Polarization-assisted Vector Magnetometry in Zero Bias Field with an Ensemble of Nitrogen-Vacancy Centers in Diamond
Franz Münzhuber, Johannes Kleinlein, Tobias Kiessling, Laurens W. Molenkamp
PPolarization-assisted Vector Magnetometry in Zero Bias Field with an Ensemble ofNitrogen-Vacancy Centers in Diamond
F. M¨unzhuber, J. Kleinlein, T. Kiessling, ∗ and L. W. Molenkamp Physikalisches Institut Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany
We demonstrate vector magnetometry with an ensemble of nitrogen-vacancy (NV) centers indiamond without the need for an external bias field. The anisotropy of the electric dipole momentsof the NV center reduces the ambiguity of the optically detected magnetic resonances upon polarizedvisible excitation. Further lifting of the remaining ambiguities is achieved via application of anappropriately linearly polarized microwave field, which enables suppression of spin-state transitionsof a certain crystallographic NV orientation. This allows for the full vector reconstruction of small( ≤ . PACS numbers: 61.72.jn, 07.55.Ge, 81.05.ug
I. INTRODUCTION
The negatively charged nitrogen-vacancy (NV) centerin diamond has attracted enormous research interest inthe last decade because of its formidable versatility inphotonic applications. In addition to its suitability asquantum information processing tool , quantum cryp-tography element and frequency standard , the defectis a prospective candidate for the realization of quan-tum sensors for a wide band of physical parameters,such as temperature , acceleration , pressure , elec-tric , and in particular magnetic fields . The singleatom-like defect embedded in a controllable, but nearlynon-interacting solid state environment enables the con-struction of magnetometers with an unprecedented com-bination of spatial and magnetic field resolution. Pre-viously proposed schemes build on established scanningprobe technology and add the magnetic field sensitivityof an atomic gas sensor, which promises insight into newphysics .Magnetometry with NV centers is based on opticallydetected magnetic resonance (ODMR) spectroscopy ofthe field-sensitive spin states of the defect . The mul-tiplicity of the electronic ground state of the NV cen-ter is S = 1. The three resulting eigenstates alongthe quantization axis, which is set by the crystallo-graphic orientation of the NV axis, are usually labeled as | m S = − , , (cid:105) . Due to crystal field splitting the |± (cid:105) -states are energetically shifted from the | (cid:105) -state. Opticalexcitation of the center as well as subsequent optical re-combination are spin conserving. As crucial ingredient,the optical intensity of the |± (cid:105) -states is weaker com-pared to the | (cid:105) -state because of an additional recom-bination channel in the infrared for this configuration .Therefore, the intensity of the optical photoluminescence(PL) signal can serve as a measure for the NV spin state.For an external magnetic field B ext parallel or antipar-allel to the NV axis, the energetic shift of the states isdescribed by the Zeeman term∆ E Zeeman = gµ B B ext S z (1) where g is the electron g-factor, µ B the Bohr magnetonand S z the projection of the spin along the quantizationaxis. (A general description for arbitrarily oriented fieldscan be found e.g. in Ref. .) Accordingly, only the ener-getic position of |± (cid:105) -states changes as a function of theexternal field.Direct transitions from | (cid:105) → |± (cid:105) can be introducedvia the application of a microwave (MW) field, whichmatches the splitting between the states in energy. Be-cause the splitting is sensitive to the magnetic field atthe site of the NV center, the local magnetic field canbe probed via the MW frequency needed to induce anODMR Signal.Two approaches for NV center based magnetometerscan be found in the literature. Using a single NV centerallows a spatial resolution given by the Bohr radius ofthe defect, which is on the nm scale . In this config-uration only the projection of the magnetic field vectoralong the NV axis can be measured. Determining theother components of the field vector thus requires morethan one sensor.The second approach employs an ensemble of NV cen-ters . The NV centers are oriented along the fourcrystallographic axes within the diamond lattice. A givenmagnetic field will obviously have different projectionsalong these distinct axes. With all four possible NV ori-entations present in the observation volume, all vectorcomponents can be measured with the same probe inone run.The high information density of the resulting ODMRspectrum is, however, not straightforward to analyze. Abare spectrum as depicted in Fig. 1 (a) contains no in-formation on which pair of the resulting eight resonancesbelongs to which NV orientation. Without further inputabout the external field, the correct reconstruction of themagnetic field vector is impossible .In principle, an additional bias field that has differentprojections along the four NV axes can be used to identifythe resonances . However, it may not always be desir-able to expose the investigated sample to a magnetic biasfield.In the following we demonstrate how the ambiguity of a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n N NNVV NV P o l . ( d e g ) P o l . ( d e g ) (b)(d) 2.80 2.85 2.90 2.95090180 MW frequency (GHz) P o l . ( d e g ) (c) 2.75 2.80 2.85 2.90 2.95 3.00 MW frequency (GHz) N o r m . OD M R - S i gn a l (a) k Laser || [111] k Laser || [110] k Laser || [100] k Laser || [111] k Laser || [110] k Laser || [100]
FIG. 1. (a)
NV ensemble ODMR spectrum in random mag-netic field with random orientation of laser polarization. Thefield orientation is chosen such that each of the eight pos-sible resonances is clearly resolved. (b)
ODMR signal as afunction of the MW frequency and the orientation angle ofthe linear polarization excitation plane. A horizontal cut re-sembles an ODMR spectrum as in (a). The pairwise appear-ance of resonances under variation of the polarization angleis characteristic for light incident along the direction of the[100]-crystallographic axis. Magnetic field ( | (cid:126)B | ≈ (c) Same as (b) for light incident alongthe [110]-axis. The contribution from two NV orientationscan be clearly assigned, the two other remain indistinguish-able. Magnetic field is oriented coarsely along [110]. (d)
Same as (b)for light incident along the [111]-axis. Each of thefour NV orientations gives an unique response to the variationof the laser polarization. Magnetic field is oriented coarselyalong [110]. Below: Schematic view of the diamond latticealong the [100]- (left), [110]- (center), and [111]- (right) crys-tallographic axis for better visualization. the resonances can be lifted by exploiting the NV centerselection rules for both the excitation in the visible regionand the induced transitions by the MW field.
II. POLARIZATION SELECTIVE VISIBLEEXCITATION OF NV CENTERS
The first prerequisite for the appearance of a specificresonance in the ODMR spectrum is optical excitation ofthe NV centers along the associated crystallographic axis.We use a 532 nm diode laser system to excite the NVcenters in a standard confocal geometry. The orientationof the linear polarization plane of the laser beam is tunedby the rotation of a λ/ ±
40 nm band pass.The excitation efficiency is crucially determined by thecoupling strength between the electric field of the laserbeam and the dipole moments of electron orbitals of theNV center, which are oriented perpendicular to the NVaxis and to each other. For instance, the dipole moments d d .The probability Γ to excite a NV center is proportionalto the projection of the electric field vector (cid:126)ε of the in-coming laser beam on its dipole moments (cid:126)d :Γ ∝ (cid:12)(cid:12)(cid:12) (cid:126)ε · (cid:126)d (cid:12)(cid:12)(cid:12) (2)For laser incidence parallel to the [100] axis, the po-larization can be rotated in the (100)-plane via the λ/ (cid:126)ε isthen described by (cid:126)ε = E sin θ pol E cos θ pol (3)with θ pol the angle between the polarization and a specificaxis in the (100)-plane. Accordingly, the probability Γ isa function of the angle of laser polarization:Γ d ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E sin θ pol E cos θ pol / √ − / √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E θ pol − cos θ pol ) = E (cid:16) θ pol + π (cid:17) , Γ d ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E sin θ pol E cos θ pol − / √ / √ / √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E θ pol + cos θ pol ) = E
12 sin (cid:16) θ pol + π (cid:17) ,I ∝ Γ d + Γ d ∝ E (cid:32) cos (cid:16) θ pol + π (cid:17) + sin (cid:0) θ pol + π (cid:1) (cid:33) . (4)Consequently, the intensity I of the optical transitionschanges upon rotation of the linear polarization of theexcitation.Looking onto a (100)-facet of diamond, one can distin-guish (Fig. 1 below) pairs of the NV axes that have paral-lel projections into the observational plane. Accordingly,the resonances of these orientations appear and quenchpairwise during the rotation of the linear polarization ofthe exciting laser beam. The number of possible NV ori-entations to which a resonance can be assigned is there-fore reduced to two. Vice versa, a specific orientation ofthe polarization allows for selective NV center excitationof certain crystallographic orientations.This principle can be further improved by using sam-ples of diamond that show (110)- or (111)-facets. If theincident laser beam is parallel to the [110]-axis, two of thefour orientations are parallel to the observational plane.According to Eq. 2, the angle of polarization can be cho-sen such that the signal from those orientations is en-tirely suppressed. These can, therefore, unambiguouslybe identified from their response to the variation of theexcitation polarization, as shown in Fig. 1 (c). However,an assignment of the two remaining resonances to thetwo out-of-plane orientations remains unfeasible.As demonstrated in Fig. 1 (d), illumination of the dia-mond along the [111]-axis allows to overcome this deficit.Three NV orientations are nearly parallel to the obser-vational plane and the angles of laser polarization forwhich they are preferentially excited are at intervals of∆ θ = 60 ◦ . Only the out-of-plane orientation is con-stantly excited at any angle of polarization, which ren-ders it easy to identify. In principle, all eight resonancescan be assigned unequivocally to their corresponding NVorientations, i.e., all components of a magnetic field vec-tor can be reconstructed (except their sign) without theneed of a bias magnetic field.In practice, this is complicated if the magnetic fieldshave equal or nearly equal projections along the out-of- MW frequency (GHz) P o l . ( d e g ) MW frequency (GHz) P o l . ( d e g ) (a)(b) (a)(b) B ext B MW FIG. 2. (a), (b) ODMR signal obtained from a [111] ori-ented diamond with laser illumination along [111]-axis and | (cid:126)B | ≈ µ m. plane NV axis and one or more additional NV axis. Inthis specific case and more general for small magneticfields, the spectral positions of the resonances heavilyoverlap. This renders a convincing evaluation of theODMR spectrum very challenging. For this reason, itis highly desirable to further reduce the number NV ori-entations which contribute to the ODMR spectrum. III. POLARIZATION SELECTIVE MICROWAVEEXCITATION OF NV CENTERS
We can further enhance the directional selectivity byutilizing the selection rules of the microwave-induced NVspin transitions. The Hamiltonian H MW of the NV cen-ter spin subjected to a driving magnetic field B MW offrequency ω in the presence of a static magnetic field B ext takes the basic structure (in the rotating wave ap-proximation) H MW = ∆ − (cid:15) − (cid:15) ∗− (cid:15) + (cid:15) ∗ + ∆ + (5)where ∆ ± = ω ± L ± ω describes the energetic detuning ofthe MW frequency from the Larmor frequency of the NVspin ω ± L = D ± gµ B (cid:126) B ext,axial , which itself is set by thecrystal field splitting D (=2.87 GHz) and the projection B ext of the static magnetic field along the the quantiza-tion axis of the NV center (for small fields). The transi-tions between the | (cid:105) and the |± (cid:105) states are driven bythe (cid:15) ± ∝ ( B MWx ± iB MWy ) components. In this picture,we immediately recognize that only MW fields whichhave a x - or y -component in the NV basis can intro-duce transitions between the eigenstates and thus causean ODMR signal.Guiding the MW via a coplanar waveguide to the di-amond, both the magnetic as well as the electric fieldcomponent of the MW emitted from the waveguide arelinearly polarized. The impact of the linear MW polar-ization is already obvious from the data shown in Fig. 1.Due to the relative orientation of the MW field to the fourpossible NV axes, the ODMR transitions are noticeablydifferent in amplitude.To further demonstrate this interplay, we exploit thespatial stray field distribution near the signal lead of thewaveguide structure. Depending on whether one probesdirectly on top of the signal or in the gap between signaland ground lines, the polarization of B MW is either par-allel to the waveguide surface or points out of the plane.This dramatically changes the efficiency with which tran-sitions of the out-of-plane NV orientation can be driven.In Fig. 2 we present a comparison of ODMR signals ac-quired at the aforesaid positions. We apply an in-planeexternal magnetic field in order to evidence the effect onthe different transitions. This keeps the contribution ofthe out-of-plane NV centers at their zero-field value (thephysics of the remaining splitting are discussed below).Their spectral position therefore correspond to the situ-ation of a low external field measurement, for which theproblem of overlapping resonances occurs.For polarization parallel to the surface we see inFig. 2 (a) that the out-of-plane orientation dominates thespectrum. The contributions of the remaining in-planeNV orientations are strongly reduced. The weakest sig-nal is observed from the NV orientation that experiencesthe strongest spectral shift. This is immediately under-stood from Eq. 5. The largest shift corresponds to thelargest projection of B ext onto one of the NV axes. As B ext is collinear with the polarization vector of B MW inthis configuration, this means that the B MWx and B MWy components are minimized. Therefore the MW drivestransitions of this NV orientation the least.The situation changes dramatically for out-of-planeMW fields. The formerly dominating contribution fromthe out-of-plane orientation is then virtually eliminated,whereas the in-plane orientations show comparable in-tensities.Combined with the controlled rotation of the excitinglaser polarization, we are now in a position to selectivelyaddress single NV orientations of an ensemble of NV cen-ters, thereby lifting the inherent ambiguities which other-wise arise when working with an ensemble of NV centers.
ODMR signal (arb. un)
M W f r e q u e n c y ( G H z ) ( a ) ( b )1 . 4 m T | | [ 1 1 1 ] 0 . 0 1 0 . 1 1 0 . 0 10 . 11
Evaluated magnetic field (mT)
E x t . m a g n e t i c f i e l d ( m T ) D Beval (µT) D B e x t ( µ T ) FIG. 3. (a) Magnetic field response of a single NV orienta-tion, isolated by combination of selective MW and laser exci-tation. The external field is applied along the [111]-direction.The contribution from the three other NV orientations to theODMR spectra is negligible. (b) Robustness of the line shapeanalysis. At small magnetic fields (
B < µ T) the influenceof the transverse crystal field limits the field resolution. Theinset shows how the reliability can be further increased byhigher frequency sampling, when the transverse crystal fieldplays a minor role (see text).
Before employing this method in a real vector mag-netometry experiment, we investigate its sensitivity. Tothis end, we apply an external magnetic field along agiven NV axis and test the accuracy of the line shapeanalysis.The response of the selectively excited NV centers isshown in Fig. 3 (a). For strong enough external field thespectrum explicitly confirms the excitation of only oneorientation, because we can observe merely two transi-tions instead of four, six, or eight. For very small fieldswe recognize a further splitting, which is not related tothe applied magnetic field and persists even at zero-fieldvalue. This splitting is well understood and arises fromthe transverse component of the crystal field to the NVcenters .The experimental spectra in Fig. 3 (a) can be per-fectly reproduced by a model function consisting of twoLorentzians with negative amplitudes. Because of thetransverse crystal field splitting the spectral shift of theirminima does not follow the external field linearly downto very small magnetic fields . The spectral position ofthe transitions can be described by∆ ν = (cid:113) E trans + ( gµ B B ext /h ) (6)where E trans corresponds to the strength of the trans-verse crystal field.Taking the above into account, our analysis results in a Pol = 140 deg
MW frequency (GHz) OD M R s i gn a l ( a r b . un . ) MW frequency (GHz) Pol = 0 deg 2.80 2.85 2.90 2.95 Pol = 70 deg
MW frequency (GHz) OD M R s i gn a l ( a r b . un . ) Pol = 140 deg Pol = 70 deg
MW frequency (GHz) Pol = 0 deg(a)(b) (c)
10 µmscan trajectory 0 deg70 deg140 deg
FIG. 4. (a) ODMR spectra as a function of laser polarization with out-of-plane MW field for suppressing contributions fromout-of-plane NV centers. The spectra for each angle of laser polarization are offset for better visualization. The angles with thehighest selectivity are 0 ◦ (green), 70 ◦ (blue), and 140 ◦ (red). (b) Schematic view of the diamond surface and the structure ontop of it. The scan trajectory, a scale bar, and the relative orientation of the in-plane NV axis to the structure are indicated.The color scheme links the NV orientation to the belonging angle of excitation polarization from (a). (c) ODMR spectra asa function of spatial position and angle of laser polarization. The step size between each measurement point is ∆ x = 500 nm.We recognize the individual evolution of the spectra at each angle, confirming the complete selectivity of the excitation. Onthe right side a reconstruction of the magnetic field vector is shown, reproducing the supposed decay and rotation of the strayfield. The length as well as the color scheme corresponds to the field strength. The direction of the arrows corresponds to thedirection of the field vector if translated to the trajectory in (b). The size of the circle corresponds to the detection area. very good agreement between the actual applied field andthe field values obtained from the line shape analysis. Asa rough estimate, we achieve a field resolution of betterthan B = 50 µ T at a spectral resolution of 1 MHz and anintegration time of 200 ms.From Fig. 3 (b) one observes a decreasing accuracy anda trend to overestimate the actual field at small fields.This is understood from the structure of Eq. 6. Forsmall fields, the E -part dominates the spectral positionand the contribution due to B ext is vanishing. For in-stance, an external field of B = 10 µ T is expected toresult in a shift of ∆ ν = 280 kHz for an undisturbed NVcenter according to Eq. 1. In contrast, having an addi-tional transverse field splitting of E = 5 MHz as in ourdiamond (comp. to Fig. 3 (a)), Eq. 6 yields a shift ofonly ∆ ν = 8 kHz.This effect limits the achievable field resolution atsmall fields in our measurements. To circumvent thisissue, homogenization and relaxation of the local latticeenvironment are required, which have been demonstratedpreviously to be feasible . The inset in Fig. 3 (b) showsthat smaller changes in the external field can be resolved,as soon as the resonance shift occurs approximately lin-early with the change of the magnetic field. The applied fields range from 1 .
20 mT to 1 .
23 mT and the samplingrate is set to 100 Hz. This enables a field resolution downto a few µ T. IV. SPATIALLY RESOLVED VECTORMAGNETOMETRY
As a demonstration of the capabilities of our techniquewe investigate the stray field of a ferromagnetic stripestructured on the surface of a processed sample of [111]diamond. The geometry of the structure is chosen suchthat change in amplitude and direction of the static B-field vector occur on a length scale comparable to ourresolution limit.We start by identifying the angles of laser polariza-tion that show the highest degree of selective excitation.As shown in Fig. 4 (a), these are found at 0 ◦ , 70 ◦ , and140 ◦ . The deviation from the expected 60 ◦ intervalsarise from artifacts induced by optical elements in thesetup. The beam splitter used for guiding the excita-tion onto the optical axis of microscope objective anddetection path, which is mandatory in confocal opticalsetups, has slightly different reflectivities for the s - and FIG. 5. Comparison between measured (a) and simulated (b) stray field distribution of the micromagnet. We recognizethe white space in the measured data to be caused by the strong fields and field gradients at these grid points, rendering ameasurement of local magnetic field unfeasible in the given configuration. The scan directions are not parallel to the symmetryaxis of the structure in order to place the micromagnet at the appropriate position relative to the waveguide. Therefore the gridof the experimental results is slightly rotated. The background in part (a) is a b/w image from the diamond surface, whereinthe microstructure is marked. p -components of the incident laser beam. Therefore, thebeam splitter acts for certain angles of polarization as anadditional rotator.The application of external reference fields enables thedetermination of the orientation of the in-plane NV axesrelative to the laboratory frame and the microstructure.Fig. 4 (b) indicates their relative alignment and furthermarks the trajectory along which we perform vector mag-netometry that is shown in Fig. 4 (c).We sample a trajectory of length of l = 12 µ m in stepsof ∆ x = 500 nm for each angle of the laser polarization.The low implantation depth ( d = 10 nm) of the NV cen-ters assures that the emitted signal stems only from theregion directly beneath the microstructure. For this rea-son we assume a perfect in-plane magnetic stray field andneglect the out-of-plane component in the further analy-sis.Using the model function described in the previous sec-tion we determine the absolute values of ∆ ν for identifiedlaser polarization angles. Being insensitive to sign of ∆ ν ,the resulting value for the local field component along aspecific NV orientation also only corresponds to the ab-solute value of component and remains ambiguous. Onlyupon the comparison of the three in-plane NV orienta-tion, the number of possible reconstructions is reduceddown to two. These vector reconstructions are identicalexcept for their sign.This allows us to map the stray field from our mi-crostructure. The reconstructed field vectors are shownat the right edge of Fig. 4 (c). We assume an evolv- ing rotation of the field vector in order to resolve thesign-ambiguity. This last remaining question could beaddressed by the application of circularly polarized MWfields, which selectively excite transitions between the | (cid:105) → | +1 (cid:105) - and the | (cid:105) → |− (cid:105) -states .We can finally expand the method to a two dimensionalscan to investigate the complete stray field distribution.The result is shown in Fig. 5 (a). The vector field rotatesaround the tip of the structure. The comparison with anumerically simulated stray field distribution of such astructure as is shown in Fig. 5 (b) strongly supportsthe reliability of our method.It further reveals some drawbacks of the procedure.When the observation spot is chosen close to the magnet,two problems occur. First, the strength of the local mag-netic field shifts the energetic position of the | (cid:105) → |± (cid:105) -transitions out of the sampled frequency interval. Sec-ond, the field gradient along the observed defect ensem-ble leads to a strong broadening of the resonances in theODMR signal. The spectra in Fig. 4 (c) at the strongestenergetic shift already mildly indicate these effects.Both issues can actually be avoided. Adjustment ofthe sampled frequencies combined with nano-patterningof diamonds will allow for further improvements ofthe presented principle of magnetometry. At the sametime, the utilization of nano-structured diamond sampleswill help to increase the overall spatial resolution.To summarize, we demonstrated how the ambiguitiesof an ODMR spectrum of an ensemble of NV centers canbe lifted by the application of properly polarized opticalexcitation. The linear polarization of the laser can reducethe number of excited NV orientation to two in case of in-cidence along the [111]-crystallographic axis. With helpof linearly polarized MW fields, a single NV orientationcan be chosen to contribute to the ODMR spectrum. Thesensitivity of the method is limited by the transverse crys-tal field splitting E, but magnetic fields small as 50 µ Tcan be reliably detected. This allows for the mappingof magnetic fields as is exemplified on the stray field of a ferromagnetic microstructure. The method can be ap-plied as well to AC magnetometry schemes where highersensitivity limits can be reached. 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