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Development of an analytical mobility model for the simulation of ultra-thin single- and double-gate SOI MOSFETs
Solid-State Electronics 48 (2004) 589–595 www.elsevier.com/locate/sse
Development of an analytical mobility model for the simulation of ultra-thin single- and double-gate SOI MOSFETs q Marco Alessandrini a, David Esseni b, Claudio Fiegna a
a,c,*
Dipartimento di Ingegneria, Universit a di Ferrara, Via Saragat 1, 44100 Ferrara, Italy b DIEGM, Universita di Udine, Italy c ARCES, Via Toffano 2/2, 40100 Bologna, Italy
Received 13 May 2003; received in revised form 4 August 2003; accepted 18 September 2003
Abstract An analytical model for the electron mobility limited by surface optical phonons is developed and applied to the simulation of ultra-thin SOI MOSFETs. The developed model reproduces the main features of experimental data recently reported in the literature and has been implemented in a conventional device simulator. An application to the analysis of technological options such as doping concentration and silicon thickness in SOI MOSFETs, is reported. 2003 Elsevier Ltd. All rights reserved. Keywords: Mobility; Electrons; MOSFET; SOI; Device simulation
1. Introduction The scaling of the conventional ‘‘bulk’’ CMOS technology causes the increase of the short channel effects (SCE) and of the tunneling gate leakage currents; furthermore, the need of high doping concentrations to counteract SCE, leads to a significant degradation of the low-field mobility. SOI devices with an almost undoped silicon layer of thickness (TSi ) less than 10 nm represent a possible solution for short channel effects and mobility degradation. The SCE can be reduced further using a doublegate structure that, thanks to the combined effect of the two gate electrodes, provides improved control over the distribution of electrostatic potential within the channel.
q Review of this paper was arranged by editor Sorin Cristoloveanu. * Corresponding author. Address: Dipartimento di Ingegneria, Universit a di Ferrara, Via Saragat 1, 44100 Ferrara, Italy. Tel.: +39-532-974832; fax: +39-532-974870. E-mail address: cfi[email protected] (C. Fiegna).
Several studies about the low-field mobility in thin SOI devices have been reported in the literature [1–4]. In particular, recent experimental studies of mobility in ultra-thin SOI MOSFETs [3,4], reported a dependence of effective mobility (leff ) on TSi , which is particularly evident at low-inversion densities (Fig. 1). This analysis has been extended to devices operated in double-gate mode [5] in which two symmetric inversion layers are obtained at the two Si–SiO2 interfaces by biasing both the front gate and the silicon substrate that acts as a back gate. The modulation of the low-field mobility by TSi is analyzed in [6], where the role of scattering with surface optical (SO) phonons, interface states and surface roughness in ultra-thin SOI is discussed; all of these mechanisms, involving the interaction of the electron wave function with scattering centers located at or close to the Si–SiO2 interfaces, are likely to produce a modulation of the effective mobility by the thickness of the silicon layer, that is the distance between the two Si– SiO2 interfaces. As shown in Fig. 1, a conventional mobility model that reproduces the experimental data for bulk devices (e.g. [7–9]) overestimates the experimental data obtained
0038-1101/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2003.09.023
electron mobility [cm2/Vs]
590
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
1000 900 800 700 600
Tsi=21 nm Tsi=9.4 nm Tsi=5.2 nm
500 400 300
200
12
13
10
10
Ninv
impact of Coulomb scattering by interface states (not included in the simulations) is negligible due to the screening of the charged interface states by the inversion charge. This paper is organized as follows: Section 2 reviews the issue of scattering with surface optical phonons and describes the derivation of the analytical model; its implementation in a commercial device simulator is described in Section 3. Section 4 reports a comparison of simulations with available experimental data for ultrathin SOI MOSFETs. The developed model is applied to the analysis of technological options in Section 5. Finally the conclusions of this work are provided.
[cm-2 ]
Fig. 1. Mobility versus inversion density (Ninv ) in SOI devices. open symbols: experimental data [3], filled symbols: conventional mobility model for bulk MOSFETs [9].
for SOI MOSFETs and is not able to adequately reproduce the dependence of leff on TSi , because it does not account for the effects of surface optical phonons and interface states. In [6], the leff of SOI devices is modeled in the frame of the ‘‘relaxation time approximation’’ exploiting an accurate description of wave functions and subband charge occupation provided by Schr€ odinger–Poisson calculations. This approach accurately accounts for the non-local nature of scattering mechanisms, taking into consideration the finite spatial extension of electron wave function, but it does not lend itself to a straightforward implementation in drift-diffusion or hydrodynamic simulators because, in order to be compatible with conventional device simulators, mobility models must enforce the following requirements: 1. they must be expressed in terms of analytical functions of local quantities such as carrier density, doping concentration and electric field; 2. they must be compatible with conventional models developed for ‘‘Bulk’’ MOSFETs. In this work we developed an analytical model for the electron mobility limited by surface optical phonons, a scattering process that produces a modulation of mobility by TSi in SOI MOSFETs [6]. The developed mobility model is combined by the Mathiessen rule with a conventional model for mobility degradation at the Si–SiO2 interface developed for bulk MOSFETs [7] and is applied in the frame of conventional drift-diffusion simulations to the calculation of the electron effective mobility in ultra-thin film single-gate and double-gate SOI MOSFETs. The developed model provides a good agreement with the available experimental data for both single- and double-gate devices, at least at large inversion layer densities, for which the
2. Scattering with surface optical phonons Surface optical polar phonons account for vibrational modes that originate in the gate dielectric and propagate in the silicon undergoing a rapid attenuation in the silicon bulk away from the Si–SiO2 interface. The role of this scattering mechanism in bulk MOSFETs has been studied in [10] by means of Monte Carlo simulations. In [6] a numerical model for SO-phonons has been implemented and applied to the case of ultra-thin SOI devices, showing that due to the presence of two Si–SiO2 interfaces on the two sides of a thin silicon slab, a dependence of the low-field mobility on the thickness TSi may be expected. The reason for the dependence of leff on TSi is related to the interaction of carriers with the phonons originated at both the front and the back oxide of ultra-thin (TSi < 20 nm) SOI structures. This effect is critically dependent on the position of the centroid of the inversion charge with respect to the two Si–SiO2 interfaces. 2.1. Derivation of the analytical model for device simulation The scattering rate of SO-phonons and its dependence on the silicon thickness in SOI devices, strictly depend on the details of the 2D subband structure associated to carrier confinement in the inversion layer. Conventional device simulators neglect the 2D nature of the inversion electron gas or account for it with simplified approaches based on corrections to the local bandgap [11] or to the electrostatic potential [12]. Therefore, the lack of detailed information about subbands and electron wave functions requires to substantially simplify numerical scattering models such as [10], by introducing some approximations. In this paper an analytical model for the electron mobility limited by SO-phonons has been developed starting from the general formulation of [10] under the following approximations:
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
1. Single parabolic unprimed subband with an analytical approximation for the electron wave function derived from the model proposed in [13] and empirically adapted to the results of numerical calculations including several subbands. 2. One constant effective value for the wave vector exchanged between the scattered electron and the surface optical phonons q ¼ jkf ki j with no angular dependence. In the remainder of the paper this effective exchanged wave vector will be denoted by qe . Under these approximations the following expression for the rate of absorption of SO-phonons at the front interface can be obtained: Z TSi 2 1 e2 xSO 2 qe z ; ¼ Nb nðzÞ e dz 2pm d 2 sSO qe h ^ 0
ð1Þ
where z is the coordinate normal to the Si–SiO2 interfaces (TSi ! 1 in the case of bulk MOSFETs); nðzÞ is the wave function for the equivalent single-subband model, xSO is the angular frequency of the surface optical phonons, md is the density of states effective mass (md ¼ mt for (1 0 0) surface), Nb is the phonons concentration given by Bose–Einstein distribution, and 1 1 ^1 ¼ ðox ðox i þ sc Þ 0 þ sc Þ . Table 1 reports the values assumed for the constants involved. A similar formula may be obtained for the case of phonons originated in the back oxide, by straightforward change of the reference system (integration has now to be performed starting from the back interface up to the front one). The equivalent single-subband electron wave function is an essential ingredient of the model and its spatial extension within the silicon layer determines the dependence of the scattering rate on the silicon thickness TSi (see the integrand function in Eq. (1)). In this model the wave function is approximated according to the formulation provided in [13] for the electrical quantum limit:
Table 1 Physical constants and empirical fitting parameters (in bold characters) Symbol
Quantity
Value
ml mt sc ox 0 ox i hxSO qe a
Long. eff. mass in Si Trans. eff. mass in Si Static permittivity in Si Static perm. in SiO2 Interm. perm. in SiO2 SO-phonon energy Effective exchanged k vector Fitt. parameter Eq. (3)
0.91 m0 0.19 m0 11.7 0 3.9 0 3.05 0 50 meV 0.55 nm1 0.88
1 nðzÞ2 ’ gðzÞ ¼ b3 z2 expðbzÞ; 2
591
ð2Þ
In the case of single-subband, nðzÞ2 represents the normalized concentration: nðzÞ=Ninv . In this work, the parameter b is empirically related to the effective field Eeff by modifying the original formulation valid in the quantum limit [13] as: b¼
12eml aEeff h2
1=3 :
ð3Þ
Eq. (3) is obtained from the original formula [13] by substituting the electric field qðNA þ 11=32Ninv Þ=Si with aEeff ; a is treated as a fitting parameter to be determined by comparison of Eqs. (2) and (3) with the normalized charge density obtained from Schr€ odinger–Poisson calculations, including several subbands. This fitting procedure aims at providing a good approximation of the distribution of the inversion density within the silicon layer that, as discussed above, is crucial in order to reproduce the experimental dependence of low-field mobility on TSi . From Eqs. (1)–(3), by combining the effects of both the front and back Si–SiO2 interfaces, the following expressions are obtained: 1 e2 xSO ¼ Nb 2pmd ðjMf j2 þ jMb j2 Þ; sSO qe h2^
ð4Þ
R TSi Mf ¼
0
eqe z gðzÞ dz ; R TSi gðzÞ dz 0
R TSi Mb ¼
0
eqe ðTSi zÞ gðzÞ dz : R TSi gðzÞ dz 0
ð5Þ
ð6Þ
The integrals in Eqs. (5) and (6) can be readily evaluated, leading to a closed-form analytical model that can be implemented in conventional device simulators. Eq. (4)–(6) describe a non-local model that relates the rate of SO-phonons absorption to the effective field. As discussed in Section 1, a mobility model compatible with conventional device simulators must be expressed as a function of local quantities. In order to make the model local, following an approach common to most of the mobility models for the MOS inversion layer [7–9], we replace Eeff with the local value of the electric field component normal to the closest Si–SiO2 interface. Finally, the electron mobility limited by SO phonons is evaluated in the frame of the relaxation time approximation: lSO ¼
e sSO ; mc
ð7Þ
592
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
where mc is the conduction effective mass (mc ¼ mt for (1 0 0) surface) and the momentum relaxation time is approximated by the rate of SO-phonons absorption.
0.8 12
Ninv =1·10 cm-2
12
Ninv =5·10 cm-2
0.5 12
Ninv =1·10 cm-2
12
Ninv =5·10 cm-2
0.4
0.2
0 0
5
0
5
z [nm]
z [nm]
Fig. 3. Comparison between normalized concentration obtained by Schr€ odinger–Poisson calculations (solid line) and gðzÞ (dashed line). Bulk MOSFET, NA ¼ 2:4 1018 cm3 . a ¼ 0:88.
15
2 3
The derivation of the analytical model for the mobility limited by surface optical phonons required the introduction of two empirical parameters: a in Eq. (3) and qe in Eqs. (4)–(6). The values of these parameters have been determined by fitting the results of the numerical self-consistent solution of Schr€ odinger–Poisson equations and of the scattering rate calculations using the complete model of [6]. As already anticipated, the parameter a has been obtained by fitting the normalized electron concentration obtained by solving the 1-D Schr€ odinger and Poisson equations for a multi-subband structure, with the analytical approximation of Eqs. (2) and (3). A good analytical fitting of the normalized electron concentration has been obtained for a single value of the parameter (a ¼ 0:88) for both the bulk and ultra-thin SOI cases. Figs. 2 and 3 report the comparison of analytical and numerical calculations for the cases of a thin SOI and bulk MOSFET, respectively; a good agreement is obtained for a single value of the empirical parameter a. The effective exchanged wave vector (qe ) is obtained by fitting the results of rigorous numerical calculation of the scattering rate for surface optical phonons [6]. Fig. 4 reports the mobility limited by SO-phonons lSO calculated according to Eqs. (4)–(7), and compares it to the results of the numerical model [6]. A good agreement is obtained for a single value of the effective exchanged wave vector qe ¼ 0:61 nm1 and for a rather large range of TSi values.
SO-phon. mobility [10 ·cm /Vs]
2.2. Comparison with the numerical model
n(z)/Ninv, g(z)
0.6
5
10
5
12
Ninv=3·1012 cm-2
Ninv =1·10 cm-2 0
5
10
15
TSI [nm]
20
0
0
5
10
15
20
TSI [nm]
Fig. 4. SO-phonons limited electron mobility versus TSi for FDSOI MOSFETs, NA ¼ 1015 cm3 . Open symbols: numerical model [6]; filled symbols: analytical model applied to electric field and carrier concentration obtained by Schr€ odinger–Poisson calculations. a ¼ 0:88, qeff ¼ 0:61 nm1 .
n(z)/Ninv, g(z)
0.4
The values obtained for a and qe have been taken as an initial guess for the comparison with experimental data, reported in Section 4.
0.3 0.2
3. Model implementation for device simulation
0.1 0 0
1
2
3
z [nm]
4
5
0
1
2
3
4
5
6
z[nm]
Fig. 2. Comparison between the normalized concentration obtained by Schr€ odinger/Poisson calculations (solid line) and analytical approximation gðzÞ (dashed line). SOI MOSFET, TSi ¼ 5:2 nm. a ¼ 0:88.
The analytical model for electron mobility limited by SO-phonons derived in Section 2 has been implemented into the device simulator DESSIS [14], taking advantage of the physical model interface (PMI). This interface allows the user to implement physical models using a C++ software module that is linked run-time with the simulator. The model for mobility degradation at the Si–SiO2
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
interface is among those that can be modified by the user. In this work the the well known model for mobility degradation at the Si–SiO2 interface proposed by Lombardi et al. [7] has been used as a starting point because it is inherently simpler, and its implementation is more straightforward compared to [8,9]. In our calculations, the carrier mobility for bulk silicon and its dependence on the carrier density and doping concentration are described by the bulk mobility model proposed by Philips (Philips unified mobility model [15]). This model provides a better description of bulk mobility compared to the simpler formulation originally adopted in [7] for two reasons: (a) it differentiates between minority and majority carrier mobility, in agreement with experiments; (b) it accounts for the effects of screening on the ionized-impurity scattering rate. Thanks to the adoption of the model [15] and to improved selection of the model parameters in DESSIS (see the documentation in [14] regarding the implementation of the Lombardi model in DESSIS), the calculated effective mobility provides improved agreement with experimental data compared to the original formulation of [7]. In particular, an excessive dependence of the mobility vs. effective field curves on doping concentration is avoided (e.g. compare Fig. 1 in [8] with Fig. 5 of the present paper). In our calculations, the parameters of the Lombardi and of the bulk mobility models have been kept at the default values assumed in DESSIS, with the only exception of the parameter d, related to the mobility degradation by surface roughness, whose value has been reduced by 10.6% compared to the DESSIS implementation of the Lombardi model, in order to slightly improve the agreement with the universal mobility curves by Takagi et al. [16] at large effective field (mobility mainly limited by surface roughness). The newly developed model for SO-phonons has been combined with 1000
µeff [cm2 /Vs]
3.9E15
7.2E16
7.7E17 3.0E17
100
10
5
2.4E18
10
6
E eff [V/cm] Fig. 5. leff versus Eeff for bulk MOSFETs, including the effects of SO phonons. Filled symbols: simulations including SO phonons; lines: experimental data [16].
593
the Lombardi model using the Mathiessen rule and has been applied to the simulation of bulk and ultra-thin SOI devices: 1 l1 ¼ l1 LOMB þ lSO :
As for the model for SO phonons limited mobility, the effective-exchanged wave vector qe was slightly modified compared to the initial value set by comparison with numerical calculations (0.55 nm1 , instead of 0.61 nm1 ), in order to reproduce the dependence of effective mobility on silicon thickness TSi (see next section). Fig. 5 reports the results obtained for conventional bulk devices; the model for SO-phonons has a limited impact on the low-field mobility of bulk MOSFETs and a good agreement with experimental data [16] has been obtained. The mobility is overestimated at low-doping concentration and low-effective field. Notice that, as discussed previously, the overall agreement with experimental data is better compared to the original formulation of the Lombardi model, due to an improved selection of the values for the fitting parameters compared to the original paper [7] and to the adoption of the Philips unified mobility model for the doping-dependence of bulk mobility.
4. Comparison with experimental data for ultra-thin SOI MOSFETs In this section the model for electron mobility limited by surface optical phonons is applied to the simulation of the effective mobility in ultra-thin SOI MOSFETs operated in both single- and double-gate mode. The model parameters are the same used for the simulation of bulk devices presented in the previous section. Fig. 6 reports a comparison of simulated mobility (right panel) with the experimental data reported in [4] (left panel) for devices obtained from SIMOX wafers with very low density of surface states and TSi of 80 and 8 nm. The results of simulation provides a slight overestimation of the low-field mobility which is consistent with the results obtained in the bulk devices for comparable doping concentration (Fig. 5, NA ¼ 3:9 1015 cm3 ), but correctly accounts for the dependence of mobility on the thickness of the silicon layer. A comparison with experimental data for thinner devices measured in [3] is reported in Fig. 7; in these devices the scattering with charged surface states plays an important role at low-carrier densities. A good agreement is obtained for large carrier densities (in excess of approximately 3 · 1012 cm2 ), for which, due to the strong screening effect by the mobile carriers, the role of Coulomb scattering becomes almost negligible. An improved agreement at low-inversion density would require to take into account the effects of Coulomb
594
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
µ eff [cm2/(Vs)]
µ eff [cm2/(Vs)]
103
1000 900 800 700 600 500 400 300
Tsi =80 [nm] Tsi =8 [nm]
102 0.01
0.1
SOI DG Tsi =5.2 [nm]
Tsi =80 [nm] Tsi =8 [nm]
1 0.01
0.1
1
E eff [MV/cm]
scattering with charged surface states that become dominant at low-carrier density. A slight overestimation of mobility at very large inversion layer density seems to indicate the need for a better description of surface roughness, in order to account for the presence of the two Si–SiO2 interfaces close to each other. The dashed line represents the results of simulations without the effects of SO-phonons, showing that the model for SOphonons improves the agreement with experimental data. Fig. 8 compares simulated and experimental mobility for the 5.2 nm thick SOI MOSFET operated in doublegate mode [5] (a back bias is applied to the bulk electrode in order to obtain two symmetric inversion layers, as in an ideal double-gate device); a good agreement is obtained between experiments and simulations at relatively large inversion density. Notice that in the case of 800 700 600
µeff [cm2/Vs]
12
10
10
Ninv
Fig. 6. Mobility versus Eeff including the effects of SO phonons. Left: experimental data for different TSi [4]; right: simulations with SO phonons scattering model.
500
300
SOI SG Tsi =5.2 [nm] 12
10
13
Ninv [cm-2 ]
10
Fig. 7. Effective electron mobility versus inversion density (Ninv ) in single-gate SOI devices; TSi ¼ 5:2 nm. Filled symbols: experimental data [3], dashed line: conventional mobility model, solid line: simulations including SO-phonons.
13
[cm-2 ]
Fig. 8. Effective electron mobility versus inversion density (Ninv ) in double-gate SOI devices; TSi ¼ 5:2 nm. Filled symbols: experimental data [3], dashed line: conventional mobility model, solid line: simulations including SO phonons.
double-gate operation, we report mobility as a function of one half of the inversion charge, that is the charge of one single inversion layer.
5. Application to the analysis of technology options This section provides an example of application of the developed mobility model to the analysis of technology options. In particular, the dependence of the low-field mobility on the silicon thickness and doping concentration of thin SOI MOSFETs with negligible interface state density at both interfaces, has been analyzed. Fig. 9 reports the electron effective mobility as a function of silicon thickness for different levels of acceptor doping concentration (NA ) in the silicon layer, and for a given value of inversion charge density Ninv . The dependence of mobility on TSi at given Ninv is non-monotonous and a maximum exists for a TSi value significantly smaller than the depletion qffiffiffiffiffiffiffiffiffiffiffi length for each considered density NA : LDEPL
400
200
200
2Si 2/F . qNA
For very thin
layers, the dependence of leff on TSi is dominated by SOphonons and leff is largely degraded. For large TSi values, leff starts decreasing with respect to the maximum because of the increase of the depletion charge with TSi , that enhances the normal electric field acting on the inversion layer (the effective field). As TSi exceeds the depletion length, the depletion charge does not increase any more with TSi and leff saturates at the value corresponding to the bulk device. It is therefore clear the existence of a NA -dependent optimum TSi value for which, at given inversion charge density, the advantage related to the reduced fixed depletion charge in fully depletion mode is balanced by
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595 NA =3x1015 [cm-3]
µeff [cm2 /(Vs)]
800 NA =1x1016 [cm-3]
600 NA =1x1017 [cm-3]
400 NA =1x1018 [cm-3]
200
1
10
100
1000
Tsi [nm] Fig. 9. Effective electron mobility versus Silicon thickness in single-gate SOI devices with different doping concentrations, for Ninv ¼ 1012 cm3 .
the degradation induced by SO-phonons. Qualitatively similar results are obtained for different levels of inversion charge density.
6. Conclusions In this work, an analytical model for the mobility limited by surface optical phonons has been developed, implemented in a conventional device simulation through a software interface and applied to the calculation of the electron effective mobility in bulk and ultrathin SOI MOSFETs. The proposed model is compatible with conventional mobility models developed for bulk devices, and allows to reproduce the main feature of recently reported mobility data for ultra-thin SOI MOSFETs operated in both single and double-gate modes. In particular, the developed model is able to reproduce the dependence of low-field mobility on the thickness of the silicon layer TSi if the effects of Coulomb scattering with charged interface states are negligible, still maintaining a good agreement with the mobility of conventional bulk MOSFETs. In order to improve the agreement with experimental data at very large inversion layer densities and for very thin silicon layers, the model for the mobility limited by surface roughness needs to be revised in order to properly account for the presence of two Si–SiO2 interfaces very close to each other.
Acknowledgements Work partially supported by the Italian MIUR (PRIN-2002 project) and by the EU (NESTOR project).
595
References [1] Choi J-H, Park Y-J, Min H-S. Electron mobility behavior in extremely thin SOI MOSFETs. IEEE Electron Dev Lett 1995;16(11):527–9. [2] Toriumi A, Koga J, Satake H, Ohata A. Performance and reliability concerns of ultra-thin SOI and ultra-thin gate oxide MOSFETs. In: Proceedings IEDM. 1995, p. 847– 50. [3] Esseni D, Mastrapasqua M, Celler G, Fiegna C, Selmi L, Sangiorgi E. Low field electron and hole mobility of SOI transistors fabricated on ultra-thin silicon films for deep submicrometer technology application. IEEE Trans Electron Dev 2001;48(12):2842–50. [4] Koga J, Takagi S, Toriumi A. Influence of buriedoxide interface on inversion-layer mobility in ultra-thin SOI MOSFETs. IEEE Trans Electron Dev 2002;49: 1042–8. [5] Esseni D, Mastrapasqua M, Fiegna C, Celler GK, Selmi L, Sangiorgi E. An experimental study of mobility enhancement in ultra-thin SOI transistors operated in double-gate mode. IEEE Trans Electron Dev 2003;50(3):802–8. [6] Esseni D, Abramo A, Selmi L, Sangiorgi E. Study of low field electron transport in ultra-thin single and double gate SOI MOSFETs. In: Proceedings IEDM, 2002, p. 719– 22. [7] Lombardi C, Manzini S, Saporito A. A physical based mobility model for numerical simulation of nonplanar devices. IEEE Trans Comput-Aided Des 1988;7(11):1164– 71. [8] Darwish M, Lentz J, Pinto M, Zeitzoff P, Krutsick T, Vuong H. An improved electron and hole mobility model for general purpose device simulation. IEEE Trans Electron Dev 1997;44(9):1529–37. [9] Reggiani S, Valdinoci M, Colalongo L, Baccarani, G. A unified analytical model for bulk and surface mobility in Si n- and p-channel MOSFETs. In: ESSDERC, 1999, p. 240– 44. [10] Fischetti M, Laux SE. Monte Carlo study of electron transport in silicon inversion layers. Phys Rev 1993;48(4): 2244–74. [11] van Dort PHWMJ, Walker AJ. A simple model for quantization effects in heavily-doped silicon MOSFETs at inversion conditions. Solid-State Electron 1994;37(3): 411–4. [12] Ancona M, Iafrate G. Quantum corrections to the equations of state of an electron gas in a semiconductor. Phys Rev B 1989;39(13):9536–40. [13] Stern F, Howard WE. Properties of semiconductor surface inversion layer in the electric quantum limit. Phys Rev 1967;163:816–35. [14] ISE TCAD release, 7.5. Part. 16. [15] Klaassen D. A unified mobility model for device simulation, I. Model equations and concentration dependence. Solid-State Electron 1992;35(7):953–60. [16] Takagi S, Toriumi A, Iwase M, Tango H. On the universality of inversion layer mobility in Si MOSFETs: Part I––Effects of substrate impurity concentration. IEEE Trans Electron Dev 1994;41:2357–62.
Development of an analytical mobility model for the simulation of ultra-thin single- and double-gate SOI MOSFETs q Marco Alessandrini a, David Esseni b, Claudio Fiegna a
a,c,*
Dipartimento di Ingegneria, Universit a di Ferrara, Via Saragat 1, 44100 Ferrara, Italy b DIEGM, Universita di Udine, Italy c ARCES, Via Toffano 2/2, 40100 Bologna, Italy
Received 13 May 2003; received in revised form 4 August 2003; accepted 18 September 2003
Abstract An analytical model for the electron mobility limited by surface optical phonons is developed and applied to the simulation of ultra-thin SOI MOSFETs. The developed model reproduces the main features of experimental data recently reported in the literature and has been implemented in a conventional device simulator. An application to the analysis of technological options such as doping concentration and silicon thickness in SOI MOSFETs, is reported. 2003 Elsevier Ltd. All rights reserved. Keywords: Mobility; Electrons; MOSFET; SOI; Device simulation
1. Introduction The scaling of the conventional ‘‘bulk’’ CMOS technology causes the increase of the short channel effects (SCE) and of the tunneling gate leakage currents; furthermore, the need of high doping concentrations to counteract SCE, leads to a significant degradation of the low-field mobility. SOI devices with an almost undoped silicon layer of thickness (TSi ) less than 10 nm represent a possible solution for short channel effects and mobility degradation. The SCE can be reduced further using a doublegate structure that, thanks to the combined effect of the two gate electrodes, provides improved control over the distribution of electrostatic potential within the channel.
q Review of this paper was arranged by editor Sorin Cristoloveanu. * Corresponding author. Address: Dipartimento di Ingegneria, Universit a di Ferrara, Via Saragat 1, 44100 Ferrara, Italy. Tel.: +39-532-974832; fax: +39-532-974870. E-mail address: cfi[email protected] (C. Fiegna).
Several studies about the low-field mobility in thin SOI devices have been reported in the literature [1–4]. In particular, recent experimental studies of mobility in ultra-thin SOI MOSFETs [3,4], reported a dependence of effective mobility (leff ) on TSi , which is particularly evident at low-inversion densities (Fig. 1). This analysis has been extended to devices operated in double-gate mode [5] in which two symmetric inversion layers are obtained at the two Si–SiO2 interfaces by biasing both the front gate and the silicon substrate that acts as a back gate. The modulation of the low-field mobility by TSi is analyzed in [6], where the role of scattering with surface optical (SO) phonons, interface states and surface roughness in ultra-thin SOI is discussed; all of these mechanisms, involving the interaction of the electron wave function with scattering centers located at or close to the Si–SiO2 interfaces, are likely to produce a modulation of the effective mobility by the thickness of the silicon layer, that is the distance between the two Si– SiO2 interfaces. As shown in Fig. 1, a conventional mobility model that reproduces the experimental data for bulk devices (e.g. [7–9]) overestimates the experimental data obtained
0038-1101/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2003.09.023
electron mobility [cm2/Vs]
590
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
1000 900 800 700 600
Tsi=21 nm Tsi=9.4 nm Tsi=5.2 nm
500 400 300
200
12
13
10
10
Ninv
impact of Coulomb scattering by interface states (not included in the simulations) is negligible due to the screening of the charged interface states by the inversion charge. This paper is organized as follows: Section 2 reviews the issue of scattering with surface optical phonons and describes the derivation of the analytical model; its implementation in a commercial device simulator is described in Section 3. Section 4 reports a comparison of simulations with available experimental data for ultrathin SOI MOSFETs. The developed model is applied to the analysis of technological options in Section 5. Finally the conclusions of this work are provided.
[cm-2 ]
Fig. 1. Mobility versus inversion density (Ninv ) in SOI devices. open symbols: experimental data [3], filled symbols: conventional mobility model for bulk MOSFETs [9].
for SOI MOSFETs and is not able to adequately reproduce the dependence of leff on TSi , because it does not account for the effects of surface optical phonons and interface states. In [6], the leff of SOI devices is modeled in the frame of the ‘‘relaxation time approximation’’ exploiting an accurate description of wave functions and subband charge occupation provided by Schr€ odinger–Poisson calculations. This approach accurately accounts for the non-local nature of scattering mechanisms, taking into consideration the finite spatial extension of electron wave function, but it does not lend itself to a straightforward implementation in drift-diffusion or hydrodynamic simulators because, in order to be compatible with conventional device simulators, mobility models must enforce the following requirements: 1. they must be expressed in terms of analytical functions of local quantities such as carrier density, doping concentration and electric field; 2. they must be compatible with conventional models developed for ‘‘Bulk’’ MOSFETs. In this work we developed an analytical model for the electron mobility limited by surface optical phonons, a scattering process that produces a modulation of mobility by TSi in SOI MOSFETs [6]. The developed mobility model is combined by the Mathiessen rule with a conventional model for mobility degradation at the Si–SiO2 interface developed for bulk MOSFETs [7] and is applied in the frame of conventional drift-diffusion simulations to the calculation of the electron effective mobility in ultra-thin film single-gate and double-gate SOI MOSFETs. The developed model provides a good agreement with the available experimental data for both single- and double-gate devices, at least at large inversion layer densities, for which the
2. Scattering with surface optical phonons Surface optical polar phonons account for vibrational modes that originate in the gate dielectric and propagate in the silicon undergoing a rapid attenuation in the silicon bulk away from the Si–SiO2 interface. The role of this scattering mechanism in bulk MOSFETs has been studied in [10] by means of Monte Carlo simulations. In [6] a numerical model for SO-phonons has been implemented and applied to the case of ultra-thin SOI devices, showing that due to the presence of two Si–SiO2 interfaces on the two sides of a thin silicon slab, a dependence of the low-field mobility on the thickness TSi may be expected. The reason for the dependence of leff on TSi is related to the interaction of carriers with the phonons originated at both the front and the back oxide of ultra-thin (TSi < 20 nm) SOI structures. This effect is critically dependent on the position of the centroid of the inversion charge with respect to the two Si–SiO2 interfaces. 2.1. Derivation of the analytical model for device simulation The scattering rate of SO-phonons and its dependence on the silicon thickness in SOI devices, strictly depend on the details of the 2D subband structure associated to carrier confinement in the inversion layer. Conventional device simulators neglect the 2D nature of the inversion electron gas or account for it with simplified approaches based on corrections to the local bandgap [11] or to the electrostatic potential [12]. Therefore, the lack of detailed information about subbands and electron wave functions requires to substantially simplify numerical scattering models such as [10], by introducing some approximations. In this paper an analytical model for the electron mobility limited by SO-phonons has been developed starting from the general formulation of [10] under the following approximations:
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
1. Single parabolic unprimed subband with an analytical approximation for the electron wave function derived from the model proposed in [13] and empirically adapted to the results of numerical calculations including several subbands. 2. One constant effective value for the wave vector exchanged between the scattered electron and the surface optical phonons q ¼ jkf ki j with no angular dependence. In the remainder of the paper this effective exchanged wave vector will be denoted by qe . Under these approximations the following expression for the rate of absorption of SO-phonons at the front interface can be obtained: Z TSi 2 1 e2 xSO 2 qe z ; ¼ Nb nðzÞ e dz 2pm d 2 sSO qe h ^ 0
ð1Þ
where z is the coordinate normal to the Si–SiO2 interfaces (TSi ! 1 in the case of bulk MOSFETs); nðzÞ is the wave function for the equivalent single-subband model, xSO is the angular frequency of the surface optical phonons, md is the density of states effective mass (md ¼ mt for (1 0 0) surface), Nb is the phonons concentration given by Bose–Einstein distribution, and 1 1 ^1 ¼ ðox ðox i þ sc Þ 0 þ sc Þ . Table 1 reports the values assumed for the constants involved. A similar formula may be obtained for the case of phonons originated in the back oxide, by straightforward change of the reference system (integration has now to be performed starting from the back interface up to the front one). The equivalent single-subband electron wave function is an essential ingredient of the model and its spatial extension within the silicon layer determines the dependence of the scattering rate on the silicon thickness TSi (see the integrand function in Eq. (1)). In this model the wave function is approximated according to the formulation provided in [13] for the electrical quantum limit:
Table 1 Physical constants and empirical fitting parameters (in bold characters) Symbol
Quantity
Value
ml mt sc ox 0 ox i hxSO qe a
Long. eff. mass in Si Trans. eff. mass in Si Static permittivity in Si Static perm. in SiO2 Interm. perm. in SiO2 SO-phonon energy Effective exchanged k vector Fitt. parameter Eq. (3)
0.91 m0 0.19 m0 11.7 0 3.9 0 3.05 0 50 meV 0.55 nm1 0.88
1 nðzÞ2 ’ gðzÞ ¼ b3 z2 expðbzÞ; 2
591
ð2Þ
In the case of single-subband, nðzÞ2 represents the normalized concentration: nðzÞ=Ninv . In this work, the parameter b is empirically related to the effective field Eeff by modifying the original formulation valid in the quantum limit [13] as: b¼
12eml aEeff h2
1=3 :
ð3Þ
Eq. (3) is obtained from the original formula [13] by substituting the electric field qðNA þ 11=32Ninv Þ=Si with aEeff ; a is treated as a fitting parameter to be determined by comparison of Eqs. (2) and (3) with the normalized charge density obtained from Schr€ odinger–Poisson calculations, including several subbands. This fitting procedure aims at providing a good approximation of the distribution of the inversion density within the silicon layer that, as discussed above, is crucial in order to reproduce the experimental dependence of low-field mobility on TSi . From Eqs. (1)–(3), by combining the effects of both the front and back Si–SiO2 interfaces, the following expressions are obtained: 1 e2 xSO ¼ Nb 2pmd ðjMf j2 þ jMb j2 Þ; sSO qe h2^
ð4Þ
R TSi Mf ¼
0
eqe z gðzÞ dz ; R TSi gðzÞ dz 0
R TSi Mb ¼
0
eqe ðTSi zÞ gðzÞ dz : R TSi gðzÞ dz 0
ð5Þ
ð6Þ
The integrals in Eqs. (5) and (6) can be readily evaluated, leading to a closed-form analytical model that can be implemented in conventional device simulators. Eq. (4)–(6) describe a non-local model that relates the rate of SO-phonons absorption to the effective field. As discussed in Section 1, a mobility model compatible with conventional device simulators must be expressed as a function of local quantities. In order to make the model local, following an approach common to most of the mobility models for the MOS inversion layer [7–9], we replace Eeff with the local value of the electric field component normal to the closest Si–SiO2 interface. Finally, the electron mobility limited by SO phonons is evaluated in the frame of the relaxation time approximation: lSO ¼
e sSO ; mc
ð7Þ
592
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
where mc is the conduction effective mass (mc ¼ mt for (1 0 0) surface) and the momentum relaxation time is approximated by the rate of SO-phonons absorption.
0.8 12
Ninv =1·10 cm-2
12
Ninv =5·10 cm-2
0.5 12
Ninv =1·10 cm-2
12
Ninv =5·10 cm-2
0.4
0.2
0 0
5
0
5
z [nm]
z [nm]
Fig. 3. Comparison between normalized concentration obtained by Schr€ odinger–Poisson calculations (solid line) and gðzÞ (dashed line). Bulk MOSFET, NA ¼ 2:4 1018 cm3 . a ¼ 0:88.
15
2 3
The derivation of the analytical model for the mobility limited by surface optical phonons required the introduction of two empirical parameters: a in Eq. (3) and qe in Eqs. (4)–(6). The values of these parameters have been determined by fitting the results of the numerical self-consistent solution of Schr€ odinger–Poisson equations and of the scattering rate calculations using the complete model of [6]. As already anticipated, the parameter a has been obtained by fitting the normalized electron concentration obtained by solving the 1-D Schr€ odinger and Poisson equations for a multi-subband structure, with the analytical approximation of Eqs. (2) and (3). A good analytical fitting of the normalized electron concentration has been obtained for a single value of the parameter (a ¼ 0:88) for both the bulk and ultra-thin SOI cases. Figs. 2 and 3 report the comparison of analytical and numerical calculations for the cases of a thin SOI and bulk MOSFET, respectively; a good agreement is obtained for a single value of the empirical parameter a. The effective exchanged wave vector (qe ) is obtained by fitting the results of rigorous numerical calculation of the scattering rate for surface optical phonons [6]. Fig. 4 reports the mobility limited by SO-phonons lSO calculated according to Eqs. (4)–(7), and compares it to the results of the numerical model [6]. A good agreement is obtained for a single value of the effective exchanged wave vector qe ¼ 0:61 nm1 and for a rather large range of TSi values.
SO-phon. mobility [10 ·cm /Vs]
2.2. Comparison with the numerical model
n(z)/Ninv, g(z)
0.6
5
10
5
12
Ninv=3·1012 cm-2
Ninv =1·10 cm-2 0
5
10
15
TSI [nm]
20
0
0
5
10
15
20
TSI [nm]
Fig. 4. SO-phonons limited electron mobility versus TSi for FDSOI MOSFETs, NA ¼ 1015 cm3 . Open symbols: numerical model [6]; filled symbols: analytical model applied to electric field and carrier concentration obtained by Schr€ odinger–Poisson calculations. a ¼ 0:88, qeff ¼ 0:61 nm1 .
n(z)/Ninv, g(z)
0.4
The values obtained for a and qe have been taken as an initial guess for the comparison with experimental data, reported in Section 4.
0.3 0.2
3. Model implementation for device simulation
0.1 0 0
1
2
3
z [nm]
4
5
0
1
2
3
4
5
6
z[nm]
Fig. 2. Comparison between the normalized concentration obtained by Schr€ odinger/Poisson calculations (solid line) and analytical approximation gðzÞ (dashed line). SOI MOSFET, TSi ¼ 5:2 nm. a ¼ 0:88.
The analytical model for electron mobility limited by SO-phonons derived in Section 2 has been implemented into the device simulator DESSIS [14], taking advantage of the physical model interface (PMI). This interface allows the user to implement physical models using a C++ software module that is linked run-time with the simulator. The model for mobility degradation at the Si–SiO2
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
interface is among those that can be modified by the user. In this work the the well known model for mobility degradation at the Si–SiO2 interface proposed by Lombardi et al. [7] has been used as a starting point because it is inherently simpler, and its implementation is more straightforward compared to [8,9]. In our calculations, the carrier mobility for bulk silicon and its dependence on the carrier density and doping concentration are described by the bulk mobility model proposed by Philips (Philips unified mobility model [15]). This model provides a better description of bulk mobility compared to the simpler formulation originally adopted in [7] for two reasons: (a) it differentiates between minority and majority carrier mobility, in agreement with experiments; (b) it accounts for the effects of screening on the ionized-impurity scattering rate. Thanks to the adoption of the model [15] and to improved selection of the model parameters in DESSIS (see the documentation in [14] regarding the implementation of the Lombardi model in DESSIS), the calculated effective mobility provides improved agreement with experimental data compared to the original formulation of [7]. In particular, an excessive dependence of the mobility vs. effective field curves on doping concentration is avoided (e.g. compare Fig. 1 in [8] with Fig. 5 of the present paper). In our calculations, the parameters of the Lombardi and of the bulk mobility models have been kept at the default values assumed in DESSIS, with the only exception of the parameter d, related to the mobility degradation by surface roughness, whose value has been reduced by 10.6% compared to the DESSIS implementation of the Lombardi model, in order to slightly improve the agreement with the universal mobility curves by Takagi et al. [16] at large effective field (mobility mainly limited by surface roughness). The newly developed model for SO-phonons has been combined with 1000
µeff [cm2 /Vs]
3.9E15
7.2E16
7.7E17 3.0E17
100
10
5
2.4E18
10
6
E eff [V/cm] Fig. 5. leff versus Eeff for bulk MOSFETs, including the effects of SO phonons. Filled symbols: simulations including SO phonons; lines: experimental data [16].
593
the Lombardi model using the Mathiessen rule and has been applied to the simulation of bulk and ultra-thin SOI devices: 1 l1 ¼ l1 LOMB þ lSO :
As for the model for SO phonons limited mobility, the effective-exchanged wave vector qe was slightly modified compared to the initial value set by comparison with numerical calculations (0.55 nm1 , instead of 0.61 nm1 ), in order to reproduce the dependence of effective mobility on silicon thickness TSi (see next section). Fig. 5 reports the results obtained for conventional bulk devices; the model for SO-phonons has a limited impact on the low-field mobility of bulk MOSFETs and a good agreement with experimental data [16] has been obtained. The mobility is overestimated at low-doping concentration and low-effective field. Notice that, as discussed previously, the overall agreement with experimental data is better compared to the original formulation of the Lombardi model, due to an improved selection of the values for the fitting parameters compared to the original paper [7] and to the adoption of the Philips unified mobility model for the doping-dependence of bulk mobility.
4. Comparison with experimental data for ultra-thin SOI MOSFETs In this section the model for electron mobility limited by surface optical phonons is applied to the simulation of the effective mobility in ultra-thin SOI MOSFETs operated in both single- and double-gate mode. The model parameters are the same used for the simulation of bulk devices presented in the previous section. Fig. 6 reports a comparison of simulated mobility (right panel) with the experimental data reported in [4] (left panel) for devices obtained from SIMOX wafers with very low density of surface states and TSi of 80 and 8 nm. The results of simulation provides a slight overestimation of the low-field mobility which is consistent with the results obtained in the bulk devices for comparable doping concentration (Fig. 5, NA ¼ 3:9 1015 cm3 ), but correctly accounts for the dependence of mobility on the thickness of the silicon layer. A comparison with experimental data for thinner devices measured in [3] is reported in Fig. 7; in these devices the scattering with charged surface states plays an important role at low-carrier densities. A good agreement is obtained for large carrier densities (in excess of approximately 3 · 1012 cm2 ), for which, due to the strong screening effect by the mobile carriers, the role of Coulomb scattering becomes almost negligible. An improved agreement at low-inversion density would require to take into account the effects of Coulomb
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M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595
µ eff [cm2/(Vs)]
µ eff [cm2/(Vs)]
103
1000 900 800 700 600 500 400 300
Tsi =80 [nm] Tsi =8 [nm]
102 0.01
0.1
SOI DG Tsi =5.2 [nm]
Tsi =80 [nm] Tsi =8 [nm]
1 0.01
0.1
1
E eff [MV/cm]
scattering with charged surface states that become dominant at low-carrier density. A slight overestimation of mobility at very large inversion layer density seems to indicate the need for a better description of surface roughness, in order to account for the presence of the two Si–SiO2 interfaces close to each other. The dashed line represents the results of simulations without the effects of SO-phonons, showing that the model for SOphonons improves the agreement with experimental data. Fig. 8 compares simulated and experimental mobility for the 5.2 nm thick SOI MOSFET operated in doublegate mode [5] (a back bias is applied to the bulk electrode in order to obtain two symmetric inversion layers, as in an ideal double-gate device); a good agreement is obtained between experiments and simulations at relatively large inversion density. Notice that in the case of 800 700 600
µeff [cm2/Vs]
12
10
10
Ninv
Fig. 6. Mobility versus Eeff including the effects of SO phonons. Left: experimental data for different TSi [4]; right: simulations with SO phonons scattering model.
500
300
SOI SG Tsi =5.2 [nm] 12
10
13
Ninv [cm-2 ]
10
Fig. 7. Effective electron mobility versus inversion density (Ninv ) in single-gate SOI devices; TSi ¼ 5:2 nm. Filled symbols: experimental data [3], dashed line: conventional mobility model, solid line: simulations including SO-phonons.
13
[cm-2 ]
Fig. 8. Effective electron mobility versus inversion density (Ninv ) in double-gate SOI devices; TSi ¼ 5:2 nm. Filled symbols: experimental data [3], dashed line: conventional mobility model, solid line: simulations including SO phonons.
double-gate operation, we report mobility as a function of one half of the inversion charge, that is the charge of one single inversion layer.
5. Application to the analysis of technology options This section provides an example of application of the developed mobility model to the analysis of technology options. In particular, the dependence of the low-field mobility on the silicon thickness and doping concentration of thin SOI MOSFETs with negligible interface state density at both interfaces, has been analyzed. Fig. 9 reports the electron effective mobility as a function of silicon thickness for different levels of acceptor doping concentration (NA ) in the silicon layer, and for a given value of inversion charge density Ninv . The dependence of mobility on TSi at given Ninv is non-monotonous and a maximum exists for a TSi value significantly smaller than the depletion qffiffiffiffiffiffiffiffiffiffiffi length for each considered density NA : LDEPL
400
200
200
2Si 2/F . qNA
For very thin
layers, the dependence of leff on TSi is dominated by SOphonons and leff is largely degraded. For large TSi values, leff starts decreasing with respect to the maximum because of the increase of the depletion charge with TSi , that enhances the normal electric field acting on the inversion layer (the effective field). As TSi exceeds the depletion length, the depletion charge does not increase any more with TSi and leff saturates at the value corresponding to the bulk device. It is therefore clear the existence of a NA -dependent optimum TSi value for which, at given inversion charge density, the advantage related to the reduced fixed depletion charge in fully depletion mode is balanced by
M. Alessandrini et al. / Solid-State Electronics 48 (2004) 589–595 NA =3x1015 [cm-3]
µeff [cm2 /(Vs)]
800 NA =1x1016 [cm-3]
600 NA =1x1017 [cm-3]
400 NA =1x1018 [cm-3]
200
1
10
100
1000
Tsi [nm] Fig. 9. Effective electron mobility versus Silicon thickness in single-gate SOI devices with different doping concentrations, for Ninv ¼ 1012 cm3 .
the degradation induced by SO-phonons. Qualitatively similar results are obtained for different levels of inversion charge density.
6. Conclusions In this work, an analytical model for the mobility limited by surface optical phonons has been developed, implemented in a conventional device simulation through a software interface and applied to the calculation of the electron effective mobility in bulk and ultrathin SOI MOSFETs. The proposed model is compatible with conventional mobility models developed for bulk devices, and allows to reproduce the main feature of recently reported mobility data for ultra-thin SOI MOSFETs operated in both single and double-gate modes. In particular, the developed model is able to reproduce the dependence of low-field mobility on the thickness of the silicon layer TSi if the effects of Coulomb scattering with charged interface states are negligible, still maintaining a good agreement with the mobility of conventional bulk MOSFETs. In order to improve the agreement with experimental data at very large inversion layer densities and for very thin silicon layers, the model for the mobility limited by surface roughness needs to be revised in order to properly account for the presence of two Si–SiO2 interfaces very close to each other.
Acknowledgements Work partially supported by the Italian MIUR (PRIN-2002 project) and by the EU (NESTOR project).
595
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