Question

For an n-type silicon, an extrinsic semiconductor, the natural logarithm of normalized conductivity (σ) is plotted as a function of inverse temperature. Temperature interval-I corresponds to the intrinsic regime, interval-II corresponds to saturation regime and interval-III corresponds to the freeze-out regime, respectively. Then

• A. the magnitude of the slope of the curve in the temperature interval-I is proportional to the bandgap, E_g
• B. the magnitude of the slope of the curve in the temperature interval-III is proportional to the ionization energy of the donor, E_d
• C. in the temperature interval-II, the carrier density in the conduction band is equal to the density of donors
• D. in the temperature interval-III, all the donor levels are ionized

Answer 1

The Correct Option is A, B, C

To understand the behavior of the conductivity in an n-type silicon semiconductor across different temperature intervals, let's analyze the given plot of the natural logarithm of normalized conductivity, \( \ln(\sigma) \), as a function of inverse temperature, \( \frac{1}{T} \).

1. The plot is divided into three intervals:
   • Interval-I (Intrinsic Regime): In this high-temperature region, the intrinsic carriers dominate. The slope of the curve in this interval is related to the intrinsic carrier concentration and is proportional to the bandgap energy, \( E_g \). This is because the intrinsic carrier concentration is given by: \(n_i \propto e^{-\frac{E_g}{2kT}}\), where \( k \) is the Boltzmann constant. Thus, the magnitude of the slope is proportional to \( E_g \).
   • Interval-II (Saturation Regime): In this medium-temperature range, the carriers from donor atoms dominate the conduction, and the carrier density in the conduction band becomes approximately equal to the density of donors. This condition holds because most of the donor atoms are ionized, but the intrinsic carrier contribution is still negligible.
   • Interval-III (Freeze-Out Regime): At low temperatures, not all donors are ionized. The slope in this part reflects the ionization energy of the donor atoms, \( E_d \). Here, the conductivity depends on the ionization of electrons from donor levels, expressed by: \(\sigma \propto e^{-\frac{E_d}{kT}}\). Hence, the magnitude of the slope is proportional to \( E_d \).

Thus, analyzing each temperature interval, we conclude that the correct interpretations for the plot are:

• The magnitude of the slope of the curve in the temperature interval-I is proportional to the bandgap, \( E_g \).
• The magnitude of the slope of the curve in the temperature interval-III is proportional to the ionization energy of the donor, \( E_d \).
• In the temperature interval-II, the carrier density in the conduction band is equal to the density of donors.

Option not chosen: "In the temperature interval-III, all the donor levels are ionized," is incorrect as not all donors are ionized at low temperatures. The electrons are bound to donor atoms, reflecting the freeze-out behavior.