Semiconductors

A semiconductor is not defined by a tiny resistor symbol. It has a mostly filled valence band, a mostly empty conduction band, a relatively small band gap, and carrier populations that depend on material, temperature, doping, and electrical boundary conditions. The same silicon sample can behave differently when those inputs change.

Think like a programmer

Model a device as state plus boundary conditions. Keep material constants, temperature, donor and acceptor concentrations, geometry, contact type, and applied bias explicit. A current value with no sign convention or contact model is not a reproducible result.

Model checklist

Inputs: Material band gap, temperature, donor and acceptor concentrations, mobilities, geometry, contact conditions, and bias.
State: Electron and hole carrier densities, electric potential, and any depletion-region approximation.
Rule: Relate carrier densities and fields to drift and diffusion current under stated assumptions.
Output: Carrier type, qualitative band diagram, current direction, or an idealized current–voltage curve.
Check: At zero applied bias, an equilibrium model should report zero net terminal current.

Carriers are a population model

Electrons promoted to the conduction band and holes left in the valence band both contribute to transport. A simple one-dimensional drift term for conventional current density is

\[J_{\mathrm{drift}}=q\left(n\mu_n+p\mu_p\right)E\]

q is the magnitude of elementary charge, n and p are electron and hole number densities, μn and μp are mobilities, and E is electric field. Units are a useful check: number density times charge times mobility times field gives current per area. The equation is a reduced transport rule, not a complete device simulator.

Doping changes which carrier type dominates. Donor dopants make an n-type region with more mobile electrons; acceptor dopants make a p-type region with more holes. Joining p-type and n-type regions produces a p–n junction. Diffusion and the resulting charge separation create a depletion region, so a simulator must distinguish equilibrium from externally biased conditions.

if (biasVolts === 0) {
  expect(netTerminalCurrentA).toBeCloseTo(0, 12);
}

That test does not say there is no microscopic motion. It says opposite contributions balance in the ideal equilibrium model.

Temperature is not decoration

Carrier populations depend strongly on temperature. For an intrinsic teaching model, the relevant energy scale includes the band gap Eg, so report temperature whenever comparing materials or carrier counts. Do not hard-code a room-temperature value and imply that it is a universal property.

Try this experiment

Prediction: At zero bias, swapping the labels p and n changes the diagram's orientation but not the requirement of zero net terminal current in an ideal equilibrium model.

Write down the sign convention for voltage and conventional current. Predict the terminal current before and after swapping the region order at zero bias, then identify which condition would need to change to produce a diode-like current.

Where this model breaks

This lesson omits contact resistance, recombination details, high-field transport, quantum tunnelling, heating, device geometry, and fabrication variation. The drift expression is not sufficient for a predictive transistor or diode design.

Summary

Semiconductor code starts with named material, carrier, temperature, and boundary inputs. Use an equilibrium check before adding bias, and keep the model's sign convention and omitted transport mechanisms visible.

Glossary

Band gap: energy separation between valence and conduction bands.
Doping: deliberately adding impurities that change dominant carrier populations.
Depletion region: junction region with mobile carriers reduced by charge separation.

Self-check

Which inputs must be recorded before comparing two semiconductor current results?
What should an ideal equilibrium terminal-current check return?
Why does a p–n junction require more state than a single resistor?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Semiconductors, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Atoms and Solids, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Semiconductors as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Semiconductors into a test

Model semiconductor devices from explicit material, doping, temperature, and boundary conditions.

Name the inputs and units that the atoms and solids model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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