An \(n\)-type semiconductor is known to have an electron concentration of \(5 \times 10^{17} \mathrm{~m}^{-3} .\) If the electron drift velocity is \(350 \mathrm{~m} / \mathrm{s}\) in an electric field of \(1000 \mathrm{~V} / \mathrm{m}\), calculate the conductivity of this material.

At the heart of modern electronics lies the semiconductor, a material that exhibits conductivity levels between that of an insulator and a conductor. An n-type semiconductor is created when a pure semiconductor, typically silicon, is doped with elements from group V of the periodic table (like phosphorus or arsenic), which have more valence electrons than silicon.

These additional electrons, or charge carriers, are free to move within the crystal lattice of the semiconductor, which increases the material's conductivity. Consequently, in n-type semiconductors, electrons are the majority charge carriers, hence the 'n' for negative charge. This type of material is fundamental in electronics, serving as a building block for diodes, transistors, and various integrated circuits.

The presence of these extra electrons is crucial to the behavior of n-type semiconductors when they're placed in an electric field. The ability to manipulate the concentration of charge carriers is what allows for the precise control over the conductivity of the material, leading to a myriad of applications in electronic devices.

When discussing electrical current in semiconductors, it's essential to understand the concept of electron drift velocity. Drift velocity refers to the average velocity that free electrons in a conductor or semiconductor obtain due to an applied electric field. Although individual electron velocities can vary greatly due to constant collisions with atoms within the material, the drift velocity gives a measure of the overall motion of charge carriers.

The drift velocity (\( u_d \)) of electrons is directly proportional to the applied electric field (\( E \)). This relationship is described by the equation:\[ u_d = \frac{J}{n \times q} \]where \( J \) is the current density, \( n \) is the charge carrier density, and \( q \) is the charge of an electron. By knowing the drift velocity, one can infer a lot about the behavior of electrons in an electric field, which is pivotal for the design and function of electronic components. Faster drift velocities mean that charge can move through the material more quickly, resulting in better electrical performance.

The conductivity formula is a fundamental expression in solid-state physics and electrical engineering. It quantitatively describes how well a material can conduct an electric current. The formula is given by:\[ \text{Conductivity} (\text{\text{σ)}) = q \times n \times u_d \]where:\( q \) is the absolute value of electric charge per charge carrier (in coulombs), \( n \) is the concentration of charge carriers per cubic meter, and \( u_d \) is the drift velocity.

For an n-type semiconductor, you would use the electron charge and the concentration of electrons and multiply it by the electron drift velocity to calculate its conductivity. A high conductivity value typically means the material has a greater capacity to conduct electrical current. The unit of measure for conductivity is siemens per meter (\text{S/m}).

Understanding and applying this formula allows engineers and scientists to engineer materials with desired electrical properties for specific applications, such as in the fabrication of electronic devices where precise conductivity levels are required.