deletions | additions       diff --git a/subsection_Reduction_to_a_scattering__.tex b/subsection_Reduction_to_a_scattering__.tex  index c98f115..f5470bf 100644  --- a/subsection_Reduction_to_a_scattering__.tex  +++ b/subsection_Reduction_to_a_scattering__.tex 
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\subsection{Reduction to a scattering problem}  To evaluate this integral, $k(x)$ should, in principle, come from the (complex) dispersion relation of the crystal, or some model thereof (eg \cite{Guan_2011}). In our approach, we will properly evaluate $k(x)$; however, it's worth noting the clever trick which Sze leverages to ignore the details of the dispersion and map the solution onto an introductory-level scattering problem. In explaining this point, we will consider only electrons with no transverse momentum, and we briefly restore the assumption of uniform electric field, $\xi(x)=\xi=E_G/q(x_c-x_v)$.  With the assumption of parabolic bands (both of effective mass $m^*$) for small $k$, one can analytically continue the dispersions $E(k)=E_v(x)-\hbar^2k^2/2m^*$, $E(k)=E_c(x)+\hbar^2k^2/2m^*$ onto imaginary $k$ to find valid solutions in the gap. (Note that this only holds near to the respective band edges because of the parabolicity assumption.) So, near $x_v$, we have $E=E_V(x)+\hbar^2|k|^2/2m^*$. Since, at the start of the tunnel region, $E_V(x_v)=E$, and the slope of $E_V$ is given by $\xi$, $E_V=E-q\xi(x-x_V)$, $E_V(x)=E-q\xi(x-x_V)$,  and $$ik(x)=\sqrt{2m^*q\xi(x-x_v)/\hbar^2}$$  The above is formally the same as the classic dispersion of an particle of energy $E$ in a barrier $U(x)$, where $U(x)-E=q\xi(x-x_v)$. Playing the same game near $x_c$, we find an effective barrier $U(x)-E=q\xi(x_c-x)$. When the electron is deep into the tunneling region, the dispersion is of course more complicated. Nonetheless, Sze interpolates the simplest algebraic functional form for a barrier $U(x)$ which fits to the above limits. That is the quadratic:  $$U(x)-E=\frac{(E_G/2)^2-(q\xi x)^2}{E_G}$$