Multi-valley envelope function equations and effective potentials for P impurity in silicon

We propose a system of real-space envelope function equations without fitting parameters for modeling the electronic spectrum and wave functions of a phosphorus donor atom embedded in silicon. The approach relies on the Burt-Foreman envelope function representation and leads to coupled effective-mass Schroedinger equations containing smooth effective potentials. These potentials result from the spatial filtering imposed on the exact potential energy matrix elements in the envelope function representation. The corresponding filter function is determined from the definition of the envelope function. The resulting effective potentials and the system of envelope functions jointly reproduce the valley-orbit coupling effect in the doped silicon. Including the valley-orbit coupling not only of the 1s, but also for 2s atomic orbitals, as well as static dielectric screening is found crucial to accurately reproduce experimental data. The measured binding energies are recovered with a maximum relative error of 1.53 %. The computed wave functions are in a good agreement with experimental measurements of the electron density provided by scanning tunneling microscopy.