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Harmonic distortion analysis using an improved charge sheet model for PD SOI MOSFETs
ARTICLE IN PRESS
Microelectronics Journal 38 (2007) 321–326 www.elsevier.com/locate/mejo
Harmonic distortion analysis using an improved charge sheet model for PD SOI MOSFETs Joaquı´ n Alvaradoa,, Antonio Cerdeiraa, Valeria Kilchytskab, Denis Flandreb a
Solid-State Electronics Section (SEES), CINVESTAV-IPN, Me´xico D.F., Me´xico Microelectronics Laboratory, Universite´ Catholique de Louvain, Louvain-la-Neuve, Belgium
b
Received 16 November 2006; accepted 9 January 2007 Available online 6 March 2007
Abstract This work presents an Improved Charge Sheet compact Model (ICSM) especially valuable for distortion analysis, where precise calculation of derivatives of at least third order is required. A new expression for the charge is used in the calculation of the current. Vertical electric field, mobility and DIBL are represented using previously reported for other purposes more precise expressions. The very good agreement obtained between experimental PD SOI MOSFETs with channel lengths from 0.32 to 10 mm and modeled currents, derivatives and distortion figures is shown. r 2007 Elsevier Ltd. All rights reserved. Keywords: MOSFET modeling; Compact modeling; Harmonic distortion; Integral Function Method
1. Introduction At present MOSFET models can be grouped in several types. Models based on threshold voltage [1], give good accuracy for digital applications but not for analog applications. Surface potential models provide good coincidence with experiment in all operation regions, as well as accurate expressions for harmonics, but the resultant equations are too complicated [2]. Both of them have large number of parameters and complicated extraction procedures. Charge sheet models, based in the normalization of the charge taking in to account the physics and symmetry of the MOSFET have less number of parameters and simpler extraction procedures [3]. Our work was focused on the nonlinear distortion analysis of partially depleted (PD) SOI MOSFET with body contact, which presents similar behavior to the bulk MOSFET. We started the analysis of harmonic distortion using published data for the EKV3.0 model [4–7], with which we could not obtain acceptable coincidence between the modeled and curves. These problems may be related to some limitations of this model previously reported by their Corresponding author. Tel.: +52 5511150188; fax: +52 5550613978.
E-mail address: [email protected] (J. Alvarado). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.01.017
authors in [8]. For comparison purposes, this model will be used as reference model (RM). In order to obtain more precise results for nonlinearity analysis we introduced a new interpolation function for the normalized inversion charge; a new representation of the channel vertical field; mobility and DIBL. The improved charge sheet model required for distortion analysis will be referred as ISCM, and was implemented in Mathematica. 2. Improved model In the charge sheet approximation approach [9], the channel current in the MOS transistor is expressed as function of the surface potential at source and drain edges, evaluated numerically. The calculation of the DC current characteristics represents a hard task. However, analytical symmetrical equations based on the linearization of the inversion charge density and the normalization of currents and voltages can be used to simplify calculation procedure, as in [3]. The normalized current i and the normalized inversion charge density q0i can be expressed as i¼
ID , IS
(1)
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
322
q0i ¼
Q0I , QS
(2)
where ID is the drain current, Q0I the inversion charge per unit area, IS and QS the specific current and charge equal to [3,5,6] I S ¼ 2nq mo C 0ox U 2T W =L,
(3)
QS ¼ 2nq C 0ox U T .
(4)
The normalization factor nq is given by Sallese [6] as pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi nq ¼ 1 þ g=ð 2Ff þ CP Þ. (5) In these expressions, C 0ox is the oxide capacitance per unit area, mo is the mobility for low vertical field, UT is the thermal voltage, g is the body factor, Ff is the bulk Fermi potential and CP is the pinch-off surface potential, defined as [6] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! V G V FB 1 1 2 þ CP ¼ V G V FB g , (6) 4 2 g2 where VFB is the flat band voltage and VG is the gate voltage. The pich-off voltage is considered as [3] VP ¼
VG VT , n
(7)
where VT is the threshold voltage and the subthreshold ideality factor n is equal to pffiffiffiffiffiffiffi n ¼ 1 þ g=2 CP . (8) Furthermore, considering an interpolation function, the normalized forward if and reverse ir currents, can be defined via the normalized inversion charges at the source q0if and at the drain q0ir , as [3] V P V SðDÞ 2 , (9) ifðrÞ ¼ q2ifðrÞ þ qifðrÞ ¼ ln 1 þ exp 2U T where VS is the source voltage and VD is the drain voltage. The total normalized current i can be expressed as the difference between a forward and a reverse normalized currents: i ¼ if ir .
velocity, channel length reduction and DIBL are considered in our model as follows. Charges at source and drain Taking into account the different effect of charges at source and drain edges, the well-known expression for vertical field [5] is changed by the following expression, which symmetrically takes into account the inversion and depleted charges near the drain and source edges [11]: E? ¼ Q0BfðrÞ
ðQ0Bf þ Q0Br Þ þ 1=2ðQ0If þ Q0Ir Þ , 2si qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ gC 0ox Cp =U T þ Q0IfðrÞ ðnq 1=nq Þ,
Q0IfðrÞ ¼ 2nq U T C 0ox q0IfðrÞ ,
(12) (13) (14)
where esi is the silicon permittivity, the depleted and inverted charge per unit area near the source are denoted by Q0Bf and Q0If , respectively and near the drain by Q0Br and Q0Ir . Field dependence mobility The mobility model in RM [5] was simplified taking into account the three main scattering effects: Coulomb, phonon and surface roughness, using the Mathiessen’s rules, in the expression: mo meff ¼ , (15) 1=3 mC þ mPh E ? þ mSR E 2? where mC is the mobility for the Coulomb scattering, mPh for the phonon scattering and mSR for the surface roughness scattering. mC, mPh and mSR are considered in the ICSM as fitting parameters. The total series resistance Rds can be included in (15) as part of the effective mobility: meff . (16) mR ¼ 1 þ ðW =LÞRds C 0ox meff ðV G V T ð1=2Þnq V D Þ
Saturation velocity and saturation voltage
(10)
In order to obtain better coincidence with the curves in the case of the PD SOI transistors analyzed in this paper, the presented ICSM introduces the following new expressions. The normalized charge is calculated using as interpolation function the Lambert’s W function [10], with a fitting parameter a ¼ 2.29: nq V P V SðDÞ q0ifðrÞ ¼ W exp , (11) a UT since nq is a function of VG, factor nq/a is not a constant. The effects of the difference of charge at source and drain, effective mobility, series resistance, saturation
Another important variable is the drift velocity as a function of the longitudinal field, vd [12]. It accurate expression was found by Yamaguchi [13], which was modified by van Langevelde et al. [14] and improved by Bucher [5] giving the next formulation for electrons: E k =E C vd ¼ vsat qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ½2ð2dÞðE k =E C Þ2 1 þ 0:025þj2ð2dÞðE k =E C Þj þ ðE k =E C Þ2
(17)
where vsat is the carrier velocity saturation, EJ is the longitudinal field, EC is the critical longitudinal field and d is a fitting parameter with values from 1 to 2. Using the relation between the mobility and the saturation velocity, for electrons in [5] where (16) and (17) are substituted and
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
integrating the longitudinal field along the channel we find the following mobility expression: mR m ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (18) ½4ð2dÞðq0if q0ir Þ2 2 0 0 1 þ 0:025þj4ð2dÞðq0 q0 Þj þ ½2ðqif qir if
ir
where e ¼ UT/(LEC). Eq. (18), however, is valid only when the saturated velocity is not reached. We can separate the channel in two parts, the first part near the source where the velocity saturation effect is neglected, and the other, near the drain with the velocity saturation effect taken into account. In order to link both parts we can use the following smoothening function [15]: V dseff ¼ V Dsat þ 12½V D V Dsat þ DV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV D V Dsat þ DV Þ2 þ 4DV V Dsat ,
ð19Þ
where DV ¼ U T =d qirsat =q0if qirsat is the small normalized charge, present when the condition of saturation velocity near the drain junction is reached. qirsat can be calculated by equating the saturated drain current I Dsat ¼ WQ0Ir vsat to the drain current expression given by (9)–(11), where m from (18) is used instead of mo. The reverse normalized inversion charge q0ir obtained solving these equations correspond to qirsat, [5]. The saturation voltage is then given by V Dsat ¼ U T ½2ðq0if qirsat Þ þ lnðq0if =qirsat Þ.
(20)
Eq. (19) is used instead of VD in all other equations. Channel length reduction When VD is larger than VDsat the velocity saturation or pinch-off point moves towards the source, causing that the channel length L decreases by DL. This reduction can be expressed as [4] V D V Dsat DL ¼ lLC ln 1 þ , (21) LC ðvsat =mo Þ where l is the channel length model parameter, LC ¼ ðsi X j =C 0ox Þ1=2 and Xj is the silicon layer thickness in the case of SOI transistors. Drain-induced barrier lowering The effect of DIBL on threshold voltage is included using the following expression [9]: V T ¼ V TO 2ðsi =ox Þðtox =LÞ½ðFf þ V S Þ þ sV D ,
(22)
where VTO is the drain bias independent threshold voltage, eox the oxide permittivity and s is the DIBL parameter. 3. Results and discussion Results from modeling using ICSM were compared with measured data and with modeling using the RM. The experimental data correspond to an analog PD SOI
323
MOSFETs fabricated with a 0.12 mm SOI CMOS process with 150 nm silicon layer, 5 nm gate oxide, 2.5 mm channel width and channel length from 0.32 to 10 mm. Common parameters required by RM and ICSM were extracted using the direct parameter extraction method proposed in [16]. The new parameters mC, mPh and mSR required by ICSM, were extracted by optimization. The extracted parameters are given in Table 1. The nonuniform doping, quantum effect and polydepletion effect included in RM do not have hard influence in the PD SOI transistors measured and then we do not considered it. Fig. 1 shows the transfer characteristic at VD ¼ 50 mV, varying VG from 0 to 1.8 V. In order to see the subthreshold and the transition regions, the log (ID) curves are shown too. As can be seen an excellent coincidence between experiment and ICSM modeling is obtained. This is even clearer in Fig. 2 where the transconductances is calculated. We can see that the gm obtained by ICSM calculating first derivative of the current (solid lines) follows with high accuracy the experimental data in all regions of vertical field and mobility expressions. Calculation using the RM differs significantly after the maximum gm, where the surface roughness scattering dominates. It is worth to remark, that according to [4], in the RM gm is calculated by an approximate derivation of the normalized current since through the expression [5]: gm ¼ ðI S =nU T Þðqif qir Þ.
(23)
The transfer characteristics in saturation, for VD ¼ 1 V, and the corresponding gm are analyzed in Fig. 3 where the ICSM again presents better coincidence for all L. The output characteristics for four lengths at fixed VG ¼ 1.5 V (Fig. 4), show very good coincidence with the experimental data again in all regions. Few models have the capability to accurately simulate the MOSFET current in the quasi-linear region for fixed VG and varying VD from negative to positive values. Fig. 5 shows again much better results in this region obtained by the ICSM with respect to the RM where the input signal amplitude is 1 V peak-to-peak. One important element of this type of compact models is how well they reproduce the derivatives of higher orders [17]. We will compare the third-order derivatives of the I–V characteristics in saturation region at VD ¼ 1 V in Fig. 6 and in the quasi-linear region at VG ¼ 1.5 V in Fig. 7. As can be seen, the ICSM offers better coincidence with the experimental data in both cases. As usual, these derivatives can be used in the harmonic distortion analysis of order 3 only in the case of low input amplitudes. The harmonic distortion analysis, done in [8] at very low signal amplitude of 22.3 mV, does not give accurate results especially for n type transistors. The analysis of MOSFET as tunable resistor was done for amplitudes lower than 22 mV, where the EKV3.0 is practically symmetrical, when in the real cases large amplitudes, and nonlinearities, are required.
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
324 Table 1 Extracted model parameters L (mm)
VTO (V)
mC (cm2/V s)
mPh (cm2/V s)
mSR (cm2/V s)
Rds (O)
vsat (cm/s)
d
l
s
10 1 0.5 0.32
0.4 0.405 0.42 0.423
1.0E16 1.0E16 1.0E16 1.0E16
7.2E3 7.2E3 7.1E3 5.6E3
1E15 17E15 11E15 29E15
200 200 200 200
1E7 1E7 1E7 1E7
1.8 1.8 1.8 1.8
0.27 0.27 0.27 0.27
1.1 1.1 1.1 1.1
Fig. 1. Transfer characteristics at VD ¼ 50 mV, VS ¼ 0 V for different channel lengths.
Fig. 2. gm characteristics for the devices of Fig. 1.
An analysis of the nonlinear harmonic distortion including higher amplitudes can be done using the Integral Function Method (IFM) [18]. In this method the determination of the Total Harmonic Distortion (THD) and the Third Order Harmonic Distortion (HD3) requires only the DC transfer characteristic. Fig. 8 shows the THD behavior and Fig. 9 shows the HD3 behavior, both obtained from the measured and modeled characteristics in saturation shown in Fig. 3, when the input sinusoidal signal has an amplitude Va ¼ 0.1 V and the bias voltage Vo(Vo ¼ VG)
Fig. 3. Transfer and gm characteristics in saturation at VD ¼ 1 V, VS ¼ 0 V for different channel lengths.
Fig. 4. Output characteristics for different channel lengths and VG ¼ 1.5 V.
varies from 0.84 to 1.66 V. We see that ICSM provides a much better approximation than the RM, especially for shorter channel lengths. For transistors used as linear resistors, the I–V characteristics in Fig. 5 at VG ¼ 1.5 V, with different channel lengths, were used to determine THD and HD3. The results are shown in Figs. 10 and 11, respectively, where signal amplitude Va varies from 50 to 300 mV (0.6 V peak-to-peak), while the bias voltage Vo is equal to zero. Again, the much better coincidence of ICSM with
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
Fig. 5. Quasi-linear I–V for different channel lengths with VG ¼ 1.5 V and VS ¼ 0 V.
Fig. 6. Third derivative of the transfer characteristics in saturation (Fig. 3) at VD ¼ 1 V.
Fig. 7. Third derivative of the quasi-linear characteristics (Fig. 5) at VG ¼ 1.5 V.
325
Fig. 8. Total harmonic distortion in saturation region (VD ¼ 1 V) as a function of the gate bias voltage.
Fig. 9. Third harmonic distortion in saturation region for the same conditions as in Fig. 8.
Fig. 10. Total harmonic distortion in linear region (VG ¼ 1.5 V) as a function of the signal amplitude applied around a drain-to-source bias of 0 V.
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J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
Acknowledgments This work was partially supported by CONACYT project 39708. References
Fig. 11. Third harmonic distortion in linear region in the same conditions as in Fig. 10.
experimental data is shown for different channel length and for all signal amplitudes.
4. Conclusions This work presents an improved charge sheet compact model which provides very good results for distortion analysis, where precise calculation of derivatives of at least third order is required. A new expression for the charge is used in the calculation of the current. Vertical electric field, mobility, DIBL and saturation mobility are represented using more precise expressions previously reported for other purposes. The model was validated for a PD SOI MOSFET with channel lengths from 0.32 to 10 mm used in analog design. An excellent agreement was obtained in the following current–voltage characteristics and their derivatives: transfer characteristics in the linear and saturation regions, output characteristics and the quasi-linear regime characteristic at fixed gate voltage as tunable resistor. Better precision in modeling of the transistor behavior, as well as in the calculation of THD and HD3 was obtained in all cases showing that the ICSM is especially valuable for distortion analysis.
[1] W. Liu, MOSFET Models for SPICE Simulation Including BSIM3v3 and BSIM4, Wiley, New York, 2001. [2] D.B.M. Klaassen, et al., The MOS Model, level 110, Philips Research Laboratories. [3] C.C. Enz, et al., An analytical MOS transistor model valid in all regions of operations and dedicated to low-voltage and low-current applications, Analog Int. Circuits Signal Proc. J. 8 (1995) 83–114. [4] M. Bucher, et al., The EKV 3.0 Compact MOS Transistor Model: accounting for deep-submicron aspects, in: MSM’2002, 2002, pp. 670–673. [5] M. Bucher, Analytical MOS transistor modeling for analog circuit simulation, Ph.D. Thesis, E´cole Polytechnique Fe´de´rale de Lausanne, 1999. [6] J.-M. Sallese, et al., Inversion charge linearization in MOSFET modeling and rigorous derivation of the EKV compact model, Solid State Electron. 47 (2003) 667–683. [7] M. Bucher, et al., Scalable gm/I based MOSFET model, Int. Semicond. Device Res. Symp. (1997) 615–618. [8] M. Bucher, et al., Analysis of harmonic distortion in deep submicron CMOS, in: 11th IEEE International Conference on Electronics Circuits and Systems (ICECS’04), Israel, December, 2004, pp. 395–398. [9] Y. Tsividis, Operation and Modeling of the MOS Transistor, WCB/ McGraw-Hill, 1999. [10] R.M. Corless, et al., On the Lambert W Function, Adv. Comput. Math. 5 (1996) 329–359. [11] B. In˜iguez, et al., Accurate compact MOSFET modeling scheme for harmonic distortion analysis, J. Semi. Tech. Sci. 4 (3) (2004). [12] D.M. Caughey, et al., Carrier mobility in silicon empirically related to doping and field, Proc. IEEE (Letters) (1967) 2192–2193. [13] K. Yamaguchi, A mobility model for carriers in the MOS inversion layer, IEEE Trans. Electron Devices ED-30 (6) (1983) 658–663. [14] R. van Langevelde, et al., Accurate drain conductance modeling for distortion analysis in MOSFETs, IEDM 1997 Technical Digest, 1997, pp. 313–317. [15] Y. Cheng, et al., BSIM3v3 Manual, UC Berkeley, California, 1995–1996. [16] M. Bucher, et al., An efficient parameter extraction methodology for the EKV MOST model, in: IEEE International CMTS, vol. 9, Trento, Italy, March 19, pp. 145–150. [17] R. van Langevelde, et al., Effect of gate-field dependent mobility degradation on distortion analysis in MOSFETs, IEEE Trans. Electron Devices 44 (11) (1997). [18] A. Cerdeira, M.A. Alema´n, M. Estrada, D. Flandre, Integral Function Method for the determination of nonlinear harmonic distortion, SSE 48 (2004) 2225–2234.
Microelectronics Journal 38 (2007) 321–326 www.elsevier.com/locate/mejo
Harmonic distortion analysis using an improved charge sheet model for PD SOI MOSFETs Joaquı´ n Alvaradoa,, Antonio Cerdeiraa, Valeria Kilchytskab, Denis Flandreb a
Solid-State Electronics Section (SEES), CINVESTAV-IPN, Me´xico D.F., Me´xico Microelectronics Laboratory, Universite´ Catholique de Louvain, Louvain-la-Neuve, Belgium
b
Received 16 November 2006; accepted 9 January 2007 Available online 6 March 2007
Abstract This work presents an Improved Charge Sheet compact Model (ICSM) especially valuable for distortion analysis, where precise calculation of derivatives of at least third order is required. A new expression for the charge is used in the calculation of the current. Vertical electric field, mobility and DIBL are represented using previously reported for other purposes more precise expressions. The very good agreement obtained between experimental PD SOI MOSFETs with channel lengths from 0.32 to 10 mm and modeled currents, derivatives and distortion figures is shown. r 2007 Elsevier Ltd. All rights reserved. Keywords: MOSFET modeling; Compact modeling; Harmonic distortion; Integral Function Method
1. Introduction At present MOSFET models can be grouped in several types. Models based on threshold voltage [1], give good accuracy for digital applications but not for analog applications. Surface potential models provide good coincidence with experiment in all operation regions, as well as accurate expressions for harmonics, but the resultant equations are too complicated [2]. Both of them have large number of parameters and complicated extraction procedures. Charge sheet models, based in the normalization of the charge taking in to account the physics and symmetry of the MOSFET have less number of parameters and simpler extraction procedures [3]. Our work was focused on the nonlinear distortion analysis of partially depleted (PD) SOI MOSFET with body contact, which presents similar behavior to the bulk MOSFET. We started the analysis of harmonic distortion using published data for the EKV3.0 model [4–7], with which we could not obtain acceptable coincidence between the modeled and curves. These problems may be related to some limitations of this model previously reported by their Corresponding author. Tel.: +52 5511150188; fax: +52 5550613978.
E-mail address: [email protected] (J. Alvarado). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.01.017
authors in [8]. For comparison purposes, this model will be used as reference model (RM). In order to obtain more precise results for nonlinearity analysis we introduced a new interpolation function for the normalized inversion charge; a new representation of the channel vertical field; mobility and DIBL. The improved charge sheet model required for distortion analysis will be referred as ISCM, and was implemented in Mathematica. 2. Improved model In the charge sheet approximation approach [9], the channel current in the MOS transistor is expressed as function of the surface potential at source and drain edges, evaluated numerically. The calculation of the DC current characteristics represents a hard task. However, analytical symmetrical equations based on the linearization of the inversion charge density and the normalization of currents and voltages can be used to simplify calculation procedure, as in [3]. The normalized current i and the normalized inversion charge density q0i can be expressed as i¼
ID , IS
(1)
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
322
q0i ¼
Q0I , QS
(2)
where ID is the drain current, Q0I the inversion charge per unit area, IS and QS the specific current and charge equal to [3,5,6] I S ¼ 2nq mo C 0ox U 2T W =L,
(3)
QS ¼ 2nq C 0ox U T .
(4)
The normalization factor nq is given by Sallese [6] as pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi nq ¼ 1 þ g=ð 2Ff þ CP Þ. (5) In these expressions, C 0ox is the oxide capacitance per unit area, mo is the mobility for low vertical field, UT is the thermal voltage, g is the body factor, Ff is the bulk Fermi potential and CP is the pinch-off surface potential, defined as [6] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! V G V FB 1 1 2 þ CP ¼ V G V FB g , (6) 4 2 g2 where VFB is the flat band voltage and VG is the gate voltage. The pich-off voltage is considered as [3] VP ¼
VG VT , n
(7)
where VT is the threshold voltage and the subthreshold ideality factor n is equal to pffiffiffiffiffiffiffi n ¼ 1 þ g=2 CP . (8) Furthermore, considering an interpolation function, the normalized forward if and reverse ir currents, can be defined via the normalized inversion charges at the source q0if and at the drain q0ir , as [3] V P V SðDÞ 2 , (9) ifðrÞ ¼ q2ifðrÞ þ qifðrÞ ¼ ln 1 þ exp 2U T where VS is the source voltage and VD is the drain voltage. The total normalized current i can be expressed as the difference between a forward and a reverse normalized currents: i ¼ if ir .
velocity, channel length reduction and DIBL are considered in our model as follows. Charges at source and drain Taking into account the different effect of charges at source and drain edges, the well-known expression for vertical field [5] is changed by the following expression, which symmetrically takes into account the inversion and depleted charges near the drain and source edges [11]: E? ¼ Q0BfðrÞ
ðQ0Bf þ Q0Br Þ þ 1=2ðQ0If þ Q0Ir Þ , 2si qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ gC 0ox Cp =U T þ Q0IfðrÞ ðnq 1=nq Þ,
Q0IfðrÞ ¼ 2nq U T C 0ox q0IfðrÞ ,
(12) (13) (14)
where esi is the silicon permittivity, the depleted and inverted charge per unit area near the source are denoted by Q0Bf and Q0If , respectively and near the drain by Q0Br and Q0Ir . Field dependence mobility The mobility model in RM [5] was simplified taking into account the three main scattering effects: Coulomb, phonon and surface roughness, using the Mathiessen’s rules, in the expression: mo meff ¼ , (15) 1=3 mC þ mPh E ? þ mSR E 2? where mC is the mobility for the Coulomb scattering, mPh for the phonon scattering and mSR for the surface roughness scattering. mC, mPh and mSR are considered in the ICSM as fitting parameters. The total series resistance Rds can be included in (15) as part of the effective mobility: meff . (16) mR ¼ 1 þ ðW =LÞRds C 0ox meff ðV G V T ð1=2Þnq V D Þ
Saturation velocity and saturation voltage
(10)
In order to obtain better coincidence with the curves in the case of the PD SOI transistors analyzed in this paper, the presented ICSM introduces the following new expressions. The normalized charge is calculated using as interpolation function the Lambert’s W function [10], with a fitting parameter a ¼ 2.29: nq V P V SðDÞ q0ifðrÞ ¼ W exp , (11) a UT since nq is a function of VG, factor nq/a is not a constant. The effects of the difference of charge at source and drain, effective mobility, series resistance, saturation
Another important variable is the drift velocity as a function of the longitudinal field, vd [12]. It accurate expression was found by Yamaguchi [13], which was modified by van Langevelde et al. [14] and improved by Bucher [5] giving the next formulation for electrons: E k =E C vd ¼ vsat qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ½2ð2dÞðE k =E C Þ2 1 þ 0:025þj2ð2dÞðE k =E C Þj þ ðE k =E C Þ2
(17)
where vsat is the carrier velocity saturation, EJ is the longitudinal field, EC is the critical longitudinal field and d is a fitting parameter with values from 1 to 2. Using the relation between the mobility and the saturation velocity, for electrons in [5] where (16) and (17) are substituted and
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
integrating the longitudinal field along the channel we find the following mobility expression: mR m ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (18) ½4ð2dÞðq0if q0ir Þ2 2 0 0 1 þ 0:025þj4ð2dÞðq0 q0 Þj þ ½2ðqif qir if
ir
where e ¼ UT/(LEC). Eq. (18), however, is valid only when the saturated velocity is not reached. We can separate the channel in two parts, the first part near the source where the velocity saturation effect is neglected, and the other, near the drain with the velocity saturation effect taken into account. In order to link both parts we can use the following smoothening function [15]: V dseff ¼ V Dsat þ 12½V D V Dsat þ DV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV D V Dsat þ DV Þ2 þ 4DV V Dsat ,
ð19Þ
where DV ¼ U T =d qirsat =q0if qirsat is the small normalized charge, present when the condition of saturation velocity near the drain junction is reached. qirsat can be calculated by equating the saturated drain current I Dsat ¼ WQ0Ir vsat to the drain current expression given by (9)–(11), where m from (18) is used instead of mo. The reverse normalized inversion charge q0ir obtained solving these equations correspond to qirsat, [5]. The saturation voltage is then given by V Dsat ¼ U T ½2ðq0if qirsat Þ þ lnðq0if =qirsat Þ.
(20)
Eq. (19) is used instead of VD in all other equations. Channel length reduction When VD is larger than VDsat the velocity saturation or pinch-off point moves towards the source, causing that the channel length L decreases by DL. This reduction can be expressed as [4] V D V Dsat DL ¼ lLC ln 1 þ , (21) LC ðvsat =mo Þ where l is the channel length model parameter, LC ¼ ðsi X j =C 0ox Þ1=2 and Xj is the silicon layer thickness in the case of SOI transistors. Drain-induced barrier lowering The effect of DIBL on threshold voltage is included using the following expression [9]: V T ¼ V TO 2ðsi =ox Þðtox =LÞ½ðFf þ V S Þ þ sV D ,
(22)
where VTO is the drain bias independent threshold voltage, eox the oxide permittivity and s is the DIBL parameter. 3. Results and discussion Results from modeling using ICSM were compared with measured data and with modeling using the RM. The experimental data correspond to an analog PD SOI
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MOSFETs fabricated with a 0.12 mm SOI CMOS process with 150 nm silicon layer, 5 nm gate oxide, 2.5 mm channel width and channel length from 0.32 to 10 mm. Common parameters required by RM and ICSM were extracted using the direct parameter extraction method proposed in [16]. The new parameters mC, mPh and mSR required by ICSM, were extracted by optimization. The extracted parameters are given in Table 1. The nonuniform doping, quantum effect and polydepletion effect included in RM do not have hard influence in the PD SOI transistors measured and then we do not considered it. Fig. 1 shows the transfer characteristic at VD ¼ 50 mV, varying VG from 0 to 1.8 V. In order to see the subthreshold and the transition regions, the log (ID) curves are shown too. As can be seen an excellent coincidence between experiment and ICSM modeling is obtained. This is even clearer in Fig. 2 where the transconductances is calculated. We can see that the gm obtained by ICSM calculating first derivative of the current (solid lines) follows with high accuracy the experimental data in all regions of vertical field and mobility expressions. Calculation using the RM differs significantly after the maximum gm, where the surface roughness scattering dominates. It is worth to remark, that according to [4], in the RM gm is calculated by an approximate derivation of the normalized current since through the expression [5]: gm ¼ ðI S =nU T Þðqif qir Þ.
(23)
The transfer characteristics in saturation, for VD ¼ 1 V, and the corresponding gm are analyzed in Fig. 3 where the ICSM again presents better coincidence for all L. The output characteristics for four lengths at fixed VG ¼ 1.5 V (Fig. 4), show very good coincidence with the experimental data again in all regions. Few models have the capability to accurately simulate the MOSFET current in the quasi-linear region for fixed VG and varying VD from negative to positive values. Fig. 5 shows again much better results in this region obtained by the ICSM with respect to the RM where the input signal amplitude is 1 V peak-to-peak. One important element of this type of compact models is how well they reproduce the derivatives of higher orders [17]. We will compare the third-order derivatives of the I–V characteristics in saturation region at VD ¼ 1 V in Fig. 6 and in the quasi-linear region at VG ¼ 1.5 V in Fig. 7. As can be seen, the ICSM offers better coincidence with the experimental data in both cases. As usual, these derivatives can be used in the harmonic distortion analysis of order 3 only in the case of low input amplitudes. The harmonic distortion analysis, done in [8] at very low signal amplitude of 22.3 mV, does not give accurate results especially for n type transistors. The analysis of MOSFET as tunable resistor was done for amplitudes lower than 22 mV, where the EKV3.0 is practically symmetrical, when in the real cases large amplitudes, and nonlinearities, are required.
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324 Table 1 Extracted model parameters L (mm)
VTO (V)
mC (cm2/V s)
mPh (cm2/V s)
mSR (cm2/V s)
Rds (O)
vsat (cm/s)
d
l
s
10 1 0.5 0.32
0.4 0.405 0.42 0.423
1.0E16 1.0E16 1.0E16 1.0E16
7.2E3 7.2E3 7.1E3 5.6E3
1E15 17E15 11E15 29E15
200 200 200 200
1E7 1E7 1E7 1E7
1.8 1.8 1.8 1.8
0.27 0.27 0.27 0.27
1.1 1.1 1.1 1.1
Fig. 1. Transfer characteristics at VD ¼ 50 mV, VS ¼ 0 V for different channel lengths.
Fig. 2. gm characteristics for the devices of Fig. 1.
An analysis of the nonlinear harmonic distortion including higher amplitudes can be done using the Integral Function Method (IFM) [18]. In this method the determination of the Total Harmonic Distortion (THD) and the Third Order Harmonic Distortion (HD3) requires only the DC transfer characteristic. Fig. 8 shows the THD behavior and Fig. 9 shows the HD3 behavior, both obtained from the measured and modeled characteristics in saturation shown in Fig. 3, when the input sinusoidal signal has an amplitude Va ¼ 0.1 V and the bias voltage Vo(Vo ¼ VG)
Fig. 3. Transfer and gm characteristics in saturation at VD ¼ 1 V, VS ¼ 0 V for different channel lengths.
Fig. 4. Output characteristics for different channel lengths and VG ¼ 1.5 V.
varies from 0.84 to 1.66 V. We see that ICSM provides a much better approximation than the RM, especially for shorter channel lengths. For transistors used as linear resistors, the I–V characteristics in Fig. 5 at VG ¼ 1.5 V, with different channel lengths, were used to determine THD and HD3. The results are shown in Figs. 10 and 11, respectively, where signal amplitude Va varies from 50 to 300 mV (0.6 V peak-to-peak), while the bias voltage Vo is equal to zero. Again, the much better coincidence of ICSM with
ARTICLE IN PRESS J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
Fig. 5. Quasi-linear I–V for different channel lengths with VG ¼ 1.5 V and VS ¼ 0 V.
Fig. 6. Third derivative of the transfer characteristics in saturation (Fig. 3) at VD ¼ 1 V.
Fig. 7. Third derivative of the quasi-linear characteristics (Fig. 5) at VG ¼ 1.5 V.
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Fig. 8. Total harmonic distortion in saturation region (VD ¼ 1 V) as a function of the gate bias voltage.
Fig. 9. Third harmonic distortion in saturation region for the same conditions as in Fig. 8.
Fig. 10. Total harmonic distortion in linear region (VG ¼ 1.5 V) as a function of the signal amplitude applied around a drain-to-source bias of 0 V.
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J. Alvarado et al. / Microelectronics Journal 38 (2007) 321–326
Acknowledgments This work was partially supported by CONACYT project 39708. References
Fig. 11. Third harmonic distortion in linear region in the same conditions as in Fig. 10.
experimental data is shown for different channel length and for all signal amplitudes.
4. Conclusions This work presents an improved charge sheet compact model which provides very good results for distortion analysis, where precise calculation of derivatives of at least third order is required. A new expression for the charge is used in the calculation of the current. Vertical electric field, mobility, DIBL and saturation mobility are represented using more precise expressions previously reported for other purposes. The model was validated for a PD SOI MOSFET with channel lengths from 0.32 to 10 mm used in analog design. An excellent agreement was obtained in the following current–voltage characteristics and their derivatives: transfer characteristics in the linear and saturation regions, output characteristics and the quasi-linear regime characteristic at fixed gate voltage as tunable resistor. Better precision in modeling of the transistor behavior, as well as in the calculation of THD and HD3 was obtained in all cases showing that the ICSM is especially valuable for distortion analysis.
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