Question

Which of the following correctly represents the density of states D(E) for quantum well :
(Here, $K_o, K_1, K_2, K_3$ are constants; $d_i$ are degeneracies; $E_{iw}$ energy of level "$i$" in potential well)

• A. $D(E) = K_2 \sum d_i$
• B. $D(E) = \frac{1}{2} K_1 \sum d_i (E-E_{iw})^{-1/2}$
• C. $D(E) = K_o \sum d_i \delta(E-E_{iw})^2$
• D. $D(E) = \frac{3}{2} K_3 E^{1/2}$

Hint:

Memorize the DOS energy dependencies: 3D is $\sqrt{E}$, 2D is a constant step $E^0$, 1D is $1/\sqrt{E}$, and 0D is a delta point $\delta(E)$.

Answer 1

The Correct Option is A

Step 1: Concept:
The question is testing knowledge of the fundamental Density of States (DOS) mathematical formulas for differently dimensioned systems (Bulk, Quantum Well, Quantum Wire, and Quantum Dot).

Step 2: Key Formula or Approach:
The energy dependency of the DOS function $D(E)$ changes dramatically depending on quantum confinement:
- 3D (Bulk): $D(E) \propto E^{1/2}$ (continuous parabolic curve).
- 2D (Quantum Well): $D(E) \propto \sum \Theta(E - E_n)$, where $\Theta$ is the Heaviside step function. The DOS is constant within a subband, rising in discrete stair-steps.
- 1D (Quantum Wire): $D(E) \propto \sum (E - E_n)^{-1/2}$ (sharp peaks that decay).
- 0D (Quantum Dot): $D(E) \propto \sum \delta(E - E_n)$ (discrete delta-function spikes).

Step 3: Step-by-step Explanation:

• Let's evaluate the given mathematical options against our standard models:

• (4) $D(E) = \frac{3{2} K_3 E^{1/2}$:} The $E^{1/2}$ dependence is the classic signature of a standard unconfined 3D bulk material.

• (2) $D(E) = \frac{1{2} K_1 \sum d_i (E-E_{iw})^{-1/2}$:} The $(E-E_{iw})^{-1/2}$ dependence contains singularities and represents the DOS of a 1D structure, a Quantum Wire.

• (3) $D(E) = K_o \sum d_i \delta(E-E_{iw})^2$: The presence of the delta function $\delta$ indicates fully discrete atomic-like energy levels, characteristic of a 0D structure, a Quantum Dot.

• (1) $D(E) = K_2 \sum d_i$: In an ideal 2D system, once an energy level $E_{iw}$ is surpassed, that subband contributes a constant value to the DOS. The total DOS is simply the sum of these constant contributions (step functions). The expression shows a summation of constant terms without energy dependency $E$ attached to the active terms, perfectly capturing the flat, step-like nature of a Quantum Well.

Step 4: Final Answer:
The correct representation for a quantum well is option (A).