# Electronic specific heat capacity using free electron theory

While considering the contribtution of the electronic specific heat capacity, it is stated that, when the temperature increases from 0 K , only the electrons with energy range of the order of $$k_bT$$ from the fermi energy are excited. How was this range determined? (Ref: Introduction to Solid State Physics, 8th Edition by Charles Kittel)

This is an approximate statement, which follows from understanding the shape of the Fermi distribution: $$n_F(E) = \frac{1}{e^{\frac{E - E_F}{k_B T}} + 1}$$ This distribution has a step-like form: it is approximately 1 for $$E < E_F$$ and approximately zero for $$E > E_F$$, except for the region around $$E_F$$ where $$|E-E_F|\sim k_B T$$. In the limit of very low temperatures this region is nearly non-existent and the distribution becomes a step-function $$n_F(E) = \theta (E_F - E).$$ Broadening of this region when the temperature increases is what one refers to as the electrons excited above the Fermi surface (i.e. some states below $$E_F$$ are not fully filled, and some states above $$E_F$$ now contain electrons. But the states away from this region are still either filled (occupancy $$1$$) or empty (occupancy $$0$$).