Phosphorus Impurities In Silicon Research Articles

Provably Trainable Rotationally Equivariant Quantum Machine Learning

Exploiting the power of quantum computation to realize superior machine learning algorithms has been a major research focus of recent years, but the prospects of quantum machine learning (QML) remain dampened by considerable technical challenges. A particularly significant issue is that generic QML models suffer from so-called barren plateaus in their training landscapes—large regions where cost function gradients vanish exponentially in the number of qubits employed, rendering large models effectively untrainable. A leading strategy for combating this effect is to build problem-specific models that take into account the symmetries of their data in order to focus on a smaller, relevant subset of Hilbert space. In this work, we introduce a family of rotationally equivariant QML models built upon the quantum Fourier transform, and leverage recent insights from the Lie-algebraic study of QML models to prove that (a subset of) our models do not exhibit barren plateaus. In addition to our analytical results we numerically test our rotationally equivariant models on a dataset of simulated scanning tunneling microscope images of phosphorus impurities in silicon, where rotational symmetry naturally arises, and find that they dramatically outperform their generic counterparts in practice. Published by the American Physical Society 2024

Comprehensive understanding on phosphorus precipitation in heavily phosphorus-doped Czochralski silicon

Heavily phosphorus-doped Czochralski (HP-CZ) silicon is an important substrate material for manufacturing power electronic devices. The high concentration of phosphorus impurities may be supersaturated during the crystal growth of HP-CZ silicon or device manufacturing. Thus, understanding phosphorus precipitation in HP-CZ silicon is of technological significance. Herein, a panoramic view of phosphorus precipitation in HP-CZ silicon is presented in terms of crystallography, thermodynamics, and kinetics. It is found that the orthorhombic SiP precipitates can form during the crystal growth of HP-CZ silicon and also during the post-anneals of HP-CZ silicon at 450–1050 °C. Along with increasing annealing temperature, the formed SiP precipitates tend to adopt the platelet, polyhedron, and sphere-like shapes. Moreover, the excess point defects, i.e., silicon self-interstitials and vacancies, are found to affect phosphorus precipitation occurring in the low and high temperature regimes in different ways. In light of the kinetics of phosphorus precipitation at different temperatures, it is deduced that phosphorus precipitation follows a growth law in compliant with Ham's theory to a large extent. As an important output of this work, the temperature-dependent phosphorus solubilities in the dislocation-free silicon, which have been hardly acquired previously, are derived on the basis of investigating phosphorus precipitation in a set of HP-CZ silicon wafers with different phosphorus concentrations. Moreover, the derived solvus line for the phosphorus impurities in silicon could be a beneficial supplement to the existing phase diagram of the Si–P binary system in the extremely silicon-rich corner.

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.

Weak probe readout of coherent impurity orbital superpositions in silicon

Pump-probe spectroscopy is the most common time-resolved technique for investigation of electronic dynamics, and the results provide the incoherent population decay time T1. Here we use a modified pump-probe experiment to investigate coherent dynamics, and we demonstrate this with a measurement of the inhomogeneous dephasing time T2* for phosphorus impurities in silicon. The pulse sequence produces the same information as previous coherent all-optical (photon-echo-based) techniques but is simpler. The probe signal strength is first order in pulse area but its effect on the target state is only second order, meaning that it does not demolish the quantum information. We propose simple extensions to the technique to measure the homogeneous dephasing time T2, or to perform tomography of the target qubit.

Multivalley envelope function equations and effective potentials for phosphorus impurity in silicon

We propose a system of real-space envelope function equations without fitting parameters for modeling the electronic spectrum and wave functions of a phosphorus donor atom embedded in silicon. The approach relies on the Burt-Foreman envelope function representation and leads to coupled effective-mass Schroedinger equations containing smooth effective potentials. These potentials result from the spatial filtering imposed on the exact potential energy matrix elements in the envelope function representation. The corresponding filter function is determined from the definition of the envelope function. The resulting effective potentials and the system of envelope functions jointly reproduce the valley-orbit coupling effect in the doped silicon. Including the valley-orbit coupling not only of the 1s, but also for 2s atomic orbitals, as well as static dielectric screening is found crucial to accurately reproduce experimental data. The measured binding energies are recovered with a maximum relative error of 1.53 %. The computed wave functions are in a good agreement with experimental measurements of the electron density provided by scanning tunneling microscopy.

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.

Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf stars

Laboratory spectroscopy of atomic hydrogen in a magnetic flux density of 10(5) T (1 gigagauss), the maximum observed on high-field magnetic white dwarfs, is impossible because practically available fields are about a thousand times less. In this regime, the cyclotron and binding energies become equal. Here we demonstrate Lyman series spectra for phosphorus impurities in silicon up to the equivalent field, which is scaled to 32.8 T by the effective mass and dielectric constant. The spectra reproduce the high-field theory for free hydrogen, with quadratic Zeeman splitting and strong mixing of spherical harmonics. They show the way for experiments on He and H(2) analogues, and for investigation of He(2), a bound molecule predicted under extreme field conditions.

Violation of a Leggett–Garg inequality with ideal non-invasive measurements

The quantum superposition principle states that an entity can exist in two different states simultaneously, counter to our 'classical' intuition. Is it possible to understand a given system's behaviour without such a concept? A test designed by Leggett and Garg can rule out this possibility. The test, originally intended for macroscopic objects, has been implemented in various systems. However to date no experiment has employed the 'ideal negative result' measurements that are required for the most robust test. Here we introduce a general protocol for these special measurements using an ancillary system, which acts as a local measuring device but which need not be perfectly prepared. We report an experimental realization using spin-bearing phosphorus impurities in silicon. The results demonstrate the necessity of a non-classical picture for this class of microscopic system. Our procedure can be applied to systems of any size, whether individually controlled or in a spatial ensemble.

Electron Spin Decoherence in Isotope-Enriched Silicon

Silicon is promising for spin-based quantum computation because nuclear spins, a source of magnetic noise, may be eliminated through isotopic enrichment. Long spin decoherence times T2 have been measured in isotope-enriched silicon but come far short of the T2=2T1 limit. The effect of nuclear spins on T2 is well established. However, the effect of background electron spins from ever present residual phosphorus impurities in silicon can also produce significant decoherence. We study spin decoherence decay as a function of donor concentration, 29Si concentration, and temperature using cluster expansion techniques specifically adapted to the problem of a sparse dipolarly coupled electron spin bath. Our results agree with the existing experimental spin echo data in Si:P and establish the importance of background dopants as the ultimate decoherence mechanism in isotope-enriched silicon.

Quantum Hall charge sensor for single-donor nuclear spin detection in silicon

We propose a novel optical and electrical hybrid scheme for the measurement of nuclear spin qubits in silicon. By combining the environmental insensitivity of the integer quantum Hall effect with the optically distinguishable hyperfine states of phosphorus impurities in silicon, our system can offer both nuclear spin measurement and robustness against environmental defects. 31P donor spins in isotopically purified 28Si are often discussed as very promising quantum memory qubits due to their extremely long decoherence times, and our proposed device offers an effective implementation for such a quantum memory system.

Electroluminescence at 7 terahertz from phosphorus donors in silicon

Terahertz (THz) emissions corresponding to intracenter transitions of phosphorus impurities in silicon have been observed up to 30K. Electrical pulses (250ns) with a repetition rate of 413Hz were used for excitation, and the peak power was calculated to be ∼20μW∕facet for a 190×120μm2 device with a peak pumping current of 400mA at 12K. THz emission intensity increased linearly with pumping current and quenched when the sample temperature was above 30K. The current–voltage characteristics suggested a conduction and excitation mechanism by injection of electrons from a Schottky barrier followed by impact ionization of the neutral impurities.

Diffusion of boron and phosphorus in silicon during high-temperature ion implantation

A theoretical model is proposed for describing the accelerated diffusion of boron and phosphorus impurity in silicon during high-temperature implantation. The model takes into account the production and diffusion of impurities and point defects and the formation and decomposition of defect-impurity pairs. The formation of dislocations and an increase (decrease) of their size during bombardment is taken into account. Dislocations are sinks for defects and impurities. The computed profiles are compared with existing data. It is shown that dislocations play an important role as trapping centers. Oscillatory behavior of the impurity profile was observed near the surface; this behavior is due to impurity implantation and diffusion processes occurring simultaneously.

Dielectric screening in semiconductors.

Intra-atomic and interatomic Coulomb interactions are incorporated into bond-orbital theory, based upon universal tight-binding parameters, in order to treat the effects of charge redistribution in semiconductor bonds. The dielectric function \ensuremath{\epsilon}(q) is obtained for wave numbers in a [100] direction. The screening of differences in average hybrid energy across a heterojunction is calculated in detail, indicating that the decay length for the potential depends upon the relative values of Madelung and intra-atomic Coulomb terms. The parameters used here predict an imaginary decay length and thus an oscillating potential near the interface. The same theory is applied to point defects by imbedding a cluster in a matrix lattice, taking charges in that lattice to be consistent with continuum theory. Illustrating the theory with a phosphorus impurity in silicon, it is seen that the impurity and its neighboring atoms have charges on the order of only one-tenth of an electronic charge, alternating in sign from neighbor to neighbor as for planar defects. Although there are shifts in the term values on the order of a volt, the difference in these shifts for neighboring atoms is much smaller so that the effect on the bonds is quite small. This behavior is analogous to the response of a dielectric continuum to a point charge: The medium is locally neutral except at the center of the cluster and there are slowly varying potentials ${e}^{2}$/\ensuremath{\epsilon}r. Because of this slow variation, free-atom term values should ordinarily suffice for the calculation of bond properties and bond lengths at impurities. Corrections are larger for homovalent substitutions such as carbon in silicon.

Resolved fine structure of exciton complexes bound to phosphorus impurities in silicon : R. R. Parsons. Solid St. Commun.22, 671 (1977)

Resolved fine structure of exciton complexes bound to phosphorus impurities in silicon : R. R. Parsons. Solid St. Commun.22, 671 (1977)

Resolved fine structure of exciton complexes bound to phosphorus impurities in silicon

A Fabry-Pérot interferometer is used to observe resolved fine structure for two of the luminescence lines associated with exciton complexes bound to phosphorus impurities in silicon. The results are interpreted in terms of j-j couplings for the electrons and holes and give support for the recently proposed shell-model. Linewidths are obtained for the four luminescence lines associated with 1-, 4-, 5-, and 6- exciton complexes. The linewidths of the 4-, 5-, and 6- exciton peaks are found to be considerably broader than that of the single bound exciton.

New results on the Zeeman effect of phosphorus in silicon

We report variational calculations of the quadratic Zeeman effect of phosphorus impurity in silicon, which are compared to our experimental data for B⊥ E and B⊥ E. It is shown that a small variation of the ratio γ of the transverse and effective masses can lead to a good agreement between theory and experiment for the 2 p 0 line. The shift of the central component of the 2 p ± line has been found to depend on the polarization of the incident radiation and this fact has not yet been explained. It is also shown that the shift observed for the line ascribed previously to a 1 s( A 1)→3 d 0 transition can be consistent with this attribution, giving experimental evidence of its electronic nature.

On the Change of Activation Energy with Impurity Concentration in Semiconductors

The existing theories of change in activation energy with impurity concentration are discussed and their limitations pointed out. The Ionization energy of a pair of impurities in a dielectric medium is calculated as a function of impurity separation Assuming a random distribution of impurities, an averaging process is employed to calculate the mean change in activation energy with concentration. This approximation should be poor for high impurity densities but good for low densities. Specific calculations have been made for the case of phosphorus impurities in silicon and satisfactory agreement is obtained with experimental results