deletions | additions       diff --git a/subsection_Reduction_to_a_scattering__.tex b/subsection_Reduction_to_a_scattering__.tex  index eba226f..4076fb1 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). 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 briefly restore the assumption of uniform electric field $\xi(x)=\xi$, $\xi(x)=\xi=E_G/q(x_C-x_V)$,  and will consider only electrons with no transverse momentum.So $q\xi=E_G/(x_C-x_V)$  With the assumption of parabolic bands (both of effective mass $m^*$) for small $k$, one can just 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 $E=E_V(x)+\hbar^2|k|^2/2m^*=E-\xi(x-x_V)+\hbar^2|k|^2/2m^*$, ie $ik=\sqrt{2mq\xi(x-x_V)/\hbar^2}$. That is the same as the regular dispersion of an particle in a barrier $U(x)$ such that $U(x)-E=q\xi(x-x_V)$. Similarly, near $x_C$, we have $ik=\sqrt{2m^*q\xi(x_C-x)/\hbar^2}$, which is equivalent to a barrier $U(x)-E=q\xi(x-x_C)$. When the electron is deep into the tunneling region, the dispersion is more complicated but Sze just interpolates the simplest algebraic functional form for a barrier $U(x)$ which reproduces the correct dispersion near the edges. That is the quadratic:  $$U(x)-E=\frac{(E_G/2)^2-(q\xi x)^2}{E_G}$$