Rising Stars in Computational Materials Science
Recent progress in atomistic modelling and simulations of donor spin qubits in silicon

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Abstract

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.

Introduction

In his famous lecture in 1982, Richard Feynman proposed the idea of a computing machine which can exploit quantum mechanical principles to simulate complex physical phenomena at atomic and molecular scale [1]. He conceptualized that nature is not classical and therefore a computing machine based on quantum mechanical properties such as entanglement, superposition and interference are needed to efficiently simulate the natural phenomena. This postulation is considered one of the early conceptions of a quantum computing device. In the following years, it turned out that quantum algorithms, such as Shor’s [2] and Grover’s [3], can be designed which could expand the reach of quantum computers beyond just the Feynman’s idea of simulation of quantum systems and towards mainstream computing tasks such as search in large unstructured databases or finding prime factors of large integer numbers with implications for cybersecurity. Today, hundreds of quantum algorithms have been proposed, indicating that quantum computing may offer efficient solutions to many computationally intensive problems, which are intractable on today’s supercomputers [4], [5]. In 2019, the Google Quantum AI team demonstrated quantum supremacy experiment by showing that their quantum computer device can perform certain tasks much more efficiently than any classical machine, marking a significant milestone in the field of quantum computing [6]. This has initiated a global race to achieve quantum advantage i.e. to solve a real-world useful problem on a quantum computer, which is intractable on a classical machine.

Quantum computing offers promising prospects for solving many complex real-life problems, with potential to impact research areas such as drug design, clean energy, industrial chemical manufacturing, big data science, and particle physics among others. Therefore, stakes to build a large-scale quantum computer are high. The initial progress to build quantum machines was very slow. It turned out that the quantum mechanical effects such as superposition and entanglement, which make a quantum machine highly powerful, are not only highly challenging to measure, control and manipulate, but are also very fragile and disappear very quickly in the presence of any noise or interaction with environment – the phenomena known as decoherence. Remarkable technological advancements in fabrication and quantum control over the last couple of decades have now enabled the development of small to medium sized quantum devices consisting of 50–100 qubits [7]. These current generation of quantum devices also known as near-term intermediate-scale noisy quantum (NISQ) devices [8] due to yet relatively low but consistently improving fidelities offer glimpse of a bright future for quantum computing and provide a test-bench for quantum algorithm development and benchmarking. The tremendous progress in the area of quantum error correction algorithms is also expected to play an important role in overcoming noise and boosting the fidelities of the available NISQ devices, allowing them to tackle larger size of problems commensurate with real-world applications [9].

The demonstration of quantum advantage may still require significant further technological advancement, however a global race to build a first fault-tolerant universal quantum computer is already under-way. Among many material platforms being pursued for the implementation of qubits and quantum gates, phosphorus impurity spin qubits in silicon (Si:P) are a promising candidate. In 1998, Kane proposed the idea of building a quantum computing device based on an array of phosphorus nuclear spin qubits fabricated in silicon [10]. This led to a remarkable progress in the fabrication and control of Si:P qubits in the following years. The Si:P qubits offer very long coherence times and promising prospect for scalability towards a large-scale fault-tolerant architecture [11]. In the last decade, significant progress has been made towards design and characterisation of Si:P spin qubits, and some of the key milestones achieved include the single shot spin readout [12], the demonstration of single qubits based on both electron [13] and nuclear [14] spins, the fabrication of donors in silicon lattice with atomic-level precision [15], [16], the post-fabrication pinpointing of their locations in silicon with exact lattice site precision [17], the two electron coherent spin oscillations [18], and a direct two-electron fast SWAP operation [19]. The amazing experimental progress has been supported and guided by major contributions from the advancements in theoretical understanding of the underpinning Si:P donor physics. A number of theoretical research groups around the world have developed modelling and simulation techniques, which have provided new physical insights and helped in the design of Si:P spin qubit devices. This article does not aim to provide a comprehensive review of the complete theoretical literature which is highly rich and diverse, rather we primarily focus on the key aspects of atomistic modelling efforts performed by our group at the University of Melbourne during the course of the last five years and highlight key related work from the other research groups.

To accurately model donor physics in silicon, we have develop a state-of-the-art theoretical framework, which can simulate phosphorus donor qubits in silicon based on multi-million-atom calculations of electron wave functions [20]. The theoretical model incorporated a comprehensive set of central-cell-corrections within the atomistic tight-binding (TB) theory which included donor potential cut-off at donor site with its long-range tail screened by non-static dielectric screening and bond-length deformation around the donor atom. The TB model reproduced a set of experimental measurements on donor energy spectra [20], the hyperfine Stark shift [21], the strain induced hyperfine shift [21], the electron g-factor in strained silicon environments [22], and the spatial profiles of the donor wave functions in real-space [17]. In the latter case, by coupling the donor wave function simulations with Bardeen’s tunnelling theory [23] and Chen’s derivative rule [24], we computed scanning tunnelling microscope (STM) images of donor wave functions [17] which demonstrated an unprecedented agreement with the experimental STM measurements [25]. Our work discovered that the brightness and spatial symmetry of the features in STM images were directly related to the position of phosphorus donor atoms inside the silicon crystal lattice. This insight led to the formulation of an atomic-precision spatial metrology technique for donor atoms up to 5 nm below the silicon surface [17]. The established metrology technique is expected to make important contribution in the design of high-fidelity two-qubit exchange-based quantum gates, as the determination of the exact donor atom locations may lead to characterisation of the exchange interaction between the donor pairs with a high-level of accuracy. We also theoretically proposed that the STM images of the shallow donor depths could provide a way to directly visualise the role of central-cell effects on the wave function spatial profiles [26].

The recent focus of our work has been on the design of high-fidelity two-qubit quantum gates and scale-up schemes. The calculation of exchange interaction between donor pairs is a computationally intensive problem and the early work on the computation of exchange interaction strength was performed by Cullis and Marko [27]. Subsequently, many research groups have theoretically investigated the exchange interaction between donor pairs in silicon in the context of design and implementation of high fidelity qubit gates [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. It is also well-known that the exchange interaction between two donor atoms is a function of their spatial locations in the silicon crystal [38], and any small variations, even of the order of one lattice site, could lead to large variations in the exchange strength. This presents a significant challenge towards implementation of high-fidelity two-qubit exchange-based quantum gates. To mitigate the effect of variation in exchange on quantum gate fidelities, the techniques already developed in the literature include the design of composite pulse schemes [42], [38], the electric field tuning [43], the application of lattice strain [37], [44], [35], [31], [33] and donor placement along the [110] crystal direction in silicon lattice [85]. In this article, we have mainly focused to review the role of external strain for exchange interaction control within the context of atomistic tight-binding framework and STM donor placement strategies. It was shown that the application of a small lattice strain (5%) can lead to an order of magnitude increase in the exchange strength [86]. The application of strain also brings uniformity in exchange with respect to the donor position axis (for example similar exchange strengths were computed along the [100] and [110] directions). Importantly, we showed that the strain drastically suppresses the exchange variations reducing them to within a factor of 5. This when accompanied with a large electric field tuning is expected to help design high-fidelity two-qubit gates irrespective of the one lattice size donor position variations. Furthermore, the strain enhanced exchange strength could allow donor placements at much larger distances (20–30 nm) which are commensurate with the pitch requirements of a large-scale surface code quantum computer architecture [11].

The true potential of quantum computers could only be realised when a large-scale fault-tolerant universal quantum computer will be available. This may require significant technological development in the next few years, however proposals for scale-up of qubit systems are being actively developed to guide experimental directions. For Si:P qubits, Kane proposed a scale-up scheme based on a linear array of phosphorus donor atoms controlled by hyperfine (A) and exchange (J) control gates [10]. In recent years, several other proposals have been developed [11], [45], [46], [47]. Among these, Hill et al. [11] reported a surface-code architecture scheme which was based on dipole couplings as the two-qubit interactions. More recently, exchange interaction has been incorporated in the surface-code architecture scheme [48], which allows faster two-qubit operations (O(μsec)) while donor placement commensurate with the pitch requirements (20 nm).

One of the key challenges in the design and operation of any large-scale architecture scheme is expected to be related to post-fabrication characterisation of millions of qubits. For Si:P qubits, we have formulated a theoretical framework, which can use a machine learning technique to find, autonomously and with high-throughput, donor locations and count by classifying STM images of qubit wave functions [49]. The training of a convolutional neural network (CNN) based on 100,000 computed STM images of qubit wave functions acquired a learning accuracy of above 99%. When tested over 17,600 computed images with simulated noise, the trained CNN characterised qubits with more than 98% fidelity, proposing that a CNN in conjunction with STM fabrication [15] and image set-up [25] could provide a reliable, fast and autonomous way to characterise qubits in large-scale architectures.

The global race to build a first large-scale fault-tolerant quantum computer now includes a large number of industry and academic groups working on many different qubit systems including trapped ions, superconductors, and spin qubits in silicon (both atomic qubits and silicon quantum dot qubits). The Si:P spin qubits in silicon offer some of the most promising properties such as long coherence times, potential for scalability and advanced manufacturing techniques inherited from decades long microelectronic industry. Our theoretical work in the last five years has provided significant new insights towards understanding the fundamental physics underpinning the Si:P qubits and developed new computational techniques to characterise two-qubit interactions. The highly accurate donor wave function modelling, the atomic-precision autonomous spatial metrology of donor atoms in silicon and the proposed full control of exchange interaction via strain/electric fields are some of the key theoretical contributions, which are expected to play an important role in the future development of donor-based quantum computing schemes.

The remainder of this review paper is organised in the following sections. Section 2 provides an overview of the role of central-cell-corrections within the tight-binding model to represent the electron wave functions bounded at the dopant atom in silicon crystal. Section 3 discusses the impact of strain on donor wave functions and the underpinning valley configurations. The high-level agreement of the computed and measured STM images which led to a theoretical proposal for the direct visualisation of the central-cell-effects in the STM images is discussed in Section 4 and the exact-atom spatial metrology of the donor atom positions in silicon is reviewed in Section 5. Section 6 highlights the role of strain on the control of two-qubit exchange interactions in the presence of one-lattice site variations in donor atom positions. Section 7 describes our machine learning framework to autonomously characterise spin qubits based on STM image classification by a CNN. The final conclusions of this paper are provided in Section 8.

Section snippets

Tight-binding wave functions and the role of central-cell-corrections

The calculation of donor bound electron wave function is a challenging problem. First, it requires incorporation of the valley-orbit interaction which has been shown to play an important role to accurately capture the experimentally measured energy spectra of donor impurities in silicon. Secondly, a high precision calculation of donor wave function spatial profile requires a proper implementation of central-cell effects (short-range potential) and the dielectric screening of the long-range

Impact of strain on donor energies and wave function

The control over donor hyperfine and exchange interaction between two dopant atoms are crucial elements of any donor-based quantum computing scheme. Strain and electric field both can be used to manipulate hyperfine and exchange interactions. While the electric field control of hyperfine through Stark effect was investigated in the literature [64], [65], [66], [58], [67], [20], [66], [68], strain control was relatively less studied and understood [69], [70]. In 2015, we applied our

Theoretical proposal for direct visualisation of central-cell corrections

In 2014, Salfi et al. developed and reported an experimental technique to directly measure the spatial profiles of the electron wave functions bounded to subsurface As donor atoms in silicon based on the low-temperature scanning tunnelling microscope (STM) measurements [25]. In order to provide a quantitative understanding of the real-space features in the measured STM image features, we established a fully quantum, large-volume treatment of the STM-dopant system [17], which was based on

Donor qubit metrology with exact atom precision

As discussed in the previous Section 4, the computed STM image of a phosphorus donor wave function demonstrated an excellent agreement with the experimental measurement as shown in the Fig. 4 [17]. One of the key outcomes of this study was that the feature symmetry and brightness of the computed STM images were directly dependent on the corresponding position of the donor atom in the silicon crystal. In order to systematically investigate the dependence of the STM images on the position of

Exchange interaction and control in coupled donor qubits

The calculation of exchange interaction between P donor atoms in silicon is a computationally intensive problem and its theoretical development goes several decades back to Cullis and Marko [27]. Over the years, many research groups have applied different methods to compute exchange interaction, with majority of the literature being based on Heitler–London (HL) formalism which is expected to provide a good estimate of exchange strength for donor separations more than 12 nm [28], [29], [30], [31]

Autonomous metrology by machine learning for scale-up

In the Section 5 above, we discussed atomic precision metrology of phosphorus donor qubits in silicon based on STM images of the electron wave functions. In the work reported in the Ref. [17], the comparison between the computed and measured STM images was performed at pixel-by-pixel level. However, for a large-scale quantum computer architecture, the number of qubits is expected to be in the range of several thousands, in which case a direct comparison at pixel-by-pixel level would be highly

Conclusions

This article provides a brief overview of the recent progress on theoretical modelling and simulations of donor spin qubits in silicon material based on our published work and relevant references from the literature. The computation of donor electron energies and wave functions was performed by solving an sp3d5s atomistic tight-binding Hamiltonian, in which the donor atom was represented by a Coulomb potential modified by a set of central-cell-corrections. The calculation explicitly involved

Data availability

The data that support the findings of this study are available within the article. Further requests can be made to the corresponding author.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The author is indebted to Lloyd Hollenberg and Charles Hill from the University of Melbourne for their strong support and contributions during all aspects of this work. The author thanks his student Yi Z. Wong who contributed in the development of machine learning framework. Many thanks to Juanita Bocquel, Joe Salfi, Benoit Voisin, Michelle Simmons and Sven Rogge from the University of New South Wales who led the experimental fabrication and STM measurements of donor qubits in silicon, and for

Dr. Muhammad Usman is a Senior Lecturer at the University of Melbourne, Australia. He obtained his Ph.D. degree from Purdue University, West Lafayette, Indiana USA in 2010. During 2010–2013 he held a postdoctoral position at the Tyndall National Institute Cork Ireland, making leading contributions in EU FP7 BIANCHO project. Since 2014, he has worked at the University of Melbourne and has also been affiliated with the ARC Center of Excellence for Quantum Computation and Communication Technology.

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  • Dr. Muhammad Usman is a Senior Lecturer at the University of Melbourne, Australia. He obtained his Ph.D. degree from Purdue University, West Lafayette, Indiana USA in 2010. During 2010–2013 he held a postdoctoral position at the Tyndall National Institute Cork Ireland, making leading contributions in EU FP7 BIANCHO project. Since 2014, he has worked at the University of Melbourne and has also been affiliated with the ARC Center of Excellence for Quantum Computation and Communication Technology. He is currently leading a quantum computing research and education program at the Faculty of Engineering and Information Technology. His theoretical work has made several breakthrough advancements towards the design and characterisation of atomic spin qubits in silicon and scale-up architectures. Dr Usman is a recipient of USA Fulbright Fellowship, German DAAD Research Fellowship and an Early Career Best Research Award at the University of Melbourne. His research interests are in the areas of computational materials science, quantum hardware development and machine learning applications in materials science. More information about his work can be found at: http://www.quantumelectronics.org/.

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