Engineering long spin coherence times of spin-orbit qubits in silicon.

Electron-spin qubits have long coherence times suitable for quantum technologies. Spin-orbit coupling promises to greatly improve spin qubit scalability and functionality, allowing qubit coupling via photons, phonons or mutual capacitances, and enabling the realization of engineered hybrid and topological quantum systems. However, despite much recent interest, results to date have yielded short coherence times (from 0.1 to 1 µs). Here we demonstrate ultra-long coherence times of 10 ms for holes where spin-orbit coupling yields quantized total angular momentum. We focus on holes bound to boron acceptors in bulk silicon 28, whose wavefunction symmetry can be controlled through crystal strain, allowing direct control over the longitudinal electric dipole that causes decoherence. The results rival the best electron-spin qubits and are 104 to 105 longer than previous spin-orbit qubits. These results open a pathway to develop new artificial quantum systems and to improve the functionality and scalability of spin-based quantum technologies.

Monolithic Three-Dimensional Tuning of an Atomically Defined Silicon Tunnel Junction.

A requirement for quantum information processors is the in situ tunability of the tunnel rates and the exchange interaction energy within the device. The large energy level separation for atom qubits in silicon is well suited for qubit operation but limits device tunability using in-plane gate architectures, requiring vertically separated top-gates to control tunnelling within the device. In this paper, we address control of the simplest tunnelling device in Si:P, the tunnel junction. Here we demonstrate that we can tune its conductance by using a vertically separated top-gate aligned with ±5 nm precision to the junction. We show that a monolithic 3D epitaxial top-gate increases the capacitive coupling by a factor of 3 compared to in-plane gates, resulting in a tunnel barrier height tunability of 0-186 meV. By combining multiple gated junctions in series we extend our monolithic 3D gating technology to implement nanoscale logic circuits including AND and OR gates.

Characterization of a Scalable Donor-Based Singlet-Triplet Qubit Architecture in Silicon.

We present a donor-based quadruple-quantum-dot device, designed to host two singlet-triplet qubits fabricated by scanning tunnelling microscope lithography, with just two leads per qubit. The design is geometrically compact, with each pair of dots independently controlled via one gate and one reservoir. The reservoirs both supply electrons for the dots and measure the singlet-triplet state of each qubit via dispersive sensing. We verify the locations of the four phosphorus donor dots via an electrostatic model of the device. We study one of the observed singlet-triplet states with a tunnel coupling of 39 GHz and a S0-to- T- decay of 2 ms at zero detuning. We measure a 5 GHz electrostatic interaction between two pairs of dots separated by 65 nm. The results outline a low-gate-density pathway to a scalable 1D building block of atomic-precision singlet-triplet qubits using donors with dispersive readout.

Ultralow-Noise Atomic-Scale Structures for Quantum Circuitry in Silicon.

The atomically precise doping of silicon with phosphorus (Si:P) using scanning tunneling microscopy (STM) promises ultimate miniaturization of field effect transistors. The one-dimensional (1D) Si:P nanowires are of particular interest, retaining exceptional conductivity down to the atomic scale, and are predicted as interconnects for a scalable silicon-based quantum computer. Here, we show that ultrathin Si:P nanowires form one of the most-stable electrical conductors, with the phenomenological Hooge parameter of low-frequency noise being as low as ≈10(-8) at 4.2 K, nearly 3 orders of magnitude lower than even carbon-nanotube-based 1D conductors. A in-built isolation from the surface charge fluctuations due to encapsulation of the wires within the epitaxial Si matrix is the dominant cause for the observed suppression of noise. Apart from quantum information technology, our results confirm the promising prospects for precision-doped Si:P structures in atomic-scale circuitry for the 11 nm technology node and beyond.

A tight-binding study of single-atom transistors.

A detailed theoretical study of the electronic and transport properties of a single atom transistor, where a single phosphorus atom is embedded within a single crystal transistor architecture, is presented. Using a recently reported deterministic single-atom transistor as a reference, the electronic structure of the device is represented atomistically with a tight-binding model, and the channel modulation is simulated self-consistently with a Thomas-Fermi method. The multi-scale modeling approach used allows confirmation of the charging energy of the one-electron donor charge state and explains how the electrostatic environments of the device electrodes affects the donor confinement potential and hence extent in gate voltage of the two-electron charge state. Importantly, whilst devices are relatively insensitive to dopant ordering in the highly doped leads, a ∼1% variation of the charging energy is observed when a dopant is moved just one lattice spacing within the device. The multi-scale modeling method presented here lays a strong foundation for the understanding of single-atom device structures: essential for both classical and quantum information processing.

Transport in asymmetrically coupled donor-based silicon triple quantum dots.

We demonstrate serial electron transport through a donor-based triple quantum dot in silicon fabricated with nanoscale precision by scanning tunnelling microscopy lithography. From an equivalent circuit model, we calculate the electrochemical potentials of the dots allowing us to identify ground and excited states in finite bias transport. Significantly, we show that using a scanning tunnelling microscope, we can directly demonstrate that a ∼1 nm difference in interdot distance dramatically affects transport pathways between the three dots.

Valley splitting in a silicon quantum device platform.

By suppressing an undesirable surface Umklapp process, it is possible to resolve the two most occupied states (1Γ and 2Γ) in a buried two-dimensional electron gas (2DEG) in silicon. The 2DEG exists because of an atomically sharp profile of phosphorus dopants which have been formed beneath the Si(001) surface (a δ-layer). The energy separation, or valley splitting, of the two most occupied bands has critical implications for the properties of δ-layer derived devices, yet until now, has not been directly measurable. Density functional theory (DFT) allows the 2DEG band structure to be calculated, but without experimental verification the size of the valley splitting has been unclear. Using a combination of direct spectroscopic measurements and DFT we show that the measured band structure is in good qualitative agreement with calculations and reveal a valley splitting of 132 ± 5 meV. We also report the effective mass and occupation of the 2DEG states and compare the dispersions and Fermi surface with DFT.

Limits to metallic conduction in atomic-scale quasi-one-dimensional silicon wires.

The recent observation of ultralow resistivity in highly doped, atomic-scale silicon wires has sparked interest in what limits conduction in these quasi-1D systems. Here we present electron transport measurements of gated Si:P wires of widths 4.6 and 1.5 nm. At 4.6 nm we find an electron mobility, µ(el)≃60 cm²/V s, in excellent agreement with that of macroscopic Hall bars. Metallic conduction persists to millikelvin temperatures where we observe Gaussian conductance fluctuations of order δG∼e²/h. In thinner wires (1.5 nm), metallic conduction breaks down at G≲e²/h, where localization of carriers leads to Coulomb blockade. Metallic behavior is explained by the large carrier densities in Si:P δ-doped systems, allowing the occupation of all six valleys of the silicon conduction band, enhancing the number of 1D channels and hence the localization length.

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.

Machine Learning-Assisted Precision Manufacturing of Atom Qubits in Silicon.

Donor-based qubits in silicon, manufactured using scanning tunneling microscope (STM) lithography, provide a promising route to realizing full-scale quantum computing architectures. This is due to the precision of donor placement, long coherence times, and scalability of the silicon material platform. The properties of multiatom quantum dot qubits, however, depend on the exact number and location of the donor atoms within the quantum dots. In this work, we develop machine learning techniques that allow accurate and real-time prediction of the donor number at the qubit site during STM patterning. Machine learning image recognition is used to determine the probability distribution of donor numbers at the qubit site directly from STM images during device manufacturing. Models in excess of 90% accuracy are found to be consistently achieved by mitigating overfitting through reduced model complexity, image preprocessing, data augmentation, and examination of the intermediate layers of the convolutional neural networks. The results presented in this paper constitute an important milestone in automating the manufacture of atom-based qubits for computation and sensing applications.

Engineering Spin-Orbit Interactions in Silicon Qubits at the Atomic-Scale.

Spin-orbit interactions arise whenever the bulk inversion symmetry and/or structural inversion symmetry of a crystal is broken providing a bridge between a qubit's spin and orbital degree of freedom. While strong interactions can facilitate fast qubit operations by all-electrical control, they also provide a mechanism to couple charge noise thereby limiting qubit lifetimes. Previously believed to be negligible in bulk silicon, recent silicon nano-electronic devices have shown larger than bulk spin-orbit coupling strengths from Dresselhaus and Rashba couplings. Here, it is shown that with precision placement of phosphorus atoms in silicon along the [110] direction (without inversion symmetry) or [111] direction (with inversion symmetry), a wide range of Dresselhaus and Rashba coupling strength can be achieved from zero to 1113 × 10-13eV-cm. It is shown that with precision placement of phosphorus atoms, the local symmetry (C2v, D2d, and D3d) can be changed to engineer spin-orbit interactions. Since spin-orbit interactions affect both qubit operation and lifetimes, understanding their impact is essential for quantum processor design.

Direct measurement of the band structure of a buried two-dimensional electron gas.

We directly measure the band structure of a buried two dimensional electron gas (2DEG) using angle resolved photoemission spectroscopy. The buried 2DEG forms 2 nm beneath the surface of p-type silicon, because of a dense delta-type layer of phosphorus n-type dopants which have been placed there. The position of the phosphorous layer is beyond the probing depth of the photoemission experiment but the observation of the 2DEG is nevertheless possible at certain photon energies where emission from the states is resonantly enhanced. This permits direct access to the band structure of the 2DEG and its temperature dependence.

Engineering independent electrostatic control of atomic-scale (∼4 nm) silicon double quantum dots.

Scalable quantum computing architectures with electronic spin qubits hosted by arrays of single phosphorus donors in silicon require local electric and magnetic field control of individual qubits separated by ∼10 nm. This daunting task not only requires atomic-scale accuracy of single P donor positioning to control interqubit exchange interaction but also demands precision alignment of control electrodes with careful device design at these small length scales to minimize cross capacitive coupling. Here we demonstrate independent electrostatic control of two Si:P quantum dots, each consisting of ∼15 P donors, in an optimized device design fabricated by scanning tunneling microscope (STM)-based lithography. Despite the atomic-scale dimensions of the quantum dots and control electrodes reducing overall capacitive coupling, the electrostatic behavior of the device shows an excellent match to results of a priori capacitance calculations. These calculations highlight the importance of the interdot angle in achieving independent control at these length-scales. This combination of predictive electrostatic modeling and the atomic-scale fabrication accuracy of STM-lithography, provides a powerful tool for scaling multidonor dots to the single donor limit.

The Use of Exchange Coupled Atom Qubits as Atomic-Scale Magnetic Field Sensors.

Phosphorus atoms in silicon offer a rich quantum computing platform where both nuclear and electron spins can be used to store and process quantum information. While individual control of electron and nuclear spins has been demonstrated, the interplay between them during qubit operations has been largely unexplored. This study investigates the use of exchange-based operation between donor bound electron spins to probe the local magnetic fields experienced by the qubits with exquisite precision at the atomic scale. To achieve this, coherent exchange oscillations are performed between two electron spin qubits, where the left and right qubits are hosted by three and two phosphorus donors, respectively. The frequency spectrum of exchange oscillations shows quantized changes in the local magnetic fields at the qubit sites, corresponding to the different hyperfine coupling between the electron and each of the qubit-hosting nuclear spins. This ability to sense the hyperfine fields of individual nuclear spins using the exchange interaction constitutes a unique metrology technique, which reveals the exact crystallographic arrangements of the phosphorus atoms in the silicon crystal for each qubit. The detailed knowledge obtained of the local magnetic environment can then be used to engineer hyperfine fields in multi-donor qubits for high-fidelity two-qubit gates.

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.

Charge sensing of precisely positioned p donors in Si.

Real-time sensing of (spin-dependent) single-electron tunneling is fundamental to electrical readout of qubit states in spin quantum computing. Here, we demonstrate the feasibility of detecting such single-electron tunneling events using an atomically planar charge sensing layout, which can be readily integrated in scalable quantum computing architectures with phosphorus-donor-based spin qubits in silicon (Si:P). Using scanning tunneling microscopy (STM) lithography on a Si(001) surface, we patterned a single-electron transistor (SET), both tunnel and electrostatically coupled to a coplanar ultrasmall quantum dot, the latter consisting of approximately four P donors. Charge transitions of the quantum dot could be detected both in time-averaged and single-shot current response of the SET. Single electron tunneling between the quantum dot and the SET island on a time-scale (τ ∼ ms) two-orders-of-magnitude faster than the spin-lattice relaxation time of a P donor in Si makes this device geometry suitable for projective readout of Si:P spin qubits. Crucial to scalability is the ability to reproducibly achieve sufficient electron tunnel rates and charge sensitivity of the SET. The inherent atomic-scale control of STM lithography bodes extremely well to precisely optimize both of these parameters.

Ramped measurement technique for robust high-fidelity spin qubit readout.

State preparation and measurement of single-electron spin qubits typically rely on spin-to-charge conversion where a spin-dependent charge transition of the electron is detected by a coupled charge sensor. For high-fidelity, fast readout, this process requires that the qubit energy is much larger than the temperature of the system limiting the temperature range for measurements. Here, we demonstrate an initialization and measurement technique that involves voltage ramps rather than static voltages allowing us to achieve state-to-charge readout fidelities above 99% for qubit energies almost half that required by traditional methods. This previously unidentified measurement technique is highly relevant for achieving high-fidelity electron spin readout at higher temperature operation and offers a number of pragmatic benefits compared to traditional energy-selective readout such as real-time dynamic feedback and minimal alignment procedures.

Coherent control of a donor-molecule electron spin qubit in silicon.

Donor spins in silicon provide a promising material platform for large scale quantum computing. Excellent electron spin coherence times of [Formula: see text] µs with fidelities of 99.9% have been demonstrated for isolated phosphorus donors in isotopically pure 28Si, where donors are local-area-implanted in a nanoscale MOS device. Despite robust single qubit gates, realising two-qubit exchange gates using this technique is challenging due to the statistical nature of the dopant implant and placement process. In parallel a precision scanning probe lithography route has been developed to place single donors and donor molecules on one atomic plane of silicon with high accuracy aligned to heavily phosphorus doped silicon in-plane gates. Recent results using this technique have demonstrated a fast (0.8 ns) two-qubit gate with two P donor molecules placed 13 nm apart in natSi. In this paper we demonstrate a single qubit gate with coherent oscillations of the electron spin on a P donor molecule in natSi patterned by scanning tunneling microscope (STM) lithography. The electron spin exhibits excellent coherence properties, with a [Formula: see text] decoherence time of 298 ± 30 µs, and [Formula: see text] dephasing time of 295 ± 23 ns.

Spin read-out in atomic qubits in an all-epitaxial three-dimensional transistor.

The realization of the surface code for topological error correction is an essential step towards a universal quantum computer1-3. For single-atom qubits in silicon4-7, the need to control and read out qubits synchronously and in parallel requires the formation of a two-dimensional array of qubits with control electrodes patterned above and below this qubit layer. This vertical three-dimensional device architecture8 requires the ability to pattern dopants in multiple, vertically separated planes of the silicon crystal with nanometre precision interlayer alignment. Additionally, the dopants must not diffuse or segregate during the silicon encapsulation. Critical components of this architecture-such as nanowires9, single-atom transistors4 and single-electron transistors10-have been realized on one atomic plane by patterning phosphorus dopants in silicon using scanning tunnelling microscope hydrogen resist lithography11,12. Here, we extend this to three dimensions and demonstrate single-shot spin read-out with 97.9% measurement fidelity of a phosphorus dopant qubit within a vertically gated single-electron transistor with <5 nm interlayer alignment accuracy. Our strategy ensures the formation of a fully crystalline transistor using just two atomic species: phosphorus and silicon.