Guidelines and Experimental Determination

Guidelines are established in terms of a flowchart shown in Fig. 5.24. There are six input parameters: a, b, d, d0, , and . The corresponding Dac and Dop can be determined according to Eq. (2.3.2.1) and (2.3.3.1), respectively. To facilitate the procedure, we first took a literature value of 26.6 eV for d0 [20] and hence the corresponding Dop of 8.510^-8 eV/cm. Then, a fit to the experimental unstrained hole

effective mobility data in Fig. 5.25 as cited elsewhere [21] was carried out, producing Δ = 0.42 nm and λ = 2.6 nm. It can be seen from the figure that a good fitting appears in the high Eeff region or the universal mobility region, valid for different substrate orientations and different transport directions. This validates the presented calculation method. Here, Eeff is the vertical effective electric field in the inversion layer, which was calculated using the empirical formula: Eeff = e(Pinv + Pdep) with  taken as 1/3 according to Takagi, et al. [22], where Pinv is the inversion-layer density and Pdep is the substrate depletion charge density. Deviations in the low Eeff region are expected because impurity Coulomb scattering was not taken into account in this work.

At this point, the number of input parameters reduces to three. Since a and b are weak or moderate in effect, we can quote the literature values: a = 2.46 eV [15] and b = 2.1 eV [16]. Then, with a guess of d, the strain induced hole mobility change was calculated, an updated d was obtained in comparison with the experimental data.

This process was iterated until a good fitting is achieved. In this way, we obtained d =

3.1 eV and Dac = 5.62 eV from a fit to hole inversion-layer mobility enhancement data under uniaxial compressive stress [10], [11], as depicted in Fig. 5.26.

Biaxial-stress mobility data [23], [24] also were quoted. Extra calculation for this case was performed. The result is shown in the inset of the figure. Fairly good agreement remains, without changing any parameters. The extracted results are listed in Table 5.2. The corresponding calculated hole mobility change at two different Eeff is shown in Fig. 5.21 for comparison.

Even making a change of b to 1.6 eV, we found that the reproduction quality is acceptable, as shown in the inset of Fig. 5.26. Strikingly, such change in b does not significantly affect the calculated hole mobility enhancement in case of uniaxial stress, as shown in Fig. 5.26. This invariability supports the published error range of 2.58

eV  b 1.5eV [14]-[17]. Above arguments hold for other avalues, as has been proved in Fig. 5.23. This can reasonably explain the commonly used values of 2.06 eV  a 2.46 eV [14], [17]. Good reproduction of the data is evident and can be found when a = 2.46 eV, b = 2.1 eV, and d = 3.1 eV. It is noteworthy that the experimentally determined d of 3.1 eV in this work is exactly that (3.1 eV) [16]

based on cyclotron resonance measurements [25].

Above results stemmed from a specific d0 of 26.6 eV. As illustrated by the guidelines in Fig. 5.24, a change in d0 may change the extracted surface roughness parameters. In fact, the quoted mobility data sources [10], [11], [21], [23], [24] came from different manufacturing processes featuring different surface roughness details.

Thus, it is clear that the uncertainty exists in d0 or equivalently the surface roughness parameters. To reflect this, we show in Fig. 5.27 the effect of varying surface roughness height . Evidently, the calculated hole mobility change in Fig. 5.27 is almost the same as Fig. 5.26. In other words, the uncertainty in  does not significantly affect the calculated hole mobility change. This also is the case for surface roughness correlation length . Therefore, the extracted parameters as listed in Table 5.2 remain valid in the presence of the uncertainty in the surface roughness parameters.

Finally, we add two interesting calculation results, as depicted in Fig. 5.28. First, the inclusion of the screening effect in surface roughness scattering will reduce the calculated hole inversion-layer mobility change, particularly in the high stress region.

Second, a change in the surface roughness model from the exponential function to the Gaussian function does not influence the result.

5.5 Conclusions

It has been found the phonon-limited mobility change is more stress-sensitive than the surface-roughness-limited one, and the mobility change ratio can be reversely proportional to the conductivity effective mass and density-of-states effective mass only in the absence of the surface roughness-limited mobility change. Individual contributions of hole mobility change have all be quantified. Calculated hole mobility change due to varying a, b, and d has been created and has accounted for 3-D uniaxial stress conditions. The primary factor d and the secondary factor b have been drawn. Guidelines have been established, followed by the experimental determination of a, b, and d. The literature errors of the Bir-Pikus deformation potentials have therefore been improved.

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Table 5.1 List of scattering parameters and Bir-Pikus potentials used in Section 5.2 and 5.3. Notice that the scattering parameters used in Section 5.2 and 5.3 are assumed to be independent of the Bir-Pikus potentials av, b, and d.

Dac

(eV) Dop

(10^-8eV/cm) ћωop

(meV) λ (nm) Δ

(nm) av

(eV) b

(eV) d (eV)

d0

(eV) Section 5.2 & 5.3

N_sub=1x10¹⁷cm^-3

T=300K 5.62 8.5 61.2 2.6 0.42 2.46 -2.1 -4.8 41.5

Table 5.2 List of the modified scattering parameters and Bir-Pikus potentials used in Section 5.4.

Dac

(eV) Dop

(10^-8eV/cm) ћωop

(meV) λ (nm) Δ

(nm) av

(eV) b

(eV) d (eV)

d0

(eV) Section 5.4

Nsub=1x10¹⁷cm^-3 T=300K

5.62 8.5 61.2 2.6 0.42 2.46 -2.1 -3.1 26.6

Table 5.3 Material parameters used in Section 5.4. γ1, γ1, and γ1 are Luttinger parameters; av, b, and d are the Bir-Pikus potentials; d0 is the optical deformation potential; split-off is the split-off hole energy; c11, c12, and c44 are the elastic coefficients; ρ and a0 are the crystal density and lattice constant of silicon; and are the longitudinal and transverse sound velocity.

γ₁ γ₂ γ₃ a_v

(eV)

b (eV)

d (eV)

d₀ (eV)

Δ_split-off (eV)

4.285 0.339 1.446 2.46 -2.1 -3.1 26.6 0.044

c₁₁ c₁₂ c₄₄ ρ

(g/cm³)

a₀ (Å)

v_l v_t

(10¹⁰N/m²) (10⁵ cm/sec)

16.6 6.41 7.94 2.329 5.43 9.04 5.41

-3 -2 -1 0 1 2 3

Fig. 5.1 Calculated 3-D uniaxial stress dependence of hole inversion-layer mobility change for different deformation potentials on (001) substrate. Comparison is done with other groups [8], [9]. The Bir-Pikus potentials av=2.1 eV, b=-1.6 eV, and d=-2.7 eV are cited in [6].

-3 -2 -1 0 1 2 3

Fig. 5.2 Calculated 3-D uniaxial stress dependence of hole inversion-layer mobility change for different deformation potentials on (110) substrate. Comparison is done with other groups [8]. The Bir-Pikus potentials av=2.1 eV, b=-1.6 eV, and d=-2.7 eV are quoted in [6].

Fig. 5.3 The device structures for (001) and (110) p-MOSFETs. The channel direction and applied stress direction are clarified. Here, three-dimensional in-plane longitudinal, transverse and out-of-plane stress are involved in this dissertation.

n‐type

(110) wafer

Trans.

< 001>

Long.

<110>

Out-of-plane <1-10>

n‐type

Channel

(001) wafer

Trans.

< 1-10>

Long.

<110>

Out-of-plane <001>

xx zz yy

p⁺ p⁺

Channel

-3 -2 -1 0 1 2 3 -200

0 200 400 600 800 1000 1200 1400

  /  (%)

在文檔中 一種新穎自行開發之快速精準量子模擬器用於三維度高應力下矽電洞能帶結構及反置層遷移率之研究 (頁 118-132)