Device Modelling in the Kane Quantum Computer Architecture Solution of the Donor Electron Schrodinger Equation

In the Kane quantum computer, information is encoded onto the nuclear spins of donor P atoms in doped silicon electronic devices. Using applied electric and magnetic fields the P donor electron density can be perturbed. By perturbing the electronic density away from the P nucleus we can tune the resonance frequency of individual P nuclear spins and the exchange interaction between adjacent donor electrons. These interactions are crucial in performing logical operations in the Kane quantum computer. My project involves calculating the P donor electron wavefunction in the Si host geometry and in the presence of externally applied fields. Furthermore we calculate and optimise the hyperfine interaction between donor electron and nucleus, and exchange interaction between adjacent donor electrons.

Principal Investigator

Louise Kettle
Chemistry/Physics
University of Queensland

Project

f19

Co-Investigator

Sean Smith
Chemistry
University of Queensland

RFCD Codes

250601, 240201

Significant Achievements, Anticipated Outcomes and Future Work

Up to date we have modelled the P donor electron wave function in the Si wafer device and in the presence of externally applied fields and interfaces using a single valley approach. We have obtained the electron wave function for varying experimental parameters: gate voltage, donor depth below the silicon oxide layer, back gate depth and inter donor separation. Once the wave function was calculated we evaluated the hyperfine interaction and exchange coupling. We performed a Heitler-London calculation of the exchange coupling between two dopant P atoms to study how the electrostatic potential at the J-gate, and the donor position in the lattice affects the exchange coupling. We calculate both the hyperfine and exchange coupling as a function of the experimental parameters: gate voltage, inter donor separation and donor depth. These calculations were performed using an anisotropic effective mass Hamiltonian.

We are trying to extend this approach to include all 6 valleys into the calculations. We started this work in Oct 2003 and are still trying to fine tune the code to reproduce the single valley results as a check. This is important work which can provide a lot more insight into our problem and a more complete basis to check our previous results using only a single valley approach. Once we have obtained the Hamiltonian matrix at the particular experimental parameters, we need to analyse this data to get the donor electron wave function using all 6 valleys in the calculation. With the new basis for our donor electron wave function we will obtain the exchange and hyperfine coupling

We are also currently concentrating on confirming these results using a more rigorous evaluation of the exchange coupling, which incorporates a larger basis in the calculation of the two donor electron wave function than the Heitler-London singlet and triplet states.

 

Computational Techniques Used

To calculate the P donor electron wave function we expand the donor wave function in a basis of deformed hydrogenic functions, and diagonalise the matrix Hamiltonian. This was all done using code I've written in Fortran 90. The matrix elements with the electric field are computed numerically, and the integrals are evaluated using a simple trapezoidal rule. The matrix elements for the interface regions are calculated using a monte carlo Vegas routine. The matrix elements for the zero field Hamiltonian, and magnetic field Hamiltonian were first evaluated analytically, and then coded. The exchange coupling was calculated using a monte carlo integration routine, Vegas for the 3D and 6D integrals

 

Publications, Awards and External Funding

External Funding and Awards

This work was supported in part by the Australian Research Council, the Australian Government, the US National Security Agency, the Advanced Research and Development Activity, and the US Army Research Office under contract number DAAD19-01-1-0653.

ARC Large Grant No: A29937112 Discovery Project Grant no: DP0211019

Publications

L.M. Kettle, H.-S. Goan, Sean C. Smith, L.C.L. Hollenberg, C.J. Wellard and C.I. Pakes, Numerical study of hydrogenic effective mass theory for an impurity P donor in Si in the presence of an electric field and interfaces, Phys. Rev. B, 68 , 2003, 075317.
C.J. Wellard, L.C.L. Hollenberg, F. Parisoli, L.M. Kettle, H.-S. Goan, J.A.L. McIntosh and D.N. Jamieson, Electron exchange coupling for single-donor solid-state spin qubits, Phys. Rev. B, 68, 2003, 195209.
L.M. Kettle, H.-S. Goan, Sean C. Smith, L.C.L. Hollenberg and C.J. Wellard, The effects of J-gate potential and interfaces on donor exchange coupling in the Kane quantum computer architecture, J.Phys: Cond. Matter, 16, 2004, 1011.