7. Diffusion and
Electrical Conductivity (Part 2)
- 3 주차 -
7.3 Electrical Conductivity
▪ Historically ceramics were exploited for their electric insulation properties, which together with their chemical and thermal stability rendered them ideal insulating materials in applications ranging from power lines to cores bearing wire-wound resistors.
▪ Today their use is much more ubiquitous (=common) − in addition to their
traditional roles as insulators, they are used as electrodes, catalysts, fuel cells, photoelectrodes, varistors, sensors, and substrates, among them many other applications.
This section deals solely with the response of ceramics to the application of a constant electric field and the nature and magnitude of the steady-state current (i.e., long-range motion of charge carriers) that results. As discussed next, the ratio of this current to the applied electric field is proportional to a material property known as conductivity, which is the focus of this section.
On the other hand, the displacement currents or non-steady-state response of solids (i.e., short-range motion of charge carriers) which gives rise to
capacitive properties is dealt with separately in Chaps. 14 and 15 which treat the linear and nonlinear dielectric properties, respectively.
Dielectric Properties
Linear
Nonlinear
Polarization: the finite displacement of bound charges of a dielectric in response to an applied electric field, and the orientation of their molecular dipoles
▪ In metals, free electrons are solely responsible for conduction. In
semiconductors, the conducting species are electrons and/or electron holes.
In ceramics, however, because of the presence of ions, the application of an electric field can induce these ions to migrate. Therefore, when dealing with conduction in ceramics, one must consider both the ionic and the electronic contributions (i.e., electrons and electron holes) to the overall conductivity.
Before one makes that distinction, however, it is important to develop the concept of conductivity. A good starting point is Ohm’s law, which states that
𝑉𝑉 = 𝑖𝑖𝑖𝑖
where 𝑉𝑉 is the applied voltage (V) across a sample, 𝑖𝑖 its resistance in ohms (Ω), and 𝑖𝑖 the current (C/s) passing through the solid.
Rearranging Eq. (1), dividing both sides by the cross-sectional area through which the current flows 𝐴𝐴, and multiplying the right-hand side by 𝑑𝑑/𝑑𝑑, where d is the thickness of the sample, one gets
𝐼𝐼 = 𝐴𝐴𝑖𝑖 = 𝑅𝑅𝐴𝐴𝑑𝑑 𝑉𝑉𝑑𝑑
(1)
where 𝐼𝐼 = 𝑖𝑖/𝐴𝐴 is the current density passing through the sample. Given that 𝑉𝑉/𝑑𝑑 is nothing but the electric potential gradient (i.e., electric field) 𝑑𝑑∅/𝑑𝑑𝑑𝑑, Ohm’s law can be rewritten as
𝐼𝐼𝑖𝑖 = −𝜎𝜎𝑖𝑖 𝑑𝑑∅
𝑑𝑑𝑑𝑑 where
𝜎𝜎 = 𝑑𝑑 𝑖𝑖𝐴𝐴
Eq. (2) states that the flux 𝐼𝐼 is proportional to 𝑑𝑑∅/𝑑𝑑𝑑𝑑. The proportionality constant 𝜎𝜎 is the conductivity of the material, which is the conductance of a cube of material of unit cross section. The units of conductivity are siemens per meter or Sm-1, where S = Ω-1.
(2)
The range of electronic conductivity (Fig. 7.6, right-hand side) in ceramics is phenomenal (=surprising) − it varies over 24 orders of magnitude, and that does even include superconductivity! Few, if any, other physical properties vary over such a wide range. In addition to electronic conductivity, some ceramics are known to be ionic conductors (Fig. 7.6 left-hand side).
In order to understand the reason behind this phenomenal range and why some ceramics are ionic conductors while other are electronic conductors, it is necessary to delve into the microscopic domain and relate the
macroscopically measurable 𝜎𝜎 to more fundamental parameters, such as carrier mobilities and concentrations.
Fig. 7.6 Range of electronic (right-hand side) and ionic (left-hand side) conductivities in Ω-1cm-1 exhibited by ceramics and some of their uses.
▪ Generalized Equations
▪ If one assumes there are 𝑐𝑐𝑚𝑚 mobile carriers per cubic meter drifting with an average drift velocity 𝑣𝑣𝑑𝑑, it follows that their flux is given by
𝐼𝐼𝑖𝑖 = 𝑧𝑧𝑖𝑖 𝑒𝑒𝑒𝑒𝑖𝑖 = 𝑧𝑧𝑖𝑖 𝑒𝑒 𝑐𝑐𝑚𝑚,𝑖𝑖𝑣𝑣𝑑𝑑,𝑖𝑖 = 𝑧𝑧𝑖𝑖 𝑒𝑒𝑐𝑐𝑚𝑚,𝑖𝑖𝑣𝑣𝑑𝑑,𝑖𝑖
The electric mobility 𝜇𝜇𝑑𝑑,𝑖𝑖 (m2/V⋅s) is defined as the average drift velocity per electric field, or
𝜇𝜇𝑑𝑑,𝑖𝑖 = −𝑣𝑣𝑑𝑑,𝑖𝑖 𝑑𝑑∅/𝑑𝑑𝑑𝑑
Combining Eqs. (2) to (4) yields the important relationship 𝜎𝜎𝑖𝑖 = 𝑐𝑐𝑚𝑚,𝑖𝑖𝑒𝑒 𝑧𝑧𝑖𝑖 𝜇𝜇𝑑𝑑,𝑖𝑖
between the macroscopically measurable quantity 𝜎𝜎 and the microscopic parameters µ𝑑𝑑 and 𝑐𝑐𝑚𝑚.
(3)
(4)
(5)
In driving this equation, it was assumed that only one type of charge carrier was present. However, in principle, any mobile charged species can and will contribute to the overall conductivity. Thus the total conductivity is given by
𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡 = �
𝑖𝑖
𝑐𝑐𝑚𝑚,𝑖𝑖𝑒𝑒 𝑧𝑧𝑖𝑖 𝜇𝜇𝑑𝑑,𝑖𝑖
The absolute value sign about 𝑧𝑧𝑖𝑖 ensures that the conductivities are always positive and additive regardless of the sign of the carrier.
The total conductivity is sometimes expressed in terms of the transference or transport number, defined as
𝑡𝑡𝑖𝑖 = 𝜎𝜎𝑖𝑖 𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡
from which it follows that 𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑡𝑡𝑖𝑖𝑡𝑡𝑖𝑖𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡, where 𝑡𝑡𝑖𝑖𝑡𝑡𝑖𝑖 is the ionic transference number and includes both anions and cations and 𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 is the electronic transference number which includes both electrons and electron holes. For any material 𝑡𝑡𝑖𝑖𝑡𝑡𝑖𝑖 + 𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1.
