The Inner Workings of Transistors

Understanding Charge Control and Semiconductor Behavior

Transistors are everywhere, which is exactly why they’re easy to take for granted. They sit quietly inside nearly every device you use, and they do their job so well that most people never need to think about what’s happening inside them.

But inside a transistor, the “simple” idea of switching and amplifying turns into something far more interesting: you’re shaping electric fields, rearranging charge populations, bending energy bands, and pushing carriers through regions where the rules change depending on how you bias the device.

This post is a guided tour of that inner machinery. We’ll build the full mental model in the correct order:

1. what a semiconductor actually is

2. how electrons and holes exist (and why energy bands matter)

3. doping: how we engineer carrier populations

4. the PN junction: the electrostatic barrier that makes everything nonlinear

5. current flow across the junction (Shockley and friends)

6. the BJT: amplification through minority-carrier injection

7. small-signal behavior and high-frequency limits

8. the MOSFET: voltage control through electrostatics and inversion

You’ll see equations, but the goal is never math for its own sake. The goal is understanding: what charge is where, what fields exist, and what changes when you move a voltage by a few hundred millivolts.

Quick symbol cheat sheet

(So you don’t have to scroll upward every time.)

• q: elementary charge (1.602×10⁻¹⁹ C)

• k: Boltzmann constant

• T: temperature (K)

• VT = kT/q: thermal voltage (≈ 25.85 mV at 300 K)

• n, p: electron and hole concentrations (cm⁻³)

• ni: intrinsic carrier concentration (cm⁻³)

• ND, NA: donor and acceptor concentrations (cm⁻³)

• μn, μp: electron and hole mobility (cm²/V·s)

• Dn, Dp: diffusion coefficients

• Eg: bandgap energy (eV)

• Vbi: built-in potential of a PN junction

• W, xn, xp: depletion width and its portions

• IC, IB, IE: collector, base, emitter currents

• β: BJT current gain

• gm: transconductance

• Cox: oxide capacitance per unit area

• Vth: MOSFET threshold voltage

1) The transistor starts with a semiconductor

A semiconductor is not “half a conductor.” It’s a material whose conductivity can be pushed around dramatically by temperature, impurities (doping), and electric fields. That controllability is the whole game.

Silicon dominates because it’s abundant, stable, and forms a fantastic native oxide (SiO₂). That oxide is not just convenient insulation. It’s the reason modern MOS technology exists at all.

Other materials matter too:

• GaAs: high electron mobility, excellent for RF and high-frequency work. More expensive and less friendly to large-scale manufacturing than silicon.

• Ge: historically important and still used in some high-performance devices (often in mixed SiGe structures), but pure Ge has thermal and leakage drawbacks compared to Si.

Key idea: transistor behavior is not “a special property of silicon.” It’s the result of engineering carrier populations and electrostatic barriers inside a crystal.

Takeaways

• Semiconductors are controllable because carrier populations are controllable.

• Silicon wins mainly due to manufacturing, oxide quality, and stability.

• The physics you learn here generalizes beyond silicon.

2) Charge carriers and energy bands

Electrons and holes

In a silicon crystal, electrons normally sit in covalent bonds. If an electron gains enough energy to leave its bond, it enters the conduction band and can move through the crystal. The missing electron state left behind behaves like a positively charged mobile carrier: a hole.

A hole is not a little physical ball of positive charge. It’s a bookkeeping device that turns out to match reality extremely well: if you track the motion of missing electron states, you get the correct current directions, forces, and device behavior.

The bandgap is the “permission barrier”

In solids, atomic energy levels broaden into bands:

• Valence band: usually full of electrons

• Conduction band: usually mostly empty

• Bandgap (Eg): forbidden energies between them

Roughly:

• Conductors: overlapping bands or no meaningful gap

• Semiconductors: small gap (around 1 eV scale)

• Insulators: large gap

Typical room-temperature bandgaps:

• Silicon: ~1.12 eV

• Germanium: ~0.66 eV

• GaAs: ~1.42 eV

Intrinsic carriers rise fast with temperature

For intrinsic silicon, carrier concentration increases roughly like:

• ni ∝ exp(−Eg / 2kT)

That exponential is why temperature matters so much. Small changes in T don’t just “nudge” carrier density. They can multiply it.

A useful more explicit form is:

• ni = √(NC NV) exp(−Eg / 2kT)

where NC and NV are effective density-of-states terms (they carry the T^(3/2) dependence and effective masses).

Mobility, diffusion, and why effective mass matters

Carriers in a crystal behave as if they have an “effective mass” m* because the lattice modifies how they accelerate under force. Effective mass matters because it affects both:

• how easily carriers move (mobility)

• how many states exist near the band edges (density of states)

Mobility:

• μ = qτ / m*

Conductivity:

• σ = q(nμn + pμp)

Transport mechanisms:

• Drift: carriers pushed by electric field

• Diffusion: carriers spread from high concentration to low

And mobility links to diffusion via the Einstein relation:

• D/μ = kT/q

Takeaways

• Holes are missing electron states treated as positive carriers because it works.

• The bandgap controls how many carriers exist at a given temperature.

• Drift and diffusion are the two fundamental transport modes.

3) Doping: engineering carrier populations

Doping is the controlled insertion of impurities that create carriers cheaply (energetically speaking).

N-type: donors add electrons

Group V dopants (P, As, Sb) have five valence electrons. Four bond into the lattice; the fifth is weakly bound and easily ionized at room temperature. Result: lots of free electrons.

P-type: acceptors create holes

Group III dopants (B, Al, Ga) have three valence electrons. They leave one bond incomplete. That missing bond behaves like a hole. Result: lots of holes.

Majority and minority carriers

At equilibrium, the mass-action law holds:

• np = ni²

So if you heavily increase n via donors, p must drop accordingly.

Approximations (room temperature, shallow dopants, complete ionization):

• n-type: n ≈ ND, p ≈ ni² / ND

• p-type: p ≈ NA, n ≈ ni² / NA

Fermi level shift: the clean quantum interpretation

Doping shifts the Fermi level:

• n-type → EF moves toward the conduction band

• p-type → EF moves toward the valence band

That shift changes carrier populations exponentially.

Degenerate doping (very heavy doping)

At very high doping (often >10¹⁹ cm⁻³), EF can move into a band. Then the semiconductor starts acting metallic in some ways. Useful for ohmic contacts and source/drain regions, but it introduces complications (mobility degradation, bandgap narrowing, reduced lifetimes).

Takeaways

• Doping is how we turn “barely conducting” silicon into “device-grade” material.

• Minority carriers don’t disappear; they become rare, but still important.

• Fermi level movement explains doping cleanly and generally.

4) The PN junction: a self-built electric field

When p-type and n-type regions touch, carriers diffuse:

• electrons diffuse from n → p and recombine

• holes diffuse from p → n and recombine

That recombination exposes fixed ion charges:

• donors left behind on the n-side become positive ions

• acceptors left behind on the p-side become negative ions

Those fixed charges create an electric field that pushes back against diffusion. Equilibrium is reached when drift and diffusion balance.

The carrier-free region is the depletion region.

Built-in potential

The built-in potential is:

• Vbi = (kT/q) ln((NA ND) / ni²)

This is the “barrier” the junction constructs by itself.

Charge neutrality condition

Inside the depletion region, total uncovered charge must balance:

• NA xp = ND xn

The depletion width expands more into the more lightly doped side.

Total depletion width:

• W = √[(2εs/q) ((NA + ND)/(NA ND)) Vbi]

Takeaways

• A PN junction is a diffusion event that creates an electric field that stops further diffusion.

• The depletion region is the core nonlinear element behind diodes and transistor junctions.

• Built-in potential depends logarithmically on doping but still ends up around “silicon-scale” voltages.

5) Current across a junction: drift, diffusion, and Shockley

Apply a voltage and you tilt the barrier:

• Forward bias lowers the effective barrier (Vbi − V)

Reverse bias raises it (Vbi + |V|)

Forward bias injects minority carriers across the junction. Those injected carriers diffuse away and recombine. Under common assumptions (uniform doping, low-level injection, negligible recombination in depletion), the current becomes:

• I(V) = IS (exp(V/VT) − 1)

This one equation explains why a diode turns “on” so sharply: the energy barrier is being adjusted in units of kT, and kT at room temperature corresponds to only ~26 mV.

Real diodes deviate due to series resistance, depletion-region recombination, high-level injection, etc. A common correction:

• I(V) = IS (exp(V/(ηVT)) − 1) with 1 ≤ η ≤ 2

Takeaways

• Exponential I–V behavior is barrier control, not magic.

• Reverse current is small because it’s tied to minority carrier generation and diffusion.

• The ideality factor tells you what non-ideal physics is dominating.

6) The BJT: amplification via minority carrier injection

A BJT is two PN junctions arranged to “inject here, collect there.”

For an NPN:

• Emitter (N+): heavily doped

• Base (P): lightly doped and very thin

• Collector (N): moderately doped

In active operation:

• base-emitter junction is forward biased

• base-collector junction is reverse biased

What actually causes gain

Forward bias injects electrons from emitter into base. These electrons are minority carriers in the base. Because the base is thin, most electrons cross it before recombining and are swept into the collector by the reverse-biased collector-base electric field.

A small base current exists because some electrons recombine in the base, and each recombination needs a hole supplied from the base terminal.

The essential relationships:

• IE = IC + IB

• IC ≈ α IE with α close to 1

• β = IC/IB = α/(1 − α)

Two physical “efficiency” ideas make α close to 1:

• Emitter injection efficiency: emitter doping is high, base doping is low, so injection is dominated by electrons from emitter rather than holes from base.

• Base transport factor: base is thin relative to diffusion length, so few injected electrons recombine before reaching the collector junction.

Takeaways

• A BJT amplifies because most injected carriers make it through the base, and only a small fraction need “replacement” via base current.

• The base is intentionally weak and thin; that is not a flaw, it is the point.

• The collector junction is not “doing amplification”; it’s acting like an efficient extractor.

7) Small-signal BJT: linearizing the exponential

Around a bias point, the exponential becomes a linear gain mechanism.

If:

• IC ≈ IC0 exp(VBE/VT)

Then the small-signal transconductance is:

• gm = ∂IC/∂VBE = IC/VT

At room temp, VT ≈ 25.85 mV, so gm is big even at modest currents. That’s why BJTs are so strong in analog design.

Hybrid-π basics:

• rπ = β/gm

• ro ≈ VA/IC (Early effect)

High-frequency limits show up as capacitors and transit times:

• Cπ (base-emitter diffusion capacitance)

• Cμ (base-collector depletion capacitance) and the Miller effect

• fT ≈ 1/(2πτF) where τF is forward transit time

Takeaways

• gm is the heart of BJT gain and it comes directly from the exponential.

• Frequency response is charge storage plus parasitics, not some mysterious “speed limit.”

• Cμ becomes nasty in common-emitter because voltage gain multiplies it.

8) The MOSFET: controlling a channel with an electric field

The MOSFET is fundamentally a capacitor that controls conduction.

For an n-channel enhancement MOSFET:

• p-type substrate

• n+ source and drain

• gate separated from silicon by oxide

As VGS increases, the surface goes through:

1. accumulation

2. depletion

3. inversion

Strong inversion forms when surface potential reaches roughly:

• ψs ≈ 2φF

Threshold voltage (one common long-channel form):

• Vth = VFB + 2φF + (√(2εs q NA 2φF))/Cox

Above threshold, inversion charge density scales approximately as:

• Qinv ≈ −Cox (VGS − Vth)

Then current in the linear region (small VDS):

• ID ≈ μn Cox (W/L) (VGS − Vth) VDS

Saturation (long-channel ideal):

• IDsat ≈ (1/2) μn Cox (W/L) (VGS − Vth)²

Real MOSFET reality check (short version)

Modern MOSFETs deviate from these clean square-law models due to:

• mobility reduction at high vertical fields

• velocity saturation (current grows more linearly than square-law)

• short-channel effects (DIBL, Vth roll-off)

• body effect (Vth depends on VSB)

A common body effect expression:

• Vth = Vth0 + γ(√(2φF + VSB) − √(2φF))

Takeaways

• MOSFETs are voltage-controlled because the gate creates an electrostatic charge sheet.

• Threshold is the point where inversion becomes strong enough to form a conducting channel.

• Modern MOSFET behavior is still charge control, but short-channel physics changes the simple equations.

What “charge control” really means

If you strip away the component packaging, the symbols, the circuit diagrams, and the folklore, transistors boil down to one theme:

You are controlling where charge can exist, how much charge exists, and how easily it can move.

• In a PN junction: the depletion field blocks carriers until you lower the barrier.

• In a BJT: you inject minority carriers and let geometry decide how many survive to be collected.

• In a MOSFET: you use a gate field to summon or banish a conducting channel at the surface.

Once that clicks, transistors stop being black boxes and start being landscapes. And landscapes can be reasoned about.

Appendix: math-heavy add-ons

A1) Drift-diffusion current densities

• Jn = qnμnE + qDn(dn/dx)

• Jp = qpμpE − qDp(dp/dx)Einstein relation: D/μ = kT/q

A2) PN junction depletion charge and Poisson

Space charge density (abrupt junction):

• ρ(x) = −qNA for −xp < x < 0

• ρ(x) = +qND for 0 < x < xn

Poisson:

• d²V/dx² = −ρ/εs

Charge neutrality:

• NA xp = ND xn

Built-in potential:

• Vbi = VT ln((NA ND)/ni²)

Total depletion width:

• W = √[(2εs/q)((NA + ND)/(NA ND))Vbi]

A3) BJT current gain relationships

• IE = IC + IB

• IC ≈ α IE

• β = IC/IB = α/(1 − α)

Small-signal:

• gm = IC/VT

• rπ = β/gm

• ro ≈ VA/IC

A4) MOSFET long-channel region equations

• Linear: ID ≈ μn Cox (W/L) [(VGS − Vth)VDS − VDS²/2]

• Saturation: IDsat ≈ (1/2) μn Cox (W/L) (VGS − Vth)²

• Channel-length modulation: ID ≈ IDsat (1 + λVDS)