Notation and amount of substance
Counting atoms seems impossible, yet it is done every day: a doping of 10¹⁶ atoms per cubic centimeter is two parts per million, and the resistivity of a wafer depends on that number.
01The mole: a very large dozen
Atoms are so small that counting them one by one is useless. Chemistry uses a package of fixed size, just as the dozen is used for eggs, only vastly larger:
The beauty of the number is that it was chosen so that the mass of one mole, in grams, coincides with the atomic mass in the periodic table. One mole of silicon weighs 28.09 g; one mole of copper, 63.55 g. That turns a balance into an atom counter.
02Saying how much of something there is
Concentration units are chosen according to the order of magnitude of the problem. In solution chemistry, molarity is used; in materials, parts per million; in semiconductors, atoms per cubic centimeter.
| Unit | Definition | Used in |
|---|---|---|
| Molarity, M | moles of solute / liter of solution | Laboratory solutions |
| Molality, m | moles of solute / kg of solvent | Colligative properties |
| % by mass | g of solute / 100 g of solution | Alloys, commercial reagents |
| ppm, ppb | Parts per million or per billion | Purity of materials |
| Atoms/cm³ | How many there are in one cubic centimeter | Doping of semiconductors |
The last two are the ones in this field, and it pays to know how to convert from one to the other. All that is needed is a single piece of data: how many silicon atoms there are per cubic centimeter, which comes from its density and its molar mass.
03From the mole to the resistivity of a wafer
With the atomic density of silicon, a doping in atoms/cm³ translates immediately into parts per million. And since each dopant atom contributes one free carrier, that concentration determines the resistivity, which is what is finally measured with an ohmmeter.
Move the dopant concentration and watch the three ways of saying the same thing, plus what the instrument reads. The scale is logarithmic because useful doping spans five orders of magnitude.
04Formulas and nomenclature
The empirical formula gives the simplest ratio between the atoms; the molecular formula, the actual number per molecule. Hydrogen peroxide is HO as an empirical formula and H₂O₂ as a molecular formula.
The compounds that show up all the time in electronics are worth having identified:
| Formula | Name | Where it appears |
|---|---|---|
| SiO₂ | Silicon dioxide | The gate oxide of a MOSFET; glass |
| Si₃N₄ | Silicon nitride | Passivation layer of integrated circuits |
| Al₂O₃ | Aluminum oxide, alumina | Ceramic substrate, power insulators |
| GaAs | Gallium arsenide | Microwave devices, infrared LEDs |
| Ta₂O₅ | Tantalum pentoxide | The dielectric of a tantalum capacitor |
| Fe₂O₃, MnZn | Iron oxides, ferrites | Cores of inductors and transformers |
| C₃H₈O | Isopropyl alcohol | Cleaning circuit boards |
To oxidize silicon and form the gate of a MOSFET, the reaction is \( \mathrm{Si} + \mathrm{O_2} \to \mathrm{SiO_2} \). For each mole of silicon consumed, one mole of oxide is formed, but the oxide takes up more volume: the thickness ratio is 0.44 to 1. That is, to grow 100 nm of oxide, 44 nm of silicon is consumed, and the surface ends up lower than where it started. That calculation, which is pure stoichiometry, defines the geometry of a transistor.
05In the lab
Weigh a piece of copper and calculate how many atoms it has. Then estimate how many conduction electrons it contributes —one per atom— and compare with the carrier density listed in the tables, \( 8.5\times10^{22}\ \mathrm{cm^{-3}} \).
For an n-type silicon wafer of \( 10^{15}\ \mathrm{cm^{-3}} \), calculate the expected resistivity and the resistance of a piece 1 cm long with a cross section of 1 mm². Measure it with a four-point ohmmeter and compare.
Calculate how many grams of silicon are consumed when a 500 nm layer of SiO₂ grows on a wafer 200 mm in diameter, using the densities and the molar masses. Verify the thickness ratio of 0.44.
06Common mistakes
- Confusing molar mass with atomic mass. They are the same number with different units: u per atom, g/mol per mole.
- Mixing up molarity and molality. One divides by the volume of solution and the other by the mass of solvent.
- Giving ppm without saying of what. It is not the same by mass as by number of atoms.
- Using liters where the problem is in cm³. In semiconductors everything is counted per cubic centimeter.
- Forgetting that the empirical formula does not identify the compound. Several different substances share the same one.
07Self-assessment
How many atoms are there in 28.09 g of silicon?
One mole: \( 6.022\times10^{23} \). The molar mass is chosen precisely for that.
How many silicon atoms are there per cm³?
About \( 5.0\times10^{22} \), which comes from the density times Avogadro's number divided by the molar mass.
How many ppm does a doping of \( 5\times10^{16}\ \mathrm{cm^{-3}} \) correspond to?
To one: \( 5\times10^{16}/5\times10^{22} = 10^{-6} \).
What happens to the resistivity if the doping is doubled?
It falls to about half: there are twice as many carriers. The proportion is not exact because the mobility also drops as the doping increases.
How much silicon is consumed when 100 nm of oxide grows?
44 nm. The ratio is 0.44 and it comes from the densities and the molar masses.
Empirical or molecular formula of hydrogen peroxide?
Empirical HO, molecular H₂O₂. The empirical formula only gives the ratio.
08Further reading
- Raymond Chang and Kenneth A. Goldsby. Chemistry. 12th ed. (Spanish edition, “Química”), McGraw-Hill, 2017. Chapters 2 and 3: the mole, molar mass, formulas and concentration calculations.
- Ben G. Streetman and Sanjay Banerjee. Solid State Electronic Devices. 7th ed., Pearson, 2015. Doping, carrier density and resistivity, with the curves used by this lab.
- S. M. Sze and Kwok K. Ng. Physics of Semiconductor Devices. 3rd ed., Wiley, 2007. The tables of mobility versus doping and the numerical data for silicon, an essential reference.