Specific heat of phonon

The energy of one phonon is defined as: $E = \hbar\omega$ where $\omega$ is the angular frequency of this phonon. In quantum statistics, phonon number, n, follows Bose-Einstein distribution: $n=\frac 1{e^{\frac {\hbar\omega}{k_BT}}-1}$ where $k_B$ is the Boltzmman constant, T is temperature. Therefore, the thermal energy contribution by phonons is expressed as: $E_{thermal}=(n+\frac 12)\hbar\omega$ Then the specific heat of, C, phonon is derived by: $\begin{equation} \begin{split} C&=\frac {\partial E_{thermal}}{\partial T} \\ &=\frac {\partial n}{\partial T}\hbar\omega \\ &=\frac {e^{\frac {\hbar\omega}{k_BT}}}{({e^{\frac {\hbar\omega}{k_BT}}-1)}^2}\frac {\hbar^2\omega^2}{k_BT^2}\\ &=n(n+1)\frac {\hbar^2\omega^2}{k_BT^2} \end{split} \nonumber \end{equation}$