Temperature and Charge Carrier Scattering

Now that we have a model for why, in general, some materials conduct electricity and some don't, we'll move onto another puzzle: heating up a conductor increases its resistance. Why?

We'll need to start with an axiom, here. Again, from Richard Feynman (Feynman, R.P., Propagation in a crystal lattice. Feynman, R. P. et al, The Feynman Lectures on Physics; World Student Series; Addison-Wesley Pub: Reading, Mass., 1963.:

You would, at first sight, think that a low-energy electron would have great difficulty passing through a solid crystal. The atoms are packed together with their centers only a few angstroms apart, and the effective diameter of the atom for electron scattering is roughly an angstrom or so. That is, the atoms are large, relative to their spacing, so that you would expect the mean free path between collisions to be of the order of a few angstroms—which is practically nothing. You would expect the electron to bump into one atom or another almost immediately. Nevertheless, it is a ubiquitous phenomenon of nature that if the lattice is perfect, the electrons are able to travel through the crystal smoothly and easily—almost as if they were in a vacuum.

In short, electrons travel smoothly and easily through a perfect crystals, accelerating uninhibited as if moving through a vacuum. However, Feynman includes a very important "if" in his statement: "if the lattice is perfect, the electrons are able to travel through the crystal smoothly and easily". Feynman was very intentional with this "if", and we, as materials scientists, know why: we never have perfect crystals.

Let's consider one type of deviation from perfections: when a material heats up, its atoms vibrate with more kinetic energy. We saw this in our molecular dynamics models earlier. These lattice vibrations are deviations from crystalline perfection as now they move atoms and electrons away from their perfect crystalline arrangement. As we'll see, these vibrations will "scatter" the electrons, changing their direction and thus lowering their average velocity as they travel from from cathode to anode.

We'll explore a model of this below, but first we'll go one level deeper and ask: why would lattice vibrations scatter electrons in the first place? We've claimed that any deviation from perfection will scatter electrons, but can we intuit why? We'll explain this from two different perspectives.

Perspective 1: Free electron vs. phonons (thermal changes in electron density)

We've looked at molecular dynamics models of atoms vibrating more intensely with increased temperature, such as in Section 4.9. In a molecular dynamics model, we treat the atoms as Newtonian point particles that interact according to an interatomic potential function. Recall, though, that atomic bonding is actually due to electron density increasing between the nuclei of atoms, as discussed in Section 3.5. So, when atoms vibrate, the density of electrons around them is changing as shown in the animation shown in Figure 13.7.1.

As temperature increases, the vibrations get more intense, and fluctuations of electron density get more intense. These fluctuations propagate through the material as waves and are called "phonons". Consider for a second, based on our understanding, why these vibrations might be wave-like in nature, as opposed to completely random in any direction... indeed we've seen these before in molecular dynamic simulations of crystalline materials earlier in the term!

Phonons are in some ways analogous to photons, having both wave-like and particle like properties. For our purposes here though, the only thing that matters is that these oscillations cause the electron density within the crystal to be constantly shifting, i.e., producing regions of high electron density that would repel, or scatter, the electron - transferring energy and momentum to the electron and altering its path through the crystal.

Phonon-electron interactions are, of course, quite complicated, but the critical aspect of this scattering interaction is associated with the motion of the electron cloud due to thermal vibrations.

Perspective 2: Disrupted Resonance

Previously, we explained why metals conduct electricity in terms of electrons being free to move between resonance structures. If the two resonance structures are totally equivalent, the electron is equally shared or averaged across the resonance structures. This is the case in a perfect metallic crystal, and it gives maximum mobility to the electrons.

If the atomic positions are distorted in some way, it will cause one resonance structure to be favored over the other. As atoms vibrate in a lattice, some bonds are momentarily getting shorter and some longer. This will make the electrons more likely to be located in the bonds near the equilibrium bond length, which has the lowest energy, and less likely to be in the stretched or compressed bonds. This makes the electrons less mobile, because they are less likely to move to those bonds. So, as temperature increases, resonance gets disrupted, and conduction goes down.

Model of electron scattering

With those explanations of why electrons scatter, we will once again simplify and ignore electron density and resonating bonds, going back to our free electron model but with the addition of electron scattering. Electrons now have a probability of scattering each time step, modeled by resetting the speed of the electron to a low value (which means the electron has transferred energy to the lattice) and changing to a random direction. The model below is the same as the one in Section 13.4 except that scatter-prob has been added which determines the probability of electrons scattering each time step.