What is intrinsic semiconductor? Obtain an expression for the intrinsic carrier concentration in an 1 intrinsic semiconductor. Under what condition will the Fermi level be in the middle of the forbidden gap?

Intrinsic carrier concentration, denoted as \(n_{\text{i}}\), is a key concept in understanding how pure semiconductors behave. In an intrinsic semiconductor, the number of free electrons in the conduction band is equal to the number of holes in the valence band. Thus, the intrinsic carrier concentration is a measure of the electron and hole pairs generated purely due to thermal energy, without any impurities affecting the process. The mathematical expression for \(n_{\text{i}}\) is given by the equation, \[n_{\text{i}} = \frac{N_{\text{c}} N_{\text{v}}}{2} e^{-\frac{E_{\text{g}}}{2kT}}\]Here:

• \(N_{\text{c}}\) and \(N_{\text{v}}\) are the effective density of states in the conduction band and valence band.
• \(E_{\text{g}}\) is the energy bandgap, which is the energy difference between the conduction band and the valence band.
• \(k\) stands for Boltzmann's constant, and
• \(T\) is the absolute temperature.

This formula implies that as the temperature increases, \(n_{\text{i}}\) increases due to the exponential dependence on \(\frac{1}{T}\). This highlights the thermal generation of carriers.

The Fermi level, denoted as \(E_{\text{F}}\), is an important concept in semiconductor physics. It describes the energy level at which the probability of finding an electron is 50%. In intrinsic semiconductors, the Fermi level is positioned in the middle of the energy bandgap when the effective masses of electrons (\(m_{\text{e}}^*\)) and holes (\(m_{\text{h}}^*\)) are equal. The Fermi level position provides insight into the electrical properties and behavior of the semiconductor. If the effective masses are equal, the Fermi level equation is given by:\[E_{\text{F}} = \frac{1}{2} E_{\text{g}}\] This indicates symmetry in the distribution of electron and hole energies. However, in real materials, slight asymmetries can exist, and temperature or doping can shift the Fermi level from this central position.

The effective density of states in the conduction band (\(N_{\text{c}}\)) and the valence band (\(N_{\text{v}}\)) are parameters reflecting how many states are available for electrons or holes to occupy at a particular energy level. These are influenced by the effective masses of the carriers and the temperature. The equations for \(N_{\text{c}}\) and \(N_{\text{v}}\) are:\[N_{\text{c}} = 2\bigg(\frac{2\backpi m_{\text{e}}^* kT}{h^2}\bigg)^{\frac{3}{2}}\]\[N_{\text{v}} = 2\bigg(\frac{2\backpi m_{\text{h}}^* kT}{h^2}\bigg)^{\frac{3}{2}}\]Here:

• \(h\) is Planck's constant
• \(m_{\text{e}}^*\) is the effective mass of the electron
• \(m_{\text{h}}^*\) is the effective mass of the hole
• \(k\) is Boltzmann's constant
• \(T\) is the absolute temperature

These densities of states tell us how many states are available for occupation by the charge carriers and directly influence the intrinsic carrier concentration.

The energy bandgap (\(E_{\text{g}}\)) is the energy difference between the top of the valence band and the bottom of the conduction band in a semiconductor. It is a crucial property as it determines whether the material behaves as an insulator, semiconductor, or conductor. For intrinsic semiconductors, this gap is what electrons need to overcome to participate in conduction. The size of the energy bandgap influences the intrinsic carrier concentration according to the relationship:\[n_{\text{i}} \backsim e^{-\frac{E_{\text{g}}}{2kT}}\]In simpler terms, a larger \(E_{\text{g}}\) implies fewer intrinsic carriers at a given temperature, making the material less conductive. This is why semiconductors with different bandgaps are suitable for various applications; for instance, silicon (\(Si\)) has a smaller bandgap than diamond, making it more conductive at room temperature.

Boltzmann's constant (\(k\)) is a fundamental physical constant that appears in many statistical mechanics equations, including those describing intrinsic carrier concentration in semiconductors. It relates energy at the particle level with temperature. The value of Boltzmann's constant is approximately \(1.38 \times 10^{-23}\) J/K. In the context of intrinsic semiconductors, \(k\) appears in the expression for carrier concentration:\[n_{\text{i}} = \frac{N_{\text{c}} N_{\text{v}}}{2} e^{-\frac{E_{\text{g}}}{2kT}}\]Here, \(k\) helps scale the thermal energy available to push the electrons across the energy bandgap. Therefore, understanding Boltzmann's constant is crucial for predicting how the intrinsic carrier concentration will behave with changes in temperature and energy band.