quantum field theory - Creation and Annihilation Operators in QFT

I have a general question about heuristic way of QFT to introduce creation and annihilation operators: The Klein-Gordon field is introduced as continuous interference of plane waves $\mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})}$ with positive energy (resp $\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})}$ with negative energy).

The annihilation operator is $\hat{b}_{\mathbf{p}}^{\dagger}$ and creation operator is $\hat{a}_{\mathbf{p}}$.

The KG field expanded in terms of creation/annihilation operators is

$$\varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[\hat{a}(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + \hat{b}^\dagger(\vec{k})\mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})}\right].$$

My question is why do the anhilation operator correspond to the plane wave with positive energy and the creation operator to the plane wave with negative energy? How can it be heuristically explained?

Or equivalently why it is told that the anhilation operates on plane waves with positive energy?