Abstract
The analytical model of random variation in drain current of the Floating Gate MOSFET (FGMOSFET) has been proposed in this research. The model is composed of two parts for triode and saturation region of operation where the process induced device level random variations of each region and their statistical correlations have been taken into account. The nonlinearity of floating gate voltage and dependency on drain voltage of the coupling factors of FGMOSFET have also been considered. The model has been found to be very accurate since it can accurately fit the SPICE BSIM3v3 based reference obtained by using Monte-Carlo SPICE simulation and FGMOSFET simulation technique with SPICE. It can fit the BSIM4 based reference if desired by using the optimally extracted parameters. By using the proposed model, the variability analysis of FGMOSFET and the analytical modeling of the variation in the circuit level parameter of any FGMOSFET based circuit can be performed. So, this model has been found to be an efficient tool for the variability aware analysis and design of FGMOSFET based circuit.
1. Introduction
FGMOSFETs have been extensively utilized in various analog/digital circuits such as [1–9]. Similarly to the MOSFET based circuits, the performances of the FGMOSFET based circuits have been deteriorated by process induced device level random variations [10–12]. This is because these device level variations yield random variations in circuit level parameters, for example, drain current () and transconductance. These variations yield variations in parameters of FGMOSFET based circuit such as transconductance of the FGMOSFET based voltage to current converter [1] and equivalent resistance of the FGMOSFET based voltage controlled resistor [2, 3]. For handling this issue, variability aware analysis/design concept has been applied in the designing of many FGMOSFET based circuits, for example, [4–6].
Similarly to MOSFET, has been found to be the key circuit level parameter of FGMOSFET as it is directly measurable and can be the basis for determining the others. According to its importance, the analytical models of process induced random variation and mismatch in of MOSFET have been proposed without regard to certain circuit in many previous researches, for example, [13–15]. So, their results are applicable to any MOSFET based circuit. For the FGMOSFET, most of the previous work has been oriented to certain FGMOSFET circuits, for example, [16–19]. Some previous studies have not been devoted to any circuit, for example, [20, 21]. However, the results of these works are incomplete since the statistical correlations between device level random variations have been neglected. Moreover, nonlinearity of floating gate voltage and dependency on drain voltage of the coupling factors, which are the important features of FGMOSFET [22], have not been considered.
So, the analytical model of process induced random variation in () of the FGMOSFET has been proposed in this research. The proposed model is composed of two parts for triode and saturation regions of operation where the process induced device level random variations of each region and their statistical correlations have been taken into account. The nonlinearity of floating gate voltage and dependency on drain voltage of the coupling factors of FGMOSFET have also been considered. This model has been formulated without regard to any circuit. So, it is applicable to all FGMOSFET based circuits. It has been found to be very accurate since it can fit the SPICE BSIM3v3 based reference obtained by using FGMOSFET simulation technique with SPICE [23] and Monte-Carlo SPICE simulation with very high accuracy. If desired, it can fit the BSIM4 based reference by applying the optimum parameters extracted by using the optimization algorithm [24]. The variability analysis of FGMOSFET and the analytical modeling of process induced random variation in circuit level parameter of any FGMOSFET based circuit can be performed by using the proposed model as the basis. Hence, this model has been found to be an efficient tool for the variability aware analysis/design of any circuit involving FGMOSFET.
2. The Overview of FGMOSFET
FGMOSFET is a special type of MOSFET with an additional gate isolated within the oxide, namely, the floating gate [2]. A cross-sectional view of an N-type FGMOSFET with inputs implemented as discrete input gates where can be shown in Figure 1.
It should be mentioned here that the dimension (width and length) of any input gate determines the magnitude of its corresponding input capacitance. The symbol and equivalent circuit of inputs FGMOSFET are shown in Figure 2. Such equivalent circuit is composed of a MOSFET, input capacitances (), overlap capacitance between floating gate and drain (), overlap capacitance between floating gate and source (), and parasitic capacitance between floating gate and substrate () [2].
(a)
(b)
Let and let any th input capacitance be denoted by ; the floating gate voltage, , can be given by [2] where is the input voltage at any th input gate, is the drain voltage, is the source voltage, and is the bulk voltage. Moreover, denotes the total capacitance of the floating gate which can be defined as [2]
Let , , , and denote the coupling factor of any th input gate, drain, source, and bulk and let them be defined as , , , and , respectively; can be alternatively given as follows:
From either (1) or (3), it can be seen that depends on , , , and . According to [22], of FGMOSFET in triode is a nonlinear function of and depends on as does. So, , , , and are dependent on since they are functions of . For FGMOSFET in saturation region, is constant and so are , , , , and . As a result, become a linear function of .
3. Formulation of the Proposed Model
3.1. Triode Region Part
Firstly, of FGMOSFET in triode region will be formulated by letting the gate to source voltage () in the analytical model of of MOSFET in such region be replaced by . The second order effects such as mobility degradation and short channel effect have been taken into account for making the model applicable to deeply scaled technology. In order to do so, the linear model of mobility degradation [15] has been adopted. Hence, of the MOSFET in triode can be given by where , , , and denote drain to source voltage, threshold voltage, current factor, and mobility degradation coefficient, respectively.
Since of FGMOSFET in this region is a nonlinear function of , can be given by where , , , and , which are, respectively, dependent , , , and , can be modeled as power series of as follows: where , , , and are coefficients of the power series representation of , , , and , respectively.
As can be very low in recent CMOS technology, high order terms of , , , and can be neglected. So, (6)–(9) become (11)–(13) where , , , and and , , , and are , , , and with and :
By using (4), (10)–(13), and the methodology stated above, of the triode region operated FGMOSFET is given by (14) as :
It can be seen from (14) that the process induced device level random variations of FGMOSFET in triode are random variations in , , , , , , , , and denoted by , , , , , , , , and , respectively. So, of FGMOSET in triode region can be given by (15) where all derivatives given by (16)–(24) can be found by using (14):
Since , , , , , , , , and are random variables, so are and the statistical behavior of must be analytically modeled for its complete modeling. In order to do so, its average () and variance () must be formulated. As a result, = 0 similar to those of , , , , , , , , and and of FGMOSFET in triode can be given by taking the statistical correlations of , , , , , , , , and into account as follows: where , for example, , , and , denotes the correlation coefficient of and and displays their degree of statistical correlation. Moreover, , , , , , , , , and are nonzero and, respectively, denote the variances of , , , , , , , , and . Finally, it can be seen that the 1st up to 9th terms of (25) have been contributed by , , , , , , , , and where the others have been caused by the statistical correlations.
3.2. Saturation Region Part
Firstly, of FGMOSFET in this region will be formulated in a similar manner to that of FGMOSFET in triode where the second order effects have been taken into account as well. With the linear model of mobility degradation, of MOSFET in saturation can be given by where stands for the channel length modulation coefficient which must be taken into account for MOSFET in saturation if the short channel effect has been considered.
For FGMOSFET in saturation region, the model formulation is simpler than that of triode FGMOSFET since , , , and are constant. So, can be given by (3) and can be determined in a similar manner to that of FGMOSFET in triode by using (3) and (26) as given in (27) since :
As the process induced random variations in must also be considered for FGMOSET in saturation region, is given by (28). By using (27), all derivatives can be found as (29)–(35):
Similarly to of the FGMOSFET in triode, of the FGMOSFET in saturation is also a random variable. So, its statistical behavior must be analytically modeled as well. Since , , , , , , and have zero means, it has also been found that = 0. Moreover, can be given by taking the statistical correlations of , , , , , , and into account as given in (36) where the first seven terms have been, respectively, contributed by , , , , , , and . On the other hand, the other terms have been contributed by the statistical correlations.
Before proceeding to the model verification, it should be mentioned here that can be obtained either by using the scatter plot [25, 26] or by calculation as given in (37) which can be obtained based on the prior knowledge that the averages of and are 0 as they are process induced device level random variations, that is, , , and . Moreover, stands for the expectation operator:
4. Model Verification
Before proceeding further, it should be mentioned here that the model verification has been performed by assuming that , , as has been assumed in many previous studies on FGMOSFET, for example, [2, 7–9, 20]. As a result, , , and can be neglected as , , and is now independent of because which causes the dependency of on can be neglected. So, , which depends on , is now constant for both triode and saturation regions as also assumed in [22].
Moreover, the model verification has been performed based on FGMOSFET of both N-type and P-type with and for both triode and saturation regions, W/L = 20/0.25, SPICE BSIM3v3, and 0.25 μm level CMOS process technology of TSMC where all necessary parameters have been provided by MOSIS. For performing the verification, the root mean square (rms.) value of calculated by using the model () has been graphically compared to its SPICE BSIM3v3 based reference () obtained by using the Monte-Carlo SPICE simulation with 1000 runs. For convenience, , , , , , and have been assumed to be normally distributed and all ’s have been assumed to be 0.5 which is a reasonable estimation [26]. Since correlations must be taken into account in the simulations, each of , , , , , and has been expressed as a weighted sum of its correlated and uncorrelated components which are both normally distributed [26] and which have equal weights given by due to the assumed ’s.
Finally, the SPICE BSIM3v3 based modelling of FGMOSFET with can be performed by using the two inputs’ version of the equivalent circuit of FGMOSFET in Figure 2 where the core MOSFET has been modelled by using the SPICE BSIM3v3 and the simulation methodology proposed in [23] has been adopted for solving the convergence problem of the simulator. Both and have been expressed as percentage of deterministic and comparatively plotted against the magnitude of the voltage of the 1st and 2nd inputs of FGMOSFET which are ranged from 3 to 8 V and denoted by and , respectively. It should be mentioned here that = 0 V in the comparative plots against and vice versa. At this point, verification of the proposed model will be presented.
4.1. Verification of Triode Region Part
Since and of course have not been considered in triode region, only Δ, , , , and have been taken into account. We let for making the device level variations be bias-free and let these quantities be 0.01 as it has been assumed that , , , , and have 1% variation. For N-type FGMOSFET, the comparative plots of and against where = 0 and vice versa can be, respectively, shown in Figures 3 and 4 whereas those of P-type FGMOSFET are shown in Figures 5 and 6. In these figures, highly strong agreements between and which are, respectively, drawn as blue normal curves and red dotted curves can be observed and the average deviations of from have been found as 3.74725% and 3.70765% for N-type and P-type FGMOSFET, respectively. These deviations which can be determined from triode region N-type and P-type FGMOSFET based comparative plots shown in Figures 3 and 4 and in Figures 5 and 6, respectively, are notably very small. So, it can be seen that triode region part of the proposed model has been found to be very accurate.
4.2. Verification of Saturation Region Part
In saturation region, and must be considered. For making the device level variations be bias-free and following the assumption that , , , , , and have 1% variation, we let . The N-type FGMOSFET based comparative plots of and against where = 0 and vice versa can also be, respectively, depicted in Figures 3 and 4 where those based on P-type device are shown in Figures 5 and 6 as well. However, and have been, respectively, drawn as green normal curves and black dotted curves instead in this case. From Figures 3–6, highly strong agreements between and can also be seen in this case. The average deviations between and have been, respectively, found as 2.6901% and 3.6681% for N-type and P-type FGMOSFET. These deviations which can be determined from the saturation region FGMOSFET based comparative plots are also considerably very small. As a result, the accuracy of saturation region part of the model has been verified.
From Figures 3–6, it has been found that both and are increased with decreasing and because they have been expressed as percentage of their corresponding deterministic ’s which are increased with increasing and but with larger degree than the gross values of both rms. values do. Such increasing of and with decreasing and means that become more critical in low voltage/low power condition.
By considering the magnitude of both and , it has also been found that the FGMOSFET in triode is more robust than that in saturation region when and become sufficiently large. This can be obviously seen in P-type device. On the other hand, the device in saturation is more robust when and become adequately low. This is more obvious in the N-type FGMOSFET. If desired, can fit simulated by using BSIM4. In order to do so, the optimum parameters of MOSFET’s equations, that is, (6) and (14), extracted from BSIM4’s model by using the optimization algorithm [24] must be used.
5. Discussions
Firstly, insight into device level variability of FGMOSFET will be discussed. By using the proposed model, the rate of change of per unit () of FGMOSFET in triode region with respect to per unit , , , , , , , , and denoted by , , , , , , , , and , respectively, can be approximately given under low voltage/low power condition as follows:
For the saturation region, a similar rate of changes can be approximately found under a similar condition as
From these equations, it can be seen that all process induced device level random variations are relatively insignificant compared to , , and under low voltage/low power condition. So, , , and must be mostly considered in the variability aware design of low voltage/low power FGMOSFET based circuit. Moreover, the effects of and can be reduced by lowering either or or both as can be seen from (38), (40), (48), and (49).
Secondly, application of the proposed model will be discussed. This model can be used as the basis for the analytical modeling of process induced random variation in the circuit level parameter of any FGMOSFET based circuit. Let the circuit level parameter of interest of any FGMOSFET based circuit which is composed of FGMOSFETs be . The process induced random variation in () caused by the combined effect of of each FGMOSFET can be given by where and denote the and of any th FGMOSFET. Since can be analytically determined by using the proposed model due to its definition, so does via (54).
Since the average of is zero as those of ’s are, it is convenient to analytically model the statistical behavior of by using its variance () as those of ’s are nonzero. According to the above definition of , the variance of () can be determined by using the proposed model. So, can be determined by using the proposed model as well as with the aid of the following equation:
As a practical illustrative example, let be the transconductance () of the FGMOSFET based voltage to current converter (VIC) [1]. The equivalent circuit of its core circuit which is the FGMOSFET based transconductor can be depicted as in Figure 7 where and are the feedback and input capacitance of the transconductor [1]. Moreover, , , and are, respectively, , , and of .
Obviously, can be given by where and denote and of .
Since is in saturation region and has , can be given as follows:where , , and denote , , and of .
Moreover, , , and can be defined as
At this point, the analytical model of process induced random variation in () can be given by using (54) and (56) as follows: where Δ denotes the process induced random variation in and can be determined by using (28)–(35) which belong to the saturation region part of the proposed model, with .
Moreover, it is necessary to model the statistical behavior of by using its variance () because its mean is equal to zero similarly to that of . By using (55) and (56), can be formulated as where denotes the variance of and can be analytically determined by using (29)–(36) which belong to the saturation region part proposed model, with . It should be mentioned here that there is no correlation related term in (60) since is solely contributed by . Finally, we can plot against as shown in Figure 8 where has been expressed as percentage of the nominal .
6. Conclusion
In this research, the analytical model of of FGMOSFET has been proposed. The model is composed of two parts for triode and saturation regions of operation. The process induced device level random variations of each region, their statistical correlations, nonlinearity of , and dependency on of the coupling factors have been taken into account. The proposed model has been found to be very accurate since it can fit the SPICE BSIM3v3 based reference obtained by using Monte-Carlo simulation and FGMOSFET simulation technique with SPICE with very high accuracy. If desired, it can also fit the BSIM4 based reference if the optimally extracted parameters obtained by using optimization algorithm have been applied.
By using the proposed model, the insight into the device level variability of FGMOSFET can be provided and the analytical model of process induced random variation in circuit level parameter of any FGMOSFET based circuit can be obtained. It has been shown that become more critical in low voltage/low power operating condition where the influences of , , and become relatively large compared to those of other variations. Furthermore, the effects of and can be reduced by lowering either or or both. So, this model has been found to be an efficient tool for the variability aware analysis/designing of any FGMOSFET based circuit.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author would like to acknowledge Mahidol University, Thailand, for the online database service.