Phosphorus atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat treatments; the background concentration of \(\mathrm{P}\) in this silicon material is known to be \(5 \times 10^{19}\) atoms \(/ \mathrm{m}^{3}\). The predeposition treatment is to be conducted at \(950^{\circ} \mathrm{C}\). for 45 minutes; the surface concentration of \(P\) is to be maintained at a constant level of \(1.5 \times 10^{26}\) atoms \(/ \mathrm{m}^{3}\). Drive-in diffusion will be carried out at \(1200^{\circ} \mathrm{C}\) for a period of \(2.5 \mathrm{~h}\). For the diffusion of \(\mathrm{P}\) in \(\mathrm{Si}\), values of \(Q_{d}\) and \(D_{0}\) are \(3.40 \mathrm{eV} /\) atom and \(1.1 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), respectively. (a) Calculate the value of \(Q_{0}\). (b) Determine the value of \(x_{j}\) for the drive-in diffusion treatment. (c) Also for the drive-in treatment, compute the position \(x\) at which the concentration of \(P\) atoms is \(10^{24} \mathrm{~m}^{-3}\).

Phosphorus diffusion into silicon is a critical process in the manufacturing of semiconductor devices. The process involves the incorporation of phosphorus atoms into the silicon lattice to create regions of increased positive charge, or 'n-type' regions, within the semiconductor. This modification alters the electrical properties of the silicon, making it a fundamental step in creating diodes, transistors, and integrated circuits.

The process entails two major steps: predeposition and drive-in heat treatment. During predeposition, phosphorus atoms are introduced at the silicon surface, typically via a gas phase containing the dopant atoms at high temperatures. The surface concentration of phosphorus is maintained at a constant level through control of the environment and temperature, allowing for a predictable and uniform diffusion into the substrate.

Understanding phosphorus diffusion requires a grasp of the variables that affect it, such as temperature and time, as these determine how deep and how fast the phosphorus atoms will diffuse into the silicon wafer.

Silicon wafer predeposition is the initial phase in the doping process where dopant atoms, such as phosphorus, are deposited onto the silicon surface. This stage aims to establish a high concentration of dopants at the surface, from which they will subsequently diffuse into the wafer. The predeposition step is performed at elevated temperatures, and the temperature must be precisely controlled to ensure that the dopant atoms do not diffuse too deeply.

For example, in the exercise, the predeposition treatment is conducted at a temperature of 950°C for 45 minutes. A surface concentration of phosphorus is maintained at a constant level, which lays the groundwork for the subsequent diffusion process. It is the control over these predeposition conditions that determines the initial distribution of phosphorus atoms on the wafer surface and sets the stage for the following drive-in diffusion step.

Following the predeposition, drive-in heat treatment is the next crucial step. During this phase, the wafer is subjected to a higher temperature for an extended period, leading to additional diffusion of phosphorus deeper into the silicon substrate.

In the context of the provided exercise, the drive-in diffusion is performed at 1200°C for 2.5 hours, which allows the phosphorus atoms to migrate from the high concentration surface layer into the bulk silicon wafer, creating a graded dopant profile. This high-temperature phase 'drives' the dopants in to achieve the desired junction depth and concentration levels throughout the silicon material.

The resulting electrical characteristics of the silicon are tailored by carefully controlling the drive-in conditions, making it an essential part of semiconductor device fabrication. The complete process, including predeposition followed by drive-in, forms the profile of the dopant distribution, which directly impacts the performance of the final electronic devices.

The diffusion coefficient is a crucial parameter in the analysis of diffusion processes, as it quantifies the rate at which atoms or molecules spread out due to random motion. In the context of phosphorus diffusion into silicon, the diffusion coefficient (D) can be calculated using the equation:
\[D = D_0 \mathrm{e}^{-\frac{Q_d}{kT}}\]
where \(D_0\) is the pre-exponential factor representing the diffusion coefficient at infinite temperature, \(Q_d\) is the activation energy for diffusion, \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature in Kelvin.

Determining the correct diffusion coefficient is necessary for predicting how quickly and how far phosphorus atoms will diffuse within a given time at a particular temperature. By calculating this for different temperatures, as shown in the exercise for both predeposition and drive-in treatments, engineers can optimize diffusion processes to create the desired doping profiles for semiconductor devices.