Screening procedure

To reduce the possibility of accidental exclusion of viable host materials, the screening of the Materials Project’s 125,223 inorganic compound entries was conducted in four distinct stages, as shown in Fig. 3.

Fig. 3: Four-tier screening process to evaluate the potential viability of materials as hosts for quantum defects.

Stage 1: Beginning with all known experimental inorganic phases, those with a calculated nonzero net magnetic moment, crystallization in a polar space group and/or containing atomic species with <50% I = 0 isotopes were removed. Stage 2: Phases containing radioactive Th and U, toxic Cd and Hg, and the often magnetically impure rare-earth elements were removed. The one phase containing a noble gas element that passed this sieve, orthorhombic XeO3, was also removed due to its known instability under standard conditions. Stage 3: The calculated, predicted, and measured experimental band gaps of the remaining phases were recorded from the Materials Project, the JARVIS-DFT database, the AFLOW repository, the AFLOW-ML API, and the existing literature. Those materials that could not be reasonably predicted to have a band gap larger than 1.1 eV were dismissed. Stage 4: The intrinsic diamagnetic (DM) character and phase stability under standard conditions of the remaining candidates were confirmed using a combination of standard electron-counting principles and literature reports. Remaining phases with calculated E Above Hull values >0.2 eV/atom (from the Materials Project database) were dismissed. All database searches were conducted in February and March of 2020.

Stage 1

The first screen was applied utilizing the Materials Project’s built-in query API29, and admitted all entries corresponding to phases crystallizing in nonpolar space groups, derived from an experimental ICSD entry, containing only those elements with a >50% natural abundance of zero nuclear spin isotopes, and calculated to be nonmagnetic systems.

Stage 2

The second screen was applied by hand, and included the removal of all phases containing uranium, thorium, cadmium, and mercury, due to the radioactive and/or toxic nature of the majority of stable phases that each may form, as well as those containing noble gases, as none exist as stable solids under standard conditions. All rare-earth-element-containing phases were also removed, due to the difficulty in obtaining sufficiently pure starting materials free of nuclear spin.

Stage 3

The third and most important filter was applied utilizing calculated band gaps for all remaining phases reported by three publicly available DFT computation repositories; the Materials Project, the Joint Automated Repository for Various Integrated Simulations (JARVIS), and the Automatic Flow of Materials Discovery (AFLOW) library, each of which employs a different band gap calculation and/or prediction method. In addition, where available, the machine-learning-predicted band gap reported by Isayev et al. in their predictive modeling studies was also taken into consideration.

At this stage, those phases with a Materials Project band gap of 2.0 eV were deemed likely to possess a large enough band gap to accommodate the necessary optical transitions of useful quantum defects, and, so as not to miss any potentially viable candidates, only those with calculated band gaps <0.5 eV were removed. For all other phases, any reports of experimental or otherwise calculated band gaps in the literature were taken into consideration. Various higher-order compounds for which the calculated band gaps were insufficient, such as in the Ba-Hf-S family, were approved based on general band gap trends. Those phases with calculated and/or experimentally derived band gaps consistently in the range of that of Si, which would represent the lower bound of viability for potential host materials, are grayed in Supplementary Tables 14.

The Materials Project30 utilizes the DFT-based Vienna ab initio simulation package (VASP) software31 to calculate a wide variety of structural, energetic, and electronic properties for all reported inorganic structures in the ICSD. An initial relaxation of cell and lattice parameters is performed using a 1000 k-point mesh to ensure that all properties calculated are representative of the idealized unit cell for each material in its respective crystal structure. To calculate band structures for these materials, the generalized gradient approximation (GGA) functional is applied to the relaxed structures. For structures containing one of several transition-metal elements such as Cr, Fe, Mo, and W, the +U correction is also applied to correct for correlation effects in occupied d– and f-orbital systems that are not addressed by pure GGA calculations32. The authors caution that due to the high computational cost of more sophisticated calculation methods, those employed often produce severely underestimated band gaps relative to the experimentally derived values. The energy above the hull (E Above Hull), which indicates the thermodynamic stability for each phase, with respect to decomposition, is also computed. An E Above Hull value of zero is defined as the most stable phase at a given composition, while large positive values indicate increased instability with respect to the stable phase(s).

JARVIS-DFT33, originally compiled as a database for functional materials, with a focus on the discovery of novel two-dimensional systems, also employs the DFT-based VASP software to perform a variety of material property calculations. As opposed to the Materials Project, JARVIS-DFT employs the OptB88vdW (OPT) functional, which was initially designed to better approximate the properties of two-dimensional van der Waals materials, and has since also been shown to be effective for bulk systems34,35. Structures are first sourced from the Materials Project database, and then re-optimized using the OPT functional. A representative band gap is then calculated through a combination of the OPT and modified Becke–Johnson (mBJ) functionals. The mBJ and combined OPT-mBJ functionals have both been shown to predict band gap sizes with more accuracy than other DFT-based calculation methods36.

The AFLOW37 repository relies on a highly sophisticated and automatic framework for the calculation of a wide array of inorganic material properties38. The GGA-based Perdew–Burke–Ernzerhof functional with projector-augmented wavefunction (PAW) potentials is first used within the VASP software to twice relax and optimize the ICSD-sourced structure using a 3000–6000 k-point mesh. The increased k-point mesh density, compared to that employed by the Materials Project, is indicative of a more computationally expensive calculation. The band structure is then calculated with an even higher-density k-point mesh, as well as with the +U correction term for most occupied d– and f-orbital systems, and the standard band gap (Egap) is extracted39. A “fitted” band gap (Egfit) is then calculated by applying a standard fit, derived from a selective study of DFT-computed vs. experimentally measured band gap widths, to the initially calculated value40.

AFLOW-ML41, a machine-learning API designed to predict thermomechanical and electronic properties based on chemical composition alone, further builds upon the entries present in the ICSD and calculated through the AFLOW framework. Using only provided atomic compositional and positional information, so-called “fragment descriptors”, the system first applies a binary metal/insulator classification model. For materials predicted to be insulators, an additional regression model is applied to predict the band gap width. Each model was subjected to a fivefold cross validation process, in which it was trained to more accurately predict properties in an independent data set. The initial binary classification model is shown to have a 93% prediction success rate, with the majority of misclassified materials being narrow-gap semiconductors. While the accuracy of the predicted band gap sizes relative to experimental values is not mentioned by the authors, roughly 93% of the machine-learning-derived values are found to be within 25% of the DFT + U-calculated gap width. Only those phases identified in the authors’ initial cross-validated test set were used for comparison.

Stage 4

The final stage. The criteria by which potential host species were excluded in the first three stages were largely derived computationally, with subsequent manual checks. In contrast, the final stage of screening involved the confirmation of various fundamental properties of the materials found in the existing literature.

First, for those remaining stoichiometries for which multiple phases appeared to be viable, the relative stability of each polymorph at STP was recorded. All recognized high-pressure and high-temperature phases that were not reported to be quenchable to a stable state under standard conditions were removed. The relative thermodynamic stability of each identified phase was also considered. As larger calculated E Above Hull values are indicative of a thermodynamically unstable material, phases for which the Materials Project predicted an E Above Hull >0.2 eV/atom were excluded from consideration, since they would likely be too prone to decomposition to be viable (Fig. 4). A total of 39 (largely ternary and quaternary) entries were deemed nonviable.

Fig. 4: Distribution of Materials Project-calculated E Above Hull values for otherwise viable host phases.

The distribution of calculated E Above Hull values found for unary (orange), binary (green), ternary (purple), and quaternary or higher-order (yellow) phases which appeared to be otherwise viable at the beginning of the Stage 4 screening. (Inset) Zoomed view of the distribution from 0.08–0.80 eV/atom above the hull.

The intrinsic magnetic character of each pure phase was then confirmed through a combination of standard electron-counting rules and literature reports. As many of the potentially viable materials contained oxygen, particular care was taken to consider whether reported paramagnetic character could be due to oxygen defects, especially in the various molybdate, platinate, and palladate phases. Any pure phase that was reported to deviate from diamagnetic character was removed.

Where the true size of a material’s band gap could not be reasonably assumed suitable based on calculations alone, other reported computationally and experimentally derived band gap values were also taken into consideration when available. While experimental band gaps were either not available or not recorded for the majority of phases considered, those that were available were compared to the calculated values from each database, in order to better judge the viability of the selected materials with small calculated gaps (Fig. 5). The standard band gap calculations performed by the Materials Project, JARVIS-DFT, and AFLOW databases were found to be underestimated by roughly 40–50%, relative to measured values, consistent with the long-known inaccuracies of DFT estimates, often due to the presence of significant spin-orbit coupling effects42. AFLOW-ML’s machine-learning-based predictions were underestimated to a similar degree. On average, the OPT-mBJ hybrid functional employed by the JARVIS-DFT database for some phases was found to reduce this underestimation to 18.4%, while the “fitted” band gap calculated by AFLOW was actually observed to be overestimated by about 1.8% relative to experimental values. However, the latter figure is heavily influenced by several outlying underestimations in the AFLOW “fitted” gap data set, suggesting that on average, band gaps calculated in this manner will be overestimated to a greater degree. The determination of band gap suitability was thus made with these findings in mind, as well as with the understanding that the database-calculated band gap widths for heavier element-containing phases are often significantly underestimated, sometimes due to the failure of standard DFT calculation methods to account for spin-orbit coupling effects42.

Fig. 5: Distribution of database-calculated and experimentally-measured band gap widths for a representative sample of phases.

Comparison of reported experimental band gaps to those calculated by a the Materials Project, b AFLOW-ML, c JARVIS-DFT (OPT only), d JARVIS-DFT (OPT-mBJ), e AFLOW (unfit), and f AFLOW (fit). On average, the standard DFT calculations a, c, e and machine-learning predictions b were found to underestimate the experimental band gaps by roughly 40–50%. JARVIS-DFT OPT-mBJ calculations d were found to underestimate measured values by about 18% on average, while AFLOW fitted band gap calculations f were found to overestimate by about 2% on average, with several significant outlying underestimations. The red lines show the linear least squares fits to the data.

Additional considerations

It should be noted that despite appearing in the literature, the most stable phases of several, long-recognized materials were found to lack an experimental crystal structure in the ICSD, and subsequently were also absent from the Materials Project database. Several of these, such as the STP-stable phases of BaGeO3, BaGe2O5, and C70 would be potentially viable host species, but due to their absence from any of the databases studied are not included. However, while comparing computed band gaps with reported literature values, three additional phases were identified that lacked a corresponding Materials Project entry, but did appear in at least one of the other databases. fcc-C60 is the only included phase with a corresponding ICSD entry that did not appear in any of the databases considered.

While suitably sized single crystals are necessary for the fabrication of functional devices for quantum information systems, few materials have been reported to be easily grown as large, defect-free, and optically clear single crystals. As such, the existence of a published single-crystal synthesis, regardless of product size, is denoted in the “SC” column in the tables of results by an asterisk. Noted air and moisture instability in the literature was also considered, but was not exclusionary, as additional post-synthesis processing may allow otherwise viable host materials to be utilized.


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