Administrivia. Lab notebooks from Lab 0 are due at 2 PM in Hebb 42 at the start of Lab 1.

Transcription

1 Administrivia Lab notebooks from Lab 0 are due at 2 PM in Hebb 42 at the start of Lab 1. You can trade lab partners for the next lab, by mutual agreement of all the parties concerned. No unilateral abandonments, please.

2 Diode Lab vs Lab 0 In Lab 0, the data was from a nearly perfect sine wave of large amplitude from a signal generator. The function you were fitting was a sine wave with an offset, an amplitude, a frequency, and a phase. You should have gotten a very good fit, with an amplitude and frequency that agreed very well with the signal generator setting, an offset near zero, and a phase that was different every time you pressed the Acquire button. You looked at the residuals of the fit, and they probably looked like random noise. Gnuplot s fit calculated the RMS (standard deviation) of the residuals. You drew error bars equal to that value, and probably found that the residuals were consistent with zero within the error about 2/3 of the time. The point was to learn how to use the data acquisition computers, and to use Gnuplot to fit some data. The situation was artificially simple, both to make it easy to get started, and also to show you what good agreement between data and a model looks like. This is the last time in the course where it will be that easy to fit data and get good agreement with the model. In the diode lab, the data will push the instruments to the limit, and the models won t be so perfect.

3 Lab0 looked trivially simple, but was not! Is there any systematic behavior in the residuals? YES, there is!

4 Gnuplot Replot Replot followed by additional stuff is treated like an extended plot command: plot file1.dat u 1:3 replot f(x) is just the same as plot file1.dat using 1:3, f(x) Typing replot f(x) again at this point is the same as plot file1.dat using 1:3, f(x), f(x) which will plot f(x) on top of itself and add it to the key, which is probably not what you want. To change the function parameters and recreate the plot, just type replot by itself, not replot f(x) This is great for building up a plot command with stuff from many files. The drawback is that you can t use uparrow to edit the long line you are building. The workaround is the command show plot which does show you the full long plot command, which you can edit. You can copy and paste in the Gnuplot window (and nearly any Linux or Physics window) by dragging to highlight something (which implicitly copies it to the clipboard), then left-clicking to indicate the destination, then clicking both buttons simultaneously (or middle button on a 3-button mouse) to paste at the destination. Then you can backspace over the line to edit or remove things.

5 More Gnuplot Tricks You can abbreviate any command to the fewest unambiguous letters: replot can be rep, using can be u, with can be w, errorbars can be e, xrange can be xr You can see your function definitions by show fun You can see your variable values by show var You can save your functions, variables, plot settings, the most recent plot command to a file named filename by save filename You can then quit out of Gnuplot, and get exactly the same functions, variables, and recreate the same plot by load filename The file that gets created is just a text file that you can look at and edit and ship from machine to machine. The file contains lots of lines that define every single internal variable of Gnuplot, most of which you don t care about. The functions, variables, and plot are in the last few lines. The file also contains the most recent fit command, although it is commented out so the fit will not be re-run when you load the file. But you can edit out the # characters. Note that the plot may not work on a different machine unless all the same data files exist in the same places. So you either have to arrange that, or edit the file.

6 Meaning of Error to a Physicist Physicists use the word error to mean the accuracy of a measurement, independent of expectation. It means how well you think you measured it It DOES NOT mean the difference between what you got and what you should have got If you mean the difference between what you measured and what you expected, please use a different word, like discrepancy or disagreement. If the discrepancy is less than the error, then there is no significant problem. If the discrepancy is much greater than the error, then there is a problem. The problem might be that you didn t do what you thought you did your equipment is broken or not set up right your equipment doesn t do quite what you thought your expectation of the physics was wrong.

7 Fit Parameter Errors A typical Gnuplot fit parameter result might look like Final set of parameters Asymptotic Standard Error ======================= ========================== a b = = / / (18.2%) (695.1%) c = / (0.9844%) d = e-05 +/ (1040%) The parameter errors represent how well the data and its errors constrain the values of the parameters. This also is the amount that we expect the fit parameters to vary if we took many identical sets of data. The number in parenthesis is the error divided by the parameter value expressed as percent. Don t be alarmed by that fact that one parameter has 695% error and another has an error over 1000%! In this case, that just means that the values of those two parameters are statistically consistent with zero (which happens to be true for the data that is being fit).

8 Fit Parameter Errors (2) How does Gnuplot calculate these errors? To do the fit, it finds the parameter values that minimize the sum of the squares of the residuals. This involves calculating the derivatives of the function with respect to the fit parameters at each data point, building a matrix, inverting it, and iterating. It uses a standard algorithm for this. The sum of the squares of the residuals at the minimum is then used to estimate the noise in the data, on the assumption that the noise is the same for each data point. Then it uses standard error-propagation formulas to calculate the error on the fit parameters due to the noise it sees in the data. If the noise on each data point is not the same, then thisprocedure isn t the right thing to do. But you can tell Gnuplot what you know about the errors in your data by including a third column in the fit command, containing your estimate of the errors of the data points. Gnuplot will weight the data by these errors, so points with smaller errors will get more weight in the fit. It will then estimate a correction factor to your error bars, and use the corrected errorbars to calculate the fit errors

9 Fit Parameter Errors (3) The fit parameter errors depend on the details of the particular problem, but also on the amount of data (more is better), and the the size of the data errors (smaller is better). All else being equal, if you have twice as much data, the errors on the fit parameters will be divided by 2. All else being equal, if you double the errors on the data points, you should double the fit parameter errors. Actually, because of the correction factor calculation mentioned above, Gnuplot does this wrong! If you double the errors in your data file, gnuplot will give the same errors in the fit! Gnuplot always uses the RMS residuals to calculate its own errors on the data points, even when you give it errors on the data points! It seems to only use the data errors in a relative-weight sense, and ignores their absolute size. You can recover the correct behavior by dividing the Gnuplot errors by the square-root of what Gnuplot calls reduced WSSR. Since this factor is typically close to 1 if the errors on the data are accurate, this normally is a fairly small change.

10 Gnuplot Fitting Issues Gnuplot has a bug in its fitting package: if you give any initial parameter value of exactly zero, the fit may give the wrong answer. You can tell that this happened by a message Singular matrix in InvertRvR but it s easy to miss it. You can also tell by the absence of any fit parameter errors. Never use initial parameters of exactly zero, use some very small number like instead. There is another bug that shows up most often if your parameter guesses are not very good. The fit may say that it has converged and give parameter errors but still be wrong if the lambda value reported for the final iteration is not much less than 1.0. If you are using Windows, or the Physics Sun, type in FIT_START_LAMBDA=0.01 before doing any fits. The spelling (all caps, with underscores) is critical. You only have to do this once per session. This makes the lambda problem happen less often. If a fit gives a large lambda value, improve the parameter guesses and start again.

11 This lab has several goals Overview of Diode Lab Learn about a circuit component that is more complicated than a resistor or capacitor Learn about a circuit component that is less engineered than a logic gate or op-amp but more physics-driven Learn about recognizing the limitations of your equipment (and overcoming them) Learn about recognizing the limitations of a theory or model of how something works by careful examination of your data An ideal diode is a circuit device that has a zero resistance in one direction and infinite resistance in the other. A real semiconductor diode doesn t have zero resistance in the forward direction, and doesn t have infinite resistance in the backward direction. Saying it has finite but constant resistances that are different in the two directions isn t a particularly good model either.

12 Junction Diode Current-Voltage Law The actual current-voltage relationship for a junction diode for reasonably small voltages and currents is I( V) = I 0 [ e V V 0 1] where I 0 is a small current determined by the geometry and doping of the silicon, and V 0 is determined by more fundamental physics. For positive voltage V >> V 0, the current is exponential with the voltage. For negative voltage V << V 0, the current is I 0. For zero voltage, the exponential is 1, the bracket is 0, and the current is zero. Current exp(x) Voltage

13 On a Log Scale If we neglect the 1 and take the log of both sides, [ ] I 0 exp V V 0 I = I 0 exp( V V 0 ) 1 { ( )} = log{ I 0 } + log exp( V V 0 ) = log{ I 0 } +V V 0 log{ I} = log I 0 exp V V 0 { } ( ) Plot the current on a log scale and the voltage on a linear scale, and we should approach a straight line: Current exp(x)-1 exp(x) Voltage The slope is 1 V 0 and the intercept is I 0. The current at negative voltage is negative, so it can t be plotted on a log scale.

14 Log Plots and Errors Log plots can be very revealing for some functions, but they can also be very misleading. A constant absolute error will look very different depending on where the point is plotted. Here s some fake diode data with random fluctuations, where the reverse current was 10 times smaller than the measurement errors. The absolute value of the function and of the data was plotted, so the negative voltage and current region can be seen on the log plot e-05 'fake.dat' using 1:(abs($2)):(1.e-8) abs(1.e-9*(exp(x/0.05)-1)) 1e-06 1e-07 1e-08 1e-09 1e-10 1e All the points have the same errorbar, but it looks much bigger on some than on others!

15 Our Diode Circuit Coax Coax V0 V1 Diode Signal Generator Resistor Coax V2 We want to measure the diode voltage and current. We make the diode current flow through a resistor and use Ohm s Law to infer the current from the voltage. We subtract the voltages on the two sides of the diode to determine the voltage across the diode. For negative and small-positive voltage, the diode current is tiny, so V2 is nearly zero. For substantial positive voltage, the V2 is about half a volt less than V1. But how do diodes work?

16 Semiconductors and Doping Silicon has the same chemistry as carbon with 4 valence electrons that like to form 4 covalent bonds to other atoms. The crystal structure is just like diamond. And very pure silicon is a rather good insulator, like diamond. Diamonds don t conduct because under normal conditions, all the electrons are tied up in covalent bonds. It s not because electrons can t move through through the crystal. If you shine X-rays or other radiation on a diamond, the photons will free some electrons from the covalent bonds, and those electrons will move if you apply an electric field. If you add an impurity like phosphorus with 5 valence electrons to silicon, it turns out that the extra electron is fairly mobile, but the impurity atom stays fixed. This makes the silicon into a fairly good conductor, and is called N-type doping. An impurity like aluminum with 3 valence electrons leaves a hole in the pattern of bonds, and it turns out that this hole is also mobile, and makes the silicon conduct. This is called P-type doping. A chunk of P-type silicon does not have a net positive charge. There are mobile positive holes, and an equal number of immobile net-negative atoms, so the silicon is overall neutral.

17 Semiconductor Junction If you put an N-type crystal and a P-type crystal into atomic contact, the electrons from the N-type crystal diffuse into the P-crystal, and the holes diffuse into the N- crystal. In both cases, the ions of opposite charge stay put. P N Depletion Layer There is a gradient in the concentrations of electrons and holes across the junction. log concentration Holes Electrons There is a layer where both the electron and hole concentrations are low, called the depletion layer. The depletion layer has very few charge carriers, so it is very high resistance.

18 Semiconductor Junction (2) Why doesn t the diffusion make the electron and hole concentrations flat through the joined crystals? Positive holes diffusing to the right are a positive electric current to the right. Negative electrons diffusing to the left are also a positive current to the right. So the diffusion causes an electric current to the right. But because the oppositely charged ions don t diffuse, there is a net separation of charges across the junction. This produces an electric field across the junction. The electric field causes a conduction current, but in the direction opposite to the diffusion current. In the steady state, the diffusion current and the conduction current are equal. The diffusion increases the electric field until the electric conduction current cancels the diffusion current. The conduction current is due to minority carriers : the small number of mobile electrons at the P-side of the depletion layer, and the small number of mobile holes at the N-side of the depletion layer. The concentration of minority carriers at the edges of the depletion layer is controlled by the doping and the temperature, but not by the electric field.

19 Junctions and Voltages The electric field across the junction means that there is a potential or voltage difference across the junction. The potential difference means that a hole moving to the right is going uphill. A negative electron moving to the left is also going uphill. Let s call this energy difference from one side of the junction to the other ΔE = U 0. Holes Electrons Electric Potential The potential is flat in the P and N regions because they are good conductors with low electric fields. If we apply an external voltage, all of the voltage appears across the junction. If the positive voltage is on the left, the net energy difference for a charge carrier is then ΔE = U 0 qv

20 Statistical Mechanics In statistical mechanics, the probability of observing a component of a system (like an electron or hole) at a state of one energy compared to another energy, due to a thermal fluctuation, is proportional to the Boltzman factor exp ΔE kt where ΔE is the energy difference between the states, k is Boltzmann s constant, and T is the absolute temperature. When the concentration of something varies within a system, there is an effective energy differencerelated to the concentration ratio, called the chemical potential difference. In the case of a junction, the chemical potential difference between the P and N sides is exactly equal to U 0, the negative of the energy difference caused by the equilibrium between the diffusion and conduction currents with zero applied voltage. Including the chemical potential and the total electrical potential, the Boltzman factor is exp U 0 + U 0 qv kt +qv = exp kt which depends only on the applied voltage.

21 Diode Equation The diffusion current is proportional to the number of carriers that can jump the total potential barrier at the junction, which is proportional to the Boltzman factor. At zero applied voltage, the total current is zero, because the diffusion current is cancelled by the conduction current. If the applied voltage is large and negative, the Boltzman factor becomes negligible. But there is still the same conduction current, controlled by the doping and temperature but not the voltage. The net result is that the sum of the diffusion current and the conduction current is I( V,T) = I 0 ( T) exp +qv kt 1 The current at fixed temperature is exponential with voltage (as long as qv is a few times kt). The temperature dependence at fixed voltage is partly from the Boltzman term (which decreases with temperature for positive voltage), and partly from the I 0 ( T) term (which increases strongly with temperature because it is proportional to a different Boltzman term with a large negative energy difference).