Subsections

• 4.3.1 Piezoresistance Model
• 4.3.2 Physically Based Mobility Model for Strained Si
  • 4.3.2.1 Electron Mobility in Unstrained and Undoped Si
  • 4.3.2.2 Electron mobility in Strained and Undoped Si
  • 4.3.2.3 Electron Mobility in Unstrained and Doped Si
  • 4.3.2.4 Electron Mobility in Strained and Doped Si
  
  
• 4.3.3 Doping Dependence
• 4.3.4 Mobility under Shear Stress

4.3 Bulk Mobility of Strained Si

For low values of the electric fields, the carriers are almost in equilibrium with the lattice vibrations and the low-field mobility is mainly affected by phonon and Coulomb scattering. For device simulation purposes, several mobility models have been suggested for unstrained Si which are mostly semi-empirical in nature. Established models are due to Caughey and Thomas  [Caughey67], Arora [Arora82], and Klaassen [Klaassen92] who suggested a unified low-field mobility model.

As was discussed in Section 3.3.3, the presence of mechanical strain modifies the relative position of the different valleys in the conduction band. A first order estimate of these effects on the mobility can be obtained using the piezoresistance model.

4.3.1 Piezoresistance Model

Piezoresistivity is the phenomena referring to the coupling between electrical conductivity (resistivity) and mechanical stress. Fig. (4.2) shows how the resistance of a n-type Si sample varies with hydrostatic stress. The resistance decreases linearly with stress until 20GPa. This change in resistance is related to a change in the resistivity, $\rho$ and can be expressed as

$$\frac{\Delta R}{R} = \frac{\Delta \rho}{\rho} \propto \mathrm{Stress}$$ (4.33)

Figure 4.2: Resistance of n-type Si sample as a function of the hydrostatic pressure. Figure adapted from [Seeger88](Fig.4.26).

The variation of the mobility with applied stress can be obtained from the piezoresistive coefficients originally measured by Smith [Smith54]. In the presence of stress, the conductivity of the unstrained semiconductor,

$$\sigma_n = \sum_{i=1}^{3}\mathrm{q} n^{(i)} \mu_n^{(i)} ,$$ (4.34)

gets modified to

$$\sigma_n = \sum_{i=1}^{3}\mathrm{q} \Delta n^{(i)} \mu_n^{(i)} + \sum_{i=1}^{3}\mathrm{q} n^{(i)} \Delta \mu_n^{(i)}$$ (4.35)

Here $n^{(i)}$ and $\mu_n^{(i)}$ denote respectively, the carrier concentration and mobility in the $i^{th}$ valley pair in strained Si. Assuming a doped semiconductor, ( $\Delta n=0$), the change in conductivity is given as [Manku93a].

$$\frac{\Delta\sigma_{nij}} {\sigma_{n0}} = \frac{\Delta\mu_{nij}} {\mu_{n0}} = -\sum_{k,l=1}^{3} {\pi_{ijkl} T_{kl}}.$$ (4.36)

Here $T_{kl}$ are the components of the strain tensor. The quantity $\pi_{ijkl}$ is a fourth rank tensor of piezoresistance coefficients which are a measure of the first order change in normalized resistivity per unit applied stress for different stress directions. It has 81 elements which upon the application of the point group symmetry operations reduces to only 3 piezoresistance coefficients, $\pi_{11}$,$\pi_{12}$ and $\pi_{44}$. Table 4.3.1 lists the values of these piezoresistance coefficients for n-type and p-type Si. Using the contracted notation shown in (3.16), the change in mobility can be expressed as

	$\pi_{11}$	$\pi_{12}$	$\pi_{44}$
p-type Si	$6.6$	$-1.1$	$138.1$
n-type Si	$-102.2$	$53.4$	$-13.6$

Table 4.1: Measured values of piezoresistance coefficients taken from Smith [Smith54]

$$\frac{1}{\mu} \begin{pmatrix}\Delta \mu_1 \Delta \mu_2 \Del... ...matrix} \begin{pmatrix}T_1 T_2 T_3 T_4 T_5 T_6 \end{pmatrix}.$$ (4.37)

4.3.2 Physically Based Mobility Model for Strained Si

To develop an electron mobility model for Si under different strain conditions, the relative electron population of the different valleys in Si have to be properly considered [Manku92,Egley93].

As suggested in [Manku92], the anisotropic electron mobility in strained Si can be computed as the weighted average of the unstrained electron mobility tensor, $\ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,uns}}}^{(i)}$, of the $i^{th}$ conduction band valley pair in Si with the corresponding electron population, $p^{(i)}$ in the $i^{th}$ pair,

$$\displaystyle\ensuremath{{\underline{\mu}}}_{\ensuremath{{\mathrm... ...{(i)} \cdot \ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,uns}}}^{(i)}.$$ (4.38)

The relative populations of each valley pair is given by

$$p^{(i)} = \frac{n_\ensuremath{{\mathrm{str}}}^{(i)} } {\displaystyle\sum_{i=1}^{3}n_\ensuremath{{\mathrm{str}}}^{(i)}}$$ (4.39)

where $n_\ensuremath{{\mathrm{str}}}^{(i)}$ is calculated for non-degenerate doping concentrations using Boltzmann statistics with $N_{C}^{(i)}$ as the effective density of states.

$$n_\ensuremath{{\mathrm{str}}}^{(i)} = N_C^{(i)} \cdot\exp \left[\frac{\Delta{\epsilon _C^{(i)}} }{ k_{B}T }\right]$$ (4.40)

The $\Delta{\epsilon _{C}^{(i)}}$ denote the strain-induced energy shifts which can be computed from deformation potential theory discussed in Section 3.3.1.1. The $\mu_\mathrm{n,uns}^{(i)}$ refer to the unstrained electron mobilities in the $i^{th}$ valley pair.

$$\ensuremath{{\underline{\mu}}}_\mathrm{n,uns}^x = \begin{pmatrix}... ...0 \displaystyle 0 &\mu_t & 0 \displaystyle 0 & 0 & \mu_l \end{pmatrix}$$ (4.41)

Here $\mu_l$ and $\mu_t$ denote the mobilities along the major and minor axes in each ellipsoid, respectively. The $\ensuremath{{\underline{\mu}}}^{(i)}_\ensuremath{{\mathrm{n,uns}}}$ denote the electron mobility tensors for Si for the [100], [010], and [001] valley pairs corresponding to directions $x$, $y$, and $z$, respectively. Using  (4.38) to (4.41), the in-plane (x-component) and perpendicular components (z-component) of the electron mobility in strained Si on a (001) SiGe substrate can be expressed as

$$\mu_{\parallel} = \frac{ (\mu_l + \mu_t)\exp{[-\Delta \epsilon ^x... ...BT)]} }{2\exp{[-\Delta \epsilon ^x/(k_BT)]+\exp{[-\Delta \epsilon ^z/(k_BT)]}}}$$ (4.42)

$$\mu_{\perp} = \frac{ 2\mu_t \exp{[-\Delta \epsilon ^x/(k_BT)]} + ... ...BT)]} }{2\exp{[-\Delta \epsilon ^x/(k_BT)]+\exp{[-\Delta \epsilon ^z/(k_BT)]}}}$$ (4.43)

In arriving at (4.42) and (4.43), the relation $\Delta \epsilon ^x = \Delta \epsilon ^y$ has been used which is a consequence of the biaxial tensile strain resulting from growing Si on SiGe (see Section 3.3.3.1). The unstrained mobility can be obtained by setting $\Delta \epsilon ^x = \Delta \epsilon ^y = \Delta \epsilon ^z = 0$.

$$\mu_\mathrm{uns} = \frac{2\mu_t + \mu_l}{3}.$$ (4.44)

Figure 4.3: In-plane and perpendicular electron mobilities in undoped strained Si versus the Ge content in the [001] oriented SiGe buffer layer calculated using (4.42) and (4.43). Also shown is the mobility obtained from piezoresistance coefficients [Smith54] [Kanda82] as described in Section 4.3.1 and from Monte Carlo simulations.

Equations (4.42) and (4.43) represent the model in [Manku92]. The resulting in-plane and out-of-plane electron mobilities in a strained Si layer as a function of the Ge content $y$ in the SiGe (001) substrate is shown in Fig. 4.3. The model reproduces the linear increase (decrease) in the in-plane (out-of-plane) electron mobility component for low strain followed by mobility saturation for high strain levels. However, the model shows a too high value of the unstrained mobility if the saturation mobility values are fixed. This is due to the fact that it does not consider the effect of inter-valley scattering which is present in unstrained Si and results in a lower mobility. The model in (4.42) and (4.43) uses only the two parameters, $\mu_t$ and $\mu_l$, with which it is not possible to match three quantities simultaneously such as the unstrained, and the strained in-plane and perpendicular electron mobilities.

To improve the electron mobility model for strained Si, the effect of inter-valley scattering is included. Equation (4.38) is modified as follows

$$\displaystyle\ensuremath{{\underline{\mu}}}_{\ensuremath{{\mathrm... ...{(i)} \cdot \ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,str}}}^{(i)}.$$ (4.45)

Here the $\ensuremath{{\underline{\mu}}}^{(i)}_{n,str}$ denote the electron mobility tensors of strained Si for the [100], [010], and [001] valleys pairs. In (4.45) a mobility tensor is modeled as a product of a scalar mobility and the scaled inverse mass tensor.

$$\ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,str}}}^{(i)} = \mu \cdot \ensuremath{{\underline{m}}}_{(i)}^{-1}, \hspace{5mm}i = x, y, z$$ (4.46)

The scaled inverse effective mass tensors for the $x$, $y$ and $z$ directions are given as

$$\ensuremath{{\underline{m}}}_{x}^{-1} = \begin{pmatrix}\displayst... ...ac{m_{c}}{m_{t}} & 0 0 & 0 & \displaystyle \frac{m_{c}}{m_{l}} \end{pmatrix}$$ (4.47)

with $m_{t}$ and $m_{l}$ denoting the transversal and longitudinal masses, respectively for the ellipsoidal $\Delta$-valleys in Si. The mass tensors are scaled to a dimensionless form by the conductivity mass $m_{c}$

$$\displaystyle m_{c} = \frac{3}{\displaystyle\frac{2}{m_{t}} + \displaystyle\frac{1}{m_{l}}}.$$ (4.48)

From this scaling it follows that $\ensuremath{{\underline{m}}}_{x}^{-1} + \ensuremath{{\underline{m}}}_{y}^{-1} + \ensuremath{{\underline{m}}}_{z}^{-1} = 3I$, where $I$ denotes the identity matrix. The scalar mobility $\mu$ includes the dependences on the energy shifts $\Delta{\epsilon _C^{(i)}}$ and the doping concentration $N_{I}$ of the strained Si layer.

$$\mu(N_{I},\Delta{\epsilon _C^{(i)}})=\displaystyle\frac{e}{m_{c}\... ...u_{\text{neq}}(\Delta{\epsilon _C^{(i)}})} + \frac{1}{\tau_{I}(N_{I})} \right)}$$ (4.49)

In (4.49) the following momentum relaxation times are assumed:

1. $$\tau_{\text{equiv}}$$ for acoustic intra-valley scattering and inter-valley scattering between equivalent valleys ($g$-type).
2. $$\tau_{\text{neq}}(\Delta{\epsilon _C^{(i)}})$$ for inter-valley scattering between non-equivalent valleys ($f$-type scattering).
3. $$\tau_{I}(N_{I})$$ for impurity scattering.

Figure 4.4: Inter and intra valley transitions within the 3 valley pairs in Si.

Fig.  illustrates the f-type and g-type scattering mechanisms. The effect of the different scattering mechanisms on the total mobility is estimated by Matthiessen's rule, see (4.49). To arrive at a formal description of the mobility components in strained Si in terms of measurable macroscopic quantities, the following cases are considered.

4.3.2.1 Electron Mobility in Unstrained and Undoped Si

The scalar electron mobility for unstrained and undoped Si, also referred to as lattice mobility $\mu^{L}$, can be derived by dropping the impurity scattering rate in (4.49).

$$\ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,uns}}} = \f... ...e\left(\frac{1}{\tau_{\text{equiv}}} + \frac{1}{\tau_{\text{neq}}^{0}} \right)}$$ (4.50)

Here $\tau_{\text{neq}}^0$ denotes the inter-valley relaxation time for $f$-type phonon scattering in unstrained Si. Since in the unstrained case all three valley pairs are equally populated, we have $p^{x} = p^{y} = p^{z} = 1/3$. Using (4.49) the total unstrained mobility can then be written as

$$\ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,uns}}} = \f... ...suremath{{\underline{m}}}_{(i)}^{-1} = \mu^{L}\cdot\ensuremath{{\underline{I}}}$$ (4.51)

Note that the sum evaluates to the identity matrix $\ensuremath{{\underline{I}}}$.

4.3.2.2 Electron mobility in Strained and Undoped Si

When dealing with in-plane biaxial tensile strain, the 6-fold degenerate $\Delta_6$-valleys in Si are split into 2-fold degenerate $\Delta _2$ valleys (lower in energy) and 4-fold degenerate $\Delta_4$ valleys (higher in energy) with electrons preferentially occupying the lower energy levels. Under such conditions, the electron mobilities in the in-plane and perpendicular directions saturate. The saturation values can be obtained from (4.45) by setting $\tau_{\text{neq}}^{-1}(\Delta{\epsilon _C^{(i)}}) = \tau^{-1}_{I}(N_{I}) = 0$.

$$\mu_{t}^{\text{sat}} = \displaystyle\frac{e\cdot\tau_{\text{equiv... ...} \mu_{l}^{\text{sat}} = \displaystyle\frac{e\cdot\tau_{\text{equiv}}} {m_{l}}.$$ (4.52)

Here it is assumed that the strain-induced valley splitting is large enough such that the lowest valley is fully populated and the inter-valley scattering to higher valleys is suppressed.

The ratio of the fully-strained mobility $\mu_{t}^{\text{sat}}$ to the unstrained mobility $\mu^L$ defines the mobility enhancement factor

$$f = \frac{\mu_{t}^{\text{sat}}}{\mu^L} = \frac{m_{c}}{m_{t}}\left(1 + \frac{\tau_{\text{equiv}}}{\tau_{\text{neq}}^0}\right).$$ (4.53)

4.3.2.3 Electron Mobility in Unstrained and Doped Si

In analogy with (4.49) the electron mobility for unstrained Si with doping concentration $N_{I}$ can be written as

$${\mu}^{LI} = \displaystyle\frac{e}{m_{c}\displaystyle\left(\frac{... ...)}\cdot\ensuremath{{\underline{I}}} = \mu^{LI}\cdot\ensuremath{{\underline{I}}}$$ (4.54)

where ${\mu}^{LI}$ signifies the lattice mobility including the effect of impurity scattering. Multiplying the RHS of (4.54) with $\tau_{\text{equiv}}$ gives

$${\mu}^{LI} = \displaystyle\frac{e\cdot \tau_{\text{equiv}}}{m_{c}... ...}}{\tau_{\text{neq}}^0} + \frac{\tau_{\text{equiv}}}{\tau_{I}(N_{I})} \right)}.$$ (4.55)

Rearranging (4.55), we can express the ratio $$\tau_{\text{equiv}}/\tau_{I}(N_{I})$$ as

$$\frac{\tau_{\text{equiv}}}{\tau_{N_{I}}} = \frac{e\tau_{\text{equ... ...{1}{\mu^{LI}} - \left(1+\frac{\tau_{\text{equiv}}}{\tau_{\text{neq}}^0}\right).$$ (4.56)

From (4.51), the lattice mobility $\mu^L$ can be rewritten as

$$\mu^L = \frac{e\cdot \tau_{\text{equiv}}}{m_{c}\displaystyle\left(1 + \frac{\tau_{\text{equiv}}}{\tau_{\text{neq}}^{0}} \right)}.$$ (4.57)

Substituting the value of $$\frac{e\cdot \tau_{\text{equiv}}} {m_{c}}$$ from (4.57) into (4.56) gives

$$\frac{\tau_{\text{equiv}}}{\tau_{N_{I}}} = \left(\frac{\mu^L}{\mu... ...^0}\right) = \left(\frac{\mu^L}{\mu^{LI}} - 1\right)\frac{m_{t}}{m_{c}}\cdot f.$$ (4.58)

The last relation in (4.58) is obtained using (4.53).

4.3.2.4 Electron Mobility in Strained and Doped Si

The inter-valley scattering rate is a function of the strain-induced splitting of the valleys. It can be expressed by using a dimensionless factor $h^{(i)}$, such that

$$\frac{1}{\tau_{\text{neq}}^{(i)}} = \frac{h^{(i)}}{\tau_{\text{neq}}^0}.$$ (4.59)

In strained Si, the total rate for electrons to scatter from initial valley $i$ to final valleys $j$ and $l$ is given by

$$\frac{1}{\tau_{\text{neq}}^{(i)}} = \frac{1}{\tau_{(i\to j)}} + \frac{1}{\tau_{(i \to l)}}$$ (4.60)

For low electric fields an equilibrium distribution function can be assumed and $\tau_{(i \to j)}$ can be calculated [Conwell67]:

$$\hspace{1mm} \frac{1}{\tau_{(i \to j)}} \hspace{1mm}= \frac{\disp... ...\epsilon} {\displaystyle\int_{0}^{\infty}\displaystyle f(\epsilon ) d\epsilon},$$ (4.61)

with the inter-valley scattering rate $S$ defined as

$$S(\epsilon,\Delta_{ij}) =C\cdot \left[(\epsilon - \Delta_{ij}^{\t... ...{\text{opt}}}{k_{B}T}\right)(\epsilon - \Delta_{ij}^{\text{abs}})^{1/2} \right]$$ (4.62)

$$\Delta_{ij}^{\text{emi}} =\Delta{\epsilon _C^{(j)}} - \Delta{\epsilon _C^{(i)}} - \hbar\omega_{\text{opt}}$$ (4.63)

$$\Delta_{ij}^{\text{abs}} =\Delta{\epsilon _C^{(j)}} - \Delta{\epsilon _C^{(i)}} + \hbar\omega_{\text{opt}}$$ (4.64)

and the Boltzmann distribution function $f(\epsilon )$.

$$f(\epsilon ) = \exp{\left(\frac{-\epsilon }{k_BT}\right)}$$ (4.65)

Here $\hbar\omega_{\text{opt}}$ denotes the phonon energy and $\Delta{\epsilon _C^{(i)}}$ the strain induced splitting, and C is a constant. A more accurate formalism for calculating the scattering rate would be through the incorporation of the density of states $D(\epsilon )$ into (4.61), giving [Seeger88]

$$\hspace{1mm} \frac{1}{\tau_{(i \to j)}} \hspace{1mm}=\frac{\displ... ...isplaystyle\int_{0}^{\infty}\displaystyle f(\epsilon ) D(\epsilon ) d\epsilon}.$$ (4.66)

The calculation of the scattering rate using (4.61) and (4.66) is shown in Appendix A. For our modeling purpose, the definition in (4.61) leading to simpler expressions was found to be sufficient.

Using these expressions, $\tau_{\text{neq}}^{(i)}$ can be expressed as

$$\frac{1}{\tau_{\text{neq}}^{(i)}} = C \left[ g\left(\displaysty... ...(\displaystyle \frac{ \Delta_{ij}^{\text{abs}}}{k_{B}T}\right)\right ] \right].$$ (4.67)

The function $g$ is defined as

$$g(z) = \left\{ \begin{array}{ll} \exp(-z)\cdot\Gamma(\frac{3}{2})... ...a(\frac{3}{2},-z) \hspace{2mm} &\forall \hspace{2mm} z < 0. \end{array} \right.$$ (4.68)

Here $\Gamma(\frac{3}{2}) = \frac{\displaystyle \sqrt\pi}{2}$ and $\Gamma(\frac{3}{2},-z)$ denotes the incomplete Gamma function. In the unstrained case

$$\Delta_{ij}^{\text{emi}} = - \hbar\omega_{\text{opt}}, \qquad \Delta_{ij}^{\text{abs}} =+ \hbar\omega_{\text{opt}},$$ (4.69)

and therefore the unstrained inter-valley relaxation time $\tau_{\text{neq}}^{0}$ can be obtained as

$$\frac{1}{\tau_{\text{neq}}^{0}} = 2 C \left[g\left(\displaysty... ...{opt}}} {k_{B}T}\right) + \displaystyle \Gamma\left(\frac{3}{2}\right) \right].$$ (4.70)

The factor $h^{(i)}$ in (4.59) is thus determined from (4.67) and (4.70).

Multiplying the RHS of (4.49) with $\tau_{\text{equiv}}$ gives

$$\mu^{(i)} = \displaystyle\frac{e\cdot \tau_{\text{equiv}}}{m_{c}\... ...\tau_{\text{neq}}^{(i)}} + \frac{\tau_{\text{equiv}}}{\tau_{I}(N_{I})} \right)}$$ (4.71)

Using the relations in (4.53), (4.57) and (4.58), the electron mobility for the $i^{th}$ valley in strained Si can be written as

$$\ensuremath{{\underline{\mu}}}_\ensuremath{{\mathrm{n,str}}}^{(i)... ...(\frac{\mu^{L}}{\mu^{LI}}-1\right)}\cdot\ensuremath{{\underline{m}}}_{(i)}^{-1}$$ (4.72)

where $\ensuremath{{\underline{m}}}_{(i)}^{-1}$ denotes the scaled effective mass tensor for the $i^{th}$ valley pair in (4.47) and $\beta = \displaystyle {f \cdot m_{t}/{m_{c}} }$. Equation (4.72) is plugged into (4.45) to give the total mobility tensor for electrons in strained Si as a function of doping concentration $N_{I}$ and strain. The tensor in (4.72) is given in the principal coordinate system and has diagonal form.

4.3.3 Doping Dependence

The doping and strain dependence of the in-plane and perpendicular electron mobilities in Si is calculated using (4.72) by adopting any suitable expression describing the doping dependence in unstrained Si. A model distinguishing between the doping dependence, $\mu^{\mathrm{LI}}$ of the majority and minority electrons in $\mathrm{Si}$ has been suggested in [Palankovski04,Kosina98]

$$\mu^{\mathrm{LI}}_{\ensuremath{{\mathrm{n}}},\ensuremath{{\mathrm... ...hrm{min}}}}\right)}^{\lambda}}} + \mu^{\mathrm{hi}}_\ensuremath{{\mathrm{min}}}$$ (4.73)

$$\mu^{\mathrm{LI}}_{\ensuremath{{\mathrm{n}}},\ensuremath{{\mathrm... ...rm{maj}}}}\right)}^{\lambda}}} + \mu^{\mathrm{hi}}_\ensuremath{{\mathrm{maj}}},$$ (4.74)

where, $\mu^{\mathrm{L}}_\ensuremath{{\mathrm{n}}}$ is the mobility for the undoped material, $\mu^{\mathrm{hi}}$ is the mobility at the highest doping. All other parameters are used as fitting parameters. Although initially proposed for the majority electron mobility in Si, equation (4.74) offers enough flexibility to model also the minority electron mobility in Si. The difference between majority and minority electron mobilities [Masetti83] is caused by effects such as degeneracy and the different screening behavior of electrons and holes in the semiconductor. Equation (4.74) describes a mathematical function with two extreme values and can deliver a second maximum or minimum at very high doping concentrations depending on the sign of $\mu_1$. Thus, it allows both majority and minority carrier mobilities to be properly modeled.

4.3.4 Mobility under Shear Stress

The effect of shear stress on the band structure was discussed in Section 3.3.4. For tensile stress along [110] the energy dispersion of the lowest conduction band is influenced as follows:

1. The $\Delta_4$ valleys located along the $[100]$ and $[010]$ directions move up in energy with respect to the $\Delta _2$ valleys located along the $[001]$ direction.

2. The shape of the valleys located along the $[001]$ direction is distorted which results in a variation of the effective masses.

3. The band minima of the $\Delta _2$ valley pair along the $[001]$ direction move towards the zone boundary $X$ points, ${\mathbf{k}}_X =\frac{2\pi}{a_0} (0,0,\pm1)$.

The model presented in Section 4.3.2 relies on a) a model for the momentum relaxation time, b) the relative populations of the different valley pairs as a result of the energy shifts, and c) an effective mass tensor (with constant $m_t$ and $m_l$), which basically provides the tensorial description to the mobility. However, as discussed in Section 3.3.4, in the presence of a uniaxial tensile stress along $\langle 110 \rangle$ the two-fold degenerate $\Delta _2$-valleys which are lowered in energy experience a change in the effective masses.

For the calculation of the mobilities in the presence of shear strain, the effective mass tensors in (4.47) have to be modified. Using (3.35) the energy dispersion relation for the lower lying $\Delta _2$ valleys for stress along the $[110]$ direction can be rewritten as

$$\epsilon({\mathbf{k}}) = \frac{\hbar^2}{2}\left( \frac{k_{[110]}^... ...ert}} + \frac{k_{[\overline{1}10]}^2} {m_{t\perp}} + \frac{k_z^2}{m_l}. \right)$$ (4.75)

Utilizing the transformation (3.66), the dispersion relation in the principal coordinate system ($x, y, z$) modifies to

$$\epsilon({\mathbf{k}}) = \frac{\hbar^2}{2}\left( k_x^2\left[\frac... ...rac{1}{m_{t\vert\vert}}-\frac{1}{m_{t\perp}}\right] + \frac{k_z^2}{m_l} \right)$$ (4.76)

which can be expressed as

$$\epsilon(k) = \frac{\hbar^2}{2} {\mathbf{k}}^T \cdot \ensuremath{{\underline{m}}}^{-1} \cdot {\mathbf{k}}.$$ (4.77)

Here ${\mathbf{k}} = (k_x,k_y,k_z)$ and the matrix $\ensuremath{{\underline{m}}}^{-1}$ denotes the inverse mass tensor,

$$\ensuremath{{\underline{m}}}_{z}^{-1} = m_c \begin{pmatrix}\displ... ..._{\Delta}^{-1} & \! m_{t}^{-1} & \! 0 0 & \! 0 & \! m_{l}^{-1} \end{pmatrix}$$ (4.78)

$$m_t^{-1} = \displaystyle \frac{m_{t\parallel }^{-1} + m_{t\perp }^{-1}}{2}$$ (4.79)

$$m_\Delta^{-1} = \displaystyle \frac{m_{t\parallel }^{-1} - m_{t\perp }^{-1}}{2}$$ (4.80)

$$m_{c}^{-1} = \displaystyle \frac{m_{t\parallel}^{-1} + m_{t\perp}^{-1}+m_{l}^{-1}} {3}.$$ (4.81)

The transversal masses along the $[110]$ and $[\overline{1}10]$ directions of the $\Delta _2$-valleys are defined by $m_{t\parallel}$ and $m_{t\perp}$, respectively. The variation of these masses can be expressed as a function of strain as described in (3.73) and (3.74). Substituting (4.78) into (4.72) gives the mobility tensor which now has a non-diagonal form in the principal coordinate system.

The scaled inverse mass tensor in (4.78) was derived assuming only $\varepsilon _{xy}$ to be non-zero. For non zero values of the $\varepsilon _\ensuremath{{\mathrm{yz}}}$ and $\varepsilon _\ensuremath{{\mathrm{xz}}}$ shear strain components, the scaled inverse mass tensor in (4.78) can be permutated to obtain the scaled inverse mass tensors for the $x$ and $y$ valleys as

$$\ensuremath{{\underline{m}}}_{x}^{-1} = m_c \begin{pmatrix}\displ... ... 0 & \! m_{t}^{-1} & 0 m_{\Delta}^{-1} & \! 0 & \! m_{l}^{-1} \end{pmatrix}.$$ (4.82)