device distance Fig. 2-6 illustr

an take this graph as a part of Fig. 2-7, which is the same kind of graph but in longer distance. Analogy to the small signal analysis in the circuit theory, the macro variation just likes the range near the bias point and appears in piecewise linear form, while the micro variation can be taken as the noise. In order to identify the effects of macro and micro variation, the parameters differences of mutual devices under certain distance are divided with several groups according to the distance between two devices. In previous studies [5], the averages of parameters differences stand for macro variation of LTPS TFTs, while the standard deviation of parameter differences shows the micro variation in the devices. These figures, from Fig. 2-8(a) to Fig. 2-9(c), show the average and the standard deviation of parameters differences of N-type and P-type TFTs. As the mutual device distance increases, the deviations of these parameter differences almost do not change with the device distance. It can be explained that the micro variation will merely vary with distance as we expect. As for the macro variation, these figures show the diverse results. In the graph of Vth difference and S.S. difference, the averages are increasing with device distance.

However, the average of the Mu difference is decreasing when the distance of mutual devices is increasing. Although the averages of the differences of these parameters show different behaviors, they still appear in linear form. On the other hand, the

effects of variation in a range are still minor than those of the micro variation under short device distance.

In general, the macro variation results from the issues of process control, such as gate

2-3-3. The models of distributions

ehaviors by above statistical analysis, how insulator thickness, LDD length fluctuation and ion implantation uniformity. This non-uniformity of process control will lead to the common shift for device parameters.

The solution of macro variation is well-controlled in process. On the other hand, micro variation may come from the difference of the defect site, defect density in the active region and the activation efficiency. Since these conditions differ from device to device, the micro variation will lead to the random distribution of device parameters. Therefore, owing to describe the micro variation and evaluate the effects, the statistical analysis is need. Generally, the distance between devices is not too long for the layout of electric circuits, the major variation source is micro variation. To get more accurate simulation results, establishing a micro variation model is required.

Since we know the device variation b

to apply these results to evaluate the effects of variation on the circuit performance is a topic we are interested in. Because the distance between two devices will not be too long for the layout of the circuit, the macro variation is not our concern. A better approach is to find the proper mathematical expression for the distribution of the differences of these parameters. Firstly, we introduce the coefficient of determination (R square) to evaluate the fitness of our work, which is defined as

²

SSR

SSE

SST

∑ ( y

y )

SSE

∑ e ˆ

∑ ( y

ˆ y )

Generally speaking, the values of R square above 0.7 represnent the good fitness for the chosen funcion.

For the distribution of the difference of Vth, Gaussian-Lorentzian cross product is apply to the fitting, which is function ,while 1 is a pure Lorentzian distribution

Fig. 2-10(a) ~ (f) are shown respcetively the Vth difference distributions of N-type and P-type TFT with different device distance.

As for the distribution of the difference of Mu, the Lorentzian distribution is apply to the fitting, which is

The Mu difference distributions of N-type and P-type TFT with different device distance are shown in Fig. 2-11(a) ~ (f).

The Gaussian function is chosen to fit the distribution of the difference of S.S, which can be expressed as

The S.S. difference distributions of N-type and P-type TFT with different device distance are shown in Fig. 2-12(a) ~ (f). The values of R squre of the above fittng curves both higher than 0.85. It clearly shows the good fitness of our proposed mathemtical model and most of the fitting parameters slightly changing with distance, which supports the effects of macro variation are minor than those of micro variation we mentioned before. The values of R squre are so high that the device micro varion behavior can be described in these proposed distribution models. Therefore, the more accurate simulation results will be obtained with these proposed models and the effects of these proposed models on circuit performance will be discussed in 3

chatper.

In additon, although theses parameter difference distributions of N-type and P-type TFT can be expressed in the same mathematical function, those distributions are obvious different to some degree. The reasons of these phenomena will be discussed in next section.

2-4. The distribution comparison between N-type and P-type TFTs

The distributions of Vth and SS differences of N-type TFT are narrower than those of P-type TFT. This phenomena might result from channel doping of N-type TFT during process. In order to avoid obtaining the negative Vth of N-type TFT, N-type TFT should be dealt with extra process, channel doping. The extra process step might also increase an uncertain factor causing the device variation difference between N-type and P-type TFT.

Fig. 2-15 is the Mu difference distributions of N-type and P-type TFT. The Mu difference distribution of N-type TFT are wider than that of P-type TFT. This phenomenon might due to the different device structure between N-type and P-type TFT. The degradation of hot-carrier effects is a serious problem and these effects are induced by the presence of intense electric fields at the drain junction. The electric field at the drain junction is determined by the ion doping and the activation process used by impurities. Therefore, the TFT with a light-doped drain (LDD) are attractive for used with N-type TFTs. However, the N-type TFT with a light-doped drain (LDD) also increases an uncertain factor causing device variation. Therefore, the Mu difference distribution of N-type TFT are wider than that of P-type TFT.

Furthermore, the mathematical model for the distributions of the parameters differences is established, the applications for these models for circuit simulation will be discussed in the following chapter.

Chapter 3