Figure 4.6.
20
A scatter plot between the peak of error Er,max and the corresponding twofold lowest subband population po (under
ph=
sr) for different substrate dopingconcentrations (1014 to 1018 cm-3) with temperature as a parameter is shown in Figure 4.7. Because the separation of subband is strong with high doping
concentration, more inversion carrier occupies on lowest subband with high doping concentration and low temperature. We found that the peak of error Er,max increase for increasing temperatures and decreasing substrate doping concentrations. Obviously, there is a unique relationship existing. We can figure out a power-law relationship different temperatures corresponding to different values of a, and different temperatures corresponding to different values of as depicted in Figure 4.9.
However, there is a fitting line that can be drawn in Figure 4.8 and Figure 4.9, yielding a =0.024 + 2.4910-4 T and
= 1/ (0.026 910-
4 T) in Figure 4.8 and Figure 4.9, respectively, regardless of the doping concentrations.At this point, we are able to establish a semi-empirical model in the context of
ph > sr, respectively. The validity of the error calculated by Eq.(4.2.3) and Eq.(4.2.4) has been confirmed by the simulation for different substrate doping concentrations (1014 to 1018 cm-3) at 300 K.
The left hand side of Figure 4.10 reveals21
such results for fivedifferent substrate doping concentrations at 300 K as
ph < sr, and the results for
ph > sr as shown in the right hand side of Figure 4.10. Note that under the critical situation of ph =sr, Er in Eq. (4.2.4) reduces to its peak value Er,max. Finally, we want to highlight that the validity of the errors Er between 𝜇𝑢𝑛𝑖 and 𝜇𝑢𝑛𝑖,𝑀 which did not consider the mobility of ionized impurity at high Eeffregion in
this work is adequate. Therefore, we calculated the total mobility 𝜇𝑡𝑜𝑡 consist of phonon limited mobility, surface roughness limited mobility, and ionized impurity mobility asThe corresponding error
E
r tot, of total mobility caused by using Matthiessen’s rule is, shows a scatter plot between the peak of error for Eeff
larger than 1 MV/cm versus the
corresponding twofold lowest subband population po (under
ph=
sr) for different substrate doping concentrations for comparison with result in Figure 4.6. The comparison results pointed out that although using universal curves (𝜇𝑢𝑛𝑖) to calculate the error by Eq. (4.2.1) may influence the value of Er,max, it is insignificant to compare the difference between the Er,max and the peak of Er,tot. Thus, the effect of ionized impurity mobility can be suppressed evidently in the high vertical electric field region.Therefore, the validity of the peak error in this work is adequate.
However, it should be noticed that the critical Eeff
is smaller than 1 MV/cm when
22
temperature decreasing to 100K as shown in Figure 4.5. Because phonon limited mobility increase as temperature decreasing and surface roughness limited mobility is less dependent on temperature, the critical Eeff under
ph=
s would move into low vertical electric field region, thus the effect of ionized impurity mobility should be considered.4.3 Correction Model of Matthiessen’s Rule
Based on the above analysis, we will show how to correct Matthiessen’s rule in the high vertical effective electric field in this section. The error-free version of Matthiessen’s rule is reached by combining Eq. (4.2.3) and Eq. (4.2.4):
1 1 1
( )(1 r)
uni ph sr
E
(4.3.1)
In executing this method, only the self-consistent solving of coupled Poisson
equations and Schrödinger’s equations is needed with aim to determine the critical Eeff
under ph = sr
,
which in turn determines the peak of Er, and hence the corresponding twofold lowest subband population po. Once po is known, we can readily determine the maximum error Er,max via Eq. (4.2.3). As a consequence, the Er in Eq. (4.3.1) becomes a function of only the ratio of ph and sr according to Eq. (4.2.4). Therefore, we can directly obtain the universal mobility 𝜇𝑢𝑛𝑖 for given
ph and
sr by using Eq.(4.2.4); otherwise, the value of universal mobility will be overestimated as in Figure 2.1 in terms of 𝜇𝑢𝑛𝑖,𝑀.Reciprocally speaking, this method of Eq.(4.3.1) can work for
ph and
srassessment for case of given universal mobility 𝜇𝑢𝑛𝑖 data. To demonstratethis, one may quote the mobility extraction study by Takagi, et al. [4] and Hauser [5] in terms of their empirical models of
ph and
sr. Since these experimentally-determined23
models were obtained based on the conventional use of Matthiessen’s rule and according to Eq. (4.3.1), the resulting
ph and
sr are definitely underestimated and must be further multiplied by a factor of (1 + Er), as shown in Figure 3.3.Finally, we want to stress that the proposed method can work for other situations like strain effect of mobilities. In Figure 4.13, we show a scatter plot of the peak Er that is the maximum error Er,max and the twofold lowest subband population po, which were created via simulations for <110> uniaxial tensile stress of 500 MPa. Noticeably, the effect of strain is primarily to increase po. In this strain case, the power-law relationship Eq. (4.2.4) again holds, as depicted in this figure.
24
Chapter 5 Conclusion
In this work, we have shown that the universal mobility produced by Matthiessen’s rule may not be considered as the result of experimental data, because the error between the universal mobility of simulations and the apparent universal mobility calculated by Matthiessen’s rule is worse. It may even cause the wrong trends of mobility characterization.
We also quoted the detailed formula to calculate the Coulomb-limited mobility due to ionized impurity atoms in substrate region; the simulated result is comparable with D. Esseni, et al [6]. The extracted ionized impurity mobility by using Matthiessen’s rule also exhibits a large discrepancy as compared with simulated one.
The analysis results in this thesis point out that overlooking the error of Matthiessen’s rule only leads to poor extraction of individual mobility components.
Therefore, through the experimentally-validated universal mobility simulation, a semi-empirical model for the errors of Matthiessen’s rule has been established in this work. As a consequence, the conventional extraction error can be corrected using an error-free version of Matthiessen’s rule which has been created in this thesis.
25
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27
Figure 2.1 The energy band diagram of a poly gate/SiO2/p-substrate system.
28
Figure 2.2 The flowchart of Poisson and Schrödinger self-consistent solving procedure.
29
0.4 0.8 1.2 1.6
0 200 400 600 800
Npoly=1x1020cm-3 T=300K