Density and $T_1$ of surface and bulk spins in diamond in high magnetic field gradients

We report on surface and bulk spin density measurements of diamond, using ultra-sensitive magnetic force microscopy with magnetic field gradients up to 0.5 T/$\mu$m. At temperatures between 25 and 800 mK, we measure the shifts in the resonance frequency and quality factor of a cantilever with a micromagnet attached to it. A recently developed theoretical analysis allows us to extract a surface spin density of 0.072 spins/nm$^2$ and a bulk spin density of 0.4 ppm from this data. In addition, we find an increase of the $T_1$ time of the surface spins in high magnetic field gradients due to the suppression of spin diffusion. Our technique is applicable to a variety of samples other than diamond, and could be of interest for several research fields where surface, interface or impurity bulk spin densities are an important factor.


I. INTRODUCTION
Noise coming from paramagnetic impurities is a widespread phenomenon that is relevant to fields ranging from magnetometry to solid-state qubits [1,2]. An example is negative nitrogen-vacancy (N-V) centers in diamond (from now on referred to as "N-V centers"), which have become one of the workhorses in quantum technology. Interaction with paramagnetic impurities is considered one of the main factors that induce decoherence of the N-V center [3]. This decoherence is faster for shallow N-V centers close to the surface and slower for N-V centers in the bulk of the diamond sample, because shallow N-V centers are under the influence of a layer of electron spins at the surface of the diamond [1,4]. Understanding and potentially eliminating this source of decoherence has been a longstanding goal of the field [5]. Here we present a method to measure the impurity spin density, where the sensor is decoupled from the diamond sample. We use an ultrasoft cantilever with an attached micromagnet that couples to the spins via dipole-dipole interaction. The method is easily transferable to a wide range of samples [6].
Multiple experiments have been conducted to measure the diamond surface impurity spin density and to characterize the properties of this two-dimensional electron spin bath, such as correlation times of the fluctuating spins [4,[7][8][9][10]. The measured spin densities differ and range * oosterkamp@physics.leidenuniv.nl † Both authors contributed equally to this work. from 0.01 to 0.5 μ B /nm 2 . Most of these experiments were done at room temperature, except for one measurement at 10 K [8]. All mentioned studies used N-V centers to probe the surface electron spin bath. The technological challenge of measuring surface or bulk spin densities on samples other than diamond can be met by use of a scanning N-Vcenter approach [7]. Unfortunately, the detection range of a scanning N-V-center approach is limited to a few nanometers. Our method is capable of sensing spins at micrometer distances. We do our experiments at millikelvin temperatures, where no surface spin-density measurements on diamond have been performed yet. The low temperature in combination with a high magnetic field gradient allows us to measure with an extremely low force noise [11]. In addition, it allows us to interact with electron spins that can easily be polarized by small magnetic fields and to disregard all physical processes involving phonons. This makes our method suitable for measuring spin densities in very dilute spin systems. In particular, it is of interest for the fields of quantum computation devices [12,13], magnetometry [14], and magnetic resonance force microscopy (MRFM) [15], as surface and bulk impurity spins play an important role there.
Our group has previously demonstrated surface spindensity measurements of dangling bonds on a silicon oxide surface [6]. Here we present spin-density measurements of paramagnetic impurities on a diamond surface and also expand our method to probe impurity spins in the bulk of the sample. We show that strong magnetic field gradients influence the T 1 relaxation time of the impurity spins and that this effect is an important ingredient to understand the system.

A. Experimental setup
In our experiments we use a commercially available diamond sample with a size of 2.6 × 2.6 × 0.3 mm 3 and that is specified to have less than 1 ppm of nitrogen impurities [16]. One surface is polished twice to a surface roughness R a < 5 nm [17]. We clean the diamond subsequently in acetone, 2-propanol, fuming nitric acid, hydrofluoric acid, and water to start the fabrication process with a clean surface and without oxides. On the surface we fabricate a niobium titanium nitride pickup loop and rf wire, the latter of which is not used in the present experiment [18]. After fabrication, the sample is exposed to air for several months. Before mounting of the sample, it is ultrasonically cleaned in acetone, and thereafter in 2-propanol to remove organics and dust.
The measurements are performed with a MRFM setup comparable to the one used in earlier experiments [6]. To establish the magnetic interaction, we use a spherical (Nd,Fe)B particle (from now on simply referred to as the "magnet") with a diameter of 2.99 μm. This magnet is glued with platinum by electron-beam-induced deposition to the end of an ultrasoft cantilever with a length of 166 μm, width of 5 μm, and thickness of 100 nm [19]. This geometry leads to an intrinsic spring constant k 0 = 5.0 × 10 −5 N/m, with a corresponding natural frequency of 2850 Hz (see Supplemental Material Fig. S1 for the properties of the cantilever measured versus temperature at 5-μm height, far away from the sample [20]). After the magnet is attached, it is placed in an external field of 5 T, leading to a magnetic moment m of 1.5 × 10 −11 A m 2 pointing along the direction of movement of the cantilever [ Fig. 1(a)]. The magnetic particle is responsible for the B field, which polarizes the spins in the sample and also creates large magnetic field gradients of up to 0.5 T/μm. The magnetized cantilever is mounted above the sample and can be moved with respect to the sample with use of a modified piezoKnob-based cryogenic positioning stage [21]. The absolute tip position is measured with three capacitive sensors, while the precise distance between the surface of the magnet and the surface of the diamond is calibrated by gently lowering the magnet until the two touch, using the piezoKnobs.
The motion of the cantilever is measured with a superconducting-quantum-interference-device-(SQUID-) based read-out [22], where we detect the changing magnetic flux in the pickup loop [yellow in Fig. 1 the moving magnet. We can determine the linear response of the cantilever by driving a small piezoelectric element at the base of the cantilever. When we sweep the drive frequency and measure the cantilever response using a lock-in amplifier, we obtain the resonance frequency and quality factor by fitting the square of the SQUID output with a Lorentzian, as seen in Fig. 2.
The full experimental setup is mounted at the mixing chamber of a cryogen-free dilution refrigerator with vibration isolation [23], and with a base temperature of 10 mK. The gold-plated copper sample holder is thermally connected to the mixing chamber with a silver strip. A heater and calibrated low-temperature thermometer are used to control the temperature of the sample holder. Because of the limited thermal conductance between the mixing chamber, the sample holder, and the diamond sample itself, the sample temperature typically saturates at approximately 25-30 mK.

B. Spin-bath-cantilever coupling
When the tip of the cantilever is positioned close to the sample, it couples to the electron spins via the magnetic field that originates from the magnet. The motion of the cantilever changes the direction and strength of this field. The electron spins will follow the varying magnetic field, but with a lag due to their relaxation times T 1 and T 2 . This effect will, in return, change the motion of the cantilever. De Voogd et al. [24] investigated the complete spincantilever system while taking into account the intrinsic damping of the cantilever and the spin-lattice relaxation. They derived an expression for the change of the resonance frequency f = f res − f 0 , where f res is the resonance frequency taking into account the interaction with the spins, and f 0 is the resonance frequency of the cantilever. Only spins with T 1 1/ω 0 , with ω 0 = 2π f 0 , contribute to the frequency shift. A similar expression is found for the inverse quality factor (1/Q) = 1/Q − 1/Q 0 , with Q 0 the cantilever's bare quality factor. The spin-mediated dissipation 1/Q is significant only when T 1 ≈ 1/ω 0 . We now give the expressions for f and (1/Q) for our specific sample.
In our sample, we expect two main sources for the signal: spins in the bulk and spins on the surface of the diamond. First, we expect a contribution from the free electron spins associated with the nitrogen impurities in the bulk of diamond (P1 and P2 centers). The T 1 relaxation time of the dilute electron spins in the bulk of diamond has been reported to increase to several seconds at low temperatures [25]. Since this means that the spin-lattice relaxation is negligible on timescales comparable to the cantilever period, there is no dissipation path for the cantilever through the spin to the lattice at the cantilever's resonance frequency. Hence, we expect the bulk spininduced shift of the quality factor to be zero. Taking these assumptions into account, we find the contributions of the spins in the bulk to be where ρ is the bulk spin density, μ B is the Bohr magneton, and T is the temperature of the spin bath. The integral is over the entire volume of the sample. Since our sample is exposed to air before the experiment, we expect a second contribution from a layer of surface spins that can be expected on any surface that has been exposed to air for extended times [26]. As these are dilute paramagnetic spins, we expect these spins to have T 1 times in the millisecond range. In the case that T 1 is similar to 1/ω 0 , these spins create an additional dissipation path for the cantilever. Therefore, these spins should cause additional shifts given by and (5) where σ is the surface spin density. We have placed the term containing ω 0 T 1 inside the integral to reflect the fact that T 1 may depend on the magnetic field gradient. The integral is over the entire surface of the sample. 064045-3 de WIT, WELKER, de VOOGD, and OOSTERKAMP PHYS. REV. APPLIED 10, 064045 (2018) To calculate the expected frequency shift and additional dissipation, accurate values are needed for the magnetic moment and the shape and size of the magnetic field. In our experiment, since the magnetic particle is almost perfectly spherical, we can calculate the field as if it originates from a magnetic dipole. In the coordinate-free form, this is given by [27] where m is the magnetic moment of the magnet. From this field, we can calculate all relevant derivatives as required for Eq. (3).

C. Spin diffusion in high magnetic field gradients
The theory presented so far describes the spin-cantilever interaction for a constant T 1 of the spins. For most applications (e.g., in bulk techniques with homogeneous external fields), this is a good approximation. However, this approximation does not hold when dilute spins are placed in large magnetic field gradients, as is the case in our experiment. These gradients can increase the relaxation times by suppressing spin diffusion, a concept first derived by Bloembergen [28]. Spin diffusion in diamond was studied by Cardellino et al. [29].
In this model, it is assumed that different spins can have different relaxation times depending on their local environment. This results in the presence of fast-relaxing spins that can rapidly thermalize to the lattice, and slowrelaxing spins that are badly coupled to the lattice. After a perturbation of the thermal equilibrium, relaxation of the polarization of this sample to equilibrium occurs via spin diffusion, which couples the slower-relaxing spins to the faster-relaxing spins through flip-flop interaction, reducing the overall relaxation time of the sample.
However, spin diffusion can be suppressed by application of a large magnetic field gradient, which reduces the probability of two spins exchanging energy by introducing a difference in field felt by neighboring spins. An ansatz for the suppression of the spin diffusion can be obtained by calculation of the normalized overlap interal between the line shapes of two spins [30]: where G is the gradient of the magnetic field strength at the position of the spins,ā 0.5r is the approximate average spacing between spins in the radial direction, withr the nearest-neighbor distance under the assumption of a cubic lattice, and F(B) is the resonance line shape of the spins. We look at the spacing in the radial direction, since this direction has the largest magnetic field strength gradients, and therefore the highest suppression of spin diffusion.
Since we are considering a layer of spins on the surface of the diamond, the total number of spins is too small to measure the actual spectra of the surface spins by bulk techniques such as ESR, so we assume these spins have a Lorentzian profile with a dipolar linewidth given by [ [31], p. 128] where γ e /2π = 28.0 GHz/T is the electron gyromagnetic ratio. Because the convolution of two Lorentzian profiles with width is itself a Lorentzian with width 2 , we can evaluate Eq. (7), from which we find that where G * = B dd /ā is a measure for the gradient when the quenching becomes significant, from now on called the "critical gradient." In short, it is the gradient for which the difference in the field at neighboring spins becomes larger than the spin linewidth. (āG) can be seen as a flip-flop suppression factor. When (āG) is 0, flip-flops are fully suppressed, and when (āG) approaches 1, spins can exchange energy and flip-flops are possible. Thus, we find that the relaxation time is given by where T ff 1 is the reduced T 1 time due to flip-flops between neighboring spins and T * 1 is the intrinsic relaxation time of the system when the flip-flops are completely quenched. This is only a heuristic description of the effects of spin diffusion, and does not take into account the direction of the gradient or the effects of the spin-bath polarization on the flip-flop rate.

III. RESULTS AND DISCUSSION
For our experiments, we change the height (distance between the two surfaces of the diamond and the magnet) and vary the temperature from 25 to 800 mK. At every height-temperature combination, the resonance frequency and quality factor are measured as described in Sec. II A.

A. Frequency shift and dissipation
The results of the measurements of the frequency shift are shown in Fig. 3. The solid lines indicate the results of the fits according to Eqs. (1) and (4), with the total frequency shift given by f = f bulk + f surf . We calculate f 0 at each height by extrapolating the measured frequency-shift data to higher temperatures. The two-dimensional and three-dimensional integrals over C(r) are calculated with the magnetic field distribution defined by Eq. (6). As mentioned before, both integrals are calculated over the entire surface and volume, respectively, but they converge within several micrometers from the magnet due to the strong distance dependence of C(r). The only free parameters remaining in the model are the two spin densities ρ and σ for the bulk and the surface, respectively, and the T 1 time of the surface spins, which for now is fixed at 0.5 ms. As the term (ω 0 T 1 ) 2 /[1 + (ω 0 T 1 ) 2 ] converges to 1 for ω 0 T 1 1, the effect of the T 1 time on the total frequency shift can be ignored, so the precise value for the T 1 time is important only in the analysis of the temperature-dependent change of the quality factor.
A complication in fitting the values for the two spin densities is that the functions for f bulk and f surf are not independent. To determine the precise values, we fix ρ and fit σ as a shared fit parameter over the temperature traces for all heights. Next we vary ρ to minimize the average fitting error. This method yields global values of ρ = 0.4 ppm, compatible with the specifications of the diamond sample, and σ = 0.072 spins/nm 2 , in line with previously measured surface spin densities [9]. The dashed line in Fig. 3 shows the frequency shift due to the bulk spins at a height of 20 nm for this concentration, signifying that even very low spin densities have a substantial effect on the total frequency shift.
The measured changes of the quality factor for each height and temperature are shown in Fig. 4. The total value for the inverse quality factor is given by 1/Q = 1/Q 0 + (1/Q surf ), where Q 0 is the quality factor of the resonator without coupling to the spin bath. Q 0 can be obtained by extrapolation of the measured dissipation to high temperatures. For a height of 3.4 μm, we find that Q 0 is 24 000, and for a height of 2.4 μm, we find Q 0 is 18 500. For all smaller heights we fix Q 0 to 18 500. These values are lower than the vacuum quality factor (at a height of 5 μm, we find Q 0 is 35 000; see Supplemental Material Fig. S1), and we ascribe this to some other long-range effect (e.g., electrostatic interactions [32,33]).
To fit the data to Eq. (5), we fix the spin densities of both the surface and the bulk to the values obtained from the frequency-shift analysis. Our attempt to fit these data using Eq. (5) with a constant T 1 time independent of position did not yield a good match with the data. This is illustrated by the dashed line in Fig. 4, which shows the result of the calculation at a height of 20 nm, with T 1 = 1.3 ms. A clear deviation between the data and calculated values at low temperatures is visible. Repeating the calculation for each available height separately results in different T 1 times. We find that T 1 increases from 0.4 ms at a large height to 1.3 ms at a small height (see Supplemental Material Fig. S2 for the fitting results obtained with constant T 1 times [34]). This observation is a strong indication of the suppression of the spin diffusion by the high magnetic field gradient.
We include this effect by inserting Eq. (11) into Eq. (5), yielding a position-dependent T 1 time bound by T * 1 in the high gradients close to the magnet, and T ff 1 for spins far away from the magnet. Using the surface spin density σ = 0.072 spins/nm 2 obtained from the frequency-shift data, we find that in our caser = σ −1/2 = 3.7 nm, resulting in a linewidth of B dd = 0.14 mT according to Eq. (9). This leads to a critical gradient G * = 73 mT/μm, a value smaller than the maximum field gradients in our setup as indicated in Fig. 1(a). The resulting dependence of the T 1 time on the magnetic field strength gradient is shown in Fig. 5.
To obtain reliable values for the relaxation times T ff 1 and T * 1 , we use an interesting feature of the coupling between the spins and the magnet. Figure 6 shows the spatial distribution of C for various temperatures, calculated at a constant tip-sample separation of 20 nm, indicating the position of the spins contributing most to the signal. It is clear that as the temperature of the sample decreases, the average location of contributing spins moves away from the cantilever. This immediately implies that at low temperatures most of the contributing spins are located in a region with a magnetic field gradient below G * , which means that spin diffusion is not suppressed, and thus their relaxation time approaches T ff 1 . Equivalently, at high temperatures, the spins that contribute the most are close to the magnet in a high magnetic field gradient, meaning flipflops are quenched and T 1 ≈ T * 1 . This allows us to fit T ff 1 and T * 1 almost independently. The solid lines in Fig. 4 show the final calculations including the effects of spin diffusion using T * 1 = 10 ms and T ff 1 = 0.45 ms.
We select a value of 10 ms for T * 1 . Higher values for T * 1 do not significantly change the dissipation, because T * 1 becomes too far away from the cantilever period. In other words, the dissipation of the cantilever mediated by the spins peaks when T 1 matches the cantilever period, so we are only sensitive to T 1 times of up to several milliseconds. Spins with a T 1 time greater than several milliseconds do not contribute to the enhanced dissipation, but they do change the resonance frequency.
There are still some unexplained features in the data. First, there is a clear difference between data and theory for large tip-surface separations at low temperatures. It seems that the quality factor of the silicon cantilever increases when the temperature decreases (see Supplemental Material Fig. S1 for the data showing the increasing Q factor at low temperatures measured at a height of 5 μm [35]), presumably due to the freezing out of the quantum fluctuators on the surface of the silicon beam [36]. Furthermore, the measurements at a height of 1.5 μm also strongly deviate from the fit for both the resonance frequency and the quality factor. This could be because this measurement is performed directly above a superconducting line of the pickup loop, which might lead to a lower density of paramagnetic electron spins on and beneath the superconductor. The low quality factor can then be explained by the increased coupling with the pickup loop, which leads to additional dissipation of the cantilever energy via the inductive coupling to resistive elements. We did not take the data measured at this height into account in our final analysis.
As the measured data as a function of temperature (for each height) show a clear nontrivial curve that matches the magnetic interaction as described by de Voogd et al., we are confident that the observed effects can be ascribed to paramagnetic spins. Other dissipation channels, such as dielectric fluctuations, are estimated to be smaller than what we find. In an improved version of our experiment, we would like to apply an external magnetic field, although this is a large technical challenge because the SQUID detection is very sensitive to noise in the applied external magnetic field.

IV. SUMMARY AND OUTLOOK
In conclusion, by using our MRFM setup as an ultrasensitive, long-range magnetic force microscope, we are able to measure the amount of nitrogen impurities in our diamond sample, resulting in a bulk spin density of only 0.4 ppm. This shows that our method allows us to characterize samples containing low spin densities over a field of view of several micrometers. Furthermore, we characterize the paramagnetic electronlike spins on the surface of the diamond, yielding a density of 0.072 spins/nm 2 , and T 1 times of several milliseconds, heavily influenced by the presence of spin diffusion. As it is the fluctuation of these spins that is typically held responsible for the reduced performance of a variety of nanodevices such as qubits and superconducting resonators, we believe that our technique offers a useful tool to characterize the properties of the surface spin system and understand the resulting dissipation in these devices.
As the flip-flop interaction between the surface spins on the diamond can be reduced by use of a high gradient, it could be possible to improve the coherence of various diamond-based devices. The idea of suppressing flip-flopinduced spin-bath fluctuations for this purpose has been demonstrated before by increase of the polarization of the spin bath to more than 99% [25]. However, this works only for low temperatures and high magnetic fields. This is not the case for gradient-based quenching of flip-flops. Furthermore, since the required magnitude of the critical gradient depends on the spin density, relatively modest magnetic field strength gradients are required to isolate a single spin from its environment in very pure samples. For example, to suppress spin diffusion in a diamond sample with a nitrogen spin density of 1 ppm, it is sufficient to have a gradient of 1 mT/μm.
A potential near-future application of this technique could be the testing of various sample preparation steps that are typically used to enhance the performance of nanodevices. As an example, we expect that a short chemical wet etch of diamond using hydrofluoric acid should reduce the density of the unpaired spins on the surface, resulting in the case of MRFM in a higher quality factor of the resonator close to the surface and in the case of shallow N-V centers in enhanced correlation times. Our technique would allow us to test the effect of this etch in any intermediate state of the fabrication of one of these devices, allowing better optimization of the fabrication process.