How does the spin-state of Co ions affect the insulator-metal transition in Bi2A2Co2O8 (A = Ca, Sr, Ba)?

The misfit layered Bi2A2Co2O8 (A = Ca, Sr, Ba) compounds experience an insulator to metal transition as A’s ionic radius increases. This feature is contradictory to the conventional wisdom that larger lattice constant favors insulating rather than metallic state, and is also difficult to be reconciled using the Anderson weak localization theory. In this paper, we show from the first-principles calculation that an insulator-metal transition takes place from a nonmagnetic low-spin state of Co3+ ions to a hexagonally arranged intermediate-spin low-spin mixed-state in CoO2 plane when ionic radius increases from Ca to Ba. The predicted low-spin state of Bi2Ca2Co2O8 and Bi2Sr2Co2O8 and intermediate-spin low-spin mixed-state of Bi2Ba2Co2O8 are consistent not only with their measured transport properties, but also with the magnetic-field suppressed specific-heat peak observed at the transition temperature. In agreement with experiments, strong electronic correlation is required to stabilize the low-spin insulator and intermediate-spin low-spin metal.

(1) The convergence checks on k-points samplings for the total energy of LS-state of Bi Ba Co O . (U=8 eV，JH=1eV) In the total energy calculations we use 8μ8μ2 and 4μ8μ2 -centered k-points samplings for the unit cell (28 atoms) and double-cell (56 atoms), and the energy convergence checks are displayed in Table S1 and Table S2. An absolute error in energy is less than 0.1meV per unit cell using the parameter settings above. Therefore, excellent numerical convergences have been achieved on total energies. out the best candidates for the two electronic states below and above the insulator-metal transition temperature, we search for the possible ground states of the compounds (unit cell of Figure S1) in following steps.
(1) The uniform spin-states of the four Co ions in unit cell: As shown in Table S3, the uniform LS-state has the lowest energy among uniform LS-, IS-, and HS-states.
(2) Single IS-or HS-Co ions embedded in the background of LS-Co ions in unit cell: As Co-3 and Co-4 positions are equivalent in the P-1 crystal group (see Figure S1), we only consider six spin structures, i.e. single IS-or HS-Co at Co-1, Co-2, or Co-3 positions. The results are summarized in Table   S4. It is seen that the energies of single IS-Co cases are always lower than those of single HS-Co cases, thus we focus on IS-state only below for more complex magnetic structures. In fact, single IS-Co -ion embedded in the background of LS-Co ions can have even lower energy than that of the uniform LS-state when U is larger while the opposite is true when U is smaller. To check whether single IS-Coion embedded in the background of LS-Co ions is the most favored magnetic state, we have also considered double and triple IS-Co -ions embedded in LS-Co ions and their energies are listed in Tables S5-S7. From Table S5 and Table S6, double IS-Co -ion at Co-1 and Co-2 positions with parallel spins yields the lowest energy among its class, but the energy is higher by 161.1meV than that of single IS-Co -ion at Co-1.
Also, triple IS-Co -ion case is higher in energy by 463.6meV than that of single IS-Co -ion at Co-1. Thus, we conclude that single IS-Co -ion at Co-1 position has the lowest energy for one unit-cell configuration.  To restore the hexagonal symmetry of the Fermi surface observed experimentally, we double the unit cell along a-axis. Because the single IS Co-1 and Co-2 configurations are very close in energy (2.8 meV at U=8.0 eV), we interchange the spin states of Co-1 and Co-2 ions of the second unit cell (see Figure S2). In this way, we arrived at the hexagonally arranged IS-LS mixed-state whose energy is further lowered by 52 meV (at U=8.0 eV) per unit cell. Another possible hexagonally arranged IS-LS mixed-state is depicted in Figure S3 with IS-Co takes the Co-3 position of the first unit cell and Co-4 position in the second unit-cell. However, this state is higher in energy by 95.6meV per unit cell than the Co-1 Co-2 configuration. Thus, we conclude that Co-1 Co-2 hexagonally arranged IS-LS mixed state is the spin-state competing with the uniform LS-state.

Figure S2
The hexagonally arranged IS-LS mixed-state with IS-Co ions at Co-1 and Co-2 positions.

Figure S3
The hexagonally arranged IS-LS mixed-state with IS-Co ions at Co-3 and Co-4 positions. To illustrate the typical electronic spectra for the three phases studied in this paper, the LS state metal (marked green in Fig. 3 are shown in Figure S4-S6, respectively.