# Shockley and Arrhenius Diagrams in Semiconductor Physics > A focused lecture for undergraduates in electronic physics on the two logarithmic diagrams that turn semiconductor measurements into energies. The Arrhenius plot linearises a thermally activated rate against inverse temperature, and a worked deep-level emission example extracts a trap depth of 0.40 eV from the slope and a capture cross section from the intercept, with the same construction applied to diffusion and carrier freeze-out. The Shockley diagram plots carrier and charge concentrations against the Fermi level, locates the Fermi level from charge neutrality, and places the measured trap on the same axes to read its occupancy and the Shockley-Read-Hall quantities n1 and p1. The lecture closes on generation and recombination through that trap, why midgap defects dominate, and how an Arrhenius plot of diode leakage tells generation from diffusion. - Canonical watch page: [Shockley and Arrhenius Diagrams in Semiconductor Physics](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Physics - Published: 2026-10-16T18:02:45.703Z - Updated: 2026-10-16T18:02:45.703Z - Duration: PT468S (7 minutes 48 seconds) - Chapters: 4 - Views: 2 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M52X7J8PNG1Y697ARRPE5Z64/0/dark/master.m3u8) - Embed: [Player](https://academa.ai/embed/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M52X7J8PNG1Y697ARRPE5Z64/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M52X7J8PNG1Y697ARRPE5Z64/0/dark/poster.jpg) ## Description Arrhenius plots and Shockley diagrams for semiconductors: activation energies from slopes, Fermi levels from neutrality, and midgap trap generation. ## Chapters - [00:00–00:48.179 · Two Diagrams, Two Questions](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=0) - [00:48.179–03:6.63 · The Arrhenius Plot](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=48.17902083333334) - [03:6.63–05:34.927 · The Shockley Diagram](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=186.629875) - [05:34.927–07:48 · Generation, Recombination and Leakage](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=334.92724999999996) ## Transcript ### [00:00 · Two Diagrams, Two Questions](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=0) 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. ### [00:48.179 · The Arrhenius Plot](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=48.17902083333334) 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. ### [03:6.63 · The Shockley Diagram](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=186.629875) 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. ### [05:34.927 · Generation, Recombination and Leakage](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=334.92724999999996) 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. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M52X7J8PNG1Y697ARRPE5Z64/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Two Diagrams, Two Questions](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=0) Span: 00:00–00:48.179 (0s–48.17902083333334s). #### Objects - arr\_axes: an Axes (x\_range=(2.5, 5.0), y\_range=(-10.0, 4.0), x\_label='1/T') - arr\_label: a Tex \[text\] that says "Arrhenius: rate against $1/T$" - arr\_line: a FunctionPlot \[yellow\] drawn in arr\_axes (function=\) - heading: a Heading that says "Two Straight-Line Diagrams" - n\_line: a FunctionPlot \[blue\] drawn in sho\_axes (function=\) - p\_line: a FunctionPlot \[green\] drawn in sho\_axes (function=\) - point: a Point \[yellow\] drawn in sho\_axes (location=(0.547, 9.83)) - point\_2: a Point \[yellow\] drawn in sho\_axes (location=(0.547, 9.83)) - sho\_axes: an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_label='E\_F') - sho\_label: a Tex \[text\] that says "Shockley: concentration against $E\_F$" #### Beats ##### [00:00](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=0) Narration: 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. Board: Empty. Actions: - [00:00](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=0): heading is shown on the screen, written out. ##### [00:12.105](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=12.105) Narration: 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. Board: heading — a Heading that says "Two Straight-Line Diagrams" Actions: - [00:13.24](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=13.24): arr\_axes is shown on the screen, written out. - [00:13.24](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=13.24): arr\_label is shown on the screen, written out. - [00:21.8](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=21.799999999999997): arr\_line is shown on the screen, written out. ##### [00:24.3](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=24.3) Narration: 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. Board: arr\_label — a Tex \[text\] that says "Arrhenius: rate against $1/T$"; arr\_axes — an Axes (x\_range=(2.5, 5.0), y\_range=(-10.0, 4.0), x\_label='1/T'); heading — a Heading that says "Two Straight-Line Diagrams"; arr\_line — a FunctionPlot \[yellow\] drawn in arr\_axes (function=\) Actions: - [00:25.52](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=25.52): sho\_axes is shown on the screen, written out. - [00:25.52](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=25.52): sho\_label is shown on the screen, written out. - [00:28.32](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=28.32): n\_line is shown on the screen, written out. - [00:29.28](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=29.279999999999998): p\_line is shown on the screen, written out. - [00:34.4](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=34.39999999999999): point is shown on the screen, grown. - [00:36.591](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=36.59068095238095): point is hidden from the screen. ##### [00:37.038](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=37.0375) Narration: 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. Board: arr\_label — a Tex \[text\] that says "Arrhenius: rate against $1/T$"; arr\_axes — an Axes (x\_range=(2.5, 5.0), y\_range=(-10.0, 4.0), x\_label='1/T'); sho\_label — a Tex \[text\] that says "Shockley: concentration against $E\_F$"; sho\_axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_label='E\_F'); heading — a Heading that says "Two Straight-Line Diagrams"; arr\_line — a FunctionPlot \[yellow\] drawn in arr\_axes (function=\); n\_line — a FunctionPlot \[blue\] drawn in sho\_axes (function=\); p\_line — a FunctionPlot \[green\] drawn in sho\_axes (function=\) Actions: - [00:39.4](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=39.4): The segment (2.6, 2.581) to (4.9, -8.096) in arr\_axes is lit up. - [00:42.12](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=42.12): point\_2 is shown on the screen, grown. - [00:42.12](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=42.12): arr\_axes: retire a lit segment (unemphasize\_line). - [00:44.248](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=44.24841855352825): point\_2 is hidden from the screen. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): arr\_axes is hidden from the screen — left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): arr\_line is hidden from the screen — arr\_axes left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): arr\_label is hidden from the screen — left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): heading is hidden from the screen — left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): sho\_axes is hidden from the screen — left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): n\_line is hidden from the screen — sho\_axes left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): p\_line is hidden from the screen — sho\_axes left the board. - [00:47.137](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=47.13735416666667): sho\_label is hidden from the screen — left the board. ### Scene 2: [The Arrhenius Plot](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=48.17902083333334) Span: 00:48.179–03:6.63 (48.17902083333334s–186.629875s). #### Objects - axes: an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0) - caveat: a Text \[text\] that says "A curved Arrhenius plot means competing mechanisms or an uncorrected prefactor." - e0: a Math \[text\] that says "$e\_n = sigma\_n v\_upright("th") N\_c exp(-frac(E\_c - E\_t, k T))$" - e1: a Math \[text\] that says "$frac(e\_n, T^2) = sigma\_n gamma\_n exp(-frac(E\_c - E\_t, k T))$" - e2: a Math \[text\] that says "$ln frac(e\_n, T^2) = ln(sigma\_n gamma\_n) - frac(E\_c - E\_t, k) dot frac(1, T)$" - extension: a Line \[gray\] drawn in axes (start=(0.0, 14.65), end=(2.5, 3.045), dashed=True) - fit: a FunctionPlot \[yellow\] drawn in axes (function=\, x\_range=(2.5, 5.0)) - h1: a Heading that says "The Arrhenius Plot" - h2: a Heading that says "Reading Slope and Intercept" - h3: a Heading that says "The Same Line Everywhere" - intercept\_pt: a Point \[green\] labelled "ln(sigma\_n gamma\_n)" drawn in axes (location=(0.0, 14.65)) - m0: a Math \[text\] that says "$m = frac(Delta y, Delta x) approx -4.64 times 10^3 thin upright("K")$" - m1: a Math \[text\] that says "$E\_c - E\_t = -k m$" - m2: a Math \[text\] that says "$ln(sigma\_n gamma\_n) approx 14.7 arrow.r sigma\_n approx 10^(-15) thin upright("cm")^2$" - points: a Point \[red\] drawn in axes (location=(2.8, 1.9024)) - points\_2: a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)) - points\_3: a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)) - points\_4: a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)) - points\_5: a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)) - points\_6: a Point \[red\] drawn in axes (location=(4.8, -7.7316)) - secant: a SecantLine \[yellow\] drawn in axes (target='fit', x1=3.0, x2=4.5) - table: a Table \[text\] that says "Process Plotted Slope gives trap emission $ln(e\_n / T^2)$ $E\_c - E\_t$ dopant diffusion $ln D$ migration energy carrier freeze-out $ln n$ $E\_D / 2$ intrinsic regime $ln n\_i$ $E\_g / 2$" (rows=(('Process', 'Plotted', 'Slope gives'), ('trap emission', '$ln(…, header=True) #### Beats ##### [00:48.179](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=48.17902083333334) Narration: 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. Board: Empty. Actions: - [00:48.179](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=48.17902083333334): h1 is shown on the screen, written out. - [00:48.179](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=48.17902083333334): axes is shown on the screen, written out. - [00:58.739](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=58.73902083333334): points is shown on the screen, written out. - [00:58.894](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=58.89428174009125): points\_2 is shown on the screen, written out. - [00:59.042](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=59.041706889791854): points\_3 is shown on the screen, written out. - [00:59.18](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=59.179716584687775): points\_4 is shown on the screen, written out. - [00:59.327](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=59.32707900270887): points\_5 is shown on the screen, written out. - [00:59.534](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=59.534093545052755): points\_6 is shown on the screen, written out. ##### [01:3.317](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=63.31652083333334) Narration: 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. Board: axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); h1 — a Heading that says "The Arrhenius Plot"; points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)) Actions: - [01:5.819](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=65.81902083333334): axes moves to a new place on the board. - [01:5.819](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=65.81902083333334): e0 is shown on the screen, written out. - [01:7.899](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=67.89902083333334): e0 (the "sigma\_n" part) is emphasized. - [01:9.979](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=69.97902083333334): e0 (the "sigma\_n" part) is no longer emphasized. - [01:9.979](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=69.97902083333334): e0 (the "v\_upright("th")" part) is emphasized. - [01:11.659](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=71.65902083333334): e0 (the "N\_c" part) is emphasized. - [01:11.659](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=71.65902083333334): e0 (the "v\_upright("th")" part) is no longer emphasized. - [01:13.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=73.41902083333335): e0 (the "N\_c" part) is no longer emphasized. - [01:13.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=73.41902083333335): e0 (the "exp(-frac(E\_c - E\_t, k T))" part) is emphasized. - [01:15.382](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=75.38152083333334): e0 (the "exp(-frac(E\_c - E\_t, k T))" part) is no longer emphasized. ##### [01:15.982](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=75.98152083333333) Narration: 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. Board: e0 — a Math \[text\] that says "$e\_n = sigma\_n v\_upright("th") N\_c exp(-frac(E\_c - E\_t, k T))$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); h1 — a Heading that says "The Arrhenius Plot"; points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)) Actions: - [01:19.779](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=79.77902083333333): e0 (the "v\_upright("th")" part) is indicated — a transient flash. - [01:21.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=81.41902083333333): e0 (the "N\_c" part) is indicated — a transient flash. - [01:26.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=86.41902083333333): e1 is shown on the screen, written out. ##### [01:30.372](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=90.37152083333334) Narration: 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. Board: e0 — a Math \[text\] that says "$e\_n = sigma\_n v\_upright("th") N\_c exp(-frac(E\_c - E\_t, k T))$"; e1 — a Math \[text\] that says "$frac(e\_n, T^2) = sigma\_n gamma\_n exp(-frac(E\_c - E\_t, k T))$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); h1 — a Heading that says "The Arrhenius Plot"; points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)) Actions: - [01:31.739](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=91.73902083333334): e2 is shown on the screen, written out. - [01:38.819](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=98.81902083333333): e2 (the "frac(E\_c - E\_t, k)" part) is emphasized. - [01:41.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=101.41902083333332): e2 (the "frac(E\_c - E\_t, k)" part) is no longer emphasized. - [01:41.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=101.41902083333332): e2 (the "ln(sigma\_n gamma\_n)" part) is emphasized. ##### [01:42.767](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=102.76652083333333) Narration: And the points obey it: one straight line runs through them all, the first sign that a single activated process is at work. Board: e0 — a Math \[text\] that says "$e\_n = sigma\_n v\_upright("th") N\_c exp(-frac(E\_c - E\_t, k T))$"; e1 — a Math \[text\] that says "$frac(e\_n, T^2) = sigma\_n gamma\_n exp(-frac(E\_c - E\_t, k T))$"; e2 — a Math \[text\] that says "$ln frac(e\_n, T^2) = ln(sigma\_n gamma\_n) - frac(E\_c - E\_t, k) dot frac(1, T)$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); h1 — a Heading that says "The Arrhenius Plot"; points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)) Actions: - [01:42.767](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=102.76652083333333): e2 (the "ln(sigma\_n gamma\_n)" part) is no longer emphasized. - [01:45.299](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=105.29902083333332): fit is shown on the screen, written out. - [01:50.304](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=110.30402083333334): e2 moves to a new place on the board. - [01:50.304](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=110.30402083333334): e0 is hidden from the screen — left the board. - [01:50.304](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=110.30402083333334): e1 is hidden from the screen — left the board. - [01:50.304](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=110.30402083333334): h1 is hidden from the screen — left the board. ##### [01:50.904](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=110.90402083333333) Narration: 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. Board: e2 — a Math \[text\] that says "$ln frac(e\_n, T^2) = ln(sigma\_n gamma\_n) - frac(E\_c - E\_t, k) dot frac(1, T)$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)); fit — a FunctionPlot \[yellow\] drawn in axes (function=\, x\_range=(2.5, 5.0)) Actions: - [01:50.904](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=110.90402083333333): h2 is shown on the screen, written out. - [01:52.139](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=112.13902083333333): secant is shown on the screen, written out. - [01:56.139](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=116.13902083333332): m0 is shown on the screen, written out. ##### [02:0.159](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=120.15902083333333) Narration: Multiply by minus Boltzmann's constant, and the trap depth comes out: zero point four zero electron volts below the conduction band. Board: e2 — a Math \[text\] that says "$ln frac(e\_n, T^2) = ln(sigma\_n gamma\_n) - frac(E\_c - E\_t, k) dot frac(1, T)$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)); fit — a FunctionPlot \[yellow\] drawn in axes (function=\, x\_range=(2.5, 5.0)); m0 — a Math \[text\] that says "$m = frac(Delta y, Delta x) approx -4.64 times 10^3 thin upright("K")$"; h2 — a Heading that says "Reading Slope and Intercept"; secant — a SecantLine \[yellow\] drawn in axes (target='fit', x1=3.0, x2=4.5) Actions: - [02:0.419](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=120.41902083333333): m1 is shown on the screen, written out. - [02:4.579](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=124.57902083333333): m1 becomes "$E\_c - E\_t = -k m approx 0.40 thin upright("eV")$". - [02:7.779](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=127.77902083333333): A box is drawn around m1. ##### [02:8.909](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=128.90902083333333) Narration: 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. Board: e2 — a Math \[text\] that says "$ln frac(e\_n, T^2) = ln(sigma\_n gamma\_n) - frac(E\_c - E\_t, k) dot frac(1, T)$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)); fit — a FunctionPlot \[yellow\] drawn in axes (function=\, x\_range=(2.5, 5.0)); m0 — a Math \[text\] that says "$m = frac(Delta y, Delta x) approx -4.64 times 10^3 thin upright("K")$"; m1 — a Math \[text\] that says "$E\_c - E\_t = -k m$"; h2 — a Heading that says "Reading Slope and Intercept"; secant — a SecantLine \[yellow\] drawn in axes (target='fit', x1=3.0, x2=4.5) Actions: - [02:9.499](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=129.4990208333333): extension is shown on the screen, written out. - [02:14.579](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=134.57902083333335): intercept\_pt is shown on the screen, written out. - [02:17.299](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=137.29902083333332): m2 is shown on the screen, written out. ##### [02:23.774](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=143.77402083333334) Narration: 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. Board: e2 — a Math \[text\] that says "$ln frac(e\_n, T^2) = ln(sigma\_n gamma\_n) - frac(E\_c - E\_t, k) dot frac(1, T)$"; axes — an Axes (x\_range=(0.0, 5.0), y\_range=(-10.0, 16.0), x\_ticks\_every=1.0); points — a Point \[red\] drawn in axes (location=(2.8, 1.9024)); points\_2 — a Point \[red\] drawn in axes (location=(3.2, -0.4044000000000015)); points\_3 — a Point \[red\] drawn in axes (location=(3.6, -1.9112000000000013)); points\_4 — a Point \[red\] drawn in axes (location=(4.0, -4.168000000000001)); points\_5 — a Point \[red\] drawn in axes (location=(4.4, -5.574800000000004)); points\_6 — a Point \[red\] drawn in axes (location=(4.8, -7.7316)); fit — a FunctionPlot \[yellow\] drawn in axes (function=\, x\_range=(2.5, 5.0)); m0 — a Math \[text\] that says "$m = frac(Delta y, Delta x) approx -4.64 times 10^3 thin upright("K")$"; m1 — a Math \[text\] that says "$E\_c - E\_t = -k m$"; m2 — a Math \[text\] that says "$ln(sigma\_n gamma\_n) approx 14.7 arrow.r sigma\_n approx 10^(-15) thin upright("cm")^2$"; h2 — a Heading that says "Reading Slope and Intercept"; secant — a SecantLine \[yellow\] drawn in axes (target='fit', x1=3.0, x2=4.5); extension — a Line \[gray\] drawn in axes (start=(0.0, 14.65), end=(2.5, 3.045), dashed=True); intercept\_pt — a Point \[green\] labelled "ln(sigma\_n gamma\_n)" drawn in axes (location=(0.0, 14.65)) Actions: - [02:26.739](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=146.7390208333333): intercept\_pt is indicated — a transient flash. - [02:31.979](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=151.97902083333332): m0 is indicated — a transient flash. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): points is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): points\_2 is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): points\_3 is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): points\_4 is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): points\_5 is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): points\_6 is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): secant is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): fit is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): extension is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): intercept\_pt is hidden from the screen. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): axes is hidden from the screen — left the board. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): e2 is hidden from the screen — left the board. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): h2 is hidden from the screen — left the board. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): m0 is hidden from the screen — left the board. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): m1 is hidden from the screen — left the board. - [02:32.972](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=152.97152083333333): m2 is hidden from the screen — left the board. ##### [02:34.172](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=154.17152083333332) Narration: Nothing here is special to traps. Any process with a Boltzmann factor gives the same straight line; only the axis label changes. Board: Empty. Actions: - [02:34.172](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=154.17152083333332): h3 is shown on the screen, written out. - [02:34.172](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=154.17152083333332): table is shown on the screen, written out. - [02:35.619](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=155.6190208333333): table is shown on the screen, written out. ##### [02:42.387](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=162.38652083333332) Narration: 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. Board: h3 — a Heading that says "The Same Line Everywhere" Actions: - [02:42.619](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=162.6190208333333): table is shown on the screen, written out. - [02:46.939](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=166.9390208333333): table is shown on the screen, written out. - [02:52.779](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=172.77902083333333): table is shown on the screen, written out. ##### [02:56.397](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=176.3965208333333) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [02:56.899](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=176.8990208333333): caveat is shown on the screen, written out. - [03:5.588](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=185.5882083333333): caveat is hidden from the screen — left the board. - [03:5.588](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=185.5882083333333): h3 is hidden from the screen — left the board. - [03:5.588](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=185.5882083333333): table is hidden from the screen — left the board. ### Scene 3: [The Shockley Diagram](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=186.629875) Span: 03:6.63–05:34.927 (186.629875s–334.92724999999996s). #### Objects - answer: a Math \[text\] that says "$E\_c - E\_F approx 0.21 thin upright("eV")$" - axes: an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2) - hA: a Heading that says "The Shockley Diagram" - hB: a Heading that says "A Defect Level on the Diagram" - line: a Line \[yellow\] drawn in axes (start=(0.915, 16.0), end=(0.0, 16.0), dashed=True) - line\_2: a Line \[yellow\] drawn in axes (start=(0.915, 16.0), end=(0.915, 0.0), dashed=True) - n1\_dot: a PlotPoint \[yellow\] labelled "n\_1" drawn in axes (target='n\_plot', x=0.72) - n\_dot: a PlotPoint \[blue\] drawn in axes (target='n\_plot', x=\) - n\_hot: a FunctionPlot \[gray\] drawn in axes (function=\) - n\_plot: a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\) - nd\_dot: a PlotPoint \[red\] labelled "0.55" drawn in axes (target='nd\_plot', x=\) - nd\_plot: a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\) - p1\_dot: a PlotPoint \[yellow\] labelled "p\_1" drawn in axes (target='p\_plot', x=0.72) - p\_hot: a FunctionPlot \[gray\] drawn in axes (function=\) - p\_plot: a FunctionPlot \[green\] labelled "p" drawn in axes (function=\) - point: a Point \[yellow\] drawn in axes (location=(0.547, 9.83)) - point\_2: a Point \[yellow\] drawn in axes (location=(0.915, 16.0)) - point\_3: a Point \[yellow\] drawn in axes (location=(0.915, 3.65)) - point\_4: a Point \[yellow\] drawn in axes (location=(0.539, 13.9)) - point\_5: a Point \[yellow\] drawn in axes (location=(0.745, 16.0)) - probe\_line: a Line \[gray\] drawn in axes (start=(\, 0.0), end=(\, 20.0), dashed=True) - s0: a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$" - s1: a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$" - s2: a Math \[text\] that says "$N\_D^+ = frac(N\_D, 1 + 2 exp(frac(E\_F - E\_D, k T)))$" - s3: a Math \[text\] that says "$n = p + N\_D^+$" - t0: a Math \[text\] that says "$f\_t = frac(1, 1 + exp(frac(E\_t - E\_F, k T)))$" - t1: a Math \[text\] that says "$n\_1 = N\_c exp(-frac(E\_c - E\_t, k T))$" - t2: a Math \[text\] that says "$p\_1 = N\_v exp(-frac(E\_t - E\_v, k T))$" - trap: a Line \[yellow\] labelled "E\_t" drawn in axes (start=(0.72, 0.0), end=(0.72, 20.0), dashed=True) - trial: a VariableNumber (initial\_value=0.55, format\_spec='.2f') #### Beats ##### [03:6.63](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=186.629875) Narration: 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. Board: Empty. Actions: - [03:6.63](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=186.629875): hA is shown on the screen, written out. - [03:6.63](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=186.629875): axes is shown on the screen, written out. ##### [03:19.675](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=199.674875) Narration: 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. Board: axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram" Actions: - [03:19.95](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=199.949875): axes moves to a new place on the board. - [03:19.95](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=199.949875): s0 is shown on the screen, written out. - [03:27.63](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=207.629875): n\_plot is shown on the screen, written out. ##### [03:30.867](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=210.86737499999998) Narration: 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. Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\) Actions: - [03:31.11](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=211.109875): s1 is shown on the screen, written out. - [03:31.91](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=211.909875): p\_plot is shown on the screen, written out. - [03:35.99](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=215.98987499999998): point is shown on the screen, grown. - [03:38.736](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=218.73599151745165): point is hidden from the screen. ##### [03:40.905](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=220.904875) Narration: 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. Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; s1 — a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\) Actions: - [03:41.79](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=221.789875): s2 is shown on the screen, written out. - [03:52.03](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=232.029875): nd\_plot is shown on the screen, written out. ##### [03:53.562](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=233.562375) Narration: Neutrality demands that negative charge balance positive charge: n equals p plus the ionized donors. So where must the Fermi level sit? Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; s1 — a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$"; s2 — a Math \[text\] that says "$N\_D^+ = frac(N\_D, 1 + 2 exp(frac(E\_F - E\_D, k T)))$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\) Actions: - [03:53.83](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=233.82987500000002): s3 is shown on the screen, written out. ##### [04:3.175](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=243.174875) Narration: 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. Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; s1 — a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$"; s2 — a Math \[text\] that says "$N\_D^+ = frac(N\_D, 1 + 2 exp(frac(E\_F - E\_D, k T)))$"; s3 — a Math \[text\] that says "$n = p + N\_D^+$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\) Actions: - [04:3.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=243.469875): probe\_line is shown on the screen, written out. - [04:3.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=243.469875): nd\_dot is shown on the screen, written out. - [04:3.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=243.469875): n\_dot is shown on the screen, written out. - [04:6.19](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=246.18987499999997): The segment (0.55, 9.87) to (0.55, 16.0) in axes is lit up. - [04:10.83](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=250.829875): axes: retire a lit segment (unemphasize\_line). ##### [04:12.19](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=252.18987499999997) Narration: Try one electron volt. Now electrons outnumber donors, too much negative charge, so it must come back down. Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; s1 — a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$"; s2 — a Math \[text\] that says "$N\_D^+ = frac(N\_D, 1 + 2 exp(frac(E\_F - E\_D, k T)))$"; s3 — a Math \[text\] that says "$n = p + N\_D^+$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\); probe\_line — a Line \[gray\] drawn in axes (start=(\, 0.0), end=(\, 20.0), dashed=True); nd\_dot — a PlotPoint \[red\] labelled "0.55" drawn in axes (target='nd\_plot', x=\); n\_dot — a PlotPoint \[blue\] drawn in axes (target='n\_plot', x=\) Actions: - [04:12.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=252.46987499999997): probe\_line is redrawn as the numbers it depends on change. - [04:12.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=252.46987499999997): nd\_dot is redrawn as the numbers it depends on change. - [04:12.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=252.46987499999997): n\_dot is redrawn as the numbers it depends on change. - [04:12.47](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=252.46987499999997): trial ticks to 1.0. ##### [04:20.217](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=260.217375) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [04:21.87](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=261.869875): probe\_line is redrawn as the numbers it depends on change. - [04:21.87](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=261.869875): nd\_dot is redrawn as the numbers it depends on change. - [04:21.87](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=261.869875): n\_dot is redrawn as the numbers it depends on change. - [04:21.87](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=261.869875): trial ticks to 0.915. - [04:22.99](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=262.989875): answer is shown on the screen, written out. - [04:26.91](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=266.909875): point\_2 is shown on the screen, grown. - [04:26.91](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=266.909875): line is shown on the screen, drawn. - [04:26.91](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=266.909875): line\_2 is shown on the screen, drawn. - [04:28.785](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=268.78493301080505): point\_2 is hidden from the screen. - [04:28.785](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=268.78493301080505): line is hidden from the screen. - [04:28.785](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=268.78493301080505): line\_2 is hidden from the screen. ##### [04:30.985](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=270.984875) Narration: And the hole line, down here, is twelve orders of magnitude lower. That is how the diagram tells you which terms may be dropped. Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; s1 — a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$"; s2 — a Math \[text\] that says "$N\_D^+ = frac(N\_D, 1 + 2 exp(frac(E\_F - E\_D, k T)))$"; s3 — a Math \[text\] that says "$n = p + N\_D^+$"; answer — a Math \[text\] that says "$E\_c - E\_F approx 0.21 thin upright("eV")$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\); probe\_line — a Line \[gray\] drawn in axes (start=(\, 0.0), end=(\, 20.0), dashed=True); nd\_dot — a PlotPoint \[red\] labelled "0.55" drawn in axes (target='nd\_plot', x=\); n\_dot — a PlotPoint \[blue\] drawn in axes (target='n\_plot', x=\) Actions: - [04:31.51](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=271.50987499999997): point\_3 is shown on the screen, grown. - [04:34.011](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=274.01066172732135): point\_3 is hidden from the screen. - [04:38.975](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=278.974875): probe\_line is hidden from the screen. - [04:38.975](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=278.974875): n\_dot is hidden from the screen. - [04:38.975](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=278.974875): nd\_dot is hidden from the screen. ##### [04:39.575](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=279.574875) Narration: 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. Board: s0 — a Math \[text\] that says "$n = N\_c exp(-frac(E\_c - E\_F, k T))$"; s1 — a Math \[text\] that says "$p = N\_v exp(-frac(E\_F - E\_v, k T))$"; s2 — a Math \[text\] that says "$N\_D^+ = frac(N\_D, 1 + 2 exp(frac(E\_F - E\_D, k T)))$"; s3 — a Math \[text\] that says "$n = p + N\_D^+$"; answer — a Math \[text\] that says "$E\_c - E\_F approx 0.21 thin upright("eV")$"; axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); hA — a Heading that says "The Shockley Diagram"; n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\) Actions: - [04:39.87](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=279.869875): n\_hot is shown on the screen, written out. - [04:39.87](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=279.869875): p\_hot is shown on the screen, written out. - [04:46.83](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=286.829875): point\_4 is shown on the screen, grown. - [04:48.51](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=288.50987499999997): point\_5 is shown on the screen, grown. - [04:48.856](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=288.8562655805928): point\_4 is hidden from the screen. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): n\_hot is hidden from the screen. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): p\_hot is hidden from the screen. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): answer is hidden from the screen — left the board. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): hA is hidden from the screen — left the board. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): s0 is hidden from the screen — left the board. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): s1 is hidden from the screen — left the board. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): s2 is hidden from the screen — left the board. - [04:50.197](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.19737499999997): s3 is hidden from the screen — left the board. - [04:50.872](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=290.8724102941176): point\_5 is hidden from the screen. ##### [04:51.397](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=291.397375) Narration: 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. Board: axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\) Actions: - [04:51.397](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=291.397375): hB is shown on the screen, written out. - [04:57.71](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=297.709875): trap is shown on the screen, written out. - [05:0.91](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=300.909875): t0 is shown on the screen, written out. ##### [05:4.667](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=304.667375) Narration: 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. Board: axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\); t0 — a Math \[text\] that says "$f\_t = frac(1, 1 + exp(frac(E\_t - E\_F, k T)))$"; hB — a Heading that says "A Defect Level on the Diagram"; trap — a Line \[yellow\] labelled "E\_t" drawn in axes (start=(0.72, 0.0), end=(0.72, 20.0), dashed=True) Actions: - [05:8.15](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=308.149875): The segment (0.72, 16.0) to (0.915, 16.0) in axes is lit up. - [05:9.43](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=309.429875): t0 (the "exp(frac(E\_t - E\_F, k T))" part) is emphasized. - [05:12.63](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=312.62987499999997): t0 (the "exp(frac(E\_t - E\_F, k T))" part) is no longer emphasized. - [05:12.63](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=312.62987499999997): axes: retire a lit segment (unemphasize\_line). ##### [05:13.61](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=313.609875) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [05:15.55](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=315.549875): n1\_dot is shown on the screen, written out. - [05:17.07](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=317.069875): t1 is shown on the screen, written out. - [05:22.43](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=322.429875): p1\_dot is shown on the screen, written out. - [05:23.55](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=323.549875): t2 is shown on the screen, written out. ##### [05:24.817](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=324.81737499999997) Narration: 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. Board: axes — an Axes (x\_range=(0.0, 1.12), y\_range=(0.0, 20.0), x\_ticks\_every=0.2); n\_plot — a FunctionPlot \[blue\] labelled "n" drawn in axes (function=\); p\_plot — a FunctionPlot \[green\] labelled "p" drawn in axes (function=\); nd\_plot — a FunctionPlot \[red\] labelled "N\_D^+" drawn in axes (function=\); t0 — a Math \[text\] that says "$f\_t = frac(1, 1 + exp(frac(E\_t - E\_F, k T)))$"; t1 — a Math \[text\] that says "$n\_1 = N\_c exp(-frac(E\_c - E\_t, k T))$"; t2 — a Math \[text\] that says "$p\_1 = N\_v exp(-frac(E\_t - E\_v, k T))$"; hB — a Heading that says "A Defect Level on the Diagram"; trap — a Line \[yellow\] labelled "E\_t" drawn in axes (start=(0.72, 0.0), end=(0.72, 20.0), dashed=True); n1\_dot — a PlotPoint \[yellow\] labelled "n\_1" drawn in axes (target='n\_plot', x=0.72); p1\_dot — a PlotPoint \[yellow\] labelled "p\_1" drawn in axes (target='p\_plot', x=0.72) Actions: - [05:27.99](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=327.989875): t1 is indicated — a transient flash. - [05:29.75](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=329.74987500000003): t2 is indicated — a transient flash. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): axes is hidden from the screen — left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): n\_plot is hidden from the screen — axes left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): p\_plot is hidden from the screen — axes left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): nd\_plot is hidden from the screen — axes left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): trap is hidden from the screen — axes left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): n1\_dot is hidden from the screen — axes left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): p1\_dot is hidden from the screen — axes left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): hB is hidden from the screen — left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): t0 is hidden from the screen — left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): t1 is hidden from the screen — left the board. - [05:33.886](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=333.88558333333333): t2 is hidden from the screen — left the board. ### Scene 4: [Generation, Recombination and Leakage](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=334.92724999999996) Span: 05:34.927–07:48.128 (334.92724999999996s–468.1283125s). #### Objects - dot: a PlotPoint \[yellow\] labelled "0.16" drawn in gaxes (target='gcurve') - gaxes: an Axes (x\_range=(-0.5, 0.5), y\_range=(-9.0, 0.5), x\_ticks\_every=0.25) - gcurve: a FunctionPlot \[red\] drawn in gaxes (function=\) - hA: a Heading that says "Shockley, Read and Hall" - hB: a Heading that says "Generation Against Trap Position" - hC: a Heading that says "Leakage Read Two Ways" - hD: a Heading that says "What Each Diagram Gives" - j0: a Math \[text\] that says "$J\_upright("gen") prop n\_i prop T^(3/2) exp(-frac(E\_g, 2 k T))$" - j1: a Math \[text\] that says "$J\_upright("diff") prop n\_i^2 prop T^3 exp(-frac(E\_g, k T))$" - probe: a VariableNumber (initial\_value=0.16, format\_spec='.2f') - sign\_note: a Text \[text\] that says "$U \> 0$: net recombination. $U \< 0$: net generation." - summary: a Block \[text\] that says "Arrhenius plot: $ln$ rate against $1/T$; slope $-E\_a / k$, intercept gives the prefactor. Shockley diagram: $log$ concentration against $E\_F$; the neutrality crossing fixes $E\_F$. A trap at $E\_t$ sets $n\_1$ and $p\_1$; generation peaks when…" - u0: a Math \[text\] that says "$U = frac(n p - n\_i^2, tau\_p (n + n\_1) + tau\_n (p + p\_1))$" - u1: a Math \[text\] that says "$G = frac(n\_i^2, tau\_p n\_1 + tau\_n p\_1)$" - u2: a Math \[text\] that says "$G = frac(n\_i, 2 tau cosh(frac(E\_t - E\_i, k T)))$" - verdict: a Text \[text\] that says "Slope near $E\_g / 2$: midgap generation. Slope near $E\_g$: diffusion." #### Beats ##### [05:34.927](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=334.92724999999996) Narration: 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. Board: Empty. Actions: - [05:34.927](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=334.92724999999996): hA is shown on the screen, written out. - [05:36.607](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=336.60724999999996): u0 is shown on the screen, written out. - [05:40.687](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=340.68724999999995): u0 (the "n p - n\_i^2" part) is emphasized. - [05:46.687](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=346.68724999999995): u0 (the "n p - n\_i^2" part) is no longer emphasized. - [05:46.687](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=346.68724999999995): u0 (the "n\_1" part) is emphasized. - [05:47.407](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=347.40725): u0 (the "n\_1" part) is no longer emphasized. - [05:47.407](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=347.40725): u0 (the "p\_1" part) is emphasized. - [05:48.795](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=348.79474999999996): u0 (the "p\_1" part) is no longer emphasized. ##### [05:49.395](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=349.39474999999993) Narration: 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. Board: u0 — a Math \[text\] that says "$U = frac(n p - n\_i^2, tau\_p (n + n\_1) + tau\_n (p + p\_1))$"; hA — a Heading that says "Shockley, Read and Hall" Actions: - [05:49.927](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=349.92724999999996): sign\_note is shown on the screen, written out. - [06:3.23](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=363.22974999999997): u0 moves to a new place on the board. - [06:3.23](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=363.22974999999997): hA is hidden from the screen — left the board. - [06:3.23](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=363.22974999999997): sign\_note is hidden from the screen — left the board. ##### [06:3.83](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=363.82974999999993) Narration: 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. Board: u0 — a Math \[text\] that says "$U = frac(n p - n\_i^2, tau\_p (n + n\_1) + tau\_n (p + p\_1))$" Actions: - [06:3.83](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=363.82974999999993): hB is shown on the screen, written out. - [06:3.83](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=363.82974999999993): gaxes is shown on the screen, written out. - [06:6.287](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=366.28725): u1 is shown on the screen, written out. - [06:15.407](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=375.40725): u2 is shown on the screen, written out. ##### [06:17.617](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=377.61724999999996) Narration: 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. Board: u0 — a Math \[text\] that says "$U = frac(n p - n\_i^2, tau\_p (n + n\_1) + tau\_n (p + p\_1))$"; u1 — a Math \[text\] that says "$G = frac(n\_i^2, tau\_p n\_1 + tau\_n p\_1)$"; u2 — a Math \[text\] that says "$G = frac(n\_i, 2 tau cosh(frac(E\_t - E\_i, k T)))$"; gaxes — an Axes (x\_range=(-0.5, 0.5), y\_range=(-9.0, 0.5), x\_ticks\_every=0.25); hB — a Heading that says "Generation Against Trap Position" Actions: - [06:18.447](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=378.44724999999994): gcurve is shown on the screen, written out. - [06:24.327](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=384.32724999999994): The segment (0.15, -2.22) to (0.45, -7.26) in gaxes is lit up. - [06:28.522](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=388.52225): gaxes: retire a lit segment (unemphasize\_line). ##### [06:29.122](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=389.12224999999995) Narration: 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. Board: u0 — a Math \[text\] that says "$U = frac(n p - n\_i^2, tau\_p (n + n\_1) + tau\_n (p + p\_1))$"; u1 — a Math \[text\] that says "$G = frac(n\_i^2, tau\_p n\_1 + tau\_n p\_1)$"; u2 — a Math \[text\] that says "$G = frac(n\_i, 2 tau cosh(frac(E\_t - E\_i, k T)))$"; gaxes — an Axes (x\_range=(-0.5, 0.5), y\_range=(-9.0, 0.5), x\_ticks\_every=0.25); hB — a Heading that says "Generation Against Trap Position"; gcurve — a FunctionPlot \[red\] drawn in gaxes (function=\) Actions: - [06:30.047](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=390.04724999999996): dot is shown on the screen, written out. ##### [06:38.437](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=398.43724999999995) Narration: 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. Board: u0 — a Math \[text\] that says "$U = frac(n p - n\_i^2, tau\_p (n + n\_1) + tau\_n (p + p\_1))$"; u1 — a Math \[text\] that says "$G = frac(n\_i^2, tau\_p n\_1 + tau\_n p\_1)$"; u2 — a Math \[text\] that says "$G = frac(n\_i, 2 tau cosh(frac(E\_t - E\_i, k T)))$"; gaxes — an Axes (x\_range=(-0.5, 0.5), y\_range=(-9.0, 0.5), x\_ticks\_every=0.25); hB — a Heading that says "Generation Against Trap Position"; gcurve — a FunctionPlot \[red\] drawn in gaxes (function=\); dot — a PlotPoint \[yellow\] labelled "0.16" drawn in gaxes (target='gcurve') Actions: - [06:38.727](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=398.72724999999997): dot is redrawn as the numbers it depends on change. - [06:38.727](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=398.72724999999997): probe ticks to 0.4. - [06:43.447](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=403.44724999999994): dot is redrawn as the numbers it depends on change. - [06:43.447](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=403.44724999999994): probe ticks to 0.0. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): gaxes is hidden from the screen — left the board. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): gcurve is hidden from the screen — gaxes left the board. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): dot is hidden from the screen — gaxes left the board. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): hB is hidden from the screen — left the board. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): u0 is hidden from the screen — left the board. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): u1 is hidden from the screen — left the board. - [06:50.922](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=410.92224999999996): u2 is hidden from the screen — left the board. ##### [06:52.122](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=412.12224999999995) Narration: 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. Board: Empty. Actions: - [06:52.122](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=412.12224999999995): hC is shown on the screen, written out. - [06:55.447](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=415.44724999999994): j0 is shown on the screen, written out. - [07:1.607](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=421.60724999999996): j0 (the "frac(E\_g, 2 k T)" part) is emphasized. ##### [07:3.457](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=423.45724999999993) Narration: 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. Board: j0 — a Math \[text\] that says "$J\_upright("gen") prop n\_i prop T^(3/2) exp(-frac(E\_g, 2 k T))$"; hC — a Heading that says "Leakage Read Two Ways" Actions: - [07:3.457](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=423.45724999999993): j0 (the "frac(E\_g, 2 k T)" part) is no longer emphasized. - [07:3.727](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=423.7272499999999): j1 is shown on the screen, written out. - [07:7.887](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=427.88724999999994): j1 (the "frac(E\_g, k T)" part) is emphasized. - [07:9.007](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=429.00724999999994): verdict is shown on the screen, written out. - [07:9.007](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=429.00724999999994): j1 (the "frac(E\_g, k T)" part) is no longer emphasized. - [07:14.377](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=434.37724999999995): hC is hidden from the screen — left the board. - [07:14.377](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=434.37724999999995): j0 is hidden from the screen — left the board. - [07:14.377](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=434.37724999999995): j1 is hidden from the screen — left the board. - [07:14.377](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=434.37724999999995): verdict is hidden from the screen — left the board. ##### [07:14.977](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=434.9772499999999) Narration: 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. Board: Empty. Actions: - [07:14.977](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=434.9772499999999): hD is shown on the screen, written out. - [07:15.927](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=435.92724999999996): summary is shown on the screen, written out. - [07:19.207](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=439.20724999999993): summary (the "Arrhenius plot" part) is emphasized. ##### [07:26.457](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=446.45724999999993) Narration: 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. Board: summary — a Block \[text\] that says "Arrhenius plot: $ln$ rate against $1/T$; slope $-E\_a / k$, intercept gives the prefactor. Shockley diagram: $log$ concentration against $E\_F$; the neutrality crossing fixes $E\_F$. A trap at $E\_t$ sets $n\_1$ and $p\_1$; generation peaks when…"; hD — a Heading that says "What Each Diagram Gives" Actions: - [07:26.847](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=446.84725): summary (the "Arrhenius plot" part) is no longer emphasized. - [07:26.847](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=446.84725): summary (the "Shockley diagram" part) is emphasized. ##### [07:35.45](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=455.44974999999994) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [07:36.727](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=456.72724999999997): summary (the "Shockley diagram" part) is no longer emphasized. - [07:36.727](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=456.72724999999997): summary (the "generation peaks" part) is emphasized. - [07:46.837](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=466.83664583333325): summary (the "generation peaks" part) is no longer emphasized. - [07:47.087](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=467.08664583333325): hD is hidden from the screen — left the board. - [07:47.087](https://academa.ai/@mapaor/lectures/shockley-and-arrhenius-diagrams-in-semiconductor-physics?t=467.08664583333325): summary is hidden from the screen — left the board.