Electronic specific heat of a solid at temperature T is C = 𝛾T, where 𝛾 is a
constant related to the thermal effective mass (m) of the electrons. Then which
of the following statements are correct?
Options:
(A) $\gamma \propto m_{\text{eff}}$
(B) $m_{\text{eff}}$ is greater than free electron mass for all solids
(C) Temperature dependence of $C$ depends on the dimensionality of the solid
(D) The linear temperature dependence of $C$ is observed at $T \ll \text{Debye temperature}$
Official Answer Key:
A,D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.51)**
**1. Solid State Physics Principles (Electronic Specific Heat)**:
- Electronic heat capacity of a Fermi gas at low temperatures:
$C_e = \frac{\pi^2}{3} D(E_F) k_B^2 T = \gamma T$
- Density of states at Fermi energy for a 3D electron gas:
$D(E_F) = \frac{m^*}{\hbar^2 \pi^2} k_F \propto m^* = m_{\text{eff}}$
**2. Step-by-Step Statement Evaluation**:
- **(A) [TRUE]**: Since $\gamma = \frac{\pi^2}{3} k_B^2 D(E_F)$ and $D(E_F) \propto m_{\text{eff}}$, the Sommerfeld coefficient is directly proportional to effective mass: $\gamma \propto m_{\text{eff}}$.
- **(B) [FALSE]**: In many semiconductors (e.g., GaAs, InSb) and semimetals, $m_{\text{eff}}$ is significantly smaller than the bare electron mass ($m_e^* < m_0$).
- **(C) [FALSE]**: The linear temperature dependence $C_e \propto T$ arises from the Fermi-Dirac step distribution at the Fermi surface and is fundamentally linear in any dimension.
- **(D) [TRUE]**: At low temperatures $T \ll \Theta_D$, the electronic contribution $C_e = \gamma T$ dominates over the lattice vibrational contribution $C_{\text{lattice}} \propto T^3$.
**3. Official Verification**:
The verified official IIT Master Answer is **Options (A, D)**.
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