Error-Free Matthiessen's Rule in the MOSFET Universal Mobility Region

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Error-Free Matthiessen’s Rule in the MOSFET

Universal Mobility Region

Ming-Jer Chen, Senior Member, IEEE, Wei-Han Lee, Student Member, IEEE, and Yi-Hui Huang

Abstract—Through the experimentally validated

inversion-layer mobility simulation, we devise an error-free version of Matthiessen’s rule for a single-gate n-channel bulk MOSFET in the universal mobility region. The core of the new rule lies in a semi-empirical model, which explicitly expresses the errors due to the conventional use of Matthiessen’s rule as a function of both the lowest subband population and the relative strength of individual mobility components. The model holds under practical conditions (with temperatures up to 400 K) and in a broad range of substrate doping concentrations (1014to 1018cm−3). To make the error-free proposal more general, we elaborate on several issues, including strain, impurity Coulomb scattering, and remote scattering. The thin-film case can be treated accordingly.

Index Terms—Matthiessen’s rule, metal–oxide–semiconductor

field-effect transistors (MOSFETs), mobility, model, scattering, simulation, strain, universal mobility.

I. INTRODUCTION

T

O PROBE individual scattering mechanisms in the in-version layers of MOSFETs, Matthiessen’s rule may be favored because of its additive property of reciprocal mobility components. As pointed out earlier by Stern [1], however, there will be errors of more than 15% due to the use of Matthiessen’s rule for temperatures over 40 K. Since then, there have been four fundamentally different methods published in the literature concerning the validity and applicability of Matthiessen’s rule [2]–[7]. First, Matthiessen’s rule must be carried out under ex-treme or impractical conditions such as very low temperatures (near absolute zero) [2]. Second, sophisticated numerical sim-ulations on individual mobility components were instead used, with no need to account for Matthiessen’s rule [3]. Third, for the engineering purpose, the errors caused by Matthiessen’s rule were overlooked while assessing mobility components individ-ually [4], [5]. Fourth, mobility simulations were performed to deliver the errors of the mobility components extracted using the rule, with [6] and without [3], [7] the inclusion of ionized impurity Coulomb scattering.

It is noteworthy that Stern [1] had suggested the relative strength of individual mobility components as one origin of the

Manuscript received August 23, 2012; revised November 8, 2012; accepted December 4, 2012. Date of publication January 1, 2013; date of current version January 18, 2013. This work was supported by the National Science Council of Taiwan under Contract NSC 101-2221-E-009-057-MY3. The review of this paper was arranged by Editor H. Shang.

The authors are with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: chenmj@faculty.nctu.edu.tw).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2012.2233202

errors. The other origin in terms of the subband population had also been put forward by Fischetti et al. [3]. The combination of these two origins should enable the creation of a fifth method in the field.

Herein, we propose such a new method in terms of an error-free version of Matthiessen’s rule. The establishment of the method is demonstrated in a single-gate bulk structure in the universal mobility region, with the practical situations (different temperatures and different doping concentrations) taken into account. To make possible the general applications of the error-free proposal, we elaborate on such issues as strain, impurity Coulomb scattering, and remote scattering. The thin-film case is also addressed.

II. SIMULATIONFRAMEWORK ANDERRORDEFINITION

We previously established a sophisticated simulation pack-age [7]–[10] in the context of a silicon conduction-band struc-ture with six constant-energy surfaces [11]. This package was dedicated to a single-gate bulk structure, consisting of a self-consistent Poisson and Schrödinger’s equations solver in the confinement direction [8] and an inversion-layer mobility cal-culation program [7], [9], [10]. The outcomes of the former in-clude the subband level, the Fermi level, and the wave function, all of which serve as inputs to the latter to calculate the electron inversion-layer mobility.

In the mobility calculation, the momentum relaxation time approximation [12], [13] was made. Thus, for the thick gate oxide case where three well-known scattering mechanisms prevail, the total energy-dependent relaxation time τtot can be expressed as a function of the relaxation time τph due to phonon scattering alone, the relaxation time τsr due to surface roughness scattering alone, and the relaxation time τimpdue to impurity Coulomb scattering alone, i.e.,

1 τ_totij(E) = 1 τ_phij(E)+ 1 τsrij(E) + 1 τ_impij (E) (1) where j = 1 represents the twofold valley Δ2, j = 2 represents the fourfold valley Δ4, and i represents the corresponding sub-band number. The literature formalisms for τph [14], τsr [15], and τimp[16], [17] were quoted. The corresponding mobility of subband i of valley j can be calculated [12] as

μij_x =

q_Eij∞ (E− Eij)τxij(E)

∂f ∂E dE mcj _∞ Eij(E− Eij) ∂f ∂E dE (2)

where x stands for ph, sr, or tot in (1), Eij represents the

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relaxation time approximation have all be used in this paper dedicated primarily to the single-gate bulk structure. Extension to the thin-film structure is possible, as will be explained later.

By going through all subbands and valleys, the total mobility and the individual mobility components can be obtained [12] as follows:

μx=

ij

μij_x pij (3)

where pij _{is the fractional population of subband i of valley j.}

Once individual mobility components are known, the apparent total mobility can be calculated according to Matthiessen’s rule as follows: 1 μtot, M = 1 μph + 1 μsr + 1 μimp . (4)

This leads to the error of applying Matthiessen’s rule to the total mobility reproduction, i.e.,

Er=

μtot, M− μtot

μtot

. (5)

III. SEMI-EMPIRICALMODEL

To illustrate the establishment of a semi-empirical model for the error Erof Matthiessen’s rule, we took Takagi et al.’s

bulk data [4] of electron inversion-layer mobility, as plotted in Fig. 1 versus the effective electric field Eeﬀ. Here, Eeﬀ =

q(0.5Ninv+ Ndep)/εs, where Ninv is the inversion-layer den-sity, Ndepis the substrate depletion density, and εsis the silicon

permittivity. Through the aforementioned mobility simulation, a fairly good reproduction of the data, i.e., both the substrate doping concentration dependence and the temperature depen-dence, was achieved over Eeﬀ. The corresponding material and/or physical parameters as labeled in the figure are all reasonable compared with those of Takagi et al. [14]. This experimentally validates the mobility simulation work here.

Fig. 1 clearly shows the presence of a universal mobility curve. Particularly, the larger the substrate doping concentration

Nsub, the narrower the range of Eeﬀdominated by both phonon scattering and surface roughness scattering. Corresponding calculated values of phonon-limited mobility μph, surface-roughness-limited mobility μsr, and impurity-Coulomb-limited mobility μimp, are shown in Fig. 2 for Nsub= 1017 cm−3. Straightforwardly, the apparent total mobility and, hence, error

Er, can be obtained. Remarkably, we found that the largest error Er, maxoccurs at critical Eeﬀ, where μph is nearly equal to μsr, and apart from this point, the errors decrease gradually, as shown in Fig. 2. Specifically, this criterion lies in the

high-Eeﬀ region (> 1 MV/cm) where impurity Coulomb scattering is weak. This indicates that the relative strength [1] responsible for the errors stem primarily from two distinct components: μph and μsr. The other origin [3], namely the fractional population

Fig. 1. (Symbols) Electron effective mobility data [4] for (a) six substrate doping concentrations at 300 K and (b) four temperatures for a fixed sub-strate doping concentration plotted versus vertical effective electric field Eeﬀ.

(Lines) Simulated total mobility curves are shown. Dacis the acoustic

defor-mation potential, Dkis the deformation potential of the kth intervalley phonon,

λ is the surface roughness correlation length, and Δ is the surface roughness

RMS height.

Fig. 2. Simulated total mobility, phonon-limited mobility, surface-roughness-limited mobility, and ionized-impurity-surface-roughness-limited mobility versus Eeﬀ for Nsub= 1017 cm−3 at 300 K. The apparent total mobility obtained by

Matthiessen’s rule and, hence, the errors, are together plotted. The arrow indi-cates the critical Eeﬀ, where phonon-limited mobility and

surface-roughness-limited mobility have the same value. The inset shows the corresponding population of two lowest subbands.

po[i.e., p11in (3)] of the twofold lowest subband, can be drawn

under μph= μsr, as shown in the inset of the figure. We also found that an increase in temperature and a decrease in Nsub can both increase Er, max. In Fig. 3, we show a scatter plot

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Fig. 3. (Symbols) Scatter plot of the simulated peak error and the corre-sponding lowest subband population, created from different substrate dop-ing concentrations (1014 _{to 10}18 _cm−3_{), with temperature as a parameter.}

(Lines) The calculated results using (6) are shown.

Fig. 4. (Symbols) Extracted power-law exponent γ in (6) versus temperature. (Line) The best fitting is shown. The inset depicts the case of the prefactor a.

between peak error Er, max and the corresponding po for all

substrate doping concentrations (1014to 1018cm−3), with the temperature as a parameter. Clearly, a power-law relationship exists between the two, i.e.,

Er, max= apγo (6)

where a is the prefactor, and γ is the power-law exponent. Different temperatures correspond to different values of a and

γ, as depicted in Fig. 4. Two fitting lines can be drawn therein,

yielding a =−0.024 + 2.49 × 10−4 T and γ = 1/(0.026− 9× 10−4T), regardless of doping concentrations.

Then, taking into account the relative strength of μphand μsr, a semi-empirical model results

Er= Er, max 1− α exp βmin(μph, μsr) max(μph, μsr) . (7) Equation (7) was obtained by the observation of the distribution of Eraround the critical point. As shown in Fig. 2, the

decreas-ing rate of Eris faster for μph< μsr than μph> μsr. This is reflected in prefactor α. The falling trend can be treated with an exponent of β times μph/μsr or μsr/μph. Through a best fitting, we obtained β =−5, and α = 1 and 2 for μph < μsr and μph> μsr, respectively. The errors calculated using (6) and

Fig. 5. Comparison of (symbols) simulated and (lines) calculated errors for five different substrate doping concentrations at 300 K, plotted as a function of the ratio of phonon-limited mobility and surface-roughness-limited mobility.

(7) were all confirmed by simulation, as shown in Fig. 5, for a temperature of 300 K. Note that under the critical situation of

μph = μsr, Erin (7) reduces to its peak value Er, max.

IV. ERROR-FREEMETHOD ANDDISCUSSION

Combining (4) and (5), we reach the goal in terms of the error-free version of Matthiessen’s rule as follows:

1 μtot = 1 μph + 1 μsr + 1 μimp (1 + Er). (8)

As justified above, only in the universal mobility region can the effect of impurity Coulomb scattering on Erbe neglected. This

means that Erin (8) can be obtained using (6) and (7), for the

case that μph, μsr, and μimpare known. In doing so, only the self-consistent solving of coupled Poisson and Schrödinger’s equations is needed, with an aim to determine the critical Eeﬀ under μph= μsrand, hence, po. Once pois known, maximum

error Er, max and error Er both can be readily determined.

This leads to the actual inversion-layer mobility in the universal mobility region according to (8).

Reciprocally speaking, the method can hold in the assess-ment of μph and μsr, given the universal mobility data [4]. To facilitate the process, we assume that μph is independent of Eeﬀ and μsr follows a power-law dependence on Eeﬀ. In this situation, error Erwas calculated accordingly while at the

same time solving (8). This process is iterated. The result is shown in Fig. 6. In addition, shown in the figure is the case of the conventional use of Matthiessen’s rule. It can be seen that large discrepancies occur in the conventional extraction.

At this point, we want to stress the general aspects of the error-free proposal. First, to make possible the application of the method in strain case, we used a previously established strain quantum simulator [10]. The same procedure was again executed, resulting in a scatter plot of simulated Er, max and po, as shown in Fig. 7 for a uniaxial tensile stress of 500 MPa.

Noticeably, the effect of strain is primarily to increase po. In

this strain case, the power-law relationship (6) again holds, as depicted in the figure.

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Fig. 6. Phonon-limited mobility and surface-roughness-limited mobility ex-tracted from the universal mobility data [4], using two different methods of applying Matthiessen’s rule: (solid lines) the error-free one and (dashed lines) the conventional one.

Fig. 7. (Symbols) Scatter plot corresponding to Fig. 3 but under a uniaxial tensile stress of 500 MPa. (Lines) The calculation results came from (6) with

a =−0.018 + 2.69 × 10−4T and γ = 1/(0.042− 9 × 10−4T).

Fig. 8. Simulated total mobility, phonon-limited mobility, surface-roughness-limited mobility, and ionized-impurity-surface-roughness-limited mobility versus Eeﬀ for Nsub= 1016 cm−3 at 300 K. The apparent total mobility obtained by

Matthiessen’s rule and the corresponding errors are together plotted.

Second, to examine the errors of Matthiessen’s rule in the im-purity Coulomb scattering dominant region, we show in Fig. 8 the simulated mobility components for Nsub= 1016 cm−3

Fig. 9. Comparison of (symbols) simulated and (lines) calculated errors for four different substrate doping concentrations at 300 K, plotted as a function of the ratio of phonon-limited mobility to impurity-Coulomb-limited mobility.

at 300 K, along with those from Matthiessen’s rule and the corresponding errors. Obviously, in the low-Eeﬀ region (< 0.1 MV/cm), the error increases with decreasing Eeﬀ. The corresponding errors at 300 K are plotted in Fig. 9 versus the ratio of μph to μimp, with the substrate doping concentra-tion as a parameter. We found that the same coefficients can work: β =−5 and α = 1. The corresponding semi-empirical model is

Er= Er, max(1− exp(−5μph/μimp)) (9) with Er, max= 3× 106N−0.26_sub . The fitting is good, thus

veri-fying the applicability of the proposed model. Further, Fig. 10 shows the simulated impurity-Coulomb-limited mobility μimp, the apparent impurity-Coulomb-limited mobility μimp,M from the conventional use of Matthiessen’s rule, and the error of extracting μimp,M (see [6] for the definition of the extraction error). Reasonable agreements with those of Esseni and Driussi [6] are evident, again supporting the simulation work here. Indeed, devising another model for the extraction error is challenging and needs to be investigated further. In this sense, our proposed model in this paper may be helpful because, according to the work by Esseni and Driussi [6], a certain relationship exists between the global error [i.e., Er in (5)]

and the local error (i.e., the extraction error of the individual mobility component).

Third, in the case of high-k/metal-gate (HKMG) bulk MOSFETs, it can be intuitively drawn that in the universal

mobility region, the proposed semi-empirical error model can

work, provided that remote scattering is insignificant as in the case of the impurity Coulomb scattering aforementioned. If that is not the fact, however, a sophisticated numerical simulation task is needed to reproduce HKMG mobility data, with remote scattering mechanisms taken into account. In this sense, (1) is augmented to be 1 τ_totij (E) = 1 τ_phij(E) + 1 τsrij(E) + 1 τ_impij (E)+ 1 τ_addij (E) (10) where τadd is the relaxation time due to additional scattering alone. One of the τadd expressions due to optical phonons

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Fig. 10. Comparison of (a) simulated and extracted impurity-Coulomb-limited mobility and (b) corresponding errors in this paper with those of [6], plotted as a function of substrate doping concentration.

in high-k dielectric can be found elsewhere [18]. The corre-sponding error-free Matthiessen’s rule may be devised in a similar way.

Finally, we want to state that the presented semi-empirical error model should be, in principle, applicable to the double-gate case or the FinFET case. Particularly, the model predicts an increase in the error of Matthiessen’s rule if the doping concentration of the film is decreased. However, as the film thickness decreases, the mobility expressions used in this paper fail and deteriorate the quality of the model. To overcome this, one has to turn to the literature sources [13], [19]–[21] concerning the mobility expressions suitable for thin-film struc-tures and extra scatterers. Then, the corresponding error-free Matthiessen’s rule should be able to be produced in the context of the key origins of the errors [1], [3].

V. CONCLUSION

Through the experimentally validated electron inversion-layer mobility simulation, a semi-empirical model for the errors of Matthiessen’s rule has been established for a single-gate bulk MOSFET in the universal mobility region. The error-free version of Matthiessen’s rule has straightforwardly been obtained. To make the error-free proposal more general, we have elaborated on several challenging issues.

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Wei-Han Lee (S’09) received the B.S. degree in electrophysics and the Ph.D. degree in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2005 and 2012, respectively.