channel, when the average channel doping is higher. Each graph is composed from a thousand different simulations using MinimosNT to have a statistically meaningful sample size. |
| Figure 5.13: | The data from Figure 5.12 at Vg = Vth compared to the step heights obtained if the doping is continuous and not discrete. Clearly visible is the need for RDD in order to correctly describe the large dispersion in step heights, which cannot be obtained using a continuous doping. It is also remarkable that the macroscopic doping concentration has virtually no influence on the maximum step height for continuous channel doping. Each graph is composed from a thousand different simulations using MinimosNT. |
In real devices there are often a number of oxide traps. In sub-deca micrometer devices the oxide volumes are so small that only a few traps, if any, can be observed in a single device [105]. When considering multiple traps and RDD the question arises whether or not two discrete traps have an influence on the ΔVth caused by one another [4]. It was found by [106] that two or more traps act independent of each other, at least electrostatically. The CDFs of ΔVth for multiple traps are comparatively plotted in Figure 5.14.
| Figure 5.14: | The cumulative distribution of step heights for multiple evenly distributed traps in a 35nm p-channel MOSFET with a channel doping of ND = 1018cm-3. It is obvious that the maximum step height increases with the number of traps. More interestingly no ΔVth step above 30mV could be observed even for 21 oxide traps, which corresponds to a trap concentration of 1.7⋅1019cm-3. |
Comparing simulation data (DG and DD), to measurement data it was observed that the doping dependent mobility model needs to be slightly adjusted. Most authors have used constant homogeneous mobilities in their random dopant studies, see for example [107]. It was later found in [98, 108] through classical Monte Carlo simulations that random discrete dopants not only cause potential fluctuations, but also spatial fluctuations in the scattering rates. For moments based charge transport models it was shown by [109] that it is sufficient to adjust the doping dependent mobility model parameters such that the bulk-mobility of a resistor is the same when simulated with continuous and discrete doping. In the course of this thesis it was found that this is not quite correct. It is possible to reproduce measured mean and standard deviation in Vth due to random discrete dopants only [110]. However, it was found impossible to reproduce measurements taken on an ensemble of p-channel MOSFETs by simulation with quantum corrected drift diffusion taking into account discrete traps and random discrete dopants. The devices were p-channel MOSFET with a channel doping of ND = 1018cm-3 produced by imec. The measurements have been taken with great care to ensure that only a single trap is created per device. The microscopic device for simulation was calibrated to the mean Id - Vgcurve of the undegraded imec devices. From both sets of data, namely measurement and simulation, the cumulative distribution functions of distinct step heights are shown in Figure 5.15. To highlight the amplification of the defect impact due to the random dopants and to be able to compare MOSFETs of slightly different geometries, we normalize the step height by the theoretical value obtained through the charge-sheet approximation assuming that each trap is located directly at the interface:
 | (5.3) |
where Cox is the nominal gate oxide capacitance. Surprisingly the simulation cannot reproduce the measurement by a factor of 5 in ηr. This suggests that the purely electrostatic picture to determine step heights is not entirely correct as previously reported, for example in [98].
| Figure 5.15: | Comparison of simulation (solid lines) and measurement data (dashed lines) for a p-channel MOSFET with a single occupied trap, where ΔVth has been normalized using the charge sheet approximation. Various CDFs as a function of gate overdrive as well as the CDF for 21 traps from Figure 5.14 are plotted. None of the simulation data, even the one for 21 traps, are close to the measurement and are off by a factor 5. This is a hint that the purely electrostatic picture to determine step heights is not entirely correct. The measurement data has been provided by imec. |
Not even the largest step heights, caused by a single trap, can be reproduced. Large step heights are, in the simulation, due to huge fluctuations in the potential leading to a dominant percolation, which can be ‘blocked’ by a single trap causing a huge step in drain current. Recalling that ΔVth is estimated by comparing Id -Vgcurves or by calculating ΔVth from measured steps in the drain current ΔId by a SPICE level 1 model [111], the only parameter that has not been accounted for is charge carrier mobility μ. As stated initially, in [98] it was shown that a quantum corrected drift diffusion simulation underestimates the standard deviation in current fluctuations caused by random discrete dopants or random discrete charges by at least 20%, when compared to solutions obtained from a calibrated Monte Carlo simulator incorporating RDD. Thus deviations seen in Figure 5.15 can be minimized by incorporating fluctuations in mobility on a macroscopic level or ionized impurity scattering on a microscopic level. This deviation of 20% explains the difference between measurement and simulation using DD and DG models.