Note: this problem can be used as a substantial assignment.] Consider two GaAs layers separated by an Al0.3Ga0.7As barrier, 5 nm thick, which has a potential height for electrons of ~ 231 meV and an effective mass of ~0.0919 mo. Presume the electron effective mass in GaAs is 0.067 mo. The temperature is 300K, and the carrier density on the left of the barrier is such that the Fermi energy is at the bottom of the conduction band. (You may use the Maxwell-Boltzmann approximation to the thermal distribution of electrons.) There are presumed to be no carriers on the right hand side of the barrier, though the GaAs is assumed to be at the same uniform potential as the GaAs on the left hand side of the barrier (i.e., assume the bottom of the GaAs conduction band is at the same constant energy on both sides).
(i) Plot the transmission probability for an electron incident on the barrier from the left as a function of the energy of the electron, from 0 to 0.5 eV.
(ii) Calculate the current density of electrons (in A/cm2 ) moving from the left of the barrier to the right that we would expect on a simple classical calculation (i.e., the Richardson-Dushman equation).
(iii) Calculate the current density now using the quantum mechanical calculation with the transmission probability included.
(iv) Now replace the single barrier with a pair of Al0.3Ga0.7As barriers each 3 nm thick around an empty GaAs "well" of thickness 4 nm. Plot the transmission probability for an electron incident on the barrier from the left as a function of the energy of the electron, from 0 to 0.5 eV.
(v) With this new structure, calculate the current density now using the quantum mechanical calculation with the transmission probability included