A universal and ultrasensitive vectorialnanomechanical sensor for imaging 2D force fieldsLaure Mercier de Lépinay1,2, Benjamin Pigeau1,2, Benjamin Besga1,2, Pascal Vincent3,Philippe Poncharal3 and Olivier Arcizet1,2*

The miniaturization of force probes into nanomechanical oscillators enables ultrasensitive investigations of forces ondimensions smaller than their characteristic length scales. It also unravels the vectorial character of the force field andhow its topology impacts the measurement. Here we present an ultrasensitive method for imaging two-dimensionalvectorial force fields by optomechanically following the bidimensional Brownian motion of a singly clamped nanowire. Thisapproach relies on angular and spectral tomography of its quasi-frequency-degenerated transverse mechanicalpolarizations: immersing the nanoresonator in a vectorial force field not only shifts its eigenfrequencies but also rotatesthe orientation of the eigenmodes, as a nanocompass. This universal method is employed to map a tunable electrostaticforce field whose spatial gradients can even dominate the intrinsic nanowire properties. Enabling vectorial force fieldimaging with demonstrated sensitivities of attonewton variations over the nanoprobe Brownian trajectory will have astrong impact on scientific exploration at the nanoscale.

The nanosciences were revolutionized by the invention ofatomic force microscopy1, which enabled the first perceptionsof the nanoworld by measuring the proximity forces exerted

by a sample on a micromechanical oscillator. The subsequentemergence of nanomechanical oscillators and evolution in readouttechniques2–4 lead to impressive improvements in force sensitivity5,enabling the detection of collective spin dynamics6–8 and single elec-tron spins9, mass sensing of atoms10–12 or inertial sensing13. There isalso great potential in nanoresonators hybridized to single quantumsystems such as molecular magnets14, spin or solid state qubits15–19.This reduction of the probe size naturally motivates the explorationof force fields on dimensions smaller than their characteristic lengthscales, where a great physical richness is expected—particularly fornanoscale imaging or investigations of fundamental interactionssuch as proximity forces or near-field couplings. In this situation,measurements are performed in presence of strong force field gradi-ents whose vectorial structure becomes crucially relevant and shouldbe fully accounted for to describe the measurement process itself 20.

However, most of the existing force sensing scanning probeexperiments are based on oscillators such as cantilevers, whosemotion remains quasi-monodimensional, even in the case ofmulti-mode operation21. Thus only scalar quantities can bemeasured, giving access to the projection of the force field gradientalong the measured eigenmodes, which only provides partialinformation on the force field under investigation.

In this Article we present a novel ultrasensitive measurementtechnique that goes beyond scalar measurements and permits vec-torial two-dimensional (2D) force fields to be directly imaged. Itis based on a singly clamped nanowire (NW) which can oscillatealong two perpendicular transverse directions20,22–26. Its vibratingextremity is immersed in the force field under investigation whileits two-dimensional (2D) Brownian motion is optically detectedand reconstructed in real time. A novel measurement principle,based on the angular and spectral tomography of the Brownianmotion of the NW, provides access to the eigenfrequencies andeigenvectors of the two quasi-degenerated fundamental vibrational

modes. The latter are hybridized by the external force field gradients,which can be subsequently deduced from a direct mathematicalinversion of the tomographic parameters after having understoodthe bidimensional coupling mechanism. The strength of themethod relies on the use of quasi-frequency-degenerated NWs,whose eigenmodes are likely to be cross-coupled by the shear com-ponents of the external force field, generating large changes in theeigenmode orientations as the force field gradients easily takes pre-cedence over the intrinsic mechanical properties of the NW.We firstintroduce the principle of angular and spectral tomography of thenanowire 2D thermal noise and demonstrate its capability toprobe the in-plane 2D electrostatic force field gradients generatedby a voltage-biased nanotip that is brought into the vicinity of theNW extremity. The choice of using point source-like attractivecentral force fields permits the panorama of dual mode couplingin 2D to be explored. Finally, we realize a cartography of the vector-ial structure of the electrostatic force field generated by the tip,which illustrates the strength of the method for future applicationsin vectorial force field scanning probe imaging.

Similar NWs were previously employed20 to analyse the optome-chanical back-action in 2D generated by a tightly focused laser field.To do so the optical force field was measured by means of a coherentpump-probe response measurement where the laser intensity wasmodulated to resonantly drive the NW with optical forces. Theforce measurements were realized at low optical powers so thatthe dynamical response of the NW was only weakly perturbed.Beyond the demonstrated novelty, this methodology lacked univer-sal applicability as the vast majority of the forces under investigationin nanoscience cannot be time modulated and a novel approach thatdoes work in presence of strong force field gradients, as expected atthe nanoscale, had to be developed. The strategy consists in exploit-ing a quasi-frequency-degenerated NW, whose eigenmodes aredressed by the external force field under investigation. The dimen-sionality of the problem requires a novel analysis of mechanicalmode coupling phenomenology in 2D, which is at the heart of thepresented measurement protocol.

1University Grenoble Alpes, Institut Néel, F-38000 Grenoble, France. 2CNRS, Institut Néel, F-38000 Grenoble, France. 3University Lyon, Université ClaudeBernard Lyon 1, CNRS, Institut Lumière Matière, Villeurbanne F-69622, France. *e-mail: olivier.arcizet@neel.cnrs.fr

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Bidimensional optomechanical readoutOur nanoresonators are typically 50 µm-long silicon carbide NW(±20 µm) with diameters of approximately 200 nm that are sus-pended at the extremity of a sharp tungsten tip. The vibrating extremityof the NW is immersed in a strongly focused laser beam that is used toprobe its vibrations. The focusing is achieved by a ×100/0.75 NAmicroscope objective, producing an optical waist of approximately330 nm. Its long working distance (4 mm) bestows a convenientbuffer space between the NW and the first optical element andprovides spatial access to the vibrating extremity of the NW toapproach the sample under test. In this work, we investigate theelectrostatic in-plane force field exerted on the NW by a 17 µm-high platinum-coated pyramidal tip with a radius of curvature ofapproximately 100 nm. Although the whole set-up is grounded,this tip is electrically insulated from the rest of the set-up and canbe voltage biased up to a few volts. The tip position can be piezo-scanned in the vicinity of the NW extremity (see Fig. 1b) torealize force field cartography. The whole set-up, NW and tipsample (along with its positioning piezostage) can also be piezo-scanned with respect to the measuring laser beam to set aworking position. The experiment is carried out at room tempera-ture in a vacuum chamber operated at a static pressure below10−2 mbar to suppress acoustic damping. Its temperature is stabilizedwithin 0.01 °C, which limits thermal drifts below 2 nm h–1.

An optomechanical readout is employed to fully characterize thevibration properties of the NW in 2D and to access the spatial struc-ture of its Brownian motion. To do so, the light backscattered on theNW extremity is collected and focused onto an amplified splitphotodiode that provides the sum/difference (V⊕ /V⊖) of the quad-rant output voltages, which permits the precise micropositioning ofthe NW with respect to the laser beam in the waist area, see Fig. 1c.The temporal vibrations δr(t) of the NWextremity in the horizontaloscillating plane around its rest position r0 are converted into tem-poral fluctuations in the photodetector outputs as V⊖,⊕(r0 + δr(t)).Thus both signals convey a projective measurement of the NWdeflection δr⊖,⊕(t) ≡ δr(t) · e⊖,⊕ along a unitary measurementvector defined by e⊖,⊕ ≡ ∇V⊖,⊕ / ∇V⊖,⊕

∣∣ ∣∣ evaluated at the nanore-sonator rest position r0. By selecting a working position where bothspatial gradients are at least non-collinear and ideally perpendicular(highlighted in Fig. 1c by an orange circle, see SupplementaryInformation for the treatment of a generic configuration of spatialgradients), a fully 2D vectorial readout of δr(t) in real time can beachieved. Moreover, this working point lies on the optical axiswhich ensures the nullity of the non-conservative components ofthe laser dynamical back-action20, which is further circumventedby working at sufficiently low optical powers (≈20–60 µW). Oncethe optical working position is determined, a 2D feedback loop isactivated to stabilize the optomechanical probe against thermaldrift and static deflections (see Supplementary Information).

The 2D Brownian motion trajectory reconstructed from thesimultaneous acquisition of δr⊖,⊕(t) is shown in Fig. 1e. It presents aGaussian distribution that spreads over Δx =

��������������kBT /MeffΩ

2m

√≈ 5 nm

at room temperature T, where kB is the Boltzmann constant andMeff is the effective mass. This is around 2.5% of the NW diameter.Corresponding vibration noise spectra Sδr⊖,⊕

[Ω] (Ω being the ana-lysed pulsation) are shown in Fig. 1f, restricted to the fundamentallongitudinal mode family oscillating around Ωm/2π = 77 kHzwith quality factors Q ≈ 1,000 at 5 × 10−5 mbar and Meff = 600 fg(stiffness ki ≈ 150 µN m−1). A slight asymmetry in the NW geome-try lifts the frequency degeneracy between the two mechanicalpolarizations to Ω2−Ω1 ≈ 2π × 400 Hz whose orientations e1,2 canbe visualized in the 2D representation under resonant actuation ofthe NW, see Fig. 1d. To fully exploit the NW sensing capacity, itis necessary to analyse its 2D thermal noise and detect modificationsof effective temperatures, eigenfrequencies and eigenmode orien-tations. To do so the recorded 2D NW thermal noise trajectory δr(t)

is numerically projected along an arbitrary unitary measurementvector eβ oriented along the angle β, providing the projected displa-cement δrβ(t)≡ δr(t)·eβ, whose spectral density is computed andevaluated at the eigenfrequencies, yielding Sδrβ [Ω1,2]. Repeatingthis sequence for various measurement angles allows the reconstruc-tion of the angular and spectral tomography of the NW thermalnoise in 2D, as shown in Fig. 1g. This representation permits anidentification of the eigenmodes orientations e1,2 as measurementdirections that maximize the heights of the noise peaks and an esti-mation of Meff and temperatures, see Supplementary Information.

2D force field imagingThe modification of the NW 2D thermal noise in presence of anexternal force field gradient is at the heart of the measurement prin-ciple, which is now explained in detail. The NW dynamics,restricted to the two fundamental eigenmodes, is described by 2Dpoint-like oscillator dynamics:

δr = −Ω2 · δr − Γδr +F(r0 + δr)

Meff

+δFthMeff

(1)

where

Ω2 ≡ Ω21 00 Ω2

2

( )

is the intrinsic restoring force matrix for the e1,2 basis, Γ is the mech-anical damping rate and δFth represents the 2D Langevin forcevector that independently drives each eigenmode27 with a whiteforce noise of spectral density SthF = 2MeffΓkBT. The use of thenormal mode expansion is justified by the low (≲10%) dampingasymmetry observed on the uncoupled eigenmodes28. F(r0 + δr)represents the action of the force field under investigation on thetwo fundamental eigenmodes, evaluated at the vibrating extremityposition. In this work we investigate situations where the forcefield experienced by the NW (here of electrostatic origin) instan-taneously follows—on mechanical timescales—the change in theNW position. Delays in the establishment of the force25,29,30 couldbe treated with the same formalism31.

The force experienced by the NW can be linearized asF(r0 + δr) ≈ F(r0) + δr · ∇( )F|r0 . The first term represents the meanforce experienced by the NW, which is responsible for a static deflec-tion that is compensated by the 2D position feedback maintaining theNWextremity at themeasurement location32. As a side note, recordingthe correction of the feedback loop in both directions δrstat for varyingpositions of the sample under test (see Fig. 4c) enables a first staticmethod to probe the 2D in-plane vectorial force field according toF =MeffΩ

2·δrstat. A static force larger than�������������MeffΩ

2mkBT

√≈ 1 pN

will induce a NW deflection that exceeds the spatial spreading of itsthermal noise. As described below, a dynamical measurement of theNW Brownian motion around its rest position allows for muchmore sensitive and robust sensing.

The second linear term in the force field expansion is responsiblefor a modification of the restoring forces experienced by the NW,which affects its dynamics and the spatial architecture of itsBrownian motion. In Fourier space, the NW deflection can beexpressed as δr[Ω] = χ[Ω]·δFth[Ω] where we have introducedthe mechanical susceptibility matrix, whose inverse reads in theunperturbed basis e1,2:

χ−1[Ω] =MeffΩ2

1 −Ω2 − iΩΓ − g11 −g21−g12 Ω2

2 −Ω2 − iΩΓ − g22

( )

(2)

where χ−11,2 ≡ Meff (Ω21,2 − Ω2 − iΩΓ) are the unperturbed inverse

mechanical susceptibilities. Owing to the immersion in the external

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eβ

eβ

a5 μm

b

f

−25 0 25 0−80

1 μm

c

S δr⊖

,⊕ (m

2 ·HZ−1

)

V⊖ V⊕

V⊕ (mV)V⊖ (mV)

e⊕

e⊖

x

z

2 nm10 nm

x

z

ee1

e11 2

e2

e2

d

x

z

Frequency (kHz)7876 80

gz

x

10−20

10−22

10−24

δr(t)

eβ = e⊕ eβ = e⊖

Microscopeobjective

Opticalreadout

y

x zLaser

Nanowire

W tip

V⊖

V⊕

e1e2

Sample

Figure 1 | 2D optomechanical readout. a, Schematic of the experiment. The transverse vibrations of a singly clamped SiC NW are optically read out using adual photodiode amplifier that records the reflection of a laser beam tightly focused on the NW extremity. b, CCD image of a voltage biased sharp metallictip piezopositioned below the NW extremity to generate the d.c. electrostatic force field under investigation. c, Reflection maps of the d.c. differential/sumsignals V⊕/⊖(r) obtained by piezoscanning the NW in the laser waist. The measurement location indicated by an orange circle provides orthogonalprojective measurement vectors, e⊖,⊕ allowing a 2D reconstruction of the NW trajectories δr(t). d, Driven trajectories obtained by resonant electrostaticactuation of both fundamental eigenmodes. e, Thermal noise trajectory recorded over 100 ms (100 ns sampling). f, Calibrated projected displacement noisespectra Sδr⊖,⊕

[Ω]. The red (blue) open diamonds point noise amplitude at Ω1 (Ω2) for eβ = e⊖,⊕. g, Angular spectral tomography of the 2D thermal noise,showing the relative variations of Sδrβ [Ω1,2] with the analysis vector eβ, a generic example of which is drawn as an orange arrow. The red (blue) opendiamonds point noise amplitude at Ω1 (Ω2) for the example vector eβ.

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force field, the NW dynamics now depends on the four gradients ofthe 2D force field: gij(r0) ≡ 1/Meff∂iFj|r0 whose shear components(i≠ j) control the cross-coupling between eigenmodes. The suscep-tibility matrix is diagonalized into new eigenmodes of vibration,labelled with ± indices with eigenfrequencies:

Ω2± ≡

Ω21∥ + Ω2

2∥

2±12

������������������������(Ω2

2∥ − Ω21∥)

2+ 4g12g21

√(3)

and unitary eigenvectors:

e− ≡1�����������

g212 + ΔΩ2⊥

√ ΔΩ2⊥

g12

( )

and

e+ ≡1�����������

g221 + ΔΩ2⊥

√ −g21ΔΩ2

⊥

( )

where we have introduced Ω2i∥ ≡ Ω2

i − gii and ΔΩ2⊥ ≡ Ω2

2∥ − Ω2−=

Ω2+ − Ω2

1∥. The impact of the force field gradients on the NWthermal noise is illustrated in Fig. 2 in a simplified representation,and the full calculation of the 2D thermal noise spectra is carriedout in the Supplementary Information. The force field divergenceg11 + g22 is responsible for a global frequency shift, while thePoisson contribution g11 − g22 modifies the frequency splitting. Onthe contrary, shear components gi≠j are responsible for the rotationof the eigenmode orientations. In a conservative force field, verifyingg12 = g21, both eigenmodes are equally rotated, which preserves theirinitial orthogonality as e− · e+ ∝ roty(F). The situation is qualitativelydifferent in a non-conservative force field that breaks eigenmodesorthogonality and increases their effective temperatures, seeSupplementary Information for the full calculation. This lastobservation implies that a 2D vectorial readout is essential to fullydetermine a 2D force field. Indeed, a measurement that is basedon a single projective readout is unable to distinguish between

eigenmode rotation and effective temperature increase. As can bemathematically derived, see Supplementary Information, it is suffi-cient to determine the eigenfrequencies (Ω±) and eigenvectors (e±) ofboth fundamental eigenmodes to fully determine the four componentsof the force field gradients. Those are precisely the quantities thatcan be extracted from the spectral and angular tomography methodintroduced above, see Supplementary Information.

Probing electrostatic force field gradientsTo illustrate the strength of the method and demonstrate its validitywith a well-controlled force field, we employ the NW to probe theelectrostatic force field generated by the metallic tip. The latter ispiezopositioned at a location where large shear components are gen-erated to enhance hybridization of the eigenmodes. Thermal noisespectra obtained for a varying bias voltage V applied on the tipare shown in Fig. 3a, and the eigenfrequencies and eigenvectorsorientations (e± · ex) are reported in Fig. 3b,c, respectively. Botheigenmodes rotate while preserving their initial orthogonality,which is a direct experimental verification of the conservative char-acter of the electrostatic force field. Strikingly, a rotation of almost60° can be achieved at large biases, indicating that the externalforce field prevails over the intrinsic properties of the NW in deter-mining its eigenmodes structure. The force field gradients aresubsequently extracted from eigenfrequencies and eigenmodesorientations for each bias voltage, permitting the verificationof the quadratic scaling of the force gradients according togij = g0ij + αij(V − V0)

2, as expected for electrostatic actuation; seethe fits in Fig. 3b,c following this scaling of force parameters. Thevery good agreement observed across the full bias voltage rangedemonstrates the validity of the method even in extremely largeforce field gradients. The corresponding interpolated force field(Meff gij δri) evaluated for a bias voltage of V−V0 = 300 mV isshown in Fig. 3d. The extreme sensitivity of the NW is revealed:the relative variations of the force field magnitude over the spatialspreading of its thermal noise are only 30 aN. The intrinsic low/high frequency eigenmodes (1/2) are dressed by the external forcefield, defining the new set of eigenmodes (−/+) that tend to align

Noi

se s

pect

rum

Forc

e gr

adie

ntTo

mog

raph

y

a

b

c

−+

2

2

e2 = e+ e2 = e+

e1 = e− e1 = e−

2 2 21 1 1

g11 + g22 g11 − g22 g12 + g21 g12 − g21

1

1−

++

−

− +

e2

e− e−

e1e1

e+e+

e2

Figure 2 | Measurement principle of 2D force field gradients. a, Any 2D force field gradient can be fully expanded on the geometric basis shown. b,c, Theirrespective impacts on thermal noise spectra (b, measured along one single measurement vector) and on angular and spectral tomography representations (c).The knowledge of the modified orientations of the eigenvectors and eigenfrequencies allows a complete derivation of the local external force field gradients.

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their orientations with the eigenaxes of the external force field gra-dient matrix. The eigenvectors e∞± of the gij matrix are also shownand they match the eigenmode orientations measured at large elec-trostatic biases. This representation also provides an understandingof the change of curvature that is visible in the frequency shifts (seeSupplementary Information). For example, at low bias voltages

(≲300mV) the low-frequency mode (1/−) still oscillates along itsintrinsic orientation e1 and experiences a trapping force field thatpoints towards the centre, which results in a frequency increase.However, at larger bias voltages it rotates and aligns with e∞− ,where it experiences an antitrapping force field, which lowersits oscillation frequency. This dynamical electrostatic force

−3 −1.5 0 1.5 3

110

130

150

Voltage (V)

−3 −1.5 0 1.5 3Voltage (V)

Orie

ntat

ion,

θi (

°)

40

41

Freq

uenc

y (k

Hz)

a b

c

39 40 41Frequency (kHz)

V = 3 V

V = −3 V

d

1 nm 0

30 aN

x

z

ei

e1

e∞−

e∞+

e2

e⊖

Sδr⊕ Sδr⊖

e⊕

30 dB

θi

θ1, θ2−90°

θ+−90°

Ω−/2π

Ω+/2π

Ω2/2π

Ω1/2π

θ−

1 nm

Figure 3 | Mapping electrostatic force field gradients. A voltage bias is applied on the electrostatic tip to tune the external force field gradient experiencedby a 40 kHz NW for a nanotip position that is chosen to generate large cross-coupling between eigenmodes. a, Thermal noise spectra Sδr⊖,⊕

[Ω] obtained forvarying voltage bias. The inset shows the experimental configuration, see Supplementary Information: orange arrows indicate the measurement directionse⊖,⊕ and the red arrow indicates the direction of a generic mode. b,c, Deduced frequency shifts (b) and eigenmode absolute orientations (c, computed witha typical uncertainty of 5°), knowledge of which permits a full derivation of the force field gradients ∂iFj. For comparison, 90° were removed to θ+. Solid linesare fits according to the theory, based on a quadratic dependence of the electrostatic force field using Meffgij = {−146,100,114,14} nN m−1 V−2 (i,j= 1,2) and anoffset voltage of 1 V, see text and Supplementary Information. d, Linearly extrapolated cartography of the local force field (F(r)−F(r0)) drawn over the spatialextension of the NW thermal noise for a 300 mV bias voltage. At large bias voltages, the NW eigenvectors are aligned with the eigenvectors e∞± of theelectrostatic force field gradient matrix gij.

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measurement33,34 also permits access to the local electrostatic zero,V0 ≈ 1 V here, whose spatial variations play an essential role insurface quality analysis and proximity force measurements22,35.

Qualitatively, the longitudinal force field gradient requiredto shift eigenfrequencies by their mechanical linewidthsamounts to ∂1F1 = k1/Q ≈ 10−7 N m−1, while the shear force fieldgradients required to rotate eigenmodes by 45° amounts to∂2F1 = k1(Ω2 − Ω1)/Ω1 ≈ 2 × 10−7 Nm−1 for a 0.5% intrinsic fre-quency splitting andQ = 1,000. These expressions underline the impor-tance of working with soft NWs, which have large aspect ratios, largequality factors and weak intrinsic asymmetries, to achieve largesensitivities to both longitudinal and transverse force field gradients.

Cartography of electrostatic force field gradientsThe force field gradients generated by the nanotip can be spatiallymapped by piezoscanning its position with respect to the NWextremity, see Fig. 4a. The electrostatic tip was horizontallyscanned over 40 × 40 positions in a 6 × 6 µm2 area at a verticaldistance of around 500 nm below the NW extremity. In eachposition, the bias voltage was scanned between −2 and 2 Vwhile simultaneously recording the 2D thermal noise andanalysing modifications in the frequency, orientations and staticdeflection (3 s acquisition time). The map of the low-frequencymode rotation angle is shown in Fig. 4b, illustrating the nanocom-pass effect, as this mode always rotates towards the softer direction(e∞− ), keeping track of the nanotip position. Reproducing in eachdata point the analysis exposed above, the spatial map of theforce field gradients ∂iFj(r), i,j = 1,2 can be reconstructed asshown in Fig. 4a. For our quasi-point-like attractive central forcefield, unperturbed eigenmode orientations appear as symmetryaxes in the divergence-like components (∂iFi) maps and as anti-symmetry axes for the shear components. The shear componentmaps are identical (within experimental uncertainties) as expectedfromtheconservativenatureof theelectrostaticactuation(∂1F2 = ∂2F1).Atilt in the tip scanning plane with respect to the NW axis isresponsible for the slight top/bottom asymmetry observed. A phe-nomenological evaluation based on a radial approximation36 is ingood agreement with our measurements (see SupplementaryInformation). The attractive pattern of the force, computed from

the 2D feedback loop error on the d.c. position of the wire in thelaser beam, is shown in Fig. 4c. The dynamically measured 2Dforce field gradients can be compared to the spatial gradientsmeasured from of the static NW deflection map (seeSupplementary Information): Fig. 4d compares the force field 2Ddivergence ∝ g11 + g22 obtained from both methods. Althoughboth results are in quantitative agreement, the dynamical measure-ments provide the highest measurement quality, as expected. TheNW extremity can also be scanned in the vertical plane and dueto its large sensitivity, force field gradients can still be measuredmicrometres away from the surface, see Fig. 4e. Then the 2D forcefield can be reconstructed through a 2D integration of the fourforce field gradients maps, including a constant force vectorchosen to cancel the force field above the tip, see SupplementaryInformation. Its attractive pattern is shown in Fig. 4c, allowing fora detailed investigation of the electrostatic behaviour of electrostaticforce microscopes tips. Beyond demonstrating the scanning probeimaging capacity of the system, this measurement also permits thegreat richness of strong dual mode coupling in 2D to be explored:the mode crossing/anticrossing phenomenology37 can be tuned bycarefully positioning the tip, see Supplementary Information. Thisalso allows the 2D force probe to be engineered to maximize its sen-sitivity to specific components of the force field gradients. As anexample, its sensitivity to the shear component can be maximizedby electrostatically minimizing its intrinsic frequency splitting.The measurements accumulated at varying distances (both horizon-tally and vertically) and bias voltages have permitted to verify thelinearity of our probe for a large dynamical range that spans from afew nanonewtons per metre to tens of micronewtons per metre.

ConclusionsWe have demonstrated a novel approach that allows vectorial forcefield imaging at the nanoscale, especially suited for probing in-plane2D force fields. By bringing an additional dimension to surfaceimaging techniques based on mechanical signals with demonstratedforce sensitivities in the attoNewton range at room temperature -several orders of magnitude better than commercial AFM appar-atus- we anticipate that our approach will quickly bring novel per-spectives in scientific imaging at the nanoscale. It will contribute

e

μN m−1

yxz

100 fN

cd

x

1 μm

0 −2

−2μN m−1

0 0

4

y (μm)

0

−21 μm

0

−2

2

0

a

x

z

−2

−4 −2

2

0

0

−2

−4

μN m

−1μN

m−1

μN m

−1

∂1 F2 ∂2 F2

∂1 F1 ∂2 F1

∂1 F1

+ ∂2 F2

μN m

−1 μN m

−1

b

−90°

90°

0

1 μm

1 μm

δθ

e1

Figure 4 | Scanning probe measurements of electrostatic force field gradients. a, Map of the force field gradients obtained for a bias voltage of −2 V whilescanning the tip approximately 500 nm below the NW extremity in a 6 × 6 µm2 area (150 nm grid). b, Map of the relative eigenmode rotation (δθ−)illustrating the nanocompass mechanism: the e1/− eigenmode is pointing towards the softer electrostatic gradient, which is generally oriented towards thenanotip. Scale bar, 1 µm. c, Map of the 2D force field computed by integrating the measured 2D gradients for a 1 V bias using an integration constant (forcevector of (−37, +4) fN) that cancels out the force field just above the tip, see Supplementary Information. The maximum static deflection measured wasaround 4 nm, see Supplementary Information, which is small compared with the characteristic length over which the force field varies. d, Comparison ofstatic (blue) and dynamical (red) measurements of the force field divergence (∂xFx + ∂zFz). e, Cartography in the vertical (x–y) and horizontal (x–z) planes ofthe force field divergence measured for a 1 V bias. See the diagram in the bottom right corner for conventions. Inset, the minimum in-plane divergence of theforce field is well fitted (red line) with a y−3 power law, see Supplementary Information.

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to shed new light on the field of probe-surface interaction22,38–40.Developments of novel measurement protocols based on driven2D trajectories will permit both increasing the measurementspeed and improving the sensitivity, while simultaneous observationof several longitudinal modes families21 will further enrich theimaging capacity of our vectorial force probe. In particular, itcould be straightforwardly employed to explore proximity forcesat the nanoscale, such as Casimir forces in nano-structuredsamples where novel phenomenology can be expected41,42 (R.Guérout, A. Lambrecht and S. Reynaud, personal communication).The method is also compatible with non-conservative force fieldimaging and will permit further exploration of fluctuation theoremsin 2D. Finally, our 2D analysis can also be transposed to 3D, using3D angular and spectral analysis with for instance trapped ions43 orlevitated nano-particles44,45.

Received 9 March 2016; accepted 31 August 2016;published online 17 October 2016

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AcknowledgementsWe thank the PNEC group at ILM, A. Gloppe, B. Canals, J. Chevrier, S. Reynaud,A. Lambrecht, R. Guérout, J.P. Poizat, G. Bachelier, J. Jarreau, C. Hoarau, E. Eyraud andD. Lepoittevin for theoretical, experimental and technical assistance. This project issupported by the ANR (FOCUS-13-BS10-0012), the ERC Starting GrantStG-2012-HQ-NOM and Lanef (CryOptics).

Author contributionsAll authors contributed to all aspects of this work.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to O.A.

Competing financial interestsThe authors declare no competing financial interests.

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