A Quasi-2D Model for Reverse Short Channel Effect
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SI
(1)
The "÷1" model [5, 6] was used to determine initial concentration of interstitial. The effective plus number of one was used. The parameters used in our calculations such as binding energy of BI pairs, equilibrium concentration of interstitials, diffusivities of interstitials and BI pairs are from ab initio [7] and tight-binding [8] calculations. According to ab initio calculations boron diffuses by kick-out mechanism [7]. Since we want to reduce the number of species, the specie of boron interstitials is not included in the present model. In this case an effective diffusivity of BI pairs has to be used. Let To, •-• and DB• be kick-out time, kick-in time and diffusivity of boron interstitiais, respectively. After kicking out of a boron atom, the atom will diffuse for a time of •-• until it kicks in. The corresponding diffusion length, lB,, is given by 12 -- 6DB•vi. For a BI pair to diffuse a Bi
distance Iu,, it takes a time of To to kick-out the boron in the pair and a time of •-i for the 141 Mat. Res. Soc. Symp. Proc. Vol. 532 ©1998 Materials Research Society
boron to finish diffusing. Therefore, a BI pair diffusing in a distance of IB 3 will take a total time of (T- + Ti). The effective BI diffusivity of DBJ is DBrTi/(Tr+-i). From the ab initio data [7], we obtain that DBI = DoB1 exp(-0.7eV/kT), where Dt1 is a constant. T and k
represent absolute temperature and Boltzmann constant. The present form of the equations in the model is one-dimensional in the lateral direction. It is straightforward to extend the model to a 2D structure [9]. To consider diffusion in the depth direction, we use a layer structure, as shown in Fig. 1, to approximate a real 2D system which is the reason why we call it as a quasi-2D model. The layer structure consists of an interface of Si/Si0 2 and three Si layers below the interface. Since the boron pile-up in the channel of devices is mostly within a 400 A region near the Si/SiO2 interface, a thickness of 400 A was chosen for each layer. The diffusion flux between the adjacent layers are determined by first Fick's law. There are three reaction-diffusion equations for the interface and each layer. So there are totally twelve reaction-diffusion equations needed to be solved. We have used ALAMODE [10], a PDE solver developed in Stanford University, to numerically solve the model. To model the interface as a sink of interstitials, a damping term of (Ct-itf V f was implemented in the equation describing the interstitial concentration in the Si/Si0 2 interface aCt-itf
t
CI _ C~eq T--
koC-tC-t+keC kforCl-i
-t tf ±
+Ct-1
±
- Ct-itf -itf - 1
2
(2)
where C1 -itf, Ceq, and r--itf are the interstitial concentration in the interface, interstitial concentration at equilibrium and the lifetime of interstitials in the interface. The quantities of k1fo, and krev are the reaction rates for the forward and reverse reactions expressed by eq.(1) taking place in the interface. The fourth term on right-hand-side of eq.(2) describes interstitial flux between the interface and the layer belo
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