Question

A 100mm crystalline silicon wafer of 1mm thickness is coated with arsenic. The as-coated silicon wafer is annealed in a furnace to allow arsenic to diffuse into the silicon. Under these conditions, the solubility of arsenic is 2.3×10²¹ atoms/cm³, and the diffusion constant is 5×10⁻¹³cm²/s. There is an arsenic impurity in the silicon at 2.3×10¹⁷ atoms/cm³. The goal is to have a concentration of arsenic atoms of 2.065×10²⁰ atoms/cm³ at a depth of 0.5 microns into the silicon.

What is the time required for this concentration to occur?

Answer 1

The problem asks for the time required for an arsenic diffusion concentration to reach a specific level in silicon, but cannot be solved without further information or assumptions. Relevant parameters include the diffusion coefficient and the target concentration at a set depth.

Step-by-step explanation:

The problem involves determining the time required for a concentration of arsenic to diffuse to a specific depth in a silicon wafer during an annealing process. This is a classic diffusion problem which can be approached using Fick's second law for non-steady-state diffusion since the concentration at a specific depth changes over time.

However, calculating the time explicitly requires additional information or simplifying assumptions, such as the initial concentration profile and the boundary conditions. Typically, the solution would involve the use of complementary error functions or similar methods to express the concentration as a function of time and position.

Without these simplifications or additional data, we cannot solve this mathematical problem completely. Nevertheless, we can discuss the relevant parameters such as the diffusion coefficient (5×10⁻¹³ cm²/s) and the desired concentration (2.065×10²⁰ atoms/cm³) at a specific depth (0.5 microns).

Under theoretical circumstances, one would set up the appropriate differential equation with the given boundary conditions and solve for time, which typically involves error function solutions for the concentration profile in the wafer. Since the problem doesn't provide sufficient specifics, we must acknowledge that a clear numerical answer cannot be calculated from the information given.