Theoretical design of highly correlated electron states in delafossite heterostructures

Delafossites represent natural heterostructures which can host rather different electronic characteristics in their constituting layers. The design of novel heterostructure architectures highlighting the competition between such varying layer properties is promising from the viewpoint of basic research as well as for future technological applications. By means of the combination of density functional theory and dynamical mean-field theory, we here unveil the formation of highly correlated electron states in delafossite heterostructures build from metallic PdCrO$_2$ and insulating AgCrO$_2$. Due to the sophisticated coupling between layers of strong and of weak internal electron-electron interaction, correlation-induced semimetals at ambient temperature and doped Mott-insulators at lower temperature are predicted. The unique electronic structure of delafossite heterostructures opens a door to research on novel challenging quantum matter.

Delafossites represent natural heterostructures which can host rather different electronic characteristics in their constituting layers. The design of novel heterostructure architectures highlighting the competition between such varying layer properties is promising from the viewpoint of basic research as well as for future technological applications. By means of the combination of density functional theory and dynamical mean-field theory, we here unveil the formation of highly correlated electron states in delafossite heterostructures build from metallic PdCrO2 and insulating AgCrO2. Due to the sophisticated coupling between layers of strong and of weak internal electron-electron interaction, correlation-induced semimetals at ambient temperature and doped Mott-insulators at lower temperature are predicted. The unique electronic structure of delafossite heterostructures opens a door to research on novel challenging quantum matter.
Delafossite oxides 1 ABO 2 , where A and B are different metallic elements, have gained ample renewed interest, mainly from two directions. First, the dominant part of these compounds is insulating, and technologically promising due to their potential as transparent conducting oxides 2 as well as their possibly multifunctional character, e.g. by showing magnetic/ferroelectric response 3,4 and high potential for photocatalytic applications 5 . Second, the small subgroup of metallic delafossites usually displays an enormously high electric conductivity, belonging to the most-conductive metals at room temperature 6,7 . General key signatures of delafossite crystal lattices are the triangular-coordinated planes and especially the dumbbell O−A−O bonds along the c-axis. The layered R3m architecture of alternating A +and B 3+ O 2 planes gives rise to a 'natural heterostructure', i.e. the different planes may host a rather different electronic character. Tuning and designing the different aspects of delafossites by further technological heterostructuring thus may open a route to challenging new materials with exceptional properties.
Among these fascinating materials, the PdCrO 2 compound stands further out as it unites high metallicity from the Pd + (4d 9 ) planes with Mott-insulating CrO 2 layers [8][9][10][11] . Replacing Pd by neighboring Ag formally leads to a closed 4d 10 shell and nonsurprisingly, AgCrO 2 is indeed insulating with a technologically interesting band gap of 1.68 eV 5 . In both delafossites, the localized Cr spins order in an antiferromagnetic 120 • manner below Néel temperatures 37.5 K (PdCrO 2 ), 21 K (AgCrO 2 ). Moreover, the palladium delafossite shows an unconventional anomalous Hall effect 12 at low temperature T , whereas the silver delafossite displays multiferroic properties 3 . While both individual compounds are of prominent interest of their own, we want to open a new path in delafossite research in this work. By using the oxide-heterostructure concept, the goal is to theoretically further engineer the unique layer-triggered competition between itinerant and insulating tendencies in these CrO 2 -based delafossites. As we wil show, this manipulation enables the stabilization of even more intriguing electronic phases, and drives the challenge of the com- mon metal-versus-insulator paradigm to another level. We study delafossite heterostructures Pd n Ag m CrO 2 with n(m)=1,2 planes of Pd(Ag) in the primitive cell, and as in the conventional bulk compounds, separated by CrO 2 layers. Thus, three different artificial heterostructures are realized mediating between the natural heterostructures PdCrO 2 and AgCrO 2 , namely Pd 2 AgCrO 2 (P2A), PdAgCrO 2 (PA) and PdAg 2 CrO 2 (PA2) (see Fig. 1). In order to cope with the demanding problem involving the competition between itinerant, band-insulating as well as Mott-insulating layers, the elaborate charge self-consistent combination [13][14][15] of density functional theory (DFT) and dynamical mean-field theory (DMFT) is put into practise to elucidate the correlated electronic structure, respectively. At room temperature, the three designed heterostructures are predicted to be correlationinduced semimetals with singular electronic characteristics. Lowering the temperature leads to a doped-Mott insulator scenario at low energy.

I. THEORETICAL APPROACH
A charge self-consistent combination of density functional theory and dynamical mean-field theory (DMFT) is employed 15 .
The mixed-basis pseudopotential method 16,17 , based on norm-conserving pseudopotentials with a combined basis of localized functions and plane waves is used for the DFT part with the generalizedgradient approximation in form of the PBE functional 18 . Within the mixed basis, localized functions for Cr(3d), Pd(4d) and Ag(4d) states as well as for O(2s) and O(2p) are utilized to reduce the plane-wave energy cutoff. The latter is chosen E cut = 20 Ryd for the bulk structures and E cut = 16 Ryd for the heterostructures. A k-point mesh of 13×13×13 partition is utilized for the bulk structures and of 11×11×3 partition for the heterostructures.
The correlated subspace for the DMFT part consists of the effective Cr(3d) Wannier-like functions as obtained from the projected-local-orbital formalism 19,20 , using as projection functions the linear combinations of atomic 3d orbitals, diagonalizing the Cr(3d) orbital-density matrix. A five-orbital Slater-Kanamori Hubbard Hamilto- nian governs the interacting electrons, parametrized by a Hubbard U and a Hund's exchange J H . The coupled single-site DMFT impurity problems in the given structures are solved by the continuous-time quantum Monte Carlo scheme 21,22 as implemented in the TRIQS package 23,24 . A double-counting correction of fully-localizedlimit type 25 is applied. No Hubbard interactions are applied to Pd(4d) and Ag(4d). Such interactions are already expected much smaller than on Cr(3d) and then furthermore their effect onto nominal d 9 (d 10 ) of Pd(Ag) should be weak in the metallic-layer architecture. Thus restricting the explicit DMFT-impurity treatment to the Cr sites is sound to study key correlation effects.
To obtain the spectral information, analytical continuation from Matsubara space via the maximum-entropy method as well as the Padé method is performed. In the former case, the k-integrated Bloch Green's functions are continued, whereare in the latter case the local Cr self-energies are continued. If not otherwise stated, the maximum-entropy scheme is chosen to reveal kintegrated spectral functions and the Padé scheme is chosen to obtain k-resolved spectral functions. Note moreover that the k-resolved spectral functions are computed on a dense mesh along high-symmetry lines and in the k z = 0 plane of reciprocal space. For this purpose, the converged real-frequency local self-energies from the Padé scheme, and the self-consistent projected-local-orbitals recomputed on the dense mesh, are utilized to obtain well-resolved Green's functions in k-space.
The DFT+DMFT calculations are conducted by setting the system temperature to room temperature T = 290 K, if not otherwise stated. Paramagnetism is as- sumed in all the computations. Furthermore in the following, if not otherwise stated, all shown spectral data is obtained from charge self-consistent DFT+DMFT calculations.

A. PdCrO2 and AgCrO2
To set the stage, we first briefly focus on the electronic structure of the natural bulk systems PdCrO 2 and AgCrO 2 . Experimental lattice parameters 1,5 a = 2.930Å and c = 18.087Å for PdCrO 2 as well as a = 2.985Å and c = 18.510Å for AgCrO 2 are used. The internal degree of freedom z, governing the oxygen distance to the A + plane is obtained from DFT structural optimization, reading z = 0.1101 (0.1095) for PdCrO 2 (AgCrO 2 ). Figures 2, 3 present a summary of the spectral characteristics. For PdCrO 2 , local Coulomb interactions on the chromium site of U = 3 eV and J H = 0.7 eV prove adequate as discussed in Ref. 10. The system is metallic with a single conducting Pd-dominated (cPd) quasiparticle dispersion at the Fermi level ε F and Mott-insulating CrO 2 layers (see Fig. 2a). The cPd dispersion is dominantly formed by a linear combination of in-plane d x 2 −y 2 and d xy , as well as by some d z 2 contribution (see Ref. 10 for more details). Note that the k-integrated projected spectral function A proj (ω) still displays sizable Cr-weight at low energy, associated with the cPd band. Thus intricate hybridization between the conducting and the Mottinsulating layers is at play 10 . The single-sheet Fermi surface, shown in Fig. 2b, is of hole kind and of hexagonal shape, in line with photoemission 9,26 . It is furthermore instructive to visualize the spectral functions A ν (ω) with Bloch index ν. The function A ν (ω) amounts to k-integrated spectral weight of the correlated system per Bloch state, i.e. it provides the spectral information of the correlated material as represented in Bloch Hilbert space. The total spectral function then reads A tot (ω) = ν A ν (ω). Figure 3c shows that dominantly, there is a single-ν low-energy Pd-dominated (LE-Pd) contribution at the Fermi level. Be still aware that the complete cPd spectral weight consists of the low-energy ν-sum of all Bloch spectra.
In the case of AgCrO 2 , the quasi closed-shell character of Ag reduces the screening within the CrO 2 subsystem and we therefore increased the Hubbard interaction on Cr to U = 4 eV. Figure 3a,b exhibits that the CrO 2 layers remain Mott insulating, but the previously conducting band becomes completely filled and thus the system as a whole insulating. Hence AgCrO 2 is a combined band-Mott insulator. The obtained charge gap of ∼ 1.8 eV is in good agreement with experiment. Note that the valence-band maximum is of mixed Ag, O character, while the conduction-band minimum of dominant Cr upper-Hubbard band character. Notably, whereas the Pd(4d) states in PdCrO 2 align with the Cr lower Hubbard band, the Ag(4d) states align with O(2p).
Let us comment on another important aspect, namely the evolution from the DFT electronic structure towards the fully-interacting one. When turning on interactions, the electronic spectrum in CrO 2 -based delafossites does not evolve from a non-interacting(-like) band structure to a Mott-insulating spectrum, as observed for common Mott insulators 27 . For instance in PdCrO 2 , the metallic band structure evolves to another metallic spectrum, both spectra with weakly-correlated-looking dispersions at low energy. This interaction evolution takes place in a highly nontrivial way, as depicted in Fig. 4. In the DFT (U = J H =0) case, the three Cr-based t 2g bands cross the Fermi level and the later cPd band is still completely filled well below ε F , as visualized from the orbital-resolved spectral-function weights at the bottom of Fig. 4a. For U = 1.5 eV, J H = 0.35 eV, (i.e. half the interaction strength of the true system) the CrO 2 layers are not yet Mott insulating, and the former Crbased t 2g bands are now strongly incoherent even close to the Fermi level. Furthermore, the later cPd band also becomes very incoherent when entering the low-energy regime (cf. Fig. 4b). These incoherent features may be observed from the spreading of the spectral weight (i.e. "bluriness") leading to a (near) disappearance of the original well-defined dispersions. Thus intriguingly, PdCrO 2 at hypothetical smaller U is a very incoherent metal (or "bad metal"), apparently evading Fermi-liquid characteristics and seemingly in conflict with Luttinger's theorem. Coherence in the cPd band is reached at larger U when the CrO 2 layers are Mott insulating, which again under-lines the intricate coupling between the Pd and the CrO 2 layers. Substitutional doping of the Cr site by isovalent Mo could be a way to reach a lower effective U value in the system.

B. Designed heterostructures
For the heterostructures Pd n Ag m CrO 2 , supercells of 1 × 1 × 3 kind (12 basis atoms) for n = m and of 1 × 1 × 2 kind (6 basis atoms) for n = m = 1 are constructed (see  In order to first address the phase stability, the electronic formation energy where E tot are the total energies either in DFT or DFT+DMFT, may be considered. Note that here, we aligned the plane-wave energy cutoff E cut to obtain comparable total energies. Within DFT, all designed heterostructures yield a negative E form , thus the considered phases are stable against decomposition into a mixture of the bulk phases. Within DFT+DMFT, only the P2A structure has a negative formation energy, but the absolute values of the order of ∼ 10 meV/atom are rather small troughout the series. Note that the given E (nm) form should only be interpreted as a rough guide for phase stability, since the latter depends furthermore on the temperature window, the entropy, the vibrational free energy and further nano-structuring effects, i.e. given through thin-film architectures. Therefore in summary, we consider the given Pd n Ag m CrO 2 phases as indeed serious candidates for stable heterostructures in experiment.
In bulk PdCrO 2 , the CrO 2 layers apparently do not severely harm the itinerancy of the free Pd electron, the established Mott-insulating layers even tend to stabilize the weakly-correlated cPd dispersion (see last paragraph of section II A). The present heterostructuring then formally poses a rather unique canonical problem: the single highly-itinerant electron, derived from the Pd layer, faces additional strong driving towards non-itinerant behavior imposed outside its hosting layer via the blocking layers of coupled Ag-CrO 2 kind. Standard mechanisms of creating "heaviness" for such an electron are not easily applicable, yet simply ignoring those new boundary conditions is neither an option. How does the electron cope with the new situation at room temperature? The spectral information for the three heterostructures is summarized in Fig. 5. From the k-resolved low-energy dispersions in Figs. 5a,d,g nothing spectacular seems to happen, there are still similar "low-energy bands" at ε F , reminiscent of the PdCrO 2 bulk case. However surprisingly, the k-integrated spectra (see also Figs. 5c,f,i) of all three designed heterostructures show a pseudogaplike structure reaching zero spectral weight at ε F . This gap feature is of size ∼ 0.2 eV for P2A and PA, and about ∼ 0.1 eV for PA2. It is important to point out that this at first seemingly contradicting findings of kresolved vs. k-integrated spectra are not an artifact of the analytical continuation from Matsubara to real frequencies, but well-reproducable both with maximumentropy as well as with Padé schemes. This is shown in Fig. 6, depicting a comparison of the maximum-entropy vs. Padé-derived spectrum for bulk PdCrO 2 and heterostructure PdAgCrO 2 . While the straightforward kintegrated Padé-based spectrum is more 'wiggly', the key result at low energy is identical: whereas in PdCrO 2 a quasiparticle feature crosses ε F , PdAgCrO 2 shows a gap(like) feature at ω = 0.
In fact, there is no contradiction between the kresovled and the k-integrated spectrum. The physical scenario is best coined by the term correlation-induced semimetal (CIS), where electronic correlations suppress the low-energy spectral weight in a non-trivial dispersion setting. In a correlated system, a dispersion always carries spectral weight less than unity (contrary to e.g. a DFT picture), and in the present case the spectral weight associated with the dispersion crossing ε F is exceptionally small at room temperature. Figures 5c,f,i show that most of the expected low-energy spectral weight piles up in sidebands at ∼ ±0.2 eV. Note that this specific lowenergy behavior is due to the unique coupling of originally weakly correlated Pd-based and strongly correlated Crbased contributions in the (Pd,Ag)CrO 2 system. This coupling leads to low-energy dispersions of very weak spectral weight, resulting in neglible Fermi-level contribution upon k integration.
Each Fermi-surface sheet, two in the case of P2A, still effectively encompasses one electron, respectively (cf. Figs. 5b,e,h). Only for the PA2 heterostructure the Γ-centered hole sheet, intriguingly, appears slightly enlarged, and we will come back to that observation in the next section.
A further aspect is noteworthy, and can be elaborated best from inspection of the P2A electronic structure with a subtle twofold dispersion crossing ε F (cf. Fig. 7b).  Fig. 7c. The low-energy Ag(4d) weight accumulates in an occupied flat-band part along K-M with no relevant contribution right at ε F (see Fig. 7d). While the Cr' weight is balanced between the twofold dispersion crossing ε F , the Cr" weight breaks this symmetry and attains somewhat larger weight on the FS sheet with slightly larger k F (see Fig. 7e,f). Hence within the challenging correlation physics of the present delafossite heterostructures, there is furthermore an intricate hybridization component.

Below room temperature
The CIS state at room temperature is expectedly the result of a very low coherence scale of an underlying effectively doped Mott-insulating state of the CrO 2 layers. The DFT+DMFT calculations at lower temperature are numerically especially demanding for the multi-site fiveorbital correlated subspaces in the heterostructure design. But indeed, below room temperature, a low-energy resonance emerges in the total spectral function of the delafossite heterostructures (see Fig. 8a). Notably, this resonance is now of dominating Cr character, and carries sizable Pd weight only for the P2A structure (see Fig. 8, right). Thus the given delafossite heterostructures are rather sensitive to temperature and display a semimetalto-doped-Mott-insulator transition around 190 K.
A minimal picture behind these numerical results may be as follows. Heterostructuring of PdCrO 2 and AgCrO 2 leads to an effective doping of the CrO 2 layers, thus charge fluctuations between the Pd layers and the latter increase. Since the hopping within the Pd layers does not become heavily dressed in the Hubbard sense, the cPd dispersion is not strongly renormalized. In other words, there is no strong cPd band-narrowing effect, as seen in Fig. 5a,d,g. Instead, spectral-weight transfer away from low energy and loss of coherence occurs for an only weakly modified cPd dispersion. Still, the scattering with the doped CrO 2 layers is effective in marking a semimetallic scenario. At lower temperatures, the excitations in the doped-Mott layers eventually become coherent and a resonance appears at low energy.
Finally, let us remark on the fermiology of PA2. As obvious from Fig. 8, the PA2 structure develops a lowenergy resonance at T lower than the other two heterostructures. At room temperature, the PA2 structure is furthest away from the doped-Mott coherence state. Low-energy coherence of the Mott layers, in the sense previously discussed for Fig. 4, is thus also not yet fully established at that higher T . Therefore, the PA2 Fermi surface in Fig. 5h shows the deviation from the expected sheet size. Only at lower temperature, when coherence is reinforced, the expected sheet size for one electron is established. This is visualized in the temperature comparison of the low-energy dispersions in Figs. 8b,c.

III. CONCLUSIONS AND DISCUSSION
Delafossite materials, and especially the metallic ones, are not only highly interesting in stoichiometric bulk form.
In this paper we have shown that tailored heterostructuring of metallic PdCrO 2 and insulating AgCrO 2 gives rise to a plethora of challening physics. The explicit aspect of electronic correlation, which is present but to a large extent hidden within the bulk compounds, takes over in the heterostructure phases Pd n Ag m CrO 2 and renders the demanding rivalry between metallicity and insulating tendencies strikingly obvious. Correlation-induced semimetallic states at room temperature and a transition to a doped Mott-insulating regime with low coherence scale at lower temperature have been revealed. Furthermore, layer-selective hybridization may play a decisive role in order to select the stable many-body states at low energy. Eventual com-petition with antiferromagnetic spin-ordering tendencies upon further lowering T will even enlarge the scenario of competing many-body instabilities. But already without even considering possible magnetic ordering, there are many relevant degrees of freedom enabling a fascinating playground for very rich physics in designed delafossite heterostructures. An interesting route for investigations could be given by uniaxial-pressure experiments [28][29][30] . Either to modify the Hubbard U on Cr to reach the intricate non-Fermi-liquid regime at smaller U (cf. Fig. 4), or to effectively metallize the Mott layers by effective bandwidth enlargement. Moreover, the construction of explicit topological states within a Mott-critical background may be possible from further tailored heterostructure design.
Still, the question concerning experimental realization of these and further delafossite heterostructures arises. The representation of delafossites with standard growth techniques has been proven challenging. However the recent growth of PdCoO 2 by molecular beam epitaxy 31 is encouraging and should pave the road for a future realization of delafossite heterostructures. In addition, single-layer deposition onto suitable terminated bulk delafossites may already prove helpful in creating exciting electron states on a surface (see e.g. Ref. 32 for work in that direction).
Let us conclude by the final remark that the key metaldelafossite physics of a crucial coupling between weakly itinerant layers and strongly correlated (Mott) layers connects also to the recent finding of superconductivity in infinite-layer nickelate upon hole doping 33 , where similar scenarios might be at play at stoichiometry.

ACKNOWLEDGMENTS
We thank A. P. Mackenzie and V. Sunko for helpful discussions. Financial support from the DFG LE-2446/4-1 project "Design of strongly correlated materials" is acknowledged. Computations were performed at the JUWELS Cluster of the Jülich Supercomputing Centre (JSC) under project number hhh08.