4. Modeling of the mobility

#### 4.1. Silicon wafers with a (100) crystallographic orientation

The modeling of the hole mobility using Eq. (9) with the data reported in Table 1 (μ<sup>0</sup> = 115 cm2 /Vs and θ = 0.35 V�<sup>1</sup> ) has been carried out and compared with the experimental data of the effective mobility for Si(100) p-MOSFETs. The result is reported with the dashed line in Figure 5 and demonstrates the great accuracy of the extraction method and of the model provided by Eq. (9) when the effective electric field Eeff is above 0.3 MV/cm. Below this value, the model is inaccurate since the effective mobility is limited by the Coulomb scatterings, scatterings that are not taken into account in Eq. (9). The conduction parameters (μ<sup>0</sup> = 115 cm<sup>2</sup> /Vs, θ = 0.35 V�<sup>1</sup> , Racc = 70 Ω, ΔL = �0.33 μm, ΔW = �0.13 μm, Vd = 50 mV and W = 20 μm) have been implemented in Eqs. (8) and (9) to model the drain current Id in p-MOSFETs with different gate length L and the transconductance gm has been calculated afterwards. The results are shown in Figure 6 with the thick full lines. At the exception of Vg<Vth, the modeling is greatly fitting the experimental data for either Id or gm. The maximum of the transconductance gm cannot be estimated because Eq. (9) does not model the Coulomb scatterings

Figure 5. Effective mobility μeff as a function of the effective electric field Eeff for Si(100) and Si(110) p-MOSFETs. The dashed lines report the modeling carried out with Eq. (9), μ<sup>0</sup> = 115 cm<sup>2</sup> /Vs and θ = 0.35 V�<sup>1</sup> for the Si(100) wafers and μ<sup>0</sup> = 285 cm2 /Vs, θ = 0.038 V�<sup>1</sup> for the Si(110) ones. The full line reports the modeling with Eq. (10) and μ<sup>0</sup> = 280 cm2 /Vs, θ<sup>1</sup> =0V�<sup>1</sup> , θ<sup>2</sup> = 0.05 V�<sup>2</sup> , and α = 0.04 for Si(110) wafers.

Figure 6. Drain current Id (left) and transconductance gm (right) as a function of the gate overdrive voltage Vg–Vth for Si (100) p-MOSFETs featuring different gate length. The lines are the modeling carried out with μ<sup>0</sup> = 115 cm<sup>2</sup> /Vs, θ = 0.35 V�<sup>1</sup> , Eqs. (9) and (8) with (full thick lines) and without (dashed lines) taking into account the parasitic access resistances Racc = 70 Ω. ΔL = �0.33 μm, ΔW = �0.13 μm, Vd = 50 mV, W = 20 μm.

mechanisms. A second simulation has been calculated without taking into account the parasitic access resistances Racc in Eq. (8), and the results are shown with the dashed line in Figure 6. The fact to neglect the parasitic access resistances Racc leads to a discrepancy between the model and the experimental data that is enhanced when the size of the device is shrinked.

#### 4.2. Silicon wafers with a (110) crystallographic orientation

(100) p-MOSFETs. However, the main result is the impossibility to extract the parasitic access resistances Racc and the mobility attenuation factor θ with the Ghibaudo method, whereas the method has been successfully employed for Si(100) p-MOSFETs [23]. Indeed, as shown in

wafers. The linear fitting of the curve allows the extraction of the intermediate parameter Gm in the frame of the Ghibaudo

of the low field mobility μ0, the gate length reduction ΔL and the gate width reduction ΔW, the

method, the same procedure as previously described has been carried out and a behavior

The modeling of the hole mobility using Eq. (9) with the data reported in Table 1 (μ<sup>0</sup> = 115 cm2

effective mobility for Si(100) p-MOSFETs. The result is reported with the dashed line in Figure 5 and demonstrates the great accuracy of the extraction method and of the model provided by Eq. (9) when the effective electric field Eeff is above 0.3 MV/cm. Below this value, the model is inaccurate since the effective mobility is limited by the Coulomb scatterings, scatterings that are not taken into account in Eq. (9). The conduction parameters (μ<sup>0</sup> = 115

have been implemented in Eqs. (8) and (9) to model the drain current Id in p-MOSFETs with different gate length L and the transconductance gm has been calculated afterwards. The results are shown in Figure 6 with the thick full lines. At the exception of Vg<Vth, the modeling is greatly fitting the experimental data for either Id or gm. The maximum of the transconductance gm cannot be estimated because Eq. (9) does not model the Coulomb scatterings

similar to the one noticed for Si(100) p-MOSFETs has been acknowledged.

method. L = 10 μm, Vd = 50 mV for Si(100) and Vd = 100 mV for Si(110) transistors.

4.1. Silicon wafers with a (100) crystallographic orientation

�0.5 versus the gate voltage could not be done. Concerning the Schreutelkamp

�0.5 as a function of the gate overdrive voltage Vg–Vth for transistors fabricated on Si(100) and Si(110)

) has been carried out and compared with the experimental data of the

, Racc = 70 Ω, ΔL = �0.33 μm, ΔW = �0.13 μm, Vd = 50 mV and W = 20 μm)

0.5 has been possible, and in turn the extraction

/Vs

Figure 4, whereas the extraction of Gm from Id/gm

10 Different Types of Field-Effect Transistors - Theory and Applications

4. Modeling of the mobility

linear fitting of gm

Figure 4. Plot of gm

and θ = 0.35 V�<sup>1</sup>

/Vs, θ = 0.35 V�<sup>1</sup>

cm<sup>2</sup>

The modeling of the mobility has been carried out for the Si(110) wafers with the parameters obtained in Table 2. μ<sup>0</sup> = 285 cm<sup>2</sup> /Vs and θ = 0.038 V�<sup>1</sup> have been implemented in Eq. (9), and the result is shown with the dashed line in Figure 5 and does not provide a great accuracy like it was the case for Si(100) wafers. Nevertheless, the procedure has been moved forward, and the simulation of the drain current Id and the transconductance gm has been calculated with Eq. (8) and the following parameters: μ<sup>0</sup> = 285 cm2 /Vs and θ = 0.038 V�<sup>1</sup> , ΔL = �0.4 μm, ΔW = �0.4 μm and Racc = 60 Ω. The results for the drain current Id are shown with the dashed lines in Figure 7, while the results for the transconductance gm are shown in Figure 8 with the dashed lines. The modeling is accurate at first but strongly divert from the experimental data when the gate overdrive voltage is increased. The use of Eq. (9) does not give at all satisfactory

Figure 7. Drain current Id as a function of the gate overdrive voltage Vg–Vth for Si(110) p-MOSFETs featuring different gate length. The dashed lines are the modeling carried out with Eqs. (8), and (9), μ<sup>0</sup> = 285 cm<sup>2</sup> /Vs and θ = 0.038 V�<sup>1</sup> . The full lines are the modeling carried out with Eqs. (1), and (10), μ<sup>0</sup> = 280 cm2 /Vs, θ<sup>1</sup> =0V�<sup>1</sup> , θ<sup>2</sup> = 0.05 V�<sup>2</sup> , and α = 0.04. ΔL = �0.4 μm, ΔW = �0.4 μm and Racc = 60 Ω.

Figure 8. Transconductance gm as a function of the gate overdrive voltage Vg–Vth for Si(110) p-MOSFETs featuring different gate length. The dashed lines are the modeling carried out with Eqs. (8), and (9), μ<sup>0</sup> = 285 cm<sup>2</sup> /Vs and θ = 0.038 V�<sup>1</sup> . The full lines are the modeling carried out with Eq. (8), and (10), μ<sup>0</sup> = 280 cm2 /Vs, θ<sup>1</sup> =0V�<sup>1</sup> , θ<sup>2</sup> = 0.05 V�<sup>2</sup> , and α = 0.04. ΔL = �0.4 μm, ΔW = �0.4 μm and Racc = 60 Ω.

agreement with the experiment, especially concerning the transconductance gm. The wellestablished model that is Eq. (9) cannot be used to simulate the mobility and thus the drivability of Si(110) p-MOSFETs. Other models [24, 25] have been implemented but did not give enough satisfactory results. Contrary to the (100) orientation for which the single phonon scattering mechanism is limiting the hole mobility over the working range, the hole mobility for the (110) orientation is limited by the Coulomb, phonon and surface roughness scatterings mechanism over the whole measurement range. Thus, a model able to take into account these three mechanisms is required. While Eq. (9), which models only the phonon scattering, is sufficient to simulate the Si(100) p-MOS transistors, a new model including all three scattering mechanisms is needed for the Si(110) wafers.

the result is shown with the dashed line in Figure 5 and does not provide a great accuracy like it was the case for Si(100) wafers. Nevertheless, the procedure has been moved forward, and the simulation of the drain current Id and the transconductance gm has been calculated

ΔW = �0.4 μm and Racc = 60 Ω. The results for the drain current Id are shown with the dashed lines in Figure 7, while the results for the transconductance gm are shown in Figure 8 with the dashed lines. The modeling is accurate at first but strongly divert from the experimental data when the gate overdrive voltage is increased. The use of Eq. (9) does not give at all satisfactory

Figure 7. Drain current Id as a function of the gate overdrive voltage Vg–Vth for Si(110) p-MOSFETs featuring different

Figure 8. Transconductance gm as a function of the gate overdrive voltage Vg–Vth for Si(110) p-MOSFETs featuring

different gate length. The dashed lines are the modeling carried out with Eqs. (8), and (9), μ<sup>0</sup> = 285 cm<sup>2</sup>

. The full lines are the modeling carried out with Eq. (8), and (10), μ<sup>0</sup> = 280 cm2

0.04. ΔL = �0.4 μm, ΔW = �0.4 μm and Racc = 60 Ω.

/Vs, θ<sup>1</sup> =0V�<sup>1</sup>

gate length. The dashed lines are the modeling carried out with Eqs. (8), and (9), μ<sup>0</sup> = 285 cm<sup>2</sup>

full lines are the modeling carried out with Eqs. (1), and (10), μ<sup>0</sup> = 280 cm2

�0.4 μm, ΔW = �0.4 μm and Racc = 60 Ω.

V�<sup>1</sup>

/Vs and θ = 0.038 V�<sup>1</sup>

, ΔL = �0.4 μm,

/Vs and θ = 0.038 V�<sup>1</sup>

, and α = 0.04. ΔL =

/Vs and θ = 0.038

, and α =

, θ<sup>2</sup> = 0.05 V�<sup>2</sup>

, θ<sup>2</sup> = 0.05 V�<sup>2</sup>

/Vs, θ<sup>1</sup> =0V�<sup>1</sup>

. The

with Eq. (8) and the following parameters: μ<sup>0</sup> = 285 cm2

12 Different Types of Field-Effect Transistors - Theory and Applications

µCoul = ACoulT�<sup>1</sup> Eeff<sup>β</sup>Coul [26] with βCoul ≥ 0 and µsr=AsrEeff�<sup>2</sup> [27] are simple ways to model the Coulomb scatterings and surface roughness scatterings, respectively. ACoul and βCoul are constants associated with the Coulomb scattering mechanism, while Asr is a constant associated with the surface roughness scattering mechanism. T is the temperature. Assuming that the effective electric field Eeff is proportional to the gate overdrive voltage Vg – Vth, the dependence of the several scattering mechanisms can be introduced into Eq. (9), which is already modeling the Coulomb scatterings mechanisms to finally give [21]

$$
\mu\_{\rm eff} = \mu\_0 \frac{\mathbf{A}\_a (\mathbf{V\_g} - \mathbf{V\_{th}})^a}{1 + \Theta\_1 (\mathbf{V\_g} - \mathbf{V\_{th}}) + \Theta\_2 (\mathbf{V\_g} - \mathbf{V\_{th}})^2}. \tag{10}
$$

μ<sup>0</sup> is the low field mobility. θ<sup>1</sup> corresponds to the conventional mobility attenuation factor seen in Eq. (9) and is related to the contribution coming from the phonon scatterings. θ<sup>2</sup> is a quadratic mobility attenuation factor related to the surface roughness scatterings. α is a parameter related to the Coulomb scatterings, while A<sup>α</sup> equals to 1 and is introduced to maintain the uniformity of the unit system. As shown in Figure 6 with the full line, Eq. (10) greatly matches the experimental data. The fitting parameters are as follows μ<sup>0</sup> = 280 cm2 /Vs, θ<sup>1</sup> =0V�<sup>1</sup> , θ<sup>2</sup> = 0.05 V�<sup>2</sup> , and α = 0.04. The simulation of the drain current Id and the tranconductance gm has been carried out for Si(110) p-MOSFETs featuring different gate length by implementing Eq. (10) into Eq. (8). μ<sup>0</sup> = 280 cm2 /Vs, θ<sup>2</sup> = 0.05 V�<sup>2</sup> , α = 0.04, ΔL = �0.4 μm, ΔW = �0.4 μm, Racc = 60 Ω, Vd = 100 mV and W = 20 μm. The results are reported with the full lines in Figures 7 and 8. The modeling of the drain current Id is greatly accurate even for short gate length. The modeling of the transconductance gm in Figure 8 is also fairly accurate. Both, the results regarding the drain current Id and the transconductance gm testify of the good agreement of Eq. (10). In Figure 8, it seems that the maximum of the transconductance gm, result of α the parameter related to the Coulomb scatterings, can be also calculated. This statement must be taken with care since the maximum of the transconductance is obtained for biases that do not correspond to the linear region, making Eq. (8) obsolete. Actually, the parameter α does not solely reflect the Coulomb scatterings. Indeed, the hole mobility in Si (110) wafers has a peculiar behavior in the form of the inter-subband phonon scatterings. Contrary to the acoustic phonon scatterings that are more and more limiting the mobility with an increase of the effective electric field, the inter-subband phonon scatterings have the specificity to decrease when the effective electric field is increased [21, 28] as sketched in Figure 5.
