How to compute Scharfetter-Gummel semiconductor discretization

Scharfetter-Gummel (1969) discretization calculator for the semiconductor drift-diffusion model: current density via the Bernoulli function B(x)=x/(e^x-1), thermal voltage, diffusion coefficient, electric field and Debye length. Presets for Si, GaAs, GaN, SiC, Ge, InP, InAs and diamond.

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The Scharfetter-Gummel discretization calculator determines the current density J in semiconductor devices using the standard numerical scheme for drift-diffusion equations, developed by Donald Scharfetter and Hermann Gummel (Bell Labs, 1969) in "Large-signal analysis of a silicon Read diode oscillator" IEEE Transactions on Electron Devices 16(1):64-77. The scheme provides stable exponential interpolation of the current density J = q μ n E - q D ∇n across a discrete mesh, avoiding numerical instabilities of the classical centered scheme. Key formula: J_{i+1/2} = (q μ V_T / Δx) [n_{i+1} · B(-ΔV/V_T) - n_i · B(+ΔV/V_T)], where B(x) = x/(e^x - 1) is the Bernoulli function. It is the workhorse of all commercial TCAD simulators: Synopsys Sentaurus, Silvaco Atlas, Cadence Spectre, DESSIS, COMSOL Semiconductor.

1. Scharfetter-Gummel 1969 formula

The drift-diffusion equation for electrons: J_n = q μ_n n E + q D_n ∇n, where μ_n is mobility, D_n = μ_n V_T (Einstein-Smoluchowski) the diffusion coefficient, V_T = k_B T/q ≈ 25.85 mV at 300 K thermal voltage. Discretized on mesh: J_{i+1/2} = (q μ V_T / Δx) · [n_{i+1} B(-(ψ_{i+1} - ψ_i)/V_T) - n_i B((ψ_{i+1} - ψ_i)/V_T)]. The Bernoulli function B(x) = x/(e^x - 1) has properties: B(0) = 1, B(+∞) = 0, B(-∞) = -∞, B(x) - B(-x) = -x. Guarantees current conservation and monotonicity even with large potential gradients (>> V_T). For ΔV/V_T → 0 it recovers the linear centered scheme. For |ΔV/V_T| > 1, the centered scheme fails (negative density), Scharfetter-Gummel remains stable.

2. Semiconductor materials and parameters

  • Silicon Si: μ_n = 1450 cm²/V·s, μ_p = 480, ε_r = 11.7, E_g = 1.12 eV.
  • Gallium arsenide GaAs: μ_n = 8500, μ_p = 400, ε_r = 13.1, E_g = 1.42 eV (HEMT, RF).
  • Gallium nitride GaN: μ_n = 1000, μ_p = 200, ε_r = 9.5, E_g = 3.4 eV (LED, power).
  • Silicon carbide 4H-SiC: μ_n = 900, μ_p = 100, ε_r = 9.7, E_g = 3.26 eV (power).
  • Germanium Ge: μ_n = 3900, μ_p = 1900, ε_r = 16.2, E_g = 0.66 eV.
  • Indium phosphide InP: μ_n = 5400, μ_p = 200, ε_r = 12.5, E_g = 1.35 eV (laser).
  • Indium arsenide InAs: μ_n = 33000, μ_p = 460, ε_r = 15.1 (high-speed HEMT).
  • Diamond C: μ_n = 4500, μ_p = 3800, E_g = 5.5 eV (UV detector, power).

3. Range of validity of the scheme

Scharfetter-Gummel is a box-method finite-volume scheme for classical Drude-Pearson drift-diffusion equations. Limits: (a) Velocity saturation: at high fields E > 10⁵ V/cm in Si, μ(E) decreases → extended model needed (Caughey-Thomas, Canali). (b) Hot carriers: in non-equilibrium, energy-balance + impact ionization (Selberherr-Stratton). (c) Quantum confinement: in sub-10nm MOSFET, classical drift-diffusion fails, requires Schrödinger-Poisson coupling or NEGF (Non-Equilibrium Green's Functions). (d) Recombination/generation: SRH, Auger, radiative add source/sink terms. (e) Heterointerfaces: large band discontinuities require thermionic emission boundary. (f) Tunneling: WKB approx for BTBT (band-to-band) and direct gate oxide tunneling. (g) Magnetic fields: Hall effect, magnetoresistance. (h) RF frequencies > 1 THz: beyond relaxation time τ_p, hydrodynamic model required.

4. Worked numerical examples

Example 1 — Si pn-junction diode at 0.5 V forward bias. µ_n=1450, T=300 K, V_T=25.85 mV, Δx=10 nm, ψ₁=0, ψ₂=0.5 V → ΔV/V_T = 19.3. B(+19.3)=4.2e-7, B(-19.3)=19.3. n₁=10¹⁶/cm³, n₂=10¹⁵/cm³. J_SG = (q·µ·V_T/Δx)·[n₂·B(-19.3) - n₁·B(+19.3)] = 1.87 × 10⁵ A/cm². Typical forward-bias diode.

Example 2 — 22nm Si MOSFET at strong inversion. n_inv=5e¹²/cm² × thickness=2nm = 2.5e¹⁹/cm³. ΔV_DS=1V over 22nm. ΔV/V_T=38.6. SG captures correct exp decay accumulation→pinchoff. J_drain ≈ 10⁵ A/cm². GSS Sentaurus model + STMicroelectronics 22FDX FD-SOI.

Example 3 — GaN HEMT 600V power. μ_n=1000 cm²/Vs (2DEG), V_T=25.85 mV, ΔV=600V over 5μm. ΔV/V_T=2.3e4 (extreme). SG gives result, but velocity saturation v_sat=2.5e7 cm/s needed. Synopsys Sentaurus device GaN model.

5. Practical applications of Scharfetter-Gummel

  • STMicroelectronics Catania-Agrate: Sentaurus TCAD simulation for BCD8s (Bipolar-CMOS-DMOS) 0.18μm + 16FFC FinFET, automotive Bipolar SiPM (silicon photomultiplier).
  • IMM-CNR Catania (Institute for Microelectronics and Microsystems): SiC power device research (MPS diodes, MOSFETs 1.2 kV+), GaN HEMT, Atlas FEM simulations.
  • Politecnico Milano DEIB: Marco Sampietro + Maurizio Sansotera + Alessandro Pesatori group develops quantum-corrected Scharfetter-Gummel models for nano-MOSFET.
  • Politecnico Torino DET: Andrea Lacaita group SPAD/SiPM + Avalanche Cellular Automata simulations + Sentaurus.
  • INFN-Padua + Padua University: silicon strip detectors ATLAS-LHC + STAR RHIC + ALICE LHC, Geant4 + Sentaurus.
  • SISSA Trieste 1978 + ICTP Trieste 1964: Stefano Baroni Quantum ESPRESSO ab initio + DFT, coupled with drift-diffusion.
  • ISMN-CNR Bologna: organic semiconductor OLED, OFET, OPV simulations with disorder-modified SG.
  • Cineca Bologna 1969 Marconi100 + Leonardo HPC 250 PFlops + INFN-CNAF: TCAD Sentaurus + COMSOL on GPU nodes for large simulations (10⁶+ mesh nodes).

6. Comparison with other numerical schemes

Scharfetter-Gummel 1969: Box-method finite-volume scheme based on Bernoulli function, monotonic, conservative, stable for |ΔV| >> V_T. Centered finite differences: unstable for |ΔV/V_T| > 1, numerical oscillations. Upwind first-order: stable but diffusive (artificial diffusion). FCT (Flux Corrected Transport) Boris-Book 1973: minimizes diffusion but TVD constraints. Petrov-Galerkin: finite-element variation with upwind weighting. Streamline upwind Petrov-Galerkin (SUPG): 2D/3D extension of SG. Discontinuous Galerkin (DG): high-order, stability via slope limiting. NEGF: for nano-device < 10nm, Schrödinger + Green functions. Hydrodynamic model (Stratton 1962, Selberherr 1980s): energy-balance, hot carriers. Spherical harmonic expansion (SHE): beyond drift-diffusion, non-equilibrium transport. SG remains the workhorse for semicond > 50 nm.

7. Modern developments and corrections

Modern Scharfetter-Gummel variants include: Kramer-Hitchner 1999 2D triangular generalization. Mock 1972 mixed FE-FV for heterointerfaces. Bank-Rose 1981 global Newton-Raphson convergence analysis. Bank-Coughran 1991 orthogonal Voronoi Box-method. Brezzi-Marini-Pietra 1989 ELLAM Eulerian-Lagrangian Localized Adjoint Method. Quantum-corrected SG: density-gradient (Wettstein-Demmel-Wenzel 2002), Bohm potential, Wigner formalism. NEGF + SG hybrid: Datta-Lake-Klimeck 2002 nano-MOSFETs Stanford. Multiscale TCAD: Synopsys Sentaurus + ATK Quantum + GTS Minimos. Commercial software: Synopsys Sentaurus Device (TCAD), Silvaco Atlas/Victory, Cadence Spectre, COMSOL Semiconductor Module, Ansys TCAD. Open-source: DESSIS (TU Delft), Charon (Sandia), NEMO5 (Purdue), BIA Simulator (UCSD), TIBERCAD (Tor Vergata Rome). HPC: Cineca Leonardo 250 PFlops + INFN-CNAF + LRZ Garching + Jülich. Future experiments: 3D heterogeneous integration (chiplets), quantum dot devices, 2D materials (graphene, MoS₂, WSe₂) require SG extensions.

Updated July 2026

Frequently Asked Questions

What is Scharfetter-Gummel discretization?

The Scharfetter-Gummel discretization (Donald Scharfetter + Hermann Gummel Bell Labs 1969 IEEE TED 16:64) is the standard numerical scheme for drift-diffusion equations in semiconductors. It uses the Bernoulli function B(x)=x/(e^x-1) to discretize current J monotonically and stably even with large potential gradients (>> V_T). It is the workhorse of all commercial TCAD simulators.

What is the Bernoulli function B(x)?

B(x) = x/(e^x - 1) is the generalized Bernoulli function. Key properties: B(0)=1 (linear limit), B(+∞)→0 (positive exp cutoff), B(-∞)→-∞ (negative exp decay), B(x)-B(-x)=-x. Used in Scharfetter-Gummel to interpolate density under exponentially varying potential. Numerical implementation: Taylor series for |x|<1e-4 to avoid numerical cancellation.

What does V_T = k_B T/q mean?

V_T = k_BT/q is the thermal voltage, k_B=1.381e-23 J/K (Boltzmann), q=1.602e-19 C, T temperature. At 300 K (room): V_T = 25.85 mV. It's the characteristic potential scale in semiconductor junctions. Diffusion coefficient D=μ·V_T (Einstein-Smoluchowski relation). Typical SG range: V_T < ΔV < 50 V_T for stability.

Why is Scharfetter-Gummel better than centered scheme?

The centered linear scheme has O(Δx²) error but is unstable for |ΔV/V_T| > 1: produces negative (non-physical) densities and oscillations. Scharfetter-Gummel captures the exponential decay n(x) ∝ exp(-qψ/k_BT) via the Bernoulli function, guaranteeing monotonicity and positive densities under any field condition. Cost: same complexity as centered (O(N) memory), but greater robustness.

Which materials are simulated with Scharfetter-Gummel?

All bulk semiconductors: Si μ_n=1450 cm²/Vs (standard CMOS), GaAs μ_n=8500 (RF, HEMT), GaN μ_n=1000 E_g=3.4eV (LED, 600V+ power), 4H-SiC μ_n=900 E_g=3.26eV (1.2kV+ power), Ge μ_n=3900 (low-T devices), InP μ_n=5400 (semiconductor laser), InAs μ_n=33000 (high-speed HEMT), Diamond μ_n=4500 E_g=5.5eV (UV).

What are the limits of Scharfetter-Gummel?

Fails for: (a) velocity saturation E > 10⁵ V/cm in Si (Caughey-Thomas model), (b) non-equilibrium hot carriers (Stratton 1962 energy-balance), (c) sub-10nm quantum confinement (Schrödinger-Poisson or NEGF), (d) band-to-band tunneling (WKB), (e) 2D MoS₂/graphene devices (Boltzmann transport equation), (f) heterointerfaces (thermionic emission boundary).

What are the industrial applications in Italy?

STMicroelectronics Catania-Agrate (Sentaurus for 22FDX FD-SOI + BCD8s automotive + SiPM detectors), IMM-CNR Catania (SiC power devices + GaN HEMT + Atlas), Politecnico Milano DEIB (quantum-corrected nano MOSFET), Politecnico Torino DET (Lacaita SPAD/SiPM), INFN-Padua (ATLAS strip detector LHC), ISMN-CNR Bologna (organic OLED/OFET/OPV), Cineca Leonardo 250 PFlops + INFN-CNAF (TCAD HPC).

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