Semiconductor Physics Essentials: Carriers, Doping and the PN Junction
Every diode, transistor, solar cell and integrated circuit relies on a few ideas from semiconductor physics: how carriers are created, how doping controls their numbers, how they move, and what happens when p-type and n-type regions meet. This tutorial covers those essentials for silicon with numbers you can calculate yourself: carrier concentrations from the mass action law, the built-in potential and depletion width of a PN junction, and the diode current from the Shockley equation. These are the calculations behind the 0.7 V rule of thumb you use in circuit courses.
Intrinsic and Doped Semiconductors
In a pure (intrinsic) semiconductor, thermal energy breaks a few covalent bonds, creating free electrons in the conduction band and an equal number of holes in the valence band. The energy needed is the band gap: about 1.12 eV for silicon, 0.66 eV for germanium and 1.42 eV for gallium arsenide at room temperature. The intrinsic carrier concentration ni of silicon at 300 K is about 1e10 per cubic centimetre; many older textbooks use 1.5e10. That is tiny compared with roughly 5e22 silicon atoms per cubic centimetre, which is why pure silicon is a poor conductor.
Doping changes this dramatically. Adding a group V element such as phosphorus (a donor) gives one extra electron per atom, making n-type material. Adding a group III element such as boron (an acceptor) creates holes, making p-type material. At room temperature nearly all dopant atoms are ionised.
Worked example: carrier concentrations
Mass action law (thermal equilibrium): n . p = ni^2 n-type silicon, ND = 1e16 cm^-3, ni = 1e10 cm^-3 Majority electrons : n ≈ ND = 1e16 cm^-3 Minority holes : p = ni^2 / n = 1e20 / 1e16 = 1e4 cm^-3 Doping raised electrons by a factor of a million and lowered holes by the same factor.
How Carriers Move: Drift and Diffusion
Carriers move by two mechanisms. Drift is motion caused by an electric field, with average velocity equal to mobility times field. Electrons in silicon are roughly two to three times more mobile than holes, which is why NPN transistors and NMOS devices are faster than their PNP and PMOS counterparts. Diffusion is motion from a region of high concentration to low concentration, just as ink spreads in water, and needs no field at all. The diffusion constant and mobility are linked by the Einstein relation D/μ = kT/q, where kT/q is the thermal voltage VT, about 25.9 mV at 300 K.
| Mechanism | Driven by | Current density (electrons) | Dominant where |
|---|---|---|---|
| Drift | Electric field E | q n μn E | Resistors, channels of MOSFETs |
| Diffusion | Concentration gradient dn/dx | q Dn dn/dx | Forward-biased junctions, BJT base |
The PN Junction in Equilibrium
When p-type and n-type regions meet, electrons diffuse from the n side into the p side and holes diffuse the other way. They recombine near the junction, leaving behind fixed ionised dopants: positive donors on the n side and negative acceptors on the p side. This region, empty of mobile carriers, is the depletion region. Its charge creates an electric field that pushes carriers back, and equilibrium is reached when drift balances diffusion. The resulting potential step is the built-in potential.
Built-in potential: Vbi = VT . ln( NA . ND / ni^2 ) Example: NA = 1e17 cm^-3 (p side), ND = 1e16 cm^-3 (n side), VT = 25.85 mV Vbi = 0.02585 x ln(1e33 / 1e20) = 0.02585 x 29.93 ≈ 0.77 V Depletion width: W = sqrt( (2 εs / q) . (Vbi - V) . (1/NA + 1/ND) ) εs (silicon) = 11.7 x 8.854e-14 F/cm, q = 1.602e-19 C At V = 0 : W ≈ 0.33 µm At V = -5 V : W ≈ 0.91 µm (reverse bias widens it) Charge balance NA.xp = ND.xn -> depletion extends 10 times further into the lightly doped n side: xn ≈ 0.30 µm, xp ≈ 0.03 µm
Two practical lessons come from this example. The depletion region lives mostly in the lightly doped side, which is why the light doping sets the breakdown voltage of a power diode. And because W changes with applied voltage, the junction behaves as a voltage-dependent capacitor. Varactor diodes used for tuning in VCOs exploit exactly this effect.
Forward and Reverse Bias
| Condition | Effect on barrier | Depletion width | Current |
|---|---|---|---|
| Forward bias (p positive) | Lowered to Vbi - V | Narrows | Rises exponentially with voltage |
| Zero bias | Vbi | Equilibrium value | Zero net current |
| Reverse bias (n positive) | Raised to Vbi + |V| | Widens | Small saturation current |
| Breakdown | Field exceeds critical value | Wide or very thin | Large; Zener tunnelling or avalanche |
The forward current follows the Shockley diode equation, where Is is the saturation current, which depends on doping, junction area and temperature, and n is the ideality factor, between 1 and 2.
I = Is ( e^(V / (n VT)) - 1 ) Example: Is = 1e-14 A, n = 1, VT = 25.85 mV V = 0.6 V -> I ≈ 0.12 mA V = 0.7 V -> I ≈ 5.7 mA Current rises about 10 times for every 60 mV (VT x ln 10 = 59.5 mV).
This is the origin of the 0.7 V rule: across several decades of useful current, the forward voltage of a small silicon diode stays in a narrow band around 0.6 to 0.7 V. The forward voltage at constant current also falls by roughly 2 mV per degree Celsius, which is how a diode or transistor junction works as a temperature sensor.
Under strong reverse bias the junction breaks down. In heavily doped junctions with thin depletion regions, electrons tunnel directly across, which is Zener breakdown and dominates at low voltages, typically below about 5 V. In lightly doped junctions, carriers gain enough energy to knock out new electron-hole pairs, an avalanche process that dominates at higher voltages. Zener diodes are built to use either mechanism in a controlled way.
Common Mistakes
- Assuming majority and minority concentrations add up to the doping. Use n.p = ni^2 for the minority carriers.
- Using 0.7 V for every diode. Schottky diodes, germanium diodes and LEDs have very different forward voltages.
- Ignoring temperature: ni rises steeply with heat, and a common rule of thumb is that the reverse saturation current of a silicon diode roughly doubles for every 10 degree rise.
- Placing the depletion region equally on both sides. It extends mostly into the lightly doped side.
Summary
Doping sets majority carrier concentration, and the mass action law gives the minority concentration. Carriers move by drift in a field and by diffusion along a gradient. At a PN junction, diffusion creates a depletion region and a built-in potential of about 0.7 to 0.8 V for typical silicon doping. Reverse bias widens the depletion region, forward bias narrows it and lets current rise exponentially according to the Shockley equation. These results underpin every device in electronics, from rectifiers to the MOSFETs in a CMOS inverter.