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.
The diffusivity determined experimentally over a range of diffusion temperatures can be expressed as :
\(D=D_{0}\exp\left(-\dfrac{E}{RT}\right)\)
Diffusivity follows the Arrhenius law — option 3:
\(D=D_{0}\exp\left(-\dfrac{E}{RT}\right)\)
Why the exponent must be negative. A dopant atom moves only by surmounting an energy barrier — it must break free of its lattice site or wait for a vacancy. The fraction of atoms with enough thermal energy to do so is the Boltzmann factor \(e^{-E/RT}\), which rises as T rises because the exponent becomes less negative. Option 1's positive exponent would make D fall with temperature, and its argument \(RT/q\) is a voltage rather than a dimensionless ratio in any case.
Why the exponent must be dimensionless. Both E and RT are energies per mole, so their ratio is a pure number, as an exponent must be. Option 2 divides \(D_{0}\), a diffusivity, by an energy, and option 4 divides an energy by a temperature — neither is dimensionless, so both fail before any physics is considered.
How the constants are measured. Taking logarithms turns the law into a straight line:
\(\ln D=\ln D_{0}-\dfrac{E}{R}\left(\dfrac{1}{T}\right)\)
Plotting \(\ln D\) against \(1/T\) gives a slope of \(-E/R\) and an intercept of \(\ln D_{0}\). This Arrhenius plot is exactly how the tabulated values for each dopant are obtained — which is what the question means by "determined experimentally over a range of diffusion temperatures".
| Symbol | Meaning |
|---|---|
| D0 | Pre-exponential factor — the extrapolated diffusivity at infinite temperature |
| E | Activation energy, typically 3 – 4 eV in silicon |
| R | Gas constant (or k, with E in eV per atom) |
How steep the dependence is. With \(E\approx3.5\) eV, a rise from 1000 °C to 1100 °C — less than 8 % in absolute temperature — multiplies D by roughly a factor of ten. This is why diffusion furnaces are controlled to within a degree, and why the junction depth \(x_{j}\propto\sqrt{Dt}\) is set far more sensitively by temperature than by time.
Hence, the correct expression is D = D0 exp(−E/RT).
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Important Questions from Semiconductor Process Flow
Which one of the following processes is preferred to form the gate dielectric (SiO2) of MOSFETs?
Arrange the following in the correct sequence of steps for MOS fabrication
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B. Photolithography
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Choose the correct answer from the options given below:Which of the following statements are true for optical photoresists
A. Solubility of a photo-resist changes on exposure to light in developer solution.
B. Solubility of a photo-resist does not change on exposure to light in developer solution.
C. A negative photo-resist on exposure to light becomes less soluble in developer solution.
D. A positive photo-resist on exposure to light becomes more soluble in developer solution.
Choose the correct answer from the options given below:- In particle beam lithography, the resolution depends upon :
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:
