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Basic Concepts

Understanding the key concepts behind CarrierCapture will help you use it effectively.

Overview

CarrierCapture calculates carrier capture rates for defects in semiconductors using multiphonon theory (Huang-Rhys formalism). The key idea: charge carriers can be captured non-radiatively via emission of multiple phonons.

Key Concepts

1. Potential Energy Surfaces (PES)

A potential energy surface describes how the total energy of a system varies with atomic configuration.

Mathematical form:

\[E(Q) = \text{energy as function of configuration coordinate } Q\]

Where \(Q\) is the mass-weighted displacement coordinate (units: amu\(^{0.5}\)·Å).

Example - Harmonic potential:

\[E(Q) = E_0 + \frac{1}{2}k(Q - Q_0)^2\]

Where:

  • \(E_0\): minimum energy
  • \(Q_0\): equilibrium position
  • \(k\): force constant (related to phonon frequency)

In CarrierCapture:

from carriercapture.core import Potential

# Create harmonic potential
pot = Potential.from_harmonic(
    hw=0.008,  # ℏω phonon energy (eV)
    Q0=5.0,    # Equilibrium at Q = 5
    E0=0.0     # Ground state energy
)

2. Configuration Coordinate Diagrams

A configuration coordinate (CC) diagram plots the potential energy surfaces for different electronic states on the same axes.

Physical picture:

  • Horizontal axis: Configuration coordinate \(Q\) (atomic positions)
  • Vertical axis: Total energy \(E\)
  • Curves: PES for different charge states

Example:

E (eV) │
       │     Initial state (excited)
   1.5 │    ╱‾‾‾╲
       │   ╱     ╲
   1.0 │  ╱       ╲
       │ ╱         ╲
   0.5 │╱           ╲___
       │              Final state (ground)
   0.0 │              ╲___╱
       └────────────────────────────── Q
            Q₀        Q₁

Key features:

  1. Displacement: \(\Delta Q = Q_1 - Q_0\) (shift between minima)
  2. Energy difference: \(\Delta E = E_{\text{initial}}(Q_1) - E_{\text{final}}(Q_1)\)
  3. Crossing point: Where curves intersect (barrier for capture)

3. Vibrational Wavefunctions

At each electronic state, the atoms vibrate around equilibrium. These vibrations are quantized with discrete energy levels.

Schrödinger equation:

\[\hat{H}\psi_n(Q) = \varepsilon_n \psi_n(Q)\]

Where:

  • \(\psi_n(Q)\): wavefunction for state \(n\)
  • \(\varepsilon_n\): energy eigenvalue
  • \(n = 0, 1, 2, ...\): quantum number

Harmonic oscillator solutions:

\[\varepsilon_n = E_0 + \hbar\omega(n + \tfrac{1}{2})\]

In CarrierCapture:

# Solve for wavefunctions
pot.solve(nev=60)  # Get first 60 states

# Access results
energies = pot.eigenvalues   # Shape: (60,)
wavefunctions = pot.eigenvectors  # Shape: (60, N_Q)

4. Franck-Condon Principle

Statement: Electronic transitions are vertical on a CC diagram (atoms don't move during transition).

Implication: After a sudden electronic transition, atoms are in a non-equilibrium configuration and must relax via:

  1. Radiative: Photon emission
  2. Non-radiative: Phonon emission (what CarrierCapture calculates)

5. Carrier Capture Coefficient

The capture coefficient \(C(T)\) quantifies how fast charge carriers are captured (units: cm³/s).

Physical meaning:

\[\text{Capture rate} = C(T) \times n\]

Where \(n\) is the carrier concentration.

Temperature dependence:

  • Low T: Thermally activated (exponential increase with T)
  • High T: Multiphonon cascade (can decrease with T)

Typical values:

Capture Type \(C\) (cm³/s) Example
Very slow 10⁻²⁰ Deep traps at low T
Slow 10⁻¹⁵ Shallow defects
Moderate 10⁻¹² Typical defects at 300K
Fast 10⁻⁸ Resonant capture

6. Electron-Phonon Coupling

The coupling strength \(W\) describes how strongly electronic and vibrational degrees of freedom interact.

Physical meaning:

  • Large \(W\): Strong coupling → fast non-radiative capture
  • Small \(W\): Weak coupling → slow capture (more radiative)

Typical range: \(W \sim 0.01\) to \(0.5\) eV

In multiphonon theory:

\[C \propto W^2\]

So capture rate scales quadratically with coupling!

7. Overlap Integrals

The overlap between initial and final wavefunctions determines transition probability.

Mathematical form:

\[S_{ij} = \langle \psi_i | \hat{O} | \psi_j \rangle = \int \psi_i^*(Q) \cdot \hat{O} \cdot \psi_j(Q) \, dQ\]

Where \(\hat{O}\) is typically \((Q - Q_0)\) for dipole coupling.

Physical interpretation:

  • Large overlap: Easy transition
  • Small overlap: Forbidden/unlikely transition

Selection rules: Dictate which \(i \to j\) transitions are allowed.

Putting It Together: The Full Calculation

Step-by-Step Process

  1. Define potentials for initial and final electronic states
  2. Solve Schrödinger equation to get vibrational states
  3. Calculate overlaps between vibrational wavefunctions
  4. Apply energy conservation (delta function)
  5. Thermally weight contributions from different initial states
  6. Sum over all pathways to get total capture coefficient

Mathematical Formula

The capture coefficient at temperature \(T\) is:

\[C(T) = \frac{V \cdot 2\pi}{\hbar} \cdot g \cdot W^2 \cdot \sum_{i,j} p_i |S_{ij}|^2 \delta(\varepsilon_i - \varepsilon_j)\]

Where:

  • \(V\): supercell volume
  • \(g\): degeneracy factor (often = 1)
  • \(W\): electron-phonon coupling
  • \(p_i = \frac{e^{-\beta\varepsilon_i}}{Z}\): thermal occupation (Boltzmann)
  • \(S_{ij}\): overlap integral
  • \(\delta(\Delta E)\): energy-conserving delta function (Gaussian)

In Code

from carriercapture.core import Potential, ConfigCoordinate

# 1. Define potentials
pot_i = Potential.from_harmonic(hw=0.008, Q0=0.0, E0=0.5)
pot_f = Potential.from_harmonic(hw=0.008, Q0=10.0, E0=0.0)

# 2. Solve for states
pot_i.solve(nev=180)
pot_f.solve(nev=60)

# 3-6. Calculate capture coefficient
cc = ConfigCoordinate(pot_i=pot_i, pot_f=pot_f, W=0.068)
cc.calculate_overlap(Q0=5.0)
cc.calculate_capture_coefficient(volume=1e-21, temperature=[300])

# Result
print(f"C(300K) = {cc.capture_coefficient[0]:.3e} cm³/s")

Common Terminology

Term Symbol Units Meaning
Configuration coordinate \(Q\) amu\(^{0.5}\)·Å Mass-weighted displacement
Phonon energy \(\hbar\omega\) eV Vibrational quantum
Displacement \(\Delta Q\) amu\(^{0.5}\)·Å Shift between states
Energy difference \(\Delta E\) eV Vertical separation
Coupling \(W\) eV Electron-phonon interaction
Capture coefficient \(C\) cm³/s Capture rate constant
Huang-Rhys factor \(S\) - Effective phonon number
Reorganization energy \(\lambda\) eV Energy to relax

Physical Intuition

Why Multiphonon?

For large \(\Delta E \gg \hbar\omega\), a single phonon can't bridge the gap. Instead:

  1. Cascade emission: Many phonons emitted sequentially
  2. Effective phonon number: \(n_{\text{eff}} \sim \Delta E / \hbar\omega\)

Example: \(\Delta E = 1\) eV, \(\hbar\omega = 0.01\) eV \(\Rightarrow\) need ~100 phonons!

Temperature Effects

Low Temperature:

  • Only ground state occupied (\(p_0 \approx 1\))
  • Needs thermal activation to reach crossing point
  • \(C \propto e^{-E_a/k_BT}\)

High Temperature:

  • Many initial states populated
  • More pathways available
  • But overlap factors decrease (orthogonality)
  • Complex temperature dependence

Material Dependence

Good light emitters (LEDs):

  • Small \(\Delta Q\) → small overlap
  • Weak coupling \(W\) → slow non-radiative
  • \(\Rightarrow\) Radiative process wins!

Efficient carrier capture (detectors, solar cells):

  • Large \(\Delta Q\) → large overlap
  • Strong coupling \(W\) → fast non-radiative
  • \(\Rightarrow\) Carriers captured quickly!

Assumptions and Limitations

Static Coupling Approximation

Assumption: Potential surfaces are independent of electronic state.

Limitation: Breaks down for strongly correlated electrons.

Harmonic Approximation

Assumption: Potentials are parabolic near minimum.

Limitation: Anharmonic effects matter for large displacements.

CarrierCapture can use spline or Morse potentials to go beyond harmonic!

Single Configuration Coordinate

Assumption: One effective coordinate \(Q\) captures all relaxation.

Limitation: Real systems have many vibrational modes.

Justification: Often one "breathing mode" dominates.

Next Steps

Now that you understand the concepts:

Further Reading