Crystal-chemical origins of the ultrahigh conductivity of metallic delafossites

Despite their highly anisotropic complex-oxidic nature, certain delafossite compounds (e.g., PdCoO2, PtCoO2) are the most conductive oxides known, for reasons that remain poorly understood. Their room-temperature conductivity can exceed that of Au, while their low-temperature electronic mean-free-paths reach an astonishing 20 μm. It is widely accepted that these materials must be ultrapure to achieve this, although the methods for their growth (which produce only small crystals) are not typically capable of such. Here, we report a different approach to PdCoO2 crystal growth, using chemical vapor transport methods to achieve order-of-magnitude gains in size, the highest structural qualities yet reported, and record residual resistivity ratios ( > 440). Nevertheless, detailed mass spectrometry measurements on these materials reveal that they are not ultrapure in a general sense, typically harboring 100s-of-parts-per-million impurity levels. Through quantitative crystal-chemical analyses, we resolve this apparent dichotomy, showing that the vast majority of impurities are forced to reside in the Co-O octahedral layers, leaving the conductive Pd sheets highly pure (∼1 ppm impurity concentrations). These purities are shown to be in quantitative agreement with measured residual resistivities. We thus conclude that a sublattice purification mechanism is essential to the ultrahigh low-temperature conductivity and mean-free-path of metallic delafossites.


Introduction
Complex oxide materials have proven to be fertile ground for the discovery of new physical phenomena and the advancement of technologically important device function, the ABO3 perovskites being a prime example [1][2][3][4] . Many related classes of complex oxides offer similarly substantial potential but remain less extensively studied. ABO2 delafossites are one example, where two-dimensional triangular sheets of typically monovalent A ions are separated by layers of edge-shared BO6 octahedra [5][6][7][8][9][10] . Most delafossite compounds are insulating or semiconducting, such as the CuFeO2 studied for frustrated magnetism 11,12 and the CuAlO2 considered for transparent conductive oxide applications 13 . It has been known since 1971, however [5][6][7][8] , that some delafossites exhibit metallic character, exemplified by PdCoO2 and PtCoO2. These metallic delafossites received little attention until the relatively recent discovery that, in bulk-single-crystal form, they are the most conductive oxides known 5,10 . While their c-axis resistivities (c) are more than 100 times higher 5,8,9,[14][15][16] , the room-temperature a-b-plane resistivities (ab) of PdCoO2 and PtCoO2 are only 3.1 and 1.8 cm, respectively 17 , comparable to or lower than Au 5 . Their lowtemperature (T) residual ab values fall as low as 8.1 ncm 17 , implying residual resistivity ratios (RRRs) up to 376 17 and low-T mean-free-paths of 20 m 5,15 . These are astonishing values, particularly in complex oxides, in such highly anisotropic structures, and given the relatively little materials development performed. 3 The electronic structure of metallic delafossites is similarly noteworthy. Taking PdCoO2 as an example, the established R-3m structure of the 3R polymorph [5][6][7]9,10,18 results in a strikingly simple electronic structure where a single Pd d band crosses the Fermi level, with only modest contributions from Co, and negligible O character 5,9,10,[18][19][20][21] . The triangular Pd 1+ planes thus generate a Fermi surface closely approximating a hexagonal-cross-section cylinder filling half the hexagonal Brillouin zone in the a*-b* reciprocal-space plane 5,15,[18][19][20][21][22][23] . The remarkable simplicity of this electronic structure is underscored by electron effective masses of only 1-1.7 me 5,15,18,24 .
The edge-shared CoO6 octahedra also have short Co-O bond lengths and large crystal field splitting, stabilizing low-spin (S = 0) Co 3+ 18 . PdCoO2 is thus a model metallic delafossite, which can be thought of as triangular metallic sheets of Pd 1+ separated by insulating nonmagnetic layers of Co 3+ O6 edge-shared octahedra.
The above attributes, particularly the exceptional low-T mean-free-path and simple hexagonal Fermi surface, have enabled a remarkable string of recent achievements with bulk crystals of PtCoO2 and PdCoO2. These include observations of: various forms of quantum oscillation 14,15,25,26 , very large positive magnetoresistance along the c-axis 27 , potential phonon-drag effects in resistivity 15 , possible hydrodynamic electron transport 25 , and itinerant surface ferromagnetism 23,28,29 . Yet more recently, non-local electrodynamics 30 , directional ballistic transport 31 , supergeometric electron focusing 32 , and non-local transport 33 have been reported in PdCoO2 crystals and micron-scale structures, exploiting the long mean-free-path and anisotropic Fermi surface. The addition of magnetic moments in PdCrO2 adds further richness, encompassing magnetic frustration [34][35][36] , a complex antiferromagnetic spin structure [34][35][36][37] , possible chirality 37 , an anomalous Hall effect 38,39 , and T-linear resistivity 40,41 potentially due to magneto-elastic scattering 41 . Metallic delafossites are thus making their mark in condensed matter and materials physics, and potential applications are being explored, in nanoelectronic interconnects 42 , harshenvironment electronic devices [43][44][45] , electrocatalysis 43,46 , ballistic transport devices 31-33,43,44 , etc. Despite this progress, the origins of the ultrahigh conductivities and mean-free-paths in metallic delafossites remain unclear. While progress has been made with understanding thermal properties of PdCoO2 18 and related phonon spectra 47,48 , much remains to be elucidated regarding electronphonon scattering and coupling 5,14 and thus the low room-T resistivities 5 . Yet more urgently given the above advances 14,15,[25][26][27][29][30][31][32][33] , the origin of the outstanding a-b-plane residual resistivities and low-T mean-free-paths remain incompletely understood 5,43 . It is widely accepted that ultrahigh purity and ultralow defect density are necessary for such behavior 5,14,15,25,49,50 , but, confusingly, the crystal growth methods applied to metallic delafossites 6,14,16,[51][52][53][54] are not typically capable of such 5,49 . The prevailing method for PdCoO2 and PtCoO2 is a metathesis-based flux growth 6,14,16,52,53 , which is not well understood, typically limited to 1-mm lateral sizes, and not performed under conditions that would be expected to realize ultrahigh purity. Multiple authors have in fact commented on the crude synthesis methods for metallic delafossites relative to other systems with comparable mean-free-paths 5,49 . On the other hand, recent electron microscopy on single-crystal PdCoO2 supports sufficiently low defect densities that only upper limits could be placed 49,55 . An electronic transport study of metallic delafossites as a function of irradiationinduced point defect density also evidenced extraordinarily low initial defect densities on the Pd/Pt planes, insensitive to the (unknown) disorder level on the B-O layers 49 . It is thus unclear what the impurity and defect densities are in single-crystal metallic delafossites, how they are divided between A and B-O planes 49 , if/how defect densities are minimized by current growth methods 49 , or whether some form of defect mitigation or tolerance occurs 49 . One proposal for the latter is the hidden kagome picture recently advanced to explain reduced electron-impurity scattering 56 .
The current work seeks to elucidate some of the above through advances that are thus far absent from the PdCoO2 literature: improvements in bulk crystal growth in terms of mechanistic understanding, crystal size, defect density, and purity; more extensive structural and chemical characterization; and comprehensive trace impurity analysis. The latter is particularly conspicuous in its absence from the metallic delafossite literature, likely because the premier methodinductively coupled plasma mass spectrometry (ICP-MS) -typically involves acid digestion.
Here, we first elaborate on a new crystal growth method for PdCoO2 based on chemical vapor transport (CVT), for which a simple mechanism is proposed. CVT is then compared to the established metathesis/flux method and is shown to enable orders-of-magnitude gains in crystal size and mass, the highest structural quality yet reported, and record RRR. Nevertheless, particleinduced X-ray emission (PIXE) and low-T magnetometry suggest significant impurity concentrations. Enabled by microwave digestion methods, the first comprehensive trace impurity analyses of metathesis/flux-grown and CVT-grown PdCoO2 crystals were thus performed by ICP-MS, with remarkable conclusions. Specifically, these metallic delafossite crystals are definitively not ultrapure in a general sense. Standard metathesis/flux crystals typically have 250 ppm metalsbasis impurities; CVT crystals are purer but still harbor 50 ppm impurities. Through a detailed crystal-chemistry-based analysis, however, we show that the vast majority of these impurities must populate Co-O octahedral layers, while as few as four elements can conceivably substitute for Pd.
The concentration of these elements is of order 1 ppm, 100 times lower than the total impurity concentration, which is shown to be in quantitative agreement with the measured residual ab. We thus propose that a "sublattice purification" mechanism is central to the extraordinary low-T conductivity of metallic delafossites, enabling ultrahigh purity in the conductive Pd/Pt sheets, largely unaffected by substantial impurity concentrations in the CoO6 octahedra.

Chemical vapor transport crystal growth
As noted in the Introduction, metathesis-based flux growth has become standard for single-crystal PdCoO2 and PtCoO2 6,14,16,52,53 . This is based on metathesis reactions such as PdCl2 + Pd + Co3O4 → 2PdCoO2 + CoCl2, i.e., PdCl2, Pd, and Co3O4 reagents (99.995-99.999% purity in our case) reacting to form the target PdCoO2 plus a CoCl2 by-product, which is chemically removed (see Methods for more details). In the current study, dwell temperatures of 700-750 C were employed for metathesis growth, followed by cooling to 400 C at 40-60 C/h. As shown in the scanning electron microscopy (SEM) image in Fig. 1b, the resulting product is a mass of PdCoO2 crystals and crystallites. The optical microscope image in the inset to Fig. 1b highlights one of the largest crystals from a typical growth, measuring 0.8  0.6 mm 2 laterally, and 0.017 mm thick. This corresponds to 0.008 mm 3 and 0.065 mg, comparable to literature reports 16,25,31,49,50 . Typical powder X-ray diffraction (PXRD) data from such crystals are shown in Fig. 1d (green), displaying a good match with a PdCoO2 standard (grey), and yielding lattice parameters a = 2.8308  0.0002 Å and c = 17.747  0.002 Å, in agreement with accepted values 5,6 .
While such crystals have enabled exciting advances 14,15,17,[22][23][24][25][26][27][29][30][31][32][33] , as discussed in the Introduction, it is nevertheless true that little progress in PdCoO2 crystal size or quality has been reported since ca. 2007. There is also limited understanding of the growth mechanism. Most publications describe the mechanism as "flux" or "self-flux" and employ specific temperature-time trajectories, but with little explicit justification 14,16,[51][52][53] . Seeking improvement over this, we explored CVT crystal growth of PdCoO2. CVT is not commonly applied to complex oxides but the fact that PdCoO2 decomposes (at 800 C in air, for example 45 ) hints at CVT as a possibility, as does the relatively low decomposition temperature of PdCl2 (600 C in low Cl2 partial pressure 57,58 ), a potential transport agent. CVT was thus performed in a set-up shown schematically in Fig. 1a, and elaborated upon in Methods and Supplementary Information Sections A and B. The solid precursors are coarse PdCoO2 powder derived from the above metathesis growth, along with 99.999%-pure PdCl2. At typical hot and cold zone temperatures of 760 and 710 C, we propose that PdCl2 decomposition is followed by a reaction such as PdCoO2(s) + 2Cl2(g) ↔ PdCl2(g) + CoCl2(g) + O2(g), i.e., Pd chloride and Co chloride vapor transport across the temperature gradient in a Cl2(g) and O2(g) atmosphere. The reaction runs to the left in the growth (cold) zone, resulting in PdCoO2 crystals and chloride by-products, which are chemically removed (see Methods).
As shown in Fig. 1c, after optimization of hot and cold zone temperatures, relatively large PdCoO2 crystals and multicrystals derive from this process, at high yield (>90%). Specifically, the reaction product is a mix of single crystals and larger multicrystals, the latter likely forming due to nucleation of Pd or PdO seeds during the initial inverted-temperature-gradient period of the growth (see Methods and Supplementary Information Section A). Multicrystals up to 12  12 mm 2 laterally and 0.3 mm thick are grown (Fig. 1c), corresponding to 43.2 mm 3 and 345 mg. Gently breaking apart such multicrystals isolates single crystals up to 6.0  4.0 mm 2 laterally and 0.17 mm thick, corresponding to 4.08 mm 3 and 32.6 mg. A representative example is shown in the inset to Fig. 1c. These CVT PdCoO2 multicrystals are thus larger than our metathesis/flux crystals by factors of 15 in maximum lateral dimension, 18 in thickness, and 5300 in volume and mass.
Correspondingly, the CVT single crystals are larger than metathesis/flux single crystals by factors of 8 in maximum lateral dimension, 10 in thickness, and 500 in volume and mass.
As illustrated in Fig. 1e [59][60][61][62][63][64] , including in complex oxides [60][61][62][63][64] . Minimization of unit cell volume has in fact emerged as a guiding principle for synthesis of highly perfect complex oxides with minimized densities of non-stoichiometry-accommodating defects [60][61][62][63][64] . These data thus hint at lower defect densities in CVT-grown crystals, a conclusion that is reinforced below. As a final comment on structure, note that powdered samples of these CVT crystals were recently used in a 12-1000 K structural study via PXRD and accompanying refinement, confirming their singlephase nature and the thermal stability of the R-3m structure 18 .

Structural and chemical characterization
With CVT growth established to provide substantial improvements in PdCoO2 crystal size and mass, and with improved defect density suggested by the lattice parameters, it is important to more rigorously assess structural quality and purity. Preliminary prior characterization established single crystallinity via two-dimensional single-crystal X-ray diffraction 18 . In Fig. 2a this is improved on by presenting Laue diffraction data conclusively confirming single crystallinity. The six-foldsymmetric pattern of diffraction spots down this [001] zone axis, interspersed with a second sixfold pattern of weaker spots (seen upon close examination of Fig. 2a), is as expected, as illustrated by the simulated pattern in Fig. 2b based on the accepted R-3m structure and lattice parameters 6 .
Wide-range specular high-resolution X-ray diffraction (HRXRD) was also performed on 001oriented crystals (Fig. 2c), showing only sharp 003 through 0015 reflections, further confirming single crystallinity and phase purity. Of particular significance, a HRXRD rocking curve through the 006 reflection is shown in the inset to Fig. 2c, revealing a full-width-at-half-maximum of only 0.0089. To the best of our knowledge, this is the lowest rocking curve width reported for a metallic delafossite, demonstrating very low mosaic spread, another strong indicator of low defect density.
Turning to chemical characterization, Fig. 2d shows an energy-dispersive X-ray spectroscopy (EDS) scan from a representative PdCoO2 CVT crystal. Only Pd L and Co K peaks are clearly visible in this wide-energy-range scan, their relative intensities giving a Pd/Co ratio of 0.98  0.05, consistent with the nominal stoichiometry. The inset to Fig. 2d focuses on the low-energy region shaded gray in the main panel, revealing, with the exception of a small detector-artifact peak and the inevitable C contamination in SEM/EDS, only peaks from Pd, Co, and O. Unsurprisingly, EDS is thus not sufficiently sensitive to detect any trace impurities in these PdCoO2 crystals, which was the motivation for the PIXE data in Fig. 2e. PIXE is analogous to EDS except that the excitation is achieved with MeV-range He ions rather than electrons, vastly decreasing the broad background due to bremsstrahlung. As can be seen in Fig. 2e, which shows spectra from two different CVT crystals (orange and blue), PIXE thus exposes a plethora of Pd, Co, and sum peaks, at high signalto-noise ratio. Aside from some instrumentation artifacts (labeled with asterisks in Fig. 2e), almost all aspects of these spectra are reproduced by a GUPIXWIN 65 simulation for PdCoO2, shown as the black line. The inset to Fig. 2e, however, shows a close-up of the 4.5 to 7.3 keV region (shaded gray in the main panel), highlighting two peaks that are not accounted for by Pd, Co, or O. These peaks also exhibit crystal-to-crystal variations in intensity, strongly suggesting that they derive from trace impurities. The 5.4 keV peak very likely indicates Cr impurities, while the 6.4 keV peak is very likely due to Fe. Full quantification of impurity concentrations at this level is challenging for PIXE but our best estimates based on GUPIXWIN fits to Pd L, Co K, and Fe K peaks suggest Fe concentrations of ≳ 50 ppm. (As for all impurity concentrations in this paper, this value is on a weight basis, i.e., it is in g/g). While this is only approximate, this is a first indication that even CVT-grown PdCoO2 crystals may not actually be ultrapure. As a final comment on these PIXE data, we note that Cl was not detected; the estimated limit of detection is of order 1000 ppm, however, due to overlap of Cl K and Pd L peaks.

Electronic and magnetic properties
Electronic transport is an excellent relative probe of total defect and impurity densities, and thus T-dependent ab measurements were made on the new CVT-grown crystals. It is essential to acknowledge at this point that such measurements are nontrivial for single-crystal PdCoO2 17,25,49,52 , because of the extremely small low-T resistances (due to the very low residual ab), and the sizable c-axis/a-b plane resistance anisotropy (350-750, depending on T 8,14-16 ). The most accurate measurements in the literature in fact use focused ion beam techniques to pattern structures with long channel lengths to increase resistance, at the same time fully constraining the current path in the a-b plane 17,25,49,52 . In our case, the availability of larger crystals than in prior work enabled cleavage into relatively long and thin bar-shaped samples for simpler, in-line, topcontact, four-wire measurements. The inset to Fig. 3a shows an example of such an in-line geometry on a 1.2-mm-long, 23-m-thick crystal. Even in this case, residual resistances were found to be 20 , necessitating low-noise measurements and careful accounting for instrumental offset voltages. Full details are provided in Methods and Supplementary Information Section C. Briefly, an AC resistance bridge with a pre-amplifying channel scanner was employed, using parallel measurements on superconducting V thin films to determine and account for offsets.
Top-contact geometries such as those shown in the inset to Fig. 3a are also subject to systematic error due to the large c/ab 8,14-16 . As described in Supplementary Information Section C, finiteelement simulations of current flow and potential drop were thus performed in our specific measurement geometry. These confirm overestimation of ab due to the large c that generates potential drop through the crystal thickness. This issue worsens with cooling (as c/ab increases), meaning, importantly, that RRR values in this geometry represent lower bounds on true values (see Supplementary Information Section C for details).
Based on the above procedures, presented in Fig. 3a is ab(T) for the representative CVT-grown PdCoO2 crystal shown in the inset; Fig. 3b shows the same data on a log10-log10 scale. The 300 K resistivity of this representative crystal is 3.65 cm, which can be compared to reported values of 2.0-6.9 cm using standard contact methods 8,14-16 and 3.1 cm using more accurate focusedion-beam-patterned geometries 17 . Our value of 3.65 cm is thus indeed slightly larger than the most accurate determination of 3.1 cm 17 , but within the bounds expected for overestimation of ab due to the top-contact geometry (see Supplementary Information Section C). On cooling, the ab(T) of this crystal saturates below 10-20 K, reaching a measured value of 8 n cm. Taking this at face value, and considering the residual resistivity measurement uncertainty (dominated by offset subtraction), a RRR of 436  25 is indicated. As noted above and discussed in detail in Supplementary Information Section C, however, this RRR represents a lower bound due to the top-contact geometry. Finite element simulations suggest a true residual ab of 4.4 n cm, yielding an actual RRR up to 670 (see Supplementary Information Section C). Even without this correction, however, our measured RRR of 440 is a record in PdCoO2, which can be compared to 376 in prior metathesis/flux-grown crystals measured with focused-ion-beam patterning methods 17 . In addition to realizing order-of-magnitude gains in crystal size and mass, our new CVT approach thus also results in measurably lower disorder in electronic transport.
Magnetometry measurements provide another probe of crystal quality and impurity concentrations in PdCoO2 and were thus performed on CVT crystals. A typical result for the T-dependent molar susceptibility (mol) is shown in Fig. 3c. As illustrated by the black solid line, these data are well

, a sum of T-independent Pauli-related and T-dependent
Curie-Weiss contributions, where C is the Curie constant and  is the Curie-Weiss temperature.
The Pauli contribution derives from delocalized Pd d electrons, the Pauli = 0 = 1.65 × 10 -4 emu mol -1 Oe -1 (the factor of 3/2 accounts for Landau diamagnetism) in this case being near the middle of the distribution of literature values (0.9 × 10 -4 to 2.9 × 10 -4 emu mol -1 Oe -1 , see Table 1). The low-T Curie-Weiss tail, however, which should not ideally occur in PdCoO2 (the Co 3+ ions are low-spin, S = 0), very likely arises from local moments due to magnetic impurities. As shown in Table 1, the  = -2.6 K extracted from the fit in Fig. 3c is indeed small, indicating weak intermoment interactions, as would be expected of dilute impurities. This is reinforced by the inset to  (Table 1). Further insight into this is provided in Fig. 3d, which shows the applied magnetic field (H) dependence of the magnetization (M) at 2 K, after correcting for 0. The solid black line Brillouin fit captures the data moderately well, yielding a reasonable J = 1.2, along with a concentration of local moments corresponding to 130 ppm. (This is again on a weight basis; the atomic mass of Fe was assumed to convert from a number density of impurities (from the fit in Fig. 3d) to a weight-basis concentration). Taking this J and combining it with the Curie constants from the fits in Fig. 3c then yields local moment concentrations of 200 ppm. The low-T paramagnetism in Figs. 3c, d, can thus be consistently described by dilute, non-interacting local atomic moments, at concentrations of order 100 ppm.
While some fraction of this Curie-tail magnetism no doubt arises from surface contamination, this is a second indication, along with PIXE data (Fig. 2e), that even CVT-grown PdCoO2 crystals host substantial impurity concentrations. Nevertheless, the eff values from the Curie-Weiss tails of CVT-grown crystals do represent a significant improvement over metathesis/flux crystals. This is emphasized in Table 1, which includes data from our metathesis/flux crystals, in addition to prior literature 14,16,53 . The Curie-Weiss eff of our CVT crystals is over 4 times lower than our metathesis/flux crystals and is the lowest reported for PdCoO2. Literature values on metathesis/flux-grown crystals in fact span 0.15 to 0.78 B per formula unit (Table 1) 14,16,53 , implying substantial magnetic impurity concentrations, even in crystals with large RRR 14 . This apparent contradiction -clear indications of significant densities of magnetic impurities from magnetometry, in crystals that exhibit low disorder in transport -appears not to have been addressed in the literature. While CVT crystals are improved in this regard (they have the lowest eff), and surface contributions are likely, the implied level of magnetic impurities is concerning.
For this reason, and to augment the chemical characterization in Figs. 2d, e, we thus undertook trace impurity analysis via ICP-MS.
Such substantial impurity concentrations, which are broadly consistent with PIXE and magnetometry (Figs. 2e and 3c, d), frame a central question for the remainder of this work: How can the record residual ab of these materials, and their exceptional RRR, be reconciled with such impurity concentrations? To underscore the importance of this question, note that the reported 20-m low-T mean-free-path of PdCoO2 would require impurity concentrations several orders-ofmagnitude lower than the above ICP-MS values, a point that we return to quantitatively below.
Analysis of the distribution of the detected impurity elements casts much light on this issue. To this end, we first elaborate a simple classification scheme for the various substitutional elemental impurities that can be accommodated in PdCoO2. Fig. 4a depicts the crystal structure of PdCoO2, along with a classification for potential A-and B-site impurities in the ABO2 delafossite structure.
The critical point here is that, very unlike ABO3 perovskites for example, there is a massive asymmetry in the number of elements that can conceivably populate A vs. B sites. This arises because the B-site in metallic ABO2 delafossites involves a common cation valence in oxides (3+) and a common coordination with O 2-(octahedral); this is as in perovskites, which can accommodate much of the periodic table on the B site. The A-site in metallic ABO2 delafossites, on the other hand, involves a relatively uncommon valence in oxides (1+) and a far less common coordination with O 2ions (linear, see the structure in Fig. 4a). This situation, which contrasts starkly with ABO3 perovskites, greatly reduces the possible accommodation of A-site vs. B-site substitutional impurities in ABO2 compounds.
Considering this in more detail, as shown in Fig. 4a, potential A-site-substituting impurities can be classified in terms of elements that are known to adopt the A-site in ABO2 compounds, that either can (dark yellow) or cannot (light yellow) feasibly substitute for Pd in PdCoO2. We determine this feasibility by applying Goldschmidt-type criteria, requiring <30% ionic size mismatch, an ability to adopt the required coordination, and valence within 1 of the host element. with their valence, coordination number, and ionic radii in the state most closely matching the delafossite structure. As shown in the figure, Li + , Na + , K + , Rb + , Cu + , Ag + , Pt + , and Hg 2+ (shaded yellow) are known to adopt the A-site in ABO2 compounds, but in most cases cannot feasibly substitute for Pd in PdCoO2. In particular, Li + , Na + , K + , and Rb + (light yellow) only form ABO2 compounds with distinctly different structure to delafossite PdCoO2, with large coordination number with O 2-. In most cases their ionic radii in this coordination are also far beyond the 30% mismatch for feasible substitution for Pd. In fact, the elements that can feasibly substitute for Pd in PdCoO2 (dark yellow) number only four: Cu, Ag, Pt, and Hg, highlighted via the bold border in Fig. 4b. Hg substitution would also be a special case, as the 2+ valence (Hg does not typically adopt 1+ valence in 2-fold coordination) would require an additional charge-balancing defect. This general concept is supported by prior work on deliberately Mg-substituted PdCoO2, where A-site substitution with Mg 2+ was argued against based on similar reasoning 66 .
The situation on the B site is very different. As shown in Fig. 4a, potential B site substituents can be classified in terms of those that are known to adopt the B site in delafossites and can feasibly substitute for Co (darkest blue), those that are known to adopt the B site in ABO2 compounds but cannot feasibly substitute for Co (lightest blue), and those for which it is unknown whether they form ABO2 compounds but do fit the Co site in PdCoO2 (intermediate blue). Again, feasibility is based on <30% size mismatch, 31 valence, and octahedral coordination. As shown in Fig. 4b, the total group of possible B-site substituents is numerous, totaling at least 40 (all blue shades), remaining as high as 20 even after applying our fit criteria (darkest blue shades). In contrast to ABO3 perovskites, the fundamental crystal chemistry of the ABO2 delafossites thus results in a large asymmetry between the number of elements that can exist as substitutional impurities on the A and B sites. Substantial concentrations are found of elements that are likely to populate the Co site in PdCoO2 (e.g., Mn, Fe, Ni, Al, Cr, Ir, Sn; the darker blue shades in Fig. 4b), but not of elements that can feasibly populate the Pd site. Neither Cu, Ag, Pt, or Hg (in the dark border in Figs. 4b, c) reach even 1 ppm in CVT crystals. These findings are further quantified in Table 2, where the top panel summarizes the total impurity concentrations of the elements likely to populate the Pd and Co sites in PdCoO2. In metathesis/flux crystals, of the total impurity concentration of 251 ppm (far right, Table 2), at least 132 ppm are elements that would be expected to populate the Co site (darkest blue shades in Fig. 4b), whereas only 4.4 ppm are elements that could feasibly populate the Pd site (Cu, Ag, Pt, Hg, dark yellow in Fig. 4b). We thus define a "sublattice purification ratio" of 4.4/251 = 1.8% of the total impurity concentration that could feasibly substitute for Pd. As also shown in the top panel of Table 2, the equivalent for CVT PdCoO2 is 1.9/47 = 4.0% of the total average impurity concentration that could feasibly substitute for Pd, in this case only 1.9 ppm.
The middle and bottom panels of Table 2 Table 1. Quantitatively, the estimated magnetic impurity concentration from the bottom panel of Table 2 is 31 ppm in CVT crystals, compared to order 100 ppm from magnetometry. This is reasonable agreement considering the likelihood of additional surface contamination in magnetometry. In terms of comparisons to PIXE, ICP-MS on CVT crystals indicates 9  2 ppm of Fe, also comparing reasonably to the order-50-ppm approximate estimate from PIXE (Fig. 2e). Finally, ICP-MS was also used to determine the Pd/Co ratio of CVT crystals, yielding 1.08  0.02. This is inconsistent with the EDS result of 0.98  0.05 at first sight, but EDS data were acquired from 25  25 m 2 regions free of surface particles. These particles are of course not excluded from ICP-MS, and thus remnant surface Pd and PdCl2 likely skews the ICP-MS Pd/Co ratio.

Discussion
The conclusion from the above is that the prototypical metallic delafossite PdCoO2 is by no means ultrapure in general. Total metals-basis impurity concentrations decrease from 250 ppm in metathesis/flux crystals to 50 ppm in CVT crystals, but remain substantial. Crystal-chemical principles, however, dictate that the great majority of substitutional impurities must reside on the Co site, the relatively rare valence and coordination on the Pd site resulting in few elemental impurities (particularly Cu, Ag, and Pt) being capable of substitution. Sublattice purification ratios of 1% thus arise, limiting the substitutional impurities on the Pd site to 1 ppm. Our simple conclusion from these findings is that the electronic conduction in metallic delafossites such as PdCoO2, which is strongly restricted to Pd d states at the Fermi level 5,10,18-21 , thus occurs in highly pure Pd planes, with minimal substitutional impurities. This sublattice purification naturally explains how ultralow residual ab and extraordinary mean-free-paths can arise in metallic delafossite single crystals, despite the relatively dirty methods employed in their synthesis 5,49 .
Directly addressing a question posed in recent work 49 , the BO6 octahedral planes in metallic delafossites thus do host substantial disorder, in contrast to the Pd/Pt sheets, which were deduced to be highly perfect 49 . The apparent contradiction of sizable magnetic impurity concentrations in crystals with high RRR is thus naturally resolved, as the majority of substitutional impurities, including magnetic ones, populate only the CoO6 octahedra.
The above picture is predicated, however, on two key assumptions: that the 1 ppm deduced impurity concentrations in the Pd planes are sufficiently low to facilitate the observed residual ab and mean-free-paths, and that the significant impurity concentrations in the BO6 octahedral planes have sufficiently little impact on conduction in the Pd sheets. The first assumption can be addressed using the 2D unitary scattering approach recently applied by Sunko et al., which was validated in irradiated metallic delafossite crystals 49 . This essentially assumes the strongest possible s-wave scattering 49 in a Drude model, resulting in ℏ , where ℏ and e are the reduced Planck constant and electronic charge, nd is the areal defect density, and n is the volume carrier density.
Based on this, our deduced residual ab of 4 nΩcm would require a 2D defect density of 6  10 9 cm -2 in CVT-grown PdCoO2. Remarkably, this is in striking agreement with our determined Pdsheet impurity density, which corresponds to 5  10 9 cm -2 based on the 1.9 ppm of A-site impurities in Table 2. (This calculation takes into account the exact A-site impurity elements in Table 2, and their relative masses). This indicates quantitative agreement between our deduced A-site impurity concentration and measured residual resistivity. This extends also to flux/metathesis crystals, where the A-site impurity concentration of 4.4 ppm in Table 2 would be expected to generate a residual ab of 7 nΩcm, remarkably close to measured values 17 . (Again, this calculation takes into account the exact A-site impurity elements in Table 2, and their relative masses; the distribution of impurities is different for CVT and metathesis/flux crystals). The second assumption is also supported by the recent work of Sunko et al., which explicitly concluded that disorder in the B-O layers must be efficiently screened 49 ; the extent of this disorder was not determined in that paper, but is now known to be substantial (of order 100's of ppm).
Density functional theory (DFT) calculations provide further support for the above. These were  Fig. 5a, and the atom-projected DOS in Fig. 5d is of very similar form for Pd and Pt. Note here that the simulated Pd0.963Pt0.037CoO2 has 26-times more Pd than Pt but no normalization is applied in Figs. 5d-f, simply to promote visibility of the impurity DOS.
As might also be anticipated, the situation for Fe impurities on the B site is more complex. A narrow impurity band forms below EF (Fig. 5b), generating a clear peak in the Fe-projected DOS in Fig. 5e. Nevertheless, this main impurity band peak is centered at 0.2 eV below EF (many times kBT at the temperatures of interest in this work), and is quite narrow, meaning that the impact at It should be emphasized that the above analysis focuses on substitutional impurities. While interstitial impurities should also be considered, the delafossite structure is unlikely to host interstitial sites in which the ions in Fig. 4b can notably the X-ray rocking curve widths (Fig. 2c inset). Substitutional impurities thus appear to be the key defects in metallic delafossites, which have been comprehensively elucidated here. Future work to more completely understand the reasons for the low density of other point, line, and areal defects in these compounds is nevertheless clearly important.
In summary, this work establishes CVT as a highly promising approach to bulk single crystal growth of PdCoO2, with potential applicability to other metallic delafossites such as PdCrO2 and (typically a total of 1 ppm of Cu, Ag, and Pt). The "sublattice purification ratio" is thus 1%, resulting in ultrahigh purity in the Pd planes in which electronic conduction takes place, largely unaffected by the impurities in the CoO6 octahedra. This sublattice purification mechanism is therefore central to the ultrahigh low-T conductivity of metallic delafossites, resolving the apparent contradiction that they appear impure from magnetometry yet highly pure in electronic transport.
These results significantly demystify the outstanding low-T conductivity and mean-free-path in metallic delafossites, setting the stage for further advances with this extraordinary materials class.
As an example, the analyses in Refs. 25,30 place PdCoO2 in a situation where the momentumconserving scattering rate is larger, but not much larger, than the momentum-relaxing scattering rate, leaving the system in a crossover regime between ballistic and viscous electronic transport.
The momentum-conserving scattering is unlikely to be electron-electron scattering at the temperatures of interest 30

Crystal growth
PdCoO2 crystals were grown by two methods: the established metathesis/flux process 6  Minor unreacted Pd and Co3O4 phases were often detected in these products by PXRD (see Supplementary Information Section B). The largest crystals found in the products were 0.8  0.6 mm 2 laterally and 0.017 mm thick (Fig. 1b, inset).
For CVT, 1 g of coarse-grained PdCoO2 product from the above metathesis reaction was mixed with ~120 mg of PdCl2 powder, then loaded and vacuum sealed (at 10 -6 Torr) in quartz ampoules.
Ampoules were then placed in a two-zone tube furnace with the reagents at one end of the ampoule (Fig. 1a). CVT was performed at a range of temperatures, spanning 740-800 C for the hot zone and 680-750 C for the cold zone. Hot-and cold-zone temperatures of 760 and 710 C were found optimal in terms of crystal size and complete transport (i.e., yield). For the first 3 days, the empty end of the tubes was maintained as the hot end, to clean the growth zone. The temperature gradient was then inverted for 13 days to establish the reagent-filled end of the tubes as the hot zone and the originally empty end as the cold/growth zone (Fig. 1a). We believe CVT to proceed via decomposition of PdCl2 (above 600 C 57,58 ), followed by the reaction PdCoO2(s) + 2Cl2(g) ↔ PdCl2(g) + CoCl2(g) + O2(g) (Fig. 1a). Multicrystals up to 12  12 mm 2 laterally and 0.3 mm thick result from this process (Fig. 1c). Single crystals were then isolated by gently breaking the multicrystals apart, generating up to 6.0  4.0 mm 2 lateral sizes, with thickness up to 0.17 mm (see Fig. 1c, inset, for an example crystal). Excess chlorides were then removed by washing PdCoO2 crystals in boiling ethanol. We are aware of no prior report of such CVT growth of metallic delafossites, although one approach to PdRhO2 growth did employ a CVT-like temperature gradient 51 but with clear qualitative differences with our methods and findings.

Structural and chemical characterization
Optical microscope images of crystals were acquired with a Zeiss Axiovert A1 MAT inverted microscope, while SEM was performed in a JEOL JSM-6010PLUS/LA. EDS data were taken with an EDS detector integrated in the SEM, at 20 kV accelerating voltage. PIXE spectra were acquired in a MAS 1700 pelletron tandem ion accelerator (National Electrostatic Corp.) at 50 μC dose from a 4 MeV He 2+ beam; PIXE data were analyzed using GUPIXWIN 65 . Laue diffraction, PXRD, and single-crystal HRXRD (including rocking curves) were taken using Photonic Science backreflection Laue, Rigaku MiniFlex 600, and Rigaku SmartLab XE diffractometer systems, respectively, the latter two with CuK  radiation. PXRD analysis employed JADE for whole pattern fits, which were used to determine a and c lattice parameters. Laue analysis employed Lauesim 67 , modified for hexagonal symmetry.

Electronic transport and magnetometry measurements
Single crystals for electronic transport were first cleaved into bar shapes (Fig. 3a, inset) and affixed with GE varnish to Al2O3 wafers. Al contact wires were attached in a four-point in-line geometry to the top surface of the PdCoO2 using ultrasonic wire bonding. T-dependent (4-300 K) resistance measurements were then performed in a Janis cryostat/superconducting magnet, using a Lakeshore 370 AC resistance bridge and Lakeshore 3708 preamplifier/channel scanner (at 13.7 Hz and 10 mA). As alluded to in the main text and described in detail in Supplementary Information Section C, great care was taken to accurately determine and account for instrumental voltage offsets, which are important due to the very low low-T resistance of PdCoO2. A specialized protocol for this was developed and tested on zero-resistance superconducting V films. As also described in Supplementary Information Section C, finite element COMSOL simulations were performed to support the interpretation of the transport results, particularly with respect to the issue of systematic errors associated with the c/ab anisotropy. To minimize contamination for magnetometry measurements, crystals were first ground with thoroughly cleaned nonmagnetic tools then washed in ethanol to remove CoCl2. The latter could occur not only at the original crystal surfaces but also at internal inclusions/voids 16

. Magnetometry was then done in a Quantum Design Physical
Property Measurement System equipped with vibrating sample magnetometry (VSM) and high-T options, from 2 to 600 K in magnetic fields to 7 T.

DFT calculations
First-principles density functional theory calculations were performed with the Quantum ESPRESSO package using norm-conserving pseudopotentials, with the exchange-correlation energy approximated by the Perdew-Burke-Ernzerhof functional [68][69][70] . The energy cutoff for the plane waves was set to 100 Ry. Dilute impurities on the Pd-and Co-sites (i.e., Pd1-xZxCoO2 and PdCo1-xZxO2) were simulated in a 3 × 3 × 3 supercell with 108 atoms, with one Pd or Co atom replaced, corresponding to x = 0.037. Pt substituted for Pd, and Fe and Al substituted for Co were considered. Both the unit cell shape and atomic positions were relaxed for electronic structure calculations. The band structure and DOS were calculated on a Γ-centered Monkhorst-Pack grid of 4 × 4 × 4 k points in the Brillouin zone of the supercell. The unfolded band structures in Fig. 5 were obtained using the bandUPpy code [71][72][73] . The unfolding method projects a wave vector Ḵ in the supercell Brillouin zone onto a corresponding wave vector ḵ in the primitive cell Brillouin zone via a supercell reciprocal lattice vector. This unfolding evaluates the spectral weight or probability of finding a set of primitive-cell Bloch states at wave vector ḵ that are contributing to the supercell eigenstates at wave vector Ḵ at the same energy. The specific method adopted here 72 finds the effective band structure for doped PdCoO2 by using the spectral function to calculate the weight, which is given by the number of primitive cell bands that are crossing within a small energy window around the band center.

Data availability
All data needed to evaluate the conclusions presented in the paper are included in the paper and/or the Supplementary Information. Digital data are available from the corresponding author upon reasonable request.

Author contributions
CL conceived the study and supervised its execution. YZ and FT grew the crystals, which were

Competing interests
The authors declare no competing interests.

Additional information
Supplementary information: The online version contains supplementary information available at XXX.
Correspondence and requests for materials should be addressed to Chris Leighton.  The peaks labeled with an "*" are known instrumental artifacts in our ion beam accelerator/PIXE system. The inset is a close-up of the 4.5 to 7.3 keV region shaded grey; note the different intensities of the Cr and Fe K  peaks in the two crystals. The Cr peak at 5.4 keV has partial overlap with an artifact peak but the variation in intensity from crystal-to-crystal strongly suggests a contribution from Cr. The Fe peak at 6.4 keV has no such overlap and also varies from crystal-tocrystal.   Also shown for each element are the valence, coordination number, and ionic radius 75

CVT Setup and Optimization.
As noted in the main text, while CVT has not been reported for PdCoO2, there are several reasons to consider this growth approach, including the fact that PdCoO2 decomposes 1-3 and that PdCl2 (which decomposes at 600 C in low Cl pressure 4,5 ) is a potentially convenient transport agent. CVT crystal growth of Co3O4 6 , Pd 6 , and PdO 6 have also been reported.
A CVT process was thus designed based on the known properties of PdCoO2, PdCl2, and CoCl2.
CVT growth of Co3O4, Pd, and PdO using PdCl2 as a transport agent have been reported at growth temperatures in the 400-1000 °C range 6 , based on reactions such as 3 ⇌ The choice of transport agent was also considered, i.e., whether to use PdCl2, PdBr2, or PdI2, for example. For PdCoO2 growth, the co-stability of halides of Co and Pd is essential. PdBr2 and PdI2 have very low decomposition points (250 and 360 °C, respectively 7 ), much lower than the CoBr2 and CoI2 melting points (678 and 520 °C, respectively) 7 . The bromides or iodides of Pd and Co having comparable vapor pressures at reasonable temperatures is thus unlikely. PdCl2, however, decomposes at 600-740 °C (the range is likely due to variable Cl2 pressures) 4,5 , with a melting point at 680 °C 7 , while CoCl2 has a similar melting point of 737 °C 7 ; this suggests a reasonable likelihood of co-stability of their vapors at 700 °C. PdCl2 was therefore chosen as the transport agent, taking advantage of its partial thermal decomposition to generate Cl2 at ~600 °C, without the use of hazardous Cl2 gas, and avoiding potential impurities from Cl2-generating transport agents such as HgCl2 6 .
As shown in Fig. S1a, for CVT growth, vacuum-sealed quartz ampoules (see Methods) were held horizontally in a multi-zone furnace, with the precursor powders (1 g of metathesis/flux-grown PdCoO2 and 0.12g of PdCl2) loaded at the right end of the ampoule (in Zone 2), and the empty end of the ampoule in Zone 1. For the first 3 days, Zone 1 was the hot zone and Zone 2 the cold zone, while for the remaining 13 days the temperatures gradient was reversed; the rationale for this is provided below. A typical ampoule after the end of a CVT growth is shown in Fig. S1b. At the end of a successful growth, essentially all of the PdCoO2 is transported to the growth (cold) zone, forming PdCoO2 single crystals, PdCoO2 multicrystals, and excess chlorides. The other ampoule end is essentially empty, indicating high yield.
A 50 °C temperature difference was chosen for most growths in this work. Smaller differences, such as 30 °C, led to slow crystal growth rates and incomplete transport in 13 days, while larger differences, such as 60 °C, led to smaller crystal sizes, likely indicating too high a transport rate.
Most growths were done with 710/760 °C cold/hot zone temperatures (such as the ampoule in Fig.   S1b), which were found to yield the largest crystals, with reflective surfaces, and near-complete transport. Higher temperature conditions were experimented with, using 750/800 °C and 800/850 °C (the ampoules in Figs. S1c, d, respectively), for example. At these higher temperatures, the yield of PdCoO2 crystals decreased, with no improvement in crystal size. A magenta coating also developed on the inner ampoule walls, as in metathesis/flux growth at 750 °C, which was found by X-ray diffraction to be mainly Co2SiO4. This likely results from high-temperature side reactions between cobalt compounds and quartz. Eventually, in the 800/850°C growth, no PdCoO2 crystals were grown, and PdO, along with <10 wt.% untransported PdCoO2 precursor was found to remain in the hot zone. At these temperatures, PdCoO2 likely decomposes into binaries at 850 °C, the resulting cobalt compounds reacting with the quartz to form the Co2SiO4 tube coating in Fig. S1d.
CVT Mechanisms. Based on the above, and on additional observations below, various conclusions can be reached regarding the mechanisms of CVT growth. The growth starts with the initial conditions in Fig. S2a, with the mix of precursor powders located in Zone 2 and the nominally empty ampoule end in Zone 1. Note here that the "empty" end of the ampoule is inevitably unintentionally dusted with precursor powder particles during loading, as shown in Fig. S2a. As it is common for metathesis/flux-grown PdCoO2 (one of the precursors here) to have small amounts of impurity phases, including Pd (see Section B below), there are thus microscopic quantities of Pd present, as also shown in Fig. S2a. In the first stage of the growth, illustrated in Fig. S2b, the ampoule end filled with precursors is the cold zone (Zone 2) and the empty end is the hot zone (Zone 1), for an inverted-temperature-gradient period of 3 days. The purpose of this first stage is to transport any PdCoO2 powder particles in the "empty" end of the ampoule to the other end, hence "cleaning" the growth zone, i.e., ridding it of stray nuclei. During this stage however, which was found to be essential, Pd (whether from phase impurities in the metathesis/flux-grown PdCoO2 or partial thermal decomposition of PdCl2) will be transported to the empty ampoule end, as Pd is known to transport via Cl2 from cool to hot 6 ; this is due to the exothermic → reaction 6 . The significance of this point is returned to below. The initial 3 days of inverted temperature gradient are followed by the main (13-day) growth stage shown in Fig. S2c.
At the start of this period, as explained above, the growth zone (Zone 1) can be expected to contain Pd crystallites (whether there originally, or transported from Zone 2). These Pd crystallites may then act as nucleation centers for PdCoO2 crystal growth. Notably, the interatomic distance within the (111) planes of Pd is 0.275 nm 8 , similar (within -2.8%) to the PdCoO2 lattice parameter of a = 0.283 nm 1,8,9 . Pd particles can thus likely seed the PdCoO2 CVT crystal growth in this process, which we propose as an explanation for the multicrystal formation that we found common. In essence, we propose that PdCoO2 crystals can grow in multiple directions from different (111)family facets of Pd crystallites, generating multicrystals. As noted in the main text, these multicrystals can be easily separated into single crystals, and some single crystals also form, presumably due to non-Pd-seeded growth. As a final comment on this mechanism, note that PdO may also play a similar role in nucleation.

Section B. Specific Growth Parameters of Characterized Crystals
Tables S1 and S2 summarize the metathesis/flux and CVT growth conditions used to grow the specific PdCoO2 crystals characterized in this paper. Powder X-ray diffraction (PXRD) was performed on almost all metathesis/flux and CVT crystals, and the other characterization/ measurement techniques applied are listed in each case.
In addition to the characterization in the main text, Fig. S3 shows additional PXRD from four further metathesis/flux crystal batches and four further CVT batches. The phase purity is seen to significantly improve via CVT. As shown in Fig. S3a, Co3O4 and Pd minor phase impurities were commonly found in our metathesis/flux crystals. In CVT crystals (Fig. S3b), only a minor Pd phase impurity was ever detected.  Table S2. Growth parameters of CVT-grown crystals. Crystal growth parameters of CVT batches that produced samples characterized in this paper. Each batch corresponds to one ampoule.
Listed are the batch numbers, cold and hot zone temperatures (Tcold and Thot), and characterization methods. Characterizations that appear in the main text are bolded with the corresponding figure/table in parentheses. PXRD is powder X-ray diffraction, ICP-MS is inductively coupled plasma mass spectrometry, PIXE is particle-induced X-ray emission, EDS is energy-dispersive Xray spectroscopy, HRXRD is high-resolution X-ray diffraction, and RC is X-ray rocking curve analysis.

Section C. Transport Measurement Details
Offset Correction and Signal-to-noise Ratio. As noted in the main text, the very low residual value of the a-b plane resistivity (ab) in PdCoO2 single crystals renders low-temperature (T) transport measurements challenging. Specifically, signal-to-noise ratio becomes an issue, as do instrumental offsets. These issues were alleviated in this work through the study of relatively long bar-shaped crystals (Fig. 3a, inset), but must still be carefully addressed. The signal-to-noise ratio was maximized through the use of an AC resistance bridge (a Lakeshore 370 operating at 13.7 Hz) in tandem with a preamplifier/channel scanner (a Lakeshore 3708). Currents of 10 mA were employed and self-heating was rigorously ruled out by estimations of the (very low) deposited power, and careful low-T checks of resistance vs. current and resistance vs. T at various excitation currents. The resulting noise performance is illustrated below and in the data of Figs. 3a, b.
With respect to offsets, which are critical given the 20  residual resistances, these were first carefully characterized via parallel measurements of zero-resistance superconducting V thin films.
The latter were Si/Si-N/V(100 nm) films deposited in ultrahigh vacuum in a molecular beam epitaxy system. As shown in the inset to Fig. S4, the same top-contact, in-line, four-wire geometry as was used for PdCoO2 crystals (Fig. 3a, inset) was employed for the V films. The four contacts are labeled A, B, C, D, and a standard notation for four-wire resistance (R) measurements is followed by using the first two letters to refer to the I+ and I-terminals, respectively, and the last two letters to refer to the V+ and V-terminals, respectively, e.g., RADBC. As shown in Fig. S4, the nominally zero-magnetic-field RADBC(T) of such films reveals a superconducting transition with an onset temperature of 5.2 K and an endpoint of 5.1 K, in agreement with expectations 11 . Such samples were thus measured at 3.9 K, safely in the zero-resistance regime, to characterize the voltage and resistance offsets in our measurement set-up.
Offset measurements were made both with and without the preamplifying channel scanner. Table   S3 shows typical results for two separate sets of contacts on V films at 3.9 K. The first three data columns are without the preamplifier, and the last three data columns are with the preamplifier.
Considering the data without the preamplifier first, measurements of RADBC and RDACB were found to be near identical, at -0.4 to -0.3  for Contact Set 1 and 0.3 to 0.7  for Contact Set 2. Such findings are representative. Over many measurements, using different wire and contact sets, different bridge ranges and currents (10 mA was used in Table S3), etc., the resistances offsets from zero were found to be approximately ±1 μΩ, with noise of approximately ±0.5 μΩ. Most importantly, the approximately ±1 μΩ offset indicates that measurements of 20 μΩ resistance can be made with reasonable accuracy. Adding the preamplifier to the set-up (right side of Table S3) was found to result in differences. The noise dropped to approximately ±0.1 μΩ (by roughly a factor of 5), but the offsets from zero resistance were found to grow to approximately ±5 μΩ (also by roughly a factor of 5). Interestingly, however, it was found empirically that the sign of the offset reproducibly inverted on going from RADBC to RDACB, i.e., with a 180 phase shift of the excitation (Table S3). Averaging between RADBC and RDACB thus yielded offsets of only ±1-2 μΩ, over many measurements, using different wire and contact sets, different bridge ranges and currents, etc.
Thus, regardless of whether the preamplifier was included, offsets from zero resistance could be kept to a workable ±1-2 μΩ. and RDACB(T) were measured, as shown in Fig. S5. Based on the above analysis, these values were then averaged to the solid line in Fig. S5, thereby accounting for the offsets to within ±1-2 μΩ. ion-beam-patterned crystals 16 . Although it impacts the results negligibly, the contacts were simulated using typical resistivities for Cu. All of the 300-K and low-T resistivity values are listed in Table S4. As shown by  Table S4 (right columns) for the side-contact and top-contact geometries. The side-contact results are as expected, reproducing the input RRR from resistivity to good accuracy. The top-contact simulated R values are 50-90% larger than those for side current contacts, however, due to the effect of the large c/ab. This effect worsens with decreasing T as c/ab grows on cooling (also shown in Table S4). The top-contact RRR of 278 is thus 20% smaller than the side-contact (true)

Finite-element Simulations of Current Flow and Voltage
value. As stated in the main text, measured RRR values in this work (440 for CVT crystals) are thus lower bounds due to overestimation of the residual ab.
Further quantification of the extent of underestimation of the RRR was achieved by running additional top-contact simulations that reproduce the observed RRR of 440. This was done by finding the required true value of residual ρab to reproduce the experimental results, while keeping the 300-K value of ρab (which is dominated by phonon, not impurity, scattering) constant. (In the absence of additional information, ρc was also maintained at the same values as in prior simulations). As shown in Table S5, the true residual ρab required to reproduce a top-currentcontact RRR of 436 is 4.4 nΩcm, close to half the measured value. The measured RRR of 436, by     Table S4. Contacts B and C are the voltage contacts, placed 1 mm apart, edge-to-edge. The resulting simulated 300-K voltage drops are shown as color maps based on the scale to the right. Note the through-thickness color gradient near the contacts in (a).

Table S4. Input resistivity values and extracted resistances from finite element simulations.
Shown are 300-K and residual low-T values of the PdCoO2 a-b plane resistivity, c-axis resistivity, their ratio, and the Cu contact resistivity 13,16,17 , which were used as inputs for the simulations. As described above, the PdCoO2 values shown are from metathesis/flux-grown crystals 13,16 , measured on focused-ion-beam patterned samples for the a-b-plane case 16

Section D. ICP-MS Trace Impurity Analysis Details
As noted in the main text, 54 elements were selected for ICP-MS trace impurity analysis of both metathesis/flux-and CVT-grown PdCoO2 crystals. The selection of these 54 elements was based on their being either known to form ABO2 compounds, or being known impurities in the growth reagents. A full listing of these elements, and their corresponding analyte and analysis mode is provided in Table S6. Elements that are known to form at least one ABO2 compound were first identified, in part using Ref. 18 . Purity certifications for the commercial reagents used in the crystal growth processes were then examined. Table S7 shows the grades and predominant impurities of the reagents used in most crystal growth batches in this work, including the crystals that were used for ICP-MS and transport measurements. The least pure reagent, Pd, is >99.995% pure, corresponding to <55 ppm impurities. Based on Table S7, the detected presence of Ni, Mn, Fe, Sn, Ir, and Ag in the final PdCoO2 crystals (see Table 2) is thus expected at some level. Additionally, Pt would not be surprising as an impurity in a Pt-group reagent element such as Pd, and the presence of Ni, Mn, Fe, and Zn in the reagents suggests that the first-row transition metals Cu and Cr would also not be unexpected. All the elements in the middle and bottom panels of Table 2 can thus be reasonably accounted for based on the reagents, with the exception of Al. The latter may arise from Al-containing sample handling apparatus, particularly weighing boats.

Table S6. Tested elements and ICP-MS analytical conditions
Analytical conditions for all the trace elements quantified in this work by ICP-MS. The analyte for each element is listed in the second column, with the corresponding analysis mode in the third column. Kinetic energy discrimination helium collision cell (KED-He) or triple quadrupole oxygen collision cell (TQ-O2) mode were used.

Element Analyte
Analysis mode Element Analyte Analysis mode Li Table S7. Reagent grades and impurities Purity grades and predominant impurity elements (> 1 ppm) from the manufacturers' certificates of analyses for the commercial reagents used to grow the crystals characterized here by ICP-MS.

Manufacturer
Reagent grade Predominant impurity elements found