ECE Homework -

Problem 1 -

In this problem we consider the tunneling of electrons from the conduction band of a semiconductor, across a triangular barrier, into a dielectric or wide band gap semiconductor.  A schematic diagram is shown below.  We wish to consider the current flow associated with the tunneling, and we would like to compare the results for a 1D semiconductor (such as a nanotube or nanowire, with only one transverse mode); a 2D semiconductor (such as a MoS2 or graphene-like films, with only one transverse mode); and a 3D semiconductor.  We can assume parabolic bands (E ~k²), and we can neglect reverse tunneling back into the semiconductor.

a) Write down an appropriate integral (or set of integrals) that expresses the forward current J as a function of incident kinetic energy for the electrons ε_inc (as discussed in the notes, this involves the kinetic energy associated with k_inc, the wave vector component in the direction of the barrier), for the 1D case, the 2D case and the 3D case. Two of these are actually already provided in the notes!  The barrier tunneling integral expressing the dependence on ε_inc can be taken to be the one stated in the notes.

b) Use your favorite numerical tool to plot the tunnel integral and the remaining factor(s) in the integrand in your expression for J, as a function of ε_inc. Show the energy ε_inc for which the integrand is largest.

c) Compare the values that you get for the integrand factors for the different dimensionalities considered.

d) Calculate the currents for the 3 cases.

For numerical results, use the following parameters:

Barrier height: 0.2eV (measured relative to the conduction band minimum)

Electric field applied:  500 KV/cm

Fermi level in semiconductor: 0.1eV above the bottom of the conduction band

Electron effective mass in the barrier: 0.5 mo

Electron effective mass in the semiconductor: 0.1 mo

Room temperature

Problem 2 -

Compute the energy levels for a benzene molecule using the simple Huckel model (considering only the p_z orbitals). The benzene molecule is planar and hexagonal, with C-C distance of 1.39A. Assume the value of the overlap integral ("beta") for carbon-carbon atoms is 2.5eV; you may take the reference energy ("alpha") equal to 0 for simplicity.  What levels correspond to the LUMO and the HOMO?  Can you show what is the alignment of the p_z orbitals among the different atoms (ie indicate which orbitals have + sign and - sign for a given energy level)?

Problem 3 -

In this problem you are asked to compute the maximum efficiency that you might expect to get making a solar cell from a new hypothetical material whose band gap energy is 1.3 eV.  Such a number is given in the notes, but here you should substantiate the calculation directly!

a) To do so, first make an estimate of the open circuit voltage.  You make take this to be the built-in potential of a p-n junction in the material.  To compute this built-in potential, estimate that the effective density of states in the conduction band is NC = 1e19 cm-3, and the n side doping is ND = 1e17 cm-3.  Similarly, the effective density of states in the valence band is NV=1e19cm-3 and the doping is NA=1e17cm-3 (these quantities allow you to compute the distance of the Fermi levels from the conduction and valence band edges).

b) Estimate the short circuit current by assuming one electron per absorbed photon. You may use the curves below that indicate 1) the fraction of the incident solar photon flux that corresponds to photons with energy above a threshold energy E; and 2) the fraction of the incident solar power (=energy flux) corresponding to photons with energy above a threshold energy E. The 2^nd curve weights the fractions of the 1^st curve by the respective energies per photon.

Guess the fill factor to be 0.8.  This depends on "non-ideal" mechanisms such as recombination- and you may assume that these issues have been largely overcome. 

c) Suppose the overall active thickness of your solar cell is 3 um.  In order to get absorption of >80% of the photons at the energies that you wish to collect, what is the minimum optical absorption coefficient (in cm-1) that you need to have?

Attachment:- Notes.rar