Suppression of magnetism and Seebeck effect in Na_0.875CoO₂ induced by Sb_Co dopants

Abstract

We examined the electronic property of Sb-doped Na_0.785CoO₂ using density functional calculations based on GGA+U formalism. We demonstrated that Sb dopants were the most stable when replacing Co ions within the complex Na_0.875CoO₂ lattice structure. We also showed that the Sb_Co dopants adopted the + 5 oxidation state introducing two electrons into the host Na_0.875CoO₂ compound. The newly introduced electrons recombined with holes that were borne on Co⁴⁺ sites that had been created by sodium vacancies. The elimination of Co⁴⁺ species, in turn, rendered Na_0.875(Co_0.9375Sb_0.0625)O₂ non-magnetic and diminished the compound’s thermoelectric effect. Furthermore, the Sb_Co dopants tended to aggregate with the Na vacancies keeping a minimum distance. The conclusions drawn here can be generalised to other highly oxidised dopants in Na_xCoO₂ that replace a Co.

Introduction

Sodium cobaltate (Na_xCoO₂) 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]. Na_xCoO₂′s unit cell consists of two alternating Na layers and edge-sharing CoO₆ octahedra. Co ions’ mixed valency in Na deficient Na_xCoO₂ (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 Na_xCoO₂ an advantage over other thermoelectric materials [8, 9].

Controlling Na concentration (x) has been the primary technique to push the ZT of Na_xCoO₂ 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 Na_xCoO₂ [12,13,14,15]. Additionally, doping other elements in Na_xCoO₂ has also been utilised to improve sodium cobaltate’s thermoelectric properties [16]. In spite of the efforts, doping Na_xCoO₂, 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 (k_l) and thus improving ZT [17, 18]. Doping Ru and Mn in Na_0.5CoO₂, however, increased the resistivity to ~ 300 µΩ m, three times higher than that of the undoped Na_0.5CoO₂ [19]. Similarly, co-doping Ti and Bi almost halved the Seebeck coefficient in Na_0.6CoO₂ to ~ 60 μK/V [20].

The mixed experimental results indicate that further progress in realising the functional applications of Na_xCoO₂ requires an accurate understanding of the effects of Na concentration and the presence of dopants on the electronic property of Na_xCoO₂. More specifically, the effect of the dopants on the delicate magnetic interactions in Na_xCoO₂ is still under debate [21] and needs careful investigation. In this work, therefore, the behaviour of Sb dopants in Na_0.875CoO₂ is examined using density functional theory. Na_xCoO₂ 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 Na_3–xSn_2–xSb_xNaO₆ [25], all of which share considerable structural features with Na_xCoO₂. Furthermore, although the Sb-doped Na₁CoO₂, Na_0.75CoO₂ and Na_0.5CoO₂ compounds have been previously examined [26, 27], Sb doping in Na_0.875CoO_2, 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 U_eff = 4 eV (U_eff = 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]. U_eff 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 Na_0.875CoO₂ supercell, we first optimised the P6₃/mmc unitcell of the Na₁CoO_2, which acts as the building block for the Na_0.875CoO₂ 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 Na_0.875CoO₂ 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 Na_0.875CoO₂ unit cell had a primitive monoclinic (P2) representation in which all Na ions were at Na2 sites. Although the lattice parameters of Na_0.875CoO₂ 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 Na_0.875CoO₂ structure, we fixed the lattice parameters of the doped supercell at the theoretical values of the pristine Na_0.875CoO₂, 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

The spin density of the 2a × 4a × 1c supercell used for simulating Sb-doped Na_0.875CoO₂ compound in the most stable configuration. Co and O ions occupy the Wyckoff 2a and 4f sites of the hexagonal lattice structure, respectively. In Na_0.875CoO₂, all Na ions occupy 2c (Na2) sites. For smaller Na concentrations, some Na ions occupy 2b (Na1) sites. The distance between V_Na and Sb⁵⁺ is marked with dotted brown lines. The spin density isosurfaces were drawn at ρ = 0.035 e/ų. Yellow and cyan surfaces indicate spin-up and spin-down, respectively

Results and discussion

We first have to consider the formation energy (E^f) of Sb dopants in Na_xCoO₂. 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 μ_O = E^t(O₂)/2 in which the E^t(O₂) is the total energy of the O₂ molecule [55]. Accordingly, the most stable configuration was found Sb_Co with an E^f of 1.175 eV followed by Sb_Na1 with an E^f of 4.547 eV, Sb_Na2 with an E^f of 4.805 eV and finally Sb_Int with an E^f of 5.515 eV. This stability sequence is similar to that of Sb doping in Na₁CoO₂ (x = 1.00) for which Sb_Co had an E^f of 1.356 eV while Sb_Na1, Sb_Na2 and Sb_Int each had an E^f of 4.798 eV, 4.893 eV and 11.243 eV, respectively.

In the undoped Na_0.875CoO₂ compounds, Co⁴⁺ species are produced when the original Co³⁺ ions in the parent Na₁CoO₂ compound are oxidised to bear the holes created by V_Nas. As a result, the Na₁₄Co₁₆O₃₂ supercell of the Na_0.875CoO₂ 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 Na_0.875CoO₂:Sb_Co 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 Na_0.875CoO₂ compounds in which the Co⁴⁺ ions were antiferromagnetically aligned across the CoO₂ planes [56, 57]. Moreover, we have earlier shown that in Na₁CoO₂ and Na_0.75CoO₂, Sb_Co dopant also adopts the higher oxidation state [27].

Fig. 2

The partial density of states (PDOS) of Sb_Co-doped Na_0.875CoO₂. 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 Sb_Co-doped Na_0.875CoO₂ [Na_0.875(Co_0.9375Sb_0.0625)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 2p_x, 2p_y, and 2p_z 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 3d_xy, 3d_xz, and 3d_yz orbitals point away from the O 2p orbitals and, therefore, have no significant σ overlap with O 2p orbitals, constituting the nonbonding t_2g 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⁺t_2g states at the top of the valence band and the empty antibonding e_g 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 Na_0.875CoO₂. Additionally, Sb’s empty 5 s states are located at the top of the conduction band at ~ 2.5 eV hybridising with the empty e_g 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 Sb_Co’ high oxidation state, instead of neutralising all hole introduced by V_Nas in Na deficient compounds, reduce the Co³⁺ species to Co²⁺ in Na_0.785CoO₂, while leaving all or some of the Co⁴⁺ ions unaltered. Co²⁺ and Co⁴⁺ are magnetic with a different spin and orbital degeneracies, implying that Na_0.875CoO₂: Sb_Co might potentially exhibit a different magnetic and thermoelectric behaviour. We examined the possibility of alternative manifestation of oxidation states for Co ions in Na_0.875CoO₂: Sb_Co 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 Na_0.875CoO₂ would be non-magnetic—all Co ions at $t_{2g}^{6} e_{g}^{0}$ for a wide range of temperatures.

Fig. 3

The spin density of the 2a × 4a × 1c supercell used for simulating Na_0.875CoO₂:Sb_Co compound. Here, Sb⁵⁺ is at the closest distance to the V_Nas as in Fig. 1, however, the spin state of Co ions corresponds to an excited state

Fig. 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 Sb_Co-doped Na_0.875CoO₂ system [Na_0.875(Co_0.9375Sb_0.0625)O₂], Sb dopants can occupy few distinct positions when substituting Co. For instance, either close to V_Nas, as shown in Fig. 1, which corresponds to the most stable arrangement of Sb dopant, or far from the V_Nas, as shown in Fig. 5. The distance between Sb_Co and V_Nas for the configuration presented in Fig. 1 is 2.23 Å for the V_Na at Z = 0 and 2.23 Å for the V_Na at Z = 0.5. These distances for the configuration presented in Fig. 5 were 5.23 Å for the V_Na at Z = 0 and 5.26 Å for the V_Na 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 V_Nas 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 V_Na results in the cancellation of internal lattice stress leading to greater stability.

Fig. 5

The spin density of the 2a × 4a × 1c supercell used for simulating Sb-doped Na_0.875CoO₂ compound when Sb⁵⁺ is not in the immediate vicinity of the V_Nas. The distance between V_Na 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 V_Nas 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 V_Nas in the undoped compound [45], doping ${\text{Sb}}_{{{\text{Co}}^{3 + } }}^{5 + }$ in the vicinity of V_Nas (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 V_Nas, 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 Na_xCoO_2, 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]:

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

(1)

in which k_B 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 (g_orbital) and spin (g_spin) degeneracies of electrons for a given transition metal ion. In low spin, octahedral coordination for which Co³⁺ and Co⁴⁺ electrons occupy t_2g orbitals only, g_orbital is the total number of permutation of electrons and is calculated as:

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

(2)

in which n^↑ and n_↓ are the number of spin-up and spin-down electrons, respectively. g_spin, 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 Na_0.875CoO₂ 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 Na_0.875CoO₂. 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 Sb_Co dopant into the Na₁₄Co₁₆O₃₂ supercell, however, eliminates the Co⁴⁺ species and brings the hole concentration to zero resulting in a band insulating character for Na_0.875CoO₂:Sb_Co. 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, Na_0.875CoO₂:Sb_Co 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 Na_xCoO₂, 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 Na_0.875CoO₂: Sb_Co, ineffective doping cases on thermoelectric compounds have been reported in the literature. For example, Al doping plays no effect on Mg₂Si_0.75Sn_0.25 [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].

Conclusions

In this work, using density functional theory, the formation energy of Sb dopants in Na_0.875CoO₂ was examined. The main findings of this investigation can be summarised as follow: (1) Sb_Co is the most stable configuration Na_0.875CoO₂ 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 Sb_Co configuration increases against those other configurations in which Sb is located in the Na layer; (2) the formation energy of Sb_Co decreases with decreasing x—E^f(Na₁CoO₂:Sb_Co) = 1.356 eV and E^f(Na_0.875CoO₂:Sb_Co) = 1.175 eV. This trend implies that doping Na deficient systems with Sb is practically easier; (3) in Na_0.875CoO₂:Sb_Co, Sb’s 5 s states are located above the Fermi level hybridising with Co’s empty e_g states while Sb’s 5p states are located even higher along with Na’s 3 s states implying an oxidation state of 5 + for Sb_Co; (4) in Na_0.875CoO₂:Sb_Co, Sb_Co tends to aggregate with the V_Nas; (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 Na_xCoO₂; (6) in an excited state, in Na_0.875CoO₂:Sb_Co, 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 Na_xCoO₂.