PHTN1300: Lasers and Light Sources
Determining Planck's Constant (2018F)

Introduction

In photonics, few constants are so widely utilized as Planck's constant, the constant which relates the energy of a photon to it's frequency according to E=hf where E is the photon's energy in Joules, h is Planck's constant, and f is the photon's frequency. By observing the wavelength of light emitted from semiconductor LEDs, and knowing the energy lost by electrons in the downward transition, this constant may be accurately determined.

In the case of this experiment, we will exploit the fact that for a light-emitting semiconductor diode, the energy of an emitted photon is related to the energy required to jump the bandgap between the semiconductor materials. In the case of a diode, the bandgap energy is seen as the constant voltage across the device while current is flowing. If the voltage across the device is below that of the bandgap, no current flows (and in the case of an LED, no light is emitted). If we increase the voltage gradually, we will reach a point where we have supplied electrons with enough energy to jump the gap: current will flow and an LED will emit light.

Knowing the voltage required to cause photon emission, and knowing the frequency of the light (i.e. by measuring the wavelength of this emission using a spectroscope and/or by using the value from a datasheet), we can calculate the energy of photons emitted and then Planck's constant.

This is a two period lab - it is expected that it will require more than one period to complete all observations.

Prelab (To be done prior to the lab, due upon entry to the first lab period)

WARNING: Arrival to the lab without your parts kit including a multimeter and jumper cables will result in being marked ABSENT with the accompanying penalty (including a deduction in marks and eventually being placed on course condition) - you will NOT be permitted time to "run home" to obtain your kit.

    Submit the following upon entry to the first lab period:

  1. Compute, according to the formulae in Csele section 2.10 (equation 2.10.1), the expected bandgap voltage for this typical LED device. Bandgap voltage is, of course, just the bandgap energy in eV. As per the discussion in the lectures, one compensate for thermal energy (kT). Parameters will be required from the datasheet to do this - check the entire datasheet to be certain you understand the parameters you are getting. Compute TWO values: one value assuming a junction temperature of 37C (this temperature assumes the thermal conductivity of the device and the dissipated power from the datasheet as Tj = Ta + Rθ * P), and one value assuming a "worst case" maximum junction temperature as stated in the datasheet (you need to look this up). As a "check" of your answer, refer to values from the datasheet: yours should fall into the expected range given there.
  2. Compute the expected "most red" wavelength for this typical LED ("λr" as defined in Csele section 2.10). Again, two values will result (as per above), one corresponding to each assumed junction temperature above.
  3. Compute "λv" in each case, and then the expected FWMH of the output spectrum in both cases above. State, as well, the value from the datasheet for comparison.
  4. Assuming a spectroscope with a grating of 7500 lines/inch, calculate (showing all work) the expected angle for both wavelengths in question 2 (the "most red wavelength"). Be sure to convert the units of lines/inch to lines/m for the grating formula. You will likely have to review your notes from last year or use the Fundamentals text to do this.
  5. The prelab assignment will consist, then, of four questions. In each case, show all work, all substituted values, and all calculations. Numeric answers with no calculations and no substitutions shown will receive a mark of zero.

    The prelab assignment is worth 15% of the total lab mark and is due at the beginning of the first lab period. Late marks are not assigned if the prelab is not received at the beginning of the lab: you lose 15% of the total lab marks immediately with no recourse if it is not received upon entering the lab (extensions will NOT be given to "print it out" in the lab ... be prepared with the hardcopy already printed).

    Unlike the lab, prelabs may be handwritten or word processed (your choice), and may be submitted as stapled papers. You can use a report cover if you wish (optional) but do NOT use a BINDER.

    The prelab will be returned and reviewed in class before the lab is due allowing students to correct and problems in the methodology of calculating bandgap energy/voltage given only peak wavelength.

The Experiment (to be done in the lab)

Part A: Determining Visible LED Characteristics

First, prepare by setting-up a manual spectroscope as this will be required to observed the spectrum of each visible LED. Check the calibration of the spectroscope at the beginning of this lab by using a mercury lamp (in the same manner as you did in labs last year). Observe and record the angle of several (at least three) key mercury emission lines. Calculate the wavelength of each observed line (using the same formula as you did in the prelab which was corrected and returned to you in class) to ensure it is reasonably close to the known mercury emission wavelengths. You only need to do this once at the beginning of the lab before determining the spectra of all the LEDs used in this lab. This should be a fairly quick procedure and you MUST accomplish this within a half hour or so (COME PREPARED TO THE LAB: you did this last year so refer to your notes before coming to the lab if required to avoid wasting time).

Wire the a red LED in a circuit as below: in series with a 330Ω to 1000Ω resistor, a digital multimeter used to measure current (set to the 20mA range), an Agilent 3620A variable power supply (set its meter to V1), and the digital bench meter directly across the LED to measure the bandgap. Jumper clips are used for all connections. A digital voltmeter must be connected directly across the LED in order to measure the bandgap voltage directly as the voltmeter on the power supply is useless (it does not measure directly across the LED as required). Set the voltage control on the power supply to zero volts and turn ON the supply. Note the voltage across the device at which light first appears (despite the fact that current will read zero at this point) and then increase the voltage gradually so that current increases in steps of approximately 1mA as seen on the ammeter.

Record the voltage and actual current at each step (Read both from their respective meters - not the ones on the power supply which are not particularly accurate). Be sure to set the RANGE on the bench voltmeter so that the maximum number of digits is displayed (e.g. x.xxxx Volts).

Experiment Setup
Figure 1: Schematic of the LED test circuit
Experiment Setup
Figure 2: Pictorial diagram of the LED test circuit

You will note that at a certain voltage, the LED begins to light but below that, no light is emitted. Continue recording V & I pairs exactly (with current in steps of approximately 1mA) until a current of 10mA flows through the LED at which point the LED will glow brightly. Mount the LED in a suitable 3-point holder on the breadboard, point it towards the slit on the spectroscope, and observe the output wavelength of the LED at a current of 10mA. It will be required to adjust the alignment of the LED as well as the slit on the spectroscope. Observe the longest ("most red") wavelength of emission as seen in the spectroscope (record the angle, and convert to wavelength as you did last year as well as during your check of the spectroscope in this lab using the mercury source). Only one observation of output wavelength per LED is required (with the LED operating at 10mA).

Repeat this procedure of determining I&V points and determining the "most red" emission wavelength for five 'regular' visible LEDs (Red, Orange, Yellow, Green, and Blue). Note that the BLUE LED is static sensitive and can be destroyed by stray charges: be sure to ground yourself before touching these components and carry the device with both terminals shorted.

For EACH of the five visible LEDs be sure you have recorded the following:

  1. Ten Voltage/Current pairs (at appproximately 1mA intervals)
  2. An observation through the spectroscope of the "most red" wavelength seen

Part B: White LEDs and Fluorescence

Using the same setup as part A (with the manual spectroscope), observe the output of a white LED at 10mA. It should become evident that it will be impossible to determine the "most red" wavelength of the pump diode. From the assigned lecture reading you should know how a white LED works and what the dominant wavelength should be for the actual source LED inside the device. Record the V/I curve of this LED (in approximately 1mA steps) so that you can later determine the bandgap voltage of this LED.

Part C: Determining the wavelength of an unknown diode

Choose an infrared LED - these come in a variety of wavelengths ranging from 700nm to 1500nm - record the serial number on your IR LED as there are several types in the class, identified only by this number (choose only one). Measure the V/I characteristics of this diode in the same manner as you did for Part A. Observation of the spectrum is not possible since it is an infrared LED.

BEFORE YOU LEAVE THE LAB, be sure you have the following observations ...

  1. For EACH of the five visible LEDs be sure you have recorded ten Voltage/Current pairs (at appproximately 1mA intervals)
  2. For EACH of the five visible LEDs be sure you have recorded an observation through the spectroscope of the "most red" wavelength seen
  3. For the white LED be sure you have recorded ten Voltage/Current pairs (at appproximately 1mA intervals)
  4. For the infrared LED be sure you have recorded the serial number and ten Voltage/Current pairs (at appproximately 1mA intervals)

Analysis (to be done after the lab, at home)

Finding the Bandgap voltage from V/I Observations

I-V Curve for two LEDs

When analyzing the data (after the lab is complete), graph voltage vs. current for each LED in a manner similar to that seen to the left (the same as that outlined in Csele section 2.9). In ExcelTM this is a scatter graph. You will notice the 'knee' in the curve where light first appears. This is the voltage at which photons are emitted - the gap voltage. Find the gap voltage accurately by drawing an asymptote (line) to the linear (upper) part of the I/V curve and extrapolating to the zero intercept on the y-axis - in a spreadsheet the easiest way to do this is to add a second series of data covering only the upper (linear) portion of the graph (determined by inspecting the original graph) and adding a trendline to that data. Extend that line (forecast backwards) until it crosses the x-axis. This best-fit line, and the equation, MUST be shown directly on each graph submitted, and be sure to show where it intercepts the x-axis as this is the bandgap voltage which will be used in the rest of the experiment.

To determine intercept, display the equation of the line (y=mx+b) and solve for y=0 ... the value of x found will be the intercept with the axis as required.

Use of a spreadsheet to determine this intercept is required in this lab: manually-drawn "best fit" lines are not acceptable at all and will result in a large penalty.


Finding Planck's constant using observed wavelengths (Method One)

Now, each photon produced originates from the downward transition of a single electron. That electron, with electronic charge e, acquires an energy of V volts. The energy of each electron is therefore Vgap in electron-volts as discussed in section 2.10 and in class (an exact match is not expected though due to thermal energy and the distribution of charge carriers, as discussed in the lectures). Convert each gap energy in eV (one for each LED as found using the graphs above) into Joules.

Now, convert the longest wavelength observed (that will be the closest emission expected from the edges of the bands as seen in figure 2.10.1 of the text) to a frequency (in Hz) and plot bandgap energy in Joules on the y-axis (as obtained from voltage observations) and frequency in Hz on the x-axis (as obtained from spectroscope observations). Draw a best fit line (a "trendline", if using Excel) which includes an intercept of (0,0). The slope (rise/run) of this line will have units of Joules/Hz or Joule-seconds, the units in which Planck's constant is usually expressed, and this slope _is_ Planck's constant as required in this experiment.

When drawing the best-fit line in the analysis, you might have to omit the data point for the blue LED .... analyze your data with, and without the blue LED to see how it "fits" (some blue LEDs use a quantum well structure and so the bandgap voltage does not correspond to the frequency of the light emitted in the way other LEDs do - it is impossible to predict without viewing the resulting graph). Analyze your data early and ask the professor if required (but don't wait until the night before it is due to consider this).

In the report include five graphs showing the V/I characteristic for each LED (this may be a single graph with five lines - see the report requirements below), the bandgap voltage as found using each graph (and subsequent conversion from eV to Joules), the wavelength observations for each LED, a graph of gap energy vs. frequency used to determine Planck's constant, and finally report Planck's constant as determined from this experiment along with an uncertainty (±) figure.

Finding Planck's constant using datasheet wavelengths (Method Two)

Observations of bandgap voltage should be quite accurate, however wavelength observations in the first method might be erroneous due to difficulties in defining the "most red" emission seen (as discussed in the text, real devices do not exhibit the expected sharp cutoff). For this reason, the predicted emission corresponding to the bandgap edge will also be computed theoretically from available datasheet parameters (usually, peak wavelength).

First, research on DIGIKEY the data for the following LEDs used in the lab:

LED ColourDigi-Key PN
Green754-1285-ND
Blue67-1750-ND
Orange754-1271-ND
Yellow365-1183-ND
Red516-1379-ND

As per Fundamentals of Light Sources and Lasers (by Csele), equation 2.10.1, peak emission occurs at a wavelength corresponding to the bandgap energy plus thermal energy (kT) and so, by subtracting thermal energy from the energy of a photon at peak wavelength, the energy of the bandgap itself (in Joules) may be found.

Furthermore, the frequency corresponding to the bandgap energy (i.e. the frequency of a photon with exactly the bandgap energy) is simply E=hf so f=E/h. This should be comparable to the observations of the "red end" of each LED spectrum although it is derived from the datasheet now (directly from the known peak wavelength).

This leaves only one issue: what is the actual junction temperature (as opposed to the assumed values in the prelab)? Electrical power dissipation of the device (in Watts) is found by multiplying the bandgap voltage (measured in this experiment) times the current (10mA = 0.010A). The junction temperature is now found according to the formula:

Tjunction = Tambient + Rtheta * Power

where Tambient is the ambient temperature, Rtheta the "thermal resistance" of the device to ambient (found on the prelab datasheet and in units of "degrees C per Watt" and assumed to be the same for all devices even though it is not listed in all datasheets), and Power is the electrical dissipation of the LED we just found (i.e. current times voltage) in Watts. Note how the units cancel (°C/W * W = °C). Knowing the actual junction temperature (convert it to Kelvin degrees), the predicted wavelengths follow that of the prelab identically (which illustrates why it is so important to do the prelab assignments).

Expect the "most red" wavelength to be within 10nm or so of the peak wavelength - this small difference is the result of the small power dissipation of the LED resulting in very little heat produced (and hence "kT" is a small quantity, much smaller than the energy of the photon itself).

Finding the wavelength of the White and Infrared LEDs

For the white and infrared LEDs, determine the voltage at the 'knee' and using the same method above (i.e. drawing an asymptote and finding the intercept with the axis) and, using the two values for Planck's constant which you determined in this experiment (not the "accepted" value), determine a predicted wavelength (in nm) for each LED. The wavelength you predict will be the longest wavelength though (at the edge of the bandgap): Predict the PEAK emission wavelength according to that outlined in 2.10 of Csele by adding a thermal energy to the determined energy of the bandgap - calcualte this temperature by first finding power dissipation (current times voltage) and using the same thermal resistance as the prelab (and the rest of the lab). Peak emission wavelength is how commercial LEDs are rated.

Tutorial: Finding an Intercept using ExcelTM

Excel FInding an Intercept Using Excel to find an intercept as required to find the bandgap voltage for each LED: (This example assumes Excel 2013 ... actual procedure may vary slightly if a different version is used)

  1. Graph data in the usual manner: enter all experimental data into to columns (representing X and Y), highlight all the data points, and plot a graph. This is done for the example data given for the entire range x=1 through x=6.
  2. Determine the ranges required for the linear line - it does NOT include all data but rather only data lying on a linear line. In the example graph given, this range is defined by the x=4 through x=6 values. This is done by inspection so that ther range covers only the appropriate linear region (read your old notes on diodes to understand this).
  3. Right click on the existing graph, click "Select Data", then "ADD" a series. Select the data for the new series using the range you determined, in this case the x=4 through x=-6 values.
  4. Add a trendline though the new series. Be sure to forecast backwards one period (or more, as required), so that the line crosses the axis. Display the line equation on the graph as well.
  5. Finally, to find the intercept of the lines with the axis, set y=0 in the equation for the line and solve for x algebraically. The answer will be in terms of x= which is precisely what is desired. In the example given, (X-3)=0 so X=3 is the intercept. If this is an I/V graph, it is the bandgap voltage of the device.

Tutorial: Linear Regression using ExcelTM

In this lab, the slope of a line (rise over run) represents Planck's constant. Using ExcelTM to do this, we can determine several useful parameters about this line-of-best-fit including R-squared and standard error by analyzing the data statistically (which makes sense when presented with many points of data, each of unknown uncertainty). One important parameter we can determine, given a large array of data, is standard error.

What does standard error really mean? 95% of all observations will fall within ± (2* standard error). If you took reading from another LED, one would presumably find that point within that range ... it gives us a "prediction interval" which is really confidence in our value. Regardless, it offers a useful and easy analysis of our data in terms of how tightly grouped out data point were.

One can now, logically, express Planck's constant as h ± (2*standard error) which will give a confidence inerval of 95%. This is required in this lab.

Procedure:

  1. Graph your data
  2. Add a trendline and display both (a) the equation and (b) the R-squared value on the chart. R-squared represents how well these points fit the line and the slope of the line is h.
  3. Use either the LINEST function or the Data Analysis function (depending on your version of Excel) to analyze the data for the trendline.

Lab Report

Hand In a WORD PROCESSED (not handwritten) lab report with contents as outlined below.

Lab Report The FIRST PAGE must be a title page containing nothing more than the title of the lab, the course, the student's name and ID number, and your lab partner's name.

Answer each question as "1", "2", etc with each new question starting on a NEW PAGE so that question 2 starts on the top of a new page and question 3 starts at the top of a different page, etc. You'll have, therefore, at _least_ eight pages in this report and probably a few more than that.

The lab must be submitted in a report cover (either a three-hole punched cover or one with a clamp on the left side, not a binder), and NEVER as a stapled mass of loose papers.

Failure to follow this simple outline, used for all condensed labs in this course, will result in deduction of marks.

PLAGIARISM WARNING: All labs are to be done individually so while the data will (and must) be the same as your lab partner, each lab and each analysis presented must be your own unique work. Data presented must be your own as well, not "borrowed" from another group. Where two or more labs are copied, each student involved will receive a mark of zero and will be required to meet with the associate dean who will assess further penalties (including expulsion for repeat offenders).

Lab Report Binder Lab reports are to be submitted in a small, thin folder, NOT A BINDER.

LAB SUBMISSION:

Show all values such as bandgap voltages to FOUR digits of precision (e.g. 1.xxx volts). For each question below, show one complete set of calculations since part marks cannot be assigned where work is not shown (these calculations include all numeric values used).

    Determining Bandgap Energy:

  1. For each of the five visible LEDs used in this experiment do the following:
    1. Show all I/V data as a chart/table showing current and voltage pairs collected in the lab.
    2. Show a graph of the V/I characteristic for each LED. Identify the bandgap voltage as found using each graph (a minimum of four digits of precision is required). On each graph show the equation of the trendline covering ONLY the linear upper portion of the data as well as the equation for each trendline (in the form y=mx+b) plus the calculated intercept with y=0 (i.e. at zero current). This is covered in the tutorial above and must be done for each LED.
  2. You will have five scatter graphs (x/y), one per visible LED, each with all collected I/V points in a chart/table and each with a trendline to the linear portion (crossing the x-axis). Directly on the graph, display the equation for the trendline as well as the R-squared value (all with four significant digits).

  3. Show one example calculation (from one of the five LEDs observed) of an observed bandgap energy (from the graphs above, found using the equation of the line you placed on the graph), converted into Joules. Include a single chart/table with the headings "LED Colour", "BANDGAP VOLTAGE (V)", and "BANDGAP ENERGY (J)" summarizing the numbers calculated for all five LEDs investigated.
  4. Lab Report Chart
    These bandgap energies will be used in all further calculations (both methods).

    Method ONE Using observed wavelengths:

  5. Outline wavelength observations from the spectroscope (of the "most red" wavelength) for each LED in the form of a single chart/table with the headings "LED Colour", "Observed Angle (degrees)", "WAVELENGTH (nm)", and "FREQUENCY (Hz)". Since you checked the calibration of the spectroscope in the lab, these should be relatively reasonable observations
  6. Show a single graph of bandgap energy (in J, from Q2) vs. frequency (in Hz, from Q3) used to determine Planck's constant, including the best-fit line (a "trendline" in Excel) and finally a value of Planck's constant (which is, of course, the slope of this line in units of J/Hz or J-s). Include a table of the five data points (with headings "Energy (J)" and "Frequency (Hz)"), then plotted on the graph in addition to the "best fit" line. For accuracy, ensure the trendline intercepts (0,0) since a frequency of zero implies an energy of zero as well (in Excel, simply select the intercept option when formatting the trendline). Include a trendline label on the graph showing the equation of the line (which displays slope) and be sure to format the trendline label as a "scientific" number format with four significant digits of accuracy so that an answer of the form "x.xxx E xx" will be reported for slope (and hence the value of Planck's constant reported here). The value of "h" must be reported in this manner. An UNCERTAINTY figure is also required so the final answer would be in the form "h ± Δh" as discussed in the "linear regression" tutorial above. Explain, in words, how this uncertainty figure was found (and give numerical values). Ensure the graph is complete, like all other graphs, with axes titles, and several major/minor gridlines to show the values.
  7. You will have a graph with five Energy/Frequency points (one per LED) and with a single trendline fitting through the points. Display the equation for the trendline directly on the graph to four significant digits as well as the R-squared value. The final answer reported in each case will be in the form "x.xxx *10-xx ± y.yyy *10-yy J-s". Use of this method is required (i.e. do NOT compute a value of h for each LED separately).

    Method TWO Using datasheet wavelengths:

  8. Research then outline the theoretical wavelength for each LED as derived from datasheet parameters. Since the datasheet lists only peak wavelength (careful: it is NOT "dominant" wavelength) one must find the wavelength corresponding to the bandgap edge by subtracting thermal energy (kT) in the same manner as the prelab (see "Analysis" in the lab above which outlines the method). Compute the expected "most red" emission frequency from each LED (showing a complete set of calculations for one LED as an example of the method used: include calculation of junction temperature for this device, it will be different for each) then summarize the "theoretical/datasheet" parameters for all LEDs in the form of a single chart/table with the headings "LED Colour", "DATASHEET PEAK WAVELENGTH (nm)", "Thermal Energy kT (J)", "Bandgap Energy (J)", and "FREQUENCY (Hz)".
  9. Show a graph of bandgap energy (in J, from Q2) vs. frequency (in Hz, from Q5) used to determine Planck's constant, including the best-fit line (a "trendline" in Excel) and finally a value of Planck's constant (which is, of course, the slope of this line in units of J/Hz or J-s). Include a table of the five data points (with headings "Energy (J)" and "Frequency (Hz)"), then plotted on the graph in addition to the "best fit" line. For accuracy, ensure the trendline intercepts (0,0) since a frequency of zero implies an energy of zero as well (in Excel, simply select the intercept option when formatting the trendline). Include a trendline label on the graph showing the equation of the line (which displays slope) and be sure to format the trendline label as a "scientific" number format with four significant digits of accuracy so that an answer of the form "x.xxx E xx" will be reported for slope (and hence the value of Planck's constant reported here). The value of "h" must be reported in this manner. An UNCERTAINTY figure is also required so the final answer would be in the form "h ± Δh" as discussed in the "linear regression" tutorial above. Explain, in words, how this uncertainty figure was found (and give numerical values). Ensure the graph is complete, like all other graphs, with axes titles, and several major/minor gridlines to show the values.
  10. You will have a graph with five Energy/Frequency points (one per LED) and with a single trendline fitting through the points. Display the equation for the trendline directly on the graph as well as the R-squared value. The final answer reported in each case will be in the form "x.xxx *10-xx ± y.yyy *10-yy J-s". Use of this method is required (i.e. do NOT compute a value of h for each LED separately).

    Application:

  11. The analysis of the white LED - find the wavelength of the 'pump' LED. Include the V/I graph, determination of the bandgap energy from this graph (i.e. include the trendline with intercept, then find the bandgap in eV and then Joules), calculation of the energy of maximum photon emissions (by adding thermal energy as per the method in the prelab you submitted earlier), and calculation of peak emission wavelength by using the values of Planck's constant you determined in this experiment (NOT the accepted value). You may use either value of Planck's constant as determined above, which ever is closest to the accepted value. Report, as well, the optical observations of the output of this LED (especially the shortest and longest wavelengths seen). A little research into how a white LED operates will go a long way to understanding what you are seeing and why the bandgap voltage is higher than other LEDs observed.
  12. As usual, show a complete set of calculations for the above

  13. The analysis of the unknown IR diode - find the wavelength of peak emission. Include the V/I graph, determination of the bandgap energy from this graph (trendline, then intercept in eV and then Joules), calculation of the energy of maximum/peak photon emissions (by adding thermal energy as per the method in the prelab you submitted earlier which will be handed-back in class before the lab is due so that corrections can be made if your method was incorrect), and calculation of peak emission wavelength by using the values of Planck's constant you determined in this experiment (NOT the accepted value). You may use either value of Planck's constant as determined above. Be sure to report the serial number of the unknown diode (IRx).
  14. As usual, show a complete set of calculations for the above