Impurities and Excitons

Abstract

Selected impurities are frequently introduced into semiconductors to make them n–type or p–type. The introduction of impurities into a crystal lattice not only shifts the Fermi level, but also results in a perturbation to the periodic potential, giving rise to bound impurity levels which often occur in the band gap of the semiconductor.

Problems

20.1

This problem considers the density of Frenkel defects. By considering $N'$ as the density of possible interstitial sites, show that the density of occupied interstitial sites is

$$\begin{aligned} n_i\simeq (NN')^{\frac{1}{2}} \; \exp \biggr [{-\frac{E_i}{2k_BT}} \biggr ] \end{aligned}$$

(20.26)

where $E_i$ is the energy needed for removing an atom from a lattice site to form an interstitial defect site.

20.2

The band gaps of $\mathrm{WSe}_2$ and $\mathrm{MoSe}_2$ are 1.64 and 1.61 eV, respectively, and their conduction band offset is 228 meV with $\mathrm{WSe}_2$ higher in energy than for $\mathrm{MoSe}_2$.

• (a) Estimate the maximum energy that an interlayer exciton could have (assuming zero binding energy)?

• (b) The observed value of the interlayer exciton emission is 1.350 eV. Based on this, what is the exciton binding energy of this interlayer exciton?

• (c) Based on this binding energy, do you expect these interlayer excitons to be stable at room temperature?

• (d) Explain why the interlayer exciton binding energy is different from the binding energy of intralayer excitons 400 meV.

20.3

The exciton binding energy of an exciton in monolayer $\mathrm{MoS}_2$ is 400 meV. Assuming that this binding energy is dominated by the ground particle-in-a-box energy corresponding to the 3.0 Å layer thickness of this material, with an effective mass of $m^*=0.6m_o$, estimate the binding energy of bilayer $\mathrm{MoS}_2$ (thickness $=$ 9.16 Å). (See Fig. 20.21)

Fig. 20.21

Ball and stick model of $\mathrm{MoS}_2$. (Taken from Liang et al. 2008)

20.4

Calculate the bound state energies of excitons in GaAs (with both heavy and light holes) for

• (a) bulk GaAs, assuming a hydrogenic model.

• (b) a 50 Å quantum well, assuming the binding energy is dominated by the particle-in-a-box energy corresponding to the finite quantum well thickness.

• (c) a 5 Å quantum well, assuming the binding energy is dominated by the particle-in-a-box energy corresponding to the finite quantum well thickness.

20.5

Repeat Problem 20.4 for germanium.

20.6

• (a) By what factor (approximately) does a quantum well with a width of 50  Å increase the binding energy of an exciton for a quantum well of GaAs? The lattice constant for GaAs is 5.65  Å and the exciton radius of bulk GaAs is about 150  Å.

• (b) The spin-orbit interaction for GaAs results in a splitting of the valence band , with the split-off band extremum at an energy 0.34 eV below the valence band extremum. Using 0.34 eV as a measure of the size of the spin-orbit interaction in GaAs , estimate the effect of the spin-orbit interaction on the binding energy of the exciton. [Hint: Consider the effect of the large Bohr radius on the magnitude of the spin-orbit interaction.] The donor impurity radius for GaAs is 136  Å. Use values of $m_{hh} = 0.62$ and $m_{\ell h} = 0.074$ for the hole masses and a dielectric constant of 15.

• (c) If an exciton has no net charge, why is an exciton attracted to a charged impurity center to form a bound exciton?

20.7

Is the binding energy for an exciton in a semiconducting wire of 100 Å diameter increased or decreased relative to bulk values if the Bohr radius of the exciton for this material in the bulk is 30 Å? Under which conditions would the effect of confining the exciton within a small diameter wire be large? What is the reason for your answer?