Microscopic imaging of elastic deformation in diamond via in-situ stress tensor sensors
D.A. Broadway, B.C. Johnson, M.S.J. Barson, S.E. Lillie, N. Dontschuk, D.J. McCloskey, A. Tsai, T. Teraji, D.A. Simpson, A. Stacey, J.C. McCallum, J.E. Bradby, M.W. Doherty, L.C.L. Hollenberg, J.-P. Tetienne
MMicroscopic imaging of elastic deformation in diamond via in-situ stress tensor sensors
D. A. Broadway,
1, 2
B. C. Johnson,
1, 2
M. S. J. Barson, S. E. Lillie,
1, 2
N. Dontschuk,
1, 2
D. J. McCloskey, A. Tsai, T. Teraji, D. A. Simpson, A. Stacey,
2, 5
J. C. McCallum, J. E. Bradby, M. W. Doherty, L. C. L. Hollenberg,
1, 2, ∗ and J.-P. Tetienne † School of Physics, University of Melbourne, Parkville, VIC 3010, Australia Centre for Quantum Computation and Communication Technology,School of Physics, University of Melbourne, Parkville, VIC 3010, Australia Laser Physics Centre, Research School of Physics and Engineering,Australian National University, Canberra, ACT 2601, Australia National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan Melbourne Centre for Nanofabrication, Clayton, VIC 3168, Australia Electronic Materials Engineering, Research School of Physics and Engineering,Australian National University, Canberra, ACT 2601, Australia
The precise measurement of mechanical stress at the nanoscale is of fundamental and technologicalimportance. In principle, all six independent variables of the stress tensor, which describe thedirection and magnitude of compression/tension and shear stress in a solid, can be exploited totune or enhance the properties of materials and devices. However, existing techniques to probethe local stress are generally incapable of measuring the entire stress tensor. Here, we make use ofan ensemble of atomic-sized in-situ strain sensors in diamond (nitrogen-vacancy defects) to achievespatial mapping of the full stress tensor, with a sub-micrometer spatial resolution and a sensitivityof the order of 1 MPa (corresponding to a strain of less than 10 − ). To illustrate the effectivenessand versatility of the technique, we apply it to a broad range of experimental situations, includingmapping the elastic stress induced by localized implantation damage, nano-indents and scratches.In addition, we observe surprisingly large stress contributions from functional electronic devicesfabricated on the diamond, and also demonstrate sensitivity to deformations of materials in contactwith the diamond. Our technique could enable in-situ measurements of the mechanical response ofdiamond nanostructures under various stimuli, with potential applications in strain engineering fordiamond-based quantum technologies and in nanomechanical sensing for on-chip mass spectroscopy. Pressure is a powerful thermodynamic variable oftenused to modify a material’s properties [1]. Most notably,strain enhances the charge carrier mobility in modernelectronics [2–4]. It allows the tuning of the optical prop-erties of materials [5–7], can confer them with a ferroelec-tric nature [8, 9], or even make them better supercon-ductors [10]. Despite these successes, strain engineeringis still a largely unexplored field considering the hugeparameter space available; indeed, stress is character-ized by six parameters (three axial components and threeshear components, defining the stress tensor) which, inprinciple, can be continuously and independently variedover many orders of magnitude to optimize the functionalproperties of materials [1].Key to further innovations is the ability to characterisestress at the nanoscale, and in particular to quantita-tively determine the six components of the stress ten-sor. Existing methods to probe stress in solids typicallyrely on the interaction between a beam of probe par-ticles (usually electrons or photons) with the stressedmaterial [11–19]. However, these techniques are gener-ally sensitive to only one or a convolution of the stresscomponents. Exceptions include off-axis electron holog-raphy [20], but at the cost of sample destruction to pro- ∗ [email protected] † [email protected] duce suitably thin lamellae (under 200 nm thickness,which may cause strain release), and off-axis Ramanspectroscopy [21], but at the expense of a spatial reso-lution limited to millimeter scales.In this work, we describe a radically different approach,which relies on atomic-sized strain sensors embeddedinto the material to characterise the local strain. In re-cent years, several materials have been found to hostsuch in-situ strain sensors, in the form of optically ad-dressable point defects that exhibit strain-dependent en-ergy levels. These include diamond [22–24], silicon car-bide [25] and silicon [9]. To date, however, real-spaceimaging of the full stress tensor using in-situ sensors hasremained elusive, in part due to the difficulty of ad-dressing a sufficiently dense ensemble of sensors whileretaining the capability to extract all the tensor com-ponents. In the case of the nitrogen-vacancy (NV) de-fect in diamond [26], employed in this work, previousworks have achieved two-dimensional (2D) imaging of asingle strain component [27, 28] and outlined a methodto extract the entire tensor [22]. Single NV defects havealso been proposed for nanomechanical sensing, includ-ing vector force sensing and mass spectroscopy [24]. Herewe demonstrate wide-field 2D mapping of the full stresstensor with sub-micrometer spatial resolution, at roomtemperature. We apply the technique to spatially non-uniform strain features induced by scratches, implan-tation damage, nano-indents and devices fabricated on a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec S N sCMOSLaser ObjectiveDiamondMagnetCoverslipDichoricMirrorLensPL MW
No stressAxial stressShear stress (c)(b)(a) (d) (h) (g) P L ( a . u . ) Frequency (MHz) NV NV NV NV NV NV NV V N z xy V V V Diamond
Fab.NV layer ImplantScratch /Indent
C N CV i C X i Y i Z i -15 -10 -5 0 5 Height, z (nm) (e)(f)
10 μm
Figure 1. Stress-tensor mapping with nitrogen-vacancy centres. (a) Diagram of the experimental set-up, depicting a diamondsensing chip mounted on a glass coverslip with a microwave (MW) resonator. The layer of nitrogen-vacancy (NV) centres inthe diamond is illuminated by a green laser and imaging of the red NV photoluminescence (PL) is captured with a sCMOScamera. (b) Fine structure of the NV electronic ground state showing the two spin transitions with frequencies ω ± that canbe probed experimentally by virtue of spin-dependent PL (symbolized by light bulbs of different brightness). (c) Schematic ofthe unit cell of the diamond crystal under different stress conditions: no stress (blue), axial stress (orange), and shear stress(green). (d) Unit cell with the four NV orientations depicted as different vacancy locations V i ( i = 1 , , , xyz ). (e) Schematic cross-section of the diamond illustratinghow the NV layer (green band) is used to probe the stress produced by local features. (f) Defect structure of the NV andits native coordinate system ( X i Y i Z i ). (g) Optically-detected magnetic resonance spectra under an external magnetic field ofstrength B ∼
50 G, recorded from NV centres at three different locations in a scratched diamond indicated by circles in (h):away from scratch damage (blue), underneath a scratch (orange), and adjacent to a scratch (green). Each resonance is labelledaccording to its corresponding NV orientation. (h) Atomic force microscopy (AFM) image of the diamond surface showingscratches created with a diamond scribe. diamond. These results establish NV-based wide-fieldstress mapping as a powerful technique for nanoscalemechanical studies of diamond, which is an interestingmaterial for nanomechanical applications with extrememechanical properties and sometimes unexpected strain-stress relationships [29]. The technique may also facili-tate strain engineering for diamond-based quantum ap-plications [30–36], and find applications in multiplexednanomechanical sensing for mass spectroscopy and mi-crofluidics [37–40].
RESULTS AND DISCUSSION
The experimental set-up is depicted in Figure 1a. Itconsists of a wide-field fluorescence microscope equippedwith microwave excitation to allow the spin transitionfrequencies of the NV centres ( ω ± , Figure 1b) to beprobed via optically-detected magnetic resonance spec-troscopy, thanks to their spin-dependent photolumines-cence (PL) [41]. These frequencies depend on the spatialdistribution of the unpaired spin density and so are sen-sitive to the local strain in the lattice, or equivalently [24]to the local stress which can have axial and shear compo-nents (Figure 1c). As we will see, the use of multiple NV orientations (Figure 1d) enables the full reconstructionof the local stress tensor. By using a thin layer of NVcentres near the diamond surface, we can then spatiallymap the stress tensor in two dimensions and study theeffect of localized strain-inducing features (Figure 1e).We note that the presence of the NV centres and otherunavoidable defects such as substitutional nitrogen mayalso contribute to the net measured stress [42].The fine structure of the NV electronic ground state(Figure 1b) is governed by the spin Hamiltonian [43], H i = ( D + M Z i ) S Z i + γ NV (cid:126)B · (cid:126)S − M X i ( S X i − S Y i ) + M Y i ( S X i S Y i + S Y i S X i ) , (1)where D ≈ .
87 GHz is the temperature-dependent zero-field splitting parameter, γ NV = 28 . − isthe NV gyromagnetic ratio [26], (cid:126)S i = ( S X i , S Y i , S Z i ) arethe spin-1 operator, (cid:126)B is the applied magnetic field, and (cid:126) M i = ( M X i , M Y i , M Z i ) is the effective electric field as-sociated with mechanical stress, where we neglect anyresidual true electric field caused by charge effects [44, 45](Supporting Information, Section IV). Here ( X i Y i Z i ) isthe coordinate system of the NV defect structure (Fig-ure 1f), and the index i = 1 , , , (cid:126) M i and the local stress is mostconveniently expressed by defining the stress tensor, ←→ σ ,with respect to the diamond unit cell coordinate system( xyz ) defined in Figure 1d [24, 46]. This gives M X i = b Σ axial X + c Σ shear X i M Y i = √ b Σ axial Y + √ c Σ shear Y i (2) M Z i = a Σ axial Z + 2 a Σ shear Z i where the stress susceptibility parameters are a =4 . a = − . b = − . c = 3 . − [24], and we introduced the quanti-ties Σ axial X = − σ xx − σ yy + 2 σ zz Σ axial Y = σ xx − σ yy (3)Σ axial Z = σ xx + σ yy + σ zz , which capture the contribution of the axial stress com-ponents and are independent of the NV orientation [47],and Σ shear X i = 2 f i σ xy + g i σ xz + f i g i σ yz Σ shear Y i = g i σ xz − f i g i σ yz (4)Σ shear Z i = f i σ xy − g i σ xz − f i g i σ yz , which describe the effect of the shear stress components.In Equation (4), the functions f i and g i evaluate to ± i = 1 , , , f , f , f , f ) = (+1 , − , − , +1) and( g , g , g , g ) = ( − , +1 , − , +1). We note that the spin-mechanical interaction of the NV is invariant under in-version of the nitrogen and the vacancy, therefore thereare only four orientations to consider in contrast to trueelectric fields [48].Because the stress tensor is described by six indepen-dent parameters, a measurement relying on a single NVcentre is not sufficient to infer the full stress tensor evenif the effective field (cid:126) M i is completely determined [22].However, by using a small ensemble of NV centres withmultiple orientations, it is possible to determine all sixstress components in the corresponding volume, assum-ing a uniform stress within this volume. To see that, weconsider the limit of small magnetic fields | (cid:126)B | (cid:28) D , forwhich the spin transition frequencies are given by [24, 44]( ω ± ) i = D + M Z i (5) ± (cid:112) ( γ NV B Z i ) + ( M X i ) + ( M Y i ) . By using Equation (2-4), we find that the sum fre-quencies ( ω + + ω − ) i = D + M Z i of the four possi-ble orientations can be used to uniquely determine thethree shear stress components ( σ xy , σ xz , σ yz ) as well asthe sum of the axial components, ( σ xx + σ yy + σ zz ).The axial components are then determined individu-ally by using the difference frequencies ( ω + − ω − ) i =2 (cid:112) ( γ NV B Z i ) + ( M X i ) + ( M Y i ) , which in general is anover-determined problem when the magnetic field (cid:126)B is known. This is the basis of our method to determinethe full stress tensor (see further details in SupportingInformation, Section I and II).Experimentally, we measured the frequencies ( ω ± ) i forthe four possible NV orientations by recording optically-detected magnetic resonance (ODMR) spectra under asmall applied magnetic field (cid:126)B aligned so that all eighttransitions can be resolved simultaneously [45, 49], asshown in Figure 1g. As a preliminary test, we inves-tigated the stress induced from scratching the diamondwith a diamond tipped scribe [50], generating cuts thatare less than a micron wide and range from 5 to 20 nmin depth (Figure 1h), with the NV layer extending fromabout 5 to 30 nm below the surface [51]. A referenceODMR spectrum (i.e. away from any scratch) is shownin blue in Figure 1g, which is used to infer the magneticfield, here ( B x , B y , B z ) ≈ (22 , − , −
40) G. An ODMRspectrum taken from underneath the scratch is shownin orange and exhibits a shift of about 10 MHz that isrelatively uniform across all the resonances, suggesting astress field in the GPa range that is dominated by axialstress components. An ODMR spectrum taken from aregion adjacent to the scratch is shown in green, and ex-hibits shifts that are markedly different for different NVorientations. This is the signature of a large contributionfrom shear stress.To test our method for reconstructing the full stresstensor, we first implanted C molecules into the diamondto create localized regions of damage extending 5-10 nmbelow the diamond surface. Implantation is commonlyused in diamond to introduce dopants for electrical de-vices [52] or produce buried graphitic electrical wires [53].The resultant confined amorphous carbon has a differentdensity to diamond thus introducing an embedded forcethat pushes in all directions (Figure 2a). It also causes areduction in PL from the NV defects by about 25%, asseen in Figure 2b. Here, the implant pattern is a circularring of 1 . µ m width and 50 µ m diameter. The embeddedforce from the damage is sufficient to cause a bulging ofthe diamond surface [54], which was measured by atomicforce microscopy (AFM) to be on the order of 2 nm inthis case (Figure 2c,d).The spatial maps of the six stress tensor componentsnear this circular ring are shown in Figure 2e, revealingring-shaped patterns with stress values of up to tens ofMPa (corresponding to ODMR frequency shifts in the100 kHz range), much smaller than for the scratches dis-cussed above. The pixel-to-pixel noise is about 1 MPa(standard deviation) for the three shear stress compo-nents, against 10 MPa for the axial stress components,with a total acquisition time of about 10 hours. This dif-ference originates from the presence of the magnetic fieldprojection B Z i in the last term of Equation (5) used toseparate the individual axial components, which resultsin a reduced sensitivity when ( γ NV B Z i ) (cid:29) ( M X i ) +( M Y i ) (Supporting Information, Section III). This lossof sensitivity could be mitigated by performing sequen-tial measurements minimising B Z i for each NV orienta- zzyyxx xy -20 0 20 (MPa) CrDiamond C (a) (b)(d) (e)Damage PL (a.u.)20 μm H e i gh t, z ( n m ) Position, x (µm)(c) -10 -5 0 5
Height, z (nm)2 μm NV xy xz yz
20 μm ij Figure 2. Stress induced by implantation damage. (a) Schematic of the stress induced by locally implanting C molecules in thediamond. (b) NV PL image of an annulus-shaped implanted region. (c) AFM image of a segment of the annulus (dashed boxin (b)). (d) AFM profile across the implanted region (dashed line in (c)). (e) Spatial maps of the six stress tensor componentsmeasured with the NV sensors in the same region as in (b). The amplitude of the σ xz and σ yz components has been multipliedby a factor of 5. tion. Nevertheless, all stress components exhibit resolv-able features, and the different symmetries indicate theyhave been sensibly separated. We note that the axialcomponents have a positive value near the damaged re-gion indicating a compressive stress, consistent with anexpansion of the lattice caused by the implantation [54].The smallest spatial features in Figure 2e have a sizeof the order of 1 µ m. Moreover, in the Supporting In-formation (Figure S2) we show that implanted regionsseparated by less than 1 µ m can be resolved in the stressmaps, indicating a sub- µ m spatial resolution close to theoptical diffraction limit ( ≈
400 nm [55]).The values of stress measured here are consistent withRaman spectroscopy [54] performed on a similar sample(Supporting Information, Figure S4), which indicates anaxial stress of 7 ± σ xx , σ yy , σ xy ), the other compo-nents being much weaker indicating that the deformedmaterial pushes predominantly in the directions parallelto the surface of the undeformed diamond.To facilitate the interpretation of the stress maps, itis convenient to plot the body force instead, where theCartesian components are derived from the stress tensorvia f j = − (cid:88) i ∂ i σ ji (6)with i, j = x, y, z ( ∂ i denotes the partial derivative withrespect to Cartesian coordinate i ). The x and y deriva-tives can be readily computed from the stress maps whilethe z derivatives can be estimated through a suitable ap-proximation (Supporting Information, Section V). Thebody force near the indent is shown in Figure 3e, reveal-ing a complicated pattern. This pattern indicates thatthe tips of the impression push mostly outwards whilethere is an inward force at its flat edges. This illustratesthat Berkovich indentation results in a significant shearstress component which is known to aid phase transfor-mation [57]. xx xy xzyy yzzz (a)(b) (e) Height, z (nm)2 μm -200 -100 1000 200 (MPa) xy PL (a.u.) (c) (d)
20 806040 (N μm -3 ) H e i gh t, z ( n m ) Position (µm) 5 μm ij Figure 3. Stress induced by a nano-indent. (a) AFM image of a representative nano-indent in the diamond surface. (b) AFMprofile across the indent (dashed line in (a)). (c) NV PL image of a region containing a nano-indent. (d) Spatial maps ofthe six stress tensor components in the same region as in (b). The black shapes represent the contour of the nano-indent asdetermined from the PL image. The amplitude of the σ xz and σ yz ( σ xy ) components has been multiplied by a factor of 10 (2).(e) Map of the body force deduced from the stress tensor using Equation (6). The colour denotes the magnitude | (cid:126)f | while theoverlaid arrows indicate the projected force in the xy plane. We now return to scratches made with a diamondscribe to map the full stress tensor. Scratching the dia-mond surface can result in plastic deformation [57]. Sim-ilar to the indent case, there is so sign of fracture of thediamond. A PL image of a scratched region is shown inFigure 4a, where the scratch appears as a dark streakdue to the reduction in crystal quality. The body forcederived from the measured stress tensor (Supporting In-formation, Figure S7) is plotted in Figure 4b. For thisshallow scratch (5 nm deep at most, see AFM profile inFigure 4c and image in inset), the magnitude of the bodyforce reaches about 40 N µ m − corresponding to a stressof up to 0.2 GPa. Deeper scratches like those depictedin Figure 1g,h can produce in excess of 1 GPa of stress(Supporting Information, Figure S7). The body force de-cays from the center of the cut over distances of severalmicrometers, much larger than the width of the physicalcut ( ≈
300 nm FWHM, see orange line in Figure 4c)and the width of the PL quenching feature ( ≈ µ mFWHM, black line in Figure 4c). This illustrates thateven a very shallow cut can elastically deform the dia-mond over distances significantly larger than the size ofthe cut itself. To gain more insight into the direction ofthe force, we plot the three force components across thescratch in Figure 4d, represented as arrows overlaid onthe diamond cross-section in Figure 4e. Interestingly, theforce remains mostly perpendicular to the diamond sur-face over several micrometers about the cut even thoughthe NV sensors are located at a distance of ≈
15 nm fromthe surface. This behaviour suggests a highly anisotropic propagation of the stress along the diamond surface.As a final illustration of the technique, we mapped thestress resulting from the fabrication of electronic deviceson diamond, which is of interest for high-power and high-frequency electronics applications [59] and for studies ofcharge transport using NV-based magnetometry [49, 60].First, we fabricated a strip of Ti/Au (thickness 10/100nm) on the bare diamond surface by thermal evapora-tion and lift-off (Figure 5a,b), and found that the stripinduces a small but measurable compressive stress in thediamond (Figure 5c,d) especially near the edges of thestrip ( ∼
10 MPa). Here, we plot the sum of the axialstress components, Σ axial Z = σ xx + σ yy + σ zz , as this givesa significantly improved signal-to-noise ratio over the in-dividual components (Supporting Information, SectionIII), as well as one of the shear stress components, σ xy .We note that the weight of the deposited metal amountsto a pressure of about 0.02 Pa, and as such cannot ac-count for the measured stress. Also visible in Figure 5care some polishing marks on the bare diamond surface,producing an axial stress of up to Σ axial Z ∼
40 MPa.Next, we fabricated TiC/Pt/Au contacts that extendabout 15 nm into the diamond, through evaporation of aTi/Pt/Au stack (thickness 10/10/70 nm) and subsequentannealing [45] (Figure 5e,f). Such embedded contactsare typically used to form low-resistance ohmic contactswith the two-dimensional hole gas (2DHG) present ona hydrogen-terminated diamond surface [45, 61] as wellas in high-power electronics [62]. As seen in the stressmaps (Figure 5g,h), there is a large compressive stress in m
300 nm0.20.61.0 P L ( a . u . ) -5-4-3-2-10 H e i gh t, z ( n m ) (b)(c)(a) 10 μm PL (a.u.) -20-4002040 50403020100
Position, x (µm) f x f z f y f i ( N μ m - ) z ( n m ) (N μm -3 )
10 20 30 40 50
10 μm(d)(e)
Figure 4. Stress induced by a superficial scratch. (a) PLimage of a region containing a scratch. (b) Map of the bodyforce magnitude for the region imaged in (a), with the overlaidarrows indicating the force projected in the xy plane. (c) Linecuts of the PL (black) and AFM profile (orange) taken alongthe red dashed line shown in (a,b). The left inset is a zoom-in; the right inset is an AFM image of the angled section ofthe scratch. (d) Corresponding line cuts of the three bodyforce components (in and out of plane). (e) Schematic of thediamond cross-section (not to scale), with the overlaid arrowsindicating the body force projected in the xz plane as derivedfrom (d). the diamond below the TiC contact (Σ axial Z ∼
100 MPa).This is expected because TiC has a lower atomic densitythan diamond. It is relatively uniform across the contactexcept for some small spots (a few µ m in size) where thereis no stress, indicative of imperfections in the formed TiClayer.We then fabricated a transistor device based on the 2DHG at the diamond surface [61, 63], by first patterningthe surface termination to form a conductive (hydrogen-terminated) channel on an otherwise insulating (oxygen-terminated) surface, covering the whole surface with a 50-nm insulating layer of Al O , and evaporating a metallictop gate (Al, 50 nm thick). While operating the device,we observed the formation of a defect in the Al O layerabove the conductive channel next to the gate, as illus-trated in Figure 5i and seen in the PL image (Figure 5j).The stress maps (Σ axial Z in Figure 5k and σ xz in Fig-ure 5l) reveal a compressive stress under the oxide defect(Σ axial Z ∼
20 MPa) and a tensile stress just outside thedefect. The presence of the metallic gate on top of the ox-ide induces a stress of similar magnitude in the diamond,although the odd parity of Σ axial Z is in contrast with thefully compressive stress observed in Figure 5c suggest-ing a non-trivial mediation by the oxide layer, possiblythrough permanent deformation of the oxide under thegate while operating the device. These experiments il-lustrate that the technique can be applied to monitormechanical deformations in materials and devices out-side the diamond, through the elastic stress applied tothe diamond substrate as a result, which could find ap-plications as a diagnostic tool for device variability orfailure analysis. CONCLUSION
In summary, we presented a method to quantitativelyimage the full stress tensor below the surface of a dia-mond, at room temperature, using a layer of NV strainsensors. We illustrated the versatility of the techniqueby experimentally imaging the stress under a number ofscenarios, from localized implantation damage to nano-indents to devices fabricated on the diamond surface.With our current experimental set-up, the spatial res-olution is limited by the diffraction limit ( ≈
400 nm)but could be improved by implementing super-resolutiontechniques [64, 65]. The measurement sensitivity is of theorder of 1 MPa per 1 × µ m pixel for typical acquisitiontimes ( ∼
10 hours), corresponding to a strain of less than10 − [24], and could be improved by an order of mag-nitude through futher optimisation of the NV-diamondsamples [66].The technique could be directly applied to characterisethe residual stress in various types of diamond nanostruc-tures commonly used for quantum sensing and quantuminformation science, such as solid-immersion lenses [67],nano-pillars [68, 69], nano-beams [36, 70], and photonicwaveguides and cavities [71–74], which could help im-prove the sensing accuracy or optimise the fabricationprocesses. Moreover, the technique is compatible withstroboscopic measurements, capable for instance of imag-ing the time evolution of stress in mechanically driven di-amond cantilevers [32, 33, 75], which are a testbed for hy-brid spin-mechanical quantum systems. More generally,the ability to measure the complete stress tensor at the Axial
20 μm xy xy
30 μm
PL (a.u.)
Axial
30 μm -25 -12.5 12.50 25 (MPa) xy xy Axial
20 μm xz xy Gate Damage H - t e r m
20 μm(e) (f) (g)(a) (b) (c) (h)(i) (j) (k) (l)20 μm (d)AuTi Evaporated DiamondNV layer PtAuTi x CEmbeddedAl O DamagedCompression Tensile Al Gate ij Figure 5. Stress induced by device fabrication. (a) Schematic cross-section of a device consisting of an evaporated metallic wire(Ti/Au). (b) PL image of a typical device. (c,d) Corresponding maps of the total axial stress Σ axial Z = σ xx + σ yy + σ zz (c) andof the shear stress component σ xy . (e-h) Same as (a-d) but for an embedded TiC/Pt/Au contact. (i-l) Same as (a-d) but for atransistor device where the conductive channel is formed by a hydrogen-terminated (‘H-term.’) region of the diamond surface(delimited by red dashed lines in (j) but giving no PL contrast), insulated via a Al O oxide layer and controlled by a top Algate (appears dark in the PL). A defect formed in the oxide layer above the conductive channel while operating the device,visible through a fringe pattern in the PL. (l) shows the σ xz map. All the stress maps share the same color bar ranging from-25 to +25 MPa, except (g) where the stress values have been divided by a factor of 4 (i.e. ranging from -100 to +100 MPa). nanoscale opens up the field of multiplexed nanomechan-ical sensing, where structures with multiple mechanicaldegrees of freedom could be designed and measured (con-sider, for example, a cantilever with a paddle on the end),with possible applications in on-chip mass spectroscopyand microfluidics. METHODS
Diamond Samples
The NV-diamond samples used inthese experiments were made from 4 mm × × µ m electronic-grade ([N] < { } edges and a (001) top facet,purchased from Delaware Diamond Knives. The plateswere used as received (i.e. polished with a best sur-face roughness < µ m ofCVD diamond ([N] < C-enriched (99.95%)methane [76], leaving an as-grown surface with a rough-ness below 1 nm [50]. All the plates were laser cut intosmaller 2 mm × × µ m plates and acid cleaned (15 minutes in a boiling mixture of sulphuric acid andsodium nitrate), and then implanted with either N + or N + ions (InnovIon) at an energy of 4 −
30 keV and afluence of 10 − ions cm − , with a tilt angle of 7 ◦ .Following implantation, the diamonds were annealed in avacuum of 10 − − − Torr to form the NV defects [51]and acid cleaned (as before). The relevant parametersfor the six diamonds used in this work, labelled
Preparation of stress-inducing features
The scratchesexamined in Figure 1g,h and in Figure 4 were formedby manually dragging the tip of a diamond scribe acrossthe diamond surface (diamond molecules at an energy of 15 keVand a fluence of 3 . × molecules cm − (in diamond O. The average penetra-tion depth of the implanted carbon atoms is 10 nm as pre-dicted by Stopping and Range of Ions in Matter (SRIM)simulations, shown in the Supporting Information (Fig-ure S1), with the peak of the vacancy distribution (i.e.of the damage) at about 6 nm depth. In comparison,the average depth of the nitrogen atoms previously im-planted to create the NVs was 7 nm in this diamondaccording to SRIM. The fluence of the carbon implantwas such that amorphous carbon was formed (confirmedby Raman spectroscopy) at the peak of the damage, inwhich NV defects no longer exist, explaining the reduc-tion in PL in the implanted regions (by about 25%). Theremaining PL (75% of the non-damaged case) is likely tocome from NVs that are located under the peak of thedamage, i.e. 15-20 nm from the surface, in the tail of thenitrogen implant which extends further than predictedby SRIM due to ion channelling [77]. In the SupportingInformation (Figure S3-S5), we show additional measure-ments on a similarly prepared sample except for a higherC fluence of 3 . × molecules cm − (in diamond ◦ C for 1 h to induce graphitization of the amorphizedimplanted regions.The indents shown in Figure 3 were formed by apply-ing a load of 1 N using a Berkovich tip (three-sided pyra-mid), corresponding to a maximum vertical displacementof about 130 nm, and a residual displacement of about15 nm (plastic deformation). Loads of 500 mN (70 nmmaximum displacement) and under did not plasticallydeformed the diamond. A series of 10 indents were madewith a 1 N load for each and a lateral spacing of 20 µ m,showing good reproducibility in the load-unload curve.The devices imaged in Figure 5 were made as fol-lows. The metallic wire in Figure 5a-d (diamond ◦ C for 20minutes in hydrogen gas (10 Torr) to favor inter-diffusionof Ti and C atoms, which formed a TiC layer extend-ing about 15 nm into the diamond [45]. The transistordevices in Figure 5i-l were made on the same diamond(diamond O on the whole sample via atomiclayer deposition (ALD), and finally patterning an evapo-rated layer of Al (thickness 50 nm) to form the top gate. Experimental apparatus
The diamonds were placedon a glass cover slip patterned with a metallic waveguidefor microwave (MW) delivery and connected to a printedcircuit board (PCB). A layer of immersion oil was usedto ensure good optical transmission between the diamondand the glass cover slip, with the NV side of the diamond exposed to ambient air.The NV measurements were performed using a custom-built wide-field fluorescence microscope [45, 55]. Op-tical excitation from a λ = 532 nm continuous-wave(CW) laser (Coherent Verdi) was gated using an acousto-optic modulator (AA Opto-Electronic MQ180-A0,25-VIS), beam expanded (5x) and focused using a wide-fieldlens ( f = 200 mm) to the back aperture of an oil im-mersion objective lens (Nikon CFI S Fluor 40x, NA =1.3). The CW laser power entering the objective was300 mW, corresponding to a maximum power density ofabout 5 kW cm − at the sample given the ≈ µ m 1 /e beam diameter. The photoluminescence (PL) from theNV defects was separated from the excitation light witha dichroic mirror and filtered using a bandpass filter be-fore being imaged using a tube lens ( f = 300 mm) ontoan sCMOS camera (Andor Zyla 5.5-W USB3). MW ex-citation was provided by a signal generator (Rohde &Schwarz SMBV100A) gated using the built-in IQ modu-lation and amplified (Mini-Circuits HPA-50W-63) beforebeing sent to the PCB. A pulse pattern generator (Spin-Core PulseBlasterESR-PRO 500 MHz) was used to gatethe excitation laser and MW and to synchronise the im-age acquisition.The optically-detected magnetic resonance (ODMR)spectra of the NV layer were obtained by sweeping themicrowave frequency while repeating the following se-quence: 10 µ s laser pulse, 1 . µ s wait time, 300 ns mi-crowave pulse; and alternating MW on/off to removethe common-mode PL fluctuations, with total acquisitiontimes of at least 10 hours typically. All measurementswere performed in an ambient environment at room tem-perature, under a bias magnetic field (cid:126)B generated usinga permanent magnet. ACKNOWLEDGEMENTS
We acknowledge support from the Australian Re-search Council (ARC) through grants DE170100129,CE170100012, FL130100119 and DP170102735. Thiswork was performed in part at the Materials Charac-terisation and Fabrication Platform (MCFP) at theUniversity of Melbourne and the Victorian Node of theAustralian National Fabrication Facility (ANFF). Thiswork was performed in part at the Melbourne Centrefor Nanofabrication (MCN) in the Victorian Node ofthe Australian National Fabrication Facility (ANFF).We acknowledge the AFAiiR node of the NCRIS HeavyIon Capability for access to ion-implantation facilities.D.A.B., S.E.L., D.J.M. and A.T. are supported by anAustralian Government Research Training ProgramScholarship. T.T. acknowledges the support of Grants-in-Aid for Scientific Research (Grant Nos. 15H03980,26220903, and 16H06326), the “Nanotechnology Plat-form Project” of MEXT, Japan, and CREST (GrantNo. JPMJCR1773) of JST, Japan.
SUPPORTING INFORMATIONI. ANALYSIS OF THE ODMR DATA
The ODMR data is analyzed by first fitting the ODMR spectrum at each pixel with a sum of eight Lorentzianfunctions with free frequencies, line widths and amplitudes. The resulting frequencies { ω ± ,i } (with i = 1 . . .
4) are thenused to infer the Hamiltonian parameters (denoted as vector (cid:126)p ) by minimising the root-mean-square error function ε ( (cid:126)p ) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i =1 (cid:88) j = ± (cid:2) ω j,i − ω calc j,i ( (cid:126)p ) (cid:3) (7)where { ω calc ± ,i ( (cid:126)p ) } are the calculated frequencies for a given set of parameters (cid:126)p obtained by numerically computing theeigenvalues of the spin Hamiltonian for each NV orientation, H i = ( D + M Z i + k (cid:107) E Z i ) S Z i + γ NV ( B X i S X i + B Y i S Y i + B Z i S Z i ) − ( M X i + k ⊥ E X i ) ( S X i − S Y i )+ ( M Y i + k ⊥ E Y i ) ( S X i S Y i + S Y i S X i ) , (8)where for completeness we added the effect of the true electric field, (cid:126)E = ( E X i , E Y i , E Z i ) expressed in the NVframe ( X i Y i Z i coordinate system), with electric susceptibility parameters k (cid:107) = 0 . − and k ⊥ = 17(3)Hz cm V − [43, 44]. We note that the additional strain terms derived in Ref. [78] were neglected as they are an orderof magnitude smaller.In general, the set of four Hamiltonians { H , H , H , H } is characterized by a total of 13 parameters: thetemperature-dependent zero-field splitting parameter D , the magnetic field (cid:126)B = ( B x , B y , B z ), the electric field (cid:126)E = ( E x , E y , E z ), and the stress tensor ←→ σ = { σ xx , σ yy , σ zz , σ yz , σ xz , σ xy } (in Voight notation), all expressed inthe crystal frame ( xyz coordinate system). The relationships between these parameters and the quantities enteringthe Hamiltonians are given by Equation (2-4) of the main text for the effect of stress, and by B X i = 1 √ − f i g i B x − g i B y − f i B z ) B Y i = 1 √ f i g i B x − g i B y ) (9) B Z i = 1 √ f i g i B x + g i B y − f i B z )for the magnetic field (which simply expresses the rotation of the coordinate system), with identical expressions forthe electric field.To allow inference of these 13 parameters from the 8 ODMR frequencies, we must make assumptions, which mayintroduce systematic errors in the inferred stress values (see Section IV). First, we assume that there is no net electricfield, i.e. (cid:126)E = 0. In reality, an electric field may be present in the NV layer as a result of charge effects in the diamondbulk and band bending at the diamond surface [45, 79]. We emphasize that the effect of stress and electric field cannotbe combined in a single entity when measuring multiple NV orientations, even though they both effectively act as anelectric field in the Hamiltonian, Equation (8). This is because the effective electric field associated with stress differsfor different NV orientations since it derives from a tensor, see Equation (2) of the main text, while the true electricfield is a vector that is simply projected onto the NV axes, see Equation (9). For the same reason, an extra four NVorientations (inverting the nitrogen and the vacancy) would need to be considered in the presence of a true electricfield, whereas the effective electric field associated with stress, (cid:126) M i , is invariant under this inversion.Second, we assume that the stress is null except near the deliberately introduced features (scratches, implantationdamage, indents etc.). In reality, there may exist a residual stress across the sample [28] (see Section IV). Thisassumption allows us to use the ODMR data far from those features to estimate D and (cid:126)B , by minimising ε ( D, (cid:126)B ).Doing so to the full field of view (excluding the stressed regions) gives a value of D that is relatively uniform acrossthe image (typically between 2869.5 and 2870.5 MHz), whereas (cid:126)B exhibits slow variations due to the magnetic fieldproduced by the permanent magnet being non uniform. A multi-polynomial fit was applied to the (cid:126)B maps, and thisfit was then used as a known input for the second fitting step. In the second step, we refit the data (now includingthe stressed regions) with the six stress parameters as the only unknown, i.e. we minimise ε ( ←→ σ ), while holding D and (cid:126)B to the values determined previously.0 II. UNIQUENESS OF THE SOLUTION
When minimising ε ( ←→ σ ) to infer the best-fit stress tensor ←→ σ at each pixel, we rely on the fact that there is a uniquesolution to the optimisation problem. Here we investigate the conditions in which this is true. We first consider thesmall magnetic field limit ( | (cid:126)B | (cid:28) D ) leading to Equation (5) of the main text, allowing us to obtain simple expressionslinking the NV frequencies ( ω ± ) i to the stress components. Namely, for the sum frequencies S i = ( ω + + ω − ) i we have S i = D + a ( σ xx + σ yy + σ zz )+ 2 a ( f i σ xy − g i σ xz − f i g i σ yz ) (10)which gives a set of four independent linear equations with four unknowns: σ xy , σ xz , σ yz and the sum D + a Σ axial Z where Σ axial Z = σ xx + σ yy + σ zz . Therefore, these four quantities can always be uniquely retrieved.On the other hand, the difference frequencies D i = ( ω + − ω − ) i are given by D i = 4 γ f i g i B x + g i B y − f i B z ) + 4[ b ( − σ xx − σ yy + 2 σ zz )+ c (2 f i σ xy + g i σ xz + f i g i σ yz )] + 12[ b ( σ xx − σ yy ) + c ( g i σ xz − f i g i σ yz )] . (11)This gives a set of four equations with only two unknown, for instance σ xx and σ yy , where σ zz can be expressed as afunction of ( σ xx , σ yy ) and the measured { S i } , whereas the shear stress components have been completely determinedfrom { S i } . However, these four equations are not linear and not necessarily independent with respect to the unknownparameters. In particular, in the situation where all the shear components are null, σ xy = σ xz = σ yz = 0, Equation (11)reduces to D i = 4 γ f i g i B x + g i B y − f i B z ) + 4 b ( − σ xx − σ yy + 2 σ zz ) + 12 b ( σ xx − σ yy ) . (12)Clearly, in this situation only a single combination of σ xx and σ yy can be inferred from the measured { D i } , whichmeans that these axial stress components cannot be resolved individually, i.e. there an infinite number of solutionsfor the axial stress components.When the shear components are non-zero, however, Equation (11) forms an overdetermined system that is likely tohave a unique solution. To test this, we numerically explored the full parameter space for the six stress componentsand evaluated the error ε comparing the NV frequencies calculated with a given ←→ σ versus those calculated with atrial solution ←→ σ trial , while holding the other parameters D and (cid:126)B constant. We found that there is indeed a uniquesolution that gives a vanishing error, ←→ σ trial = ←→ σ , except when two of the three axial shear components are null (inwhich case there are two solutions, where the incorrect solution simply swaps two of the axial components) and ofcourse when all three shear components are null (as discussed above). This conclusion holds regardless of the value ofthe magnetic field or of the other stress components. Thus, our method enables the full stress tensor to be determinedwith no ambiguity, as long as at least two of the shear stress components are not exactly null. III. SENSITIVITY OF THE METHOD
The sensitivity of the stress measurements, which dictates the smallest detectable change (in space or time) for agiven integration time, is limited by statistical errors arising from the statistical noise in the measured NV frequencies( ω ± ) i . The latter originates from photon count noise in the ODMR data, which has contributions from both thephoton shot noise and dark current noise from the camera. To characterise this noise, we calculate the standarddeviation from an ensemble of pixels in a region of the image where the stress is relatively uniform, i.e. the pixel-to-pixel noise. We find this noise to be typically of the order of 1 MPa for the shear components and 10 MPa for theaxial components for a 1 × µ m pixel and a total integration time of 10 hours, with no significant improvement withlonger acquisition indicating that this is close to the technical noise floor. As expected, the error is larger on the axialstress components than on the shear components, because of the difference in sensitivity of the NV frequencies to1these different components. Indeed, the shear components are obtained from Equation (11) as a linear combinationof the sum frequencies { S i } , with a sensitivity ∂S i ∂σ xy ∼ a ≈ − . (13)In contrast, the individual axial components are derived from the difference frequencies { D i } which has a sensitivity,in the limit γ NV B Z i (cid:29) M ⊥ ,i [28], ∂D i ∂σ xx ∼ b M ⊥ γ NV B Z i ≈ − × M ⊥ γ NV B Z i , (14)where M ⊥ . = (cid:113) M X i + M Y i . The suppression factor M ⊥ γ NV B Zi depends on the exact conditions but is typically below0.1, which results in a statistical noise on the axial components (as obtained by solving the overdetermined problem)typically an order of magnitude larger than that on the shear components. We note that the sum of the axialcomponents, Σ axial Z = σ xx + σ yy + σ zz , is fixed by the sum frequencies { S i } and therefore is not affected by thissuppression factor, yielding a statistical noise down to 1 MPa. This was employed in Figure 5 of the main text,where we plotted Σ axial Z in order to detect weak stress features. For applications where measuring the individual axialstress components with maximum sensitivity is desirable, we could perform four consecutive ODMR measurementswith an external magnetic field optimized to minimise B Z i for each NV orientation sequentially. Such a sequentialmeasurement strategy would also allow much larger stress values to be measured (up to 100 GPa), currently limitedto a few GPa by the minimum spacing between adjacent ODMR lines (20-30 MHz) [80]. IV. ACCURACY OF THE METHOD
Systematic errors in the stress values may be present as a result of the assumptions made in the model convertingthe frequencies ( ω ± ) i into stress. One assumption is the neglect of the true electric field, (cid:126)E . In Ref. [45] it wasreported that in samples comparable to those used here, an electric field perpendicular to the diamond surface existsowing to surface band bending, with values up to E z ∼
500 kV cm − . We first examine the effect of a non-zero E z on the determination of the shear stress components. Extending Equation (10) to include E z gives S i = D + a Σ axial Z − k (cid:107) √ f i E z + 2 a ( f i σ xy − g i σ xz − f i g i σ yz ) . (15)We can then write the following linear combinations S − S − S + S = 8 a σ xy − k (cid:107) √ E z S − S + S − S = 8 a σ xz S + S − S − S = 8 a σ yz S + S + S + S = 4 D + 4 a Σ axial Z . (16)Thus, the presence of E z will affect only the σ xy component. Precisely, the assumption E z = 0 in the stress recon-struction amounts to offsetting σ xy by − k (cid:107) a √ E z where a <
0. For instance, the ≈
10 MPa background observedin the σ xy map in Figure 2d of the main text can be attributed to an electric field of E z ≈ +350 kV cm − uniformacross the field of view, in line with the values reported in Ref. [45]. Furthermore, since the magnetic field doesnot enter Equation (16), the assumption that there is no background stress away from the local features should notaffect the outcome for these four quantities, which then correspond to the net stress including a non-zero backgroundcontribution if present.This is in contrast with the individual axial components, which are determined through Equation (11) which involvesin a non-trivial way both the electric field and the magnetic field, which itself is determined under the assumptionof no background electric field or stress. For illustration purpose, let us consider the situation where σ xx is the onlynon-zero stress component. Equation (11) then reads D i = 4( γ NV B Z i ) + 4( − bσ xx − f i √ k ⊥ E z ) + 4( √ bσ xx ) (17)2Here, an error on B Z i would act as a simple offset on σ xx , while E z has a more subtle effect as it can add or subtractto the σ xx term depending on the NV orientation. Moreover, when the other stress components are present, an erroron the magnetic field may also act to bias the estimated stress values in a way that depends on the other stress values.In other words, the assumptions made may introduce not just a global offset on the stress maps but also a distortionof the spatial features.To test these effects, we used a set of experimental data and compared the stress maps produced by different analysisprocedures. Namely, instead of assuming no electric field and background stress, we estimated the magnetic field (farfrom the local stress features) by including an electric field along z , i.e. minimising ε ( D, (cid:126)B, E z ), or by including abackground axial stress along z , i.e. minimising ε ( D, (cid:126)B, σ zz ), or by including both at the same time. The stresstensor was then determining by fitting the whole image again (but including the stressed regions) while holding D , (cid:126)B and E z to the previously determined values. As expected, the shear components as well as Σ axial Z are essentiallyunchanged by the analysis procedure, except for a global offset in σ xy due to the electric field. On the other hand,these various methods affect the axial stress components not only as a global offset, but also by changing the valuesnear local features. For example, in Figure 2e of the main text, the increase in σ xx near the implanted region can varyfrom 20 MPa to nearly zero depending on the initial assumptions, compensated by an opposite variation in σ yy or σ zz (since the sum is well determined). This means that the uncertainty on these individual axial components can beup to 100%, calling for caution when extracting quantitative information from the corresponding maps. Nevertheless,the fact that the stress maps in Figure 2e of the main text exhibits the expected symmetry (e.g., σ xx maximum onthe x axis and minimum on the y axis) indicates that the different axial components have been sensibly separated.Finally, we note that the uncertainty on the spin-mechanical coupling parameters adds an additional uncertaintyon the stress values. Namely, the 5% uncertainty on a translates into a similar uncertainty on the shear stresscomponents, whereas the axial components are affected by the >
10% uncertainty on b . V. BODY FORCE APPROXIMATIONS
The body force equations are given by − f x = ∂ x σ xx + ∂ y σ xy + ∂ z σ xz − f y = ∂ x σ xy + ∂ y σ yy + ∂ z σ yz (18) − f z = ∂ x σ xz + ∂ y σ yz + ∂ z σ zz . The measured stress components are averaged over the thickness of the NV layer¯ σ ij ( x, y ) = 1∆ (cid:90) z − ∆ / z − ∆ / σ ij dz (19)where z and ∆ are the mean depth and thickness of the NV layer, assuming that the NV defects are uniformlydistributed within the slab. Our aim is to use the stress equations to express the averaged body forces¯ f i ( x, y ) = 1∆ (cid:90) z − ∆ / z − ∆ / f i dz (20)in terms of the measured averaged stress components ¯ σ ij so that we can use the measurements to obtain the averagebody forces and interpret their origin.To do this, let’s integrate the stress equations − (cid:90) z +∆ / z − ∆ / f x dz = 1∆ (cid:90) z +∆ / z − ∆ / ∂σ xx ∂x + ∂σ xy ∂y + ∂σ xz ∂z dz − (cid:90) z +∆ / z − ∆ / f y dz = 1∆ (cid:90) z +∆ / z − ∆ / ∂σ xy ∂x + ∂σ yy ∂y + ∂σ yz ∂z dz − (cid:90) z +∆ / z − ∆ / f z dz = 1∆ (cid:90) z +∆ / z − ∆ / ∂σ xz ∂x + ∂σ yz ∂y + ∂σ zz ∂z dz − ¯ f x = ∂ x ¯ σ xx + ∂ y ¯ σ xy + 1∆ [ σ xz ( z + ∆2 ) − σ xz ( z − ∆2 )] − ¯ f y = ∂ x ¯ σ xy + ∂ y ¯ σ yy + 1∆ [ σ yz ( z + ∆2 ) − σ yz ( z − ∆2 )] − ¯ f z = ∂ x ¯ σ xz + ∂ y ¯ σ yz + 1∆ [ σ zz ( z + ∆2 ) − σ zz ( z − ∆2 )] . If ∆ is sufficiently small (i.e. NV layer sufficiently thin), then we could immediately apply the above by making theapproximation that each third term of the right hand side is negligible, − ¯ f x = ∂ x ¯ σ xx + ∂ y ¯ σ xy − ¯ f y = ∂ x ¯ σ xy + ∂ y ¯ σ yy − ¯ f z = ∂ x ¯ σ xz + ∂ y ¯ σ yz . (21)This is equivalent to stating that the out-of-plane ( z ) derivatives of the respective stress components are negligible at z . Alternatively, we can appeal to Leibniz integral rule to identify ddz ¯ σ ij = 1∆ ddz (cid:90) z +∆ / z − ∆ / σ ij dz = 1∆ ( σ ij ( z + ∆) − σ ij ( z )) . Thus, the neglect of the third term in the above may instead be interpreted as stating that the average stresscomponents depend little upon the depth of the NV layer z (i.e. ∆ is sufficiently large to capture all z dependence).So, in either limit – ∆ much smaller than z variations or ∆ much larger than z variations – we can adopt theapproximate equation, Equation (21).In intermediate situations, one can derive an estimate for the z derivatives as follows. If we assume that the stressdecays in the z direction as σ ij ( z ) = σ ij (0) exp( − z/L ) where L is the characteristic decay length, then for L (cid:29) z , ∆we can approximate the z derivative term as1∆ [ σ ij ( z + ∆2 ) − σ ij ( z − ∆2 )] = σ ij ( z )∆ [ e − ∆2 L − e + ∆2 L ] (22) ≈ − ¯ σ ij L (23)hence − ¯ f x = ∂ x ¯ σ xx + ∂ y ¯ σ xy − ¯ σ xz L − ¯ f y = ∂ x ¯ σ xy + ∂ y ¯ σ yy − ¯ σ yz L − ¯ f z = ∂ x ¯ σ xz + ∂ y ¯ σ yz − ¯ σ zz L . (24) L is expected to be similar to the characteristic decay length observed in the xy plane, i.e. a few microns. Inpractice, we find that the third term in the right hand side of Equation (24) is negligible in calculating f x and f y ,but is dominant in calculating f z , i.e. we have ¯ f z ≈ ¯ σ zz L . In Figure 3 and 4 of the main text, the body force werecalculated using Equation (24), taking L = 3 µ m. VI. DIAMOND SAMPLES
The relevant parameters for the six diamonds used in this work, labelled
VII. SUPPLEMENTARY DATA: IMPLANTATION DAMAGEA. SRIM simulations
We performed full cascade Stopping and Range of Ions in Matter (SRIM) Monte Carlo simulations to estimate thedepth distribution of the damage created by C implantation with respect to the NV layer, assuming a diamond density4
Diamond Surface Energy Fluence Figure(keV) (ions/cm )
1, 4 & S7
2, S2 & S3 × S3, S4 & S5TABLE I. List of diamonds used in this work, indicating the type of surface (polished ‘P’, or as-grown ‘A’), the implantationenergy and fluence of the nitrogen implant used to create the NV defects, and the figure in which each sample appears in themain text and in this document. Samples and10 carbon cm − , respectively. of 3.51 g cm − and a displacement energy of 50 eV. Figure 6a shows the depth distribution of the N atoms implantedat 4 keV as in diamond molecule implant at 15 keV (fluence 3 . × molecules cm − )was performed to achieve a 5 keV C implant to a fluence of 10 atoms cm − . Figure 6b shows the vacancy and C depthdistributions from this implant which peak at about 6 and 10 nm, respectively. Due to the large vacancy concentration,amorphous carbon forms at the peak of the damage (the critical vacancy concentration for amorphisation in diamondis 2 × vacancies cm − [81]), as confirmed by Raman spectroscopy (not shown). We note that although SRIMis able to model the general shape of the implant profile well, it does not account for channeling effects and maytherefore underestimate the ion range [77]. Depth (nm) [ V ] ( c m ‐ ) [ N ] ( c m ‐ ) [ C ] ( c m ‐ ) nitrogen implant carbon implant Figure 6. Stopping and Range of Ions in Matter (SRIM) simulations of the N implant used to create the NV defects (a) and ofthe C implant used to create the damage (b). Plotted are the nitrogen concentration (a), the vacancy concentration (b, bluedata) and the implanted carbon concentration (b, red data), as a function of depth.
B. Stress from different carbon implant shapes
Various structures were fabricated with the C implant in diamond σ xy , which has the largest signal-to-noise ratio) and a line cut for different shapes. The square ring(Figure 7a,b) has a very similar stress pattern to the circular ring and has a maximum shear stress magnitude of | σ xy | ≈
15 MPa localised at the edges of the damaged region. Figure 7c,d show the case of closed squares of differentsizes, indicating that the maxima coincide with the edges of the carbon implant with a small increase of the magnitudeof shear stress with increasing size. The smallest squares (1 × µ m) are still resolvable on the right of the image. Linesof different widths are imaged in Figure 7e,f, showing more clearly the size dependence. A 1 µ m-wide line followinga zigzag pattern is imaged in Figure 7g,h, clearly showing that lines separated by less than 1 µ m can be resolved,allowing us to conclude that the spatial resolution of the technique is at least better than 1 µ m.5
20 μm P L ( a . u . ) P L ( a . u . ) P L ( a . u . ) P L ( a . u . )
20 μm 20 μm -10-50 σ xy ( M P a ) Position, x (µm)20 μm20 μm(a)(b) (c)(d) (e)(f) (g)(h) 20 μm20 μm -40-200 σ xy ( M P a ) Position, x (µm) -15-10-50 σ xy ( M P a ) Position, x (µm) -20-1001020 σ xy ( M P a ) -20-1001020 σ xy ( M P a ) -20-1001020 σ xy ( M P a ) σ xy ( M P a ) -20-100 σ xy ( M P a ) Position, x (µm)20 μm Figure 7. Stress induced from different shapes of carbon implant in diamond µ m. (b) Measured stress component, σ xy , from the square ring shown in (a) with a line cut along the dashed line.(c) NV PL of a series of solid squares and circles varying in width/radius from 10 µ m (left) to 1 µ m (right). (d) σ xy stressfrom (c) with a line cut along the dashed line. (e) NV PL of a series of solid lines varying in width from 0.1 µ m (left) to 5 µ m(right). (f) σ xy stress from (e) with a line cut along the dashed line. (g) NV PL of a solid lines with a width of 1 µ m in azigzag pattern. (h) σ xy stress from (g) with a line cut along the dashed line. (a) (c)(b) -20 -10 0 10 20 (MPa) -20 -10 0 10 20 (MPa) ( M P a ) Position, y (µm)Low fluence
High fluence xy xyxy
20 μm 20 μm xy xy
Figure 8. Effect of carbon implant fluence. (a,b) σ xy map of diamond to a fluence of 3 . × and 3 . × molecules cm − , respectively. (c) Line cuts along the dashed lines in (a,b). C. Effect of carbon implant fluence and post-annealing
We fabricated similar structures in a separate diamond (diamond implant to 3 . × molecules/cm . Despite this fluence being ten times as large as previously, we found the measuredstress to be actually smaller than in the low fluence sample, by roughly a factor of 2, as shown in Figure 8. This canbe explained by a difference in stress relief via the free surface, which was confirmed by AFM (Figure 9a) where thebulge of the diamond surface in diamond − [16, 42, 54]. The volume of the lattice probed by Raman is defined by the 100x objective (NA = 0.95) pointspread function rather than, as in the case of NV sensing, the position of the quantum probes and their own range ofview. This volume thus encompasses the region of the implant as well as the underlying substrate where a range of6 (a) (b) (d) Height, z (nm)20 μm H e i gh t, z ( n m ) Position, x (µm) Raman s hift (cm -1 ) Δ ω ( c m - ) Position, x (µm)(c)
Figure 9. AFM imaging and Raman spectroscopy in diamond µ m-wide implanted strip (horizontal). (d) Linecut of the relative shift in the diamond Raman line along the dashed line in (c). -10 0 10 (MPa) xyxy -10 0 10 (MPa) ( M P a ) Position, x, y (µm)(a) (c)(b)20 μm 20 μm xy xy xy Hor.Vert.
Figure 10. Effect of annealing in diamond σ xy map of an implanted region in diamond σ xy map of the same region after annealing at 800 ◦ C for 1 hour. (c) Line cuts along the colour coded dashed lines in (a) and (b). strains may exist that act to broaden the Raman signal and limit its accuracy. For simplicity, we assume a hydrostaticstress, σ h = σ xx = σ yy = σ zz , which can be linked to the change in the Raman shift position, ∆ ω , via the empiricalrelation σ h = ∆ ω/α h , where α h = 2 .
83 cm − GPa − [82]. The Raman measurement was performed with a RenishawInVia Raman system equipped with a 2400 l/mm grating and a 532 nm wavelength laser which was scanned across thesurface of the diamond to generate an image. A spatial map of the diamond line shift across a 5- µ m-wide implantedstrip in diamond ± .
005 cm − (Figure 9d).This leads to a stress of σ h = 7 ± σ xy ) of the sameregion before and after annealing the sample at 800 ◦ C for 1 hour are shown in Figure 10a and 10b, respectively. Thecomparison (Figure 10c) indicates that there is a small increase in the stress upon annealing of the order of 20%,similar to previous results [54].
VIII. SUPPLEMENTARY DATA: INDENTS
The impressions such as the one studied in Figure 3 of the main text were made by nano-indentation using aBerkovich tip. Figure 11a shows a typical load-unload curve, indicating a continuous deformation up to the maximumload of P max = 1 N. The maximum depth beneath the free surface is approximately 130 nm although this could beaffected by simultaneous deformation of the Berkovich indenter. The depth of the residual impression was 15 nm infair agreement with the AFM results. Similar load-unload curves were obtained for the ten indents we performed inthe same diamond. Figure 11b shows AFM images of three different indents. From these images, we can get a roughestimate of the contact area of the indent, A ∼ µ m , from which we derive the hardness H V = P max A ∼
200 GPa.7This is larger than the values typically reported for diamond which fall in the range 80-120 GPa [58, 83, 84], but thiscan be explained by the indentation size effect arising from the relatively small load used in our experiments [85, 86].We also note that the sub-surface defects and damage introduced by the N implant (used to create the NVs) mayinfluence the mechanical response of our diamonds. (a) (b)
Figure 11. Characterisation of indents in diamond
IX. SUPPLEMENTARY DATA: SCRATCHES
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