= Solution
In a periodic crystal potential, atomic levels broaden into bands of allowed <bloch states>. Coupling between states of the same symmetry opens <avoided crossings>, including Bragg-coupled states in the nearly free-electron picture. Bonding and antibonding combinations provide a complementary tight-binding picture. A <semiconductor> has a filled <valence band> and an empty <conduction band> at zero <temperature> separated by a forbidden <energy> interval; thermal excitation, light and doping provide mobile <electrons> and holes. The gap depends on the crystal's <periodic potential>, bonding and symmetry, not on a classical collision barrier. In a direct gap, the relevant extrema coincide in <wavevector>.
For the stated parabolic edges, take $E_v(k)=-\hbar^2k^2/(2m_h^*)$ and $E_c(k)=E_g+\hbar^2k^2/(2m_e^*)$. Negligible <photon> <momentum> and no <phonon> make the transition nearly vertical in $E$ versus $k$. <parabolic-band optical energy sharing> divides the excess <photon> <energy> $0.32\ \mathrm{eV}$ in inverse proportion to the effective masses:
$$
K_e=0.32\frac{0.4}{0.14+0.4}=0.2370\ \mathrm{eV},\qquad
K_h=0.32\frac{0.14}{0.14+0.4}=0.0830\ \mathrm{eV}.
$$
Thus the initial and final <electron> energies, relative to the valence-band maximum, are
$$
\boxed{E_i=-0.0830\ \mathrm{eV},\qquad E_f=0.4170\ \mathrm{eV}}.
$$
Their difference is $0.5000\ \mathrm{eV}$. The selected $|k|$ is about $9.33\times10^8\ \mathrm{m^{-1}}$; either sign gives the same <energy> in this symmetric approximation.
A shallow donor supplies an <electron> in a screened impurity potential, so its localized level lies just below the conduction edge. A shallow acceptor binds a hole and has an electron-occupation level just above the valence edge; thermal ionization supplies mobile holes. The level positions are measured relative to the relevant band edge, and deep impurities need not obey this hydrogenic approximation.
For the supplied bound-carrier <mass> and unit effective ionic charge, <dielectric screening> and effective <mass> change the hydrogen binding <energy> to
$$
\boxed{E_b=13.6\frac{0.1}{12^2}\ \mathrm{eV}=9.44\ \mathrm{meV}}.
$$
This corresponds to $E_b/k_B\simeq110\ \mathrm K$ and an effective Bohr radius about $6.35\ \mathrm{nm}$. The binding is shallow compared with the <band gap> and typical atomic binding. Trivalent denotes chemical valence, not an unscreened hydrogenic charge of three. An acceptor physically binds a hole; the question's stated $0.1m_e$ is used as its stipulated bound-carrier <mass> rather than silently replacing it by the separate band-edge $0.4m_e$.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-2-semiconductor-bands.png]
{title=Original direct-gap vertical transition with mass-dependent energy sharing, and separate schematic shallow donor and acceptor levels}
Back to article page
