of the conjugated length. Since the single and double bonds, with bond lengths of 154 pm and 130 pm, respectively, are approximately 120o from each other, one can approximate the length of the box as(E2.4.1)2 Calculation of the energy difference between n = 3 and n = 4. Use me = 9.1×10−31 [kg] and h = 6.6 × 10−34 [Js] for the electron mass and Planck's constant. Since the length of the box was estimated to 2 significant figures, the entire computation is carried out with 2 significant figures:
Analysis of units:
(E2.4.2)
ΔE = 3.7 × 10−19 [J]
Figure 2.7 Absorption spectra of nanoparticles as a function of particle size. As expected, the larger particles exhibit lower energy (longer wavelength) transitions.
1 ΔE = hc/λ or λ = hc/ΔE(E2.4.3)
2.5.2 Quantum Dots
Certain quantum dot structures can also be modeled by a 2D particle in a box. Quantum dots may be manufactured by creating small circular or square semiconductor deposits on a substrate that is an electric insulator. The electrons of the semiconductor spots are free to move over the entire size of the dot, and the energy levels of the free electrons follow a 2D PiB model [3]. Consequently, the color of electronic transitions can be tuned by varying the size of the quantum dot.
Closely related to these 2‐dimensional quantum dots are 3‐dimensional nanoparticles, such as spheres of metallic or semiconductor materials. The free electrons on the surface of such spheres can assume wave patterns known as “spherical harmonic functions” that are the solutions for particle on a sphere. Similar functions will be discussed in Chapter 7 during the treatment of the hydrogen atom wavefunctions. Again, the optical properties of such nanoparticles can be tuned by adjusting the size of the nanoparticle. This is shown in Figure 2.7.
2.5.3 Quantum Cascade Lasers
Finally, an example of a commercial application of the particle in a box will be discussed, namely, that of a solid‐state infrared laser known as the quantum cascade laser (QCL) [4]. In QCLs, the “box” is constructed by creating a semiconductor “superlattice” potential functions that mimics a PiB with finite potential energy barriers. Furthermore, the bottom of the energy well is not flat, but at a slant, as shown in Figure 2.8a. Both the barriers and the slant of the bottom of the energy well can be achieved in the fabrication process by vaporizing different composition of semiconductor materials (doping).
Figure 2.8 (a) An individual energy well with finite barrier height and a sloping energy bottom with the two lowest energy states and wavefunctions. (b) The superlattice structure in a quantum cascade laser modeled by successive PiB potential energy levels.
The slant of the energy well bottom has the effect of distorting the PiB wavefunctions as shown exaggerated for the two lowest energy states in Figure 2.8a. (The computation of wavefunctions in the presence of a sloping baseline will be discussed in Appendix 2, perturbation methods.) The distortion causes the amplitude of the wavefunction to shift toward lower potential energy with the consequence that an electron in the ground state of the well has a higher probability of tunneling through the barrier, due to their higher amplitude at the right side of the well. The “superlattice” is formed by having a large number of these energy wells arranged as shown in Figure 2.8b.
During the operation of the QCL, electrons are injected, via an electric current, at a potential energy marked by the * symbol in Figure 2.8b into a highly excited energy state and undergo a transition as indicated by the leftmost down arrow. During this transition, an (infrared) photon is emitted. Subsequently, the electron in the ground state may tunnel through the finite‐height barrier and arrives in the next quantum well and undergoes another transition. The emission and tunneling processes are repeated as many times as there are quantum wells in the superstructure. The term “cascade” in QCL is due to the fact that one electron can undergo many consecutive emission processes in the superlattice structure. By placing the superlattice crystal into an optic cavity, stimulated emission (see Chapter 3) from the excited states into the ground states of each well can be achieved.
References
1 1 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon.
2 2 Shoemaker, D.P., Garland, C.W., and Nibler, J.W. (1996). Experiments in Physical Chemistry, 6e. New York: McGraw‐Hill.
3 3 Banin, U. et al. (1999). Identification of atomic‐like electronic states in indium arsenide nanocrystal quantum dots. Nature 400 (6744): 542–544.
4 4 Faist, J. et al. (1994). Quantum Cascade Laser. Science 264 (5158): 553–556.
Problems
The following trigonometric integral relationships are needed for these problems:
