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A 2D analytic field-dependent-mobility model for the I-V characteristics of thin film fully-depleted SOI MOSFETs
Solid-State ElectronicsVoi. 38. No. I. DD.261-264. 1995 Copyright 0 1995 ElscYierSC& Ltd
Pergamon
Printed in
Great
Britain. All rights reserved 0038-I101/95 89.50+ 0.00
NOTE A 2D ANALYTIC FIELD-DEPENDENT-MOBILITY MODEL FOR THE Z-V CHARACTERISTICS OF THIN FILM FULLY-DEPLETED SO1 MOSFETs (Received 28 January 1994; in revised _form I I May 1994)
INTRODUCTION
and
Silicon-on-insulator (SOI) technology is useful for high performance three-dimensional integrated circuits. Fully depleted SOI MOSFETs are advantageous over partially depleted ones in advanced integrated technologies[ 141. The ultrathin SO1 film has been used to reduce the short-channel effects[5] and to reduce electric field at the channel/gate oxide interface, thereby enhancing inversion layer mobilities[t%q. The carrier mobility in the surface inversion layer of a MOSFET is dependent on the electric field perpendicular to the gate oxide[l]. Several models have been proposed[9,10] to be tested for silicon bulk MOSFETs with either uniform doping or step profile. In practice, however, the thin film does not have a uniform doping profile; instead a Gaussian distribution more closely resembles the real doping profile. Lim and Fossum[l 1] calculate the drain current-voltage characteristics for large devices neglecting the possible small geometry effects. Aggarwal et al. have presented a two-dimensional analytical model[l2] for short channel fully depleted SO1 MOSFETs which calculates the potential distribution in silicon film as a function of terminal voltages and model parameters. In the present note an analytical mobility model has been derived for a fully depleted SO1 MOSFET where we assume a Gaussian distribution in thin silicon film as it results from deep implants, commonly used in the n-channel MOSFETs to suppress the back surface leakage. The dependence of mobility on terminal voltages has been analysed. The values of drain current and drain saturation voltages are calculated for the constant mobility model as well as the field dependent mobility model. The present model is verified with experimental data.
0,
1 af
(6)
v + $8 [ is the built in potential, Es is silicon bandgap and VT is the thermal voltage. The assumptions for the derivation of drain current in terms of front gate voltage, drain voltage and back substrate bias are negligible diffusion current, gradual channel approximation, non-uniform doping and field dependent mobility. In practical devices, thermal oxidation causes impurity redistribution resulting in a non-uniform distribution[l3]. The impurity profile resembles a Gaussian distribution and is taken as:
NA(x) = *exp[
--WI
(7)
where Q, R, and u are channel implant dose, projected range and standard deviation respectively. Their values are given in Table 1. This distribution defines a full Gaussian profile over the range x = 0 to 1000 A. For a strongly inverted n-channel enhancement mode SO1 MOSFET, current in the front channel is given by:
r, = ~~.rlQn(y)l
d@,(y)
(f-3)
dy.
Here Q,(y) is the charge in the inversion layer and is given by:
PI: exp(-yJK)+gexp(y&),
Vdt
.I
& = Vrln
The structure of the n-channel enhancement mode SO1 MOSFET to be analysed is illustrated in Fig. 1. The front surface potential distribution in the silicon film is obtained by considering appropriate boundary conditions --ar+A
V,-Vtif)+(V,,b-
Here I,, t,, t, are silicon film thickness, front oxide thickness and buried oxide thickness respectively. 6, is the dielectric permittivity of silicon and c, is the dielectric permittivity of oxide. N*(O) is the dopant concentration at the front interface, V, is the gat&ource voltage, V,“, is the back substrate bias, and V,,,, and V,, are front channel and back channel flat band voltages respectively. V, is the drain-source voltage, L is the channel length.
ANALYSIS
WY)= where:
(0)
fir=?-
Q.(Y) = Q,(Y) - Qs(Y)
(1)
(9)
with
(Ai + Of)- k&i + fJf + V,,)eXP( - r,)l [I - ev(-2r,)l I exp(_rf) B = (hn + ‘ds + uf) - bh + ufr)exp(-rf)l * =
11 -
ew(
uf=$
-
2rf
Q,(Y) = -W’,
(2)
v,,-~~,(y) +
vsy,,],
(3)
Q&I=
)I
Tr=Lfir
-
the surface charge density, -q[j;N,Wd+i
the depletion layer charge density, where C, = c,,/rf is the front gate oxide capacitance and x&=
261
%(hi
+ vsub) OS sN.,(‘3 1
(10)
Note
Si 01
0 Fig 1. Cross-sectional view of an n-channel enhancement mode fully depleted SO1 MOSFET. The shaded area denotes the depletion region. is the depletion region depth at the onset of strong inversion in a bulk SO1 MOSFET. Here we assume that the SOI film is thin enough to be fully depleted so that the inversion (areal) charge density is a linear function of surface potential. However for the partially depleted devices the inversion charge density is calculated in analogy to bulk devices and is a non-linear function of surface potential. We are presenting a current-voltage model for a fully depleted SO1 MOSFET. The mobility of electrons in the inversion layer, called surface mobility is smaller than the bulk mobility. The mobility degradation is induced by the electric field component perpendicular to the silicon surface. A MOSFET operates under the influence of gate field and drain field. The vertical electric field tends to accelerate the electrons towards the semiconductor-oxide interface, where they suffer collisions in addition to the collisions with the crystal lattice and ionized impurity atoms. This additional effect on the mobility is called the surface scattering which causes the mobility degradation[l4]. As VBsincreases, more electrons are accumulated at the silicon surface. The surface scattering process becomes the dominant mechanism limiting the electron mobility. Since the normal field, in general, varies along the channel, mobility will also vary. For a given temperature, mobility is found to be only a function of average normal field in the inversion layer and is expressed by the following formula:
PnedP)=
P”O
,+
WY) L-1 EC
0.4
0.8 Channel
I.2 position
_ 1.6
-2.0
-. 2.4
(pm )
Fig. 2. Field dependent mobility along the channel with drain voltage as a parameter for VP = 0.5V for a SOI MOSFET with f, = 0.1 pm, r, = 0.025 pm, W/L = 50 pm/ 2 pm. E, = 6.01 x T’.55x 10e2 V/m.
The variation of p&y) along the channel at the front interface is shown in Fig. 2. It is seen that the channel may be divided into a portion (adjacent to the source and drain) in which the surface mobility is field dependent, another (intermediate channel region), in which it is constant. The surface potential increases from & (at source) to 4si + V,, (at drain) passing through a minimum, accordingly electric field varies from source to drain obeying eqn (12). Having a higher value at source, mobility decreases to a constant value in the channel and then increases again at the drain for low values of drain voltage i.e. in the linear region. The effects of drain and gate voltages on surface electron mobility are shown in Figs 3 and 4 respectively, for different channel positions. We divide the channel in three regions: (1) Centre of the channel: The mobility is almost independent of drain and gate-source voltages as shown in Figs 3 and 4. (2) Source end: Figure 3 shows that mobility does not change with Vds for V,, = 0.5 V. However it decreases with increasing gate voltage in the linear and saturation region, as shown in Fig. 4(a) and (b) respectively. (3) Drain end: The variation of mobility is quite dramatic at drain end. Figure 3 shows that with VP = 0.5 V, surface mobility is low for drain voltages, then it increases abruptly
(11)
where, pno is the low field mobility, ECis a constant, whose value is given in Fig. 2 and the normal field at the surface is:
v
vgs=o.5
Vsub = 0 V
960
As it has been explained earlier that with increasing gate field, more carriers are induced in the channel which causes more collisions with the ionized impurity atoms reducing the mobility of charge carriers. Table I 0
RP Q
(I 65, 6”.
3.4x IO-*pm
I.2 X IO’bin2 2.6867x lO~‘flrn 1.04x 10~‘°F/m 0.34 x IO-” F/m
I
0.5
I I.0 Drain
I
I
I
1.5
2.0
2.5
voltage
3
0
(VI
Fig. 3. Dependence of surface electron mobility on drain voltage for Vgs= 0.5 V. Model parameters are same as in Fig. 2.
Note
‘L””
For low drain voltages [Fig. 4(a)], mobility first increases, then remains constant in the region of weak inversion and then decreases gradually with increasing gate voltage. On the other hand, for high V.,, [Fig. 4(b)], mobility increases gradually and almost linearly with VP. Replacing p”, by p,,a(y) in eqn (8), and integrating from source [mr(y) = &] to drain [Q&y) = & + Vdr], we obtain: Z +a,+ V& (13) IQ,(y)l~eff(~)d~r(y), Id& = x s +h where Z is the channel width. The drain saturation voltage is calculated by substituting Q.(L) = 0 in (9):
(a) Vdt= 0.2
_ 1000
263
V
-
vd,, = v,, -
0
0.2
0.4 Gate
0.6
voltage
(V
0.8
1.0
1
1200
(b) _ 1000
Vds =I.5
v
I-
>” “E ’ 800 2 600 ZI C 5 400
Vfir - &, + cub -
Qe@ ).
(14)
We compare the model with the experimental data originally given by Colinge[rl] to verify the present model for the drain current-voltage characteristics with consideration of field dependent mobility. The experimental data together with the values of oxide fixed charge density, Q, and VIubfor the theoretical predictions (for field dependent mobility and constant mobility models) are given in Fig. 5. The present model is in good agreement with experimental results for low values of V, while the large variation of measured drain current from Id, [with pnr = p’noin (8)] clearly indicates the inadequacy of the constant mobility model. The reason is evident from Fig. 2, which shows that surface mobility is constant at a value < pLno over a channel region whose length is smaller than L. CONCLUSIONS
s 200
0 0
I 0.2
I
I
0.4
0.6
Gate voltage
I
0.8
1
(VI
Fig. 4. Dependence of surface electron mobility on gate voltage for (a) Vds= 0.2 V, (b) Vds= 1.5 V, for the parameters defined in Fig. 2. to a maximum value at V,,, = 0.5 V for the parameters of a given model and finally decreases for increasing drain voltage. Figure 4 shows that the behaviour of mobility in the linear region is different from that in the saturation region.
5.0
%ub= ov,6 _3 &(O)= 8x IO cm
t
The major feature of the proposed model is that the mobility of electrons vary along the channel due to its dependence on vertical electric field at the front interface. The study shows that the phenomenon of surface scattering plays an important role in limiting the electron mobility in the regions adjacent to the source and drain. The effect of terminal voltages on surface electron mobility is studied and it is found that mobility changes its behaviour as channel changes its state from weak inversion to strong inversion. The present model has been verified by comparing the calculated drain current-voltage characteristics with the experimental data. The study reveals that the constant mobility model overestimates the drain current and saturation voltages. This can be accounted for, as the effective length of the channel region in which mobility is constant at a value
VANEETA AGGARWAL R. S. GUPTA
REFERENCES
0
I
2
Drain
voltage
3
4
5
(VI
Fig. 5. Drain current as a function of drain voltage with gate voltage as a parameter. (0) experimental data, (-) field dependent mobility model and (---) constant mobility model for a SOI MOSFET with z,, = 0.1 pm, I, = 0.025 pm, Z = 50 pm and L = 2 pm. N,(O) = 8 x 10z2ions/m3.
1. S. Veeraghavan and J. G. Fossum, IEEE Trans. Electron Devices 35, 1866 (1988). 2. L. A. Akers and J. J. Sanchez, Solid-S!. Electron. 25, 621 (1982). 3. T. Kekigawa and Y. Hayashi, Solid-S/. Electron. 27, 827 (1984). 4. J. P. Colinge, IEEE Electron Device Left. 19, 97 (1988). 5. K. K. Young. IEEE Trans. Electron Devices 36, 399 (1989).
264
Note
6. K. K. Young, IEEE Trans. Elecrron Devices 36, 504 11989). 7. k Ybshimi, H. Hazama, M. Takahashi, S. Kambyashi and H. Tango, Electron. L&t. 24, 1078 (1988). 8. C. Turchetti and G. Masetti, IEEE Trans. Electron Devices ED-32, 773 (1985). 9. K. Yamaguchi, IEEE Trans. Electron Devices ED-26,
1068 (1979). 10. C. Y. Wu and Y. W. Daih, Solid-St. Electron. 28, 1271 (1985).
11. H. K. Lim and J. G. Fossum, IEEE Devices ED-31, 401 (1984).
Trans. Electron
12. Vaneeta Aggarwal, ‘Manoj Khanna, Rachna Sood, Subhasis Haldar and R. S. Gupta, Solid-St. Electron. 37, 1537 (1994).
13. S. M. Sze, Physics of Semiconductor Devices, 2nd edn, p. 469. Wiley, New York (1981). 14. Y. P. Tsividis, Operation and Modeling of the MOS Transistor, p. 141. McGraw-Hill, New York (1987).
Pergamon
Printed in
Great
Britain. All rights reserved 0038-I101/95 89.50+ 0.00
NOTE A 2D ANALYTIC FIELD-DEPENDENT-MOBILITY MODEL FOR THE Z-V CHARACTERISTICS OF THIN FILM FULLY-DEPLETED SO1 MOSFETs (Received 28 January 1994; in revised _form I I May 1994)
INTRODUCTION
and
Silicon-on-insulator (SOI) technology is useful for high performance three-dimensional integrated circuits. Fully depleted SOI MOSFETs are advantageous over partially depleted ones in advanced integrated technologies[ 141. The ultrathin SO1 film has been used to reduce the short-channel effects[5] and to reduce electric field at the channel/gate oxide interface, thereby enhancing inversion layer mobilities[t%q. The carrier mobility in the surface inversion layer of a MOSFET is dependent on the electric field perpendicular to the gate oxide[l]. Several models have been proposed[9,10] to be tested for silicon bulk MOSFETs with either uniform doping or step profile. In practice, however, the thin film does not have a uniform doping profile; instead a Gaussian distribution more closely resembles the real doping profile. Lim and Fossum[l 1] calculate the drain current-voltage characteristics for large devices neglecting the possible small geometry effects. Aggarwal et al. have presented a two-dimensional analytical model[l2] for short channel fully depleted SO1 MOSFETs which calculates the potential distribution in silicon film as a function of terminal voltages and model parameters. In the present note an analytical mobility model has been derived for a fully depleted SO1 MOSFET where we assume a Gaussian distribution in thin silicon film as it results from deep implants, commonly used in the n-channel MOSFETs to suppress the back surface leakage. The dependence of mobility on terminal voltages has been analysed. The values of drain current and drain saturation voltages are calculated for the constant mobility model as well as the field dependent mobility model. The present model is verified with experimental data.
0,
1 af
(6)
v + $8 [ is the built in potential, Es is silicon bandgap and VT is the thermal voltage. The assumptions for the derivation of drain current in terms of front gate voltage, drain voltage and back substrate bias are negligible diffusion current, gradual channel approximation, non-uniform doping and field dependent mobility. In practical devices, thermal oxidation causes impurity redistribution resulting in a non-uniform distribution[l3]. The impurity profile resembles a Gaussian distribution and is taken as:
NA(x) = *exp[
--WI
(7)
where Q, R, and u are channel implant dose, projected range and standard deviation respectively. Their values are given in Table 1. This distribution defines a full Gaussian profile over the range x = 0 to 1000 A. For a strongly inverted n-channel enhancement mode SO1 MOSFET, current in the front channel is given by:
r, = ~~.rlQn(y)l
d@,(y)
(f-3)
dy.
Here Q,(y) is the charge in the inversion layer and is given by:
PI: exp(-yJK)+gexp(y&),
Vdt
.I
& = Vrln
The structure of the n-channel enhancement mode SO1 MOSFET to be analysed is illustrated in Fig. 1. The front surface potential distribution in the silicon film is obtained by considering appropriate boundary conditions --ar+A
V,-Vtif)+(V,,b-
Here I,, t,, t, are silicon film thickness, front oxide thickness and buried oxide thickness respectively. 6, is the dielectric permittivity of silicon and c, is the dielectric permittivity of oxide. N*(O) is the dopant concentration at the front interface, V, is the gat&ource voltage, V,“, is the back substrate bias, and V,,,, and V,, are front channel and back channel flat band voltages respectively. V, is the drain-source voltage, L is the channel length.
ANALYSIS
WY)= where:
(0)
fir=?-
Q.(Y) = Q,(Y) - Qs(Y)
(1)
(9)
with
(Ai + Of)- k&i + fJf + V,,)eXP( - r,)l [I - ev(-2r,)l I exp(_rf) B = (hn + ‘ds + uf) - bh + ufr)exp(-rf)l * =
11 -
ew(
uf=$
-
2rf
Q,(Y) = -W’,
(2)
v,,-~~,(y) +
vsy,,],
(3)
Q&I=
)I
Tr=Lfir
-
the surface charge density, -q[j;N,Wd+i
the depletion layer charge density, where C, = c,,/rf is the front gate oxide capacitance and x&=
261
%(hi
+ vsub) OS sN.,(‘3 1
(10)
Note
Si 01
0 Fig 1. Cross-sectional view of an n-channel enhancement mode fully depleted SO1 MOSFET. The shaded area denotes the depletion region. is the depletion region depth at the onset of strong inversion in a bulk SO1 MOSFET. Here we assume that the SOI film is thin enough to be fully depleted so that the inversion (areal) charge density is a linear function of surface potential. However for the partially depleted devices the inversion charge density is calculated in analogy to bulk devices and is a non-linear function of surface potential. We are presenting a current-voltage model for a fully depleted SO1 MOSFET. The mobility of electrons in the inversion layer, called surface mobility is smaller than the bulk mobility. The mobility degradation is induced by the electric field component perpendicular to the silicon surface. A MOSFET operates under the influence of gate field and drain field. The vertical electric field tends to accelerate the electrons towards the semiconductor-oxide interface, where they suffer collisions in addition to the collisions with the crystal lattice and ionized impurity atoms. This additional effect on the mobility is called the surface scattering which causes the mobility degradation[l4]. As VBsincreases, more electrons are accumulated at the silicon surface. The surface scattering process becomes the dominant mechanism limiting the electron mobility. Since the normal field, in general, varies along the channel, mobility will also vary. For a given temperature, mobility is found to be only a function of average normal field in the inversion layer and is expressed by the following formula:
PnedP)=
P”O
,+
WY) L-1 EC
0.4
0.8 Channel
I.2 position
_ 1.6
-2.0
-. 2.4
(pm )
Fig. 2. Field dependent mobility along the channel with drain voltage as a parameter for VP = 0.5V for a SOI MOSFET with f, = 0.1 pm, r, = 0.025 pm, W/L = 50 pm/ 2 pm. E, = 6.01 x T’.55x 10e2 V/m.
The variation of p&y) along the channel at the front interface is shown in Fig. 2. It is seen that the channel may be divided into a portion (adjacent to the source and drain) in which the surface mobility is field dependent, another (intermediate channel region), in which it is constant. The surface potential increases from & (at source) to 4si + V,, (at drain) passing through a minimum, accordingly electric field varies from source to drain obeying eqn (12). Having a higher value at source, mobility decreases to a constant value in the channel and then increases again at the drain for low values of drain voltage i.e. in the linear region. The effects of drain and gate voltages on surface electron mobility are shown in Figs 3 and 4 respectively, for different channel positions. We divide the channel in three regions: (1) Centre of the channel: The mobility is almost independent of drain and gate-source voltages as shown in Figs 3 and 4. (2) Source end: Figure 3 shows that mobility does not change with Vds for V,, = 0.5 V. However it decreases with increasing gate voltage in the linear and saturation region, as shown in Fig. 4(a) and (b) respectively. (3) Drain end: The variation of mobility is quite dramatic at drain end. Figure 3 shows that with VP = 0.5 V, surface mobility is low for drain voltages, then it increases abruptly
(11)
where, pno is the low field mobility, ECis a constant, whose value is given in Fig. 2 and the normal field at the surface is:
v
vgs=o.5
Vsub = 0 V
960
As it has been explained earlier that with increasing gate field, more carriers are induced in the channel which causes more collisions with the ionized impurity atoms reducing the mobility of charge carriers. Table I 0
RP Q
(I 65, 6”.
3.4x IO-*pm
I.2 X IO’bin2 2.6867x lO~‘flrn 1.04x 10~‘°F/m 0.34 x IO-” F/m
I
0.5
I I.0 Drain
I
I
I
1.5
2.0
2.5
voltage
3
0
(VI
Fig. 3. Dependence of surface electron mobility on drain voltage for Vgs= 0.5 V. Model parameters are same as in Fig. 2.
Note
‘L””
For low drain voltages [Fig. 4(a)], mobility first increases, then remains constant in the region of weak inversion and then decreases gradually with increasing gate voltage. On the other hand, for high V.,, [Fig. 4(b)], mobility increases gradually and almost linearly with VP. Replacing p”, by p,,a(y) in eqn (8), and integrating from source [mr(y) = &] to drain [Q&y) = & + Vdr], we obtain: Z +a,+ V& (13) IQ,(y)l~eff(~)d~r(y), Id& = x s +h where Z is the channel width. The drain saturation voltage is calculated by substituting Q.(L) = 0 in (9):
(a) Vdt= 0.2
_ 1000
263
V
-
vd,, = v,, -
0
0.2
0.4 Gate
0.6
voltage
(V
0.8
1.0
1
1200
(b) _ 1000
Vds =I.5
v
I-
>” “E ’ 800 2 600 ZI C 5 400
Vfir - &, + cub -
Qe@ ).
(14)
We compare the model with the experimental data originally given by Colinge[rl] to verify the present model for the drain current-voltage characteristics with consideration of field dependent mobility. The experimental data together with the values of oxide fixed charge density, Q, and VIubfor the theoretical predictions (for field dependent mobility and constant mobility models) are given in Fig. 5. The present model is in good agreement with experimental results for low values of V, while the large variation of measured drain current from Id, [with pnr = p’noin (8)] clearly indicates the inadequacy of the constant mobility model. The reason is evident from Fig. 2, which shows that surface mobility is constant at a value < pLno over a channel region whose length is smaller than L. CONCLUSIONS
s 200
0 0
I 0.2
I
I
0.4
0.6
Gate voltage
I
0.8
1
(VI
Fig. 4. Dependence of surface electron mobility on gate voltage for (a) Vds= 0.2 V, (b) Vds= 1.5 V, for the parameters defined in Fig. 2. to a maximum value at V,,, = 0.5 V for the parameters of a given model and finally decreases for increasing drain voltage. Figure 4 shows that the behaviour of mobility in the linear region is different from that in the saturation region.
5.0
%ub= ov,6 _3 &(O)= 8x IO cm
t
The major feature of the proposed model is that the mobility of electrons vary along the channel due to its dependence on vertical electric field at the front interface. The study shows that the phenomenon of surface scattering plays an important role in limiting the electron mobility in the regions adjacent to the source and drain. The effect of terminal voltages on surface electron mobility is studied and it is found that mobility changes its behaviour as channel changes its state from weak inversion to strong inversion. The present model has been verified by comparing the calculated drain current-voltage characteristics with the experimental data. The study reveals that the constant mobility model overestimates the drain current and saturation voltages. This can be accounted for, as the effective length of the channel region in which mobility is constant at a value
VANEETA AGGARWAL R. S. GUPTA
REFERENCES
0
I
2
Drain
voltage
3
4
5
(VI
Fig. 5. Drain current as a function of drain voltage with gate voltage as a parameter. (0) experimental data, (-) field dependent mobility model and (---) constant mobility model for a SOI MOSFET with z,, = 0.1 pm, I, = 0.025 pm, Z = 50 pm and L = 2 pm. N,(O) = 8 x 10z2ions/m3.
1. S. Veeraghavan and J. G. Fossum, IEEE Trans. Electron Devices 35, 1866 (1988). 2. L. A. Akers and J. J. Sanchez, Solid-S!. Electron. 25, 621 (1982). 3. T. Kekigawa and Y. Hayashi, Solid-S/. Electron. 27, 827 (1984). 4. J. P. Colinge, IEEE Electron Device Left. 19, 97 (1988). 5. K. K. Young. IEEE Trans. Electron Devices 36, 399 (1989).
264
Note
6. K. K. Young, IEEE Trans. Elecrron Devices 36, 504 11989). 7. k Ybshimi, H. Hazama, M. Takahashi, S. Kambyashi and H. Tango, Electron. L&t. 24, 1078 (1988). 8. C. Turchetti and G. Masetti, IEEE Trans. Electron Devices ED-32, 773 (1985). 9. K. Yamaguchi, IEEE Trans. Electron Devices ED-26,
1068 (1979). 10. C. Y. Wu and Y. W. Daih, Solid-St. Electron. 28, 1271 (1985).
11. H. K. Lim and J. G. Fossum, IEEE Devices ED-31, 401 (1984).
Trans. Electron
12. Vaneeta Aggarwal, ‘Manoj Khanna, Rachna Sood, Subhasis Haldar and R. S. Gupta, Solid-St. Electron. 37, 1537 (1994).
13. S. M. Sze, Physics of Semiconductor Devices, 2nd edn, p. 469. Wiley, New York (1981). 14. Y. P. Tsividis, Operation and Modeling of the MOS Transistor, p. 141. McGraw-Hill, New York (1987).