A pure Si crystal has 5×10^28 atoms per metre cube. It is doped by 1 ppm concentration of As. Find number of electrons and holes. Given that number of intrinsic charge carriers is 1.5×10^16 per metre cube.

To determine the number of electrons and holes in a doped silicon (Si) crystal, we first need to understand the effects of doping and how it influences the charge carrier concentration. In this case, we have a pure silicon crystal with a concentration of 5×10^28 atoms per cubic meter, which is then doped with arsenic (As) at a concentration of 1 part per million (ppm).

Doping and Charge Carriers

Doping silicon with arsenic introduces extra electrons into the crystal structure. Arsenic is a donor impurity, meaning it donates an extra electron to the conduction band of silicon. The concentration of arsenic at 1 ppm can be converted into a number of atoms per cubic meter.

Calculating the Concentration of Arsenic

1 ppm means 1 part of arsenic per 1 million parts of silicon. To find the number of arsenic atoms in the crystal, we can use the following calculation:

• Concentration of Si = 5×10^28 atoms/m³
• 1 ppm = 1×10^-6
• Concentration of As = 1×10^-6 × 5×10^28 = 5×10^22 atoms/m³

Determining the Number of Electrons

Each arsenic atom donates one electron. Therefore, the number of extra electrons contributed by the arsenic doping is equal to the concentration of arsenic:

• Number of electrons from As = 5×10^22 electrons/m³

Calculating the Total Electron Concentration

The total electron concentration in the doped silicon can be found by adding the intrinsic electron concentration to the number of electrons contributed by the doping:

• Intrinsic electron concentration = 1.5×10^16 electrons/m³
• Total electron concentration = 1.5×10^16 + 5×10^22 = 5.000015×10^22 electrons/m³

Finding the Hole Concentration

In a semiconductor, the relationship between electron concentration (n), hole concentration (p), and intrinsic carrier concentration (n_i) is given by the equation:

n * p = n_i²

Where:

• n = total electron concentration
• p = hole concentration
• n_i = intrinsic carrier concentration

We can rearrange this equation to solve for the hole concentration:

• p = n_i² / n

Substituting the known values:

• n_i = 1.5×10^16 electrons/m³
• n = 5.000015×10^22 electrons/m³

Now, calculating p:

• p = (1.5×10^16)² / (5.000015×10^22)
• p = 2.25×10^32 / 5.000015×10^22 ≈ 4.5×10^9 holes/m³

Summary of Results

In summary, after doping the pure silicon crystal with arsenic at a concentration of 1 ppm, we find:

• Total number of electrons = 5.000015×10^22 electrons/m³
• Total number of holes = 4.5×10^9 holes/m³

This analysis shows how doping can significantly alter the charge carrier concentrations in semiconductors, which is crucial for their applications in electronic devices.