Read the following passage and answer the questions that follow : Diffusion of impurity atoms in silicon is important in silicon integrated circuit processing. The idea of using diffusion techniques to alter the type of conductivity in silicon or germanium was disclosed in a patent by Pfann. Since then various ways of introducing dopants into silicon by diffusion have been studied with the goal of controlling dopant distribution, total dopant concentration, its uniformity reproducibility, and of processing a large number of device wafers in a batch to reduce the manufacturing cost. The diffusion is used to form bases, emitters and resistors in bipolar devices technology, to form source and drain regions and to dope polysilicon in MOS device technology. Dopant atoms that span a wide range of concentrations can be introduced into silicon in many ways.
A thin layer of dopant is deposited onto silicon surface with a fixed total amount of dopants QT per unit area. If D is the diffusivity of dopants, then the surface concentration is :
\(C_{S}=\dfrac{Q_{T}}{\sqrt{\pi Dt}}\)
This is the limited-source or drive-in case, whose profile is Gaussian — and the surface concentration is that Gaussian evaluated at x = 0.
The profile. With a fixed total dose QT deposited in a thin layer and then driven in, the solution of the diffusion equation is
\(C(x,t)=\dfrac{Q_{T}}{\sqrt{\pi Dt}}\exp\left(\dfrac{-x^{2}}{4Dt}\right)\)
Setting x = 0 gives the surface value, since the exponential becomes 1:
\(C_{S}=\dfrac{Q_{T}}{\sqrt{\pi Dt}}\)
— option 3. Option 2 is the trap: it is the complete profile \(C(x,t)\) rather than the surface concentration, so it still carries the x-dependent exponential. The question asks specifically for \(C_{S}\).
Two checks that confirm the form.
Dimensions. QT is a dose per unit area, in cm–2. Since Dt has units of cm2, \(\sqrt{\pi Dt}\) is a length, and dividing gives cm–3 — a concentration, as required. Option 4 divides by Dt rather than its square root and yields the wrong units; option 1 equates a volume concentration to an areal dose.
Conservation. Integrating the Gaussian over all depth returns exactly QT, which is the defining property of this case: no dopant enters or leaves, it merely spreads.
The physical consequence is important. Because QT is fixed, the surface concentration falls as \(1/\sqrt{t}\) while the profile deepens as \(\sqrt{Dt}\). Driving in longer spreads the same dopant further and dilutes the surface — exactly what is wanted when a lightly doped, deep region such as a well or a bipolar base is needed.
Contrast with the other standard case. A constant-source diffusion holds the surface at the solid solubility limit and lets dopant keep entering; its profile is a complementary error function, its surface concentration is fixed, and the dose grows as \(\sqrt{t}\). Real processes use both in sequence — a short constant-source predeposition to meter the dose, then a limited-source drive-in to place it.
Hence, CS = QT/√(πDt).
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Arrange the following in the correct sequence of steps for MOS fabrication
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D. A positive photo-resist on exposure to light becomes more soluble in developer solution.
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C. Wet etching is normally anisotropic
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E. Bird's beak formation is a characteristic of LOCOS isolation
Choose the correct answer from the options given below:
