First-Principles Study of Charge Diffusion between Proximate Solid-State Qubits and Its Implications on Sensor Applications

Solid-state qubits from paramagnetic point defects in solids are promising platforms to realize quantum networks and novel nanoscale sensors. Recent advances in materials engineering make it possible to create proximate qubits in solids that might interact with each other, leading to electron spin or charge fluctuation. Here we develop a method to calculate the tunneling-mediated charge diffusion between point defects from first principles and apply it to nitrogen-vacancy (NV) qubits in diamond. The calculated tunneling rates are in quantitative agreement with previous experimental data. Our results suggest that proximate neutral and negatively charged NV defect pairs can form a NV-NV molecule. A tunneling-mediated model for the source of decoherence of the near-surface NV qubits is developed based on our findings on the interacting qubits in diamond.

Figure 1

Nitrogen-vacancy (NV) defect in diamond. (a) Optimized geometry in the core of NV center. (b) Single particle defect levels in the fundamental band gap in the ground state of NV defects. Empty (shaded) arrows depict holes (electrons). (c) Many-body levels at room temperature and decay from the optically active ${}^{3}E$ excited state to the ${}^3{A}_2$ ground state of $\mathrm{NV}(-)$. Radiative (nonradiative) transitions are depicted by straight (dashed-curved) arrows. Bright (dark) lamp illustrates the most (less) intense fluorescence of NV center that is the base of optical spin polarization and readout of the $ms$ electron spin state.

Figure 2

Defect wave functions in $10\times 10\times 5$ hexagonal supercell of diamond with $25.1\times 25.1\times 30.7{\AA }^3$ lattice constants. (a) Top and side view of the $e_x$ and $e_y$ defect wave functions of $\mathrm{NV}(-)$. For the sake of clarity, the atoms are not shown in diamond. The real space wave functions are visualized: the green and yellow lobes represent the isosurface of the wave function at values of $+1.89\times {10}^{-6}$ and $-1.89\times {10}^{-6}1/{\AA }^3$, respectively. (b) The wave function amplitude profile along the dashed line in (a). The wave function completely decays at the edge of the supercell boundaries. The origin is set to the middle of the dashed line.

Figure 3

The band structure of NV defect in a $4\times 4\times 4$ cubic cell along $\mathrm{\Gamma }(0,0,0)$ point to $X(1/2,0,0)$ point with half extra charge. The valence band maximum is aligned to zero. The calculations were carried by a PBE DFT functional that provides an inaccurate band gap, but the Kohn-Sham wave functions should be accurate. Because of the half charge the symmetry is tilted a bit to $C_{1h}$ symmetry; thus, $e_x$ and $e_y$ levels slightly split, but it is very close to the $C_{3v}$ symmetry. The spin polarized PBE DFT calculations result in separate spin-up (black straight curve) and spin-down (red dotted curve) bands and levels. The valence and conduction band regions are green regions. The orange and white arrows represent the occupied and unoccupied defect states, respectively. The six panels show the dispersion of each in-gap defect level in the region $\mathrm{\Gamma }\text{−}X$. The hopping integral was inferred from the dispersion of the half electron ($1/2e$) filled $e_y$ level.

Figure 4

(a) Schematic illustration of two NV centers with the separation of $r$. The wave function of NV is converted into a three-dimensional equidistant grid. The maximum of separation $r$ in the 3000-atom hexagonal cell is 5.2 nm in the (111) plane. (b) The calculated hopping rate of electrons of all possible configurations at a given distance $r$. The average hopping rate values are shown as a blue line. We note that all the possible configurations could be taken into account for $r\le 4.5\mathrm{nm}$. For larger $r$ we have only a subset of configurations because of the constraint of the size and shape of the hexagonal cluster. The value of the hopping rate can be extrapolated at larger $r$ by assuming an overall exponential decay. The dashed horizontal line shows a critical rate at $(2\pi )13.2\mathrm{MHz}$, which is the deduced rate of the radiative decay (see Ref. [42]).