Energy Bands in Crystals — Revision Notes

⚡ 30-Second Revision

Energy Bands — Continuous ranges of allowed electron energies in crystals.
Forbidden Energy Gap ($E_g$) — Energy range where no electron can exist.
Valence Band (VB) — Highest filled/partially filled band at $0,\text{K}$. Electrons are bound.
Conduction Band (CB) — Lowest empty/partially filled band. Electrons are free carriers.
Conductors — $E_g \approx 0$ (bands overlap or CB partially filled). High conductivity.
Semiconductors — Moderate $E_g$ ($0.5-1.5,\text{eV}$). Conductivity increases with $T$.

- Si: $E_g \approx 1.12,\text{eV}$ - Ge: $E_g \approx 0.67,\text{eV}$

Insulators — Large $E_g$ ($>3,\text{eV}$). Very low conductivity.
Photon energy — $E = h\nu = hc/\lambda$. For absorption/emission, $E \ge E_g$.
Shortcut — $\lambda (\text{nm}) = 1240 / E_g (\text{eV})$

2-Minute Revision

Energy bands are formed in crystalline solids when discrete atomic energy levels split and broaden due to interatomic interactions, a consequence of the Pauli Exclusion Principle. These bands are separated by forbidden energy gaps ($E_g$).

The two critical bands are the valence band (VB), containing bound electrons, and the conduction band (CB), containing free electrons. The width of $E_g$ dictates a material's electrical properties: conductors have zero or overlapping $E_g$, allowing free electron movement; insulators have a large $E_g$ (e.

g., $>3,\text{eV}$), preventing conduction; semiconductors have a moderate $E_g$ (e.g., $0.5-1.5,\text{eV}$), allowing some electrons to jump to the CB with thermal energy, thus increasing conductivity with temperature.

This band theory is crucial for understanding all semiconductor devices.

5-Minute Revision

The concept of energy bands is central to solid-state physics, explaining the electrical behavior of materials. When individual atoms form a crystal, their discrete electron energy levels interact and split into a vast number of closely spaced levels, forming continuous 'energy bands'. This is due to the Pauli Exclusion Principle. These allowed energy bands are separated by 'forbidden energy gaps' ($E_g$).

Key bands are the Valence Band (VB), the highest band filled with bound electrons at $0,\text{K}$, and the Conduction Band (CB), the lowest empty band. Electrons in the CB are free to conduct electricity.

Material Classification based on $E_g$:

1
Conductors (e.g., metals) — VB and CB overlap, or CB is partially filled. $E_g \approx 0$. Abundant free electrons, high conductivity.
2
Semiconductors (e.g., Si, Ge) — Moderate $E_g$ ($0.5-1.5,\text{eV}$). At $0,\text{K}$, they act as insulators. At room temperature, thermal energy excites some electrons from VB to CB, creating electron-hole pairs and enabling moderate conductivity. Conductivity increases with temperature.

* Silicon (Si): $E_g \approx 1.12,\text{eV}$ * Germanium (Ge): $E_g \approx 0.67,\text{eV}$

1
Insulators (e.g., Diamond) — Large $E_g$ (typically $>3,\text{eV}$). Very few electrons can cross the gap, resulting in extremely low conductivity.

Important Formulas:

Energy of a photon: $E = h\nu = \frac{hc}{\lambda}$
For electron excitation across band gap: $E_g = \frac{hc}{\lambda_{max}}$ (where $\lambda_{max}$ is the maximum wavelength that can be absorbed)
Useful constant: $hc \approx 1240\,\text{eV nm}$
So, $\lambda (\text{nm}) = \frac{1240}{E_g (\text{eV})}$

Example: If a semiconductor has $E_g = 2.0,\text{eV}$, the maximum wavelength of light it can absorb is $\lambda = \frac{1240}{2.0} = 620,\text{nm}$. This light would be visible red-orange light. If it emits light, it would be at this wavelength or shorter.

Prelims Revision Notes

1
Origin of Energy Bands — In isolated atoms, electrons have discrete energy levels. In a crystal, due to close atomic proximity and the Pauli Exclusion Principle, these discrete levels split into numerous closely spaced levels, forming continuous 'energy bands'.
2
Forbidden Energy Gap ($E_g$) — Regions of energy between allowed bands where electrons cannot exist. Its width is crucial for material classification.
3
Valence Band (VB) — The highest energy band completely or partially filled with electrons at $0,\text{K}$. Electrons here are typically bound in covalent bonds and do not contribute to conduction.
4
Conduction Band (CB) — The lowest energy band that is empty or partially filled. Electrons in the CB are free to move and carry electric current.
5
Conductors — VB and CB overlap or CB is partially filled. $E_g \approx 0$. High conductivity at all temperatures. Conductivity decreases with increasing temperature due to increased electron scattering.
6
Semiconductors — Moderate $E_g$ (e.g., $0.5,\text{eV}$ to $1.5,\text{eV}$). At $0,\text{K}$, they behave as insulators. At room temperature, thermal energy excites some electrons from VB to CB, creating electron-hole pairs. Conductivity increases significantly with increasing temperature.

* Silicon (Si): $E_g \approx 1.12,\text{eV}$ at $300,\text{K}$. * Germanium (Ge): $E_g \approx 0.67,\text{eV}$ at $300,\text{K}$.

1
Insulators — Large $E_g$ (typically $>3,\text{eV}$ to $6,\text{eV}$). Electrons cannot gain enough energy to jump to the CB. Very low conductivity, almost independent of temperature.
2
Doping — Adding impurities to semiconductors creates new energy levels within the forbidden gap.

* n-type: Donor impurities (e.g., Phosphorus in Si) create donor levels just below the CB, easily donating electrons to CB. * p-type: Acceptor impurities (e.g., Boron in Si) create acceptor levels just above the VB, easily accepting electrons from VB, creating holes.

1
Photon Energy and Band Gap — For a photon to be absorbed and create an electron-hole pair, its energy must be at least equal to the band gap: $E_{photon} \ge E_g$. Conversely, when an electron and hole recombine, a photon of energy $E_g$ (or less) can be emitted.

* $E = h\nu = \frac{hc}{\lambda}$ * Useful constant for quick calculations: $hc \approx 1240\,\text{eV nm}$. So, $\lambda (\text{nm}) = \frac{1240}{E_g (\text{eV})}$.

Mains Revision Notes

As NEET UG does not feature a separate 'Mains' exam, the 'Prelims Revision Notes' comprehensively cover all the essential factual and conceptual recall required for the single objective-type examination.

Students should focus on internalizing these points for quick and accurate recall during the exam. The depth of understanding required for NEET is well-addressed by the detailed explanations and revision materials provided, ensuring readiness for both straightforward recall questions and application-based problems related to energy bands in crystals.

Vyyuha Quick Recall

To remember the order of conductivity based on band gap: Conductors Semiconductors Insulators. Think: Can Someone Ignore? (Smallest to Largest Band Gap). Or, Conductors Surely Ignite (meaning, they are active/conductive). Smallest $E_g$ means highest conductivity.