$Withionmove

When the system has ion to perform as drift-diffusion equations, we solve the time dependent drift-diffusion for slow ion move simulation. Since the ion moves may not be governed fermi-level concept. We simply treat is a tradiational drift-diffusion equations.

 $\frac{\partial M_{i o n}}{\partial t} = \nabla ((q_{s i g n}) e μ M_{i o n} \vec{E} - q D_{M} \nabla M_{i o n})$

Ideally, the ion density is given by initial setting. The total ion number should be fixed. The program is aim to model multi-ions drift-diffusion. The command is as following.

$Withionmove
 $N_{s w e e p}$   $N_{o u t p u t}$ 
 $T_{1, s t o p}$    $d t_{1}$   $T_{2, s t o p}$    $d t_{2}$  ...  $T_{N_{s w e e p}, s t o p}$    $d T_{M_{s w e e p}}$ 
 $N_{i o n s} q_{s i g n, 1} q_{s i g n, 2} q_{s i g n, 3} . . . . q_{s i g n, N_{i o n s}}$ 
 $P_{t y p e, 1} M_{i o n s, 1} μ$  p3 p4 Parameters of the 1 layer
 $P_{t y p e, 2} M_{i o n s, 2} μ$  p3 p4 Parameters of the 2 layer
 $P_{t y p e, 3} M_{i o n s, 3} μ$  p3 p4 Parameters of the 3 layer
 ....
 .....
 $P_{t y p e, N} M_{i o n s, N} μ$  p3 p4 Parameters of the $totalregion layer

 $N_{s w e e p}$ : The number of runs for the time step
 $N_{o u t p u t}$ : The number of output results for each run
 $T_{1, s t o p}$  : The time for the first run.  $T_{2, s t o p}$ The time for the 2nd run. 
 $d t_{1}$  is the  $δ t$  for each sweep. 
 $N_{i o n s}$  How many ions are considered. If we only want to consider 1 negative ion,we can put 1
 $q_{s i g n}$  The sign of ions. only accept  $\pm 1.0$

For example: Consider 2 ions, 1st is negative charges, 2nd is positive charges, total 5 Regions we can

$Withionmove
1 1000
1.00  1.0d-4 
2 -1.0 1.0
1 0.0e17 0.0      0.0e17 0.0   
1 1.0e17 1.0e-11  2.0e17 1.0e-12
1 1.0e17 1.0e-11  2.0e17 1.0e-12
1 1.0e17 1.0e-11  2.0e17 1.0e-12
1 1.0e17 1.0e-11  2.0e17 1.0e-12

 $P_{t y p e, 1}$  is the parameter type:  it depends on ion number  $N_{i o n s}$ 
1:  $M_{i o n s, 1} μ$ ,  $M_{i o n s, 2} μ$ ,........  $M_{i o n s, N_{i o n s}}, μ_{N_{i o n s}}$ 
2:  $M_{i o n s, 1} μ D_{M}$ ,  $M_{i o n s, 2} μ D_{M}$ ,........  $M_{i o n s, N_{i o n s}}, μ_{N_{i o n s}} D_{M}$

When type 1 is chosen, we only put mobility $μ$ , and the diffusion coefficient $D_{M}$ is calculated with Einstein relation, where

  $D_{M} = μ k_{B} T / q$

$Withionmove

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