Linearity of Conduction in Metals.
This is 'work in progress' and may be updated as further sources of information are consulted.
Metal conductors are known to be highly linear, but although attempts to measure harmonic distortion in cables at audio frequencies invariably fail, successful measurements have been made of distortion in NiCr metal films with a thickness less than 0.001mm, [ Ref.1 ], as used in metal film resistors. A resistance of 250 kohm with a 1 mA 10 kHz signal applied had third harmonic voltage 0.1 mV. The fundamental was at 250 volts, so the distortion was 0.00004 % (-128dB). Other harmonics were much lower. Further measurement showed that the distortion voltage was proportional to film length, to current density to the power 2.2 to 2.8, and inversely proportional to film thickness to the power 3. This dependence is unlikely to be maintained up to 1mm thickness, but in the absence of any better figures we can estimate that for example a current of 1 amp in a 1mm diameter wire will have distortion about 0.0000000001% (-200dB). This result is probably highly inaccurate, but it does suggest the sort of levels that could be involved, in which case it would be pointless trying to measure them. Measurements at much higher frequencies are referred to later, but again trying to estimate figures at audio frequencies based on these results is difficult, and the most we can conclude is that they are extremely low. I have personally measured distortion components below -140dB in amplifier circuits, while using ordinary copper cables.
It is interesting to ask why metal wires are so linear. There are certainly all sorts of obstacles to the passage of current, such as impurities, dislocations of the crystal structure and so on, and it is sometimes suggested that these could introduce non-linearity, but I will try to present an explanation of why these effects are generally insignificant.
First of all a few figures:
In copper there are about 8.47 x 1028 conduction electrons per cubic metre.
Each electron has charge 1.6 x 10-19 coulombs.
From this we can calculate that in a 1 metre length of 1mm diameter wire there are 6.68 x 1022 conduction electrons with a total charge about 10,000 coulombs. A current of 1 amp is a flow of 1 coulomb per second, so at this rate it would take 10,000 seconds for the total conduction charge in 1 metre to flow past a given point, and so the average drift velocity of the electrons is just 0.1 mm per second. As mentioned in the previous transmission line article the electromagnetic energy between a pair of wires travels close to the speed of light, c. (dielectric effects are being ignored in both articles, these will be covered later, if at all, there being little of interest involved at audio frequencies). The excess electron density responsible for the field also travels close to c, but average drift velocity of the electrons is very small. The total charge of the conduction electrons just mentioned is of course almost exactly balanced by positive charges in the atoms, and even with hundreds of volts between a pair of wires the excess charge in the negative conductor will only be around 1 part in 1012 of the total conduction charge.
It is wrong to conclude from the small drift velocity that individual electrons have low velocity. The classical theory of thermodynamics proposed that at temperature T (degrees Kelvin) an electron would have thermal energy 3kT/2 where k is Boltzmann's constant. At 20 degrees C this corresponds to a thermal velocity of 105 m/sec. In the absence of a current these thermal velocities are in random directions, and the average of all electron velocities is close to zero relative to the conductor. (At any instant the average will not be exactly zero, and this contributes to the noise voltage and current of the conductor.) The classical theory was known to be incorrect, and eventually the correct quantum theory showed that the thermal velocity of conduction electrons has a much lower temperature dependence, and the correct value, known as the Fermi Velocity is much higher than the classical value at 20 deg. C in copper, and is 1.57 x 106 m/sec. ( i.e. about the distance from London to Rome in 1 sec. ), so the thermal velocity is typically over 1010 times larger than the average drift velocity of a 1 amp current.
So why do the electrons not travel this sort of distance along a wire in a second? It is because they undergo collisions and are therefore constantly being deflected from a straight line path. A simplified QM (quantum mechanical) theory of conduction in a crystal lattice is known as the Kronig-Penney model, and the periodic potential experienced by an electron travelling through a crystal is approximated by a regular series of rectangular potential wells. Solving the Schrodinger equation for the conduction electrons gives the result that there is a continuous band of possible conduction electron energy, and the electrons are free to move without obstruction through the crystal. It is believed that a more exact analysis will give similar results, but there is one important detail left out, and that is the thermal vibration of the atoms. The classical description is that the crystal structure at any given time is not exactly regular because of this constant thermal motion, and the electrons can be deflected as a result of interaction with the irregular potential. In the more correct QM version the lattice vibrations are quantised and behave like particles (called phonons), and then electron-phonon interactions cause the deflections.
This interaction is also responsible for the positive temperature coefficient of resistance of metals. As the temperature rises so does the vibration of the atoms, and so the probability of an electron being deflected increases. The 'mean free path' between collisions falls, and so the acceleration from the electric field has less effect before the electron is deflected in a more or less random direction. The current produced by a given field reduces, and this corresponds to a higher resistance.
But what about all the other defects such as impurities and dislocations mentioned earlier? The effect of these can be estimated because they have a far lower temperature dependence than the thermal scattering effect. As temperature is reduced towards zero (but not low enough for superconduction to occur ) the resistance falls along with the thermal scattering, but not towards zero, only to a residual value determined by the impurities etc. For one sample of 'oxygen free' copper the residual resistance at 4 deg.K was found to be 70 times smaller than that at 293 deg.K (20 deg.C ) and so in this case the impurities etc. make a contribution to electron scattering about 70 times lower than the thermal scattering effect. Even the thermal scattering is a fairly small effect, the mean free path of conduction electrons in copper at 20 deg.C is about 5 x 10-8 m. This is about 200 times the distance between atoms, so on average an electron passes 200 atoms without interaction before being deflected from its straight line path.
So what can we conclude from all this about the linearity of metals? The important point is that the motion of electrons is almost entirely a thermal effect. The conduction effect is a minute modulation of the motion. The time between deflections is changed by typically less than one part in 1010. Even if the motion was in some way non-linear when the drift velocity was significant compared to the thermal velocity, e.g. 1% of its value, this is unlikely to be more complicated than a simple power series, and at worst the distortion would fall in proportion to the drift velocity, and would be 108 times lower at the 1 amp level previously considered, and therefore unmeasurably low. In fact what evidence there is suggests that the harmonic distortion is almost entirely third order, and it would then fall in proportion to drift velocity squared and be 1016 times lower.
Attempting to eliminate impurities in the belief that these could cause non-linearity may be pointless, the scattering from these impurities being generally just a small addition, e.g. 1 part in 70 to scattering from thermal vibrations, which have similar effect. At best the resistance will be reduced by 1 part in 70 by the total elimination of the remaining impurities and discontinuities from standard oxygen-free copper. A temperature rise of just 2 or 3 degrees would add more resistance than the elimination of all impurities would remove.
Reducing crystal boundary dislocations, often called 'grain boundaries', and other dislocations beyond a certain level is also probably pointless because even if a perfect single crystal copper wire could be produced, the effect of just bending the wire beyond its elastic limit (the point up to which it will spring back to its original shape) will be to create massive dislocations of the crystal structure, so such a perfect structure would be difficult to maintain in practice. The dislocations produced by bending are responsible for an effect called 'work hardening'. Bending a piece of soft copper makes it harder because the resulting dislocations tend to lock the crystal structure and make further bending more difficult. A residual resistance ratio of 40,000 times lower has been claimed for one single crystal measured [Ref4], but an almost equally impressive bulk figure of 36,000 suggests that crystal boundaries were of little effect. These boundaries can be just a discontinuity in the regular cubic structure just one or two atoms in thickness. More complex grain boundary structures are possible, and the properties and effects can then be more varied. A classic paper on crystal structure [Ref.3] used bubble rafts to illustrate the nature of dislocations, impurities and grain boundaries.
Copper oxide is a semiconductor, and it is possible to make a rectifying junction between copper and copper oxide, but we cannot conclude from this that oxygen impurities in copper wire will create rectifying junctions. Individual atoms of oxygen, in common with single atoms of copper, are neither a conductor nor a semiconductor. The electrons are confined to discrete energy levels, and application of a small electric field will not generate a current. It is only when many copper atoms are present in a regular lattice that the interaction between them causes the electron energy levels to form a continuous band and unobstructed flow becomes possible in an applied field. Adding an impurity atom in place of one of the copper atoms in the lattice will in general add some degree of irregularity because the potential is different in magnitude or shape. It is the regularity of the lattice which permits conduction, and the irregularity at a given location which interacts with conduction electrons to cause scattering. There are differences between impurity scattering, thermal scattering and scattering from dislocations in the lattice, for example the temperature dependence, but the effects are very similar. At normal room temperatures and common levels of impurity it is the thermal effects which dominate, and other effects are relatively insignificant. If we actually wanted to produce rectification effects using a metal oxide we need to produce a significant region of fairly pure metal oxide so that it is the properties of the oxide which determined the electron energy bands such that the Fermi energy level was somewhere between the bands and the conduction band contained only a low density of electrons. For copper with a random distribution of oxygen at low levels it will be the copper which determines the energy bands. Only for an extended region of fairly pure copper oxide could semiconducting effects occur, and then only if it extended across the full area of the conductor so that no other lower resistance current path was available. Copper oxide can occur as a precipitate, e.g. in small spheres in the grain boundaries when cast as anodes for electrolytic purification, and to a lesser extent in later stages of production, but it is found that precipitates have little effect on conductivity compared to impurities in solution in the copper. The current will just flow round the high impedance region, so it will have little effect unless it is large compared to the cross-section of the wire.
It is commonly assumed that oxygen impurity is a bad thing, but the oxygen in 'electrolytic tough pitch' copper (ETP) is added intentionally to act as a 'scavenger' to extract hydrogen and sulphur impurities, which combine with oxygen to form gas bubbles of H2O and SO2, which are later eliminated during hot rolling. [Ref.5]. Other impurities can also reduce conductivity of copper when in solution, but the effects are much smaller when the impurities are oxidised and form precipitates, so oxygen also reduces these effects. The electrical conductivity of ETP copper is a maximum for oxygen content of 200 ppm. Reducing oxygen below this level actually reduces conductivity.
There may be some point in avoiding large levels of magnetic impurities, because of their non-linear response to magnetic fields. Just avoiding high magnetic fields may be a good idea to avoid other effects such as magnetoresistance, but I know of no evidence that this is at a measurable level in ordinary copper wire at room temperature.
There appears to be some concern about 'skin depth effect' in audio circles. The greatest effect this has is to slightly increase the resistance of the cable at high frequency which will produce small amplitude and phase changes. All real speakers have far greater amplitude and phase errors than any reasonably good cable of a few metres in length.
As for non-linearity, there is good evidence to show that there is none associated with the skin depth in situations where cable linearity is genuinely important. The transmitters used for cellular phone base stations operate in the GHz range where skin depths really are small, typically 0.002mm, and a wide range of frequencies are being transmitted at a given time. Intermodulation products could be produced at the same frequency as the much smaller signals being recieved from the phones, and even distortion at -150dB can cause problems, so reducing non-linearity in the transmitter cables and connectors is of great importance, and considerable effort has been made to measure and eliminate these effects. Measurements are commonly made more than -160dB relative to the signal level, [Ref.2] and a rather surprising finding is that there is little frequency dependence of intermodulation distortion in cables. The level at 1800MHz is little different to that at 900MHz even though the skin depth is about 30% less at the higher frequency. So, even in a situation where skin depth is very small and measurements beyond -160dB are possible, reducing the skin depth adds no significant non-linear distortion. It is common to use copper conductors and connectors in these applications because of the good linearity. It is frequently mentioned that magnetic impurities such as iron and nickel should be avoided to keep distortion low. Silver plating is often added, and both copper and silver are regarded as excellent low distortion conductors. Because of the small skin depth and high transmitter powers, current densities are high, and yet distortion is still at a very low level.
The observation that reducing skin depth appears to have no effect on linearity seems to contradict the results from Ref 1 concerning metal films where reducing film thickness gave a large increase in distortion. One possibility is that a real physical thickness acts in a different way to the skin depth, which does not give the same sharp cut-off. Also, some articles on distortion at high frequencies suggest that one of the magnetic impurities to avoid is nickel, yet the metal films measured in ref.1 are NiCr, so maybe the effects measured are magnetic in origin and different to those in the high frequency intermodulation tests using copper and silver, and different dependence on conductor thickness is not then so unlikely, and the relatively high distortion in the NiCr film compared to the high frequency figures for copper and silver is also explained. Ref.2 states that the addition of nickel plating to a conductor worsens the distortion by about 40dB. So why use NiCr films in resistors? Because -128dB third harmonic distortion at 250 volts is unlikely to be a problem, and is far better than some other commonly used materials such as carbon, and the temperature coefficient of resistance is typically 50 ppm/deg.C compared to 6800 ppm/deg.C for copper.
Suggestions that audio speaker cables will give different results if reversed in direction are easily dismissed as impossible because of the alternating current, so that charge flows alternately in opposite directions in different half-cycles of a signal. It is only the electromagnetic field which has energy flow in one direction, but even then for partly reactive loads the direction of flow can reverse over part of a cycle. It may still be true that there is some assymmetry with regard to direction if there are small levels of even order distortion detectable. Ref.1 only gives figures for carbon resistors, for example the 4th harmonic is about 1000 times lower than the 3rd for a carbon film. A similar ratio for metal films would make the 4th harmonic too low to detect, which perhaps explains why this was not included in the results. (For unknown reasons the second harmonic was not mentioned, but it seems safe to assume that this also is very low.) The presence of even order harmonics would suggest some small directional effect, but there would be no point reversing cables because all this will do is reverse the phase of the distortion, not change its level. A level 60dB further below a -160dB 3rd harmonic in a cable would not be something to worry about anyway, even if it existed and could somehow be detected.
Another effect which is frequently mentioned, but turns out to be unimportant, is the possibility of reflections in a speaker cable if the characteristic impedance is not matched to the speaker. In practice such matching is impossible because the speaker impedance will vary considerably with frequency, and using an 8 ohm cable will have the effect of adding unnecessary capacitance.
There is one more topic to consider, which is not strictly a non-linearity of conduction in a wire, but of course wires end somewhere, and the method of termination may cause problems. Ref.1 suggested that the method of termination of the metal film resistors had some effect on the distortion, and postulated that this was determined by the 'work function' of the metal used in the termination. The 'work function' in metals is the energy difference between the Fermi level and the vacuum potential, but it is known that the effect of bringing together two clean metal surfaces with different work functions is that charge is transferred from one to the other sufficient to equalise the Fermi levels, and since there are energy states above and below the original levels this only changes which states are occupied, there is no gap in levels, no depletion zone, and no rectification or non-linearity as in the case of semiconductor junctions. (There may be exceptions for some metals, e.g. in zinc the Fermi energy is close to the conduction band edge, and conduction appears to be a flow of 'holes' rather than electrons, but most metals are better behaved) Metal to metal junctions are highly linear unless there is contamination of the surfaces e.g. by heavy oxidation, but simple precautions can usually avoid such extreme effects. Ref.2 suggests that high contact pressure is the best way to make a good contact and avoid non-linearity, but where high pressure is not possible gold plating helps prevent oxidation causing problems.
 "Nonlinearity of Resistors and Its Geometric Factor", Takahisa et al, Electronics and Communications in Japan, Vol. 56-C, No.6, 1973.
 'Passive Intermodulation Distortion in Connectors, Cables and Cable Assemblies' by David Weinstein, Amphenol Corporation. (pdf file)
'A Dynamical Model of a Crystal Structure', Bragg and Nye, Proc. Royal Society of London, Vol.190, Sept 1947, pp.474-481. (Also reprinted in 'The Feynman Lectures on Physics' Vol.2, pp.30-10 to 30-26)
'International Motivation and Cooperation for Research in the Ultra-High Purification of Metals', Kekesi, European Integration Studies, Miskolc, Vol.1, No.2, 2002, pp.109-126.
Copper Development Association 'The Metallurgy of Copper Wire' by Dr. Horace Pops.
A search on Google will find plenty of information ranging from elementary introductions to advanced research. Generally searching for technical terms will lead to reliable scientific/technical sources, but just searching for 'audio cable' for example is more likely to find cable manufacturer sources, which are inevitably trying to sell a product, so their explanations and claims should be approached with caution.