532 |
M.E. Law et al. |
built the RG terms, and then added or subtracted them from the appropriate continuity equations, allowing easy changing of reactions or reaction energetics and eventually 20 reactions all at once.
16.3.3.3Formulation of the Reaction Rates
Following [30], the reaction rates used above are calculated based on the energetics of 12 and 13 as calculated by first principles calculations. There are three cases of interest: (1) a mobile species combines with a static species, overcoming a reaction barrier that is larger than diffusion barrier; (2) a mobile species combines with a static species, but the reaction barrier is lower than the diffusion barrier; and (3) a mobile species escapes from a static species. The potential energy diagrams for the three cases are shown in Fig. 16.10.
The equations for each case are as follows:
ð |
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kBT |
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D exp |
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Ebar Ed |
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k ¼ |
2LCD |
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k ¼ f exp kBTi |
(16.30) |
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E |
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LC is an estimated “critical length” or critical distance beyond which the mobile particle cannot “see” the trap, but less than which the mobile particle will definitely be captured by the trap. This distance is on the order of a few atomic spacings for uncharged interactions. If one or more of the particles is charged, then this distance can be estimated to be larger. This value can be estimated because, as LC affects k on a
Fig. 16.10 Energy diagram showing diffusion and reaction energies in the forward and reverse directions
16 Reliability Simulation |
533 |
linear scale, it is not nearly as important as E, which affects k on an exponential scale. D is the diffusivity of the mobile species. The Einstein relation is used to convert between mobility and diffusivity. Ebar is the reaction barrier obtained from first principles calculations. Ed is the diffusion energy of the mobile species. It is subtracted from Ebar in order to avoid “double-counting” diffusion through both D and the reaction kinetics. Ei is the trap energy, or trap depth. f is the “attempt-to-escape” frequency of the trapped particle. kB is Boltzmann’s constant, and T is temperature.
16.3.3.4Preliminary Solves: Poisson-Only Solution, DC elec/hole/DevPsi Only
Before a transient simulation can be carried out, following the experimental conditions of [22] and [31], the system is first solved in equilibrium (DC). Obtaining convergence of electrons and holes in the oxide can be a challenge due to the very small concentrations there. Solving Poisson’s equation alone first, then using this solution as an initial guess for the full DC solve may be required. That is, first, solve only electrostatic potential as an independent variable, and then use the Boltzmann relation to directly calculate electrons and holes:
solution add name¼DevPsi pde damp continuous
#using the Poisson equation as defined above for DevPsi solution add Silicon name¼Elec const val¼(Nc*exp( Ec- 0.0/kT))
solution add Silicon name¼Hole const val¼(Nv*exp( 0.0- Ev/kT))
solution add Oxide name¼Elec const val¼(Nc*exp( Ec-0.0/ kT))
solution add Oxide name¼Hole const val¼(Nv*exp( 0.0-Ev/ kT))
where Nc, Nv, Ec, and Ev are appropriately defined for each material. The 0.0 represents the value of the Fermi level when the silicon is biased at 0 V. If the 0 of potential is defined differently, or if the silicon is biased at a different value, then this value will have to be changed. A simple initial guess will suffice for a Poissononly solution:
sel z¼ 4.1 name¼DevPsi device
The solutions in this simplified case are saved by FLOORS and can now serve as an initial guess for the full DC solution:
solution add name¼DevPsi pde damp continuous solution add name¼Elec pde !negative solution add name¼Hole pde !negative