Subsections

• 5.2.1 Analytical High Field Velocity Model
• 5.2.2 Total Velocity for Arbitrary Field Direction

5.2 High-Field Electron Transport in Strained Si

In this section, the electron high-field transport in Si under biaxial and uniaxial stress conditions is studied and some peculiarities of the velocity-field characteristics in strained Si are presented. The velocity-field characteristics have been obtained by full-band Monte Carlo simulations using VMC [IuE06] and compared with the analytical model developed in Section 4.6. Results of full-band Monte Carlo simulations have been calibrated with the existing theoretical and experimental data.

It was discussed in Section 5.1 that the enhancement of the bulk low-field electron mobility saturates at around 1.7 [Dhar05b]. In order to maintain the desired mobility enhancement, the g-type and f-type coupling constants had to be adjusted to the values stated in Table 5.1. In addition, it was required to adjust the acoustic deformation potential from its original value of 8.9 eV [Jungemann03a] for analytical band Monte Carlo to 8.5 eV for full-band Monte Carlo simulation. The effect of impact ionization was neglected for the field regime investigated.

Fig. 5.11 presents the velocity-field characteristics for unstrained and strained Si for different field directions as obtained from Monte Carlo simulations. Also displayed are the results from Bufler [Bufler97], Canali [Canali71], Smith [Smith80], Fischer [Fischer99], and Ismail [Ismail93]. The simulation results agree well with measured data from Smith for the [111] field direction and with Canali for the [100] field direction for the unstrained case and with Jungemann [Jungemann03a], Bufler, and Ismail for the strained case.

Figure 5.11: Comparison of electron velocity versus field characteristics in unstrained and strained ($^\star$) Si on Si$_{0.7}$Ge$_{0.3}$ for [100] and [111] field directions.

Fig. 5.12 depicts the velocity-field characteristics as obtained from Monte Carlo simulations for biaxially strained Si grown on a relaxed SiGe substrate for different Ge content and field along the in-plane ([100]) and out-of-plane ([001]) direction, respectively. The total velocity increases with strain for a field along the [100] direction and it decreases for a field along the [001] direction. For the in-plane electric field ([100]) the electron velocity shows a region of small negative differential mobility.

The velocity-field characteristics for field along the [001] direction exhibit an untypical form for high strain levels. This phenomenon can be explained as follows. For field along [001] direction the $\Delta _2$-valleys are lowered in energy with increasing strain and have the longitudinal mass in the field direction. These valleys are located at a scaled distance of 0.85 and 1.15 from the center of the first Brillouin zone and are separated by an energy barrier of 129 meV at the X-point (Fig. 5.13). The average velocity in the left and right valley and also the average of these velocities are shown in Fig. 5.14. For low-fields, electrons in both valleys are slightly displaced with respect to the valley minima. This results in the initial velocity increase for both valleys shown in Fig. 5.14. However, as the field increases, electrons in both valleys gain energy, and electrons from one valley can surpass the energy barrier and drift to the valley in the next Brillouin zone. As sketched in Fig. 5.13, there are more electrons populating the right side of the double valley than the left side, giving rise to a slight increase in average velocity. If only the left valley is considered, there are more electrons populating the left edge of the single valley resulting in a negative valley velocity, as shown in Fig. 5.14.

Figure 5.12: Electron velocity in strained Si on SiGe with Ge content as a parameter for field along the [100] and [001] directions.

Figure 5.13: Asymmetric electron populations of the double valley close to the equilibrium state (top) and at high-field (bottom). Solid circles indicate electrons with positive group velocity (right side of each valley). Open circles refer to electrons with negative group velocity (left side of each valley).

Figure 5.14: Velocity versus field for the left valley and right valley together with the average valley velocity, computed by full-band Monte Carlo simulations.

5.2.1 Analytical High Field Velocity Model

For the empirical model, the parameters in (4.108) to (4.112) have been obtained using the optimization framework of MATLAB [MATLAB04]. A multidimensional unconstrained nonlinear minimization (Nelder-Mead) technique was adopted for obtaining the parameter set. The optimized values of the parameters for these field directions are listed in Tables 5.2.1 to 5.2.1. It should be noted that the optimization technique is sensitive to the initial conditions of the parameters and thus a small variation in the initial conditions can result in a slightly varied parameter set.

Parameter	Units	E [100]	E [110]	E [11$\sqrt{2}$]
		E [001]	E [101]
$v_{s1}$	[10$^7$ cm s$^{-1}$]	1.026	1.058	1.042
$\beta_{1}$	[1]	1.085	1.2475	1.273

Table 5.4: Parameter values for the parallel velocity component $v_E$ in unstrained Si.

Parameter	Units	E [100]	E [001]	E[110]	E [101]	E [11$\sqrt{2}$]
$v_{s2}$	[10$^5$ cms$^{-1}$eV$^{-1}$]	$-5.5691$	33.731	1.4988	11.739	11.067
$\beta_{2}$	[eV$^{-1}$]	$-0.33235$	$-5.2879$	0.22885	$-0.30235$	$-0.39907$
$\xi_{1}$	[eV]	0.37994	$-0.22859$	0.45615	$-0.84676$	$-0.76303$
$\xi_{2}$	[1]	1.6239	1.0333	1.5468	6.3401	4.7611
$\eta_{1}$	[10$^4$ Vcm$^{-1}$]	2.1254	6.3369	0.6651	4.2133	5.4664
$\eta_{2}$	[10$^5$ Vcm$^{-1}$eV$^{-1}$]	$-1.13$	$-2.748$	$-0.86273$	$-2.0402$	$-1.5317$
$\gamma_{1}$	[1]	1.3707	2.6051	1.3869	2.4453	3.4612
$\gamma_{2}$	[eV$^{-1}$]	$-0.73185$	$-6.3392$	0.61215	$-13.938$	$-7.1773$

Table 5.5: Parameter values for the parallel velocity component $v_E$ for $\Delta \epsilon <0$.

Parameter	Units	E [100]	E [001]	E [110]	E [101]	E [11$\sqrt{2}$]
$v_{s2}$	[10$^5$ cms$^{-1}$eV$^{-1}$]	$-20.608$	10.822	$-14.625$	2.2239	3.5825
$\beta_{2}$	[eV$^{-1}$]	0.472	0.5135	0.21785	$-0.41762$	$-0.27011$
$\xi_{1}$	[eV]	0.47701	$-0.29814$	1.0876	$-1.3718$	1.9381
$\xi_{2}$	[1]	3.1569	2.1639	$-8.5962$	$-4.2752$	5.5323
$\eta_{1}$	[10$^4$ Vcm$^{-1}$]	7.6075	3.7613	5.8913	0.25071	1.1382
$\eta_{2}$	[10$^5$ Vcm$^{-1}$eV$^{-1}$]	1.471	2.7214	1.3928	0.92226	$-0.19962$
$\gamma_{1}$	[1]	3.815	1.163	4.7754	1.4471	0.7351
$\gamma_{2}$	[eV$^{-1}$]	2.9118	4.8595	5.2425	0.14618	5.2995

Table 5.6: Parameter values for the parallel velocity component $v_E$ for $\Delta \epsilon >0$.

Parameter	Units	E [101]	E [101]	E [11$\sqrt{2}$]	E [11$\sqrt{2}$]
		$\Delta \epsilon <0$	$\Delta \epsilon >0$	$\Delta \epsilon <0$	$\Delta \epsilon >0$
$v_{s2}$	[10$^6$ cms $^{-1}$eV$^{-1}$]	4.0975	2.8429	3.53	3.4858
$\beta_{2}$	[eV$^{-1}$]	$-1.7085$	$-0.22173$	$-1.7571$	$-0.26191$
$\xi_{1}$	[eV]	$-0.16917$	$-0.3354$	$-0.17293$	$-0.36817$
$\xi_{2}$	[1]	0.75896	$-1.9529$	0.76209	2.2891
$\eta_{1}$	[10$^4$ Vcm$^{-1}$]	3.9982	5.9335	4.223	6.5366
$\eta_{2}$	[10$^5$ Vcm$^{-1}$eV$^{-1}$]	$-2.2147$	1.9583	$-2.1683$	1.8587
$\gamma_{1}$	[1]	1.8204	1.9209	2.0921	1.668
$\gamma_{2}$	[eV$^{-1}$]	$-4.6684$	3.5664	$-3.5816$	5.4713

Table 5.7: Parameter values for the [001] velocity component $v_3$.

Fig. 5.15a shows the $v_E(E)$ characteristics for a 1GPa stress along [001] and the field along the [100] and [001] directions. Application of uniaxial compressive stress enhances the velocity along the [100] direction in the same way as biaxial tensile strain does. Conversely, applying uniaxial tensile stress results in an enhanced velocity along the [001] direction. For the perpendicular velocity the fitting of the parameters in (4.114) is performed such that the error in $v_\perp$ is minimized. The values of the other parameters for the field directions [101] and [11$\sqrt{2}$] are listed in Table 5.2.1. Fig. 5.15b shows the perpendicular electron velocity ${\mathbf{v}}_\theta$ for field along the [101] direction for increasing stress level as obtained from Monte Carlo simulations. The component $\vec{v}_\theta$, although small for low stress levels, has a significant magnitude for intermediate field regimes and can result in a total velocity different from the parallel velocity.

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

Figure 5.15: (a) Parallel electron velocity component versus field for Si under uniaxial stress (1GPa) along [001] and field along [100] and [001], respectively. (b) Perpendicular velocity versus field for Si under increasing uniaxial stress along [001] and field along [101].

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

Figure 5.16: (a) Parallel (par) and perpendicular (prp) velocity components and total (tot) velocity versus field for Si under $-1$ GPa uniaxial stress along [001] and field along [11$\sqrt{2}$]. (b) Interpolated parallel (par), perpendicular (prp) velocity and the total (tot) velocity versus field for Si under 3GPa uniaxial stress along [001] and field along [111].

5.2.2 Total Velocity for Arbitrary Field Direction

It was found that interpolation of the quantities $\Omega = v_E^2$ and $\Omega = v_3^2$ gives good agreement to the Monte Carlo data. Fig. 5.16a shows a comparison of the velocity components and total velocity for $-1$ GPa stress for field along the [11$\sqrt{2}$] direction, as obtained from Monte Carlo simulations and the analytical model. The results from the model for the sample field direction chosen are in good agreement with the Monte Carlo data and demonstrate the validity of the model.

Fig. 5.16b shows a comparison of the velocity components and the total velocity as obtained from the interpolation and Monte Carlo simulations for field along the [111] direction for uniaxial tensile stressed Si. In this case the perpendicular velocity is negative.