Microscopic origin for the orientation dependence of NV centers in chemical-vapor-deposited diamond

The (1 1 0) surface is a popular substrate for growing NV centers by CVD [12–14]. Under H-rich growth condition, each C on the surface is passivated by one H and the degree of relaxations of surface C are highly reduced after passivation. The fact that the surface C atoms are close in their positions to their bulk counterparts indicates that the surface is fully stabilized by H passivation [21].

During CVD growth, impurities are formed more easily at surface than in bulk, with qualitatively different formation mechanisms [22, 23]. We consider three types of impurities, N_C, V_C and NVH complex, as the most important near-surface precursors for NV incorporation. On the H saturated surface layer, the most stable N_C removes a C and a passivating H. The most stable V_C has all three C dangling bonds passivated by H. This is different from bulk where the limited space prevents the H passivation of all four dangling bonds. A surface NVH, which is the most likely precursor to the NV centers, on the other hand, is composed of an N_C, a V_C and an extra H. The atomic structures and formation energies of surface impurities are shown in figure 1.

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In the bulk, the NV centers are usually charged depending upon the position of the Fermi level of the system. In the initial growth at the surfaces, the most favorable structures of the N-related defects are usually neutralized by hydrogen atoms. Thus, charged NV complexes are not considered in this study.

The formation energies of surface impurities are indeed noticeably lower than those in bulk ( $E_{{\rm f}}^{{\rm N}_{{\rm C}}} = 4.20\,{\rm eV}$, $E_{{\rm f}}^{{\rm V}_{{\rm C}}} =5.81\,{\rm eV}$ and $E_{{\rm f}}^{{\rm NV}} = 5.42\,{\rm eV}$ when μ_H = 0 eV). This is an indication that the incorporation of impurities during CVD growth should be more effective. Among the three impurities/defects, V_C has the highest energy, N_C has the lowest, while NVH is in the middle. If we assume that the growth mode of diamond is layer-by-layer, we can expect that at the first step of the NV center incorporation, an N_C is formed favorably. Neither NVH nor V_C will form in the first step. In other words, the growth of NV center depends on further deposition.

Figure 2 shows the possible defect structures when the next layer is grown on the already N_C incorporated (1 1 0) surface. These structures either have low formation energies or are closely related to the formation of NV centers at the end. As shown in figure 2(a), the most stable structure is an NVH complex, composed of the N_C (now in the first sublayer) and a V_C in the newly grown surface layer. Typically, a surface layer containing no impurities (such as those in figures 2(b) and (c)) is not energetically favored. Hence, horizontal NV center in the first sublayer (namely, figure 2(b)) is also not favored. Increasing the number of passivating H could reduce the energy for the complex in figure 2(b) but it is still much higher than the one in figure 2(a), as shown in figure 2(d). This result suggests that the most stable NVH complex at this stage must be formed in two steps: first, in depositing the first layer, an N substitutes a surface C. Second, with the deposition of the second layer, a V_C is formed in the newly formed surface layer as the nearest neighbor of the preceding N_C.

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In figure 2(a), double arrows represent the four C–C bonds in diamond. On the (1 1 0) surface, two of them (the solid ones) give dangling bond directions out of the surface and the other two (dashed ones) are within the surface. Therefore, the discussion above asserts that possible orientations of the NVH complexes on the (1 1 0) surface are those along the solid arrows only. This result explains the observed preferential orientations of NV centers in [12]. It also indicates that near surface NVH complexes are responsible for the NV centers formed inside bulk (although the latter maybe only a fraction of the former).

To better understand the role of surface, figure 3 shows the local density of states (LDOS) of C and N in the vicinity of impurities in the bulk and at surface. We note that in the bulk, V_C has deep gap states originated from C dangling bonds; N_C is a relatively shallow donor with occupied anti-bonding states close to the conduction band minimum (CBM). These high-lying states are the reasons why isolated V_C and N_C have relatively high energies. Figure 3(e) shows that an NV center also consists of deep gap states but the N-related states are now near the valence band maximum (VBM). The net result of NV formation is thus to significantly lower the energy of occupied N states to lower system total energy. It explains why N_C attracts V_C in bulk diamond.

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The situation is completely different at the surface. For the V_C, figure 3(b) shows that the deep gap states have completely gone. This is because the dangling bonds of the V_C have all been passivated by H. For N_C, figure 3(d) shows that the originally near-CBM states are now below the VBM. This is because the N_C has a dangling bond to accommodate the one extra electron. The resulting surface NVH also has no gap states, as revealed by figure 3(f). It is this surface effect that makes the formation of N_C and V_C, as well as that of NVH complex in CVD diamond considerably more favorable. Note that figure 1(d) shows that a fully passivated V_C is still relatively high in energy than an N_C. This could be a strain effect as V_C has a limited space to host three H atoms.

Until now, the incorporation of NV centers is not completed yet. With more layers deposited, the atomic structure of the N-related impurities may still change, e.g. an incoming C could occupy a V_C site in NVH, thereby reducing the complex back to the original N_C. Figure 4 shows the representative atomic structures of the formed possible complexes, and the corresponding formation energies. We see that an NV center straddling the first and second sublayers without any H, has the highest energy. With passivating H (denoted as NVH_X, where X = 1–3 is the number of H), the energy could be lowered. Generally, the higher the H chemical potential, the lower the formation energies of NVH_X, among which NVH₂ is the most stable. When the H chemical potential is low, on the other hand, reduced N_C can have the lowest energy. These suggest that, in CVD growth, there is a delicate balance for the choice of μ_H to avoid having either too many hydrogenated NV centers or too many reduced N_C. Note that, unlike at the top surface, here the dangling bonds of NVH₂ are not fully passivated due to limited vacancy space. Experiments showed that NV centers in diamond are always accompanied by other defects, of which the most popular ones are N_C and NVH_X [12–14, 16] with a typical concentration ratio N_C : NVH_X : NV = 300 : 30 : 1. It qualitatively agrees with our calculated results at low μ_H in figure 4.

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The properties of the NVH_X complexes have been studied experimentally [12, 13, 16]. Usually, hydrogenated NV centers are expected to contain only one H (bonded to a C). Commonly used experimental techniques for studying NV centers usually involve electron paramagnetic resonance or optical spectroscopic techniques. However, these techniques are not so sensitive to local structures of the NVH_X. Based on the calculated formation energy, over a wide range of general growth conditions, we find X = 2 is most stable, followed by X = 3 and then by X = 1. The energy differences (∼ 1 eV) are not small. Of course, at elevated temperature for diamond growth, one may also need to take into account the effect of entropy which would favor X = 1.

Besides the (1 1 0), one may also use (1 0 0) or (1 1 1) surfaces for CVD growth [12–14, 19, 20]. For (1 0 0) surface, the most stable structure contains 2 × 1 dimer rows, in which every surface C binds to one H [23]. For (1 1 1) surface, the most stable structure is 1 × 1 where every surface C binds to one H [19]. Similarity of these hydrogenated surfaces suggests that the above (1 1 0) surface-assisted multi-step growth model for NV centers may also apply to (1 0 0) and (1 1 1) surfaces. Indeed, figures 5(a) and (c) show that an N_C can easily form at the topmost (1 0 0) and (1 1 1) surfaces similar to (1 1 0) surface. Figure 5(b) shows that, as the growth proceeds on the (1 0 0) surface, a vacancy can be easily formed adjacent to the N_C now at the first sublayer. The (1 1 1) surface is, however, slightly different, as growth may proceed in a double-layer-by-double-layer fashion. Figure 5(d) shows that under the double-layer growth mode, once the N_C is formed (now at the top part of the first sub bilayer), both NVH_X and NV could form in subsequent growth.

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On the (1 0 0) surface, the dangling bonds have four directions, as shown by double arrows in figure 5(a). At any time, two equivalent directions appear on the top surface, but as the growth proceeds, the other two directions take over. This process repeats. Hence, over time, all four directions are equivalent, so one can expect that there are four equivalent NV axes. This result agrees with experiment [14]. On the other hand, on the (1 1 1) surface, due to the doubly-layer growth mode, the direction of the dangling bonds is only along one axis, namely, along [1 1 1] at all times.

Thus, to incorporate the NV centers during CVD growth, on the (1 1 0) and (1 0 0) surfaces, one needs three separate steps, involving three sequential atomic layers. On the (1 1 1) surface, one needs, on the other hand, only two steps, involving two sequential atomic double-layers. Although detailed growth kinetics could be different on different surfaces to result in different concentration ratios among N_C, NV and NVH, the orientation dependence of the NV centers, which is analyzed here solely based on local energetic consideration with demonstrated success when compared with available experiments, should remain the same, irrespective of the kinetics.

NV centers with only one orientation have many advantages in terms of technological applications, such as the improvement of magnetic sensitivity [14]. Therefore, CVD growth of NV centers on (1 1 1) surface is highly desirable [24, 25]. The growth model developed here should also apply to other semiconductors, as a means to control the incorporation of desirable impurities. In this regard, we note that Si-related complexes in diamond also show surface-dependent properties [15].