E/K and the Band Gap1

e/k and the band gap1

Bryn Mawr College

Department of Physics

Undergraduate Teaching Laboratories

e/k and the Band Gap of a Semiconductor

Introduction

The current-voltage relationship of a p-n junction is measured at different temperatures, permitting a determination of the ratio of e (the magnitude of the charge of an electron) to k (Boltzmann's constant). The measurements display significant physical properties of semiconductor junctions and a very large range of currents can be measured. Accuracy and error analysis are an important part of the learning experience in this experiment.

Theory

Under the appropriate conditions the measured current is given by

(1)

where V is the applied voltage. When eVkT, this equation reduces to

(2)

In equations 1 and 2, I and V are taken as positive quantities. Also, the charge of the electron is q = e where e is a positive constant. So, eV/kT is a positive quantity. The parameters Io and e/(kT) can be determined from a measurement of I versus V. If you have a very sensitive picoammeter and can go to very low currents, you may find that these equations break down completely. Also, for sufficiently large voltages, saturation effects may occur.

Procedure

Build a variable negative dc voltage source using a 1 k helipot and a 1.5 V battery as shown below. Check that the voltage varies as the helipot dial is turned (careful of the polarity, wherever necessary). As your helipot goes from 0 to 10, the voltage should go from near 0 to near 1.5V.

PLEASE DISCONNECT THE BATTERY AT THE END OF THE DAY.

The currents in this experiment will be measured with a picoammeter. If the scales on the picoammeter are set by external buttons (like the Keithley Model 480), then it should be on the least sensitive scale when turning on or off or when making changes in the circuit. If the scales are set by "soft key" control (like the Keithley Model 486), then set it to its least sensitive scale after you turn it on. On the more sensitive scales, be careful not to overload the instrument. Although these instruments are protected against overload, the calibration can be ruined. Always watch the picoammeter when changing the voltage level to ensure it does not go off scale. Never leave it on a sensitive scale if you leave the apparatus or make changes in the circuit. Always turn it to the least sensitive scale when you are not making measurement.

Connect the circuit as shown below. The transistor is an n-p-n. (There is a very interesting and helpful paper by Inman and Miller in the folder. Figure 1 in this paper is incorrect. The figure in this write-up is correct.) E, C and B refer to emitter, collector and base. Be sure to observe that one side of the picoammeter must be grounded. (Some newer picoammeters will have a third ground lead.)

You want to use the same semiconductor for the entire experiment. Begin with the semiconductor in a test tube filled with oil that has been at room temperature for some reasonable time.

Experimental procedure at Room Temperature

Put a mercury thermometer in the oil in the tube with the semiconductor. If there are several thermometers lying around, don't choose one that clearly has a systematic error. Make a quick measurement of the I-V curve at room temperature covering as many orders of magnitude in I as possible. Note that although you are observing negative voltages and, perhaps, negative currents, you should treat them as positive quantities when you plot them. If you are using a Keithley 486 picoammeter, you must turn off the "zero check" to measure the current. When you make changes to the circuit, put it back on "zero check."

Make a semi-log plot (why?) of this quick-measurement data using many-decade graph paper. Please do not use a computer at this stage. There is a very fast and simple way of finding the slope on a quick hand-made semi-log plot. The procedure is in the folder. Determine a rough value of e/kT and therefore of e/k. Sort out all the units. Compare it with the theoretical value of e/k knowing that e = 1.602 176 487 (40) C and that k = 1.380 6504 (24) J/K where the parentheses indicate the uncertainties in the last two digits. These values are from the NIST web site.

Make detailed measurements of the I-V characteristics. Monitor the temperature now and again. Extend the range of measurements as much as possible but don't go above a current of 2 mA which is the upper range of most picoammeters. (One could measure higher currents with a conventional digital multimeter but there is no point since most of these transistors will begin to saturate by 2 mA anyway. Also, one begins to heat the transistor.) When you make a careful measurement, space the I values logarithmically. Why? (This is an unusual example of an experiment where you set the values of the "dependent" variables and note the resulting values of the "independent" variables. So much for word usage.)

Use KaleidaGraph or Origin and perform a Simplex fit to the data, thus determining Ioand e/kT and their uncertainties. There are some precautions here which have to do with the algorithms of the Simplex fits used by these programs. If you fit a semi-log plot to the resident exponential fit, it will work very well but it will not give uncertainties. The resident fits do not use the Simplex method. If you write your own Simplex fit, you must either fit ln(Io) versus V to a linear fit, or, you must weight the data in an exponential fit appropriately. The trick is to determine what "appropriately" means here. For sure, the simplest procedure and the easiest to defend is to make an additional data column, use the Formula Entry (in KaleidaGraph) to compute ln(Io), and perform a linear fit.

You may not want to fit the entire range of voltages, since you only want the exponential behavior for the fit. (Your data should be "clean enough" that you can clearly see the departures from exponential behavior. That means a departure from linear behavior on a ln(Io) versus V plot. Equivalently, this means that equation 2 is breaking down.) However, you do want to plot all the data so don't use the Mask procedure (in KaleidaGraph). Instead, make a separate column for the data you don't want to fit and, if necessary, move Iodata (or ln(Io) data) at the highest and lowest voltages from the regular column into the new column. You can plot both, but just fit to one. In KaleidaGraph, make sure the Curve Fit option is set to extend the fit to the limits of the graph. You can then see the departure of the data from the best fit at the highest and lowest voltages. You can move data back and forth between the "fitted" and "not fitted" columns until you are happy with the visual result. You can also monitor how the slope and its uncertainty changes and you maneuver through this procedure.

Make sure you present a thorough discussion of what you are doing and why you are making the choices you make. From the slope, determine e/kT (and therefore e/k) and from the intercept determine Io [or ln(Io)], and their uncertainties. Don't forget to fold in the uncertainty in temperature T when you compute e/k from e/kT. Taking into account your uncertainty, is your measured value consistent with the accepted value? If not, take note and return to this problem after you have made measurements at other temperatures.

Experiments at Other Temperatures

Repeat the I-V measurements at other standard temperatures.

Water/ice mixture273 K

Dry ice/isopropyl alcohol (bubbling)195 K

Liquid Nitrogen77 K

Boiling Water373 K

When collecting data at 273 K (ice and water) and 373 K (boiling water), the transistor should be submerged in oil in a test tube so it might make sense to do these temperatures first since the transistor is already in the oil bath from the room temperature measurement.

When using dry ice (solid carbon dioxide), liquid nitrogen, and isopropyl alcohol, you should wear safety glasses.

Water ice is available from the Solid State NMR lab or from the Chemistry Department and a dewar of ice should be on the desk. Dry ice (solid carbon dioxide) is obtained from the Chemistry Department and should on the desk when you need it. Mixing dry ice and isopropyl alcohol can be a little dangerous and you should make this mixture carefully. Use the small dewar in a stand which makes the assembly more stable. When you have finished using the isopropyl alcohol/dry ice mixture, it is best to leave it overnight and let the dissolved dry ice all evaporate. A dewar of liquid nitrogen should be on the desk when you need it.

To collect data at 195 K and 77 K, remove the transistor from the oil, wipe it off, and place it directly in the isopropyl/CO2 mixture or the liquid nitrogen.

Use the same KaleidaGraph data file for the entire experiment with V always in column zero. Just start the experiments at each temperature at a new row (and, of course, with a new set of columns). Leave two columns for each temperature since you may want both a "fitting column" and a "non-fitting column." You will have a large data file, using many columns, by the time you are finished. That's okay. Make sure you label the columns (by double clicking on the column headings) so you can keep everything straight.

Make a different plot for each temperature T. Remember, that when you save a plot in KaleidaGraph, the data is saved with it. Never, after saving a plot, subsequently save the data as well. When it asks you, just say "no." For visual comparison only, make another plot with all the data on it. Provide a general discussion of this data. Point out and discuss the departures from exponential behavior.

Make a new KaleidaGraph data file with T in column zero (C0), its uncertainty in C1, e/(kT) in C2, its uncertainty in C3, ln(Io) in C4, and its uncertainty in C5. There will only be five rows, one for each temperature. You cannot use Ioand ask these programs to compute ln(Io). The numbers are simply too small. Use Formula Entry to compute e/k in C6 and devise some procedure to determine its uncertainty. Compute 1/T in the last column.

Discuss the e/k values and their uncertainties.

Determining the Transistor Band Gap

Plot e/(kT) with error flags versus 1/T and find a best value of e/k.

Note the incredible range in Io values. Plot ln(Io) with error flags versus 1/T and find the value of Egap/k (in volts) (the band-gap energy in silicon). The dependence of Io on T is described by Kirkup and Placido in the folder. Note that even though there is an f(T) in Io = f(T)exp(Egap/kT), the T-dependence of f(T) can be ignored. Or can it? Why? Why not?

e/k 2008