4.4 Bulk Mobility of Strained Ge

The model equations derived in Section 4.3.2 for calculating the mobility in strained Si can also be applied to estimate the mobility in strained Ge. This could be very useful since recently the possibility of utilizing Ge as a channel material [Liu05] [Low04] [Yeo05] is being explored for the next generation CMOS technologies. This interest stems from the significantly higher carrier mobilities in Ge in comparison to Si.

In Ge the conduction band minima consist of four degenerate pairs of $L$-valleys located along the $\langle111\rangle$ directions. Application of strain lifts the degeneracy of the valleys. The valley splitting for the $i^{th}$ valley-pair can be calculated using (3.42) and the mobility tensor can be expressed as

$$\ensuremath{{\underline{\mu}}}_{\ensuremath{{\mathrm{n}}}}^{\text... ...*{1cm} \normalsize i = [111],[\overline{1}11],[11\overline{1}],[1\overline{1}1]$$ (4.83)

Here the scaled inverse mass tensors given by

$$\ensuremath{{\underline{m}}}_{(i)}^{-1} = Z^T \cdot S \cdot Z, \h... ...tyle m_{t}^{-1} & \! 0 0 & \! 0 & \! \displaystyle m_{t}^{-1} \end{pmatrix}.$$ (4.84)

The masses are $m_t = 0.081 \cdot m_0$ and $m_l = 1.88 \cdot m_0$ and the transformation matrices are given as,

$$Z_{[111]} = \begin{pmatrix}\frac{1}{\sqrt{3}} & \! \frac{1}{\sqrt... ...\sqrt{6}} & \! \frac{-1}{\sqrt{6}} & \! \frac{\sqrt{2}}{\sqrt{3}} \end{pmatrix}$$ (4.85)

$$Z_{[11\overline{1}]} = \begin{pmatrix}\frac{-1}{\sqrt{3}} & \! \f... ...{\sqrt{6}} & \! \frac{1}{\sqrt{6}} & \! \frac{\sqrt{2}}{\sqrt{3}} \end{pmatrix}$$ (4.86)

For uniaxial compressive strain along the $[111]$ direction, the valley pairs located along the $[111]$ direction ($L_1$) are lowered in energy, while the remaining three valley pairs ($L_2$, $L_3$, $L_4$) move up in energy and remain degenerate. By this effect, the transport mass in the (111) plane is lowered and inter-valley phonon scattering is reduced, which results in a mobility enhancement.

The temperature dependence of the mobility for the strained case can be fit using a power law expression.

$$\mu = \mu_{300} \left(300/T\right)^\alpha$$ (4.87)

Here $\mu_{300}$ is the bulk mobility at 300K and $\alpha$ is a parameter. The temperature dependence is introduced into the analytical model through the enhancement factor $f$ as

$$f = f_{300} \left(300/T\right)^\beta$$ (4.88)

where $f_{300}$ is the mobility enhancement in unstrained Ge at 300K. The lattice temperature also affects the mobility through the inter-valley scattering rate (4.61) and the valley populations (4.39).