Estimate the temperature at which GaAs has an electrical conductivity of \(3.7 \times 10^{-3}\) \((\Omega \cdot \mathrm{m})^{-1}\), assuming the temperature dependence for \(\sigma\) of Equation 18.36. The data shown in Table \(18.3\) may prove helpful.

Gallium Arsenide (GaAs) is a compound semiconductor material that has distinctive properties distinguishing it from traditional silicon semiconductors. Due to its efficient conversion of electricity into light, GaAs is commonly used in light-emitting diodes (LEDs), laser diodes, and solar cells.

Its electron mobility is higher than silicon, meaning charge carriers (electrons and holes) can move faster in GaAs, leading to its prevalent use in high-speed electronic devices and integrated circuits. Unlike silicon which operates well under standard conditions, GaAs particularly excels in high-frequency and high-temperature environments.

This semiconductor also exhibits a direct bandgap, implying that it can efficiently emit light when an electron transitions from the conduction band to the valence band. This makes it an ideal substance for optoelectronic applications where the conversion of electrical signals to light is crucial.

Semiconductor materials are the backbone of modern electronics. They have electrical conductivity levels between conductors (like metals) and insulators (like glass). This unique property stems from their atomic structure, which allows for the controlled addition of impurities, a process known as doping, to modify their electrical characteristics.

These materials are divided into two main types: intrinsic and extrinsic. Intrinsic semiconductors are pure forms of the material, while extrinsic semiconductors have been doped with impurities to increase the number of charge carriers (either electrons or holes).

Semiconductors support the flow of electrons but only under certain conditions, typically influenced by factors such as temperature or the presence of light. This controllable conductivity is essential for the operation of electronic devices like diodes, transistors, and integrated circuits.

Electrical resistivity, often denoted by the symbol \rho, is a fundamental property that quantifies how strongly a given material opposes the flow of electric current. The inverse of resistivity is conductivity, symbolized as \(\sigma\), which measures how easily electric current can pass through a material.

For semiconductors like GaAs, the resistivity is highly sensitive to temperature, the presence of impurities, and the intrinsic properties of the material. As temperature increases, semiconductors typically become more conductive because the thermal energy overcomes the bandgap energy, freeing more charge carriers to conduct electricity.

The exercise provided explores this relationship, showing that to find the specific temperature that corresponds to a particular conductivity, one must manipulate the formulas connecting resistivity, temperature, and the material's characteristics.

Thermal activation energy, represented as \(E_g\), relates to the energy needed to move an electron from the valence band to the conduction band in a semiconductor. This transition is essential for electrical conduction, as it determines the number of charge carriers available to conduct current.

In the context of the exercise, the activation energy is part of the formula used to calculate the temperature-dependent electrical conductivity of GaAs. A higher thermal activation energy indicates that the material needs more energy (in the form of heat, for instance) for its electrons to overcome the energy barrier and become free charge carriers.

The exponential factor in the equation \(\sigma(T) = \sigma_0 e^{-\frac{E_g}{2kT}}\) shows the temperature's effect on conductivity, where \(E_g\) denotes the thermal activation energy, \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature. This exemplifies how critical the principle of thermal activation is in determining the behavior of semiconductor materials under different thermal conditions.