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A physical compact model for direct tunneling from NMOS inversion layers
Solid-State Electronics 45 (2001) 1705±1716
A physical compact model for direct tunneling from NMOS inversion layers R. Clerc a, P. O'Sullivan b,*, K.G. McCarthy c, G. Ghibaudo a, G. Pananakakis a, A. Mathewson b a
Laboratoire de Physique des Composants a Semiconducteurs, ENSERG, 23 rue des Martyrs, BP 257, 38016 Grenoble Cedex 1, France b National Microelectronics Research Center (NMRC), University College Cork, Lee Maltings, Prospect Row, Cork, Ireland c Department of Electrical and Electronic Engineering, University College Cork, Cork, Ireland Received 19 January 2001; received in revised form 23 May 2001; accepted 13 June 2001
Abstract This paper presents a physically based, analytical, circuit simulation model for direct tunneling from NMOS inversion layers in a MOS structure. The model takes account of the eect of quantization on surface potential in the silicon, the supply of carriers for tunneling and the oxide transmission probability. The inclusion of quantum eects is based on a variational approach to the solution of the Poisson and Schr odinger equations in the silicon inversion layer [Rev Modern Phys 54 (1982) 437]. Usually the variational approach requires iterative solution of equations which is computationally prohibitive in a circuit simulation environment. In this paper, it is shown that by considering the dominant eects in weak and strong inversion, it is possible to formulate a set of equations which give all required quantities for the calculation of quantization in the inversion layer, without the requirement for iterative solution. The tunneling model is based on the concept of transparency. Improved formulae for the transparency and the escape frequency are used. Comparisons with coupled Poisson and Schr odinger simulations and with measurements are demonstrated. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Quantum eects; Direct tunneling; MOS model; Gate current; Circuit simulation
1. Introduction Oxide current due to direct tunneling of carriers is increasingly becoming an important aspect of device operation in decananometer MOS technology. As dielectric thickness is scaled down, leakage current through the gate dielectric due to direct tunneling becomes a problem even at low operating voltages, especially for analog circuit applications. Therefore, a compact model for gate current due to direct tunneling is required, which is suitable for circuit simulation.
*
Corresponding author. Tel.: +353-21-4904392; fax: +35321-4270271. E-mail address: [email protected] (P. O'Sullivan).
Direct tunneling has been analytically modeled in the past, using classical considerations for the behavior of the electrodes [2,3]. There has been increasing interest in recent times in the simulation of direct tunneling for thin oxides on highly doped substrates [4±7]. It is clear from the recent literature, that as oxides are scaled into the ultra-thin regime the eects of the quantization of energy levels in the silicon inversion layer become important in the calculation of the gate tunneling current. There have been several circuit simulation models for gate current proposed in the recent past [8±10]. These models have been based on either purely classical treatment of the electrodes as in Ref. [8] or have used a hybrid of classical and quantum considerations [9,10]. For example, Choi et al. [9] used the classically derived model of Ref. [3] and corrected the surface potential in
0038-1101/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 1 0 1 ( 0 1 ) 0 0 2 2 0 - 9
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strong inversion for quantum eects using the method proposed by van Dort et al. [11,12]. In this paper, quantum equations are used as a starting point for the derivation of the circuit simulation model rather than classical equations. The eect of quantization on each of the component quantities of the gate current is calculated in a consistent fashion. This results in a very physically based model. Further the proposed model allows the explicit calculation of surface potential as a function of gate voltage including quantum eects. This is an important feature and is a requirement for a circuit simulation model. The ®rst part of this paper deals with the basic assumptions made in the quantum modeling of the MOS structure. In particular a variational approach to the calculation of the quantized energy levels in the inversion layer is discussed and compared to exact solution of the Poisson and Schr odinger equations. Having demonstrated that this variational approximation approximates well to the exact solution of Poisson and Schr odinger equations in the oxide thickness regime of interest, a form of this variational approximation which does not require iterative solution of the equations is obtained. This formulation allows explicit calculation of the surface potential and the energy of the ®rst quantized level in the silicon from the applied gate voltage. With these two quantities it is then possible to calculate the number of carriers available for tunneling, the impact frequency at the interface and the transmission probability through the oxide. These are the required quantities for the calculation of the tunnel current. Agreement between the analytical model derived here and the more rigorous solution of the variational equations, which requires iterative solution of a transcendental equation is demonstrated. In the ®nal part of the paper, agreement with measured gate and drain currents is demonstrated. 2. Basic assumptions The eective mass approximation and constant substrate doping are assumed. Thermodynamic equilibrium is also assumed. That is, the number of carriers lost by tunneling through the gate is expected to be small compared to the number of carriers coming into the channel by the drift-diusion mechanism. Fig. 1 shows the calculated tunneling time versus oxide thickness compared to the supply time for dierent channel lengths. It is clear that for oxides down to 1.5 nm the condition of thermodynamic equilibrium will hold as long as the channel length of the device is <10 lm and indeed for thicker oxides the inversion layer is sustainable at channel lengths of 100 lm or more. The goal of this work is a model which is suitable for circuit simulation. In developing a model for use
Fig. 1. Simulated tunneling time and time to supply the inversion layer versus oxide thickness. The symbols are the calculated tunneling time at Vg 1:5 V (using a parabolic eective mass mox 0:5) versus thickness. The horizontal lines are time to supply the inversion layer by drift diusion mechanism for various channel lengths (L 100, 10, 1 lm).
in circuit simulation, computational eciency is of paramount importance. Therefore, a circuit simulation model must identify and include the most signi®cant eects. While quantization results in the splitting of the carriers in the silicon into many discrete energy levels, in general, the contribution of the carriers in the lowest energy sub-band is dominant. Fig. 2 shows the density of electrons available for tunneling versus gate voltage calculated using full self-consistent solution of the Poisson and Schr odinger equations. The density of electrons, in each energy level is shown and it is clear from this that the dominant contribution is from the lowest energy level labeled EL0 over most of the bias range. Hence the assumption that the lowest energy level is dominant is a reasonable one for a circuit simulation model. In this model, the quantization in the silicon and the transmission of the carriers through the barrier region are assumed to be decoupled. The quantization in the silicon is calculated using a variational wave function which does not penetrate the barrier region. The concept of transparency of the silicon dioxide is then used for the calculation of tunneling time. It has been veri®ed in Ref. [13], that the transparency approach used in this work yields similar results for the tunneling time as a more rigorous variational approach which allows penetration of the wave function into the oxide. The transparency concept is easily adapted to the compact modeling application and is directly comparable to previously ex-
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quired. The basis for the model in this work is the variational approximation to the exact solution of the Poisson and Schr odinger equations proposed by Stern and co-workers [1,17]. In the variational approximation, the wave function fb x of the lowest energy level is of the form: r b3 bx 2 fb x x exp 2 2 This expression can be used with the Poisson equation to calculate the self-consistent potential energy. The value of the parameter b is chosen to minimize the ®rst energy level, for each surface potential, ws , and is given by b ws Fig. 2. Comparison between coupled Poisson and Schr odinger simulations and the variational approach. Solid line is total density of electrons in inversion layer versus gate voltage simulated using coupled Poisson and Schr odinger. The density of electrons in the ®rst four energy levels are also shown (EL0 ®rst energy level for the longitudinal valleys, ET0 for the transversal valleys, EL1 second energy level for the longitudinal valleys, ET1 second energy level for the transversal valleys). The total electron density obtained using the variational calculation is also shown ( ). The simulation parameters are: VFB 1 V, Na 3 1017 cm 3 , tox 23 A.
isting models for direct and Fowler±Nordheim tunneling. Using these assumptions, the tunneling current density can be estimated as J qfnT
1
where q is electronic charge, f is the impact frequency against the barrier, T is the oxide transparency, and n is the density of electrons available for tunneling. The calculation of these quantities requires an analytical expression for the ®rst energy level in the conduction band. The transparency of the oxide depends on the oxide voltage and so an explicit equation for surface potential versus gate voltage is required.
3. Quantum modeling of the ®eld eect 3.1. The variational approximation If equilibrium conditions are satis®ed as discussed in the previous section, the charge available for tunneling may be calculated by self-consistent solution of the Poisson and Schr odinger equations as in Refs. [14±16]. However, for a circuit simulation model, an analytical approximation to the solution of these equations is re-
E
1=3 12 2 n ws N mq W w a d s 3 e h2
3 h2 b2 8m
3
4
where E is the energy of the ®rst sub-band (Joules), m is the longitudinal eective mass in silicon m 0:98 [17], e is the dielectric constant of silicon, n ws is the density of electrons at the Si/SiO2 interface and is a function of the surface potential ws , Na is the average doping density, and Wd is the depletion width, calculated using the depletion approximation [17]. The use of the depletion approximation for the calculation of the depletion width assumes that the quantum length Lq is much smaller than the depletion width Wd . The expected doping levels in the next few technology generations are of the order of 1±5 1018 [18]. Even for Na 1019 , the quantum length is of the order of 1 nm at Vg 1:5 V while the depletion width is of the order of 13 nm, so this approximation is still valid in the expected channel doping regime of the MOS transistor. The relationship between the surface potential and the gate voltage Vg is given by Vg VFB ws
qn ws qNa Wd ws Cox
5
where VFB is the ¯at band voltage, and Cox the value of the oxide capacitance. The density of electrons for both weak and strong inversion regimes can be obtained using Fermi±Dirac statistics and the two dimensional density of states: gmd E Efs n ws 2 kT ln 1 exp 6 kT p h where g is the degeneracy factor (g 2 for the ®rst energy level), md is the density of states eective mass (md 0:19 [17]), T is the temperature, k is Boltzmann's constant and Efs is the Fermi level at the interface:
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Efs qws kT ln
Nv Na
Eg
7
where Nv is the eective density of states in the valence band, and Eg is the energy gap in the silicon. There is an implicit relationship between the number of electrons and the energy level and so the solution of these equations requires numerical iteration. For simplicity, these equations have been expressed for the case where there is zero applied bulk bias and drain bias, however, extension to the MOSFET case where the eective bulk bias varies along the channel can be done by replacing Vg in Eq. (5) by Vgb , expressing the depletion width as a function of the applied reverse bias to the channel and replacing Ef in Eq. (6) by the electron quasi-Fermi level Efn . Fig. 2 compares the electron density calculated using the variational approach with the total electron density using the self-consistent solution of the Poisson and Schr odinger equations. It can be seen that good agreement is obtained between the self-consistent Poisson and Schr odinger and the variational solution. There is a small discrepancy in the weak inversion regime, mainly because the energy level of the ®rst transversal valley begins to make a more signi®cant contribution to the total inversion charge. However, even in this regime, the error associated with the variational approximation is small and the results of the variational calculation give a reasonable approximation to the results obtained by coupled solution of the Poisson and Schr odinger equations.
3.2. Approximate solution of the variational approach in weak inversion In the weak inversion regime, the contribution to the total charge of the electron at the interface is small compared to the depletion charge. As a consequence, n ws Na Wd , and using Eqs. (3) and (4), the following expression is obtained for the ®rst energy level versus surface potential: 1=3 9 h2 E ws 3q Na ws 16 me
8
Because the inversion charge is negligible, the relation between the surface potential and the gate voltage can be explicitly calculated by solving Eq. (5) in weak inversion, using the approach of van Langevelde and Klaasen [19]. Therefore it is possible to calculate the ®rst energy level and the density of electrons for tunneling explicitly as a function of gate voltage in this region.
3.3. Approximate solution of the variational approach in strong inversion Strong inversion begins when the concentration of electrons at the interface is equal to the doping level. The surface potential for the onset of strong inversion wst can be estimated to a ®rst order by wst 2/f
E 2/f q
9
where 2/f is the classical value of the surface potential at threshold [20] and /f is the midgap potential. In a quantum model, this classical value of the strong inversion surface potential is augmented by the value of the ®rst energy level, to take into account the eective increase of the bandgap due to the quantization of the energy. The threshold voltage is calculated from Eq. (5) using the threshold value of surface potential (wst ) as de®ned above. This is similar to the approach used by van Dort et al. [11,12]. In the strong inversion regime, the inversion layer charge becomes the most important contribution to the total charge. In classical modeling, the surface potential is approximately constant in the strong inversion regime, and a linear relationship between the inversion charge and the gate voltage holds as follows: n Vg
Cox Vg q
Vth
10
where Vth is the threshold voltage. In the presence of quantum eects, the relationship between the inversion layer charge and the gate voltage can still be modeled as linear but the increase in the threshold voltage due to quantization and the decreased coupling with the gate must be taken into account. The increase in threshold voltage can be calculated from Eqs. (5) and (9). The decrease in the gate coupling can be interpreted in term of parasitic depletion layer between the oxide and the peak charge concentration, due to the annulment of the wave function at the interface [21]. The thickness of this depletion layer Lq , is de®ned as the centroid of the charge concentration. An expression for Lq can be found by ®tting an empirical formula to the results of the Poisson±Schr odinger solution as in Ref. [21]. However, the variational approximation also gives an analytical formula for the centroid of the inversion layer charge as follows: Z 1 3 Lq zjfb zj2 dz 11 b 0 Therefore, Eq. (10) may be modi®ed to include quantum eects by using a ®eld dependent eective oxide capacitance and by rede®ning the threshold condition using Eq. (9).
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
n Vg
1 eox Vg q tox eox =e 3=b Vg
Vth
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12
where eox is the dielectric constant of silicon dioxide, and tox the oxide thickness and a ®rst approximation of b Vg is calculated using the physical oxide thickness. This gives an explicit relationship between the inversion layer charge and the gate voltage. By using Eq. (12) in Eq. (5), a relationship between the surface potential and the gate voltage in strong inversion can be obtained. The details are given in Appendix A. 3.4. Transition between weak and strong inversion In Fig. 3, the number of electrons available for tunneling calculated using a numerical solution of the variational approximation is plotted versus gate voltage, for two dierent doping levels. The weak and strong inversion approximations to the inversion charge are also shown. Clearly, there is good agreement between the approximations and the variational calculation except in the transition region between weak and strong inversion. The strong inversion approximation can be more clearly seen in Fig. 4, where the number of electrons is now plotted in a linear scale, for two dierent oxide thicknesses. The impact of the quantum eect on the eective capacitance is clearly shown. It is particularly signi®cant for the thinner oxide. In Figs. 3 and 4, the analytical approximations described above and the full iterative variational approximation have been compared for vari-
Fig. 4. Strong inversion approximation to the variational calculation. Density of electrons calculated using the variational approach (symbols) for two dierent oxide thicknesses (( ) tox 1:2 nm, ( ) tox 2:3 nm). Solid lines are the strong inversion approximation of Eq. (12). The dashed lines show the classical calculation of inversion charge. These curves are calculated for ¯atband voltage VFB 1 V and substrate doping Na 6:5 1017 cm 3 .
ous doping levels and oxide thickness showing that the approximations proposed hold for dierent technological parameters. An analytical model for circuit simulation requires continuity of the equations from weak to strong inversion. Therefore, a method to match the approximated solution in both weak and strong inversion must be found. To obtain a continuous equation for the inversion charge, a smoothing function of the following form is used: fA x; y
A ln1 exp x=A 1 A=y exp x=A
13
where A is a positive constant. Smoothing functions are widely used in compact MOS models [19,22,23]. This function fA has the following properties: · when x A and y=A exp x=A, f x; y x · when x A and y A, f x; y y
Fig. 3. Weak and strong inversion approximations to the variational calculation. Density of electrons in the inversion layer versus gate voltage, calculated by the variational approach (symbols) for two dierent doping levels (( ) Na 1017 cm 3 and ( ) Na 1018 cm 3 ) compared to the analytical approximations for weak (dashed lines) and strong inversion (solid lines).
Hence, the number of electrons available for tunneling n Vg can be calculated in a continuous way as n Vg fA ns Vg ; nw Vg
14
where ns Vg and nw Vg are the strong and week approximations respectively. Both approximations are explicit functions of gate voltage.
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The parameter A is a smoothing parameter, which de®nes the shape of the transition between weak and strong inversion. The following formula is used to calculate A: A
ns Vth Vo b
15
The inversion charge is calculated at an oset voltage Vo from threshold to ensure that strong inversion holds. The b parameter governs the smoothness of the transition. For doping levels greater than 1017 cm 3 , and oxide the following choice thickness in the range of 12±25 A, of parameters gives good agreement: b 10;
Vo 0:4 V
16
Fig. 5 compares the inversion charge calculated using the numerical variational approach compared with the explicit approximated solutions made continuous using Eq. (14). Even in the threshold region, the ®tting between the simulated model and the approximation is now very good, for a wide range of oxide thicknesses. Using the continuous expression for inversion charge versus gate voltage of Eq. (14) it is possible to calculate the surface potential versus gate voltage continuously from weak to strong inversion as follows:
Fig. 6. Calculation of surface potential. Surface potential ( ) and oxide voltage ( ) versus gate voltage, calculated using the numerical variational model (symbols) compared to the analytical model of Eq. (17) (solid lines). Also shown are the surface potential and oxide voltage calculated using the classical model (dash lines) (VFB 1 V, doping level Na 6:5 1017 cm 3 ).
ws Vg Vg
c2 qn Vg Vfb Cox 2 s 2 c qn Vg c Vg Vfb 4 Cox
17
p where c 2qeNa =Cox is the body eect coecient. The calculation of the surface potential is essential for the correct calculation of the oxide ®eld. Fig. 6 shows that the surface potential and the oxide ®eld calculated using this analytical model agrees well with the variational calculation. Fig. 6 also shows the classically calculated surface potential and oxide ®eld compared to the quantum model. 3.5. Eect of polysilicon depletion
Fig. 5. Impact of the smoothing procedure. Density of electrons calculated using the variational approach versus gate voltage for two dierent oxide thicknesses (( ) tox 1:2 nm, ( ) tox 3:0 nm). The symbols are the exact results using the variational approach, and the lines are the continuous analytical solution proposed in this work, using the smoothing function of Eq. (13). The curves are calculated using Na 6:5 1017 cm 3 , b 10, Vo 0:4 V.
The impact of the depletion region in the polysilicon gate has to be also taken into account for the modeling of gate current, especially in the strong inversion regime. This can be done in a simple but ecient way using the depletion approximation, and making a distinction between the applied gate voltage Vgapp and the eective gate voltage Vg [22]: Vgapp Vg Vp
18
where Vp is the voltage drop in the polysilicon interface. The relation between the eective gate voltage Vg and the applied gate voltage Vgapp can be expressed as
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
1 fT s
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20
where f is the impact frequency against the barrier and T is the oxide transparency (or transmission probability). There are many numerical approaches to the calculation of the tunneling time [24±30], but for an analytical model the simplest approach is to adopt the concept of the barrier transparency or transmission probability. 4.1. Transparency
Fig. 7. Impact of polysilicon depletion on density of electrons available for tunneling in strong inversion. Density of electrons versus gate voltage calculated (a) classically (dotted line) (b) including quantum eects but without polysilicon depletion (dashed line) (c) including both quantum eects and polysilicon depletion (symbols) (d) using analytical model solution proposed in this work, where the polysilicon eect is taken into account using Eq. (19).
qeNp 2 Cox (s 2C 2 Vgapp VFB wst 1 ox qeNp
Vg Vgapp VFB wst
) 1 19
where Np is the polysilicon gate doping level. For simplicity, the surface potential in strong inversion has been approximated here by its value at threshold wst . Fig. 7 shows the impact of the polysilicon depletion on the inversion charge in strong inversion. It is clear that the eect of polysilicon depletion is signi®cant and must be included for accurate calculation of the charge available for tunneling and the surface potential. The ®gure also compares the result of applying Eqs. (18) and (19) with a numerical iterative solution, showing that the approximation of the polysilicon depletion by pinning the surface potential at the wst value works well.
4. Direct tunneling current equations In the previous section, formulae which allow the continuous calculation of the number of electrons available for tunneling as a function of gate voltage were obtained. From Eq. (1), to calculate the tunnel current the tunneling time s, must be calculated:
Transparency is usually estimated using the WKB approximation [31]. However, this approximation only takes into account the attenuation of the wave function through the energy potential barrier, and does not take account of the re¯ections at the oxide interface. Register et al. [32] proposed an improvement to the transparency by introducing a correction term C E to take into account re¯ections both at the Si/SiO2 and SiO2 /polysilicon interface. T E C ETwkb E
21
In this work, a one band parabolic dispersion relation in the oxide is used and so: p 4 2mox tox Twkb E; Vox exp q/ox E3=2 3q hVox q/ox qVox E3=2 22 It is also possible to obtain, a simple expression for the WKB transparency using a two band single eective mass approximation as in Refs. [32,33]. For electron tunneling from the conduction band of the silicon, both relationships can give similar results with suitable choice of eective mass. It should be noted that it is also possible to use the approach developed here for the calculation of the quantization in the silicon with transparency based on a two band approximation for the dispersion in the oxide. From Ref. [32]: C E; Vox
4t Etox E t E2 tox E2 4t E qVox tox E qVox t E qVox 2 tox E qVox 2
23
p p where t E 2E=2m and tox E 2 q/ox E=2mox in the case of a parabolic dispersion relationship in the oxide, m is eective mass in the silicon and mox is eective mass in the oxide. In order, to evaluate this corrective term for the case of a trapezoidal barrier which occurs in the direct
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the bounded wave function (2), that the impact frequency for the ®rst energy level can be calculated as f
hb2 2 fF 4pm 3
25
5. Comparison with experiments
Fig. 8. Impact of Register's correction [32] on the transparency. Transparency of the oxide versus gate voltage (tox 1:2 nm) calculated using Gundlach's exact solution for the trapezoidal barrier [34,35] (symbols), compared to WKB approximation (dashed line) and WKB approximation including Register's correction (solid line).
tunneling situation, the transparency calculated using the WKB approximation and the correction proposed by Register et al. [32] were compared to the exact expression for the transparency of a trapezoidal barrier calculated by Gundlach using Airy functions [34,35]. The dierent eective masses in the oxide and silicon were taken into account in the calculation. Fig. 8 shows the results of these calculations. The correction proposed by Register is extremely eective and signi®cantly reduces the error associated with the WKB approximation.
Capacitance±voltage (C±V ) measurements and current±voltage (I±V ) measurements were performed on large area NMOS transistor test structures (W 300 lm, L 100 lm). A procedure was used to correct the impact of series resistance and tunneling on the C±V measurements at high ®eld [38]. Oxide thickness was extracted using an optimized Maserjian procedure in the accumulation regime. The ¯at band voltage and the doping level in the substrate and in the polysilicon gate was also extracted. More details of the various extraction procedures are given in Ref. [24]. Fig. 9 compares the measured gate current versus gate voltage with the ®t obtained using the analytical model. The usual values of eective mass (mox 0:5) and barrier height were used (/ox 3:1 eV) in the calculation of the transparency [39]. The ¯at band voltage (VFB 0:98 V) was extracted from C±V measurements. A series resistance has been also extracted in the high
4.2. Impact frequency The last parameter required for the calculation of the tunneling current is the impact frequency or the ¯ux of particles against the barrier per unit time. For a plane wave (i.e. free electrons), this frequency fF can be calculated as a function of the energy as (only for the ®rst energy level): fF
2E h
24
This formula has already been used in previous work [7,32,36]. But in MOS tunneling, electrons are no longer free, but con®ned in a quantum well, which would be expected to change the impact frequency. It has been shown [37], using the decomposition in plane wave of
Fig. 9. Comparison of the gate current model with experimental results. Symbols are measured gate current density versus gate voltage for an oxide thickness of 2.3 nm. The solid line is the ®t obtained using this model including series resistance correction. Also shown is the ®t which would be obtained without series resistance correction (dotted line) and the ®t without Register's correction to the transparency (dashed line). Parameters used for the ®t are mox 0:5, /ox 3:1 eV, VFB 0:98 V, tox 2:28 nm, Na 4:2 1017 cm 3 .
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
®eld region (Vg > 2 V) of the gate current curve, where it presents a clear linear dependency with the gate voltage. The impact of this series resistance correction on the ®tted curves is also shown in Fig. 9. The polysilicon depletion has been also taken into account for this ®tting, using the polysilicon doping level extracted from C±V measurement. The substrate doping level and the oxide thickness extracted from C±V were re-optimized for the ®tting of the gate current characteristic but do not dier signi®cantly from the original values extracted from C±V . A substrate doping of Na 4:2 1017 cm 3 was used for the current ®t compared to Na 6:0 1017 cm 3 extracted from C±V and an oxide thickness of 2.28 nm was used compared to 2.30 nm extracted from C±V . It is clear from Fig. 9 that the quality of the ®t obtained using this model is good and is certainly of comparable quality to those obtained by numerical simulation [4,5]. In particular the model follows the shape of the curve particularly well from weak to strong inversion. This is principally because the decay in the supply of carriers for tunneling in the weak inversion regime is included in the model and also because of the correction to the WKB transparency. Fig. 10 shows the ®ts obtained using this model to measured data taken from a range of oxide thicknesses again showing good agreement. The data shown in Fig. 10 was measured from structures fabricated in dierent processes with dierent substrate and polysilicon doping levels. In obtaining these ®ts, the parameters of the model, were maintained close to their physical values and no ®tting of eective mass or barrier height was performed.
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6. Conclusion An analytical model suitable for use in circuit simulation has been presented for direct tunneling of electrons from inversion layer of an NMOSFET to the gate. The model includes the eect of quantization in the silicon inversion layer using the variational approximation to the solution of the Poisson and Schr odinger equations. The use of the variational approximation usually requires iterative solution of a set of equations. In this work, the dominant terms and eects in each region of operation have been identi®ed. The key to the model is an analytical equation which ®ts the inversion charge from weak to strong inversion. This removes the requirement for an iterative solution of the equations. It also allows the calculation of all of the other quantities required for the tunneling current (the surface potential, the oxide ®eld and the position of the ®rst energy level in the silicon) in a consistent fashion. For the calculation of the tunneling time, the Register correction [32] for the transparency calculated by the WKB approximation was validated. An improved formula for the impact frequency which takes into account the con®nement of the carriers was also adopted. The model is formulated such that the component quantities of the current have a clearly identi®able physical basis. In particular the fact that the inversion charge and the surface potential are available using this methodology opens up the possibility of a compact model in which quantum eects on the gate and the drain current could be calculated in a consistent fashion.
Acknowledgements The authors want to thank LETI laboratories (Grenoble, France) for providing samples, and J.L. Autran (L2MP, Marseille, France) for very helpful discussions.
Appendix A. List of equations used for the compact model
Fig. 10. Fit of gate current model for dierent oxide thicknesses. Symbols are measured gate current density versus gate voltage for dierent oxide thicknesses (( ) tox 2:95 nm, ( ) tox 2:05 nm, ( ) tox 2:05 nm, ( ) tox 1:73 nm). Solid lines are the model ®t.
Constants h Planck's constant q elementary charge k Boltzmann constant T temperature (T 300 K for this work) e dielectric constant of silicon eox dielectric constant of silicon dioxide m longitudinal eective mass in silicon conduction band (m 0:98) md density of states eective mass (for longitudinal electrons) (md 0:19)
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g
degeneracy factor (for longitudinal electrons) (g 2) eective mass in silicon dioxide conduction band (mox 0:5) barrier height between silicon and silicon dioxide conduction band (/ox 3:1 eV) silicon bandgap (Eg 1:12 eV 1:7942 1019 J) eective density of states in the valence band (Nv 1:04 1019 cm 3 ) eective density of states in the conduction band (Nc 2:8 1019 cm 3 )
mox /ox Eg Nv Nc
Parameters extracted form C±V measurements (extraction procedure: see Ref. [37]) VFB ¯at band voltage Na doping level in the substrate Np doping level in the polysilicon gate tox oxide thickness Notation surface potential ws Vgapp applied gate voltage Vox oxide voltage Vg eective gate voltage corrected for polysilicon depletion Equations: (A) Quantum model for the charge available for tunneling ni intrinsic concentration: p Eg ni Nc Nv exp 2kT Bulk potential: kT Na ln /f q ni Fermi level in the bulk: Nv Ef kT ln Na Oxide capacitance: Cox
eox tox
First energy level in weak inversion: 1=3 9 h2 Na ws Ew ws 3q 16 me Surface potential versus gate voltage in weak inversion: r c2 c2 wsw Vg Vg Vfb c Vg Vfb where 2 4 p c 2qeNa =Cox
Density of electrons in weak inversion regime as a function of surface potential: gmd qws Ef Ew ws Eg nwo ws 2 kT exp kT p h Density of electrons in weak inversion regime as a function of gate voltage: nw Vg nwo wsw Vg Surface potential at threshold: wst 2Uf
Ew 2Uf q
Depletion Width: s 2ews Wd ws qNa b parameter: b n; ws
i1=3 h 12 2 n mq W w N a d s 3 e h2
Threshold voltage: Vth VFB wst
qnwo wst qNa Wd wst Cox
First approximation to density of electrons in strong inversion regime: nso Vg
Cox Vg q
Vth
Eective oxide capacitance: Coxeff Vg
eox tox eox =e 3=b nso Vg ; wst
Density of electrons in strong inversion regime: ns Vg
Coxeff Vg Vg q
Vth
Smoothing parameters: b 10;
Vo 0:4 V;
A
ns Vth Vo b
Density of electrons in both weak and strong inversion: n Vg
A ln1 expns Vg =A A exp ns Vg =A 1 nw Vg
Surface potential versus gate voltage in both weak and strong inversion:
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
ws Vg Vg
c2 qn Vg Vfb Cox 2 s 2 c qn Vg Vg Vfb c 4 Cox
In¯uence of the polysilicon depletion eect on the gate voltage: Vg Vgapp VFB wst (s qeNp 2C 2 Vgapp VFB wst 2 1 ox qeNp Cox
) 1
(B) Tunnel current equations Voltage drop across oxide: Vox Vg Vg
VFB
ws Vg
First energy level: 2 3h2 b n Vg ; ws Vg E Vg 8m Impact frequency: 2 h b n Vg ; ws Vg f Vg 4pm WKB formula for direct tunneling transparency: p 4 2mox tox q/ox E3=2 Twkb E; Vox exp 3qhVox q/ox qVox E3=2 Register correction to WKB transparency: s r E q/ox E ; tox E t E ; 2m 2mox C E; Vox
4t Etox E 2
4t E qVox tox E qVox 2
t E tox E t E qVox 2 tox E qVox 2
Direct tunneling transparency: T Vg C E Vg ; Vox Vg Twkb E Vg ; Vox Vg Tunneling gate current: I Vg qf Vg T Vg n Vg
References [1] Ando T, Fowler AB, Stern F. Electronic properties of 2D systems. Rev Modern Phys 1982;54:437±672. [2] Simmons J. Generalized formula for the electric tunnel eect between similar electrodes separated by a thin insulating ®lm. J Appl Phys 1963;34(6):1793±803.
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[3] Depas M, Vermeire B, Mertens PW, Van Meirhaeghe RL, Heyns MM. Determination of tunneling parameters in ultra thin oxide layer poly Si/SiO2 /Si structures. Solid-State Electron 1995;38(8):1465±71. [4] Lo SH, Buchanan DA, Taur Y. Quantum mechanical modeling of electron tunneling current from the inversion layer of ultra-thin-oxide nMOSFETs. IBM J Res 1999; 43(3):209±11. [5] Rana F, Tiwari S, Buchanan DA. Self consistent modelling of accumulation layers and tunneling currents through very thin oxides. Appl Phys Lett 1996;69(8):1104±6. [6] Magnus W, Schoenmaker W. Full quantum mechanical treatment of charge leakage in MOS capacitors with ultrathin oxide layers. Proceedings of European Solid State Device Research Conference (ESSDERC), 1999. p. 248± 51. [7] Yang N, Kirklen HW, Hauser JR, Wortman JJ. Modelling study of ultrathin gate oxides using direct tunneling current and capacitance±voltage measurements in MOS devices. IEEE Trans Electron Dev 1999;46(7):1464±71. [8] Lee WC, Hu C. Modeling gate and substrate currents due to conduction and valence-band electron and hole tunneling. Symp on VLSI Tech, Digest of Technical Papers, 2000. p. 198±9. [9] Choi CH, Oh KH, Goo JS, Yu Z, Dutton RW. Direct tunneling current model for circuit simulation. Proceedings of International Electron Devices Meeting (IEDM), 1999. p. 734±8. [10] O'Sullivan P, Fox A, McCarthy KG, Mathewson A. Towards a compact model for MOSFETS with direct tunneling gate dielectrics. Proceedings of European Solid State Device Research Conference (ESSDERC), 1999. p. 488±91. [11] Van Dort MJ, Woerlee PH, Walker AJ. A simple model for quantisation eects in heavily doped silicon at inversion conditions. Solid-State Electron 1994;37(3):411±4. [12] Van Dort MJ, Woerlee PH, Walker AJ, Juermans CAH, Lifka H. In¯uence of high substrate doping levels on the threshold voltage and the mobility of deep submicrometer MOSFET's. IEEE Trans Electron Dev 1992;39(4): 932±7. [13] Clerc R, Ghibaudo G, Pananakakis G, Bardeen's approach for tunneling in MOS structure. Proceedings of ULIS Workshop 2001, January 2001, Grenoble, France, SolidState Electron, submitted for publication. [14] Ohkura K. Quantum eect in Si-nMOS inversion layer at high substrate concentration. Solid-State Electron 1990; 33:1581. [15] Moglestue C. Self-consistent calculation of electron and hole inversion charges at silicon±silicon dioxide interfaces. J Appl Phys 1986;59(9):3175±83. [16] Janik T, Majkusiak B. Analysis of the MOS transistor based on the self-consistent solution to the Schr odinger and Poisson equations and on the local mobility model. IEEE Trans Electron Dev 1998;45(6):1263±71. [17] Matthieu, Physique des dispositifs a semiconducteur, third edition, Masson. [18] Skotnicki T. Heading for decananometer CMOS-Is navigation among icebergs still a viable strategy? Proceedings of European Solid State Device Research Conference (ESSDERC), 2000. p. 19±33.
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[19] Van Langevelde R, Klaasen FM. An explicit surfacepotential-based MOSFET model for circuit simulation. Solid-State Electron 2000;44(3):409±18. [20] Nicollian EH, Brews JR. MOS Physics and Technology, Wiley, New York, 1982. [21] Liu W, Jin X, King Y, Hu C. An ecient and accurate compact model for thin oxide MOSFET intrinsic capacitance considering the ®nite charge layer thickness. IEEE Trans Electron Dev 1999;46(5):1070±2. [22] http://www.device-eecs.berkeley.edu/bsim3/. [23] http://www-eu3.semiconductors.com/Philips_Models/. [24] Clerc R, Devoivre T, Ghibaudo G, Caillat C, Guegan G, Reimbold G, Pananakakis G. Capacitance±voltage (C±V ) thick gate oxide: parameter excharacterization of 20 A traction and modeling. Microelectron Reliability 2000; 40(4):571±5. [25] Landau L, Lischitz L. Quantum Mechanics, MIR, Moscow, 1977. [26] Gildenblat G, Gelmont B, Vatannia S. Resonant behavior symmetry, and singularity of the transfer matrix in asymmetric tunneling structrure. J Appl Phys 1995;77(12):6327± 31. [27] Cassan E. On the reduction of direct tunneling leakage through ultrathin gate oxides by a one dimensional Schrodinger±Poisson solver. J Appl Phys 2000;87(11): 7931±9. [28] Dalla SA, Abramo A, Palestri P, Selmi L, Widdershoven F. A comparison between semi classical escape times for gate current calculations. Proceedings of European Solid State Device Research Conference (ESSDERC), 2000. p. 340±3. [29] Bardeen J. Tunnelling from a many-particle point of view. Phys Rev Lett 1961;6:57.
[30] Wettstein A. Quantum Eects in MOS Devices. In: Fichtner W, Guggenbuhl W, Melchior H, Moschytz GS, editors. Series in Microelectronics, vol. 94, publ. Hartung Gorre. [31] Schi L. Quantum Mechanics, McGraw-Hill, New York, 1968. [32] Register LF, Rosenbaum E, Yang K. Analytic model for direct tunneling current in polycrystalline silicon-gate metal-oxide-semiconductor devices. Appl Phys Lett 1999;74(3):457±9. [33] Majkusiak B. Gate tunnel current in an MOS transistor. IEEE Trans Electron Dev 1990;37(4):1087±92. [34] Gundlach KH. Zur Berechnung des tunnelstroms durch eine trapezf ormige potentialstufe. Solid-State Electron 1966;9:949±57. [35] Schenk A, Heiser G. Modelling and simulation of tunneling through ultra-thin gate dielectrics. J Appl Phys 1997;81(12):7900±5. [36] Weinberg ZA. Tunneling of electrons from Si into thermally grown SiO2 . Solid-State Electron 1977;20(11):11±8. [37] Clerc R, DeSalvo B, Caillat C, Ghibaudo G, Pananakakis G. Electrical characterisation and modelling of MOS structures with ultra thin oxide. In: Grenoble LPCS, editor. Proceedings of the First European Workshop on the Ultimate Integration of Silicon (ULIS), 2000. p. 81. Solid-State Electron, submitted for publication. [38] Yang KJ, Hu C. MOS capacitance measurements for high leakage thin dielectrics. IEEE Trans Electron Dev 1999;46(7):1500±1. [39] Weinberg ZA. On tunneling in metal-oxide-silicon structures. J Appl Phys 1982;53(7):5052±6.
A physical compact model for direct tunneling from NMOS inversion layers R. Clerc a, P. O'Sullivan b,*, K.G. McCarthy c, G. Ghibaudo a, G. Pananakakis a, A. Mathewson b a
Laboratoire de Physique des Composants a Semiconducteurs, ENSERG, 23 rue des Martyrs, BP 257, 38016 Grenoble Cedex 1, France b National Microelectronics Research Center (NMRC), University College Cork, Lee Maltings, Prospect Row, Cork, Ireland c Department of Electrical and Electronic Engineering, University College Cork, Cork, Ireland Received 19 January 2001; received in revised form 23 May 2001; accepted 13 June 2001
Abstract This paper presents a physically based, analytical, circuit simulation model for direct tunneling from NMOS inversion layers in a MOS structure. The model takes account of the eect of quantization on surface potential in the silicon, the supply of carriers for tunneling and the oxide transmission probability. The inclusion of quantum eects is based on a variational approach to the solution of the Poisson and Schr odinger equations in the silicon inversion layer [Rev Modern Phys 54 (1982) 437]. Usually the variational approach requires iterative solution of equations which is computationally prohibitive in a circuit simulation environment. In this paper, it is shown that by considering the dominant eects in weak and strong inversion, it is possible to formulate a set of equations which give all required quantities for the calculation of quantization in the inversion layer, without the requirement for iterative solution. The tunneling model is based on the concept of transparency. Improved formulae for the transparency and the escape frequency are used. Comparisons with coupled Poisson and Schr odinger simulations and with measurements are demonstrated. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Quantum eects; Direct tunneling; MOS model; Gate current; Circuit simulation
1. Introduction Oxide current due to direct tunneling of carriers is increasingly becoming an important aspect of device operation in decananometer MOS technology. As dielectric thickness is scaled down, leakage current through the gate dielectric due to direct tunneling becomes a problem even at low operating voltages, especially for analog circuit applications. Therefore, a compact model for gate current due to direct tunneling is required, which is suitable for circuit simulation.
*
Corresponding author. Tel.: +353-21-4904392; fax: +35321-4270271. E-mail address: [email protected] (P. O'Sullivan).
Direct tunneling has been analytically modeled in the past, using classical considerations for the behavior of the electrodes [2,3]. There has been increasing interest in recent times in the simulation of direct tunneling for thin oxides on highly doped substrates [4±7]. It is clear from the recent literature, that as oxides are scaled into the ultra-thin regime the eects of the quantization of energy levels in the silicon inversion layer become important in the calculation of the gate tunneling current. There have been several circuit simulation models for gate current proposed in the recent past [8±10]. These models have been based on either purely classical treatment of the electrodes as in Ref. [8] or have used a hybrid of classical and quantum considerations [9,10]. For example, Choi et al. [9] used the classically derived model of Ref. [3] and corrected the surface potential in
0038-1101/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 1 0 1 ( 0 1 ) 0 0 2 2 0 - 9
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R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
strong inversion for quantum eects using the method proposed by van Dort et al. [11,12]. In this paper, quantum equations are used as a starting point for the derivation of the circuit simulation model rather than classical equations. The eect of quantization on each of the component quantities of the gate current is calculated in a consistent fashion. This results in a very physically based model. Further the proposed model allows the explicit calculation of surface potential as a function of gate voltage including quantum eects. This is an important feature and is a requirement for a circuit simulation model. The ®rst part of this paper deals with the basic assumptions made in the quantum modeling of the MOS structure. In particular a variational approach to the calculation of the quantized energy levels in the inversion layer is discussed and compared to exact solution of the Poisson and Schr odinger equations. Having demonstrated that this variational approximation approximates well to the exact solution of Poisson and Schr odinger equations in the oxide thickness regime of interest, a form of this variational approximation which does not require iterative solution of the equations is obtained. This formulation allows explicit calculation of the surface potential and the energy of the ®rst quantized level in the silicon from the applied gate voltage. With these two quantities it is then possible to calculate the number of carriers available for tunneling, the impact frequency at the interface and the transmission probability through the oxide. These are the required quantities for the calculation of the tunnel current. Agreement between the analytical model derived here and the more rigorous solution of the variational equations, which requires iterative solution of a transcendental equation is demonstrated. In the ®nal part of the paper, agreement with measured gate and drain currents is demonstrated. 2. Basic assumptions The eective mass approximation and constant substrate doping are assumed. Thermodynamic equilibrium is also assumed. That is, the number of carriers lost by tunneling through the gate is expected to be small compared to the number of carriers coming into the channel by the drift-diusion mechanism. Fig. 1 shows the calculated tunneling time versus oxide thickness compared to the supply time for dierent channel lengths. It is clear that for oxides down to 1.5 nm the condition of thermodynamic equilibrium will hold as long as the channel length of the device is <10 lm and indeed for thicker oxides the inversion layer is sustainable at channel lengths of 100 lm or more. The goal of this work is a model which is suitable for circuit simulation. In developing a model for use
Fig. 1. Simulated tunneling time and time to supply the inversion layer versus oxide thickness. The symbols are the calculated tunneling time at Vg 1:5 V (using a parabolic eective mass mox 0:5) versus thickness. The horizontal lines are time to supply the inversion layer by drift diusion mechanism for various channel lengths (L 100, 10, 1 lm).
in circuit simulation, computational eciency is of paramount importance. Therefore, a circuit simulation model must identify and include the most signi®cant eects. While quantization results in the splitting of the carriers in the silicon into many discrete energy levels, in general, the contribution of the carriers in the lowest energy sub-band is dominant. Fig. 2 shows the density of electrons available for tunneling versus gate voltage calculated using full self-consistent solution of the Poisson and Schr odinger equations. The density of electrons, in each energy level is shown and it is clear from this that the dominant contribution is from the lowest energy level labeled EL0 over most of the bias range. Hence the assumption that the lowest energy level is dominant is a reasonable one for a circuit simulation model. In this model, the quantization in the silicon and the transmission of the carriers through the barrier region are assumed to be decoupled. The quantization in the silicon is calculated using a variational wave function which does not penetrate the barrier region. The concept of transparency of the silicon dioxide is then used for the calculation of tunneling time. It has been veri®ed in Ref. [13], that the transparency approach used in this work yields similar results for the tunneling time as a more rigorous variational approach which allows penetration of the wave function into the oxide. The transparency concept is easily adapted to the compact modeling application and is directly comparable to previously ex-
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
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quired. The basis for the model in this work is the variational approximation to the exact solution of the Poisson and Schr odinger equations proposed by Stern and co-workers [1,17]. In the variational approximation, the wave function fb x of the lowest energy level is of the form: r b3 bx 2 fb x x exp 2 2 This expression can be used with the Poisson equation to calculate the self-consistent potential energy. The value of the parameter b is chosen to minimize the ®rst energy level, for each surface potential, ws , and is given by b ws Fig. 2. Comparison between coupled Poisson and Schr odinger simulations and the variational approach. Solid line is total density of electrons in inversion layer versus gate voltage simulated using coupled Poisson and Schr odinger. The density of electrons in the ®rst four energy levels are also shown (EL0 ®rst energy level for the longitudinal valleys, ET0 for the transversal valleys, EL1 second energy level for the longitudinal valleys, ET1 second energy level for the transversal valleys). The total electron density obtained using the variational calculation is also shown ( ). The simulation parameters are: VFB 1 V, Na 3 1017 cm 3 , tox 23 A.
isting models for direct and Fowler±Nordheim tunneling. Using these assumptions, the tunneling current density can be estimated as J qfnT
1
where q is electronic charge, f is the impact frequency against the barrier, T is the oxide transparency, and n is the density of electrons available for tunneling. The calculation of these quantities requires an analytical expression for the ®rst energy level in the conduction band. The transparency of the oxide depends on the oxide voltage and so an explicit equation for surface potential versus gate voltage is required.
3. Quantum modeling of the ®eld eect 3.1. The variational approximation If equilibrium conditions are satis®ed as discussed in the previous section, the charge available for tunneling may be calculated by self-consistent solution of the Poisson and Schr odinger equations as in Refs. [14±16]. However, for a circuit simulation model, an analytical approximation to the solution of these equations is re-
E
1=3 12 2 n ws N mq W w a d s 3 e h2
3 h2 b2 8m
3
4
where E is the energy of the ®rst sub-band (Joules), m is the longitudinal eective mass in silicon m 0:98 [17], e is the dielectric constant of silicon, n ws is the density of electrons at the Si/SiO2 interface and is a function of the surface potential ws , Na is the average doping density, and Wd is the depletion width, calculated using the depletion approximation [17]. The use of the depletion approximation for the calculation of the depletion width assumes that the quantum length Lq is much smaller than the depletion width Wd . The expected doping levels in the next few technology generations are of the order of 1±5 1018 [18]. Even for Na 1019 , the quantum length is of the order of 1 nm at Vg 1:5 V while the depletion width is of the order of 13 nm, so this approximation is still valid in the expected channel doping regime of the MOS transistor. The relationship between the surface potential and the gate voltage Vg is given by Vg VFB ws
qn ws qNa Wd ws Cox
5
where VFB is the ¯at band voltage, and Cox the value of the oxide capacitance. The density of electrons for both weak and strong inversion regimes can be obtained using Fermi±Dirac statistics and the two dimensional density of states: gmd E Efs n ws 2 kT ln 1 exp 6 kT p h where g is the degeneracy factor (g 2 for the ®rst energy level), md is the density of states eective mass (md 0:19 [17]), T is the temperature, k is Boltzmann's constant and Efs is the Fermi level at the interface:
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Efs qws kT ln
Nv Na
Eg
7
where Nv is the eective density of states in the valence band, and Eg is the energy gap in the silicon. There is an implicit relationship between the number of electrons and the energy level and so the solution of these equations requires numerical iteration. For simplicity, these equations have been expressed for the case where there is zero applied bulk bias and drain bias, however, extension to the MOSFET case where the eective bulk bias varies along the channel can be done by replacing Vg in Eq. (5) by Vgb , expressing the depletion width as a function of the applied reverse bias to the channel and replacing Ef in Eq. (6) by the electron quasi-Fermi level Efn . Fig. 2 compares the electron density calculated using the variational approach with the total electron density using the self-consistent solution of the Poisson and Schr odinger equations. It can be seen that good agreement is obtained between the self-consistent Poisson and Schr odinger and the variational solution. There is a small discrepancy in the weak inversion regime, mainly because the energy level of the ®rst transversal valley begins to make a more signi®cant contribution to the total inversion charge. However, even in this regime, the error associated with the variational approximation is small and the results of the variational calculation give a reasonable approximation to the results obtained by coupled solution of the Poisson and Schr odinger equations.
3.2. Approximate solution of the variational approach in weak inversion In the weak inversion regime, the contribution to the total charge of the electron at the interface is small compared to the depletion charge. As a consequence, n ws Na Wd , and using Eqs. (3) and (4), the following expression is obtained for the ®rst energy level versus surface potential: 1=3 9 h2 E ws 3q Na ws 16 me
8
Because the inversion charge is negligible, the relation between the surface potential and the gate voltage can be explicitly calculated by solving Eq. (5) in weak inversion, using the approach of van Langevelde and Klaasen [19]. Therefore it is possible to calculate the ®rst energy level and the density of electrons for tunneling explicitly as a function of gate voltage in this region.
3.3. Approximate solution of the variational approach in strong inversion Strong inversion begins when the concentration of electrons at the interface is equal to the doping level. The surface potential for the onset of strong inversion wst can be estimated to a ®rst order by wst 2/f
E 2/f q
9
where 2/f is the classical value of the surface potential at threshold [20] and /f is the midgap potential. In a quantum model, this classical value of the strong inversion surface potential is augmented by the value of the ®rst energy level, to take into account the eective increase of the bandgap due to the quantization of the energy. The threshold voltage is calculated from Eq. (5) using the threshold value of surface potential (wst ) as de®ned above. This is similar to the approach used by van Dort et al. [11,12]. In the strong inversion regime, the inversion layer charge becomes the most important contribution to the total charge. In classical modeling, the surface potential is approximately constant in the strong inversion regime, and a linear relationship between the inversion charge and the gate voltage holds as follows: n Vg
Cox Vg q
Vth
10
where Vth is the threshold voltage. In the presence of quantum eects, the relationship between the inversion layer charge and the gate voltage can still be modeled as linear but the increase in the threshold voltage due to quantization and the decreased coupling with the gate must be taken into account. The increase in threshold voltage can be calculated from Eqs. (5) and (9). The decrease in the gate coupling can be interpreted in term of parasitic depletion layer between the oxide and the peak charge concentration, due to the annulment of the wave function at the interface [21]. The thickness of this depletion layer Lq , is de®ned as the centroid of the charge concentration. An expression for Lq can be found by ®tting an empirical formula to the results of the Poisson±Schr odinger solution as in Ref. [21]. However, the variational approximation also gives an analytical formula for the centroid of the inversion layer charge as follows: Z 1 3 Lq zjfb zj2 dz 11 b 0 Therefore, Eq. (10) may be modi®ed to include quantum eects by using a ®eld dependent eective oxide capacitance and by rede®ning the threshold condition using Eq. (9).
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
n Vg
1 eox Vg q tox eox =e 3=b Vg
Vth
1709
12
where eox is the dielectric constant of silicon dioxide, and tox the oxide thickness and a ®rst approximation of b Vg is calculated using the physical oxide thickness. This gives an explicit relationship between the inversion layer charge and the gate voltage. By using Eq. (12) in Eq. (5), a relationship between the surface potential and the gate voltage in strong inversion can be obtained. The details are given in Appendix A. 3.4. Transition between weak and strong inversion In Fig. 3, the number of electrons available for tunneling calculated using a numerical solution of the variational approximation is plotted versus gate voltage, for two dierent doping levels. The weak and strong inversion approximations to the inversion charge are also shown. Clearly, there is good agreement between the approximations and the variational calculation except in the transition region between weak and strong inversion. The strong inversion approximation can be more clearly seen in Fig. 4, where the number of electrons is now plotted in a linear scale, for two dierent oxide thicknesses. The impact of the quantum eect on the eective capacitance is clearly shown. It is particularly signi®cant for the thinner oxide. In Figs. 3 and 4, the analytical approximations described above and the full iterative variational approximation have been compared for vari-
Fig. 4. Strong inversion approximation to the variational calculation. Density of electrons calculated using the variational approach (symbols) for two dierent oxide thicknesses (( ) tox 1:2 nm, ( ) tox 2:3 nm). Solid lines are the strong inversion approximation of Eq. (12). The dashed lines show the classical calculation of inversion charge. These curves are calculated for ¯atband voltage VFB 1 V and substrate doping Na 6:5 1017 cm 3 .
ous doping levels and oxide thickness showing that the approximations proposed hold for dierent technological parameters. An analytical model for circuit simulation requires continuity of the equations from weak to strong inversion. Therefore, a method to match the approximated solution in both weak and strong inversion must be found. To obtain a continuous equation for the inversion charge, a smoothing function of the following form is used: fA x; y
A ln1 exp x=A 1 A=y exp x=A
13
where A is a positive constant. Smoothing functions are widely used in compact MOS models [19,22,23]. This function fA has the following properties: · when x A and y=A exp x=A, f x; y x · when x A and y A, f x; y y
Fig. 3. Weak and strong inversion approximations to the variational calculation. Density of electrons in the inversion layer versus gate voltage, calculated by the variational approach (symbols) for two dierent doping levels (( ) Na 1017 cm 3 and ( ) Na 1018 cm 3 ) compared to the analytical approximations for weak (dashed lines) and strong inversion (solid lines).
Hence, the number of electrons available for tunneling n Vg can be calculated in a continuous way as n Vg fA ns Vg ; nw Vg
14
where ns Vg and nw Vg are the strong and week approximations respectively. Both approximations are explicit functions of gate voltage.
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The parameter A is a smoothing parameter, which de®nes the shape of the transition between weak and strong inversion. The following formula is used to calculate A: A
ns Vth Vo b
15
The inversion charge is calculated at an oset voltage Vo from threshold to ensure that strong inversion holds. The b parameter governs the smoothness of the transition. For doping levels greater than 1017 cm 3 , and oxide the following choice thickness in the range of 12±25 A, of parameters gives good agreement: b 10;
Vo 0:4 V
16
Fig. 5 compares the inversion charge calculated using the numerical variational approach compared with the explicit approximated solutions made continuous using Eq. (14). Even in the threshold region, the ®tting between the simulated model and the approximation is now very good, for a wide range of oxide thicknesses. Using the continuous expression for inversion charge versus gate voltage of Eq. (14) it is possible to calculate the surface potential versus gate voltage continuously from weak to strong inversion as follows:
Fig. 6. Calculation of surface potential. Surface potential ( ) and oxide voltage ( ) versus gate voltage, calculated using the numerical variational model (symbols) compared to the analytical model of Eq. (17) (solid lines). Also shown are the surface potential and oxide voltage calculated using the classical model (dash lines) (VFB 1 V, doping level Na 6:5 1017 cm 3 ).
ws Vg Vg
c2 qn Vg Vfb Cox 2 s 2 c qn Vg c Vg Vfb 4 Cox
17
p where c 2qeNa =Cox is the body eect coecient. The calculation of the surface potential is essential for the correct calculation of the oxide ®eld. Fig. 6 shows that the surface potential and the oxide ®eld calculated using this analytical model agrees well with the variational calculation. Fig. 6 also shows the classically calculated surface potential and oxide ®eld compared to the quantum model. 3.5. Eect of polysilicon depletion
Fig. 5. Impact of the smoothing procedure. Density of electrons calculated using the variational approach versus gate voltage for two dierent oxide thicknesses (( ) tox 1:2 nm, ( ) tox 3:0 nm). The symbols are the exact results using the variational approach, and the lines are the continuous analytical solution proposed in this work, using the smoothing function of Eq. (13). The curves are calculated using Na 6:5 1017 cm 3 , b 10, Vo 0:4 V.
The impact of the depletion region in the polysilicon gate has to be also taken into account for the modeling of gate current, especially in the strong inversion regime. This can be done in a simple but ecient way using the depletion approximation, and making a distinction between the applied gate voltage Vgapp and the eective gate voltage Vg [22]: Vgapp Vg Vp
18
where Vp is the voltage drop in the polysilicon interface. The relation between the eective gate voltage Vg and the applied gate voltage Vgapp can be expressed as
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
1 fT s
1711
20
where f is the impact frequency against the barrier and T is the oxide transparency (or transmission probability). There are many numerical approaches to the calculation of the tunneling time [24±30], but for an analytical model the simplest approach is to adopt the concept of the barrier transparency or transmission probability. 4.1. Transparency
Fig. 7. Impact of polysilicon depletion on density of electrons available for tunneling in strong inversion. Density of electrons versus gate voltage calculated (a) classically (dotted line) (b) including quantum eects but without polysilicon depletion (dashed line) (c) including both quantum eects and polysilicon depletion (symbols) (d) using analytical model solution proposed in this work, where the polysilicon eect is taken into account using Eq. (19).
qeNp 2 Cox (s 2C 2 Vgapp VFB wst 1 ox qeNp
Vg Vgapp VFB wst
) 1 19
where Np is the polysilicon gate doping level. For simplicity, the surface potential in strong inversion has been approximated here by its value at threshold wst . Fig. 7 shows the impact of the polysilicon depletion on the inversion charge in strong inversion. It is clear that the eect of polysilicon depletion is signi®cant and must be included for accurate calculation of the charge available for tunneling and the surface potential. The ®gure also compares the result of applying Eqs. (18) and (19) with a numerical iterative solution, showing that the approximation of the polysilicon depletion by pinning the surface potential at the wst value works well.
4. Direct tunneling current equations In the previous section, formulae which allow the continuous calculation of the number of electrons available for tunneling as a function of gate voltage were obtained. From Eq. (1), to calculate the tunnel current the tunneling time s, must be calculated:
Transparency is usually estimated using the WKB approximation [31]. However, this approximation only takes into account the attenuation of the wave function through the energy potential barrier, and does not take account of the re¯ections at the oxide interface. Register et al. [32] proposed an improvement to the transparency by introducing a correction term C E to take into account re¯ections both at the Si/SiO2 and SiO2 /polysilicon interface. T E C ETwkb E
21
In this work, a one band parabolic dispersion relation in the oxide is used and so: p 4 2mox tox Twkb E; Vox exp q/ox E3=2 3q hVox q/ox qVox E3=2 22 It is also possible to obtain, a simple expression for the WKB transparency using a two band single eective mass approximation as in Refs. [32,33]. For electron tunneling from the conduction band of the silicon, both relationships can give similar results with suitable choice of eective mass. It should be noted that it is also possible to use the approach developed here for the calculation of the quantization in the silicon with transparency based on a two band approximation for the dispersion in the oxide. From Ref. [32]: C E; Vox
4t Etox E t E2 tox E2 4t E qVox tox E qVox t E qVox 2 tox E qVox 2
23
p p where t E 2E=2m and tox E 2 q/ox E=2mox in the case of a parabolic dispersion relationship in the oxide, m is eective mass in the silicon and mox is eective mass in the oxide. In order, to evaluate this corrective term for the case of a trapezoidal barrier which occurs in the direct
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R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
the bounded wave function (2), that the impact frequency for the ®rst energy level can be calculated as f
hb2 2 fF 4pm 3
25
5. Comparison with experiments
Fig. 8. Impact of Register's correction [32] on the transparency. Transparency of the oxide versus gate voltage (tox 1:2 nm) calculated using Gundlach's exact solution for the trapezoidal barrier [34,35] (symbols), compared to WKB approximation (dashed line) and WKB approximation including Register's correction (solid line).
tunneling situation, the transparency calculated using the WKB approximation and the correction proposed by Register et al. [32] were compared to the exact expression for the transparency of a trapezoidal barrier calculated by Gundlach using Airy functions [34,35]. The dierent eective masses in the oxide and silicon were taken into account in the calculation. Fig. 8 shows the results of these calculations. The correction proposed by Register is extremely eective and signi®cantly reduces the error associated with the WKB approximation.
Capacitance±voltage (C±V ) measurements and current±voltage (I±V ) measurements were performed on large area NMOS transistor test structures (W 300 lm, L 100 lm). A procedure was used to correct the impact of series resistance and tunneling on the C±V measurements at high ®eld [38]. Oxide thickness was extracted using an optimized Maserjian procedure in the accumulation regime. The ¯at band voltage and the doping level in the substrate and in the polysilicon gate was also extracted. More details of the various extraction procedures are given in Ref. [24]. Fig. 9 compares the measured gate current versus gate voltage with the ®t obtained using the analytical model. The usual values of eective mass (mox 0:5) and barrier height were used (/ox 3:1 eV) in the calculation of the transparency [39]. The ¯at band voltage (VFB 0:98 V) was extracted from C±V measurements. A series resistance has been also extracted in the high
4.2. Impact frequency The last parameter required for the calculation of the tunneling current is the impact frequency or the ¯ux of particles against the barrier per unit time. For a plane wave (i.e. free electrons), this frequency fF can be calculated as a function of the energy as (only for the ®rst energy level): fF
2E h
24
This formula has already been used in previous work [7,32,36]. But in MOS tunneling, electrons are no longer free, but con®ned in a quantum well, which would be expected to change the impact frequency. It has been shown [37], using the decomposition in plane wave of
Fig. 9. Comparison of the gate current model with experimental results. Symbols are measured gate current density versus gate voltage for an oxide thickness of 2.3 nm. The solid line is the ®t obtained using this model including series resistance correction. Also shown is the ®t which would be obtained without series resistance correction (dotted line) and the ®t without Register's correction to the transparency (dashed line). Parameters used for the ®t are mox 0:5, /ox 3:1 eV, VFB 0:98 V, tox 2:28 nm, Na 4:2 1017 cm 3 .
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
®eld region (Vg > 2 V) of the gate current curve, where it presents a clear linear dependency with the gate voltage. The impact of this series resistance correction on the ®tted curves is also shown in Fig. 9. The polysilicon depletion has been also taken into account for this ®tting, using the polysilicon doping level extracted from C±V measurement. The substrate doping level and the oxide thickness extracted from C±V were re-optimized for the ®tting of the gate current characteristic but do not dier signi®cantly from the original values extracted from C±V . A substrate doping of Na 4:2 1017 cm 3 was used for the current ®t compared to Na 6:0 1017 cm 3 extracted from C±V and an oxide thickness of 2.28 nm was used compared to 2.30 nm extracted from C±V . It is clear from Fig. 9 that the quality of the ®t obtained using this model is good and is certainly of comparable quality to those obtained by numerical simulation [4,5]. In particular the model follows the shape of the curve particularly well from weak to strong inversion. This is principally because the decay in the supply of carriers for tunneling in the weak inversion regime is included in the model and also because of the correction to the WKB transparency. Fig. 10 shows the ®ts obtained using this model to measured data taken from a range of oxide thicknesses again showing good agreement. The data shown in Fig. 10 was measured from structures fabricated in dierent processes with dierent substrate and polysilicon doping levels. In obtaining these ®ts, the parameters of the model, were maintained close to their physical values and no ®tting of eective mass or barrier height was performed.
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6. Conclusion An analytical model suitable for use in circuit simulation has been presented for direct tunneling of electrons from inversion layer of an NMOSFET to the gate. The model includes the eect of quantization in the silicon inversion layer using the variational approximation to the solution of the Poisson and Schr odinger equations. The use of the variational approximation usually requires iterative solution of a set of equations. In this work, the dominant terms and eects in each region of operation have been identi®ed. The key to the model is an analytical equation which ®ts the inversion charge from weak to strong inversion. This removes the requirement for an iterative solution of the equations. It also allows the calculation of all of the other quantities required for the tunneling current (the surface potential, the oxide ®eld and the position of the ®rst energy level in the silicon) in a consistent fashion. For the calculation of the tunneling time, the Register correction [32] for the transparency calculated by the WKB approximation was validated. An improved formula for the impact frequency which takes into account the con®nement of the carriers was also adopted. The model is formulated such that the component quantities of the current have a clearly identi®able physical basis. In particular the fact that the inversion charge and the surface potential are available using this methodology opens up the possibility of a compact model in which quantum eects on the gate and the drain current could be calculated in a consistent fashion.
Acknowledgements The authors want to thank LETI laboratories (Grenoble, France) for providing samples, and J.L. Autran (L2MP, Marseille, France) for very helpful discussions.
Appendix A. List of equations used for the compact model
Fig. 10. Fit of gate current model for dierent oxide thicknesses. Symbols are measured gate current density versus gate voltage for dierent oxide thicknesses (( ) tox 2:95 nm, ( ) tox 2:05 nm, ( ) tox 2:05 nm, ( ) tox 1:73 nm). Solid lines are the model ®t.
Constants h Planck's constant q elementary charge k Boltzmann constant T temperature (T 300 K for this work) e dielectric constant of silicon eox dielectric constant of silicon dioxide m longitudinal eective mass in silicon conduction band (m 0:98) md density of states eective mass (for longitudinal electrons) (md 0:19)
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R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
g
degeneracy factor (for longitudinal electrons) (g 2) eective mass in silicon dioxide conduction band (mox 0:5) barrier height between silicon and silicon dioxide conduction band (/ox 3:1 eV) silicon bandgap (Eg 1:12 eV 1:7942 1019 J) eective density of states in the valence band (Nv 1:04 1019 cm 3 ) eective density of states in the conduction band (Nc 2:8 1019 cm 3 )
mox /ox Eg Nv Nc
Parameters extracted form C±V measurements (extraction procedure: see Ref. [37]) VFB ¯at band voltage Na doping level in the substrate Np doping level in the polysilicon gate tox oxide thickness Notation surface potential ws Vgapp applied gate voltage Vox oxide voltage Vg eective gate voltage corrected for polysilicon depletion Equations: (A) Quantum model for the charge available for tunneling ni intrinsic concentration: p Eg ni Nc Nv exp 2kT Bulk potential: kT Na ln /f q ni Fermi level in the bulk: Nv Ef kT ln Na Oxide capacitance: Cox
eox tox
First energy level in weak inversion: 1=3 9 h2 Na ws Ew ws 3q 16 me Surface potential versus gate voltage in weak inversion: r c2 c2 wsw Vg Vg Vfb c Vg Vfb where 2 4 p c 2qeNa =Cox
Density of electrons in weak inversion regime as a function of surface potential: gmd qws Ef Ew ws Eg nwo ws 2 kT exp kT p h Density of electrons in weak inversion regime as a function of gate voltage: nw Vg nwo wsw Vg Surface potential at threshold: wst 2Uf
Ew 2Uf q
Depletion Width: s 2ews Wd ws qNa b parameter: b n; ws
i1=3 h 12 2 n mq W w N a d s 3 e h2
Threshold voltage: Vth VFB wst
qnwo wst qNa Wd wst Cox
First approximation to density of electrons in strong inversion regime: nso Vg
Cox Vg q
Vth
Eective oxide capacitance: Coxeff Vg
eox tox eox =e 3=b nso Vg ; wst
Density of electrons in strong inversion regime: ns Vg
Coxeff Vg Vg q
Vth
Smoothing parameters: b 10;
Vo 0:4 V;
A
ns Vth Vo b
Density of electrons in both weak and strong inversion: n Vg
A ln1 expns Vg =A A exp ns Vg =A 1 nw Vg
Surface potential versus gate voltage in both weak and strong inversion:
R. Clerc et al. / Solid-State Electronics 45 (2001) 1705±1716
ws Vg Vg
c2 qn Vg Vfb Cox 2 s 2 c qn Vg Vg Vfb c 4 Cox
In¯uence of the polysilicon depletion eect on the gate voltage: Vg Vgapp VFB wst (s qeNp 2C 2 Vgapp VFB wst 2 1 ox qeNp Cox
) 1
(B) Tunnel current equations Voltage drop across oxide: Vox Vg Vg
VFB
ws Vg
First energy level: 2 3h2 b n Vg ; ws Vg E Vg 8m Impact frequency: 2 h b n Vg ; ws Vg f Vg 4pm WKB formula for direct tunneling transparency: p 4 2mox tox q/ox E3=2 Twkb E; Vox exp 3qhVox q/ox qVox E3=2 Register correction to WKB transparency: s r E q/ox E ; tox E t E ; 2m 2mox C E; Vox
4t Etox E 2
4t E qVox tox E qVox 2
t E tox E t E qVox 2 tox E qVox 2
Direct tunneling transparency: T Vg C E Vg ; Vox Vg Twkb E Vg ; Vox Vg Tunneling gate current: I Vg qf Vg T Vg n Vg
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