Electronics360: "A Better Material than Silicon for Transistors"

I do not understand the few sentences "So the solution is to replace the SiO₂ in the gate with an insulator with a high dielectric constant (high-k) in order to be able to reduce the capacitance of the gate-channel (to reduce the leakage current). With a high-k insulator, we can afford a thicker gate insulator, as is indicated in the figure."

Higher K raises the capacitance, so how does the higher capacitance itself lower the leakage? What we want to do is increase the thickness of the gate insulator, thus lowering the intensity of the electric field between the gate electrode and the underlying transistor structure. The final conclusion seems in the right direction, one can increase the oxide insulator thickness when the insulator has higher K. The reduced field strength then reduces the tunneling which is believed to be the primary driver of leakage.

The main problem is probably the overall wording.