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Device Physics, TCAD, and Mathematical Modeling

Keywords: device physics tcad,tcad,device physics,semiconductor device physics,band theory,drift diffusion,poisson equation,boltzmann transport,carrier transport,mobility models,recombination models,process tcad

Device Physics, TCAD, and Mathematical Modeling

1. Physical Foundation

1.1 Band Theory and Electronic Structure

Energy bands arise from the periodic potential of the crystal lattice

Conduction band (empty states available for electron transport)

Valence band (filled states; holes represent missing electrons)

Bandgap $E_g$ separates these bands (Si: ~1.12 eV at 300K)

Effective mass approximation

Electrons and holes behave as quasi-particles with modified mass

Electron effective mass: $m_n^*$

Hole effective mass: $m_p^*$

Carrier statistics follow Fermi-Dirac distribution:

f(E) = \frac{1}{1 + \exp\left(\frac{E - E_F}{k_B T}\right)}

Carrier concentrations in non-degenerate semiconductors:

n = N_C \exp\left(-\frac{E_C - E_F}{k_B T}\right)

p = N_V \exp\left(-\frac{E_F - E_V}{k_B T}\right)

Where:

$N_C$, $N_V$ = effective density of states in conduction/valence bands
$E_C$, $E_V$ = conduction/valence band edges
$E_F$ = Fermi level

1.2 Carrier Transport Mechanisms

Mechanism	Driving Force	Current Density
Drift	Electric field $\mathbf{E}$	$\mathbf{J} = qn\mu\mathbf{E}$
Diffusion	Concentration gradient

abla n$ |

Thermionic emission	Thermal energy over barrier	Exponential in $\phi_B/k_BT$
Tunneling	Quantum penetration	Exponential in barrier

Einstein relation connects mobility and diffusivity:

D = \frac{k_B T}{q} \mu

1.3 Generation and Recombination

Thermal equilibrium condition:

$$np = n_i^2$$

Three primary recombination mechanisms:

1. Shockley-Read-Hall (SRH) — trap-assisted 2. Auger — three-particle process (dominant at high injection) 3. Radiative — photon emission (important in direct bandgap materials)

2. Mathematical Hierarchy

2.1 Quantum Mechanical Level (Most Fundamental)

Time-Independent Schrödinger Equation

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

Where:

$\hbar$ = reduced Planck constant
$m^*$ = effective mass
$V(\mathbf{r})$ = potential energy
$\psi$ = wavefunction
$E$ = energy eigenvalue

Non-Equilibrium Green's Function (NEGF)

For open quantum systems (nanoscale devices, tunneling):

G^R = [EI - H - \Sigma]^{-1}

$G^R$ = retarded Green's function
$H$ = device Hamiltonian
$\Sigma$ = self-energy (encodes contact coupling)

Applications:

Tunnel FETs
Ultra-scaled MOSFETs ($L_g < 10$ nm)
Quantum well devices
Resonant tunneling diodes

2.2 Boltzmann Transport Level

Boltzmann Transport Equation (BTE)

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

Where:

$f(\mathbf{r}, \mathbf{k}, t)$ = distribution function in phase space
$\mathbf{v}$ = group velocity
$\mathbf{F}$ = external force
RHS = collision integral

Solution Methods:

Monte Carlo (stochastic particle tracking)
Spherical Harmonics Expansion (SHE)
Moments methods → leads to drift-diffusion, hydrodynamic

Captures:

Hot carrier effects
Velocity overshoot
Non-equilibrium distributions
Ballistic transport

2.3 Hydrodynamic / Energy Balance Level

Derived from moments of BTE with carrier temperature as variable:

\frac{\partial (nw)}{\partial t} + abla \cdot \mathbf{S} = \mathbf{J} \cdot \mathbf{E} - \frac{n(w - w_0)}{\tau_w}

$w$ = carrier energy density
$\mathbf{S}$ = energy flux
$\tau_w$ = energy relaxation time
$w_0$ = equilibrium energy density

Key feature: Carrier temperature $T_n eq$ lattice temperature $T_L$

2.4 Drift-Diffusion Level (The Workhorse)

The most widely used TCAD formulation — three coupled PDEs:

Poisson's Equation (Electrostatics)

abla \cdot (\varepsilon abla \psi) = -\rho = -q(p - n + N_D^+ - N_A^-)

$\psi$ = electrostatic potential
$\varepsilon$ = permittivity
$\rho$ = charge density
$N_D^+$, $N_A^-$ = ionized donor/acceptor concentrations

Electron Continuity Equation

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

Hole Continuity Equation

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

Current Density Equations

Standard form:

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

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

Quasi-Fermi level formulation:

\mathbf{J}_n = q\mu_n n abla E_{F,n}

\mathbf{J}_p = q\mu_p p abla E_{F,p}

System characteristics:

Coupled, nonlinear, elliptic-parabolic PDEs
Carrier concentrations vary exponentially with potential
Spans 10+ orders of magnitude across junctions

3. Numerical Methods

3.1 Spatial Discretization

Finite Difference Method (FDM)

Simple implementation
Limited to structured (rectangular) grids
Box integration for conservation

Finite Element Method (FEM)

Handles complex geometries
Basis function expansion
Weak (variational) formulation

Finite Volume Method (FVM)

Ensures local conservation
Natural for semiconductor equations
Control volume integration

3.2 Scharfetter-Gummel Discretization

Critical for numerical stability — handles exponential carrier variations:

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

Where the Bernoulli function is:

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

Properties:

Reduces to central difference for small $\Delta\psi$
Reduces to upwind for large $\Delta\psi$
Prevents spurious oscillations
Thermal voltage: $V_T = k_B T / q \approx 26$ mV at 300K

3.3 Mesh Generation

2D: Delaunay triangulation
3D: Tetrahedral meshing

Adaptive refinement criteria:

Junction regions (high field gradients)
Oxide interfaces
Contact regions
High current density areas

Quality metrics:

Aspect ratio
Orthogonality (important for FVM)
Delaunay property (circumsphere criterion)

3.4 Nonlinear Solvers

Gummel Iteration (Decoupled)

repeat: 1. Solve Poisson equation → ψ 2. Solve electron continuity → n 3. Solve hole continuity → p until convergence

Pros:

Simple implementation
Robust for moderate bias
Each subproblem is smaller

Cons:

Poor convergence at high injection
Slow for strongly coupled systems

Newton-Raphson (Fully Coupled)

Solve the linearized system:

\mathbf{J} \cdot \delta\mathbf{x} = -\mathbf{F}(\mathbf{x})

Where:

$\mathbf{J}$ = Jacobian matrix $\partial \mathbf{F}/\partial \mathbf{x}$
$\mathbf{F}$ = residual vector
$\delta\mathbf{x}$ = update vector

Pros:

Quadratic convergence near solution
Handles strong coupling

Cons:

Requires good initial guess
Expensive Jacobian assembly
Larger linear systems

Hybrid Methods

Start with Gummel to get close
Switch to Newton for fast final convergence

3.5 Linear Solvers

For large, sparse, ill-conditioned Jacobian systems:

Method	Type	Characteristics
LU (PARDISO, UMFPACK)	Direct	Robust, memory-intensive
GMRES	Iterative	Krylov subspace, needs preconditioning
BiCGSTAB	Iterative	Non-symmetric systems
Multigrid	Iterative	Optimal for Poisson-like equations

4. Physical Models in TCAD

4.1 Mobility Models

Matthiessen's Rule

Combines independent scattering mechanisms:

\frac{1}{\mu} = \frac{1}{\mu_{\text{lattice}}} + \frac{1}{\mu_{\text{impurity}}} + \frac{1}{\mu_{\text{surface}}} + \cdots

Lattice Scattering

\mu_L = \mu_0 \left(\frac{T}{300}\right)^{-\alpha}

Si electrons: $\alpha \approx 2.4$
Si holes: $\alpha \approx 2.2$

Ionized Impurity Scattering

Brooks-Herring model:

\mu_I \propto \frac{T^{3/2}}{N_I \cdot \ln(1 + b^2) - b^2/(1+b^2)}

High-Field Saturation (Caughey-Thomas)

\mu(E) = \frac{\mu_0}{\left[1 + \left(\frac{\mu_0 E}{v_{\text{sat}}}\right)^\beta\right]^{1/\beta}}

$v_{\text{sat}}$ = saturation velocity (~$10^7$ cm/s for Si)
$\beta$ = fitting parameter (~2 for electrons, ~1 for holes)

4.2 Recombination Models

Shockley-Read-Hall (SRH)

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

Where:

$\tau_n$, $\tau_p$ = carrier lifetimes
$n_1 = n_i \exp[(E_t - E_i)/k_BT]$
$p_1 = n_i \exp[(E_i - E_t)/k_BT]$
$E_t$ = trap energy level

Auger Recombination

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

$C_n$, $C_p$ = Auger coefficients (~$10^{-31}$ cm$^6$/s for Si)
Dominant at high carrier densities ($>10^{18}$ cm$^{-3}$)

Radiative Recombination

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

$B$ = radiative coefficient
Important in direct bandgap materials (GaAs, InP)

4.3 Band-to-Band Tunneling

For tunnel FETs, Zener diodes:

G_{\text{BTBT}} = A \cdot E^2 \exp\left(-\frac{B}{E}\right)

$A$, $B$ = material-dependent parameters
$E$ = electric field magnitude

4.4 Quantum Corrections

Density Gradient Method

Adds quantum potential to classical equations:

V_Q = -\frac{\hbar^2}{6m^*} \frac{ abla^2\sqrt{n}}{\sqrt{n}}

Or equivalently, the quantum potential term:

\Lambda_n = \frac{\hbar^2}{12 m_n^* k_B T} abla^2 \ln(n)

Applications:

Inversion layer quantization in MOSFETs
Thin body SOI devices
FinFETs, nanowires

1D Schrödinger-Poisson

For stronger quantum confinement: 1. Solve 1D Schrödinger in confinement direction → subbands $E_i$, $\psi_i$ 2. Calculate 2D density of states 3. Compute carrier density from subband occupation 4. Solve 2D Poisson with quantum charge 5. Iterate to self-consistency

4.5 Bandgap Narrowing

At high doping ($N > 10^{17}$ cm$^{-3}$):

\Delta E_g = A \cdot N^{1/3} + B \cdot \ln\left(\frac{N}{N_{\text{ref}}}\right)

Effect: Increases $n_i^2$ → affects recombination and device characteristics

4.6 Interface Models

Interface trap density: $D_{it}(E)$ — states per cm$^2$·eV
Oxide charges:

Fixed oxide charge $Q_f$

Mobile ionic charge $Q_m$

Oxide trapped charge $Q_{ot}$

Interface trapped charge $Q_{it}$

5. Process TCAD

5.1 Ion Implantation

Monte Carlo Method

Track individual ion trajectories
Binary collision approximation
Accurate for low doses, complex geometries

Analytical Profiles

Gaussian:

N(x) = \frac{\Phi}{\sqrt{2\pi}\Delta R_p} \exp\left[-\frac{(x - R_p)^2}{2\Delta R_p^2}\right]

$\Phi$ = dose (ions/cm$^2$)
$R_p$ = projected range
$\Delta R_p$ = straggle

Pearson IV: Adds skewness and kurtosis for better accuracy

5.2 Diffusion

Fick's First Law:

\mathbf{J} = -D abla C

Fick's Second Law:

\frac{\partial C}{\partial t} = abla \cdot (D abla C)

Concentration-dependent diffusion:

D = D_i \left(\frac{n}{n_i}\right)^2 + D_v + D_x \left(\frac{n}{n_i}\right)

(Accounts for charged point defects)

5.3 Oxidation

Deal-Grove Model:

x_{ox}^2 + A \cdot x_{ox} = B(t + \tau)

$x_{ox}$ = oxide thickness
$A$, $B$ = temperature-dependent parameters
Linear regime: $x_{ox} \approx (B/A) \cdot t$ (thin oxide)
Parabolic regime: $x_{ox} \approx \sqrt{B \cdot t}$ (thick oxide)

5.4 Etching and Deposition

Level-set method for surface evolution:

\frac{\partial \phi}{\partial t} + v_n | abla \phi| = 0

$\phi$ = level-set function (zero contour = surface)
$v_n$ = normal velocity (etch/deposition rate)

6. Multiphysics and Advanced Topics

6.1 Electrothermal Coupling

Heat equation:

\rho c_p \frac{\partial T}{\partial t} = abla \cdot (\kappa abla T) + H

Heat generation:

H = \mathbf{J} \cdot \mathbf{E} + (R - G)(E_g + 3k_BT)

First term: Joule heating
Second term: recombination heating

Thermoelectric effects:

Seebeck effect
Peltier effect
Thomson effect

6.2 Electromechanical Coupling

Strain effects on mobility:

\mu_{\text{strained}} = \mu_0 (1 + \Pi \cdot \sigma)

$\Pi$ = piezoresistance coefficient
$\sigma$ = mechanical stress

Applications: Strained Si, SiGe channels

6.3 Statistical Variability

Sources of random variation:

Random Dopant Fluctuations (RDF) — discrete dopant positions
Line Edge Roughness (LER) — gate patterning variation
Metal Gate Granularity (MGG) — work function variation
Oxide Thickness Variation (OTV)

Simulation approach:

Monte Carlo sampling over device instances
Statistical TCAD → threshold voltage distributions

6.4 Reliability Modeling

Bias Temperature Instability (BTI):

Defect generation at Si/SiO$_2$ interface
Reaction-diffusion models

Hot Carrier Injection (HCI):

High-energy carriers damage interface
Coupled with energy transport

6.5 Noise Modeling

Noise sources:

Thermal noise: $S_I = 4k_BT/R$
Shot noise: $S_I = 2qI$
1/f noise (flicker): $S_I \propto I^2/(f \cdot N)$

Impedance field method for spatial correlation

7. Computational Architecture

7.1 Model Hierarchy Comparison

Level	Physics	Math	Cost	Accuracy
NEGF	Quantum coherence	$G = [E-H-\Sigma]^{-1}$	$$$$$	Highest
Monte Carlo	Distribution function	Stochastic DEs	$$$$	High
Hydrodynamic	Carrier temperature	Hyperbolic-parabolic PDEs	$$$	Good
Drift-Diffusion	Continuum transport	Elliptic-parabolic PDEs	$$	Moderate
Compact Models	Empirical	Algebraic	$	Calibrated

7.2 Software Architecture

┌─────────────────────────────────────────┐
│           User Interface (GUI)          │
├─────────────────────────────────────────┤
│         Structure Definition            │
│    (Geometry, Mesh, Materials)          │
├─────────────────────────────────────────┤
│          Physical Models                │
│  (Mobility, Recombination, Quantum)     │
├─────────────────────────────────────────┤
│         Numerical Engine                │
│  (Discretization, Solvers, Linear Alg)  │
├─────────────────────────────────────────┤
│        Post-Processing                  │
│   (Visualization, Parameter Extraction) │
└─────────────────────────────────────────┘

7.3 TCAD ↔ Compact Model Flow

┌──────────┐    calibrate    ┌──────────────┐
│   TCAD   │ ──────────────► │ Compact Model│
│(Physics) │                 │   (BSIM,PSP) │
└──────────┘                 └──────────────┘
      │                             │
      │ validate                    │ enable
      ▼                             ▼
┌──────────┐                 ┌──────────────┐
│ Silicon  │                 │   Circuit    │
│   Data   │                 │  Simulation  │
└──────────┘                 └──────────────┘

Equations:

Fundamental Constants

Symbol	Name	Value
$q$	Elementary charge	$1.602 \times 10^{-19}$ C
$k_B$	Boltzmann constant	$1.381 \times 10^{-23}$ J/K
$\hbar$	Reduced Planck	$1.055 \times 10^{-34}$ J·s
$\varepsilon_0$	Vacuum permittivity	$8.854 \times 10^{-12}$ F/m
$V_T$	Thermal voltage (300K)	25.9 mV

Silicon Properties (300K)

Property	Value
Bandgap $E_g$	1.12 eV
Intrinsic carrier density $n_i$	$1.0 \times 10^{10}$ cm$^{-3}$
Electron mobility $\mu_n$	1450 cm$^2$/V·s
Hole mobility $\mu_p$	500 cm$^2$/V·s
Electron saturation velocity	$1.0 \times 10^7$ cm/s
Relative permittivity $\varepsilon_r$	11.7

Source: ChipFoundryServices — Search this topic — Ask CFSGPT

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