3.3: Conduction

There are three objectives in this section. The first is to have a microscopic picture of the carrier motions to associate with ohmic or unipolar conduction models. The second is to illustrate how constitutive laws for media in motion can be derived from models based on particular microscopicmodels, or (on the basis of the field transformations) found by generalizing empirically determined laws established in the laboratory for materials at rest. Finally, a byproduct of the discussion is an introduction to Hall effect.

Consider the carrier motions represented by Eqs. 3.2.3, with the magnetic field \(\overrightarrow{H} = H_o \overrightarrow{i}_x\) externally imposed. The components of these equations then respectively become

\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & \mp b_{\pm} \mu_o H_o \\ 0 & \pm b_{\pm} mu_o H_o & 1 \end{bmatrix} \begin{bmatrix} v_{x \pm} \\ v_{y \pm} \\ v_{z \pm} \end{bmatrix} = \begin{bmatrix} \pm b_{\pm} E_x \\ \pm b_{\pm} E_y \pm b_{\pm} v_z mu_o H_o \\ \pm b_{\pm} E_z \mp b_{\pm} v_y mu_o H_o \end{bmatrix} \label{1} \]

where particle mobilities are defined as \(b_{\pm} = q_{\pm} / m_{\pm} v_{\pm}\).

These three equations can be inverted to find the relative carrier velocities in terms of \((\overrightarrow{E}, \overrightarrow{H}, \overrightarrow{v})\):

\[ \begin{bmatrix} v_{x \pm} \\ v_{y \pm} \\ v_{z \pm} \end{bmatrix} = \frac{1}{\Delta_{\pm}} \begin{bmatrix} \frac{ \pm b_{+}}{\Delta_{\pm}} & 0 & 0 \\ 0 & \pm b_{\pm} & b^2_{\pm} \mu_o H_o \\ 0 & -b_{\pm}^2 \mu_o H_o & \pm b_{\pm} \end{bmatrix} \begin{bmatrix} E_x \\ E_y + v_z \mu_o H_o \\ E_z - v_y \mu_o H_o \end{bmatrix} \label{2} \]

where \(\Delta_{\pm} = 1 + (\mu_o H_o b_{\pm})^2\).

These velocity components can now be introduced into Equation 3.2.5 to express the free current density as

\[ \begin{align} \overrightarrow{J}_f = (n_{+} q_{+} b_{+} + n_{-} q_{-} b_{-}) \overrightarrow{E}^{'}_x \overrightarrow{i}_x &+ \Bigg (\frac{n_{+} q_{+} b_{+}}{\Delta_{+}} + \frac{n_{-} q_{-} b_{-}}{\Delta_{-}} \Bigg ) (\overrightarrow{E}^{'}_y \overrightarrow{i}_y + \overrightarrow{E}^{'}_z \overrightarrow{i}_z) \nonumber \\ &+ \Bigg (\frac{n_{+} q_{+} b_{+}^2}{\Delta_{+}} + \frac{n_{-} q_{-} b_{-}^2}{\Delta_{-}} \Bigg ) \mu_o \overrightarrow{E}^{'} \times \overrightarrow{H}_o + \rho_f \overrightarrow{v} \nonumber \end{align} \label{3} \]

where \(\overrightarrow{E}^{'} \equiv \overrightarrow{E} + \overrightarrow{v} \times \mu_o \overrightarrow{H} \) is the electric field in a frame of reference moving with the material (for a magnetoquasistatic system).

From Equation \ref{3}, it is clear that there are two components to the current density, one in the direction of the imposed electric field and the second perpendicular to it. The latter term is called the Hall current and is due to the tendency of the particles to move perpendicular to their own velocity and to the imposed magnetic field intensity. This last term is ignorable if

\[ \mu_o H_o b_{\pm} << 1 \label{4} \]

A typical magnetic flux density is \(\mu_o H_o = 1\) (\(10,000\, gauss\), which is in the range where magnetic materials saturate). Electrons in copper Rave a mobility on the order of \(3 x 10^-3\, m^2/volt\, sec\), so that the parameter on the left is then much less than \(1\). Ions in liquids have mobilities that are typically \(5 x 10^{-8}\, m^2/volt\, sec\) and the approximation is even better. But in silicon or germanium, where the electron mobility is in the range of \(10^{-1} \, m^2/volt\, sec\), the Hall effect is coming into play by the time\(\mu_o H_o\) is of the order of unity. With the inequality of Equation \ref{4} satisfied, Equation \ref{3} reduces to the familiar form

\[ \overrightarrow{J}_f = (n_{+} q_{+} b_{+} + n_{-} q_{-} b_{-}) \overrightarrow{E}^{'} + \rho_f \overrightarrow{v} \label{5} \]

If the number density of charge carriers \(n_{+}\) and/or \(n_{-}\) remains essentially the same in spite of the application of \(\overrightarrow{E}\), then the factor multiplying \(\overrightarrow{E}\) in Equation \ref{5} is usefully regarded as a parameter characterizing the material, the electrical conductivity \(\sigma\). This case of ohmic conduction is displayed by materials ranging from metallic conductors, where the carriers are electrons and essentially immobile ions,to electrolytes, where ions of at least two species participate in the conduction. In any of these cases, for the ohmic model to be valid, the conduction must involve at least two species with both \(n_{+}q_{+}\) and \(n_{-}q_{-}\) greatly exceeding the net charge \(\rho_f\). By introducing the conductivity as a parameter, the detailed analysis necessary to determine the self-consistent distributions of the individual
carriers is avoided. But to examine the conditions under which the conductivity model is valid, it is necessary to formulate the laws that govern the self-consistent carrier motions. This is best done in the context of molecular diffusion (Chapter 10) so that other important limitations on the model can
also be identified.

Even though in accounting for conduction it is useful to have in mind microscopic mechanisms, it is also important to recognize the far-reaching implications of empirical relations. Given any conduction law based on laboratory measurements made with a fixed sample, effects of material motion can be brought in by using the transformation laws. For example, if it is known that the conductor obeys Ohm's law when stationary, then in a primed inertial frame moving with the velocity \(\overrightarrow{v}\) of the conductor, the experiment shows that

\[ \overrightarrow{J}_f^{'} = \sigma \overrightarrow{E}^{'} \label{6} \]

In an electroquasistatic system, including polarization, \(\overrightarrow{J}_f^{'} = \overrightarrow{J}_f - \rho_f \overrightarrow{v} \) (Equation 2.5.12a) and \(\overrightarrow{E}^{'} = \overrightarrow{E} \) (Equation 2.5.9a). Hence, Equation \ref{6} becomes Equation \ref{5}. In a magnetoquasistatic system, including magnetization, \(\overrightarrow{J}_f^{'} = \overrightarrow{J}_f \)(Equation 2.5.11b) and \(\overrightarrow{E}^{'} = \overrightarrow{E} + \overrightarrow{v} \times \mu_o \overrightarrow{H} \) (Equation 2.5.12b). Substitution in Equation \ref{6} now gives Equation \ref{5}, except for the charge convection term \(\rho_f \overrightarrow{v}\). In a magnetoquasistatic system, this term is second-order, as will be argued in the next section.

Fundamental to the use of an empirical law determined for the stationary material is the assumption that material acceleration and deformation do not influence the conduction. In any case, if acceleration did effect the conduction, the close tie between conduction and the Lorentz force density,illustrated in this and the previous section, calls into question the notion that the electromechanicscan be modeled by a single continuum subject to the Lorentz force density.