Role of N defects in paramagnetic CrN at finite temperatures from first principles

Simulations of defects in paramagnetic materials at high temperature constitute a formidable challenge to solid-state theory due to the interaction of magnetic disorder, vibrations, and structural relaxations. CrN is a material where these effects are particularly large due to a strong magnetolattice coupling and a tendency for deviations from the nominal 1:1 stoichiometry. In this work, we present a first-principles study of nitrogen vacancies and nitrogen interstitials in CrN at elevated temperature. We report on formation energetics, the geometry of interstitial nitrogen dimers, and the impact on the electronic structure caused by the defects. We find a vacancy formation energy of 2.28 eV with a small effect of temperature, i.e., a formation energy for N interstitial in the form of a $\langle 111\rangle$-oriented split bond of 3.77 eV with an increase to 3.97 at 1000 K. Vacancies are found to add three electrons, while split-bond interstitial adds one electron to the conduction band. The band gap of defect-free CrN is smeared out due to vibrations, although it is difficult to draw a conclusion about the exact temperature at which the band gap closes from our calculations. However, it is clear that at 900 K there is a nonzero density of electronic states at the Fermi level. At 300 K, our results indicate a border case where the band gap is about to close.

Figure 1

Potential energy of paramagnetic CrN as a function of simulation time calculated at $T=300$ K using the DLM-MD method. The figure also includes the potential energy of cubic CrN in the presence of N vacancy and N interstitial $\left(I_N^{111}\right)$ calculated at the same temperature.

Figure 2

Possible N interstitial initial positions in the CrN unit cell. The blue circles show nitrogen and the gray circles are Cr atoms. The figure demonstrates N interstitial in the (a) $I_N^{110}$ position, (b) $I_N^{111}$ position, and (c) $I_N^{\mathrm{tet}}$ position. (d) The spherical coordinates related to the coordinates and lines depicted in Fig. 3.

Figure 3

Nitrogen pair [$\text{N}\left(I_N\right)$-N(CrN)] dynamics in paramagnetic CrN as a function of simulation time at $T=300$ K (left column) and $T=900$ K (right column). Panels from top to bottom show split-bond $\langle 110\rangle$, split-bond $\langle 111\rangle$, and tetrahedral configurations, respectively. The spherical coordinate system used here is shown in Fig. 2. The blue color is the radial distance between the nitrogen pair in $\AA$. The angles show the orientation of the N-N bond vector and the dashed lines represent all of the symmetry directions equivalent to the directions $\langle 111\rangle ,\langle \overline{1}11\rangle ,\langle 11\overline{1}\rangle ,\langle 1\overline{1}1\rangle$, etc. The green color shows the azimuthal angle $\phi$. The dashed green lines are different values of $\phi$ between 0 and $2\pi$. The red solid and dashed lines show the polar angle $\theta$ which varies between 0 and $\pi$. In the case of $I_N^{110}$, the value of $\theta$ (red solid line) should be constant, $\pi /2$. However, as can be seen in the figure, $\theta$ changes to $\pi /3$ and vibrates around this value, indicating that the pair is in the $\langle 111\rangle$ direction.

Figure 4

Nitrogen pair [$\text{N}\left(I_N\right)$-N(CrN)] probability density in paramagnetic CrN at $T=300$ K (left column) and $T=900$ K (right column). Panels from top to bottom show split-bond $\langle 110\rangle$, split-bond $\langle 111\rangle$, and tetrahedral geometries, respectively. The figure includes only the neighboring Cr atoms (gray). Red shows the interstitial nitrogen and blue is the nitrogen from the CrN matrix. The starting geometry is shown by a black circle. The blue and red colors are densities that show where the two nitrogens in the pair spend most of their time during the simulation. The densities follow the trend shown also in Fig. 3. For instance, the nitrogen in the tetrahedral configuration at $T=300$ K transforms to the $I_N^{111}$ position and remains there, as can be seen in the related panel in Fig. 3.

Figure 5

The evolution of the magnitude of the Cr magnetic moments, neighbor to the defect and further away from the defect as a function of simulation time for (a) nitrogen vacancy case and (b) nitrogen interstitial in the $\langle 111\rangle$ case, carried out at $T=300$ K.

Figure 6

(a) Total density of states of CrN at $T=0$ K obtained from static supercell DLM calculations. In order to have a detailed view of the region around the Fermi level, in (b)–(d) we show only the area that is marked with a red rectangle in (a). (b) DOS obtained in static calculations. We also show the total density of states of CrN obtained from DLM-MD calculations at temperatures (c) 300 K and (d) 900 K. The results are based on a 64-atom supercell with 32 Cr atoms and 31 (vacancy), 32 (defect-free), or 33 (interstitial) N atoms. The Fermi level, shown by the vertical dashed line, is aligned to zero.