Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

A Appendix

A.1 WKB Approximation

The WKB approximation was developed in 1926 independently by Gregor Wentzel, Hendrik Kramers and Léon Brillouin [178]. It can be used for solving quantum mechanical eigen-value problems iteratively. Thus, it can also be used for approximating the time independent Schrödinger equation

(A.1) \{begin}{align} -\frac {\hbar ^2}{2m}\frac {\dd ^2\psi (x)}{\dd x^2} + V(x)\psi (x) = E\psi (x), \{end}{align}

which can be rewritten as

(A.2) \{begin}{align} \frac {\dd ^2\psi (x)}{\dd x^2} + k^2(x)\psi (x) =0 \label {equ:SEQ2} \{end}{align}

with \( k^2(x) = \frac {2m}{\hbar ^2}(E-V(x)) \). Solving this equation can be done using the ansatz

(A.3) \{begin}{align} \psi (x) = c \mathrm {e}^{\frac {i}{\hbar }W(x)}. \label {equ:ansatz} \{end}{align}

Plugging the ansatz (A.3) into the Schrödinger equation (A.2) it can be transfered to an inhomogeneous non-linear differential equation of second order for \( W \) which results in

(A.4) \{begin}{align} (W’(x))^2-i\hbar W”(x) = \hbar ^2 k^2(x). \label {eq:SEQ3} \{end}{align}

Based on the principle that quantum mechanical relations should correspond to classical relations when \( \hbar \rightarrow 0 \) the phase \( W(x) \) can be expanded in \( \hbar \)

(A.5) \{begin}{align} W(x) = \sum \limits _{n=0}^{\infty } (i\hbar )^nW_n(x). \label {eq:expansion} \{end}{align}

Plugging (A.5) into (A.4) and sorting the result by the order in \( \hbar \) we get

(A.6) \{begin}{align} (W_0’^2-\hbar ^2k^2) +i\hbar (2W_0’W_1’-W_0”)+\mathcal {O}(\hbar ^2) = 0 \{end}{align}

where \( \hbar ^2k^2 \) is of order zero in \( \hbar \) because \( k^2\propto 1/\hbar ^2 \). From the zeroth order term the differential equation

(A.7) \{begin}{align} W_0’(x)=\pm \hbar k(x) \{end}{align}

can be extracted and solved:

(A.8) \{begin}{align} W_0(x) = \pm \hbar \int \limits ^x k(x’)\dd x’. \{end}{align}

From the first order \( \hbar \) term the corresponding differential equation

(A.9) \{begin}{align} W_1’(x)=\frac {1}{2}\frac {W_0”(x)}{W_0’(x)} \{end}{align}

can be solved

(A.10) \{begin}{align} W_1(x) = \ln \sqrt {k(x)}. \{end}{align}

Using these results in the ansatz (A.3) the WKB-wavefunction has the from

(A.11) \{begin}{align} \psi (x) = \frac {c_+}{\sqrt {k(x)}}\mathrm {e}^{i\int \limits ^x\sqrt {k(x’)}\dd x’} + \frac {c_-}{\sqrt {k(x)}}\mathrm {e}^{-i\int \limits ^x\sqrt {k(x’)}\dd x’}. \{end}{align}