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For CdSe quantum dots (bulk band gap 1.74 eV), assume a simple model in which the confinement energy increase is inversely proportional to the square of the radius R, and that it is 0.30 eV for R = 3.0 nm. Which value is closest to the effective band gap of a quantum dot with R = 2.0 nm? Neglect the Coulomb interaction between electron and hole.

  1. AAbout 2.19 eV
  2. BAbout 2.42 eV
  3. CAbout 2.75 eV
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Answer: B. About 2.42 eV

The confinement energy is proportional to 1/R², so going from R = 3.0 nm to 2.0 nm multiplies it by (3.0/2.0)² = 2.25, giving 0.30 × 2.25 = 0.675 eV. The effective band gap is 1.74 + 0.675 ≈ 2.42 eV, corresponding to an emission wavelength of roughly 1240/2.42 ≈ 513 nm (green). At R = 3.0 nm it is 2.04 eV (about 608 nm, orange to red), so smaller dots emit at shorter wavelengths. About 2.19 eV assumes proportionality to 1/R, and about 2.75 eV to 1/R³. In real quantum dots, the effective mass approximation breaks down at small sizes (band nonparabolicity), and the confinement energy usually increases more gently than 1/R², so this value is only a guide.

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