Question

State whether each statement is true or false, and why. (a) A typical band gap energy for an insulator is \(400 \mathrm{~kJ} / \mathrm{mol}\) (b) The conduction band is higher in energy than the valence band. (c) Electrons can conduct well if they are in a filled valence band. (d) Holes refer to empty atomic sites in a solid crystal.

Short answer

(a) True. The given band gap energy of \(4.14 \text{ eV}\) lies within the typical range for insulators (2-10 eV). (b) True. The conduction band is associated with a higher energy level than the valence band. (c) False. Electrons in a filled valence band cannot move freely or change their energy states, resulting in low or no conductivity. (d) False. Holes refer to the absence of an electron in a filled energy state in the valence band, not empty atomic sites in a solid crystal.

Step-by-step solution

(a) Analyzing band gap energy

A typical band gap energy for an insulator is 400 kJ/mol. To verify if this statement is true or false, let's first understand the concept of band gap energy. In solid-state physics, the band gap (energy gap) is the energy range in which no electron states can exist. Insulators have a large band gap, which prevents electrons from moving across the bands and conducting electricity. The given value of 400 kJ/mol is equivalent to \(6.63 \times 10^{-20}\) Joules. When converted to electron volts (eV), it's about \( (6.63\times 10^{-20}\, \text{J} / 1.6\times 10^{-19}\, \text{Coulomb}) = 4.14 \, \text{eV} \). Considering the typical band gap for insulators is in the range of 2-10 eV, this statement is true.

(b) Analyzing the conduction band and valence band

The conduction band is higher in energy than the valence band. This statement is true. The valence band is the highest energy band in a solid in which electrons can reside. When an electron gains energy and moves to the conduction band, it becomes free and can participate in conduction. The conduction band is associated with a higher energy level than the valence band, and thus it is true the conduction band is of higher energy.

(c) Checking for conductivity in filled valence bands

Electrons can conduct well if they are in a filled valence band. This statement is false. In a filled valence band, all the electron states are occupied, so the electrons cannot move freely or change their energy states. In this case, there's no conductivity or very low conductivity. For good conduction, electrons must be in an almost empty conduction band or have available energy states to move into.

(d) Understanding holes in solid crystals

Holes refer to empty atomic sites in a solid crystal. This statement is false. Holes in solid-state physics do not refer to empty atomic sites. Instead, they refer to the absence of an electron in a filled energy state in the valence band. Holes can play a significant role in the conduction process as electrons can move to fill these holes, allowing for electrical current flow.

Key concepts

Band Gap Energy

Imagine electrons in a material as marathon runners at a starting line. The 'band gap energy' is like a tall wall separating the runners from the finish line. In solid-state physics, band gap energy represents the energy difference between the top of the valence band (where electrons 'live' when they're not conducting electricity) and the bottom of the conduction band (where electrons can move freely and conduct electricity).

For insulators, this 'wall' is usually very high, with a band gap typically between 2 to 10 electron volts (eV). Electrons in insulators need a significant amount of energy to climb over the wall to reach the conduction band. If a statement mentions a band gap energy of around 4.14 eV, it's plausible for an insulator, since it falls within the common range.

Conduction Band

Now, picture a high diving platform at a pool—the 'conduction band' is like this platform. It's situated above where electrons normally hang out (the valence band). Electrons need energy to climb up to this platform. Once there, they’re like divers ready to jump and swim freely—that is, conduct electricity.

The conduction band is a range of electron energy levels where if an electron reaches this band, it can conduct electricity through the material. This band is key for understanding the electrical properties of different materials. In semiconductors, having electrons in the conduction band allows for the fascinating electronics we use every day.

Valence Band

Back on the ground, the 'valence band' is the electron's usual hangout spot. Electrons here are like people lounging on their couches—they're comfortable, but they're not going anywhere, and thus, aren't doing much work.

The valence band contains the energy levels that electrons occupy at room temperature. When the valence band is completely filled with electrons, they're effectively 'locked in place' and cannot contribute to electrical conductivity. It's essential to understand that for electrons to help in electricity conduction, they need some room to move—which they won't have in a filled valence band.

Electrical Conductivity

Imagine a highway—electrical conductivity is the measure of how easily traffic (electrons) can flow. Just like a clear highway allows cars to move swiftly, materials with high electrical conductivity let electrons move freely and carry charge, which translates into electric current.

Conductors, such as metals, have lots of 'open lanes' for electron traffic because their valence band overlaps with the conduction band or they have an empty conduction band close in energy. Insulators, however, have 'traffic jams' due to a full valence band and a significant gap before the next open lane (conduction band). Semiconductors are in the middle—they can behave like conductors when some external energy is provided to push electrons into the 'open lanes' of the conduction band.

Holes in Semiconductors

Holes in semiconductors are not actual voids in the material, but rather the absence of an electron in a spot where one could exist—think of it like an empty seat on a bus. This 'empty seat' can move around as electrons from neighboring seats jump in to fill it. In electronic terms, these holes act like positive charge carriers because the movement of an absence of negative charge can be equated to the movement of a positive charge.

The mobility of these holes is crucial for semiconductor devices. When an electron gets excited to the conduction band, it leaves behind a hole in the valence band that can also contribute to electrical conductivity, especially in materials like silicon, germanium, and certain compound semiconductors used in diodes, transistors, and solar cells.