Dopant Profile in Silicon Processing
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DOPANT PROFILE IN SILICON PROCESSING Kal. Renganathan Sharma Ph D PE Professor I/C, Research Vellore Institute of Technology (Deemed University) Vellore, TN, India 632 014 Tel: 91 – 416 – 243091, Fax: 91 – 416 – 243092 Email: [email protected]
Abstract: Non-Fickian effects are accounted for in dopant diffusion by the solution of hyperbolic mass wave propagative equation. The surface flux is represented by a modified Bessels composite function of first kind of 0th order in the open interval of τ>x.
Background: Doping is a general term which refers to the introduction of (1) impurities into a semiconductor medium. Solid state Diffusion and Ion Implantation are two techniques of doping. Solid state diffusion which is the oldest of the two methods consists of 2 steps. There are pre-deposition and drive-in. The study of transient diffusion is of interest in the VLSI industry. Improved models from experimental data to predict diffusion results from theoretical analysis are being developed. The ultimate goal (Sze, 2) of diffusion studies is to calculate the electrical characteristics of a semiconductor device from the processing parameter. The continuum theory of Fick’s simple diffusion equation can be used to explain the diffusivity of a dopant element as can be determined from experimental measurement. The other approach is the atomistic theory. Surface concentration, junction depth or concentration profiles may be obtained upon solution of continuum equations. Fick’s law of diffusion is valid only for low rate steady state transfer process. Physically, there exists a time where the linear mass diffusion relationship is not valid. This is because the Fick’s law implicitly assumes an infinite speed of propagation of mass in any media. In reality, mass travels at a large but finite velocity. Mathematically this manifests as singularity in concentration for short B4.30.1 Downloaded from https://www.cambridge.org/core. Access paid by the UCSB Libraries, on 06 Sep 2017 at 11:17:13, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1557/PROC-716-B4.30
times and short distances Maxwell (3) proposed the relaxation concept and used it to predict the Visco elasticity of materials. He assumed that there are no essential differences between the mechanical properties of viscous fluids and solids on the basis of relaxation concept. Relaxation is a progressive reduction of elastic stresses with a constant presented elastic strain. Shear stresses in a viscoelastic body release over a time interval, τr, at a certain finite rate. In a elastic body the shear stresses remain constant, the relaxation time becomes infinite and liquid behavior as a amorphous solid. When relaxation time approaches zero, the body behaves as on ordinary viscous liquid. Based on analogy with momentum and heat transfer a modified Ficks expression for mass transfer is proposed. J = -D
∂C ∂x
- τr
∂J ∂t
(1)
It was shown in such expressions (4) that the second term in equation is mathematica
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