Thermal diffusion and trapping of vacancies for the formation of optical centers in diamond

Abstract

Since its introduction, quantum physics has revolutionized our understanding of the fundamental laws governing the universe. Originally employed to address problems unsolvable by classical mechanics, quantum physics has gradually found a wide variety of applications in modern life, many of which are based on the principle of quantum coherence, such as lasers. Despite these advancements, understanding the behavior of complex quantum systems remains an enduring challenge, primarily due to the exponential growth in complexity as the number of system components increases. This difficulty is largely attributed to the limited computational power of classical computers. As Richard Feynman famously remarked, ”Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly, it’s a wonderful problem, because it doesn’t look easy”, highlighting the intrinsic link between the complexity of quantum phenomena and the need for computational paradigms that can inherently handle such complexity. Contrary to Gordon Moore’s 1965 prediction, recent years have seen a slowdown in the reduction of transistor size as physical limits have been approached, thereby obstructing further advances in classical computational power. Ongoing research into alternative methods to enhance computational performance has thus shifted focus towards quantum computers. Proposed initially in 1988 by Y. Yamamoto and K. Igeta, quantum computers are based on qubits as units of information. Unlike classical bits that exist strictly as either 0 or 1, qubits are quantum systems that can exist in any arbitrary superposition of two distinct states (|0⟩ and |1⟩). This property enables quantum computers to perform computational tasks through the temporal evolution of these systems via logical operations (defined as gates), potentially offering exponential speed-ups for problems intractable with classical computation. Over the years, advancements in the production and control of various quantum systems have led to the exploration of applications such as nanoscale quantum sensing. Among several types of quantum technologies, including single photons and trapped ions, devices based on spin impurities in solid-state systems have been demonstrated exceptionally effective for both quantum computing and quantum sensing. These impurities meet the DiVincenzo criteria for quantum systems, featuring spins that not only define discrete, individually addressable energy levels but also allow for initialization, coherent manipulation, and measurement. Furthermore, these systems are sensitive to physical quantities like electric and magnetic fields due to Stark and Zeeman interactions, respectively. Among the various spin impurities, the negatively charged nitrogen-vacancy (NV) center in diamond is extensively studied. This atomic-scale spin system is composed of a nitrogen atom located in a neighboring lattice site of a vacancy and surrounded by carbon atoms. Recent advances in material engineering have enabled the controlled synthesis of diamond crystals with a high degree of purity, allowing NV centers to function as wellisolated spin systems. The NV center is characterized by triplet spin energy configuration, including a spin-selective relaxation via inter system crossing, that enables to initialize, manipulate, and optically read out the ground state spin. Moreover, the energy levels of the NV center are sensitive to variations in magnetic fields, electric fields, temperature, and strain, making this center a versatile sensor for various physical phenomena. Its atomic size and spin properties render the NV suitable for quantum sensing applications. Unlike other competing sensing technologies that require high-energy systems, the NV center can operate under standard temperature and pressure conditions. Furthermore, the chemically inert nature of diamond renders NV-based devices biocompatible, allowing for their placement within a few nanometers from the field sources and, thus, enabling imaging of magnetic field at nanometer scale. The most common technique for creating single NV centers in diamond involves ion implantation followed by thermal annealing. This method allows the creation of NVs at various depths within diamond with nanometer precision. Despite its spatial resolution, low-energy implantation, necessary for near-surface NV positioning (relevant for sensing applications), results in a low yield of NVs even when the concentration of vacancies induced by implantation is significantly higher than that of atomic nitrogen included within the diamond lattice. This inefficiency is related to the tendency of vacancy to aggregate into di-vacancies (V2) or multi-vacancy complexes (Vn) in ion-damaged areas rather than getting trapped at nitrogen lattice site and forming NV centers. Although a higher implantation dose could potentially increase the yield, it would also result in greater lattice damage, which would reduce the spin coherence time (T2) of the NV, compromising its sensitivity as a quantum sensor. In addition to the challenges associated with low formation efficiency, the proximity of NV centers to the diamond surface (less than 10 nm) introduces further complications. Surface-induced noise and electronic defects can degrade spin coherence and destabilize the NV charge state, significantly impacting the performance and reliability of these shallow defects. Addressing these issues, while simultaneously maximizing NV concentration and preserving T2 and signal contrast, remains a critical challenge in the field of quantum sensing. Recent advancements in material processing and NV creation strategies have provided promising solutions. Co-implantation of nitrogen and helium ions, with precise optimization of implantation energies (below 10 keV) and fluences, has been shown to minimize lattice damage and enhance the efficiency of NV formation. Helium co-implantation, in particular, facilitates the introduction of vacancies at controlled depths, improving the nitrogen-to-NV conversion efficiency. This technique has proven particularly effective for generating shallow NV centers with narrow linewidths in optically detected magnetic resonance (ODMR) measurements, a key requirement for quantum sensing applications. In parallel, high-temperature annealing protocols, typically at temperatures exceeding 1200◦C, have been employed to mitigate the formation of vacancy complexes and restore lattice integrity, further enhancing NV coherence times. Furthermore, surface treatments, such as hydrogen and oxygen termination prior to implantation, have shown promise in addressing surface-induced noise. These treatments not only enhance the nitrogen-to-NV conversion efficiency but also improve the stability of shallow NV centers by reducing charge-state conversion to neutral NV configurations. Additionally, pre-doping diamond substrates with nitrogen prior to ion implantation has emerged as a powerful method to enhance NV yield. Molecular dynamics simulations suggest that nitrogen concentrations of approximately 1000 ppm optimize vacancy diffusion and NV formation, achieving yields up to 10%. These techniques, combined with advancements in chemical vapor deposition (CVD) diamond growth, allow for controlled nitrogen incorporation and defect alignment along specific crystallographic orientations, critical for achieving reproducible quantum performance. In addition to quantum sensing, the domain of quantum information processing has also gathered increasing attention. Although NV centers meet the DiVincenzo necessary criteria for quantum applications, they lack some optical properties ideal for quantum information processing, such as a high Debye-Waller factor and minimal spectral diffusion. In contrast, color centers in diamond based on group-IV elements are considered suitable candidates. These centers not only exhibit higher optical performances than NV centers but also show inherent compatibility with nanostructures usually employed for enhancing optical properties. Like NVs, these group-IV color centers can be precisely engineered through ion implantation followed by annealing. However, since nitrogen is intrinsically present within the diamond lattice, the formation of unwanted NV centers is inevitable. Developing methods to suppress the formation of unwanted NVs is crucial for advancing the application of group-IV color centers in quantum information processing. This thesis primarily examines the formation of color centers in diamond, achieved through the trapping of diffusing vacancies. Specifically, it explores the use of irradiation and annealing techniques designed to generate vacancies and promote their diffusion within the diamond lattice. A critical aspect of this study involves identifying the inherent limitations associated with these techniques and understanding the underlying reasons of these constraints. The aim is to develop novel methodologies that can overcome these limitations. One such innovative method has been applied for the synthesis of tin-vacancy (SnV) centers in diamond. These developments are crucial for improving the production of color centers in diamond, thereby broadening their utility in quantum technology applications. Impurity centers in diamond lattice. The present work starts with an overview of diamond as host material for color centers employed for quantum applications, focusing particularly on the NV. The discussion includes description of the chemical vapor deposition (CVD) and high temperature high pressure (HPHT) growth techniques of diamond, pivotal in synthesizing substrates with controlled level of the impurities. This control is significant as the presence of any paramagnetic impurities within the lattice can compromise the spin properties of targeted color centers. The chapter further focuses on the NV center physical properties, providing a basic introduction of the associated spin manipulation techniques. Moreover, detailed attention is given to the methods employed to create NV centers, with a specific focus on ion implantation followed by annealing, and on the limitations associated with these technique. Vacancy diffusion and defect formation in a crystalline solid. The second chapter addresses the diffusion of vacancies and their trapping by defects within the diamond lattice. Here, a novel model of vacancy diffusion based on probabilistic atomic jumps in crystal lattice, is developed to investigate the limit in the NV formation due to trapping of diffusing vacancies induced by irradiation. The NV formation within the model is considered as competing mechanism of vacancy trapping between nitrogen atoms, divacancies (V2) and multi-vacancies complexes (Vn). A critical parameter of the described model is the so-defined capture cross-section, which, quantifying the probability for a vacancy to be trapped by a specific defect, can be related to the formation energy of the corresponding defect. The model, developed for different vacancies distributions scenarios, has been validated through Monte Carlo simulations. The efficiency of NV formation by irradiation techniques: estimates by the model and the experimental validation. The third chapter starts with the description of the so-called ”Scanning Protocol” developed to collect profiles of NV centers within the diamond bulk with a nanometric precision. In this protocol, photoluminesce (PL) confocal scans of fixed area have been collected at different depth with a step 0.1 µm. Within each scan, the position and the fluorescence intensity of NV centers have been evaluated by employing a 2-D Gaussian fit. An additional 1-D Gaussian fit of the NV PL intensity as function of depth result in the localization of the NV in the diamond bulk. NV depth distributions by helium implantation followed by annealing have been collected for different annealing temperatures. A fit of these distributions through the model developed in chapter 2, provides an activation energy for vacancies diffusion in diamond of 1.7 eV. This value has subsequently been used into the model to evaluate the capture cross section ratio between NV and V2 in a range of 0.1 to 0.5. This result demonstrates the tendency of vacancies to preferentially aggregate rather than being trapped by nitrogen, defining the limit (low formation yield) in the NV formation (as for other color center in diamond) by ion implantation or electron irradiation followed by annealing. The ability of the model developed in chapter 2 to predict NV center concentrations resulting from the aforementioned techniques enables the formulation of strategies aimed at enhancing the formation of NV centers associated with these techniques. This enhancement can be achieved while preserving the spin properties of the NV centers. Furthermore, the model probabilistic approach holds promise for wider applications in the engineering of additional vacancy-related defects in diamond. This extends the model utility significantly within the realm of quantum engineering. Planar p-n junction structures on diamond for controlling the vacancy diffusion. The final chapter explores an advanced engineering technique to control vacancy diffusion at thermal annealing. Specifically, vacancies created by ion implantation within the depletion region of a diamond p+ junction get charged, and their long-range diffusion has been proved to be suppressed due to repulsive forces from ionized donors in the depleted region of the n-doped substrate. This mechanism is demonstrated to reduce the formation of NV centers by limiting vacancy trapping at nitrogen sites in the diamond bulk. The effectiveness of this technique is verified through the fabrication of p+-n junctions in high purity single-crystal diamond substrates. Indeed, C and He implantations (tuned to create vacancies as described above) across such junction, followed by annealing at 1200◦ resulted in a strong reduction of NVs compared to not doped areas subjected to the same implantation and annealing procedure. The method described has been successfully applied to the implantation of tin to produce SnV centers in diamond. This approach resulted in both an enhanced yield of SnV centers and a reduction of unwanted NVs along the implantation paths of the Sn atoms. The utility of this method extends beyond the creation of SnV centers; it is also applicable to the formation of other color centers in diamond. By providing control over vacancy diffusion within semiconductor materials, this technique possesses substantial potential for a variety of applications in the field of quantum technologies.