QB_PH_124: Bose-Einstein and Fermi-Dirac statistics

For a non-relativistic two-dimensional electron gas of surface number density $, how does the Fermi energy $ scale with 127

Correct Answer

### 1. Conceptual Framework & Core Principles In a two-dimensional electron gas (2DEG), the density of states (E)$ per unit area is energy-independent (constant): 12696D_{2D}(E) = rac{g_s m}{2\pi \hbar^2} = rac{m}{\pi \hbar^2}12696 where = 2$ is the spin degeneracy factor. ### 2. Analytical & Mathematical Derivation The total number of electrons per unit area at = 0$ K is obtained by integrating the density of states up to the Fermi energy $: 12696n_s = \int_0^{E_F} D_{2D}(E) dE = \int_0^{E_F} rac{m}{\pi \hbar^2} dE = rac{m}{\pi \hbar^2} E_F12696 Solving directly for $: 12696E_F = rac{\pi \hbar^2}{m} n_s12696 Hence, $ is strictly directly proportional to $ ( \propto n_s$). ### 3. Formula Breakdown - In 3D: \propto n^{2/3}$ - In 2D: \propto n_s^1 = n_s$ - In 1D: \propto n_{1D}^2$ ### 4. Student Traps & Exam Tips **Common Trap:** Students reflexively apply the 3D free-electron result \propto n^{2/3}$ without noticing that the problem specifies a 2D electron gas where the density of states is constant.