Carrier Dynamics Slide Deck Optoelectronic Materials • Part I
Hours 1–4 • Advanced Physics
Fundamental Carrier Thermodynamics
Bandgap Engineering & Carrier Dynamics
Non-equilibrium carrier injection, quasi-Fermi level splitting, recombination channels, and optical gain.
01. Bandgap & Lattice
Direct vs indirect gaps and critical thickness.
02. Quasi-Fermi Levels
Splitting \(\Delta F = F_n - F_p\) under carrier injection.
03. Gain Threshold
Bernard-Duraffourg condition and ABC model.
01 • Structural Properties Epitaxial Compatibility
Bandgap Tuning vs. Lattice Mismatch
Vegard's Law & Band Bowing
Linear lattice parameter in ternary alloy \(A_x B_{1-x} C\):
\[ a(x) = x \cdot a_{AC} + (1-x) \cdot a_{BC} \]
Fundamental bandgap with bowing parameter \(b\):
\[ E_g(x) = x E_{AC} + (1-x) E_{BC} - b \cdot x(1-x) \]
Strain & Critical Thickness
Misfit strain \(\epsilon_0 = (a_s - a_e)/a_e\) imposes critical limit \(h_c\):
\[ h_c \approx \frac{b_{\text{burg}}}{2\pi \epsilon_0 (1+\nu)} \left[ \ln\left(\frac{h_c}{b_{\text{burg}}}\right) + 1 \right] \]
Layers exceeding \(h_c\) relax via misfit dislocations, introducing non-radiative recombination centers.
Direct bandgap semiconductors provide high radiative efficiency without phonon mediation. Optoelectronics Core
02 • Carrier Statistics Non-Equilibrium Steady State
Quasi-Fermi Level Splitting (\(\Delta F\))
Intra-band Thermalization
Carrier-carrier scattering (\(\sim 10^{-13}\text{ s}\)) establishes separate quasi-Fermi levels \(F_n\) and \(F_p\) before recombination (\(\sim 10^{-9}\text{ s}\)).
\[ f_c(E) = \left[ 1 + e^{(E - F_n)/k_B T} \right]^{-1} \]
\[ f_v(E) = \left[ 1 + e^{(E - F_p)/k_B T} \right]^{-1} \]
Carrier Population Product
Excess carrier injection dramatically amplifies the \(n \cdot p\) density product:
\[ n \cdot p = n_i^2 \exp\left( \frac{F_n - F_p}{k_B T} \right) \gg n_i^2 \]
Separation \(\Delta F = F_n - F_p > 0\) quantifies optical excitation energy stored in electronic subsystems.
High injection drives \(F_n\) toward the conduction band and \(F_p\) toward the valence band. Carrier Equilibrium
03 • Dynamic Losses Carrier Lifetime
Recombination Channels: The ABC Model
Shockley-Read-Hall
\[ R_{\text{SRH}} = A \cdot n \]
Defect-mediated trapping in the forbidden band. Dominates at low \(n\).
Radiative Emission
\[ R_{\text{rad}} = B \cdot n^2 \]
Direct electron-hole annihilation producing photons. Essential for LEDs.
Auger Process
\[ R_{\text{Auger}} = C \cdot n^3 \]
Energy transferred to a 3rd carrier. Dominates at high injection (droop).
Internal Quantum Efficiency (IQE):
Peak efficiency is achieved at carrier density \(n_{\text{opt}} = \sqrt{A/C}\).
\[ \eta_{\text{int}} = \frac{B n^2}{A n + B n^2 + C n^3} \]
Total carrier lifetime is given by \(\tau^{-1} = A + B n + C n^2\). Kinetics Overview
04 • Optical Gain Lasing Criterion
Net Optical Gain & Bernard-Duraffourg Condition
Gain Spectrum \(g(\hbar\omega)\)
Net amplification when stimulated emission exceeds absorption:
\[ g(\hbar\omega) = \alpha_0(\hbar\omega) \left[ f_c(E_2) - f_v(E_1) \right] \]
Positive net gain requires inversion:
\[ f_c(E_2) > f_v(E_1) \]
The Bernard-Duraffourg Criterion
Evaluation with Fermi distributions yields the necessary condition:
\[ E_g < \hbar\omega < (F_n - F_p) \]
Lasing threshold balances confinement factor \(\Gamma\) and cavity loss:
\[ \Gamma g_{\text{th}} = \alpha_i + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right) \]
Quasi-Fermi splitting sets the strict thermodynamic ceiling for stimulated emission energy. Lesson 1 Review
Carrier Dynamics Learning Pack Physics Honours • Optoelectronic Materials (Hours 1–4)
Carrier Dynamics & Optical Gain
Pack 01
Student Name: ____________________________________
Date: ________________________
Theoretical Framework • Non-Equilibrium Statistics & Net Gain
Under steady-state optical or electrical injection, intra-band scattering establishes independent quasi-Fermi levels \(F_n\) (conduction band) and \(F_p\) (valence band). The net stimulated emission rate per unit volume between state \(E_2\) and \(E_1\) (\(E_2 - E_1 = \hbar\omega\)) is governed by:
\( r_{\text{stim}}(\hbar\omega) - r_{\text{abs}}(\hbar\omega) = B_{21} \rho(\hbar\omega) [f_c(E_2) - f_v(E_1)] \)
Optical gain requires \( f_c(E_2) > f_v(E_1) \), which directly establishes the Bernard-Duraffourg condition : \( E_g < \hbar\omega < (F_n - F_p) \).
Problem 1 • Formal Proof of Bernard-Duraffourg Inversion (15 pts)
Starting from the Fermi-Dirac occupation functions \(f_c(E_2) = [1 + \exp((E_2 - F_n)/k_B T)]^{-1}\) and \(f_v(E_1) = [1 + \exp((E_1 - F_p)/k_B T)]^{-1}\), prove algebraically that the condition for net optical amplification \(f_c(E_2) - f_v(E_1) > 0\) requires the quasi-Fermi level separation to strictly exceed the transition photon energy: \(\Delta F \equiv F_n - F_p > \hbar\omega\).
Optoelectronic Materials • Physics Core Page 1 of 2
Quantitative Dynamics & Structural Limits Pack 01 • Page 2
Problem 2 • ABC Kinetics & Optimum Efficiency in GaAs (20 pts)
At \(T = 300\text{ K}\), a \(\text{GaAs}\) active layer has recombination coefficients: \(A = 1.0 \times 10^7\text{ s}^{-1}\) (SRH), \(B = 1.2 \times 10^{-10}\text{ cm}^3/\text{s}\) (radiative), and \(C = 3.0 \times 10^{-30}\text{ cm}^6/\text{s}\) (Auger).
(a) Optimum Carrier Density
Derive \(n_{\text{opt}}\) where Internal Quantum Efficiency \(\eta_{\text{int}}(n)\) reaches its maximum, and compute its numerical value in \(\text{cm}^{-3}\).
(b) Maximum IQE & Lifetime
Evaluate \(\eta_{\text{int}}^{\text{max}}\) and calculate the differential carrier lifetime \(\tau\) at this peak operating point.
Problem 3 • Critical Epitaxial Thickness (15 pts)
An \(\text{In}_{0.2}\text{Ga}_{0.8}\text{As}\) epilayer (\(a_e = 5.734\text{ \AA}\)) is grown pseudomorphically on a \(\text{GaAs}\) substrate (\(a_s = 5.653\text{ \AA}\)). Using Poisson's ratio \(\nu = 0.31\) and Burgers vector magnitude \(b_{\text{burg}} = 4.0\text{ \AA}\), calculate the in-plane misfit strain \(\epsilon_0\), and estimate the Matthews-Blakeslee critical thickness \(h_c\) (in nm). State the physical consequence of exceeding \(h_c\) on LED radiative efficiency.
Carrier Dynamics & Optical Gain • Practice & Derivations Page 2 of 2
Quantum Structures Slide Deck Optoelectronic Materials • Part II
Hours 5–8 • Advanced Physics
Low-Dimensional & Molecular Photophysics
Quantum Structures & Organic Semiconductors
Inter-band and intra-band transitions, polarization selection rules, and conjugated \(\pi\)-electron molecular systems.
01. Confinement & DOS
3D bulk continuum to step-like 2D density of states.
02. Optical Transitions
Inter-band (\(\Delta n = 0\)) vs intra-band (\(z\)-polarized).
03. Organic Systems
Frenkel excitons, HOMO-LUMO, and spin statistics.
01 • Dimensional Physics Density of States (DOS)
Density of States in Reduced Dimensions
Bulk (3D) vs. Quantum Well (2D)
3D parabolic continuum:
\[ g_{\text{3D}}(E) = \frac{1}{2\pi^2}\left(\frac{2m^*}{\hbar^2}\right)^{3/2} \sqrt{E - E_c} \]
2D staircase with subband thresholds \(E_n\):
\[ g_{\text{2D}}(E) = \sum_n \frac{m^*}{\pi \hbar^2 L_z} \Theta(E - E_n) \]
Optoelectronic Advantages
• Sharper DOS at band edge: Massive carrier concentration concentrated at lowest subband.
• Reduced transparency current: Drastically suppresses threshold current density in laser diodes.
• Differential gain: Much higher \(dg/dn\) leads to ultrafast modulation bandwidths.
Confinement energy in infinite well: \(E_n = \frac{\hbar^2 \pi^2 n^2}{2 m^* L_z^2}\). Quantum Well Foundations
02 • Transition Selection Rules Valence Subband Splitting
Inter-band Transitions & Polarization
Envelop Overlap & Parity
Transition matrix element factorizes into atomic dipole \(\mathbf{p}_{cv}\) and envelope overlap:
\[ M_{\text{inter}} \propto \langle u_c | \mathbf{e} \cdot \mathbf{p} | u_v \rangle \int \psi_{e,n}^*(z) \psi_{h,m}(z) \, dz \]
In symmetric wells, envelope orthogonality imposes: \(\Delta n \equiv |n - m| = 0, 2, 4\dots\) (dominantly \(\Delta n = 0\)).
Heavy-Hole vs. Light-Hole
Confinement lifts the \(\Gamma_8\) valence degeneracy at \(k_\parallel = 0\):
\[ \Delta E_{\text{split}} = E_{\text{HH}1} - E_{\text{LH}1} > 0 \]
• C1 – HH1: Purely TE-polarized (\(\mathbf{E} \perp \hat{z}\)).
• C1 – LH1: Contributes strongly to TM (\(\mathbf{E} \parallel \hat{z}\)).
Compressive strain shifts HH further up, enforcing pure TE optical gain. Inter-band Dynamics
Quantum Structures Learning Pack Physics Honours • Optoelectronic Materials (Hours 5–8)
Quantum Structures & Transitions
Pack 02
Student Name: ____________________________________
Date: ________________________
Theoretical Framework • Reduced Dimensional Transitions
For an infinite quantum well of thickness \(L_z\), electronic subbands follow \(E_n(k_\parallel) = E_n + \frac{\hbar^2 k_\parallel^2}{2 m^*}\) where \(E_n = \frac{\hbar^2 \pi^2 n^2}{2 m^* L_z^2}\). The inter-band joint density of states per unit area is:
\( J_{\text{2D}}(\hbar\omega) = \sum_n \frac{\mu_{r}}{\pi \hbar^2} \Theta\left(\hbar\omega - E_g - E_{e,n} - E_{h,n}\right) \quad \text{where} \quad \mu_r^{-1} = m_e^{*-1} + m_h^{*-1} \)
Valence band subband splitting separates heavy-hole (HH) and light-hole (LH) levels, yielding distinct optical edges for TE (\(\mathbf{E} \perp \hat{z}\)) and TM (\(\mathbf{E} \parallel \hat{z}\)) polarizations.
Problem 1 • 2D Heavy-Hole vs Light-Hole Quantization (15 pts)
Consider a \(\text{GaAs}\) quantum well of width \(L_z = 10\text{ nm}\) surrounded by infinite barriers. Given: \(E_g = 1.424\text{ eV}\), electron effective mass \(m_e^* = 0.067 m_0\), heavy hole \(m_{\text{HH}}^* = 0.45 m_0\), and light hole \(m_{\text{LH}}^* = 0.082 m_0\) (\(m_0 = 9.109 \times 10^{-31}\text{ kg}\)).
Tasks: (a) Calculate ground state subband energies \(E_{e,1}\), \(E_{\text{HH},1}\), and \(E_{\text{LH},1}\) in meV. (b) Determine the fundamental transition threshold energies (in eV) for C1 \(\to\) HH1 and C1 \(\to\) LH1, and specify which transition determines the optical absorption edge.
Optoelectronic Materials • Physics Core Page 1 of 2
Intersubband Physics & Molecular Organics Pack 02 • Page 2
Problem 2 • Quantum Cascade / Intersubband Dipole Selection (18 pts)
In a conduction-band quantum well designed for a quantum cascade laser (QCL), electrons undergo radiative intra-band transitions between states \(\psi_2(z)\) and \(\psi_1(z)\).
(a) Dipole Selection Rule
Show that the intra-band dipole matrix element \(\langle \psi_2 | \hat{x} | \psi_1 \rangle = 0\), while \(\langle \psi_2 | \hat{z} | \psi_1 \rangle \neq 0\). Why does normal incidence radiation fail to induce transitions?
(b) Emission Wavelength
If the subband separation \(E_2 - E_1 = 124\text{ meV}\), calculate the free-space emission wavelength \(\lambda\) (\(\mu\text{m}\)) and designate its atmospheric transmission window.
Problem 3 • Frenkel vs Wannier Excitons & Spin Multiplicity (17 pts)
An organic semiconductor film has dielectric constant \(\epsilon_r = 3.2\) and effective carrier mass \(m^* = 1.0 m_0\), whereas inorganic \(\text{GaAs}\) has \(\epsilon_r = 12.9\) and reduced mass \(\mu = 0.059 m_0\).
Photonic Modulation Slide Deck Optoelectronic Materials • Part III
Hours 9–12 • Advanced Physics
Wave Propagation & Dynamic Modulation
Propagation, Modulation & Exciton Quenching
Anisotropic media, linear and quadratic electro-optic effects, Franz-Keldysh, QCSE, and field-induced exciton dissociation.
01. Wave Propagation
Dielectric tensor, index ellipsoid, and birefringence.
02. Electro-Optics
Pockels and Kerr phase shifts, Mach-Zehnder \(V_\pi\).
03. Electro-Absorption
Bulk Franz-Keldysh vs QCSE and exciton ionization.
01 • Wave Propagation Anisotropic Dielectric Tensor
Anisotropy & The Optical Indicatrix
Constitutive Relations
In anisotropic crystals, \(\mathbf{D}\) and \(\mathbf{E}\) are non-collinear:
\[ D_i = \epsilon_0 \epsilon_{ij} E_j \quad \implies \quad \eta_{ij} = \epsilon_0 (\epsilon^{-1})_{ij} \]
The index ellipsoid (optical indicatrix) in principal axes:
\[ \frac{x^2}{n_x^2} + \frac{y^2}{n_y^2} + \frac{z^2}{n_z^2} = 1 \]
Phase Velocity & Birefringence
For wavevector \(\mathbf{k}\), the cross-section of the indicatrix perpendicular to \(\mathbf{k}\) defines two orthogonal normal modes:
• Ordinary wave (\(n_o\)): Polarization experiences isotropic index regardless of propagation angle.
• Extraordinary wave (\(n_e(\theta)\)): Phase velocity depends on angle \(\theta\) relative to optical axis.
Birefringence \(\Delta n = n_e - n_o\) causes spatial walk-off between Poynting vector \(\mathbf{S}\) and \(\mathbf{k}\). Crystal Optics
02 • Electro-Optic Effects Pockels & Kerr Phase Modulation
Electro-Optic Modulation & \(V_\pi\)
Pockels (Linear) vs. Kerr (Quadratic)
Perturbation of the impermeability tensor:
\[ \Delta \left(\frac{1}{n^2}\right)_{\!ij} = r_{ijk} E_k + s_{ijkl} E_k E_l \]
• Pockels \(r_{ijk}\): Requires non-centrosymmetric crystals (e.g., \(\text{LiNbO}_3\), \(\text{GaAs}\)). \(\Delta n \propto E\).
• Kerr \(s_{ijkl}\): Present in all media (centrosymmetric isotropic). \(\Delta n \propto E^2\).
Mach-Zehnder Interferometer (MZI)
Phase shift converted to intensity modulation:
\[ \Delta\phi(V) = \frac{\pi}{\lambda} n^3 r E \cdot L = \pi \frac{V}{V_\pi} \]
Half-wave voltage for transverse configuration:
\[ V_\pi = \frac{\lambda d}{n^3 r L} \]
Push-pull electrode configurations halve the required driving voltage \(V_\pi\). Electro-Optic Devices
Photonic Modulation Learning Pack Physics Honours • Optoelectronic Materials (Hours 9–12)
Propagation, Modulation & Quenching
Pack 03
Student Name: ____________________________________
Date: ________________________
Theoretical Framework • Electro-Optic Waveguide Modulation
The linear electro-optic (Pockels) effect perturbs the optical indicatrix: \(\Delta (1/n^2)_i = \sum_j r_{ij} E_j\). For a \(z\)-cut, \(y\)-propagating \(\text{LiNbO}_3\) waveguide with field \(E_z = V/d\), the extraordinary index changes as \(\Delta n_e = -\frac{1}{2} n_e^3 r_{33} \frac{V}{d}\).
\( \Delta\phi = \frac{2\pi}{\lambda_0} \Delta n_e L = \pi \frac{V}{V_\pi} \quad \text{where} \quad V_\pi = \frac{\lambda_0 d}{n_e^3 r_{33} L} \)
In a balanced push-pull Mach-Zehnder interferometer, optical power output is modulated according to \(I_{\text{out}}(V) = I_{\text{in}} \cos^2\left(\frac{\pi V}{2 V_\pi}\right)\).
Problem 1 • Pockels Mach-Zehnder Modulator Design (16 pts)
A telecom modulator operates at \(\lambda_0 = 1.55\,\mu\text{m}\) using a \(\text{LiNbO}_3\) waveguide with \(n_e = 2.14\) and electro-optic coefficient \(r_{33} = 30.8 \times 10^{-12}\text{ m/V}\). The coplanar electrode gap is \(d = 8.0\,\mu\text{m}\).
(a) Derive the interaction length \(L\) required to achieve a half-wave voltage \(V_\pi = 3.5\text{ V}\).
(b) If a DC bias \(V_B = V_\pi / 2\) (quadrature point) is applied with a small sinusoidal signal \(v(t) = V_m \sin(\omega_m t)\) where \(V_m \ll V_\pi\), prove that the output intensity is linearly proportional to \(v(t)\).
Optoelectronic Materials • Physics Core Page 1 of 2
Electro-Absorption & Quenching Dynamics Pack 03 • Page 2
Problem 2 • QCSE Stark Shifts vs Bulk Franz-Keldysh (18 pts)
In bulk \(\text{GaAs}\), an applied electric field \(F = 1.0 \times 10^5\text{ V/cm}\) broadens the absorption edge via the Franz-Keldysh effect and field-ionizes the exciton (\(E_B \approx 4.2\text{ meV}\), \(a_B \approx 13\text{ nm}\)). In contrast, a \(10\text{ nm}\) \(\text{GaAs}\) quantum well maintains sharp excitonic resonance under the same field due to the Quantum-Confined Stark Effect (QCSE).
(a) Ionization Criterion
Calculate the classical potential drop \(e F a_B\) across the bulk exciton Bohr diameter and compare it to \(E_B\). Why is the bulk exciton dissociated?
(b) QCSE Confinement
Explain physically why the potential barriers prevent carrier ionization in the quantum well, and estimate the direction and sign of the Stark shift \(\Delta E\).
Problem 3 • Stern-Volmer Exciton Quenching Kinetics (16 pts)
Excitons in a conjugated polymer film undergo radiative decay (\(k_{\text{rad}}\)), non-radiative internal conversion (\(k_{\text{nr}}\)), and bimolecular quenching with fullerenes \([Q]\) at rate \(k_q [Q]\).
