**Problem C2.1 (Transmission through an array of barriers)**

Use the propagation matrix approach to calculate the transmission coefficient T for an array of N rectangular potential barriers, each having width L = 1 nm, height V_{0} = 0.3 eV and a barrier-barrier separation a = 1 nm. Plot T as a function of energy for an increasing number of barriers. Do you see any oscillations in the transmission? Any regions where the transmission is heavily reduced? What is the underlying physical explanation? Play with the parameters, and compare your results with those of theory (e.g. the Kronig-Penney potential model). In what limit is the theoretical model valid? Can you predict where the regions of high transmission (bands) will appear? Tip: Instead of just plotting the transmission T (E), also try to plot the damping (defined as −ln(T (E)).

**Problem C2.2(**Transmission through a parabolic potential**)**

Use (a modified version of) your program and calculate the transmission through the potential step defined by

V (x) =(x^{2}/L^{2} ) if |x| <= L,

0 elsewhere.

Use L = 5 nm. Look at the bound state energies and compare your results with those of the harmonic oscillator. How does the quality of your result depend on the number of barriers used (the potential ”roughness”)? Tip: If needed, plot your transmission using a logarithmic scale.

Answer is attached below with pdf file along with MATLAB Program.

Fund of Nanoscience – Project_II_Ravi Sharma Dulal

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