Conductivity in solids

Simplified band diagram showing the differences between a conductor, a semiconductor, and an insulator. The main difference has to do with the band gap energy, or separation between the top of the valence band (VB) and the bottom of the conduction band (CB). The Fermi level is indicated by a dashed line on the diagram. 

A simplistic rationalization for the electrical conductivity of the elements in the periodic table based on their band structures. 

The conductivity (tr) of a material is given by Equation (11.28), where p is the resistivity of the metal. The resistivity has units of iim and is given by Equation (11.29), which is a simple algebraic rearrangement of Pouillet s law, given by Equation (I 1.30), where R is the electrical resistance (measured in ohms, Q), A is the cross-sectional area of the material, and / is the length of the material. 

The hydraulic analogy between water in a pipe filled with sand and the electrical resistance in a material. 

The Fermi-Dirac distribution as a function of temperature in a semiconductor. 

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Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Mott N F and M J Littleton 1938. Conduction in Polar Crystals. I. Electrolytic Conduction in Solid Salts. Transactions of the Faraday Society 34 485-499. 

The heat conductivity in solids occurs via phonons. This conductivity is ideal in single crystals and is considerably reduced in porous solids, by one to two orders of magnitude. Therefore thermal insulation materials are built up of small particles which should touch each other at only a few points. This effect is of course enhanced by a low density of the material. 

The conductivity of solid salts and oxides was first investigated by M. Faraday in 1833. It was not yet known at that time that the nature of conduction in solid salts is different from that in metals. A number of fundamental studies were performed between 1914 and 1927 by Carl Tubandt in Germany and from 1923 onward by Abram Ioffe and co-workers in Russia. These studies demonstrated that a mechanism of ionic migration in the lattice over macroscopic distances is involved. It was shown that during current flow in such a solid electrolyte, electrochemical changes obeying Faraday s laws occur at the metal-electrolyte interface. 

In an ideal ionic crystal, all ions are held rigidly in the lattice sites, where they perform only thermal vibratory motion. Transfer of an ion between sites under the effect of electrostatic fields (migration) or concentration gradients (diffusion) is not possible in such a crystal. Initially, therefore, the phenomenon of ionic conduction in solid ionic crystals was not understood. 

The point defects are decisive for conduction in solid ionic crystals. Ionic migration occurs in the form of relay-type jumps of the ions into the nearest vacancies (along the held). The relation between conductivity o and the vacancy concentration is unambiguous, so that this concentration can also be determined from conductivity data. 

R. Berman Thermal Conduction in Solids, Clarendon, Oxford (1976)... 

Mott N. F and Littleton M. S. (1938). Conduction in polar crystals, I Electrolitic conduction in solid salts. Trans. Faraday Soc., 34 485-499. 

Thermal Properties of Metallic Solids. In the preceding sections, we saw that thermal conductivities of gases, and to some extent liquids, could be related to viscosity and heat capacity. For a solid material such as an elemental metal, the link between thermal conductivity and viscosity loses its validity, since we do not normally think in terms of solid viscosities. The connection with heat capacity is still there, however. In fact, a theoretical description of thermal conductivity in solids is derived directly from the kinetic gas theory used to develop expressions in Section 4.2.1.2. 

One of the most important aspects of point defects is that they make it possible for atoms or ions to move through the structure. If a crystal structure were perfect, it would be difficult to envisage how the movement of atoms, either diffusion through the lattice or ionic conductivity (ion transport under the influence of an external electric field) could take place. Setting up equations to describe either diffusion or conductivity in solids is a very similar process, and so we have chosen to concentrate here on conductivity, because many of the examples later in the chapter are of solid electrolytes. 

Electrical conductance in solids (other than ionic conductors) depends on the availability of delocalised orbitals close enough together in energy to form bands. 

Bottger H (1985) Hopping conduction in solids. VCH, Deerfield Beach, Fla... 

Following the introduction of basic kinetic concepts, some common kinetic situations will be discussed. These will be referred to repeatedly in later chapters and include 1) diffusion, particularly chemical diffusion in different solids (metals, semiconductors, mixed conductors, ionic crystals), 2) electrical conduction in solids (giving special attention to inhomogeneous systems), 3) matter transport across phase boundaries, in particular in electrochemical systems (solid electrode/solicl electrolyte), and 4) relaxation of structure elements. 

We begin our discussion by characterizing the electrical conduction in solid electrolytes. These are solids with predominantly ionic transference, at least over a certain range of their component activities. This means that the electronic transference number, defined as... 

The theoretical treatment of ionic conductivity in solids is very similar to that of diffusion, the main difference is the superimposition of the potential field upon the potential barrier to migration (Fig. 3). 

The classical theory for electronic conduction in solids was developed by Drude in 1900. This theory has since been reinterpreted to explain why all contributions to the conductivity are made by electrons which can be excited into unoccupied states (Pauli principle) and why electrons moving through a perfectly periodic lattice are not scattered (wave-particle duality in quantum mechanics). Because of the wavelike character of an electron in quantum mechanics, the electron is subject to diffraction by the periodic array, yielding diffraction maxima in certain crystalline directions and diffraction minima in other directions. Although the periodic lattice does not scattei the elections, it nevertheless modifies the mobility of the electrons. The cyclotron resonance technique is used in making detailed investigations in this field. 

The equation of thermal energy (Eq. 2.9-16) for transient conduction in solids without internal heat sources reduces to... 

In Section 8.2, we have studied oxygen and proton conduction in solids here we emphasize materials and performance of the cathode and the anode. 

H. Bottger, V.V. Bryksin Hopping Conductions in Solids (Akademie Verlag, Berlin 1985)... 

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