Self-regulated growth of candidate topological superconducting parkerite by molecular beam epitaxy

The ongoing search for materials that exhibit exotic quantum phenomena has led to a surge of research in low-dimensional chalcogen- and pnictogen-based compounds and intermetallics.¹ These materials possess the promising ingredients for novel behavior, including magnetism,^2–9 topology,^10–14 and superconductivity,^15–22 and as such, they may possess desirable properties, such as quantized anomalous Hall effect,^23,24 Weyl and Dirac nodes,^3,25–27 negative thermal expansion,^28,29 and topological superconductivity,^30–32 and may be key to realizing room-temperature applications³³ of topological materials. One structural family of interest that has been sparsely explored is the parkerites with the formula unit A₃B₂X₂ (B₂A₃X₂) (A = Ni, Rh, Pt, Pd; B = Bi; and X = S, Se).^34–38 Numerous compounds of these exhibit superconductivity, including Ni₃Bi₂S₂,³⁹ Ni₃Bi₂Se₂,³⁹ Pd₃Bi₂Se₂,³⁵ and Rh₃Bi₂Se₂.³⁴ Interestingly, all these possess a lower symmetry monoclinic structure, which opposes one of the Matthias rules for superconductivity.^40,41 Rh₃Bi₂Se₂ includes a charge density wave occurring below 250 K, which coexists with the superconducting state. Furthermore, Pd₃Bi₂S₂ is a semimetal that is predicted to host exotic nontrivial band structures,¹ and recent bulk crystal and thin film experiments show evidence of topological surface states.^42,43 In the same family of materials are the shandites, which also possess the A₃B₂X₂ stoichiometry (A = Ni, Co, Rh, Pd; B = Pb, In, Sn, Tl; and X = S, Se).^38,44 Many of these materials exhibit exotic transport phenomena, in part derived from the kagome network of A-site atoms arranged in the structure, most notably in the ferromagnetic Weyl semimetal Co₃Sn₂S₂.^38,45,46 Thus, this family of structurally and chemically similar materials exhibits a tantalizing interplay of topology, magnetism, and superconductivity and could serve as an ideal playground to explore novel quantum heterostructures and interfacial phenomena.

Both the parkerites and shandites are subclasses of the half-antiperovskites, which themselves are vacancy derivatives of the simple perovskite structure.^47,48 The structural evolution from the perovskite to half-antiperovskite parkerites and shandites is outlined in Fig. 1. In a typical cubic ABX₃ perovskite, A and B are the cations that sit at the unit cell corners and center, respectively, while X is an anion that occupies the unit cell face centers, forming a 3D connected octahedral network, which is shown in Fig. 1(a). Here, X is most commonly O, but, for example, it can also be F, N, and S, and the bonding is mostly ionic in nature. The structure is stable across a wide range of cations and anions with different atomic radii and charge states. This chemical diversity is accommodated through displacements of the cations, as well as distortions and rotations of the oxygen octahedra, and is made possible by a large stability window geometrically captured by the Goldschmidt tolerance factor. As shown in Fig. 1(b), the antiperovskites are characterized by simply switching the A-site cation position with the anion, leading to a chemical formula A₃BX, for example, in Mn₃GaN. Although there is no loss of symmetry between these two classes and, in fact, the same structural distortions can be found in both, it is useful to classify them as different materials since perovskites are typically ionically bonded, whereas the bonding in antiperovskites is a mixture of predominately metallic and covalent.⁴⁹ The half-antiperovskite structures are derived by leaving half the A-site atoms vacant, giving a formula of A_3/2BX or equivalently A₃B₂X₂. Furthermore, the nature of the ordering of the vacancies dictates if they fall into the shandites or parkerites, which can drastically affect their ground state properties. The shandite compound is made by ordering the A-site vacancies in every other plane in the (111) direction, leading to the creation of a well-separated network of kagome planes that can drive exotic magnetic and electronic phenomena.^3,38,50–53 In the parkerites, the vacancies order in two ways, as shown in Figs. 1(d) and 1(e). The A-sites can be vacant along every other (110) plane, leading to a monoclinic 2D layered-like structure (space group 12), or with alternate A-sites vacant along each (110) plane, leading to a cubic interconnected 3D structure (space group 199).^36,44,54,55 Interestingly, superconductivity has primarily been observed in the 2D-type monoclinic parkerites but not in the cubic parkerites and only reported in the Pd₃Pb₂Se₂ shandite compound at high pressures.⁵⁶ This suggests, again, a strong interplay between structure, chemical composition, and quantum phenomena.

A key challenge toward understanding and ultimately utilizing these novel materials is growing high-quality thin films of ternary compounds by molecular beam epitaxy (MBE).^57,58 In contrast, the growth of high-quality binary chalcogenide thin films has been relatively simple, given the existence of stoichiometric growth windows. This technique, also known as the three-temperature method, is widely utilized in the III–V semiconductor industry, as well as enabling GaAs/Ga_1−xAl_xAs heterostructures to achieve mobilities in excess of 40 000 000 cm² V⁻¹ s⁻¹,^59,60 and is also the case of many binary chalcogenide topological insulators, such as Bi₂Se₃ or Bi₂Te₃.⁵⁷ In principle, the high vapor pressure element (Se or Te) is supplied in excess compared to the low vapor pressure element (Bi). The substrate is kept above the sublimation temperature of the high vapor pressure element such that the excess material is desorbed rather than being incorporated into the film, leaving only perfect stoichiometric material behind.^61–63 Key challenges for extending this to the ternary systems are that a growth window may only exist for a single element but could emerge for multiple elements due to reduced/enhanced volatility of bonded components;⁶⁴ an example of the latter is the hypothetical shandite Co₃Sn₂Se₂, where the production and sublimation of highly volatile Sn–Se on the growing surface result in Sn-free Co₇Se₈ films.^57,65 Here, we report the synthesis of thin films of the ternary chalcogenide parkerite material Pd₃Bi₂Se₂ by MBE. We show that by tuning the growth temperature and elemental flux ratios supplied to the growing surface, a self-regulated growth window emerges, which is controlled by the Pd flux due to the higher volatility of Bi and Se. This results in the high-quality single phase Pd₃Bi₂Se₂, which exhibits a superconducting transition of 800 mK as confirmed by transport and magnetometry measurements. The information gained here will enable understanding and navigating the growth of this class of ternary chalcogenides and enable observations of new phases and new applications.

Samples were grown using a home-built MBE system operating at a base pressure lower than 5 × 10⁻¹⁰ Torr. Pd, Bi, and Se were all supplied via thermal effusion cells. The cell temperatures were adjusted before growth to supply the desired flux, which was calibrated by a quartz crystal microbalance. The Pd flux was kept at ∼2.5 × 10¹³ cm⁻² s⁻¹, while the Bi flux was adjusted to change the Pd:Bi flux ratio and tune stoichiometry within the film. Se was supplied in excess of 20 × 10¹³ cm⁻² s⁻¹ to overcome the high volatility and prevent formation of defects related to Se vacancies. Samples were grown on both SrTiO₃ (001) and SrTiO₃ (110) substrates. X-ray diffraction measurements (XRD) were performed on a Malvern Panalytical X’Pert³ with a four-circle goniometer using Cu k_α1 radiation. Normal state transport measurements were performed in the van der Pauw geometry using pressed indium contacts in a quantum design physical property measurement system down to a base temperature of 2 K. Millikelvin transport was probed in an Oxford Triton cryogen-free dilution refrigerator using Stanford Research SR860 lock-in amplifiers. Magnetometry measurements were performed with Quantum Design MPMS3 with the iQuantum He3 option. Rutherford backscattering spectroscopy (RBS) measurements were conducted at Auburn University using a 6HDS-2 tandem, National Electrostatics Corporation Pelletron, with two sources for ions, an alphatross (RF source for production of He⁺) and SNICS source (source of negative ions by cesium sputtering), using a He²⁺ ion beam energy of 1.972 MeV, an incident angle of α = 0°, an exit angle of β = 10°, and a scattering angle of θ = 170°. Fits to the experimental data were completed using the analysis software SIMNRA (simnra.com). First-principles density functional theory (DFT) calculations were performed using the Vienna ab initio simulation package (VASP);⁶⁶ more details can be found in the supplementary material. Scanning transmission electron microscopy (STEM) measurements were taken using a Nion ultraSTEM C3/C5 aberration corrected microscope operating at 100 keV. The sample was prepared using a FEI Nova dual beam focused ion beam (FIB) using a Ga⁺ liquid ion source.

MBE growth of ternary chalcogenide thin films is expected to be highly sensitive to the growth parameters, particularly, deposition temperature, and the relative flux ratio among all three elements. For the current case of Pd₃Bi₂Se₂, the relative volatility of Se is much larger than that of Bi, which, in turn, is much larger than that of Pd. For example, the cell temperatures used, which reflect relative volatility, are of order 200 °C for Se, 600 °C for Bi, and 1300 °C for Pd. Furthermore, all Bi–Se compounds are also relatively volatile in comparison to Pd.⁶⁷ This can be seen since Bi–Se can be used as a congruently sublimating binary source where the temperature is of order 400–500 °C.^67,68 We conjectured that a stoichiometric growth window can be found when Se is supplied well in excess of both Bi and Pd and Bi is supplied at or in excess of the stoichiometric ratio with Pd. Specifically, on the growing surface, Se and Bi will physically adsorb onto the surface and then (i) bond directly into the Pd₃Bi₂Se₂ structure, (ii) form a Bi–Se phase, which can then be incorporated into the Pd₃Bi₂Se₂ structure or evaporate as a compound, or (iii) can directly evaporate off the growing surface. As such, excess Se (>10× Bi) is necessary to overcome the high rate of evaporation of Se and enable both the reactions to Bi–Se compounds and Pd₃Bi₂Se₂ to go forward. This method of excess Se flux has been explored in great detail in binary selenide compounds and is necessary for the formation of stoichiometric thin films with minimal Se vacancies.^69–71 Therefore, fixing the Se flux at >10× Bi enables the growth mechanisms to be systematically elucidated by varying only two parameters: the growth temperature and the Bi flux relative to the Pd flux (i.e., Bi:Pd > 3:2), which we discuss next.

To explore the temperature dependence of the growth, four films were grown at 200, 300, 400, and 500 °C at roughly the stoichiometric Pd:Bi flux ratio of 3:2 on SrTiO₃ 001 to a thickness of 40 nm. The resulting XRD 2θ–θ scans are shown in Fig. 2(a). Single phase Pd₃Bi₂Se₂ films could be achieved between 300 and 400 °C, where the predominate set of peaks is the monoclinic 00L planes (orthogonal to the monoclinic a–b plane), which indicates the films preferentially nucleated with the 2D-like vacancy-ordered planes parallel to the substrate surface. The additional peaks that can be observed off the main 002 (29.41°) peak are due to the $22029.89°,222̄29.99°,and402̄(30.4°)$ orientations of the film, which are all structurally similar to the perovskite 110 surface but with different vacancy orderings. At 200 °C, films segregated into Bi₂Se₃, Pd₃Bi₂Se₂, and other unidentified phases, which indicates that additional Se is absorbed due to the insufficient thermal energy. At 500 °C, in contrast, films became insulating and optically transparent rather than reflective, and XRD 2θ–θ scans revealed a large misorientation of the Pd₃Bi₂Se₂ film and emergence of an unidentified peak due to the formation of a secondary phase. This can, in part, be understood as the inability to form a conformal film of Pd. This is likely since 400–500 °C coincides with the so-called 3/8-rule for growth of a metal film. At or below 3/8 of the melting point of a metal, a conformal film can be achieved, and above this temperature, the material will coalesce into islands.⁷² For the case of the ternary oxide PdCoO₂, this has been similarly found to delineate the growth of conformal films vs islands.⁷³ This hints that the reluctance for Pd to bond (i.e., Pd is a noble metal) may result in the dominance of this simple metal-like behavior but requires future study.

Together, these data show that across the approximate temperature range of 300–400 °C, the Pd₃Bi₂Se₂ phase is stabilized in excess Se with stoichiometric Bi. Using this information, a series of 40 nm thick Pd₃Bi₂Se₂ films were grown to map out the growth mechanisms in excess Bi to show if there exists a stoichiometric growth window. Furthermore, the substrate orientation was changed to SrTiO₃ 110, which more closely matches faces of Pd₃Bi₂Se₂. This can be seen by comparing the unit cells for perovskite 110 to those of parkerite 001, as shown in Fig. 1(g). Here, it can be seen that the Bi sites overlay well with Sr sites for the 110 SrTiO₃ surface, whereas the match to SrTiO₃ 001 is not nearly as good. Experimentally, it was found that the growth on SrTiO₃ 110 reduced misoriented nucleation. Moreover, as is commonly used to aid nucleation and promote conformal coverage at a heterointerface, an initial 2 nm layer was grown at a stoichiometric flux ratio of 3:2 at 300 °C^69,74 (these conditions were chosen since it is the lowest temperature where there were no Bi-rich phases found). The sample was then heated to 400 °C, at which time the remaining 38 nm film was deposited with the desired Pd:Bi flux ratio. Films were grown over a range of Pd:Bi = 3:1.8–3:5, and the resulting XRD 2θ–θ scans are shown in Fig. 2(b). Films grown at low Bi flux again showed secondary phases and misoriented domains, similar to the film grown at 500 °C, which suggests that the previously grown film was Bi-deficient in addition to not being conformal. Increasing the Bi flux to well above the stoichiometric ratio to 3:5 led to fully 001 oriented films. We hypothesize that the orientation control through Bi flux may be related to the effects of surface energy and nucleation (see below), which are likely highly dependent on stoichiometry.

A zoom-in of the 2θ–θ scan about the 002 Pd₃Bi₂Se₂ peak for the film grown at 3:3.5, shown in Fig. 2(c), reveals Laue-oscillations about the main peak, which indicates that the films are near atomically flat. Furthermore, the rocking curve scan shown in Fig. 2(d) exhibits a narrow full-width at half-maximum (FWHM) of 0.033° to 118 arc sec, indicating a large coherence among the 00L planes and, therefore, a very high structural quality. The 00L lattice spacing was extracted, which for all films was between 6.054 and 6.061 Å, varying by less than 0.5% and was relaxed close to the bulk parameter of 6.069 Å. Furthermore, all films were textured with grain sizes on the order of >1 µm with a slight degree of in-plane (IP) twinning and rotations from the substrate, as can be seen in the atomic force microscopy (AFM) images (supplementary material, Fig. S1). Furthermore, high-angle annular dark-field imaging scanning transmission electron microscopy (HAADF-STEM) measurements were performed, and the results are shown in Fig. 2(e) where there are several notable features. First, the wide scale image confirms that the lateral scale of grains is much larger than the viewing window. Second, the zoom-in confirms that vacancy-ordered planes lay perpendicular to the substrate surface (parallel to the 110 planes of SrTiO₃) and the high coherency among the 00L planes. This can be seen by comparing the zoomed-in image to the corresponding structural model and composite unit cell shown in Fig. 2(f). Altogether, this confirms that the films grown are of high structural quality but do not directly address the films’ stoichiometry.

RBS, a direct probe of film stoichiometry, was performed on two different samples, and the results are shown in Table I. The first sample was grown at the stoichiometric flux ratio of 3:2 Pd:Bi, while the second was grown at the Pd:Bi flux at a ratio of 3:3.6. Despite this, both samples had nearly identical film stoichiometries of Pd_3(±0.06)Bi_1.81(±0.04)Se_2.16(±0.11) and Pd_3(±0.06)Bi_1.84(±0.04)Se_1.97(±0.10), respectively. This confirms that excess Bi and Se are not incorporated into the film as impurity phases or defects, but rather are desorbed off the growing surface. Interestingly, the results suggest that the films are slightly Bi-deficient as increasing Bi flux does not change the composition (i.e., Pd₃Bi_2(1−δ)Se₂, with δ ≲ 10%). However, it is not clear if this is due to vacancies on the Bi site, if Pd is occupying some of the vacant sites in the half-antiperovskite structure, or some other mechanism. Remarkably, these results when combined with XRD show that film orientation is highly dependent on the Pd:Bi flux ratio, while the stoichiometry is nearly independent of this ratio when sufficient Bi is supplied at the proper growth temperature.

Altogether, this information enabled the creation of a schematic phase diagram, shown in Fig. 3, which highlights the conditions needed for self-regulated growth by MBE. The phase diagram was plotted with the vertical axis being the Pd:Bi ratio (with increasing Bi and fixed Se) and the horizontal axis being the temperature. It was found that at and below 250 °C, Bi₂Se₃ and Pd containing compounds were the primary phases where the Bi₂Se₃ constituents increase with the increasing Bi flux. For temperatures above 300–450 °C, there is sufficient thermal energy to create the Pd₃Bi₂Se₂ phase via desorption of excess Bi and Se, which confirms that the Pd flux controls the growth. This adsorption-controlled growth window (shaded region) is delineated by the solid to dashed lines that trend upward with increasing temperature, which qualitatively capture the increased propensity for desorption at increased temperature [i.e., for the upper curve, at higher temperature, the phase boundary between Pd₃Bi₂Se₂ and Bi-rich phases increases to higher Bi-vapor pressure (effectively flux), and for the lower curve, at higher temperature, the phase boundary between Pd₃Bi₂Se₂ and Pd-rich phases similarly increases to higher Bi-vapor pressures]. Within the growth window, the films are extremely flat and of high structural quality. Above 450 °C, Bi and Se volatilize out of the film faster than it could be incorporated, leading to insulating, nonstoichiometric material. Thus, the optimal deposition strategy using this diagram was found to be a thin stoichiometric buffer layer at 300 °C, followed by deposition of the remainder of the film at 400 °C in excess Bi.

The XRD results for the stoichiometric series point to a relation between nucleation kinetics and the amount of Bi supplied to the growth front. Specifically, films grown at a stoichiometric flux ratio are single-phased but possess multiple crystalline orientations, and single domain films are only achievable at the highest Bi fluxes. First-principles density functional theory (DFT) calculations were employed to give insight into this novel effect (see the supplementary material for calculation details). Surface energy calculations were performed for the four observed film orientations for each surface being either mixed-terminated or Pd-terminated, with the atomic supercell structures used in the calculation. This mimics the likely scenario where Bi-deficient/Pd-rich conditions may favor mixed-terminated surfaces and drive nucleation of these other orientations of Pd₃Bi₂Se₂. The results are given in Fig. 4, where Figs. 4(a)–4(h) show the surface structures in both plane-view (top) and cross section (bottom) and Fig. 4(i) shows the relative surface energy for the Pd chemical potential, μ_Pd. It was found the mixed-terminated 001 surface was the most stable for all investigated chemical potentials, while the Pd-terminated 001 surface exhibited the highest energy. This is consistent with the bonding geometry of Pd₃Bi₂Se₂ where the mixed termination splits the structure at the layered-like boundary and the Pd termination has the split within a layered unit (breaking more chemical bonds). This is further consistent with the experimental results, which show that the predominate orientation is always 001, which indicates that Pd₃Bi₂Se₂ prefers growing as a layered unit assembled parallel to the surface. Similarly, for the other surfaces, the calculations show that a Pd-terminated surface is higher in energy than the same surface with mixed termination. Furthermore, these are significantly lower in energy than a Pd-terminated 001 surface. To understand how these correlate with the experimental observations, consider the growth in Pd-rich conditions where the lowest energy surface is 001. The excess Pd is likely accommodated in some portion as a combination of Pd antisite defects (likely on the Bi site) or as Bi-vacancies. However, for Pd-rich conditions, the formation of a full 001 Pd-terminated surface is very unlikely. Thus, the DFT results suggest that it may be far more likely to nucleate alternate surfaces that are lower in energy, which correlates well with XRD analysis. These calculation and experimental observations point out an interesting observation where film orientation may be controlled not simply by choosing a specific substrate but through the overall growth conditions and is an interesting open question going forward.

Electrical transport and magnetization measurements give information about intrinsic properties, such as the character of superconductivity and aspects of the electronic band structure, and extrinsic properties, such as defects incorporated during the growth. The resistivity is shown in Fig. 5(a) for a sample grown with Pd:Bi = 3:2.5, which reveal metallic transport behavior down to 2 K where the resistivity decreases with decreasing temperature. Overall, the residual resistivity ratios (RRR, ratio of the resistance at 300 K and resistance at 2 K) for all films in this series were found to be around 20 with no clear relation to the growth conditions, as shown in the inset of Fig. 5(a). This shows that growth conditions are not the limiting factor for the transport mean-free-path and points to an additional mechanism. This could be grain boundaries, dislocations, film thickness, or possibly the Bi deficiency. Regarding the latter, however, the percent-level of native Bi-deficient defects found in RBS would likely severely limit the overall mean-free-path to a few nanometers and, thereby, RRR. This will be further addressed next in the context of Hall effect measurements.

Magnetic field dependent Hall effect measurements can reveal additional insights into the electronic properties of these films. The Hall effect was measured at a temperature of 2 K and a magnetic field of up to 14 T. The data were then anti-symmetrized to remove the unintentional mixing of the longitudinal component into the Hall component. The resulting data, shown in Fig. 5(b), revealed a dramatic non-linear character. This indicates intrinsic multicarrier behavior in the films due to both electron conduction and hole conduction. This can be seen by a change in the sign of the slope of the Hall effect in going from the low-field to the high-field limits. Data were then fit using a two-carrier electron and hole model to determine the carrier type and mobility (see the supplementary material for more details). The dominate carrier type was found to be p-type (hole-like), with ∼2 to 4 × 10²² cm⁻³ with a mobility of ∼100 cm² V⁻¹ s⁻¹, and a small electron pocket contributed between 1 and 3 × 10¹⁹ cm⁻³ and a mobility of ∼1000 cm² V⁻¹ s⁻¹. There was no discernible dependence on the growth conditions. Overall, the mixed character found in the Hall effect data is likely not anomalous. It is quite typical that good metals exhibit multiband transport due to band filling and subsequent zone-folding.^75,76 To observe such multicarrier behavior requires a large mobility such that the inverse mobility is less than the maximum field scale used (i.e., μ⁻¹ < B_max). Here, we observe the crossover to be of order several tesla, indicating a mobility of order of a 1000 cm² V⁻¹ s⁻¹. This is consistent with the low-density, higher mobility electron pocket dominating the response at a low-field, which gives way to the high-field response being dominated by the lower mobility hole. From this mobility scale (several-hundred to thousand), the transport mean-free-path can be estimated from a simple free electron model where l_mfp = ℏμ/e(3π²n_3d)^1/3, where ℏ is Planck’s reduced constant, μ is the mobility, e is the electron charge, and n_3D is the carrier density. This yields l_mfp ≈ 100–1000 nm, which is consistent with the scale of both the grains (∼μm) and the thickness (depending on the level of specular vs non-specular scattering, surfaces can be the dominant scattering mechanism with l_mfp ≫ thickness).⁷⁷ 

Finally, polycrystalline bulk Pd₃Bi₂Se₂ samples have been reported to be a low temperature, type-II s-wave superconductor.³⁵ Low-temperature AC-magnetic susceptibility magnetometry measurements were carried out on a stoichiometric film down to 300 mK for in-plane (IP) and out-of-plane (OOP) geometries, measured on cooling in a zero applied DC field, with an AC field of 2.5 Oe and a frequency of 75.7 Hz. DC superconducting quantum interference device (SQUID) measurements were performed using the same instrument with ZFC (zero field cooled) followed by FC (field cooled) cooling at each field. The results of the AC-magnetometry measurements are shown in Figs. 6(a) and 6(b), respectively. Superconductivity can be seen to arise where there is a large drop in the in-phase component and a rise in the out-of-phase component, which indicates the onset of diamagnetic screening.⁷⁸ This occurs at approximately T_C ≈ 800 mK for both the IP and OOP orientations. The unusual temperature dependence observed in the OOP geometry can likely be attributed to the extreme demagnetization effects in this geometry.⁷⁹ Similarly, resistance measurements were performed down to 25 mK in an OOP orientation, and the results are shown in Fig. 6(c). Here, the resistivity sharply drops into the zero-resistivity state at 800 mK, which agrees well with magnetization measurements. Further insights into the superconducting properties were gained by detailed temperature sweeps at different magnetic fields. In Fig. 6(c), the results are shown for the transport measurements. Here, with the increasing magnetic field, the transition into the superconducting phase is successively pushed to lower temperatures with fields >16 mT, quenching the superconductivity down to 25 mK. In addition, as the magnetic field was increased, the width of the transition into the superconducting phase increased. Specifically, for zero-field, the width is less than 10 mK, and above 14 mT, the width increased to about 100 mK consistent with weak-link behavior observed in textured thin films.⁸⁰ The field dependence of the superconductivity transition is captured by plotting T_C vs μ₀H, shown in Fig. 6(d), where T_C was taken at the onset and at half the normal state resistance in transport and from the onset in DC SQUID measurements. The slope of μ₀H_C near T_C was found to follow the Werthamer–Helfand–Hohenberg (WHH) theory for a type-II superconductor in the dirty limit.^35,81 The critical field at zero temperature [H_C2(0)] is extracted using the WHH theory for the slope of μ₀H_C2(T) as T → T_C,

Here, μ₀H_C2(0) was extracted to be 17.7 ± 0.5 mT. From this, the functional form for T_C vs H, given by

is plotted as a dashed curve in Fig. 6(d), with H_C2(0) given by the WHH theory. It can be seen that the data are in excellent agreement with this function in the zero-temperature limit. Furthermore, the coherence length at zero temperature [ξ_GL(0)] is estimated to be around 140 nm from the following relation:

where ϕ₀ is the magnetic flux quanta. This value is significantly larger than previous experiments on bulk polycrystalline samples (32.3 nm).³⁵ Additionally, this coincides with the order of magnitude of the transport mean-free-path estimated from the Hall effect observed in the film, which, again, suggests either grain boundaries or the surfaces are the limiting defects in these samples and points to a high crystalline quality within each grain.

In conclusion, we have demonstrated thin film synthesis of the strongly spin–orbit coupled superconductor Pd₃Bi₂Se₂ using molecular beam epitaxy. By carefully controlling growth conditions, we have shown the existence of a self-regulated growth window, resulting in highly reproducible thin film growth with high structural quality. This is confirmed via XRD and RBS measurements. Films grown within this window exhibit metallic conductivity with relatively high residual resistivity ratios, superconductivity below 800 mK, and a relatively large coherence length of ∼140 nm. Synthesizing high-quality thin film materials is the first step in the scientific process, and key questions remain for Pd₃Bi₂Se₂ and the parkerites and shandites, in general. Of particular importance are understanding and characterizing the topology of the band structure. This can be answered by a combination of first-principles calculations and photoemission spectroscopy, which also will give insights into the strong multiband behavior observed here. From a materials perspective, the role of defects is of critical importance. It remains unknown why Pd₃Bi₂Se₂ is found to be intrinsically Bi-deficient and if this is the stable state or if this is controlled by the volatility of Bi and Se specific to MBE growth. Growth of high-quality bulk crystals will enable illumination of this question. Second, the observed stoichiometric-dependent nucleation suggests that this may be used to create films or nanostructures with different orientations that are independent of the substrates’ surface structure. Finally, Pd₃Bi₂Se₂ is found to nucleate better on the SrTiO₃ (110) surface compared to the (001) surface due to the better surface match. However, there, the films are still found to be composed of in-plane grains, which may be the limiting factor for transport. Therefore, understanding what determines how the parkerites nucleate on these surfaces and their overall interfacial structure, as well as finding if there exists a better substrate, is of critical importance for future work, which will push toward higher film quality. Overall, this work opens the potential for thin film synthesis of the parkerites and shandite classes of materials, which when coupled with their chemical and structural compatibilities make a promising playground to explore the interplay of spin–orbit coupling, topology, magnetism, and superconductivity.

See the supplementary material for additional structural measurements, first-principles calculation methods, and details regarding the multicarrier transport analysis, which also contains Refs. 66, 82, and 83.

This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division (growth, structure, and electron microscopy), and the National Quantum Information Science Research Centers, Quantum Science Center (transport). This research used resources of the Oak Ridge Leadership Computing Facility and the National Energy Research Scientific Computing Center, DOE Office of Science User Facilities. RBS measurement at Auburn University was supported by the Air Force Office of Scientific Research under Award No. FA9550-20-1-0034.

This article was authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).