1RR 9/00 SS 12/01 Physics 342 Laboratory The Electronic Structure of Solids: Electrical Resistance as a Function of Temperature Objective: To measure the temperature dependence of the electrical resistance of a metal and semiconductor and to interpret the observed behavior in terms of the underlying band structure of the solids. Apparatus: Electrical furnace, NiCr-Ni thermocouple, variac power supply for furnace, CASSY power/interface, current module (524-031), thermocouple module (524-045), computer, Pt resistor, semiconductor resistor. References: 1. W. Pauli, Z. Physik 31, 373 (1925). 2. E. Fermi, Z. Physik 36, 902 (1926). 3. P.A.M. Dirac, Proc. Roy. Soc. London A 115, 483 (1926). 4. E. Wigner and F. Seitz, Phys Rev. 43, 804 (1933) and Phys. Rev. 46, 509 (1934). 5. N.F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, Oxford University Press, Oxford, 1936. 6. D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics; 5th Edition, Wiley and Sons, New York, 1997; Part 5, pgs. 1053-69. 7. K. Krane, Modern Physics, 2nd Ed., Wiley and Sons, New York, pgs. 309-29 and pgs. 344-62. Introduction An understanding of how much current flows through a conductor for a given applied voltage resulted from Georg Ohm’s thorough work in 1827. The empirical relationship known as Ohm’s Law has remained valid over the years and is still widely used today. Although Ohm’s work focussed primarily on metals, studies by Seebeck in 1821 and by Faraday in 1833 reported anomalies in current flow through a class of materials we now know as semiconductors. Interestingly, the temperature dependence of current flow measured by Faraday in semiconductors was quite different than the temperature dependence of current flow in metals first reported by Davy in 1820. The fundamental origin of this difference remained unexplained for about a century until the development of quantum mechanics. Following the successful quantum theory of electronic states in isolated atoms, attention turned toward a better understanding of electronic states in molecules and solids. Only with the completion of this effort was it possible to understand the implications of the simple observations about the temperature dependence of current flow made in the early 1800s. It is now well established that any property of a solid, including its electrical resistance, is in some way controlled by the electronic states of that solid. As a way of introducing the important differences between the electronic structure of metals and semiconductors, you will measure the temperature dependence of the electrical resistance2of samples made from these two important classes of materials. Before beginning these measurements, it is useful (without paying undue attention to many of the details) to review i) the modifications to electron states as we move from the atomic to the molecular to the solid state and ii) a simple physical model for current flow in solids. Theoretical Considerations A. Electronic Structure The important features of an isolated atom are a nucleus surrounded by a complement of electrons that are associated with it in a specifically defined manner. The Pauli exclusion principle requires these electrons to be non-uniformly distributed around the nucleus in regions of space, forming ‘shells’ of charge known as atomic orbitals. The total negative charge of the electrons exactly balances the total positive charge contained in the nucleus. Most importantly, the electrons, because they are confined to a limited region of space, acquire quantized energy levels. As atoms are brought together to form a molecule, the outermost electrons from one atom will interact with the outermost electrons of a neighboring atom. This interaction is subtle and a variety of theories have been devised to explain it accurately. The end result is a profound modification to the allowed energies and spatial arrangement of the electronic states. Figure 1: In a), a schematic diagram of a butadiene molecule C4H6. The bonding electrons are indicated by the heavy sticks between atoms. The delocalized electrons, which extend both above and below the plane of the diagram, are schematically indicated by the dotted path along the length l of the butadiene molecule. In b), the allowed energies for the electrons in the molecule assuming l is 0.55 nm. The two lowest states are filled. Higher vacant energy states (n=3, 4, 5 . . ) are available for occupation. The HOMO (highest occupied molecular orbital) and the LUMO (lowest unoccupied molecular orbital) are also labeled. To understand the nature of these modifications, it is useful to briefly consider a simple molecule like butadiene (C4H6). This molecule is a coplanar arrangement of 4 carbon atoms combined with six hydrogen atoms. Each carbon atom contributes 4 electrons; each hydrogen atom contributes 1 electron. During the synthesis of this3molecule, interactions between electrons cause a significant rearrangement of negative charge. Many of the electrons become localized in regions of space that lie between two atoms, forming states known as σ bonds. These states are covalent in nature and are fully occupied, containing a charge equivalent of two electrons. The negative charge carried by these σ bonds effectively screens the electrostatic repulsion that is present between the atomic nuclei. Each carbon atom brings one more electron than required to form the 9 σ bonds in butadiene. These extra electrons assume the lowest energy configuration possible which results in a delocalized occupied state referred to as a π orbital in the molecule. A schematic picture of these two different electron states is given in Fig. 1(a). As will become clear below, because of the delocalized nature of these π states, one might conclude that the butadiene molecule forms an extremely simple example of a tiny one-dimensional metal. If one could somehow connect clip leads to either end and apply a potential across it, one might expect current to flow through a single butadiene molecule in much the same way as it does through a copper wire! As suggested in Fig. 1(a), the π electrons