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Body factor conscious modeling of single gate fully depleted SOI MOSFETs for low power applications
Solid-State Electronics 49 (2005) 997–1001 www.elsevier.com/locate/sse
Body factor conscious modeling of single gate fully depleted SOI MOSFETs for low power applications Anil Kumar *, Toshiharu Nagumo, Gen Tsutsui, Tetsu Ohtou, Toshiro Hiramoto Institute of Industrial Science, University of Tokyo, Hiramoto Lab, 4-6-1 Komaba, Meguro-Ku, Tokyo 153-8505, Japan Received 10 November 2004; received in revised form 25 February 2005; accepted 16 March 2005 Available online 26 April 2005
The review of this paper was arranged by Prof. S. Cristoloveanu
Abstract Degradation of body factor (c) and subthreshold factor (S) of single gate fully depleted SOI MOSFETs due to short channel effects has been studied analytically. The effect of source/drain fringing fields in buried oxide is found to play a more significant role in the reduction of body factor at smaller gate lengths. Present work provides the analytical expressions of effective back gate voltage, body factor and subthreshold factor of short channel fully depleted SOI MOSFETs. The results obtained are found in good approximation with 2D simulation. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: SOI MOSFETs; Body factor and subthreshold factor
1. Introduction Fully depleted (FD) silicon-on-insulator (SOI) CMOS are better candidate for low power applications due to its superior short channel immunity, low junction capacitance and steep subthreshold slope [1–3]. Also FD SOI MOSFETs have a freedom of the application of the back gate bias which allows the control of threshold voltage. It is known that VTCMOS scheme [4,5] is one of the most powerful ways to control the subthreshold leakage current, where threshold voltage is controlled by substrate bias utilizing the body effect. As body factor is the key parameter in VTCMOS scheme, its degradation affects the applicability of this scheme. To take the full advantage of this scheme the body factor should be large as well as constant. However, in simulation it is found that body factor of FD SOI MOSFETs degrades in
*
Corresponding author. Tel.: +81 3545 26264; fax: +81 3545 26265. E-mail address: [email protected] (A. Kumar).
0038-1101/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2005.03.017
shorter gate lengths due to fringing fields in the buried oxide [6]. When buried oxide (BOX) is thick enough and gate length is short, the potential in buried oxide does not vary linearly due to the influence of source and drain fringing fields in the buried oxide as shown in Fig. 1. V 0 /b Therefore boundary condition, Eback ¼ gbtbox , for electric field at back Si/SiO2 interface which is used in earlier models [7–12], does not remains valid. So to maintain the linearity of this boundary condition, potential at the bottom of buried oxide must be determined in such a way that the potential variation results in the same electric field at the Si/BOX interface. There have been many reports on modeling of these devices [7–15]. Some [14,15] of the papers are based on infinite solutions of Fourier series. Though these methods allow the calculation of the 2D PoissonÕs equation without fitting parameters, solution does not provide any physical insight of these devices. The papers [7,8] assumed the zero electric field at back interface hence models are independent of the buried oxide thickness and the effects
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001
Potential in BOX (V)
998 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
18
Na = 1x10 cm tsi = 30nm Vds = 0.1V
which includes the effect of source/drain fringing field into buried oxide and is given as
-3
V beff ¼ /b þ tbox electric field at Si=BOX Vbeff
0.04
0.06
0.08
0.10
0.12
depth x, (µm) Fig. 1. Vertical potential in buried oxide.
generated in buried oxide. Joachim et al. tried to include two-dimensional effects in the buried oxide empirically and fit to a set of parameters. However this model lacks the physical meaning and computationally inefficient. Analytical models developed so far either ignored or included empirically the effect of the fringing fields. In this work, the effective back gate voltage is calculated analytically incorporating the fringing fields in buried oxide. Furthermore, Banna et al.Õs threshold voltage model which ignores the effective back gate voltage effect has been extended using developed expression of effective back gate voltage. It is found that reduction in threshold voltage at shorter gate lengths is a function of back gate bias. As the threshold voltage reduction due to short channel effects is a function of back gate bias, therefore body factor also degrades. None of the papers has reported or analytically studied this fact in SOI MOSFETs so far. The expression of body factor is obtained analytically and the relationship of S and c factors in short channel single gate fully depleted SOI MOSFET is also clarified. Finally the model results have been verified with 2D Medici simulation.
2. Numerical simulations The 2D simulator Medici is performed in order to validate the analytical results. The source and drain regions are rectangular and uniformly doped with NSD of 1020 cm3. The p type channel is uniformed of 1017 cm3. The p+ base substrate is also uniformly doped of 1020 cm3 and this work is restricted to the highly doped substrates. Therefore no depletion of the substrate is considered in the present analysis. Threshold voltage is extracted by constant current method.
ð1Þ
where /b is the potential at back interface. A simple analytical model for electrical potentials in buried oxide based on Schwarz conformal mapping has been reported in [6] and potential induced by drain junction in buried oxide is given as V bi þ V ds V bL p WD ¼ Re ln 1 þ exp ðy L tbox p þiðtsi þ tbox xÞ ð2Þ where VbL is the long channel potential at the Si/BOX interface. The potential induced by the source end can also be obtained by similar approach. Electric field at back interface can be calculated by differentiating the potential at Si/BOX with respect to y and putting y = tsi and x = xmin. Using (1), calculated potential and electric field at Si/SiO2, the Vbeff is obtained as V beff ¼ V bs þ þ
V V bL bi p exp tbox ðxmin Þ 1
V bi þ V ds V bL p exp tbox ðL xmin Þ 1
ð3Þ
where second and third term reflect the influence of fringing fields. As the channel length is reduced, Vbeff increases and this effect is more prominent in deep submicrometer devices.
4. c and S factors in long channel FD SOI MOSFETs In this study, body factor c is defined by [5] dV th c¼ dV
ð4Þ
bs
instead of the usual definition of body factor in the text books [16] which is applicable to only the bulk MOSFET with uniformly doped channel profile. The definition of Eq. (4) can be applied to all the MOS structures with any channel impurity profile including retrograde MOSFET, delta doped MOSFET. Then the threshold voltage shift in VTCMOS is given by dV th ¼ cjdV bs j
ð5Þ
3. Expression of effective back gate voltage
In long channel single gate FD SOI MOSFETs, body factor c is given by
Here, effective back gate voltage Vbeff has been defined as the voltage at substrate/buried oxide interface
c0 ¼
C 0box C fox
ð6Þ
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001
where Cfox is front gate oxide capacitance. C 0box is simply the combination of silicon and buried oxide capacitances and is given by 1 1 1 ¼ þ 0 C box C box C si
ð7Þ
999
Eq. (11) can also be written as S ¼ lnð10Þ
oV gs oV s min oV gs ¼ 60 oV s min o logðI ds Þ oV s ðxmin Þ
ð12Þ
where Vs min is the minimum potential and xmin is the position of minimum potential in the channel. Using Eq. (12) and [11], subthreshold factor, S, of short channel MOSFET is derived as,
Since S factor in long channel case at room temperature is given by C 0box S 0 ¼ 60 1 þ ð8Þ C fox
with
therefore relation between c and S factors is simply given by [5],
1 !1 x L xmin L min cs ¼ 1 sin h þ sin h sin h 1 1 1
S 0 ¼ 60ð1 þ c0 Þ
ð9Þ
where S0 and c0 are S factor and body factor in long channel. However, Eq. (9) is not valid in short channel because c decreases and S increases due to short channel effects.
S ¼ 60cs ð1 þ c0 Þ
ð13Þ
where cs is correction factor due to short channel effects and l is characteristic length [11]. In long channel case cs reduces to unity and Eq. (13) becomes Eq. (9). Eq. (13) is the universal expression of relationship between S and c factors in FD SOI MOSFETs as it is valid in short as well as long channel devices.
5. Expression of short channel body factor Short channel threshold voltage model for FD SOI MOSFETs was developed by Banna et al. [11] using quasi-two-dimensional approach. It is in simple in functional form and no charge partitioning is assumed. The model is also suitable for circuit simulation and device design. Using Eq. (4), threshold voltage model by Banna et al. and above calculated effective back gate voltage, the expression of short channel body factor of single gate FD SOI MOSFETs has been obtained as 0 0 C 0box @ C box @ 1 c¼ 1 p C fox C box þ C si xmin 1 exp tbox 11 1 AA ð10Þ þ p ðL xmin Þ 1 exp tbox where first term represents the long channel body factor and second and third terms represent the reduction due to short channel effects. From (10), it can be seen that source/body and drain/body are the cause of degradation of body factor.
6. Modified relation between c and S factors Subthreshold factor S, is defined as the change in gate bias required changing the subthreshold drain current by one decade, and is given as S¼
oV gs o logðI ds Þ
ð11Þ
7. Quantum effects of body factor The body factor in Eq. (6) is approximately given by tfox/tbox. However, if we improve the degree of approximation using quantum mechanical effects, this ratio turns out to be determined not only by tfox and tbox, but also by the distribution of the wave functions of carriers in the SOI regions. And the body factor now can be 1
defined as [17] CBGeff/CGeff where C Geff ¼ tefox þ tsi hbi , esi ox 1 hbi tbox C BGeff ¼ esi þ eox , and hbi is the average distance of electrons from the interface between the SOI and BOX. More details can be found in [17].
8. Results and discussion The effective back gate voltage Vbeff has been calculated analytically using (3) and also extracted from 2D Medici simulation. Fig. 2 shows the variation of effective back gate voltage with gate length. The Vbeff is almost constant in long channel devices and starts increasing as gate length approaches to deep submicron regime. When the gate length is large enough, the potentials from source and drain does not affect the device characteristics. However in short channel devices contribution of these potentials is more. This clearly explains how the Vbeff is related to the fringing fields induced by the source and drain junctions in buried oxide. The effective back gate voltage as a function of silicon thickness is shown in Fig. 3. Effective back gate voltage increases with silicon thickness. Model results show the good agreement with simulated data.
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001 1.5 Circles: MEDICI Solid lines: Model
1.0 0.5
tfox = 3nm, tbox = 100nm tSi = 30nm Vds = 0V, Vgs = 0V
0.0
Vbs = 0V
-0.5 -1.0
Vbs = -1V
-1.5 -2.0
Vbs = -2V
0.24
Threshold voltage (V)
Effective back-gate voltage, Vbeff (V)
1000
-2.5
0.20
Vbs = -2V
0.16
Vbs = -1V Vbs = 0V
0.12
0.2
0.4
0.6
0.8
tfox = 3nm, tbox = 100nm, tSi = 30nm
0.08
Vds = 0.1V, Na = 1x1017cm-3
0.0
0.0
0.2
1.0
0.40 0.35
0.032
0.8
1.0
Circles: MEDICI Solid line: Model
Body factor
0.030 Vgb = 0 to -2V tfox = 3nm tbox = 100nm tSi = 30nm Vds = 0.1V
0.028
0.026
Na = 1x1017cm-3 0.024 0.0
0.2
0.4
0.6
0.8
1.0
Gate length (µm) Fig. 5. Variation of body factor with gate lengths.
100 95
Circles: MEDICI Solid line: Model
90 85
75
tfox = 3nm tbox = 100nm tSi = 30nm Vds = 0.1V
70
Na = 1x1017cm-3
80
65 60 0.0
Circles: MEDICI Solid line: Model
0.6
Fig. 4. Variation of threshold voltage with gate lengths.
Subthreshold factor (mV/dec)
Effective back-gate voltage, Vbeff (V)
Fig. 4 shows the variation of threshold voltage with gate length at back gate bias as a parameter. Threshold voltage is constant at longer gate lengths and degrades at shorter gate lengths. It can be noticed that the degradation of threshold voltage in short gate lengths due to short channel effects is a function of back gate voltage also. This illustrates the effect of Vbeff on threshold voltage. Variation of c with gate length is shown in Fig. 5. The model results are in good agreement with 2D simulation, indicating the validity of our model. c degrades at short gate lengths due to short channel effects caused by source and drain potentials and affects the performance of VTCMOS scheme. However in case of bulk MOSFET the body factor degrades due to charge sharing in the channel [18]. It can be noted that at larger gate lengths Dc is zero and starts increasing as approaches shorter gate lengths. This increase is the result of short channel effects in the device. The dependence of S on gate length is shown in Fig. 6. At larger gate lengths, short channel effect is not so important and cs is nearly unity and S factor is simply 60(1 + c0). On the other hand, when gate length is short
0.4
Gate length (µm)
Gate length (µm) Fig. 2. Variation of effective back gate voltage with gate lengths.
Vbs= -3V
0.04
Vbs = -3V
-3.0
Circles: MEDICI Solid lines: Model
0.2
0.4
0.6
0.8
1.0
Gate length (µm) Fig. 6. Variation of S factor with gate lengths.
0.30 0.25
enough, cs increases and degrades S factor. Analytical results are in very good agreement with 2D simulation.
tfox = 3nm, tbox = 100nm,
0.20
Vds = 0V, Na = 1x1017cm-3 0.15 0
10
20
30
40
50
60
SIlicon thickness, tSi (nm) Fig. 3. Variation of effective back gate voltage with silicon thickness.
9. Conclusion An analytical expression of effective back gate voltage has been obtained. Developed expression has been used
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001
to extend an earlier threshold voltage model. The body factor of short channel single gate FD SOI MOSFETs has also been calculated which includes the effect of source and drain fringing fields in buried oxide. It is found that degradation of body factor for FD SOI MOSFETs is due to fringing fields from source and drain into buried oxide however in bulk MOSFETs it is due to the charge sharing in the channel. Furthermore, the modified relation between S factor and body factor has been found. The obtained results have good proximity with two-dimensional Medici simulations.
Acknowledgements The authors are thankful to Japan Society of Promotion of Science (JSPS), MEXT, Japan to provide the financial support to this work. This work was also partly supported by a program for the ‘‘Promotion of Leading Researches’’ in Special Coordination Funds for Promoting Science and Technology by MEXT, Japan.
References [1] Colinge JP. Thin-film SOI technology: The solution to many submicron problems. IEDM Tech Dig. 1989. p. 817–20. [2] Imai K, Onishi H, Yamaguchi K, Inoue K, Matsubura Y, Ono A, et al. A 0.18 lm fully depleted CMOS on 30 nm thick SOI for sub-1.0 V operation. VLSI Symp Tech Dig. 1998. p. 116–7. [3] Chen J, Maiti B, Connelly D, Mendicino M, Huang F, Adetitu O, et al. 0.18 lm metal gate fully depleted SOI MOSFETs for advanced CMOS applications. VLSI Symp Tech Dig. 1999. p. 25–6. [4] Kuroda T et al.. A 0.9-V, 15-MHz, 10-mV, 4 mm2, 2-D discrete cosine transform core processor with variable threshold-voltage (VT) scheme. IEEE J Solid State Circuit 1996;31:1770–9.
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[5] Hiramoto T, Takamiya M. Low power and low voltage MOSFETs with variable threshold voltage controlled by back bias. IEIEC Trans Electron 2000;E82C:161–9. [6] Ernst T, Cristoloveanu S. Buried oxide fringing capacitance: A new physical model and its implication on SOI device scaling and architecture. IEEE Int SOI Conf 1999:38–9. [7] Young KK. Short channel effect in fully depleted SOI MOSFETÕs. IEEE Trans Electron Dev 1989;36:399–401. [8] Yan RH, Ourmazd A, Lee KF. Scaling the Si MOSFET: From bulk to SOI to bulk. IEEE Trans Electron Dev 1992;39:1704–10. [9] Joachim HO, Yamaguchi Y, Ishikawa K, Inoue Y, Nishimura T. Simulation and two-dimensional modeling of subthreshold slope in ultrathin-film SOI MOSFETÕs down to 0.1 lm gate length. IEEE Trans Electron Dev 1993;40:1812–7. [10] Suzuki K, Tosaka Y, Tanaka T, Horie H, Arimoto Y. Scaling theory for double gate SOI MOSFETÕs. IEEE Trans Electron Dev 1993;40:2326–9. [11] Banna SR, Chan PCH, Ko PK, Nguyen CT, Chan M. Threshold voltage model for deep-submicrometer fully depleted SOI MOSFETÕs. IEEE Trans Electron Dev 1995;42:1949–55. [12] Frank DJ, Taur Y, Wong HSP. Generalized scale length for two dimensional effects in MOSFETÕs. IEEE Electron Dev Lett 1998;19:385–7. [13] Yeh PC, Fossum JG. Physical subthreshold MOSFET modeling applied to viable design of deep-submicrometer fully depleted SOI low-voltage CMOS technology. IEEE Trans Electron Dev 1995;42:1605–13. [14] Woo JCS, Terrill KW, Vasudev PK. Two-dimensional analytic modeling of very thin SOI MOSFETÕs. IEEE Trans Electron Dev 1990;37:1999–2006. [15] Guo JY, Wu CY. A new 2D analytic threshold voltage model for fully depleted short-channel SOI MOSFETÕs. IEEE Trans Electron Dev 1993;40:1653–61. [16] Tsividis Y. Operation and modeling of the MOS transistors. Japan: McGraw-Hill; 1999. [17] Horiguchi S, Fujiwara A, Inokawa H, Tkahashi Y. Analysis of back-gate voltage dependence of threshold voltage of thin siliconon-insulator metal-oxide-semiconductor field-effect transistor and its application to Si single-electron transistor. Jpn J Appl Phys 2004;43:2036–40. [18] Kumar A, Nagumo T, Tsutsui G, Hiramoto T. Analytical model of body factor in short channel bulk MOSFETs for low voltage applications. Solid State Electron 2004;48:1763–6.
Body factor conscious modeling of single gate fully depleted SOI MOSFETs for low power applications Anil Kumar *, Toshiharu Nagumo, Gen Tsutsui, Tetsu Ohtou, Toshiro Hiramoto Institute of Industrial Science, University of Tokyo, Hiramoto Lab, 4-6-1 Komaba, Meguro-Ku, Tokyo 153-8505, Japan Received 10 November 2004; received in revised form 25 February 2005; accepted 16 March 2005 Available online 26 April 2005
The review of this paper was arranged by Prof. S. Cristoloveanu
Abstract Degradation of body factor (c) and subthreshold factor (S) of single gate fully depleted SOI MOSFETs due to short channel effects has been studied analytically. The effect of source/drain fringing fields in buried oxide is found to play a more significant role in the reduction of body factor at smaller gate lengths. Present work provides the analytical expressions of effective back gate voltage, body factor and subthreshold factor of short channel fully depleted SOI MOSFETs. The results obtained are found in good approximation with 2D simulation. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: SOI MOSFETs; Body factor and subthreshold factor
1. Introduction Fully depleted (FD) silicon-on-insulator (SOI) CMOS are better candidate for low power applications due to its superior short channel immunity, low junction capacitance and steep subthreshold slope [1–3]. Also FD SOI MOSFETs have a freedom of the application of the back gate bias which allows the control of threshold voltage. It is known that VTCMOS scheme [4,5] is one of the most powerful ways to control the subthreshold leakage current, where threshold voltage is controlled by substrate bias utilizing the body effect. As body factor is the key parameter in VTCMOS scheme, its degradation affects the applicability of this scheme. To take the full advantage of this scheme the body factor should be large as well as constant. However, in simulation it is found that body factor of FD SOI MOSFETs degrades in
*
Corresponding author. Tel.: +81 3545 26264; fax: +81 3545 26265. E-mail address: [email protected] (A. Kumar).
0038-1101/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2005.03.017
shorter gate lengths due to fringing fields in the buried oxide [6]. When buried oxide (BOX) is thick enough and gate length is short, the potential in buried oxide does not vary linearly due to the influence of source and drain fringing fields in the buried oxide as shown in Fig. 1. V 0 /b Therefore boundary condition, Eback ¼ gbtbox , for electric field at back Si/SiO2 interface which is used in earlier models [7–12], does not remains valid. So to maintain the linearity of this boundary condition, potential at the bottom of buried oxide must be determined in such a way that the potential variation results in the same electric field at the Si/BOX interface. There have been many reports on modeling of these devices [7–15]. Some [14,15] of the papers are based on infinite solutions of Fourier series. Though these methods allow the calculation of the 2D PoissonÕs equation without fitting parameters, solution does not provide any physical insight of these devices. The papers [7,8] assumed the zero electric field at back interface hence models are independent of the buried oxide thickness and the effects
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001
Potential in BOX (V)
998 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
18
Na = 1x10 cm tsi = 30nm Vds = 0.1V
which includes the effect of source/drain fringing field into buried oxide and is given as
-3
V beff ¼ /b þ tbox electric field at Si=BOX Vbeff
0.04
0.06
0.08
0.10
0.12
depth x, (µm) Fig. 1. Vertical potential in buried oxide.
generated in buried oxide. Joachim et al. tried to include two-dimensional effects in the buried oxide empirically and fit to a set of parameters. However this model lacks the physical meaning and computationally inefficient. Analytical models developed so far either ignored or included empirically the effect of the fringing fields. In this work, the effective back gate voltage is calculated analytically incorporating the fringing fields in buried oxide. Furthermore, Banna et al.Õs threshold voltage model which ignores the effective back gate voltage effect has been extended using developed expression of effective back gate voltage. It is found that reduction in threshold voltage at shorter gate lengths is a function of back gate bias. As the threshold voltage reduction due to short channel effects is a function of back gate bias, therefore body factor also degrades. None of the papers has reported or analytically studied this fact in SOI MOSFETs so far. The expression of body factor is obtained analytically and the relationship of S and c factors in short channel single gate fully depleted SOI MOSFET is also clarified. Finally the model results have been verified with 2D Medici simulation.
2. Numerical simulations The 2D simulator Medici is performed in order to validate the analytical results. The source and drain regions are rectangular and uniformly doped with NSD of 1020 cm3. The p type channel is uniformed of 1017 cm3. The p+ base substrate is also uniformly doped of 1020 cm3 and this work is restricted to the highly doped substrates. Therefore no depletion of the substrate is considered in the present analysis. Threshold voltage is extracted by constant current method.
ð1Þ
where /b is the potential at back interface. A simple analytical model for electrical potentials in buried oxide based on Schwarz conformal mapping has been reported in [6] and potential induced by drain junction in buried oxide is given as V bi þ V ds V bL p WD ¼ Re ln 1 þ exp ðy L tbox p þiðtsi þ tbox xÞ ð2Þ where VbL is the long channel potential at the Si/BOX interface. The potential induced by the source end can also be obtained by similar approach. Electric field at back interface can be calculated by differentiating the potential at Si/BOX with respect to y and putting y = tsi and x = xmin. Using (1), calculated potential and electric field at Si/SiO2, the Vbeff is obtained as V beff ¼ V bs þ þ
V V bL bi p exp tbox ðxmin Þ 1
V bi þ V ds V bL p exp tbox ðL xmin Þ 1
ð3Þ
where second and third term reflect the influence of fringing fields. As the channel length is reduced, Vbeff increases and this effect is more prominent in deep submicrometer devices.
4. c and S factors in long channel FD SOI MOSFETs In this study, body factor c is defined by [5] dV th c¼ dV
ð4Þ
bs
instead of the usual definition of body factor in the text books [16] which is applicable to only the bulk MOSFET with uniformly doped channel profile. The definition of Eq. (4) can be applied to all the MOS structures with any channel impurity profile including retrograde MOSFET, delta doped MOSFET. Then the threshold voltage shift in VTCMOS is given by dV th ¼ cjdV bs j
ð5Þ
3. Expression of effective back gate voltage
In long channel single gate FD SOI MOSFETs, body factor c is given by
Here, effective back gate voltage Vbeff has been defined as the voltage at substrate/buried oxide interface
c0 ¼
C 0box C fox
ð6Þ
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001
where Cfox is front gate oxide capacitance. C 0box is simply the combination of silicon and buried oxide capacitances and is given by 1 1 1 ¼ þ 0 C box C box C si
ð7Þ
999
Eq. (11) can also be written as S ¼ lnð10Þ
oV gs oV s min oV gs ¼ 60 oV s min o logðI ds Þ oV s ðxmin Þ
ð12Þ
where Vs min is the minimum potential and xmin is the position of minimum potential in the channel. Using Eq. (12) and [11], subthreshold factor, S, of short channel MOSFET is derived as,
Since S factor in long channel case at room temperature is given by C 0box S 0 ¼ 60 1 þ ð8Þ C fox
with
therefore relation between c and S factors is simply given by [5],
1 !1 x L xmin L min cs ¼ 1 sin h þ sin h sin h 1 1 1
S 0 ¼ 60ð1 þ c0 Þ
ð9Þ
where S0 and c0 are S factor and body factor in long channel. However, Eq. (9) is not valid in short channel because c decreases and S increases due to short channel effects.
S ¼ 60cs ð1 þ c0 Þ
ð13Þ
where cs is correction factor due to short channel effects and l is characteristic length [11]. In long channel case cs reduces to unity and Eq. (13) becomes Eq. (9). Eq. (13) is the universal expression of relationship between S and c factors in FD SOI MOSFETs as it is valid in short as well as long channel devices.
5. Expression of short channel body factor Short channel threshold voltage model for FD SOI MOSFETs was developed by Banna et al. [11] using quasi-two-dimensional approach. It is in simple in functional form and no charge partitioning is assumed. The model is also suitable for circuit simulation and device design. Using Eq. (4), threshold voltage model by Banna et al. and above calculated effective back gate voltage, the expression of short channel body factor of single gate FD SOI MOSFETs has been obtained as 0 0 C 0box @ C box @ 1 c¼ 1 p C fox C box þ C si xmin 1 exp tbox 11 1 AA ð10Þ þ p ðL xmin Þ 1 exp tbox where first term represents the long channel body factor and second and third terms represent the reduction due to short channel effects. From (10), it can be seen that source/body and drain/body are the cause of degradation of body factor.
6. Modified relation between c and S factors Subthreshold factor S, is defined as the change in gate bias required changing the subthreshold drain current by one decade, and is given as S¼
oV gs o logðI ds Þ
ð11Þ
7. Quantum effects of body factor The body factor in Eq. (6) is approximately given by tfox/tbox. However, if we improve the degree of approximation using quantum mechanical effects, this ratio turns out to be determined not only by tfox and tbox, but also by the distribution of the wave functions of carriers in the SOI regions. And the body factor now can be 1
defined as [17] CBGeff/CGeff where C Geff ¼ tefox þ tsi hbi , esi ox 1 hbi tbox C BGeff ¼ esi þ eox , and hbi is the average distance of electrons from the interface between the SOI and BOX. More details can be found in [17].
8. Results and discussion The effective back gate voltage Vbeff has been calculated analytically using (3) and also extracted from 2D Medici simulation. Fig. 2 shows the variation of effective back gate voltage with gate length. The Vbeff is almost constant in long channel devices and starts increasing as gate length approaches to deep submicron regime. When the gate length is large enough, the potentials from source and drain does not affect the device characteristics. However in short channel devices contribution of these potentials is more. This clearly explains how the Vbeff is related to the fringing fields induced by the source and drain junctions in buried oxide. The effective back gate voltage as a function of silicon thickness is shown in Fig. 3. Effective back gate voltage increases with silicon thickness. Model results show the good agreement with simulated data.
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001 1.5 Circles: MEDICI Solid lines: Model
1.0 0.5
tfox = 3nm, tbox = 100nm tSi = 30nm Vds = 0V, Vgs = 0V
0.0
Vbs = 0V
-0.5 -1.0
Vbs = -1V
-1.5 -2.0
Vbs = -2V
0.24
Threshold voltage (V)
Effective back-gate voltage, Vbeff (V)
1000
-2.5
0.20
Vbs = -2V
0.16
Vbs = -1V Vbs = 0V
0.12
0.2
0.4
0.6
0.8
tfox = 3nm, tbox = 100nm, tSi = 30nm
0.08
Vds = 0.1V, Na = 1x1017cm-3
0.0
0.0
0.2
1.0
0.40 0.35
0.032
0.8
1.0
Circles: MEDICI Solid line: Model
Body factor
0.030 Vgb = 0 to -2V tfox = 3nm tbox = 100nm tSi = 30nm Vds = 0.1V
0.028
0.026
Na = 1x1017cm-3 0.024 0.0
0.2
0.4
0.6
0.8
1.0
Gate length (µm) Fig. 5. Variation of body factor with gate lengths.
100 95
Circles: MEDICI Solid line: Model
90 85
75
tfox = 3nm tbox = 100nm tSi = 30nm Vds = 0.1V
70
Na = 1x1017cm-3
80
65 60 0.0
Circles: MEDICI Solid line: Model
0.6
Fig. 4. Variation of threshold voltage with gate lengths.
Subthreshold factor (mV/dec)
Effective back-gate voltage, Vbeff (V)
Fig. 4 shows the variation of threshold voltage with gate length at back gate bias as a parameter. Threshold voltage is constant at longer gate lengths and degrades at shorter gate lengths. It can be noticed that the degradation of threshold voltage in short gate lengths due to short channel effects is a function of back gate voltage also. This illustrates the effect of Vbeff on threshold voltage. Variation of c with gate length is shown in Fig. 5. The model results are in good agreement with 2D simulation, indicating the validity of our model. c degrades at short gate lengths due to short channel effects caused by source and drain potentials and affects the performance of VTCMOS scheme. However in case of bulk MOSFET the body factor degrades due to charge sharing in the channel [18]. It can be noted that at larger gate lengths Dc is zero and starts increasing as approaches shorter gate lengths. This increase is the result of short channel effects in the device. The dependence of S on gate length is shown in Fig. 6. At larger gate lengths, short channel effect is not so important and cs is nearly unity and S factor is simply 60(1 + c0). On the other hand, when gate length is short
0.4
Gate length (µm)
Gate length (µm) Fig. 2. Variation of effective back gate voltage with gate lengths.
Vbs= -3V
0.04
Vbs = -3V
-3.0
Circles: MEDICI Solid lines: Model
0.2
0.4
0.6
0.8
1.0
Gate length (µm) Fig. 6. Variation of S factor with gate lengths.
0.30 0.25
enough, cs increases and degrades S factor. Analytical results are in very good agreement with 2D simulation.
tfox = 3nm, tbox = 100nm,
0.20
Vds = 0V, Na = 1x1017cm-3 0.15 0
10
20
30
40
50
60
SIlicon thickness, tSi (nm) Fig. 3. Variation of effective back gate voltage with silicon thickness.
9. Conclusion An analytical expression of effective back gate voltage has been obtained. Developed expression has been used
A. Kumar et al. / Solid-State Electronics 49 (2005) 997–1001
to extend an earlier threshold voltage model. The body factor of short channel single gate FD SOI MOSFETs has also been calculated which includes the effect of source and drain fringing fields in buried oxide. It is found that degradation of body factor for FD SOI MOSFETs is due to fringing fields from source and drain into buried oxide however in bulk MOSFETs it is due to the charge sharing in the channel. Furthermore, the modified relation between S factor and body factor has been found. The obtained results have good proximity with two-dimensional Medici simulations.
Acknowledgements The authors are thankful to Japan Society of Promotion of Science (JSPS), MEXT, Japan to provide the financial support to this work. This work was also partly supported by a program for the ‘‘Promotion of Leading Researches’’ in Special Coordination Funds for Promoting Science and Technology by MEXT, Japan.
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