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4.3 Quantization in UTB Films for Primed Subbands
As pointed out before a shear strain component in the
direction does not affect the primed valleys along
and
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
.
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
as a function of
and vice versa:
![$\displaystyle \mathcal{H}= \left( \begin{array}{cc} \frac{\hbar^{2}k_{z}^{2}}{2...
...{2}k_{y}^{2}}{2 m_{t}}+\frac{\hbar^{2}}{M} k_{x} k_{y} \end{array}\right)\quad.$](img626.png) |
(4.27) |
Starting with the transformation to dimensionless form according to:
![$\displaystyle X=\frac{k_{x}}{k_{0}}\:, \quad Y=\frac{k_{y}}{k_{0}}\:, \quad Z=\frac{k_{z}}{k_{0}}\:, \quad E_{0}=\frac{\hbar^{2}k_{0}^{2}}{m_{t}}\quad,$](img627.png) |
(4.28) |
and some further rearrangements
![$\displaystyle \mathscr{E}=\frac{E}{E_{0}}-\frac{m_{t}}{m_{l}}Z^{2}/2-Y^{2}/2\:, \quad \nu=\frac{m_{t}}{m_{l}}Z\quad,$](img628.png) |
(4.29) |
the eigenvalue problem takes the following form:
![$\displaystyle \left( \begin{array}{cc} \frac{X^{2}}{2}-\frac{m_{t}}{M} X Y - \m...
...u & \frac{X^{2}}{2}+\frac{m_{t}}{M} X Y - \mathscr{E} \end{array} \right)\quad.$](img629.png) |
(4.30) |
Setting the determinant of (4.30) to zero allows to obatin
as a function of
![\begin{displaymath}\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}\end{displaymath}](img630.png) |
(4.31) |
Like before for the unprimed subbands, the obtained fourth order equation,
![$\displaystyle X^{4}-4\left( \mathscr{E}+\frac{m_{t}^{2}}{M^{2}}Y^{2}\right) X^{2}+4\left(\mathscr{E}^{2}-\nu^{2}\right) = 0\quad,$](img631.png) |
(4.32) |
can be reformulated into two second order equations:
![\begin{displaymath}\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}\end{displaymath}](img632.png) |
(4.33) |
The identities
![\begin{displaymath}\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}\end{displaymath}](img633.png) |
(4.34) |
allow to introduce a
dependence in (4.33) and formulate the problem as
and vice versa
![\begin{displaymath}\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}\end{displaymath}](img636.png) |
(4.35) |
So
as a function of
is described by the following equation:
![$\displaystyle X_{1}^{2}=\left( \frac{m_{t}^{2}}{M^{2}}Y^{2} + \frac{\left( X_{1...
...^{2}-X_{2}^{2}}{4}\right)^{2}\right)-\frac{\nu^{2}M^{2}}{m_{t}^{2}Y^{2}} \quad.$](img637.png) |
(4.36) |
Substituting
into (4.36) results in:
![$\displaystyle 0=X_{1}^{4}-2\left(4 \Upsilon + X_{2}^{2}\right)X_{1}^{2}+\left(16\left(\Upsilon^{2}-\nu^{2}\right)-8\Upsilon X_{2}^{2}+X_{2}^{4}\right)\quad,$](img639.png) |
(4.37) |
and enables the derivation of
as a function of
![\begin{displaymath}\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}\end{displaymath}](img640.png) |
(4.38) |
In order to obtain
the minus branch of (4.35) is used:
![$\displaystyle X_{2}^{2}=\left( \frac{m_{t}^{2}}{M^{2}}Y^{2} - \frac{\left( X_{1...
...^{2}-X_{2}^{2}}{4}\right)^{2}\right)-\frac{\nu^{2}M^{2}}{m_{t}^{2}Y^{2}} \quad.$](img642.png) |
(4.39) |
After analogous treatment of (4.39) the relation of
as a function of
is derived:
![\begin{displaymath}\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}\end{displaymath}](img643.png) |
(4.40) |
The corresponding scaled energy dispersion relation is given by:
![$\displaystyle \tilde{\mathscr{E}}\left(X,Y,Z\right) = \frac{X^{2}}{2} + \frac{Y...
...qrt{\frac{m_{t}^{2}}{M^{2}} Y^{2} X^{2}+\frac{m_{t}^{2}}{m_{l}^{2}}Z^{2}}\quad.$](img644.png) |
(4.41) |
Next: 4.4 Effective Mass of
Up: 4. Quantum Confinement and
Previous: 4.2 Effective Mass of
T. Windbacher: Engineering Gate Stacks for Field-Effect Transistors