Band Theory: Metals, Insulators, and Semiconductors

Hello! Welcome to our lesson on the electronic classification of solids.

In the previous lesson, we explored the concept of the electronic density of states (DOS), $g(E)$, which quantifies the number of available electronic states at a given energy. We saw how features in the band structure, $E(\mathbf{k})$, such as flat bands and critical points, give rise to sharp features in the DOS called van Hove singularities.

Today, we will use this knowledge to address a fundamental question in condensed matter physics: what makes a material a metal, an insulator, or a semiconductor? We will see that the answer lies in the interplay between the band structure (via the DOS) and the number of electrons available to occupy these states. Our goal is to distinguish between these material classes based on how the energy bands are filled and the resulting value of the DOS at the Fermi level.

1. Band Filling and the Fermi-Dirac Distribution

Imagine the energy bands of a solid as a set of "containers" for electrons, with the shape and capacity of these containers at each energy level described by the density of states, $g(E)$. To determine the properties of the material, we need to fill these containers with the available valence electrons.

At absolute zero temperature ($T=0$), electrons occupy the lowest available energy states, filling them up to a specific energy level known as the Fermi energy, $E_F$. All states with energy $E<E_F$ are occupied, and all states with $E>E_F$ are empty.

At any finite temperature $T>0$, the sharp cutoff at $E_F$ is smoothed out. The probability that a state with energy $E$ is occupied is given by the Fermi-Dirac distribution:

Here, $\mu$ is the chemical potential, which is equal to the Fermi energy at $T=0$. For typical temperatures, this distribution still closely resembles a step function, with significant changes in occupation occurring only in a narrow energy window of a few $k_{B}T$ around $\mu$.

The crucial factor that distinguishes different types of materials is the position of the Fermi level relative to the energy bands. There are two fundamental possibilities:

1. The Fermi level lies within an energy band.
2. The Fermi level lies in the gap between two energy bands.

This distinction is the basis for classifying materials as metals, semiconductors, or insulators.

2. Metals: Partially Filled Bands

A material is classified as a metal if, at $T=0$, its highest occupied energy band is only partially filled. As shown in the image above, this means the Fermi energy $E_F$ lies within a continuous band of states.

The defining characteristic of a metal is a non-zero density of states at the Fermi level: $g(E_F)\ne 0$.

Consequences for Conductivity:
Because there are empty states available at energies infinitesimally above $E_F$, it takes a vanishingly small amount of energy to excite an electron from an occupied state just below $E_F$ to an unoccupied state just above it. When an external electric field is applied, these electrons can easily gain energy and move into these adjacent states, resulting in a net flow of charge—an electric current. This is why metals are good conductors.

This situation typically arises in two scenarios:

• Odd number of valence electrons: If the crystal's unit cell contains an odd number of valence electrons (e.g., alkali metals like Na, K), the bands can never be completely filled, as each band (including spin) can hold an even number of electrons per unit cell.
• Band overlap: In many materials (e.g., alkaline earth metals like Mg, Ca), the top of a filled band can overlap in energy with the bottom of the next empty band. This creates a single, continuous, partially-filled band of states, making the material metallic even with an even number of valence electrons.

3. Insulators and Semiconductors: The Band Gap

A material is an insulator or a semiconductor if, at $T=0$, it has just enough valence electrons to exactly fill one or more energy bands, leaving all higher-energy bands completely empty.

• The highest filled band is called the valence band.
• The lowest empty band is called the conduction band.
• The energy difference between the top of the valence band ($E_v$) and the bottom of the conduction band ($E_c$) is the band gap, $E_g=E_c-E_v$.

In this case, the Fermi level $E_F$ is located somewhere within the band gap. The defining characteristic is a zero density of states at the Fermi level: $g(E_F)=0$.

Consequences for Conductivity:
At $T=0$, no current can flow in a weak electric field because there are no available empty states for electrons to move into. To conduct electricity, an electron must be given enough energy to jump from the valence band, across the entire band gap, into the conduction band. This requires an energy of at least $E_g$.

Distinguishing Semiconductors from Insulators

So, what is the difference between a semiconductor and an insulator? The distinction is not sharp but is one of degree, based on the magnitude of the band gap $E_g$.

• Insulators: Have a large band gap (e.g., $E_g>3$ eV).
• Semiconductors: Have a relatively small band gap (e.g., $E_g\approx 0.1-3$ eV).

This quantitative difference has profound consequences at finite temperatures. The probability of an electron being thermally excited from the valence band to the conduction band depends on the Boltzmann factor, $e^{-E_g/k_{B}T}$.

• For an insulator like diamond ($E_g\approx 5.5$ eV), at room temperature ($k_{B}T\approx 0.025$ eV), this probability is astronomically small. Practically no electrons are in the conduction band, so the material does not conduct.
• For a semiconductor like silicon ($E_g\approx 1.1$ eV), the probability is small but significant. A measurable number of electrons are thermally excited into the conduction band, leaving an equal number of empty states, or holes, in the valence band. Both the electrons in the conduction band and the holes in the valence band can move and contribute to conductivity.

This leads to the characteristic temperature-dependent conductivity of semiconductors.

The key insight is that a semiconductor is effectively an insulator at $T=0$ but becomes a modest conductor at finite temperatures.

4. A Practical Look at Band Diagrams

Let's ground these ideas by looking at the band structures and DOS of real materials.

This analysis of real materials solidifies the concepts we've discussed. The band diagram, calculated using methods like tight-binding, combined with a determination of the Fermi level, provides a complete picture of a material's electronic nature.

Conclusion

Today, we've established the fundamental principles behind the electronic classification of solids. By combining the concept of the density of states with the rules of electron filling, we can understand why some materials conduct electricity easily while others do not.

Key Takeaways:

• Metals: The Fermi level $E_F$ lies within a partially filled band, leading to a non-zero DOS at $E_F$. This allows for easy excitation of electrons and high conductivity.
• Insulators & Semiconductors: The Fermi level $E_F$ lies within a band gap $E_g$ between a filled valence band and an empty conduction band. The DOS at $E_F$ is zero.
• The distinction: The magnitude of the band gap is the key difference. Insulators have large gaps ($E_g>3$ eV), while semiconductors have smaller gaps.
• Temperature's role: In semiconductors, thermal energy at room temperature is sufficient to excite a significant number of electrons into the conduction band, creating mobile charge carriers (electrons and holes) and enabling moderate conductivity. This effect is negligible in insulators.

Preview of the Next Lesson:
We've briefly touched upon direct and indirect band gaps. In the next lesson, we will delve deeper into this distinction, exploring its origins in the $E(\mathbf{k})$ diagram and its critical consequences for the optical properties of semiconductors, such as light absorption and emission. This will be highly relevant to your experimental background in spectroscopy.