device physics tcad

**Device Physics, TCAD, and Mathematical Modeling**\n\nEvery transistor is governed by the same physics — the drift and diffusion of charge carriers through a doped crystal under electrostatic control — but no single equation is solved in practice. Device engineering is a ladder of approximations: the atomistic quantum picture is exact but unaffordable, the compact SPICE model is instant but only a calibrated fit, and the real work of technology computer-aided design (TCAD) is choosing the coarsest level that still captures the effect you care about. The map below is the spine of the whole field; everything that follows fills in one rung at a time.\n\n```svg\n\n```\n\n## 1. Physical Foundation\n\n### 1.1 Band Theory and Electronic Structure\n\n- **Energy bands** arise from the periodic potential of the crystal lattice — the conduction band holds empty states available for transport, the valence band holds filled states whose vacancies act as holes, and the bandgap $E_g$ separates them (Si: ~1.12 eV at 300 K).\n- **Effective mass approximation** — electrons and holes move as quasi-particles with a modified mass, electron $m_n^*$ and hole $m_p^*$, that folds the lattice potential into a single scalar.\n- **Carrier statistics** follow the Fermi–Dirac distribution:\n\n$$f(E) = \frac{1}{1 + \exp\left(\frac{E - E_F}{k_B T}\right)}$$\n\nIn non-degenerate semiconductors the carrier concentrations reduce to Boltzmann form:\n\n$$n = N_C \exp\left(-\frac{E_C - E_F}{k_B T}\right)$$\n\n$$p = N_V \exp\left(-\frac{E_F - E_V}{k_B T}\right)$$\n\nWhere:\n\n- $N_C$, $N_V$ = effective density of states in the conduction / valence bands\n- $E_C$, $E_V$ = conduction / valence band edges\n- $E_F$ = Fermi level\n\n### 1.2 Carrier Transport Mechanisms\n\n| Mechanism | Driving Force | Current Density |\n|-----------|---------------|-----------------|\n| Drift | Electric field $\mathbf{E}$ | $\mathbf{J} = qn\mu\mathbf{E}$ |\n| Diffusion | Concentration gradient | $\mathbf{J} = qD\nabla n$ |\n| Thermionic emission | Thermal energy over a barrier | Exponential in $\phi_B / k_B T$ |\n| Tunneling | Quantum penetration | Exponential in barrier width |\n\nThe **Einstein relation** ties mobility and diffusivity together, so a single measurement fixes both:\n\n$$D = \frac{k_B T}{q}\, \mu$$\n\n### 1.3 Generation and Recombination\n\nAt thermal equilibrium the mass-action law $np = n_i^2$ holds. Away from equilibrium, three mechanisms restore it: **Shockley–Read–Hall (SRH)** trap-assisted recombination, **Auger** recombination (a three-particle process that dominates at high injection), and **radiative** recombination (photon emission, important in direct-bandgap materials such as GaAs and InP).\n\n## 2. The Mathematical Hierarchy\n\n### 2.1 Quantum Mechanical Level (most fundamental)\n\nThe time-independent Schrödinger equation sets the states available to a confined carrier:\n\n$$\left[-\frac{\hbar^2}{2m^*}\nabla^2 + V(\mathbf{r})\right]\psi = E\psi$$\n\nFor open systems — tunnel FETs, ultra-scaled MOSFETs with $L_g < 10$ nm, resonant tunneling diodes — the **Non-Equilibrium Green's Function (NEGF)** formalism handles contacts and coherence:\n\n$$G^R = [EI - H - \Sigma]^{-1}$$\n\nHere $H$ is the device Hamiltonian and the self-energy $\Sigma$ encodes coupling to the contacts. This is the most physically complete and the most expensive rung on the ladder.\n\n### 2.2 Boltzmann Transport Level\n\nThe Boltzmann Transport Equation (BTE) evolves the full carrier distribution in phase space and captures hot-carrier effects, velocity overshoot, and ballistic transport that the continuum models miss:\n\n$$\frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla_{\mathbf{r}} f + \frac{\mathbf{F}}{\hbar}\cdot\nabla_{\mathbf{k}} f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}}$$\n\n**Solution methods:** stochastic Monte Carlo particle tracking, spherical-harmonics expansion (SHE), and moment methods — the last of which is exactly what produces the drift-diffusion and hydrodynamic models below.\n\n### 2.3 Hydrodynamic / Energy-Balance Level\n\nTaking moments of the BTE with carrier energy as a variable yields an energy-balance equation whose signature feature is that the carrier temperature is allowed to decouple from the lattice, $T_n \neq T_L$:\n\n$$\frac{\partial (nw)}{\partial t} + \nabla\cdot\mathbf{S} = \mathbf{J}\cdot\mathbf{E} - \frac{n(w - w_0)}{\tau_w}$$\n\nWhere $w$ is the carrier energy density, $\mathbf{S}$ the energy flux, and $\tau_w$ the energy-relaxation time.\n\n### 2.4 Drift-Diffusion Level (the workhorse)\n\nThe overwhelming majority of production TCAD runs solve three coupled PDEs. **Poisson's equation** sets the electrostatics:\n\n$$\nabla\cdot(\varepsilon\nabla\psi) = -\rho = -q\,(p - n + N_D^+ - N_A^-)$$\n\nThe **continuity equations** conserve each carrier species:\n\n$$\frac{\partial n}{\partial t} = \frac{1}{q}\nabla\cdot\mathbf{J}_n + G_n - R_n$$\n\n$$\frac{\partial p}{\partial t} = -\frac{1}{q}\nabla\cdot\mathbf{J}_p + G_p - R_p$$\n\nAnd the **current-density equations** close the system, either in drift-plus-diffusion form:\n\n$$\mathbf{J}_n = q\mu_n n\,\mathbf{E} + qD_n\nabla n$$\n\n$$\mathbf{J}_p = q\mu_p p\,\mathbf{E} - qD_p\nabla p$$\n\nor, more compactly, as a gradient of the quasi-Fermi level $\mathbf{J}_n = q\mu_n n\,\nabla E_{F,n}$. The system is coupled, nonlinear, and elliptic-parabolic, and because carrier concentrations vary exponentially with potential it spans more than ten orders of magnitude across a junction — which is what makes the discretization below non-trivial.\n\n## 3. Numerical Methods\n\n### 3.1 Spatial Discretization\n\n- **Finite Difference (FDM)** — simple, but limited to structured rectangular grids.\n- **Finite Element (FEM)** — handles complex geometry through basis-function expansion and a weak variational form.\n- **Finite Volume (FVM)** — integrates over control volumes to guarantee local conservation, which is the natural fit for the semiconductor equations.\n\n### 3.2 Scharfetter–Gummel Discretization\n\nThe single most important trick for numerical stability: it interpolates carrier density exponentially between nodes so the current stays smooth despite huge potential swings.\n\n$$J_{n,i+\frac{1}{2}} = \frac{qD_n}{h}\left[n_i B\left(\frac{\psi_i - \psi_{i+1}}{V_T}\right) - n_{i+1} B\left(\frac{\psi_{i+1} - \psi_i}{V_T}\right)\right]$$\n\nwhere the Bernoulli function is $B(x) = x / (e^x - 1)$. It reduces to central differencing for small $\Delta\psi$ and to upwinding for large $\Delta\psi$, suppressing the spurious oscillations that a naive scheme produces. The thermal voltage $V_T = k_B T / q \approx 26$ mV at 300 K sets the scale.\n\n### 3.3 Nonlinear and Linear Solvers\n\n**Gummel iteration** decouples the system — solve Poisson, then electron continuity, then hole continuity, and repeat to convergence. It is robust and cheap per step but converges slowly under strong coupling or high injection. **Newton–Raphson** solves the fully coupled linearized system $\mathbf{J}\cdot\delta\mathbf{x} = -\mathbf{F}(\mathbf{x})$ with quadratic convergence near the solution, at the cost of assembling a Jacobian and solving a larger system. In practice a **hybrid** strategy starts with Gummel to get close, then switches to Newton for fast final convergence. The resulting sparse, ill-conditioned Jacobians are solved with direct factorizations (PARDISO, UMFPACK) or preconditioned Krylov methods (GMRES, BiCGSTAB), with multigrid reserved for the Poisson-like blocks.\n\n## 4. Physical Models\n\n### 4.1 Mobility\n\nIndependent scattering mechanisms combine through Matthiessen's rule, $1/\mu = 1/\mu_\text{lattice} + 1/\mu_\text{impurity} + 1/\mu_\text{surface} + \cdots$. Lattice (phonon) scattering falls with temperature as $\mu_L = \mu_0 (T/300)^{-\alpha}$ ($\alpha \approx 2.4$ for Si electrons), while ionized-impurity scattering follows the Brooks–Herring model. At high field the velocity saturates via the Caughey–Thomas form:\n\n$$\mu(E) = \frac{\mu_0}{\left[1 + \left(\frac{\mu_0 E}{v_\text{sat}}\right)^\beta\right]^{1/\beta}}$$\n\nwith $v_\text{sat} \approx 10^7$ cm/s for silicon.\n\n### 4.2 Recombination\n\n**Shockley–Read–Hall** (trap-assisted), **Auger** (high-density), and **radiative** (direct-gap) recombination each get an explicit rate:\n\n$$R_\text{SRH} = \frac{np - n_i^2}{\tau_p(n + n_1) + \tau_n(p + p_1)}$$\n\n$$R_\text{Auger} = (C_n n + C_p p)(np - n_i^2)$$\n\n$$R_\text{rad} = B(np - n_i^2)$$\n\n### 4.3 Tunneling and Quantum Corrections\n\n**Band-to-band tunneling** — the mechanism behind tunnel FETs and Zener breakdown — scales as $G_\text{BTBT} = A\,E^2 \exp(-B/E)$. For inversion-layer quantization in scaled MOSFETs, FinFETs, and nanowires, the **density-gradient method** adds a quantum potential $V_Q = -\frac{\hbar^2}{6m^*}\frac{\nabla^2\sqrt{n}}{\sqrt{n}}$, while stronger confinement calls for a self-consistent **1D Schrödinger–Poisson** loop that solves for subbands and iterates the quantum charge into Poisson. At high doping, **bandgap narrowing** $\Delta E_g = A\,N^{1/3} + B\ln(N/N_\text{ref})$ raises $n_i^2$ and feeds back into recombination.\n\n## 5. Process TCAD\n\nThe same numerical machinery models how the device is *built*, not just how it operates. **Ion implantation** is captured either by Monte Carlo trajectory tracking or by analytic Gaussian / Pearson-IV profiles. **Diffusion** obeys Fick's laws, $\partial C/\partial t = \nabla\cdot(D\nabla C)$, with a concentration-dependent $D$ that accounts for charged point defects. **Oxidation** follows the Deal–Grove relation $x_\text{ox}^2 + A\,x_\text{ox} = B(t + \tau)$, linear for thin oxides and parabolic for thick. **Etch and deposition** surfaces evolve by the level-set equation $\partial\phi/\partial t + v_n|\nabla\phi| = 0$, where the zero contour of $\phi$ is the moving surface.\n\n## 6. Multiphysics and Reliability\n\nReal devices are never purely electrical. **Electrothermal coupling** feeds Joule and recombination heating $H = \mathbf{J}\cdot\mathbf{E} + (R - G)(E_g + 3k_BT)$ into a lattice heat equation. **Strain engineering** shifts mobility as $\mu_\text{strained} = \mu_0(1 + \Pi\cdot\sigma)$ — the basis of strained-Si and SiGe channels. **Statistical variability** from random dopant fluctuations, line-edge roughness, and metal-gate granularity is swept by Monte Carlo over device instances to produce threshold-voltage distributions. And **reliability** models — bias-temperature instability (BTI) and hot-carrier injection (HCI) — track interface-defect generation over the device lifetime, while thermal, shot, and 1/f noise set the analog floor.\n\n## 7. Computational Architecture\n\n### 7.1 Model Hierarchy — Cost vs. Accuracy\n\n| Level | Physics captured | Governing math | Cost | Accuracy |\n|-------|------------------|----------------|------|----------|\n| NEGF | Quantum coherence | $G = [EI - H - \Sigma]^{-1}$ | Highest | Highest |\n| Monte Carlo | Full distribution function | Stochastic BTE | High | High |\n| Hydrodynamic | Carrier temperature | Hyperbolic-parabolic PDEs | Medium | Good |\n| Drift-Diffusion | Continuum transport | Elliptic-parabolic PDEs | Low | Moderate |\n| Compact | Empirical fit | Algebraic | Lowest | Calibrated |\n\n### 7.2 The TCAD ↔ Compact-Model Flow\n\nTCAD does not replace circuit simulation — it *feeds* it. Physics-based TCAD is calibrated against silicon measurements, then distilled into a compact model (BSIM, PSP) whose algebraic I–V equations are what SPICE actually evaluates a billion times per chip. Silicon data validates the TCAD; the compact model enables the circuit. That two-way loop — physical rigor upstream, computational speed downstream — is the reason the hierarchy at the top of this page exists at all.\n\n## 8. Reference Values\n\n| Symbol | Name | Value |\n|--------|------|-------|\n| $q$ | Elementary charge | $1.602 \times 10^{-19}$ C |\n| $k_B$ | Boltzmann constant | $1.381 \times 10^{-23}$ J/K |\n| $\hbar$ | Reduced Planck | $1.055 \times 10^{-34}$ J·s |\n| $\varepsilon_0$ | Vacuum permittivity | $8.854 \times 10^{-12}$ F/m |\n| $V_T$ | Thermal voltage (300 K) | 25.9 mV |\n\n| Silicon property (300 K) | Value |\n|--------------------------|-------|\n| Bandgap $E_g$ | 1.12 eV |\n| Intrinsic carrier density $n_i$ | $1.0 \times 10^{10}$ cm⁻³ |\n| Electron mobility $\mu_n$ | 1450 cm²/V·s |\n| Hole mobility $\mu_p$ | 500 cm²/V·s |\n| Electron saturation velocity | $1.0 \times 10^7$ cm/s |\n| Relative permittivity $\varepsilon_r$ | 11.7 |\n\nRead device physics through a *quantitative* lens rather than a purely qualitative one: the transistor is not a schematic symbol but a boundary-value problem, and every design decision — channel material, doping profile, gate stack, thermal budget — is ultimately a choice about which term in these equations you are willing to pay to solve exactly and which you can afford to approximate.\n