Central-cell Corrections Research Articles

Room Temperature Quantum Control of N‐Donor Electrons at Si/SiO2Interface

AbstractThe manuscript theoretically discusses various important aspects for donor atom based single qubit operations in silicon (Si) quantum computer architecture at room temperature using a single nitrogen (N) deep‐donor close to the Si/SiO2interface. Quantitative investigation of room temperature single electron shuttling between a single N‐donor atom and the interface is the focus of attention under the influence of externally applied electric and magnetic field. To apprehend the realistic experimental configurations, central cell correction along with effective mass approach is adopted throughout the study. Furthermore, a detailed discussion currently explores the significant time scales implicated in the process and their suitability for experimental purposes. Theoretical estimates are also provided for all the external fields required to successfully achieve coherent single electron shuttling and their stable maintenance at the interface as required. The results presented in this work offer practical guidance for quantum electron control using N‐donor atoms in Si at room temperature.

Central-cell corrections for hydrogenic, silicon (Si), selenium (Se), sulfur (S), and germanium (Ge) donor impurities and pressure–temperature effects on the optical properties of the GaAs/GaAlAs multi-layer quantum disk

GaAs/GaAlAs quantum dots can be doped with various impurities to modify their optoelectronic properties. Common impurities used for doping include: Hydrogenic impurity (HI), Silicon (Si), Selenium (Se), Germanium (Ge), and Sulfur (S) are common n-type dopant in GaAs-based materials. This work has investigated the effects of temperature and pressure on the refractive index changes (RICs) and optical absorption coefficients (OACs) in the multi-layer quantum disk, accounting for central cell correction. We have used the finite element method to calculate the eigenvalues and their corresponding wave-functions. These values are utilized in the computation of the OACs and RICs. Our outcomes can be summarized as follows: (i) The applied pressure induces a red-shift of the OACs peaks, and their magnitude is reduced. Similarly, temperature causes a blue-shift of the OACs peaks, and their magnitude is enhanced. (ii) The presence and type of impurity lead to significant changes in both OACs and RICs.

Gaussian-process-regression-based method for the localization of exceptional points in complex resonance spectra

Resonances in open quantum systems depending on at least two controllable parameters can show the phenomenon of exceptional points (EPs), where not only the eigenvalues but also the eigenvectors of two or more resonances coalesce. Their exact localization in the parameter space is challenging, in particular in systems, where the computation of the quantum spectra and resonances is numerically very expensive. We introduce an efficient machine learning algorithm to find EPs based on Gaussian process regression (GPR). The GPR-model is trained with an initial set of eigenvalue pairs belonging to an EP and used for a first estimation of the EP position via a numerically cheap root search. The estimate is then improved iteratively by adding selected exact eigenvalue pairs as training points to the GPR-model. The GPR-based method is developed and tested on a simple low-dimensional matrix model and then applied to a challenging real physical system, viz., the localization of EPs in the resonance spectra of excitons in cuprous oxide in external electric and magnetic fields. The precise computation of EPs, by taking into account the complete valence band structure and central-cell corrections of the crystal, can be the basis for the experimental observation of EPs in this system.

Simple accurate model of silicon donor arrays

Donors in silicon can now be positioned with an accuracy of about one lattice constant, making it possible to form donor arrays for quantum computation or quantum simulation applications. However, the multivalley character of the silicon conduction band combines with central cell corrections to the donor states to convert small atomic scale imperfections in donor placement into strong interdonor hybridization disorder. We present a simple model that is able to account for central cell corrections accurately, and use it to assess the impact of donor positional disorder on donor array properties in both itinerant and localized limits. Our results show that donor arrays in silicon simulate strongly disordered one-dimensional electrons.

Recent progress in atomistic modelling and simulations of donor spin qubits in silicon

Electron or nuclear spins associated with dopant atoms, such as phosphorus impurities in silicon (Si:P), have been shown to form excellent qubits with promising potential for scale-up towards a fault-tolerant quantum computer architecture. The remarkable progress in the design and characterisation of Si:P qubits and quantum gates has been led by recent experimental demonstrations. Equally importantly, advances in theoretical modelling and simulations over a number of years have underpinned the experimental efforts through the fundamental understanding of dopant physics and by providing crucial interpretation of the experimental evidence. This brief review article provides highlights of our research on developing atomistic theoretical methods and their application to the understanding, characterisation and scale-up of Si:P qubits in silicon. We have established a state-of-the-art theoretical framework which is capable of performing electronic structure simulations over millions of atoms. This includes a comprehensive set of central-cell corrections within atomistic tight-binding theory to simulate dopant energy spectra and electronic wave functions with high precision. When integrated with Bardeen’s tunnelling formalism and Chen’s derivative rule, the theoretical simulations were able to reproduce the measured spatially resolved scanning tunnelling microscope (STM) images of dopant wave functions, providing an unprecedented access to the dopant physics in silicon. A systematic examination of the STM image features (brightness and symmetry) allowed pinpointing of the dopant atom positions in silicon lattice with an exact atom precision and for dopant depths up to 5 nm below the silicon surface. The scale-up of the metrology technique was demonstrated by training a machine learning algorithm such as convolutional neural network. For the design and implementation of high-fidelity two-qubit quantum gates, we investigated exchange interaction between dopant pairs and showed that the application of a small lattice strain could provide a full control in the presence of one-lattice site donor position variations. The state-of-the-art computational capability developed by our team is a culmination of more than five years of research efforts – it has been well-benchmarked against several different experimental measurements and is expected to play an important role in design and characterisation of quantum gates and scale-up architectures in the coming years.

Exchange interaction in the yellow exciton series of cuprous oxide

We experimentally and numerically investigate the exchange interaction of the yellow excitons in cuprous oxide. By varying the material parameters in the numerical calculations, we can interpret experimental findings and understand their origin in the complex band structure and central-cell corrections. In particular, we experimentally observe the reversal of the ortho- and paraexciton for the $2S$ yellow exciton, and explain this phenomenon by an avoided crossing with the green $1S$ orthoexciton in a detailed numerical analysis. Furthermore, we discuss the exchange splitting as a function of the principal quantum number $n$ and its deviation from the $n^{-3}$ behavior expected from a hydrogenlike model. We also explain why the observed exchange splitting of the green $1S$ exciton is more than twice the splitting of the yellow $1S$ state.

Exciton mass and exciton spectrum in the cuprous oxide

Excitons with a radius of a few lattice constants can be affected by strong central-cell corrections, leading to significant deviations of the optical spectrum from the hydrogen-like Rydberg series, and also to an enhancement of the exciton mass. We present an approach to this situation based on a lattice model that incorporates the effects of a non-parabolic band structure, short distance corrections to the Coulomb interaction between electrons and holes, spin–orbit and exchange coupling. The lattice model allows for observation of the crossover from large radius Wannier to small-radius Frenkel excitons without invoking a continuum approximation. We apply the lattice model approach especially to the yellow exciton series in the cuprous oxide, for which the optical spectrum and exciton mass enhancement are obtained through adaptation of only a few model parameters to material-specific values. Our results predict a strongly anisotropic ortho-exciton mass.

Exciton-polaritons in cuprous oxide: Theory and comparison with experiment

The observation of giant Rydberg excitons in cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ up to a principal quantum number of $n=25$ by T.~Kazimierczuk \emph{et al.} [Nature \textbf{514}, 343, (2014)] inevitably raises the question whether these quasi-particles must be described within a multi-polariton framework since excitons and photons are always coupled in the solid. In this paper we present the theory of exciton-polaritons in $\mathrm{Cu_{2}O}$. To this end we extend the Hamiltonian which includes the complete valence band structure, the exchange interaction, and the central-cell corrections effects, and which has been recently deduced by F.~Schweiner \emph{et al.} [Phys.~Rev.~B \textbf{95}, 195201, (2017)], for finite values of the exciton momentum $\hbar K$. We derive formulas to calculate not only dipole but also quadrupole oscillator strengths when using the complete basis of F.~Schweiner \emph{et al.}. Very complex polariton spectra for the three orientations of $\boldsymbol{K}$ along the axes $[001]$, $[110]$, and $[111]$ of high symmetry are obtained and a strong mixing of exciton states is reported. The main focus is on the $1S$ ortho exciton-polariton, for which pronounced polariton effects have been measured in experiments. We set up a $5\times 5$ matrix model, which accounts for both the polariton effect and the $K$-dependent splitting, and which allows treating the anisotropic polariton dispersion for any direction of $\boldsymbol{K}$. We especially discuss the dispersions for $\boldsymbol{K}$ being oriented in the planes perpendicular to $[1\bar{1}0]$ and $[111]$, for which experimental transmission spectra have been measured. Furthermore, we compare our results with experimental values of the $K$-dependent splitting, the group velocity, and the oscillator strengths of this exciton-polariton.

Ground-state splitting of ultrashallow thermal donors with negative central-cell corrections in silicon

Ultrashallow thermal donors (USTDs), which consist of light element impurities such as carbon, hydrogen, and oxygen, have been found in Czochralski silicon (CZ Si) crystals. To the best of our knowledge, these are the shallowest hydrogen-like donors with negative central-cell corrections in Si. We observed the ground-state splitting of USTDs by far-infrared optical absorption at different temperatures. The upper ground-state levels are approximately 4 meV higher than the ground-state levels. This energy level splitting is also consistent with that obtained by thermal excitation from the ground state to the upper ground state. This is direct evidence that the wave function of the USTD ground state is made up of a linear combination of conduction band minimums.

Even exciton series in Cu2O

In $\mathrm{Cu_{2}O}$ parity is a good quantum number and thus the exciton spectrum falls into two parts: The dipole-active exciton states of negative parity and odd angular momentum, which can be observed in one-photon absorption ($\Gamma_4^-$ symmetry) and the exciton states of positive parity and even angular momentum, which can be observed in two-photon absorption ($\Gamma_5^+$ symmetry). The unexpected observation of $D$ excitons in two-photon absorption and of $F$ excitons in one-photon absorption has given evidence that the dispersion properties of the $\Gamma_5^+$ orbital valence band is giving rise to a coupling of the yellow and green exciton series. In this paper we show that the even and odd parity exciton system can be consistently described within the same theoretical approach. However, the Hamiltonian of the even parity system needs, in comparison to the odd exciton case, modifications to account for the very small radius of the $1S$ exciton. In the presented treatment we take special care of the central-cell corrections, which comprise a reduced screening of the Coulomb potential at distances comparable to the polaron radius, the exchange interaction being responsible for the exciton splitting into ortho and para states, and the inclusion of terms in the fourth power of $p$ in the kinetic energy. Since the yellow $1S$ exciton state is coupled to all other states of positive parity, we show how the central-cell corrections affect the whole even exciton series. The close resonance of the $1S$ green exciton with states of the yellow exciton series has a strong impact on the energies and oscillator strengths of all implied states. The consistency between theory and experiment with respect to energies and oscillator strengths for the even and odd exciton system in $\mathrm{Cu_{2}O}$ is a convincing proof for the validity of the applied theory.

Donors in Ge as qubits —Establishing physical attributes

Quantum electronic devices at the single-impurity level demand the understanding of the physical attributes of dopants with an unprecedented accuracy. Germanium-based technologies have been developed recently, creating the necessity to adapt the latest theoretical tools to the unique electronic structure of this material. We investigate basic properties of donors in Ge which are not known experimentally, but are indispensable for qubit implementations. Our approach provides a description of the wave function at multiscale, associating microscopic information from density functional theory and envelope functions from state-of-the-art multivalley effective mass calculations, including a central-cell correction designed to reproduce the energetics of all group-V donor species (P, As, Sb and Bi). With this formalism, we predict the binding energies of negatively ionized donors ( state). Furthermore, we investigate the signatures of buried donors to be expected from scanning tunneling microscopy (STM). The naive assumption that attributes of donor electrons in other semiconductors may be extrapolated to Ge is shown to fail, similarly to earlier attempts to recreate in Si qubits designed for GaAs. Our results suggest that the mature techniques available for qubit realizations may be adapted to germanium to some extent, but the peculiarities of the Ge band structure will demand new ideas for fabrication and control.

Linewidths in excitonic absorption spectra of cuprous oxide

We present a theoretical calculation of the absorption spectrum of cuprous oxide $\left({\mathrm{Cu}}_{2}\mathrm{O}\right)$ based on the general theory developed by Y. Toyozawa. An inclusion not only of acoustic phonons but also of optical phonons and of specific properties of the excitons in ${\mathrm{Cu}}_{2}\mathrm{O}$ like the central-cell corrections for the $1S$ exciton allows us to calculate the experimentally observed linewidths in experiments by T. Kazimierczuk et al. [T. Kazimierczuk, D. Fr\"ohlich, S. Scheel, H. Stolz, and M. Bayer, Nature (London) 514, 343 (2014)] within the same order of magnitude, which demonstrates a clear improvement in comparison to earlier work on this topic. We also discuss a variety of further effects, which explain the still observable discrepancy between theory and experiment but can hardly be included in theoretical calculations.

Electronic spectrum of non-tetrahedral acceptors in CdTe:Cl and CdTe:Bi,Cl single crystals

The electronic spectra of complex acceptors in compensated CdTe:Cl, CdTe:Ag,Cl, and CdTe:Bi,Cl single crystals are studied using low-temperature photoluminescence (PL) measurements under both nonresonant and resonant excitation of distant donor–acceptor pairs (DAP). The wavelength modulation of the excitation source combined with the analysis of the differential PL signal is used to enhance narrow spectral features obscured because of inhomogeneous line broadening and/or excitation transfer for selectively excited DAPs. For the well-known tetrahedral (TD) AgCd acceptor, the energies of four excited states are measured, and the values obtained are shown to be in perfect agreement with the previous data. Moreover, splitting between the 2P3/2 (Г8) and 2S3/2 (Г8) states is clearly observed for AgCd centers located at a short distance (5–7 nm) from a hydrogen-like donor (ClTe). This splitting results from the reduction of the TD symmetry taking place when the acceptor is a member of a donor–acceptor pair. For the Cl-related complex acceptor with an activation energy of ∼121 meV (A-center), the energies of eight excited states are measured. It is shown that this defect produces low-symmetry central-cell correction responsible for the strong splitting of S-like TD shells. The energy spectrum of the Bi-related shallow acceptor with an activation energy of ∼36 meV is measured as well. The spectrum obtained differs drastically from the hydrogen-like set of levels, which indicates the existence of repulsive low-symmetry perturbation of the hydrogen-like Coulomb potential. It is also shown that the spectra of selectively excited PL recorded for a macroscopic ensemble of distant donor–acceptor pairs allow one to detect the low symmetry of acceptors of a given type caused by their complex nature or by the Jahn–Teller distortion. This method does not require any additional (external) field and is applicable to acceptors in diverse zinc-blende compound semiconductors.

Multivalley effective mass theory simulation of donors in silicon

Last year, Salfi et al. made the first direct measurements of a donor wave function and found extremely good theoretical agreement with atomistic tight-binding [Salfi et al., Nat. Mater. 13, 605 (2014)]. Here, we show that multi-valley effective mass theory, applied properly, does achieve close agreement with tight-binding and hence gives reliable predictions. To demonstrate this, we variationally solve the coupled six-valley Shindo-Nara equations, including silicon's full Bloch functions. Surprisingly, we find that including the full Bloch functions necessitates a tetrahedral, rather than spherical, donor central cell correction to accurately reproduce the experimental energy spectrum of a phosphorus impurity in silicon. We cross-validate this method against atomistic tight-binding calculations, showing that the two theories agree well for the calculation of donor-donor tunnel coupling. Further, we benchmark our results by performing a statistical uncertainty analysis, confirming that derived quantities such as the wave function profile and tunnel couplings are robust with respect to variational energy fluctuations. Finally, we apply this method to exhaustively enumerate the tunnel coupling for all donor-donor configurations within a large search volume, demonstrating conclusively that the tunnel coupling has no spatially stable regions. Though this instability is problematic for reliably coupling donor pairs for two-qubit operations, we identify specific target locations where donor qubits can be placed with scanning tunneling microscopy technology to achieve reliably large tunnel couplings.

Strain and electric field control of hyperfine interactions for donor spin qubits in silicon

Control of hyperfine interactions is a fundamental requirement for quantum computing architecture schemes based on shallow donors in silicon. However, at present, there is lacking an atomistic approach including critical effects of central-cell corrections and non-static screening of the donor potential capable of describing the hyperfine interaction in the presence of both strain and electric fields in realistically sized devices. We establish and apply a theoretical framework, based on atomistic tight-binding theory, to quantitatively determine the strain and electric field dependent hyperfine couplings of donors. Our method is scalable to millions of atoms, and yet captures the strain effects with an accuracy level of DFT method. Excellent agreement with the available experimental data sets allow reliable investigation of the design space of multi-qubit architectures, based on both strain-only as well as hybrid (strain+field) control of qubits. The benefits of strain are uncovered by demonstrating that a hybrid control of qubits based on (001) compressive strain and in-plane (100 or 010) fields results in higher gate fidelities and/or faster gate operations, for all of the four donor species considered (P, As, Sb, and Bi). The comparison between different donor species in strained environments further highlights the trends of hyperfine shifts, providing predictions where no experimental data exists. Whilst faster gate operations are realisable with in-plane fields for P, As, and Sb donors, only for the Bi donor, our calculations predict faster gate response in the presence of both in-plane and out-of-plane fields, truly benefiting from the proposed planar field control mechanism of the hyperfine interactions.

Confinement in thickness-controlled GaAs polytype nanodots.

Polytype nanodots are arguably the simplest nanodots than can be made, but their technological control was, up to now, challenging. We have developed a technique to produce nanowires containing exactly one polytype nanodot in GaAs with thickness control. These nanodots have been investigated by photoluminescence, which has been cross-correlated with transmission electron microscopy. We find that short (4-20 nm) zincblende GaAs segments/dots in wurtzite GaAs confine electrons and that the inverse system confines holes. By varying the thickness of the nanodots we find strong quantum confinement effects which allows us to extract the effective mass of the carriers. The holes at the top of the valence band have an effective mass of approximately 0.45 m0 in wurtzite GaAs. The thinnest wurtzite nanodot corresponds to a twin plane in zincblende GaAs and gives efficient photoluminescence. It binds an exciton with a binding energy of roughly 50 meV, including central cell corrections.

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, D0, D−, and 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.

Donor hyperfine Stark shift and the role of central-cell corrections in tight-binding theory

Atomistic tight-binding (TB) simulations are performed to calculate the Stark shift of the hyperfine coupling for a single arsenic (As) donor in silicon (Si). The role of the central-cell correction is studied by implementing both the static and the non-static dielectric screenings of the donor potential, and by including the effect of the lattice strain close to the donor site. The dielectric screening of the donor potential tunes the value of the quadratic Stark shift parameter (η2) from −1.3 × 10−3 µm2 V−2 for the static dielectric screening to −1.72 × 10−3 µm2 V−2 for the non-static dielectric screening. The effect of lattice strain, implemented by a 3.2% change in the As–Si nearest-neighbour bond length, further shifts the value of η2 to −1.87 × 10−3 µm2 V−2, resulting in an excellent agreement of theory with the experimentally measured value of −1.9 ± 0.2 × 10−3 µm2 V−2. Based on our direct comparison of the calculations with the experiment, we conclude that the previously ignored non-static dielectric screening of the donor potential and the lattice strain significantly influence the donor wave function charge density and thereby leads to a better agreement with the available experimental data sets.

An improved effective-mass-theory equation for phosphorus doped in silicon

A new multi-valley effective-mass-theory (EMT) equation is derived for the phosphorus doped in silicon. This equation admits solutions which agree with the measured ground state energy and the square modulus of the ground-state wavefunction |ΨA1(0)|2 at the donor site accurately. This avoids the use of the so-called “central-cell correction” approximation method to calculate the hyperfine constant at the donor site. Furthermore, the energy levels for the upper lying states of T2 and E can also be predicted relatively accurately. The newly derived EMT equation has applications in the characterization of semiconductor or spintronics devices.

Modification to the central-cell correction of germanium acceptors

In this paper, we report a correction to the model potential of the Ga acceptor in germanium, evidenced by high-magnetic-field photoconductivity measurements. We found that under high magnetic fields the chemical shift of the binding energy of Ga acceptors vanishes, contrary to the results given by the generally accepted theory. To fit our data, we found that the central-cell correction should contain a repulsive part (i.e., it must be bipolar), in contrast to the purely attractive screened point-charge potential widely used in the literature.