(a) Calculate the drift velocity of electrons in germanium at room temperature and when the magnitude of the electric field is \(1000 \mathrm{~V} / \mathrm{m}\). (b) Under these circumstances, how long does it take an electron to traverse a \(25-\mathrm{mm}\) (1-in.) length of crystal?

Germanium is a group IV element and plays a significant role in the semiconductor industry. Its properties make it an important material for devices like transistors and diodes. One of the key characteristics of germanium is its intrinsic semiconductor capability at room temperature. This means that without any doping, germanium has a balance of electrons and holes—which are the positively charged counterparts to electrons—that can carry current.

Compared to silicon, another popular semiconductor, germanium has a higher atomic number and density, resulting in a different crystal lattice structure. This unique structure gives germanium a relatively high electron mobility, making it more responsive to electric fields. However, germanium's sensitivity to temperature changes and its relatively high level of intrinsic impurities limit its widespread use compared to silicon.

Intrinsic semiconductor properties make germanium valuable for certain applications, especially when low-temperature operation or high-speed signal processing is required. Because electron mobility is an important feature in germanium, it directly influences the performance of germanium-based electronic components.

Electron mobility is a significant property when discussing semiconductors like germanium. It refers to how quickly an electron can move through a material when subjected to an electric field. The speed at which these charge carriers (electrons) can move is vital for the efficiency and speed of semiconductor devices.

For germanium, the electron mobility is exceptionally high, with values such as the given example of \(3900 \mathrm{~cm}^2 / \mathrm{V} \cdot \mathrm{s}\) at room temperature. High mobility allows electrons to move more rapidly and with less resistance through the material, which is beneficial for fast-switching electronic devices and high-speed computing.

Given that mobility can be affected by temperature, impurities, and the crystalline structure of the semiconductor, achieving high purity and quality in the manufacturing process is crucial. This ensures that the innate high mobility of electrons in germanium is harnessed effectively for technological applications.

The electric field has a profound effect on the velocity of electrons within a semiconductor. Drift velocity, in particular, is the average velocity attained by charged particles, such as electrons, in response to an electric field. This relationship is depicted by the equation \(v_d = \mu_e \cdot E\), where \(v_d\) is the drift velocity, \(\mu_e\) the electron mobility, and \(E\) the electric field intensity.

When a voltage is applied, the electric field causes electrons to accelerate. However, these electrons frequently collide with the atoms in the semiconductor lattice, which limits their acceleration and results in a constant average velocity—this is the drift velocity. The higher the electric field, the greater the force on the electrons, leading to a higher drift velocity up to a certain point beyond which the material may become saturated or damaged.

The ability of germanium to handle a high electric field before reaching saturation makes it a choice material for certain high-speed electronic applications. It's remarkable how the intrinsic properties of germanium, combined with the effects of an electric field, work together to influence the behavior of electrons, which is the foundation of modern electronics.