Basic semiconductor theory: carrier transport
A silicon wafer with one phosphorus atom per ten million conducts more than a hundred thousand times better than pure silicon. Every device is born from controlling how many carriers there are and how they move.
01From material to device
The page on semiconductor physics established how many carriers there are: the intrinsic concentration \( n_i \), on the order of \( 10^{10} \) cm⁻³ in silicon at room temperature, the law of mass action \( n\,p = n_i^2 \), and how doping fixes one of the two concentrations. A device, on the other hand, is defined by how those carriers move: what current flows, in which direction, and how long it takes to appear or disappear.
There are only two mechanisms that produce current in a semiconductor, drift and diffusion, and two processes that change the number of carriers, generation and recombination. With those four pieces and one equation that ties them together you can explain the diode, the bipolar transistor and the solar cell.
02Drift and mobility
An electric field accelerates the carriers, but collisions with the crystal lattice and with impurities slow them down again and again. The net result is an average velocity proportional to the field, and the constant of proportionality is the mobility \( \mu \):
In lightly doped silicon at 300 K, \( \mu_n \approx 1400 \) and \( \mu_p \approx 470 \) cm²/V·s: electrons are about three times more mobile than holes, and that asymmetry reappears in every device (a PMOS transistor has to be wider than an NMOS to conduct the same current). With 1 V applied across 10 µm, the field is 1000 V/cm and an electron drifts at about 14 km/s.
Mobility drops as doping increases, because each ionized impurity is one more scattering center. A widely used empirical fit is the Caughey–Thomas one:
\( v = \mu E \) holds for moderate fields. Above about \( 10^4 \) V/cm the velocity stops growing and saturates near \( 10^7 \) cm/s. In a modern MOS transistor, with channels tens of nanometers long, almost the whole channel works in velocity saturation: it is one of the short-channel effects covered in the unit on MOS technology.
You choose the impurity type and its concentration. The plot shows the resistivity of silicon at 300 K for both types (the Irvin curve, which fabs use to measure the doping of wafers) and the point you chose. With the lifetime it also calculates how far a minority carrier travels before it recombines.
03Diffusion and the Einstein relation
With no applied field there can still be current. If the carrier concentration is not uniform, their random thermal motion carries more carriers from the crowded region to the sparse one than the other way around: this is diffusion, the same process that spreads a drop of ink in a glass of water. The current is proportional to the concentration gradient:
Drift and diffusion are not independent: both are limited by the same collisions. Einstein showed that the ratio between the diffusion coefficient and the mobility is the thermal voltage:
The total current of each carrier is the sum of the two contributions. These two expressions are the starting point of all device theory:
The thermal voltage \( V_T \) appears here for the first time and never goes away: it is in the diode equation, in the transconductance of the bipolar transistor and in the subthreshold slope of the MOSFET.
04Generation, recombination and lifetime
At equilibrium, electron-hole pairs are generated by thermal agitation at the same rate at which they recombine, and \( n\,p = n_i^2 \). If something injects extra carriers (light, a forward-biased junction), the excess recombines and the concentration returns to equilibrium. Under low injection —the excess is much smaller than the majority-carrier concentration— the minority carrier sets the rate, and recombination is proportional to the excess:
Silicon has an indirect band gap: an electron cannot fall directly into a hole emitting a photon, and it almost always recombines through defects with levels inside the band gap (the Shockley–Read–Hall mechanism). That is why the lifetime depends more on purity than on the material: it ranges from milliseconds in very pure silicon to nanoseconds when it is deliberately contaminated with gold or platinum to make fast diodes. With \( D_n = 36.6 \) cm²/s and \( \tau_n = 1 \) µs, the diffusion length is about 60 µm.
| Device | What is wanted | Why |
|---|---|---|
| Solar cell | Long \( \tau \), \( L \) of hundreds of µm | Carriers generated by light have to reach the junction before they recombine |
| Bipolar transistor | Base much thinner than \( L \) | Electrons injected by the emitter have to cross the base without being lost |
| Fast switching diode | Very short \( \tau \) | The stored charge has to disappear quickly when switching from forward to reverse |
05The continuity equation
The previous pieces come together in a balance: in a small piece of semiconductor, the number of electrons changes by what comes in minus what goes out, plus what is generated, minus what recombines:
The most useful case is the steady state, with no field, with minority carriers injected at one end. The equation becomes \( D\,\dfrac{d^2 \Delta p}{dx^2} = \dfrac{\Delta p}{\tau} \), and its solution is an excess that decays exponentially with the diffusion length:
That profile is exactly what happens on each side of a forward-biased junction, and from its slope at the edge comes the diode current, as seen in the next unit.
06Where it appears in electronics
| Case | The mechanism |
|---|---|
| Integrated resistors | A diffused strip of known sheet resistance: the resistivity divided by the thickness |
| Measuring the doping of a wafer | The four-point probe measures the resistivity and the Irvin curve gives the concentration |
| Hall-effect sensor | The drift of carriers deflected by a magnetic field; the sensitivity grows with mobility |
| Photoconductor | Light generates pairs and the conductivity rises for as long as their lifetime lasts |
| Fast semiconductors (GaAs, InP) | Electron mobilities several times that of silicon, for radio frequency and microwaves |
07In the lab
Use four collinear, equally spaced probes to measure a silicon wafer and calculate \( \rho = 2\pi s\,V/I \) (valid for thicknesses large compared with the spacing \( s \); otherwise, apply the correction factor). Deduce the doping from the curve in this page's lab.
Bias a photoresistor (LDR) with a source and a series resistor, illuminate it with a pulsed LED and use the oscilloscope to see how long the resistance takes to return to its dark value when the light is turned off.
Measure the resistance of an NTC thermistor between 0 °C and 80 °C and plot the logarithm of \( R \) against \( 1/T \). Estimate the activation energy from the slope and compare it with the band gap of the material.
08Common mistakes
- Thinking that more doping always gives proportionally more conduction. The conductivity grows less than the doping, because the mobility drops.
- Using the same mobility for electrons and holes. In silicon they differ by a factor of three.
- Forgetting diffusion where there is no field. In the base of a bipolar transistor almost all of the current is diffusion current.
- Applying \( v = \mu E \) at high fields. The velocity saturates near \( 10^7 \) cm/s.
- Confusing lifetime with transit time. The first is how long a minority carrier lives; the second, how long it takes to cross a region.
09Self-assessment
What are the two current mechanisms in a semiconductor and what drives them?
Drift, driven by the electric field, and diffusion, driven by the concentration gradient.
Why does mobility decrease with doping?
Because each ionized impurity is an additional scattering center for the carriers.
If \( \mu_p = 470 \) cm²/V·s, what is \( D_p \) at 300 K?
\( D_p = \mu_p V_T = 470 \times 0.02585 \approx 12.2 \) cm²/s.
With \( D = 12 \) cm²/s and \( \tau = 10 \) µs, what is the diffusion length?
\( L = \sqrt{12 \times 10^{-5}} \approx 1.1 \times 10^{-2} \) cm, about 110 µm.
Why is the silicon of a fast diode contaminated with gold?
Gold introduces levels in the middle of the band gap that speed up recombination and shorten the lifetime, so the stored charge disappears sooner.
10Further reading
- Robert F. Pierret. Semiconductor Device Fundamentals. Addison-Wesley, 1996. Drift, diffusion and the continuity equation developed patiently, with worked problems.
- Donald A. Neamen. Semiconductor Physics and Devices. 4th ed., McGraw-Hill, 2012. The most widely used course text; its chapters on transport and excess carriers are the basis of this unit.
- Simon M. Sze and Kwok K. Ng. Physics of Semiconductor Devices. 3rd ed., Wiley, 2007. The reference book, with the measured mobility and resistivity curves.