Determine the amount (in grams) of boron that when substitutionally incorporated into 1 kg of silicon will establish a charge carrier density of 3.091×1017 carriers/cm3.
Answer : 2.4227*10^-3
i.e. 0.002423
thanks
Information
density Si=2.33 g/cc
Mw(P)=10.81 g/mole
NA=6.022*10^23 at/mole
V (Si)=1 Kg *1000 g/Kg /(2.33 g/cc = 4.29*10^2 cc
moles B/cc = 3.091*10^17 carriers/cc /6.022*10^23 =5.133*10^-7 moles B/cc
moles B= 5.133*10^-7 moles/cc *4.29*10^2 cc= 2.203*10^-4 moles B
g B= 2.203*10^-4 moles * 10.81 g/mole = 2.4227*10^-3 g
To determine the amount of boron in grams that would establish a specific charge carrier density in silicon, we need to consider the following steps:
Step 1: Convert the charge carrier density from cm³ to m³.
Given that the charge carrier density is 3.091×10^17 carriers/cm³, we need to convert it to carriers/m³ by dividing it by 10⁶, as there are 10⁶ cm³ in 1 m³.
So the charge carrier density becomes:
3.091×10^17 carriers/m³.
Step 2: Calculate the number of boron atoms required per unit volume.
Boron is commonly used as a dopant in silicon to modify its electrical properties. Silicon has a lattice constant of 5.43 Å (angstroms), and each silicon atom has a volume of (5.43 Å)³.
Thus, one silicon atom occupies a volume of (5.43 × 10^-10 m)³ = (1.827 × 10^-29 m³).
For every silicon atom that we substitute with boron, we introduce an additional charge carrier. Therefore, the number of boron atoms required per unit volume to achieve the given charge carrier density is given by the formula:
Number of boron atoms per m³ = (Charge carrier density) / (Number of charge carriers introduced per atom × Volume per silicon atom)
In this case, since every substitutional boron atom introduces one charge carrier, the formula becomes:
Number of boron atoms per m³ = (Charge carrier density) / (Volume per silicon atom).
Step 3: Convert the number of boron atoms to grams.
We need to convert the number of boron atoms per m³ to grams by considering the molar mass of boron (atomic weight) and Avogadro's number. The molar mass of boron is approximately 10.81 g/mol, and Avogadro's number is 6.022 × 10^23 atoms/mol.
Thus, the number of boron atoms per unit volume in grams is given by the formula:
Number of boron atoms (in grams) = (Number of boron atoms per m³) × (Molar mass of boron / Avogadro's number).
By following these steps, we can calculate the amount of boron in grams that would establish the desired charge carrier density in silicon.