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3.6.2 Strain-Induced Lifting of Degeneracy at X point

In the case of degenerate bands strain does not only shift the band as a whole, but may also split bands as a result of partial or complete removal of degeneracy upon the reduction of symmetry. In the diamond crystal structure, the lowest two conduction bands $ \Delta _1$ and $ \Delta _{2'}$ touch at the zone boundary $ X$ due to a special symmetry of the diamond structure, namely the presence of three glide reflection planes, given by $ x=a_0/8, y=a_0/8$, and $ z=a_0/8$ [Yu03]. For example, the plane $ z=a_0/8$ is a glide plane since diamond is invariant under a translation by $ \frac{a_0}{4} (1,1,0)$ followed by a reflection on this plane. Whenever the strain tensor in the crystal system contains a shear component $ {\ensuremath {\varepsilon _{xy}}}$ (for example, as a result from stress along the [110] direction), the strained lattice belongs to an orthorombic crystal system (see Section 3.5.5). The shear component removes the glide reflection plane $ z=a_0/8$ and consequently the degeneracy of the two lowest conduction bands $ \Delta _1$ and $ \Delta _{2'}$ at the symmetry points $ X=\frac{2\pi}{a_0} (0,0,\pm 1)$ is lifted [Hensel65,Bir74] (compare Section 3.5.5). It should be noted that the glide reflection symmetry is preserved in biaxially strained Si layers grown on {001} Si$ _{1-y}$Ge$ _{y}$ substrate, as well as in Si uniaxially strained/stressed along a fourfold rotation axis $ \langle 100 \rangle$.

From kp theory (see Section 3.7) including terms of third order Bir and Pikus found that by lifting the degeneracy at a zone boundary $ X$ point a comparatively large change in the energy dispersion of the conduction band minimum located close to this $ X$ point is induced [Bir74]. In Si this effect was verified experimentally by Hensel and Hasegawa [Hensel65], who measured the change in effective mass for stress along $ \langle110\rangle$, and by Laude [Laude71] who measured the indirect exciton spectrum.

Equation (3.47) has to be modified as follows when taking into account the lifting of the degeneracy of the two lowest conduction bands $ \Delta _1$ and $ \Delta _{2'}$ at the $ X$ points $ \frac{2\pi}{a_0} (0, 0,\pm 1)$ [Hensel65]

$\displaystyle \begin{pmatrix}\delta E_0 & \delta E_1\\ \delta E_1 & \delta E_0\... ...} \\ \end{pmatrix} = \delta E \begin{pmatrix}\xi \\ \hat{\xi} \\ \end{pmatrix},$ (3.49)

where

$\displaystyle \delta E_0$ $\displaystyle = \ensuremath {\Xi_\mathrm{d}}^{\Delta}\,$Tr$\displaystyle (\ensuremath{{\underaccent{\bar}{\varepsilon}}}) + \ensuremath {\Xi_\mathrm{u}}^{\Delta}{\ensuremath{\varepsilon_{zz}}}\ ,$ (3.50)
$\displaystyle \delta E_1$ $\displaystyle = \Xi_{u'} e_{xy} = 2 \Xi_{u'} {\ensuremath{\varepsilon_{xy}}}\ .$ (3.51)

Here, a new deformation potential $ \Xi _{u'}$ is introduced. The solutions of the eigenvalue problem (3.49) are

$\displaystyle \delta E = \delta E_0 \pm \delta E_1 \quad\mathrm{for}\quad \hat{\xi} = \pm \xi\ .$ (3.52)

Thus, at the $ X$ points $ \frac{2\pi}{a_0} (0, 0,\pm 1)$ the bands shift by an amount of $ \delta E_0$, which is the shift given by (3.47). Additionally, the degeneracy is lifted with a splitting of $ 2\delta E_1$. From (3.51) it follows that the splitting is proportional to the shear strain $ \varepsilon_{xy}$ is given by

$\displaystyle (E_{\Delta_1} - E_{\Delta_{2'}}) {\Big\vert}_{X_{[001]}} = 2\delta E_1 = 4 \Xi_{u'} {\ensuremath{\varepsilon_{xy}}}\ .$ (3.53)

For the shear deformation potential $ \Xi_{u'}^X$ a value of 5.7 $ \pm$ 1 eV [Hensel65] has been predicted from cyclotron resonance experiments. From measurements of the indirect exciton spectrum of Si a similar value (7.5 $ \pm$ 2 eV) has been obtained [Laude71].

Figure 3.12 shows the splitting for three levels of strain $ {\ensuremath {\varepsilon _{xy}}}$. The splitting is very pronounced even for relatively small strain ($ < 1\%$). One can also observe that the $ \Delta _1$ conduction band is deformed in the vicinity of the symmetry points $ X=\frac{2\pi}{a_0} (0,0,\pm 1)$ due to the lifting of the degeneracy.

Figure: Structure of the $ \Delta _1$ and $ \Delta _{2'}$ conduction bands near the zone boundary $ X$ points. The solid lines indicate a splitting of the bands due to shear strain $ {\ensuremath {\varepsilon _{xy}}}$ for three levels of strain. $ \Delta $ denotes the band separation of unstrained Si at the conduction band edge $ {\ensuremath{\mathitbf{k}}}_\mathrm{min}=\frac{2\pi}{a_0} (0,0,0.85)$. It can be observed that the minimum of the conduction band $ \Delta _1$ along [001] (highlighted by an open circle) moves towards the zone boundary as $ {\ensuremath {\varepsilon _{xy}}}$ increases. The conduction bands along [100] and [010] are not affected by $ \varepsilon_{xy}$.
\resizebox{90mm}{!}{\includegraphics{inkscape/XsplittingSchematicsZeroStrain.eps}} \resizebox{63mm}{!}{\includegraphics{inkscape/XsplittingSchematicsZeroStrain2.eps}}
\resizebox{90mm}{!}{\includegraphics{inkscape/XsplittingSchematics.eps}} \resizebox{63mm}{!}{\includegraphics{inkscape/XsplittingSchematicsZeroStrain3.eps}}
\resizebox{90mm}{!}{\includegraphics{inkscape/XsplittingSchematicsLargeStrain.eps}} \resizebox{63mm}{!}{\includegraphics{inkscape/XsplittingSchematicsZeroStrain3.eps}}
\resizebox{90mm}{!}{\includegraphics{inkscape/XsplittingSchematicsLargeStrain2.eps}} \resizebox{63mm}{!}{\includegraphics{inkscape/XsplittingSchematicsZeroStrain3.eps}}

In unstrained Si the constant energy surfaces of the six conduction band valleys have a prolate ellipsoidal shape, where the semi-axes are characterized by $ \ensuremath{m_\mathrm{l}}$ and $ \ensuremath{m_\mathrm{t}}$, denoting the longitudinal and transverse electron masses, respectively. The minima of the three valley pairs are located along the three equivalent $ \langle 100 \rangle$ directions and have the same energies (see Section 3.4.1).

From Figure 3.12 it can be observed that a non-vanishing shear component $ {\ensuremath {\varepsilon _{xy}}}$ in the strain tensor affects the energy dispersion of the lowest conduction band in three ways:

  1. The band edge energy of the valley pair oriented along the [001] direction moves down with respect to the four valleys oriented along [100] and [010].
  2. Since the shape of the conduction band around its minima is considerably deformed, the effective mass of the valley pair along [100] is expected to change as $ \varepsilon_{xy}$ grows.
  3. The positions of the conduction band minima along [001] move towards the zone boundary $ X$ points at $ \frac{2\pi}{a_0} (0, 0,\pm 1)$.
Within the presented model $ \varepsilon_{xy}$ has no effect on the conduction bands near the zone boundaries $ X=\frac{2\pi}{a_0} (1,0,0)$ and $ X=\frac{2\pi}{a_0} (0,1,0)$ (compare Figure 3.12). A nonzero component $ {\ensuremath{\varepsilon_{xz}}}$ and $ {\ensuremath{\varepsilon_{yz}}}$, however, will lift the degeneracy at $ X=\frac{2\pi}{a_0} (0,\pm 1, 0)$ and $ X=\frac{2\pi}{a_0} (\pm 1, 0, 0)$, respectively.

If the splitting of the conduction bands is different at the different zone boundaries (for example, $ {\ensuremath{\varepsilon_{xy}}}\neq{\ensuremath{\varepsilon_{xz}}}\neq{\ensuremath{\varepsilon_{yz}}}$), the conduction band minima along the $ \langle 001\rangle$ axes have different energies. This may result in a repopulation between the six conduction band valleys. Note that such an effect cannot be explained via equation (3.47) alone, where a possible lifting of the degeneracy at the $ X$ point induced by shear strain is neglected and application of shear strain yields no valley repopulation.

An analytical expression for the valley shift along the $ \Delta $ direction can be derived using a degenerate kp theory at the zone boundary $ X$ point [Bir74,Hensel65]. A shear strain $ {\ensuremath {\varepsilon _{xy}}}$ causes an energy shift between the conduction band valleys along [001] with respect to the valleys along [100] and [010]. The shift is given by

$\displaystyle \delta E_{\mathrm{shear}} = \left\{ \begin{array}{ll} -\frac{\Del... ...\quad $\vert{\ensuremath{\varepsilon_{xy}}}\vert>1/\kappa$} \end{array} \right.$ (3.54)

Here, a dimensionless parameter $ \kappa=(4\Xi_{u'})/\Delta$ has been introduced, and $ \Delta $ is the band separation between the two lowest conduction bands at the conduction band edge

$\displaystyle \Delta = (E_{\Delta_{2'}} - E_{\Delta_1}) {\Big\vert}_{{\ensuremath{\mathitbf{k}}} = {\ensuremath{\mathitbf{k}}}_{\mathrm{min}}}\ .$ (3.55)

The position of the band edge in the unstrained lattice is $ {\ensuremath{\mathitbf{k}}}_{\mathrm{min}} = \frac{2\pi}{a_0} (0,0,0.85)$ (compare Figure 3.12). The derivation of (3.54) is given in Section 3.7.2 after outlining the basic framework of the kp method.


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E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology