Simulation Study of Coulomb Mobility in Strained Silicon - diegm

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developed by Stern and Howard [29]. ... energy model, the magnitude k of the wave vector is k = ..... 100 K, the lowest Δ2 subbands carry more than 90% of the.
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 9, SEPTEMBER 2009

Simulation Study of Coulomb Mobility in Strained Silicon Francesco Driussi and David Esseni, Senior Member, IEEE

Abstract—This paper presents a detailed simulation analysis aimed at assessing and explaining the dependence on the biaxial strain of the Coulomb-limited mobility in n-type silicon MOSFETs. By using a model based on the momentum relaxation time (MRT) approximation, we first show that we can reproduce fairly well a wide set of published experimental data, and then, we use our model to discuss the dependence on the strain of the mobility limited by either interface states or substrate impurities. Different from the experiments, in the simulations, the MRT approach allows us to analyze the different mobility components without resorting to the Matthiessen’s rule, whose use may result in large errors in the extracted mobility components. Our simulations indicate that the interface-state-limited mobility is reduced in strained devices; this is in qualitative agreement with the experiments, and we discuss its interpretation in terms of physically transparent arguments. Our analysis also suggests that the strain-induced changes of the substrate-impurity-limited mobility are instead very small, and we provide a clear interpretation of such a result. Recent experiments, however, have reported a strain-induced improvement of the substrate-impurity-limited mobility, which has been unavoidably extracted by using the Matthiessen’s rule. We argue that the systematic errors produced by the Matthiessen’s rule can help reconcile the simulation and the experimental results. Index Terms—Coulomb mobility, interface states, Matthiessen’s rule, modeling, scattering, strained Si (s-Si), substrate impurities.

I. INTRODUCTION

W

ITH THE geometrical scaling that is not as rewarding as in the past in terms of performance improvements, the strained-Si (s-Si) technology has become one of the most important technology boosters. In this respect, s-Si n-type MOSFETs have already demonstrated drain-current enhancements of 20%–30% for sub-50-nm channel lengths [1]–[3], where a tight link still exists between the drain current and the low field mobility [4], [5]. This clear correlation between the strain-induced mobility and IDS enhancements has also been predicted by numerical simulations [6]–[9] and observed in experimental results [1]–[3], [7], [10]. Since the on-current is evaluated at VGS = VDS = VDD , much work has been devoted to investigate the effects of the strain on the mobility at high inversion densities and effective fields, where the mobility is essentially limited by the surfaceManuscript received February 24, 2009; revised May 28, 2009. Current version published August 21, 2009. This work was supported by the EU through the PULLNANO IP Project under Grant FP6 IST-026828-IP and the NANOSIL NoE Project under Grant FP7 IST-216171. The review of this paper was arranged by Editor V. R. Rao. The authors are with the DIEGM, University of Udine, 33100 Udine, Italy, and also with IU.net, 40125 Bologna, Italy (e-mail: francesco.driussi@ uniud.it). Digital Object Identifier 10.1109/TED.2009.2026394

roughness and the phonon scattering mechanisms [7], [8], [11]– [14]. Nevertheless, it is very important to analyze in detail the effect of the strain engineering on the Coulomb-scattering (CS)limited mobility for several reasons. In fact, the Coulomb or neutral defects are indicated as the possible culprits for the mobility degradation of very short MOSFETs [15], and furthermore, the importance of the CS is unfortunately emphasized in high-κ CMOS technologies [16]. Finally, for the Coulomblimited mobility in s-Si MOSFETs, some inconsistencies between the theoretical predictions and the experiments have been recently pointed out and demand a clarification. As for the latter point, the behavior of the mobility μit and μsub limited by interface-state or substrate-impurity scattering, respectively, has been investigated in [17]–[21]. The Monte Carlo simulations reported in [17] suggested that μsub should stay essentially the same while μit should be improved in the presence of biaxial strain. An insensitivity of μsub to the strain has been actually inferred from experiments in [18] (with no explicit discussion for μit ), whereas the recent experiments in [20] and [21] indicate that μit is reduced and that μsub is improved by the strain. The recent data reported in [19] show a μsub modulation even for the case of uniaxial stress. It is worth noticing that the μit and μsub values have been extracted by using the Matthiessen’s rule in all the aforementioned studies [17], [19]–[21]. The possible inaccuracies related to the Matthiessen’s rule have been pointed out a long time ago [22], [23] and critically reconsidered in [21]. In this paper, the μit and μsub in biaxially strained nMOSFETs have been analyzed in detail by means of numerical simulations based on the momentum relaxation time (MRT) approximation, which allows us to determine μit and μsub without resorting to the Matthiessen’s rule. Starting from a quantitative comparison with the recent experimental data, the mobility is studied for different dopant concentrations, magnitudes of the biaxial strain, and operating temperatures. Our results indicate that, throughout the studied range of parameters, the strain degrades the μit , whereas it hardly affects the μsub . All the results are explained by presenting a systematic discussion of the strain-induced subband repopulation and of the features of the CS matrix elements and relaxation times. II. DESCRIPTION OF THE MODEL The model for the mobility calculation has been described in detail in previous publications [24]–[26]. The intravalley acoustic phonon scattering is included according to the elastic equipartition energy approximation, which makes the scattering mechanism isotropic [25]. The intervalley phonons are

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DRIUSSI AND ESSENI: SIMULATION STUDY OF COULOMB MOBILITY IN STRAINED SILICON

described by means of the widely employed 3f and 3g phonon model [27]. The surface roughness for bulk MOSFETs is treated according to the approach originally proposed in [28]. In the following parts of this section, we provide more details about the model for the Coulomb-limited mobility and for the calculation of the overall MRT. The aspects of the models hereafter addressed are those that will be used in Sections IV and V as the basic ingredients for the interpretation of the simulation results.

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where e is the positive electron charge, while si and ox denote the oxide and silicon dielectric constants, respectively. Equation (2) is valid for any value of z0 , and in particular, for the Coulomb centers located at the interface (i.e., z0 = 0), we have φunscr (q, z, 0) =

e e−q|z| . q(si + ox )

(3)

The qualitative z and z0 dependence of the screened potential φ(q, z, z0 ) is very similar to that of the unscreened potential φunscr (q, z, z0 ).

A. Scattering With Coulomb Centers The determination of CS mobility in our model consists of two steps. We first calculate the scattering potential produced by the point charge by accounting for the screening produced by the electrons in the inversion layer, and then, we determine the MRT and the mobility by accounting for the spatial distribution of the scattering centers [24]. Let us indicate with R = (r, z) the total real space vector, where r = (x, y) is the position in the transport plane and z is the abscissa in the quantization direction, which is positive in silicon and has the origin at the silicon-oxide interface. Hereafter, φ(q, z, z0 ) denotes the 2-D Fourier component (calculated in the xy plane) of the CS potential produced by a point charge located at (0, 0, z0 ), which is a function of z and of the magnitude q of the 2-D wave vector q. For any given z0 and q values, the φ(q, z, z0 ) profile accounting for the screening effect of the electrons in the inversion layer was obtained by solving the integral equation stemming from the perturbative approach developed by Stern and Howard [29]. More details concerning the calculation procedure can be found in [24]. Once the screened potential φ(q, z, z0 ) of the point charge 0 (q, z0 ) between has been determined, the matrix element Mi,j the ith and the jth subband can be expressed as [24], [30]–[32]

Mi,j (q, z0 ) =

e A

∞

B. Calculation of the Relaxation Time and the Mobility As for the calculation of the MRT τm , we first notice that, in our calculations, we embraced a parabolic isotropic electron energy dispersion. Within the parabolic band approximation, the isotropic dispersion is correct for the unprimed subbands (which have a quantization mass mz = 0.916 m0 and a density √ of state mass md = mx my = 0.19 m0 ), whereas it neglects the anisotropy of the primed subbands (which have mz = 0.19 m0 and md  0.417 m0 ) [35]. The unprimed subbands are two-time degenerate and are correspondingly denoted as Δ2 ; similarly, the primed subbands are denoted as Δ4 because they are four-time degenerate. The approximation of the isotropic energy dispersion is not expected to introduce significant errors in the determination of the low field mobility in silicon inversion layers [36], and it is very useful in the calculation of MRTs. Our model employs an isotropic formulation of the phonon scattering mechanisms, so that an explicit expression for the relaxation time τm is easily obtained [24]. As for the anisotropic scattering mechanisms, the expression for the relaxation time τm of the intrasubband transitions in the subband i is given by (i)

m 1 = d3 τm,i (E) π

π dθ|Mi,j |2 (q)(1 − cos θ).

(4)

0

dzφ(q, z, z0 )ξi (z)ξj (z)

(1)

0

where ξi (z) is the envelope wave function and A is just the normalization area in the transport plane. All the calculations were performed by using the eigenvalues and eigenfunctions obtained with the self-consistent Schrödinger–Poisson solver described in [33]. The overall squared matrix elements for the CS are finally obtained by summing the product of |Mi,j (q, z0 )|2 times the appropriate density of Coulomb centers at z0 [24], [30]–[32]. The scattering rate is then obtained by using Fermi’s golden rule. All the mobility calculations presented in this paper have been obtained by accounting for the screening; however, in order to introduce simple expressions for the scattering potential useful for the physical insight, we notice that the unscreened scattering potential φunscr (q, z, z0 ) in silicon (i.e., for z > 0) is given by [34]   si −ox e −q(z+|z0 |) e −q|z−z0 | e + e φunscr (q, z, z0 ) = 2qsi si +ox 2qsi (2)

Equation (4) has been very frequently used in the literature [31], and θ denotes the angle between the before and after scattering wave vector, so that the magnitude q of the exchanged wave vector is given by q = 2k sin(θ/2). In the isotropic energy model, the magnitude k of the wave vector is k =  2md (E − εi )/, where εi is the subband bottom energy and (E − εi ) is the kinetic energy in the subband. The calculation of the relaxation time is much more complicated for intersubband transitions. In fact, in this case, one cannot find an explicit expression for the relaxation time in a given subband, but rather an implicit definition that links the relaxation time of the different subbands [24], [36]. As discussed in detail in [24], since our model uses parabolic isotropic bands, the relaxation times are independent of the direction of the wave vector k. This simplification allows us to calculate the relaxation times for Coulomb and surface-roughness scattering by appropriately accounting for the coupling between the τm values of the different subbands produced by the intersubband transitions [24]. When different scattering mechanisms are included in the (tot) calculations, the total MRT τm,i is obtained by summing the

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inverses of the MRTs. Then, for a parabolic energy model, the average value of the relaxation time necessary for the mobility calculation is simply given by [31] ∞ τm,i  =

εi

(tot)

(E − εi )τm,i (E)f0 (E) [1 − f0 (E)] dE ∞ KB T εi f0 (E)dE

(5)

where f0 (E) is the equilibrium Fermi–Dirac occupation function. The mobility μi in the subband i and the effective mobility μeff in the inversion layer are finally expressed as μi =

eτm,i  mc,i

μeff =

1  Ni μi Ninv i

(6)

where mc,i denotes the effective conduction mass in the plane of transport, while Ni and Ninv denote the inversion density in the subband i and in the total inversion layer, respectively. The effective mobility μeff is thus the average of the subband mobilities weighted by their corresponding electron densities. It is understood that, in (6), the sum over the subband i includes both the unprimed and the primed subbands; for simplicity of notation, we did not introduce an index for the valleys. The effective conduction mass is given by mc = 2(m−1 x + −1 ) , and it evaluates to 0.19 m for the Δ , i.e., the m−1 0 2 y unprimed subbands, and to 0.315 m0 for the Δ4 , i.e., the primed subbands. By including one single scattering mechanism, (5) and (6) can be naturally used to calculate the mobility limited by the single mechanism at the study. This is in fact how we calculated the μit and μsub values discussed in Section IV, without resorting to the Matthiessen’s rule. III. COMPARISON BETWEEN SIMULATIONS AND E XPERIMENTS In Fig. 1, the model previously described is used to analyze the experimental mobility data for relaxed Si and s-Si reported in [20] and [21]. In order to simulate the mobility of s-Si devices, we followed the same procedure used in [6] and [7], where the valley splitting induced by strain is calculated as a function of the Ge content of the SiGe virtual substrate [7, eq. (6)], and the rms value of the surface-roughness spectrum (ΔSR ) has been reduced to fit the mobility data at large effective fields (Eeff ). The other model parameters have the same values as in [7, Table 1]. Fig. 1(a) shows that our simulations (symbols) well reproduce the experimental μeff values (lines) for different substrate doping concentrations ranging from NA = 3 × 1016 to 2.2 × 1018 cm−3 (doping values are taken from [20] and [21]). Fig. 1(b) shows the experimental mobility data (taken from [20, Fig. 6]) for relaxed and strained transistors before and after a gate current stress that is able to produce a significant amount of surface interface states. The stress clearly reduces the experimental mobility in the case of both relaxed Si and s-Si. The stress-induced mobility degradation is reproduced fairly well by the simulations by using interface-state densities

Fig. 1. (Symbols) Simulations corresponding to (lines) the experimental mobility data reported in [20]. (a) Mobility values of relaxed Si and s-Si for different substrate doping values. (Dashed line) The universal mobility [37] is reported as reference. (b) Mobility data before and after Fowler–Nordheim stress. Interface-state densities of Nit = 5 × 1010 eV−1 cm−2 and Nit = 5 × 1011 eV−1 cm−2 are used in the simulation to fit the virgin and the stressed data, respectively. The same couple of Nit values is used for both the relaxed-Si and the s-Si cases.

of Nit = 5 × 1010 eV−1 cm−2 and Nit = 5 × 1011 eV−1 cm−2 for the virgin and the stressed devices, respectively. In this case, Nit has been used as a fitting parameter. The same couple of Nit values has been used for both the relaxed Si and the s-Si. It is worth pointing out that the simulated strain-induced mobility improvements at large Ninv ’s in Fig. 1 are essentially due to the smaller ΔSR value used for surface-roughness scattering in s-Si [6], [7]. However, the discussion of the μit and μsub carried out in the following sections is totally independent of the phonon-limited or the surface-roughness-limited mobilities, because μit and μsub are directly calculated with the MRT approach, without resorting to the Matthiessen’s rule. IV. ANALYSIS OF THE COULOMB-LIMITED MOBILITY The previous section illustrated a good consistency between our simulations and the experiments for both the relaxed and the strained cases and for different channel-doping and interfacestate concentrations. Such an agreement legitimates the use of our model to analyze how the strain affects the Coulomblimited mobility, even because, owing to the MRT approach, it is possible to directly determine the μit and μsub values, without resorting to the Matthiessen’s rule (whose reliability will be further discussed in Section V).

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DRIUSSI AND ESSENI: SIMULATION STUDY OF COULOMB MOBILITY IN STRAINED SILICON

Fig. 2. Simulated interface-state-limited mobility (μit ) as a function of the inversion charge density for s-Si and relaxed-Si MOSFETs. The s-Si exhibits reduced μit with respect to relaxed Si.

Fig. 3. (a) Simulated interface-state-limited mobility (μit ) of the lowest and the second lowest unprimed (Δ2 ) and primed (Δ4 ) subbands versus the Ge content of the SiGe virtual substrate. (Open symbols) Δ2 valleys exhibit lower μit values than (filled symbols) Δ4 valleys, and μit is the lowest for the lowest Δ2 subband. The dashed line is the average μit in the inversion layer. (b) Simulated population for the two lowest Δ2 and the two lowest Δ4 subbands versus the Ge content in the virtual substrate.

A. Interface-State-Limited Mobility Fig. 2 shows the simulated μit for a small channel-doping concentration (NA = 1016 cm−3 ) for the relaxed as well as different strain conditions. We observe a systematic reduction of the μit with the strain, which is consistent with the experimental data of [21]. In order to explain the results in Fig. 2, Fig. 3(a) shows, for Ninv = 1012 cm−2 , the μit mobility for the two lowest Δ2 unprimed subbands and the two lowest Δ4 primed subbands versus the Ge content of the virtual substrate. As it can be seen, for all the strain conditions, the mobility is significantly smaller in the Δ2 than in the Δ4 subbands, and it is the smallest in the lowest Δ2 subband. This result may seem surprising since the Δ2 subbands have a smaller conduction mass; however, it will be convincingly explained later on in this section. The changes produced by the strain in the μit of the subbands are relatively modest compared with the μit differences among the subbands. The subband occupation, instead, is remarkably varied by the strain. In this respect, Fig. 3(b) shows that, in the relaxed case, the subband carrying the largest Ninv is the lowest Δ4 subband rather than the lowest Δ2 subband [13], [14].

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Fig. 4. (Left y-axis) Calculated scattering potential φ(q, z, 0) for a point charge located at the silicon-oxide interface and (right y-axis) squared eigenfunctions for the lowest Δ2 and Δ4 subbands along the vertical direction of a MOS structure. Wave vector magnitude q = 0.2 nm−1 .

This is because, at the relatively low inversion density Ninv = 1012 cm−2 , the lower occupation of the Δ4 due to the larger eigenvalue is more than compensated by the almost four times larger density of the states. The strain-induced splitting between the Δ2 and the Δ4 subbands drastically reduces the occupation of the Δ4 subbands, and Fig. 3(b) shows that the relative population in the Δ4 subbands becomes vanishing for a progressively larger Ge content. This repopulation of the Δ2 subbands (and, in particular, of the lowest Δ2 subband) has been identified as the mechanism mainly responsible for the overall μit degradation induced by the strain shown in Fig. 2. In fact, Fig. 3(a) shows that the lowest Δ2 subband has the smallest mobility; hence, its repopulation degrades the average mobility determined according to (6). In this respect, Fig. 3(b) shows that, for Ge fractions larger than about 15%, the lowest Δ2 subband carries almost 60% of the entire inversion density. Returning to the μit values in the Δ2 and Δ4 subbands, Fig. 4 shows the explanation why μit is the smallest in the lowest Δ2 subband. The figure shows the scattering potential φ(q, z, z0 ) produced by a point charge at the silicon-oxide interface for a −1 comparable with the electron therwave vector q = 0.2 nm√ mal wave vector kth = 2mc KB T /; for the Δ2 subbands, we have mc = 0.19m0 ; hence, kth ≈ 0.36 nm−1 . In the same figure, we also see the squared magnitude of the wave functions for the Δ2 and the Δ4 lowest subbands obtained with the Schrödinger–Poisson solver [33]. By recalling the formulation of the scattering matrix element given in (1), Fig. 4 shows very neatly that the lowest Δ2 has the largest matrix element with the scattering potential produced by an interface charge (as confirmed by the numerical calculations—not shown). In fact, the large quantization mass results in a stronger confinement of the wave function toward the silicon-oxide interface. The proximity of the wave function to the interface is critical for the scattering with the interface states, and therefore, the Δ2 subbands are unfavorable with respect to the Δ4 subbands. Such an effect results in a smaller μit for the Δ2 subbands despite their smaller effective transport mass mc . The fact that the strain-induced μit degradation is essentially due to the repopulation of the Δ2 subbands is further confirmed by the mobility behavior as a function of the temperature, as

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Fig. 5. (a) Simulated μit as a function of temperature. At low T , the mobilities of relaxed Si and s-Si are similar. (b) Calculated subband population versus temperature. At low T , essentially all the electrons belong to the Δ2 subband in both relaxed Si and s-Si.

Fig. 6. Simulated mobility limited by substrate impurities as a function (a) of inversion density Ninv and (b) of the substrate doping concentration NA . Although the mobilities are very similar, in s-Si devices, μsub is slightly larger than in the relaxed-Si case.

shown in Fig. 5. In fact, the simulations in Fig. 5(a) reveal that the μit is significantly reduced for decreasing temperatures for both the strained and the unstrained cases, even if the temperature dependence of the μit of each single subband is quite modest (not shown). Fig. 5(a) shows that the overall μit degradation with decreasing temperatures is again due to the temperature-induced repopulation of the lowest Δ2 subband. It is also noteworthy that, for T values below approximately 100 K, the lowest Δ2 subbands carry more than 90% of the Ninv even in the unstrained device [13]. Fig. 5(b) shows that, correspondingly, the strain effect on μit tends to disappear at low temperature, in agreement with the experimental results reported in [21, Fig. 9]. B. Substrate-Impurity-Limited Mobility Fig. 6 shows that the simulated μsub is very similar in the strained and unstrained devices for both different inversion densities and different doping concentrations. The μsub is slightly larger in the strained case, but the difference is very small. We also verified that this result holds for different strain magnitudes, i.e., for different Ge contents in the virtual substrate (not shown). To understand this result, also for the μsub , we performed an analysis based on the mobility in the different subbands and

Fig. 7. (a) Simulated mobility limited by substrate impurities versus the inversion charge density. (Circles) Δ2 valleys have a slightly larger μsub than (squares) Δ4 ones. This difference increases with Ninv , and it does not depend on the Si strain level. s-Si: 30% Ge. (b) Calculated population of Δ2 and Δ4 subbands.

on the subband repopulation. In this respect, Fig. 7(a) shows that the unstrained μsub (filled symbols) is somewhat larger for the lowest Δ2 than it is for the lowest Δ4 subband. This is an opposite behavior with respect to what was observed for the μit in Fig. 3; however, the μsub difference between the lowest Δ2 and the lowest Δ4 subbands is small for Ninv below approximately 1012 cm−2 . Fig. 7(a) also shows that the μsub in the subbands is not much affected by the strain. The aforementioned considerations explain why the strain-induced subband repopulation shown in Fig. 7(b) has an overall modest effect on the μsub . Moreover, since, in the relaxed Si, only approximately 30% of the inversion charge is in the lowest Δ4 subband (a smaller fraction with respect to Fig. 3 due to the larger doping concentration and the correspondingly larger confining electric field), this effect of the repopulation is further reduced [14]. Therefore, as shown in Fig. 6, the μsub is very similar in the relaxed-Si and s-Si devices. The larger μsub values for the Δ2 subbands with respect to the Δ4 subbands are by no means a surprising result, because the Δ2 subbands feature a smaller effective conduction mass mc . Such an mc difference is the main explanation for the larger phonon-limited mobility in the Δ2 subbands discussed in detail in [25]. However, the μsub advantage for the Δ2 subbands shown in Fig. 7(a) is small compared with the phonon-limited mobility. This is a delicate point for the interpretation of the μsub results, because, if the advantage of the lowest Δ2 subband were larger, then the strain-induced subband repopulation could result in a more appreciable mobility improvement than that shown in Fig. 6. In order to further analyze this latter point, we return to the main constituents of the Coulomb-limited mobility discussed in Section II. The form of the CS potential suggests that the matrix elements are larger for smaller magnitudes q of the exchanged wave vector. As already said in the discussion of (4), for an intrasubband transition, we have  2 2md (E − εi ) sin(θ/2) (7) q=  where θ is the angle between the before- and after-scattering wave vector. At relatively small inversion densities, around

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DRIUSSI AND ESSENI: SIMULATION STUDY OF COULOMB MOBILITY IN STRAINED SILICON

Fig. 8. Calculated MRT τm for the lowest Δ2 and the lowest Δ4 subband as a function of the kinetic energy E . At the thermal energy E = KB T , the τm limited by substrate impurities is larger for the Δ2 valleys compared with the Δ4 valleys.

1012 cm−2 , the electron gas is not appreciably degenerate, and the average energy in each subband is KB T , so that (7) indicates that, statistically speaking, the q values are smaller in the Δ2 subbands than in the Δ4 subbands, because the former have a smaller value for md (i.e., 0.19 m0 instead of 0.417 m0 ). The picture at small q values is complicated by the screening, which tends to reduce the scattering rates particularly at small q values [24]. However, the scattering events with small θ and, thus, small q values are less effective in the relaxation of the momentum, as clearly pointed out by (4), so that, overall, the smaller md of the Δ2 subbands is expected to yield smaller values of relaxation times, at least in the energy range of most practical importance, namely, around KB T . The reduction in the τm values of the Δ2 lowest subband is confirmed by the numerical calculations performed at room temperature and shown in Fig. 8. These smaller τm values tend to counteract the better effective conduction mass of the Δ2 with respect to the Δ4 subbands. Such an antagonist effect with respect to the conduction mass is responsible for the modest improvement of the μsub in Δ2 with respect to Δ4 subbands, which, in turn, results in a very modest strain-induced mobility increase. V. DISCUSSION AND CONCLUSION The simulated results in Fig. 2 are consistent with the experiments of [21] in showing a μit degradation in strained devices, and this conclusion cannot be disputed on the basis of the possible inaccuracies in the experimental μit due to the use of the Matthiessen’s rule. In fact, the Matthiessen’s rule yields a systematic underestimate of μit [11], [22], [23], and the smaller is the error, the larger is the reference mobility curve with respect to the μit [21]. At small inversion densities, the reference curve is essentially the phonon-limited mobility, which is significantly improved in strained devices. Consequently, the underestimate of μit due to the Matthiessen’s rule is larger in relaxed devices than it is in strained devices. These arguments demonstrate that the μit degradation in strained devices cannot possibly be an artifact of the Matthiessen’s rule, but instead, the μit degradation is likely to be underestimated in the experiments because of the extraction procedure [21].

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This latter statement is also supported by the comparison of Fig. 2 of this paper with the data of [21] (in particular, see [21, Fig. 9]). In this respect, we first notice that the absolute values of the simulated μit are well larger than the corresponding experimental values. We believe that this discrepancy is mainly due to the systematic errors produced by the Matthiessen’s rule used in the experiments, because the simulated and measured values of the total effective mobility are instead very similar [see Fig. 1(b)]. Moreover, it is also worth noticing that the strain-induced mobility degradation is larger in our simulations than it is shown in [21, Fig. 9]; we think that this happens mainly because, in the experiments, the Matthiessen’s rule underestimates the mobility difference between the relaxed and the strained devices. As for the substrate-impurity-limited mobility, our simulations have shown a very modest improvement of μsub in strained devices, because the lower conduction mass of the Δ2 valleys is compensated by a larger scattering rate with respect to the Δ4 valleys. In particular, the strain-induced μsub changes are smaller than those experimentally shown in [21, Fig. 7]. We think that the quantitative discrepancy is mainly due to the use of the Matthiessen’s rule in the experimental data. In this respect, it is useful to remember that, in order for the Matthiessen’s rule to be accurate, it is necessary that 1) the carriers are confined in a single subband [11], [21] and 2) the different scattering mechanisms contributing to μeff have the same energy dependence of the MRT τm [22], [23]. It has been argued that the errors of the Matthiessen’s rule related to the subband occupation are negligible in the case of strained devices because the strain-induced valley splitting confines the carriers in the Δ2 subbands [see Fig. 7(b)] [21]. The authors of [21] also noticed that such errors are expected to be small even for the unstrained silicon if the relaxation time τm of the Δ4 subbands is lower than the one of Δ2 subbands. However, Fig. 8 shows that τm is slightly larger in Δ4 than in Δ2 valleys, which can lead to significant errors in the μsub values extracted with the Matthiessen’s rule. Aside from the issue of the subband occupation, the accuracy of the Matthiessen’s rule for the extraction of the Coulomblimited mobility is seriously hampered by the fact that the Coulomb and the phonon scattering mechanisms have a very different energy dependence of the relaxation time. For a given subband, the phonon scattering rate is roughly proportional to the density of the allowed final states [24], [25]. This implies that the relaxation time τm typically tends to reduce for increasing energy, at least in the range of small energy values which determine the mobility, namely, up to 3 or 4KB T . Fig. 8 shows that, for the CS, instead, the relaxation time tends to increase with the kinetic energy inside the subband. This is essentially due to the dependence of the scattering potential on the magnitude q of the wave vector [see (2) and (3)], which favors scattering events with small q values, which, in turn, correspond to electrons with small kinetic energies. The screening effect tends to reduce the scattering rates for very small q and energy values; however, the energy dependence in the range of practical interest is the one shown in Fig. 8, where the screening effect is, in fact, accounted for.

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once the model has been validated by a comparison with the experiments, the different mobility components can be distinguished without resorting to the Matthiessen’s rule. ACKNOWLEDGMENT The authors would like to thank Prof. L. Selmi for the many helpful discussions. R EFERENCES

Fig. 9. Simulated mobility limited by substrate impurities versus Ninv . (Solid lines) μsub is extracted by applying the Matthiessen’s rule to the simulated μeff in Fig. 1(a) following the procedure in [12]. (Dashed lines) The application of Matthiessen’s rule lead to large errors with respect to the simulated μsub and shows larger μsub in the case of s-Si.

The aforementioned arguments suggest that errors in the mobility extracted by using the Matthiessen’s rule can be unfortunately large even for the s-Si case, particularly at room temperature [22]. In order to quantitatively address this point, we applied the Matthiessen’s rule to our simulated μeff curves in order to emulate the procedure of μsub extraction used in the experiments [21]. Fig. 9 shows the μsub values obtained from the simulated μeff in Fig. 1(a). As it can be seen, the μsub values extracted by the Matthiessen’s rule (solid lines) are remarkably smaller than the correct values (dashed lines) and much closer to the experimental data reported in [21, Fig. 10]. The comparison of the curves in Fig. 9 shows, by using simulative data, the large errors produced by the use of the Matthiessen’s rule [22], [23]. Furthermore, the most important thing for the purpose of this paper is that such errors are quite different for the relaxed-Si and the s-Si cases. This is the origin of the apparent μsub improvement for s-Si shown in Fig. 9 (solid lines), which is not observed in the exact results for μsub in Fig. 6. For our simulation results, it is possible to certainly state that the strain-induced μsub enhancement shown in Fig. 9 is an artifact of the Matthiessen’s rule. In our opinion, the μsub enhancements for strained devices observed in the experiments of [21] are similarly due, to a large extent, to the inaccuracies in the μsub determination due to the unavoidable use of the Matthiessen’s rule. This interpretation, on the one hand, helps to reconcile our simulation results with the experiments of [21] and, on the other hand, underlines that the Matthiessen’s rule should not be considered a quantitatively reliable procedure for the extraction of the different mobility components from the measured mobility characteristics. In this sense, we think that the mobility components obtained from the measurements by using the Matthiessen’s rule (such as the CS mobility) should not be considered as experimental results; in fact, the extraction procedure relies on heavily restricting modeling assumptions which, unfortunately, are not satisfied in most practical cases. We thus believe that a quantitative analysis and interpretation of the experimental mobility curves demands a more theoretically sound basis than the Matthiessen’s rule. In this sense, the use of the MRT model is particularly appealing because,

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DRIUSSI AND ESSENI: SIMULATION STUDY OF COULOMB MOBILITY IN STRAINED SILICON

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Francesco Driussi received the Laurea and Ph.D. degrees in electronic engineering from the University of Udine, Udine, Italy, in 2000 and 2004, respectively, where he worked on the reliability of Flash memory devices. From October 2002 to September 2003, he held a student position at Philips Research Leuven, Leuven, Belgium. In 2005, he became a Research Associate with the University of Udine, where he is currently with DIEGM. He is also with IU.net, Bologna, Italy. His research activities are mainly in the field of nonvolatile memory cell reliability, and in particular, his interests have been in the characterization of device degradation and gate dielectric reliability. In particular, he has worked on substrate-enhanced hot-electron phenomena, and he has investigated on their implications for Flash EEPROM devices. More recently, he worked on the development of physical and statistical models for the study of SILC in large Flash memory arrays and of the oxide trap generation and distribution. He is also working on the characterization and modeling of SONOS memory cells and of the trapping properties of silicon nitride. Part of his current activities are also in the field of experimental characterization and modeling of the carrier mobility in MOSFET devices featuring strained silicon and high-κ gate stacks.

David Esseni (S’98–M’00–SM’06) received the Laurea and Ph.D. degrees in electronic engineering from the University of Bologna, Bologna, Italy, in 1994 and 1998, respectively. During 2000, he was a Visiting Scientist at Bell Labs–Lucent Technologies, Murray Hill, NJ. Since 2005, he has been an Associate Professor with the DIEGM, University of Udine, Udine, Italy. He is also with IU.net, Bologna, Italy. His research interests are mainly focused on the characterization, modeling, and reliability of MOS transistors and nonvolatile memories (NVMs). In the field of NVMs, he has worked on the low-voltage and the substrate-enhanced hot-electron phenomena and on several aspects of Flash EEPROM memories, including innovative programming techniques and reliability issues related to the statistical distribution of stress-induced leakage current. Starting from year 2000, he has been involved in the field of advanced or innovative CMOS devices. In particular, he has experimentally investigated low field mobility in ultrathin SOI MOS transistors and then started an activity of semiclassical transport modeling in advanced n-MOS and p-MOS transistors. In this field, his research interests also include quantization models beyond the effective mass approximation as well as modeling and characterization of stress effects on the CMOS technologies. Dr. Esseni served as a member of the technical committee of the International Electron Devices Meeting in 2003 and 2004. He is currently in the technical committee of the European Solid-State Device Research Conference and the IEEE International Reliability Physics Symposium and is a member of the Technology Computer Aided Design Committee of the IEEE Electron Devices Society. He is the Associate Editor for the IEEE TRANSACTIONS ON ELECTRON DEVICES (TED) and has been one of the Guest Editors of a Special Issue of the IEEE TRANSACTIONS ON ELECTRON DEVICES devoted to the simulation and modeling of nanoelectronics devices.

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