Nanoscale electrical conductivity imaging using a nitrogen-vacancy center in diamond
The electrical conductivity of a material can feature subtle, nontrivial, and spatially-varying signatures with critical insight into the material’s underlying physics. Here we demonstrate a conductivity imaging technique based on the atom-sized nitrogen-vacancy (NV) defect in diamond that offers local, quantitative, and noninvasive conductivity imaging with nanoscale spatial resolution. We monitor the spin relaxation rate of a single NV center in a scanning probe geometry to quantitatively image the magnetic fluctuations produced by thermal electron motion in nanopatterned metallic conductors. We achieve 40-nm scale spatial resolution of the conductivity and realize a 25-fold increase in imaging speed by implementing spin-to-charge conversion readout of a shallow NV center. NV-based conductivity imaging can probe condensed-matter systems in a new regime, and as a model example, we project readily achievable imaging of nanoscale phase separation in complex oxides.
The motion of electrons in the solid-state provides important insight into a material’s multiple interacting degrees of freedom, and understanding the complexity of these interactions is at the heart of condensed matter physics. Measurements of a material’s conductivity and its signatures as a function of temperature or magnetic field, for instance, often provide the best evidence for the microscopic mechanisms at play. In recent years, significant interest has turned to condensed matter phenomena in which the electrical conductivity exhibits nontrivial spatial variations. Examples include topological insulators Moore:2010ig; Langbehn2017, which host conducting surfaces and an insulating interior, and Mott insulators that at critical temperatures and magnetic fields exhibit nanoscale phase separation, where pocket-like metallic regions form in an insulating matrix Mattoni2016; Imada1998. As another example, anomalous domain wall conductivity has been observed in various complex oxides, such as multiferroics Seidel2010 and iridates Ma2015; Ueda2014a, with predictions of exotic Weyl semimetal behavior at these domain walls Yamaji:2014fa. These phenomena are challenging to probe with standard transport measurements that average over a macroscopic area of the sample, and detecting these spatial signatures is the goal of many advanced techniques.
Many spatially-resolved probes of local electron dynamics exist, including optical conductivity probes and scanning probe microscopy (SPM)-based probes. Optical probes are limited in their spatial resolution by the diffraction limit (with specialized near-field probes approaching resolutions of tens of nanometers Atkin:2012cm; Fei:2012bi), probe conductivity at optical frequencies only, and are generally surface sensitive. SPM-style probes, such as conducting atomic force microscopy (AFM) Murrell1993 and microwave impedance microscopy (MIM) Lai2009, offer high spatial resolution but are limited in other respects. A prominent drawback is that the signal they produce is convolved with the geometry of the probe and its interface with the sample under study, thus making quantitative measurements challenging. Importantly, all of these local conductivity probes measure the response of a system to some driving perturbation, which could induce a nonlinear response or void the subtle effects one hoped to study Altshuler2009.
On the other hand, the nitrogen-vacancy (NV) center in diamond presents a quantitative, noninvasive, and nanoscale sensor capable of measuring electrical conductivity via directly sensing the magnetic fields produced by thermal electron motion in the material. An atom-like quantum sensor, the NV center’s signal is analytically related to the sample conductivity through fundamental constants. And because the NV monitors the fluctuating magnetic fields produced by electrons in thermal equilibrium, the NV sensing mechanism involves neither a driving perturbation nor sample contact, which also allows for subsurface sensing. Third, due to its atomic-size, the NV center affords very high spatial resolution Taylor2008; Rugar2014, with recent magnetic imaging reaching sub-10 nm resolution Pelliccione2016; Haberle2015. Lastly, the versatility of the NV center is an attractive feature; it is capable of operating over a wide range of temperatures Bar-Gill2013; Toyli2012 and in a wide range of imaging modalities, offering the possibility of combining conductivity imaging with simultaneous magnetic, electric, and thermal imaging. Recent work by Kolkowitz et al. used the NV center to sense spectral and thermal signatures of electron behavior in metallic films, elucidating transitions from diffusive to ballistic transport regimes with changes in temperature Kolkowitz2015.
In this work we integrate an NV sensor with a scanning probe microscope to spatially image local electron conductivity in nanopatterned metal films with 40-nm scale spatial resolution. By monitoring the relaxation rate of the NV center spin state while the NV is scanned in nanometer proximity to a metal, we quantitatively measure the local conductivity of several metals. By using spin-to-charge conversion readout techniques on a shallow NV center, we demonstrate a 25-fold reduction in imaging time. With the sensitivity and spatial resolution demonstrated here, we project NV-based conductivity imaging of nanoscale phase separation in complex oxides, a model example of a spatially inhomogeneous phenomenon in a condensed matter system.
Relaxation model and experimental setup. The NV center in diamond is a point defect comprising a substitutional nitrogen atom and a nearby vacancy in the carbon lattice. The two unpaired electrons of the defect center form a ground state spin triplet that features long energy relaxation times () at room temperature. The conductivity imaging experiments described here utilize the sensitivity of the spin to fluctuating magnetic fields produced by electrons moving in a nearby conductor.
The Hamiltonian of the NV ground state spin triplet in the presence of a magnetic field is given by
where GHz is the zero field splitting between the and triplet states, is the electron gyromagnetic ratio, are the spin operators and are the spin raising and lowering operators , and the direction is chosen to point along the NV axis Doherty2012. Incoherently oscillating magnetic fields that are at the frequency of the transition and perpendicular to the NV axis induce transitions between the two states, thus speeding up the NV center’s relaxation rate Schafer-Nolte2014. This sensitivity to fluctuating magnetic fields can then be used to detect stochastic electron motion in a conductor.
The magnetic fluctuations emanating from a conductor can be related to the material’s conductivity by first invoking the Biot-Savart law, which gives the magnetic field produced by current densities as where is the electron position and is the vacuum permeability, and then the Johnson-Nyquist formula, , where is the spectral density of the current density fluctuations, is the Boltzmann constant, and is temperature. Note that for GHz.
For the conductor geometry studied in this work, a thin-film of thickness a distance away from the NV sensor, a volume integral over the conductor yields the -component of the magnetic spectral density
and (a full derivation is given in supplementary information SI 1.1) Henkel1999a; Langsjoen2014. The spectral density component perpendicular to the NV axis is, for the cut diamond used in this work, , which induces transitions at a rate . Applying perturbation theory to the NV center spin triplet (details in SI 1.3) yields the metal-induced relaxation rate
Thus a metal induces relaxation proportional to its conductivity . Further, the relaxing effect goes as for and for . In this work nm and varies from 10 to 1000 nm, thus spanning both regimes.
The conductivity imaging setup consists of a laser scanning confocal microscope integrated with a tuning fork-based atomic force microscope (AFM) (Fig. 1a) Pelliccione2014. All experiments are performed in ambient conditions in a small applied magnetic field of G. The AFM scans an NV center within nanometer-scale proximity of the surface of a conducting sample and the confocal microscope is used to optically initialize and readout the spin state of the NV center. Optical access is through the 150-m thick diamond plate. A waveguide patterned on the diamond is used to transmit microwaves to coherently drive transitions between the spin states. NV centers reside nm below the surface of the bulk piece of diamond and are formed by N implantation and subsequent annealing (details in Methods). To enhance photon collection efficiency, the diamond sample is patterned with 400-nm diameter nanopillars; only pillars containing 1 NV center are used here. Conducting samples are patterned onto custom-fabricated scanning probes with flat plateau-tips that have diameters of several micrometers (Fig. 1a). The fabrication procedure (details in Methods) allows for a variety of sample geometries and materials, several of which we image in this work. The probes are then mounted onto a quartz tuning fork for AFM feedback and scanning. To minimize relative position drift between the conducting sample and the NV center, we implement temperature stabilization to 1 mK/day in concert with active drift correction that utilizes AFM-based image registration.
Electrical conductivity measurement. To measure a metal’s conductivity we measure the NV center relaxation rate as a function of where
where is the intrinsic relaxation rate of the NV (), which is Hz for the NVs in this study. The is measured by initializing the NV into its spin state with a 10-s pulse of 532 nm light and then allowing the NV to decay for a dark time toward a thermally mixed state; this decay is measured via a subsequent spin-state-dependent photoluminescence (PL) measurement. For each , two measurements are performed: in the first, we readout the PL of the NV ; in the second, we insert a resonant microwave pulse after to swap the and populations and then readout the PL (see SI Fig. S3a) Myers2017. The difference corresponds to the difference in population between the and states, which decays to 0 with as plotted in Fig. 1c (details in SI 1.2). The data in Fig. 1c show a 5-fold reduction in NV in the presence of a metal film, corresponding to 60 Hz.
We now demonstrate quantitative electrical conductivity measurements using the NV center. Figure 2 plots as a function of for three different 85-nm thick metal films of Ag, Al, and Ti, each measured with a different NV center. The films are deposited via thermal evaporation onto a -m diameter flat AFM tip as depicted in Fig. 1a. Here, however, the film is continuous across the full extent of the plateau tip. Two qualitative observations can immediately be drawn from the data: first, increases for small NV-metal separations; second, the highly conducting Ag and Al induce faster relaxation rates than Ti, a relatively poor conductor. These observations are consistent with Eq. 3.
We quantitatively determine by performing a least squares regression on the data in Fig. 2 to Eq. 3. The extracted values of are , , and . These values are smaller than their bulk values by factors of 2.7, 1.7, and 4, respectively. A reduced conductivity is expected for metal films whose thickness is on the order of the electron mean free path ( nm), and our measurements are consistent with experimental and theoretical values for 85-nm thick Ag, Al, and Ti films Lacy2011; Zhang2004. We note that Eq. 3 assumes a slab of infinite extent, but here we study slabs with m diameter. However, this finite-size effect contributes deviations that are only 10% of the experimental error and hence is negligible for this study (details in SI 1.4).
In measuring we first contact the conducting sample to the diamond and then retract a known distance . The NV-metal separation in contact, , is treated as a free fit parameter in the total NV-metal separation . Different values of are obtained for the three different NV-metal combinations, likely due to different tilt angles between the diamond and metal surfaces.
We also note that the data in Fig. 2 cannot be fit by a simple or dependence, as would be expected for or , respectively, indicating that the NV’s distance-dependent response is also sensitive to thickness of the conducting region.
Nanoscale conductivity imaging. We now demonstrate nanoscale imaging of spatially inhomogeneous conductivity by laterally scanning an NV center over an array of Al pads, as pictured in the scanning electron microscope (SEM) image in Fig. 3a. This sample is formed by thermally evaporating 85 nm of Al onto 400 nm x 400 nm pads in silicon, fabricated by etching a grid of 350-nm deep, 100-nm wide trenches in the silicon. Figure 3b plots the NV as the nanopatterned sample is scanned laterally over the NV. The is reduced when the NV is directly above the Al blocks and then recovers when above the gaps, clearly resolving the conducting features of our nanopatterned sample.
To expedite imaging we implement an adaptive, single- algorithm which sets the single point to be 0.7 of the previously measured pixel. The point is also measured. This measurement method reduces the per-pixel measurement time to the order of a minute (details in SI 3). At these time scales, however, thermal drifts can still play a significant role and we perform active NV-sample drift correction via image registration. Topographic AFM images and NV PL images can both provide highly repeatable and sharp features for drift correction. In Fig. 3b we use PL-based image registration every two hours to correct for 10 nm NV-sample drifts with 1 nm error (details in SI 2) Guizar-Sicairos2008.
In Fig. 3c we perform a high-resolution, 5-nm point spacing line scan of the dotted orange line in Fig. 3a, demonstrating the nanoscale spatial resolution of our NV conductivity imaging technique. A different NV is used than for the image in Fig. 3b. Topographic-based image registration is performed once per hour. In addition to using an adaptive single- measurement, this measurement also implements a spin-to-charge readout sequence, which further reduces imaging time by a factor of 25 compared to standard spin-state dependent photoluminescence measurements and brings us to 5x the spin projection noise limit (details in SI 4).
We now turn to a discussion of the spatial resolution and sensitivity of NV-based electrical conductivity imaging. From the line scan in Fig. 3c, the metal-induced magnetic fluctuations at two points separated by 5 nm is resolved within the measured error, thus demonstrating 5 nm spatial resolution in the imaging plane. However, this does not necessarily correspond to the smallest resolvable conducting feature in the material. The ultimate resolution will also depend on AFM stability and NV-metal separation. The thermally-induced NV-sample drifts in Fig. 3 are 10 nm due to infrequent drift corrections, whose frequency could be increased to minimize thermal drifts, and ultimately, picometer-scale stability could be achieved with active drift compensation techniques King:2009gq. For the data in Fig. 3, the closest NV-metal separation is 40 nm, which sets a conductivity spatial resolution of 40 nm.
For the mechanically and thermally stable imaging apparatus used here, the NV-metal separation limits both the spatial resolution and sensitivity of conductivity imaging. The smallest achievable separation is set by the NV depth in the diamond, 7 nm in this work. NV centers at few nm depths that exhibit several-ms times have also been measured Rosskopf:2014io. The increased NV-metal separation we observe is likely dominated by an angular misalignment between the faces of the 3 m diameter tip and the 400 nm diameter diamond pillar; a 5 misalignment gives an NV-metal separation of 25 nm for an NV at the center of the pillar. For the Ti data in Fig. 2 we measure nm, demonstrating the feasibility of close contact. With more controlled tilting and shallower NV centers, a 5-nm NV-metal separation should be achievable.
An important advance presented here is the implementation of NV spin-to-charge conversion (SCC) readout techniques Shields:2015ik in imaging with shallow NV centers. In doing so we significantly reduce imaging time by a factor of 25, which is particularly relevant for relaxation imaging, an inherently long measurement due to the ms-scale times and point-by-point scanning. Notably, we find that the SCC readout technique is highly robust for shallow NV centers; all measured photostable NVs exhibit a significant enhancement in sensitivity, with a typical 20-30x reduction in measurement time (details in SI 4). Thus SCC should find ubiquitous utility for shallow NV sensing and imaging. By enhancing sensitivity, SCC techniques also extend the measurable range of conductivities and hence the range of accessible condensed matter phenomena such as phase separation in complex oxides Qazilbash:2007cm, as we show next.
To illustrate the feasibility of resolving spatial conductivity variations in a relevant material system, in Fig. 4 we plot a simulated line scan for an NV center scanned across a material with pockets of metallic phases inside an insulating matrix, as is seen in complex oxides across a metal-to-insulator transition Yee:2015jn; Qazilbash:2007cm. The two nanoscale metallic regions and the 5-nm wide insulating barrier separating them are clearly resolved, both in the theoretical curve (black line) and in the simulated measurement (orange circles). The simulated measurement accounts for the expected measurement error with 1 minute of averaging per point, an adaptive- SCC readout technique, = 5 nm, an intrinsic NV of 6 ms, and . This conductivity value is typical for a Mott insulator Imada1998, and is only half the conductivity of Ti measured in Fig. 2. To simulate the magnetic fluctuations from the conducting pockets we implement a Monte Carlo simulation of the electron trajectories (details in SI 1.4) Bulashenko1992. This simulation demonstrates the feasibility of noninvasive, nanoscale, NV-based imaging of inhomogeneous electrical conductivity in Mott insulators.
The NV center in diamond is emerging as a versatile quantum sensor capable of imaging magnetism Rondin2013; Maletinsky2012, temperature Kucsko2013, thermal conductivity Laraoui2015, and DC currents Tetienne2017; Chang2017. In this paper we add conductivity to the arsenal of NV imaging modalities. Future studies can, for example, combine DC magnetic field and conductivity sensing in one integrated tool to yield unique insight into materials with multiple order parameters, such as NdIrO with the possibility of domain wall conductivity at magnetic domain walls Ma2015, and buried LAO/STO interfaces with coexisting superconductivity and ferromagnetism Bert2011.
Experimental setup and sample preparation. The experimental setup consists of a home-built confocal microscope and atomic force microscope. The experiments are done in ambient conditions with active temperature control to within 1 mK. A continuous wave laser at 532 nm is used for optical pumping and readout of the NV spin, and is gated with an acousto-optic modulator (AOM). For spin-to-charge readout, continuous wave lasers at 594 nm and 637 nm gated with AOMs are also used. Photons emitted by the NV are collected into a single-mode fiber and directed to a fiber-coupled avalanche photodiode. The collection efficiency is amplified by a factor of 5 due to waveguiding from the 400-nm wide, 500-nm tall, truncated-cone diamond pillars. Microwaves used for resonant spin manipulation are delivered via a 300-nm thick waveguide evaporated onto the diamond surface closest to the NVs. Timing of the pulse sequences is controlled by a Spincore Pulseblaster ESR-Pro 500 MHz card.
Nanopatterned metal samples are fabricated in a top-down process starting with Silicon on Insulator wafers with a 10-m thick device layer. Nanopatterns are defined in the device Si layer using electron beam lithography and reactive-ion etching. Subsequently, a standard AFM probe fabrication process flow is carried out. The flat-faced tip is formed by stopping the KOH anisotropic Si etch such that the apex of the Si pillar retains a 2 m diameter with the nanopattern untouched. Cantilevers are released via a backside Si deep-etch and an HF SiO etch. 85 nm of metal is then thermally evaporated onto the tip face.
To perform AFM, these cantilevers are then glued onto tuning forks using a micromanipulator. To minimize angular misalignments between the tip face and the diamond surface, we first use an SEM to measure the relative tilt between the tip face and the tuning fork mount. We then tilt the entire tuning fork assembly with respect to the diamond to make them parallel to within several degrees, which is limited by the SEM resolution of nonconducting silicon. Existing SEM technologies specifically designed for semiconductors could improve this resolution and tilt adjustment. We operate in tapping mode AFM, in which we electrically drive the quartz tuning fork to amplitudes of 1 nm, and we measure the amplitude with a lock-in amplifier for feedback control (Zurich Instruments). XYZ positioning is controlled by piezoelectric scanners.
The diamond is prepared via growth of a 50-nm thick 99.99% C isotopically purified thin film on a commercial Element 6 electronic grade (100) diamond substrate. Prior to growth, the diamond is etched with ArCl plasma (1 m) to mitigate polishing damage and cleaned in boiling acid HNO:HSO. NV centers are formed by N ion implantation with a dosage of ions / cm at 4 keV and a 7 tilt, which yields an expected depth of 7 nm (calculated by Stopping and Range of Ions in Matter (SRIM)). The sample is then annealed in vacuum ( Torr at max temperature) at 850 C for 2.5 hours with a 40-minute temperature ramp. After annealing, the sample is cleaned in HClO:HNO:HSO 1:1:1 for 1 hour at 230-240 C.
Error analysis. Errors in measured are given by the standard error in the exponential fit where the decay constant and amplitude are the only free parameters, as in Fig. 1c. In Fig. 2 a full measurement is done at each point and error bars correspond to the standard error in the fit. In the case of the single- measurement in Fig. 3, is explicitly calculated and the error results from propagating the measured standard error of the photoluminescence. Error in the measured conductivities is given as the standard error in the fit to the data in Fig. 2.
Data availability. All relevant data are available upon request from A.C.B.J.
We thank Shimon Kolkowitz and Nathalie de Leon for helpful discussions. We also thank Matthew Pelliccione for constructing the scanning probe setup. This work was supported by a PECASE award from the Air Force Office of Scientific Research. A.A. acknowledges funding from the Elings Postdoctoral Prize Fellowship from the UCSB California NanoSystems Institute.
A.A. fabricated the samples. A.A. and D.B. performed the experiments and data analysis. B.A.M. developed most of the experiment infrastructure. A.A., D.B., and A.C.B.J wrote the paper. A.C.B.J. supervised the project. All authors contributed to the design of the experiment and discussions during the course of the measurements and analysis.
Competing interests: The authors declare no competing financial interests.
Supplementary Note 1: Theoretical Model of Nv Relaxation
1.1: Magnetic spectral density emanated by a thin conducting film
To quantitatively measure a material’s conductivity using an NV center in diamond, we use theory developed by Henkel1999a; Langsjoen2014 to describe the magnetic Johnson Noise produced by an infinite conducting slab of finite thickness . The experiments presented here probe polycrystalline metals at room temperature, and therefore we do not consider non-local effects such as those observed in Kolkowitz2015. We then develop an analytic solution to the magnetic spectral density from a conductor with finite thickness . For consistency with referenced works, in this supplement we define the metal thickness as and the distance of the NV to the conductor surface as , instead of and , as used in the main text, respectively.
First, consider the commonly presented solution as in Henkel1999a; Kolkowitz2015 for the magnetic spectral density emanated by a metallic half-space
Where is the vacuum permeability, the Boltzmann constant, the temperature, and the metal conductivity. A half-space a distance away is equivalent to the summation of a slab of thickness a distance away and a half-space a distance away.
This elegant solution neglects the effect of the boundary at . To confirm its validity, we rigorously derive the same result using Fresnel coefficients for a finite-thickness film.
Derived using the fluctuation-dissipation theorem and a magnetic Green’s tensor, the magnetic spectral density tensor at angular frequency and temperature can be expressed as
Where has units of magnetic spectral density (Tesla / Hz), and is a dimensionless tensor. For simplicity we consider a coordinate system with the -axis perpendicular to the metal, and being the distance to the nearest face of the metal film, such that becomes a diagonal tensor with elements
Where we use and Fresnel coefficients and . For simplicity we drop the double index on the diagonal elements such that As shown in Langsjoen2014, for a metal of finite thickness these Fresnel coefficients are
Where are the relative dielectric functions of the two media, which in this case are and . In the frequency regime of the NV level splitting 2.8 GHz, and for metal conductivity
Thus, the above expressions relate the magnetic noise spectral density at the NV produced by a metal with conductivity .
As described in Henkel1999a the integrals are dominated by values of for our experiment where nm and GHz. For the metallic conductivities studied here, - , and thus . This is equivalent to stating that the skin depth , where . We now make the critical approximation that and set for vacuum
Multiplying the numerator and denominator by the complex conjugate of the denominator
Applying to the integral in Eq. S4 yields
is a real, positive number always greater than 2, thus . Taking the real part,
Noting that , and substituting in
Using the same approximations as above one finds , and then that . Further, at room temperature and 2.8 GHz, . Calculation of the dimensionless tensor then yields the magnetic spectral density tensor elements
As expected, Eq. S14 simplifies to the half-space solution as one takes film thickness , and the calculation performed here matches our geometric derivation in Eq. S2. Importantly, the Johnson noise is white for the relevant GHz frequencies.
We now complete this analysis by relating the magnetic spectral density to the ensuing relaxation rate from to for a spin-1 system like the NV.
1.2: Population dynamics of the NV three-level system
As described in Myers2017, the relaxation rate of to is approximately the same as the relaxation rate of to . Denoting this rate as , and denoting the double-quantum transition rate between and as , the population dynamics are thus described by
Solving the eigenvalue-eigenvector problem, we see that
As described in the main text, we optically polarize the NV into . Ideally and , but we account for imperfect optical polarization by defining such that and , where is nominally 0.05 Shields:2015ik. With these initial conditions, we see that
In a spin-dependent photoluminescence (SDPL) measurement sequence, such as shown in Fig. S3a, optical polarization is followed by evolution for a dark time , and then the NV photoluminescence (PL) signal is measured, where is given by
and describe the brightness of the and states, respectively. The and states are taken to be equally bright. Background designates any non-NV signal that may come from e.g. APD dark counts, light leakage, and any population in the neutral NV due to imperfect charge state polarization. We then repeat this experiment, but immediately before readout a resonant microwave pulse swaps the population of the and states. This second measurements yields the signal :
We then take the difference between the two measurements and to calculate the quantity
where we include the imperfect spin polarization in the definition of contrast . In Fig. 1c of the main text we plot a normalized version of , where the contrast is normalized to 1 by means of reference measurements of . We fit to an exponential decay with decay rate
Double-quantum relaxation, imperfect spin polarization, and any background signal do not modify the measured . However, AOM leakage during long, ms-scale dark times, can repolarize the NV at a rate , which changes the form of the decay curve to . is an inevitable offset since the dark, steady-state population with a slow polarization rate will not be an even thermal mixture of the 3 spin states. We find that several nW of laser leakage can repolarize the NV at a rate of 50 Hz. Thus we take care to reduce laser leakage to below 1 nW and confirm that as .
1.3: NV relaxation rate near a conductor
We now calculate via perturbation theory applied to the NV Hamiltonian, with NV coordinates . The NV’s magnetic moment axis, the -axis, makes an angle with the -axis. We can further select this axis to lay entirely in the - plane without any loss of generality, since the and components of the magnetic spectral density tensor are equivalent.
Hence, and . Note that the components of the magnetic field are all uncorrelated, and thus and . We use the formalism described in Henkel1999a to describe the relaxation rate under the interaction Hamiltonian , as described in the main text with NV gyromagnetic ratio and raising and lowering operators and
The diamonds used in this work are oriented, and thus all four possible NV orientations make an angle with the -axis, making our analysis independent of the NV orientation.
We now finish our derivation of relaxation rate by using our derived expression for the magnetic spectral density tensor.
Note that in the main text we define and .
1.4: Calculation of the magnetic spectral density emanated by a finite-geometry conductor
The magnetic fluctuations emanating from a conductor can be derived in an alternate method to that presented in section 1.1. This method is the one presented in the main text and it offers much physical intuition. However, we warn that the results for the and components of the magnetic spectral density are overestimated by a factor of 3 because we do not account for boundary conditions at the surface. Nevertheless, this method has further utility as it allows us to estimate the magnetic noise from finite geometries Henkel2005.
In the Drude model, the component () of an electron’s velocity will be correlated in time by , where is the mean electron collision time. With the Wiener-Khinchin theorem, taking a Fourier transform of this velocity autocorrelation function yields a two-sided velocity spectral density
The velocity noise produced by an electron with mass and mean thermal energy is thus
The Biot-Savart law gives the magnetic field from a moving electron as , where is the electronic charge and is the distance to the electron. From the autocorrelation function we calculate . Note that using the Wiener-Khinchin theorem to find the velocity spectral density is equivalent to the argument in the main text where we start at the Johnson-Nyquist expression for current spectral density.
We assume all of the electrons in the metal are uncorrelated, and thus their magnetic field spectral densities add incoherently. Summing up the mean electron contributions from volume elements with electron density , we then calculate from an infinite film of thickness a distance away,
where is the electrical conductivity with for GHz. This calculation method yields the correct expression for , which was rigorously derived in section 1.1 using the fluctuation-dissipation theorem. However, because we did not account for boundary conditions at the surface and would be overstated by a factor of 3 with this method of calculation Henkel2005.
In Fig. 2 of the main text we study the noise from plateau tips of m radius, which we approximate to be infinite films. We can conveniently estimate the deviation from the infinite-film approximation by performing the radial integral in Eq. S31 to 1.5 m, instead of . Similarly, we calculate integral expressions for and and integrate to 1.5 m. We then compare to . Although the simplistic Biot-Savart calculation ignores surfaces and thus is slightly skewed in magnitude, by considering the volume integral we closely estimate the relative deviation from the infinite-film model. We find that this relative deviation is approximately 10% of the experimental error for for Fig. 2, which we deem to be negligible for this study.
This Biot-Savart method of calculation also allows us to form the basis for the simulation in Fig. 4 of the main text. We employ a Monte Carlo simulation of a single electron in a metal, which has previously been developed to estimate the velocity autocorrelation function and spectral density inside finite geometries Bulashenko1992. We simulate an electron with a given velocity and a certain probability to scatter dictated by the bulk, mean collision time . The electron also scatters at boundaries, however, and this effect modifies the velocity spectral density: near the boundary the effective collision time will decrease, which stretches the velocity spectral density (Eq. S28) and suppresses the magnitude of the noise at GHz.
For the simulation in Fig. 4 of the main text, we perform a Monte Carlo simulation of the electron trajectory, but instead of calculating the velocity noise we use the Biot-Savart law to explicitly calculate the magnetic autocorrelation spectral density for every point in free space. This simulation allows us to account for the finite size of the conductor as well as the approximate noise suppression at boundaries. Since the noise from two conductors will be uncorrelated, we can sum the spectral densities from the two conducting blocks in Fig. 4. This can be done for any geometry. In order to ensure that the magnitude of the magnetic spectral density is properly estimated, we employ the same logic as for the estimation described above: we compare the simulated to to estimate the relative deviation from an infinite film and we then apply Eq. S14 to obtain the absolute magnitude.
Supplementary Note 2: Temperature Control and Spm Stability
Temperature fluctuations play an important role in our system’s stability; a 1 mK change results in 1 nm of drift of the SPM cantilever relative to the NV, as a consequence of using materials with different coefficients of thermal expansion. To mitigate thermal drifts, we implement several layers of thermal isolation and active temperature feedback. Our experimental apparatus is enclosed in an insulating box on an optical table, with the entire optical table isolated by curtains. The laboratory temperature is stabilized to within 1K. A PID loop measures voltage across a thermistor with a sensitivity 0.1 mK / Hz and then heats thin resistance wire in feedback control. Thin resistance wire is optimal, with a small thermal mass and large surface area relative to its volume, yielding responsive feedback. This also allows us to distribute the heating sources evenly around the insulating box, which is critical in minimizing temperature gradients.
We achieve temperature stability to within 1 mK on the timescale of weeks. However, ambient changes in temperature outside the box can cause the temperature control system to introduce changing temperature gradients inside the box, which are the present limiting factor for drift.
In order to correct for these drifts, we employ image registration every 1-2 hours, using either a topographic AFM image or an NV PL image (the one with the sharper features is chosen for image registration). Two example PL images taken an hour apart in time are shown in Fig. S1. We use an image registration algorithm that allows for subpixel correction to 1 nm precision Guizar-Sicairos2008; Pelliccione2014.
Supplementary Note 3: Optimal Measurement
For the exponentially decaying signal , the most time-efficient method of measuring is by acquiring data at and , which we show here.
Consider a weighted, least-squares linear fit for data with relationship with weights , where is the standard deviation of the measured values. One can calculate, then, that the variance on the fitted value of B is
If we linearize our signal , referred to now as for brevity, we find that
We consider , where is the number of repetitions and is the per-shot measurement error, which we assert is the same for each . Consider a per-shot overhead time and dark times such that , with the total measurement time. Thus, one finds that
In Fig. S2 we plot the relative speed for linearly spaced values between to , as a function of and . The plot shows a clear maximum in measurement speed at . Fig. S2 thus emphasizes the importance of an adaptive- algorithm when imaging over an area where varies. For example, in the area imaged in Fig. 3 of the main text varies from 0.5 to 5 ms.
Supplementary Note 4: Spin-to-Charge Readout on a Shallow Nv Center
For Fig. 3c of the main text we implement spin-to-charge conversion (SCC) readout Shields:2015ik on a shallow NV center with a measurement sensitivity 5x the spin projection noise (SPN) limit. Fig. S3b and Fig. S3c illustrate the SCC measurement scheme and show an example SCC measurement that is 5x the SPN limit.
The SPN limit is the result of a Bernoulli distribution: each projective measurement will result in a success or a failure . We make a measurement at , for which the spin population is roughly split in half between and . If the population is split in half between the two outcomes, this results in a standard deviation of 1/2. Since each measurement of is the difference of two measurements ( and ), whose errors add in quadrature, the SPN-limited standard error of measurements of is
More carefully accounting for the 3-level system yields a 5% smaller SPN limit.
In Fig. 3c of the main text and in Fig. S3 we experimentally measure . For spin-dependent photoluminescence (SDPL) readout, we typically measure , which is consistent with photon shot noise for the experimental parameters of a 400-ns long readout, -state PL of 180 kCounts/s, and PL contrast of 30% between and states. Further, note that the SPN value quoted above assumes perfect initial polarization into NV and ; in practice imperfect spin and charge polarization increases our experimental error by a factor of . Thus, in reality our measured readout error .