Compare silicon (Eg = 1.12 eV) and germanium (Eg = 0.67 eV) at 300 K…

Step 1. Thermal energy. $kT = 8.617 \times 10^{-5}\ \text{eV/K} \times 300\ \text{K} = 0.02585\ \text{eV} = 25.9$ meV.

Step 2. The exponents. For silicon, $\dfrac{E_g}{2kT} = \dfrac{1.12\ \text{eV}}{2 \times 0.02585\ \text{eV}} = 21.7$. For germanium, $\dfrac{0.67\ \text{eV}}{2 \times 0.02585\ \text{eV}} = 13.0$.

Step 3. The ratio. The factors are $e^{-21.7} = 3.9 \times 10^{-10}$ and $e^{-13.0} = 2.4 \times 10^{-6}$. The ratio is $e^{21.66 - 12.96} = e^{8.70} = 6.03 \times 10^3$.

Why. The smaller gap of germanium lets heat make about six thousand times as many carriers (by this factor alone). In a device, these carriers give a leakage current that grows with temperature, so germanium is less steady than silicon.