A solution to electric-field screening in diamond quantum electrometers

Nanoscale charge imaging has been employed for diverse purposes including high-sensitivity biological and chemical sensorsCui et al. (2001); Patolsky et al. (2006), detectors within quantum devicesEizerman et al. (2004), and investigating fundamental physical phenomenaMartin et al. (2008); Yoo et al. (1997). Many techniques have been employed for precision electrometry with nanometer spatial resolutionYoo et al. (1997); Martin et al. (2008); Henning et al. (1995); Williams et al. (1989), elementary charge detectionYoo et al. (1997); Martin et al. (2008); Devoret and Schoelkopf (2000); Schönenberger and Alvarado (1990); Martin et al. (1988); Cleland and Roukes (1998); Bunch et al. (2007); Salfi et al. (2010); Lee et al. (2008), and the ability to operate at ambient temperatures and pressuresBunch et al. (2007); Lee et al. (2008). However, no device yet possess all three of these properties simultaneously. This capability would be extremely valuable for investigating biological systems, such as neuronsBarry et al. (2016); Hanlon et al. (2019), and as a critical characterization tool for the emerging field of two-dimensional electronicsSchwierz (2010); Radisavljevic et al. (2011); Mak and Shan (2016). For example, atomic-resolution imaging of silicene to aid development of a room-temperature transistorTao et al. (2015), probing novel charged quasiparticles in MOS₂ filmsMak et al. (2012), and detection of polarization skyrmionsDas et al. (2019).

The nitrogen-vacancy (NV) centerDoherty et al. (2013) is currently the only system capable of nanoscale resolution electrometry of elementary charges under ambient conditions. This point defect in diamond consists of a substitutional nitrogen atom ($\text{N}_{\text{S}}$) situated adjacent to a carbon vacancy. Single NV centers have demonstrated room-temperature a.c. (d.c.) electric-field sensitivities reaching 202 $\text{V cm}^{-1}\text{ Hz}^{-1/2}$ (891 $\text{V cm}^{-1}\text{ Hz}^{-1/2}$)Dolde et al. (2011, 2014). Ensembles of NV centers have achieved shot-noise limited a.c. sensitivities on the order of 1 $\text{V cm}^{-1}\text{ Hz}^{-1/2}$Chen et al. (2017) and also been employed as in-situ electric field sensors within semiconductor heterojunctionsIwasaki et al. (2017).

The NV center’s proficiency for quantum sensing is due to a unique combination of capabilities. Firstly, the NV exhibits bright optical fluorescence allowing for identification of single defects that can be employed for measurements with nanoscale resolution. Secondly, the NV possesses a mechanism for optical spin initialization and readout which permits spin resonances of individual defects to be measured with high fidelityDoherty et al. (2013). Finally, the NV boasts the longest room-temperature coherence time for any solid-state defectBalasubramanian et al. (2009) allowing for high-resolution detection of spin resonances when combined with optical readout. In addition to electrometry these properties have been applied for precision nano-magnetometryTetienne et al. (2014); Thiel et al. (2016); Häberle et al. (2015); Arai et al. (2015), thermometryKucsko et al. (2013); Neumann et al. (2013); Toyli et al. (2013) and quantum computingWaldherr et al. (2014), as well as proposed for investigating fundamental physical phenomena such as magnetic phase changesCai et al. (2013) and coherent quantum transportOberg et al. (2019).

While the NV center can exist in several charge states, including neutral (NV⁰) and negative (NV^-), only the latter possesses the aforementioned properties needed for electrometry. In particular the spin resonances of the NV^- triplet ground state are susceptible to electric field induced changes in its electron spin-spin interactions. The resulting Stark shifts can be detected using optically detected magnetic resonance (ODMR) and used to determine the field magnitude. Furthermore, vector components of the electric field can be measured through rotation of a bias transverse magnetic fieldDolde et al. (2014).

Although single NV centers possess sufficiently high sensitivities, elementary charges external to diamond have not been detected with nanoscale resolution. We hypothesize that this is because of electric field screening and charge-state quenching of NV centers. For the former, recent experimental works have identified multiple screening sources inherent to diamond systems. These include Debye screening from bulk defectsBroadway et al. (2018), charge reorganization in primal sp² surface defectsStacey et al. (2019) and polarization of adsorbed water vaporMertens et al. (2016). Moreover, p-type defects within bulk diamondLiao et al. (2008), surface acceptor defectsStacey et al. (2019) and surface terminations with negative electron affinitiesHauf et al. (2011) are known to quench the negative charge state, particularly for near-surface NV centersCui and Hu (2013); Ohno et al. (2012). However, the extent that these sources impact charge detection are unknown, and a comprehensive theoretical treatment is needed.

The first aim of this paper is to develop a physical model of screening due to the external environment, internal diamond, and diamond surface. This is performed in Section I where we identify that screening due to charge rearrangement amongst sp² surface defects is the greatest impediment to electrometry. We propose a solution by saturating the charge traps through a sacrificial $\delta$-doped layer of $\text{N}_{\text{S}}$. The effectiveness of this idea is explored in Section II in which an analytical toy model is developed for a simplified electrometer. Finally, in Section III this toy model is adapted into a more sophisticated device compatible with NV quantum sensing. The electrostatic properties of the device are modeled computationally, and the physical parameters optimized for charge detection. We conclude that this design successfully mitigates screening for concentrations of sp² surface defects below approximately $10^{16}$ $\text{m}^{-2}$, two orders of magnitude lower than that currently demonstrated on fluorine-passivated diamondStacey et al. (2019).

Screening and quenching effects within diamond can be decomposed into three coupled systems; the external atmosphere, the surface and the internal diamond. These three environments and their associated screening/quenching sources are depicted in Figure 1. In the following subsections we model each system individually and assess the associated impacts for diamond electrometry.

A solution to both surface-induced screening and charge instability is to saturate the surface traps using sacrificial donors within the diamond. We propose fabricating a $\delta$-doped layer of $\text{N}_{\text{S}}$ which introduces a donor level 3.8 eV above the valence band. In the limit that the concentration of $\text{N}_{\text{S}}$ exceeds that of the sp² defects, the Fermi-level will be pinned to the donor level and prevent quenching of the NV^-. This is possible as present doping techniques allow for precision control of the $\delta$-layer height and defect concentrations up to 1000 ppmOhno et al. (2012); Chandran et al. (2016). In this section we demonstrate the effectiveness of this idea through an analytical toy model that explores the complex interactions of screening and quenching within a highly coupled system.

Consider the schematic for a simplified electrometer presented in Figure 4 (a). The $\delta$-doped layer is positioned at a depth $D$ below the diamond surface ($z=0$) while the NV spin-probe is placed between them at a depth $d<D$. At electrostatic equilibrium the occupation of the sp² defects leads to the accumulation of an isotropic and homogeneous charge density on the surface, $\rho_{S}$. This charge density subsequently generates a surface potential $V_{S}(z=0)$ which is related as per

where $C$ is the device capacitance, $q$ is the electron charge, $\sigma_{T}$ is the density of surface traps, $f$ is the Fermi-Dirac distribution, $E_{N}=3.8$ eV is the energy of $\text{N}_{\text{S}}$ and $E_{T}\approx 2.2$ eV is the energy of an sp² surface defectStacey et al. (2019). The two layers – surface and $\delta$-doping – effectively form a parallel plate capacitor and hence $C=\epsilon_{D}/D$ where $\epsilon_{D}=5.7\epsilon_{0}$ is the dielectric permittivity of diamond. This induces a linear potential between the capacitor plates such that the potential at the NV center is given by

The magnitude of the potential induced at the surface has major ramifications for NV^- charge stability. Figure 4 (b) depicts the energies of $\text{N}_{\text{S}}$, NV and sp² defects within the simplified electrometer. Upon charging, the sp² defect and NV energies are raised by an amount $qV_{S}(0)$ and $qV_{S}(d)$ respectively. To avoid transition to NV⁰, the condition

must be maintained. Inserting equation (7) into the inequality (8) places a constraint on the maximum surface potential that prevents quenching, given by

The surface potential also has major implications for screening. Note that $V_{S}(0)$ is ultimately limited above by $0\leq qV_{S}(0)\leq E_{N}-E_{T}\approx 1.6$ eV. When $qV_{S}(0)\approx E_{N}-E_{T}$ the defect energy is pinned to that of the $\text{N}_{\text{S}}$ donors and screening effects dominate; equation (6) indicates that $\rho_{S}=q\sigma_{T}/2$ and hence the surface is effectively conducting. Clearly, electrometry requires that

for under such conditions $f(E_{T}+qV_{S}(0)-E_{N})\approx 1$ (the linear regime) and the surface charge cannot reorganize in response to an external electric field. Fortunately, the inequality (10) can be determined precisely and the robustness of the linear regime to charge screening can be demonstrated quantitatively.

Consider the screening field induced by a perturbing potential at the diamond surface, $\delta V$. Within the linear regime we have that

where we have denoted $\Delta\equiv(E_{T}+qV_{S}(0)-E_{N})$ and expanded to first order in $\delta V$. Suppose that the perturbing potential is due to a point charge $Q$ positioned a height $h$ above the electrometer surface and aligned with the NV center. Then equation 11 indicates that the induced charge density is given by

as a function of radial distance from the point charge $r$. This produces a screening field at the NV center given by

Considering that the field generated by the point charge at a depth $d$ in the absence of screening is

the magnitude of the screening ratio is given by

Equation 15 demonstrates a further bound on the maximum surface potential compatible with electrometry. Taking $(d+h)=200$ nm and $\sigma_{T}=10^{18}$ $\text{m}^{-2}$ we obtain a screening ratio of $R\approx 10^{-22}e^{\beta qV_{S}(0)}$. Screening effects will therefore dominate if the surface potential is not sufficiently deep within the linear regime. In this specific example, a ratio $R\leq 1\%$ only occurs for $qV_{S}(0)\leq E_{N}-E_{T}-15k_{B}T=1.2$ eV at room temperature.

To summarize, our toy model has identified the fundamental limitations to precision electrometry. The potential of the surface must be controlled such that all charge traps are saturated while the NV energy is maintained below the Fermi level. In general, one should aim to reduce $V_{S}(0)$ to avoid the detrimental effects of screening and quenching. Equation 6 indicates that the surface potential is a function of only two variables; the capacitance and the density of charge traps. For the simplified electrometer design, $V_{S}(0)\propto C^{-1}$ within the linear regime and therefore $D$ should be minimized. However, $D$ is limited below by physical constraints such as the thickness of the $\delta$-doped layer and its proximity to the surface and NV center. Hence the surface potential is largely dictated by the density of surface traps. Whereas the capacitance can be controlled, surface defects are an undesirable byproduct of diamond surface passivationStacey et al. (2019).

The performance of a diamond electrometer is strongly dependent on the surface trap density. This relationship is demonstrated in Figure 5 where $D=100$ nm, $d=60$ nm and $T=300$ K have been chosen as a realistic example of device parameters. Three different electrometer operating regions can be distinguished. Region (i) represents values of $\sigma_{T}$ which correspond to surface voltages within the linear regime. Here electrometry is viable as the charge traps are fully saturated and the NV maintains its negative charge state. Region (ii) also represents values of $\sigma_{T}$ for which the NV^- center remains charge stable. However, the sp² defects are only partially occupied and so electrometry is impossible due to surface screening. Similarly, in region (iii) $\sigma_{T}$ is so large that the NV center has been quenched.

Consequently, the presence of a $\text{N}_{\text{S}}$ $\delta$-doped layer can simultaneously maintain NV^- stability and prevent surface screening within the linear regime. However, the capabilities of the device are fundamentally limited by the density of surface traps. As capacitance is a geometric property, equation (6) indicates that this result is universal to any device which employs donors to saturate surface defects. The parallel plate capacitor has one of the greatest capacitances over small length scales and hence Figure 5 demonstrates that electrometry is only compatible with surface densities on the order of $\sigma_{T}\lesssim 10^{16}$ $\text{m}^{-2}$. This is two orders of magnitude lower than defect densities currently observed on fluorine terminated surfaces passivated using SF₆ plasma. However, fluorine is a relatively new surface termination and many alternative passivation techniques exist and continue to be developedCui and Hu (2013); Rietwyk et al. (2013); Salvadori et al. (2010). The sp² defect yields on these surfaces are yet to be characterized and may well be low enough to permit precision electrometry.

The simple electrometer presented in the previous section has several deficiencies which make it impractical for quantum sensing. Fortunately, these can be overcome with a simple modification to the electrometer design. In this section we discuss the shortcomings of the toy-model electrometer and their solutions, culminating in the presentation of an effective and physically realizable device.

Initialization and readout of the NV center requires optical control using a 532 nm green laserDoherty et al. (2013); Dolde et al. (2011, 2014). As depicted in Figure 6, performing electrometry with the toy device requires the optical path to pass through the $\delta$-doped $\text{N}_{\text{S}}$ layer. This introduces several complications. Firstly, $\text{N}_{\text{S}}$ layers typically contain some density of erroneous NV centers. These are capable of producing background counts during read-out which decrease measurement contrast and lead to lower sensitivity. Secondly, the 532 nm laser ionizes $\text{N}_{\text{S}}$ defects to form $\text{N}_{\text{S}}^{+}$ which modulates the charge density within the $\delta$-doped layerIakoubovskii and Adriaenssens (2000). During the optical steady state this results in a local positive potential which reduces charge stability of the $\text{NV}^{-}$ center and further diminishes signal contrast. Moreover, following initialization of the NV^- spin-state the laser is deactivated and the induced charge relaxes back to the equilibrium it adopted before the laser was turned on. If this relaxation time is slow ($\approx 100$ ns) compared to the NV sensing period ($>1$ $\mu$s)Dolde et al. (2011, 2014), this relaxation will influence the sensing measurement. Ionization of $\text{N}_{\text{S}}$ also introduces free charge carriers into the lattice, and hence there also exists a low probability of Auger electrons scattering against the NV^- and causing quenching. Finally, the $\delta$-doped $\text{N}_{\text{S}}$ layer acts as a spin-bath which can cause decoherence of the NV spin at close proximitiesBalasubramanian et al. (2009).

One possible solution to these issues is to spatially separate the $\text{N}_{\text{S}}$ layer and the NV center. The optical focus can then be maintained on the NV while the $\delta$-doping is subject to a negligible laser intensity. We estimate this would require a separation distance of approximately $0.5$ $\mu$m. However, this severely limits the sensitivity of the spin-probe to external charges as equation (9) necessitates that $D\gtrsim 1$ $\mu$m and therefore $d\gtrsim 0.5$ $\mu$m to maintain $\text{NV}^{-}$ charge stability. A more practical solution is to introduce of a hole within the $\delta$-doped layer. Consider the schematic presented in Figure 7 in which a disk of pure diamond has been fabricated around the NV center. This hole permits optical access to the spin probe while simultaneously minimizing the number of ionized $\text{N}_{\text{S}}$ defects. Furthermore, the hole reduces the probability of optically addressing multiple NV centers and so increases the yield of electrometer fabrication.

The capabilities of this realistic electrometer design for elementary charge detection were investigated using COMSOL Multiphysics software. The potential at the NV center and surface as well as the occupation of charge traps were simulated as a function of the device parameters; the NV depth $d$, $\text{N}_{\text{S}}$ layer depth $D$ and hole radius $r$. This was achieved by solving Poisson’s equation self-consistently for a surface charge density given by equation 6 and assuming a grounded $\delta$-doped layer. The device parameters were then optimized to identify the greatest possible trap density compatible with precision electrometry. As discussed in Section II, these criterion are charge stability of the NV ($qV_{S}(d)\ll E_{N}-E_{NV}=0.9$ eV) and a surface potential which leads to less than 1% surface screening as per equation (15).

Figure 8 presents the largest viable surface trap density as a function of the $\delta$-doping depth and hole radius. Trap densities greater than those presented lead to field screening in excess of 1%. For the parameters sampled here ($30$ $\text{nm}<D<100$ nm and $80$ $\text{nm}<r<150$ nm) we find that electrometry is feasible for sp² surface densities within the regime of $10^{15}$ $\text{m}^{-2}$ and that no charge quenching occurs for NV centers level with the $\delta$-doped layer. Devices with smaller radii and $\delta$-doping depth possess a greater capacitance and are therefore compatible with larger defect densities. Note that the optical spot size is diffraction limited by $\approx 200$ nm, and hence some degree of $\text{N}_{\text{S}}$ ionization will occur for hole radii less than 100 nm. The impact of the induced positive charge density for NV quantum sensing is unknown and left as an avenue for future work. If the effects are significantly detrimental, Figure 8 indicates that the hole size can simply be increased beyond the optical spot size without drastically reducing the achievable sp² defect concentrations.

Sub-nanometer resolution electrometry of elementary charges under ambient conditions would allow for investigation of diverse electrical phenomenon ranging from biological systems to fundamental physics. The NV center is the only known system capable of such a feat, however measurements are currently limited to charges internal to diamond. In this paper we have applied theoretical modeling to conclusively demonstrate that external charge detection is not yet feasible due to field screening. While screening due to the atmosphere and internal defects can be mitigated using fluorine-passivated and ultra-pure diamond, electrometry is ultimately frustrated by charge rearrangement amongst surface defects.

We have proposed a solution to surface screening through introduction of a sacrificial $\text{N}_{\text{S}}$ $\delta$-doped layer. Fabrication of a $\text{N}_{\text{S}}$ deficit hole surrounding the NV center allows for optical access to the spin probe while minimizing the read-out of erroneous NV centers and ionization of free charges. This electrometry device is technologically feasible and computational simulations have demonstrated that it can successfully mitigate screening effects for surface trap densities up to $\approx 10^{16}$ $\text{m}^{-2}$. Although this is two orders of magnitude below currently observed sp² defect densities on fluorine-terminated diamond, the outcome of this work is a clear pathway towards nanoscale imaging of external charges at ambient conditions. Electrometry cannot be achieved until surface passivation technologies realize lower defect concentrations on fluorine-terminated diamond.

We acknowledge funding from the Australian Research Council (DP170102735). We thank Patrick Maletinsky for providing insightful feedback on this manuscript.