# Are insulators and conductors arbitrary categories?

I have seen charts showing the transition from insulator to semi-conductor is at $10^{-8}~\frac{\text{S}}{\text{cm}}$ and between semi-conductor and conductor is $10^{3}~\frac{\text{S}}{\text{cm}}$. Are these switch over points arbitrary man-made conventions or is there a physical differences that sharply divides insulators, semiconductor and conductors?

I've been telling myself for example that all materials conduct electricity, it's just that some do it radically less well, up to 18 orders of magnitude less well. I'm wondering if I'm correct.

• One key insight is that resistivity is a dimensional quantity: asking whether it's high or low is not particularly well defined if you don't say what sort of voltages you'll be applying and what sort of current leaks you find acceptable. – Emilio Pisanty Apr 25 '16 at 0:58
• One point to note: Within a given temperature range, if resistance decreases with increase in temperature the material is acting as a semiconductor; if it increases it is acting as a conductor. – Pieter Geerkens Apr 25 '16 at 8:39
• It's also telling that there are materials which conduct too well to be considered an insulator, but not well enough to normally be considered a conductor. – WhatEvil Apr 25 '16 at 15:53

As you have expected, there is no sharp divide between the groups. The divide is man made.

Since all conductors have some resistance, (except superconductors - follow this link to find out more) and all insulators will conduct some current if they are forced to, this means there is no absolute dividing line between conductors and insulators. Since some metals are very, very good conductors with only a very small resistance, and some non-metals are very, very good insulators, the terms are convenient when we are dealing with the usual voltages encountered in a laboratory. (source)

In practice the definitions are helpful because the dynamic range between good conductors and good insulators is very large, and there are not very many situations where the middle ranges between conductor, semiconductor, and insulator prove useful.

Needless to say, those who work with high voltages (hundreds of kilovolts and above) will define the boundary between insulator and not-insulator very differently than a hobbyist tinkering with 5-10V. At higher voltages, we find the resistance of insulators starts to have more of an impact on our design, and we become more picky about how high that resistance needs to be before we call it an insulator.

• Thanks, this is what I expected, but great to confirm it. – Philip Apr 25 '16 at 13:45
• Not many uses for the middle ranges of conductivity? Isn't that exactly what a resistor is? I'd say that resistors are pretty darn useful. – Darrel Hoffman Apr 25 '16 at 16:26
• @DarrelHoffman That's true. I had a feeling I'd have a hard time with that wording =) I think I was approaching it from a materials point of view. There's not a lot of reasons to pay attention to the different materials in that region. To the best of my understanding, most non-extreme resistors are all made of the same few materials. The value of, say, moving to a material with double the resistivity appears to be less important than the other material properties, such as its behavior when heated. – Cort Ammon Apr 25 '16 at 20:00
• I once worked with a radio engineer who dealt with very high voltage AM antenna amplifiers. I've never forgotten this little gem he said one day: "At 10,000 volts, everything is a conductor". – Todd Wilcox Apr 25 '16 at 20:12

Another way of distinguishing conductors and non-conductors or insulators is with band gap - for good conductors the fermi level of electrons is inside a band - semiconductors have a small band gap and good insulators have large band gaps...

Electrons in solids lie in energy bands, whereas in atoms and molecules they have generally sharp levels.

If you have a partially filled band then electrons can reorganize themselves within a band and easily allow a current to pass if a voltage is applied.

If bands are completely full no reorganization is possible, but above the highest filled band will be an empty band. - the filled is the valence band the empty one is the conduction band. If the gap between these is small then electrons from the valence can move to the conduction and then electrons can reorganize themselves a bit in both levels to allow current to flow.

If the band gap is large then it is hard to move electrons up and get a current to flow.

• This is fine, but the usual disclaimer for band theory applies- it is only true when interactions are weak, and can break down completely in cases like Mott insulators. – Rococo Apr 26 '16 at 1:01
• With the band gap perspective I'd like to add that there indeed is a clearer distinction between the groups: semiconductors and insulators are in the same category with only the size of the band gap as the difference. But as conductors have overlapping bands and therefore no band gap at all, these may be considered distinctly as their own category – Steeven Apr 26 '16 at 7:00

Metals (often copper) can be used as insulating support structures in superconducting magnets. Compared to ~0 resistivity of the coil, the resistivity of metals makes for very good insulating properties!

Or, going in the other direction, a 50-watt VandeGraaff-type power supply may output 500KV at 100uA. Such a supply has an internal series resistance of 5e9 ohms. So, a million-ohm resistor placed across the generator's output will act as a short circuit, pulling the 500KV down to 0.02 percent; 100V. Or, we could use some long strings of 1-megohm resistors as wires. They'd serve as excellent conductors; having little effect on the 500KV output voltage, even at the maximum supply current. For this VandeGraaff machine, objects having resistance under ~500 meg-ohms are "good conductors."

So, obviously the definition of "conductor" and "insulator" varies depending on the system we're dealing with. As a rule of thumb, the line separating the two is roughly the same as the internal resistance of any "power supplies" present.

• Very interesting real-world examples. – Philip Apr 26 '16 at 17:09

As is often the case, the answer to this is actually slightly context-dependent. For many everyday purposes, the answer of Cort Ammon that it is all a matter of degree is correct.

However, one other context worth mentioning is when condensed matter physicists speak of whether a particular state of matter is a "conducting" (or "metallic") state or an "insulating" state. In this case there is a precise distinction: a conducting state is one that is gapless, meaning that as the size of the material tends towards infinity (the so-called "thermodynamic limit"), the energy difference between the ground and first excited state of the material also tends towards zero*. One can use this to classify various states of matter unambiguously and study transitions between them.

For many typical materials the classification according to this scheme will line up with the "everyday" classification, but in principle there could be exceptions. One could imagine, for example, a material that has a persistent gap but that gap is much smaller than the scale of thermal excitations, so at room temperature it is effectively conducting.

See also this blog post for some nice entry-level details on this classification.

*slightly more precisely, the spectrum near the ground state becomes continuous. The distinction becomes important in Anderson insulators, which are insulators but have a pointlike discrete energy spectrum.