Quantum-Dot-Photodetector · PbS colloidal quantum dots · short-wave infraredHow small must a PbS dot be to see a given wavelength?

Choose the wavelength a detector should absorb. The dot diameter follows from the measured sizing curve; the page then shows what the textbook models predict for that dot, how many of its atoms sit at the surface, what the smaller gap costs in dark current, and how the ligand length sets the current between dots. Everything is computed in the browser.

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Band gap against dot diameter

Three models of the same dot; the dashed line is the gap the wavelength asks for

Atoms at the surface

Counted on the rock-salt lattice, and a continuum shell one Pb–S distance thick

Dot size and band gap

measured sizing curve [7]: E₀(d) = 0.41 + 1 / (0.0252 d² + 0.283 d) (d in nm, E in eV); the diameter for a wavelength solves E₀(d) = hc/λ

effective-mass approximation [5]: E = E_g + ħ²π²/(2R²) (1/m_e* + 1/m_h*) − 1.786 e²/(4πε₀ε_∞R)

hyperbolic bands [18]: E = [E_g² + 2E_għ²/m* (π/R)²]^½ − 1.786 e²/(4πε₀ε_∞R)

PbS: E_g = 0.41 eV at 300 K [6, 7], m_e* = m_h* = 0.085 m₀ [18], ε_∞ = 17.2. Both models assume isotropic masses and infinite walls; PbS has an anisotropic, fourfold-degenerate L-point band edge and finite ligand barriers, so neither is used for sizing.

Surface atoms

Rock salt puts Pb and S on a simple-cubic lattice of spacing h = a/2 (a = 0.5936 nm [27]); every atom has six nearest neighbours. All lattice sites within d/2 of the centre are counted, and an atom is at the surface when fewer than six of its neighbours are inside. The count depends on where the centre sits, so it is averaged over a sphere centred on an atom and one centred in a cube of atoms. Shell model: f = 1 − (1 − a/d)³. The count is purely geometric: no facets, reconstruction or ligand coverage.

Dark current and tunnelling

ideal diode, gap constant in T [22]: J_diff ∝ T³ e^−E_g/kT (n = 1), J_gen ∝ T^1.5 e^−E_g/2kT (n = 2); ratio of two dots = e^ΔE_g/nkT; halving step = ln 2 / (d ln J/dT)

tunnelling: rate ∝ e^{−β d}, β = 2κ = 2(2m*φ)^½/ħ for a rectangular barrier

β of the dot film is the range measured by Liu et al. [19] for PbSe dots with alkanedithiols of different lengths, 1.08–1.10 Å⁻¹ (the midpoint is used); the alkanethiol monolayer has β = 0.79 Å⁻¹ with φ = 1.39 eV [21]; the vacuum gap assumes a 4.5 eV work function. The presentation this repository supports used 0.96 Å⁻¹ (a factor 2.6 per Å, from a review); see the README, Correction. Hopping mobility also contains thermal activation, so β is an effective value.

Checks

Sources

[5] L. E. Brus, J. Chem. Phys. 80, 4403 (1984).
[6] F. W. Wise, Acc. Chem. Res. 33, 773 (2000).
[7] I. Moreels et al., ACS Nano 3, 3023 (2009).
[18] Y. Wang, A. Suna, W. Mahler, R. Kasowski, J. Chem. Phys. 87, 7315 (1987).
[19] Y. Liu et al., Nano Lett. 10, 1960 (2010).
[21] W. Wang, T. Lee, M. A. Reed, Phys. Rev. B 68, 035416 (2003).
[22] S. M. Sze, K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (2007).
[23] M. Vafaie et al., Matter 4, 1042 (2021): PbS CQD photodiode. [24] J.-X. Zhang et al., Front. Phys. 9, 678192 (2021): InGaAs focal-plane array.
[27] Handbook of Mineralogy: galena.

Code: github.com/Shuruyue/Quantum-Dot-Photodetector. The physics block of this page is checked against the Python package by scripts/verify_web_page.py. Style shared with the ethanol–water page.