Transition metal impurities in silicon: computational search for a semiconductor qubit

Quantum information science aims to revolutionize information processing and transmission. Semiconductor qubits are a promising platform, with existing examples like gate-controlled nanostructures, shallow dopants in silicon, and optically-addressable point defects (e.g., nitrogen-vacancy centers in diamond). However, challenges remain, particularly regarding scalability and operating temperature. While nitrogen-vacancy (NV) centers in diamond operate at room temperature, their scalability is limited. Conversely, 31P in silicon offers long coherence times but requires extremely low operating temperatures (<10K). This research seeks to find an analogous system in silicon, ideally with optical triplet-triplet transitions within the silicon band gap, mirroring the NV centers' advantageous properties. Previous studies on transition metal (TM) impurities in silicon have been limited by the accuracy of the methods used, particularly regarding band gap prediction and structural relaxation. This study addresses these limitations by using advanced computational techniques to systematically evaluate the whole 3d series of TM impurities and selected 4d and 5d TMs.

Extensive research exists on transition metal impurities in silicon, driven by both scientific and technological interests. Early work, notably by Ludwig and Woodbury, proposed a model based on tetrahedral crystal field and Hund’s rule to explain electron paramagnetic resonance measurements. Beeler et al. employed density functional theory (LDA) to investigate the electronic properties of the 3d TM impurities, finding a breakdown of Hund’s rule for certain cases. However, LDA's limitations in accurately predicting silicon band edges and neglecting structural relaxation hindered the accuracy of their predictions. Generalized density functional theory, specifically HSE06, offers improved band gap prediction, but it tends to overlocalize TM d orbitals. This paper addresses this issue by employing the approach of Ivády et al., which corrects for deviations from the generalized Koopmans' condition (gKC). Previous work has also explored color centers in silicon, such as those based on 77Se+, Er3+, and defects associated with particle irradiation. While some of these exhibit in-gap optical transitions, none have demonstrated the triplet-triplet transitions crucial for mimicking NV centers in diamond.

The authors utilize a state-of-the-art computational approach based on HSE06, incorporating corrections for band edge issues and structural relaxation. To overcome the overlocalization issue of HSE06 for TM impurities, they adopt the method developed by Ivády et al., applying an occupation-dependent potential to the TM d orbitals when the deviation from gKC exceeds 0.2 eV. This approach is applied to the entire 3d series and selected 4d and 5d TM impurities. The workflow involves defect calculations to determine defect formation energies, thermodynamic charge transition levels (CTLs), and one-particle defect-level diagrams (DLDs) for substitutional, interstitial, and vacancy/TM-substitutional complexes. Optical absorption spectra are calculated to confirm the allowance of in-gap optical transitions. The selection criteria for identifying promising candidates include the presence of at least two CTLs within the silicon band gap, indicating the presence of both triplet ground and excited states, and the allowance of optical transitions. Defect formation energy is calculated using a standard supercell approach, accounting for size effects, potential alignment, and band edge problems. Total energy calculations are performed using spin-polarized generalized hybrid density functional theory (HSE06). A 216-atom supercell with gamma-only k-point sampling is employed. Thermodynamic charge transition levels (CTLs) are determined by finding the Fermi energy at which the defect formation energies for two different charge states are equal. The deviation from the generalized Koopmans' condition (gKC) is evaluated, and the HSE06(+U) method is used to correct for significant deviations. Optical absorption coefficients are calculated using the complex refractive index and the linear response approach.

The authors successfully identify seven TM impurities as promising candidates for optically active spin qubits in silicon. These include three substitutional and four interstitial defects. The selection criteria were based on the presence of at least two charge transition levels (CTLs) within the silicon band gap to accommodate both a spin-triplet ground state and a spin-triplet excited state, and the allowance of optical transitions. The electron counting method is utilized to explain the charge state of both substitutional and interstitial candidates and their locations in the periodic table. Substitutional candidates are located near the Zn column, while interstitial candidates are found on the left side of the periodic table. This finding is consistent with the observation that TMs are portable among different host materials with similar lattice symmetry and bond length. The authors highlight the importance of the relative energy position of the outermost s and d orbitals with respect to the host band structure. The predicted charge transition levels (CTLs) from the HSE06(+U) calculations generally agree well with experimental values where available. However, some discrepancies exist, potentially due to the formation of defect complexes with other impurities not considered in this study. The predicted optical transitions fall in the mid-infrared range. This makes them unsuitable for long-distance quantum communication via optical fibers due to high attenuation but suitable for free-space communication within atmospheric windows.

The findings of this study address the limitations of current silicon-based semiconductor qubits, namely, the lack of established spin-photon interfaces and low operating temperatures. The identification of TM impurities in silicon with optically allowed triplet-triplet transitions provides a potential solution. The mid-infrared wavelengths of the optical transitions could be used for free-space communication. The similar operational scheme to NV centers in diamond suggests potential for higher operating temperatures in quantum sensing applications, although limitations concerning thermal excitation and material properties are expected to constrain operating temperature compared to NV centers. The computational approach employed in this study provides a powerful tool for discovering new candidate qubit systems in materials. Future studies incorporating zero-field splitting calculations, zero-phonon line and Debye-Waller factor calculations, and intersystem crossing rate predictions can further refine the list of candidate systems and determine their suitability for specific applications.

This study presents a computational search for TM impurities in silicon that could serve as optically active spin qubits. Seven promising candidates are identified, paving the way for silicon-based qubits with enhanced operating temperatures and spin-photon interfaces. Future research should focus on refining these candidates through additional calculations and experimental verification. The results could have far-reaching implications for quantum sensing and free-space quantum communication.

The study relies on computational predictions, and experimental verification is needed to confirm the existence and properties of the identified TM impurities. The model used does not account for the effects of other impurities or defects in the silicon lattice, which could influence the properties of the TM impurities. The consideration of intersystem crossing, zero-field splitting, and zero phonon lines are needed for a complete evaluation of these defect systems for quantum computing applications. The study focuses solely on optical triplet-triplet transitions; other transitions (singlet-singlet or doublet-doublet) may also be relevant.