Question

Match List - I with List - II. 

Choose the correct answer from the options given below : 
 

• A. A-IV, B-II, C-I, D-III
• B. A-II, B-III, C-IV, D-I
• C. A-I, B-II, C-III, D-IV
• D. A-III, B-II, C-IV, D-I

Hint:

Pay close attention to the axes! In these specific textbook diagrams, Energy is plotted on the vertical y-axis, which can make the curves look rotated $90^\circ$ compared to standard mathematical plots where the independent variable is on the x-axis.

Answer 1

The Correct Option is D

Step 1: Concept:
The Density of States (DOS) function $N(E)$ dictates how many electron states are available at a given energy level. The shape of the DOS curve changes drastically depending on the number of dimensions in which the electrons are confined (quantum confinement).

Step 2: Key Formula or Approach:
The energy dependencies of the Density of States for different dimensionalities are:
- 3D (Bulk): $N(E) \propto E^{1/2}$
- 2D (Quantum Well): $N(E) \propto \text{Step Function } (E^0)$
- 1D (Quantum Wire): $N(E) \propto E^{-1/2}$
- 0D (Quantum Dot): $N(E) \propto \delta(E)$ (Discrete Delta functions)

Step 3: Step-by-step Explanation:
Let's analyze the graphs (with $E$ on the y-axis and $N(E)$ on the x-axis):

• Graph III: The curve shows $N(E)$ increasing proportionally to the square root of $E$ (a parabola opening to the right). This represents a 3D Bulk semiconductor. (A $\rightarrow$ III)

• Graph II: The graph looks like a staircase. As energy increases, $N(E)$ jumps by discrete constant amounts. This step-function behavior represents a 2D Quantum well. (B $\rightarrow$ II)

• Graph IV: The graph features sharp peaks that tend toward infinity and then decay before the next peak. This $1/\sqrt{E}$ inverse relationship represents a 1D Quantum wire. (C $\rightarrow$ IV)

• Graph I: The graph shows only discrete horizontal lines. This means $N(E)$ only exists at specific, quantized energy levels, representing delta functions. This complete confinement represents a 0D Quantum dot. (D $\rightarrow$ I)
The matched sequence is A-III, B-II, C-IV, D-I.

Step 4: Final Answer:
The correct option is (D).