A process engineer wants to reconstruct the dopant concentration profile in a silicon…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

A process engineer wants to reconstruct the dopant concentration profile in a silicon wafer of thickness L=1.0×10−4 mL=1.0\times10^{-4}\,\mathrm{m} (100 μm100\,\mu\mathrm{m}) as it stood t=3600 st=3600\,\mathrm{s} (one hour) earlier. Today's measured deviation of concentration from a uniform background is held (by the contacts) at zero at both faces x=0,Lx=0,L and is expanded in the sine modes sin⁡(nπx/L)\sin(n\pi x/L). The dopant diffusivity is D=1.0×10−13 m2 s−1D=1.0\times10^{-13}\,\mathrm{m^2\,s^{-1}}. ((a)) Compute the dimensionless group π2Dt/L2\pi^2Dt/L^2. ((b)) Compute the amplification factor for modes n=1,2,3n=1,2,3. ((c)) State which of Hadamard's three conditions hold for this backward reconstruction, and why.

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