Electromagnetism

Electromagnetism Mathematics Modeling

A comprehensive guide to the mathematical frameworks used in semiconductor device simulation, covering electromagnetic theory, carrier transport, and quantum effects.

1. The Core Problem

Semiconductor device modeling requires solving coupled systems that describe:

How electromagnetic fields propagate in and interact with semiconductor materials
How charge carriers (electrons and holes) move in response to fields
How quantum effects modify classical behavior at nanoscales

Key Variables:

Symbol	Description	Units
$\phi$	Electrostatic potential	V
$$n$$	Electron concentration	cm⁻³
$$p$$	Hole concentration	cm⁻³
$\mathbf{E}$	Electric field	V/cm
$\mathbf{J}_n, \mathbf{J}_p$	Current densities	A/cm²

2. Fundamental Mathematical Frameworks

2.1 Drift-Diffusion System

The workhorse of semiconductor device simulation couples three fundamental equations.

2.1.1 Poisson's Equation (Electrostatics)

\nabla \cdot (\varepsilon \nabla \phi) = -q(p - n + N_D^+ - N_A^-)

Where:

$\varepsilon$ — Permittivity of the semiconductor
$\phi$ — Electrostatic potential
$$q$$ — Elementary charge ( $1.602 \times 10^{-19}$ C)
$$n, p$$ — Electron and hole concentrations
$$N_D^+$$ — Ionized donor concentration
$$N_A^-$$ — Ionized acceptor concentration

2.1.2 Continuity Equations (Carrier Conservation)

For electrons:

\frac{\partial n}{\partial t} = \frac{1}{q}\nabla \cdot \mathbf{J}_n - R + G

For holes:

\frac{\partial p}{\partial t} = -\frac{1}{q}\nabla \cdot \mathbf{J}_p - R + G

Where:

$$R$$ — Recombination rate (cm⁻³s⁻¹)
$$G$$ — Generation rate (cm⁻³s⁻¹)

2.1.3 Current Density Relations

Electron current (drift + diffusion):

\mathbf{J}_n = q\mu_n n \mathbf{E} + qD_n \nabla n

Hole current (drift + diffusion):

\mathbf{J}_p = q\mu_p p \mathbf{E} - qD_p \nabla p

Einstein Relations:

D_n = \frac{k_B T}{q} \mu_n \quad \text{and} \quad D_p = \frac{k_B T}{q} \mu_p

2.1.4 Recombination Models

Shockley-Read-Hall (SRH):

R_{SRH} = \frac{np - n_i^2}{\tau_p(n + n_1) + \tau_n(p + p_1)}

Auger Recombination:

R_{Auger} = (C_n n + C_p p)(np - n_i^2)

Radiative Recombination:

R_{rad} = B(np - n_i^2)

2.2 Maxwell's Equations in Semiconductors

For optoelectronics and high-frequency devices, the full electromagnetic treatment is necessary.

2.2.1 Maxwell's Equations

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}

\nabla \cdot \mathbf{D} = \rho

\nabla \cdot \mathbf{B} = 0

2.2.2 Constitutive Relations

Displacement field:

\mathbf{D} = \varepsilon_0 \varepsilon_r(\omega) \mathbf{E}

Current density:

\mathbf{J} = \sigma(\omega) \mathbf{E}

2.2.3 Frequency-Dependent Dielectric Function

\varepsilon(\omega) = \varepsilon_\infty - \frac{\omega_p^2}{\omega^2 + i\gamma\omega} + \sum_j \frac{f_j}{\omega_j^2 - \omega^2 - i\Gamma_j\omega}

Components:

First term ( $\varepsilon_\infty$ ): High-frequency (background) permittivity
Second term (Drude): Free carrier response
$\omega_p = \sqrt{\frac{nq^2}{\varepsilon_0 m^*}}$ — Plasma frequency
$\gamma$ — Damping rate
Third term (Lorentz oscillators): Interband transitions
$\omega_j$ — Resonance frequencies
$\Gamma_j$ — Linewidths
$$f_j$$ — Oscillator strengths

2.2.4 Complex Refractive Index

\tilde{n}(\omega) = n(\omega) + i\kappa(\omega) = \sqrt{\varepsilon(\omega)}

Optical properties:

Refractive index: $n = \text{Re}(\tilde{n})$
Extinction coefficient: $\kappa = \text{Im}(\tilde{n})$
Absorption coefficient: $\alpha = \frac{2\omega\kappa}{c} = \frac{4\pi\kappa}{\lambda}$

2.3 Boltzmann Transport Equation

When drift-diffusion is insufficient (hot carriers, high fields, ultrafast phenomena):

\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_\mathbf{r} f + \frac{\mathbf{F}}{\hbar} \cdot \nabla_\mathbf{k} f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}}

Where:

$f(\mathbf{r}, \mathbf{k}, t)$ — Distribution function in 6D phase space
$\mathbf{v} = \frac{1}{\hbar}\nabla_\mathbf{k} E(\mathbf{k})$ — Group velocity
$\mathbf{F}$ — External force (e.g., $q\mathbf{E}$ )

2.3.1 Collision Integral (Relaxation Time Approximation)

\left(\frac{\partial f}{\partial t}\right)_{\text{coll}} \approx -\frac{f - f_0}{\tau}

2.3.2 Scattering Mechanisms

Acoustic phonon scattering:

\frac{1}{\tau_{ac}} \propto T \cdot E^{1/2}

Optical phonon scattering:

\frac{1}{\tau_{op}} \propto \left(N_{op} + \frac{1}{2} \mp \frac{1}{2}\right)

Ionized impurity scattering (Brooks-Herring):

\frac{1}{\tau_{ii}} \propto \frac{N_I}{E^{3/2}}

2.3.3 Solution Approaches

Monte Carlo methods: Stochastically simulate individual carrier trajectories
Moment expansions: Derive hydrodynamic equations from velocity moments
Spherical harmonic expansion: Expand angular dependence in k-space

2.4 Quantum Transport

For nanoscale devices where quantum effects dominate.

2.4.1 Schrödinger Equation (Effective Mass Approximation)

\left[-\frac{\hbar^2}{2m^*}\nabla^2 + V(\mathbf{r})\right]\psi = E\psi

2.4.2 Schrödinger-Poisson Self-Consistent Loop

2.4.3 Non-Equilibrium Green's Function (NEGF)

Retarded Green's function:

[EI - H - \Sigma^R]G^R = I

Lesser Green's function (for electron density):

G^< = G^R \Sigma^< G^A

Current formula (Landauer-Büttiker type):

I = \frac{2q}{h}\int \text{Tr}\left[\Sigma^< G^> - \Sigma^> G^<\right] dE

Transmission function:

T(E) = \text{Tr}\left[\Gamma_L G^R \Gamma_R G^A\right]

where $\Gamma_{L,R} = i(\Sigma_{L,R}^R - \Sigma_{L,R}^A)$ are the broadening matrices.

2.4.4 Wigner Function Formalism

Quantum analog of the Boltzmann distribution:

f_W(\mathbf{r}, \mathbf{p}, t) = \frac{1}{(\pi\hbar)^3}\int \psi^*\left(\mathbf{r}+\mathbf{s}\right)\psi\left(\mathbf{r}-\mathbf{s}\right) e^{2i\mathbf{p}\cdot\mathbf{s}/\hbar} d^3s

3. Coupled Optoelectronic Modeling

For solar cells, LEDs, and lasers, optical and electrical physics must be solved self-consistently.

3.1 Self-Consistent Loop

3.2 Key Coupling Equations

Optical generation rate:

G(\mathbf{r}) = \frac{\alpha(\mathbf{r})|\mathbf{E}(\mathbf{r})|^2}{2\hbar\omega}

Free carrier absorption (modifies permittivity):

\Delta\alpha_{fc} = \sigma_n n + \sigma_p p

Band gap narrowing (high injection):

\Delta E_g = -A\left(\ln\frac{n}{n_0} + \ln\frac{p}{p_0}\right)

3.3 Laser Rate Equations

Carrier density:

\frac{dn}{dt} = \frac{\eta I}{qV} - \frac{n}{\tau} - g(n)S

Photon density:

\frac{dS}{dt} = \Gamma g(n)S - \frac{S}{\tau_p} + \Gamma\beta\frac{n}{\tau}

Gain function (linear approximation):

g(n) = g_0(n - n_{tr})

4. Numerical Methods

4.1 Method Comparison

Method	Best For	Key Features	Computational Cost
Finite Element (FEM)	Complex geometries	Adaptive meshing, handles interfaces	Medium-High
Finite Difference (FDM)	Regular grids	Simpler implementation	Low-Medium
FDTD	Time-domain EM	Explicit time stepping, broadband	High
Transfer Matrix (TMM)	Multilayer thin films	Analytical for 1D, very fast	Very Low
RCWA	Periodic structures	Fourier expansion	Medium
Monte Carlo	High-field transport	Stochastic, parallelizable	Very High

4.2 Scharfetter-Gummel Discretization

Essential for numerical stability in drift-diffusion. For electron current between nodes $$i$$ and $$i+1$$ :

J_{n,i+1/2} = \frac{qD_n}{h}\left[n_i B\left(\frac{\phi_i - \phi_{i+1}}{V_T}\right) - n_{i+1} B\left(\frac{\phi_{i+1} - \phi_i}{V_T}\right)\right]

Bernoulli function:

B(x) = \frac{x}{e^x - 1}

4.3 FDTD Yee Grid

Update equations (1D example):

E_x^{n+1}(k) = E_x^n(k) + \frac{\Delta t}{\varepsilon \Delta z}\left[H_y^{n+1/2}(k+1/2) - H_y^{n+1/2}(k-1/2)\right]

H_y^{n+1/2}(k+1/2) = H_y^{n-1/2}(k+1/2) + \frac{\Delta t}{\mu \Delta z}\left[E_x^n(k+1) - E_x^n(k)\right]

Courant stability condition:

\Delta t \leq \frac{\Delta x}{c\sqrt{d}}

where $$d$$ is the number of spatial dimensions.

4.4 Newton-Raphson for Coupled System

For the coupled Poisson-continuity system, solve:

\begin{pmatrix} \frac{\partial F_\phi}{\partial \phi} & \frac{\partial F_\phi}{\partial n} & \frac{\partial F_\phi}{\partial p} \\ \frac{\partial F_n}{\partial \phi} & \frac{\partial F_n}{\partial n} & \frac{\partial F_n}{\partial p} \\ \frac{\partial F_p}{\partial \phi} & \frac{\partial F_p}{\partial n} & \frac{\partial F_p}{\partial p} \end{pmatrix} \begin{pmatrix} \delta\phi \\ \delta n \\ \delta p \end{pmatrix} = - \begin{pmatrix} F_\phi \\ F_n \\ F_p \end{pmatrix}

5. Multiscale Challenge

5.1 Hierarchy of Scales

Scale	Size	Method	Physics Captured
Atomic	0.1–1 nm	DFT, tight-binding	Band structure, material parameters
Quantum	1–100 nm	NEGF, Wigner function	Tunneling, confinement
Mesoscale	10–1000 nm	Boltzmann, Monte Carlo	Hot carriers, non-equilibrium
Device	100 nm–μm	Drift-diffusion	Classical transport
Circuit	μm–mm	Compact models (SPICE)	Lumped elements

5.2 Scale-Bridging Techniques

Parameter extraction: DFT → effective masses, band gaps → drift-diffusion parameters
Quantum corrections to drift-diffusion:

n = N_c F_{1/2}\left(\frac{E_F - E_c - \Lambda_n}{k_B T}\right)

where $\Lambda_n$ is the quantum potential from density-gradient theory:

\Lambda_n = -\frac{\hbar^2}{12m^*}\frac{\nabla^2 \sqrt{n}}{\sqrt{n}}

Machine learning surrogates: Train neural networks on expensive quantum simulations

6. Key Mathematical Difficulties

6.1 Extreme Nonlinearity

Carrier concentrations depend exponentially on potential:

n = n_i \exp\left(\frac{E_F - E_i}{k_B T}\right) = n_i \exp\left(\frac{q\phi}{k_B T}\right)

At room temperature, $k_B T/q \approx 26$ mV, so small potential changes cause huge concentration swings.

Solutions:

Gummel iteration (decouple and solve sequentially)
Newton-Raphson with damping
Continuation methods

6.2 Numerical Stiffness

Doping varies by $10^{10}$ or more (from intrinsic to heavily doped)
Depletion regions: nm-scale features in μm-scale devices
Time scales: fs (optical) to ms (thermal)

Solutions:

Adaptive mesh refinement
Implicit time stepping
Logarithmic variable transformations: $u = \ln(n/n_i)$

6.3 High Dimensionality

Full Boltzmann: 7D (3 position + 3 momentum + time)
NEGF: Large matrix inversions per energy point

Solutions:

Mode-space approximation
Hierarchical matrix methods
GPU acceleration

6.4 Multiphysics Coupling

Interacting effects:

Electro-thermal: $\mu(T)$ , $\kappa(T)$ , Joule heating
Opto-electrical: Generation, free-carrier absorption
Electro-mechanical: Piezoelectric effects, strain-modified bands

7. Emerging Frontiers

7.1 Topological Effects

Berry curvature:

\mathbf{\Omega}_n(\mathbf{k}) = i\langle\nabla_\mathbf{k} u_n| \times |\nabla_\mathbf{k} u_n\rangle

Anomalous velocity contribution:

\dot{\mathbf{r}} = \frac{1}{\hbar}\nabla_\mathbf{k} E_n - \dot{\mathbf{k}} \times \mathbf{\Omega}_n

Applications: Topological insulators, quantum Hall effect, valley-selective transport

7.2 2D Materials

Graphene (Dirac equation):

H = v_F \begin{pmatrix} 0 & p_x - ip_y \\ p_x + ip_y & 0 \end{pmatrix} = v_F \boldsymbol{\sigma} \cdot \mathbf{p}

Linear dispersion:

E = \pm \hbar v_F |\mathbf{k}|

TMDCs (valley physics):

H = at(\tau k_x \sigma_x + k_y \sigma_y) + \frac{\Delta}{2}\sigma_z + \lambda\tau\frac{\sigma_z - 1}{2}s_z

7.3 Spintronics

Spin drift-diffusion:

\frac{\partial \mathbf{s}}{\partial t} = D_s \nabla^2 \mathbf{s} - \frac{\mathbf{s}}{\tau_s} + \mathbf{s} \times \boldsymbol{\omega}

Landau-Lifshitz-Gilbert (magnetization dynamics):

\frac{d\mathbf{M}}{dt} = -\gamma \mathbf{M} \times \mathbf{H}_{eff} + \frac{\alpha}{M_s}\mathbf{M} \times \frac{d\mathbf{M}}{dt}

7.4 Plasmonics in Semiconductors

Nonlocal dielectric response:

\varepsilon(\omega, \mathbf{k}) = \varepsilon_\infty - \frac{\omega_p^2}{\omega^2 + i\gamma\omega - \beta^2 k^2}

where $\beta^2 = \frac{3}{5}v_F^2$ accounts for spatial dispersion.

Quantum corrections (Feibelman parameters):

d_\perp(\omega) = \frac{\int z \delta n(z) dz}{\int \delta n(z) dz}

Constants:

Constant	Symbol	Value
Elementary charge	$$q$$	$1.602 \times 10^{-19}$ C
Planck's constant	$$h$$	$6.626 \times 10^{-34}$ J·s
Reduced Planck's constant	$\hbar$	$1.055 \times 10^{-34}$ J·s
Boltzmann constant	$$k_B$$	$1.381 \times 10^{-23}$ J/K
Vacuum permittivity	$\varepsilon_0$	$8.854 \times 10^{-12}$ F/m
Electron mass	$$m_0$$	$9.109 \times 10^{-31}$ kg
Speed of light	$$c$$	$2.998 \times 10^{8}$ m/s

Material Parameters (Silicon @ 300K):

Parameter	Symbol	Value
Band gap	$$E_g$$	1.12 eV
Intrinsic carrier concentration	$$n_i$$	$1.0 \times 10^{10}$ cm⁻³
Electron mobility	$\mu_n$	1400 cm²/V·s
Hole mobility	$\mu_p$	450 cm²/V·s
Relative permittivity	$\varepsilon_r$	11.7
Electron effective mass	$$m_n^*/m_0$$	0.26
Hole effective mass	$$m_p^*/m_0$$	0.39

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