C.8.4.1. Overview
Electrical conductivity of materials plays a crucial role in determining the level of internal charging in dielectrics. If a charging current is constantly applied, the internal electric field rises only until an equilibrium is achieved, in which deposited and conducted currents are equal. For a layer of material of thickness d, the maximum field at equilibrium, Emax, can be found using Ohm’s Law:
V=IR i.e. Emax = V/d = (I/A).R A/d = j/σ
where
• V is potential ,
• I is current,
• R is resistance,
• j is current density, and
• σ is conductivity,
since for an area A, σ =d/RA, and j=I/A.
It is an unfortunate complication of internal charging calculations that the conductivities of dielectrics are not constant. They can be strongly affected by temperature, electric field, and radiation. Standard measurement techniques for conductivity have been defined (see [28] and [29]).
C.8.4.2. Temperature dependence
Temperature has a large effect on conductivity in dielectric materials. Higher temperatures increase the energy available to trapped electrons, enabling more of them to jump into conduction band quantum states. Hence conductivity increases with temperature, the reverse of the dependence observed in conductors. The dependence of conductivity on temperature is generally represented by the following equation:
where
• Ea is the material dependent activation energy,
• k is Boltzman’s constant,
• T is temperature (K), and
• σ∞ is the maximum conductivity as T approaches infinity.
It is important to note that Eais not the band-gap associated with the excitation of electrons from the valance band into the conduction band which is far larger (for polythene Ea is around 1 eV and the band gap is around 8,8 eV). Clearly a more subtle process is at work than a straightforward excitation of electrons. Ea is found from experimental studies and normally lies close to 1 eV for most dielectrics. Some values were calculated (see [30]) and are listed in Table C-5.