{"version":1,"lectureId":"01M52X7J8PNG1Y697ARRPE5Z64","attempt":0,"publication":{"slug":"shockley-and-arrhenius-diagrams-in-semiconductor-physics","title":"Shockley and Arrhenius Diagrams in Semiconductor Physics","subject":"physics","summary":"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.","metaDescription":"Arrhenius plots and Shockley diagrams for semiconductors: activation energies from slopes, Fermi levels from neutrality, and midgap trap generation.","transcript":"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.","watch":{"version":1,"scenes":[{"title":"Two Diagrams, Two Questions","start":0,"end":48.17902083333334,"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=<function>)","heading":"a Heading that says \"Two Straight-Line Diagrams\"","n_line":"a FunctionPlot [blue] drawn in sho_axes (function=<function>)","p_line":"a FunctionPlot [green] drawn in sho_axes (function=<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":[{"start":0,"say":"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.","live":[],"does":[[0,"heading is shown on the screen, written out."]]},{"start":12.105,"say":"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.","live":["heading"],"does":[[13.24,"arr_axes is shown on the screen, written out."],[13.24,"arr_label is shown on the screen, written out."],[21.799999999999997,"arr_line is shown on the screen, written out."]]},{"start":24.3,"say":"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.","live":["arr_label","arr_axes","heading","arr_line"],"does":[[25.52,"sho_axes is shown on the screen, written out."],[25.52,"sho_label is shown on the screen, written out."],[28.32,"n_line is shown on the screen, written out."],[29.279999999999998,"p_line is shown on the screen, written out."],[34.39999999999999,"point is shown on the screen, grown."],[36.59068095238095,"point is hidden from the screen."]]},{"start":37.0375,"say":"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.","live":["arr_label","arr_axes","sho_label","sho_axes","heading","arr_line","n_line","p_line"],"does":[[39.4,"The segment (2.6, 2.581) to (4.9, -8.096) in arr_axes is lit up."],[42.12,"point_2 is shown on the screen, grown."],[42.12,"arr_axes: retire a lit segment (unemphasize_line)."],[44.24841855352825,"point_2 is hidden from the screen."],[47.13735416666667,"arr_axes is hidden from the screen — left the board."],[47.13735416666667,"arr_line is hidden from the screen — arr_axes left the board."],[47.13735416666667,"arr_label is hidden from the screen — left the board."],[47.13735416666667,"heading is hidden from the screen — left the board."],[47.13735416666667,"sho_axes is hidden from the screen — left the board."],[47.13735416666667,"n_line is hidden from the screen — sho_axes left the board."],[47.13735416666667,"p_line is hidden from the screen — sho_axes left the board."],[47.13735416666667,"sho_label is hidden from the screen — left the board."]]}]},{"title":"The Arrhenius Plot","start":48.17902083333334,"end":186.629875,"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=<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":[{"start":48.17902083333334,"say":"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.","live":[],"does":[[48.17902083333334,"h1 is shown on the screen, written out."],[48.17902083333334,"axes is shown on the screen, written out."],[58.73902083333334,"points is shown on the screen, written out."],[58.89428174009125,"points_2 is shown on the screen, written out."],[59.041706889791854,"points_3 is shown on the screen, written out."],[59.179716584687775,"points_4 is shown on the screen, written out."],[59.32707900270887,"points_5 is shown on the screen, written out."],[59.534093545052755,"points_6 is shown on the screen, written out."]]},{"start":63.31652083333334,"say":"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.","live":["axes","h1","points","points_2","points_3","points_4","points_5","points_6"],"does":[[65.81902083333334,"axes moves to a new place on the board."],[65.81902083333334,"e0 is shown on the screen, written out."],[67.89902083333334,"e0 (the \"sigma_n\" part) is emphasized."],[69.97902083333334,"e0 (the \"sigma_n\" part) is no longer emphasized."],[69.97902083333334,"e0 (the \"v_upright(\"th\")\" part) is emphasized."],[71.65902083333334,"e0 (the \"N_c\" part) is emphasized."],[71.65902083333334,"e0 (the \"v_upright(\"th\")\" part) is no longer emphasized."],[73.41902083333335,"e0 (the \"N_c\" part) is no longer emphasized."],[73.41902083333335,"e0 (the \"exp(-frac(E_c - E_t, k T))\" part) is emphasized."],[75.38152083333334,"e0 (the \"exp(-frac(E_c - E_t, k T))\" part) is no longer emphasized."]]},{"start":75.98152083333333,"say":"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.","live":["e0","axes","h1","points","points_2","points_3","points_4","points_5","points_6"],"does":[[79.77902083333333,"e0 (the \"v_upright(\"th\")\" part) is indicated — a transient flash."],[81.41902083333333,"e0 (the \"N_c\" part) is indicated — a transient flash."],[86.41902083333333,"e1 is shown on the screen, written out."]]},{"start":90.37152083333334,"say":"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.","live":["e0","e1","axes","h1","points","points_2","points_3","points_4","points_5","points_6"],"does":[[91.73902083333334,"e2 is shown on the screen, written out."],[98.81902083333333,"e2 (the \"frac(E_c - E_t, k)\" part) is emphasized."],[101.41902083333332,"e2 (the \"frac(E_c - E_t, k)\" part) is no longer emphasized."],[101.41902083333332,"e2 (the \"ln(sigma_n gamma_n)\" part) is emphasized."]]},{"start":102.76652083333333,"say":"And the points obey it: one straight line runs through them all, the first sign that a single activated process is at work.","live":["e0","e1","e2","axes","h1","points","points_2","points_3","points_4","points_5","points_6"],"does":[[102.76652083333333,"e2 (the \"ln(sigma_n gamma_n)\" part) is no longer emphasized."],[105.29902083333332,"fit is shown on the screen, written out."],[110.30402083333334,"e2 moves to a new place on the board."],[110.30402083333334,"e0 is hidden from the screen — left the board."],[110.30402083333334,"e1 is hidden from the screen — left the board."],[110.30402083333334,"h1 is hidden from the screen — left the board."]]},{"start":110.90402083333333,"say":"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.","live":["e2","axes","points","points_2","points_3","points_4","points_5","points_6","fit"],"does":[[110.90402083333333,"h2 is shown on the screen, written out."],[112.13902083333333,"secant is shown on the screen, written out."],[116.13902083333332,"m0 is shown on the screen, written out."]]},{"start":120.15902083333333,"say":"Multiply by minus Boltzmann's constant, and the trap depth comes out: zero point four zero electron volts below the conduction band.","live":["e2","axes","points","points_2","points_3","points_4","points_5","points_6","fit","m0","h2","secant"],"does":[[120.41902083333333,"m1 is shown on the screen, written out."],[124.57902083333333,"m1 becomes \"$E_c - E_t = -k m approx 0.40 thin upright(\"eV\")$\"."],[127.77902083333333,"A box is drawn around m1."]]},{"start":128.90902083333333,"say":"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.","live":["e2","axes","points","points_2","points_3","points_4","points_5","points_6","fit","m0","m1","h2","secant"],"does":[[129.4990208333333,"extension is shown on the screen, written out."],[134.57902083333335,"intercept_pt is shown on the screen, written out."],[137.29902083333332,"m2 is shown on the screen, written out."]]},{"start":143.77402083333334,"say":"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.","live":["e2","axes","points","points_2","points_3","points_4","points_5","points_6","fit","m0","m1","m2","h2","secant","extension","intercept_pt"],"does":[[146.7390208333333,"intercept_pt is indicated — a transient flash."],[151.97902083333332,"m0 is indicated — a transient flash."],[152.97152083333333,"points is hidden from the screen."],[152.97152083333333,"points_2 is hidden from the screen."],[152.97152083333333,"points_3 is hidden from the screen."],[152.97152083333333,"points_4 is hidden from the screen."],[152.97152083333333,"points_5 is hidden from the screen."],[152.97152083333333,"points_6 is hidden from the screen."],[152.97152083333333,"secant is hidden from the screen."],[152.97152083333333,"fit is hidden from the screen."],[152.97152083333333,"extension is hidden from the screen."],[152.97152083333333,"intercept_pt is hidden from the screen."],[152.97152083333333,"axes is hidden from the screen — left the board."],[152.97152083333333,"e2 is hidden from the screen — left the board."],[152.97152083333333,"h2 is hidden from the screen — left the board."],[152.97152083333333,"m0 is hidden from the screen — left the board."],[152.97152083333333,"m1 is hidden from the screen — left the board."],[152.97152083333333,"m2 is hidden from the screen — left the board."]]},{"start":154.17152083333332,"say":"Nothing here is special to traps. Any process with a Boltzmann factor gives the same straight line; only the axis label changes.","live":[],"does":[[154.17152083333332,"h3 is shown on the screen, written out."],[154.17152083333332,"table is shown on the screen, written out."],[155.6190208333333,"table is shown on the screen, written out."]]},{"start":162.38652083333332,"say":"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.","live":["h3"],"does":[[162.6190208333333,"table is shown on the screen, written out."],[166.9390208333333,"table is shown on the screen, written out."],[172.77902083333333,"table is shown on the screen, written out."]]},{"start":176.3965208333333,"say":"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.","live":null,"does":[[176.8990208333333,"caveat is shown on the screen, written out."],[185.5882083333333,"caveat is hidden from the screen — left the board."],[185.5882083333333,"h3 is hidden from the screen — left the board."],[185.5882083333333,"table is hidden from the screen — left the board."]]}]},{"title":"The Shockley Diagram","start":186.629875,"end":334.92724999999996,"objects":{"answer":"a Math [text] that says \"$E_c - E_F approx 0.21 thin upright(\"eV\")$\"","axes":"an Axes (x_range=(0.0, 1.12), y_range=(0.0, 20.0), x_ticks_every=0.2)","hA":"a Heading that says \"The Shockley Diagram\"","hB":"a Heading that says \"A Defect Level on the Diagram\"","line":"a Line [yellow] drawn in axes (start=(0.915, 16.0), end=(0.0, 16.0), dashed=True)","line_2":"a Line [yellow] drawn in axes (start=(0.915, 16.0), end=(0.915, 0.0), dashed=True)","n1_dot":"a PlotPoint [yellow] labelled \"n_1\" drawn in axes (target='n_plot', x=0.72)","n_dot":"a PlotPoint [blue] drawn in axes (target='n_plot', x=<VariableNumber trial = 0.915>)","n_hot":"a FunctionPlot [gray] drawn in axes (function=<function>)","n_plot":"a FunctionPlot [blue] labelled \"n\" drawn in axes (function=<function>)","nd_dot":"a PlotPoint [red] labelled \"0.55\" drawn in axes (target='nd_plot', x=<VariableNumber trial = 0.915>)","nd_plot":"a FunctionPlot [red] labelled \"N_D^+\" drawn in axes (function=<function>)","p1_dot":"a PlotPoint [yellow] labelled \"p_1\" drawn in axes (target='p_plot', x=0.72)","p_hot":"a FunctionPlot [gray] drawn in axes (function=<function>)","p_plot":"a FunctionPlot [green] labelled \"p\" drawn in axes (function=<function>)","point":"a Point [yellow] drawn in axes (location=(0.547, 9.83))","point_2":"a Point [yellow] drawn in axes (location=(0.915, 16.0))","point_3":"a Point [yellow] drawn in axes (location=(0.915, 3.65))","point_4":"a Point [yellow] drawn in axes (location=(0.539, 13.9))","point_5":"a Point [yellow] drawn in axes (location=(0.745, 16.0))","probe_line":"a Line [gray] drawn in axes (start=(<VariableNumber trial = 0.915>, 0.0), end=(<VariableNumber trial = 0.915>, 20.0), dashed=True)","s0":"a Math [text] that says \"$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":[{"start":186.629875,"say":"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.","live":[],"does":[[186.629875,"hA is shown on the screen, written out."],[186.629875,"axes is shown on the screen, written out."]]},{"start":199.674875,"say":"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.","live":["axes","hA"],"does":[[199.949875,"axes moves to a new place on the board."],[199.949875,"s0 is shown on the screen, written out."],[207.629875,"n_plot is shown on the screen, written out."]]},{"start":210.86737499999998,"say":"Holes are the mirror image, falling with the same slope the other way. 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So where must the Fermi level sit?","live":["s0","s1","s2","axes","hA","n_plot","p_plot","nd_plot"],"does":[[233.82987500000002,"s3 is shown on the screen, written out."]]},{"start":243.174875,"say":"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.","live":["s0","s1","s2","s3","axes","hA","n_plot","p_plot","nd_plot"],"does":[[243.469875,"probe_line is shown on the screen, written out."],[243.469875,"nd_dot is shown on the screen, written out."],[243.469875,"n_dot is shown on the screen, written out."],[246.18987499999997,"The segment (0.55, 9.87) to (0.55, 16.0) in axes is lit up."],[250.829875,"axes: retire a lit segment (unemphasize_line)."]]},{"start":252.18987499999997,"say":"Try one electron volt. Now electrons outnumber donors, too much negative charge, so it must come back down.","live":["s0","s1","s2","s3","axes","hA","n_plot","p_plot","nd_plot","probe_line","nd_dot","n_dot"],"does":[[252.46987499999997,"probe_line is redrawn as the numbers it depends on change."],[252.46987499999997,"nd_dot is redrawn as the numbers it depends on change."],[252.46987499999997,"n_dot is redrawn as the numbers it depends on change."],[252.46987499999997,"trial ticks to 1.0."]]},{"start":260.217375,"say":"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.","live":null,"does":[[261.869875,"probe_line is redrawn as the numbers it depends on change."],[261.869875,"nd_dot is redrawn as the numbers it depends on change."],[261.869875,"n_dot is redrawn as the numbers it depends on change."],[261.869875,"trial ticks to 0.915."],[262.989875,"answer is shown on the screen, written out."],[266.909875,"point_2 is shown on the screen, grown."],[266.909875,"line is shown on the screen, drawn."],[266.909875,"line_2 is shown on the screen, drawn."],[268.78493301080505,"point_2 is hidden from the screen."],[268.78493301080505,"line is hidden from the screen."],[268.78493301080505,"line_2 is hidden from the screen."]]},{"start":270.984875,"say":"And the hole line, down here, is twelve orders of magnitude lower. That is how the diagram tells you which terms may be dropped.","live":["s0","s1","s2","s3","answer","axes","hA","n_plot","p_plot","nd_plot","probe_line","nd_dot","n_dot"],"does":[[271.50987499999997,"point_3 is shown on the screen, grown."],[274.01066172732135,"point_3 is hidden from the screen."],[278.974875,"probe_line is hidden from the screen."],[278.974875,"n_dot is hidden from the screen."],[278.974875,"nd_dot is hidden from the screen."]]},{"start":279.574875,"say":"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.","live":["s0","s1","s2","s3","answer","axes","hA","n_plot","p_plot","nd_plot"],"does":[[279.869875,"n_hot is shown on the screen, written out."],[279.869875,"p_hot is shown on the screen, written out."],[286.829875,"point_4 is shown on the screen, grown."],[288.50987499999997,"point_5 is shown on the screen, grown."],[288.8562655805928,"point_4 is hidden from the screen."],[290.19737499999997,"n_hot is hidden from the screen."],[290.19737499999997,"p_hot is hidden from the screen."],[290.19737499999997,"answer is hidden from the screen — left the board."],[290.19737499999997,"hA is hidden from the screen — left the board."],[290.19737499999997,"s0 is hidden from the screen — left the board."],[290.19737499999997,"s1 is hidden from the screen — left the board."],[290.19737499999997,"s2 is hidden from the screen — left the board."],[290.19737499999997,"s3 is hidden from the screen — left the board."],[290.8724102941176,"point_5 is hidden from the screen."]]},{"start":291.397375,"say":"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.","live":["axes","n_plot","p_plot","nd_plot"],"does":[[291.397375,"hB is shown on the screen, written out."],[297.709875,"trap is shown on the screen, written out."],[300.909875,"t0 is shown on the screen, written out."]]},{"start":304.667375,"say":"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.","live":["axes","n_plot","p_plot","nd_plot","t0","hB","trap"],"does":[[308.149875,"The segment (0.72, 16.0) to (0.915, 16.0) in axes is lit up."],[309.429875,"t0 (the \"exp(frac(E_t - E_F, k T))\" part) is emphasized."],[312.62987499999997,"t0 (the \"exp(frac(E_t - E_F, k T))\" part) is no longer emphasized."],[312.62987499999997,"axes: retire a lit segment (unemphasize_line)."]]},{"start":313.609875,"say":"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.","live":null,"does":[[315.549875,"n1_dot is shown on the screen, written out."],[317.069875,"t1 is shown on the screen, written out."],[322.429875,"p1_dot is shown on the screen, written out."],[323.549875,"t2 is shown on the screen, written out."]]},{"start":324.81737499999997,"say":"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.","live":["axes","n_plot","p_plot","nd_plot","t0","t1","t2","hB","trap","n1_dot","p1_dot"],"does":[[327.989875,"t1 is indicated — a transient flash."],[329.74987500000003,"t2 is indicated — a transient flash."],[333.88558333333333,"axes is hidden from the screen — left the board."],[333.88558333333333,"n_plot is hidden from the screen — axes left the board."],[333.88558333333333,"p_plot is hidden from the screen — axes left the board."],[333.88558333333333,"nd_plot is hidden from the screen — axes left the board."],[333.88558333333333,"trap is hidden from the screen — axes left the board."],[333.88558333333333,"n1_dot is hidden from the screen — axes left the board."],[333.88558333333333,"p1_dot is hidden from the screen — axes left the board."],[333.88558333333333,"hB is hidden from the screen — left the board."],[333.88558333333333,"t0 is hidden from the screen — left the board."],[333.88558333333333,"t1 is hidden from the screen — left the board."],[333.88558333333333,"t2 is hidden from the screen — left the board."]]}]},{"title":"Generation, Recombination and Leakage","start":334.92724999999996,"end":468.1283125,"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=<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":[{"start":334.92724999999996,"say":"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.","live":[],"does":[[334.92724999999996,"hA is shown on the screen, written out."],[336.60724999999996,"u0 is shown on the screen, written out."],[340.68724999999995,"u0 (the \"n p - n_i^2\" part) is emphasized."],[346.68724999999995,"u0 (the \"n p - n_i^2\" part) is no longer emphasized."],[346.68724999999995,"u0 (the \"n_1\" part) is emphasized."],[347.40725,"u0 (the \"n_1\" part) is no longer emphasized."],[347.40725,"u0 (the \"p_1\" part) is emphasized."],[348.79474999999996,"u0 (the \"p_1\" part) is no longer emphasized."]]},{"start":349.39474999999993,"say":"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.","live":["u0","hA"],"does":[[349.92724999999996,"sign_note is shown on the screen, written out."],[363.22974999999997,"u0 moves to a new place on the board."],[363.22974999999997,"hA is hidden from the screen — left the board."],[363.22974999999997,"sign_note is hidden from the screen — left the board."]]},{"start":363.82974999999993,"say":"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.","live":["u0"],"does":[[363.82974999999993,"hB is shown on the screen, written out."],[363.82974999999993,"gaxes is shown on the screen, written out."],[366.28725,"u1 is shown on the screen, written out."],[375.40725,"u2 is shown on the screen, written out."]]},{"start":377.61724999999996,"say":"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.","live":["u0","u1","u2","gaxes","hB"],"does":[[378.44724999999994,"gcurve is shown on the screen, written out."],[384.32724999999994,"The segment (0.15, -2.22) to (0.45, -7.26) in gaxes is lit up."],[388.52225,"gaxes: retire a lit segment (unemphasize_line)."]]},{"start":389.12224999999995,"say":"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.","live":["u0","u1","u2","gaxes","hB","gcurve"],"does":[[390.04724999999996,"dot is shown on the screen, written out."]]},{"start":398.43724999999995,"say":"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.","live":["u0","u1","u2","gaxes","hB","gcurve","dot"],"does":[[398.72724999999997,"dot is redrawn as the numbers it depends on change."],[398.72724999999997,"probe ticks to 0.4."],[403.44724999999994,"dot is redrawn as the numbers it depends on change."],[403.44724999999994,"probe ticks to 0.0."],[410.92224999999996,"gaxes is hidden from the screen — left the board."],[410.92224999999996,"gcurve is hidden from the screen — gaxes left the board."],[410.92224999999996,"dot is hidden from the screen — gaxes left the board."],[410.92224999999996,"hB is hidden from the screen — left the board."],[410.92224999999996,"u0 is hidden from the screen — left the board."],[410.92224999999996,"u1 is hidden from the screen — left the board."],[410.92224999999996,"u2 is hidden from the screen — left the board."]]},{"start":412.12224999999995,"say":"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.","live":[],"does":[[412.12224999999995,"hC is shown on the screen, written out."],[415.44724999999994,"j0 is shown on the screen, written out."],[421.60724999999996,"j0 (the \"frac(E_g, 2 k T)\" part) is emphasized."]]},{"start":423.45724999999993,"say":"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.","live":["j0","hC"],"does":[[423.45724999999993,"j0 (the \"frac(E_g, 2 k T)\" part) is no longer emphasized."],[423.7272499999999,"j1 is shown on the screen, written out."],[427.88724999999994,"j1 (the \"frac(E_g, k T)\" part) is emphasized."],[429.00724999999994,"verdict is shown on the screen, written out."],[429.00724999999994,"j1 (the \"frac(E_g, k T)\" part) is no longer emphasized."],[434.37724999999995,"hC is hidden from the screen — left the board."],[434.37724999999995,"j0 is hidden from the screen — left the board."],[434.37724999999995,"j1 is hidden from the screen — left the board."],[434.37724999999995,"verdict is hidden from the screen — left the board."]]},{"start":434.9772499999999,"say":"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.","live":[],"does":[[434.9772499999999,"hD is shown on the screen, written out."],[435.92724999999996,"summary is shown on the screen, written out."],[439.20724999999993,"summary (the \"Arrhenius plot\" part) is emphasized."]]},{"start":446.45724999999993,"say":"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.","live":["summary","hD"],"does":[[446.84725,"summary (the \"Arrhenius plot\" part) is no longer emphasized."],[446.84725,"summary (the \"Shockley diagram\" part) is emphasized."]]},{"start":455.44974999999994,"say":"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.","live":null,"does":[[456.72724999999997,"summary (the \"Shockley diagram\" part) is no longer emphasized."],[456.72724999999997,"summary (the \"generation peaks\" part) is emphasized."],[466.83664583333325,"summary (the \"generation peaks\" part) is no longer emphasized."],[467.08664583333325,"hD is hidden from the screen — left the board."],[467.08664583333325,"summary is hidden from the screen — left the board."]]}]}]},"durationSeconds":468,"chapters":[{"title":"Two Diagrams, Two Questions","startSeconds":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. 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."},{"title":"The Arrhenius Plot","startSeconds":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. 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."},{"title":"The Shockley Diagram","startSeconds":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. 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."},{"title":"Generation, Recombination and Leakage","startSeconds":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. 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."}]}}