Measuring Silicon and Germanium Band Gaps using Diode Thermometers Haris Amin Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: April 11, 2007) This paper reports the band gaps of silicon and germanium diodes using diode thermometers. We used diode thermometers to measure the temperature and voltage relationship at a constant current. From this we found the band gaps to be 0.63±0.01 ev and 1.29±0.03 ev for germanium and silicon diodes respectively. The uncertainties in our data do not fall in the range of accepted values of band gaps for intrinsic germanium (0.67 ev) and silicon (1.12 ev) at room temperature (300 K) [4]. We were also able to confirm the linear relationship between the temperature and voltage in a diode at a constant current. Semiconductors are an integral part of modern electronic devices. From computers and cellphones to calculators and digital watches, almost all the technology we use in our every day lives is associated with a semiconductor. So what makes a semiconductor so special? An essential electrical characteristic of semiconductors is that they do not allow energy ranges between the valence and conduction bands of the device. This energy gap between the two bands is also known as the band gap of a semiconductor. Semiconductor diodes, in conjunction with a constant current source, can also be used as thermometers. It has been shown experimentally that, within a certain temperature range, the relationship between temperature and voltage is almost linear given a constant current source flowing through the diode [1]. In this Letter, we will be testing the validity of this model by measuring the band gaps of silicon and germanium diodes and comparing them with accepted values for the intrinsic band gaps for germanium and silicon. We will be using a least-squares fit to the experimental data to determine the parameters of our model yielding to experimental values for the band gaps. The p-n junction bears some interesting properties which have useful applications in modern electronics. The p-doped semiconductor, which contains a number of free positive charge carriers (holes), is relatively conductive. The same is true of n-doped semiconductor, which contains a number of free negative charge carriers (electrons). However, the junction between the two is a nonconductor. The nonconducting layer, known as the depletion zone, occurs because the electrons and holes attract and eliminate each other. Diodes are essentially just these p-n junctions which are manipulating the depletion zone. Diodes allow a flow of electricity in one direction but not in the opposite direction [3]. The current-voltage relationship of a p-n junction can be described by the ideal diode equation found in many texts, I = I 0 [e ev/kt 1], (1) where I is the current through the diode, I 0 is the maximum current fora large reverse bias voltage, e in the ex-
ponent is the electron charge, V is the voltage across the diode, k is Boltzmann s constant, and T is the absolute temperature in Kelvin [4]. We need an expression for the reverse current I 0, which depends strongly on temperature but not on V. It can be shown that I 0 is proportional to the Boltzmann factor e ev/kt and to T 3+γ/2, where γ is constant [1]. γ is a constant that depends upon the temperature dependence of the mobility, lifetime, and diffusion coefficient of minority carriers which are particular to a diode [2]. Thus we are considering γ to be constant assuming that we will not be putting the diode under extreme temperatures ( near 0 K or at very high temperatures well exceeding the melting point of the material) that will change the aforementioned characteristics of the minority carriers of the diode [1]. We can then write I 0 as, 2 I 0 = AT 3+γ/2 e Eg/kT. (2) Neglecting the T 3+γ/2 dependence of I 0 in comparison to the exponential dependence on T and treating B = AT 3+γ/2 as almost a constant [1], we can rewrite Equation 2 as, I 0 = Be Eg/kT. (3) If we then combine Equation 1 and Equation 3 and neglect 1 in Equation 1 assuming that e ev/kt >> 1 [1][2] we get, I = Be Eg/kT +ev/kt. (4) We then maintain a constant current I and can write C = ev/kt E g /kt, (5) where C = ln(i/b). We can then then write the the energy ev as a function of temperature T, ev = kct E g, (6) which is the linear relationship we are looking for [1]. Remember that e here is the charge of an electron. We can then perform a least-squares fit on our T V curve to obtain the intercept. This intercept is our band gap E g. We will now evaluate the value of E g for silicon and germanium using their respective diodes. Figure 1 illustrates the schematics of the circuit used in our setup. It consists of a low voltage power supply connected in series with an ammeter and forward biased diode. Connected in parallel to the diode is a voltmeter which measures the potential difference of the current flowing through the diode. The diode connected to the circuit will be placed in a system where its temperature can be varied and monitored. Figure 2 illustrates the
this system. The diode in question is placed in a 100 ml pyrex beaker where a thermometer and a thermocouple are placed to measure the temperature of the diode. The beaker is also filled with pump oil in order to insure that the thermocouple is also insulated from any charge. To further insure proper insulation the diodes were coated with quick drying epoxy glue. The 100 ml pyrex beaker is then placed in a larger 500 ml pyrex beaker. In order to vary the temperature of the diode, the 500 ml beaker is heated via a hot plate for temperatures above room temperature. In order to vary the temperature below room temperature the 100 ml beaker is submerged in an ice bath and for temperatures below 0 C an ice and CaCl bath inside the 500 ml beaker. 3 FIG. 1: The current flows from the power supply to the ammeter. Here the ammeter measures how much current will be flowing through the circuit right before the current passes through the diode. The voltmeter attached to the two ends of the diode tells us what the potential difference is in the diode. We took voltage readings for a current range of 7 µa to 12 µa at each temperature setting for the diode. In order to account for a constant current reading for each temperature, we extrapolated our data for the value of 10 µa for each temperature setting.the range of the low voltage power supply was an ungrounded 0 to 24 V.The ammeter was in the range of 200µA. The voltmeter was in the range of 2 V. The uncertainty for the measurement of the current is ±0.005µA and ±0.0001V for the potential. As Figure 1 mentions, instead of providing a constant current source to the diode, we can take voltage readings for a range of current (7 µa to 12 µa), extrapolate our data, and then take a common current reading (10 µa) at each temperature to form our T V curve. Figure 3 displays the T V curves for the silicon and germanium diodes that we obtained from our results. Note that at a constant current we are indeed observing a linear relationship between the temperature and voltage in our diodes.
4 FIG. 2: The diode has been coated with a couple of layers of epoxy glue to make sure it is properly insulated. It is also immersed in pump oil so that the thermocouple is also insulated from any surrounding charges. For temperatures near 0 C, the 100 ml beaker was immersed in a larger 500 ml beaker with an ice bath. For temperatures below 0 C, the 500 ml beaker contained a CaCl-ice bath. For higher temperatures the 100 ml beaker was heated with a hot plate. Before each voltage reading was taken, we stirred the pump oil in which the diode was immersed to insure that thermal equilibrium had been reached for each measurement. Care was also made to insure that neither the thermocouple or the thermometer were in direct contact with the walls of the beaker or the diode. We also calibrated the thermocouple and thermometer before taking measurements. The thermocouple was observed to be not not as precise as the thermometer in measuring the exact temperature. However, the thermocouple, in agreement with the thermometer, helped us confirm that the temperature of the pump oil the diode was immersed in was constant. We found E g to be 0.63 ± 0.01 ev for the germanium diode and 1.29±0.03 ev for the silicon diode. Our values are not quite in agreement with the accepted E g values for intrinsic germanium, 0.67 ev, and intrinsic silicon, 1.14 ev, at room temperature (300 K). Our values are close, but even taking our uncertainties into account, we do not fall quite in the region of the accepted values. However, it is important to to note that we are comparing our E g for each diode to the accepted E g value of it s intrinsic material. The silicon and germanium material in our diodes have actually been doped. In semiconductor production, doping refers to the process of intentionally introducing impurities into an extremely pure (also referred to as intrinsic) semiconductor in order to change its electrical properties [3]. Our diodes went under the same process when they were produced. Thus their E g will and
should not match the E g of their intrinsic cousins. We may have been able to reduce the error by having more control over the temperature of the system and being able to carefully monitor the response to current change more quickly. However, the fact that we are comparing the E g of doped materials to their intrinsic materials would still cause a disparity between the two values. We were successful in measuring the E g values for silicon and germanium diodes relatively close to the E g values of their intrinsic materials. Using diode thermometers, we were also successful in illustrating the linear relationship between the temperature and voltage in diodes given a constant current. Our method uses simple electronics to demonstrate these two observations. Other methods often opt to use a more complex method for controlling the current current source using Field Effect Transistors (FET) [2]. Though our method may have lacked great precision, it does provide a coherent and simple approach to learning about the nature of semiconductors. 5 [1] Precker, Jurgen; da Silva, Marcilio, Experimental estimation of the band gap in silicon and germanium from the temperature-voltave curve of diode thermometers, American Journal of Physics, Issue 70, November, 2002. [2] Kirkup, L. ; Placido, F., Undergraduate experiment: Determination of the band gap in germanium and silicon, American Journal of Physics, Issue 54, October,1986. [3] Thornton, Stephen T. ; Rex, Andrew, Modern Physics for Scientists and Engineers Third Edition, Thomson Books, 2006. [4] Diefenderfer, Holton, Principles of Electronic Instrumentation Third Edition, Harcourt Brace and Company, 1994.
6 0.8 A Energy [ev] 0.7 0.6 0.5 0.4 0.3 0.2 0.1 E_g = 0.63 ± 0.01 ev Least Squares Curve Fit Data 0 1.4 1.2 0 80 160 240 320 400 E_g = 1.29 ± 0.03 ev B Energy[eV] 1 0.8 0.6 0.4 0.2 Least Squares Curve Fit Data 0 0 80 160 240 320 400 Temperature (K) FIG. 3: A)This figure the illustrates T V relationship for a germanium diode gathered from our results. The temperature and voltage indeed do seem to be linearly related. The intercept of the plot represents the E g of the germanium diode. We found it to be 0.63 ± 0.01 ev which is not quite in the range of 0.67 ev given our uncertainties. Our uncertainties were crudely measured by taking steep and shallow possible fits to our data. More careful analysis for the uncertainties might be needed. B)This figure the illustrates T V relationship for a silicon diode gathered from our results. Again, the temperature and voltage indeed do seem to be linearly related. The intercept of the plot represents the E g of the silicon diode. We found it to be 1.29 ± 0.03 ev which is not quite in the range of 1.14 ev, given our uncertainties. Our uncertainties again were crudely measured by taking steep and shallow possible fits to our data. More careful analysis for the uncertainties might be needed.