4.1.4 Substrate Orientation (111)

Figure 4.4: (a) Alignment of constant-energy surfaces of the Si conduction band with respect to the (111) substrate surface. (b) Projection of constant-energy surfaces onto the (111) plane. Since the quantization mass is equal for all valleys, only one sixfold degenerate subband ladder is formed.

         [a] [b]

The substrate normal ${\ensuremath{\mathitbf{e}}}_n^{(111)}$ for substrate orientation (111) is $\smash{\frac{1}{\sqrt 3}(1,1,1)}$. Thus, the axes of the principal crystallographic systems have to be rotated by the angles $\smash{\cos \theta = 1 / \sqrt 3}$ ( $\theta \approx 54.74^\circ$) and $\phi=45^\circ$. The inverse effective mass tensor for the three valley pairs in the coordinate system with the $z$ axis perpendicular to the substrate surface are

$$\ensuremath{{\underaccent{\bar}{\nu}}}^{(1)}= \begin{pmatrix}\fra... ...\mathrm{l}}}{3\ensuremath{m_\mathrm{t}}\ensuremath{m_\mathrm{l}}}\end{pmatrix},$$ (4.25)

$$\ensuremath{{\underaccent{\bar}{\nu}}}^{(2)}= \begin{pmatrix}\fra... ...\mathrm{l}}}{3\ensuremath{m_\mathrm{t}}\ensuremath{m_\mathrm{l}}}\end{pmatrix},$$ (4.26)

$$\ensuremath{{\underaccent{\bar}{\nu}}}^{(3)}= \begin{pmatrix}\fra... ...mathrm{l}}}{3\ensuremath{m_\mathrm{t}}\ensuremath{m_\mathrm{l}}} \end{pmatrix}.$$ (4.27)

It can be seen that all valleys have the same quantization mass $m^{(1,2,3)}_3 = \frac{3\ensuremath{m_\mathrm{t}}\ensuremath{m_\mathrm{l}}}{\ensuremath{m_\mathrm{t}}+2\ensuremath{m_\mathrm{l}}}$ and therefore belong to the same subband ladder. To determine the transport masses, the eigenvalues of $\ensuremath{{\underaccent{\bar}{M}}}^v$ have to be calculated

$$\ensuremath{{\underaccent{\bar}{M}}}^{(1,2)}= \begin{pmatrix}\fra... ...h{m_\mathrm{t}}} & 0\\ 0 & \frac{1}{\ensuremath{m_\mathrm{t}}} \end{pmatrix}\ .$$ (4.28)

Solving the secular equation (4.15) the transport masses evaluate to

$$m^{(1,2,3)}_{\shortparallel,1}=\ensuremath{m_\mathrm{t}}\ ,\quad\... ...shortparallel,2}=\frac{2\ensuremath{m_\mathrm{l}}+\ensuremath{m_\mathrm{t}}}{3}$$ (4.29)

for all three valley pairs.

The constant-energy lines of the subbands for a (111) substrate are shown in Figure 4.4b. The unprimed ladder is sixfold degenerate. Two ladders have their major principal axes parallel to the $[11\bar{2}]$ direction. The principal axes of the other ladders is inclined by an angle of $+30^\circ$ and $-30^\circ$ from $[\bar{1}10]$.

In Table 4.1 the principal effective masses of Si for the three discussed surface orientations are summarized. In unstrained Si the six conduction band valleys form a set of two subband ladders for substrate orientation (001) and (110), whereas for (111) oriented substrate only one subband ladder with sixfold degeneracy is formed. The energy alignments of the Si conduction subband ladders constituting on (001), (110), and (111) oriented substrate are shown Figure 4.5a and Figure 4.5b.

Table 4.1: Principal effective masses of the six Si conduction band minima along $\Delta$ for three surface orientations. Here, $m_{\shortparallel ,1}$, and $m_{\shortparallel ,2}$ denote the transport masses, and $m_\perp$ is the quantization mass.

surface orientation	degeneracy	$m_{\shortparallel ,1}$	$m_{\shortparallel ,2}$	$m_{\perp}$	ladder
	2	$\ensuremath{m_\mathrm{t}}$	$\ensuremath{m_\mathrm{t}}$	$\ensuremath{m_\mathrm{l}}$	unprimed
		4	$\ensuremath{m_\mathrm{l}}$	$\ensuremath{m_\mathrm{t}}$	$\ensuremath{m_\mathrm{t}}$
	4	$\ensuremath{m_\mathrm{t}}$	$\frac{\ensuremath{m_\mathrm{t}}+\ensuremath{m_\mathrm{l}}}{2}$	$\frac{2\ensuremath{m_\mathrm{t}}\ensuremath{m_\mathrm{l}}}{\ensuremath{m_\mathrm{t}}+\ensuremath{m_\mathrm{l}}}$	unprimed
		2	$\ensuremath{m_\mathrm{l}}$	$\ensuremath{m_\mathrm{t}}$	$\ensuremath{m_\mathrm{t}}$
(111)	6	$\ensuremath{m_\mathrm{t}}$	$\frac{2\ensuremath{m_\mathrm{l}}+\ensuremath{m_\mathrm{t}}}{3}$	$\frac{3\ensuremath{m_\mathrm{t}}\ensuremath{m_\mathrm{l}}}{\ensuremath{m_\mathrm{t}}+2\ensuremath{m_\mathrm{l}}}$	unprimed

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