First-principles calculations of structural and electronic properties of layered A_xRhO₂ (A = Li, Na, K, Rb, Cs)

Layered transition metal oxides A_xMO₂ (A = Li, Na, K, Rb, Cs, M = Co, Rh) show many interesting physical properties. Their crystal structures are constructed by the edge-sharing MO₆ octahedron layers and alkali ion layers, stacking alternately along the c direction. According to the stacking period, they possess two phases: P2 and O3.¹ The physical properties of A_xMO₂ usually strongly depend on the concentration of alkali ions (i.e., the value of x), which can be tuned chemically² or electrochemically.³ For example, the P2 phase Na_xCoO₂ has demonstrated variable physical phenomena with various Na concentrations, including a large thermoelectric power (x ≈ 0.5),⁴ metal-insulator transition (x ≈ 0.5),⁵ and hydration-induced superconductivity (x ≈ 0.35).⁶ The Fermi surface topology of Na_xCoO₂ is associated with Co’s t_2g bands.⁷ Some rich emergent phenomena in Na_xCoO₂ originate from its pudding-mold shaped band structure with a local minimum at the Γ point. Hole doping could tune its Fermi level and trigger a Lifshitz transition.^8–10 

Compared with the 3d transition metal oxide Na_xCoO₂, the 4d isostructural rhodate compounds A_xRhO₂ (A = Li, Na, K, Rb, Cs) could have similar physical properties due to the same crystal structure. However, some difference must be expected due to the different strengths of spin–orbit coupling (SOC) and electron correlation.^11–19 The concentration of K ions in K_xRhO₂ can also be tuned by chemical depotassiation, and the x value could be changed from 0.62 to 0.32.¹² Most of the works showed that K_xRhO₂ (x < 1.0) crystals have a P2 phase,^12,18 although one earlier report in 1987 suggests stoichiometric KRhO₂ is the O3 phase.¹³ K_xRhO₂ has a Seebeck coefficient of about 46 μV/K¹¹ or 40 μV/K¹⁴ and low electrical resistivity at room temperature.¹⁵ K_0.5RhO₂ was predicted by first-principles calculations to possess a quantum topological Hall effect originating from its non-coplanar antiferromagnetic structure.¹⁶ Chen et al. compared the band structure and thermopower between Na_xCoO₂ and K_xRhO₂ by angle-resolved photoemission spectroscopy measurements, and they concluded that the thermopower in K_xRhO₂ can be quantitatively explained with the quasiparticle framework after including an electron–phonon mass enhancement, while the doubled thermopower in Na_xCoO₂ is well accounted for by additional band renormalization from electron correlation.¹⁷ Recently, Ito et al. have found an anomaly in the Seebeck coefficient at x ≈ 0.65 in K_xRhO₂, which is probably due to the filling-induced Lifshitz transition, i.e., a sudden shape change of the Fermi surfaces when the Fermi energy varies due to the change in x.¹⁸ 

Besides K_xRhO₂, there are seldom works on other A_xRhO₂ (A = Li, Na, Rb, Cs) compounds. Na_xRhO₂ (x ≥ 0.25) with controllable Na intercalation for P2 and O3 phases could be synthesized chemically and electrochemically.^20–22 Zhang et al. discovered a metal-insulator transition slightly above x = 0.5 and attributed it to carrier-filling controlled Mott transition by an Ioffe–Regel criterion.²¹ Verger et al. determined structural changes between P2 and O3 phases of Na_xRhO₂ in electrochemical cycles and confirmed the semiconducting behavior of O3–NaRhO₂.²² Saeed et al. studied the thermoelectric properties of Na_xRhO₂ with first principles calculations.²³ Few reports of LiRhO₂ involved the charge performance²⁴ and simultaneously structural transformation of LiRhO₂ as a battery cathode.²⁵ 

To the best of our knowledge, there has been only one report on the materials Rb_xRhO₂ and Cs_xRhO₂, more than 20 years ago,²⁶ which found that both materials have the same structure as that of K_xRhO₂, i.e., the P2 phase. Recently, our group has synthesized single crystals of Rb_xRhO₂ and Cs_xRhO₂, both of which show metallic properties and will be reported elsewhere.

In this work, we present a systematical study for the structural and electronic properties of A_xRhO₂ (A = Li, Na, K, Rb, and Cs) by first-principles calculations. Our calculated structural properties, such as lattice constants, are, in general, consistent with the present experimental works. The calculated band structures and electron density of states (DOSs) with hybrid functionals indicate that the stoichiometric ARhO₂ are all semiconductors with band gaps ranging from 3 eV to 3.6 eV. The possible Lifshitz transition due to the deficiency of alkali ions in A_xRhO₂ is also discussed.

The structural and electronic properties of A_xRhO₂ (A = Li, Na, K, Rb, and Cs) are calculated by the Vienna ab initio simulation package (VASP)^27,28 based on the density functional theory (DFT). The projected augmented wave (PAW) method,^29,30 together with the generalized gradient approximation (GGA) based on the Perdew–Burke–Ernzerhof (PBE)³¹ and Heyd–Scuseria–Ernzerhof (HSE06)³² exchange-correlation functionals are used. The plane-wave cutoff energy is 520 eV. The band structure in the HSE06 calculations is obtained with the assistance of the maximally localized Wannier functions in the Wanner90 code.³³ 

In order to simulate the alkali ions deficient structure (x < 1.0) in A_xRhO₂, we have constructed 3 × 3 × 1 supercells and remove a certain number of alkali ions randomly by hand. Then, both the atomic positions and lattice constants are allowed to relax simultaneously until the maximal residual forces are smaller than 0.02 eV/Å.

The optical conductivity of KRhO₂ is calculated by the OpenMX code, which also implements the DFT using the norm-conserving pseudopotentials and pseudo-atomic basis functions.^34–36 The basis functions of K, Rh, and O ions are K10.0-s4p3d2, Rh7.0-s2p2d2f1, and O5.0-s2p2d1, respectively. The cutoff energy is 300 Ry. The optical conductivity is calculated with a k-mesh of 18 × 18 × 6.

Although A_xRhO₂ materials include 4d transition metals, the SOC effect is not included in our calculations. First, we believe that SOC is not important for the structural properties, such as lattice constants. Second, we will show that SOC has an evident effect on the band structures in these rhodates. However, we also find that it does not change the band gap, the Fermi surfaces, and the Lifshitz transition discussed. Furthermore, we did not consider magnetism in A_xRhO₂ because there is no any experimental work, which indicates that A_xRhO₂ is magnetic.

A_xRhO₂ (A = Li, Na, K, Rb, Cs) has layered structures stacked by the edge-sharing RhO₆ octahedron layers and alkali ion layers. According to different stacking orders, they usually possess two kinds of phases. One is the O3 phase, which contains three RhO₆ octahedra and three alkali ion layers in the conventional cell, shown in Fig. 1(a). The alkali ions in the O3 phase have six neighboring oxygen ions and form AO₆ octahedra. The O3 phase is a rhombohedral structure with a space group of $R3¯m$ (No. 166). Its conventional cell in Fig. 1(a) contains three primitive cells. The other is the P2 phase, which has two RhO₆ octahedra layers and two alkali ion layers in the unit cell, shown in Fig. 1(b). The alkali ions also have six neighboring oxygen ions, but form AO₆ triangular prisms. The P2 phase is a hexagonal structure with a space group of P6₃/mmc (No. 194). In experiments, stoichiometric LiRhO₂ and NaRhO₂ belong to the O3 phase,^21,25 while K_xRhO₂, Rb_xRhO₂, and Cs_xRhO₂ belong to the P2 phase.¹⁸ However, one earlier work in 1987 by Mendiboure et al. found that stoichiometric KRhO₂ has an O3 phase.¹³ Although, many other experiments suggested that the upper limit of x in K_xRhO₂ is about 0.7, and they all have the P2 structures.^12,18,22

We first construct O3 and P2 structures for five kinds of stoichiometric ARhO₂ crystals and optimize their lattice constants and atomic positions, simultaneously. Then, the total energies of the two phases for each crystal are compared and listed in Table I. It is obvious that the O3 phases in Li_xRhO₂ and Na_xRhO₂ have lower energies than their P2 phases, which is consistent with the experimental results.^22,25 On the other hand, the total energies of P2 phases of Rb_xRhO₂ and Cs_xRhO₂ are lower than those of O3 phases, which is also consistent with the experiments. While for KRhO₂, we find that the total energy of the P2 phase is a little lower than its O3 phase, which is not consistent with Mendiboure’s experiment.¹³ However, several other experiments showed that K_xRhO₂ (x from 0.32 to 0.67) has a P2 phase.^12,18 It possibly indicates that the K_xRhO₂ could have a structural phase transition depending on the concentration of K ions.

We also present the optimized lattice constants (a and c), interlayer spacing d, as well as some experimental results of A_xRhO₂ in Table II. It is found that our calculated lattice constants of LiRhO₂ and NaRhO₂ are well consistent with the experimental ones, but there is a large discrepancy in the case of O3–KRhO₂. The reason is unknown yet. However, we also present the theoretical lattice constants of P2 phase K_0.61RhO₂, which is found to be roughly consistent with the recent experiment for the P2 phase K_0.65RhO₂.¹⁸ It is noted that the experimental lattice constants of the P2 phase RbRhO₂ and CsRhO₂ are not found. Nevertheless, it is obvious that the interlayer spacing d between the adjacent RhO₆ layers increases as the ionic radii of intercalated ions increase from Li to Cs.

Tuning the concentration of intercalated ions can significantly influence the lattice constants, in particular, the interlayer spacing d, which was observed in experiments.^3,12,21,25 Here, we also examine the changes in equilibrium crystal parameters due to the variance of x by building supercells of A_xRhO₂ with specific alkali ion concentrations. As shown in Fig. 2(a), the in-plane lattice constant a of all materials increases as x increases, except for O3–Li_xRhO₂. This is quite reasonable since it needs larger spacing to accommodate more alkali ions as x increases. The increment of a in Cs_xRhO₂ is the largest since the ionic radius of Cs is the largest among the alkali ions. However, the lattice constant a in Li_xRhO₂ changes little with the value of x, which is due to the small radius of Li ions. The in-plane lattice constant a in Li_xRhO₂ is mainly determined by the RhO₆ layers. On the other hand, the interlayer spacing d decreases as x increases, as shown in Fig. 2(b), which can be attributed to the increase of Coulomb interactions between alkali layers and RhO₆ layers.

Although the concentration of alkali ions in A_xRhO₂ could be tuned in experiments,^12,21,25 it is supposed that there should be an optimized x for each material. The relative stability for different values of x, in particular, in A_xRhO₂, could be calculated by the binding energy³⁷ ΔE(x), which is defined as ΔE(x) = E(A_xRhO₂) − xE(ARhO₂) − (1 − x)E(RhO₂), where E(A_xRhO₂), E(ARhO₂), and E(RhO₂) are the total energies per formula unit of A_xRhO₂, ARhO₂, and primitive phase of RhO₂ (P4₂/mnm, a = b = 4.489 Å, c = 3.090 Å).³⁸ The calculated binding energies for all materials are shown in Fig. 3. It shows that the most stable deintercalated x is around 0.55 ∼ 0.6 for P2 phase compounds, which approximately matches that in the experiments, i.e., the as-grown crystals K_xRhO₂ have a medium alkali ion concentration of about 0.5–0.6.^11,12,22 Of course, the most stable x in Rb_xRhO₂ and Cs_xRhO₂ is not clear due to the lack of sufficient experiments. For O3–Na_xRhO₂, we have observed an almost constant binding energy within the range x = 0.5–1. The experimental electrochemical curve of O3–Na_xRhO₂ suggests a constant phase of O3–Na_xRhO₂, when x > 0.57, and a phase transition when x near 0.5,²² which is consistent with our binding energy calculations. For O3–Li_xRhO₂, the fully intercalated structure (x = 1.0) is the most stable configuration in our calculation, which is also consistent with the recent experiment.²⁵ 

After studying the structural properties of A_xRhO₂, we then focus on their electronic properties. As an example, we present the electronic band structures of stoichiometric O3–NaRhO₂ and P2–KRhO₂ calculated by the GGA-PBE functional in Figs. 4(a) and 4(c), respectively. The Brillouin zones and high symmetry points in the band structures are given in Fig. S1 of the supplementary material. The band structures of other materials are similar, which are given in the supplementary material (Fig. S2). We found that all the five stoichiometric materials are indirect band-gap semiconductors. From Table III, the indirect band gaps of all materials calculated by the GGA-PBE functional are about 1.0 eV–1.5 eV. However, it is well known that the LDA or GGA based DFT calculations usually underestimate the band gap of materials significantly. Therefore, we also calculated their band structures by the HSE06 hybrid functionals as shown in Figs. 4(b) and 4(d). It is found that the overall shape of the band structures calculated by the HSE06 hybrid functional is very similar to that calculated by the GGA-PBE functionals, except for a significant upward shift of the valence bands. As a result, the hybrid functional gives much larger band gaps for all materials, which range from 3 eV to 3.6 eV.

The band structures of stoichiometric ARhO₂ are similar to that of the isostructural NaCoO₂.³⁹ The d orbital of Rh splits into fully empty e_g and fully occupied t_2g orbitals in the octahedron crystal field. The t_2g orbital could further split into a non-degenerate a_1g orbital and twofold degenerate $eg′$ orbitals due to the trigonal distortion of the RhO₆ octahedron. The a_1g orbital forms the pudding-mold like band structures,⁸ i.e., the flat bands at the valence band maximum (VBM) around the Γ point, as shown in Fig. 4 and Fig. S2 (supplementary material). However, we can find some difference between the five materials in the enlarged band structures near the VBM given in Figs. S3 and S4 (supplementary material). In Fig. S3 (supplementary material), it is found that the pudding-mold like bands at the Γ point in LiRhO₂ and NaRhO₂ are a little below the VBM, which is located around the Z point. However, for other three materials, the VBM are all around the Γ point as shown in Fig. S4 (supplementary material). Furthermore, the pudding-mold like band in KRhO₂ has a local minimum at the Γ point, while it does not exist in RbRhO₂ and CsRhO₂.

We also present the electron DOSs of O3–NaRhO₂ and P2–KRhO₂ in Fig. 5. The DOSs of other materials are similar, which are given in the supplementary material (Fig. S5). It clearly shows that the d orbitals of Rh mainly contribute to the top of the valence bands (from −2 eV to 0 eV) and bottom of the conduction bands (from 1 eV to 3 eV in the GGA-PBE calculations or from 3 eV to 5 eV in the HSE06 hybrid functional calculations). The O 2p bands mainly locate at around −7 eV to −2 eV below the VBM. We also can find a notable contribution from Rh d orbitals in this region, which indicates the strong hybridization between the O 2p and Rh 4d orbitals. Nevertheless, the DOSs of the alkali ions are always far away from the band gap. Therefore, it is expected that the band structures near the band gap will change little when the concentration of alkali ions (x) changes, except for the change in the Fermi energy.

All the stoichiometric ARhO₂ materials show the semiconducting behavior because the Rh³⁺ ions have six valence electrons and they fully occupy the t_2g orbitals. However, if the alkali ions are deintercalated, then the t_2g orbitals are partially occupied and the material A_xRhO₂ should become metallic. In other words, the deintercalation of alkali ions equivalently dopes holes in the materials, and the current carrier should be mainly contributed from the Rh a_1g orbital. Thus, electronic properties of A_xRhO₂ are governed by RhO₆ octahedra layers, but the concentration of the alkali ions (x) could manipulate the Fermi energy and change the shape of the Fermi surface.

The calculated electronic properties of KRhO₂ could be compared with the optical conductivity measured in the experiment by Okazaki et al.¹⁹ Thus, we have also calculated the real part of the optical conductivity σ_xx of KRhO₂ by the OpenMX code. Due to the underestimation of the bandgap in the optical calculations, we have used the scissors correction in the optical conductivity. We find that a scissors correction of 0.9 eV and a large smearing of 1.0 eV could well reproduce the experimental optical conductivity as shown in Fig. 6 of this work and Fig. 3 in Okazaki’s work.¹⁹ In Fig. 6, the calculated optical conductivity shows two main peaks p1 and p2 with the photon energy of 3.5 eV and 6.1 eV, respectively. The p1 corresponds to the β′ peak, while the p2 corresponds to the α′ peak in the experiment.¹⁹ Of course, the γ′ peak in the experiment cannot be found in our calculations since we calculated the optical conductivity of semiconducting KRhO₂, while the experiment measured the metallic K_0.49RhO₂. By analyzing the energy of the optical peaks and the electron DOS, we can conclude that the optical peak p1 comes from the transition between the Rh t_2g and e_g orbitals, while the peak p2 comes from the transition between the Rh e_g and O 2p orbitals. It confirms the experimental schematic diagram of the optical β′ and α′ peaks.¹⁹ 

A recent experimental work has found an anomalous enhancement of the Seebeck coefficients below x^* < 0.65 at low temperatures in K_xRhO₂, which is possibly due to a Lifshitz transition owing to the electron filling.¹⁸ Here, we also examine the shape of the Fermi surface of K_xRhO₂ with different x values by shifting the Fermi energy downward based on the band structure of stoichiometric KRhO₂ calculated with the HSE06 hybrid functionals. As seen in Fig. 7(a), when the Fermi energy is 0.148 eV below the VBM [i.e., region (1) x < x^*], it crosses two bands near the Γ point, resulting in two cylinders in the Brillouin zone, as shown in Fig. 7(b). The inner cylinder is from the lower a_1g band [red one in Fig. 7(a)], while the outer one is from the higher a_1g band [green one in Fig. 7(a)]. The two cylinders connect at the k_z = ±π/c plane in the Brillouin zone due to the two fold degeneration of the bands. When the Fermi energy is 0.03 eV ∼ 0.12 eV below the VBM [i.e., region (3) x > x^*], it only crosses one band (green one) as shown in Fig. 7(a). Thus, the corresponding Fermi surface contains only one drum-like pocket as shown in Fig. 7(d), which is totally different from the Fermi surface in region (1) as shown in Fig. 7(b). It is obvious that in the k_z = 0 plane, the Fermi surface consists of two circles in region (1), while it only has one circle in region (3). This is just the filling-induced Lifshitz transition proposed in Ito’s experiment work.¹⁸ However, we note that the Fermi surfaces have a different shape at a k_z = ±π/c plane in both regions. We also find that there exists a small region [i.e., region (2) x ≈ x^* with yellow shadow] between regions (1) and (3), which is the critical point during the Lifshitz transition of the Fermi surface. In this region, the Fermi energy crosses the green band, which results in a cylinder Fermi surface as shown in Fig. 7(c). However, it also crosses the peak of the red band, which results in a very small pocket around the Γ point shown in Fig. 7(c).

From the enlarged band structure in Fig. S4 (supplementary material), the similar Lifshitz transition could also be observed in Rb_xRhO₂ and Cs_xRhO₂ due to the similarity in the band structure compared with the K_xRhO₂. However, it is found that the critical point of the transition is very close to the VBM, which implies that the critical value of x^* in Rb_xRhO₂ and Cs_xRhO₂ should be quite close to 1.0. However, according to our binding energy calculation, the P2 phase Rb_xRhO₂ and Cs_xRhO₂ favor a moderate alkali concentration (x ∼ 0.5). Therefore, it is quite possible that experiments could not observe the Lifshitz transition in Rb_xRhO₂ and Cs_xRhO₂. As for Li_xRhO₂ and Na_xRhO₂, there are only one pudding-mode like band near the Γ point. Therefore, it is not possible to find the same Lifshitz transition discussed in K_xRhO₂. Nevertheless, since the VBM in LiRhO₂ and NaRhO₂ is near the Z point and their pudding-mode like bands are lower than the VBM, we can also expect some kinds of change of the Fermi surface due to the change in x, which will possibly affect the electronic properties of Li_xRhO₂ and Na_xRhO₂ significantly.

Finally, we have to note that the SOC is not considered in the above calculations because we find that the bands near the band gap are not affected by the SOC as shown in Figs. S6 and S7 of the supplementary material. It is found that in all the materials, the SOC will induce evident band splittings of about −0.5 eV below the VBM, which is not considered in this work.

We have systematically studied and compared the structural and electronic properties of layered 4d transition metal oxides A_xRhO₂ (A = Li, Na, K, Rb, Cs) by the first-principles calculations. We demonstrate the tendency of structural properties of A_xRhO₂ depending on the type and concentration of alkali ions. It is found that LiRhO₂ and NaRhO₂ favor the O3 phase, while the other three compounds (i.e., KRhO₂, RbRhO₂, and CsRhO₂) prefer the P2 structure. In general, the in-plane lattice constant (a) decreases and the out-of-plane one (c) increases as the concentration (x) of alkali ions increases. We also found that the three P2 phase crystals tend to have medium alkali ion concentration with x = 0.5 ∼ 0.6, but the two O3 phase crystals tend to be fully intercalated (x ≈ 1). Then, we also studied the electronic properties of five materials. It is found that they are all indirect band-gap semiconductors with band gaps between 3 eV and 3.6 eV. The electron bands near the band gap are mainly obtained from the Rh 4d and O 2p orbitals. Moreover, we demonstrate a Lifshitz transition of the Fermi surface in K_xRhO₂, which is proposed in a recent experiment. A possible similar Lifshitz transition of other four materials is also discussed. Our work has obtained some useful structural and electronic properties of A_xRhO₂ and could give some hints on discovering new properties of these less-studied compounds in future experiments.

See the supplementary material for other information: (1) the Brillouin zones and high symmetry point of the O3 and P2 crystal structure, (2) the detailed band structures and DOS of ARhO₂ (A = Li, Na, K, Rb, and Cs), and (3) their band structures with and without spin–orbit couplings.

This work was supported by the National Key R&D Program of China (Grant No. 2016YFA0201104), the National Natural Science Foundation of China (Grant Nos. 11974163, 11890702, and 51721001), and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20171343). The use of the computational resources in the High Performance Computing Center of Nanjing University for this work is also acknowledged.