Effective Mass Concepts Explained

This question asks to identify the true statements regarding the concept of effective mass in solid-state physics.

Statement Analysis:

• Statement A: Effective mass arises due to interaction of electrons with periodic potential of lattice.
  • Analysis: True. The periodic potential of the crystal lattice modifies the electron's response to external forces, leading to the concept of effective mass, which differs from the free electron mass.
• Statement B: The curvature of the band determines the electron effective mass.
  • Analysis: True. The effective mass ($m^*$) is inversely related to the curvature of the energy band ($E(k)$) diagram. Specifically, $m^{*} \propto 1 / \frac{d^2E}{dk^2}$.
• Statement C: Effective mass cannot be negative.
  • Analysis: False. Near the top of an energy band, the curvature $d^2E/dk^2$ is negative, resulting in a negative effective mass.
• Statement D: The calculation of effective mass takes into account the shape of energy bands in 3-D k-space.
  • Analysis: True. Effective mass is derived from the second derivative of the energy dispersion relation $E(k)$ and depends on the band structure in 3D k-space.
• Statement E: Effective mass of an electron is given by $\text{m}^{*} = \frac{\hbar}{\left( \frac{\text{d}^{2}\text{E}}{\text{d}\text{k}^{2}} \right)}$.
  • Analysis: False. The correct formula for effective mass is $m^{*} = \frac{\hbar^2}{\left( \frac{d^2E}{dk^2} \right)}$. The formula provided is missing the square on $\hbar$.

Identifying True Statements

Based on the analysis, the true statements are A, B, and D.

Conclusion

The correct option includes statements A, B, and D.

Correct Answer: Option 2 (A, B, D only)