Charge Relaxation Theory
COMSOL Multiphysics includes physics interfaces for the modeling of static electric fields and currents. Deciding what specific physics interface and study type to select for a particular modeling situation requires a basic understanding of the charge dynamics in conductors.
Physics interfaces for the modeling of dynamic, quasi-static (that is, without including wave propagation effects) electric fields and currents are available with the AC/DC Module and MEMS Module.
The different physics interfaces involving only the scalar electric potential can be interpreted in terms of the charge relaxation process. The fundamental equations involved are Ohm’s law for the conduction current density
the equation of continuity
and Gauss’ law
By combining these, one can deduce the following differential equation for the space charge density in a homogeneous medium
This equation has the solution
where
is called the charge relaxation time. For a good conductor like copper, τ is of the order of 10−19 s, whereas for a good insulator like silica glass, it is of the order of 103 s. For a pure insulator, it becomes infinite.
When modeling real-world devices, there is not only the intrinsic time scale of the charge relaxation time but also an external time scale t at which a device is energized or the observation time. It is the relation between the external time scale and the charge relaxation time that determines what physics interface and study type to use. The results are summarized in Table 10-1 below,
Table 10-1: Suitable Physics Interface and Study Type for Different Time-scale Regimes.
case
Physics interface
study type
τ>>t
Electrostatics
Stationary
τ<<t
Electric Currents
Stationary
τ~t
Electric Currents
Time Dependent or, with the AC/DC Module, MEMS Module, or Semiconductor Module, Frequency Domain
First Case: τ >> t
If the external time scale is short compared to the charge relaxation time, the charges do not have time to redistribute to any significant degree. Thus the charge distribution can be considered as a given model input. The best approach is to solve the Electrostatics formulation using the electric potential V.
By combining the definition of the potential with Gauss’ law, you can derive the classical Poisson’s equation. Under static conditions, the electric potential V is defined by the equivalence E = −∇V. Using this together with the constitutive relation D = ε0E + P between D and E, you can rewrite Gauss’ law as a variant of Poisson’s equation
This equation is used in the Electrostatics interface. It is worth noting that Gauss’ law does not require the charge distribution to be static. Thus, provided dynamics are slow enough that induced electric fields can be neglected and hence a scalar electric potential is justified, the formulation can be used also in the Time Dependent study type. That typically involves either prescribing the charge dynamics or coupling a separate formulation for this.
Such separate charge transport formulations can be found in the Plasma Module, the Semiconductor Module, and the Chemical Reaction Engineering Module.
Second Case: τ <<t
If the external time scale is long compared to the charge relaxation time, the stationary solution to the equation of continuity has been reached. In a stationary coordinate system, a slightly more general form of Ohm’s law than above states that
where Je is an externally generated current density. The static form of the equation of continuity then reads
To handle current sources, the equation can be generalized to
This equation is used in the static study type for the Electric Currents interface.