BACKGROUND AND THEORY

The results of Experiment 3 appear to indicate that, in some cases, positive charges can carry the current in a semiconductor. On the other hand, there is much that leads us to believe that electrons carry the current in conducting media. Before giving up this notion, an attempt should be made to formulate a model in which these seemingly contradictory pieces of information might be compatible. Any model which could perform this task would very likely contain a host of other predictions. By checking these predictions we can check the model and thus obtain an understanding of electrical conduction itself. The model described below constitutes our present understanding of this field. When it is analyzed in detail, it predicts with remarkable accuracy the physical properties which are observed experimentally.

First, we must consider the basic structure of a solid, i.e., the regular periodic arrangement in space of its constituent atoms. That the atoms do indeed form a periodic array is well verified by the observation of diffraction patterns (similar to those of Experiment 4) which are produced when radiation with wavelength comparable to the distance between the atoms (X-rays) is passed through the solid. Now, we are supposing in this model that the outermost atomic electrons can be free to move, as seems to be the case for electrical conduction, even though we know that in the isolated atoms these electrons are tightly bound to the atom. We can imagine these outermost valence electrons as forming a "sea" of negative charges within the material The neutral atoms are ionized when they lose their valence electrons to this sea, and it is these positively charged ions which make up the regular array or "lattice" observed by the X-ray diffraction experiments.

In order to solve for the motion of the electrons of this sea, the problem can be divided into two parts. The first considers the interaction forces between the electrons and the lattice of ions; the second considers the interactions of the electrons with each other. Only the first part will be discussed below and it will be found that many accurate predictions concerning the behavior of electrons in solids can be made without ever considering the much more difficult second part.

The interaction between electrons and ions is described by a potential energy which is a minimum at each ion, where the attractive force is greatest, and is therefore periodic in space. In the microscopic domain of electrons and atoms we know that the principles of quantum mechanics must be applied in order to correctly describe the behavior of particles. Two features of quantum physics are of particular importance in this problem. One is that the electrons can not assume arbitrary values of energy and momentum. Depending on the boundaries of and forces inside the solid, a set of discrete energy or momentum values is found, and these are the only values of energy or momentum which the electron can assume. The other feature is the startling condition that only one electron can be in a given state at a given time, i.e., only one electron in the solid can have a given momentum. (Actually, to specify the "state" of an electron more than just the momentum must be known, and it turns out that two electrons - but no more than two - can have the same momentum.)

With these two features in mind the solution for the electron in a periodic potential can be discussed. In particular, we would like to know what are the allowed values of energy and momentum in a specimen of material. To actually solve the equations of motion would take too much time for the purposes of this course, but the result of such a calculation is shown in Fig. la (with a gross exaggeration of the separation between the individual energy levels).

Figure 1

Fig. 1b shows the allowed energy levels for the same specimen which would be obtained if the ions were smeared out into a uniform background of positive charge rather than a periodic array of charges. Notice that the "gaps" labeled on Fig. la have disappeared and that the quantum energy states E are related to the allowed momentum states p by the equation

which happens in this case to be the same as the familiar classical (i.e., Newtonian and not quantum-mechanical) relation between energy and momentum for particles in a uniform potential. Note that the plot in Fig. la for electrons subject to a periodic potential, as in a real solid, is similar to Fig. 1b when not too close to the energy gaps. The fine level structure indicated by the dots making up the curves is a consequence of confining the electrons to a limited region of space (finite sample size), but the most striking feature of Fig. la, the appearance of wide gaps in the spectrum of allowed energy states, is due solely to interaction of the electrons with the periodic potential of the ions.

In reality, on the scale of Fig. 1, the dots representing the discrete allowed energy states would be so close together that they would appear to form a continuous curve - except for the gaps. The allowed states are seen to cluster into "bands", where the discrete energy levels are so close as to form a nearly continuous set, separated from each other by gaps in which there are no allowed energy levels.

The fact that there are such gaps in the energy spectrum is a surprising but very important result. The gaps are many orders of magnitude larger than the normal spacing between successive allowed energy levels.

Now, if there is no energy source present to excite the electrons out of their lowest possible energy states, then each state starting from zero energy will be occupied by one electron until All of the electrons are accounted for. If, at this point, a band of states is fully occupied with no electrons left over, then no current can flow. To understand this, consider the following. Any attempt to accelerate electrons to new states by applying a voltage across the sample can only interchange electrons between states of the filled band, unless enough energy can be provided to excite the electrons across the gap into the next higher (empty) band. However, the energy gap is usually so large compared to the energy acquired by the electrons for ordinary applied voltages that virtually no electrons can overcome the energy gap, and thus no electrons are available to participate in a flow of current. The total electron momentum of the filled band is zero to begin with and, since no net change of momentum can result from a mere interchange of electrons, it must remain zero.

The above is a description of the behavior of an insulator. Thus unexpectedly the model for "free" electrons offers an explanation of an electrical property of insulators. If, on the other hand, the number of electrons is such that one of the bands is only partly filled, then small voltages can easily accelerate electrons to adjacent energy states within the same band, and metallic conductivity will follow.

We now consider the intermediate case between an insulator and a metallic conductor, namely a semiconductor. Normally we deal with materials at finite temperatures in which case the electrons will not remain in their lowest energy states. This is analogous to the more familiar case of the molecules of a gas in which the average velocity of the molecules, and thus also their energy, increases as the temperature increases. For the electrons in a metal, for example, at finite temperatures, it is not true that all states are filled up to a maximum energy beyond which all states are empty. Instead there is a region around this "maximum" energy (referred to as the Fermi energy) in which some levels are occupied and others unoccupied. The size of this region increases and therefore the average energy gets larger as the temperature is raised. In the case of the completely filled band (insulators) considered earlier, the possibility exists that the gap is small enough so that at finite temperature some electrons will have energy above the gap and a corresponding number of unoccupied states will be left in the otherwise filled band. Thus a small number of charges can be moved with the application of a potential and we have described a semiconductor. We see that a semiconductor is an insulator at absolute zero, but the gap between the last filled band and the first empty one is small enough that an appreciable number of electrons are thermally excited over the gap at room temperature. Furthermore, the apparently improbable results of Experiment 3 (referred to in the first paragraph of this introduction) can be explained on the basis of this model of a semiconductor at finite temperature. As described above, a band with every state occupied can carry no current. However, if only one state with finite momentum is unoccupied then there will be one less electron moving in the direction of that momentum state than in the opposite direction and a net current will flow. The application of a potential will result in the acceleration of all electrons and of the unoccupied state or "hole" as well. The motion of all of the electrons is most simply described in terms of the motion of the hole. The hole, being an absence of negative charge, appears to be a positive charge. A more detailed analysis confirms this model in every way, and so even though only the negative electrons are free to move, it appears that the charge carriers have a positive sign.

Rather than attempt an understanding of these details it would be more valuable for the purposes of this course to verify experimentally the basic idea of the energy gap. It is known from both theory and experiment that the number of electrons which will have an energy greater than E above their lowest possible energy state will be, on the average, proportional to , where k is Boltzmann's constant and T the absolute temperature. In the case of a pure semiconductor, a detailed calculation based on the hypothesis of an energy gap of width E_g gives the result that the number of electrons per unit volume n excited above the gap is proportional to . The extra factor 1/2 in the exponent is a purely quantum effect which has been included for completeness; the essence of the result lies in the exponential dependence of the excited electron density n on l/T which characterizes a system with an energy barrier or gap. The proportionality factor also depends on T, but its variation is very much slower than the exponential function. Thus, any property of the semiconductor which depends on the number of electrons on top of the gap should exhibit an exponential dependence on temperature. Observation of such a temperature dependence would not only confirm the existence of the gap but would determine its value as well.

The electrical conductivity is just such a property. Intuitively one would expect that the number of electrons ( and holes ) which are involved in the conduction process will be an important factor in determining the current which will flow when a voltage is applied (remember that the current is the charge on each carrier times the number of charge carriers per unit time which flow through the material). quantitatively, the number of charge carriers (electrons or holes) per unit time which cross unit area perpendicular to their average velocity is . Now, does not depend strongly on n so that the current which flows for a given voltage is proportional to n. Since I = V/R, the resistance R is inversely proportional to the density of electrons.

Therefore R.