SaM - Continuity Equation for Charge Carriers

• What is the Continuity Equation for Charge Carriers?:
  • The continuity equation for charge carriers in an electrical current is a fundamental concept in electromagnetism.
  • It describes the conservation of charge in a conducting medium.

The formula for the Continuity Equation for Charge Carriers (specifically in this case electrons), is the following:$\frac{\partial n^{\prime }}{\partial t}=\frac{1}{q_e}\nabla \cdot {\vec{J}}_n+G+\frac{n^{\prime }}{{\tau }_e}$(==Take it as true==). Where:

• $n^{\prime }=n-n_0$ and it’s the variation of density of charge carriers (or density of free electrons)
• $\nabla \cdot {\vec{J}}_n$ is the divergence of the current density or transport current (${\vec{J}}_n$), indicating the net flow of charge out of or into a volume.
  • REMEMBER that a divergence of a vector $\vec{v}$ is defined as:$\nabla \cdot \vec{v}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}$
• $G$ is the generation of charge due to thermal energy.
• $\frac{n^{\prime }}{{\tau }_e}$ is the recombintion.

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• What is the Transport Current?:

${\vec{J}}_n={\sigma }_n\vec{E}+q{D}_n(\nabla n)$Where:

• The first term: ${\sigma }_n\vec{E}$, represents the drift current density.
  This part of the current is due to the motion of electrons in response to the electric field $\vec{E}$.
  It follows Ohm’s law, where current is directly proportional to the electric field.
• The second term: $q{D}_n(\nabla n)$, represents the diffusion current density.
  This part of the current is due to the concentration gradient of electrons.
  When there is a concentration gradient, electrons tend to move from regions of higher concentration to regions of lower concentration due to diffusion.
• More specifically each term is:
  • ${\vec{J}}_n$: This is the electron current density vector. (A/m²).
  • ${\sigma }_n$: This term represents the electrical conductivity of the material for electrons.
    It’s a material-specific property and measures how well the material conducts electric current. (S/m, where S stands for Siemens).
  • $\vec{E}$: This is the electric field vector. It represents the electric force acting on the charge carriers (electrons) in the material. (V/m).
  • $q$: This represents the charge of an electron, which is approximately $1.602\times 10^{-19}$ Coulombs.
  • $D_n$: This term represents the electron diffusion coefficient.
    It describes how quickly electrons move through the material due to diffusion. (m²/s).
  • $\nabla n$ is the gradient of the density of free electrons, which measures how the electron concentration changes in space.
    It is usually measured in electrons per cubic meter (1/m³).
• REMEMBER by definition the gradient of $f$ is:$\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$