Band Theory of Metals — Revision Notes

⚡ 30-Second Revision

Energy Bands: — Formed by overlapping atomic orbitals in solids.
Valence Band (VB): — Highest occupied/partially occupied band at $0,\text{K}$.
Conduction Band (CB): — Lowest unoccupied band at $0,\text{K}$.
Forbidden Gap ($E_g$): — Energy range between VB and CB where electrons cannot exist.
Metals: — VB and CB overlap or VB is partially filled. $E_g \approx 0$. High conductivity. Conductivity decreases with temperature.
Insulators: — Fully filled VB, empty CB. Large $E_g > 5,\text{eV}$. Very low conductivity. Negligible temp effect.
Semiconductors: — Fully filled VB, empty CB. Small $E_g$ ($0.5 - 3,\text{eV}$). Moderate conductivity. Conductivity increases with temperature.
Doping: — Creates new energy levels within $E_g$, increasing charge carriers.

2-Minute Revision

The Band Theory of Metals explains the electrical properties of solids by considering energy bands rather than discrete energy levels. When atoms form a solid, their atomic orbitals merge into continuous energy bands: the Valence Band (VB) and the Conduction Band (CB). The VB contains electrons involved in bonding, while the CB contains free, mobile electrons responsible for conduction. These bands are separated by a Forbidden Gap ($E_g$), an energy range where electrons cannot exist.

For metals, the VB and CB either overlap or the VB is partially filled, meaning $E_g \approx 0$. This allows electrons to move freely, resulting in high conductivity. Their conductivity decreases with increasing temperature due to increased lattice vibrations.

Insulators have a completely filled VB and an empty CB, separated by a very large $E_g$ (typically $> 5,\text{eV}$). This prevents electron movement, leading to extremely low conductivity, largely unaffected by temperature.

Semiconductors also have a filled VB and empty CB, but with a smaller $E_g$ ($0.5 - 3,\text{eV}$). At room temperature, some electrons gain enough thermal energy to jump to the CB, increasing conductivity.

Crucially, their conductivity *increases* with increasing temperature, a key distinction from metals. Doping semiconductors introduces impurity energy levels within the forbidden gap, further enhancing conductivity.

5-Minute Revision

The Band Theory is a quantum mechanical model that describes the electronic structure of solids, explaining their electrical conductivity. It posits that in a solid, the discrete atomic energy levels broaden into continuous energy bands due to the interaction of a vast number of atomic orbitals.

The two most important bands are the Valence Band (VB), which is the highest energy band occupied by electrons (either fully or partially) at absolute zero, and the Conduction Band (CB), the lowest energy band that is typically empty at absolute zero.

Electrons in the CB are delocalized and free to move, contributing to electrical current.

Between the VB and CB lies the **Forbidden Gap ($E_g$)**, an energy range where electrons cannot exist. The width of this gap is the primary determinant of a material's electrical classification:

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Metals (Conductors): — In metals, the VB and CB either overlap, or the VB itself is only partially filled. This means there's no effective forbidden gap ($E_g \approx 0$), and electrons can easily move into available higher energy states within the band, leading to very high electrical conductivity. Increasing temperature causes increased lattice vibrations, which scatter electrons and *decrease* conductivity.

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Insulators: — Insulators have a completely filled VB and an empty CB, separated by a very large forbidden gap (typically $E_g > 5,\text{eV}$). The thermal energy at room temperature is insufficient to excite electrons across this large gap, so virtually no electrons reach the CB, resulting in extremely low conductivity. Temperature has a negligible effect on their conductivity.

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Semiconductors: — Semiconductors also have a completely filled VB and an empty CB at $0,\text{K}$, but their forbidden gap is much smaller ($0.5,\text{eV} < E_g < 3,\text{eV}$). At room temperature, thermal energy is sufficient to promote a small number of electrons from the VB to the CB. These electrons, along with the 'holes' left behind in the VB, act as charge carriers. A key characteristic is that the conductivity of semiconductors *increases* significantly with increasing temperature, as more electrons gain enough energy to jump the gap.

Example: Silicon ($E_g \approx 1.12,\text{eV}$) is a semiconductor. At $0,\text{K}$, it's an insulator. At $300,\text{K}$, some electrons jump to the CB, making it conductive. If you heat it further, more electrons jump, and conductivity rises. Copper, a metal, has overlapping bands. Its conductivity is high at $0,\text{K}$ and slightly decreases as temperature rises due to increased scattering.

Doping is a crucial process for semiconductors, where impurities are added to create new energy levels within the forbidden gap, either just below the CB (n-type) or just above the VB (p-type), thereby increasing the concentration of charge carriers and enhancing conductivity.

Prelims Revision Notes

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Origin of Bands: — When $N$ atoms form a solid, their $N$ atomic orbitals combine to form $N$ closely spaced molecular orbitals, which merge into continuous energy bands.
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Key Bands:

* Valence Band (VB): Highest energy band containing valence electrons. Can be partially or completely filled. * Conduction Band (CB): Lowest energy band capable of accepting free electrons for conduction. Typically empty at $0,\text{K}$.

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Forbidden Gap ($E_g$): — Energy difference between the top of VB and bottom of CB. No electron states exist here.
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Material Classification:

* Metals: $E_g \approx 0$ (VB and CB overlap, or VB is partially filled). High electron mobility. High conductivity. * Insulators: Large $E_g > 5,\text{eV}$. Filled VB, empty CB. Very low conductivity. * Semiconductors: Small $E_g$ ($0.5 - 3,\text{eV}$). Filled VB, empty CB at $0,\text{K}$. Moderate conductivity at room temperature.

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Temperature Effect on Conductivity:

* Metals: Conductivity *decreases* with increasing temperature (due to increased lattice vibrations and electron scattering). * Insulators: Conductivity remains *very low* (negligible change). * Semiconductors: Conductivity *increases* with increasing temperature (due to more electrons gaining thermal energy to jump to CB, creating more charge carriers).

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Doping: — Intentional addition of impurities to semiconductors to create new energy levels within the forbidden gap, increasing charge carrier concentration (e.g., n-type with donor levels below CB, p-type with acceptor levels above VB).

Vyyuha Quick Recall

To remember the conductivity trend with temperature: Metals Decrease, Semiconductors Increase. (MDI - 'Medical Doctor's Institute' - a common coaching name, helps recall).