Doped Semiconductors — Question 4

Question 4

Screening and effective mass make a shallow donor large

Use the hydrogenic donor model $E_D=13.6\,\mathrm{eV}(m^*/m_e)/\varepsilon_r^2$ and $a_D=a_0\varepsilon_r/(m^*/m_e)$, with $a_0=.0529\,\mathrm{nm}$. Let $m^*/m_e=.200$ and $\varepsilon_r=12.0$.

1. Evaluate the binding energy and donor radius.

2. Compare the radius with lattice spacing .500 nm and find the longest wavelength able to ionize the ideal donor using $hc=1240\,\mathrm{eV\,nm}$.

Question 4 - Solution

Screening and effective mass make a shallow donor large

(a) Method and calculation

$$\boxed{E_D=13.6(.200)/12^2=.01889\,\mathrm{eV}}.$$$$\boxed{a_D=.0529(12)/.200=3.174\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{a_D/a=6.348}.$$$$\boxed{\lambda_{\max}=hc/E_D=6.565\times10^4\,\mathrm{nm}=65.65\,\mu\mathrm m}.$$

Physical check

The envelope extends over many lattice sites, supporting an effective-mass description. Central-cell effects and anisotropy are omitted by the specified model.