Subsections

• 5.1.1 Doping Dependence
• 5.1.2 Strained Ge

5.1 Low-Field Mobility Modeling Results

In order to validate the analytical model, Monte Carlo simulations were performed, the parameters of which have been fit to experimental data which are available mainly in the form of piezoresistance coefficients [Smith54]. For low strain levels, the increase in the in-plane electron mobility is linear [Kanda82], characterized by the piezoresistance coefficients. It was observed that changing the uniaxial deformation potential $\Xi_{u}$ from its original value of $9.29$ eV [Rieger93] to $7.3$ eV gave good agreement for low strain levels, because varying $\Xi_{u}$ results in a variation of the valley shifts through (3.42) and hence of the valley populations. Table 5.1 lists the reported values of the uniaxial deformation potential.

Numerical calculations in literature exhibit a considerable quantitative difference in the saturation value of the in-plane electron mobility in biaxially-strained Si layers. Enhancement factors ranging from 56 % [Bufler97] up to 180% [Yamada94] have been simulated while measurements indicate a value of around 97% [Ismail93]. We have adopted a somewhat conservative value of 70% for the enhancement factor, which lies in-between other reported values [Fischetti96] [Vogelsang93].

Values	8.86	8.47	9.2	7.3	9.29	10.5
Reference	A	B	C	D	E	F

Table 5.1: Deformation potential $\Xi _u$ (in eV) reported in various references. A [Tserbak93], B [Friedel89], C [Balslev66,Jacoboni83,de Walle86], D [Yoder94,Yoder93], E [Rieger93], F [Fischetti96]

Type of scattering	Ref A	Ref B	Ref C	Ref D	This work
$g_1$	0.5	0.5	-	19.2	0.4716
$g_2$	0.8	0.8	4.0	-	0.7574
$g_3$	11	11	8.0	-	10.42
$f_1$	0.3	0.3	2.5	19.2	0.348
$f_2$	2.0	2.0	-	-	2.32
$f_3$	2.0	2.0	8.0	-	2.32

Table 5.2: The coupling constants for inter-valley scattering in Si are in $[$10$^8$ eV/cm$]$. A [Bufler97], B [Jacoboni83], C [Yamada94], D [Takagi96]

To achieve such values, it was required to adjust the coupling constants of the $g$-type and $f$-type phonons. The reported values of the coupling constants for inter-valley scattering in Si are shown in Table 5.1. Since at high strain values the $f$-type phonon scattering is completely suppressed in the $\Delta _2$ valleys, the only way to obtain an increase in the saturation mobility is by reducing the $g$-type coupling constants. This adjustment however would result in an increased value of the unstrained electron mobility. To restore the latter, it was required to increase the $f$-type phonon coupling constant. The coupling constants of the $g$-type phonons were reduced by a factor of 1.06 and that for $f$-type phonons increased by a factor of 1.16, as compared to the original values proposed by Jacoboni [Jacoboni83]. The only parameter of the inter-valley scattering model (4.67) is the phonon energy. A value of $\hbar\omega_{\text{opt}} = 60 \mathrm{meV}$ has been assumed. While the model based on piezoresistance coefficients assumes a linear relationship between mobility enhancement and strain, with the proposed model the mobility enhancement saturates at large strain values as observed experimentally.

The electron mobility components for different orientations of the underlying SiGe layer is obtained by transformation of the mobility tensor from the principle coordinate system to the interface coordinate system, using the transformation matrix (3.19). Fig. 5.1 shows the electron lattice mobility components obtained using (4.72) for substrate orientations (001) and (110), respectively. For substrate orientation (001) the two in-plane components of the electron mobility are equal with the maximum mobility saturating at a value above 2400 cm$^2/$Vs at about 30% Ge content. For (110) orientation (Fig. 5.1b), the Monte Carlo simulation results demonstrate that the in-plane component ( $[\overline{1}10]$) of the electron mobility is equal to the perpendicular component ($[110]$). This feature is also reproduced by the analytical model.

        
$\parbox{2.8in}{(a) \hfill (b)}$

Figure 5.1: In-plane and perpendicular electron mobility components in undoped strained Si versus the Ge content in a (a) (001) and (b) (110) oriented SiGe buffer layer. The mobilities calculated using (4.72) are compared with those obtained from piezoresistance coefficients and Monte Carlo simulations. A [Bufler97], B [Vogelsang93].

For the (123) oriented SiGe substrate three distinct components of the mobility can be seen in Fig. 5.2. The analytical model shows excellent agreement with the Monte Carlo simulation results for a large range of Ge content $y$ in the SiGe layer. The deviation for very large strain levels (Ge content $\ge$ 0.85) is due to the fact that the proposed model does not consider the population of the L valleys.

Figure 5.2: In-plane and perpendicular electron mobility components in undoped strained Si versus the Ge content in a (123) oriented SiGe buffer layer. The mobilities calculated using (4.72) are compared with Monte Carlo simulations.

        
$\parbox{2.5in}{(a)\hfill (b)}$

Figure 5.3: Angular dependence of the in-plane electron mobility components (in cm$^2$/Vs) in undoped strained Si on a (a) (110) and (b) (123) oriented SiGe buffer layer calculated using (4.72). The Ge content is $30\%$.

The dependence of the electron mobility in strained Si on the in-plane angle $\gamma$ can be obtained by taking the projection of the mobility tensor $\ensuremath{{\underline{\mu}}}_{\ensuremath{{\mathrm{n}}}}^{\text{tot}}$ in the direction of the in-plane vector ${\mathbf{a}}$.

$$\mu(\gamma) = \displaystyle {\mathbf{a}}^{T}(\gamma) \cdot \ensur... ...line{\mu}}}_{\ensuremath{{\mathrm{n}}}}^{\text{tot}} \cdot {\mathbf{a}}(\gamma)$$ (5.1)

Fig. 5.3 shows the variation of the mobility as a function of the angle $\gamma$ for (110) and (123) SiGe substrate calculated using (5.1). The in-plane vectors ${\mathbf{u}}$ and ${\mathbf{v}}$ at which the electron mobility assumes its maximum and minimum values can be obtained by setting the derivative of (5.1) to zero. Alternatively one may choose any two orthogonal in-plane vectors ${\mathbf{m}}$ and ${\mathbf{l}}$ and calculate the projection of the mobility tensor.

$$\displaystyle \mu_{ml} = \displaystyle {\mathbf{m}}^{T}\cdot \ens... ...h{{\underline{\mu}}}_{\ensuremath{{\mathrm{n}}}}^{\text{tot}}\cdot {\mathbf{l}}$$ (5.2)

The vectors ${\mathbf{u}}$ and ${\mathbf{v}}$ can then be obtained by calculating the eigenvalues and eigenvectors for the following matrix.

$$\begin{pmatrix}\displaystyle \mu_{mm} & \displaystyle \mu_{ml} \displaystyle\mu_{lm} & \displaystyle \mu_{ll}\end{pmatrix}\quad$$ (5.3)

        
$\parbox{2.9in}{(a)\hfill (b)}$

Figure 5.4: Effect of uniaxial $\langle 110 \rangle$ and biaxial tensile strain on (a) the valley splitting. The strain component in the stressed direction is plotted. (b) variation of the transversal and longitudinal masses of the $\Delta _2$-valleys in Si.

It was shown in Section 3.3.4 that for shear stresses an additional shift in the energy of the lowest lying $\Delta _2$ valleys occurs. Fig. 5.4a shows the splitting, $\Delta\epsilon$, for biaxially strained and uniaxially stressed Si. It is observed that biaxial tension is more effective in splitting the conduction band valleys than $\langle 110 \rangle$ uniaxial tension.

The variation of the in-plane masses parallel ( $m_{t\parallel}$) and perpendicular ( $m_{t\perp}$) to the stress, and the longitudinal ($m_l$) masses for the $\Delta _2$-valleys as a function of the strain as calculated using (3.73) to (3.75) is shown in Fig. 5.4b. The values of the parameters $\Xi_{u'}$ and $\Delta$ were chosen as 7.0 eV and 0.53 eV, respectively. It can be seen that there is a significant change in $\Delta m^*$ for increasing strain along $\langle 110 \rangle$.

In Fig. 5.5 the anisotropy of the mobility is compared for $\langle 110 \rangle$ and $\langle100\rangle$ stress directions with and without the change in the effective masses. It can be clearly seen that $\Delta m_t$ cannot be neglected for $\langle 110 \rangle$ uniaxial stress. For uniaxial compression, on the other hand, there is a negligible change in the effective masses of the lowered $\Delta_4$-valleys.

Figure 5.5: In-plane mobility as a function of the in-plane angle for 1.5 GPa uniaxial tensile stress along $\langle100\rangle$ and $\langle 110 \rangle$ . $-\star$: $\langle 110 \rangle$ stress without mass correction; $- -$: $\langle 110 \rangle$ stress with mass correction; $-$: 1.5GPa uniaxial tensile $\langle100\rangle$ stress.

Fig. 5.6 compares the electron mobility components with increasing compressive stress along $[110]$ as obtained from Monte Carlo simulations and the analytical model. Since the effective masses do not change under compressive stress, two components of mobility are identical. Three different components of the mobility, $\mu_{[110]}$, $\mu_{[\overline{1}10]}$ and $\mu_{[001]}$, appear for the case when uniaxial tensile stress is applied along the $[110]$ direction. This can be seen in Fig. 5.6b. The model shows good agreement with the Monte Carlo simulation results.

        
$\parbox{2.8in}{(a) \hfill (b)}$

Figure 5.6: Comparison of parallel ($\mu _{\vert\vert})$, perpendicular ($\mu _\perp$) and out-of-plane ( $\mu _\ensuremath {{\mathrm {oop}}}$) electron mobility components obtained from Monte Carlo simulations (symbols) and the analytical model (lines) for uniaxial $\langle 110 \rangle$ (a) compressively and (b) tensile strained Si. In (a) $m_{t\parallel } = m_{t\perp } = m_t$.

.

5.1.1 Doping Dependence

The doping dependence was calculated as described in Section 4.3.3 with the parameter values stated in Table 5.1.1. Figure 5.7 shows the doping dependence of the in-plane minority and majority electron mobility in strained Si layers for different Ge content of the underlying (001) SiGe substrate.

Parameter	Unit	Si
$\mu^{\mathrm{L}}_n$	[cm $^2V^{-1}s^{-1}$]	1430
$\mu^{\mathrm{mid}}_\ensuremath{{\mathrm{maj}}}$	[cm $^2V^{-1}s^{-1}$]	44
$\mu^{\mathrm{hi}}_\ensuremath{{\mathrm{maj}}}$	[cm $^2V^{-1}s^{-1}$]	57
$\mu^{\mathrm{mid}}_\ensuremath{{\mathrm{min}}}$	[cm $^2V^{-1}s^{-1}$]	141
$\mu^{\mathrm{hi}}_\ensuremath{{\mathrm{min}}}$	[cm $^2V^{-1}s^{-1}$]	218
$\eta$	1	0.65
$\lambda$	1	2.0
$C^\ensuremath{{\mathrm{mid}}}$	[cm$^{-3}$]	1.12 $\times 10^{17}$
$C^\ensuremath{{\mathrm{hi}}}_\ensuremath{{\mathrm{maj}}}$	[cm$^{-3}$]	1.18 $\times 10^{20}$
$C^\ensuremath{{\mathrm{hi}}}_\ensuremath{{\mathrm{min}}}$	[cm$^{-3}$]	4.35 $\times 10^{19}$

Table 5.3: Parameter values for the doping dependence of the electron mobility in Si at 300 K

The solid lines depict the results as obtained from the analytical model (4.72), while the symbols indicate the Monte Carlo simulation results. The model shows good agreement with the Monte Carlo data. It reproduces the slight increase in the minority electron mobility for high doping concentrations for all strain levels, see Fig 5.7a.

        
$\parbox{2.9in}{(a)\hfill (b)}$

Figure 5.7: Doping dependence of in-plane electron mobility in strained Si calculated using (4.72) for different Ge content in SiGe [001] substrate. (a) minority electron mobility, (b) majority electron mobility.

5.1.2 Strained Ge

The electron mobility for unstrained and strained Ge were calculated from Monte Carlo simulations and compared with the analytical model descried in Section 4.4 for different stress values and temperatures. For the analytical model the values of the dilatation and uniaxial deformation potentials for the L-valleys was chosen to be $\Xi_\mathrm{d}=-4.43$ eV and $\Xi_\mathrm{u}=16.8$ eV [Fischetti96], respectively. Fig. 5.8a shows the valley splittings $\Delta_{ij}$, between initial valley $i$ and final valley $j$, calculated using (3.42) as a function of stress along the $[111]$, $[11\overline{2}]$ and $[\overline{1}10]$ directions. It is observed that for stress along $[11\overline{2}]$ all four valleys are at different energy levels. The figure also indicates that uniaxial stress along $[111]$ is most effective in splitting the L-valleys.

        
$\parbox{2.7in}{(a) \hfill (b)}$

Figure 5.8: (a) Effect of uniaxial stress on the valley splitting ( $\Delta_{ij} = \Delta \epsilon_C^{(j)} - \Delta \epsilon_C^{(i)}$) in Ge for stress ($T$) along the $[111]$(-), $[11\overline{2}]$(-o-) and $[\overline{1}10]$(- -) directions. $T_{hkl}$ denotes the direction of the applied stress. (b) Variation of electron mobility in the (111) plane with the in-plane angle for $-1$ GPa stress along different directions, at room temperature. $[\overline{1}11]$(-+-) directions.

Fig. 5.8b shows the variation of the in-plane electron mobility in uniaxial compressively stressed Ge in the $(111)$ plane as obtained from Monte Carlo simulations. The simulations delivered an unstrained value of bulk electron mobility of approximately 3850 cm$^2$/Vs which is close to previously reported values [Prince53] [Morin54].

Fig. 5.9 shows the analytically calculated electron mobility components for uniaxial compressive and tensile stresses, respectively. Also shown for comparison are the Monte Carlo simulation results (symbols). Fig. 5.9a indicates that compressive stress increases the electron mobility in the $(111)$ plane. The mobility saturates at 6900 cm$^2$/Vs for stress values greater than $-1.5$ GPa. This mobility improvement is nearly 2.8 times the strain enhanced bulk mobility in Si. On the contrary, applying uniaxial tensile stress along [111] results in larger out-of-plane mobilities, as shown in Fig. 5.9b.

The temperature dependence of the electron mobility is shown in Fig. 5.10a. The value of the parameter $\alpha=1.78$ in (4.87) has been extracted from the Monte Carlo data and the values of $f_{300}$ and $\beta$ in (4.88) were chosen as 1.79 and $-0.12$, respectively. Fig. 5.10b shows a comparison of the mobility components in the $(111)$ plane as a function of compressive stress along $[111]$ at T = 200K, as obtained from Monte Carlo simulations and the analytical model. Very good agreement is observed.

        
$\parbox{2.7in}{(a) \hfill (b)}$

Figure 5.9: Electron mobility components in the $(111)$ plane versus uniaxial (a) compressively and (b) tensile stress along $[111]$ at room temperature. MC $_{\parallel}$ indicates the isotropic in-plane mobility, while MC$_{\perp}$ denotes the out-of-plane mobility.

        
$\parbox{2.7in}{(a) \hfill (b)}$

Figure 5.10: (a) Electron mobility in the $(111)$ plane as a function of temperature for unstrained and $-3$ GPa along [111] direction strained Ge. (b) Electron mobility in the $(111)$ plane versus uniaxial compressive stress along $[111]$ at 200K. A Comparison of the analytical model and the Monte Carlo simulation results is shown. MC $_{\parallel}$ indicates the isotropic in-plane mobility, while MC$_{\perp}$ denotes the out-of-plane mobility.