THE EXPERIMENT

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

  1. Digital millivoltmeter;
  2. Power supply with meter (0-200ma, 0-30V);
  3. Resistance decade box;
  4. Sample of InSb (indium antimonide, a semi-conductor) mounted in a protective holder;
  5. Thermistor;
  6. Thermoelectric cooler/heater;
  7. Semilog graph paper;
  8. Power supply for thermoelectric cooler/heater.

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.

! VERY IMPORTANT !

Do not apply more than 1.8 amperes to the thermoelectric cooler. Higher currents will burn out the elements.

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?

Resistance vs. Temperature for the sensor thermistor

  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