Electrical properties of matter
Pure silicon and doped silicon are the same substance, and their resistivity differs by a factor of almost a million. The only thing that changed was one phosphorus atom for every ten million atoms.
01Twenty-four orders of magnitude
No other physical property spans such a large range. Between silver and quartz there is a factor of \( 10^{24} \), and in the middle is silicon, which can be placed almost wherever you want.
| Material | \( \rho \) in Ω·m | Class |
|---|---|---|
| Silver | \( 1.6\times10^{-8} \) | The best conductor there is |
| Copper | \( 1.7\times10^{-8} \) | Almost the same and much cheaper |
| Nichrome | \( 1.1\times10^{-6} \) | Deliberately poor conductor |
| Doped silicon \( 10^{16} \) | \( 5\times10^{-3} \) | Working semiconductor |
| Intrinsic silicon | \( 2.3\times10^{3} \) | Pure semiconductor |
| Glass | \( 10^{12} \) | Insulator |
| Fused quartz | \( 10^{16} \) | The best common insulator |
The fourth and fifth rows are the same substance. Between them there is a factor of almost a million, and the only thing that changed is one phosphorus atom for every five million silicon atoms. That is the property that makes electronics possible, and no other family of materials has it.
02What temperature does to it
Here metals and semiconductors behave in opposite ways, and the reason lies in what limits conduction in each case.
| Carriers | On heating | Result | |
|---|---|---|---|
| Metal | Fixed: one or two per atom | The lattice vibrates more and scatters them | \( \rho \) rises |
| Semiconductor | Few, and they have to be created | Thermal agitation promotes many more | \( \rho \) drops, exponentially |
Metals, alloys and semiconductors on the same logarithmic scale. Choose the material and move the temperature; below, the resistance of a specific length of wire is calculated.
Constantan —copper with nickel— has an almost zero temperature coefficient, about twenty parts per million per degree, hence its name. That is why it is used in standard resistors, in measurement shunts and in strain gauges: you need the resistance to change with what you want to measure and with nothing else. Nichrome shares that virtue and also withstands red heat, which is why it is used in ovens and toasters.
03On the insulator side
An insulator has no free charges, but its molecules do respond to the field: they polarize. There are three mechanisms and they act on different time scales, which explains why permittivity depends on frequency.
| Polarization | What moves | Up to what frequency it responds |
|---|---|---|
| Electronic | The electron cloud inside the atom | Optical: \( 10^{15} \) Hz |
| Ionic | The ions within the lattice | Infrared: \( 10^{13} \) Hz |
| Orientation | Polar molecules that rotate | Microwave: \( 10^{10} \) Hz |
The third row is the microwave oven: at 2.45 GHz water molecules can still rotate following the field, and that forced rotation against the friction of their neighbors is what heats. And it is also the reason the permittivity of water falls from 80 at DC to 1.8 at optical frequencies: above a certain frequency the molecules can no longer follow the field.
That lag between the field and the polarization is what produces dielectric losses, which are reported as \( \tan\delta \). In an FR-4 laminate they are about 0.02, and that is why beyond a few gigahertz it is no longer usable: the trace turns into a heater. Radio-frequency boards use materials with a \( \tan\delta \) ten times lower.
04The other extreme
Below a critical temperature, some materials lose all resistance. It is not a very small resistance: it is exactly zero, and a current set flowing in a superconducting ring keeps going for years without decaying.
| Material | \( T_c \) | Cooled with |
|---|---|---|
| Aluminum | 1.2 K | Liquid helium |
| Niobium-titanium | 9.3 K | Liquid helium; it is the one in MRI magnets |
| Ceramic YBCO | 93 K | Liquid nitrogen, which is cheap |
The third row changed everything in 1987: going from needing helium to being able to use liquid nitrogen, which costs less than milk, moved superconductors from the laboratory to application. Today there are superconducting cables in some urban power grids, and the Josephson junction —two superconductors separated by a very thin barrier— is the basis of the primary voltage standard and of a good part of quantum computing.
05In the lab
Using a four-wire measurement, measure the resistance of a copper coil immersed in a bath, between 20 and 90 °C. Plot \( R \) against \( T \), obtain \( \alpha \) from the slope and compare it with the tabulated value.
Repeat the measurement with an NTC thermistor in the same bath. Plot both on a logarithmic scale and discuss why one rises and the other falls.
Using a four-wire measurement, measure the resistance of a copper trace of known geometry on a board with 35 µm copper. Calculate the resistivity and compare it with that of pure copper. Discuss the concept of resistance per square.
06Common mistakes
- Confusing resistance with resistivity. One depends on the geometry; the other belongs to the material.
- Believing every material increases its resistance when heated. Semiconductors and carbon do the opposite.
- Using the linear formula for \( \alpha \) far from 20 °C. It is an approximation valid over a moderate range.
- Assuming permittivity is a constant. It depends on frequency, and strongly.
- Measuring low resistances with two probes. The leads and the contacts weigh more than what you want to measure: use four wires.
- Thinking a superconductor has a very low resistance. It has zero, which is something else.
07Self-assessment
Why does the resistivity of a metal rise with temperature?
Because the number of carriers is fixed and the lattice, vibrating more, scatters them more often.
And why does that of a semiconductor fall?
Because thermal agitation creates new carriers, and that effect dominates over scattering.
What is constantan for?
For standard resistors, shunts and gauges: its temperature coefficient is almost zero.
Why does the microwave oven work at 2.45 GHz?
Because at that frequency water molecules still rotate following the field, and that forced rotation heats.
What does \( \tan\delta \) mean?
The losses of the dielectric, due to the lag between the field and the polarization. In FR-4 it is about 0.02.
Why was YBCO so important?
Because its critical temperature of 93 K allows cooling it with liquid nitrogen, which is cheap, instead of helium.
08Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). Current, resistivity, the conduction model and superconductivity.
- William D. Callister and David G. Rethwisch. Materials Science and Engineering: An Introduction. 9th ed., Reverté, 2019 (Spanish edition, “Introducción a la ciencia e ingeniería de los materiales”). Electrical properties of metals, semiconductors and dielectrics, with the materials-science treatment.
- Charles Kittel. Introduction to Solid State Physics. 8th ed. (Spanish edition, “Introducción a la física del estado sólido”, Reverté). Where metallic conduction, phonon scattering and superconductivity come from.