6.1.2 Strain-Induced Change of the Shape of the Conduction Band Edge


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6.1.2 Strain-Induced Change of the Shape of the Conduction Band Edge

In Figure 6.5 the energy dispersion of the two lowest conduction bands along the $\Delta$ symmetry lines are given for 2 GPa uniaxial stress along four stress directions. The different splitting can be interpreted from the strain tensor resulting from stress as given in (3.18):

Since the off-diagonal elements of the strain tensor vanish if stress is applied along [100], the degeneracy of the conduction bands at the -points is not lifted. Stress shifts the $\Delta _{[001]}$ -valleys up in energy with respect to the other four valleys.
For stress direction [110] the non-zero $\varepsilon_{xy}$ component lifts the degeneracy of the conduction bands at the $[00\!\pm\!1]$ -points. Thus, a noticeable effect of strain on the effective masses and the mobility is expected.
Stress along [111] is not expected to give rise to electron mobility enhancement. The non-zero off-diagonal components equally lift the degeneracy of the conduction bands in all six -points, however, the valley degeneracy remains.
For [120] stress all three valley-pairs are shifted with respect to each other. Additionally, the $\Delta _{[001]}$ -valley is affected by the splitting of the two conduction bands along [001] due to the non-zero $\varepsilon_{xy}$ component.

Figure: The two lowest conduction bands along the $\Delta$ -symmetry lines with respect to valence band edge at 2 GPa uniaxial stress for various stress directions.

$\resizebox{77mm}{!}{\includegraphics{xcrv-scipts/XSplit100_bw.eps}}$	$\resizebox{77mm}{!}{\includegraphics{xcrv-scipts/XSplit110_bw.eps}}$
$\resizebox{77mm}{!}{\includegraphics{xcrv-scipts/XSplit111_bw.eps}}$	$\resizebox{77mm}{!}{\includegraphics{xcrv-scipts/XSplit120_bw.eps}}$

**Figure:** (a) Calculated energy dispersion of the $\Delta _1$ and $\Delta _{2'}$ conduction bands of Si along the [001] direction in the vicinity of the zone boundary at various levels of shear strain $\varepsilon_{xy}$ . (b) Calculated energetic split between the two lowest conduction bands at zone boundary from EPM calculations (symbols) and from analytic expression (3.53) using 7.0 eV for the deformation potential $\Xi _{u'}$ (line).
[a] $\includegraphics[width=7.8cm]{xcrv-scipts/XSplittingEpsxy_bw.eps}$ [b] $\includegraphics[width=7.8cm]{xcrv-scipts/XSplitDup_bw1.eps}$

In Figure 6.6 the lifting of the degeneracy of the two lowest conduction bands is analyzed as a function of $\varepsilon_{xy}$ . The splitting is linear with $\varepsilon_{xy}$ and can be approximated using the analytic expression $\delta E = 4 \Xi_{u'} {\ensuremath{\varepsilon_{xy}}}$ using 7.0 eV for the shear deformation potential.

The change of position of the minima of the conduction band edge and the $\Delta$ -valley shifts resulting from shear strain $\varepsilon_{xy}$ are plotted in Figure 6.7, where results from EPM calculations are compared to the analytical expressions (3.92) and (3.100).

**Figure:** (a) Position of the conduction band minimum as a function of shear strain $\varepsilon_{xy}$ evaluated numerically by EPM calculations (symbols) and analytically (lines) from (3.92). (b) Shift of the minima of the $\Delta _{[100]}$ and $\Delta _{[010]}$ -valley pairs with respect to the $\Delta _{[001]}$ -valleys induced by shear strain $\varepsilon_{xy}$ . Comparison with analytical result from (3.100).
[a] $\includegraphics[width=7.8cm]{xcrv-scipts/kMin_bw.eps}$ [b] $\includegraphics[width=7.8cm]{xcrv-scipts/Delta4_2_epsxy_bw.eps}$

The shear strain-induced effective mass change of the transverse and longitudinal mass characterizing the $\Delta _{[001]}$ -valley pair of Si is plotted in Figure 6.8. Again, results from the EPM are compared to the analytical expressions, (3.94), (3.98), and (3.99). Good agreement can be observed for ${\ensuremath{\varepsilon_{xy}}}<1.5\%$ . For larger values of shear strain, the change of the effective masses as obtained from kp theory is smaller than that from the empirical pseudopotential method.

**Figure:** Comparison of EPM calculations for the effective masses (a) along the directions [001], and (b) along [110] and $[\,\bar{1}10]$ with analytical expressions (3.94),(3.98), and (3.99).
[a] $\includegraphics[width=7.8cm]{xcrv-scipts/mlCompEPM_KP_bw.eps}$ [b] $\includegraphics[width=7.8cm]{xcrv-scipts/mtCompEPM_KP_bw.eps}$

The calculated change of the effective mass induced by shear strain has been compared to values extracted from cyclotron resonance measurements. Good agreement is achieved as can be seen in Figure 6.9.

**Figure 6.9:** Calculated anisotropy of the cyclotron resonance effective mass, $m^{\ast } = \sqrt {m_tm_l}$ , of the (001) ellipsoid for 176 MPa tensile stress along [110] compared to measurements [Hensel65]. The effective mass change in the (001) plane is consistent with experimental data.
$\includegraphics[width=11cm]{xcrv-scipts/henselMassComp.eps}$

Finally, in Figure 6.10 the constant-energy lines in the plane $k_z = k_\mathrm{min}$ are shown. For increasing $\varepsilon_{xy}$ the evolving ellipsoid is characterized by two different transverse masses given in (3.98) and (3.99). The principal axes of the ellipses are [110] and $[1\bar{1}0]$ .

**Figure:** Constant-energy lines (units of eV) of the lowest conduction band valley along [001] in the $k_z = k_\mathrm{min}$ plane at four levels of shear strain.
[ ${\ensuremath {\varepsilon _{xy}}}=0 \%$ ] $\includegraphics[scale=1.2]{gnuplot/condBandMin001_exy0.0.eps}$ [ ${\ensuremath {\varepsilon _{xy}}}=0.5 \%$ ] $\includegraphics[scale=1.2]{gnuplot/condBandMin001_exy0.005.eps}$ [ ${\ensuremath {\varepsilon _{xy}}}=1.0 \%$ ] $\includegraphics[scale=1.2]{gnuplot/condBandMin001_exy0.01.eps}$ [ ${\ensuremath {\varepsilon _{xy}}}=1.5 \%$ ] $\includegraphics[scale=1.2]{gnuplot/condBandMin001_exy0.015.eps}$

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