Solids and conductivity
If this configuration is most stable, it will be so everywhere, and we will have the same configuration over the whole sample: called long range order.
We don’t need to study the different configurations, leading to crystals of different symmetry. The only property we need for our study of conductivity is the fact that the molecules are fixed in a regular pattern.
Assume that in some materials that there are some electrons that are not attached to a particular molecule, but can move more or less freely in the material. We call them non-localized. These must be electrons that sit in the outer shells of atoms, and can’t decide which atom they are attached to.
Then the total current would be where A is the area of the conductor. This is not Ohm’s law, which tells us that the current varies as E. One concludes that electrons don’t accelerate freely all the way across the material, but only for a short distance, presumably until they hit one of the fixed atoms and bounce off in a random direction (“scatter”):
Then the average “drift” velocity of a random electron, making a collision after the average time ? is We can express ? in terms of the mean free path and the average electron velocity:
PPT Slide
From the picture of the crystal, it is clear that since the spacing of atoms is such that that just “touch” in some sense, the value of ? must be about equal to the lattice spacing between atoms in the crystal, a few tenths nanometer.
Look at resistivity over a wide range of temperature
Another question, is there “channeling?”
Another hint as to what is wrong comes from the heat capacity of metals and insulators. In both, we should see the heat capacity of the nuclei vibrating around their locations on the crystalline lattice (3R in gas law constant units) and in insulators, no more, if there are no free electrons to move around. This is observed. In metals, we should expect an additional ½kT for motion in each direction (xyz). Not observed! There is an excess of only about 0.02R. Evidently, there are very few free electrons, but the conductivity at low temperatures is way greater than that given by our classical theory! Those few must be working very hard!
The only way an electron can work harder to make more conductivity is to move further before it scatters, but it has to move in this “forest of atoms.” How?
This long distance before the electron explains how we can get so much conductivity with few free electrons and wave doesn’t channel. Now let us look at the distribution of the electrons. The key is going to be the Pauli Principle. Use the “particle in a box” again to get a one-dimensional version of the energy distributions of the electron.
Calculate the energy for N electrons in the box, which we call the Fermi energy:
Now we must look more carefully at the wave properties of the electrons in the periodic potential of the lattice of atoms, to find out the story on conductors, insulators and semiconductors. There are two ways: 1. Start with free atoms and calculate what happens when N atoms are brought close together. (this is done in the book.) 2. Start with free electrons, calculate the solutions in a periodic potential.
That n=3, l=0 “band” of the outermost electron could hold two electrons, one spin up, one spin down. Na has only one electron in that state, so only half the states are filled. We say that the band is half filled. Then, since the states in the band are essentially a continuum, the slightest increase in energy of the topmost electron in the band can move it into a new state with no objection by the Pauli Principle. It can be moved easily by an electric field in particular, so Na is a conductor. In contrast take NaCl in solid form. The outer electron goes in the outer shell of Cl for binding the crystal, so we have a Na+ ion and a Cl- ion. Both are filled shells, so there is no nearby state:
To move an electron up to the next state, or band in the crystal, we have to go to the state of (Na+)*, and excited state, and that takes several eV. In between there are no states, and we say that there is a band gap or forbidden band. We draw the two energy level pictures:
The free electron method uses the Bloch-Floquet theorem which says that the solutions of the wave equation in a periodic potential must be of the form
What is special about ka=n? ?
Semiconductors are just insulators with a small forbidden band gap, if they are “intrinsic” superconductors with NO impurities, or “extrinsic” with impurities giving states with energies in the forbidden band:
In the n-type a little thermal exitation of the impurity (often Arsenic) introduces an electron into the conduction band, so we get electron conduction
What if we make a junction of a p-type and an n-type??
Field Effect Transitor
Almost 100 years ago, K. Onnes succeeded in liquefying Helium, finding it boiled at 4.2K. Many remarkable discoveries of quantum properties of Liquid Helium followed, but one discovery was Superconductivity. He found that below a certain termperature, lead and tin, and some other metals lost all resistance. How can it be? BCS (three guys) explained it as one electron distorting the lattice slightly, and another one with opposite spin joins (and aids) it, to form a pair: “Cooper pair.” Pauli Principle tells us that no more than a pair can play this game.
Distortion of the lattice creates a well:
Thousands of atoms across: that is a macroscopic state. It is not so easy to break that pair of electrons apart, that is why the critical temperature is high. It is very hard for just one electron to be scattered by an irregularity in the lattice, since the electrons are averaging over billions of atoms in the lattice. We saw before in normal metals that the mean free path, and the conductivity, go to infinity except for scattering. There is no scattering in the superconducting state, so the resistance is zero. Experiments have looked for decay of currents for years and seen none in good superconductors.
Since these Cooper pairs are both pure quantum states and macroscopic, it is easier to see real quantum states than with microscopic states, and relatively easy to build applied quantum devices.
Superconducting Quantum Interference Device
This finishes our exploration of quantum physics and engineering. (Nuclei and particles have interaction that are quantal, but the particles usually have high energy and act like particles indeed.)
Email: willis@nevis1.columbia.edu
Home Page: www.nevis.columbia.edu/~willis
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