Abstract
Pushing the frontiers of condensedmatter magnetism requires the development of tools that provide realspace, fewnanometrescale probing of correlatedelectron magnetic excitations under ambient conditions. Here we present a practical approach to meet this challenge, using magnetometry based on single nitrogenvacancy centres in diamond. We focus on spinwave excitations in a ferromagnetic microdisc, and demonstrate local, quantitative and phasesensitive detection of the spinwave magnetic field at ∼50 nm from the disc. We map the magneticfield dependence of spinwave excitations by detecting the associated local reduction in the disc’s longitudinal magnetization. In addition, we characterize the spin–noise spectrum by nitrogenvacancy spin relaxometry, finding excellent agreement with a general analytical description of the stray fields produced by spin–spin correlations in a 2D magnetic system. These complementary measurement modalities pave the way towards imaging the local excitations of systems such as ferromagnets and antiferromagnets, skyrmions, atomically assembled quantum magnets, and spin ice.
Introduction
Correlatedelectron systems support a wealth of magnetic excitations, ranging from conventional spin waves to exotic fractional excitations in lowdimensional or geometrically frustrated spin systems^{1,2}. Probing such excitations on nanometre length scales is essential for unravelling the underlying physics and developing new spintronic nanodevices^{3,4,5,6}. Despite recent progress with realspace techniques^{7,8,9,10,11,12,13}, a wide range of interesting magnetic phenomena in correlatedelectron materials remains experimentally inaccessible because of the required combination of resolution, magneticfield sensitivity and environmental compatibility.
The S=1 electronic spin of the nitrogenvacancy (NV) centre in diamond is an atomsized magnetic field sensor that can be brought within a few nanometres of a sample and readily interrogated with optically detected magnetic resonance^{14}. NV centre magnetometry^{15,16} has provided unprecedented roomtemperature magnetic imaging with nanometrescale resolution^{14,17,18,19} and singleprotonspin sensitivity^{20}, and has been used to study nanoscale biomagnetism^{21,22}. However, NV centres have only recently emerged as probes of collective spin dynamics in correlatedelectron systems^{19,23}. In this work, we demonstrate that singleNV magnetic imaging is a powerful tool for nanometrescale, quantitative, and nonperturbative detection of spinwave excitations. We present complementary measurement techniques to study spinwave excitations over a broad range of magnetic fields and frequencies, as well as a method to characterize spin–spin correlations. These methods may be directly applied to open problems of current interest, such as realspace imaging of skyrmion core dynamics^{24} or imaging spinwave excitations in atomically assembled magnets^{10} as a function of temperature.
Results
Spin waves in a ferromagnetic microdisc
As a model system, we consider a ferromagnetic microdisc (Ni_{81}Fe_{19}) fabricated on top of a diamond chip that contains NV centres implanted at ∼50 nm below the surface (Fig. 1a,b). We use an onchip coplanar waveguide to generate microwave magnetic fields to control the NV spin state and to drive spinwave excitations in the disc. We optically address individual NV centres using a scanning confocal microscope (Fig. 1b) and read out the NV spin state through spindependent photoluminescence (Supplementary Note 1).
Characterizing the static magnetization
Characterization of the static magnetization forms the basis for understanding the excitations of a magnetic system. Using individual NV centres close to the disc, we locally characterize the magnetization as a function of an externally applied static magnetic field B_{ext} (see Methods). We measure the electron spin resonance (ESR) frequency of an NV centre close to the disc (NV_{A} in Fig. 1b) and a reference NV centre (NV_{ref}) far from the disc (Fig. 1c). By comparing these ESR frequencies and knowing the NV gyromagnetic ratio γ=2.8 MHz G^{−1}, we determine the stray magnetic field of the disc at the location of NV_{A} (see Methods). Figure 1d shows the projection of this disc stray field onto the NV axis, B_{∥}, as a function of B_{ext}.
The local nature of the disc’s magnetization becomes clear by comparing the measured disc stray field at two locations (NV_{A} and NV_{B}, Fig. 1b). At both NV_{A} and NV_{B}, this field opposes the external field (Fig. 1d), as expected from a numerically calculated spatial field profile based on a micromagnetics simulation of the disc’s magnetization (Fig. 1e, see also Supplementary Note 2 and Supplementary Fig. 1). However, as B_{ext} is decreased, the change in the disc stray field at NV_{A} is remarkably opposite to that at NV_{B} (Fig. 1d). This behaviour is qualitatively in good agreement with numerical simulations of the disc’s magnetization and the associated disc stray field as a function of B_{ext} (Fig. 1f). These calculations indicate that as B_{ext} is decreased, the magnetization becomes less homogeneous, with spins at the disc’s edge reorienting first. The opposite behaviour of the disc stray field at NV_{A} and NV_{B} is a direct result of the differently varying local magnetization (Methods), and would not be observed in a farfield measurement.
Resonant detection of spinwave excitations
Spinwave excitations consist of collectively precessing spins in a magnetically ordered system. It was recently proposed^{25} that detection of the timevarying stray magnetic fields generated by spinwave excitations in small ferromagnets may be exploited to strongly amplify the sensitivity of single NVcentre magnetometry. Here we employ a resonant detection technique to locally sense the spinwave stray magnetic field, demonstrating the first singlespin detection of onchip magneticfield amplification by a ferromagnet. We apply a microwave (MW) magnetic field to drive spinwave excitations in the disc, choosing the MW frequency such that it is resonant with the ESR frequency of a target NV centre. The spins in the disc respond and generate a magnetic field at the site of NV_{i} which interferes with the drive field B_{D}. Transformed into a frame, rotating at the ESR frequency f, these fields are represented by and b_{D} respectively, and sum (inset Fig. 2b) to give the total field driving spin rotations (Rabi oscillations) of NV_{i} at a rate . As we tune the NV ESR frequency using B_{ext}, we observe a striking difference between the Rabi frequency of nearby NV centres (NV_{i=A,B,C}, see Fig. 1b) and the Rabi frequency of a faraway, reference NV_{ref} (Fig. 2a). This difference becomes even clearer by plotting the ratio , which corrects for any frequencydependent delivery of MWs through our setup (Fig. 2b).
Numerical calculations of the spinwave spectrum of the disc (Supplementary Note 2) indicate that the resonances in Fig. 2b occur when the NV centre ESR frequency matches the frequency of the lowest order spinwave resonance of the disc (Fig. 2c). This mode—the ferromagnetic resonance (FMR)—is efficiently excited because the driving field is spatially uniform (Supplementary Note 2 and Supplementary Fig. 2). The observed resonance is described by
where , and θ( f ) is the angle between and b_{D} determined by the dynamic susceptibility of the ferromagnet and the location of the NV centre. To illustrate the validity of this model, we fit the resonances in Fig. 2b using equation (1), assuming a simple, singlemode damped oscillator response for r( f ) and θ( f ) (Supplementary Note 3). The resulting Fanolineshape accurately describes the observed interference for NV_{A} and NV_{C}, demonstrating that this technique is sensitive to both the amplitude and phase of the spinwave magnetic field. Possible deviations from this model, such as the doublepeak structure of NV_{B}, may result from frequency dependence of the spatial profile of the spinwave excitation or fabricationrelated imperfections. The amplification of the MW field also explains the power broadening of the ESR spectra at low applied magnetic fields B_{ext}, as observed in Fig. 1c.
Nonresonant detection of spinwave excitations
Characterizing the magnetic excitation spectrum in a correlatedelectron system, as well as addressing other problems of interest such as imaging magnetic vortex^{8,9} or skyrmion core dynamics^{24}, requires a detection scheme that operates over a broad frequency range. To this end, we developed an offresonant detection technique that can detect a sample’s spin dynamics even when the NV centre ESR frequency is far detuned from the frequency of these dynamics. The idea is to drive spinwave excitations in the sample with a microwave magnetic field and detect the resulting change in the stray magnetic field by applying a multipulse sensing sequence to the NV centre (Fig. 3a)^{26}.
To understand the offresonant detection scheme in Fig. 3a, it is crucial to realize that during the excitation of a spinwave resonance, the timeaveraged longitudinal magnetization of the disc is reduced (because the precessing spins are tilted away from their equilibrium state), causing a change in the timeaveraged disc stray field (Supplementary Fig. 3). By applying the MW driving only during the central 2τ period (Fig. 3a), the disc stray field is modulated in sync with the multipulse sensing sequence applied to the NV centre, leading to a phase shift ϕ on the final NV spin state. At the end of the sequence, we read out this phase and relate it to an effective magnetic field B_{eff}=ϕ/(γT) oriented along the NV axis and averaged over the duration T of the MW driving. We note that the excitation of a spinwave resonance also generates rapidly oscillating magnetic fields with typical frequencies in the GHz range (recall Fig. 2). However, such frequencies are above the detection capabilities of the scheme in Fig. 3a, because it would require applying the NV spincontrol pulses at GHz repetition rates^{26}. Although an exceptional situation occurs for frequencies close to the NV ESR frequency, where the NV spin may pick up a phase through the dynamical (that is, a.c.) Stark effect^{27}, the Stark effect quickly diminishes for increasing detuning with the NV ESR frequency and we estimate it to play a minor role in our measurements (Supplementary Note 4 and Supplementary Fig. 4).
On the application of the measurement scheme in Fig. 3a to NVA and NVB, we observe a clear resonance that agrees well with numerical calculations of the FMR frequency of the disc (Fig. 3b, Supplementary Note 2). In Fig. 3b, B_{eff} is normalized by the square of the drive field b_{D}^{2} measured onchip using NV_{ref} to correct for a frequency dependence in the setup transmission (Methods and Supplementary Note 5). The resonance follows a Kittellike law , where A is a free parameter, characteristic of spinwave excitations in a thin ferromagnet with inplane magnetization. We therefore conclude that the observed resonance corresponds to the FMR of the disc. Furthermore, we observe striking differences in the lineshape of the resonances detected with NV_{A} and NV_{B} (Fig. 3b). To gain more insight into the origin of these differences, we now analyse the influence of the NV centre spatial location in these measurements.
Importantly, the reduction in timeaveraged longitudinal magnetization associated with the excitation of a spinwave mode is not homogeneous in space, but occurs within a specific spatial region of the disc in accordance with the spinwave mode profile^{7}. The location of an NV centre with respect to this profile determines the sign and magnitude of the corresponding change in magnetic field ΔB_{∥}(f) that is felt by the NV centre (parallel to its axis). Because of the close proximity of the NV centres, ΔB_{∥}(f) strongly depends on the NVcentre location (Fig. 3c). In addition, the spatial mode profile depends on frequency (Fig. 3c), affecting the lineshape of ΔB_{∥}(f). At B_{ext}=450 G we find a remarkably good agreement of the sign, width, and shape of the measured FMR signal with calculations (Fig. 3d) given geometrical uncertainties related to the optical resolution (∼400 nm), disc fabrication, NV implantation depth, and oxidation. However, we note that these calculations and/or our model do not account for the change in FMR lineshape observed at NV_{A} as we decrease B_{ext}. Such strong sensitivity on location highlights the unique possibilities NV centres offer to study spin dynamics quantitatively and with nanometrescale resolution.
Detection of spin noise
Spin noise contains valuable information about a system’s magnetic excitation spectrum and is present even in the absence of driving^{28,29}. Here we spectrally probe spin noise in the disc by measuring the spin relaxation rates of a proximal NV centre (NV_{A}), which depend on the strength of the magnetic field generated by the spin noise at the NV centre ESR frequencies^{30,31}. As we lower B_{ext} and thereby change the NV ESR frequencies relative to the spin–noise spectrum, we first find the m_{s}=0↔+1 and then the m_{s}=0↔−1 relaxation rate (where m_{s} denotes the projection of the spin state onto the NV axis) to increase by over an order of magnitude (Fig. 4a,b), indicating a marked increase in the noise at the ESR frequencies.
Qualitatively, this behaviour can be understood by realizing that at high magnetic field, the NV ESR frequencies are below the FMR frequency (recall Fig. 2c) and therefore in the gap of the spinwave spectrum. In contrast, at low magnetic field the ESR frequencies are above the FMR frequency where spin waves do exist and generate noise. For a more quantitative understanding, we calculate the magneticnoise spectrum at a distance d from an infinite, twodimensional (2D) magnetic plane (Fig. 4c, see Methods). We use a general framework describing the noise spectrum at the site of a sensor spin in terms of the spin–spin susceptibility and a kspace filter function associated with the dipolar coupling to the spins in the magnet (see Methods). This filter function contains a kernel ∼k^{2}e^{−2kd} that peaks at k=1/d, where d is the NV–disc distance and k is a wavenumber characterizing spatial fluctuations of the magnetization. This kernel reflects that a homogeneous magnetization (k=0) does not generate a magnetic field anywhere outside the plane. Likewise, the magnetic field generated by a spatially rapidly varying magnetization characterized by k>>d is exponentially suppressed. Clearly, the noise at the site of the NV centre is dominated by spin–spin correlations on the scale of d. Since the NV centre is far away from the disc’s edges compared with d, we can approximate the disc by an infinite plane. Furthermore, we approximate the dynamic susceptibility as being dominated by exchange interactions because of the small value of d (Supplementary Note 6). This model excellently describes the measured increase in spin noise at the NV ESR frequencies as we lower B_{ext} (Fig. 4b). From fitting, we obtain d=35(5) nm (Supplementary Note 6). However, we note that the model underestimates the disc’s thickness by more than an order of magnitude as detailed in Supplementary Note 6, possibly resulting from the model’s 2D nature. It would be interesting to perform further experiments in which the NVmagnet distance and/or magnet thickness are varied to further develop and test the concepts of NVrelaxometry of spin waves. These relaxation measurements can be extended with and T_{2} spectroscopy techniques^{32} to characterize a spin–noise spectrum over a range of frequencies at a fixed value of magnetic field.
Discussion
In this work, the ferromagnetic microdisc was fabricated directly on top of the surface of a diamond containing NV centres. As such, the NV centres were fixed in space with respect to the disc. This configuration enables a determination of the lateral NV position to about 400 nm (set by the optical diffraction limit), and allows local detection of magnetic excitations with spatial variations on the scale of the ∼50nm NV–disc distance. A variety of techniques may be used to improve the lateral imaging resolution to the fewnanometre scale: for example, optical superresolution methods^{33}, realspace magnetic field gradients created by scanned magnetic tips^{34}, and Fourierimaging techniques similar to conventional MRI^{35}. Because of the pointlike nature of the NV centre, the ultimate imaging resolution is given by how close one can bring an NV centre to a sample and how well one can control its position with respect to the sample. As shown in several recent studies (see, for example, refs 20, 32), NV centres can readily exist at just a few nanometres below the diamond surface. Although the proximity of a metallic sample may quench fluorescence for emittermetal distances below ∼10 nm (ref. 36), this should allow studies of, for example, a skyrmion lattice. Looking ahead, the complementary NV magnetometry techniques, demonstrated here for spin waves in a ferromagnetic disc, open up exciting possibilities to explore a wide variety of magnetic excitations in nanoscopic systems under ambient conditions. The techniques are directly applicable to imaging highly localized spinwave excitations such as edge modes in nanomagnets^{37} or, when combined with THz sources^{38}, highenergy excitations in patterned highcoercivity ferromagnets or in antiferromagnets. We envision nanometrescale studies of magnetic vortices and skyrmions, atomically engineered quantum magnets^{10} and spin ice. In addition, these techniques can be applied to characterize the magnetic fields generated by edge currents in quantum Hall systems and topological insulators^{13}.
Methods
Application of B_{ext}
We apply the static external field B_{ext} along the axis of target NV centres to assure good optical spin contrast (Supplementary Note 7). B_{ext} is thus oriented at an angle of 54° with respect to the plane of the disc. Throughout this work, we select NV centres with equally oriented crystal axes. To avoid hysteresis in the disc, in all measurements we first apply a large field (B_{ext}>700 G) and then sweep the field down in small steps.
D.c. magnetometry
The ESR frequencies of an NV centre in a magnetic field B are determined by the Hamiltonian , where S_{i=x,y,z} are Pauli spin matrices for a spin 1, D is the zerofield splitting, and z denotes the direction of the NV centre crystal axis. We use this Hamiltonian to calculate the projection of the magnetic field onto the NVaxis from the measured ESR frequencies (Fig. 1c,d), as described in detail in Supplementary Note 7.
Normalization procedure
To obtain the signal in Fig. 3, we apply the pulse sequence in Fig. 3a and normalize the photoluminescence (PL) on spin readout using two reference measurements. In these measurements, we apply the sequence of Fig. 3a without the MW drive field and with the final π/2pulse around the x or the –x axis, which yields minimum and maximum PL values. Using these bounds we normalize the PL measured at the end of the pulse sequence in Fig. 3a to obtain B_{eff} (Supplementary Fig. 5). We then divide B_{eff} by the square of the driving field b_{D}^{2}, which we independently determine by measuring the Rabi frequency of NV_{ref} as a function of the ESR frequency (Supplementary Fig. 6). The measured linear scaling of B_{eff} with the MWsource power validates this normalization procedure (Supplementary Fig. 7). The normalization is described in detail in Supplementary Note 5.
Strayfield characterization of magnetization and spin noise
In this section, we describe the properties of strayfield magnetometry of magnetization and spin noise which are relevant for the experiments presented in this work. In particular, we will show that the NV centre probes the spatial variations in the magnetization on the scale of the NV–disc distance, and we will derive the model used for the calculations of the fielddependent magnetic noise spectrum at the NVsite shown in Fig. 4 (for more details, see Supplementary Note 6).
An NV spin at a distance d from the surface of the disc is mostly sensitive to local variations in the magnetization on the scale of d. Intuitively, this can be easily understood: on one hand, a homogeneously magnetized infinite plane generates no stray field. On the other hand, the stray field generated by variations in the magnetization on a scale much smaller than d averages out at a distance d. More formally, it can be shown that the stray field B(r_{0}) at position r_{0}=(ρ_{0},d) (Supplementary Fig. 7) produced by a certain 2D spin texture S(ρ_{j}) is given by:
with D(ρ_{j} ρ_{0},d) being the dipolar tensor. We note that by ‘twodimensional spin texture’ we imply a spin texture that varies in the plane but not along the thickness of the film. We consider a thin magnetic film having a saturation magnetization M_{s} and thickness t. We move to the continuous limit by calling Γ=M_{S}t/(g_{L}μ_{B}S) the number of magnetic dipoles per unit surface. Here μ_{B} is the Bohr magneton, g_{L} the Landé gfactor of the local spin S. We obtain:
where k is a 2D vector in reciprocal space, and φ_{0}–φ_{k} is the angle between the inplane ρ_{0} and the k vector. Provided with such a formalism, we adopt cylindrical coordinates and compute the Fourier transform of the components of the dipolar tensor:
where S(k) is the spatial Fourier transform of S(ρ), φ–φ_{k} is the angle between ρ and the k vector, and φ_{k} is the angle between k and z (Supplementary Fig. 8). We see that strayfield detection works as a spatial Fourier filter, with a kernel given by:
The NVcentre strayfield sensor is pointlike, contrary to, for example, nanoSQUIDs or MagneticResonance Force Microscopy probes. Therefore, the stray field computed with equation (3), after a proper projection along the NV axis, directly describes how spatial modulations of the local magnetization couple to the NV spin. We obtain the following general set of conclusions, in principle valid for any S(ρ) distribution. First, all the elements in the kernel contain the term k exp(dk), which peaks at k=1/d. Strayfield magnetometry is insensitive to Fourier components of the magnetization whose spatial frequency coincides with the condition D(k,d)=0. Accordingly, an NV center cannot detect stray field originating from a uniformly magnetized (k=0) surface or from a spin structure that varies in space within distances much shorter than d. The region of maximum sensitivity corresponds to wavevectors k∼1/d, which in a 2D region of kspace defines an annulus. We refer to this region in kspace as detection annulus of the technique. Second, the inplane strayfield component orthogonal to the wavevector k is always zero. We note that in equation (3) we have assumed that all the magnetic moments along the thickness t have the same distance d from the NV centre.
The formalism just described can be used to derive a general expression for the stray magneticfield noise generated by spin noise in a thin magnetic film. Spin fluctuations δS_{j}(t) in the disc will produce a timedependent field δB′(r_{0},t) at the NV site, which can be written as:
Here we inserted a rotation matrix to express the field in an x′y′z′ frame which has the z′ axis parallel to the NV axis (that is, the x′y′z′ frame is in general rotated by an angle θ around the y axis with respect to the xyz frame, see Supplementary Fig. 8). We first compute the spectral density of the stray magneticfield noise at energy ω_{α,β} along a general direction η:
which has units of T^{2}Hz^{−1}. Here denotes an ensemble average over the magnet’s spin degree of freedom. An expression for the stray magneticfield noise can be obtained by inserting equation (6) into equation (7):
where we have defined S={S^{x,x},S^{y,y},S^{z,z}}, with:
and
and the are the matrix elements of equation (5).
The matrix N(k,d) filters in kspace the spin fluctuations of the magnetic thin film. Note that the integral in equation (8) contains a kernel k^{2}exp(−2dk) for all the components of the N(k,d) matrix. The detection annulus changes therefore slightly with respect to the case of static magnetometry.
The model linking the relaxation rates of the NV centre to the spin–noise in the disc is discussed in Supplementary Note 6. In particular, the fielddependent noise spectrum and associated NVrelaxation rates shown in Fig. 4c are obtained from the expression:
where T=300 K is the temperature, , W is the width of the FMR excitation, Δ its fielddependent gap, D the spin stiffness and S the value of the local spin.
Additional information
How to cite this article: van der Sar, T. et al. Nanometrescale probing of spin waves using single electron spins. Nat. Commun. 6:7886 doi: 10.1038/ncomms8886 (2015).
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Acknowledgements
We acknowledge support of the DARPA QuASAR program and the National Science Foundation. F.C. acknowledges support from the Swiss National Science Foundation.
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T.S., F.C. and A.Y. conceived and designed the experiments. T.S. and F.C. fabricated the samples, performed the experiments and processed the data. R.W. and A.Y. supervised the work. All authors analysed the results and contributed in writing the manuscript.
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Supplementary Figures 111, Supplementary Notes 17 and Supplementary References (PDF 5317 kb)
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van der Sar, T., Casola, F., Walsworth, R. et al. Nanometrescale probing of spin waves using single electron spins. Nat Commun 6, 7886 (2015). https://doi.org/10.1038/ncomms8886
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DOI: https://doi.org/10.1038/ncomms8886
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