- Email: [email protected]
Analytical modeling of the transconductance of short channel MOSFETs in the saturation region
Solid-State Electronics Vol. 32, No. 1, pp. 81-89, Printedin Great Britain.All rightsreserved
0038-I
1989
101/89
$3.00 + 0.00
Copyright0 1989PergamonPressplc
ANALYTICAL MODELING OF THE TRANSCONDUCTANCE OF SHORT CHANNEL MOSFETS IN THE SATURATION REGION GERARD GHIBAUDO* Sachs and Freeman Associates Inc., 1401 McCormick Drive, Landover, MD 20785, U.S.A. (Received 8 June 1988; in revised form 29 June 1988) Abstract-A
simple model for the transconductance of a MOS transistor operating in the saturation region is presented. This model based on an inversion charge formulation of the gradual channel
approximation provides a good representation of the saturation transconductance long and short channel devices.
I. INTRODU~ION
From the gate charge conservation
The modeling of short-channel MOS transistors has been the subject of much research during the last decade[l-71. Nevertheless, no satisfactory analytical expressions giving the drain current I,, and/or the transconductance g,,,,, in the saturation region are presently available. In this work, we aim to establish an expression for the transconductance as a function of gate voltage valid in the case of short-channel devices in which both the transversal and longitudinal electric field influence on the carrier mobility is taken into account. For this, we first develop within the gradual channel approximation a compact formulation of the current based on an inversion charge approach. Then, we readily deduce an expression for the saturation transconductance g,,,,, as a function of I’, that enable a consistent representation of the g,,,,, (V,) characteristics to be made for channel lengths down to 0.35 pm.
2. DRAIN
CURRENT
in strong inversion for
MODEL
v* = VKi + A
in which V,, is the flat band voltage, Q,, the absolute depletion charge, Q, the absolute interface state charge and C,,Xthe gate oxide capacitance, it is easy to prove that the increments of surface potential, d4,, and quasi-Fermi potential d4,, are linked together by the capacitive relationship[9]: Ci, + Ci d4o d4, = (20% + Cd+ Ci, + Ci
(3)
in which C, = dQ,/d4,, Ci, = dQ,/d4, and Ci = dQ,db, are the depletion, the interface state and the inversion charge capacitances respectively. The increment of inversion charge, dQi, is given by:
dQi= ci (dds- d+c).
(4)
Finally, combining eqns (3) and (4) allowing the drain current to be expressed as a function of the inversion charge in the integral form: I.$/ QiW ‘“=-‘L
In the gradual channel approximation, the drain current of a MOS transistor is generally obtained from the expressiot@]:
where Qi is the absolute inversion charge, pee is the effective mobility, 4c is the quasi-Fermi potential shift between source and drain, V, is the source to drain voltage, W is the channel width and L the channel length.
equation:
I”erQi s QiWl
where Qi(S) and Qi(o) are the inversion charges at source and drain respectively. Note that eqn (5) is valid from weak to strong inversion. For instance, in weak inversion, Ci<<(CoX+ Cd + C,) so that eqn (5) can be reduced, assuming a constant mobility, to the usual Van Overstrataen expression[lO]: W kTCox+Cd+Ci, Id= -L/&)Cm + cd 4
*Permanent address: Laboratoire de Physique des Composants g Semiconducteurs, UA-CNRS 840, ENSERG, 23 rue des Martyrs, 38031 Grenoble, France.
s
Pi@)
X
Qils)
87
dQ,-!!CbQi(s) 7 [1 IL
I
exp(-~vd)I%
(6)
GERARD GHIBAUDO
88
where C = q(C,, + Cd)/[(C,, + Cd + Ci,)RT], kT/q being the thermal voltage. In strong inversion, C, >>(CoX+ C, + C,,) and eqn (5) becomes: Id = -
W
s
Q,(D)
P~UQ, dQi.
(7)
L (& + cd) Q,(.Sl
The saturation drain current I,,,, of long channel devices, i.e. for devices in which the velocity saturation effects can be neglected, can be obtained from eqn (7) by demanding that the inversion charge at drain Qi(D) is very small i.e. practically zero. Moreover, considering as is usual for strong inversion that Qi N C,, ( V, - V,), V, being the charge threshold voltage, which is not to be confused with the extrapolated threshold voltage deduced from the &( V,) characteristics[ 111, and: Pelr” &lo/U+ @(v, -
VA1
(Bbeing the mobility reduction factor) yields for the saturation drain current (putting Q,(D) = 0):
log(1 + e(I’,-
tax(nm)
1
The saturation transconductance for long devices can be shown from eqns (2) and (5) to take the generic form: g
“‘=c,,
CL?, G dr
I
I
(9)
where Cd = W/L ‘pen Q,(S) is the ohmic channel conductance. Note that eqn (9) holds in weak as well as in strong inversion and that is more general than the usual expression of the classical model [8]: &lS,t= WPUC0X( vs - Vt)lL. In the case of short-channel devices, the velocity saturation effects prevails on the usual pinch-off phenomenon that induces the depletion of the drain channel region. As a result, the saturation drain voltage Vdsatis no longer equal to (V, - V,) (as for long devices) but becomes smaller[5,9]. Therefore, the saturation of the drain current does arise primarily from the velocity saturation and not from the pinchoff effect. So, the drain saturation current and in turn the saturation transconductance g,,, have to be calculated using eqns (5) or (7) without considering that the charge at drain Q,(o) goes to zero. For example, in strong inversion, the inversion charge at drain is such that: Q,(D) = C,, [V, - V, - Vdsa, (V,)], in which V,,, (I’,) is the saturation drain voltage involving the velocity saturation. Besides, taking as is usual for the effective mobility the first order longitudinal electric field dependence[9]:
10
100 ClassIcal
3. TRANSCONDUCTANCEMODEL
1.(8)
Vt)
It should be noted that eqn (8) has not been deduced assuming a constant mobility as in the classical model that predicts a ( I’s - V,)2 dependence for IdSa,. Figure 1 shows a typical variation of the drain saturation current as function of 0 and of the corresponding oxide thickness. It is clear that for scaled down devices with oxide thicknesses of the order of 10 nm, the discrepancy between this model and the classical one in which 8 is neglected can exceed 40%, demonstrating thereby the necessity of taking into
1000
account the gate voltage dependence of the effective mobility for the evaluation of I,&,. For short-channel devices, the above approximation no longer holds and, in this case, one cannot consider that the inversion charge at drain Qi(D) goes to zero. This is to be discussed below for the transconductance.
PO0
@O=I+Kv,’
model
(10)
in which h is the low field mobility and K = pJ(Lu,,) (usat being the velocity saturation), allow the saturation transconductance g,,,, to be finally obtained at first order as: g
,w CL PCQ msat- L C,,, + Cd 1 + K V,,,, V
Fig. 1. Theoretical variation of the drain saturation current I dm, with the mobility reduction factor 0 (parameters: V, - V, = 4 V, low field mobility h = 500 cm2/Vs, oxide thickness t,, = 22 nm, W/L = 10, substrate doping N, = 1016/cm3).
where the saturation drain voltage Vdsa,(V,) has the actual gate voltage dependence. In a first approximation, the saturation drain voltage due to velocity saturation V&t (v,) can be derived using the mobility law of eqn (10) provided that
Modeling transconductance of short-channel MOSFETs
2.5
L
I
I
-3
-2
-1
0
Log e t/v)
Fig. 2. Theoretical variations of the drain saturation voltage Vbat with the mobility reduction factor 0 as obtained numerically using eqns (7) and (10) (parameters: V,- V,=4V, ~=5OOcm*/Vs, f,,=22nm, W=20ym, N. = 10r6/cm3,o,~ = 6 x lo6 cm/s. the influence of 0 is neglected. A straight forward calculation using (1) yields for Vdsat: V da
-1 +[I +2K(V,=
V,)]“2 (12)
K
It can he checked numerically that the impact of 0 on V,,, is less pronounced than that on IdSat(Fig. 1). In fact, the Vdsat shift due to 0 does not exceed 15% for oxide thicknesses up to 10 nm as can he seen from Fig. 2. Therefore, eqn (12) can he advantageously used in eqn (11) to provide a fairly good analytical
89
description of vd=l (V,) and, in turn, of g,,, (V,). It is worth mentioning that, in the long channel case, K tends to zero so that Vdsa, N (V, - V,) and, therefore, eqn (11) reduces to eqn (9). Figure 3 shows typical variations of the normalized saturation transconductance with gate voltage, L.g,,, (V,) that can be satisfactorily described by eqns (11) and (12). For the shorter device, the threshold voltage has been adjusted, during the curve fitting, to account for a pronounced charge sharing effect. It is worth noting that the fitting of Fig. 2 has been achieved using only one extra parameter i.e. the velocity saturation u,~ which arises in eqn (10) via the value of K. The other MOSFET parameters such as h, V, and 0 have been determined separately from the ohmic Id (V,) characteristics using a standard parameter extraction technique[ 12,131. Note that the corresponding Z,,,, (V,) characteristics may then be calculated numerically by a further integration of gnl,t (V,). 4. CONCLUSION An analysis of the current and transconductance of a MOS transistor based on an inversion charge approach in the gradual channel approximation has been achieved. As a result, a simple analytical expression for the saturation transconductance has heen established. It allows a good representation of the saturation transconductance charactersitics g,,,,, (V,) in strong inversion for devices as short as 0.35pm to he obtained. Acknowledgement-The author is indebted to the Naval Research Laboratory (Washington, DC.) for having pro-
L (pm):
vide working facilities during this sabbatical year. Tlnory
2.35
REFERENCES
L Qunx 2.35 Ezprlmrm
,
Fig. 3. Theoretical and experimental (by courtesy of B. Cabon [14]) variations of the saturation tramconductance g,,,,,, with pte voltage V, for different channel lengths (parameters: b = 500 crn*/Vs, I,, = 26 nm, e,, = 6 x lo6 cm/s, V, = 0.6 V, W = 20 pm, N, = 10’6/cm3).
1. Y. El Mansy and A. R. Boothroyd, IEEE Trans. Elecfron Dev. ED-24, 241 (1977). 2. M. White, F. Van de Wiele and J. P. Lambot, IEEE Trans. Electron Dev. ED-27, 899 (1980). 3. C. Sodini, P. K. Ko and J. L. Moll, IEEE Trans. Elecfron Dev. ED-31, 1386 (1984). 4. T. Grotjohn and B. Hoethinger, IEEE Trans Electron Dev. ED-31, 234 (1984). 5. T. Y. Chan, P. K. Ko and C. Hu, IEEE Electron Dev. Lerr. EDL6, 551 (1985). 6. T. Krutsick, M. White, H. S. Wong and R. Booth, IEEE Trans. Electron Dev. ED-34, 1676 (1987). 7. P. Felix and J. L. Pelloie, Solid-St. Electron. 30, 463 (1987). 8. S. M. Sze, Physics of Semiconductors Devices, Chap. 8. Wiley, New York (1981). 9. G. Ghibaudo, Physica Status Solidi 999, Kl49 (1987). 10. R. Van Overstraeten, G. Declerck and P. Muls, IEEE Trans. Electron Dev. ED-22, 282 (1975). 11. G. Ghibaudo, Physica Status Solidi 950, 323 (1986). 12. Chu-Hao, B. Cabon, S. Cristoloveanu and G. Ghibaudo, Solid-St. Electron. 28, 1025 (1985). 13. G. Ghibaudo, Electron. Letr. 22, 543 (1988). 14. B. Cabon, PhD dissertation, INP Grenoble, France (1986); S. Cristoloveanu, B. Cabon, K. Kang, P. Gentil and J. Gautier, Rev. Phys. Appl. 19, 923 (1984).
0038-I
1989
101/89
$3.00 + 0.00
Copyright0 1989PergamonPressplc
ANALYTICAL MODELING OF THE TRANSCONDUCTANCE OF SHORT CHANNEL MOSFETS IN THE SATURATION REGION GERARD GHIBAUDO* Sachs and Freeman Associates Inc., 1401 McCormick Drive, Landover, MD 20785, U.S.A. (Received 8 June 1988; in revised form 29 June 1988) Abstract-A
simple model for the transconductance of a MOS transistor operating in the saturation region is presented. This model based on an inversion charge formulation of the gradual channel
approximation provides a good representation of the saturation transconductance long and short channel devices.
I. INTRODU~ION
From the gate charge conservation
The modeling of short-channel MOS transistors has been the subject of much research during the last decade[l-71. Nevertheless, no satisfactory analytical expressions giving the drain current I,, and/or the transconductance g,,,,, in the saturation region are presently available. In this work, we aim to establish an expression for the transconductance as a function of gate voltage valid in the case of short-channel devices in which both the transversal and longitudinal electric field influence on the carrier mobility is taken into account. For this, we first develop within the gradual channel approximation a compact formulation of the current based on an inversion charge approach. Then, we readily deduce an expression for the saturation transconductance g,,,,, as a function of I’, that enable a consistent representation of the g,,,,, (V,) characteristics to be made for channel lengths down to 0.35 pm.
2. DRAIN
CURRENT
in strong inversion for
MODEL
v* = VKi + A
in which V,, is the flat band voltage, Q,, the absolute depletion charge, Q, the absolute interface state charge and C,,Xthe gate oxide capacitance, it is easy to prove that the increments of surface potential, d4,, and quasi-Fermi potential d4,, are linked together by the capacitive relationship[9]: Ci, + Ci d4o d4, = (20% + Cd+ Ci, + Ci
(3)
in which C, = dQ,/d4,, Ci, = dQ,/d4, and Ci = dQ,db, are the depletion, the interface state and the inversion charge capacitances respectively. The increment of inversion charge, dQi, is given by:
dQi= ci (dds- d+c).
(4)
Finally, combining eqns (3) and (4) allowing the drain current to be expressed as a function of the inversion charge in the integral form: I.$/ QiW ‘“=-‘L
In the gradual channel approximation, the drain current of a MOS transistor is generally obtained from the expressiot@]:
where Qi is the absolute inversion charge, pee is the effective mobility, 4c is the quasi-Fermi potential shift between source and drain, V, is the source to drain voltage, W is the channel width and L the channel length.
equation:
I”erQi s QiWl
where Qi(S) and Qi(o) are the inversion charges at source and drain respectively. Note that eqn (5) is valid from weak to strong inversion. For instance, in weak inversion, Ci<<(CoX+ Cd + C,) so that eqn (5) can be reduced, assuming a constant mobility, to the usual Van Overstrataen expression[lO]: W kTCox+Cd+Ci, Id= -L/&)Cm + cd 4
*Permanent address: Laboratoire de Physique des Composants g Semiconducteurs, UA-CNRS 840, ENSERG, 23 rue des Martyrs, 38031 Grenoble, France.
s
Pi@)
X
Qils)
87
dQ,-!!CbQi(s) 7 [1 IL
I
exp(-~vd)I%
(6)
GERARD GHIBAUDO
88
where C = q(C,, + Cd)/[(C,, + Cd + Ci,)RT], kT/q being the thermal voltage. In strong inversion, C, >>(CoX+ C, + C,,) and eqn (5) becomes: Id = -
W
s
Q,(D)
P~UQ, dQi.
(7)
L (& + cd) Q,(.Sl
The saturation drain current I,,,, of long channel devices, i.e. for devices in which the velocity saturation effects can be neglected, can be obtained from eqn (7) by demanding that the inversion charge at drain Qi(D) is very small i.e. practically zero. Moreover, considering as is usual for strong inversion that Qi N C,, ( V, - V,), V, being the charge threshold voltage, which is not to be confused with the extrapolated threshold voltage deduced from the &( V,) characteristics[ 111, and: Pelr” &lo/U+ @(v, -
VA1
(Bbeing the mobility reduction factor) yields for the saturation drain current (putting Q,(D) = 0):
log(1 + e(I’,-
tax(nm)
1
The saturation transconductance for long devices can be shown from eqns (2) and (5) to take the generic form: g
“‘=c,,
CL?, G dr
I
I
(9)
where Cd = W/L ‘pen Q,(S) is the ohmic channel conductance. Note that eqn (9) holds in weak as well as in strong inversion and that is more general than the usual expression of the classical model [8]: &lS,t= WPUC0X( vs - Vt)lL. In the case of short-channel devices, the velocity saturation effects prevails on the usual pinch-off phenomenon that induces the depletion of the drain channel region. As a result, the saturation drain voltage Vdsatis no longer equal to (V, - V,) (as for long devices) but becomes smaller[5,9]. Therefore, the saturation of the drain current does arise primarily from the velocity saturation and not from the pinchoff effect. So, the drain saturation current and in turn the saturation transconductance g,,, have to be calculated using eqns (5) or (7) without considering that the charge at drain Q,(o) goes to zero. For example, in strong inversion, the inversion charge at drain is such that: Q,(D) = C,, [V, - V, - Vdsa, (V,)], in which V,,, (I’,) is the saturation drain voltage involving the velocity saturation. Besides, taking as is usual for the effective mobility the first order longitudinal electric field dependence[9]:
10
100 ClassIcal
3. TRANSCONDUCTANCEMODEL
1.(8)
Vt)
It should be noted that eqn (8) has not been deduced assuming a constant mobility as in the classical model that predicts a ( I’s - V,)2 dependence for IdSa,. Figure 1 shows a typical variation of the drain saturation current as function of 0 and of the corresponding oxide thickness. It is clear that for scaled down devices with oxide thicknesses of the order of 10 nm, the discrepancy between this model and the classical one in which 8 is neglected can exceed 40%, demonstrating thereby the necessity of taking into
1000
account the gate voltage dependence of the effective mobility for the evaluation of I,&,. For short-channel devices, the above approximation no longer holds and, in this case, one cannot consider that the inversion charge at drain Qi(D) goes to zero. This is to be discussed below for the transconductance.
PO0
@O=I+Kv,’
model
(10)
in which h is the low field mobility and K = pJ(Lu,,) (usat being the velocity saturation), allow the saturation transconductance g,,,, to be finally obtained at first order as: g
,w CL PCQ msat- L C,,, + Cd 1 + K V,,,, V
Fig. 1. Theoretical variation of the drain saturation current I dm, with the mobility reduction factor 0 (parameters: V, - V, = 4 V, low field mobility h = 500 cm2/Vs, oxide thickness t,, = 22 nm, W/L = 10, substrate doping N, = 1016/cm3).
where the saturation drain voltage Vdsa,(V,) has the actual gate voltage dependence. In a first approximation, the saturation drain voltage due to velocity saturation V&t (v,) can be derived using the mobility law of eqn (10) provided that
Modeling transconductance of short-channel MOSFETs
2.5
L
I
I
-3
-2
-1
0
Log e t/v)
Fig. 2. Theoretical variations of the drain saturation voltage Vbat with the mobility reduction factor 0 as obtained numerically using eqns (7) and (10) (parameters: V,- V,=4V, ~=5OOcm*/Vs, f,,=22nm, W=20ym, N. = 10r6/cm3,o,~ = 6 x lo6 cm/s. the influence of 0 is neglected. A straight forward calculation using (1) yields for Vdsat: V da
-1 +[I +2K(V,=
V,)]“2 (12)
K
It can he checked numerically that the impact of 0 on V,,, is less pronounced than that on IdSat(Fig. 1). In fact, the Vdsat shift due to 0 does not exceed 15% for oxide thicknesses up to 10 nm as can he seen from Fig. 2. Therefore, eqn (12) can he advantageously used in eqn (11) to provide a fairly good analytical
89
description of vd=l (V,) and, in turn, of g,,, (V,). It is worth mentioning that, in the long channel case, K tends to zero so that Vdsa, N (V, - V,) and, therefore, eqn (11) reduces to eqn (9). Figure 3 shows typical variations of the normalized saturation transconductance with gate voltage, L.g,,, (V,) that can be satisfactorily described by eqns (11) and (12). For the shorter device, the threshold voltage has been adjusted, during the curve fitting, to account for a pronounced charge sharing effect. It is worth noting that the fitting of Fig. 2 has been achieved using only one extra parameter i.e. the velocity saturation u,~ which arises in eqn (10) via the value of K. The other MOSFET parameters such as h, V, and 0 have been determined separately from the ohmic Id (V,) characteristics using a standard parameter extraction technique[ 12,131. Note that the corresponding Z,,,, (V,) characteristics may then be calculated numerically by a further integration of gnl,t (V,). 4. CONCLUSION An analysis of the current and transconductance of a MOS transistor based on an inversion charge approach in the gradual channel approximation has been achieved. As a result, a simple analytical expression for the saturation transconductance has heen established. It allows a good representation of the saturation transconductance charactersitics g,,,,, (V,) in strong inversion for devices as short as 0.35pm to he obtained. Acknowledgement-The author is indebted to the Naval Research Laboratory (Washington, DC.) for having pro-
L (pm):
vide working facilities during this sabbatical year. Tlnory
2.35
REFERENCES
L Qunx 2.35 Ezprlmrm
,
Fig. 3. Theoretical and experimental (by courtesy of B. Cabon [14]) variations of the saturation tramconductance g,,,,,, with pte voltage V, for different channel lengths (parameters: b = 500 crn*/Vs, I,, = 26 nm, e,, = 6 x lo6 cm/s, V, = 0.6 V, W = 20 pm, N, = 10’6/cm3).
1. Y. El Mansy and A. R. Boothroyd, IEEE Trans. Elecfron Dev. ED-24, 241 (1977). 2. M. White, F. Van de Wiele and J. P. Lambot, IEEE Trans. Electron Dev. ED-27, 899 (1980). 3. C. Sodini, P. K. Ko and J. L. Moll, IEEE Trans. Elecfron Dev. ED-31, 1386 (1984). 4. T. Grotjohn and B. Hoethinger, IEEE Trans Electron Dev. ED-31, 234 (1984). 5. T. Y. Chan, P. K. Ko and C. Hu, IEEE Electron Dev. Lerr. EDL6, 551 (1985). 6. T. Krutsick, M. White, H. S. Wong and R. Booth, IEEE Trans. Electron Dev. ED-34, 1676 (1987). 7. P. Felix and J. L. Pelloie, Solid-St. Electron. 30, 463 (1987). 8. S. M. Sze, Physics of Semiconductors Devices, Chap. 8. Wiley, New York (1981). 9. G. Ghibaudo, Physica Status Solidi 999, Kl49 (1987). 10. R. Van Overstraeten, G. Declerck and P. Muls, IEEE Trans. Electron Dev. ED-22, 282 (1975). 11. G. Ghibaudo, Physica Status Solidi 950, 323 (1986). 12. Chu-Hao, B. Cabon, S. Cristoloveanu and G. Ghibaudo, Solid-St. Electron. 28, 1025 (1985). 13. G. Ghibaudo, Electron. Letr. 22, 543 (1988). 14. B. Cabon, PhD dissertation, INP Grenoble, France (1986); S. Cristoloveanu, B. Cabon, K. Kang, P. Gentil and J. Gautier, Rev. Phys. Appl. 19, 923 (1984).