Phosphorus Donors In Silicon Research Articles

Electronic structure and terahertz intersubband absorption spectra of phosphorus donor in silicon using a variationally optimized diagonalization method

The electronic structure and terahertz (THz) intersubband absorption spectra of phosphorus (P) donor in silicon (Si) are theoretically investigated using a variationally optimized diagonalization method based on the three-dimensional (3D) nonorthogonal associated Laguerre basis set, the basis set of eigenfunctions of 3D hydrogen atom and the basis set of eigenfunctions of 3D isotropic harmonic oscillator. A fully analytical expression of the matrix elements of the Hamiltonian is derived. Due to the large band anisotropy of Si, the donor electron series appear in an anomalous energy-level order and interval of the orbital angular momentum. The donor electron energy levels with even a small number of basis functions are in excellent agreement with those reported in the previous literature. Compared to two different basis sets of eigenfunctions of 3D hydrogen atom and isotropic harmonic oscillator, my proposed variationally optimized diagonalization method based on the 3D nonorthogonal associated Laguerre basis set is proven to be more efficient and reliable in calculating highly accurate impurity states of anisotropic 3D materials. The intersubband absorption spectra of P donor electron involving transitions from the 1s to excited states are strongly dependent on the orientations of the valley and the polarization vector of the incident light, which results from the fact that the large band anisotropy of Si induces strong mixing of the donor electron states. Their absorption frequencies are found to fall into the THz part of the electromagnetic spectrum. These results show that P donor in Si provides a material platform that enables intersubband devices operating in the THz frequency domain. • A fully analytical expression of the matrix elements of the donor Hamiltonian is derived. • The donor electron energy levels are in excellent agreement with those reported in the previous literature. • The intersubband absorption spectra of P donor electron are strongly dependent on the orientations of the incident light polarization. • P donor in Si provides a material platform that enables optoelectronic devices operating in the THz frequency domain. • The 3D nonorthogonal Laguerre basis is more efficient and reliable in calculating impurity states of anisotropic 3D materials.

Bardeen's tunneling theory applied to intraorbital and interorbital hopping integrals between dopants in silicon

We utilize Bardeen's tunneling theory to calculate intra- and interorbital hopping integrals between phosphorus donors in silicon using known orbital wave functions. While the two-donor problem can be solved directly, the knowledge of hoppings for various pairs of orbitals is essential for constructing multi-orbital Hubbard models for chains and arrays of donors. To assure applicability to long-range potentials, we rederive Bardeen's formula for the matrix element without assuming non-overlapping potentials. Moreover, we find a correction to the original expression allowing us to use it at short distances. We also show that accurate calculation of the lowest donor-pair eigenstates is possible based on these tunnel couplings, and we characterize the obtained states. The results are in satisfactory quantitative agreement with those obtained with the standard H\"uckel tight-binding method. The calculation relies solely on the wave functions in the barrier region and does not explicitly involve donor or lattice potentials, which has practical advantages. We find that neglecting the central correction potential in the standard method may lead to qualitatively incorrect results, while its explicit inclusion raises severe numerical problems, as it is contained in a tiny volume. In contrast, using wave functions obtained with this correction in the proposed method does not raise such issues. Nominally, the computational cost of the method is to calculate a double integral along the plane that separates donors. For donor separation in directions where valley interference leads to oscillatory behavior, additional averaging over the position of the integration plane is needed. Despite this, the presented approach offers a competitive computational cost as compared to the standard one. This work may be regarded as a benchmark of a promising method for calculating hopping integrals in lattice models.

Flopping-Mode Electric Dipole Spin Resonance in Phosphorus Donor Qubits in Silicon

Single spin qubits based on phosphorus donors in silicon are a promising candidate for a large-scale quantum computer. Despite long coherence times, achieving uniform magnetic control remains a hurdle for scale-up due to challenges in high-frequency magnetic field control at the nanometre scale. Here, we present a proposal for a flopping-mode electric dipole spin resonance qubit based on the combined electron and nuclear spin states of a double phosphorus donor quantum dot. The key advantage of utilizing a donor-based system is that we can engineer the number of donor nuclei in each quantum dot. By creating multidonor dots with antiparallel nuclear spin states and multielectron occupation we can minimize the longitudinal magnetic field gradient, known to couple charge noise into the device and dephase the qubit. We describe the operation of the qubit and show that by minimizing the hyperfine interaction of the nuclear spins we can achieve $\ensuremath{\pi}/2\ensuremath{-}X$ gate error rates of about ${10}^{\ensuremath{-}4}$ using realistic noise models. We highlight that the low charge noise environment in these all-epitaxial phosphorus-doped silicon qubits will facilitate the realization of strong coupling of the qubit to superconducting microwave cavities, allowing for long-distance two-qubit operations.

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(T2) and 1s(E) states, finding them to be 32 and 31 meV respectively.

Electron nuclear double resonance with donor-bound excitons in silicon

We present Auger-electron-detected magnetic resonance (AEDMR) experiments on phosphorus donors in silicon, where the selective optical generation of donor-bound excitons is used for the electrical detection of the electron spin state. Because of the long dephasing times of the electron spins in isotopically purified $^{28}$Si, weak microwave fields are sufficient, which allow to realize broadband AEDMR in a commercial ESR resonator. Implementing Auger-electron-detected ENDOR, we further demonstrate the optically-assisted control of the nuclear spin under conditions where the hyperfine splitting is not resolved in the optical spectrum. Compared to previous studies, this significantly relaxes the requirements on the sample and the experimental setup, e.g. with respect to strain, isotopic purity and temperature. We show AEDMR of phosphorus donors in silicon with natural isotope composition, and discuss the feasibility of ENDOR measurements also in this system.

Determining the quantum-coherent to semiclassical transition in atomic-scale quasi-one-dimensional metals

Individual dopants in semiconductors are attracting considerable attention due to the prospect of using them in a wide range of applications. In particular, phosphorus donors in silicon have attracted significant interest owing to their weak interaction with the host crystal. However, harnessing their attributes toward the construction of scalable circuitry will require low resistive interconnects at a comparable scale as the dopant atoms. In this Rapid Communication, the authors investigate the transition from quantum coherent to the semiclassical diffusive transport in a 4.6-nm quasi-one-dimensional Si:P metal wire. Analyzing the temperature dependence of universal conductance fluctuations (UCF) the authors show that electron transport evolves from a quantum coherent to a semiclassical regime at temperatures as low as ~4 K and confirm that concepts of UCF and weak localization remain valid in metallic conductors at the atomic-scale.

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 T1 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.

Radio frequency measurements of tunnel couplings and singlet–triplet spin states in Si:P quantum dots

Spin states of the electrons and nuclei of phosphorus donors in silicon are strong candidates for quantum information processing applications given their excellent coherence times. Designing a scalable donor-based quantum computer will require both knowledge of the relationship between device geometry and electron tunnel couplings, and a spin readout strategy that uses minimal physical space in the device. Here we use radio frequency reflectometry to measure singlet–triplet states of a few-donor Si:P double quantum dot and demonstrate that the exchange energy can be tuned by at least two orders of magnitude, from 20 μeV to 8 meV. We measure dot–lead tunnel rates by analysis of the reflected signal and show that they change from 100 MHz to 22 GHz as the number of electrons on a quantum dot is increased from 1 to 4. These techniques present an approach for characterizing, operating and engineering scalable qubit devices based on donors in silicon.

Impact of nuclear spin dynamics on electron transport through donors

We present an analysis of electron transport through two weakly coupled precision placed phosphorus donors in silicon. In particular, we examine the (1,1) to (0,2) charge transition where we predict a new type of current blockade driven entirely by the nuclear spin dynamics. Using this nuclear spin blockade mechanism we devise a protocol to readout the state of single nuclear spins using electron transport measurements only. We extend our model to include realistic effects such as Stark shifted hyperfine interactions and multi-donor clusters. In the case of multi-donor clusters we show how nuclear spin blockade can be alleviated allowing for low magnetic field electron spin measurements.

Broadband electrically detected magnetic resonance using adiabatic pulses

We present a broadband microwave setup for electrically detected magnetic resonance (EDMR) based on microwave antennae with the ability to apply arbitrarily shaped pulses for the excitation of electron spin resonance (ESR) and nuclear magnetic resonance (NMR) of spin ensembles. This setup uses non-resonant stripline structures for on-chip microwave delivery and is demonstrated to work in the frequency range from 4MHz to 18GHz. π pulse times of 50ns and 70μs for ESR and NMR transitions, respectively, are achieved with as little as 100mW of microwave or radiofrequency power. The use of adiabatic pulses fully compensates for the microwave magnetic field inhomogeneity of the stripline antennae, as demonstrated with the help of BIR4 unitary rotation pulses driving the ESR transition of neutral phosphorus donors in silicon and the NMR transitions of ionized phosphorus donors as detected by electron nuclear double resonance (ENDOR).

Optimal signal processing for continuous qubit readout

The measurement of a quantum two-level system, or a qubit in modern terminology, often involves an electromagnetic field that interacts with the qubit, before the field is measured continuously and the qubit state is inferred from the noisy field measurement. During the measurement, the qubit may undergo spontaneous transitions, further obscuring the initial qubit state from the observer. Taking advantage of some well known techniques in stochastic detection theory, here we propose a novel signal processing protocol that can infer the initial qubit state optimally from the measurement in the presence of noise and qubit dynamics. Assuming continuous quantum-nondemolition measurements with Gaussian or Poissonian noise and a classical Markov model for the qubit, we derive analytic solutions to the protocol in some special cases of interest using It\={o} calculus. Our method is applicable to multi-hypothesis testing for robust qubit readout and relevant to experiments on qubits in superconducting microwave circuits, trapped ions, nitrogen-vacancy centers in diamond, semiconductor quantum dots, or phosphorus donors in silicon.

Resonant ionization of shallow donors in electric field

In this paper, we report on our experimental observations of the resonant ionization of a phosphorus donor in silicon in a homogeneous electric field, which is expressed in the sudden rise of the conductivity of the sample at a low temperature when the electric field approaches the critical value of ∼3.2 MV m−1. The effect is discussed in terms of the field-induced interaction of the states using a simplified model based on the effective-mass theory. The results from our model are qualitatively similar to the previously published advanced model base, which is based on the first principles; this predicts the ionization thresholds at approximate fields of 2.45 and 3.25 MV m−1, the latter being in very good agreement with our experiment. The possibility of observing more than one resonance is also discussed.

Robust Two-Qubit Gates for Donors in Silicon Controlled by Hyperfine Interactions

We present two strategies for performing two-qubit operations on the electron spins of an exchange-coupled pair of phosphorus donors in silicon, using the ability to set the donor nuclear spins in arbitrary states. The effective magnetic detuning of the two electron qubits is provided by the hyperfine interaction when the $^{31}$P nuclei are prepared in opposite spin states. This can be exploited to switch on and off SWAP operations with modest tuning of the electron exchange interaction. Furthermore, the hyperfine detuning enables high-fidelity conditional rotation gates based on selective resonant excitation. The latter requires no dynamic tuning of the exchange interaction at all, and offers a very attractive scheme to implement two-qubit logic gates under realistic experimental conditions.

Spin blockade and exchange in Coulomb-confined silicon double quantum dots.

Electron spins confined to phosphorus donors in silicon are promising candidates as qubits because of their long coherence times, exceeding seconds in isotopically purified bulk silicon. With the recent demonstrations of initialization, readout and coherent manipulation of individual donor electron spins, the next challenge towards the realization of a Si:P donor-based quantum computer is the demonstration of exchange coupling in two tunnel-coupled phosphorus donors. Spin-to-charge conversion via Pauli spin blockade, an essential ingredient for reading out individual spin states, is challenging in donor-based systems due to the inherently large donor charging energies (∼45 meV), requiring large electric fields (>1 MV m(-1)) to transfer both electron spins onto the same donor. Here, in a carefully characterized double donor-dot device, we directly observe spin blockade of the first few electrons and measure the effective exchange interaction between electron spins in coupled Coulomb-confined systems.

Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient

Magnetic-field gradients are important for single-site addressability and electric-dipole spin resonance of spin qubits in semiconductor devices. We show that these advantages are offset by a potential reduction in coherence time due to the nonuniformity of the magnetic field experienced by a nuclear-spin bath interacting with the spin qubit. We theoretically study spins confined to quantum dots or at single donor impurities, considering both free-induction and spin-echo decay. For quantum dots in GaAs, we find that, in a realistic setting, a magnetic-field gradient can reduce the Hahn-echo coherence time by almost an order of magnitude. This problem can, however, be resolved by applying a moderate external magnetic field to enter a motional-averaging regime. For quantum dots in silicon, we predict a crossover from non-Markovian to Markovian behavior that is unique to these devices. Finally, for very small systems such as single phosphorus donors in silicon, we predict a breakdown of the common Gaussian approximation due to finite-size effects.

Transport through a single donor in p-type silicon

Single phosphorus donors in silicon are promising candidates as qubits in the solid state. Here, we present low temperature scanning probe microscopy and spectroscopy measurements of individual phosphorus dopants deliberately placed in p-type silicon ∼1 nm below the surface. The ability to image individual dopants combined with scanning tunnelling spectroscopy allows us to directly study the transport mechanism through the donor. We show that for a single P donor, transport is dominated by a minority carrier recombination process with the surrounding p-type matrix. The understanding gained will underpin future studies of atomically precise mapping of donor-donor interactions in silicon.

Spin readout and addressability of phosphorus-donor clusters in silicon

The spin states of an electron bound to a single phosphorus donor in silicon show remarkably long coherence and relaxation times, which makes them promising building blocks for the realization of a solid-state quantum computer. Here we demonstrate, by high-fidelity (93%) electrical spin readout, that a long relaxation time T1 of about 2 s, at B=1.2 T and T≈100 mK, is also characteristic of electronic spin states associated with a cluster of few phosphorus donors, suggesting their suitability as hosts for spin qubits. Owing to the difference in the hyperfine coupling, electronic spin transitions of such clusters can be sufficiently distinct from those of a single phosphorus donor. Our atomistic tight-binding calculations reveal that when neighbouring qubits are hosted by a single phosphorus atom and a cluster of two phosphorus donors, the difference in their electron spin resonance frequencies allows qubit rotations with error rates ≈10(-4). These results provide a new approach to achieving individual qubit addressability.

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.

Effect of pulse error accumulation on dynamical decoupling of the electron spins of phosphorus donors in silicon

Dynamical decoupling (DD) is an efficient tool for preserving quantum coherence in solid-state spin systems. However, the imperfections of real pulses can ruin the performance of long DD sequences. We investigate the accumulation and compensation of different pulse errors in DD using the electron spins of phosphorus donors in silicon as a test system. We study periodic DD sequences (PDD) based on spin rotations about two perpendicular axes, and their concatenated and symmetrized versions. We show that pulse errors may quickly destroy some spin states, but maintain other states with high fidelity over long times. Pulse sequences based on spin rotations about $x$ and $y$ axes outperform those based on $x$ and $z$ axes due to the accumulation of pulse errors. Concatenation provides an efficient way to suppress the impact of pulse errors, and can maintain high fidelity for all spin components: pulse errors do not accumulate (to first order) as the concatenation level increases, despite the exponential increase in the number of pulses. Our theoretical model gives a clear qualitative picture of the error accumulation, and produces results in quantitative agreement with the experiments.

Electrically detected spin echoes of donor nuclei in silicon

The ability to electrically probe the spin properties of solid state systems underlies a wide variety of emerging technologies. Here, we demonstrate the electrical readout of the nuclear spin states of phosphorus donors in silicon in the coherent regime with modified Hahn echo sequences. We find that while the nuclear spins have electrically detected phase coherence times exceeding 2 ms, they are nonetheless limited by the artificially shortened lifetime of the probing donor electron.