Assignment_13 08 revised - Physics 214 Assignment 13...

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Physics 214 Assignment 13 Concepts: Particle in a box Atomic transitions and light emission Harmonic oscillator Hydrogen atom Correspondence principle Lasers Uncertainty principle Quantum tunneling Reading: Y&F, Vol. 3, Chapters 38, 39, 40 and 41. Assignment: Due Friday, May 2. Please turn in this sheet stapled to the top of your work. Physics Problems: 1. Y&F, Chapter 40, Question 11 2. Y&F, Chapter 40, Question 14 3. Semiconductor quantum wells and dots. The formulas derived for electrons in boxes can be applied to electrons and holes in semiconductors, with one important correction: the electrons and holes in the semiconductor each have an effective mass m eff that is different than the mass of a free electron. It is this mass that must be used in all formulas. (a) Semiconductor lasers are usually formed from multiple layers of semiconductor, each layer having a different composition. One or more of these layers is often made very thin to create a quantum well, which can increase laser efficiency (photons out/electrical energy in). In an AlGaAs-GaAs quantum well laser, the electrons are confined in a 10 nm wide GaAs layer, in which m eff,e =0.067m e . What is the ground state energy E 1,e of an electron in this well? (b) The transitions that give rise to laser light are between the quantum well states in the conduction band and in the valence band, not between quantum well states within the same band. The energy of the emitted photons is hf = E g + E 1,e + E 1,h , where E g is the separation between the conduction and valence band of bulk GaAs, E 1,e is the amount by which confinement raises the ground state energy of electrons in the conduction band, and E 1,h is the amount by which confinement lowers the ground state energy of holes in the valence band. Thus, confinement increases the emitted light frequency. What is the ratio (E 1,e + E 1,h )/E g ? Use E g = 1.43 eV, E 1,e from (a), and calculate E 1,h using m eff,h =0.45 m e . This roughly gives the fractional increase in photon energy due to confinement in the quantum well. Physics 214, Spring 2008 1 Cornell University
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Bonus: (c) The brightly colored liquids shown in lecture contain colloidal CdSe semiconductor nanocrystals. By carefully timing the growth of these crystals from solution, crystals of well defined and controllable size can be obtained. The resulting quantum confinement increases the energy of the emitted photons compared with bulk CdSe, as in the case of quantum wells. For CdSe, E g = 1.73 eV, m eff,e =0.13m e and m eff,h =0.45 m e . What must be the diameter of a nanocrystal to produce light emssion at 600 nm (red), 560 nm (yellow) and 490 nm (blue)? 4. YF Chapter 40, Problem 56. Add: (d) Take the limit U 0 in your equation from part (c) and show that the allowed k’s and energies are what we already found for the infinite well.
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Assignment_13 08 revised - Physics 214 Assignment 13...

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