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Next: 4. Mobility Modeling Up: 3. Strain Effects on Previous: 3.2 Structure of Relaxed


3.3 Effect of Strain

A comprehensive study of the effect of strain on the band structure has been performed Bir and Pikus [Bir74]. A direct result of straining the crystal is the transformation of the unstrained lattice vectors ($ {x_i}$) to the strained lattice vectors ( $ {x_{i}^{'}}$), through the relation

$\displaystyle x_i^{\prime j} = x_i^j + \sum_j \varepsilon_{ij} x_i^j,$ (3.39)

where $ \varepsilon _{ij}$ denote the strain tensor components. The effect of strain can be incorporated into $ {\bf {k\cdot p}}$ band structure calculations by introducing an additional perturbation term into the unstrained potential [Manku93b]. For the empirical pseudopotential method, an interpolation of the pseudopotential form factors is required. Moreover, in the presence of an arbitrary strain condition, there occurs a movement of the vertex atoms as well as the central atom, which results in an ambiguity in the exact location of the central atom in the bulk strained primitive unit cell. This effect can be captured by taking into account an additional internal displacement parameter [de Walle86].


Figure 3.7: The concept of pseudopotential where the true potential $ (V(r)$ and the true wave function $ \Psi$ are replaced by a pseudopotential $ V_g$ and a pseudo wave function $ \psi$.

The effect of strain on the conductivity of Si was first investigated by Smith [Smith54]. The principal finding of his experimental work was the observation of a change in the Si resistivity on applying uniaxial tensile stress. This change occurs due to a modification of the electronic band structure. Microscopically, the modification stems from a reduction in the number of symmetry operations allowed, which in turn depends on the way the crystal is stressed. This breaking of the symmetry of the fcc lattice of Si can result in a shift in the energy levels of the different conduction and valence bands, their distortion, removal of degeneracy, or any combination. In the following sections these effects are discussed in detail.

3.3.1 Strain-induced Energy Splitting

Strain induces a shift in the energy levels of the unstrained conduction and valence bands. While hydrostatic strain merely shifts the energy levels of a band, uniaxial and biaxial strain removes the band degeneracy. These energy shifts can either be extracted from the full-band structure calculated numerically including strain, or obtained analytically using the linear deformation potential theory. Deformation Potential Theory

Deformation potential theory originally developed by Bardeen and Shockley [Bardeen50] was used to investigate the interaction of electrons with acoustic phonons. It was later generalized to include different scattering modes by Herring and Vogt [Herring56]. The technique was applied to strained material systems by Bir and Pikus [Bir74].

Within the framework of this theory, the energy shift of a band extremum $ l$ is expanded in terms of the components of the strain tensor $ \varepsilon _{ij}$.

$\displaystyle \Delta \epsilon^{(l)} = \sum_{ij} \Xi_{ij}^{(l)} \varepsilon_{ij}$ (3.40)

The tensor quantity $ \Xi_{ij}$ is called the deformation potential. The symmetry of the strain tensor is also reflected in that of the deformation potential tensor, giving

$\displaystyle \Xi_{ij}^{(l)} = \Xi_{ji}^{(l)}.$ (3.41)

The maximum number of independent components of this tensor is six which is reduced to two or three for a cubic lattice. These are usually denoted by $ \Xi_{u}$, the uniaxial deformation potential constant, and $ \Xi_{d}$, the dilatation deformation potential constant. The deformation potential constants can be calculated using theoretical techniques such as density functional theory [de Walle86], the non-local empirical pseudopotential method [Fischetti96], or ab-initio calculations. However, a final adjustment of the potentials is obtained only after comparing the calculated values with those obtained from measurement techniques.

The general form of the strain-induced energy shifts of the conduction band valleys for an arbitrary strain tensor can be written as

$\displaystyle \hspace*{0mm}\Delta \epsilon_{C}^{(i,j)} = \Xi_{d}^{(j)}\hspace{1...
...remath{{\underline{\varepsilon}}}\cdot{\bf {a}}_i = \Delta^{(d)} + \Delta^{(u)}$ (3.42)

where $ {\bf {a}}_i$ is a unit vector of the $ i^{\ensuremath{{\mathrm{th}}}}$ valley minimum for the $ j^{\ensuremath{{\mathrm{th}}}}$ ($ j = X,L$) valley type. The first term, $ \Delta^{(d)}$ in (3.42) shifts the energy level of all the valleys equally and is proportional to the hydrostatic strain, $ Tr(\varepsilon ) = (\varepsilon_{xx} +
\varepsilon_{yy} + \varepsilon_{zz})$. The difference in the energy levels of the valleys arises from the second term in (3.42). The analytical expressions for this term for different stress directions is listed in Table 3.4.

Direction of Stress
Valleys $ [100]$ $ [110]$ $ [111]$
$ [100]$ $ \varepsilon _{11}$ $ \frac{1}{2}(\varepsilon_{11} + \varepsilon_{12})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$
$ [010]$ $ \varepsilon _{12}$ $ \frac{1}{2}(\varepsilon_{11} + \varepsilon_{12})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$
$ [001]$ $ \varepsilon _{12}$ $ \varepsilon _{12}$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$
$ [111]$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} + \varepsilon_{44})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} + 2\varepsilon_{44})$
$ [11\overline{1}]$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} + \varepsilon_{44})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} - \frac{2}{3}\varepsilon_{44})$
$ [1\overline{1}1]$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} - \varepsilon_{44})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} - \frac{2}{3}\varepsilon_{44})$
$ [\overline{1}11]$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} - \varepsilon_{44})$ $ \frac{1}{3}(\varepsilon_{11} + 2\varepsilon_{12} - \frac{2}{3}\varepsilon_{44})$

Table 3.4: Analytical expressions for stress-induced valley shifts for uniaxial stress to be multiplied by the deformation potential $ \Xi_{u}^{(j)}$.

3.3.2 Valence Band Splitting

Due to the degeneracy of the valence bands, strain induced splitting of the bands cannot be calculated as straight forwardly as for the conduction bands. The energy shifts can be calculated from the $ {\bf {k\cdot p}}$ theory in conjunction with the deformation potential theory. The energy dispersion of the valence bands is obtained from of a perturbation matrix $ \ensuremath{{\underline{S}}}$ and a deformation potential matrix $ \ensuremath{{\underline{D}}}$

$\displaystyle \ensuremath{{\underline{S}}} = \begin{pmatrix}L k_x^2 + M(k_y^2 +...
...k_y k_z  N k_x k_z & N k_y k_z & L k_z^2 + M(k_y^2 + k_x^2)  \end{pmatrix},$ (3.43)

$\displaystyle \ensuremath{{\underline{D}}} = \begin{pmatrix}l \varepsilon_{xx} ...
...& l \varepsilon_{zz} + m(\varepsilon_{xx} + \varepsilon_{yy})  \end{pmatrix}.$ (3.44)

The parameters $ L, M, N$ are related to the Luttinger parameters [Yu03] while $ l, m, n$ denote valence band deformation potentials. The effect of spin-orbit coupling is taken into account by introducing a spin-orbit interaction term, $ \ensuremath{{\underline{H}}}_{so}$, dependent on the unstrained energy level $ \Delta$, of the split off band, as an additional perturbation into the Hamiltonian,

$\displaystyle \ensuremath{{\underline{H}}} = \ensuremath{{\underline{S}}} + \ensuremath{{\underline{D}}} + \ensuremath{{\underline{H}}}_{so}$ (3.45)

After performing a unitary transformation on $ \ensuremath{{\underline{H}}}$ followed by some mathematical manipulations, Manku [Manku93b] arrived at the following form to describe the valence band spectrum.

\begin{gather*}\begin{split}h_{11}'h_{22}'h_{33}' + 2h_{12}'h_{23}'h_{13}' - h_{...
...+ h_{22}'h_{33}' - h_{12}^2 - h_{13}^2 - h_{23}^2 ) = 0.\end{split}\end{gather*} (3.46)

Here the $ h_{ij}$ and the $ h_{ii}'$ are related to the matrices $ \ensuremath{{\underline{S}}}$ and $ \ensuremath{{\underline{D}}}$ through the relation

$\displaystyle h_{ij} = S_{ij} + D_{ij} \hspace*{1cm} h_{ii}' = h_{ii} + \frac{\hbar^2{\bf k} ^2}{2m} - \epsilon _{\bf k} .$ (3.47)

Equation (3.47) can be simplified to

$\displaystyle \sum_{i=0}^{3} \sum_{j=0}^{3-i} a_{ij} \epsilon _{k}^{j} k^{2i} = 0,$ (3.48)

which is a cubic equation in $ \epsilon $. Its solutions give the energies for the HH, LH and the SO bands for a particular k value and arbitrary strain. The components $ a_{ij}$ are functions of the strain tensor. Unlike the work of Bir and Pikus where the spin orbit coupling was neglected, the expression in (3.49) has been derived taking into account this interaction as an additional perturbation. The splitting of the bands can be obtained by setting $ {\bf k} =0$ in (3.49) to give

$\displaystyle \sum_{i=0}^{3} a_{i} \epsilon ^{i} = 0,$ (3.49)

where the coefficients $ a_i$ are given as

$\displaystyle a_0 = \frac{\Delta}{3}(pq + pr + qr -n^2\varepsilon _{T}^{2} + n^...
...z} - p\varepsilon _{yz}^2 - q\varepsilon _{xz}^2 - r\varepsilon _{xy}^2 ) + pqr$ (3.50)
$\displaystyle a_1 = \frac{2\Delta}{3}(p + q + r) + (pq + pr + qr - n^2\varepsilon _{T}^{2})$ (3.51)
$\displaystyle a_2 = p+q+r -\Delta$ (3.52)
$\displaystyle a_3 = -1$ (3.53)
$\displaystyle \varepsilon _{T}^{2} = \varepsilon _{xy}^{2} + \varepsilon _{yz}^{2} +\varepsilon _{xz}^{2}$ (3.54)
$\displaystyle \begin{pmatrix}p  q  r  \end{pmatrix}= \begin{pmatrix}l & m...
...x}\varepsilon _{xx}  \varepsilon _{yy}  \varepsilon _{zz}  \end{pmatrix}.$ (3.55)

3.3.3 Different Stress Configurations

In this section stress configurations are discussed in which the stress is applied along the $ [001]$, $ [110]$ and $ [111]$ directions, as well as an academic case of stress along the $ [123]$ direction. Biaxial Stress: Tetragonal Distortion

For the case of biaxial stress in the (001) plane, the 6-fold degenerate $ \Delta_{6}$-valleys in Si are split into a 2-fold degenerate $ \Delta _2$ valley pair (located along the [001] direction) and a 4-fold degenerate $ \Delta_{4}$ valleys pair. In terms of symmetry considerations, this stress condition is equivalent to applying a uniaxial stress along the [001] direction. The cubic lattice of Si gets distorted to a tetragonal crystal system (right parallelopiped with a square base). The number of symmetry operations for this system is reduced by a factor of 3 compared to the unstrained case.

Biaxial tensile strain obtained by epitaxially growing Si on relaxed SiGe results in a lowering of the $ \Delta _2$ valleys in energy while the $ \Delta_4$ valleys move up in energy. As a result, the following effects become important: a) electron transfer from the high energy $ \Delta_4$ valleys to the low energy $ \Delta _2$ valleys resulting in increased population of the $ \Delta _2$ valleys. This is indicated by the increased size of the $ \Delta _2$ lobes in Fig. 3.8a. b) Reduced probability of electron scattering from $ \Delta _2$ to $ \Delta_4$, and c) the lowered $ \Delta _2$ valleys experience a smaller effective mass, $ m_t$, in the (001) plane.

For the valence bands, the degeneracy of the HH and LH bands at the $ \Gamma$ point is lifted. The top band moves to a lower hole energy and is HH like, while the other band moves higher in energy. A schematic of the band splitting is shown in Fig. 3.8b where the in-plane direction is denoted as $ [100]$ and the out-of-plane direction as $ [001]$. The curvature of the top band is higher in the out-of-plane direction as compared to the direction. Stress along [111]: Trigonal Distortion

Under the application of a uniaxial stress along the [111] direction, the fcc lattice of unstrained Si is modified to a trigonal system (rhombohedron with edges having arbitrary angles). The number of symmetry operations is 12, reduced by a factor 4 compared to the unstrained case. Because this direction coincides with the diagonal of the cube, all the $ \Delta_6$-valleys are symmetrically oriented with respect to this direction. The resulting strain tensor thus has equal magnitudes of the diagonal components (see (3.24)). Shear strain components are also present with $ \varepsilon _{xy} =
\varepsilon _{yz} = \varepsilon _{xz}$. Consequently, the degeneracy of the valleys is not lifted and electrons are populating all the valleys equally. In the valence bands, the energy dispersion in the [100] and [001] directions are the same. The valence band splittings can be seen in Fig. 3.9a.

\includegraphics[width=2.3in, angle= -0]{figures/Splitting_Biax.eps}          \includegraphics[width=2.8in]{figures/}

$\textstyle \parbox{0.47\linewidth}{(a) \hfill (b)}$

Figure 3.8: (a) Conduction and (b) valence band splitting under biaxial tensile strain in the (001) plane.

\includegraphics[width=2.8in]{figures/}          \includegraphics[width=2.8in]{figures/}
$\textstyle \parbox{2.8in}{(a) \hfill (b)}$

Figure 3.9: (a) Valence band splitting under uniaxial tensile strain along (a) the [111] and (b) the [110] direction. Stress along [110]: Orthorhombic Distortion

Applying uniaxial $ [110]$ stress distorts the lattice to a rectangular parallelopiped with a rhombic base. The resulting strain tensor has both diagonal and off-diagonal components, see (3.23). The number of symmetry operations is reduced to a mere 8. For tensile stress along the [110] direction, the strain induced valley splittings in the conduction bands are similar to the biaxial tensile case shown in Fig. 3.8. In the valence bands, the curvature of the top band in the direction is higher than in the out-of-plane direction as shown in Fig 3.9b. Stress along [123]: Triclinic Distortion

Stress along the [123] direction reduces the number of symmetry operations of the lattice to only 2. The resulting crystal structure is triclinic. The strain tensor has all components non-zero with the diagonal components being different. For uniaxial tensile stress, the 3 pairs of $ \Delta _2$ valleys in the conduction band are shifted to different energy levels. Also the curvature of the top valence band is increased in the [100] direction as compared to the [001] direction. The conduction and valence band splittings are shown in Fig. 3.10.

\includegraphics[width=2.1in, angle= -0]{figures/Splitting_123.eps}         \includegraphics[width=2.6in]{figures/}
$\textstyle \parbox{2.4in}{(a) \hfill (b)}$

Figure 3.10: (a) Conduction and (b) valence band splitting under biaxial tensile strain in the [123] direction.

3.3.4 Stress-induced Degeneracy Lifting

The first and second conduction bands are degenerate at the X-point. This coupling of the two bands at the X-point was understood in terms of the X-ray scattering results obtained on the diamond lattice [Bouckaert36]. The effect of strain on the degeneracy of the bands at the X-point was first examined in the theoretical study performed by Bir and Pikus  [Bir74] and later verified experimentally by Hensel [Hensel65] and Laude [Laude71].

For any stress condition which causes the strain tensor to have non-diagonal components, there is a distortion of the band structure and the degeneracy at some of the X-points is lifted. This leads to a change in the electron effective mass which has been detected using cyclotron resonance experiments [Hensel65]. We consider the case in which a uniaxial stress is applied along the [110] direction.

The band splitting at the X-point can be calculated from the solution of the eigenvalue problem stated in [Hensel65].

$\displaystyle \begin{pmatrix}\delta \epsilon _0 & \delta \epsilon _1 \delta \...
...nd{pmatrix} = \delta \epsilon \begin{pmatrix}\xi  \hat{\xi}  \end{pmatrix},$ (3.56)


$\displaystyle \delta \epsilon _0 = \Xi_d (\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}) + \Xi_u \varepsilon_{zz},$ (3.57)
$\displaystyle \delta \epsilon _1 = 2 \Xi_{u'} \varepsilon_{xy}.$ (3.58)

The constant $ \Xi_{u'}$ in (3.59) is a new deformation potential ascribed to the degeneracy lifting at X-point. Two different values of $ \Xi_{u'}$ have been suggested: $ 5.7 \pm$ $ 1$ eV predicted from cyclotron resonance experiments [Hensel65] and $ 7.5\pm2$ eV from indirect exciton spectrum measurements [Laude71]. The energy levels of the two conduction bands are thus given by

$\displaystyle \epsilon _{\Delta_1} = \delta \epsilon _0 - \delta \epsilon _1,$ (3.59)
$\displaystyle \epsilon _{\Delta_{2'}} = \delta \epsilon _0 + \delta \epsilon _1.$ (3.60)

The energy dispersion of the first ($ \Delta_1$) and second ( $ \Delta_{2'}$) conduction band can be determined from the eigenvalues of the Hamiltonian suggested by Bir and Pikus [Bir74],

$\displaystyle \epsilon _{\pm}(\ensuremath{{\underline{\varepsilon {}}}},{\mathb...
...\lambda \pm \sqrt{A_4^2k_z^2 + (2\Xi_{u'} \varepsilon _{xy} + A_3k_x k_y)^2} ,$ (3.61)

where $ \epsilon _{-}$ describes the dispersion of $ \Delta_1$ and $ \epsilon _{+}$ that of $ \Delta_{2'}$, and

$\displaystyle \lambda = A_1k_z^2 + A_2 (k_x^2 + k_y^2) + D_1\varepsilon _{zz} + D_2 (\varepsilon _{xx} + \varepsilon _{yy}).$ (3.62)

The parameters $ A_1$ to $ A_4$ have been obtained in [Ungersboeck07] and are given as

$\displaystyle A_1 = \frac{\hbar^2}{2 \ensuremath{m_\mathrm{l}}} , \hspace*{2mm...
...\hspace*{2mm} \vert A_4\vert = \frac{\hbar^2}{\ensuremath{m_\mathrm{l}}} k_0 .$ (3.63)


$\displaystyle \frac{1}{m'} = \frac{2}{\ensuremath{\mathrm{m}}_0^2} \sum_{n\neq ...
...angle n\vert p_y \vert\Delta_{2'}\rangle }{\epsilon _n-\epsilon _{\Delta_1}} ,$ (3.64)

and $ k_0 = \textstyle{0.15\frac{2\pi}{a_0}}$ denotes the distance of the conduction band minimum of unstrained Si measured from the $ X$ point. By adopting a new primed coordinate system that is rotated by 45$ ^\circ$ with respect to the crystallographic coordinate system,

$\displaystyle k_{x'} = \frac{k_x+k_y}{\sqrt{2}}, k_{y'} = \frac{k_x-k_y}{\sqrt{2}} , k_{z'} = k_{z},$ (3.65)

the energy dispersion (3.62) can be written as

$\displaystyle \epsilon _{\pm}(\hat{\varepsilon {}},{\mathbf{k}}) = \lambda \pm ...
...\left (2\Xi_{u'}\varepsilon _{xy} + \frac{A_3}{2} (k_x^2 - k_y^2)\right )^2} .$ (3.66)

It should be noted that $ \lambda$ is invariant under the transformation described by (3.66). The effective masses in the $ x'=[110]$ and $ y'=[1\bar{1}0]$ direction can be obtained using the relations

$\displaystyle \frac{1}{m_{\mathrm{t},x'}(\varepsilon _{xy})} = \frac{1}{\hbar^2...
...vert}_{\displaystyle {\mathbf{k}} = \frac{2\pi}{a_0} (0,0,k_z^{\mathrm{min}})},$ (3.67)

$\displaystyle \frac{1}{m_{\mathrm{t},y'}(\varepsilon _{xy})} = \frac{1}{\hbar^2...
...vert}_{\displaystyle {\mathbf{k}} = \frac{2\pi}{a_0} (0,0,k_z^{\mathrm{min}})},$ (3.68)

while the longitudinal mass is given by

$\displaystyle \frac{1}{\ensuremath{m_\mathrm{l}}(\varepsilon _{xy})} = \frac{1}...
...rt}_{\displaystyle {\mathbf{k}} = \frac{2\pi}{a_0} (0,0,k_z^{\mathrm{min}})} .$ (3.69)

Here $ k_z^{\mathrm{min}}$ denotes the minimum of the $ \Delta_1$ conduction band and can be obtained from (3.62). Substituting the values of $ A_1$ to $ A_4$ from (3.64) into (3.62) and setting $ k_x =
k_y = 0$, the dispersion relations becomes

$\displaystyle \epsilon _- = \frac{\hbar^2}{2 \ensuremath{m_\mathrm{l}}} k_z^2 -...
...k_0^2}{\ensuremath{m_\mathrm{l}}^2}k_z^2 + (2 \Xi_{u'})^2 \varepsilon _{xy}^2}.$ (3.70)

From $ \textstyle{\frac{\partial \epsilon _-}{\partial k_z} = 0}$, the position of the conduction band minimum $ k_z^{\mathrm{min}}$ is obtained [Ungersboeck07]

$\displaystyle k_z^{\mathrm{min}} = \left\{ \begin{array}{ll} k_0 \sqrt{1 - \eta...
...\eta\vert<1$} 0 & \textnormal{,\quad $\vert\eta\vert>1$} \end{array} \right.,$ (3.71)

where $ \displaystyle \eta = {2\Xi_{u'}\varepsilon _{xy}}/\Delta$. The expression (3.72) reveals that the minimum in the [001] direction moves closer to the X-point. In Fig. 3.11, the impact of shear strain on the shape of the $ \Delta_1$ and $ \Delta_{2'}$ conduction bands is plotted. For $ \vert\eta\vert\ge1$ the position of the minimum is located at the point $ X$, thus $ k_z^{\mathrm{min}}=0$, and remains fixed.

\includegraphics[width=3.0in,angle=0]{figures/XsplittingSchematicsZeroStrain.eps}          \includegraphics[width=3.0in,angle=0]{figures/XsplittingSchematicsLargeStrain2.eps}
$\textstyle \parbox{3.0in}{(a)\hfill(b)}$

Figure 3.11: Splitting of the $ \Delta_1$ and $ \Delta_{2'}$ conduction bands for stress along the [110] direction. $ k_z^{\text{min}}$ denotes the minimum of the conduction band. (a) Unstrained ( $ k_z^{\text{min}} = 0.85  2\pi/a $ ) and (b) Strained ( $ k_z^{\text{min}} \ne 0.85  2\pi/a$).

Evaluating the derivatives in (3.68) to (3.70), the strain dependence of the transversal and longitudinal masses is obtained as

$\displaystyle m_{\mathrm{t},x'}(\varepsilon _{xy}) = \left\{ \begin{array}{ll} ...
...t}{M} \Bigr)^{-1} & \textnormal{,\quad $\vert\eta\vert>1$} \end{array} \right .$ (3.72)

in the $ [110]$ direction,

$\displaystyle m_{\mathrm{t},y'}(\varepsilon _{xy}) = \left\{ \begin{array}{ll} ...
...t}{M} \Bigr)^{-1} & \textnormal{,\quad $\vert\eta\vert>1$} \end{array} \right..$ (3.73)

in $ [1\bar{1}0]$ direction, and

$\displaystyle \ensuremath{m_\mathrm{l}}(\varepsilon _{xy}) = \left\{ \begin{arr...
...vert} \Bigr )^{-1} \textnormal{,\quad $\vert\eta\vert>1$} \end{array} \right ..$ (3.74)

for the longitudinal mass along the $ [001]$ direction. Here, $ \mathrm{sgn}$ denotes the signum function and $ M \approx m_t/(1 - m_t/m_0)$. This modification of the band structure translates into a change in the shape of the constant energy surfaces. The constant energy surfaces of unstrained Si having a prolate ellipsoidal shape are now deformed to a scalene ellipsoidal shape, characterized by the masses $ m_{t,x'}$ and $ m_{t,y'}$. As can be seen from Fig. 3.12, the mass $ m_{t,x'}$ along the stress direction is reduced whereas $ m_{t,y'}$ perpendicular to the stress direction is increased.

\includegraphics[width=2.5in,angle=0]{figures/mt_circle.eps}      \includegraphics[width=2.5in,angle=0]{figures/mt_ellipse.eps}
$\textstyle \parbox{2.5in}{(a)\hfill (b)}$

Figure 3.12: Top view of the constant energy surfaces of the $ \Delta _2$-valleys in (a) unstrained (prolate ellipsoid) and (b) strained (scalene ellipsoid) cases. The minimum is located at $ k_z^{\text{min}} = 0.85  2\pi/a $ in the unstrained case.

Due to this shear strain, the 6-fold degenerate $ \Delta_6$ valleys experience an additional nonlinear shift. For uniaxial tensile stress along [110] direction, the total shift of the $ \Delta _2$ X-valleys is given by [Laude71]

$\displaystyle \Delta \epsilon = \Xi_d (\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}) + \Xi_u \varepsilon_{zz} + \Delta \epsilon ^{'}$ (3.75)

where $ \Delta \epsilon ^{'}$ can be calculated from (3.71) and (3.72).

$\displaystyle \Delta \epsilon ^{'} = \epsilon (\varepsilon _{xy},{\mathbf{k}}_{...
...\vert-1\right ) & \textnormal{,\quad $\vert\eta\vert>1$} \end{array} \right. .$ (3.76)

In Chapter 5 it is shown how the change in the effective masses contributes to the mobility enhancement. While the strain-induced deformation of the valence band structure leading to direction-dependent effective masses has been well known, a similar attention was not received by conduction band and the information of shear stress-induced electron effective mass change was lost, despite its discovery several decades back.

next up previous contents
Next: 4. Mobility Modeling Up: 3. Strain Effects on Previous: 3.2 Structure of Relaxed

S. Dhar: Analytical Mobility Modeling for Strained Silicon-Based Devices