Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are \(\rho_{h}=-\rho_{e}=6200 T^{1.5} e^{-7000 / T} \mathrm{C} / \mathrm{m}^{3}\). The functional dependence of the mobilities on temperature is given by \(\mu_{h}=2.3 \times 10^{5} T^{-2.7} \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}\) and \(\mu_{e}=2.1 \times 10^{5} T^{-2.5} \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}\), where the temperature, \(T\), is in degrees Kelvin. Find \(\sigma\) at: (a) \(0^{\circ} \mathrm{C} ;(b) 40^{\circ} \mathrm{C} ;(c) 80^{\circ} \mathrm{C}\).

Understanding the temperature dependence of carrier concentration is fundamental to grasping the behavior of semiconductors. Carrier concentration refers to the number of charge carriers, such as electrons and holes, present in a semiconductor. This concentration can greatly affect a material's electrical properties.

In semiconductors like silicon, as the temperature increases, the thermal energy provided to the material's atoms causes more electron-hole pairs to form. This process is reflected in the equation mentioned in the exercise, where the concentration of holes (and equivalently, the concentration of electrons in an intrinsic semiconductor) is given by \(\rho_{h} = 6200 T^{1.5} e^{-7000 / T} \mathrm{C} / \mathrm{m}^{3}\). The negative sign before \(\rho_{e}\) indicates that we are considering a negative charge for electrons.

The exponential factor \(e^{-7000 / T}\) represents the bandgap energy's influence, which must be overcome by thermal energy to generate carriers. At high temperatures, thermal energy is higher, making it easier to excite electrons into the conduction band, thus increasing the intrinsic carrier concentration.

The mobility of carriers in a semiconductor is another critical parameter that impacts its electrical properties. Mobility, denoted as \(\mu\), refers to how easily electrons and holes can move through a semiconductor when an electric field is applied. In general, higher mobility allows for better conductivity.

The mobility models for charge carriers suggest their velocities are not constant but can change with the temperature. The exercise provides two expressions for the temperature dependence of hole mobility \(\mu_{h}\) and electron mobility \(\mu_{e}\). Interestingly, both mobilities decrease with an increase in temperature, which is described by the equations \(\mu_{h} = 2.3 \times 10^{5} T^{-2.7} \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}\) and \(\mu_{e} = 2.1 \times 10^{5} T^{-2.5} \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}\).

This decrease in mobility at higher temperatures is attributed to the increase in phonon (vibration of the semiconductor lattice) activity, which leads to more frequent scattering of the carriers, thereby reducing their mobility.

The ability of a semiconductor to conduct electricity, termed as its electrical conductivity \(\sigma\), is directly related to both the carrier concentration and their mobility. The conductivity can be calculated using the formula given in the exercise: \(\sigma = q(\rho_{h}(\mu_{e} - \mu_{h}))\).

In this equation, \(q\) represents the elementary charge of an electron, which is a constant \(1.6 \times 10^{-19} \mathrm{C}\). Carrier concentrations \(\rho_{h}\) and mobilities \(\mu_{e}\) and \(\mu_{h}\) are variables provided in the problem statement, which are temperature-dependent. The product of the concentration of holes and the difference in mobilities of electrons and holes gives us the net charge per unit volume that contributes to conductivity.

By substituting the given values of temperature into the equations for \(\rho_{h}\), \(\mu_{e}\), and \(\mu_{h}\), one can calculate the conductivity at different temperatures. A key point to remember during calculations is that while \(\rho_{h}\) increases with temperature, the mobilities decrease, hence affecting the overall value of \(\sigma\).

Semiconductors, like silicon, exhibit significant changes in their electrical properties as a function of temperature. As seen in the problem provided, both the carrier concentration and mobility are temperature-dependent, which is typical of semiconductor behavior.

The effect of temperature on semiconductors is twofold: First, with increasing temperature, the intrinsic carrier concentration increases, which tends to boost conductivity. However, this is counteracted by the second effect, where the mobility of charge carriers decreases because the increased lattice vibrations lead to more scattering events.

These opposing effects imply a complex relationship between temperature and overall semiconductor performance. At low temperatures, the limited thermal energy results in a low carrier concentration, hindering conductivity. As the temperature rises, despite the increase in carrier concentration, the reduction in mobility may limit the improvements in conductivity, making it imperative to balance these factors in semiconductor design and application.