Introduction

Thermoelectric materials are required for power generation from waste heat as well as for refrigeration. However, the practical application is limited by the low efficiency of thermoelectric devices as compared to traditional fuel power generators and compressor-based refrigerators1. A high performance thermoelectric material has to be a good electrical and poor thermal conductor and, at the same time, has to possess a high Seebeck coefficient2. The efficiency is determined by the dimensionless figure of merit zT = σS2T/κ, where σ is the electrical conductivity, S is the Seebeck coefficient, T is the temperature and κ is the thermal conductivity. According to the classical semiconductor theory, heavily doped semiconductors with carrier concentrations of 1019 to 1021 cm−3 in general are promising candidates for high zT materials3.

The discovery of a high Seebeck coefficient (S ~ 100 μV/K at 300 K) in NaxCoO2 has attracted considerable attention about a decade ago4,5. It has been predicted that the two-dimensional structure with alternatively stacked Na and CoO2 layers plays an important role for the high zT of Na0.5CoO26. Later, Kuroki et al.7 have argued that the peculiar pudding mold shape of the valence band of NaxCoO2 is responsible for the high zT (with high σ), see also8. Interestingly, Takada et al.9 have demonstrated that the compounds can be readily hydrated to form NaxCoO2 · yH2O, maintaining the CoO2 layers, but with a considerably expanded crystal c-axis to accomodate the intercalated water. The electronic structures of hydrated and non-hydrated NaxCoO2 have been studied by first principles calculations in Ref. 10.

Analogous compounds with Rh in place of Co are also good thermoelectric materials, since the octahedrally coordinated ions Rh3+ (4d6) and Rh4+ (4d5) show similar electronic structures as Co3+ (3d6) and Co4+ (3d5), but exhibit reduced correlation effects11,12,13,14. Experiments could not verify any magnetic ordering in NaxRhO215. In general, the Rh 4d orbitals are spatially more dispersed than the Co 3d orbitals, which may result in an increased mobility of the charge carriers. Varela et al.16 have synthesized NaxRhO2 as solid solutions composed of nanometric particles of the α-NaFeO2 structure type (space group , no. 166) for x = 0.70 and 0.85. The 3R structure of NaxRhO2 contains three layers of RhO6 octahedra per unit cell. Using the hexagonal setting of the space group, the c-axis of the unit cell has a length of typically 18 Å, while the length of the a-axis strongly depends on the specific transition metal cation.

It is important to notice that the layered oxides LiRhO2, NaRhO2 and KRhO2 can form hydrated, i.e., water intercalated, phases17 with an elongated c-axis. For NaxRhO2 it has been shown that the water intercalation and the elongated unit cell result in a decrease of the resistivity18. Accordingly, in Ref. 19 a colossal thermoelectric powerfactor has been demonstrated theoretically for K7/8RhO2 in the case of hydration, which is nowadays triggering various experimental efforts20,21. From a practical point of view the results reported on K7/8RhO2 are limited by the fact that the optimal thermoelectric performance is achieved below usual operation temperatures19. In this context we propose to switch from the 2H phase (two RhO6 layers per unit cell with a hexagonal structure of space group P63/mmc, no. 194 and c ~ 12 Å) to the 3R phase, as one may speculate that the thermoelectric properties are modified by the different stacking (larger band gap in the 3R phases22). Accordingly, we study in the following the thermoelectric properties of both non-hydrated (c = 15.54 Å) and hydrated (c = 20.85 Å) 3R-NaxRhO2, for which we first determine the lowest energy structures for x = 0.75 and 0.83. Afterwards we analyze the effects of hydration on the transport properties in the temperature range from 300 K to 800 K and demonstrate the great potential of the 3R-NaxRhO2 class of materials as thermoelectrics. The 3R phase turns out to be clearly superior to the 2H phase in the technologically relevant temperature range.

Results

We start from the experimental lattice parameters a = 3.09 Å and c = 15.54 Å of NaRhO216 and construct a 2 × 2 × 1 supercell. Afterwards we remove two Na atoms from different sites to obtain Na0.83RhO2. These configurations are named A1, A2 and A3. Similarly, we remove three Na atoms to obtain Na0.75RhO2 and name these configurations B1, B2 and B3. All the structures under investigation are shown in Fig. 1. The formation energy is calculated for each configuration by the relation

Here EPristine is the total energy of a supercell of pristine NaRhO2. In addition, EDefect and ENa are the total energies of the defective supercell and of the isolated Na atom, respectively. The energy of an isolated atom is calculated by placing the atom in a cubic cell of 20 Å side length.

Figure 1
figure 1

Atomic structures of NaRhO2, Na0.83RhO2 and Na0.75RhO2 for different vacancy configurations.

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The formation energies of configurations A1, A2 and A3 amount to 7.83 eV, 6.73 eV and 6.65 eV, respectively and those of configurations B1, B2 and B3 to 11.54 eV, 10.34 and 9.90 eV. These results show that for Na0.75RhO2 and Na0.83RhO2, respectively, the A3 and B3 configuration has the lowest energy. The latter structures are characterized by maximal distances between the defects, which indicates that Na vacancies do not cluster in NaxRhO2, at least not in the two systems under investigation. Moreover, a minimum energy of 6.65 eV is required for creating two Na vacancies in NaRhO2 and an energy of 9.90 eV for creating three vacancies. In the following, we will discuss only the lowest energy structures A3 and B3 unless otherwise mentioned. In pristine NaRhO2 the Rh–O bond length amounts to 2.07 Å and the thickness of the RhO6 layer is 2.11 Å, which is larger than the thickness of the CoO6 layer (1.82 Å) in NaCoO2. Moreover, the bond angle ∠Rh−O−Rh = 96.5° is smaller than the bond angle ∠Co−O−Co = 98.35°. Next to vacancies the RhO6 octahedra shrink slightly (the Rh–O bond lengths decrease by 0.02–0.05 Å), because the interlayer coupling through the Na atoms is reduced and the RhO6 octahedra thus become a bit more compact, which also influences the electronic and thermoelectric properties of the material.

We determine the effects on the electronic band structure and corresponding density of states (DOS), for NaRhO2, Na0.83RhO2 and Na0.75RhO2 in Fig. 2. NaRhO2 has a indirect band gap of 1.33 eV along the Γ–M high symmetry direction. The valence band mainly is formed by the Rh 4d t2g states and shows a total width of a bit less than 2 eV. This value is comparable to the bandwidth of NaCoO2, even though 4d orbitals are more extended than 3d orbitals and thus usually exhibit higher bandwidths. The conduction band minimum belongs almost purely to the eg states. On the other hand, the valence band maximum, which is particularly important as it determines the consequences of hole doping, is due to the orbital. The electronic band structures obtained for configurations A3 and B3 are similar to the pristine case, except for the introduction of holes with densities of 8.8 · 1020 cm−3 (A3) and 8.7 · 1020 cm−3 (B3). Besides the characteristic peak approximately 0.2 eV below the Fermi energy, the DOS of prsitine NaRhO2 shows states down to −0.75 eV and between −1 eV and −1.75 eV, which is similar for Na0.83RhO2 and Na0.75RhO2 except for the states at the Fermi level, which have different effective masses and consequently result in different Seebeck coefficients.

Figure 2
figure 2

Band structures and Rh 4d DOSs of NaRhO2, Na0.83RhO2 (non-hydrated) and Na0.75RhO2 (non-hydrated).

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The transport properties of hydrated and non-hydrated Na0.83RhO2 and Na0.75RhO2 are addressed in Fig. 3. We show the Seebeck coefficient and figure of merit for the temperature range from 300 K to 800 K. The Seebeck coefficient is positive in the whole range, indicating that the carriers are holes. We obtain room temperature values of S = 63 μVK−1 and S = 37 μVK−1 for non-hydrated Na0.83RhO2 and Na0.75RhO2, respectively. For hydrated and non-hydrated Na0.83RhO2 the Seebeck coefficient behaves rather similarly as a function of the temperature, except for the high temperature range where hydration reduces S. Moreover, according to Figs. 3(c) and 3(d), zT at 300 K (room temperature) amounts to 0.16 and 0.06 in the two systems. The former value is twice that found for Na0.85CoO2 and Na0.88CoO2 in Ref. 5. Starting from almost the same value at 300 K, hydration enhances zT over the whole temperature range, such that at 800 K a value of about 0.4 is reached. At 580 K a maximal increment of 34% due to hydration is achieved. For Na0.75RhO2 the Seebeck coefficient increases at 300 K from 37 μVK−1 to 51 μVK−1 under hydration, where this increment of about 30% remains essentially constant over the whole temperature range. A similar effect is also found for zT. At 800 K again a value of about 0.4 is reached.

Figure 3
figure 3

Transport properties of non-hydrated and hydrated Na0.83RhO2 and Na0.75RhO2 as functions of the temperature.

(a), (b) Seebeck coefficient and (c), (d) figure of merit.

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The enhancement of S under hydration results in higher zT values. While the gross shape of the band structure changes little under hydration, see Fig. 4, the Rh 4d bands become flatter in a narrow region around the Fermi level. This fact causes the effective masses and therefore the resistivity to increase and results in a higher Seebeck coefficient. Moreover, the long c-axis of the hydrated phase causes the Rh 4d states to shift down in energy towards the Fermi level, because the Rh–O overlap is reduced (increased bond length of 2.2 Å). In order to study possible disorder of the vacancies we have also calculated the Seebeck coefficient (not shown) of the higher energy configurations A1, A2, B1 and B2. For A2 and B2 the results are close to the values of the respective lowest energy configurations, because both have the vacancies in different RhO6 layers. In contrast, concentration of vacancies in the same RhO6 layer results in a significant reduction of S, which is, however, highly unlikely as certified by the previously listed formation energies.

Figure 4
figure 4

Band structures and Rh 4d DOSs of hydrated Na0.83RhO2 and Na0.75RhO2.

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Discussion

The NaxRhO2 oxides are found to form a new class of materials with exciting thermoelectric features, even outperforming the 2H phases of the KxRhO2 system. In the latter the optimal thermoelectric performance is achieved at low temperature, whereas the modified stacking of the atomic layers in the novel 3R phases of NaxRhO2 results in a reduced interlayer coupling and, in turn, in a dramatically enhanced thermoelectric response in the technologically relevant high temperature range. We have also demonstrated that Na vacancies in NaxRhO2 avoid clustering and that the RhO6 octahedra are modified depending on the amount of Na deficiency. Analysis of the induced changes in the DOS close to the Fermi level indicates that mainly states control the transport properties of the compounds. A high zT value of 0.35 at 580 K is achieved in hydrated Na0.83RhO2 due to the enhanced effective mass of the charge carriers. In general, zT can be further increased by reduction of the Na vacancy concentration to increase the resistivity.

Methods

We study the effects of Na deficiency and hydration on the band structure of NaRhO2 using density function theory as implemented in the WIEN2k package23. The exchange-correlation potential is parametrized in the generalized gradient approximation24. We fully optimize the structure of NaxRhO2 for x = 1, 0.83 and 0.75, considering 2 × 2 × 1 supercells, until the atomic forces reach values of less than 10 mRyd/aB. For the wave function expansion inside the atomic spheres a maximal angular momentum of ℓmax = 12 is employed and a plane-wave cutoff of RmtKmax = 7 with Gmax = 24 is used. Self-consistency is assumed when the total energy variation is less than 10−4 Ry. The transport behavior is investigated using BoltzTraP25, which employs a methodology that previously has demonstrated to yield quantitatively accurate values for the thermoelectric properties of metals and doped semiconductors26,27,28,29. It is based on semi-classical Boltzmann theory within the constant relaxation time approximation, i.e., the scattering rate is assumed to be independent of momentum and energy. This approximation allows the thermopower to be directly calculated from the band structure as a function of the carrier concentration and temperature30. We note that it has been demonstrated for layered oxides isostructural to the present compound that at elevated temperatures the lattice part of the thermal conductivity can be neglected as compared to the electronic part31. We use a k-mesh of 7 × 7 × 2 points for the band structure and density of states (DOS) and a dense k-mesh of 18 × 18 × 18 points for the thermoelectric calculations.