Introduction

Quantum information science represents the next frontier in myriad fields including sensing, computing, and metrology1,2,3. Advances in this arena are enabled by new developments in material hosts for the smallest unit of quantum information—the quantum bit or qubit—each tailored for a specific use4,5. Many quantum computing architectures have been explored to date by materials scientists, including topological6,7, trapped ion8,9, superconducting10, molecular spin-based11, photonic12,13, and defect-based14. For the design of quantum sensors, defect-based electronic spins housed in nuclear spin-free matrices possess broad appeal for their in situ optical addresS1lity and relatively easy-to-characterize phonon structures15,16. Electronic spins in general are also appealing for this application because their relaxation properties are extremely sensitive to their chemical environment and tunable using well-established chemical principles17. Dramatic advances in exerting control over electronic spin-based qubits have been made in the last two decades18,19,20,21,22.

The viability of a qubit is a function of two related but distinct parameters: T1, related to the spin lattice relaxation time, and T2, related to the spin-spin relaxation time23. Spin-lattice relaxation is dictated by the interaction between the electronic spin and its local and extended phonon bath. Spin-spin relaxation is primarily a function of the interaction between an individual spin and its neighboring spin centers and magnetic fields. T2 is also a measure of the amount of time that coherent information can be recovered from an excited system, and for this reason, it has been the focus of studies across potential qubit hosts. It has been extensively demonstrated that the best way to lengthen T2 at liquid helium temperatures is through the rigorous exclusion of nuclear spins combined with dilution of electronic spins; both approaches work to prevent unwanted interactions21,24,25,26,27,28. At the same time, T2 has a physical maximum: T2 ≤ 2×T1. Thus, at higher temperatures lengthening T2 will necessarily require doing the same to T1.

The factors that control T1 in solid state systems are experimentally understudied relative the factors influencing T229,30,31,32,33. The development of design principles for lengthening T1 has consequently become an extremely active area of research34,35. In this work we have prioritized understanding the specific chemical and structural principles which dictate T1 and T2 in our qubit host materials. T1 is much more strongly influenced by lattice vibrations to which T2 is largely insensitive36, but not immune, as will be shown in this work. At its core, optimization of T1 requires careful engineering of the vibrational and phonon modes of the spin host material and of the interaction between the spin and its surrounding lattice. Such strategies as stiffening the vibrational modes surrounding the spin site and designing materials out of only lighter elements (with inherently less spin-orbit coupling) have proven encouraging for maximizing T1, but more general design principles remain under investigation30,37.

To more fully understand the ways that localized electronic spins interact with the phonon bath of their host matrix, we have chosen oxygen deficient double perovskites as a model system. Double perovskites are a well-studied class of materials and have been explored in a wide range of quantum materials studies38,39,40. The wide band-gap of these materials helps to energetically isolate defect sites above cryogenic temperatures41. Previously, we reported the discovery of extraordinarily long T1 times of paramagnetic sites embedded in the oxygen deficient double perovskite Ba2CaWO6-δ and hypothesized that the origin of the paramagnetic spin was W5+ 29. Using a combination of heat capacity measurements and pulse electron paramagnetic resonance (EPR) spectroscopy, the coherence properties of these defect sites were explained as a sum of Raman processes, a finding consistent with recent reports11,42,43,44,45. Furthermore, by calculating the phonon modes’ relative contributions to the relaxation dynamics in that work, we were able to derive new insight into which modes promote decoherence. Specifically, we determined that the vibrations of lighter elements contributed more to decoherence than those of heavier elements. This conclusion has since been further supported by numerous studies and has become a design principle in constructing solid-state electronic spin-based qubits46,47,48.

Building on these findings, here we set out to expand the study of structure-property correlations to other systems, starting with Sr2CaWO6-δ, a lower mass chemical analogue to Ba2CaWO6-δ. We hoped that the smaller spin-orbit coupling present in the strontium analogue would render the spin centers less sensitive to motion of the lattice and thereby prolong relaxation times. We were further encouraged by the fact that Sr, like Ba, has a low natural abundance of nuclear spin-active nuclei.

In this system we indeed observed coherence times measurable up to room temperature, a substantial improvement over the Ba2CaWO6-δ system and a validation of our initial hypothesis. Interestingly, we also noted an unusual elongation of coherence time T2 with increasing temperature. This prompted us to identify the precise ion involved in the paramagnetic defect centers, an unanswered question from our previous work. To perform these characterizations, we employed a combination of one-and two-dimensional pulse EPR, hyperfine sublevel correlation (HYSCORE) spectroscopy, heat capacity measurements, thermogravimetric analysis (TGA), powder x-ray diffraction (PXRD), single crystal x-ray diffraction (SCXRD), magnetization, attenuated total reflectance infrared spectroscopy (ATR-IR), crystal field calculations, and density functional theory (DFT). Using this suite of techniques, we show that local distortions in the environment of the paramagnetic site contribute to an unusual lengthening in T2 with increasing temperature. Finally, we validate our previous finding that double perovskite qubit hosts can house long-lived electronic spin-based qubits with measurable superposition lifetimes out to room temperature. This represents substantial progress towards using these materials in quantum sensing devices.

Results

Similar to the previously reported Ba analogue, Sr2CaWO6-δ is an A2BB’O6 double perovskite consisting of alternate corner-sharing octahedra of B = CaO6 and B’ = WO6. Sr cations fill the A sites of the framework. Rietveld refinement to the PXRD data (Fig. S2, Table S3) shows that the material adopts a monoclinic structure (P21/n, space group #14) at room temperature. See SI §S2 for additional discussion. These data are supported by SCXRD results at T = 213 K (data presented in SI §S3). Pawley fits (Fig. S3, Table S4) to PXRD data also indicate that the material undergoes no structural phase transitions from T = 12 K to room temperature (SI §S4). Geometric constraints in this simple unit cell imply that miniscule variations in lattice parameters are due to the tilting of the CaO6/WO6 octahedra. Such changes have been shown to affect paramagnetic sites in other solid-state materials49.

Oxygen vacancies produce W5+ sites on which single electronic spins are localized (Fig. 1a) as supported by spin density estimates from HYSCORE spectroscopy (data in SI §S5-S6). This technique provides a direct measure of the strength of the interaction between unpaired electronic spins and nuclear-spin active nuclei. As will be detailed below, the degree of spin localization may also be determined as a function of temperature. In order to quantify the precise quantity of oxygen vacancies, we first turned to TGA measurements combined with ATR-IR (data in SI §S7-S8) but, due to sample aging (SI §S9) analysis of these data was unsuccessful. Instead, an oxygen deficiency of about 1.18% (δ ≈ 0.07) was identified via magnetization measurements using the protocol described in the Methods section, data presented in SI §S10. Measurements on a range of controllably reduced Sr2CaWO6-δ polycrystalline powders (SI §S11) support this determination using magnetization alone. Magnetization data were also used to exclude the possibility of a magnetic phase in this material (Figs. S24 and S25).

Fig. 1: Relaxation time of paramagnetic electron at tungsten site.
figure 1

a Partial unit cell for Sr2CaWO6 illustrating the qubit site located next to a proposed oxygen vacancy labelled Va. Red = oxygen, gray = tungsten, blue = calcium, green = strontium. b T1 and T2 relative to temperature for the sharp, long-lived feature at approximately 3500 G for both Sr2CaWO6 and its chemical analogue Ba2CaWO6.

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Spin-spin relaxation—EPR and tungsten site symmetry

To determine the viability of this system as a host matrix for defect-based qubits, we measured T1 and T2 from T = 5 to 300 K using pulse EPR. Figure 1b depicts relaxation times for the dominant feature observed at B0 = 3500 G in both Sr2CaWO6 and the previously studied analogue compound Ba2CaWO6. In both materials, we observe decreasing T1 with increasing temperature. As previously reported, T2 for the barium compound also decreases as temperature increases29. In the strontium compound, we initially see a small rise in T2 at low temperature. This increase may be explained as a reducing contribution of spectral diffusion to relaxation, common in other materials below T = 10 K. Much less commonly observed, however, is the doubling of T2 between T = 30 K and 130 K that we encounter in Sr2CaWO6. T2 is, in theory, a metric for a temperature-independent process, depending only on the spin-spin interactions within the system and an upper limit set by T1. Here, T1 decreases over the range where T2 increases, suggesting that the peculiar increase in coherence time is not driven by spin-lattice dynamics nor an increase in the maximum possible value for T2.

Echo-detected field swept (EDFS) EPR spectra (Fig. 2a) of Sr2CaWO6 show two features at low temperature. This is in contrast to our previous observations for the chemical analogue Ba2CaWO6, which showed a single feature with g = 1.96 at all temperatures29. The strength of these features quantifies the electron-ligand field coupling and thus local distortions and Einstein modes in the material. An anisotropic feature is initially dominant at low temperature. Between T = 10–130 K the strength of this anisotropic feature is reduced as an isotropic feature at 3500 G (g ≅ 1.943) becomes dominant (see also Fig. S32). The increasing dominance of the isotropic signal corresponds roughly with the temperature region where we observe the T2 increase. Therefore, we posit that higher isotropy in the EPR data results in higher measured T2 values even while limited by decreasing T1.

Fig. 2: EDFS EPR spectra and calculated g tensors.
figure 2

a EDFS EPR spectra of a powder of Sr2CaWO6-δ. b g tensors produced by fitting data in (a) to a model of 3 symmetric tensors, with lines to guide the eye. Above 40 K, only two g tensors were required to capture peak shape; above 100 K, only one was required.

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As W4+ and W6+ are not EPR active, we assign the tungsten valency as W5+. The spin density estimates obtained from isotropic hyperfine coupling of 183W species in the material show that the dominant phase exhibiting the signal at g = 1.99 in the HYSCORE data (above T ~ 70 K; present as the isotropic component in Fig. 2a) corresponds to electrons localized mostly on oxygen orbitals at these sites, with at least some spin localization directly at the tungsten site. The formal oxidation state of reduced tungsten atoms in the material is W5+ with strong oxygen covalency for all temperatures measured. This assignment is detailed in Table S8 and the increasing isotropy of the HYSCORE data with increasing temperature may be observed in Figs. S6-S10. This increase signifies that either the spin density is becoming more spherically symmetric—such as spin localizing in an s-orbital, or more evenly distributing between a set of d-orbitals—or that the through-space dipolar interaction of the 183W nucleus with spin density localized on adjacent atoms is becoming weaker.

In the absence of evidence to support or deny the second hypothesis, we instead consider the idea that spin density (from an unpaired electron) is becoming more spherically symmetric as temperature increases. Spin density localizing in s-orbitals is unlikely for a 5 d transition metal such as tungsten; W5+ is expected to be 5d1. Therefore, the increasing isotropy of the HYSCORE data indicates that these 5d1 paramagnetic electrons are driving the increase in symmetry. We refer to this phenomenon as an increase in “effective” site symmetry because such spin delocalization across d-orbitals would not change the physical symmetry of the site in the lattice but would result in a more symmetrical electronic environment at that site. In the monoclinic lattice, the 5d1 unpaired electron occupies a single \({d}_{{z}^{2}}\) orbital with preferred orientation along the axis connecting the two closest strontium atoms (the body diagonal of the unit cell). Lattice vibrations in this geometry result in a variation of strontium position, leading in turn to a change in the preferred \({d}_{{z}^{2}}\) orbital orientation. When averaged over many such vibrations, this results in a nearly spherical spin density. See Fig. S34 and SI §S13 for additional discussion.

Note that these unpaired electrons are introduced to the tungsten sites by the loss of an oxygen atom from a formerly symmetric full octahedron. Such reduction of Sr2CaWO6 necessarily requires charge balancing in the form of changing oxidation states for other ions in the lattice—either additional tungsten atoms or a neighboring calcium or strontium. For simplicity, we will limit the following discussion to the tungsten atoms only, as Ca1+ or Sr1+ are chemically dubious and spin localization on these ions is not supported by our DFT results, discussed below. Rather, DFT results suggest one spin localizing on a specific tungsten atom with the other spin delocalized across other tungsten atoms.

In the fully oxidized material, all tungsten atoms are W6+ ions. W6+ is not EPR-active, so both full and incomplete octahedra are “silent” in our data. As shown in Fig. 3, the loss of a single oxygen bound to a tungsten site would yield a W4+ ion with an asymmetric ligand field. The resulting site could produce an anisotropic EPR signal but is not supported by our HYSCORE data as the origin of the detected signal. Instead, one of the extra electrons on a W4+ site hops onto a nearby W6+ with a complete octahedron, producing one W5+ site with an asymmetric ligand field and another W5+ site with a symmetric ligand field. We assign these two sites as responsible for the anisotropic and isotropic signals we detect, respectively. DFT estimates put the energy of this electron transfer at 0.1–0.3 eV. We find that the resulting vacancy is a shallow electron donor, meaning this process is likely to take place at low temperatures.

Fig. 3: Schematic of reduction of a tungsten site.
figure 3

a Process by which a W6+ site of the form present in fully oxidized Sr2CaWO6 is converted into a W5+ site with asymmetric ligand field by the loss of an oxygen atom. b Equilibrium between a pairing of W4+ and W6+ sites with two W5+ sites, one with an asymmetric and one with a symmetric ligand field.

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An additional, complementary effect is oxygen vacancy diffusion in the vicinity of an asymmetric W5+ center. An oxygen vacancy may occur at any of the six oxygen atoms coordinated to a central tungsten. As supported by our DFT results (Fig. S37), the diffusion of this vacancy from one position to another also serves to qualitatively increase the symmetry of the electronic environment around the central tungsten. Notably, the spin density maxima produced by averaging over \({d}_{{z}^{2}}\) orbital orientation align with the minima of spin density produced by averaging over different vacancy locations in the unit cell. The result is that, when taken together, these effects combine to produce higher effective W5+ site symmetry even for partial oxygen octahedra as depicted in Fig. 4.

Fig. 4: Qualitative illustration of increasing spin symmetry.
figure 4

Green = calcium, blue= strontium, gray = tungsten, red = oxygen, yellow = spin density of 5d1 electron on tungsten. To produce this qualitative image, weightings were assumed to be equal for each possible configuration. a Average spin density of four different orientations of \({d}_{{z}^{2}}\) orbitals along each unit cell body diagonal, localized on central tungsten atom. b Average spin density of six possible locations for oxygen vacancy around central tungsten atom. c Average spin density of four different orientations of \({d}_{{z}^{2}}\) orbitals and six possible locations for oxygen vacancy, illustrating the synergy of the two effects in producing a highly-symmetrical (more spherical) spin density.

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Vacancy diffusion becomes more frequent for partial octahedra as temperature increases, producing a more spherical distribution of spin on the tungsten site. Furthermore, the ligand field around an asymmetric W5+ will also experience increased thermal motion, leading to a more symmetric electronic environment for this spin. At the same time, thermal motion yields greater \({d}_{{z}^{2}}\) orbital averaging for both partial and complete octahedra. The population of paramagnetic electrons in partial octahedra also decreases as these electrons are thermally activated to hop onto sites with complete octahedra. Since T1 and T2 were measured at 3500 G, the increase in isotropic signal at 3500 G may be directly related to increasing population of complete octahedra. The end result of any of these effects is the same: an increase in the total number of paramagnetic electrons in effective high-symmetry ligand fields. Once the population of W5+ in complete octahedra is saturated, the normal effects which serve to decrease coherence time start to dominate over vacancy diffusion and thermal motion, ultimately leading to a decrease in relaxation time as the material approaches room temperature.

Spin-lattice relaxation—Heat capacity and phonon modes

To identify the mechanisms restricting T1 in this system, we next turned to a combination of heat capacity and EPR measurements. When plotted as Cp/T3 vs. T, heat capacity data collected via the protocol detailed in the Methods section roughly approximate the 1-D phonon density of states (DOS). As shown in Fig. 5a, the data for Sr2CaWO6-δ are best described by a combination of three modes: two acoustic Debye phonon modes and one low-lying Einstein optical mode. At sufficiently low temperatures Debye modes plateau at a constant value, while an Einstein mode accounts for peaking behavior50. The oscillator strength (Table 1) adds up to ~ 10.3(5): a confirmation that our model accurately accounts for the 10 atoms per formula unit. No evidence of phase transition has been found in the T = 2–300 K range.

Fig. 5: Heat capacity and EPR fits.
figure 5

a Heat capacity (Cp) divided by temperature cubed (T3) versus temperature (T) plotted on a logarithmic scale. The red line shows a fit to the experimental data. Contributions of individual components are plotted below: pink—Debye 1 phonon mode, blue—Debye 2 phonon mode, purple—low energy Einstein phonon mode. b Inverse of longitudinal relaxation time (T1) relative to temperature obtained from pulse EPR. The red line shows the fit to obtain the contributions of each of the decoherence processes (direct and local process corresponding to the Einstein mode and the Raman processes corresponding to the Debye modes). The characteristic temperature for each mode was fixed to those obtained from the heat capacity fit.

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Table 1 Characteristic temperatures and number of oscillators used in fitting Einstein and Debye modes to the specific heat for Sr2CaWO6-δ
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The presence of an Einstein mode is indicated by a peak at T ≅ 22 K in Fig. 5a. We ascribe the origin of this Einstein mode to activated local vibrational processes related to the tilting of WO6/CaO6 octahedral units (Fig. S33), a common structural distortion in perovskites. The appearance of such a low-lying Einstein mode in this system is surprising but not unprecedented in double perovskites39,51. Indeed, we observed similar features in our earlier analysis of the barium analogue29. The Einstein mode in Sr2CaWO6 occurs at a nearly identical temperature but is much weaker than that of Ba2CaWO6, which results in a lower overall heat capacity in the strontium analogue.

Additional insight into the types and effects of phonon modes is garnered by comparison to our EPR data. Figure 5b shows a plot of the inverse of T1 with respect to temperature generated from our pulse EPR data. These data were fit to a model described in the Methods section following the process pioneered in our earlier work on the barium analogue29, where Adir, Aram1, Aram2, and Aloc are coefficients representing the strength of one direct, two Raman, and one local process, respectively. Each Debye mode (corresponding to a Raman process) is associated with a characteristic Debye temperature (θD1, θD2) and the local mode with an Einstein temperature (θE) which are fixed to values obtained from the quantitative analysis of the heat capacity (Table 1).

Pulse EPR data below T = 10–15 K are well-modeled with the single-phonon direct process, as is typical29. This process is best described as a spin flip because the faster phonon-mediated processes are minimally activated at this temperature. Where T << θD, this process contributes to a linear increase in T1−1 with temperature. The orders-of-magnitude greater influence of the direct process in Sr2CaWO6 compared to Ba2CaWO6 (Table 2) provides an explanation for the lower T1 values in this material at low temperature29.

Table 2 Coupling constants used in fitting 1/T1 data for Sr2CaWO6-δ and Ba2CaWO6-δ
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In the intermediate temperature regime above 15 K, the data are better modeled using the Raman process, wherein two different phonons mediate spin-lattice relaxation. Contributions from the direct process and the local Einstein mode have a minor influence for Sr2CaWO6 in the low temperature (T < 30 K) regime, but the Raman processes dominate relaxation across almost the entire temperature range. This runs counter to the model often invoked in the fitting of the temperature dependence of T1, wherein the Raman process is expected to give way to localized phonon modes at the highest temperature extreme. Finally, quasi-local phonons provide a small contribution at temperatures above T = 70 K, analogous to local vibrations in the immediate vicinity of the spin center.

We also explored the relative influence on relaxation as a whole. The fit parameters obtained from the heat capacity and the spin relaxation lifetime data can be combined to gain quantitative information about the relative strength of spin-phonon coupling of each of the modes using the following equations:

$$\frac{{A}_{{ram}2}}{{A}_{{ram}1}}={\left(\frac{{{\rm{N}}}_{{\rm{D}}2}}{{{\rm{N}}}_{{\rm{D}}1}}\right)}^{2}\left(\frac{{{\rm{\theta }}}_{{\rm{D}}1}}{{{\rm{\theta }}}_{{\rm{D}}2}}\right){\left(\frac{{G}_{{ram}2}}{{G}_{{ram}1}}\right)}^{4}$$
(1)
$$\frac{{A}_{{loc}}}{{A}_{{ram}}}={\frac{36{\pi }^{5}}{9\cdot {6}^{\frac{10}{3}}{\pi }^{\frac{11}{3}}}\left(\frac{{{\rm{N}}}_{{\rm{E}}}}{{{\rm{N}}}_{{\rm{D}}}}\right)}^{2}\left(\frac{{{\rm{\theta }}}_{{\rm{D}}}}{{{\rm{\theta }}}_{{\rm{E}}}}\right){\left(\frac{{G}_{{loc}}}{{G}_{{ram}}}\right)}^{4}$$
(2)

where N is the oscillator strength of each of the modes, θ is the characteristic temperature and G is a parameter describing coupling between lattice vibrations and the spin of the paramagnetic electron52. Values are reported in Table 1. From this analysis we observe that the spin-phonon coupling parameter is strongest for the θD2 = 641(29) K mode and weakest for the θD1 = 242(2) K mode. This behavior is qualitatively similar to that of the barium analogue. Further insight into these results is gained from DFT simulations.

DFT and atomic origin of phonon modes

We next performed a series of calculations to determine the character of the phonon modes to corroborate our estimate of the phonon DOS from heat capacity measurements. Since the concentration of oxygen vacancies is relatively low, we considered a stoichiometric Sr2CaWO6 monoclinic phase, represented with a 40-atom periodic supercell (see SI Fig. S40, Table S11). A pseudocubic (α = β = γ = 90°) phase was also considered, details in SI §S14. Phonon DOS was determined by applying Gaussian smearing with 30 cm-1 FWHM of the DFT-computed vibrational frequencies.

Figure 6a shows the calculated vibrational DOS overlaid with the phonon density of states predicted by fits to the experimental heat capacity data. The Einstein mode is represented as a Dirac delta, zero everywhere except at the Einstein frequency, and the Debye modes are represented with Eq. 3 up to the Debye frequency. s is the oscillator strength and θD is the Debye temperature from Table 1. T is the temperature in Kelvin, kB is the Boltzmann constant, and ω is the frequency in Hz (note the conversion from wavenumber).

$${D}_{{Debye}}\left(\omega \right)=3s{\omega }^{2}{\left(\frac{\hslash }{{\theta }_{D}{k}_{B}}\right)}^{3}$$
(3)
Fig. 6: DFT DOS and calculated heat capacity.
figure 6

a Calculated (DFT) vibrational DOS for the monoclinic lattice (black) overlaid with modes fit from experimental heat capacity data (green = Einstein mode with θE = 113 K, blue = Debye mode with θD = 242 K, orange = Debye mode with θD = 641 K). Vibrational DOS as calculated at the gamma point and broadened using Gaussian smearing with FWHM of 30 cm−1. Above 450 cm−1, a dashed line is used to indicate that these modes are unpopulated at experimental temperatures. b Heat capacity (Cv) computed from DFT DOS shown in a) (lines) and Cp shown in Fig. 5a (circles) divided by temperature cubed (T3) from T = 1.9 to 300 K plotted on a logarithmic scale. Note that imaginary (negative frequency) contributions to the DOS were omitted in the integration. Computed data is scaled so that, at high temperature, Cv = 3nR where n = 10 atoms per formula unit and R is the molar gas constant. The solid line depicts Cv computed from the raw DOS shown in a) while the dashed line depicts Cv computed from DOS where a w2 approximation has been applied between the origin and first positive-frequency vibrational frequency, to better approximate the collective low energy phonon modes of a solid.

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Figure 6b shows the constant-volume heat capacity calculated from the vibrational density of states shown in Fig. 6a overlaid with the experimental constant-pressure heat capacity shown in Fig. 5a. Details of the calculation are provided in the Methods section below. At low temperatures, CV is expected to be approximately equal to Cp. The results of different values of FWHM in the Gaussian smearing of calculated vibrational frequencies are also provided in Fig. S42. All demonstrate that the calculations capture the key qualitative features of the experimental data, independent of precise choice of broadening, which suggests an accurate DFT representation of the simulated lattice. Importantly, the position of the peak resulting from the Einstein mode is reproduced in the computed data, indicating that the causative behavior—hypothesized from experimental data to be WO6/CaO6 octahedral tilting—is well-captured in this model.

The two Debye modes identified from heat capacity measurements roughly describe the motions of heavy and light atoms, respectively, within the lattice. The higher temperature Debye mode corresponds to the higher energy vibrations of atoms with lower masses and the low energy Debye mode corresponds to the low energy vibrations of atoms with high masses. To establish the precise atomic origin of the phonon modes via DFT, we assign them weights corresponding to the magnitude of atomic displacements normalized to unity and summed over atoms of the same type. The results of this analysis are provided in Figs. 7 and S41; visualization of the vibrational modes is provided in Movie S1.

Fig. 7: Heat map representation of the atomic contributions generated by DFT to each phonon mode.
figure 7

Data is shown for the monoclinic lattice, sorted by index number. Each amplitude vector was normalized to unity. The brighter and darker colors correspond to the larger and smaller amplitude values, respectively.

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Near-zero frequency modes correspond to lattice translations. In the monoclinic lattice, the lowest energy band (60–190 cm−1 Fig. 6a; phonon mode index 80–117 Fig. 7) is dominated by displacements of the Sr and O atoms and contains minor contributions due to Ca and W. This is the only band that includes non-negligible W contributions; hence, it is best described as rotations and displacements of WO6 octahedra coupled with the motion of the framework Sr atoms. This calculated band overlaps with the Einstein and first Debye modes identified via experimental heat capacity results, as shown in Fig. 6a.

We found that modes located in the interval of 200–450 cm−1 (mode index 25–79 in Fig. 7) are dominated by Ca and O atoms and the weights of their contributions anticorrelate, as shown in the Fig. S41. Accordingly, this band is attributed to symmetric and antisymmetric O-W-O and O-Ca-O bond bending, where Ca participates in the bond-bending motion and W remains stationary. This band overlaps with the second experimentally identified Debye mode, found to have the largest spin-phonon coupling parameter of all the experimentally identified phonon modes.

The high degree of spin-phonon coupling for this mode may be explained by the identification of this mode with O-W-O bending. Displacement of oxygen octahedra, as occurs in lower-frequency bands, has a relatively minor effect on the local electronic environment around the paramagnetic spin centers in Sr2CaWO6-δ. Conversely, O-W-O bond bending has a strong impact on the symmetry of the electric field experienced by the spin centers. The activation of the second Raman process—linked to this second Debye mode—is correlated with the temperature at which T2 is observed to increase and where EPR and HYSCORE data show increasing spin isotropy. Bending motions increasing the local symmetry of oxygen-deficient tungsten octahedra, or lowering the energy of electron transfer to complete octahedra, may be a source of increasing coherence time.

Finally, high-frequency bands (index 1-24 in Fig. 7) are attributed to O-W symmetric and anti-symmetric stretching and include only oxygen motion. These bands, represented with dashed lines in Fig. 6a, are not represented in experimental results as they are accessed above the temperature range of our heat capacity and EPR experiments. Sr2CaWO6 has been shown to undergo a discontinuous structural transition to a cubic structure at T ≅ 1100 K53, so above T = 300 K our simulated pseudocubic structure is expected to become more relevant. However, the DOS for both structure types are qualitatively similar in the high-frequency range (Figs. S39 and S41).

Discussion

Here we have shown the viability of Sr2CaWO6-δ as a qubit host by demonstrating millisecond T1 lifetimes at T ~ 10 K. Through HYSCORE and EPR measurements we were able to identify W5+ spin centers as the dominant defect centers in these materials with spin localization primarily on oxygen. Decreasing the spin orbit coupling constant by our design strategy led to slightly shorter spin-lattice relaxation times compared to the Ba analogue studied previously; however, the unusual increase in T2 observed between T = 50 and 100 K provides evidence that similar methods could be employed to successfully increase coherence times.

A key component of our original hypothesis was that decreased spin orbit coupling would help to decouple the spin center from lattice effects, but due to the role of the lattice sites (specifically tungsten octahedra) in determining the electronic environment of the paramagnetic spin, this was proved false. The higher g-tensor observed for the paramagnetic electrons in the barium analogue is likely due to higher symmetry of the lattice site54 and produces longer coherence times at low temperature. The decrease in T1 observed via replacement of the A-site cation barium by strontium in this work is limited only to low temperatures and appears related to the increased role of the direct process in spin-lattice relaxation. The overall magnitude of the associated Einstein mode is less than that of the barium analogue, resulting also in a lower heat capacity.

At higher temperatures, a slight reduction in spin-phonon coupling—likely due in part to the replacement of barium with lighter strontium—combined with fewer nuclear spins results in slightly higher T1 values for the new compound. This sets the limit of T2 higher for Sr2CaWO6-δ beyond about 50 K. Notably, this is an example of T2 values affected by lattice vibrations even when T2 ≪ T1—in contrast to the conventional wisdom—which reinforces the importance of studies on lattice vibration when designing optimized qubit host materials. As demonstrated here, a simple replacement of a cation can affect both T1 and T2 in a variety of interrelated ways even with otherwise minimal changes to the overall lattice.

The peculiar increase in T2 with temperature in the strontium system serves to produce much longer coherence times at higher temperatures despite low-temperature limitations stemming from lower T1. EPR measurements indicate that the spin responsible for these long coherence times occupies a highly symmetrical electronic environment (g ≅ 1.943, close to free electron ge = 2.00232) which is increasingly activated at higher temperatures while anisotropic spin effects are reduced. The importance of a symmetrical electronic environment on prolonging relaxation times has been established previously for 3d1 molecular54 and 3d9 ionic systems55, but the thermal activation of a higher-symmetry site created by oxygen vacancies in the dilute limit has not been observed before.

Local symmetry in this system increases as phonon modes are populated. This leads to a reduction of nuclear spin coupling as the W site approaches ideal octahedral symmetry. We propose that a combination of 1) electron motion into complete oxygen octahedra and 2) lattice effects serving to increase the electronic symmetry of incomplete octahedra together produce ever more symmetric environments as temperature increases. Future materials design work should incorporate these strategies to increase the symmetry of spin environments, bringing the electron g tensor closer to that of a free electron, and thus prolonging relaxation times. Finally, as-yet unexplained detrimental aging of the desired phase is also observed in this work and should be more concretely probed prior to application of this system in quantum information technologies.

Methods

All data were processed using a combination of Xepr, Python 2.7 and 3.11.4, Origin Pro 2018, RStudio 2024.04.2 running R version 4.1.2, Mathematica 14.1_WIN, and MATLAB R2018b and R2024a. A sample key is provided in SI §S1.

Synthesis of Sr2CaWO6

Polycrystalline Sr2CaWO6 powder was synthesized using solid state synthesis by reacting loose powders of SrCO3 (Strem Chemicals, 99.99%) CaCO3 (Noah Chemicals, 99.98%) and WO3 (Noah Chemicals, 99.99%) at 650 °C, then 1000 °C and finally 1250 °C for 24 hours per heating in air with a heating and cooling rate of 100 °C/hr. The resulting beige polycrystalline powder was compacted into rods using a hydrostatic press at 70 MPa followed by sintering at 1250 °C in air for another 24 hours. The resulting rods were yellow in color. In December 2019, these rods were melted at 40% laser power (5 ×200 W GaAs lasers – 976 nm) in a laser diode floating zone (LDFZ) furnace (Crystal Systems Inc FD-FZ-5-200-VPO-PC). A stable floating zone was maintained by counterrotating the rods at 10 rpm and a travelling speed of 10 mm/hr under Ar gas flowing at 2.5 L/min. The obtained crystal was dark blue and ground powder was light blue, differing from the polycrystalline growth rods, indicating the presence of oxygen vacancies in the sample.

The resulting crystal was immediately taken for PXRD. ATR-IR, heat capacity, EPR, and HYSCORE measurements were taken within six months. Approximately four years passed before SCXRD, TGA, and magnetization measurements were performed.

In an attempt to quantify uncertainties associated with natural aging of the sample (see §S9), additional polycrystalline Sr2CaWO6 powder was synthesized again in 2023 using an optimized solid state synthesis by reacting dried powders of SrCO3, CaCO3, and WO3 for 24 hours at 1000 °C followed by two successive 24 hour heatings at 1250 °C. The material was ground and mixed thoroughly between each heating step. Peaks belonging to Sr2WO5 were detected using PXRD in samples heated to 1000 °C but eliminated with 1250 °C heatings. Intermediate grinding was found to be essential to eliminate SrWO4, and in poorly ground samples an additional heating of 24 hours at 1350 °C or hotter was required to produce single-phase Sr2CaWO6 (see Fig. S28). The resulting single-phase material was analyzed via PXRD, TGA, ATR-IR, and magnetization within the next six months.

Powder X-ray diffraction

Laboratory-based room temperature x-ray diffraction patterns for phase identification were collected using a Bruker D8 Focus diffractometer with CuKα radiation in the 15–120° 2θ range (Figs. S1 and S2). Ground silicon SRM 640 d (space group Fd-3m, #227, a = 5.431179 Å)56 was added to allow accurate determination of lattice parameters (Table S3).

Low temperature PXRD patterns were collected with a Bruker D8 Advance with an Oxford Cryosystems PheniX cryocontroller with CuKα radiation and 6 mm tube and 0.6 mm detector slits. Freshly grown single-crystalline material (Sample SSC-19) was measured using a high-resolution scintillation counter from T = 12–297.6 K (room temperature) in the 26–32° and 53–79° 2θ ranges, referred to as low-angle and high-angle data, respectively. See SI §S2 for discussion. Polycrystalline material grown in 2024 (Sample SPC-24-2) was measured with a Bruker LynxEye detector over the range 17–72° 2θ within 1 week of synthesis. Ground silicon SRM 640 d (space group #227, Fd-3m, a = 5.431179 Å)56 was added to allow accurate determination of lattice parameters (Tables S4, S5, and S6).

Phase identification and unit cell determinations were carried out using the Bruker TOPAS 4.2 software (Bruker AXS)57.

Single crystal X-ray diffraction

SCXRD data for Sr2CaWO6-δ approximately four years after synthesis (Sample SSC-23) were collected at 213 K. Data were collected using a SuperNova diffractometer (equipped with an Atlas detector) with Mo Kα radiation (λ = 0.71073 Å) under the program CrysAlisPro (ver. CrysAlisPro 1.171.42.49, Rigaku OD, 2022). The same program was used to refine the cell dimensions and for data reduction. The structure was solved with the program SHELXS-2018/3 and was refined on F2 with SHELXL-2018/358. Analytical numeric absorption correction using a multifaceted crystal model was applied using CrysAlisPro (ver. CrysAlisPro 1.171.42.49, Rigaku OD, 2022). The temperature of the data collection was controlled using the Cryojet system (manufactured by Oxford Instruments). Results in SI §S3.

Q-Band (~34 GHz) Hyperfine Sublevel Correlation (HYSCORE) Spectroscopy

Q-band HYSCORE spectroscopy was performed at the Caltech EPR Facility at the California Institute of Technology in Pasadena, CA using a Bruker ELEXSYS E-580 pulse EPR spectrometer equipped with a Bruker D2 Q-band resonator. Temperature control was achieved using a ColdEdge ER 4118HV-CF5-L Flexline Cryogen-Free VT cryostat and an Oxford Instruments Mercury ITC temperature controller.

HYSCORE spectra were acquired using the 4-pulse sequence (\(\pi /2-\tau -\pi /2-{t}_{1}-\pi\) –\({t}_{2}\)– \(\pi /2\) – echo), where \(\tau\) is a fixed delay, while \({t}_{1}\) and \({t}_{2}\) are independently incremented by Δ\({t}_{1}\) and Δ\({t}_{2}\), respectively. The time domain data was baseline-corrected (third-order polynomial) to eliminate the exponential decay in the echo intensity, apodized with a Hamming window function, zero-filled to eight-fold points, and fast Fourier-transformed to yield the 2-dimensional frequency domain.

All HYSCORE spectra were simulated using the EasySpin59 simulation toolbox (version 5.2.36) with MATLAB 2022b using the following Hamiltonian:

$$\widehat{H}={\mu }_{B}{\mathop{B}\limits^{ \rightharpoonup }}_{0}g\hat{S}+{\mu }_{N}{g}_{N}{\mathop{B}\limits^{ \rightharpoonup }}_{0}\hat{I}+h\hat{S}\,{{\cdot }}\,{\boldsymbol{A}}\,{{\cdot }}\,\hat{I}$$
(4)

In this expression, the first term corresponds to the electron Zeeman interaction term where \({\mu }_{B}\) is the Bohr magneton, g is the electron spin g-value matrix with principle components g = [gx gy gz], and \(\hat{S}\) is the electron spin operator; the second term corresponds to the nuclear Zeeman interaction term where \({\mu }_{N}\) is the nuclear magneton, \({g}_{N}\) is the characteristic nuclear g-value for each nucleus (e.g. 183W) and \(\hat{I}\) is the nuclear spin operator; the third term corresponds to the electron-nuclear hyperfine term, where \({\boldsymbol{A}}\) is the hyperfine coupling tensor with principle components \({\boldsymbol{A}}\) = [Ax Ay Az].

TGA

Thermogravimetric analysis was performed with a TA SDT Q600 simultaneous TGA/DSC, weight calibrated before use for 2 °C/min, in order to determine the amount of oxygen vacancies in our samples. Each sample was heated under O2 flowing at 20 mL/min to 150 °C at 10 °C/min and held for one hour to drive off any adsorbed water. The mass after the hold was taken to be the initial sample mass. It was then heated, also under O2, to 1000 °C at 2 °C/min and held for four hours. The mass at the end of this hold was taken to be the final sample mass. Data is shown in SI §S7.

IR data

ATR-IR data was collected with a Thermo-Nicolet iN5 with iD5 accessory from 525–5000 cm−1. All samples were powders. “As purchased” samples were stored under atmospheric conditions, while all “freshly grown” or heated samples were measured within two days of synthesis or removal from a sealed tube. A combination of literature searches and measurements on various oxide and carbonate precursor materials was used to assign the peaks observed, details in SI §S7.

Magnetization

Magnetization as a function of temperature data were collected from T = 2-300 K with an applied field of μBH = 1 T using a Quantum Design MPMS 3 system. Samples were non-oriented LDFZ-grown pieces weighing in total ~15 mg (Sample SSC-23) suspended with plastic wrap in a plastic straw. A Curie-Weiss fit was performed on the magnetic susceptibility χ calculated from these data to approximate the number of magnetic sites. The T = 2–15 K region was selected as it was the most linear portion of the low-temperature data. A χ0 was determined using a binary search algorithm to make this data as linear as possible, then a least squares fit was performed to determine C and the Curie-Weiss temperature θCW according to Eq. 5.

$$\frac{1}{\chi -{\chi }_{0}}=\frac{T-{\theta }_{{CW}}}{C}$$
(5)

Since a S = 1/2 site (as from a single unpaired electron) should yield a Curie-Weiss constant C of 0.375 emu K/mol in this temperature region, it was determined that 1.18% of W sites in Sr2CaWO6-δ and 0.15% of W sites in Ba2CaWO6-δ contained unpaired electrons from W5+. See SI §10 for data fit.

$${\rm{Oxygen\; per\; formula\; unit}}=6-6\left(C\times \frac{8}{3}\right)$$
(6)

Magnetization as a function of field data were also collected from −5 to 5 T at 2 K, 20 K, and 50 K on both powder and crystalline samples. Samples were an oriented LDFZ-grown piece (Sample SSC-25) weighing ~53 mg and ground powder of the same sample weighing ~37 mg, suspended with plastic wrap in a plastic straw.

Reduction and reoxidation of freshly grown samples

Freshly grown samples were reduced using the following method for comparison to the floating zone sample grown in a reduced state. Approximately 15 mg of Sr2CaWO6 powder was added to the bottom of a 10” quartz tube. A narrow neck was made in the tube at its midpoint, such that a ¼” cylindrical pellet of Zr metal added to the tube was unable to contact the powder. The tube was backfilled with argon gas and evacuated to ~10−3 torr. The tube was inserted into a three-zone furnace and a heating program entered such that both sides of the tube reached their target temperatures at the same time, but the Zr pellet achieved a final temperature ≥150 °C hotter than the powder (Table S9). Samples were characterized with PXRD, TGA, and ATR-IR, results in SI §S11.

X-band EPR experimental protocol

X-band EPR spectroscopy was performed on crushed microcrystalline powders contained within a 4 mm OD quartz EPR tube within 6 months of synthesis. Samples were ground from LDFZ-grown Sr2CaWO6-δ (Sample SSC-19) confirmed by PXRD to be single-phase. EPR data were obtained at T = 70 K at X-band frequency (~0.3 T, 9.5 GHz) on a Bruker E580 X-band spectrometer at the University of Illinois EPR Lab (Urbana, IL) equipped with a 1 kW TWT amplifier (Applied Systems Engineering). Temperature was controlled using an Oxford Instruments CF935 helium cryostat and an Oxford Instruments ITC503 temperature controller (UIUC). T = 50 K, 60 K, and 70 K data were collected for a shorter measurement time and thus display higher noise.

T2 values as a function of temperature (T = 5–300 K) were determined via a Hahn-echo decay experiment utilizing a π/2 – τ – π – τ – echo sequence. Echo decay as a function of increasing delay time τ was measured and fit to a stretched exponential function (again necessitated by a presumed distribution of domain sizes and electron environments). The time constant associated with that decay is the T2 value. Spin-lattice relaxation times measured at 3500 G were the longest at all temperatures and were also the most persistently measurable with increasing temperature, yielding able to be integrated and manipulated at temperatures as high T = 300 K.

EasySpin v 6.05 running in MATLAB R2024a was used to fit the EPR data presented in Fig. 2a59. Below T = 50 K, the two peaks in the data were best modelled as a combination of three symmetric g tensors. From T = 50–100 K, only two g tensors were required, one for each peak. Beyond T = 100 K, only one peak is present and only one g tensor was required. When plotted as a function of temperature, the g tensor corresponding to the isotropic low-field peak remains nearly constant at g ≅ 1.943 (Fig. 2b). The g tensors responsible for the anisotropic peak, conversely, show some temperature dependence. Figure S32 shows the total contribution of the isotropic peak to the total signal, as a percentage of total peak area, and reproduces the trend visually observed in Fig. 2a.

Fitting of T 1 −1 data

To obtain the contributions of each decoherence process to the spin-relaxation lifetime relative to temperature we fit 1/T1 with respect to temperature. Equation 9 was used to fit the data, where Adir, Aram1, Aram2, and Aloc are coefficients representing one direct, two Raman, and one local process, respectively. Values are reported in Table 2. J8 is the transport integral describing the joint phonon DOS assumed by the Debye model taking the form described in Eq. 10.

$$\begin{array}{l}\frac{1}{{T}_{1}}={A}_{{dir}}{\rm{T}}+{A}_{{ram}1}{\left(\frac{T}{{{\rm{\theta }}}_{{\rm{D}}1}}\right)}^{9}{J}_{8}\left(\frac{{{\rm{\theta }}}_{{\rm{D}}1}}{T}\right)+{A}_{{ram}2}{\left(\frac{T}{{{\rm{\theta }}}_{{\rm{D}}2}}\right)}^{9}{J}_{8}\left(\frac{{{\rm{\theta }}}_{{\rm{D}}2}}{T}\right)\\\qquad\;+\,{A}_{{loc}}\frac{{{\rm{e}}}^{{{\rm{\theta }}}_{{\rm{E}}}/T}}{{\left({{\rm{e}}}^{{{\rm{\theta }}}_{{\rm{E}}}/{\rm{T}}}-1\right)}^{2}}\end{array}$$
(9)
$${J}_{8}\left(\frac{{\theta }_{D}}{T}\right)={\int }_{0}^{{\theta }_{D}/T}{x}^{8}\frac{{e}^{x}}{{\left({e}^{x}-1\right)}^{2}}{dx}$$
(10)

A simple monoexponential function was unable to adequately capture the shape of the curve. Attempting to fit the data using a correction to account for spectral diffusion resulted in unrealistic relaxation times below T = 30 K, representing an inability of the model to account for relaxation driven by spectral diffusion. This has been observed previously in systems in which spectral diffusion happens much faster than spin-lattice relaxation60. A stretched monoexponential better captured the curvature. Such stretch factors are typically attributable to a range of relaxation times across the sample61, which we here ascribe to the existence of more than one paramagnetic electron environment and the variance in domain sizes inherent to inhomogeneously ground microcrystalline samples.

Heat capacity measurements

A Quantum Design Physical Properties Measurement System (PPMS) was used for the heat capacity measurements from T = 1.9 to 300 K at µoH = 0 T using the semi-adiabatic method.

The data were fit using the equations:

$${{\rm{C}}}_{{\rm{Einstein}}}=3{sR}{\left(\frac{{\theta }_{E}}{T}\right)}^{2}\times \frac{{e}^{{\theta }_{E}/T}}{{\left({e}^{{\theta }_{E}/T}-1\right)}^{2}}$$
(11)
$${{\rm{C}}}_{{\rm{Debye}}}={9{sR}\left(\frac{T}{{\theta }_{D}}\right)}^{3}\times {\int }_{0}^{{\theta }_{D}/T}\frac{{\left(\frac{\theta }{T}\right)}^{4}{e}^{\left(\frac{\theta }{T}\right)}}{{[{e}^{\left(\frac{\theta }{T}\right)}-1]}^{2}}d\frac{\theta }{T}$$
(12)

yielding the fit shown in Fig. 5a. The parameters used in this fit, including the Debye (θD) and Einstein temperatures (θE), are shown in Table 1. R is the molar gas constant and s is the number of oscillators per formula unit.

DFT

DFT calculations were performed using the VASP package62,63,64 and the PBEsol density functional64. The projector-augmented wave potentials were used to approximate the effect of the core electrons65. Gamma-centered k-mesh was varied from 2 × 2 × 2 to 4 × 4 × 4 to ensure the convergence of the structural parameters; 3 × 3 × 3 mesh was used for the calculations of vibrational frequencies. The plane-wave basis-set cutoff was set to 600 eV. The total energy was converged to within 10–8 eV. The vibrational frequencies and phonon modes at the Gamma point were calculated using the finite differences approach and the displacement magnitude of 0.01 Å. Vibrational DOS was calculated by convoluting vibrational frequencies with Gaussian functions with the full width at half maximum (FWHM) of 30 cm–1.

To account for self-interaction of electrons in our DFT simulations, ionization energy and sphericity were estimated for a range of U values. For our d-orbital spin density estimates, an extra electron (fixed doublet state) was added and treated explicitly; a homogeneous background charge was also added to maintain charge neutrality. For our estimates of vacancy spin density, a triplet state was fixed. Additional discussion is provided in SI §S13-S15.

Calculation of constant volume heat capacity from DFT DOS results

Phonons were calculated using the finite displacement method of a 3 × 3 × 3 supercell. This is too coarse a mesh for a full Brillouin zone integration to obtain the true phonon DOS. For calculation of the heat capacity from the phonon DOS, we first approximated the DOS by applying a Gaussian smearing of 30 cm−1 to each calculated mode to capture the effects of finite bandwidth, as detailed in §S15 and pictured in Fig. 6a.

To calculate the constant volume heat capacity expected from this DOS, we next omitted all 0 energy acoustic phonons occurring at negative frequencies. Heat capacity was then calculated from the vibrational density of states using the standard equation for the heat capacity of bosons from statistical thermodynamics66:

$${C}_{V}=\int {\left(\frac{\hslash \nu }{T{k}_{B}}\right)}^{2}D\left(\nu \right)\frac{{k}_{B}{e}^{\frac{\hslash \nu }{T{k}_{B}}}}{{\left({e}^{\frac{\hslash \nu }{T{k}_{B}}}-1\right)}^{2}}d\nu$$
(13)

where T is temperature in Kelvin, ħ is the reduced Planck constant, ν is the frequency in Hz, and kB is the Boltzmann constant. D(ν) is the density of states as calculated by DFT and shown in Fig. 6a. Numerical integration was done using the trapezoidal Riemann approximation method in R version 4.1.2. The resulting value was normalized such that the heat capacity Cv at high temperature (where a plateau is observed, here T = 1000 K) satisfies the Dulong-Petit Law, Eq. 14.

$$\mathop{\mathrm{lim}}\limits_{T\to \infty }{C}_{v}=3{nR}$$
(14)

R is the molar gas constant. Here n = 10 atoms pe r formula unit.

The resulting data is shown in the solid line of Fig. 6b. Lastly, we repeated the calculations, adding a w2 dependence from 0 cm−1 to the first finite frequency mode to properly account for the collective low energy phonon modes of a solid. These results are shown in Fig. 6b as the dashed line. The calculations were repeated for additional Gaussian broadenings, Fig. S42, as confirmation that the calculations capture the key qualitative features of the experimental data independent of broadening.