minimum_density

Calling sequence

currents{ minimum_density }

Properties

  • using: \(\mathrm{\textcolor{ForestGreen}{optional\;within\;the\;scope}}\)

  • type: \(\mathrm{real\;number}\)

  • unit: \(\mathrm{cm^{-3}}\)

  • values: [0.0, 1e20]

  • default: 1e10

Functionality

A keyword allowing to improve the condition number of the matrix representing the current equation.

Minimum carrier density, \(\rho_\mathrm{min}\), is defined for both types of carriers at once (electrons and holes) as the lower limit for the respective density distributions entering the drift-diffusion current equations. If a density distribution computed based on quasi-Fermi levels and densities of states for a given carrier type, \(\rho_\mathrm{sim}\left(x\right)\), is smaller than \(\rho_\mathrm{min}\) within some region, then its values in the region are replaced by the \(\rho_\mathrm{min}\) for the equation. In other words, every carrier distribution entering the current equation, \(\rho_\mathrm{current}\left(x\right)\), is given by

\[\rho_\mathrm{current}\left(x\right) = \mathrm{max}\left[\rho_\mathrm{sim}\left(x\right), \rho_\mathrm{min}\right].\]

This operation is not visible in the output files.

As the drift-diffusion current is proportional to the charge carrier density, this keyword also indirectly sets the lower limit of the current.

Aside from the rather practical issue that real-life minority carrier densities are not in thermal equilibrium and thus never become as small as predicted, it seems nonphysical that one carrier per kilometer can be relevant in semiconductors or insulators. Therefore, the minimum density parameter as specified for the current equation typically should is not be smaller than \(10^{-10}\;\mathrm{cm}^{-3}\). This value corresponds to a conductivity 10 orders of magnitude lower than of the best insulators.

Note

The \(\rho_\mathrm{min}\) affects only the current operators (\(\nabla\;\mu\;\rho_\mathrm{current}\;\nabla\)) and the corresponding current for each type of carriers. Thus it has no direct influence on computed densities, Poisson equation, etc. Recombination processes can be affected by setting minimal_recombination to yes.

Hint

  • The \(\rho_\mathrm{min}\) might have to be increased in order to obtain convergence for the drift-diffusion current equations.

  • The \(\rho_\mathrm{min}\) should be as low as possible, depending on the problem solved.

  • The \(\rho_\mathrm{min}\) can be chosen as large as possible but should be small enough to obtain convergence with meaningful results.

  • Typically \(\rho_\mathrm{min} = 10^{12}\;\mathrm{cm}^{-3}\) seems to be already too high.

Attention

Setting the minimum density far too low may result in convergence issues or even in the matrix solvers exiting with an error message. The smallest reasonable number depends on the simulated system. Currently the algorithm allows using the value as small as \(10^{-100}\;\mathrm{cm}^{-3}\). Any smaller values are rounded up to this number.

When restricting effective densities in the current equations from below, one should consider impact on the physics of the modelled device, i.e., increasing minimum densities decreases resistivity of insulating regions.

Example
currents{
    recombination_model{}
    minimum_density = 1.0  # cm^-3
    minimum_density_factor = [ 1e10 , 1e8 ]
}
Unimportant currents in Insulators and Barriers

The computed current of a given type of carriers often varies over 10 orders of magnitude between barriers (insulators) and conducting regions as a result of extremely small carrier densities in the barriers. If the density in the latter regions reaches values below approximately \(10^3\;\mathrm{cm}^{-3}\), then the current flowing through them can be practically considered zero in comparison to the total current present in the structure. As a result the matrix representing the current equation, entering the linear solver, is not well conditioned and convergence of the drift-diffusion current equations may be strongly affected by round-off errors. If, the current running through the barriers is not important from the physical point of view, such that increasing it a number of orders of magnitude does not change the final result (e.g., I-V characteristic), then increasing the \(\rho_\mathrm{min}\) to overestimate the current in these regions is a very good way to restore or improve the convergence while preserving meaningful results.

Currents within intrinsic materials

If one requires to properly compute the currents within intrinsic regions, then the optimal \(\rho_\mathrm{min}\) should be chosen such that \(\rho_\mathrm{min} < \rho_\mathrm{sim}\left(x\right)\) in these regions. The maximum value of a properly chosen \(\rho_\mathrm{min}\) strongly depends on the band gap of the considered material.

Undoped wide-band-gap and highly-doped semiconductors

Minority carriers in highly-doped semiconductors or any carriers in undoped wide-band-gap semiconductors have extremely small equilibrium densities (much less than \(1.0\;\mathrm{cm}^{-3}\)). Computing all currents in these doped materials or for wide-band-gap semiconductor heterostructures, will typically require also considering currents over 15 orders of magnitude higher, which may lead to complete breakdown of the solvers for current equation due to underflow.