Chapter 3. A Modeling of Drain Electric Flux Passing through the BOX layer in Subthreshold
3.2.4 Relationship between the w and ζ planes
48
49
Here, ∆w! is equal to the difference of w between the boundaries of lower drain and the gate as shown in Fig. 6 (a), and it can be written as follows:
∆w! = 𝑖 v!−v! (D.4)
where v! and v! are the applied voltages in the lower drain and the gate, respectively. The (D.3) and (D.4) can be simplified as follows:
a! =∆!!.!!!!!
!∙! = !!!!! .!!!!!
! . (D.5)
a! can be erased by inserting (D.5) into (D.2) and w is rewritten as follows:
w=!!!!!!∙ −ln ζ−ζ! +ln ζ−ζ! +b!. (D.6)
To erase b!, w∓! is defined when ζ in (D.6) is ±∞.
w∓! =!!!!! !∙ −ln ±∞ +ln ±∞ +b!. (D.7)
This equation can be simplified as follows:
b! =w∓!. (D.8)
Here, w∓! can be replaced by the imaginary part of w∓! as shown in Fig.
6 (a), and it can be written as follows:
b! =𝑖∙ v!. (D.9)
50
The b! can be erased with the insertion of (C.7), (C.10) and (D.9) into (D.6), and the relationship between the w and ζ planes in the lower drain is derived as follows:
w=!!!!!!∙ −ln ζ−ζ! +ln ζ−ζ! +i∙v!
=!!!!! !∙ln !!∆!!
! +i∙v!. (D.10)
51
Fig. 3-6. Schematic views of the relationship between (a) w plane, and (b) ζ plane.
52 3.2.5 Capacitance and electric flux
The capacitance and electric flux between the boundaries can be derived from the inter-boundary stream function. First, the capacitance C!" between electrodes α and β as shown in Fig. 3-7 (a) is written as
C!" = −!!!!!
! (E.1)
where Q! is the charge in electrode α and v! is the voltage in electrode β as shown in Fig. 3-7 (b). The amount of charge can be derived from the product of the permittivity ε and the electric force line amount as derived from the difference in the stream function u between two points as shown in Fig. 3-7 (c).
Q! = −ε∙ u!!"#$%−u!!"#$ . (E.2)
Here, the real part of w is the stream function, and the charge in the gate can be derive from the difference between the stream functions when ζ in (D.10) are ∞ and ζ!, because the gate is assigned between ζ! and ∞ on the ζ plane as shown in Fig.
3-5 (b).
Q!" = ε∙Re w ∞ −w ζ!
=ε∙Re w ∞ −w ∆𝑦!
=ε∙Re !!!!! !∙ln !!∆!!
! −!!!!! !∙ln ∆!∆!!
!!∆!!
=!! ∙Re ln 1+∆!∆!!
! ∙ v!−v!
=!! ∙ln abs 1+∆!∆!!
! ∙ v!−v! .
(E.3)
53
Here, ∆y! and ∆y! can be taken as the BOX thickness TBOX and the sum of the effective SOI thickness 𝑇!"#$ and the effective gate oxide thickness 𝑇!"#$, which is based on ε!"# as shown in Fig. 3-3 (b), respectively.
∆y! = T!"#, (E.4)
∆y! =T!"#$+T!"#$. (E.5)
∆y! can be written by inserting (A.7) and (A.8) into (E.5)
∆y! =!!!"#
!"# ∙T!"#+!!!"#
!"#∙T!"#. (E.6)
Q!" can be simplified by inserting (E.4) and (E.6) into (E.3), and ε in (E.3) is
set to the value of the reference permittivity ε!"#.
Q!" = !!"#! ∙ln 1+!!"# !!"#
!!"#∙!!"#!!!!"#
!"#∙!!"# v!−v! . (E.7)
Here, the amount of electric flux between the boundaries is equal to the amount of charge in the boundary. Consequently, this is the model equation for the electric flux between the lower drain and the gate, which is assumed to be equivalent to that between the lower drain and the body. In addition, the model of capacitance between the lower drain and the gate can be derived using Q!", and
Q!" and the voltage between the gate and lower drain v!" are substituted into Q!
and v! in (E.1) respectively.
C!" =!!!!!"
!"
=!!"#! ∙ln 1+!!"# !!"#
!!"#∙!!"#!!!!"#
!"#∙!!"# . (E.8)
54
This is the model equation of capacitance between the gate and the drain.
Fig. 3-7. Schematic views of (a) electrodes, (b) potential distribution and (c) stream function distribution. The arrows between the electrodes indicate electric flux.
55 3.3 Verification of Model Validity
To verify the validity of the model, the amount of electric flux running from the drain to the body via the BOX layer was extracted using device simulation and compared with the amount of flux determined using the model equation (G.10).
The former was extracted by calculating the difference between the amounts of electric flux passing through the body/SOI interface when vd was 0.0 V and 1.0 V.
The fluxes for each vd value were determined using the stream function calculated from simulation results obtained using TCAD Sentaurus [41]. In this simulation, TGOX, vb, the acceptor density in the SOI layer, the acceptor density in the substrate and the donor density in the drain/source regions were set to 1 nm, 0.0 V, 1017 cm-3, 1019 cm-3 and 1020 cm-3, respectively. The gate voltage was assumed to be in the subthreshold region and set to 0.0 V. In addition, to cancel out the error caused by approximation in the model, a fitting parameter b was added to the electric flux determined using the model.
Figure 3-8 shows the εBOX- and TBOX-related dependences of the electric flux calculated using the model and the simulation when LG, TSOI and b were 100 nm, 5 nm and -2.5 × 10-11 C/m, respectively. The εBOX values were varied from 1 to 11.9 to incorporate permittivity in SON, SOS, SOD and silicon. Comparison of the dependence of the electric flux determined using the model on εBOX and TBOX
as shown in Fig. 3-8 (a) and the simulation dependence as shown in Fig. 3-8 (b) indicates a similar tendency. For more precise comparison of these results, Fig.
3-8 (c) shows the dependences on TBOX when εBOX was 11.9, and Fig. 3-8 (d) shows the dependences on εBOX when TBOX was 119 nm. These dependences are almost identical, which indicates the validity of the model.
56
Fig. 3-8. The εBOX- and TBOX-related dependences of electric flux as calculated using (a) the model and (b) simulation. (c) Comparison of TBOX-related dependences of flux based on the model and simulation results for an εBOX value of 11.9. (d) Comparison of εBOX-related dependences of flux based on the model and simulation results for a TBOX value of 119 nm.
57
To demonstrate the validity of the model from another aspect, Fig. 3-9 shows the TSOI- and TBOX-related dependences of the electric flux calculated from the model and from simulation when LG and b are 100 nm and - 2.0 × 10-11 C/m, respectively. Comparison of the dependence of the electric flux from the model on εBOX and TBOX as shown in Fig. 3-9 (a) and from simulation as shown in Fig.
3-9 (b) also shows a similar tendency. For more detailed comparison of these results, Fig. 3-9 (c) shows the dependences on TBOX when TSOI is 20 nm, and Fig.
3-9 (d) shows the dependences on TSOI when TBOX is 120 nm. This TSOI
dependence as shown in Fig. 3-9 (d) demonstrates close agreement with the values obtained from the model and the simulation. However, the dependence on TBOX as shown in Fig. 3-9 (c) is different from that obtained with the model and with simulation when TBOX becomes smaller. This is because approximation invalidates the model with smaller values of TBOX.
58
Fig. 3-9. The TSOI- and TBOX-related dependences of electric flux calculated using (a) the model and (b) simulation. (c) Comparison of TBOX-related dependences of flux based on the model and simulation when TSOI is 20 nm. (d) Comparison of TSOI-related dependences of flux based on the model and simulation when TBOX is 120 nm.
59
The insufficiency of the model’s validity is also observed from the LG- and TBOX-related dependences of the electric flux. Figure 3-10 shows the dependences calculated from the model and from simulation when TSOI and b are 5 nm and -1.75 × 10-11 C/m, respectively. Comparison of the dependence of the electric flux from the model as shown in Fig. 3-10 (a) and from simulation as shown in Fig.
3-10 (b) shows quite different characteristics for small LG values. For more detailed comparison of these results, Fig. 3-10 (c) shows the dependences on TBOX
when LG is 100 nm, and Fig. 3-10 (d) shows the dependences on LG when TBOX is 30 nm. In the dependence determined with the model, the amount of flux is constant when TBOX is constant. The source structure is also not considered in the model, and the dependence on LG is not included. In contrast, in the dependence determined from simulation, the amount of flux decreases when LG becomes smaller because the electric flux reaches the source rather than the body as the distance between the drain and source decreases due to the scaling of LG. The difference between the model and simulation results observed here indicates that the model is not sufficiently valid when the effect of the source on the drain is larger, and this influence is remarkable when TBOX is large. However, this lack of validity is not expected to be a serious problem for the SOI MOSFETs widely used today because both LG and TBOX are usually scaled at the same time, and smaller TBOX values will be chosen when the LG value is lower.
60
Fig. 3-10. The LG- and TBOX-related dependences of electric flux calculated using (a) the model and (b) simulation. (c) Comparison of TBOX-related dependences of flux based on the model and simulation when LG is 100 nm. (d) Comparison of LG-related dependences of flux based on the model and simulation when TBOX is 30 nm.
61 3.4 Summary
In this study, the amount of electric flux passing from the drain to the body via the BOX layer in the subthreshold region of GP SOI MOSFETs was modeled using conformal mapping, and the capacitance between the drain and the gate was modeled. The modeled flux amounts were compared with those of simulation, and the validity of the model was examined. The results showed close correspondence between the model and simulation fluxes with rational approximations of the model, which indicates that flux amounts can be estimated using this approach.
The model can be applied not only to SOI MOSFETs but also to advanced MOSFETs, which have various levels of permittivity in their insulator layers. As a result, it can be applied to determine BOX-related characteristics in DIBL and in the parasitic capacitance of these MOSFETs. The model is therefore expected to be useful in design of SOI MOSFETs and other advanced MOSFETs such as SON, SOS and SOD types.
In addition to the evaluation of BOX related disadvantages described in Chapter 2 and 3, to suppress the disadvantages new device structures are proposed in Chapter 4 and 5.
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