Pressure and temperature dependence of the zero-field splitting in the ground state of NV centers in diamond: A first-principles study

Nitrogen-vacancy centers in diamond (NV) attract great attention because they serve as a tool in many important applications. The NV center has a polarizable spin $S=1$ ground state and its spin state can be addressed by optically detected magnetic resonance (ODMR) techniques. The $m_S=0$ and $m_S=\pm 1$ spin levels of the ground state are separated by about 2.88 GHz in the absence of an external magnetic field or any other perturbations. This zero-field splitting (ZFS) can be probed by ODMR. As this splitting changes as a function of pressure and temperature, the NV center might be employed as a sensor operating at the nanoscale. Therefore, it is of high importance to understand the intricate details of the pressure and temperature dependence of this splitting. Here we present an ab initio theory of the ZFS of the NV center as a function of external pressure and temperature including detailed analysis on the contributions of macroscopic and microscopic effects. We found that the pressure dependence is governed by the change in the distance between spins as a consequence of the global compression and the additional local structural relaxation. The local structural relaxation contributes to the change of ZFS with the same magnitude as the global compression. In the case of temperature dependence of ZFS, we investigated the effect of macroscopic thermal expansion as well as the consequent change of the microscopic equilibrium positions. We could conclude that theses effects are responsible for about 15% of the observed decrease of ZFS.

Figure 1

Schematic diagram about energy levels of the NV center at room temperature.

Figure 2

Spin density and electronic structure of the nitrogen-vacancy center in diamond. (a) The orange (light gray) lobes show the spin density in the ${}^3{A}_2$ ground state on the NV center obtained by ab initio DFT calculations. The spin density is mostly originated from the dangling bonds of the carbon atoms at the vacancy site. The symmetry axis is represented with the dashed line. (b) Schematic diagram of the electronic structure where $e_X$ and $e_Y$ orbitals are localized on the carbon dangling bonds.

Figure 3

Calculated pressure dependence of parameter $D$ of the zero-field splitting tensor at zero temperature. The points show the calculated splitting at different ambient pressure applied on the defective supercell of the NV center in diamond. The line shows a linear fit with the slope of 9.52 MHz/GPa. It is clearly seen that the calculated pressure dependence is mostly linear but it has a slight curvature on the examined large interval of the pressure.

Figure 4

Calculated pressure dependence of parameter $D$ of the ZFS by using different models. The red points show the results of a complete self-consistent solution of the pressed supercells of a diamond embedding a single NV center (model 1). The blue triangles represent the results of a self-consistent solution of the KS system for the isotropically compressed supercell of the zero pressure structure (model 2). In this model the observed pressure dependence is a consequence of the macroscopic compression of the diamond lattice as well as the variation of the localization of the spin density. The blue (light gray) and red (dark gray) solid lines represent the zero-field splitting as calculated from analytical models (model 3) (see text for more details). The factor of compression $\alpha$ is in line with the pressure dependence of the lattice constant $[\alpha =a\left(P\right)/a_0]$ and with the pressure dependence of the C-C distance in the first neighbor shell of the carbon vacancy $[\alpha =d_{\text{C-C}}\left(P\right)/d_{\text{C-C}}^0]$ in the case of blue (light gray) and red (dark gray) lines, respectively. In these models only the distance of the spins is affected, while the change of the spin density distribution is neglected.

Figure 5

Comparison of the experimental result (a) with the results of different levels of theory (b), (c), and (d). In the most simplified method (b) only the effect of the macroscopic compression of the diamond lattice is taken into account. On the next level (c) the effect of the local structural relaxation is considered, however the wave functions are kept fixed. In the fully self-consistent solution (d) both the structure and the orbitals are relaxed (see text for more explanation).

Figure 6

Experimental and theoretical temperature shifts of the zero-field splitting $\Delta D$ in the ground state of the NV center. Line (a) represents a fitted curve to the experimental measurements [15], while lines (b) and (c) show the theoretically predicted curves in the two models. In the case of line (b) only global expansion is taken into account, while in the case of line (c) both global expansion and local structural relaxation are adopted. The orbitals are treated self-consistently in both models.

Figure 7

Angular integrated spin density distribution $\left[{\rho }_{\mathrm{spin}}\left(r\right)\right]$ as a function of scaled radius $r/d_{\mathrm{cell}}$, where $d_{\mathrm{cell}}$ is the lattice constant of the supercell. The spherical coordinate system is centered on the carbon vacant site. The red solid line represents the spin density distribution of the unpressed supercell while the blue dot-dashed and the black dotted lines correspond to a pressed supercell ($P=209$ GPa) of models 1 and 2, respectively. The first dip in the density distribution corresponds to the position of the first neighbor carbon atoms around the vacancy as labeled in the figure (see Fig. 2).

Figure 8

Matrix element of the zero-field splitting tensor as obtained from different model wave functions of linear combinations of $s$ and $p$ Gaussian orbitals. See text for the definition of the model wave function.