Question

Which of the following statements are true about effective mass?
A. Effective mass arises due to interaction of electrons with periodic potential of lattice
B. The curvature of the band determines the electron effective mass
C. Effective mass cannot be negative
D. The calculation of effective mass takes into account the shape of energy bands in 3-D k-space
E. Effective mass of an electron is given by $m^* = \frac{\hbar}{\left(\frac{d^2E}{dk^2}\right)}$
Choose the correct answer from the options given below :

• A. A, B, E only
• B. A, B, D only
• C. B, C, D only
• D. A, B, C, D only

Hint:

Always double-check formulas for correct exponents! A missing square on $\hbar$ makes the equation dimensionally invalid. Also, remember that a negative effective mass for an electron is exactly what gives rise to the concept of a "hole" in semiconductors.

Answer 1

The Correct Option is B

Step 1: Concept:
The effective mass ($m^*$) is a quantity that is used to simplify band structures by modelling the behavior of a free particle with a modified mass. It captures the effect of the internal periodic potential of the crystal lattice on the electron's movement.

Step 2: Key Formula or Approach:
The standard quantum mechanical formula for the effective mass of an electron in a crystal lattice is defined as:
\[ m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}} \]
Where $\hbar$ is the reduced Planck's constant, $E$ is the energy, and $k$ is the wavevector.

Step 3: Step-by-step Explanation:

• A is True: The effective mass conceptually absorbs the complex internal forces (the periodic potential of the lattice) so that the electron can be treated classically using external forces ($F = m^*a$).

• B is True: As seen in the formula, $m^*$ is inversely proportional to the second derivative of the energy-momentum dispersion relation ($d^2E/dk^2$), which represents the curvature of the energy band.

• C is False: Effective mass can be negative. Near the top of a valence band, the curvature $d^2E/dk^2$ is negative, resulting in a negative effective mass for electrons (which we conventionally treat as positive "holes" moving in the opposite direction).

• D is True: In a real crystal, the energy band shape is three-dimensional, meaning effective mass is actually a tensor that depends on the direction in 3D k-space.

• E is False: The formula provided in the statement is missing the square on $\hbar$. It shows $\hbar$ instead of the dimensionally correct $\hbar^2$.
Therefore, only statements A, B, and D are correct.

Step 4: Final Answer:
The correct combination is A, B, D only, matching option (B).