To gain more insight on the correlations between SOC and ground state in the iridates, DFT calculations are further performed to elucidate the effect of SOC on electrical transport property when Ru and Mg are doped in
O-SrIrO
3. The band structure of
O-SrIrO
3 subjected to SOC is shown in
Fig. 5a. For iridate Ir
4+ (5d
5), the stronger SOC leads to the
J1/2 and
J3/2 with a large energy difference λ
5d ~ 0.47 eV, which is in line with λ
5d ~ 0.43 eV in
O-SrIrO
3 from DFT+
U method with
Ueff (Ir) = 0 eV [
56]. Conversely, Ru
4+ (4d
4) with weaker SOC manifests small energy difference (λ
4d ~ 0.16 eV) between
J1/2 and
J3/2, as the band structure of SrRuO
3 subjected to SOC shown in
Fig. 5b. Here, SrIr
0.6Ru
0.4O
3 is opted as a reasonable structure model, which can intensify the understanding of the influence on SOC effects with Ru doping. The band structure of SrIr
0.6Ru
0.4O
3 subjected to SOC is shown in
Fig. 5c. The competition between SOC interaction and Hund's effect leads to diluted SOC in Ir site in SrIr
0.6Ru
0.4O
3 compared to that in
O-SrIrO
3. Thus, the weaker SOC combined with more effectively screened coulomb interactions between O 2p and Ru/Ir 4d/5d electrons can cause reduction of
t2g orbital difference and lower
EF, driving to the system toward a more robust metallic state. This is consistent with the above electronic transport measurements. As for the computing model of Sr
0.8Mg
0.2IrO
3, a supposed
O-SrIrO
3 with 20% Mg dopant concentration in Sr sites is constructed. The band structure of Sr
0.8Mg
0.2IrO
3 subjected to SOC is shown in
Fig. 5d. The stronger GdFeO
3-type distortion induces
J1/2 and
J3/2 states further splitting due to enhanced SOC. Compared with
O-SrIrO
3, Sr
0.8Mg
0.2IrO
3 exhibits further energy level splitting induced by the finite coulomb repulsion energy
U (~0.1 eV). Moving down the periodic table from 3d to 4d and then to 5d, the orbitals in the solids that contain the corresponding d orbitals become increasingly extended and so does the bandwidth (
W3d <
W4d <
W5d). As the bandwidth increases, the corresponding on-site Coulomb repulsion decreases in a sequential manner (
U3d >
U4d >
U5d) [
57]. So, the
5d orbits have smaller on-site Coulomb interaction
U than the 3d and 4d orbits. Thus, the electron correlation should contribute less to the energy band structure, giving rise to the metallic ground states in many 5d transition metal oxides (TMOs) described by the band theory of solid. However, some 5d TMOs, such as Sr
2IrO
4, Sr
3Ir
2O
7, and Ba
2NaOsO
6, have insulating ground states, which can be attribute to the key role of electron correlation effects
U [
58-
60]. To achieve the insulating bands, the on-site Coulomb interactions
U is considered in the Hamiltonian to drive the localization of charge carriers [
61]. Thus, the band structure is further calculated combined SOC and on-site Coulomb interactions, where electron correlation
Ueff (Ir) values are tested from 0.5 eV to 3 eV. When the
Ueff (Ir) value is 0.5 and 1.0 eV, the electronic structures are metallic from GGA+SOC+
U method (
Fig. 5e and Fig. S10a in Supporting information). As the
Ueff (Ir) value is adjusted to 2.0 and 3.0 eV, finite indirect band gaps of 0.18 and 0.50 eV can be observed in
Fig. 5f and Fig. S10 (Supporting information), respectively. It is seen that the occupied electronic states of Ir 5d electrons become isolated and localized. Thus, the band structure derived from GGA+SOC+
U calculations is consistent with our experimental results, demonstrating that the insulator behavior can be ignited by combining SOC and the on-site Coulomb interactions. That is, electron correlation
U and SOC effect are non-negligible factors in MIT phenomenon in DFT calculations. Electron-electron Coulomb repulsion interactions (
U) for Ru 4d orbitals are considered in GGA+SOC+
U method with
Ueff (Ru) = 2.9 eV as used to predict metallic behavior of SrRuO
3 [
62]. When the
Ueff (Ir) value is 2.0 eV and
Ueff (Ru) value is 2.9 eV, the electronic structure is metallic from GGA+SOC+
U method (Fig. S11 in Supporting information) in SrIr
0.6Ru
0.4O
3 case, which aligns with our experimental observations, confirming the existence of metallic behavior in the context of Ru doping. All in all, the ground state in strong correlated 5d elements systems is governed by the cooperative effect involving SOC, electron-electron correlations, and crystal field effect, which are expected to raise exotic physical properties in strong correlated iridate oxides.