Effect of the lattice dynamics on the electronic structure of paramagnetic NiO within the disordered local moment picture

Using the disordered local moments approach in combination with the ab initio molecular dynamics method, we simulate the behavior of a paramagnetic phase of NiO at finite temperatures to investigate the effect of magnetic disorder, thermal expansion, and lattice vibrations on its electronic structure. In addition, we study its lattice dynamics. We verify the reliability of our theoretical scheme via comparison of our results with available experiment and earlier theoretical studies carried out within static approximations. We present the phonon dispersion relations for the paramagnetic rock-salt (B1) phase of NiO and demonstrate that it is dynamically stable. We observe that including the magnetic disorder to simulate the paramagnetic phase has a small yet visible effect on the band gap. The amplitude of the local magnetic moment of Ni ions from our calculations for both antiferromagnetic and paramagnetic phases agree well with other theoretical and experimental values. We demonstrate that the increase of temperature up to 1000 K does not affect the electronic structure strongly. Taking into account the lattice vibrations and thermal expansion at higher temperatures have a major impact on the electronic structure, reducing the band gap from $\sim 3.5$ eV at 600 K to $\sim 2.5$ eV at 2000 K. We conclude that static lattice approximations can be safely employed in simulations of the paramagnetic state of NiO up to relatively high temperatures $(\sim 1000$ K), but as we get closer to the melting temperature vibrational effects become quite large and therefore should be included in the calculations.

Figure 1

Time evolution of the potential energy of the cubic paramagnetic NiO at $T=600$ K (blue dashed line) extracted from the DLM-MD calculation. The accumulated average of the potential energy is shown with a red solid line. The equilibration time of 1 ps is not included in calculations of the running average. The potential energy is stable and well converged as is seen from the running average.

Figure 2

Time evolution of the net magnetic moment per supercell (blue dashed line), its accumulated average (red solid line), and the local magnetic moment of a single Ni atom (green dots) for the cubic paramagnetic NiO at $T=600$ K from the DLM-MD calculation.

Figure 3

Phonon spectra of PM NiO at 500 K (green lines) and 1000 K (red lines), filled circles represents DMFT results [50]. Experimental data for AFM phase measured at 297 K [51] presented as open circles.

Figure 4

Electronic structure of antiferromagnetic NiO from experiment (top panel, red circles from Ref. [1] and blue dots from Ref. [13]) and theory (middle panel, green solid line denotes results of this work, blue dashed line is from Ref. [20]) as well as for paramagnetic NiO extracted from DLM-MSM calculations (bottom panel). The theoretical calculations in this work are carried out at $T=0$ K using the LDA+$U$ method with $U^{\text{eff}}=8$ eV (note that calculations in Ref. [20] were carried out with $U^{\text{eff}}=7$ eV). The lattice constant used in the calculations is the same as the 0 K experimental value $a=4.17$ Å. The zero energy point corresponds to the top of the valence band.

Figure 5

Top panel: Comparison of the density of states (DOS) of paramagnetic NiO from DLM-MSM calculations (blue dot-dashed line) with the DOS from the LDA+DMFT calculations from Ref. [19] (red solid line). Bottom panel: DOS of NiO calculated with the DLM-MSM method $(U_{\text{eff}}=8$ eV) projected onto $t_{2g}$ and $e_g$ orbitals of Ni, as well as on $p$ orbitals of O. The zero energy point corresponds to the top of the valence band.

Figure 6

Comparison of the density of states (DOS) of paramagnetic NiO at $T=600$ K obtained from the DLM-MD calculation with the static DLM-MSM calculations at $T=0$ K. The thermal expansion is included via experimental lattice spacing at different temperatures. The zero energy point is set to the top of the valence band.

Figure 7

Comparison of the density of states (DOS) of paramagnetic NiO obtained from the DLM-MD at three different temperatures: $T=600$ K (red dot-dashed line), $T=1000$ K (green dashed line), and $T=2000$ K (blue solid line). The thermal expansion is included via experimental lattice spacing at different temperatures. The zero energy point is set to the top of the valence band.

Figure 8

Comparison of the density of states (DOS) of paramagnetic NiO obtained from the DLM-MSM calculations at $T=0$ K using two different lattice spacings, 0 K lattice parameter $a=4.17$ Å (blue dashed-dot line) and 2000 K lattice parameter $a=4.28$ Å (red solid line). The zero energy point is set to the top of the valence band.