The Drude Model of Metals This material outlines the model of metallic conductivity developed by P. Drude ca. 1900. The model produced a number of successes in describing the properties of metals, and even today is useful for simple descriptions of metallic conductivity, as well as for roughly estimating properties. The Drude model, however, failed to describe a number of experimental observations, and also presented a number of conceptual problems that defined the directions for the theory of metallic conductivity for the next two decades. The resolution of these issues required the development of the quantum theory of solids. Basic Assumptions of the Model Drude based his model of metallic conductivity on applying the kinetic theory of gases to a metal which he regarded as a "gas" of electrons. (Note: The electron was discovered by J.J. Thompson in 1897.) The simplest form of the kinetic theory of gases treats the molecules of a gas as identical solid spheres that move in straight lines until they collide with each other (or with the walls of the containment vessel). The collision time is assumed negligible, and other than the force that acts during each collision, no other forces are assumed between the particles. For simple gases, there is only one kind of particle present, but as we have discussed a metal must possess at least two. The electron was known to be negatively charged, but the metal was neutral. Thus a compensating particle with positive charge was needed to maintain charge neutrality. Drude assumed that the positive charge was attached to very heavy particles which were immobile. At the time, there was no precise idea about the origins of either the light, mobile electrons or the heavy immobile positively charged particles. This was solved by the quantum theory of solids. In this treatment of the Drude model, however, it is simply assumed that when atoms of a metallic element are brought together, the valence electrons become delocalized and free to move through the metal, while the positively charged ion-core remains intact and fills the role of the heavy immobile particle. This is shown qualitatively below. An isolated atom of the metal has a Valence Electrons Conduction Electrons Core Electrons Core Electrons Nucleus Ion{ Nucleus Figure 1 Schematic of the isolated atom (left) and metal (right). 1 nucleus of charge eZa, where Za is the atomic number and e is the magnitude of the electron charge (taken to be a positive number). The nucleus is, of course, surrounded by Za electrons with charge -eZa. Some of these electrons, Z, are relatively weakly bound and are known as valence electrons. The remaining Za-Z electrons are more tightly bound and therefore play a much less important role in binding. These are known as core electrons and they will be regarded as fixed to the nucleus in the metal forming a metal ion. The valence electrons, on the other hand, are free to move far from their parent atoms. In a metal, these electrons are often referred to as conduction electrons. It is these electrons which Drude applied the kinetic theory of gases to with the modification to the theory of the presence of a background of immobile positive metal ions. The density of the electron gas is determined as follows. An element contains 23 3 6.0210 atoms per mole (Avogadro's number), and m/A moles per cm , where m is the mass density (g/cm3) and A is the atomic mass. Since each atom contributes Z conduction electrons, the conduction electron density is 3 23 n(electrons/cm ) 6.02 10 ( Zm / A ). Typical conduction electron densities are between 1022-1023 electrons/cm3. Another useful measure of electron density is given by the radius of a sphere with volume equal to the volume per conduction electron 1 1 4 3 3 rr3 ;. nn34ss Typical values of rs lie between 1-2 Å. These densities are much greater (about a thousand times greater) than molecular gas densities at normal temperatures and pressures. Nonetheless, and in spite of the presence of strong electromagnetic interactions for such model systems, Drude boldly applied the kinetic theory of a dilute neutral gas to this dense conduction electron "gas" using the following assumptions: 1) Between collisions, the conduction electrons do not interact either with each other or with the metal ions. In the absence of an externally applied field, the electrons move in straight lines with constant velocity. In the presence of an applied field the conduction electrons move according to Newton's laws of motion in the presence of the externally Figure 2 Conduction electron trajectory applied field. Neglect of the according to Drude. electron-electron interactions is known as the independent electron approximation, and neglect of the electron-ion interactions is known as the free electron approximation. 2) Collisions are instantaneous and abruptly alter the electron velocity. Drude attributed these collisions to conduction electron scattering from the metal ion cores. (The true 2 collision analogue to kinetic gas theory would be electron-electron collisions.) We will learn later that electron-electron scattering is indeed usually an unimportant scattering mechanism in a metal, however, the picture of a conduction electron scattering off the ion cores (shown above) is incorrect. However, a qualitative (and sometimes quantitative) understanding of metallic conduction is possible by simply assuming some scattering mechanism without being precise about the exact nature of this mechanism. 3) An electron undergoes a collision with probability per unit time 1/. This means that the probability of an electron experiencing a collision in any (infinitesimal) time interval dt is dt/. This time τ is known as the relaxation or collision or mean free time. This collision time is taken to be independent of the electrons position or velocity. 4) The conduction electrons achieve thermal equilibrium with their surroundings only through collisions (given the approximations in 1 no other mechanism remains). These collisions preserve local thermodynamic equilibrium as follows: when the electrons emerge from a collision, they do so with a speed that is entirely determined by the local temperature and in a random direction. This velocity is assumed to be entirely independent of the electron velocity prior to collision. Applications The remainder of this discussion will concentrate on several important applications of the Drude model, and its successes and failures in each case. DC Electrical Conductivity Any model of DC electrical conductivity must reproduce Ohm's Law which says that the current, I, flowing in a wire is proportional to the potential drop, V, along the wire: V = IR, where R is the resistance of the wire. The geometrical dependence of the resistance is usually eliminated by the introduction of a quantity characteristic of the wire material only. This quantity is the resistivity, ρ, defined as the constant of proportionality between the electric field E and the current density j induced by the field Ej . The current density is parallel to the flow of charge with a magnitude given by the total charge per unit time crossing a unit area perpendicular to the charge flow. For example, if a uniform current I flows in a wire of length L and cross-sectional area A, then the current density is j=I/A. The potential drop along the wire is given by V = EL, and we therefore have V = ρjL= ρIL/A. Thus, in this case R = ρL/A. Now, if we have n electrons per unit volume all with velocity v , then the current density is parallel to v , and in time dt the electrons move a distance vdt along this direction. Therefore, n(vdt)A electrons cross area A perpendicular to the flow, and since each carries charge -e the charge crossing A in time dt is -nevAdt. The current density is then given by j nev. Since at any point in the metal electrons move with a variety of velocities, the net current density is given by the above equation provided v is the average electronic velocity. In the absence of an electric field, the electrons are just as likely to be moving in one direction as any other and v averages to zero (there is no net current density). However, in the presence 3 of a field E , there will be a mean electronic velocity, known as the drift velocity, directed opposite the field (the electron charge is negative) which we can compute. Consider an electron at time t that underwent its last collision at t = 0. Its velocity at time t is then the velocity immediately after the collision v0 plus an additional term -e E t/m that it acquired by moving in the field for time t. The direction of the electron motion after a collision is assumed to be random so there is no net contribution to the current density from v0 , and the current density must be determined by the average of -e E t/m which is simply -e E /m. Therefore ne2 v eE /;. m j E avg m We introduce the conductivity = 1/ρ and rewrite this as ne2 jE;. m This verifies that we have a linear relationship between E and j (as required by Ohm's Law) and allows us to estimate the conductivity in terms of known quantities (with the significant exception of ). Alternatively, we could use measured resistivities to estimate the relaxation time. If we express the resistivity in units of microhm-cm we can write the relaxation time as 3 0.22 r 14 s 10 sec. a0 The quantity ao is the Bohr radius and is approximately 0.5 Å.