Brus equation

The Brus equation or confinement energy equation can be used to describe the emission energy of quantum dot semiconductor nanocrystals in terms of the band gap energy E_gap, the Planck constant h, the radius of the quantum dot r, as well as the effective mass of the excited electron m_e* and of the excited hole m_h*. The equation was named after Louis E. Brus who independently discovered it.

The radius of the quantum dot affects the wavelength of the emitted light due to quantum confinement, and this equation describes the effect of changing the radius of the quantum dot on the wavelength λ of the emitted light (and thereby on the emission energy ΔE = hc/λ, where c is the speed of light). This is useful for calculating the radius of a quantum dot from experimentally determined parameters.

The overall equation is

$\Delta E(r)=E_{\mathrm {gap} }+{\frac {h^{2}}{8r^{2}}}\left(1/m_{\mathrm {e} }^{*}+1/m_{\mathrm {h} }^{*}\right).$

E_gap, m_e*, and m_h* are unique for each nanocrystal composition. For example, with cadmium selenide (CdSe) nanocrystals:

E_gap (CdSe) = 1.74 eV = 2.8×10⁻¹⁹ J,

m_e* (CdSe) = 0.13 m_e = 1.18×10⁻³¹ kg,

m_h* (CdSe) = 0.45 m_e = 4.09×10⁻³¹ kg.