EEE4450: Modeling and Simulation of Semiconductor Devices

Course Description

EEE 4450 Modeling and Simulation of Semiconductor Devices is the course that sits between device physics and the numerical methods needed to make that physics computable. The equations governing carrier transport in a semiconductor — Poisson’s equation coupled to the electron and hole continuity equations — have closed-form solutions only for idealised one-dimensional cases. Everything else is solved numerically, and this course teaches how.

The Statewide Course Numbering System description is precise: the course “covers various numerical techniques for the modeling and simulation of semiconductor devices, such as pn-junctions, metal-oxide semiconductor contacts, metal-oxide-semiconductor field-effect-transistors, and bipolar devices. Special emphasis is on the description and simulation of electron and hole transport in semiconductor devices.”

Florida A&M University carries the course at 3 credits. The FAMU–FSU College of Engineering bulletin lists it with a prerequisite of EEE 3300 and describes it in matching terms.

What kind of course this is. This is a computational course. Students spend their time discretising differential equations, implementing iterative solvers, and diagnosing why a solution failed to converge — work that resembles a numerical methods course more than a circuits course. The reward is a genuine understanding of what a TCAD tool is doing, which is what separates an engineer who can interpret a simulation from one who can only run it.

Learning Outcomes

Required Outcomes

• State the semiconductor equations — Poisson’s equation and the electron and hole continuity equations with drift-diffusion current relations — and explain the physical meaning of each term and each coupling.
• Explain why the system is non-linear and coupled, and why that forces an iterative numerical solution.
• Discretise the semiconductor equations on a one-dimensional mesh using finite differences, and apply appropriate boundary conditions at contacts and interfaces.
• Apply the Scharfetter–Gummel discretisation for the current continuity equations and explain why naive finite differencing fails at realistic bias.
• Implement and compare the Gummel (decoupled) and Newton (fully coupled) iteration schemes, and state the convergence and cost trade-off between them.
• Apply appropriate scaling and normalisation to the variables and explain why unscaled formulations produce ill-conditioned systems.
• Solve the resulting linear systems using direct and iterative methods, and assess conditioning.
• Design a computational mesh, applying refinement where gradients are steep, and demonstrate that a result is mesh-independent.
• Simulate a p-n junction and extract its I–V and C–V characteristics, comparing against analytical depletion-approximation results and explaining the differences.
• Simulate a MOS capacitor through accumulation, depletion and inversion, and extract threshold voltage from the computed solution.
• Simulate MOSFET output and transfer characteristics and identify short-channel effects in the numerical results.
• Simulate bipolar device behaviour and extract current gain and the Early voltage.
• Select physical models appropriate to a simulation — mobility, recombination, bandgap narrowing — and justify the selection.
• Validate simulation results against analytical limits or measured data, and diagnose non-convergence systematically.

Optional Outcomes

• Extend the drift-diffusion formulation to two dimensions and apply finite-element or finite-volume discretisation.
• Apply hydrodynamic and energy-balance transport models and state when drift-diffusion becomes inadequate.
• Apply Monte Carlo particle simulation of carrier transport.
• Perform transient and small-signal AC device simulation.
• Perform process simulation and couple its output to device simulation.
• Extract compact model (SPICE) parameters from simulated device characteristics.
• Simulate heterostructure or wide-bandgap devices.
• Use a commercial TCAD package and compare its results against a self-written solver.

Major Topics

Required Topics

• The semiconductor equations — Poisson’s equation, electron and hole continuity, drift-diffusion current relations, quasi-Fermi levels, and the boundary conditions at ohmic and Schottky contacts, insulators and free surfaces.
• Physical models — field- and doping-dependent mobility, velocity saturation, Shockley–Read–Hall, Auger and surface recombination, impact ionisation, bandgap narrowing, and Fermi–Dirac versus Boltzmann statistics.
• Scaling and normalisation — De Mari scaling, dimensionless variables, and the effect of scaling on conditioning and convergence.
• Discretisation — finite-difference approximation on uniform and non-uniform meshes, truncation error, mesh generation and refinement strategies, and mesh-independence testing.
• The Scharfetter–Gummel scheme — the exponential fitting argument, the Bernoulli function, and why central differencing of the current equation produces unphysical oscillation at moderate bias. This is the central numerical idea of the course.
• Iterative solution schemes — the Gummel decoupled iteration and its linear convergence; Newton’s method and the Jacobian for the fully coupled system, with its quadratic convergence and higher per-step cost; damping and continuation strategies.
• Linear algebra — sparse matrix storage, direct solution by LU with sparse ordering, iterative Krylov methods and preconditioning, and matrix conditioning as a diagnostic.
• Convergence behaviour — residual and update criteria, bias ramping and continuation, initial-guess construction, and the systematic diagnosis of divergence.
• p-n junction simulation — equilibrium solution and band diagram, forward and reverse bias, computed I–V and C–V, and comparison against the depletion approximation and the ideal diode equation.
• MOS capacitor and contact simulation — accumulation, depletion and inversion; surface potential; computed C–V curves; threshold voltage extraction; metal–semiconductor contact modelling.
• MOSFET simulation — output and transfer characteristics, the two-dimensional nature of the problem, subthreshold behaviour, channel-length modulation, drain-induced barrier lowering and other short-channel effects as they emerge numerically.
• Bipolar device simulation — minority carrier profiles, Gummel plots, current gain and its bias dependence, base-width modulation and high-injection roll-off.
• Validation — checking against analytical limits, against measured data, and against the physics (charge neutrality, current continuity) as a self-consistency test.

Optional Topics

• Two- and three-dimensional simulation; finite-element and finite-volume methods; unstructured meshes.
• Beyond drift-diffusion: hydrodynamic and energy-balance models, and the conditions that require them.
• Monte Carlo particle transport simulation; the Boltzmann transport equation.
• Quantum corrections: density-gradient and Schrödinger–Poisson coupling for thin oxides and confined channels.
• Transient and AC small-signal simulation.
• Process simulation and process/device coupling.
• Compact model parameter extraction from simulated characteristics.
• Heterostructure, SiC and GaN device simulation.
• Commercial TCAD (Silvaco Atlas, Synopsys Sentaurus) and comparison against a self-written solver.

Resources & Tools

• Analysis and Simulation of Semiconductor Devices (Selberherr) is the foundational reference for this subject and the standard citation for the numerical formulation. Semiconductor Device Simulation and Computational Electronics (Vasileska, Goodnick and Klimeck) are the modern texts closest to this course’s scope. Fundamentals of Modern VLSI Devices (Taur and Ning) supplies the device physics being modelled.
• Numerical Recipes and a standard numerical methods text remain useful for the solver material, which many students meet for the first time here.
• MATLAB or Python for writing your own solver — the core activity of the course. Python with NumPy and SciPy (sparse matrices, scipy.sparse.linalg) is entirely adequate and is free; MATLAB’s sparse tools and profiler are convenient.
• Commercial TCAD — Silvaco Atlas and Synopsys Sentaurus Device are the industry tools. Where a course provides access, comparing your own solver against a commercial one is an unusually instructive exercise.
• Open tools — the nanoHUB platform (nanohub.org, free to academic users) hosts device simulators including PADRE and ADEPT, along with lecture series on computational electronics that map closely onto this course. DEVSIM and Charon are open-source TCAD codes worth knowing about.
• Reference material — the IEEE Electron Devices Society, IEEE Transactions on Electron Devices, the Journal of Computational Electronics, and the SISPAD conference proceedings, which is where this field publishes.

Career Pathways

• TCAD engineer — the direct destination, and a genuinely specialised role: TCAD groups exist at every major semiconductor manufacturer and at the EDA vendors, and the supply of engineers who understand both device physics and numerical methods is small.
• Device modelling engineer and compact model engineer — producing the SPICE models that circuit designers depend on, an area where the physics-to-model translation this course teaches is exactly the job.
• Device engineer and process integration engineer — simulation is how process changes are evaluated before they are run, so this skill accelerates a career in either.
• Computational engineer and scientific software developer — the numerical methods here (sparse solvers, Newton iteration, continuation) transfer directly to computational fluid dynamics, structural analysis and other simulation fields.
• Research and graduate study in computational electronics, nanoelectronics or device physics — this course is close to a prerequisite for research in the area, and it is the sort of undergraduate course that makes a graduate application distinctive.
• Florida settings include L3Harris (Palm Bay — device modelling for custom and radiation-hardened parts, including radiation-effects simulation), SkyWater Technology (Kissimmee), onsemi, and the National High Magnetic Field Laboratory in Tallahassee for computational work more broadly. ⚠ Dedicated TCAD roles are concentrated at the large device manufacturers and EDA vendors — Intel, TSMC, Samsung, GlobalFoundries, Synopsys, Silvaco, Applied Materials — and are mostly outside Florida. Plan on a national search for those specifically.
• ⚠ A note on positioning. Very few undergraduates take a device simulation course, and fewer still write their own solver. A student who can present a working drift-diffusion solver has something genuinely distinctive for both graduate admission and industry interviews — keep the code and be able to explain the Scharfetter–Gummel discretisation and your convergence strategy.

Special Information

⚠⚠ The description names numerical methods the prerequisite does not

The prerequisite is EEE 3300 Electronics I — a devices and circuits course. But the statewide description promises “various numerical techniques,” and the course delivers on that: finite differences, iterative solution of coupled non-linear systems, sparse linear algebra, and convergence analysis.

Nothing in the prerequisite chain guarantees a student has met any of this. That gap is where students fail, and it is worth naming explicitly:

• Programming fluency is assumed, not taught. You will be expected to write, structure and debug several hundred lines of MATLAB, Python or C implementing a numerical solver. Students whose programming experience is limited to short scripts find this the real barrier — not the device physics.
• Numerical methods are assumed. Discretisation, iteration, convergence and conditioning may all be new. A numerical methods course beforehand makes this course markedly easier; without one, expect to learn that material concurrently.
• Differential equations are used, not merely referenced. Boundary-value problems and their discretisation are the daily work.

Preparation that pays: if you have not taken numerical methods, spend time before the term on finite-difference approximation of a second derivative, on Newton’s method for a system of equations, and on how a sparse matrix is stored and solved. An afternoon on each will save weeks.

⚠ Course-code variation across Florida

Device simulation is offered narrowly in Florida at undergraduate level:

• EEE 4450 — Florida A&M University.
• EEE 5452 Analysis of Nanoscale Semiconductor Devices — the graduate counterpart at FAMU and FSU.
• Comparable material appears elsewhere only at graduate level or inside a broader computational course; the University of Florida’s nanodevice and memory-technology electives touch on modelling without being simulation courses.

SCNS equivalency does not cross course numbers. Because the subject is offered narrowly, expect this course to articulate as a technical elective rather than as a named requirement, which is normally what a student wants from an elective. Carry the syllabus, and keep the solver code — it is more persuasive evidence than a course description.

Position in the curriculum

EEE 4450 is a senior-level elective in the devices track, following Electronics I. It complements the analytical device courses — EEE 4351 Solid-State Electronic Devices — by supplying what happens when the analytical assumptions fail, and it complements EEE 4330 Microelectronics Engineering by explaining how a process change is evaluated before it is run. A student taking the analytical devices course first will find this one considerably easier, and that ordering is worth arranging where the schedule permits.

Difficulty and time commitment

This is a demanding course, and the difficulty is concentrated in an unusual place: debugging. A device simulator that does not converge gives you very little information about why — the residual simply fails to decrease — and the cause may be a scaling error, a boundary condition, a mesh that is too coarse in the depletion region, an initial guess that is too far from the solution, or a sign error in the Jacobian. Diagnosing this systematically is the skill the course teaches, and it takes time.

Plan on ten to twelve hours a week, weighted heavily toward implementation, and start assignments immediately. A solver that does not converge cannot be fixed the night before it is due, because the diagnosis is inherently iterative.

Two pieces of practical advice that materially reduce the pain: build the solver incrementally — get the equilibrium Poisson solution working before adding continuity equations, and get one dimension working before considering two — and check against analytical limits at every stage, because a solver that converges to a wrong answer is harder to detect than one that fails outright.

Articulation and transfer

SCNS records EEE 4450 as guaranteed to transfer to an institution offering the same course. One Florida institution carries the number, so read that narrowly. The course is upper-division, carries 3 credits, and has no general-education or Gordon Rule designation.

FE exam relevance

The NCEES Fundamentals of Engineering (Electrical and Computer) exam covers semiconductor materials and devices but does not test numerical simulation. The device physics reinforced here supports the Electronics topic area indirectly. This is an elective taken for its own value and for graduate preparation rather than for FE coverage.

AI Integration

Device simulation is an area where machine learning is being adopted seriously as an engineering method, which makes this section a description of current research practice as much as a caution.

Where AI is genuinely used in the discipline. Three developments are real and worth knowing about. Surrogate modelling trains a neural network on TCAD results so that design-space exploration can be done in seconds rather than hours — now standard practice in device optimisation. Physics-informed neural networks encode the semiconductor equations directly in the loss function, an active research direction for solving these systems. And machine learning is widely used for compact model parameter extraction, replacing laborious manual curve fitting. A student entering device modelling will meet all three.

⚠ One caution that belongs with them: a surrogate model is only valid within the region of parameter space it was trained on, and it will extrapolate confidently and wrongly outside it. The physics-based solver remains the ground truth, which is precisely why this course teaches you to write one.

Where a general-purpose assistant helps in coursework. Explaining why the Scharfetter–Gummel scheme is needed, in different words from the textbook; generating boilerplate for sparse matrix assembly; explaining a linear algebra error message; suggesting a systematic order in which to test a non-converging solver; and helping structure and document code, which matters more here than in most courses because the programs are long enough to become unmanageable.

⚠ Where it fails, and why the failure is this course’s own subject. The characteristic error of an AI tool asked to produce device simulation code is to generate a discretisation that is mathematically reasonable and physically unusable — most commonly, central-differencing the current continuity equation instead of using Scharfetter–Gummel exponential fitting. That specific failure is the central numerical lesson of the course. The naive discretisation looks correct, compiles, runs, and produces oscillatory unphysical carrier densities as soon as the bias exceeds a few thermal voltages. It does not error; it just quietly stops meaning anything.

Two further failures recur. Models omit or mis-handle scaling, producing systems so ill-conditioned that they will not converge — and then attribute the non-convergence to something else. And they supply physical model parameters (mobility coefficients, recombination lifetimes) without stating the material, temperature or doping range they apply to, which is the same conditions-free failure that afflicts every device-physics question.

There is also a subtler trap specific to computational work: a generated solver that converges is not thereby correct. Convergence means the iteration found a solution to the discretised system you wrote down; whether that system represents the device is a separate question. Validating against analytical limits — the depletion approximation, the ideal diode equation, charge neutrality — is the only check that distinguishes the two, and it is a step a generated solution will not include.

The engineer’s responsibility. A simulation result is a claim about how a physical device will behave, and engineers make expensive decisions on such claims. The discipline is validation: against analytical limits, against measurement, and against the physics itself. The habit worth forming here is to ask of every simulated result: what does it reduce to in a case I can solve by hand, and does it? A plot that has never been checked against a known limit is not evidence.

Academic integrity. FAMU and FSU both maintain academic honour policies covering AI-generated work. In a course whose deliverable is code you wrote, submitting generated code as your own is straightforwardly a violation, and it is also self-defeating: the assessed understanding is the discretisation and the convergence strategy, which is exactly what you skip by not writing it. Generated simulation results are data fabrication. Ask your instructor what assistance is permitted, and disclose its use where the syllabus requires it.