Exercise 1: Consider the case of a GaAs Q-dot. Estimate the size of the Q-dot using Heisenberg's uncertainty principle. By how much would this value change, if we used the mass of electron, instead of its effective mass in the bulk semiconductor?
Solution: from the Heisenberg uncertainty relationship we have:
dE = (p h / dx)2/2m
In the above dE is the minimum energy of the electron when dx is the maximum uncertainty in the electron position, i.e. the size of the Q-dot. So, in order to determine the size of the Q-dot we need to calculate dx:
dx = p h / (2 m dE)0.5
Now, from Table B.8 of our text we know that the effective mass of the electron in GaAs is smaller than in vacuum by a factor of 0.067. Also, the minimum energy of electron is what is required to promote it from the valence band into the conduction band. This, of course, is the band-gap-energy, which in the GaAs compound is 1.43 eV (see Table B.6 of our text). Substituting these values into above equation results in a value of
dx= 12.5 nm. What value would we get, if we were to use the electron's mass in vacuum, instead of its effective mass in the bulk semiconductor? Comment on the significance of these results.
Exercise 2: Repeat the above for an InAs Q-dot and compare your results with the Bohr radius value quoted on our class's web pages on Q-dots.
Exercise 3: Consider an exciton in an InAs Q-dot. Calculate the shortest wavelength of its emission of photons as it dexcites from its higher energy levels.
Solution: from the Bohr's model of exciton we have:
E = - 13.6 m/ ( e2 . n2) eV , where n =1,2,...
In the case of InAs we have (Tables B.8 and B.11)
m = 0.026 and e = 14.6
Using these values we obtain:
E = -1.66 / n2 meV = -(1.66x 10 -3 )(1.6 x10-19)/ n2 Joules
Now, photon's energy and wavelength are related according to:
E = h c / l
This gives us a value of 1.0 mm for the wavelength of the emitted photon. Clearly this value is not in the visible range. What would be an application of such a transition? (Hint: calculate the frequency instead.)
Last Modified: January 21, 2004 malekis@union.edu