next up previous contents
Next: Appendix B Up: Dissertation Siddhartha Dhar Previous: 6. Summary and Conclusions


Appendix A

The integrals in (4.61) are of the form

$\displaystyle I_1 = \int_\Delta^{\infty} (\epsilon-\Delta)^{1/2} \exp\left(\dis...
...+\Delta)^{1/2} \exp\left(\displaystyle\frac{-\epsilon}{k_{B}T}\right) d\epsilon$ (6.1)

Setting $ t = \displaystyle \frac{\epsilon-\Delta}{k_BT}$ in $ I_1$ and $ t =
\displaystyle \frac{\epsilon+\Delta}{k_BT}$ in $ I_2$, the integrals reduce to

$\displaystyle I_1 = (k_BT)^{3/2} \exp\left(\displaystyle\frac{-\Delta}{k_{B}T}\right)   \int_{0}^{\infty} t^{1/2} \exp{(-t)} dt,$ (6.2)
$\displaystyle I_2 = (k_BT)^{3/2} \exp\left(\displaystyle\frac{\Delta}{k_{B}T}\right)   \int_{\Delta/k_BT}^{\infty} t^{1/2} \exp{(-t)} dt$ (6.3)

which gives

$\displaystyle I_1 = (k_BT)^{3/2} \cdot\exp\left(\displaystyle\frac{-\Delta}{k_{...
...ft(\displaystyle\frac{\Delta}{k_{B}T}\right) \Gamma\left(\frac{3}{2},-z\right).$ (6.4)


Similarly, the integrals in (4.66) are of the form

$\displaystyle I_3 = \int_\Delta^{\infty} \sqrt{\epsilon (\epsilon-\Delta)} \exp...
...silon+\Delta)} \exp\left(\displaystyle\frac{-\epsilon}{k_{B}T}\right) d\epsilon$ (6.5)

Setting $ t = \displaystyle \frac{\epsilon-\Delta}{k_BT}$ in $ I_1$ and $ t =
\displaystyle \frac{\epsilon+\Delta}{k_BT}$ in $ I_2$, the integrals reduces to

$\displaystyle I_3 = (k_BT)^{2} \exp\left(\displaystyle\frac{-\Delta}{k_{B}T}\right) \int_{0}^{\infty} \sqrt{t(t+\Delta/k_BT)} \exp{(-t)} dt,$ (6.6)
$\displaystyle I_4 = (k_BT)^{2} \exp\left(\displaystyle\frac{\Delta}{k_{B}T}\right) \int_{\Delta/k_BT}^{\infty} \sqrt{t(t-\Delta/k_BT)} \exp{(-t)} dt$ (6.7)

which gives

$\displaystyle I_3 = (k_BT)^{2}\exp{\left(\frac{-\Delta}{k_BT}\right)}\left[ \fr...
...\left(\frac{\Delta}{2k_BT}\right)} K_1\left(\frac{\Delta}{2k_BT}\right)\right],$ (6.8)
$\displaystyle I_4 = (k_BT)^{2}\exp{\left(\frac{\Delta}{k_BT}\right)}\left[ \fra...
...{\left(\frac{\Delta}{2k_BT}\right)} K_1\left(\frac{\Delta}{2k_BT}\right)\right]$ (6.9)

Here, $ K_1$ denotes the modified Bessel function of the second kind.


next up previous contents
Next: Appendix B Up: Dissertation Siddhartha Dhar Previous: 6. Summary and Conclusions

S. Dhar: Analytical Mobility Modeling for Strained Silicon-Based Devices