Quantum Dots in Solar Cells · AI Generator #GEN-B7B81FE8

Physics · quantum-mechanics

Problem

Quantum dots are tiny semiconductor nanocrystals used in advanced solar cells to improve light absorption. A spherical quantum dot can be modeled as a three-dimensional infinite potential well (particle in a spherical box). For a quantum dot of radius R=2.5  nmR = 2.5\;\text{nm}, an electron is confined with energy levels given by: Enℓ=ℏ2αnℓ22mR2E_{n\ell} = \frac{\hbar^2 \alpha_{n\ell}^2}{2mR^2} where αnℓ\alpha_{n\ell} are the zeros of spherical Bessel functions. The ground state has α10=π\alpha_{10} = \pi and the first excited state has α11=4.49\alpha_{11} = 4.49. (a) Calculate the energy difference between the first excited state and ground state in eV. (b) When the electron transitions from the first excited state to the ground state, it emits a photon. What is the wavelength of this emitted light, and in which part of the electromagnetic spectrum does it lie? Use: ℏ=1.055×10−34  J⋅s\hbar = 1.055 \times 10^{-34}\;\text{J·s}, me=9.109×10−31  kgm_e = 9.109 \times 10^{-31}\;\text{kg}, h=6.626×10−34  J⋅sh = 6.626 \times 10^{-34}\;\text{J·s}, c=3.00×108  m/sc = 3.00 \times 10^8\;\text{m/s}, 1  eV=1.602×10−19  J1\;\text{eV} = 1.602 \times 10^{-19}\;\text{J}

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