Tailoring spin defects in diamond by lattice charging

Atomic-size spin defects in solids are unique quantum systems. Most applications require nanometre positioning accuracy, which is typically achieved by low-energy ion implantation. A drawback of this technique is the significant residual lattice damage, which degrades the performance of spins in quantum applications. Here we show that the charge state of implantation-induced defects drastically influences the formation of lattice defects during thermal annealing. Charging of vacancies at, for example, nitrogen implantation sites suppresses the formation of vacancy complexes, resulting in tenfold-improved spin coherence times and twofold-improved formation yield of nitrogen-vacancy centres in diamond. This is achieved by confining implantation defects into the space-charge layer of free carriers generated by a boron-doped diamond structure. By combining these results with numerical calculations, we arrive at a quantitative understanding of the formation and dynamics of the implanted spin defects. These results could improve engineering of quantum devices using solid-state systems.


Suplementary Note 1.1. Implantation process: Molecular Dynamics simulation
Similarly to Ref. [1], the Molecular Dynamics (MD) simulation is utilized to investigate events of nitrogen implantation in diamond. The ITAP MD (IMD) Program package [2,3] is used in order to study the damage radius introduced by a single nitrogen atom initially accelerated to 4.0 keV (in z direction) in a 5.5 × 5.5 × 52 nm 3 diamond structure. The atomic interaction is represented by a combination of a bond ordered Tersoff [4][5][6][7][8] and for decreasing atom separation (r < 0.3Å) a two-body Ziegler-Biersack-Littmark (ZBL) [9,10] potentials.
In a first step, the diamond system is brought into equilibrium at room temperature. Next, the nitrogen atom is placed on top of the diamond surface ([100]-orientation) at a distance d= 2.51Å, which ensures its location beyond the action range of the Tersoff potential. The nitrogen atom is then implanted with a kinetic energy of E k = 4.0 keV (normal to the surface). In order to provide enough statistics, individual implantation events with random starting positions on the xy-plane are repeated 120 times. In this way, the ion channeling effect on the stopping depth is taken into account, as can be observed by the resulting ion

Suplementary Note 1.2. Annealing process: kinetic Monte Carlo simulation
After implantation an additional annealing step is necessary to generate NV centers. In order to investigate the formation probability of NV centers and di-vacancies in their vicinity, kinetic Monte Carlo simulations are performed based on the simple hopping frequency of defects Γ = ω a e − Ea k B T at a temperature of 950 • C. Here, ω a = 10 13 s −1 denotes the frequency at which defects attempt to overcome a barrier of energy E a . The values for activation energies are 2.3 eV for vacancy (V) [11,12], 1.7 eV for interstitial nitrogen (N I ) [13] and 1.5 eV for carbon interstitial atoms (C I ) [14].
The Monte Carlo scheme models the diffusion process during annealing and investigates the corresponding influence on the conversion from implanted nitrogen atoms to NV centers (yield). The analysis includes the most important loss mechanisms of vacancies, such as migration to the surface, recombination with interstitials or the formation of di-vacancies and higher-order vacancy complexes. The process in which a self-interstitial atom replaces an on-site nitrogen at a lattice site and turns the latter into an interstitial nitrogen atom [12] was also included. In contrast to on-site nitrogen atoms which hardly move at 950 • C, interstitial nitrogen atoms are mobile during annealing [12,15]. The initial distribution after implantation are taken from MD simulations, as presented above. 100 annealing runs with different starting configurations were performed.
The simulation shows that the recombination of vacancies and interstitials happens within the first seconds of annealing since carbon interstitials are highly mobile. The number of vacancies that are left to possibly form NV centers is thus less than 50% of the number that is suggested by implantation simulations. The simulation of the vacancy diffusion within the crystal has completely come to an end after the simulated annealing time. On average, > 90% of the vacancies created by the implantation process are consumed after 2 hours of annealing (simulation time for the process). As seen in Supplementary Figure   2, approximately 30 − 40% of implantation-induced single vacancies located within defectclusters are expected to form di-vacancies, translating in 4 − 6 di-vacancies surrounding NV centers in the close vicinity to the diamond surface (depth of < 5 nm). These values are used to discuss the experimental results in the main text.

Suplementary Note 1.3. NV-V 2 interaction Hamiltonian
To gain insight into the effect of di-vacancies, we use numerical calculations to obtain the resulting spin properties and magnetic noise characteristics of NV centers. In the calculation we place N V2 di-vacancies around a central NV center. The electronic spin of individual V 2 complexes (I = 1) are assumed to be coupled to each other and the central NV center electronic spin (S) via dipole-dipole interaction. In the presence of a magnetic field of 33 mT aligned to the axis of the NV center (chosen asẑ axis), the Hamiltonian can be written as where the three terms correspond respectively to the central NV center, NV and V 2 hyperfine interactions and V 2 spins. These can be individually written as: The zero-field splittings of the NV center and V 2 are largely different, being D NV ≈ 2.87 GHz and D V 2 ≈ −0.31 GHz, respectively [16]. The hyperfine tensor A j,z and the dipole interaction tensor D i,j depend on the spatial configuration of NV-V 2 spins. In a rotating frame defined by H N V , the Hamiltonian can be rewritten as: where For the two levels m s = 0 and m s = −1 of interest here, the coherence decay is obtained through the following equations [17]: The operators U 0 (τ ) = e −iH 0 τ and U −1 (τ ) = e −iH −1 τ describes the NV center spin state conditional to the evolution of V 2 spins. τ is the interval between the π/2 pulse and the first   The correlation time of the corresponding noise is then τ c = γ −1 . For the given spin density, we deduce the probability distribution to find a nearest neighbor spin at distance d as: with p(d)dd = 1. As seen in Supplementary Figure 4b, the maximum probability occurs for a spin separation of d ∼ 1.3 nm. Alternating fields by flip-flops are produced by anti-parallel aligned surface spin pairs and their total coupling ∆ to the NV center, thus decreasing for smaller mutual distances d as: The majority of the NV centers in the reference area (approximately 70%) and a parcel of the NV centers in the sample area (approximately 20%) demonstrated low spin contrast and/or photo-blinking. These two features result in a low signal-to-noise ratio, leading to low precision and longer integration times in the needed spin measurements, specially in the T 1 times (if measurable at all). These NV centers were therefore excluded from the data presented in the main text (figure 3). We attribute this blinking behavior and/or low spin contrast to the presence of electronic states in the band gap induced by lattice defects that would change the charge state of nearby NV centers. These defects might be generated not only by the implantation and annealing processes, but also by the surface polishing [18] and during the crystal growth. In particular, charge exchange between Gd 3+ ions and the diamond surface would also result into photo-instability and low spin contrasts due to higher NV 0 /NV − rates. The T 2 and T 1 times of NV centers that fulfill the above mentioned criteria are shown in figure 3 a) and b) in the main text, respectively.
It must be emphasized that, although the data points from the reference area at depths of ∼ 3 nm in figure 3 in the main text show extremely short T 2 times, T 1 times of a few milliseconds were measured. This behavior is a typical signature from the presence of electron spins in the vicinity of NV centers [19,20]