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 $z$ 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 $v$ 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 $1$ and $2$ in Figure 4.2a and $m^{(3)}_\perp=\ensuremath{m_\mathrm{l}}$ for the valley pair with the label $3$. 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] [b]

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.

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