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Solving drift-diffusion and current continuity simultaneously
Posted Jan 16, 2015, 8:19 a.m. EST Semiconductor Devices, Materials, Modeling Tools & Definitions, Parameters, Variables, & Functions, Studies & Solvers Version 4.4, Version 5.0 2 Replies
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Hallo comsol forum !!!
I am new to comsol and I would appreciate any of your help!
I am trying to solve this drift-diffusion equation d(nd)/dt = div(D*grad(nd) - v*nd) + G (G is the generation rate), where G,v,D are expressions of the electric field E, where E= d(V,y) and V is the voltage.
{ In the model I named Voltage as variable psi} .
V satisfies also the continuity equation: div(s*grad(V)) =0.
If I apply a voltage, the density nd changes and thus the function of the voltage changes too inside my domain, which influences the density nd again. My guess is that these two equations should be solved simultaneously and after the first simulation, every output of the continuity equation must be the initial value of V in order the fisrt equation to be solved again and so on!!! In addition, I suspect that the first equation requires a time-dependant solver(or step?) and the second equation a stationary solver(or step?).How do i do this? Does anyone have some piece of advice? :)
P.S In order to construct my equations I have used the coefficient form PDE for the first equation as Physics and the Poisson's equation for the second equation
Thank you very, very much !!!!
I am new to comsol and I would appreciate any of your help!
I am trying to solve this drift-diffusion equation d(nd)/dt = div(D*grad(nd) - v*nd) + G (G is the generation rate), where G,v,D are expressions of the electric field E, where E= d(V,y) and V is the voltage.
{ In the model I named Voltage as variable psi} .
V satisfies also the continuity equation: div(s*grad(V)) =0.
If I apply a voltage, the density nd changes and thus the function of the voltage changes too inside my domain, which influences the density nd again. My guess is that these two equations should be solved simultaneously and after the first simulation, every output of the continuity equation must be the initial value of V in order the fisrt equation to be solved again and so on!!! In addition, I suspect that the first equation requires a time-dependant solver(or step?) and the second equation a stationary solver(or step?).How do i do this? Does anyone have some piece of advice? :)
P.S In order to construct my equations I have used the coefficient form PDE for the first equation as Physics and the Poisson's equation for the second equation
Thank you very, very much !!!!
2 Replies Last Post Jan 20, 2015, 8:42 a.m. EST
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Posted:
9 years ago
Asimina,
To model a drift-diffusion equation, I recommend using the "Transport of Diluted Species" physics interface. If you have access to the Chemical Reaction Engineering module, you can add migration due to an electric field as a built-in option. If not, you can still implement the term by activating convection, using the conservative form of the equation, and setting the velocity term accordingly.
If the space charge is significant, the drift-diffusion equation and the Poisson equation will be strongly coupled and must be solved simultaneously. For a time-dependent problem, you simply need to use a Time-Dependent study step. Even though it doesn't have a time-derivative, the Poisson equation will still be solved at each time step. If you are only interested in an equilibrium solution, you may still need to solve it as a time-dependent problem to get a solution.
If the space charge is not significant, you can solve the Poisson equation first using a Stationary study step and then solve the drift-diffusion equation as a second step.
Good luck!
--
Luke Gritter
AltaSim Technologies
To model a drift-diffusion equation, I recommend using the "Transport of Diluted Species" physics interface. If you have access to the Chemical Reaction Engineering module, you can add migration due to an electric field as a built-in option. If not, you can still implement the term by activating convection, using the conservative form of the equation, and setting the velocity term accordingly.
If the space charge is significant, the drift-diffusion equation and the Poisson equation will be strongly coupled and must be solved simultaneously. For a time-dependent problem, you simply need to use a Time-Dependent study step. Even though it doesn't have a time-derivative, the Poisson equation will still be solved at each time step. If you are only interested in an equilibrium solution, you may still need to solve it as a time-dependent problem to get a solution.
If the space charge is not significant, you can solve the Poisson equation first using a Stationary study step and then solve the drift-diffusion equation as a second step.
Good luck!
--
Luke Gritter
AltaSim Technologies
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Posted:
9 years ago
An even better solution here would be to look into the functionality of the Semiconductor Module:
www.comsol.com/semiconductor-module
It is correct that these equations should be solved simultaneously, as they are strongly coupled and highly nonlinear. The Semiconductor Module has several advantages over trying to set up this coupling by hand, including having an already implemented linear and logarithmic finite element formulation, as well as a finite volume formulation. There is also a special stabilization scheme (Scharfetter-Gummel upwinding) that will enhance numerical stability.
www.comsol.com/semiconductor-module
It is correct that these equations should be solved simultaneously, as they are strongly coupled and highly nonlinear. The Semiconductor Module has several advantages over trying to set up this coupling by hand, including having an already implemented linear and logarithmic finite element formulation, as well as a finite volume formulation. There is also a special stabilization scheme (Scharfetter-Gummel upwinding) that will enhance numerical stability.