Purpose
To demonstrate the existence and determine the size of the energy gap in a semiconductor by measuring its resistance as a function of temperature.
Equipment
Preparation and Lab Work
The thermoelectric cooler/heater is based on a phenomenon called the Peltier Effect. The Peltier Effect refers to the reversible heating or cooling of the junction between two dissimilar materials when a current is passed through the junction. The direction of the current determines whether heating or cooling takes place. Thus, if we have a circuit composed of materials A and B
Figure 2
junction 1 may be cooled and junction 2 heated, depending on the materials A and B and the current direction. The thermoelectric cooler/heater is simply a series of junctions of one type connected to a metal plate, i.e.
Figure 3
Thus, with the current in one direction, the plate is cooled, and vice versa. Since heat is generated in the set of junctions that are not cooled, this heat must be carried away when the current direction is such as to cool the plate. The fan at the finned heat radiator at the bottom of the thermoelectric cooler helps to accomplish this. The materials A and B in the thermoelectric cooler are impurity semiconductors of different types. One is the so-called n-type in which the current is carried by electrons in the conduction band; the other is the p-type, in which the current is carried by the positive "holes" in the filled band. Likewise, the thermistor, which is used to measure temperature, is also a semiconductor. The band-gap experiment, of course, is the measurement of the temperature dependence of a semiconductor. It turns out that by choosing the proper semiconductor material, one can use the temperature dependence of the resistivity as a thermometer. Thus one simply measures the resistance of the thermistor element and finds the equivalent temperature from the calibration chart supplied with the element.
The sample resistance will be only about one ohm at room temperature, so the resistance of the leads connecting to the sample could be a significant portion of the total resistance. To eliminate stray resistances a four-terminal connection can be made to the sample as shown in Fig. 2. This method works only if the voltmeter has an infinite input resistance so that no current flows through the potential leads and their connections at points 3 and 4. When this condition is satisfied, the current read on the meter is the same as current through the sample, and there will be no IR drops in the voltage circuit which would prevent measurement of the true potential between points 3 and 4 on the sample.
Figure 4
Also as a consequence of the small sample resistance, the potential difference established between 3 and 4 is small and the potential changes due to temperature changes may be only a fraction of this. Thus we must measure the voltage with a high resolution, sensitive, high input impedance device; a digital millivolt meter.
The sample resistance between points 3 and 4 in Fig. 4 is obtained by dividing the potential difference measured between 3 and 4 by the current flowing through the sample. Of course, the voltage changes will be increased, and thus more accuracy obtained, by increasing the current. There is a limitation on the sample current due to the I2R heating within the sample itself. This heating will lead to erroneous results and, if excessive, may even destroy the sample. We expect the resistance to decrease when the sample is heated, and thus by measuring the resistance at some fixed ambient temperature for several sample currents, you can find the maximum current which causes no noticeable heating.
The resistance measurements are greatly simplified if they can all be made at a constant value of the sample current, for then the resistance is obtained by dividing all measured voltage values by the same current value. In fact, since under these conditions the voltage is proportional to the resistance, which in turn is proportional to , it is not even necessary to calculate the resistance: the magnitude of the energy gap can be found by studying only the voltage produced across the sample at constant current. The power supply provided in the lab is designed to give either a constant voltage or constant current output. Constant voltage output resistance of the supply is vanishingly small since the IR drop across this resistance for various currents has no significant effect on the output voltage. A constant-current source, on the other hand, is characterized by a very large output resistance so that changes in the load resistance will not significantly affect the total circuit resistance, and thus the current flowing in the circuit will be independent of the load resistance. Clearly, a constant-voltage supply can be converted to a constant current supply by adding a large resistance in series with the load. You may use either the current mode control on the supply or voltage with suitable resistors. Of course, too large a series resistance will prevent enough current flowing through the sample, even at maximum power supply voltage, for accurate resistance measurements, so some compromise must be made.
The semiconductor sample whose resistance you will measure is mounted on a copper plate imbedded in a plastic disk. The current and voltage leads to the semiconductor are arranged as shown in Fig. 4 and are connected to banana plug sockets in the plastic disk. The plastic disk also has connecting sockets for the thermistor leads. Temperature is determined by measuring the resistance of the thermistor and using the resistance versus temperature chart supplied. The copper plate on which the semiconductor is mounted is thermally connected to the cooling surface of the thermoelectric cooler with a thermally conducting paste. There is a regulated power supply to apply current to the thermoelectric cooler.
You will first want to determine which current directions cause cooling and heating, respectively. Apply a small current (~0.1 amperes) to find out. The temperature range that you can safely cover with the thermoelectric cooler/heater is about -15deg. to +50deg.C. Remember not to exceed 1.8 amperes in any case. The sample will heat much more rapidly than it cools. You will be measuring the temperature (by measuring the resistance of the thermistor) as the temperature changes. Simultaneously, you will be measuring the resistance of the sample. Therefore, you don't want the temperature to change too rapidly to take down the data. There is another reason for not changing temperature too rapidly. It is important that the temperature at the thermistor is the same as the sample temperature. They should be in thermal equilibrium. Too rapid temperature changes will prevent the necessary thermal equilibrium. A good way to tell if you have thermal equilibrium is to compare data obtained while heating and cooling. Since cooling is fairly slow, you can start the cooling run with a current of ~0.7 amperes. To warm the sample, you can reduce the current until you each zero current. At that point reverse the current leads so that the thermoelectric element heats the sample. Be very careful while heating not to step up the current too rapidly and be sure not to exceed +50deg.C. Look up the resistance of the thermistor before you start heating so that you don't exceed that temperature. Your heating current steps should only be a few tenths of an ampere. You won't need 1.8 amperes to get to 50deg.C. By reducing the current, you can now cool back to room temperature. Repeat the heating/cooling cycle several times to determine reproducibility and whether you have achieved thermal equilibrium.
The simplest way to analyze your data is by means of a graph. If the energy gap model is correct, we expect the resistance (or the voltage, if data are taken at constant current) to depend on the absolute temperature T according to
.
(Hint: consider putting the data on semilog paper as log R vs T-1.) It is usual to express Eg in units of electron volts, so be sure you know how to do this with your result. Remember in analyzing your data that 0deg.C = 273.2deg.K, and that all temperatures must be expressed on the absolute scale.
Basic Lab Measurements
CAUTION
Do not connect the power supply directly to the sample because there is danger of inadvertently putting too much current through the small sample and damaging it. Start with about 100 ohms in series with the supply, and then look for the maximum current that does not heat the sample.
Include these measurements in your report.
Measure the resistance of the indium antimonide sample as a function of temperature from about -15deg.C to +50deg.C. Check and comment in your report on the thermal equilibrium between the sample and the thermometer. From your data find the energy gap of the InSb sample and give your answer in electron volts. Include some measure of the experimental uncertainty. How well do your data fit the theory? Use the calibration chart data for the thermistor to determine whether it behaves like a semiconductor. (Hint: Try graphing the data). Does it have an energy gap? If so, what is its value?
TEMP C RES TEMP C RES TEMP C RES TEMP C RES TEMP C RES 3356K 39 565.5K 9 125.5K +21 34.78K 51 11.54K 81 3147K 38 535.6K 8 119.8K 22 33.44K 52 11.15K 82 2951K 37 507.5K 7 114.5K 23 32.15K 53 10.78K 83 2769K 36 481.0K 6 109.4K 24 30.92K 54 10.42K 84 2599K 35 456.0K 5 104.5K 25 29.74K 55 10.08K 85 2440K 34 432.4K 4 100.0K 26 28.61K 56 9744 86 2292K 33 410.0K 3 95.51K 27 27.53K 57 9424 87 2154K 32 389.2K 2 91.34K 28 26.50K 58 9117 88 2025K 31 369.4K -1 87.38K 29 25.50K 59 8821 89 1904K -30 350.7K 0 83.60K 30 24.56K +60 8536 +90 1791K 29 333.1K +1 80.00K 31 23.65K 61 8261 91 1685K 28 316.4K 2 76.58K 32 22.77K 62 7996 92 1586K 27 300.6K 3 73.32K 33 21.94K 63 7741 93 1494K 26 285.7K 4 70.22K 34 21.14K 64 7496 94 1407K 25 271.6K 5 67.26K 35 20.37K 65 7259 95 1326K 24 258.3K 6 64.44K 36 19.63K 66 7030 96 1250K 23 245.7K 7 61.75K 37 18.93K 67 6810 97 1178K 22 233.8K 8 59.19K 38 18.25K 68 6598 98 1111K 21 222.5K 9 56.75K 39 17.60K 69 6393 99 1049K -20 211.9K +10 54.42K +40 16.97K +70 6195 +100 989.8K 19 201.7K 11 52.19K 41 16.37K 71 6005 +101 934.6K 18 192.2K 12 50.07K 42 15.80K 72 5821 102 882.7K 17 183.1K 13 48.04K 43 15.25K 73 5643 103 834.0K 16 174.5K 14 46.11K 44 14.72K 74 5472 104 788.2K 15 166.3K 15 44.26K 45 14.21K 75 5307 105 745.2K 14 158.6K 16 42.50K 46 13.72K 76 5147 106 704.7K 13 151.3K 17 40.81K 47 13.25K 77 4993 107 666.7K 12 144.3K 18 39.20K 48 12.79K 78 4844 108 630.9K 11 137.7K 19 37.66K 49 12.36K 79 4700 109 597.2K -10 131.4K +20 36.19K +50 11.94K +80 4561 +110