GATE 2022 PH – Question 51
Electronic heat capacity is $C=\gamma T$, where gamma relates to effective mass. Which statements are correct?
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (A) $\gamma\propto m_{eff}$; (D) Linear dependence is observed well below Debye temperature
Explanation
At low temperature the electronic specific heat of a metal is $C=\gamma T$, with the **Sommerfeld coefficient**
$$\gamma=\frac{\pi^2}{3}k_B^2\,g(E_F),$$
where $g(E_F)$ is the density of states at the Fermi level. For free electrons $g(E_F)\propto m_{eff}$ (at fixed density).
- **A. $\gamma\propto m_{eff}$.** True. ✓
- **B. The effective mass is greater than the free-electron mass in every solid.** False. Many materials (for example some semiconductors, and alkali metals in some cases) have $m_{eff}<m_e$.
- **C. The power of $T$ in $C$ depends on the dimensionality.** False. The linear dependence follows from the Fermi-Dirac statistics (only a thin shell of width $k_BT$ around $E_F$ takes part) in 1D, 2D and 3D.
- **D. The linear dependence is seen well below the Debye temperature.** True: at low temperature the electronic term dominates the $T^3$ lattice term. ✓
Answer **A and D**.