Fundamental Device Equations

1. Overview

Semiconductor device equations originate from Maxwell’s equations, however practical device modelling introduces controlled approximations, including quasistatic fields, effective‑mass band‑structure parameters, and material‑specific generation, recombination, and scattering terms, which reduce the full problem to Poisson’s equation, the carrier continuity equations with drift–diffusion transport, and the heat equation.

These coupled equations must be solved together with appropriate boundary conditions to capture realistic device operation, for example, band bending at junctions, carrier storage during conduction, and self‑heating under power stress.

2. Poisson’s Equation

Poisson’s equation provides the electrostatic backbone of a device solution, because it links the spatial variation of electrostatic potential to local charge density, which in semiconductors comprises mobile carriers, ionised dopants, and fixed or interface charges. This determines depletion widths, internal electric fields, and how bands bend under bias.

$$\begin{equation} \nabla \cdot (\varepsilon \nabla \psi) = -q\,(p - n + N_D^+ - N_A^-) - Q_f - Q_{\text{int}} - Q_{\text{bulk}} \end{equation}$$

where:

• $\varepsilon$ = permittivity (F/cm)
• $\psi$ = electrostatic potential (V)
• $q$ = elementary charge (C)
• $n$ = electron concentration (cm^-3)
• $p$ = hole concentration (cm^-3)
• $N_D^+$ = ionised donor concentration (cm^-3)
• $N_A^-$ = ionised acceptor concentration (cm^-3)
• $Q_f$ = fixed charge density (C/cm²)
• $Q_{\text{int}}$ = interface charge density, energy dependent (C/cm²)
• $Q_{\text{bulk}}$ = bulk charge density, energy dependent (C/cm³)

2.1. Carrier Concentrations

Carrier concentrations are derived from the density of states and Fermi–Dirac statistics. In non-degenerate conditions, the Fermi–Dirac integrals reduce to simple exponentials, facilitating analytic estimates and boundary conditions.

Fermi–Dirac Statistics and Approximations

$$\begin{equation} n = N_C \mathcal{F}_{1/2}(\eta_n) \approx N_C e^{\eta_n} \qquad p = N_V \mathcal{F}_{1/2}(\eta_p) \approx N_V e^{\eta_p} \end{equation}$$

Where the reduced Fermi levels ($\eta$) relate to the energy band edges and quasi-Fermi levels as follows:

$$\begin{equation} \eta_n = \frac{E_{Fn} - E_C}{k_B T} \qquad \eta_p = \frac{E_V - E_{Fp}}{k_B T} \end{equation}$$

where:

• $N_C, N_V$: Effective density of states for conduction and valence bands (cm^-3).
• $E_C, E_V$: Conduction and valence band edge energies (eV).
• $E_{Fn}, E_{Fp}$: Electron and hole quasi-Fermi levels (eV).
• $\mathcal{F}_{1/2}(\cdot)$: Fermi–Dirac integral of order $1/2$.
• $k_B T$: Thermal energy, where $k_B$ is the Boltzmann constant and $T$ is absolute temperature.

In equilibrium, $E_{Fn}$ and $E_{Fp}$ coincide with the Fermi level; however, under bias or illumination, they split, a phenomenon fundamental to modelling injection and recombination.

2.2. Effective Density of States (DOS)

The DOS counts the number of available electronic states per unit energy and volume, it is a band‑structure property rather than a transport parameter. Near band edges the dispersion can be treated as parabolic, which leads to the standard effective‑mass DOS expressions below and connects heavier effective mass with a higher DOS near the edge. This directly influences intrinsic carrier concentration and the onset of degeneracy.

$$\begin{equation} N_C = 2\left( \frac{2\pi m_n^\ast k_B T}{h^2} \right)^{3/2} \qquad N_V = 2\left( \frac{2\pi m_p^\ast k_B T}{h^2} \right)^{3/2} \end{equation}$$

where:

• $N_C$ = effective conduction‑band density of states (cm^-3)
• $N_V$ = effective valence‑band density of states (cm^-3)
• $m_n^\ast, m_p^\ast$ = electron and hole effective mass (kg)
• $k_B$ = Boltzmann constant (J/K)
• $T$ = absolute temperature (K)
• $h$ = Planck constant (J·s)
• $\pi$ = constant ≈ 3.14159 (dimensionless)

2.3. Mass Action Law

At thermal equilibrium, electron and hole populations are constrained by the mass action law, which reflects detailed balance between pair generation and annihilation.

$$\begin{equation} np = n_i^2 \end{equation}$$

where:

• $n$ = electron concentration (cm^-3)
• $p$ = hole concentration (cm^-3)
• $n_i$ = intrinsic carrier concentration (cm^-3)

2.4. Intrinsic Carrier Concentration

The intrinsic carrier concentration is set by the DOS and the bandgap, together with temperature, and it establishes the baseline against which doping and injection are measured. Wide‑bandgap materials exhibit very low $n_i$, which is central to their high‑temperature and high‑voltage advantages.

$$\begin{equation} n_i = \sqrt{N_C N_V}\,\exp\!\left( -\frac{E_g}{2 k_B T} \right) \end{equation}$$

where:

• $n_i$ = intrinsic carrier concentration (cm^-3)
• $N_C$ = effective conduction‑band DOS (cm^-3)
• $N_V$ = effective valence‑band DOS (cm^-3)
• $E_g$ = bandgap energy (eV)
• $k_B$ = Boltzmann constant (eV/K)
• $T$ = absolute temperature (K)

2.5. Charge Neutrality

Charge neutrality states that, in equilibrium and away from very small length scales, total positive and negative charge densities balance. Solving neutrality together with Poisson’s equation fixes the Fermi level and carrier densities in doped regions.

$$\begin{equation} p + N_D^+ = n + N_A^- \end{equation}$$

where:

• $p$ = hole concentration (cm^-3)
• $n$ = electron concentration (cm^-3)
• $N_D^+$ = ionised donor concentration (cm^-3)
• $N_A^-$ = ionised acceptor concentration (cm^-3)

3. Continuity Equations

The continuity equations enforce charge conservation for electrons and holes. They capture how carrier populations evolve due to current flow, generation, and recombination, which is essential for understanding conductivity modulation in bipolar structures, avalanche multiplication near breakdown, and charge‑storage effects during switching.

General FormulationThe time-evolution of electron ($n$) and hole ($p$) concentrations is given by:

$$\begin{equation} \frac{\partial n}{\partial t} = \frac{1}{q}\,\nabla \cdot \mathbf{J}_n + G_n - R_n \qquad \frac{\partial p}{\partial t} = -\frac{1}{q}\,\nabla \cdot \mathbf{J}_p + G_p - R_p \end{equation}$$

where:

• $n, p$ = carrier concentrations (cm^-3)
• $t$ = time (s)
• $q$ = elementary charge (C)
• $\mathbf{J}_n, \mathbf{J}_p$ = current density vectors (A/cm²)
• $G_n, G_p$ = generation rates (cm^-3 s^-1)
• $R_n, R_p$ = recombination rates (cm^-3 s^-1)

In steady state, the time derivatives vanish and the divergence of current is locally balanced by net generation minus recombination. In transient operation, such as turn‑off of an IGBT or reverse recovery of a diode, these equations govern how rapidly stored charge is removed, which in turn sets current decay and voltage overshoot.

3.1. Drift–Diffusion Current

Total current density is the sum of a field‑driven drift component and a gradient‑driven diffusion component. Drift reflects the action of the electric field on carriers, while diffusion arises from spatial non‑uniformity of carrier concentration.

$$\begin{equation} \mathbf{J}_n = q\,\mu_n\,n\,\mathbf{E} + q\,D_n\,\nabla n \qquad \mathbf{J}_p = q\,\mu_p\,p\,\mathbf{E} - q\,D_p\,\nabla p \end{equation}$$

where:

• $\mathbf{J}_n, \mathbf{J}_p$ = electron current density (A/cm²)
• $q$ = elementary charge (C)
• $\mu_n, \mu_p$ = carrier mobilities (cm²/(V·s))
• $n, p$ = carrier concentrations (cm^-3)
• $\mathbf{E}$ = electric field vector (V/cm)
• $D_n, D_p$ = diffusion coefficients (cm²/s)

Electric field

$$\begin{equation} \mathbf{E} = -\,\nabla \psi \end{equation}$$

where:

• $\mathbf{E}$ = electric field vector (V/cm)
• $\nabla$ = gradient operator
• $\psi$ = electrostatic potential (V)

3.2. Mobility

Mobility measures how readily carriers accelerate in response to an electric field. It is limited by scattering from phonons, ionised impurities, and interfaces, and it generally depends on temperature, doping, and field strength, which means that mobility is not a universal constant.

$$\begin{equation} \mathbf{v}_{\text{drift}} = \mu\,\mathbf{E} \end{equation}$$

where:

• $\mathbf{v}_{\text{drift}}$ = drift velocity (cm/s)
• $\mu$ = mobility, electron or hole as appropriate (cm²/(V·s))
• $\mathbf{E}$ = electric field vector (V/cm)

3.3. Einstein Relations

Diffusion and mobility are related through carrier statistics. In the degenerate case the proportionality involves ratios of Fermi–Dirac integrals, while in the non‑degenerate limit both ratios reduce to the thermal voltage.

General

$$\begin{equation} \frac{D_n}{\mu_n} = \frac{k_B T}{q}\, \frac{\mathcal{F}_{1/2}(\eta_n)}{\mathcal{F}_{-1/2}(\eta_n)} \qquad \frac{D_p}{\mu_p} = \frac{k_B T}{q}\, \frac{\mathcal{F}_{1/2}(\eta_p)}{\mathcal{F}_{-1/2}(\eta_p)} \end{equation}$$

where:

• $D_n, D_p$ = diffusion coefficients (cm²/s)
• $\mu_n, \mu_p$ = carrier mobilities (cm²/(V·s))
• $k_B$ = Boltzmann constant (J/K)
• $T$ = absolute temperature (K)
• $q$ = elementary charge (C)
• $\mathcal{F}_{1/2}(\cdot)$ = Fermi–Dirac integral of order 1/2 (dimensionless)
• $\mathcal{F}_{-1/2}(\cdot)$ = Fermi–Dirac integral of order −1/2 (dimensionless)
• $\eta_n, \eta_p$ = reduced electron / hole Fermi level (dimensionless)

Non‑degenerate limit

$$\begin{equation} \frac{D_n}{\mu_n} = \frac{k_B T}{q} \qquad \frac{D_p}{\mu_p} = \frac{k_B T}{q} \end{equation}$$

where:

• $D_n,\,D_p$ = diffusion coefficients (cm²/s)
• $\mu_n,\,\mu_p$ = carrier mobilities (cm²/(V·s))
• $k_B$ = Boltzmann constant (J/K)
• $T$ = absolute temperature (K)
• $q$ = elementary charge (C)

3.4. Recombination and Generation

Generation promotes electrons from the valence band to the conduction band, which increases $n$ and $p$, while recombination is the reverse process that removes an electron–hole pair. In power devices, generation includes impact ionisation in high‑field regions, and recombination includes Shockley–Read–Hall and Auger processes, which set lifetimes and therefore switching tails and leakage. For more information see Recombination and Generation

4. Semiconductor Heat Equation

Power devices are inherently electrothermal. Electrical work done on carriers is dissipated as lattice heat through carrier scattering, Joule heating, and recombination, and the resulting temperature rise feeds back on transport through temperature‑dependent mobility, lifetime, intrinsic carrier concentration, and ionisation rates. This bidirectional coupling motivates solving a thermal conservation equation together with the electrical equations.

$$\begin{equation} \ C_v \frac{\partial T_L}{\partial t} = \nabla \cdot \big(k_T \nabla T_L\big) + Q \end{equation}$$

where:

• $C_v$ = volumetric heat capacity (J/(m³·K))
• $T_L$ = lattice temperature (K)
• $t$ = time (s)
• $k_T$ = thermal conductivity (W/(cm·K))
• $Q$ = volumetric heat generation (W/cm³)

The left‑hand side represents the rate of change of stored thermal energy, the right‑hand side represents heat spreading by diffusion together with local heating sources.

4.1. Power Density Sources

Total local heat generation in the semiconductor comprises Joule heating of electrons and holes, recombination heating, and, when relevant, optical absorption and thermoelectric effects. These contributions are summed in $Q$, which then drives the thermal equation above.

$$\begin{equation} Q = \mathbf{J}_n \cdot \mathbf{E} + \mathbf{J}_p \cdot \mathbf{E} + Q_{\mathrm{rec}} + Q_{\mathrm{opt}} + Q_{\mathrm{thm}} \end{equation}$$

where:

• $Q$ = volumetric heat generation (W/cm³)
• $\mathbf{J}_n, \mathbf{J}_n$ = current density vectors (A/cm²)
• $\mathbf{E}$ = electric field vector (V/cm)
• $Q_{\mathrm{rec}}$ = recombination heating power density (W/cm³)
• $Q_{\mathrm{opt}}$ = optical absorption heating power density (W/cm³)
• $Q_{\mathrm{thm}}$ = thermoelectric heating or cooling power density (W/cm³)

5. Summary

The core system solved in device simulation consists of four coupled non‑linear partial differential equations, which require robust numerical methods and consistent boundary conditions.

Poisson's equation

$$\nabla \cdot (\varepsilon \nabla \psi) = -q\,(p - n + N_D^+ - N_A^-) - Q_f - Q_{\text{int}} - Q_{\text{bulk}}$$

Electron continuity equation

$$\frac{\partial n}{\partial t} = \frac{1}{q}\,\nabla \cdot \mathbf{J}_n + G_n - R_n$$

Hole continuity equation

$$\frac{\partial p}{\partial t} = -\frac{1}{q}\,\nabla \cdot \mathbf{J}_p + G_p - R_p$$

Heat equation

$$\ C_v \frac{\partial T_L}{\partial t} = \nabla \cdot \big(k_T \nabla T_L\big) + Q$$