Before You Begin

Question #43839

Difficulty: 8/10 | 🟢 Free

The Question

Consider a two-dimensional (2D) intrinsic semiconductor at thermal equilibrium. The effective masses of electrons in the conduction band and holes in the valence band are isotropic and equal ( $m_e^* = m_h^* = m^*$ ). The charge carriers can be modeled as a classical 2D ideal gas. The density of states per unit area for both bands is constant and given by $g(E) = \frac{m^*}{\pi \hbar^2}$ , where $\hbar$ is the reduced Planck constant. If the average speed of the charge carriers is measured to be $v_0$ and the bandgap energy is $E_g$ , which of the following correctly represents the intrinsic areal carrier concentration $n_i$ ?

Options

A.
$\frac{2 m^{*2} v_0^2}{\pi^2 \hbar^2} \exp\left(-\frac{\pi E_g}{4 m^* v_0^2}\right)$
B.
$\frac{m^{*2} v_0^2}{2 \pi \hbar^2} \exp\left(-\frac{E_g}{m^* v_0^2}\right)$
C.
$\frac{m^{*2} v_0^2}{8 \hbar^2} \exp\left(-\frac{4 E_g}{\pi m^* v_0^2}\right)$
D.
$\frac{m^{*2} v_0^2}{3 \pi \hbar^2} \exp\left(-\frac{3 E_g}{2 m^* v_0^2}\right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Law of Mass Action|Equipartition Theorem|RMS vs Average vs Most Probable Speed

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Solid State Physics (Law of Mass Action) and Statistical Mechanics (Maxwell-Boltzmann distribution) in a non-standard (2D) dimensionality.

Students commonly memorize the 3D formulas for average speed ( $v_{avg} = \sqrt{8k_BT/\pi m^*}$ ) or RMS speed ( $v_{rms} = \sqrt{3k_BT/m^*}$ ) and blindly apply them, failing to derive the 2D speed distribution.

Furthermore, they often confuse average speed with RMS speed when relating kinetic energy to temperature.

By requiring the integration of the 2D density of states and the 2D velocity distribution, the trap exploits the misapplication of equipartition and dimensional assumptions, forcing a first-principles derivation of the effective carrier density.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Law of Mass Action
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free