Question

The effective mass of an electron is ______________________________ in the lower part of the band and ______________________________ near the zone boundary (k\(\sim\)\(\pi\)/a).

• A. positive and negative
• B. negative and positive
• C. positive and increases
• D. negative and decreases

Hint:

Visualize the shape of a simple energy band (like a cosine curve). The bottom of the band is a "valley" (positive curvature, positive \(m^ \)). The top of the band is a "hill" (negative curvature, negative \(m^ \)). This mental image helps you instantly recall the signs of the effective mass.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The effective mass (\(m^ \)) of an electron in a crystal is a concept that accounts for the interaction between the electron and the periodic potential of the lattice. It is not the actual mass of the electron but a parameter that determines how the electron accelerates in response to an external force. It is defined by the curvature of the E-k band diagram.
Step 2: Key Formula or Approach:
The formula for effective mass is:
\[ m^ = \hbar^2 \left( \frac{d^2E}{dk^2} \right)^{-1} \]
This means the sign of the effective mass is determined by the sign of the second derivative of E with respect to k, which represents the curvature of the E-k diagram.


If the curve is concave up (like a valley), \( \frac{d^2E}{dk^2}>0 \), so \(m^ \) is positive.
If the curve is concave down (like a hill), \( \frac{d^2E}{dk^2}<0 \), so \(m^ \) is negative.

Step 3: Detailed Explanation:


In the lower part of the band (e.g., near k=0): The E-k curve starts at a minimum and curves upwards. It is concave up. Therefore, \( \frac{d^2E}{dk^2} \) is positive, and the effective mass \(m^ \) is positive. This describes how a normal electron behaves.
Near the zone boundary (k \(\sim \pi/a\)): This region corresponds to the top of the energy band. Here, the E-k curve reaches a maximum and curves downwards. It is concave down. Therefore, \( \frac{d^2E}{dk^2} \) is negative, and the effective mass \(m^ \) is negative. A negative effective mass means the particle accelerates in the opposite direction to the applied force, which is the behavior analogous to a positively charged "hole".

Step 4: Final Answer:
The effective mass is positive in the lower part of the band and negative near the zone boundary (the top of the band).