Excitons, Photoconductivity, Luminescence & Crystal Defects

1. The Nature of Excitonic Quasiparticles

When a photon with energy near the fundamental bandgap $h\nu \lesssim E_g$ is absorbed by an insulator or semiconductor, it promotes an electron from the valence band to the conduction band. The negatively charged electron ($q_e = -e$) and positively charged hole ($q_h = +e$) experience an attractive Coulomb interaction:

$$V(r) = -\frac{e^2}{4\pi\epsilon_r\epsilon_0 r}$$

Instead of escaping as free carriers, they can form a neutral composite bound state known as an exciton. Because the exciton carries zero net electrical charge, it cannot transport electrical current, but it can transport energy across the crystal without Joule heating.

2. Wannier-Mott vs Frenkel Excitons

• Wannier-Mott Excitons (Weakly Bound, Large Radius): Occur in semiconductors with large dielectric constants ($\epsilon_r \sim 10-16$) and small effective masses ($m^* \sim 0.05-0.2 m_0$). The Coulomb attraction is heavily screened. The electron and hole orbit around their common center of mass with an effective Bohr radius much larger than the lattice constant: $$a_{\text{exc}} = a_0 \epsilon_r \left( \frac{m_0}{\mu^*} \right), \quad \frac{1}{\mu^*} = \frac{1}{m_e^*} + \frac{1}{m_h^*}$$ Typically $a_{\text{exc}} \sim 5-20\text{ nm} \gg a_{\text{lattice}}$, enclosing hundreds of unit cells. The binding energy series forms hydrogen-like levels below the conduction band edge: $$E_n = E_g - \frac{R_y^*}{n^2}, \quad R_y^* = 13.6\text{ eV} \frac{\mu^* / m_0}{\epsilon_r^2} \sim 5-30\text{ meV}$$ Because $R_y^* \sim k_B T_{\text{room}}$, Wannier excitons are easily observed as sharp sub-bandgap absorption peaks at cryogenic temperatures.
• Frenkel Excitons (Tightly Bound, Small Radius): Occur in materials with small dielectric constants ($\epsilon_r \sim 2-4$) and large effective masses, such as alkali halides, rare gas solids (Ar, Kr, Xe), and molecular organic crystals (anthracene). The Coulomb screening is weak, resulting in immense binding energies $E_b \sim 0.1-1.0\text{ eV}$. The electron and hole reside essentially on the same atom or molecule ($a_{\text{exc}} \sim a_{\text{lattice}} \sim 0.3\text{ nm}$). Frenkel excitons hop from site to site as localized excitation waves, exhibiting Davydov splitting due to resonant intermolecular exchange.

1. Excitonic Optical Absorption Peaks

In the absence of excitonic interactions, the interband optical absorption coefficient for a direct bandgap semiconductor rises as a square root above the gap:

$$\alpha_0(\hbar\omega) \propto \sqrt{\hbar\omega - E_g}, \quad \hbar\omega \ge E_g$$

Coulomb attraction dramatically alters this spectrum (the Elliott formula):

1. Discrete Sub-Gap Peaks ($\hbar\omega < E_g$): A series of discrete hydrogen-like absorption lines appears at photon energies: $$\hbar\omega_n = E_g - \frac{R_y^*}{n^2}, \quad n = 1, 2, 3, \dots$$ The oscillator strength scales as $f_n \propto |\psi_n(0)|^2 \propto 1/n^3$, making the ground state ($n=1$) peak overwhelmingly prominent.
2. Continuous Sommerfeld Enhancement ($\hbar\omega \ge E_g$): Even above the ionization threshold, the Coulomb attraction pulls electron and hole wavefunctions together, enhancing the transition probability at the band edge by the Sommerfeld factor $S(E) = \frac{2\pi\eta}{1 - e^{-2\pi\eta}} \to 2\pi\eta$ as $\hbar\omega \to E_g$. The absorption steps up abruptly rather than starting smoothly from zero.

2. Exciton-Polaritons

When an optical photon couples strongly to a transverse optical exciton mode, energy oscillates coherently between the photonic and electronic states before dissipation can occur.

The coupled system cannot be described by independent photons and excitons, but forms a hybrid quasiparticle known as an exciton-polariton. The dispersion relation obeys:

$$\frac{c^2 k^2}{\omega^2} = \epsilon(\omega) = \epsilon_\infty + \frac{(\epsilon_0 - \epsilon_\infty) \omega_T^2}{\omega_T^2 - \omega^2}$$

At the crossover where the photon line $\omega = c k / \sqrt{\epsilon_\infty}$ intersects the exciton frequency $\omega_T$, an anti-crossing opens an energy gap known as the polariton stop-band (or longitudinal-transverse splitting $\omega_L - \omega_T$). Polaritons possess an ultra-light effective mass ($10^{-5} m_0$), enabling macroscopic Bose-Einstein condensation at room temperature in semiconductor microcavities!

1. Photoconductance & Excess Carrier Kinetics

When a semiconductor is illuminated by light with photon energy $h\nu \ge E_g$, optical generation creates excess electron-hole pairs at rate $G$ per unit volume per second:

$$\frac{dn}{dt} = G - \mathcal{R}_n, \quad \frac{dp}{dt} = G - \mathcal{R}_p$$

In the steady state ($dn/dt = 0$), excess carrier densities satisfy:

$$\Delta n = G \tau_n, \quad \Delta p = G \tau_p$$

where $\tau_n$ and $\tau_p$ are the excess electron and hole recombination lifetimes. The resulting change in electrical conductivity is the photoconductivity $\Delta\sigma$:

$$\Delta\sigma = e (\Delta n \mu_n + \Delta p \mu_p) = e G (\tau_n \mu_n + \tau_p \mu_p)$$

The photoconductive gain $\mathcal{G}$ is defined as the ratio of the number of electrons collected at the electrode to the number of absorbed photons:

$$\mathcal{G} = \frac{\tau_n}{t_{\text{transit}}} = \frac{\tau_n \mu_n \mathcal{E}}{L}$$

If the carrier lifetime $\tau_n$ exceeds the transit time $t_{\text{transit}} = L / v_{\text{drift}}$, electrons cycle multiple times through the external circuit before recombining, yielding gain $\mathcal{G} \gg 1$!

2. Carrier Recombination Mechanisms

• 1. Band-to-Band Radiative Recombination: An electron drops directly from the conduction band into an empty state in the valence band, emitting a photon of energy $h\nu \approx E_g$. The rate is $\mathcal{R}_{\text{rad}} = B (n p - n_i^2)$.
• 2. Shockley-Read-Hall (SRH) Trap-Assisted Recombination: Mediated by deep defect levels or impurities near mid-gap. An electron is captured by the neutral defect, followed by hole capture, releasing energy via lattice phonons. In indirect semiconductors (Si, Ge), SRH recombination dominates.
• 3. Auger Non-Radiative Recombination: A three-particle collision where an electron and hole recombine, but the released energy is transferred as kinetic energy to a third carrier (an electron in $n$-type, or a hole in $p$-type), which thermalizes via phonons. The rate scales as $\mathcal{R}_{\text{Auger}} \propto C_n n^2 p + C_p n p^2$, dominating at high carrier injection densities (concentrated solar cells and high-power laser diodes).

1. Physics of Space Charge Injection

In an ideal insulator or high-purity intrinsic semiconductor with ohmic contacts, free thermal carriers are negligible ($n_0 \approx 0$). When a voltage $V$ is applied across a sample of length $L$, the cathode injects excess electrons into the conduction band.

At low voltages, Ohm's law holds: $J = e n_0 \mu (V/L)$. However, as the voltage increases, the density of injected electrons exceeds the background thermal density. These uncompensated carriers accumulate in the bulk, forming a localized negative space charge cloud that screens the applied electric field.

2. Derivation of the Mott-Gurney Law

Assuming drift-dominated transport ($J = e n(x) \mu \mathcal{E}(x) = \text{const}$) in one dimension, Poisson's equation relates the space charge density to the electric field gradient:

$$\frac{d\mathcal{E}}{dx} = \frac{\rho(x)}{\epsilon_r\epsilon_0} = \frac{-e n(x)}{\epsilon_r\epsilon_0} = -\frac{J}{\epsilon_r\epsilon_0 \mu \mathcal{E}(x)}$$

Multiplying by $\mathcal{E}(x)$ and integrating from the injecting contact ($x = 0$, where for an ideal ohmic contact $\mathcal{E}(0) \approx 0$):

$$\mathcal{E} d\mathcal{E} = -\frac{J}{\epsilon_r\epsilon_0 \mu} dx \implies \frac{1}{2} \mathcal{E}^2(x) = -\frac{J x}{\epsilon_r\epsilon_0 \mu}$$ $$\mathcal{E}(x) = -\sqrt{-\frac{2 J x}{\epsilon_r\epsilon_0 \mu}}$$

Integrating the electric field to determine the applied voltage $V = -\int_0^L \mathcal{E}(x) dx$:

$$V = \int_0^L \sqrt{\frac{2 J x}{\epsilon_r\epsilon_0 \mu}} dx = \sqrt{\frac{2 J}{\epsilon_r\epsilon_0 \mu}} \left[ \frac{2}{3} x^{3/2} \right]_0^L = \frac{2}{3} \sqrt{\frac{2 J}{\epsilon_r\epsilon_0 \mu}} L^{3/2}$$

Squaring both sides and solving for the current density $J$:

$$V^2 = \frac{8 J L^3}{9 \epsilon_r\epsilon_0 \mu} \implies J_{\text{SCLC}} = \frac{9}{8} \epsilon_r\epsilon_0 \mu \frac{V^2}{L^3}$$

This is the celebrated Mott-Gurney Square Law (the solid-state analogue of Child's three-halves law in vacuum tubes). SCLC measurements provide a universal method for determining carrier drift mobilities in organic semiconductors, perovskites, and wide-bandgap insulators.

1. Luminescence Phenomena in Solids

Luminescence is the non-thermal emission of electromagnetic radiation from an electronically excited solid:

• Photoluminescence: Excitation by ultraviolet or visible photons.
• Electroluminescence: Excitation by direct electrical current injection (LEDs and OLEDs).
• Cathodoluminescence: Excitation by an energetic electron beam (CRTs and SEM microanalysis).

According to Stokes' Law, the emitted photon energy $h\nu_{\text{emission}}$ is almost invariably lower than the excitation photon energy $h\nu_{\text{absorption}}$:

$$\Delta E_{\text{Stokes}} = h\nu_{\text{abs}} - h\nu_{\text{em}} > 0$$

The missing energy is dissipated into the crystal lattice as vibrational heat (phonons).

• Fluorescence: Rapid emission ($\tau \sim 10^{-9}-10^{-7}\text{ s}$) originating from spin-allowed transitions ($S \to S$), terminating immediately when excitation ceases.
• Phosphorescence: Delayed, persistent afterglow ($\tau \sim 10^{-3}-10^2\text{ s}$) caused by non-radiative intersystem crossing into a metastable spin-triplet trap state ($T \to S_0$), which is quantum-mechanically dipole-forbidden by spin selection rules ($\Delta S = 0$).

2. The Configuration Coordinate Model & Franck-Condon Principle

The coupling between localized electronic transitions and lattice vibrations is modeled by the Configuration Coordinate Diagram:

The potential energy curves of the electronic ground state $|g\rangle$ and excited state $|e\rangle$ are plotted as harmonic parabolas against a generalized vibrational coordinate $Q$ representing the breathing displacement of surrounding ions:

$$E_g(Q) = \frac{1}{2} M \omega^2 Q^2, \quad E_e(Q) = E_0 + \frac{1}{2} M \omega^2 (Q - Q_0)^2$$

where $Q_0$ is the equilibrium lattice relaxation displacement caused by the altered electronic charge distribution in the excited state.

• Franck-Condon Principle: Electronic transitions occur on timescales ($10^{-15}\text{ s}$) far faster than nuclear motion ($10^{-13}\text{ s}$). Transitions are represented by vertical lines on the diagram ($\Delta Q = 0$).
• Absorption: Occurs vertically from $Q = 0$ to an excited vibrational level of $|e\rangle$ at energy $E_{\text{abs}} = E_0 + \frac{1}{2} M\omega^2 Q_0^2$.
• Lattice Relaxation: The ions relax non-radiatively to the new equilibrium minimum $Q = Q_0$, releasing relaxation energy $S \hbar\omega = \frac{1}{2}M\omega^2 Q_0^2$ (where $S$ is the dimensionless Huang-Rhys parameter).
• Emission: Occurs vertically from $Q = Q_0$ to a high vibrational state of $|g\rangle$ at energy $E_{\text{em}} = E_0 - \frac{1}{2} M\omega^2 Q_0^2$.
• Stokes Shift: $$\Delta E_{\text{Stokes}} = E_{\text{abs}} - E_{\text{em}} = 2 S \hbar\omega$$

1. Thermodynamics of Point Defects in Thermal Equilibrium

Unlike dislocations or grain boundaries, point defects are thermodynamically stable and exist in non-zero concentrations in any crystal at $T > 0\text{ K}$.

Creating $n$ point defects requires enthalpy $\Delta H = n E_d > 0$. However, distributing $n$ defects across $N$ available lattice sites introduces immense configurational entropy:

$$S_{\text{config}} = k_B \ln W = k_B \ln\left( \frac{N!}{(N - n)! n!} \right)$$

Using Stirling's approximation $\ln x! \approx x\ln x - x$:

$$S_{\text{config}} \approx k_B [N\ln N - (N - n)\ln(N - n) - n\ln n]$$

The Helmholtz free energy change is:

$$\Delta F(n) = n E_d - T S_{\text{config}}$$

Minimizing with respect to defect count $\frac{\partial \Delta F}{\partial n} = 0$:

$$E_d - k_B T \ln\left( \frac{N - n}{n} \right) = 0 \implies \frac{n}{N - n} = e^{-E_d / k_B T}$$

Since $n \ll N$ under normal conditions:

$$n = N \exp\left( -\frac{E_d}{k_B T} \right)$$

2. Schottky vs Frenkel Defects in Stoichiometric Ionic Crystals

• Schottky Defects (Paired Vacancies): Consists of a stoichiometric pair of vacancies: one cation vacancy $V_{\text{cat}}'$ and one anion vacancy $V_{\text{an}}^\bullet$, leaving the overall crystal strictly charge-neutral. The displaced ions migrate to the external crystal surface. If $E_S$ is the formation energy of a Schottky pair: $$n_S \approx N \exp\left( -\frac{E_S}{2 k_B T} \right)$$ The factor of 2 in the denominator arises because two independent vacancy configurations must be chosen. Dominates in alkali halides with similar cation and anion ionic radii (e.g. $\text{NaCl}, \text{KCl}, \text{CsCl}$), leading to a measurable reduction in crystal bulk density.
• Frenkel Defects (Vacancy-Interstitial Pairs): Consists of an ion (typically the smaller cation) displaced from its normal lattice site into an adjacent interstitial void, creating a vacancy $V_{\text{cat}}'$ and an interstitial cation $M_i^\bullet$. If $N$ is the number of regular sites and $N_i$ is the number of available interstitial sites, with formation energy $E_F$: $$n_F \approx \sqrt{N N_i} \exp\left( -\frac{E_F}{2 k_B T} \right)$$ Dominates in ionic crystals with open crystal structures and large radius mismatches (e.g. silver halides $\text{AgCl}, \text{AgBr}$, and fluorites $\text{CaF}_2$), leaving bulk density unchanged.

1. Mechanisms of Atomic Diffusion in Solids

Mass transport in crystalline solids occurs via discrete thermally activated atomic jumps across potential barriers $E_m$:

• Vacancy Mechanism: An atom jumps into an adjacent empty lattice vacancy. The jump probability is proportional to the product of vacancy probability $n_v/N \propto e^{-E_v/k_B T}$ and hopping probability $\nu e^{-E_m/k_B T}$. The diffusion coefficient obeys an Arrhenius Law: $$D(T) = D_0 \exp\left( -\frac{E_a}{k_B T} \right), \quad E_a = E_v + E_m$$ Dominates self-diffusion and substitutional solute diffusion.
• Interstitial Mechanism: Small solute atoms (H, C, N, O) hop between adjacent interstitial voids without requiring pre-existing vacancies ($E_a = E_{m,\text{int}}$), diffusing orders of magnitude faster than host atoms.

2. Color Centers: The F-Center

When an alkali halide crystal (such as pure transparent $\text{NaCl}$) is heated in an excess vapor of sodium metal, sodium atoms deposit on the surface, lose electrons to form $\text{Na}^+$ ions, and draw $\text{Cl}^-$ ions out from the interior to maintain stoichiometry.

This leaves behind empty anion vacancies inside the bulk. To preserve local charge neutrality, the freed electron becomes trapped inside the positive electrostatic potential well formed by the six surrounding alkali metal cations ($\text{Na}^+$).

This trapped-electron defect is known as an $F$-center (from the German Farbzentrum, meaning color center).

• The trapped electron behaves as a particle in a 3D finite spherical potential well.
• Its optical absorption spectrum features a strong, broad resonance corresponding to the electric dipole transition from the $1s$-like ground state to the $2p$-like excited state.
• For $\text{NaCl}$, the $F$-band absorption peak occurs at $\lambda \approx 460\text{ nm}$ (blue light absorption), tinting the normally clear crystal an intense yellow-brown. In $\text{KCl}$, it absorbs at $560\text{ nm}$, coloring the crystal violet.
• The peak wavelength follows the empirical Mollwo-Ivey Relation: $$\lambda_{\max} \propto a^n \quad (n \approx 1.8-2.0)$$ where $a$ is the lattice constant of the alkali halide.