More Electron Traps — Question 7

Question 7

Size dispersion broadens a confinement line even without a short lifetime

An ensemble of idealized nanostructures has transition energy $\Delta E=A/L^2$, where A is fixed. Widths have mean $L_0$ and a narrow Gaussian spread $\sigma_L=.0200L_0$. The central transition energy is $.200\,\mathrm{eV}$. Neglect lifetime and instrumental broadening.

1. Use linear propagation to find the energy standard deviation.

2. Find the approximate energy FWHM using $w_E=2\sqrt{2\ln2}\,\sigma_E$.

Question 7 - Solution

Size dispersion broadens a confinement line even without a short lifetime

(a) Method and calculation

$$\frac{d\Delta E}{dL}=-\frac{2A}{L^3}\Rightarrow\frac{\sigma_E}{\Delta E_0}\simeq\frac{2\sigma_L}{L_0}.$$$$\boxed{\sigma_E=.0400(.200)\,\mathrm{eV}=.00800\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$\boxed{w_E=2\sqrt{2\ln2}(.00800)\,\mathrm{eV}=.0188386\,\mathrm{eV}}.$$$$\boxed{w_E/\Delta E_0=.0941928}.$$

Physical check

This is inhomogeneous broadening from different structures, not a decay linewidth of one structure. The Gaussian approximation relies on the small fractional size spread.