Question 3: two GaAs possesses a zinc blende structure, which can be visualized as interpenetrating Bravais lattices of two-element basis with atoms located at (0,0,0) (first element) and (a, a, a) (second element), where a = 5.65 is its lattice constant and e = 13.2 is the dielectric constant. Al, Ga., As Gaas 50 X 0.28 eV 1.85 ev 1.43 eV 0.14 eV Based on this GaAs compound, one can build an Al., Ga. As/GaAs/Al...Ga.. As quantum well heterostructure, as shown below along with its energy band diagram at non-equilibrium. The band gap of the GaAs is 1.43 eV (with the Fermi level is exactly in the middle of the energy gap), while it is 1.85 eV for Al.Ga..As (electron doped with the Fermi level is 0.2 eV below the conduction band edge). The electron effective mass is m* = 0.067m, with me = 9.1 x 10-31 kg the rest mass of the electron. (a) Considering the band discontinuity and band bending, sketch the band diagram of the quantum well in the heterostructure at the equilibrium. Clearly indicate your energy labels and values. (b) Plot to scale the energy versus density of states for the quantum well. Be sure to put numbers and the units on both the axes. Calculate the minimum energy of the electrons and compare with thermal energy at room temperature. Consider the two dimensional electron gas (2DEG) in the quantum well. The 2DEG is patterned into a Hall-bar structure (as shown in the sketch below) and its longitudinal, V, and transverse (Hall), V, voltages were measured using current I = 25 A up to 8T at 1.5 K. The results show a Subnikov de-Haas oscillation and quantum Hall as depicted in the figure below. 250 10 I = 25 A 3 -0.38 mm 200 H 8 I= 4 5 150 6 6 VH (mv) V (mV) 100 4 10 12 50 2 2 0 0 0 0 1 2 3 4 5 6 7 8 Magnetic Field (T) (c) If the electron density in the well is 5 x 10"/cm, calculate the location of the Fermi energy Ep relative to the bottom of the band at o K. (d) Use the low-field (OT