Silicon has the diamond structure with a unit cell edge length of 5.43 Å and eight atoms per unit cell. (b) Suppose you dope that 1 cm3 sample of silicon with 1 ppm of phosphorus that will increase the conductivity by a factor of a million. How many milligrams of phosphorus are required?

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Convert the volume of the silicon sample from cm³ to Å³ using the conversion factor: 1 cm = 10^8 Å.

Calculate the number of unit cells in the 1 cm³ sample by dividing the total volume in Å³ by the volume of one unit cell (edge length cubed).

Determine the number of silicon atoms in the sample by multiplying the number of unit cells by the number of atoms per unit cell (8 atoms).

Calculate the number of phosphorus atoms needed for 1 ppm doping by taking 1 ppm of the total number of silicon atoms.

Convert the number of phosphorus atoms to mass in milligrams using the molar mass of phosphorus (30.97 g/mol) and Avogadro's number (6.022 x 10^23 atoms/mol).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Doping in Semiconductors

Doping is the intentional introduction of impurities into a semiconductor to modify its electrical properties. In silicon, adding elements like phosphorus creates free charge carriers (electrons), enhancing conductivity. The concentration of dopants is typically measured in parts per million (ppm), which indicates the amount of dopant relative to the total material.

Conductivity and its Measurement

Conductivity is a measure of a material's ability to conduct electric current, influenced by the number of charge carriers and their mobility. In semiconductors, conductivity can be significantly increased through doping. The relationship between conductivity and the concentration of charge carriers is crucial for understanding how doping affects a semiconductor's performance.

Unit Conversion and Mass Calculation

To determine the mass of phosphorus needed for doping, one must convert the concentration from ppm to a mass measurement. Since 1 ppm means 1 mg of solute per 1 L of solution, for a 1 cm³ sample, the calculation involves converting volume to liters and applying the ppm ratio to find the required mass in milligrams. This step is essential for quantifying the amount of dopant accurately.

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