Physics 43, Fall 1995 Lab 1 - Boltzmann's Law

Transcription

1 Physics 43, Fall 1995 Lab 1 - Boltzmann's Law Introduction Boltzmann's law states that the probability of occupation of a state of energy E, in a system in equilibrium with a heat bath at temperature T is given by Z -1 exp(-e/k B T), where Z = Sexp(-E/k B T) is the "partition function", and the sum is over all possible states. It follows that the probability p(u) of a particle overcoming an energy barrier of height U (see Figure 1) is proportional to o f(e)exp[-(e+u)/k B T]de where f(e) is the density of states in the barrier region (i.e. the number of states per unit energy interval) and e is the energy measured from the top of the barrier. This can be integrated by making the substitution x = e/k B T, giving p(u) = g(t)exp(-u/k B T) (1) where g(t) is a relatively slowly varying function which depends on the precise form of f(e). Energy U Distance Figure 1 Bipolar Transistor One example of particles overcoming a barrier by their thermal energy is the motion of electrons in an n-p-n transistor. The precise mechanism of transistor action need not concern us here: what matters is that electrons move in the potential profile sketched in Figure 2. An electron has a certain probability, given by Equation 1, of occupying a state above the barrier. There are three essential parts to a transistor: the "emitter" (E), the "base" (B) and the "collector"(c). The barrier between the collector and base is sufficiently high that the probability of an electron in the collector getting into the base is negligible. If an electron from the emitter reaches

2 Figure 2 the base, it falls into the collector region and cannot get back. Hence the current from emitter to collector, I c, is proportional to p(u) *. We connect voltage sources to the transistor, as shown in Figure 3. To protect the transistor we put a resistor in series with the power supply. It plays no role in our measurements; we measure currents and applied voltages directly at the transistor. Figure 4 shows a schematic diagram for this circuit. In the standard symbol for an n-p-n transistor, E stands for emitter, B for base and C for collector. The arrow points outwards because electrons flow from the emitter to base, and since the electron charge is defined as negative current then flows outwards. Electrons from the emitter pass over the barrier formed by the base region and fall into the deep potential valley of the collector. The height U of the barrier is controlled by varying the voltage V BE. Note that when no external voltage is applied there is still a "built in" energy barrier, and the effect of applying a voltage in the direction shown (the "forward direction") is to reduce its height. * This cannot be exact since it predicts a non-zero current for V=0. This is because we have assumed that the current is dominated by diffusion, neglecting the "generation-recombination" current which exactly cancels the forward current at V=0. This has a different dependence on U but is typically ~10-9 A or less, and its neglect is justified in your experiment, where the smallest current measured is ~10-7 A. See Grove for a full treatment.

3 Emitter Base Collector 100 ohm V EB I c - + Power supply Figure 3 Figure 4 Fig. 4 The collector current I c is proportional to the probability that an electron will overcome the barrier, so it is given by Ic = A(T)exp[-(φ-qV BE )/k B T] (2)

4 where φ is the height of the barrier in the absence of an applied voltage and A(T) is a slowly varying function of T. In the simplest model of the transistor, A(T) µ T 3/2, and is independent of U (see question 1 at the end of the writeup). Equation 2 can be written Ic = I o exp[qv BE /k B T] (3) where I o = A(T)exp[-φ/k B T] (4) Procedure A. Measure Ic as a function of V BE. Take at least 20 values of V BE between 0.3 and 0.7 V. Monitor the temperature carefully during your measurement, and if necessary use Equation 2 to correct the measured current to what it would have been at constant temperature (see question 3). Plot log(i c ) against V BE (using log-linear paper) test the validity of Equation 3. You will find that at currents greater than about 100 ma, Equation 3 no longer holds. Is the voltage higher or lower than predicted? Suggest reasons why this should be. Can you use your data, or make further measurements, to distinguish between the different possible explanations? From the temperature and the slope of your log-linear plot, obtain the ratio q/k B. Make a realistic estimate of its uncertainty, distinguishing between random and systematic errors. B. CAUTION! Hot mineral oil is used in this part of the experiment. Extreme care must be taken to avoid spilling or splashing: the oil can cause severe burns, and the alcohol can damage your eyes. Always wear goggles, and contact lenses may not be worn. Wear gloves to handle the dry ice. Never leave the beaker on the hot plate unattended. Be VERY careful not to contaminate the oil with water or alcohol, which will erupt at the boiling point. Wipe all moisture off the transistor before immersing it in the oil. Put about a pound of dry ice on the insulated container. Slowly add the alcohol to the ice until there is at least two inches of alcohol above the ice. Add the alcohol (ethanol or isopropanol) in small amounts and wait for the bubbling to subside before continuing. Suspend the transistor in the alcohol. Do not lay the transistor directly on the dry ice, but use the lid from the

5 container to suspend the transistor above the ice. Measure the temperature in the vicinity of the transistor and repeat the experiment as in part A. Make sure that the temperature has settled down and doesn't change by more than about a degree during the course of your measurements. Again, monitor it and correct for any fluctuations. Fill the beaker with mineral oil to around the 300 ml line. Place the beaker on the stand in the aluminum pot and fill the pot with water until the water level is equal to the oil level. Turn the hot plate control to "High", insert the thermometer in the mineral oil. When the temperature of the oil stabilizes (around 91 C) proceed with the measurements. Note: it will take about a half an hour for the temperature to stabilize. For both parts of B, you will have to change the range of V BE to keep the current in the range that equation 3 holds. Obtain q/k B as before. The three values of q/k B may differ from each other by more than the experimental uncertainty. Suggest reasons why this might be. Plot ln{i o T -3/2 } against 1/T, and hence obtain φ/q, the barrier height in electron volts. Since the extrapolation of your plot to V = 0 is rather unreliable, compare the values of I o that you get from these plots with those obtained by assuming the "accepted" value of q/k B. From your data, estimate the validity of the assumption that A(T) µ T 3/2. Reading Boltzmann's Law. * Kittel and Kroemer,Thermal Physics. p Reif, F., Fundamentals of statistical and thermal physics p Mandl, F., Statistical Physics sec Transistor A. S. Grove Physics and technology of semiconductor devices sections 6.6b and 7.2 (this material requires some knowledge of solid state physics, and is not essential to the understanding of this experiment. However, reading it will give you an idea of how much more complex the situation is than the simple picture given here).

6 Lab Report One writeup per lab group (if one partner works more on this than the other, you may mutually agree to split the credit unequally; let me know if that's the case. Obviously it is better to work equally). Include in your writeup a brief description of the experimental setup (equipment used and sketches of how), tables of your original data (with appropriate labels and units), plots (with labels and units), and a clear indication of how fundamental constants (or ratios thereof) were obtained from the data and/or plots. Your writeup should be typed, though sketches may hand drawn. Also include answers to the following questions: 1. Prove eq 1, and show that with the density of states for electrons which we will see in Mandl eq , f(e) µ e 1/2, it follows that g(t) µ T 3/2. There's no need to work out the constant of proportionality. 2. Physical constants can be crudely divided into "macroscopic" quantities such as the gas constant R, the Faraday F, and atomic or molecular "weight", and "microscopic" quantities such as Boltzmann's constant k B, the charge on the electron q, and the actual mass of an atom. Look up the definitions of R and F if you don't know them. Does this experiment give you the value of any microscopic quantity? What does it actually measure? A certain manufacturer of laboratory equipment describes an experiment, in principle identical to this one, as a "measurement of the charge on the electron". Is this claim justified? 3. In the silicon power transistor used here, φ ~ 1 ev and I o ~ 1 A. How large a change in temperature is needed to produce a 10% change in the current, when I ~ 1 µa? 4. Why do we use a transistor capable of handling a current ~ 10 A, when we're only going to measure current up to a few hundred ma? 5. How does the resistor in series with the emitter protect the transistor? 6. When a transistor is in practical use, a potential of a few volts is applied to the collector (in the direction to increase the barrier in Figure 2), and there is a resistor R L (the "load") in series with the collector (see Figure 5). If the voltage between the base and emitter V BE is changed by a small amount DV BE, a much larger voltage change DV CE occurs across the load resistor, so that the transistor acts as an amplifier. Why does this occur? If the collector current is I c, calculate the voltage gain G DV CE /DV BE. If I c = 10 ma, what value of R L is needed to give a gain of 100?

7 If the supply voltage is 5 V, what is the largest value of G that can be obtained from this circuit? Does this depend on the value of I c? Figure 5 Fig. 5