Nanoscale electrical conductivity imaging using a nitrogen-vacancy center in diamond

The electrical conductivity of a material can feature subtle, non-trivial, 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 non-invasive 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 not accessible to existing technologies, and as a model example, we project readily achievable imaging of nanoscale phase separation in complex oxides.


SUPPLEMENTARY NOTE 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 [1,2] to describe the magnetic Johnson noise produced by an infinite conducting slab of finite thickness a. The experiments presented here probe polycrystalline metals at room temperature, and therefore we do not consider nonlocal e↵ects such as those observed in [3]. We then develop an analytic solution to the magnetic spectral density from a conductor with finite thickness a. For consistency with referenced works, in this supplement we define the metal thickness as a and the distance of the NV to the conductor surface as z, instead of t film and d, as used in the main text, respectively.
First, consider the commonly presented solution as in [1,3] for the magnetic spectral density emanated by a metallic half-space Where µ 0 is the vacuum permeability, k B the Boltzmann constant, T the temperature, and the metal conductivity. A half-space a distance z away is equivalent to the summation of a slab of thickness a a distance z away and a half-space a distance z + a away.
This elegant solution neglects the e↵ect of the boundary at z + a. To confirm its validity, we rigorously derive the same result using Fresnel coe cients 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 T can be expressed as Where~! 3 /4⇡✏ 0 c 5 has units of magnetic spectral density (Tesla 2 Hz 1 ), and s ij is a dimensionless tensor. For simplicity we consider a coordinate system x 0 , y 0 , z 0 with the z 0 -axis perpendicular to the metal, and z being the distance to the nearest face of the metal film, such that s ij becomes a diagonal tensor with elements Where we use k = |!| /c and Fresnel coe cients r p and r s . For simplicity we drop the double index on the diagonal elements such that s z 0 ⌘ s z 0 z 0 As shown in [2], for a metal of finite thickness a these Fresnel coe cients are As expected, Supplementary Eq. 14 simplifies to the half-space solution as one takes film thickness a ! 1, and the calculation performed here matches our geometric derivation in Supplementary Eq. 2. 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 |m s = 0i to |m s = 1i for a spin-1 system like the NV.

SUPPLEMENTARY NOTE 2: POPULATION DYNAMICS OF THE NV THREE-LEVEL SYSTEM
As described in [4], the relaxation rate of |m s = 0i to |m s = 1i is approximately the same as the relaxation rate of |m s = 0i to |m s = 1i. Denoting this rate as ⌦, and denoting the double-quantum transition rate between |m s = 1i and |m s = 1i 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 |m s = 0i. Ideally ⇢ 0 (0) = 1 and ⇢ 1 (0) = ⇢ 1 (0) = 0, but we account for imperfect optical polarization by defining ⌘ such that ⇢ 0 (0) = 1 2⌘ and ⇢ 1 (0) = ⇢ 1 (0) = ⌘, where ⌘ is nominally 0.05 [5]. With these initial conditions, we see that In a spin-dependent photoluminescence (SDPL) measurement sequence, as shown in Supplementary Figure 3a, optical polarization is followed by evolution for a dark time ⌧ , and then the NV photoluminescence (PL) signal S is measured, where S is given by A and B describe the brightness of the |0i and |±1i states, respectively. The | 1i and |1i 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 0 due to imperfect charge state polarization. We then repeat this experiment, but immediately before readout a resonant microwave pulse swaps the population of the | 1i and |0i states. This second measurements yields the signal S swap : We then take the di↵erence between the two measurements S and S swap to calculate the quantity S where we include the imperfect spin polarization in the definition of contrast C. In Fig. 1c of the main text we plot a normalized version of S di↵ , where the contrast is normalized to 1 by means of reference measurements of S di↵ (0). 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 S di↵ = C exp ( (3⌦ + ) t) + y 0 . y 0 is an inevitable o↵set 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 S di↵ ! 0 as t ! 1.

SUPPLEMENTARY NOTE 3: NV RELAXATION RATE NEAR A CONDUCTOR
We now calculate ⌦ metal via perturbation theory applied to the NV Hamiltonian, with NV coordinates x, y, z. The NV's magnetic moment axis, the z-axis, makes an angle ✓ with the z 0 -axis. We can further select this axis to lay entirely in the x 0 -z 0 plane without any loss of generality, since the x 0 and y 0 components of the magnetic spectral density tensor are equivalent.
Note that the x 0 , y 0 , z 0 components of the magnetic field are all uncorrelated, and thus S x B = cos 2 (✓)S x 0 B + sin 2 (✓)S z 0 B and S y B = S y 0 B . We use the formalism described in [1] to describe the relaxation rate under the interaction Hamiltonian H 0 , as described in the main text with NV gyromagnetic ratio and raising and lowering operators S + and S The diamonds used in this work are (100) oriented, and thus all four possible NV orientations make an angle ✓ = arccos( p 1/3) ⇡ 54.7 with the z 0 -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 d ⌘ z and t film ⌘ a.

SUPPLEMENTARY NOTE 4: 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 Supplementary Note 1.This method is the one presented in the main text and it o↵ers much physical intuition. However, we warn that the results for the x 0 and y 0 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 [6].
In the Drude model, the i th component (i = x 0 , y 0 , z 0 ) of an electron's velocity v i will be correlated in time by where ⌧ c 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 m and mean thermal energy 3k B T /2 is thus The Biot-Savart law gives the magnetic field from a moving electron as , where e is the electronic charge and r 0 is the distance to the electron. From the autocorrelation function hB 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 n, we then calculate S z 0 B from an infinite film of thickness a a distance z away, where is the electrical conductivity with ⌘ Re [ (!)] ⇡ (0) for ! ⇠ 2⇡ ⇤ 2.87 GHz. This calculation method yields the correct expression for S z 0 B , which was rigorously derived in Supplementary Note 1 using the fluctuationdissipation theorem. However, because we did not account for boundary conditions at the surface S x 0 B and S y 0 B would be overstated by a factor of 3 with this method of calculation [6].
In Fig. 2 of the main text we study the noise from plateau tips of 1.5 µm 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 Supplementary Eq. 31 to 1.5 µm, instead of 1. Similarly, we calculate integral expressions for S x 0 B and S y 0 B and integrate to 1.5 µm. We then compare Biot,1 to Biot,1.5µm . 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 metal 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 [7]. We simulate an electron with a given velocity and a certain probability to scatter dictated by the bulk, mean collision time ⌧ c . The electron also scatters at boundaries, however, and this e↵ect modifies the velocity spectral density: near the boundary the e↵ective collision time ⌧ c will decrease, which stretches the velocity spectral density (Supplementary Eq. 28) and suppresses the magnitude of the noise at ! ⇠ 2⇡ ⇤ 2.87 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 sim, finite to sim,1 to estimate the relative deviation from an infinite film and we then apply Supplementary Eq. 14 to obtain the absolute magnitude.

SUPPLEMENTARY NOTE 5: 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 di↵erent coe cients 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 of 0.1 mK Hz 1/2 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 Supplementary Figure 1. We use an image registration algorithm that allows for subpixel correction to ⇠ 1 nm precision [8,9].

SUPPLEMENTARY NOTE 6: OPTIMAL T1 MEASUREMENT
For the exponentially decaying signal S di↵ = C exp( t/T 1 ), the most time-e cient method of measuring T 1 is by acquiring data at ⌧ = 0 and ⌧ ⇡ 0.7 T 1 , which we show here. Consider a weighted, least-squares linear fit for data with relationship y i = A + Bx i with weights w i = 1/ 2 yi , where yi is the standard deviation of the measured y i values. One can calculate, then, that the variance on the fitted value of B is If we linearize our signal S di↵ , referred to now as S for brevity, we find that The optimal measurement sequence is a measurement at ⌧ = 0 and at ⌧ = 0.65 T 1 . We assume a per-shot overhead time of 0.01 T 1 . Deviating from ⌧ ⇡ 0.7 T 1 greatly increases measurement time, stressing the necessity of adaptive-⌧ measurements for imaging.
We consider 2 S = 2 1 /N , where N is the number of repetitions and 2 1 is the per-shot measurement error, which we assert is the same for each ⌧ i . Consider a per-shot overhead time t extra and dark times ⌧ i such that N P (⌧ i +t extra ) = T , with T the total measurement time. Thus, one finds that In Supplementary Figure 2 we plot the relative speed (1/ 2 ) for N linearly spaced ⌧ values between ⌧ = 0 to ⌧ max , as a function of N and ⌧ max . The plot shows a clear maximum in measurement speed at ⌧ ⇡ 0.7 T 1 . Supplementary  Figure 2 thus emphasizes the importance of an adaptive-⌧ algorithm when imaging over an area where T 1 varies. For example, in the area imaged in Fig. 3 of the main text T 1 varies from 0.5 to 5 ms.

SUPPLEMENTARY NOTE 7: SPIN-TO-CHARGE READOUT ON A SHALLOW NV CENTER
For Fig. 3c of the main text we implement spin-to-charge conversion (SCC) readout [5] on a shallow NV center with a measurement sensitivity 5x the spin projection noise (SPN) limit. The principle behind SCC readout is selectively ionizing the |m s = 0i spin state into the neutral charge state (NV 0 ) while leaving the |m s = ±1i spin state in the negative charge state (NV ). The charge state is then readout with a yellow (594 nm) laser that selectively excites only the NV charge state. Supplementary Figure 3b and Supplementary Figure 3c illustrate the SCC measurement scheme and show an example SCC T 1 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 |m s = 0i or a failure |m s = ±1i. We make a measurement at ⌧ ⇡ 0.7 T 1 , for which the spin population is roughly split in half between |m s = 0i and |m s = ±1i. If the population is split in half between the two outcomes, this results in a standard deviation of 1/2. Since each measurement of S di↵ is the di↵erence of two measurements (S and S swap ), whose errors add in quadrature, the SPN-limited standard error The ⇡ pulse swaps the |0i and | 1i populations immediately before readout, as explained in Supplementary Eq. 21. b Same preparation sequence as a but readout uses SCC. To convert spin to charge, we do a ⇠ 300-µW, 60-ns green pulse to shelve |±1i into the singlet, immediately followed by a ⇠ 30-mW, 40-ns red pulse in order to discriminately ionize |0i without touching the singlet population. We then perform charge-state readout with a ⇠ 3-µW, 500-µs yellow pulse which discriminately excites NV . c SCC T 1 measurement on a shallow NV center. The measurement took 30 minutes. Plotted error bars represent the measured standard error, and are 5x larger than the spin projection noise (SPN) limit. The inset shows the relative magnitude of typical SPN, SCC, and SDPL error (1,5,25). Nominally, to achieve the same error the SCC measurement is 25x faster than SDPL.
More carefully accounting for the 3-level system yields a 5% smaller SPN limit. At saturation laser powers of ⇠ 1 mW at 532 nm, we collect ⇠ 500 kCounts s 1 from a single NV. In this work, we typically operate at ⇠ 300 µW with ⇠ 180 kCounts s 1 . Under 594 nm laser excitation (used for spin-tocharge readout), we typically measure ⇠ 5 kCounts s 1 for 3 µW incident power. In Fig. 3c of the main text and in Supplementary Figure 3 we experimentally measure SCC ⇡ 5/ p 2N = 5 SPN . For spin-dependent photoluminescence (SDPL) readout, we typically measure SDPL = 25 SPN = 5 SCC , which is consistent with photon shot noise for the experimental parameters of a 400-ns long readout, |0i-state PL of 180 kCounts s 1 , and PL contrast of 30% between |0i and | ± 1i states. Further, note that the SPN value quoted above assumes perfect initial polarization into NV and |m s = 0i; in practice imperfect spin and charge polarization increases our experimental error by a factor of ⇠ 1.  Fig. 3b-c of the main text, with T 1 converted to conductivity. a Scanning electron microscopy image of an Al nanopattern deposited onto an AFM tip, as depicted in Fig. 1a of the main text. b NV-based conductivity image of the area depicted by the 1 µm 2 blue square in a, produced by scanning the NV center over this area at a height (d 0 ) of 40 nm and with 20 nm pixel spacing. The height of 40 nm is extracted from a simulation fit with the model described in Supplementary Note 4.The conductivity is then calculated by using Eq. 3 of the main text with t film = 85 nm. c High-resolution conductivity line scan of the dotted orange line in a, using spin-to-charge readout. More data points at the left edge of the scan are included in this plot than in Fig. 3c of the main text to show the conductivity decay to 0 (not included in the main text to keep the T 1 range below 3.5 ms to focus on the conducting regions). A simulation fit here also extracts a height of 40 nm and the conductivity is calculated in the same way as in b. The intrinsic T 1 is 6 ms for both NVs. Scale bars are 400 nm. Error-weighted, light smoothing is applied to the data in b and c, for which nearest neighbors receive an additional weight reduction by a factor of 2.5. Error bars are calculated by propagating the measured standard error of the photoluminescence for the single-⌧ measurement of T 1 , and then propagating the error as prescribed by Eq. 3 of the main text. The extracted height of 40 nm from the simulation fit is approximate to ± 5 nm, which would then lead to a systematic o↵set of the plotted conductivity to ± 10%. The dark regions at the top and bottom of the AFM scan correspond to the edges of the nanopatterned sample. Neglecting these regions yields a root-mean-square variation of 5.1 nm over the entire region. For an average NV-sample separation of 40 nm these 5 nm variations correspond to 10% variations in the measured conductivity (an uncertainty of about 10 4 ⌦ 1 cm 1 at maximum points of conductivity), which is the approximate magnitude of the measured error in Supplementary Figure 4. Positive values of height correspond to retracting the nanopattern away from the diamond. Scale bar is 400 nm.