Sodium cobaltate (NaxCoO2) is an interesting and promising compound for high-efficiency thermoelectric material applications [1,2,3]. This compound also exhibits a rich magnetic and structural phase diagrams [4, 5]. NaxCoO2′s unit cell consists of two alternating Na layers and edge-sharing CoO6 octahedra. Co ions’ mixed valency in Na deficient NaxCoO2 (x < 1) creates a large spin entropy flow, and consequently, a large Seebeck coefficient [6]. Additionally, the Na layer in which Na+ ions are amorphous and liquid like at room temperature [7] strongly scatters the heat-carrying phonons. The high mobility of Na ions leads to unprecedented freedom to favourably adjust all otherwise interdependent factors of the figure of merit (ZT) independently, giving NaxCoO2 an advantage over other thermoelectric materials [8, 9].

Controlling Na concentration (x) has been the primary technique to push the ZT of NaxCoO2 to higher limits [10]. However, given the volatile nature of Na ions, accurate experimental characterisation of Na concentration and its effect on the material’s behaviour is somehow difficult [11]. This experimental difficulty motivated theoretical investigations into the structural and electronic properties of pristine NaxCoO2 [12,13,14,15]. Additionally, doping other elements in NaxCoO2 has also been utilised to improve sodium cobaltate’s thermoelectric properties [16]. In spite of the efforts, doping NaxCoO2, as a strategy to enhance the thermoelectric performance, delivered mixed results. For instance, heavier ions such as rare earth Yb and noble metal Ag were successfully used to decrease the lattice thermal conductivity (kl) and thus improving ZT [17, 18]. Doping Ru and Mn in Na0.5CoO2, however, increased the resistivity to ~ 300 µΩ m, three times higher than that of the undoped Na0.5CoO2 [19]. Similarly, co-doping Ti and Bi almost halved the Seebeck coefficient in Na0.6CoO2 to ~ 60 μK/V [20].

The mixed experimental results indicate that further progress in realising the functional applications of NaxCoO2 requires an accurate understanding of the effects of Na concentration and the presence of dopants on the electronic property of NaxCoO2. More specifically, the effect of the dopants on the delicate magnetic interactions in NaxCoO2 is still under debate [21] and needs careful investigation. In this work, therefore, the behaviour of Sb dopants in Na0.875CoO2 is examined using density functional theory. NaxCoO2 with higher Na concentration of x ≈ 0.8, as considered here, possesses excessively higher Seebeck coefficient [22] and exhibits complex Na ordering patterns [5], thus is both appealing and challenging compound to explore. Our choice of Sb dopant was motivated by the successful synthesis of Sb-doped layered metal oxide compounds such as LiMn2O4/LiSbO3 [23], Na3Ni2SbO6 [24] and Na3–xSn2–xSbxNaO6 [25], all of which share considerable structural features with NaxCoO2. Furthermore, although the Sb-doped Na1CoO2, Na0.75CoO2 and Na0.5CoO2 compounds have been previously examined [26, 27], Sb doping in Na0.875CoO2, which is thermoelectrically more critical has not yet been explored in details.

Computational settings

Spin-polarised density functional calculations were performed with VASP package [28, 29] based on the projector augmented wave method [30, 31]. The energy cutoff was set to 500 eV. Generalised gradient approximation (GGA) [32, 33] was applied for the exchange–correlation functional. We added on-site Coulomb (U) and exchange (J) interaction terms of U = 5 and J = 1 eV to Co 3d electrons using Dudarev’s approach [34]. Among many values reported for U and J in the literature [35,36,37], the chosen values reproduce the charge disproportionation [38, 39] of the Co ions more accurately. That is clearly distinguishing Co2+, Co3+ and Co4+ from one another in terms of magnetisation and partial density of states. Furthermore, the value of Ueff = 4 eV (Ueff = U − J) is at a midpoint between the low-end values of ~ 3 eV usually proposed for Co3+ [40] and the high-end values of ~ 5 eV proposed for Co4+ [41]. Ueff values close to 4 eV has been successfully applied to multivalent Co oxides [42, 43]. Brillouin zone sampling for the supercells was carried out by choosing a 4 × 2 × 2 k-point set within the Monkhorst–Pack scheme [44]. We have successfully examined the convergence of these settings earlier [45, 46]. Charges localised on cationic centres were examined with Bader charge analysis code [47]. The quoted experimental ionic radii were compiled and reported by Shannon [48].

To obtain the Na0.875CoO2 supercell, we first optimised the P63/mmc unitcell of the Na1CoO2, which acts as the building block for the Na0.875CoO2 structure. The lattice parameters of a fully optimised primitive cell of Na1CoO2 were found to be 2.87 Å for a and 10.90 Å for c, which are in good agreement with the measurements [49]. Then two Na ions, one from each Na layer, were removed from the 2a × 4a × 1c Na1CoO2 supercell to construct the undoped Na0.875CoO2 structure. The most stable arrangement of the Na vacancies was adapted from our previous work [46], which is shown in Fig. 1. To obtain the final structures, the internal coordinates of all ions in the supercell were relaxed while the lattice constants were fixed to the calculated values of pristine Na1CoO2. The resulting Na0.875CoO2 unit cell had a primitive monoclinic (P2) representation in which all Na ions were at Na2 sites. Although the lattice parameters of Na0.875CoO2 were expected to be slightly different from those of Na1CoO2, dopants’ formation energy is not significantly sensitive to this variation [12, 50]. For calculating Sb’s formation energy, after dopant’s placement in the Na0.875CoO2 structure, we fixed the lattice parameters of the doped supercell at the theoretical values of the pristine Na0.875CoO2, while all internal coordinates were relaxed. This procedure eliminates the artificial hydrostatic pressure ensuring that the final structures are in equilibrium [51, 52].

Fig. 1
figure 1

The spin density of the 2a × 4a × 1c supercell used for simulating Sb-doped Na0.875CoO2 compound in the most stable configuration. Co and O ions occupy the Wyckoff 2a and 4f sites of the hexagonal lattice structure, respectively. In Na0.875CoO2, all Na ions occupy 2c (Na2) sites. For smaller Na concentrations, some Na ions occupy 2b (Na1) sites. The distance between VNa and Sb5+ is marked with dotted brown lines. The spin density isosurfaces were drawn at ρ = 0.035 e/Å3. Yellow and cyan surfaces indicate spin-up and spin-down, respectively

Results and discussion

We first have to consider the formation energy (Ef) of Sb dopants in NaxCoO2. Cationic Sb dopants can either replace a Co or be incorporated in the Na layer. Since Sb’s ionic volume of 2.16 × 10−4 nm3 is much larger than the interstitial cavity in CoO6 layer of 1.79 × 10−7 nm3 (marked with a star in Fig. 1), this interstitial site is too small for Sb doping. Furthermore, Sb in Na layer can either be substituted for a Na site (Na1 or Na2 as marked in Fig. 1) or be incorporated interstitially, e.g. occupying a vacant Na site. For x = 0.875, Sb’s formation energy has been reported earlier [26], using the standard supercell defect formation energy approach [53] and the triangular method for obtaining the chemical potentials [54]. The reported values belonged to an O rich environment in which Sb3O4 and CoO are the competing phases. The chemical potential (μ) for the oxygen-rich environment was set at μO = Et(O2)/2 in which the Et(O2) is the total energy of the O2 molecule [55]. Accordingly, the most stable configuration was found SbCo with an Ef of 1.175 eV followed by SbNa1 with an Ef of 4.547 eV, SbNa2 with an Ef of 4.805 eV and finally SbInt with an Ef of 5.515 eV. This stability sequence is similar to that of Sb doping in Na1CoO2 (x = 1.00) for which SbCo had an Ef of 1.356 eV while SbNa1, SbNa2 and SbInt each had an Ef of 4.798 eV, 4.893 eV and 11.243 eV, respectively.

In the undoped Na0.875CoO2 compounds, Co4+ species are produced when the original Co3+ ions in the parent Na1CoO2 compound are oxidised to bear the holes created by VNas. As a result, the Na14Co16O32 supercell of the Na0.875CoO2 compound has two holes borne on two Co4+ ions, one in each CoO2 layer of the supercell. Sb dopants can adopt either + 3 or + 5 oxidation state. The Sb’s partial density of states, as presented in Fig. 2, which corresponds to the configuration in Fig. 1, shows that both 5 s and 5p states are located above the Fermi level, indicating a + 5 oxidation state. Since the supercell of the Na0.875CoO2:SbCo has a chemical composition of Na14Co15Sb1O32, charge neutrality implies that on average the oxidation state of Co ions is + 3. In the most stable spin alignment, presented in Fig. 1, all Co ions have, indeed, an oxidation state of + 3. Additionally, all Co ions were stabilised in the low spin configuration of \(t_{2g}^{6} e_{g}^{0}\), further suggesting that this compound is non-magnetic. This is in contrast to the pristine Na0.875CoO2 compounds in which the Co4+ ions were antiferromagnetically aligned across the CoO2 planes [56, 57]. Moreover, we have earlier shown that in Na1CoO2 and Na0.75CoO2, SbCo dopant also adopts the higher oxidation state [27].

Fig. 2
figure 2

The partial density of states (PDOS) of SbCo-doped Na0.875CoO2. The black and pink lines represent O 2p and Co’s 3d PDOS, respectively while the blue and green lines represent Sb 5 s and 5p states, respectively. The cyan lines represent Na 3 s states. The electronic configuration of Co3+ is schematically shown in the inset at the bottom right corner

The partial density of states of SbCo-doped Na0.875CoO2 [Na0.875(Co0.9375Sb0.0625)O2], shown in Fig. 2, has some other noticeable features. Here, as expected [58, 59], under octahedral coordination, Co \(3{\text{d}}_{{z^{2} }}\) and \(3{\text{d}}_{{x^{2} - y^{2} }}\) orbitals directly overlap with O 2px, 2py, and 2pz orbitals along the octahedral directions having σ hybridisation. This σ overlap results in low lying bonding \({\text{e}}_{g}^{b}\) states with predominantly p character and high lying antibonding \({\text{e}}_{g}^{*}\) states with predominantly d character. The remaining Co 3dxy, 3dxz, and 3dyz orbitals point away from the O 2p orbitals and, therefore, have no significant σ overlap with O 2p orbitals, constituting the nonbonding t2g states. Moreover, the overlap of O 2p and Co 3p, and O 2p and Co 4 s orbitals form \({\text{t}}_{1u}^{*}\) and \({\text{a}}_{1g}^{*}\) bands, respectively, both of which located above the \({\text{e}}_{g}^{*}\) states and have p character. Furthermore, there is a crystal field splitting of 1.01 eV (marked with a red arrow in Fig. 2) between the filled Co3+t2g states at the top of the valence band and the empty antibonding eg states at the bottom of the conduction band forcing all Co3+ in the low spin state. The bandgap of 1.01 eV is smaller than that of Na1CoO2 (~ 2.2 eV) [45], indicating that Sb doping reduces the fundamental bandgap in Na0.875CoO2. Additionally, Sb’s empty 5 s states are located at the top of the conduction band at ~ 2.5 eV hybridising with the empty eg states while Sb’s empty 5p states are located even higher at ~ 6 eV hybridising with the high lying Na’s empty 3 s states. This arrangement is different from that of Sb5+ in Sb2O5 in which both Sb 5 s and 5p states are located within the same range at ~ 4 eV above Fermi level [60].

There is another possibility that the excess electrons introduced by SbCo’ high oxidation state, instead of neutralising all hole introduced by VNas in Na deficient compounds, reduce the Co3+ species to Co2+ in Na0.785CoO2, while leaving all or some of the Co4+ ions unaltered. Co2+ and Co4+ are magnetic with a different spin and orbital degeneracies, implying that Na0.875CoO2: SbCo might potentially exhibit a different magnetic and thermoelectric behaviour. We examined the possibility of alternative manifestation of oxidation states for Co ions in Na0.875CoO2: SbCo other than its most stable arrangement as presented in Fig. 1. In doing so, we constructed the alternative spin alignments by fixing the initial magnetic moment on the Co ions in the calculations. Several different configurations were considered of which a more stable spin bearing one is presented in Fig. 3. The partial density of Co ions with distinct oxidation and spin states are presented in Fig. 4. Here, we see that high spin Co4+ (\(t_{2g}^{3} e_{g}^{2}\)) and low spin Co2+ (\(t_{2g}^{6} e_{g}^{1}\)) species were both formed at the vicinity of Sb5+ dopant. Furthermore, a larger radius of Co2+ at octahedral coordination of 0.75 Å introduced significant lattice distortions that forced several Co3+ to take high spin configurations (\(t_{2g}^{4} e_{g}^{2}\)). The total energy of this configuration was, however, higher than the non-magnetic configuration (Fig. 1) by 46.963 meV/Co, suggesting that Co ions in Sb-doped Na0.875CoO2 would be non-magnetic—all Co ions at \(t_{2g}^{6} e_{g}^{0}\) for a wide range of temperatures.

Fig. 3
figure 3

The spin density of the 2a × 4a × 1c supercell used for simulating Na0.875CoO2:SbCo compound. Here, Sb5+ is at the closest distance to the VNas as in Fig. 1, however, the spin state of Co ions corresponds to an excited state

Fig. 4
figure 4

The partial density of states of the Co 3d states of different oxidation states and spin states in the excited state presented in Fig. 3. Spin-up and spin-down channels are represented by black and red lines, respectively. The electronic configurations of various Co ions in the supercell are schematically shown in the insets at the bottom left corner

In SbCo-doped Na0.875CoO2 system [Na0.875(Co0.9375Sb0.0625)O2], Sb dopants can occupy few distinct positions when substituting Co. For instance, either close to VNas, as shown in Fig. 1, which corresponds to the most stable arrangement of Sb dopant, or far from the VNas, as shown in Fig. 5. The distance between SbCo and VNas for the configuration presented in Fig. 1 is 2.23 Å for the VNa at Z = 0 and 2.23 Å for the VNa at Z = 0.5. These distances for the configuration presented in Fig. 5 were 5.23 Å for the VNa at Z = 0 and 5.26 Å for the VNa at Z = 0.5. We found the total energy of the configuration in Fig. 1 was 27.620 meV/Co lower, and hence more stable than the configuration in Fig. 5. One possible explanation for the aggregation of Sb dopant and the VNas is that the radius of Sb5+ in octahedral coordination (0.60 Å) is ~ 8.3% larger than that of Co3+ (0.55 Å). As a consequence, the location of larger Sb ion in the vicinity of VNa results in the cancellation of internal lattice stress leading to greater stability.

Fig. 5
figure 5

The spin density of the 2a × 4a × 1c supercell used for simulating Sb-doped Na0.875CoO2 compound when Sb5+ is not in the immediate vicinity of the VNas. The distance between VNa and Sb5+ is marked with dotted brown lines

In the case where the \({\text{Sb}}_{{{\text{Co}}^{3 + } }}^{5 + }\) dopant is far away from the VNas as in Fig. 5, unlike the case of the ground state in Fig. 1, the electrons of the Sb5+ do not neutralise both of the holes that are borne on the Co4+ ions in the supercell. As shown in Fig. 5, only the Co4+ at the bottom CoO2 layer which contains the \({\text{Sb}}_{{{\text{Co}}^{3 + } }}^{5 + }\) dopant has been reduced to Co3+ while the Co4+ in the top layer has remained unaltered. The extra electron of the Sb dopant, instead, reduces a Co3+ in the vicinity of the Sb dopant to Co2+. Since the Co4+ ions are located near the VNas in the undoped compound [45], doping \({\text{Sb}}_{{{\text{Co}}^{3 + } }}^{5 + }\) in the vicinity of VNas (as in Fig. 1), facilitates the charge transfer from Sb to two cationic nearest neighbour Co4+ ions. However, when the \({\text{Sb}}_{{{\text{Co}}^{3 + } }}^{5 + }\) is far away from the VNas, as in Fig. 5, there is no cationic nearest neighbour charge transfer possible between Sb and Co4+ at the top layer. Consequently, the extra electron of \({\text{Sb}}_{{{\text{Co}}^{3 + } }}^{5 + }\) is localised on a neighbouring Co3+ in the bottom layer converting it to Co2+. We verified the stability of this spin alignment for the configuration in Fig. 5 by fixing the spin of all Co to zero (\(t_{2g}^{6} e_{g}^{0}\)) which resulted in a 112.6 meV/Co higher total energy.

In NaxCoO2, the Seebeck effect is facilitated by the spin entropy transfer between Co3+ and Co4+ ions for which the Seebeck coefficient can be estimated by the modified Heikes formula [61, 62]:

$$S\left( {T \to \infty } \right) = - \frac{{k_{B} }}{e}{\text{Ln}}\left( {\frac{{g\left( {Co^{3 + } } \right)}}{{g\left( {Co^{4 + } } \right)}} \cdot \frac{\rho }{1 - \rho }} \right),$$

in which kB and e are the Boltzmann constant and electron’s charge, respectively. g(Co3+) and g(Co4+) are the electronic degeneracies of Co3+ and Co4+ ions, respectively, and ρ is the hole concentration. The electronic degeneracies are calculated by multiplying the orbital (gorbital) and spin (gspin) degeneracies of electrons for a given transition metal ion. In low spin, octahedral coordination for which Co3+ and Co4+ electrons occupy t2g orbitals only, gorbital is the total number of permutation of electrons and is calculated as:

$$g_{{{\text{orbital}}}} = \frac{3!}{{n^{ \uparrow } !\left( {3 - n^{ \uparrow } } \right)!}} \cdot \frac{3!}{{n^{ \downarrow } !\left( {3 - n^{ \downarrow } } \right)!}},$$

in which n and n are the number of spin-up and spin-down electrons, respectively. gspin, on the other hand, equals to 2ξ + 1, where ξ is the total spin number of a given ion. Plugging the values for Co3+ and Co4+ renders g(Co3+) = 1 and g(Co4+) = 6. Additionally, since conduction in Na0.875CoO2 is achieved by holes hopping form a Co4+ site to a Co3+ site, the hole concentration is proportional to the ratio of Co4+ site to the total Co sites. Consequently, S(T → ∞) = 322.08 μK/V in pristine Na0.875CoO2. Equation 1 establishes a reverse relationship between ρ and S; so that for raising the Seebeck coefficient, one must reduce the concentration of Co4+ species [63].

The introduction of one SbCo dopant into the Na14Co16O32 supercell, however, eliminates the Co4+ species and brings the hole concentration to zero resulting in a band insulating character for Na0.875CoO2:SbCo. This is inferable from the partial density of states in Fig. 2, which shows a bandgap of 1.01 eV. As a consequence, according to Eq. 1, the Seebeck coefficient diverges. Experimentally, however, Seebeck effect measurements in insulators have indicated that the Seebeck potential approaches zero as carrier concentration diminishes [64]. Mahan has attributed this apparent contradiction to the fact that the Seebeck potential does not diminish but rather a space-charge effect in insulators screens off the Seebeck potential [65, 66]. In any case, in the absence of any carriers, Na0.875CoO2:SbCo is of no use for converting heat gradient to measurable electric energy. Although reducing the concentration of Co4+ increases S, in choosing the dopant and its concentration for NaxCoO2, one, nonetheless, must apply due diligence to avoid the catastrophic complete electron–hole recombination.

It is quite instructive to know that several times, similar to the case of Na0.875CoO2: SbCo, ineffective doping cases on thermoelectric compounds have been reported in the literature. For example, Al doping plays no effect on Mg2Si0.75Sn0.25 [67]. In Ni-doped CuInTe2 Ni-doping is rather ineffective regarding the increase of the hole concentration and the decrease of the thermal conductivity [68]. In Mg3Sb2, interstitial doping with Li, Zn, Cu, and Be is found to be ineffective for n-type doping; however, Li is identified as a good p-type dopant [69]. Bi is an ineffective dopant in Mg2Ge and precipitates into Mg2Bi3 [70]. Boron is ineffective as dopant in α-MgAgSb [71].


In this work, using density functional theory, the formation energy of Sb dopants in Na0.875CoO2 was examined. The main findings of this investigation can be summarised as follow: (1) SbCo is the most stable configuration Na0.875CoO2 as it is in Na1CoO2 as well. This trend is similar to the Sb dopant in other sodium cobaltates with smaller Na concentrations [26, 27]. Moreover, as x increases the margin of the stabilisation of SbCo configuration increases against those other configurations in which Sb is located in the Na layer; (2) the formation energy of SbCo decreases with decreasing xEf(Na1CoO2:SbCo) = 1.356 eV and Ef(Na0.875CoO2:SbCo) = 1.175 eV. This trend implies that doping Na deficient systems with Sb is practically easier; (3) in Na0.875CoO2:SbCo, Sb’s 5 s states are located above the Fermi level hybridising with Co’s empty eg states while Sb’s 5p states are located even higher along with Na’s 3 s states implying an oxidation state of 5 + for SbCo; (4) in Na0.875CoO2:SbCo, SbCo tends to aggregate with the VNas; (5) in the most stable configuration, Sb dopants reduce the magnetic Co4+ ion to non-magnetic Co3+ abating both the long-range magnetic order and the Seebeck effect which is sustained by spin entropy flow in undoped NaxCoO2; (6) in an excited state, in Na0.875CoO2:SbCo, a pair of non-magnetic Co3+ can be transformed into magnetic Co2+ and Co4+. Such magnetic state is, however, higher in energy by 46.963 meV/Co than the non-magnetic ground state. Finally, we demonstrated how a generally favourable strategy such as reducing carrier concentration for increasing the Seebeck coefficient can suppress the Seebeck effect for a specific dopant and Na concentrations in NaxCoO2.