4.3 Quantization in UTB Films for Primed Subbands

As pointed out before a shear strain component in the $\left[110\right]$ direction does not affect the primed valleys along $\left [100\right ]$ and $\left [010\right ]$ direction, except for a small shift of the minimum [186]. However, the linear combination of bulk bands method gained with the empirical pseudo-potential calculations [4] and calculations of the primed subbands based on the density functional theory (DFT) [3] uncover the relationship of the transport effective masses on the silicon film thickness $t$. Here we analyze the dependence of the primed subbands effective mass via the two-band k.p Hamiltonian utilized before (4.1). At first we have to derive analogously to the unprimed subbands an analytical expression for $k_{1}$ as a function of $k_{2}$ and vice versa:

$$\mathcal{H}= \left( \begin{array}{cc} \frac{\hbar^{2}k_{z}^{2}}{2... ...{2}k_{y}^{2}}{2 m_{t}}+\frac{\hbar^{2}}{M} k_{x} k_{y} \end{array}\right)\quad.$$ (4.27)

Starting with the transformation to dimensionless form according to:

$$X=\frac{k_{x}}{k_{0}}\:, \quad Y=\frac{k_{y}}{k_{0}}\:, \quad Z=\frac{k_{z}}{k_{0}}\:, \quad E_{0}=\frac{\hbar^{2}k_{0}^{2}}{m_{t}}\quad,$$ (4.28)

and some further rearrangements

$$\mathscr{E}=\frac{E}{E_{0}}-\frac{m_{t}}{m_{l}}Z^{2}/2-Y^{2}/2\:, \quad \nu=\frac{m_{t}}{m_{l}}Z\quad,$$ (4.29)

the eigenvalue problem takes the following form:

$$\left( \begin{array}{cc} \frac{X^{2}}{2}-\frac{m_{t}}{M} X Y - \m... ...u & \frac{X^{2}}{2}+\frac{m_{t}}{M} X Y - \mathscr{E} \end{array} \right)\quad.$$ (4.30)

Setting the determinant of (4.30) to zero allows to obatin $X$ as a function of $\mathscr{E}$

$$\begin{array}{ccc} \left( \frac{X^{2}}{2} -\mathscr{E}-\frac{... ...{2}-\frac{m_{t}^{2}}{M^{2}} X^{2} Y^{2}-\nu^{2}&=&0 \end{array}$$ (4.31)

Like before for the unprimed subbands, the obtained fourth order equation,

$$X^{4}-4\left( \mathscr{E}+\frac{m_{t}^{2}}{M^{2}}Y^{2}\right) X^{2}+4\left(\mathscr{E}^{2}-\nu^{2}\right) = 0\quad,$$ (4.32)

can be reformulated into two second order equations:

$$\begin{array}{ccc} X_{1,2}^{2}&=&2\left(\mathscr{E}+\frac{m_{... ...ight)^{2}-\frac{\nu^{2}M^{2}}{m_{t}^{2}Y^{2}}\quad. \end{array}$$ (4.33)

The identities

$$\begin{array}{ccc} \frac{X_{1}^{2}+X_{2}^{2}}{2}& = & 2 \left... ...2}}Y^{2}+\frac{\nu^{2}M^{2}}{m_{t}^{2}}Y^{2}}\quad, \end{array}$$ (4.34)

allow to introduce a $X_{1},X_{2}$ dependence in (4.33) and formulate the problem as $X_{1}(X_{2})$ and vice versa

$$\begin{array}{ccc} X_{1,2}^{2}&=&\left( \frac{m_{t}}{M} \left... ...}\right)-\frac{\nu^{2}M^{2}}{m_{t}^{2}Y^{2}} \quad. \end{array}$$ (4.35)

So $X_{1}$ as a function of $X_{2}$ is described by the following equation:

$$X_{1}^{2}=\left( \frac{m_{t}^{2}}{M^{2}}Y^{2} + \frac{\left( X_{1... ...^{2}-X_{2}^{2}}{4}\right)^{2}\right)-\frac{\nu^{2}M^{2}}{m_{t}^{2}Y^{2}} \quad.$$ (4.36)

Substituting $\Upsilon=\frac{m_{t}^{2}}{M^{2}} Y^{2}$ into (4.36) results in:

$$0=X_{1}^{4}-2\left(4 \Upsilon + X_{2}^{2}\right)X_{1}^{2}+\left(16\left(\Upsilon^{2}-\nu^{2}\right)-8\Upsilon X_{2}^{2}+X_{2}^{4}\right)\quad,$$ (4.37)

and enables the derivation of $X_{1}$ as a function of $X_{2}$

$$\begin{array}{ccc} X_{1}^{2} & = & X_{2}^{2}+4\Upsilon +4\sqr... ...} X_{2}^{2}+\frac{m_{t}^{2}}{m_{l}^{2}}Z^{2}}\quad. \end{array}$$ (4.38)

In order to obtain $X_{2}(X_{1})$ the minus branch of (4.35) is used:

$$X_{2}^{2}=\left( \frac{m_{t}^{2}}{M^{2}}Y^{2} - \frac{\left( X_{1... ...^{2}-X_{2}^{2}}{4}\right)^{2}\right)-\frac{\nu^{2}M^{2}}{m_{t}^{2}Y^{2}} \quad.$$ (4.39)

After analogous treatment of (4.39) the relation of $X_{2}$ as a function of $X_{1}$ is derived:

$$\begin{array}{ccc} X_{2}^{2} & = & X_{1}^{2}+4\Upsilon -4\sqr... ...} X_{1}^{2}+\frac{m_{t}^{2}}{m_{l}^{2}}Z^{2}}\quad. \end{array}$$ (4.40)

The corresponding scaled energy dispersion relation is given by:

$$\tilde{\mathscr{E}}\left(X,Y,Z\right) = \frac{X^{2}}{2} + \frac{Y... ...qrt{\frac{m_{t}^{2}}{M^{2}} Y^{2} X^{2}+\frac{m_{t}^{2}}{m_{l}^{2}}Z^{2}}\quad.$$ (4.41)