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## Resistivity

Our classical understanding of interactions of charge with electric fields tells us that electric fields exert forces on charges, and forces cause accelerations. In terms of energy, this means that as long as the electric field is present, the charge keeps speeding up and gaining kinetic energy. But in a conducting medium, electric fields result in electric charge moving at a *constant* rate (when averaged over all the charges). The energy given to the electrons through the work done by the electric field must go somewhere else, if it is not going into the kinetic energy of the charges.

The classical description of this transfer of energy is that the moving electrons occasionally collide with atomic sights, and when they bounce off, their average motion is slowed, while the atoms vibrate more (the conductor's thermal energy rises). The net result is that the electric field remains proportional to the current density, related by a physical property of the material called *resistivity* (\(\rho\)):

\[ \overrightarrow E = \rho \overrightarrow J\]

This relation is known as *Ohm's law*. An electric field present within a conductor will result in less current when the resistivity is higher.

The classical view of this phenomenon is actually not too far off the mark, though the classical determination of what contributes to the resistivity of a material is pretty much relegated to empirical study. Detailed quantum-mechanical examination is beyond the scope of our studies, but in a nutshell, what affects free electron acceleration in the presence of an electric field within a conductor is *irregularity* in the atomic sites.

A crude way to think about this goes back to our study of one-dimensional barriers in Section 3.5. We saw that particles which run into "speed bumps" (low barriers whose potential energy jump is less than the kinetic energy of the particle) partially reflect and partially transmit. We found that when a low barrier has just the right thickness, the particle can pass through with 100% transmission. Imagine an electron perfectly "tuned" to a long series of identical low barriers such that it passes through them easily. Now imagine that there are irregularities thrown into the series of bumps. Now some fraction of each electron's wave function will be reflected, which means that some fraction of all the electrons will not get through, reducing the total current.

The two main sources of irregularity in a lattice are impurities – other materials mixed into the otherwise regular lattice – and thermal agitation. When the atoms vibrate, their separations vary (like varying the thicknesses of the aforementioned speed bumps), and this changes conditions for the electrons in real time. Empirically, these conditions are characterized as materials and temperature.

## Valence and Conduction Bands

The discussion of resistivity mainly applies to materials that actually conduct electricity, but there are many that act as *insulators*. Resistivity relates to *how well* a material conducts electricity when an electric field is applied, but there is an entirely different question as to whether the material will conduct electricity at all. Very generally, this is decided by whether there are any free charges available to move as current.

Let's think for a moment how current in a one-dimensional lattice must work. In the absence of an applied electric field, the electrons settle into their lowest energy states. In these states, they have both potential and kinetic energy, but their motions (which can only be in the \(\pm x\) directions) are "thermal" (random). When an electric field is applied (say in the \(+x\) direction), work is done on the electrons, speeding some up, slowing others down, but overall giving them additional energy that is not so thermal – biased in a specific direction. But this energy that is given to the electrons does not break them out of their bonds with the crystal, so they muct simply be raised to new energy levels.

Suppose electron energies in the lattice were not organized in bands, but were quantized as they are in single atoms. In this case, only a minimum electric field would start driving a current, and the amount of current would jump between well-defined levels. None of this is observed – very small currents are possible, and the levels of electrical current available are pretty much continuous. This is because electrons can have their energies raised within a continuum of options within the same energy band.

Does this happen within every energy band? No! Electrons are fermions, which means that only two electrons can exist in any one of the states. It is therefore typically the case that the lower energy bands are completely filled, which means that there is no room for an electron to gain energy within the band. In fact it is only the highest energy band for which this can be possible. The energy band with the highest energy that is completely filled at zero temperature is called the *valence band*, and the energy band immediately above this is called the *conduction band*.

## Classifications

Suppose the valence band of a material is completely full, while the conduction band is completely empty. In order to elevate electrons to an energy where they can use the spare energy for (orderly) kinetic energy, they have to be pushed through the band gap. Some of these gaps are larger than others, and an arbitrary gap size of \(2eV\) has been established as sufficiently large to classify a material as an *insulator*.

Keep in mind that we are talking about the condition of these bands at zero temperature. If the temperature is raised, then the Fermi-Dirac distribution allows for some electrons to move up to higher states. So in fact insulators can get a small number of electrons into the conduction band when the temperature rises, but not enough to be significant. When the band gap is smaller than \(2eV\), however, increases in temperature can put enough electrons into the conduction band to be significant. Such materials are called *semiconductors*. Unsurprisingly, materials that naturally have electrons in their conduction band *as well as* *the space available to raise electron energy levels within that band* are called *conductors*.

## Fermi Energy

It might seem like there is a pretty smooth transition between conduction and insulation (though semiconductors). That is, it would seem like we can just take any insulator, and raise its temperature until it behaves like a semiconductor. But in fact materials that insulate do so at pretty much all temperatures. We can apply what we learned about the Fermi-Dirac distribution to show why this transition is so stark.

First of all, we defined the fermi energy as the highest energy of the collection of electrons at zero temperature, which for an insulator would be the top energy of the valence band. But this definition assumed a continuous spectrum of energies, and didn't account for a band gap. The fermi energy is the energy that electrons cross as they go from lower, occupied states, to higher, unoccupied ones, and it was defined so that it was symmetrically-placed between these conditions (by assuming that the fermi energy is where the occupation number is equal to one half). Consequently, the fermi energy is placed halfway between the top of the valence band and the bottom of the conduction band.

When the temperature is zero, the occupation number curve is a step function, and as the temperature rises, the curve flattens, as shown in Figures 7.5.4 and 7.5.5. In order for the flattening of this curve to represent a transfer of electrons across the gap, the curve needs to flatten sufficiently to span from one side of the band gap to the other. But the center of this curve is *extremely* steep. To see why, consider that at room temperature (\(300K\)), the quantity \(k_BT\) is about \(0.025eV\). If the energy of the bottom conduction band is \(5eV\) higher than the top of the valence band (as is typical for insulators), then the energy difference between the bottom of the conduction band and the fermi energy is \(2.5eV\). Even considering the number of electrons available in the valence band (say on the order of \(10^{24}\)), the exponent in the denominator of the occupation number is 100, giving a factor of about \(10^{-43}\), which means that basically no electrons move up at this temperature, and to raise the temperature sufficiently would vaporize the insulator before populating the conduction band.

One might think that the semiconductor is "close" to this amount, but that is not the case. If the band gap is (say) \(1eV\), then the exponent in the denominator is only 20, giving a factor in the denominator of less than \(10^{10}\) at room temperature, which allows for a lot of electrons when there are \(10^{24}\)) available.