Dependence of an exchange gate on valley orbit couplings in silicon quantum dots*

The mixing of conduction band valleys plays a critical role in determining electronic spectrum and dynamics in a silicon nanostructure. In this work we use Schrieffer-Wolf transformation to show that the exchange energy depends on the interdot valley phase difference as cos²(φ/2)  in a symmetric two-electron silicon double quantum dot. We show that the necessary condition for exchange gate protocol to be valid in a Si double quantum dots is that the valley splitting in each of the dots is much larger than the thermal broadening of the reservoir used for initialization. The contributions of the higher orbital states also play a vital role in determining the value of the exchange energy in general.