What causes resistance

[new299]

I've been trying to get a rough idea of how conductance works at a molecular level. I understand that in metals the metallic bonding means that there are electrons that have freedom to move throughout the metal. However I'm less clear on what causes resistance (in metals).

I assume as the current increases the electrons are moving at greater speeds through the metal, obviously resistance impedes this motion but how exactly? I also understand that resistance causes heat dissipation, but how?

I've tried googling, but didn't come up with any good information, but would appreciate links if you have them.

[Len]

http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#In_metals

[new299]

http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#In_metals

Thanks that helped me understand things a little better.

So the electrons hit other atoms, I guess this causes heat to be dissipated. Am I right in thinking that is effectively slows them down too? Because they take a longer path through the metal? And because there needs to be a more or less constant number of electrons on the metal, this limits the flow or current?

What role does the energy level of the electron play?

[helius]

this rough picture is inaccurate in several ways...
first, the free electrons in a metal are constantly in motion, whether there is current or not. the thermal energy of the moving electrons is what creates the phenomenon we call temperature.

when the electrons are in an electric field, a new motion vector is added to their thermal motion, called the drift velocity due to the field. the drift velocity is much slower than the thermal velocities, about a billion times slower! this is what relates to the current on a microscopic level.

because the thermal speed is so large compared to the drift speed, it's the thermal speed and not the drift speed that determines how much time elapses between collisions between the electron and another particle like atoms. this means that the "mean free time" of electrons in the material is independent of the electric field. the drift velocity of each separate electron is increased due to acceleration in between collisions; so the kinetic energy due to the electric field goes from zero upwards, until the next collision takes it down to zero, which also dissipates that drift kinetic energy as heat.

[new299]

because the thermal speed is so large compared to the drift speed, it's the thermal speed and not the drift speed that determines how much time elapses between collisions between the electron and another particle like atoms. this means that the "mean free time" of electrons in the material is independent of the electric field. the drift velocity of each separate electron is increased due to acceleration in between collisions; so the kinetic energy due to the electric field goes from zero upwards, until the next collision takes it down to zero, which also dissipates that drift kinetic energy as heat.

Thanks, I think I'm slowly getting a clearer picture of things.

Why is it that resistance decreases with the thickness (cross-sectional area) or the conductor? I would have thought that the probability of an electron hitting an atom would be independent of the thickness of the conductor.

[helius]

You're right, the mean free path is independent of the conductor's thickness. But that's proportional to the current density, not the current. A conductor with a thicker cross-section carries more current for the same current density.

[T3sl4co1l]

It's a little funny that it's usually phrased as "electrons hit atoms"...

Electrons propagate as waves (quantum matter waves), and to them, the crystal looks no different than, if you can imagine an underground cavern, a tight grid of tunnels, effectively an open space with periodic columns supporting the ceiling -- that is the environment around the electrons, and the electrons are waves much as acoustic or electromagnetic waves would propagate through that structure.  The columns are the atoms, with high potentials (energy barriers) around them, due to the bound (valence) electrons already "orbiting" the atoms themselves, which exclude free electrons from those areas.

So you can see that electrons, for the most part, simply wash around the atoms.  They aren't nearly small enough to "hit" an atom, in the ballistic sense.

Which is possible: that's literally what scanning electron microscopes (SEM) do -- and they need 100keV electrons to do it.  The electrons in a solid are bound with only a few eV of energy.

So what does resistance look like?  Suppose in that cavern of columns, the columns themselves are jostling about, randomly.  Instead of presenting a periodic structure for waves to diffuse (and diffract) through, they will eventually scatter the waves -- the wave fronts partially reflect and diffract around the slightly displaced barriers, and after enough of these subtle interactions, the wave fronts become randomized and diffused.  Note that the columns themselves excite waves: just as a stick in a pond creates ripples from its relative motion, the thermal motion of atoms causes ripples in the electron gas in the crystal.

Thermal motion, in turn, is analyzed in terms of phonon waves -- acoustic waves trapped within the crystal.  A superposition of these stores thermal energy, and is responsible for the heat capacity of most materials.

The coupling between electron waves and phonon waves is called electron-phonon coupling.  It would seem likely that metals with poor coupling have excellent conductivity.  I don't remember the magnitude of this effect, if it does vary with resistivity, but it's certainly a contributor.

As it turns out, at room temperature, the thermal motion of electrons is dominant, to the extent that thermal saturation velocity is on the order of 10^5 m/s for electrons in silicon, while drift velocity is on the order of 0.01 m/s!

Note that "drift" is due to the coherent motion of electrons -- not as wavefronts stimulated by an outside current, nothing like that; merely a general trend followed by that sea of rapid movement.  Just as an individual water molecule might be zipping around at the speed of sound, but a massive body of water moves cooperatively to exhibit wave effects (whether internal acoustic waves, or gravity waves on the surface).

Coherent movement is observable in fancier semiconductors; in gallium arsenide, the saturation velocity can be exceeded, and ballistic transport occurs.  The result is, at a high enough electric field, the conductivity suddenly goes up, which means the equivalent resistance goes down -- negative dynamic resistance!  Gunn diodes are not actually diodes at all, but a piece of doped GaAs (no junction at all) which exhibits this behavior.  When placed in a resonant cavity, the negative resistance can be used as an oscillator at extremely high frequencies (10s of GHz) -- the effect is atomic scale, so practically unlimited in frequency response*.

*I don't actually know what mechanism(s) limit this.  There must be a limit, since as far as I know, this effect hasn't been used for ludicrously wideband amplifiers (1THz+?).  The limits due to geometry (frequency is generally inversely proportional to the length scale, due to the speed of light and other equivalent limits within the medium) are orders of magnitude below those of a Gunn diode (i.e., ~100nm versus ~1mm!), so there must be another effect, which I don't know anything about, which prohibits this behavior!

I think the difference is, what type of semiconductor is used; silicon is an indirect bandgap semiconductor, and undergoes avalanche breakdown when critical electric field is applied (such that would give an electron velocity near the thermal saturation velocity).  Avalanche you can think of as an electron colliding with an atom to release more (free) electrons and holes; it's often a runaway cascade effect (which gives rise to switching effects in certain transistors and the noisy voltage of zener diodes).  GaAs and many other important semiconductors are direct bandgap, which means they emit light when forward-biased (yay, LEDs!); I don't know if this directly affects avalanche, or if it's purely a bandgap thing (...does cold silicon exhibit ballistic transport?), though.

(Ref: I'm a physicist.  At least, within the stuff that I remember.)

Tim

[new299]

You're right, the mean free path is independent of the conductor's thickness. But that's proportional to the current density, not the current. A conductor with a thicker cross-section carries more current for the same current density.

I see, I'm assuming the field causes a velocity drift on all electrons. So, for the same current in a larger conductor the drift velocity per electron is smaller. This means each collision results in the loss of less energy. So while the collision probability is the same, the energy loss is less. Does that sounds right?

Thanks again, extremely useful.

[helius]

I see, I'm assuming the field causes a velocity drift on all electrons. So, for the same current in a larger conductor the drift velocity per electron is smaller. This means each collision results in the loss of less energy. So while the collision probability is the same, the energy loss is less. Does that sounds right?

Well, no. Every charge in an electric field experiences a force in the same direction as the field. So if we assume that the electric field in a wire is perpendicular to the sides (technically you'd say that the wire is made of an infinite number of equipotential surfaces that are cross sections), then every electron feels a force along the wire towards the + end, and every metal nucleus feels a force towards the - end. The metal nuclei are closely bound (by atomic forces) to most of the electrons in each atom, and only loosely bound to the free electrons. So each atom less its free electrons accelerates very slowly to the - end, and the free electrons accelerate towards the + end.

The drift velocity of electrons, and their current density, is uniform at a microscopic level. It doesn't know or care if the conductor is a 34awg magnet wire or a ten-ton ingot.

edit: I think I misread your question. When you drive the same current through a thicker conductor, the E field is correspondingly weaker (as is the potential difference between the ends). This corresponds to a lower resistance. The other way that resistance varies is with the resistivity of the material. Metals with a higher density of mobile charges, or a longer mean free path, have lower resistivity.

[new299]

I see, I'm assuming the field causes a velocity drift on all electrons. So, for the same current in a larger conductor the drift velocity per electron is smaller. This means each collision results in the loss of less energy. So while the collision probability is the same, the energy loss is less. Does that sounds right?

The drift velocity of electrons, and their current density, is uniform at a microscopic level. It doesn't know or care if the conductor is a 34awg magnet wire or a ten-ton ingot.

edit: I think I misread your question. When you drive the same current through a thicker conductor, the E field is correspondingly weaker (as is the potential difference between the ends). This corresponds to a lower resistance. The other way that resistance varies is with the resistivity of the material. Metals with a higher density of mobile charges, or a longer mean free path, have lower resistivity.

Thanks. I'm confused as to why a thicker conductor has a lower resistance. While the E field is weaker, I assume causing a lower current density (and drift velocity) it's over a larger area (at the same overall current).

So for the same overall current. In a thin conductor the individual electron drift velocity is high. So at each collision a large amount of energy is dissipated. In a thicker conductor the current density is lower, and so the individual electron drift velocity is lower, so at each collision a smaller amount of energy is lost.

However, in a thicker conductor there are more collisions because there are more electrons in motion. So it would seem like the overall resistance would be the same?

I'm obviously missing something.

[T3sl4co1l]

Don't compare conductors to conductors, compare material to material.  Resistance is a geometric property that varies with scale and dimensions; resistivity is a bulk effect that is independent of geometry.  Ohm's law, for bulk material, is written as: J = E / rho, where J is current density (e.g., A/m^2), E is electric field (V/m) and rho is resistivity (ohm m).

Extremely little electric field is applied to a wire under normal conditions.  1 V/m is a lot of electric field in copper.  1V applied to 1m of #37 AWG wire (0.113 mm dia.) gives a current of 0.583 A, or a current density of 5.8 x 10^7 A/m^2.  (This wire is normally rated for about 0.03A, and makes a reasonable fuse around 3A.  Free in air, it probably wouldn't last long at 0.583A.)  Copper atoms are spaced about every 200pm, so that the energy gained per atom traveled is all of 0.2 neV.  Electrons travel about 80 atoms (1.6nm) on average before scattering to a different direction and so on, so the average energy before change is 16neV, still vanishingly small.  (If scattering were a radiative process, the light emitted would be around 3MHz!)  The corresponding peak velocity is around 50 m/s (versus a thermal velocity of 10^6 m/s).

Tim

[new299]

Don't compare conductors to conductors, compare material to material.  Resistance is a geometric property that varies with scale and dimensions; resistivity is a bulk effect that is independent of geometry.  Ohm's law, for bulk material, is written as: J = E / rho, where J is current density (e.g., A/m^2), E is electric field (V/m) and rho is resistivity (ohm m).

Thanks, this and your other reply seem to indicate that in order to correctly model resistance a wave (QM?) model is required. Is it possible to understand resistance classically? Can the Drude model (http://en.wikipedia.org/wiki/Drude_model) model the reason a thicker conductor has less resistance than a thin one?

I have no background in QM, and I'm having trouble understanding an electron as a wave. Though your description helped me understand why resistance varies with temperature in metal. Do you have a reference that might help me gain an intuitive understanding of this? Or do I need to go back to basics here?

Does this sound correct: each electron is a wave, and an irregular lattice (due to thermal effects) causes destructive interference and this is resistance. So each electron in some sense is traveling though the whole conductor? If this is the case, I'm still at a loss as to why a thicker conductor results in decreases resistance. The lattice is a thicker conductor is equally disordered isn't it?

I feel like I may not have the necessary tools to understand an electron as a wave. If you have pointers in this direction that would be most helpful.

[CatalinaWOW]

First, no one actually understands all of the details of resistance.  Some of our models, particularly the quantum models, give good approximations of behavior for a wide range of circumstances, and as mentioned above in some very specific and useful special circumstances.  It is only in the last 30 or 40 years that these models have been used to explain important phenomena like low temperature superconductivity.  Some understanding of higher temperature superconductivity is even newer.  The quantum models are too complex to directly apply to many real world applications, a carbon resistor for example.  If you want as good an understanding of resistance as is possible you should look at all of the models and understand where they are applicable and what their limitations are.   The existence of PTC resistors, NTC resistors, superconductivity and non-linear resistors should give you a strong clue that there will be no simple and universal theory of resistance.

If you want to understand wave behavior in electrons, start with baby steps.  Review the analysis of light showing both particle behavior (photons), and wave behavior (slit diffraction etc.).  Then move on to relationship of mass to energy (E=mc^2) and use the electron mass to compute energy.  Then use that energy to compute the approximate wavelength of the electron wavicle.  You will start to see some relationships and scale. 

I think it will take some fairly serious study to go much further.  "Introduction to Solid State Physics" by Kittel and "Quantum Mechanics" by Schiff are solid texts on the subject, but will require some serious grounding in math to fully understand.

[T3sl4co1l]

Thanks, this and your other reply seem to indicate that in order to correctly model resistance a wave (QM?) model is required. Is it possible to understand resistance classically? Can the Drude model (http://en.wikipedia.org/wiki/Drude_model) model the reason a thicker conductor has less resistance than a thin one?

If all you're concerned about is aspect ratio, then it doesn't matter what physics model is used, so long as Ohm's law applies in a bulk manner (i.e., J = E / rho).

A thicker conductor has less resistance because the current density J is smaller (same A, bigger /m^2), and a longer conductor has more because the potential is higher (V = E * length).  That's just geometry.

Does this sound correct: each electron is a wave, and an irregular lattice (due to thermal effects) causes destructive interference and this is resistance.

Note quite.  I think you're thinking it, but don't quite know the word.  "Destructive interference" is a coherent phenomenon, the cancellation (and converse doubling-up) of wave fronts.  We're talking incoherent (disorganized, chaotic, random) phenomena.

You can get destructive interference with laser beams, but it's much harder to do with white light (which is random, and wide band), or even filtered light (the light from an LED, for example, is fairly monochromatic, but not a coherent wave).

Scattering is the appropriate word: the redirection of waves to random directions or positions.  A polished mirror is coherent (or at least roughly so), but a matte white surface is not.  If you imagine a mirror made of many very small facets, each pointing slightly different directions, that would be a fair analogy to what's going on at the atomic level.  Except it's not a single reflective process, but a transmissive (waves traveling through a medium) process, so it would be more like a block of glass where the index of refraction varies randomly, and thus the optical paths are scattered.

I feel like I may not have the necessary tools to understand an electron as a wave. If you have pointers in this direction that would be most helpful.

If you don't know much about waves in general, you'll have to start there; suffice it to say, electrons behave as waves in much the same way as any other kind (optical, radio, acoustic, fluid gravity, etc.).

An electron's behavior isn't necessarily defined so much by its classical properties (like size, mass, position and velocity) as by the solutions to the wave equation, when applied to some system (like the simple "particle in a box" problem).  The boundary conditions which confine the particle give rise to characteristic modes and such; allowed energy states and the like.

That they can also behave as point-like particles (as well as photons, and everything else elementary) is arguably one of the curious consequences of quantum mechanics, rather than supposing that all particles are, well, particle-like, and that waves arise out of some spooky equation.  In fairness, it's an equivalence, so it doesn't actually matter: both approaches work.

Afraid I don't have any references, offhand; Wikipedia won't actually be of much help since its articles are generally written from the view of one who already understands the subject.  Textbooks wouldn't be much help for the casual reader.  Searching for tutorial or introductory websites may be helpful though.

Tim