Spatially resolving valley quantum interference of a donor in silicon

Atomically Precise Manufacturing of Silicon Electronics

Atomically precise manufacturing (APM) is a key technique that involves the direct control of atoms in order to manufacture products or components of products. It has been developed most successfully using scanning probe methods and has received particular attention for developing atom scale electronics with a focus on silicon-based systems. This review captures the development of silicon atom-based electronics and is divided into several sections that will cover characterization and atom manipulation of silicon surfaces with scanning tunneling microscopy and atomic force microscopy, development of silicon dangling bonds as atomic quantum dots, creation of atom scale devices, and the wiring and packaging of those circuits. The review will also cover the advance of silicon dangling bond logic design and the progress of silicon quantum atomic designer (SiQAD) simulators. Finally, an outlook of APM and silicon atom electronics will be provided.

Symmetry Breaking and Spin-Orbit Coupling for Individual Vacancy-Induced In-Gap States in MoS2 Monolayers

Spins confined to point defects in atomically thin semiconductors constitute well-defined atomic-scale quantum systems that are being explored as single-photon emitters and spin qubits. Here, we investigate the in-gap electronic structure of individual sulfur vacancies in molybdenum disulfide (MoS2) monolayers using resonant tunneling scanning probe spectroscopy in the Coulomb blockade regime. Spectroscopic mapping of defect wave functions reveals an interplay of local symmetry breaking by a charge-state-dependent Jahn-Teller lattice distortion that, when combined with strong (≃100 meV) spin-orbit coupling, leads to a locking of an unpaired spin-1/2 magnetic moment to the lattice at low temperature, susceptible to lattice strain. Our results provide new insights into the spin and electronic structure of vacancy-induced in-gap states toward their application as electrically and optically addressable quantum systems.

Atomic Engineering of Molecular Qubits for High-Speed, High-Fidelity Single Qubit Gates

Universal quantum computing requires fast single- and two-qubit gates with individual qubit addressability to minimize decoherence errors during processor operation. Electron spin qubits using individual phosphorus donor atoms in silicon have demonstrated long coherence times with high fidelities, providing an attractive platform for scalable quantum computing. While individual qubit addressability has been demonstrated by controlling the hyperfine interaction between the electron and nuclear wave function in a global magnetic field, the small hyperfine Stark coefficient of 0.34 MHz/MV m-1 achieved to date has limited the speed of single quantum gates to ∼42 μs to avoid rotating neighboring qubits due to power broadening from the antenna. The use of molecular 2P qubits with more than one donor atom has not only demonstrated fast (0.8 ns) two-qubit SWAP gates and long spin relaxation times of ∼30 s but provides an alternate way to achieve high selectivity of the qubit resonance frequency. Here, we show in two different devices that by placing the donors with comparable interatomic spacings (∼0.8 nm) but along different crystallographic axes, either the [110] or [310] orientations using STM lithography, we can engineer the hyperfine Stark shift from 1 MHz/MV m-1 to 11.2 MHz/MV m-1, respectively, a factor of 10 difference. NEMO atomistic calculations show that larger hyperfine Stark coefficients of up to ∼70 MHz/MV m-1 can be achieved within 2P molecules by placing the donors ≥5 nm apart. When combined with Gaussian pulse shaping, we show that fast single qubit gates with 2π rotation times of 10 ns and ∼99% fidelity single qubit operations are feasible without affecting neighboring qubits. By increasing the single qubit gate time to ∼550 ns, two orders of magnitude faster than previously measured, our simulations confirm that >99.99% single qubit control fidelities are achievable.

Influence of sample momentum space features on scanning tunnelling microscope measurements

Theoretical understanding of scanning tunnelling microscopy (STM) measurements involve electronic structure details of the STM tip and the sample being measured. Conventionally, the focus has been on the accuracy of the electronic state simulations of the sample, whereas the STM tip electronic state is typically approximated as a simple spherically symmetric s orbital. This widely used s orbital approximation has failed in recent STM studies where the measured STM images of subsurface impurity wave functions in silicon required a detailed description of the STM tip electronic state. In this work, we show that the failure of the s orbital approximation is due to the indirect band-gap of the sample material silicon (Si), which gives rise to complex valley interferences in the momentum space of impurity wave functions. Based on direct comparison of STM images computed from multi-million-atom electronic structure calculations of impurity wave functions in direct (GaAs) and indirect (Si) band-gap materials, our results establish that whilst the selection of STM tip orbital only plays a minor qualitative role for the direct band gap GaAs material, the STM measurements are dramatically modified by the momentum space features of the indirect band gap Si material, thereby requiring a quantitative representation of the STM tip orbital configuration. Our work provides new insights to understand future STM studies of semiconductor materials based on their momentum space features, which will be important for the design and implementation of emerging technologies in the areas of quantum computing, photonics, spintronics and valleytronics.

Valley interference and spin exchange at the atomic scale in silicon

Tunneling is a fundamental quantum process with no classical equivalent, which can compete with Coulomb interactions to give rise to complex phenomena. Phosphorus dopants in silicon can be placed with atomic precision to address the different regimes arising from this competition. However, they exploit wavefunctions relying on crystal band symmetries, which tunneling interactions are inherently sensitive to. Here we directly image lattice-aperiodic valley interference between coupled atoms in silicon using scanning tunneling microscopy. Our atomistic analysis unveils the role of envelope anisotropy, valley interference and dopant placement on the Heisenberg spin exchange interaction. We find that the exchange can become immune to valley interference by engineering in-plane dopant placement along specific crystallographic directions. A vacuum-like behaviour is recovered, where the exchange is maximised to the overlap between the donor orbitals, and pair-to-pair variations limited to a factor of less than 10 considering the accuracy in dopant positioning. This robustness remains over a large range of distances, from the strongly Coulomb interacting regime relevant for high-fidelity quantum computation to strongly coupled donor arrays of interest for quantum simulation in silicon.

Coupled donor wavefunctions in silicon are spatially resolved to evidence valley interference processes. An atomic-scale understanding of the interplay between interference, envelope anisotropy and crystal symmetries unveils a placement strategy compatible with existing technology where the exchange is insensitive to interference.

Scanned Single-Electron Probe inside a Silicon Electronic Device

Solid-state devices can be fabricated at the atomic scale, with applications ranging from classical logic to current standards and quantum technologies. Although it is very desirable to probe these devices and the quantum states they host at the atomic scale, typical methods rely on long-ranged capacitive interactions, making this difficult. Here, we probe a silicon electronic device at the atomic scale using a localized electronic quantum dot induced directly within the device at a desired location, using the biased tip of a low-temperature scanning tunneling microscope. We demonstrate control over short-ranged tunnel coupling interactions of the quantum dot with the device's source reservoir using sub-nanometer position control of the tip and the quantum dot energy level using a voltage applied to the device's gate reservoir. Despite the ∼1 nm proximity of the quantum dot to the metallic tip, we find that the gate provides sufficient capacitance to enable a high degree of electric control. Combined with atomic-scale imaging, we use the quantum dot to probe applied electric fields and charge in individual defects in the device. This capability is expected to aid in the understanding of atomic-scale devices and the quantum states realized in them.

Atomic-Scale Patterning of Arsenic in Silicon by Scanning Tunneling Microscopy

Over the past two decades, prototype devices for future classical and quantum computing technologies have been fabricated by using scanning tunneling microscopy and hydrogen resist lithography to position phosphorus atoms in silicon with atomic-scale precision. Despite these successes, phosphine remains the only donor precursor molecule to have been demonstrated as compatible with the hydrogen resist lithography technique. The potential benefits of atomic-scale placement of alternative dopant species have, until now, remained unexplored. In this work, we demonstrate the successful fabrication of atomic-scale structures of arsenic-in-silicon. Using a scanning tunneling microscope tip, we pattern a monolayer hydrogen mask to selectively place arsenic atoms on the Si(001) surface using arsine as the precursor molecule. We fully elucidate the surface chemistry and reaction pathways of arsine on Si(001), revealing significant differences to phosphine. We explain how these differences result in enhanced surface immobilization and in-plane confinement of arsenic compared to phosphorus, and a dose-rate independent arsenic saturation density of 0.24 ± 0.04 monolayers. We demonstrate the successful encapsulation of arsenic delta-layers using silicon molecular beam epitaxy, and find electrical characteristics that are competitive with equivalent structures fabricated with phosphorus. Arsenic delta-layers are also found to offer confinement as good as similarly prepared phosphorus layers, while still retaining >80% carrier activation and sheet resistances of <2 kΩ/square. These excellent characteristics of arsenic represent opportunities to enhance existing capabilities of atomic-scale fabrication of dopant structures in silicon, and may be important for three-dimensional devices, where vertical control of the position of device components is critical.

Electrostatic Landscape of a Hydrogen-Terminated Silicon Surface Probed by a Moveable Quantum Dot

With nanoelectronics reaching the limit of atom-sized devices, it has become critical to examine how irregularities in the local environment can affect device functionality. Here, we characterize the influence of charged atomic species on the electrostatic potential of a semiconductor surface at the subnanometer scale. Using noncontact atomic force microscopy, two-dimensional maps of the contact potential difference are used to show the spatially varying electrostatic potential on the (100) surface of hydrogen-terminated highly doped silicon. Three types of charged species, one on the surface and two within the bulk, are examined. An electric field sensitive spectroscopic signature of a single probe atom reports on nearby charged species. The identity of one of the near-surface species has been uncertain in the literature, and we suggest that its character is more consistent with either a negatively charged interstitial hydrogen or a hydrogen vacancy complex.

An orbitally derived single-atom magnetic memory

A magnetic atom epitomizes the scaling limit for magnetic information storage. Individual atomic spins have recently exhibited magnetic remanence, a requirement for magnetic memory. However, such memory has been only realized on thin insulating surfaces, removing potential tunability via electronic gating or exchange-driven magnetic coupling. Here, we show a previously unobserved mechanism for single-atom magnetic storage based on bistability in the orbital population, or so-called valency, of an individual Co atom on semiconducting black phosphorus (BP). Ab initio calculations reveal that distance-dependent screening from the BP surface stabilizes the two distinct valencies, each with a unique orbital population, total magnetic moment, and spatial charge density. Excellent correspondence between the measured and predicted charge densities reveal that such orbital configurations can be accessed and manipulated without a spin-sensitive readout mechanism. This orbital memory derives stability from the energetic barrier to atomic relaxation, demonstrating the potential for high-temperature single-atom information storage.

Intervalley Scattering of Interlayer Excitons in a MoS2/MoSe2/MoS2 Heterostructure in High Magnetic Field

Degenerate extrema in the energy dispersion of charge carriers in solids, also referred to as valleys, can be regarded as a binary quantum degree of freedom, which can potentially be used to implement valleytronic concepts in van der Waals heterostructures based on transition metal dichalcogenides. Using magneto-photoluminescence spectroscopy, we achieve a deeper insight into the valley polarization and depolarization mechanisms of interlayer excitons formed across a MoS₂/MoSe₂/MoS₂ heterostructure. We account for the nontrivial behavior of the valley polarization as a function of the magnetic field by considering the interplay between exchange interaction and phonon-mediated intervalley scattering in a system consisting of Zeeman-split energy levels. Our results represent a crucial step toward the understanding of the properties of interlayer excitons with strong implications for the implementation of atomically thin valleytronic devices.

Probing quantum coherence in single-atom electron spin resonance

Spin resonance of individual spin centers allows applications ranging from quantum information technology to atomic-scale magnetometry. To protect the quantum properties of a spin, control over its local environment, including energy relaxation and decoherence processes, is crucial. However, in most existing architectures, the environment remains fixed by the crystal structure and electrical contacts. Recently, spin-polarized scanning tunneling microscopy (STM), in combination with electron spin resonance (ESR), allowed the study of single adatoms and inter-atomic coupling with an unprecedented combination of spatial and energy resolution. We elucidate and control the interplay of an Fe single spin with its atomic-scale environment by precisely tuning the phase coherence time T₂ using the STM tip as a variable electrode. We find that the decoherence rate is the sum of two main contributions. The first scales linearly with tunnel current and shows that, on average, every tunneling electron causes one dephasing event. The second, effective even without current, arises from thermally activated spin-flip processes of tip spins. Understanding these interactions allows us to maximize T₂ and improve the energy resolution. It also allows us to maximize the amplitude of the ESR signal, which supports measurements even at elevated temperatures as high as 4 K. Thus, ESR-STM allows control of quantum coherence in individual, electrically accessible spins.

The role of the strain induced population imbalance in Valley polarization of graphene: Berry curvature perspective

Real magnetic and lattice deformation gauge fields have been investigated in honeycomb lattice of graphene. The coexistence of these two gauges will induce a gap difference between two valley points (K and K') of system. This gap difference allows us to study the possible topological valley Hall current and valley polarization in the graphene sheet. In the absence of magnetic field, the strain alone could not generate a valley polarization when the Fermi energy coincides exactly with the Dirac points. Since in this case there is not any imbalance between the population of the valley points. In other words each of these gauges alone could not induce any topological valley-polarized current in the system at zero Fermi energy. Meanwhile at non-zero Fermi energies population imbalance can be generated as a result of the external strain even at zero magnetic field. In the context of Berry curvature within the linear response regime the valley polarization (both magnetic free polarization, Π0, and field dependent response function, χ α ) in different values of gauge fields of lattice deformation has been obtained.

Towards visualisation of central-cell-effects in scanning tunnelling microscope images of subsurface dopant qubits in silicon

Atomic-scale understanding of phosphorus donor wave functions underpins the design and optimisation of silicon based quantum devices. The accuracy of large-scale theoretical methods to compute donor wave functions is dependent on descriptions of central-cell corrections, which are empirically fitted to match experimental binding energies, or other quantities associated with the global properties of the wave function. Direct approaches to understanding such effects in donor wave functions are of great interest. Here, we apply a comprehensive atomistic theoretical framework to compute scanning tunnelling microscopy (STM) images of subsurface donor wave functions with two central-cell correction formalisms previously employed in the literature. The comparison between central-cell models based on real-space image features and the Fourier transform profiles indicates that the central-cell effects are visible in the simulated STM images up to ten monolayers below the silicon surface. Our study motivates a future experimental investigation of the central-cell effects via the STM imaging technique with potential of fine tuning theoretical models, which could play a vital role in the design of donor-based quantum systems in scalable quantum computer architectures.

A minimal double quantum dot

Double quantum dots (DQDs) are a versatile platform for solid-state physics, quantum computation and nanotechnology. The micro-fabrication techniques commonly used to fabricate DQDs are difficult to extend to the atomic scale. Using an alternative approach, which relies on scanning tunneling microscopy and spectroscopy, we prepared a minimal DQD in a wide band-gap semiconductor matrix. It is comprised of a pair of strongly coupled donor atoms that can each be doubly charged. The donor excitation diagram of this system mimicks the charge stability diagram observed in transport measurements of DQDs. We furthermore illustrate how the charge and spin degrees of freedom of the minimal DQD may be used to obtain a single quantum bit and to prepare a Bell state. The results open an intriguing perspective for quantum electronics with atomic-scale structures.

Ab initio calculation of energy levels for phosphorus donors in silicon

The s manifold energy levels for phosphorus donors in silicon are important input parameters for the design and modeling of electronic devices on the nanoscale. In this paper we calculate these energy levels from first principles using density functional theory. The wavefunction of the donor electron's ground state is found to have a form that is similar to an atomic s orbital, with an effective Bohr radius of 1.8 nm. The corresponding binding energy of this state is found to be 41 meV, which is in good agreement with the currently accepted value of 45.59 meV. We also calculate the energies of the excited 1s(T ₂) and 1s(E) states, finding them to be 32 and 31 meV respectively.

Implementation of Multivariable Logic Functions in Parallel by Electrically Addressing a Molecule of Three Dopants in Silicon

To realize low-power, compact logic circuits, one can explore parallel operation on single nanoscale devices. An added incentive is to use multivalued (as distinct from Boolean) logic. Here, we theoretically demonstrate that the computation of all the possible outputs of a multivariate, multivalued logic function can be implemented in parallel by electrical addressing of a molecule made up of three interacting dopant atoms embedded in Si. The electronic states of the dopant molecule are addressed by pulsing a gate voltage. By simulating the time evolution of the non stationary electronic density built by the gate voltage, we show that one can implement a molecular decision tree that provides in parallel all the outputs for all the inputs of the multivariate, multivalued logic function. The outputs are encoded in the populations and in the bond orders of the dopant molecule, which can be measured using an STM tip. We show that the implementation of the molecular logic tree is equivalent to a spectral function decomposition. The function that is evaluated can be field-programmed by changing the time profile of the pulsed gate voltage.

Sequential reduction of the silicon single-electron transistor structure to atomic scale

Here we present an original CMOS compatible fabrication method of a single-electron transistor structure with extremely small islands, formed by solitary phosphorus dopants in the silicon nanobridge. Its key feature is the controllable size reduction of the nanobridge in sequential cycles of low energy isotropic reactive ion etching that results in a decreased number of active charge centers (dopants) in the nanobridge from hundreds to a single one. Electron transport through the individual phosphorous dopants in the silicon lattice was studied. The final transistor structure demonstrates a Coulomb blockade voltage of ∼30 mV and nanobridge size estimated as [Formula: see text]. Analysis of current stability diagrams shows that electron transport in samples after the final etching stage had a single-electron nature and was carried through three phosphorus atoms. The fabrication method of the demonstrated structure allows it to be modified further by various impurities in additional etching and implantation cycles.

A Probabilistic Finite State Logic Machine Realized Experimentally on a Single Dopant Atom

Exploiting the potential of nanoscale devices for logic processing requires the implementation of computing functionalities departing from the conventional switching paradigm. We report on the design and the experimental realization of a probabilistic finite state machine in a single phosphorus donor atom placed in a silicon matrix electrically addressed and probed by scanning tunneling spectroscopy (STS). The single atom logic unit simulates the flow of visitors in a maze whose topology is determined by the dynamics of the electronic transport through the states of the dopant. By considering the simplest case of a unique charge state for which three electronic states can be resolved, we demonstrate an efficient solution of the following problem: in a maze of four connected rooms, what is the optimal combination of door opening rates in order to maximize the time that visitors spend in one specific chamber? The implementation takes advantage of the stochastic nature of electron tunneling, while the output remains the macroscopic current whose reading can be realized with standard techniques and does not require single electron sensitivity.

From time-resolved atomic-scale imaging of individual donors to their cooperative dynamics

The key elements in the steady miniaturization process of cutting-edge semiconductor devices are the understanding and controlling of charge dynamics on the atomic scale. In detail, we address the study of charging processes of individual doping atoms and, especially, the interaction of those atoms with their surroundings. We use pulsed optical excitation in combination with scanning tunneling microscopy at the n-doped gallium arsenide [GaAs(110)] surface to investigate single donor dynamics within a nanoscaled, localized space charge region. Tuning the tunnel rate can drive the system into nonequilibrium conditions, allowing distinction between the decay of optically induced free charge carriers and the decay of donor charge states. The latter process is atomically resolved and discussed with respect to donor-level binding energies and local donor configurations.

Single-electron tunneling through an individual arsenic dopant in silicon

We report the single-electron tunneling behaviour of a silicon nanobridge where the effective island is a single As dopant atom. The device is a gated silicon nanobridge with a thickness and width of ∼20 nm, fabricated from a commercially available silicon-on-insulator wafer, which was first doped with As atoms and then patterned using a unique CMOS-compatible technique. Transport measurements reveal characteristic Coulomb diamonds whose size decreases with gate voltage. Such a dependence indicates that the island of the single-electron transistor created is an individual arsenic dopant atom embedded in the silicon lattice between the source and drain electrodes, and furthermore, can be explained by the increase of the localisation region of the electron wavefunction when the higher energy levels of the dopant As atom become occupied. The charge stability diagram of the device shows features which can be attributed to adjacent dopants, localised in the nanobridge, acting as charge traps. From the measured device transport, we have evaluated the tunnel barrier properties and obtained characteristic device capacitances. The fabrication, control and understanding of such "single-atom" devices marks a further step towards the implementation of single-atom electronics.

Time-resolved single dopant charge dynamics in silicon

As the ultimate miniaturization of semiconductor devices approaches, it is imperative that the effects of single dopants be clarified. Beyond providing insight into functions and limitations of conventional devices, such information enables identification of new device concepts. Investigating single dopants requires sub-nanometre spatial resolution, making scanning tunnelling microscopy an ideal tool. However, dopant dynamics involve processes occurring at nanosecond timescales, posing a significant challenge to experiment. Here we use time-resolved scanning tunnelling microscopy and spectroscopy to probe and study transport through a dangling bond on silicon before the system relaxes or adjusts to accommodate an applied electric field. Atomically resolved, electronic pump-probe scanning tunnelling microscopy permits unprecedented, quantitative measurement of time-resolved single dopant ionization dynamics. Tunnelling through the surface dangling bond makes measurement of a signal that would otherwise be too weak to detect feasible. Distinct ionization and neutralization rates of a single dopant are measured and the physical process controlling those are identified.

Probing individual impurities will become increasingly important as devices shrink towards the nanoscale. Here Rashidi et al., introduce a method based on time-resolved scanning tunnelling spectroscopy of surface dangling bonds to investigate the dynamics of individual dopants in silicon.

A computational workflow for designing silicon donor qubits

Developing devices that can reliably and accurately demonstrate the principles of superposition and entanglement is an on-going challenge for the quantum computing community. Modeling and simulation offer attractive means of testing early device designs and establishing expectations for operational performance. However, the complex integrated material systems required by quantum device designs are not captured by any single existing computational modeling method. We examine the development and analysis of a multi-staged computational workflow that can be used to design and characterize silicon donor qubit systems with modeling and simulation. Our approach integrates quantum chemistry calculations with electrostatic field solvers to perform detailed simulations of a phosphorus dopant in silicon. We show how atomistic details can be synthesized into an operational model for the logical gates that define quantum computation in this particular technology. The resulting computational workflow realizes a design tool for silicon donor qubits that can help verify and validate current and near-term experimental devices.

Spatial metrology of dopants in silicon with exact lattice site precision

Scaling of Si-based nanoelectronics has reached the regime where device function is affected not only by the presence of individual dopants, but also by their positions in the crystal. Determination of the precise dopant location is an unsolved problem in applications from channel doping in ultrascaled transistors to quantum information processing. Here, we establish a metrology combining low-temperature scanning tunnelling microscopy (STM) imaging and a comprehensive quantum treatment of the dopant-STM system to pinpoint the exact coordinates of the dopant in the Si crystal. The technique is underpinned by the observation that STM images contain atomic-sized features in ordered patterns that are highly sensitive to the STM tip orbital and the absolute dopant lattice site. The demonstrated ability to determine the locations of P and As dopants to 5 nm depths will provide critical information for the design and optimization of nanoscale devices for classical and quantum computing applications.

Characterizing Si:P quantum dot qubits with spin resonance techniques

Quantum dots patterned by atomically precise placement of phosphorus donors in single crystal silicon have long spin lifetimes, advantages in addressability, large exchange tunability, and are readily available few-electron systems. To be utilized as quantum bits, it is important to non-invasively characterise these donor quantum dots post fabrication and extract the number of bound electron and nuclear spins as well as their locations. Here, we propose a metrology technique based on electron spin resonance (ESR) measurements with the on-chip circuitry already needed for qubit manipulation to obtain atomic scale information about donor quantum dots and their spin configurations. Using atomistic tight-binding technique and Hartree self-consistent field approximation, we show that the ESR transition frequencies are directly related to the number of donors, electrons, and their locations through the electron-nuclear hyperfine interaction.

Valley-enhanced fast relaxation of gate-controlled donor qubits in silicon

Gate control of donor electrons near interfaces is a generic ingredient of donor-based quantum computing. Here, we address the question: how is the phonon-assisted qubit relaxation time T 1 affected as the electron is shuttled between the donor and the interface? We focus on the example of the 'flip-flop qubit' (Tosi et al arXiv:1509.08538v1), defined as a combination of the nuclear and electronic states of a phosphorus donor in silicon, promising fast electrical control and long dephasing times when the electron is halfway between the donor and the interface. We theoretically describe orbital relaxation, flip-flop relaxation, and electron spin relaxation. We estimate that the flip-flop qubit relaxation time can be of the order of 100 μs, 8 orders of magnitude shorter than the value for an on-donor electron in bulk silicon, and a few orders of magnitude shorter (longer) than the predicted inhomogeneous dephasing time (gate times). All three relaxation processes are boosted by (i) the nontrivial valley structure of the electron-phonon interaction, and (ii) the different valley compositions of the involved electronic states.

Charge-Insensitive Single-Atom Spin-Orbit Qubit in Silicon

High fidelity entanglement of an on-chip array of spin qubits poses many challenges. Spin-orbit coupling (SOC) can ease some of these challenges by enabling long-ranged entanglement via electric dipole-dipole interactions, microwave photons, or phonons. However, SOC exposes conventional spin qubits to decoherence from electrical noise. Here, we propose an acceptor-based spin-orbit qubit in silicon offering long-range entanglement at a sweet spot where the qubit is protected from electrical noise. The qubit relies on quadrupolar SOC with the interface and gate potentials. As required for surface codes, 10^{5} electrically mediated single-qubit and 10^{4} dipole-dipole mediated two-qubit gates are possible in the predicted spin lifetime. Moreover, circuit quantum electrodynamics with single spins is feasible, including dispersive readout, cavity-mediated entanglement, and spin-photon entanglement. An industrially relevant silicon-based platform is employed.

Quantum computing with acceptor spins in silicon

The states of a boron acceptor near a Si/SiO2 interface, which bind two low-energy Kramers pairs, have exceptional properties for encoding quantum information and, with the aid of strain, both heavy hole and light hole-based spin qubits can be designed. Whereas a light-hole spin qubit was introduced recently (arXiv:1508.04259), here we present analytical and numerical results proving that a heavy-hole spin qubit can be reliably initialised, rotated and entangled by electrical means alone. This is due to strong Rashba-like spin-orbit interaction terms enabled by the interface inversion asymmetry. Single qubit rotations rely on electric-dipole spin resonance (EDSR), which is strongly enhanced by interface-induced spin-orbit terms. Entanglement can be accomplished by Coulomb exchange, coupling to a resonator, or spin-orbit induced dipole-dipole interactions. By analysing the qubit sensitivity to charge noise, we demonstrate that interface-induced spin-orbit terms are responsible for sweet spots in the dephasing time [Formula: see text] as a function of the top gate electric field, which are close to maxima in the EDSR strength, where the EDSR gate has high fidelity. We show that both qubits can be described using the same starting Hamiltonian, and by comparing their properties we show that the complex interplay of bulk and interface-induced spin-orbit terms allows a high degree of electrical control and makes acceptors potential candidates for scalable quantum computation in Si.

Quantum simulation of the Hubbard model with dopant atoms in silicon

In quantum simulation, many-body phenomena are probed in controllable quantum systems. Recently, simulation of Bose-Hubbard Hamiltonians using cold atoms revealed previously hidden local correlations. However, fermionic many-body Hubbard phenomena such as unconventional superconductivity and spin liquids are more difficult to simulate using cold atoms. To date the required single-site measurements and cooling remain problematic, while only ensemble measurements have been achieved. Here we simulate a two-site Hubbard Hamiltonian at low effective temperatures with single-site resolution using subsurface dopants in silicon. We measure quasi-particle tunnelling maps of spin-resolved states with atomic resolution, finding interference processes from which the entanglement entropy and Hubbard interactions are quantified. Entanglement, determined by spin and orbital degrees of freedom, increases with increasing valence bond length. We find separation-tunable Hubbard interaction strengths that are suitable for simulating strongly correlated phenomena in larger arrays of dopants, establishing dopants as a platform for quantum simulation of the Hubbard model.

Interface effects on acceptor qubits in silicon and germanium

Dopant-based quantum computing implementations often require the dopants to be situated close to an interface to facilitate qubit manipulation with local gates. Interfaces not only modify the energies of the bound states but also affect their symmetry. Making use of the successful effective mass theory we study the energy spectra of acceptors in Si or Ge taking into account the quantum confinement, the dielectric mismatch and the central cell effects. The presence of an interface puts constraints to the allowed symmetries and leads to the splitting of the ground state in two Kramers doublets (Mol et al 2015 Appl. Phys. Lett. 106 203110). Inversion symmetry breaking also implies parity mixing which affects the allowed optical transitions. Consequences for acceptor qubits are discussed.

Electrical current through individual pairs of phosphorus donor atoms and silicon dangling bonds

Nuclear spins of phosphorus [P] donor atoms in crystalline silicon are among the most coherent qubits found in nature. For their utilization in scalable quantum computers, distinct donor electron wavefunctions must be controlled and probed through electrical coupling by application of either highly localized electric fields or spin-selective currents. Due to the strong modulation of the P-donor wavefunction by the silicon lattice, such electrical coupling requires atomic spatial accuracy. Here, the spatially controlled application of electrical current through individual pairs of phosphorus donor electron states in crystalline silicon and silicon dangling bond states at the crystalline silicon (100) surface is demonstrated using a high‐resolution scanning probe microscope operated under ultra‐high vacuum and at a temperature of 4.3K. The observed pairs of electron states display qualitatively reproducible current-voltage characteristics with a monotonous increase and intermediate current plateaus.

Optical Control of Donor Spin Qubits in Silicon

We show how to achieve optical, spin-selective transitions from the ground state to excited orbital states of group-V donors (P, As, Sb, Bi) in silicon. We consider two approaches based on either resonant, far-infrared (IR) transitions of the neutral donor or resonant, near-IR excitonic transitions. For far-IR light, we calculate the dipole matrix elements between the valley-orbit and spin-orbit split states for all the goup-V donors using effective mass theory. We then calculate the maximum rate and amount of electron-nuclear spin-polarization achievable through optical pumping with circularly polarized light. We find this approach is most promising for Bi donors due to their large spin-orbit and valley-orbit interactions. Using near-IR light, spin-selective excitation is possible for all the donors by driving a two-photon Λ-transition from the ground state to higher orbitals with even parity. We show that externally applied electric fields or strain allow similar, spin-selective Λ-transition to odd-parity excited states. We anticipate these results will be useful for future spectroscopic investigations of donors, quantum control and state preparation of donor spin qubits, and for developing a coherent interface between donor spin qubits and single photons.

A surface code quantum computer in silicon

The exceptionally long quantum coherence times of phosphorus donor nuclear spin qubits in silicon, coupled with the proven scalability of silicon-based nano-electronics, make them attractive candidates for large-scale quantum computing. However, the high threshold of topological quantum error correction can only be captured in a two-dimensional array of qubits operating synchronously and in parallel-posing formidable fabrication and control challenges. We present an architecture that addresses these problems through a novel shared-control paradigm that is particularly suited to the natural uniformity of the phosphorus donor nuclear spin qubit states and electronic confinement. The architecture comprises a two-dimensional lattice of donor qubits sandwiched between two vertically separated control layers forming a mutually perpendicular crisscross gate array. Shared-control lines facilitate loading/unloading of single electrons to specific donors, thereby activating multiple qubits in parallel across the array on which the required operations for surface code quantum error correction are carried out by global spin control. The complexities of independent qubit control, wave function engineering, and ad hoc quantum interconnects are explicitly avoided. With many of the basic elements of fabrication and control based on demonstrated techniques and with simulated quantum operation below the surface code error threshold, the architecture represents a new pathway for large-scale quantum information processing in silicon and potentially in other qubit systems where uniformity can be exploited.

Dynamic creation of a topologically-ordered Hamiltonian using spin-pulse control in the Heisenberg model

Hamiltonian engineering is an important approach for quantum information processing, when appropriate materials do not exist in nature or are unstable. So far there is no stable material for the Kitaev spin Hamiltonian with anisotropic interactions on a honeycomb lattice, which plays a crucial role in the realization of both Abelian and non-Abelian anyons. Here, we show two methods to dynamically realize the Kitaev spin Hamiltonian from the conventional Heisenberg spin Hamiltonian using pulse-control techniques based on the Baker-Campbell-Hausdorff (BCH) formula. In the first method, the Heisenberg interaction is changed into Ising interactions in the first process of the pulse sequence. In the next process of the first method, we transform them to a desirable anisotropic Kitaev spin Hamiltonian. In the second more efficient method, we show that if we carefully design two-dimensional pulses that vary depending on the qubit location, we can obtain the desired Hamiltonian in only one step of applying the BCH formula. As an example, we apply our methods to spin qubits based on quantum dots, in which the effects of both the spin-orbit interaction and the hyperfine interaction are estimated.

Spatially resolved resonant tunneling on single atoms in silicon

The ability to control single dopants in solid-state devices has opened the way towards reliable quantum computation schemes. In this perspective it is essential to understand the impact of interfaces and electric fields, inherent to address coherent electronic manipulation, on the dopants atomic scale properties. This requires both fine energetic and spatial resolution of the energy spectrum and wave-function, respectively. Here we present an experiment fulfilling both conditions: we perform transport on single donors in silicon close to a vacuum interface using a scanning tunneling microscope (STM) in the single electron tunneling regime. The spatial degrees of freedom of the STM tip provide a versatility allowing a unique understanding of electrostatics. We obtain the absolute energy scale from the thermal broadening of the resonant peaks, allowing us to deduce the charging energies of the donors. Finally we use a rate equations model to derive the current in presence of an excited state, highlighting the benefits of the highly tunable vacuum tunnel rates which should be exploited in further experiments. This work provides a general framework to investigate dopant-based systems at the atomic scale.

Theory of one and two donors in silicon

We provide here a roadmap for modeling silicon nano-devices with one or two group V donors (D). We discuss systems containing one or two electrons, that is, D(0), D(-), D(+)(2) and D(0)(2) centers. The impact of different levels of approximation is discussed. The most accurate instances--for which we provide quantitative results--are within multivalley effective mass including the central cell correction and a configuration interaction account of the electron-electron correlations. We also derive insightful, yet less accurate, analytical approximations and discuss their validity and limitations--in particular, for a donor pair, we discuss the single orbital LCAO method, the Hückel approximation and the Hubbard model. Finally, we connect these results with recent experiments on devices with few dopants.

Mapping the site-specific potential energy landscape for chemisorbed and physisorbed aromatic molecules on the Si(1 1 1)-7 × 7 surface by time-lapse STM

We present a scanning tunnelling microscope study of site-specific thermal displacement (desorption or diffusion) of benzene, toluene, and chlorobenzene molecules on the Si(1 1 1)-7 × 7 surface. Through time-lapse STM imaging and automated image analysis we probe both the chemisorbed and the physisorbed states of these molecules. For the chemisorption to physisorption transition there are distinct site-specific variations in the measured rates, however their kinetic origin is ambiguous. There is also significant variation in the competing rates out of the physisorbed state into chemisorption at the various surface sites, which we attribute to differences in site-specific Arrhenius pre-factors. A prediction of the outcome of the competing rates and pre-factors for benzene over three hours matches experiment.

Effect of strain on the electronic and optical properties of Ge–Si dome shaped nanocrystals

An atomistic study of Ge-core–Si-shell nanocrystals gives a detailed picture of how strain and confinement effect the electronic and optical properties.

Spin-lattice relaxation times of single donors and donor clusters in silicon

An atomistic method of calculating the spin-lattice relaxation times (T₁) is presented for donors in silicon nanostructures comprising of millions of atoms. The method takes into account the full band structure of silicon including the spin-orbit interaction. The electron-phonon Hamiltonian, and hence, the deformation potential, is directly evaluated from the strain-dependent tight-binding Hamiltonian. The technique is applied to single donors and donor clusters in silicon, and explains the variation of T₁ with the number of donors and electrons, as well as donor locations. Without any adjustable parameters, the relaxation rates in a magnetic field for both systems are found to vary as B⁵, in excellent quantitative agreement with experimental measurements. The results also show that by engineering electronic wave functions in nanostructures, T₁ times can be varied by orders of magnitude.