Question

The band gap of an intrinsic semiconductor increases with:

• A. Temperature
• B. Pressure
• C. Doping
• D. Time

Hint:

Remember the inverse relationship between interatomic distance and band gap for standard semiconductors: \[ \text{Volume Compression (High Pressure)} \longrightarrow \text{Smaller Interatomic Spacing} \longrightarrow \text{Larger Band Gap } (E_g) \]

Answer 1

The Correct Option is B

Concept: The energy band gap ($E_g$) of a crystalline solid is defined as the energetic separation between the top of the valence band and the bottom of the conduction band. This gap corresponds to the minimum energy required to break a covalent bond and free an electron. The magnitude of $E_g$ depends directly on the interatomic spacing ($a$) of the crystal lattice. The physical parameters alter $E_g$ through two primary mechanisms:
• Thermal Expansion Vibration: Increasing the temperature increases lattice vibrations and interatomic spacing, which typically weakens bond energy and decreases $E_g$.
• Hydrostatic Pressure: Increasing hydrostatic pressure squeezes the crystal lattice, decreasing the physical distance between adjacent atoms and modifying orbital overlaps.

Step 1: Evaluating the effect of mechanical pressure on lattice parameters.
When uniform hydrostatic pressure is applied to a semiconductor crystal, the lattice volume compresses, directly decreasing the interatomic distance $a$. According to tight-binding approximations in solid-state physics, as atoms are forced closer together, the atomic orbital overlap increases significantly. For most standard covalent and ionic semiconductors (such as Silicon, Germanium, and GaAs), this increased overlap causes a wider separation between the bonding (valence band) states and anti-bonding (conduction band) states. Consequently, the band gap energy $E_g$ expands with increasing pressure: \[ \left(\frac{\partial E_g}{\partial P}\right)_T > 0 \]

Step 2: Analyzing the effect of Temperature.
As temperature increases, two phenomena happen: the lattice undergoes thermal expansion (increasing $a$), and electron-phonon interactions become stronger. Both effects generally lead to a reduction in the energy gap. This is classically modeled by Varshni's empirical formula: \[ E_g(T) = E_g(0) - \frac{\alpha T^2}{T + \beta} \] Since $E_g$ drops as temperature increases, option (A) is completely incorrect.

Step 3: Analyzing the effect of Doping and Time.

• Doping introduces discrete impurity levels within the band gap and can lead to band-gap narrowing at high concentrations (the Burstein-Moss effect shifts the apparent optical edge, but the physical fundamental bandgap of the host shrinks due to many-body interactions). Thus, it does not increase the band gap.
• Time is an independent macroscopic state parameter and has no physical coupling with the intrinsic electronic band structure of a stable crystal. Therefore, pressure is the correct parameter that systematically causes the energy band gap to increase.