5.1 Backscattering Theory
In the backscattering theory, we use the wave concept to describe the carrier transport in the channel. As stated in backscattering theory [19]-[20], the nanoscale device performance is limited by the injection velocity and the backscattering coefficients. In this study, if the channel is under low electric field conditions, the width of the kBT layer calculated according to its definition is wide enough to be larger than the channel length L. The backscattering coefficient can be presented from
C( eff) L r low E
L λ
= + (34) where λ is the mean-free-path and L is the channel length. When the channel is under high electric field, <L, the backscattering coefficient can be estimated from
r high EC( eff)
= λ
+ (35) In our model, the channel is assumed to be operated under high electric field. In other words, we will discuss the devices operated in the saturation region. From Fig.20, we can derive the drain current easily and the drain current can be written
Ids =W J[ +(0)−J−(0)]=qW F[ +(1−R)−F−(1−R e) −qV kT/ ] (36) where R is the backscattering coefficient and T(=1-R) is the transmission coefficient. In the saturation region, the value of drain voltage will be higher than thermal voltage. So Eq. (36) can be modified as below
Ids =W J[ +(0)−J−(0)]=qWF+(1−rc) (37)
Combining (37) and (38), we can derive the formula as follows whereυinjare the thermal injection velocity at the top of source-channel junction barrier. In (39), the drain current is related to the backscattering coefficient[21]. In the mismatch model, we also can obtain the formula like Eq. (1) and Eq. (6). The mismatch model in above threshold region will be discussed later and we should characterize some parameters.
5.2 Analysis and Model
Based on backscattering theory, (39) is constructed except low drain voltage. Since the region is operated under high drain voltage. On the other hand, the main region for analog circuits is controlled in the saturation region. These two conditions will confine the region of our research. That is to say there will be some limitations when we extract the data.
In our research, there are several factors used to modify our model such as DIBL and Rsd.
Fig. 21 shows the flowchart for the procedure of extracting rc. Now we propose a new simple statistical model to quantitatively account for the above observed dependencies of the mismatch in the above threshold region on the gate-to-source bias. Eq. (40) the mismatch of the current,
Ids
σ , can be derived as a function of the coefficients of variance of the parameters :
the coefficient of the variance in the threshold voltage,
Vth
σ , the coefficient of the variance in the drain-induced-barrier-lowering σDIBL, and the coefficient of the variance in the channel backscattering coefficient
rc
σ :
( )
In the above formula, we neglect the effect of source-and-drain series resistance. However, we will show the mismatch differences between models withand without Rsd.This new formulation describes the dependence of
Ids
σ with varying gate voltage in the above threshold voltage is very small. Fig. 24 shows that we use the backscattering mismatch model to reproduce the coefficient of the variance of drain current versus gate voltage over 0.4~0.5V at drain voltage of 1V. It can be found that the differences between the calculated results and experimentally extracted values are small.
5.3 Devices Operated in above Threshold Region
From the scatter plot of the measured near-equilibrium threshold voltage versus the reciprocal of the square root of the gate area at the mask level, we can see that a well known inverse square root of area law can apply:
th The size law also remains effective for the DIBL case, as shown in Fig. 22. The physical origins of the underlying proportionality constants and can be connected to the statistical dopant fluctuation as stated previously.
Vth
A ADIBL
However, we can make sure that whether the backscattering coefficient mismatch data are able to be described by the size law or not. So we will list several cases to discuss the possible effects on the backscattering coefficient and hence determine the most fitting one in our research. First, we assume that rc mismatch obeys the size law:
c The results can be shown in Fig. 23-1. Second, rc mismatch can be presented as being reversely proportional to effective gate length:
c The corresponding results are shown in Fig. 23-2. Finally, a new dimension dependent matching relationship is produced for the backscattering coefficient:
c
The results are shown in Fig. 23-3. In the above three conditions, we will compare the accuracy of each proposal model and then select a best one. It is a straightforward task to derive a backscattering-based mismatch version of Eq. (41):
The results derived in Eq. (46) are shown in Fig. 24, Fig. 25 and Fig. 26 respectively. With the experimental means of the underlying random variables and known proportionality constants as input, the drain current mismatch was calculated using Eq. (46) and fairly reasonable agreements with the experimental data were achieved for the gate and drain voltages and effective mask gate lengths and widths under study. Obviously, as the gate length decreases the quantity of the rc term decreases through the enhanced carrier transmission across the channel.
5.4 Conclusion
In order to establish an accurate model based on backscattering theory, we try several models to obtain reasonable current mismatch. Through the comparison between three probable conditions, we can prove that the mismatch model as feasible for some conditions.
Consequently, a new dimension dependent matching relationship is produced for the rc case from Eq. (45). The corresponding proportionality constant is 0.00202μm
rc
A 1.5. With the
experimental means of the underlying random variables and the known proportionality constants as input, the drain current mismatch was calculated using Eq. (46) and fairly reasonable agreements with the experimental data were achieved for the gate and drain voltages.
The drain current model in saturation based on backscattering theory is performed more accurately than the traditional model in the nanoscale devices. We extract the parameters in a wide range of long channel to nanoscale channel MOSFETs and successfully use the new mismatch model to reproduce the experimental current mismatch.
Chapter6 Summary
In the beginning, we have discussed the MOSFETs operated in the subtrhesold region. We have found that back-gate reverse bias may cause unexpected large variations of the circuit specifications. The extracted variations in the associated process parameters have been found to follow the inverse square root of the device area.
Step by step, we have found that the mismatch coefficient can be written as a combination of
Vth
A Aγ and . Here the
Vfb
A Aγ and have been already determined according to the mismatch model in subthreshold region. We have also discussed the impact of short channel devices. Because of the importance of series resistance and overlap length, we have taken such parameters into account and have employed specific methods to gain values for both the series resistance and overlap length. Series resistance will be slightly different among the samples due to the variations of the process, especially for large channel width. The overlap length is a rough approximation to stand for the impact on the effective length. Finally, the devices operated in above threshold voltage have been addressed based on the backscattering theory. We have used this theory to establish a new mismatch model. The key point for this model is that how to determine the relationship between r
Vfb
A
c and device size.
Fortunately, we have a reasonable method to explain the mismatch model.
A more suitable model is used for our mismatch model. Indeed, we can find a satisfying result from the figure previously. The drain current model in saturation based on backscattering theory is performed more accurately than the traditional drain current model in nanoscale devices. We successfully use the new mismatch model to reproduce the experimental current mismatch.