Shockley and Arrhenius Diagrams in Semiconductor Physics

When a semiconductor misbehaves, two graphs do most of the detective work. Both use logarithmic scales, and both are useful because, on those scales, the physics becomes straight lines. The first is the Arrhenius plot. Measure a thermally activated rate at several temperatures, plot its logarithm against one over temperature, and the points fall on a line whose slope is an energy. The second is the Shockley diagram. Plot the logarithm of each carrier and charge concentration against the Fermi level, and where positive and negative charge balance, the lines cross and fix the Fermi level. So one diagram reads energies off slopes, the other reads a Fermi level off a crossing. By the end we will use them together, on a single defect in silicon.

Here is a typical measurement. A deep trap in silicon captures electrons and releases them, and transient spectroscopy gives its emission rate at several temperatures. Each red point is one temperature, hot on the left and cold on the right. To know what line these points should follow, write down the rate. It is a capture cross section, times a thermal velocity, times the effective density of states, times a Boltzmann factor in the trap depth. Two of those factors drift with temperature: the velocity like the square root of T, the density of states like T to the three halves. Together that is T squared. Divide it out, and only the exponential changes quickly. Now take the natural logarithm. The exponential becomes a straight line in one over T, with the trap depth over Boltzmann's constant as its slope, and the logarithm of the prefactor as its intercept. And the points obey it: one straight line runs through them all, the first sign that a single activated process is at work. Read the slope with a triangle. Over this run the line drops by this much, a slope of about minus four point six four thousand kelvin. Multiply by minus Boltzmann's constant, and the trap depth comes out: zero point four zero electron volts below the conduction band. Now extrapolate back to one over T equal to zero, where the exponential is one. The intercept is the logarithm of the prefactor, and dividing out the known constants gives a capture cross section near ten to the minus fifteen square centimetres. Treat that intercept with caution: it sits far outside the data, so a small error in slope moves it a long way. The slope is the robust number. Nothing here is special to traps. Any process with a Boltzmann factor gives the same straight line; only the axis label changes. Dopant diffusion: the slope gives the energy an atom needs to hop. Carrier freeze-out: the electron density falls with half the donor ionization energy. And in the intrinsic regime, the slope is half the band gap. One warning: if the points bend away from a line, two mechanisms are competing, or the prefactor hides a temperature dependence you have not divided out.

Now the second diagram. Across the bottom is the Fermi level, measured from the valence band edge of silicon up to the conduction band edge. Up the side is the logarithm of a concentration, per cubic centimetre. Electrons first. Their density grows exponentially as the Fermi level climbs toward the conduction band, so on this scale it is a straight line with slope one over k T. Holes are the mirror image, falling with the same slope the other way. Where the two lines cross, n equals p, and that height is the intrinsic density. Now add donors, ten to the sixteen per cubic centimetre, forty five milli electron volts below the conduction band. While the Fermi level stays below them they are all ionized, a flat line. Neutrality demands that negative charge balance positive charge: n equals p plus the ionized donors. So where must the Fermi level sit? Try the middle of the gap. The electron line is far below the donor line, so there is too much positive charge, and the Fermi level must rise. Try one electron volt. Now electrons outnumber donors, too much negative charge, so it must come back down. In between, the lines meet. About zero point two one electron volts below the conduction band, the charges balance, and the electron density equals the donor density. And the hole line, down here, is twelve orders of magnitude lower. That is how the diagram tells you which terms may be dropped. Heat the crystal to five hundred kelvin. Both lines flatten, since their slope is one over k T, the intrinsic crossing rises, and the neutral point slides toward midgap. Now bring back the trap we measured, zero point four zero electron volts below the conduction band: a vertical line here. Whether it holds an electron is set by Fermi statistics, comparing the trap with the Fermi level. Our Fermi level sits about two tenths of an electron volt above the trap, so the exponential is tiny and the trap is almost always filled. Read the electron line where it crosses the trap. That height is n one, the electron density if the Fermi level sat exactly at the trap. The hole line there gives p one. These two numbers measure how easily the trap trades electrons with one band and holes with the other, and they are exactly what the recombination rate needs.

Shockley, Read and Hall wrote the net rate at which one trap level recombines carriers. The numerator measures how far the crystal is from equilibrium; the denominator holds the lifetimes, and the n one and p one we just read. Its sign gives the direction. With excess carriers, n p exceeds n i squared and the trap recombines. Sweep the carriers out, as in a reverse biased depletion region, and the same trap generates them instead. In that depletion region n and p are negligible, so the rate is n i squared over the n one and p one terms. Measured from the intrinsic level, with equal lifetimes, this becomes a hyperbolic cosine. Plot its logarithm against the trap position and the Shockley picture returns: away from midgap the rate falls in straight lines, one decade for every sixty milli electron volts. Our trap sits zero point one six electron volts above midgap, so it generates about two hundred and fifty times less than a midgap level would. Push a level out to zero point four, and it loses six orders of magnitude. Bring it to the middle of the gap, and it is the most efficient generator there is, which is why midgap defects dominate leakage and lifetime. The two diagrams now finish the job together. Generation through midgap traps scales with n i, so a diode's leakage current has an Arrhenius slope of half the band gap. Diffusion current from the neutral regions scales with n i squared and shows the full gap. Measure leakage at a few temperatures, read the slope, and you know which mechanism is responsible. So, to collect it, here is what each diagram gives you. The Arrhenius plot turns a rate measured over temperature into an energy from its slope, and a prefactor from its intercept. The Shockley diagram turns charge neutrality into a position for the Fermi level, and with a trap drawn on it, into n one and p one. And those feed the recombination rate, which a defect near midgap maximises. Slope, crossing and trap level: three readings that tell you what a semiconductor is really doing.