ShaneLogic/carrier-statistics-modeling

Carrier Statistics Modeling

Use this skill when calculating carrier densities (n, p) and current densities (j_n, j_p) in semiconductor devices, particularly when:

• Simulating perovskite solar cells with organic transport layers
• Material properties suggest Boltzmann approximation may be invalid
• High doping concentrations or strong degeneracy conditions exist
• Comparing ordered (crystalline) vs disordered (organic) materials

General Carrier Density Calculation

For any statistical model, calculate carrier density using the statistical integral:

n = g_c * S((E_fn - E_c) / k_B T)
p = g_v * S((E_v - E_fp) / k_B T)

Where:

• S is the statistical integral (model-dependent)
• g_c, g_v are effective density of states
• E_fn, E_fp are quasi-Fermi levels
• E_c, E_v are band edge energies
• Account for band bending: E_c,v = const - q * φ

Select Statistical Model

1. Boltzmann Approximation (Default)

• When to use: Low carrier densities, quasi-Fermi levels > 3kT from band edge
• Statistical integral: S(xi) = exp(xi)
• Inverse: S^{-1}(y) = ln(y)

2. Parabolic Band Model (Fermi-Dirac)

• When to use: Ordered crystalline/inorganic semiconductors with high doping
• Statistical integral: Fermi-Dirac integral F_1/2(xi)
• Formula: F(xi) = (2/√π) ∫₀^∞ √(η)/(1 + exp(η - xi)) dη
• Boltzmann valid for: xi < -3 or n < 2 * g_c

3. Gaussian Band Model

• When to use: Organic/disordered materials, transport layers with hopping transport
• Statistical integral: Gauss-Fermi integral G_s(xi)
• Reference energies: LUMO (E_L), HOMO (E_H) - no defined band edges
• Disorder parameter: s (dimensionless), σ = s * k_B T
• Boltzmann form: S(xi) = exp(xi + s²/2)

Configure Non-Boltzmann Statistics

When simulating with non-Boltzmann statistics, specify parameters in parameters.m:

% ETL parameters
SE = 'F12';      % Statistical integral for ETL
SEinv = 'F12inv'; % Inverse statistical integral

% HTL parameters  
SH = 'G';        % Statistical integral for HTL
SHinv = 'Ginv';  % Inverse statistical integral

Default behavior: If not specified, uses Boltzmann approximation (exp/ln)

Calculate Current Density

Use the generalized drift-diffusion equation:

j_n = μ_n * k_B T * d/dx[ n * S^{-1}(n/g_c) - (q/k_B T) * dφ/dx ]
j_p = μ_p * k_B T * d/dx[ p * S^{-1}(p/g_v) + (q/k_B T) * dφ/dx ]

In Boltzmann limit, this reduces to standard form:

j_n = q * μ_n * n * E + k_B T * μ_n * dn/dx

Apply Transport Layer Boundary Conditions

At ETL and HTL interfaces, apply continuity conditions with equilibrium ratios:

Carrier density continuity: n|_x=0- = n|_x=0+
Current density continuity: j_n|_x=0- = j_n|_x=0+

Calculate equilibrium ratios to handle non-Boltzmann statistics across interfaces.