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.
Fick's second law of diffusion in one dimensional form can be expressed as :
\(\dfrac{\partial C(x,t)}{\partial t}=\dfrac{\partial}{\partial x}\left[D\dfrac{\partial C(x,t)}{\partial x}\right]\)
Fick's second law is a time derivative on the left and a space derivative twice on the right — option 1:
\(\dfrac{\partial C}{\partial t}=\dfrac{\partial}{\partial x}\left[D\dfrac{\partial C}{\partial x}\right]\)
Where it comes from — two statements combined.
Fick's first law says that flux is driven by a concentration gradient, and flows down it:
\(J=-D\dfrac{\partial C}{\partial x}\)
Continuity says that whatever does not flow out must accumulate:
\(\dfrac{\partial C}{\partial t}=-\dfrac{\partial J}{\partial x}\)
Substituting the first into the second gives the second law directly, and the two minus signs cancel. Seeing this derivation makes the structure inevitable: the left side is a rate of accumulation in time, and the right is the divergence of a flux in space. The two sides must therefore carry different variables, which immediately rules out options 2, 3 and 4 — each of which differentiates twice with respect to the same variable somewhere.
Checking option 1 dimensionally. With C in cm–3 and D in cm2/s:
\(\left[\dfrac{\partial C}{\partial t}\right]=\dfrac{\text{cm}^{-3}}{\text{s}},\qquad \left[\dfrac{\partial^{2}C}{\partial x^{2}}\right]=\dfrac{\text{cm}^{-3}}{\text{cm}^{2}}\)
and multiplying the second by D's cm2/s reproduces the first exactly. None of the other three balances.
Why D is left inside the derivative. If diffusivity is constant the equation simplifies to \(\partial C/\partial t=D\,\partial^{2}C/\partial x^{2}\), whose solutions are the Gaussian and error-function profiles of the previous questions. But D is not always constant: at concentrations above the intrinsic carrier density it becomes concentration dependent through the charged point defects that carry the dopant, so \(D=D(C)\) and the general form must be kept. That is why heavily doped profiles are box-shaped rather than Gaussian.
The same equation governs heat conduction, with temperature in place of concentration — which is why solutions worked out for one field transfer directly to the other.
Hence, the correct expression is option 1.
Similar Questions
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Match List - I with List - II.
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(A) Boron
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Important Questions from Semiconductor Process Flow
A. For impurity diffusion in semiconductor, concentration gradient is the only essential requirement.
B. Annealing is a mandatory step after ion-implantation.
C. Wet etching is normally anisotropic
D. High temperature is required for the growth of $SiO_2$ because oxidation reaction takes place at high temperature
E. Bird's beak formation is a characteristic of LOCOS isolation
Choose the correct answer from the options given below:
Arrange the given process steps in sequence for self-aligned gate formation.
A. Thermal oxidation to grow gate oxide.
B. Polysilicon deposition
C. Photolithography using gate mask
D. Etching of polysilicon and oxide
E. Source/drain diffusion
Choose the correct answer from the options given below:
- Match the LIST-I with LIST-IIChoose the correct answer from the options given below:
LIST-I LIST-II A. Shallow trench isolation (STI) I. Low static power dissipation B. Junction isolation II. High static power dissipation C. CMOS technology III. High component density D. Bipolar IC technology IV. Low component density Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: Impurity diffusion in semiconductor essentially needs very high temperature.
Reason R: Diffusivity of impurity atoms is negligibly small at room temperature.
In the light of the above statements, choose the most appropriate answer from the options given below
- In particle beam lithography, the resolution depends upon :
