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Modelling nanoelectronic quantum control of donor qubits in silicon

Lloyd C. L. Hollenberg, Cameron J. Wellard

Published Jan 1, 2006• 0 citations
Publisher

Abstract

The Kane concept of quantum computation using single phosphorous donors in silicon has revolutionized the way we think about P-doped silicon devices. However, because of the enormous effort required in single atom fabrication detailed modeling is required to determine and optimise the scalability, and indeed feasibility, of such a vision.

DOI
10.1109/iconn.2006.340702

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•Oct 1, 2005•0 citations
The Kane concept of quantum computation using single phosphorous donors in silicon has revolutionized the way we think about P-doped silicon devices. However, because of the enormous effort required in single atom fabrication detailed modeling is required to determine and optimise the scalability, and indeed feasibility, of such a vision.
It has been over ten years since Kane's influential proposal for a silicon-based nuclear spin quantum computer using phosphorous donors. Since then, silicon-based architectures have been refined as the experimental challenges associated with the original proposal have become better understood, while simultaneously a number of powerful and generic models for quantum computation have emerged. Here, I discuss how the cluster state or "one-way" model for quantum computing might be advantageously applied to donors in silicon, with the potential to substantially reduce the practical requirements of a successful implementation. The essence of the scheme is to use the electron spin associated with a donor to weave an entangled network between 31P donor nuclear spins. This resource has been shown to have exceptional coherence times and supports universal quantum computation through local measurements on the nuclear spins. Some of the key ingredients, such as global spin manipulation, have been robustly established, while others, such as single spin measurement, have seen much progress in recent years. A key challenge will be the demonstration of electron transfer between donors that preserves spin coherence.
In the Kane silicon-based electron-mediated nuclear spin quantum computer architecture, phosphorous 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, inter donor separation, donor depth below the silicon oxide interface and back gate depth, to explore how these variables affect the donor electro density. Second, we calculate the hyperfine interaction and the exchange coupling as a function of these parameters. These calculations were performed using an anisotropic effective mass Hamiltonian. The electric field potential was obtained using Technology Computer Aided Design software, and the interfaces were modelled as a barrier using a step function. 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.
Scalable silicon quantum computers will require a material perfection that has never been attempted. Ground state wavefunctions for conduction electrons orbiting individual phosphorous donors must be polarized electronically and coupled to nearest neighbors with great precision. Elimination of all randomizing influences can be achieved only with a fully epitaxial structure; and we believe that output circuitry must also be integrated into the qubit arrays in order to achieve the uniformity needed for large-scale integration. A process that could potentially accomplish this will be outlined, based on scanning tunneling microscope (STM) removal of individual hydrogen atoms from the H-terminated silicon surface followed by phosphine dosing and ultra-low-temperature overgrowth. Self-ordering of PH3 molecules onto extended areas of bare silicon should permit patterning of planar single-electron transistors along with P-donor qubits in the same lithographic step. Initial plans for an experiment to characterize exchange coupling under gate control will be described.
Scalable silicon quantum computers will require a material perfection that has never been attempted. Ground state wavefunctions for conduction electrons orbiting individual phosphorous donors must be polarized electronically and coupled to nearest neighbors with great precision. Elimination of all randomizing influences can be achieved only with a fully epitaxial structure; and we believe that output circuitry must also be integrated into the qubit arrays in order to achieve the uniformity needed for large-scale integration. A process that could potentially accomplish this will be outlined, based on scanning tunneling microscope (STM) removal of individual hydrogen atoms from the H-terminated silicon surface followed by phosphine dosing and ultra-low-temperature overgrowth. Self-ordering of PH3 molecules onto extended areas of bare silicon should permit patterning of planar single-electron transistors along with P-donor qubits in the same lithographic step. Initial plans for an experiment to characterize exchange coupling under gate control will be described.
The discovery of efficient quantum algorithms a decade ago has shown that a quantum computer encoding and processing information quantum mechanically, can solve important problems intractable with conventional computers. The invention of quantum error correction principle, makes quantum computation possibly fault-tolerant against the decoherence of information carriers. Thereafter, intensive research activities have been made toward the implementation of quantum computation with various realistic quantum systems. Among them, the most attractive implementation proposals are using silicon-based materials, which have the advantage of borrowing the existing ingenuity and resources accumulated during the development of modern microelectronics. In this dissertation we investigate several theoretical aspects of a silicon- based quantum computer in which qubits are represented by the spins of electrons bound to phosphorous donors in silicon. Encoding each qubit in terms of three neighboring donor electron spins, we can realize universal quantum gate operations with only the Heisenberg exchange coupling J S1 •S2 between neighboring donors. Therefore, studying the exchange coupling for a phosphorous donor pair in silicon is of central importance for providing the experimentalists with qualitative insights and quantitative guidance for building such a silicon quantum computer. After giving some general considerations on the quantum computer architecture, we develop the necessary theoretical tools. A multi-valley effective mass equation is derived and discussed, to handle impurities in a multi-valley semiconductor. Then we apply it to solve a single Si:P donor embedded in our quantum computer architecture. We show that the width of the silicon quantum well can significantly influence the energy splitting and charge distributions of the ground state. Oscillation of level splitting is observed as the quantum well width or donor position is varied at atomic scale. Elementary gate operations for 3-donor-spin qubits involve a neighboring pair of donors at each step. The exchange coupling in Heisenberg model hamiltonian, defined as the energy splitting of the lowest singlet and triplet states, has to be calculated from the realistic two-donor hamiltonian. We discuss several popular methods to solve the two-donor problem and develop an appropriate extended Hartree-Fock method that can give reliable results for a well-separated donor pair subject to tunable coupling. This method is first applied to a Si:P donor pair in the framework of hydrogenic effective mass theory, to study how to tune the exchange coupling with simple gate potentials. A followed study shows that under a parallel electric field the singlet and triplet states exhibit very different polarization behaviors. This difference can be exploited to measure the state of an electron spin. We also show that, a perpendicular electric field cannot tune the exchange coupling efficiently. Then we apply the realistic multi-valley effective mass equation, coupled with a realistic modeling of the potential generated by gate electrodes, to a pair of phosphorous donors in silicon quantum well. By varying the gate electrode voltages, the exchange coupling can be tuned with exponential efficiency in a wide range. We find that, for best performance, we need to set the quantum well width to be around 10 nm, and the donor separation to be around 10 a∗ b---- 24nm in the doping plane. We analyze in detail an adiabatic half-swap operation between neighboring donor spins. The gate operation time is estimated to be 0.2 ns, which satisfies the constraints put by the donor spin decoherence time and by the validity of adiabatic approximation. The interference between different valley components could lead to oscillations of exchange coupling as donor positions are shifted by occasion. We find the exchange oscillation persists even with only two relevant valleys for donors in a strongly-strained silicon quantum well. It is also shown that the oscillation induced by changing donor separation at atomic scale is strongly suppressed as donors approach each other. Finally, we study the entanglement issue in quantum computing context. We analyze the low-energy Hilbert space for a pair of qubits encoded by localized electron spins and suggest a suitable measure to describe the entanglement between qubits involving indistinguishable electrons and in the presence of leakage errors. The dynamics of inter-qubit entanglement during a gate operation is also studied.