In the Kane silicon-based electron-mediated nuclear spin quantum computer architecture, phosphorus is doped at precise positions in a silicon lattice, and the P donor nuclear spins act as qubits. Logical operations on the nuclear spins are performed using externally applied magnetic and electric fields. There are two important interactions: the hyperfine and exchange interactions, crucial for logical qubit operations. Single qubit operations are performed by applying radio frequency magnetic fields resonant with targeted nuclear spin transition frequencies, tuned by the gate-controlled hyperfine interaction. Two qubit operations are mediated through the exchange interaction between adjacent donor electrons. It is important to examine how these two interactions vary as functions of experimental parameters. Here we provide such an investigation. First, we examine the effects of varying several experimental parameters: gate voltage, magnetic field strength, inter donor separation, donor depth below the silicon oxide interface and back gate depth, to explore how these variables affect the donor electron density. Second, we calculate the hyperfine interaction and the exchange coupling as a function of these parameters.
These calculations were performed using various levels of effective mass theory. In the first part of this thesis we use a multi-valley effective mass approach where we incorporate the full Si crystal Bloch structure in calculating the donor electron energy in the bulk silicon. Including the detailed Bloch structure is very computationally intensive, thus when we considered the effect of the externally applied fields in the second and third part, we employ an approach where we focus on the smooth donor-modulated envelope function to determine the response of the donor electron to the applied electric and magnetic fields and qubit position in the lattice. The electric field potential was obtained using Technology Computer Aided Design software, and the interfaces were modelled as a barrier using a step function. One of the critical results of this theoretical study was finding that there exist two regimes for the behaviour of the donor electron in response to the applied gate voltage, dependent on donor distance from the gate. When the qubit is in close proximity to the gate the electron transfer to the gate is gradual. However if the qubit is located far enough from the gate, we found that the donor electron is ionised toward the gate for gate voltages above a certain threshold.
Another significant development we have made is in our calculations of the exchange coupling between two adjacent donor electrons. We extended our original Heitler-London basis to describe the two-electron system, and adopted a molecular orbital method where we included a basis of 78 singlet and 66 triplet two-electron states. In addition to calculating a more accurate exchange coupling, we also evaluated the energy spectrum of the two electron double donor system.
We aim to provide relevant information for the experimental design of these devices and highlight the significance of environmental factors other than gate potential that affect the donor electron.