Abstract
Nuclear magnetic resonance (NMR) spectroscopy is a powerful method for analyzing the chemical composition and molecular structure of materials. At the nanometer scale, NMR has the prospect of mapping the atomicscale structure of individual molecules, provided a method that can sensitively detect single nuclei and measure interatomic distances. Here, we report on precise localization spectroscopy experiments of individual ^{13}C nuclear spins near the central electronic sensor spin of a nitrogenvacancy (NV) center in a diamond chip. By detecting the nuclear free precession signals in rapidly switchable external magnetic fields, we retrieve the threedimensional spatial coordinates of the nuclear spins with subAngstrom resolution and for distances beyond 10 Å. We further show that the Fermi contact contribution can be constrained by measuring the nuclear gfactor enhancement. The presented method will be useful for mapping atomic positions in single molecules, an ambitious yet important goal of nanoscale nuclear magnetic resonance spectroscopy.
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Introduction
One of the visionary goals of nanoscale quantum metrology with NV centers is the structural imaging of individual molecules, for example proteins, that are attached to the surface of a diamond chip^{1}. By adapting and extending measurement techniques from nuclear magnetic resonance (NMR) spectroscopy, the longterm perspective is to reconstruct the chemical species and threedimensional location of the constituent atoms with subAngstrom resolution^{2,3}. In contrast to established structural imaging techniques like Xray crystallography, cryoelectron tomography, or conventional NMR, which average over large numbers of target molecules, only a single copy of a molecule is required. Conformational differences between individual molecules could thus be directly obtained, possibly bringing new insights about their structure and function.
In recent years, first experiments that address the spatial mapping of nuclear and electron spins with NVbased quantum sensors have been devised. One possibility is to map the position into a spectrum, as it is done in magnetic resonance imaging. For nanometerscale imaging, this requires introducing a nanomagnet^{4,5,6}. Another approach is to exploit the magnetic gradient of the NV center’s electron spin itself, whose dipole field shifts the resonances of nearby nuclear spins as a function of distance and internuclear angle. Refinements in quantum spectroscopic techniques have allowed the detection of up to 8 individual nuclear spins^{7,8} as well as of spin pairs^{9,10,11} for distances of up to ~30 Å^{12,13}. Due to the azimuthal symmetry of the dipolar interaction, however, these measurements can only reveal the radial distance r and polar angle θ of the interspin vector r = (r, θ, ϕ), but are unable to provide the azimuth ϕ required for reconstructing threedimensional nuclear coordinates. One possibility for retrieving ϕ is to change the direction of the static external field^{12}, however, this method leads to a mixing of the NV center’s spin levels which suppresses the ODMR signal^{14} and shortens the coherence time^{15}. Other proposed methods include positiondependent polarization transfer^{16} or combinations of microwave and radiofrequency fields^{17,18,19}.
Here we demonstrate threedimensional localization of individual, distant nuclear spins with subAngstrom resolution. To retrieve the missing angle ϕ, we combine a dynamic tilt of the quantization axes using a highbandwidth microcoil with high resolution correlation spectroscopy^{20,21}. Our method provides the advantage that manipulation and optical readout of the electronic spin can be carried out in an aligned external bias field. This ensures best performance of the optical readout and the highest magnetic field sensitivity and spectral resolution of the sensor.
Results
Theory
We consider a nuclear spin I = 1/2 located in the vicinity of a central electronic spin S = 1 with two isolated spin projections m_{S} = {0, −1}. The nuclear spin experiences two types of magnetic field, a homogeneous external bias field B_{0} (aligned with the quantization axis e_{z} of the electronic spin), and the local dipole field of the electronic spin. Because, the electronic spin precesses at a much higher frequency than the nuclear spin, the latter only feels the static component of the electronic field, and we can use the secular approximation to obtain the nuclear free precession frequencies,
Here, γ_{n} is the nuclear gyromagnetic ratio and
is the secular hyperfine vector of the hyperfine tensor \(\underline {\underline {\bf{A}} } ({\bf{r}})\) that gives rise to the hyperfine magnetic field m_{S}A_{z}(r)/γ_{n} (see Fig. 1b).
To obtain information about the distance vector r, a standard approach is to measure the parallel and transverse components of the hyperfine vector, a_{} = A_{zz} and \(a_ \bot = \left( {A_{xz}^2 + A_{yz}^2} \right)^{1/2}\), and to relate them to the field of a point dipole,
where μ_{0} = 4π × 10^{−7} TmA^{−1} is the vacuum permeability, ħ = 1.054 × 10^{−34} Js is the reduced Planck constant, γ_{e} = 2π × 28 GHz T^{−1} is the electron gyromagnetic ratio, and where we have included a Fermi contact term a_{iso} (set to zero for now) for later discussion. Experimentally, the parallel projection a_{} can be inferred from the precession frequencies \(f_{m_S}\) using Eq. (1), and the transverse projection a_{⊥} can be determined by driving a nuclear Rabi rotation via the hyperfine field of the central spin and measuring the rotation frequency^{21}. Once a_{} and a_{⊥} are known, Eqs. (3) and (4) can be used to extract the distance r and polar angle θ of the distance vector r = (r, θ, ϕ). Due to the rotational symmetry of the hyperfine interaction, however, knowledge of a_{} and a_{⊥} is insufficient for determining the azimuth ϕ.
To break the rotational symmetry and recover ϕ, we apply a small transverse magnetic field ΔB during the free precession of the nuclear spin. Application of a transverse field tilts the quantization axes of the nuclear and electronic spins. The tilting modifies the hyperfine coupling parameters a_{} and a_{⊥} depending on the angle between ΔB and A_{z}, which in turn shifts the nuclear precession frequencies \(f_{m_S}\). To secondorder in perturbation theory, the m_{S}dependent precession frequencies are given by:^{22}
where α(m_{S}) is a small enhancement of the nuclear gfactor. The enhancement results from nonsecular terms in the Hamiltonian that arise due to the tilting of the electronic quantization axis, and is given by^{22}
Here D = 2π × 2.87 GHz is the groundstate zerofield splitting of the NV center. By measuring the shifted frequencies \(f_{m_S}\) and comparing them to the theoretical model of Eqs. (5) and (6), we can then determine the relative ϕ angle between the hyperfine vector and ΔB.
Experimental setup
We experimentally demonstrate threedimensional localization spectroscopy of four ^{13}C_{1–4} nuclei adjacent to three distinct NV centers. NV_{1} is coupled to two ^{13}C spins, while NV_{2} and NV_{3} are each coupled to a single ^{13}C spin. For readout and control of the NV center spin, we use a custombuilt confocal microscope that includes a coplanar waveguide and a cylindrical permanent magnet for providing an external bias field of B_{0} ~ 10 mT applied along the NV center axis e_{z}. Precise alignment of the bias field is crucial for our experiments and is better than 0.3° (Methods section).
To dynamically tilt the external field we implement a multiturn solenoid above the diamond surface (Fig. 1d). The coil produces ~2.5 mT field for 600 mA of applied current and has a rise time of ~2 μs. We calibrate the vector magnetic field of the coil with an absolute uncertainty of less than 15 μT in all three spatial components using two other nearby NV centers with different crystallographic orientations (ref. ^{23} and Methods section).
Mapping of r and θ
We begin our 3D mapping procedure by measuring the parallel and perpendicular hyperfine coupling constants using conventional correlation spectroscopy^{21} with no coil field applied, ΔB = 0 (Fig. 2). The parallel coupling a_{} is determined from a free precession experiment (sequence 1 in Fig. 2) yielding the frequencies f_{0} and f_{−1} (Fig. 2b). The coupling constant is then approximately given by a_{}/(2π) ≈ f_{−1} − f_{0}. The transverse coupling a_{⊥} is obtained by driving a nuclear Rabi oscillation via the NV spin, using sequence 2, and recording the oscillation frequency f_{R}, where a_{⊥}/(2π) ≈ πf_{R} (Fig. 2c). Because, the Zeeman and hyperfine couplings are of similar magnitude, these relations are not exact and proper transformation must be applied to retrieve the exact coupling constants a_{} and a_{⊥} (ref. ^{21} and Methods section). Once the hyperfine parameters are known, we can calculate the radial distance r = 8.58(1) Å and the polar angle θ = 52.8(1)° of the nuclear spin by inverting the pointdipole formulas (Eqs. 3, 4). The measurement uncertainties in r and θ are very small because correlation spectroscopy provides high precision estimates of both a_{} and a_{⊥}.
Mapping of ϕ
In a second step, we repeat the free precession measurement with the coil field turned on (sequence 3), yielding a new pair of frequency values \(f_0^\prime\), \(f_{  1}^\prime\) (Fig. 2d). We then retrieve ϕ by computing theoretical values for \(f_0^{{\mathrm{(th)}}}\), \(f_{  1}^{{\mathrm{(th)}}}\) based on Eq. (5) and the calibrated fields in Table 1, and minimizing the cost function
with respect to ϕ. To cancel residual shifts in the static magnetic field and improve the precision of the estimates, we compare the frequency difference between m_{S} states rather than the absolute precession frequencies.
In Fig. 3a, we plot ξ(ϕ) for three different coil positions and opposite coil currents for ^{13}C_{1}. We use several coil positions because a single measurement has two symmetric solutions for ϕ, and also because several measurements improve the overall accuracy of the method. The best estimate ϕ = 239(2)° is then given by the least squares minimum of the cost functions (dashdotted line in Fig. 3a). To obtain a confidence interval for ϕ, we calculate a statistical uncertainty for each measurement by Monte Carlo error propagation taking the calibration uncertainties in B_{0} and ΔB, as well as the measurement uncertainties in the observed precession frequencies into account (Methods section). Values for all investigated ^{13}C nuclei are collected in Supplementary Tables 2–9.
Fermi contact interaction
Thus far we have assumed that the central electronic spin generates the field of a perfect point dipole. Previous experimental work^{22,24} and density functional theory (DFT) simulations^{25,26}, however, suggest that the electronic wave function extends several Angstrom into the diamond host lattice. The finite extent of the spin density leads to two deviations from the point dipole model: modified hyperfine coupling constants A_{ij}, and a nonzero Fermi contact term a_{iso}. In the remainder of this study we estimate the systematic uncertainty to the localization of the nuclear spins due to deviations from the point dipole model.
We first consider the influence of the Fermi contact interaction, which arises from a nonvanishing NV spin density at the location of the nuclear spin. The Fermi contact interaction adds an isotropic term to the hyperfine coupling tensor, A + a_{iso}1, which modifies the diagonal elements A_{xx}, A_{yy}, and A_{zz}. DFT simulations^{25,26} indicate that a_{iso} can exceed 100 kHz even for nuclear spins beyond 7 Å. It is therefore important to experimentally constrain the size of a_{iso}.
To determine a_{iso}, one might consider measuring the contact contribution to the parallel hyperfine parameter a_{}, which is equal to A_{zz}. This approach, however, fails because a measurement of a_{} cannot distinguish between dipolar and contact contributions. Instead, we here exploit the fact that the gyromagnetic ratio enhancement α depends on A_{xx} and A_{yy}, and hence a_{iso}. To quantify the Fermi contact coupling we include a_{iso} as an additional free parameter in the cost function (7). By minimizing ξ(ϕ, a_{iso}) as a joint function of ϕ and a_{iso} and generating a scatter density using Monte Carlo error propagation, we obtain maximum likelihood estimates and confidence intervals for both parameters (Fig. 3b). The resulting contact coupling and azimuth for nuclear spin ^{13}C_{1} are a_{iso}/(2π) = 9(8) kHz and ϕ = 238(2)°, respectively; data for ^{13}C_{2–4} are collected in Table 2. Because the gyromagnetic ratio enhancement α is only a secondorder effect, our estimate is poor, but it still allows us constraining the size of a_{iso}. By subtracting the Fermi contact contribution from a_{}, we further obtain refined values for the radial distance and polar angle, r = 8.3(2) Å and θ = 58(4)°. Note that introducing a_{iso} as a free parameter increases the uncertainties in the refined r and θ, because the error in a_{iso} is large. This leads to disproportionate errors for distant nuclei where a_{iso} is small. Once nuclei are beyond a certain threshold distance, which we set to r = 10 Å in Table 2, it therefore becomes more accurate to constrain a_{iso} = 0 and apply the simple point dipole model.
Extended electronic wave function
The second systematic error in the position estimate results from the finite size of the NV center’s electronic wave function. Once the extent of the wave function becomes comparable to r, the anisotropic hyperfine coupling constants A_{ij} are no longer described by a point dipole, but require integrating a geometric factor over the sensor spin density^{25}. While we cannot capture this effect experimentally, we can estimate the localization uncertainty from DFT simulations of the NV electron spin density. Following ref. ^{26}, we convert the calculated DFT hyperfine parameters of 510 individual lattice sites to (r, θ) positions using the pointdipole formula (Eqs. 3, 4), and compute the difference to the DFT input parameters (r_{DFT}, θ_{DFT}). The result is plotted in Fig. 4a. We find that the difference 〈Δr〉 = r − r_{DFT} decreases roughly exponentially with distance, and falls below 0.2 Å when r > 10 Å (gray dots and curve).
Discussion
Figure 4b summarizes our study by plotting the reconstructed locations for all four carbon atoms in a combined 3D chart. The shaded regions represent the confidence areas of the localization, according to Table 2, projected onto the Cartesian coordinate planes. We note that the DFT simulations are in good agreement with our experimental results.
The accuracy of our present experiments is limited by deviations from the pointdipole model, which dominate for small r (Fig. 4a). For larger r ≳ 1 nm, this systematic uncertainty becomes negligible, and the localization imprecision is eventually dictated by the NMR frequency measurement. In the present study, which probed isolated ^{13}C nuclei with a narrow intrinsic linewidth, the frequency precision was limited by the accuracy of our detection protocol to ~100 Hz. This corresponds to a radial localization error of ~0.75 Å at a distance of r = 3 nm (solid black line in Fig. 4a and Supplementary Fig. 6). Improving the frequency precision to 3 Hz^{11,27} would extend this distance to r ~ 7 nm (dashed black line).
Our work demonstrates a basic strategy for mapping spatial positions of single nuclei in 3D with high precision. Extending these experiments to single molecules outside a diamond chip poses a number of additional challenges, and overcoming them will require the combination of several strategies. To isolate single molecules, they can be embedded in a spinfree matrix layer deposited on the diamond surface^{28} or immobilized by a linker chemistry^{29}. Nuclear dipole interactions can be suppressed using homo and heteronuclear decoupling^{13}, taking advantage of the existing microcoil. Linewidths and spectral complexity can be further reduced by polarizing the constituent nuclei, and by spin dilution and isotope labeling of molecules^{30}. Alternatively, measurements of interspin couplings will allow constraining the structure and size of molecules^{31}. To sensitively detect very weakly coupled nuclei in distant molecules, spin precession can be recorded by repetitive weak measurements^{32,33}. Further improvement of sensitivity is possible by optimizing the optical detection efficiency compared to our present setup^{34}, and possibly by cryogenic operation^{11}. How well these strategies will work we do not know at present, but we believe that the prospect of a general singlemolecule MRI technique, which will have many applications in structural biology and chemical analytics, provides sufficient motivation to warrant these efforts.
Methods
Diamond sample
Experiments were performed on a bulk, electronicgrade diamond crystal from ElementSix with dimensions 2 × 2 × 0.5 mm with \(\left\langle {110} \right\rangle\) edges and a \(\left\langle {100} \right\rangle\) front facet. The diamond was overgrown with a layer structure of 20 nm enriched ^{12}C (99.99%), 1 nm enriched ^{13}C (estimated ingrown concentration ~5–10%) and a 5 nm cap layer of again enriched ^{12}C (99.99%). Nitrogenvacancy (NV) centers were generated by ionimplantation of ^{15}N with an energy of 5 keV, corresponding to a depth of ~5–10 nm. After annealing the sample for NV formation, we had to slightly etch the surface (at 580 °C in pure O_{2}) to remove persistent surface fluorescence. The intrinsic nuclear spin of the three NV centers studied in our experiments were confirmed to be of the ^{15}N isotope. Further characterizations and details on the sample can be found in a recent study (sample B in ref. ^{35}).
Coordinate systems
In Supplementary Fig. 2a both laboratory and NV coordinate system are shown in a combined schematic. The laboratory coordinate system (x_{Lab}, y_{Lab}, z_{Lab}) is defined by the normal vectors to the diamond faces, which lie along \(\left\langle {110} \right\rangle\), \(\left\langle {\bar 110} \right\rangle\) and \(\left\langle {001} \right\rangle\), respectively. The reference coordinate system of the NV center is defined by its quantization direction, which is labeled z_{NV} and lies along \(\left\langle {111} \right\rangle\). The x_{NV} and y_{NV}axis are pointing along the \(\left\langle {11\bar 2} \right\rangle\) and \(\left\langle {\bar 110} \right\rangle\) direction, respectively.
Experimental apparatus
A schematic of the central part of the experimental apparatus is shown in Supplementary Fig. 1. The diamond sample is glued to a 200 μm thick glass piece and thereby held above a quartz slide with incorporated microwave transmission line for electron spin control. Below the quartz slide we placed a high numerical aperture (NA = 0.95) microscope objective for NV excitation with a 532 nm laser and detection using a single photon counting module (SPCM). We applied static, external magnetic bias fields with a cylindrical NdFeB permanent magnet (not shown in Supplementary Fig. 1). The magnet is attached to a motorized, threeaxis translation stage. The NV control pulses were generated by an arbitrary waveform generator (Tektronix, AWG5002C) and upconverted by I/Q mixing with a local oscillator to the desired ~2.6 GHz.
Planar, highbandwidth coil
The planar coil is positioned directly above the diamond sample and attached to a metallic holder, which can be laterally shifted to translate the coil. Due to the thickness of the diamond (500 μm) and the glass slide the minimal vertical standoff of the coil to the NV centers is ~700 μm. Design parameters of the planar coil, used in our experiments, are listed in Supplementary Table 1. These were found by numerically maximizing the magnetic field at the position of the NV center, located at a planned vertical standoff of ~700–1000 μm (Supplementary Fig. 1). The coil had an inductance of ≤2.5 μH and a resistance of ≤0.5 Ω. The coil was manufactured by Sibatron AG (Switzerland) and it is mounted onto a copper plate that acts as a heatsink, using thermally conducting glue. For the coil control, a National Instruments NI PCI 5421 arbitrary waveform generator was used, to generate voltage signals that controlled a waveform amplifier (Accel Instruments TS250) which drives the coil current.
Calibration of the coil field ΔB
We calibrated the vector field generated by the coil ΔB using the target NV, coupled to nuclear spins of interest, and two auxiliary NV centers with different crystallographic orientation. All three NV centers were located in close proximity to each other, with a distance of typically ≤5 μm (Supplementary Fig. 2c). Over this separation the magnetic field of the coil can be assumed to be homogeneous. We determined the orientation of the symmetry axis of many NV centers by moving the permanent magnet over the sample and observing the ODMR splitting. The azimuthal orientation of the target NVs defines the xaxis in the laboratory and NV frame (ϕ = 0). This orientation was the same for all target NV centers investigated in this work. The auxiliary NV centers were selected to be oriented along ϕ_{a,1} = 90° and ϕ_{a,2} = 270° (Supplementary Fig. 2b). To calibrate the coil field, we removed the permanent magnet and recorded ODMR spectra for the target NV center and both auxiliary NV centers with the field of the coil activated. In this way, we record in total 6 ODMR lines, with 2 lines per NV center.
A numerical, nonlinear optimization method was used to determine the magnetic field ΔB from these ODMR resonances. For each of the three NV centers we simultaneously minimized the difference between the measured ODMR lines and the eigenvalues of the groundstate Hamiltonian:
Here, the magnetic field (ΔB)_{i} acting onto the specific NV center is obtained by a proper rotation of ΔB into the respective reference frame.
Precise alignment of the bias field B _{0}
Precise alignment of the external bias field to the quantization axis of the NV center (zaxis) is critical for azimuthal localization measurements, because residual transverse fields of B_{0} modify the precession frequencies in the same way as the field of the coil. The coarse alignment of the magnet and a rough adjustment of the magnitude of the field, to ~10 mT, was achieved by recording ODMR spectra of the target NV center for different (x, y, z)positions of the magnet. Afterwards, we iteratively optimized the alignment of the magnet. In each iteration, we reconstructed the vector field B_{0} acting on the target NV centers using the method used for the calibration of ΔB. Subsequently, we moved the magnet in the lateral (x, y)plane of the laboratory frame. The direction and step size was determined from a field map of the permanent magnet and the residual transverse components of the field B_{0}. We terminated this iterative process when the residual transverse field components were smaller than 50 μT.
Determination of hyperfine couplings (a _{}, a _{⊥}) from (f _{0}, f _{−1}, f _{R})
The hyperfine couplings a_{} and a_{⊥} in the limit \(2\pi f_0 \gg a_{},a_ \bot\) are given by:
In our experiments the hyperfine couplings and the nuclear Larmor frequency f_{0} were of similar magnitude, and we used the following transformations^{21} to obtain the hyperfine couplings.
Monte Carlo error propagation
Confidence intervals for ϕ and a_{iso} were obtained using standard Monte Carlo error propagation^{36}. The Monte Carlo simulation took calibration uncertainties in the external fields B_{0}, ΔB and in the observed precession frequencies \(f_{m_s}\) into account. All parameters subject to uncertainty were assumed to follow a normal distribution. Precession frequencies were determined using a nonlinear, leastsquares fitting algorithm and their measurement uncertainties were obtained from the fit error^{21}. The uncertainty in the magnetic field components was estimated from the residuals between calculated and measured ODMR lines in the calibration method for B_{0}, ΔB.
Nuclear gfactor enhancement
The nuclear gfactor enhancement factor α(m_{S}) given in Eq. (6) of the main text is based on the approximation of small external bias fields \(D \gg \gamma _{\mathrm{e}}B_0\). More generally the m_{S}dependent enhancement factors are given by^{37}:
which is also valid in the limit of large magnetic fields \(\gamma _{\mathrm{e}}B_0 \gg D\) and provides, in principle, more accurate theory values for small B_{0}. We have analyzed our experimental data using this expression and found deviations to Eq. (6) that are smaller than the frequency resolution in our experiments.
Code availability
Custom code was programmed to perform the Monte Carlo simulations. The code is available from the corresponding author upon request.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
The authors thank JyhPin Chou, Adam Gali, A. Nizovtsev, and Fedor Jelezko for sharing DFT data, and Julien Armijo, Kevin Chang, Nils Hauff, Konstantin Herb, Takuya Segawa, and Tim Taminiau for helpful discussions. This work was supported by Swiss NSF Project Grant 200021_137520, the NCCR QSIT, and the DIADEMS 611143 and ASTERIQS 820394 programs of the European Commission. The work at Keio has been supported by JSPS KAKENHI (S) No. 26220602, JSPS CoretoCore, and SpinRNJ.
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C.L.D. initiated the project. J.Z. conceived the microcoil. J.Z. performed the experiments with the help of J.M.B., K.S. and K.S.C. J.Z. and C.L.D. analyzed the data and cowrote the paper with the help of K.M.I. All authors discussed the results.
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Zopes, J., Cujia, K.S., Sasaki, K. et al. Threedimensional localization spectroscopy of individual nuclear spins with subAngstrom resolution. Nat Commun 9, 4678 (2018). https://doi.org/10.1038/s41467018071210
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DOI: https://doi.org/10.1038/s41467018071210
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