Question

Match List-I with List-II 

List-I	Concept	List-II	Expression / Property
A	\(1D\) density of states	III	\(n(E)\propto E^{-1/2}\)
B	\(2D\) density of states	I	Constant
C	\(3D\) density of states	II	\(n(E)\propto E^{1/2}\)
D	Effective mass	IV	\(m^*=\dfrac{\hbar^2}{d^2E/dk^2}\)

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

Hint:

DOS is \(E^{-1/2}\) in \(1D\), constant in \(2D\), and \(E^{1/2}\) in \(3D\).

Answer 1

The Correct Option is A

Concept:
Density of states depends on dimensionality, while effective mass depends on curvature of the \(E-k\) graph.

Step 1: \(1D\) density of states.
For one-dimensional system: \[ n(E)\propto E^{-1/2} \] So, \[ A\rightarrow III \]

Step 2: \(2D\) density of states.
For two-dimensional system: \[ n(E)=\text{constant} \] So, \[ B\rightarrow I \]

Step 3: \(3D\) density of states.
For three-dimensional system: \[ n(E)\propto E^{1/2} \] So, \[ C\rightarrow II \]

Step 4: Effective mass.
Effective mass is: \[ m^*=\frac{\hbar^2}{\frac{d^2E}{dk^2}} \] So, \[ D\rightarrow IV \] Therefore: \[ A-III,\ B-I,\ C-II,\ D-IV \] \[ \therefore \text{Correct Answer is (A)} \]