Diamond particles as nanoantennas for nitrogen-vacancy color centers
J.-J. Greffet, J.-P. Hugonin, M. Besbes, N.D. Lai, F. Treussart, J.-F. Roch
DDiamond particles as nanoantennas for nitrogen-vacancy color centers
J.-J. Greffet , J.-P. Hugonin , M. Besbes , N. D. Lai , F. Treussart , and J.-F. Roch Laboratoire Charles Fabry, Institut d’Optique, Univ Paris Sud,CNRS UMR 8520, 2 av Fresnel, 91127 Palaiseau Cedex, France Laboratoire de Photonique Quantique et Mol´eculaire & CNRS – UMR 8537,´Ecole Normale Sup´erieure de Cachan, 61 avenue du Pr´esident Wilson, F-94230 Cachan cedex, France
The photoluminescence of nitrogen-vacancy (NV) centers in diamond nanoparticles exhibits spe-cific properties as compared to NV centers in bulk diamond. For instance large fluctuations oflifetime and brightness from particle to particle have been reported. It has also been observed thatfor nanocrystals much smaller than the mean luminescence wavelength, the particle size sets a lowerthreshold for resolution in Stimulated Emission Depletion (STED) microscopy. We show that allthese features can be quantitatively understood by realizing that the absorption-emission of light bythe NV center is mediated by the diamond nanoparticle which behaves as a dielectric nanoantenna.
The negatively charged nitrogen-vacancy (NV − ) colorcenter in diamond is a solid-state artificial atom withunique room-temperature properties. Its perfectly sta-ble photoluminescence and an optically detectable elec-tron spin lead to a wide range of applications such asefficient on-demand generation of single photons [1] andhighly sensitive nanoscale magnetic and electric sensing[2–4]. These properties are preserved when NV centersare created in diamond particles [5, 6] with size down tothe nanometer range [7, 8]. The photoluminescence of asingle NV center in a nanocrystal was used for generat-ing single plasmons once coupled to metallic structures[9, 10]. Nanodiamonds embedding many NV centers havebeen developed for bioimaging purposes [11, 12]. More-over, due to its perfect photostability, the NV color centercan be revealed using far-field nanoscopy, based on eitherstimulated emission depletion (STED) or ground-statedepletion (GSD) microscopy techniques [13–16]. A 6-nanometer record resolution has been achieved in the caseof a NV defect in a bulk diamond sample [13]. These re-sults promise applications of NV-based luminescent nan-odiamonds for bio-imaging, with the potential of chal-lenging the use of fluorescent molecules which suffer fromlimited photostability.All these applications require to understand how theoptical properties of NV color centers are modified oncenestled in a nanodiamond. Specific nanoparticle effectshave been observed. For instance, the NV luminescencedecay time exhibits a broad statistical distribution whenrecorded from a set of nanodiamonds spincoated on acoverglass and the lifetime is on average longer thanthe value measured for a NV center in a bulk diamond[7]. This lengthening has been qualitatively attributedto modifications of the dielectric environment [5, 7]. Ithas also been noted that the lifetime distribution is notcorrelated to the brightness of the emitter [7, 15]. Fi-nally, although a 6 nm resolution was reached in bulkdiamond with STED imaging, the image of a single NVcenter in a 40 nm nanodiamond was limited by the sizeof the hosting particle [15, 16].In this letter, we show that the differences between emission by a NV center in a bulk or in a nanoparti-cle can be understood by considering that the dielectricnanoparticle acts as a dielectric nanoantenna similarly todiamonds nanopillars [17]. Furthermore, we demonstratethat the optical resolution of far-field nanoscopy tech-niques like STED imaging is intrinsically limited to thenanodiamond size when considering the regime of parti-cle dimension smaller than the emission wavelength.The optical properties of the NV center in the nanodi-amond can be described using a classical electric dipolein a lossless dielectric particle of index of refraction n .To study the influence of the nanoparticle on its emis-sion lifetime, we compute the ratio of the total poweremitted by a point-like monochromatic electric dipole ina nanosphere and in a bulk host. This ratio is equal tothe emission rate Γ np in a nanoparticle normalized bythe emission rate Γ b in the bulk. The calculations aredone using the formalism developed by Mie [18–21]. Asimilar discussion has been reported for an ensemble offluorophores embedded in a dielectric particle [20, 22].In such configurations, the question of the local field dueto the near-field environment of the fluorophore playsa key role. Here, we consider a single NV center in adiamond nanosphere. We note that the local field cor-rection is an irrelevant question as the NV center can-not be defined independently of the vacancy in the dia-mond lattice. Hence, we have normalized the emissionrate of a NV center in a particle by the emission rateof the same NV center in a bulk diamond. In the ab-sence of non-radiative decay, this ratio is equal to theratio of local density of states (LDOS) [23]. Although adiamond nanoparticle is not a microcavity sustaining asingle mode, we will refer to this normalized LDOS asPurcell factor F (cid:48) p , the prime indicating that emission inbulk diamond is taken as a reference. The Purcell factoris displayed in Fig. 1 in the case of a dipole located atthe center of the nanosphere when varying the radius a ofthe nanoparticle. We observe that F (cid:48) p < a (cid:28) λ where λ isthe vacuum wavelength at the dipole emission frequency.When the particle radius becomes larger than λ/n , the a r X i v : . [ phy s i c s . op ti c s ] J u l Purcell factor first increases and then oscillates due toMie resonances in the sphere [20, 22].
Figure 1. Purcell factor associated with radiative emission ofa NV center located at the center of a spherical nanodiamondfor different radii a.
Let us now analyse in more details the influence ofthe particle size on the emission rate. To understandthe emission rate reduction in the electrostatic regime,we invoke the reciprocity theorem. In simple words, thereciprocity theorem states that a detected amplitude doesnot change upon exchange of the positions of a source anda detector [24]. Hence, the far-field amplitude radiatedin direction u by a dipole located at r inside the particleis proportional to the field created at r by an incidentplane wave illuminating the particle propagating along − u . It follows that we can replace a far-field radiationproblem by the computation of the field in the particleilluminated by a plane wave. Fig. 2 shows the intensity | E | in the particle with 20 nm radius illuminated by aplane wave linearly polarized along Ox. xz |E| x a) b) Figure 2. Coupling between an embedded dipole in a 20 nmradius diamond nanoparticle and the far field. a) Square mod-ulus of the electric field inside the sphere excited by a planewave at 637 nm wavelength, corresponding to the zero phononline of the NV − emission. b) Square modulus of the field alongthe x axis. It is seen that the field is uniform within the sphereand much smaller than the incident field. This meansthat the Purcell factor does not depend on the positionof the dipole in the particle. Furthermore, for a dipoleat the center, the spherical symmetry entails that thePurcell factor does not depend on polarization. Not- ing that for a small particle, the electrostatic approxi-mation is valid, it follows that the internal field inten-sity is given by (cid:12)(cid:12)(cid:12)(cid:12) n + 2 E inc (cid:12)(cid:12)(cid:12)(cid:12) [24] where E inc is the in-cident field amplitude. The low value of the field ina diamond nanoparticle is a manifestation of dielectricscreening. To recover analytically the Purcell factor, westart by introducing the fictitious emission rate Γ vac ofa NV center in a vacuum as a non-physical but conve-nient intermediate parameter. The emission rate of adipole in a bulk dielectric of index of refraction n is re-lated to Γ vac by the relation Γ b = n Γ vac which accountsfor the ratio of densities of states. Using the reciprocityargument, the emission rate in the nanosphere is relatedto the emission rate in a vacuum by accounting for thedielectric screening factor Γ np = Γ vac (cid:18) n + 2 (cid:19) . It fol-lows that F (cid:48) p = 1 n (cid:20) n + 2 (cid:21) . Using n = 2 .
4, we find F (cid:48) p = 0 . . As expected, the spa-tial structure of the field in the particle reproduces themode field (not shown). According to the reciprocity the-orem, if the source dipole is located at a maximum of thefield, its radiation will be enhanced. We plot in Fig. 3(b)the field produced by the dipole when it is located at thepoint indicated by the arrow. Noting that the field struc-ture matches the spatial pattern of the resonant modefield, we conclude that the dipole radiation is mediatedby the particle mode. Hence, emission of the NV centerin the sphere is a two-step process: i) the dipole excitesresonantly a mode, ii) the mode radiates in the vacuum.The far-field angular emission pattern is shown in Fig.3(c) with an angular oscillation which again reproducesthe angular pattern of the mode in the sphere. It followsthat the brightness in a given solid angle can be almostnull although the Purcell factor can be large. Hence, fora single emitter, one should not expect a direct correla-tion between the brightness in a given direction and thelifetime in the case of a particle large enough to supportMie resonances. Obviously, this is no longer correct if allthe radiation is collected using e.g. an integrating sphereor for an ensemble of emitters in a sphere.From the reciprocity argument, we know that thestrength of the coupling of the emitting dipole to themode depends on the amplitude of the mode field at thedipole source position. Hence, we expect a strong de-pendence of the Purcell factor on the position of the NVcenter in the particle, as shown in Fig. 3(d). As ex-pected from Fig. 3(a), the Purcell factor is weak whenthe emitter is in the center of the particle and becomeslarge when the emitter is located close to a maximum ofthe mode. In summary, our analysis in the Mie regimeshows that the emission of the NV center is mediated bythe modes of the diamond particle. This nanoantennaeffect can efficiently control the lifetime and the angularemission of the point defect in the nanodiamond. a) b)c) d) Figure 3. Resonant excitation of a single mode structure ina diamond nanoparticle of a = 621 .
27 nm radius. (a) Squaremodulus of the electric field in a diamond sphere excited bya plane wave. (b) Square modulus of the electric field excitedby a dipole. The arrow indicates its position. (c) angularemission pattern for an orthoradial dipole in the particle, (d)Purcell factor versus emitter position (with orthoradial polar-ization) .
We emphasize that Fig. 3 corresponds to the specificcase for which the dipole frequency is tuned to a singleMie resonance mode inside the nanodiamond. The gen-eral case is the simultaneous excitation of several modes.An example is shown for the sake of illustration in Fig.4 for the case of a single dipole orientation with a fre-quency that does not match exactly one resonance. ThePurcell factor is then significantly lower and the angularemission pattern becomes more directional. In practice,further averaging effects are due to the existence of twoorthogonal transition dipoles in the NV center electronicstructure [25] and to its broad luminescence spectrum.Here, we have highlighted the resonant properties of thedielectric particle. This introduces the issue of the inter-play between a microcavity and a broad emitter that hasbeen discussed in the context of semiconductor quantumdots and large quality factors [26] and in the case of NVcenter coupled to a low quality Fabry-Perot microcavity[27]. The emission of the NV center in a nanodiamondsupporting Mie resonances corresponds to a broad emit-ter coupled to a multimode cavity with its free spectralrange determined by the radius of the nanoparticle. Afurther discussion of the emission spectrum modification is beyond the scope of this letter. a) b)c) d)
Figure 4. Same as Fig. 3 but with a = 621 nm. The slightdetuning drastically modifies the excitation of the modes (a,b)and consequently the angular emission pattern (c) and thePurcell factor (d). In practice, nanoparticles are usually deposited on thesurface of a dielectric substrate which modifies the envi-ronment of the particle. We have used a finite elementtechnique in order to account for the presence of the di-electric interface in the evaluation of F (cid:48) p . Following theexperiment reported in Ref. [5], we consider a particleof 45 nm radius deposited on a silica microscope cover-slip with n = 1 .
5. As the nanoparticles are embeddedin a thin polymer film deposited on the substrate, theparticle is considered in a dielectric environment of samerefractive index. Such parameters then correspond to anintermediate regime between the electrostatic case andthe Mie regime (see Fig. 1). Without accounting for theinterface, the Purcell factor F (cid:48) p is around 0 .
1. The nu-merical simulation with an interface yields larger valuesof this parameter, with a maximum (cid:39) .
39 for a dipoleparallel to the interface. Given the uncertainty on parti-cle shape and size and the fact that the model does notaccount for any nonradiative decay [28], this result can beconsidered to be in fair agreement with the approximateexperimental value (cid:39) .
47 [5].We finally address the question of imaging a NV cen-ter in the nanodiamond using the STED technique. Sincea 6 nm record resolution [13] in a bulk diamond samplehas been demonstrated, the technique is expected to beable to image a NV center in a nanoparticle. However,all attempts have failed so far, the signal appearing asdelocalized over the particle. To explain these results,we again use the view that a dielectric nanoparticle be-haves as a nanoantenna. Hence, the structured excitingfield used in the STED technique excites first the parti-cle modes which in turn, excite the defect. For a particlemuch smaller than the wavelength, only the electric dipo-lar mode is significant. Since this mode is uniform in theparticle (see Fig.2(a)) the NV center is equally excitedindependently of its location. To illustrate this behav-ior, we display the field produced by a structured beamwhich illuminates the particle. Following Ref. [13], thisbeam is generated by applying a phase shift [29] to theFourier components of an input circularly polarized beamwhich is focused on the particle with a high numericalaperture microscope objective. Figure 5 shows a simu-lation carried out for NA= 0 . a = 1000 nm, the field becomes uniform in theparticle of sub-wavelength size. This can be understoodfrom the previous discussion. Indeed the NV center ex-citation is mediated by the mode of the particle whichis the dipolar one for small particles. Since the electricfield of the dipolar mode is uniform in the particle, thespatial structure of the incident field is lost so that theSTED resolution is limited by the particle size. a) b) c)
160 nm50 nm 1000 nm
Figure 5. Square modulus of the electric field in a dielectricnanosphere illuminated by a structured field, with radii a =50 nm, a = 160 nm (b) and a = 1000 nm (c). In summary, we have discussed how the emission prop-erties of a NV center in a diamond nanoparticle can beanalysed using the concept of nanoantenna. The cou-pling of a NV center to electromagnetic fields is medi-ated by the modes of the particle in the Mie regime andis dominated by dielectric screening in the case of a sub-wavelength particle. This simple mechanism provides aunified picture of the NV optical properties as a functionof the nanodiamond size. More generally, the concept ofdielectric nanoantenna can be applied to any dielectricnanostructure containing a luminescent center.This work was supported by Triangle de la Physiquecontract 2008-057T. We are grateful to Vincent Jacquesfor many helpful discussions. [1] A. Beveratos, S. Kuhn, R. Brouri, T. Gacoin, J.-P.Poizat, and P. Grangier, Eur. Phys. J. D 18, 191 (2002)[2] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M.Taylor, P. Cappellaro, L. Jiang, M. V. Gurudev Dutt, E.Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, andM. D. Lukin, Nature 455, 644, (2008).[3] G. Balasubramanian, I.Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P.R. Hem-mer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brat-schitsch, F. Jelezko, and J. Wrachtrup, Nature 455, 648(2008). [4] F. Dolde, H. Fedder, M. W. Doherty, T. N¨obauer, F.Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L.C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, Nat.Phys. 7, 459 (2011).[5] A. Beveratos, R. Brouri, T. Gacoin, J.-P. Poizat, and P.Grangier, Phys. Rev. A 64, 061802 (2001)[6] F. Treussart, V. Jacques, E. Wu, T. Gacoin, P. Grangier,and J.-F. Roch, Physica B 376, 926 (2006)[7] J. Tisler, G. Balasubramanian, B. Naydenov, R. Kolesov,B. Grotz, R. Reuter, J.-P. Boudou, P. A. Curmi, M. Sen-nour, A. Thorel, M. Borsch, K. Aulenbacher, R. Erd-mann, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, ACSNano 3, 1959 (2009).[8] L. Rondin, G. Dantelle, A. Slablab, F. Grosshans, F.Treussart, P. Bergonzo, S. Perruchas, T. Gacoin, M.Chaigneau, H.-C. Chang, V. Jacques, and J.-F. Roch,Phys. Rev. B 82, 115449 (2010).[9] R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stohr,A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J.Wrachtrup, Nature Physics 5, 470 (2009).[10] A. Cuche, O. Mollet, A. Drezet, and S. Huant, Nano Lett.10, 4566 (2010).[11] Y. Chang, H. Lee, K. Chen, C. Chang, D. Tsai, C. Fu,T. Lim, Y. Tzeng, C. Fang, C. Han, H. Chang, and W.Fann, Nature Nanotech. 3, 284-288 (2008).[12] O. Faklaris, V. Joshi, T. Irinopoulou, P. Tauc, M. Sen-nour, H. Girard, C. Gesset, J.-C. Arnault, A. Thorel,J.-P. Boudou, P. A. Curmi, and F. Treussart, ACS Nano3, 3955-3962 (2009)[13] E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, andS. W. Hell, Nature Photonics 3,144 (2009).[14] E. Rittweger, D. Wildanger, and S. W. Hell, Europhys.Lett. 86, 14001 (2009).[15] K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C.Eggeling, and S. W. Hell, Nano Lett. 9, 3323 (2009).[16] Y.-K. Tzeng, O. Faklaris, B.-M. Chang, Y. Kuo, J.-H.Hsu, and H.-C. Chang, Angew. Chem. Int. Ed. , 2262(2011).[17] T.M. Babinec, B.J.M. Hausmann, M. Khan, Y. Zhang,J. R. Maaze, P.R. Hemmer and M. Loncar, Nature Nan-otech. , 195 (2010)[18] C.F. Bohren and D.R. Huffman, Absorption and Scatter-ing of Light by Small Particles, (Wiley, NY, 1983)[19] S.D. Druger, S.Arnold, and L.M. Folan, J. Chem. Phys. , 686 (1991)[21] S. Lange, and G. Schweiger, J. Opt. Soc. Am. B , 2444(1994)[22] H. Schniepp and V. Sandoghdar, Phys. Rev. Lett. ,257403 (2002)[23] J.-J. Greffet, M. Laroche, and F. Marquier, Phys. Rev.Lett. , 117701 (2010)[24] L.D. Landau and E.M. Lifshitz, Electrodynamics of con-tinuous media, (Pergamon Press, Oxford, 1984)[25] F. Kaiser, V. Jacques, A. Batalov, P. Siyushev, F.Jelezko, and J. Wrachtrup, arXiv:0906.3426[26] A. Auffeves, J.M. G´erard, and J.-P. Poizat, Phys. Rev.A , 053838 (2009)[27] Y. Dumeige, R. All´eaume, P. Grangier, F. Treussart, andJ.-F. Roch, New J. of Physics , 025015 (2011)[28] B. R. Smith, D. Gruber, and T. Plakhotnik, Diam. Relat.Mat. , 314 (2010)[29] A Fourier component E ( k x , k y ) of an incident Gaussianbeam is multiplied by exp[ iφ ] with tan( φ ) = k x /k yy