Step 1: The Debye temperature marks the temperature above which essentially all vibrational modes of a solid are thermally excited. It is set by the highest allowed phonon frequency $\omega_D$.
Step 2: Convert that cut-off frequency into a temperature by matching quantum energy to thermal energy: $\hbar\omega_D = k_B\theta_D$. This is the same $E = k_B T$ bookkeeping used throughout statistical physics.
Step 3: Rearranging gives $\theta_D = \dfrac{\hbar\omega_D}{k_B}$. Only a single power of $\hbar$ appears because $\hbar\omega$ is already an energy.
Step 4: Any expression carrying $\hbar^2$ would have the wrong dimensions (energy times action, not energy), so those choices cannot represent a temperature. The factor-of-two option misstates the standard definition.\[\boxed{\theta_D = \frac{\hbar\omega_D}{k_B}}\]