4.1.2 Substrate Orientation (001)

For cubic semiconductors the Miller indices of a plane $(hk\ell)$ are also the coordinates of the normal vector of this plane. For substrate orientation (001) the coordinate system in which the

axis is normal to the substrate surface coincides with the crystallographic system (see Figure 4.2a). Thus, no coordinate transformation needs to be involved and $\nu_{ij}={\nu_{ij}}'$ .

The quantization masses of the three different valleys

are $m^v_\perp=1/\nu_{33}^v$ . By comparison with (4.18) the quantization masses can be determined, yielding $m^{(1,2)}_\perp = \ensuremath{m_\mathrm{t}}$ for valleys labeled

and

in Figure 4.2a and $m^{(3)}_\perp=\ensuremath{m_\mathrm{l}}$ for the valley pair with the label

. Since $\ensuremath{m_\mathrm{l}}>\ensuremath{m_\mathrm{t}}$ , the two valleys with the large quantization mass $m^{(3)}_\perp=\ensuremath{m_\mathrm{l}}$ belong to the lowest (unprimed) subband ladder, whereas the four valleys with $m_\perp=\ensuremath{m_\mathrm{t}}$ constitute the primed subband ladder. The transport masses $m_{\shortparallel ,1}$ and $m_{\shortparallel ,2}$ for the three valleys can be found from (4.17). Since the matrix $\ensuremath{{\underaccent{\bar}{M}}}^v$ contains only diagonal entries, $M^v_{11} = \nu^v_{11}$ and $M^v_{22} = \nu^v_{22}$ , the transport masses are easily obtained from (4.18), $m^v_{\shortparallel,1}=1/\nu^v_{11}$ , and $m^v_{\shortparallel,2}=1/\nu^v_{22}$ .

**Figure 4.2:** (a) Alignment of constant-energy surfaces of the Si conduction band with respect to the substrate surface (001). (b) Projection of constant-energy surfaces onto the (001) plane. Concentric spheres indicate the twofold degeneracy of the unprimed ladder. Constant-energy lines belonging to the primed ladder (filled in dark grey) have elliptic shape.
[a] $\includegraphics[scale=1.0]{inkscape/Cut001_2.eps}$ [b] $\includegraphics[scale=1.5]{inkscape/projectionOr001.eps}$

For transport calculations not only the transport masses of a particular subband ladder are essential, but also the direction of the principal axes of the constant-energy ellipse with respect to the crystallographic axes of the wafer. In Figure 4.2b the projection of the constant-energy surfaces onto the substrate with orientation (001) is shown. The projection yields two subband ladders with spherical constant-energy lines (unprimed ladders) and four subbands with elliptic constant-energy lines (primed ladders). However, the transport properties of the four primed subband ladders are in general not equivalent since two of them are aligned along the [100] direction, whereas the other two are aligned along the [010] direction. Also, the principal axes of the constant-energy ellipsoids are interchanged.


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