Question

A semime tal (e.g. antimony, bismuth) is a material in which electrons would fill states to the top of a band the valence band- -except for the fact that the top of this band overlaps very slightly with the bottom of the nexthigher band. Explain why such a material, unlike the "real" metal copper, will have true positive charge carners and equal numbers of negative ones, even at zero temperature.

Short answer

In semimetals, even at zero temperature, the overlapping of the conduction band and the valence band allows electrons to move from the valence band to the conduction band. When an electron moves to the conduction band, it leaves behind a 'hole' in the valence band. These 'holes' act as positive charge carriers. Hence, the material ends up with an equal number of positive (holes) and negative (electrons) charge carriers.

Step-by-step solution

Understand the Electronic Bands

In a semimetal like antimony or bismuth, the valence band (which is the band of electron orbitals at a lower energy level) is fully filled up with electrons. However, its top overlaps slightly with the bottom of the next higher band which is the conduction band. This makes it easy for electrons in the valence band to move to the conduction band.

Movement of Electrons

When an electron moves from the valence band into the conduction band, it leaves behind a 'hole'. This 'hole' in the valence band acts as a virtual positive charge carrier because it's a vacancy that can be filled by an electron, which carries a negative charge.

Zero Temperature Condition

At zero temperature, the electrons occupy the lowest energy states available. However, due to the overlap of the bands, some electrons in the valence band are at a higher energy state than the ones in the conduction band. Hence, even without any external energy input (like in case of zero temperature), some of the valence band electrons will move into the conduction band creating an equal number of holes in the valence band.

Key concepts

Electronic Band Structure

The electronic band structure in solids is a fundamental concept that explains how electrons are distributed in materials. For semimetals, this involves the arrangement of energy levels into bands. Each band represents a range of energy levels that electrons can occupy. At the lower end of these energy bands is the valence band, which is generally filled with electrons. The higher energy band, known as the conduction band, is where electrons can move freely, allowing them to conduct electricity. In semimetals, the distinct characteristic is the slight overlap between the top of the valence band and the bottom of the conduction band, allowing for some unique electrical properties.

When electrons occupy these overlapping energy states, they can easily transition between bands. This behavior is critical in understanding the electrical conductivity of semimetals and differentiates them from typical insulators, where a large gap exists between the valence and conduction bands, preventing such movement.

Valence and Conduction Bands

In the context of semimetals, the valence and conduction bands play crucial roles. The valence band is the outermost band of occupied energy levels at zero temperature and contains the electrons responsible for the chemical characteristics of the material. The conduction band, on the other hand, is the band above the valence band and is essential for electrical conductivity.

Normally, there's an energy gap between these two bands. However, in semimetals, this gap is nonexistent or extremely small due to overlap. This overlap results in some electrons from the valence band having enough energy to act as if they're part of the conduction band. This property contributes to the semimetal's ability to conduct electricity but in a way that is quite different from traditional conductors like copper, wherein a significant number of free electrons exist within the conduction band.

Charge Carriers in Semimetals

Charge carriers are particles that carry electric charge, and in semimetals, both electrons and holes (the absence of electrons) act as these carriers. Because of the band overlap in semimetals, electrons can easily jump into the conduction band. Each time an electron moves out of the valence band, it leaves behind a hole. These holes are essentially positive charge carriers. This behavior is in contrast to traditional conductors, where only free electrons typically serve as charge carriers.

In semimetals, at any given moment, there is a presence of an equal number of negative charge carriers (electrons in the conduction band) and positive charge carriers (holes in the valence band). This balance is a unique feature of semimetals, leading to their distinct electrical phenomena.

Electron Movement in Semimetals

Electrons in semimetals display interesting movement because of the close proximity of the valence and conduction bands. Unlike in metals, where electrons can easily move around within a substantial conduction band, electron movement in semimetals involves transitioning between the valence and conduction bands. When an external electric field is applied, electrons can readily move to the conduction band and leave behind holes that act as positive charges.

These transitions are responsible for the electrical conductivity in semimetals. Moreover, the movement of these electrons and holes under the influence of an external field is crucial for the functioning of electronic devices that use semimetals. Understanding electron movement in semimetals is not only important for basic physics but also for practical applications in modern technology.

Zero Temperature Electron Configuration

At absolute zero temperature (0 Kelvin), the behavior of electrons in semimetals is fascinating. One might expect all electrons to reside in the lowest energy states possible, filling up the valence band completely. However, because of the overlapping band structure in semimetals, some of the lowest energy states are actually in the conduction band. As a result, there are electrons in both the valence and conduction bands even without any thermal energy to excite them.

This zero temperature condition leads to the presence of charge carriers — electrons and holes — in the absence of heat. Even at this minimal energy state, semimetals can thus conduct electricity, quite unlike conventional insulators or even metals. Providing students with such insights helps them understand the complex and often counterintuitive concepts underlying the behavior of materials at extreme conditions.