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Radiative Heat Transfer between Neighboring Particles Alejandro Manjavacas∗ and F. Javier Garc´ıa de Abajo† IQFR - CSIC, Serrano 119, 28006 Madrid, Spain (Dated: January 30, 2012) The near-field interaction between two neighboring particles is known to produce enhanced radiative heat transfer. We advance in the understanding of this phenomenon by including the full electromagnetic particle response, heat exchange with the environment, and important radiative correctionsbothinthedistancedependenceofthefieldsandintheparticleabsorptioncoefficients. Wefindthatcrossedtermsofelectricandmagneticinteractionsdominatethetransferratebetween gold and SiC particles, whereas radiative corrections reduce it by several orders of magnitude even 2 at small separations. Radiation away from the dimer can be strongly suppressed or enhanced at 1 low and high temperatures, respectively. These effects must be taken into account for an accurate 0 description of radiative heat transfer in nanostructured environments. 2 n a J Contents 6 2 I. Introduction 2 ] l l II. Description of the model 2 a h III. Results and discussion 6 - s e IV. Concluding remarks 6 m . Acknowledgments 7 t a m A. Particle polarizability 8 - d B. Validity of the dipolar approximation 9 n o C. Derivation of Eq. (1) 9 c [ D. Derivation of Eq. (2) 11 1 v E. Fluctuation-dissipation theorem 11 3 1. FDT for induced dipole fluctuations 13 7 6 F. Fluctuation-dissipation theorem for electromagnetic fields 14 5 1. Electric field fluctuations 14 . 1 2. Magnetic field fluctuations 15 0 3. Electric-magnetic field fluctuations 16 2 4. Magnetic-electric field fluctuations 17 1 : v G. Derivation of Eq. (4) 18 i X H. Symmetric and asymmetric heat transfers in inhomogeneous dimers 20 r a I. Total rate of heat loss and fraction of power exchange 20 References 21 ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] 2 I. INTRODUCTION Blackbody radiation mediates heat exchange between bodies placed in vacuum and separated by large distances d compared to the thermal wavelength λ = 2π(cid:126)c/k T. For parallel plates, this leads to a radiative heat transfer T B (RHT)rateindependentofd. However, whend(cid:28)λ , therateisenhancedbyseveralordersofmagnitudeduetothe T involvementofevanescentwaves. Pioneeringmeasurements[1,2]revealedthisphenomenon,whichwasfirstexplained intermsofnear-fieldfluctuations[3]. Afteralongseriesofexperimental[4,5]andtheoretical[3,6–14]studies,recent observationshaveaccuratelyconfirmeda1/d2 dependenceforsapphireplatesatroomtemperaturedowntod∼1µm [5],andalsoa1/ddependenceforlargesilicaspheresplacednearasilicaplatedowntod∼30nm[15],althoughthese lawscanbesubstantiallycorrectedbynonlocal[11],phonon[16,17],andphotoniccrystal[18]effects. Inthiscontext, the interaction of a particle with a plate has been explored both from experimental [4, 15, 19, 20] and theoretical [12, 16, 21–24] fronts. Modeling heat exchange between two [25–31] or more [32] particles has been the subject of intense activity as well. Magnetic polarization has been claimed to dominate RHT between metallic nanoparticles [27]. However, electromagnetic crossed terms (EMCTs, i.e., terms mixing the electric and magnetic particle responses) have been ignored so far, although they could play a leading role in transfers within heterogeneous structures. Likewise, radiative corrections in the absorption of dielectric particles deserve further consideration, as we show below. Thus, the current level of understanding of RHT between two particles appears to be incomplete. Here,weformulateacompletesolutionofRHTbetweentwonanoparticleswithintheassumptionofdipolarresponse. We show that EMCTs are dominant in combinations of metallic and dielectric particles, such as gold and SiC. We introducearelevantretardationcorrectionbeyondthecustomarytreatmentofpolarizationfluctuations,whichresults in a sizable reduction in the predicted transfer rate. Furthermore, we show that heat losses into the environment can be either dominant or negligible depending on the temperature and particle composition. An accurate description of RHT in nanostrutured environments requires incorporating these effects, for which the two-particle system discussed here provides a tutorial approach as well as an estimate of the importance of EMCTs, radiative corrections, and interaction with the environment. II. DESCRIPTION OF THE MODEL We consider two spherical particles of radius R at temperatures T and T separated by a center-to-center distance 1 2 dalongthez directionandplacedinavacuumattemperatureT ,asshowninFig.1. Wefocusonsmallparticlessuch 0 that R (cid:28) d, λ , so that their responses can be described through the polarizabilities α and α [26] (see Appendix, T 1 2 Fig. 6). RHT between the particles and the environment is produced by fluctuations in the vacuum electromagnetic field and the particle dipoles. We simplify the notation by combining electric and magnetic field components acting T 0 R T d T 1 2 FIG. 1: (color online). Description of the system under study. Two particles of radius R at temperatures T and T separated 1 2 byadistancedareplacedinavacuumattemperatureT . Eachparticleexchangesthermalenergywiththeotherparticleand 0 with the surrounding vacuum. 3 on each particle j =1,2, as well as electric (p) and magnetic (m) dipoles, in the vectors  E   p  j,x j,x E p  j,y   j,y   E   p  E = j,z , p = j,z , j H  j m   j,x   j,x   H   m  j,y j,y H m j,z j,z respectively. Likewise, the polarizability tensor becomes (cid:18)αEI 0 (cid:19) α = j 3 , j 0 αMI j 3 where I is the 3×3 identity matrix and the E (M) subscript refers to electric (magnetic) components. 3 The net power absorbed by particle 1 is the sum of dipole and field fluctuation terms, P =Pfield+Pdip. 1 1 1 More precisely (see Appendix), (cid:90) ∞ dωdω(cid:48) Pfield = e−i(ω−ω(cid:48))tω(cid:48) (1) 1 (2π)2 −∞ ×(cid:10)E+(ω(cid:48))(cid:2)iα+(ω)−(2k3/3)|α(ω)|2(cid:3)E (ω)(cid:11) 1 1 representstheworkexertedbythefluctuatingfieldonparticle1. Here,k =ω/cisthewavevectoroflightatfrequency ω. Moreover, the self-consistent fields E include the response of the system to the fluctuating source fields Efl via j j the relations E =Efl+G α E , 1 1 12 2 2 E =Efl+G α E , 2 2 21 1 1 where G is the distance-dependent dipole-dipole inter-particle interaction, 12  A 0 0 0 −C 0  0 A 0 C 0 0    0 0 B 0 0 0  G = , 12  0 C 0 A 0 0    −C 0 0 0 A 0  0 0 0 0 0 B A=exp(ikd)(k2/d+ik/d2−1/d3), B =exp(ikd)2(1/d3−ik/d2), C =exp(ikd)(k2/d+ik/d2), andG takesthesameformasG withC replacedby−C. InEq.(1),(cid:104)(cid:105)representstheaverageoverfieldfluctuations, 21 12 which we perform by applying the fluctuation-dissipation theorem (FDT) [33, 34] (see Appendix) (cid:68)(cid:2)Efl(ω)(cid:3)+Efl(ω(cid:48))(cid:69)=4π(cid:126)δ(ω−ω(cid:48))η (ω)(cid:20)n (ω)+ 1(cid:21), j j(cid:48) jj(cid:48) 0 2 whereη =(1/2)[Im{G +G }+iRe{G −G }],η =η+,η =η =(2k3/3)I ,andn (ω)=[exp((cid:126)ω/k T )− 12 12 21 12 21 21 12 11 22 6 0 B 0 1]−1 is the Bose-Einstein distribution at the vacuum temperature T . 0 4 Likewise, the contribution of fluctuating dipoles is (see Appendix) (cid:90) ∞ dωdω(cid:48) Pdip = e−i(ω−ω(cid:48))t (2) 1 (2π)2 −∞ (cid:2)iω(cid:48)(cid:10)p+(ω(cid:48))G (ω)p (ω)(cid:11)− 2ω4 (cid:10)p+(ω(cid:48))p (ω)(cid:11)(cid:3), 1 12 2 3c3 1 1 where the first term inside the square brackets accounts for the effect of the field produced by particle 2 on particle 1, while the second term describes the interaction of the particle dipole with the vacuum. The self-consistent dipoles satisfy the relations p =pfl+α G p , 1 1 1 12 2 p =pfl+α G p , 2 2 2 21 1 where pfl is the fluctuating source dipole at particle j. The relevant FDT now becomes [33, 34] (see Appendix) j (cid:68)(cid:2)pfl(ω)(cid:3)+pfl(ω(cid:48))(cid:69)=4π(cid:126)δ(ω−ω(cid:48))δ χ (ω)(cid:20)n (ω)+ 1(cid:21), j j(cid:48) jj(cid:48) j j 2 where we use χ =Im{α }−(2k3/3)|α |2I , (3) j j j 6 rather than Im{α } in order to prevent non-absorbing particles from undergoing unphysical fluctuations. We set the j polarizabiltiy to αν =(3/2k3)tν (ν =E,M), where tν is the dipole Mie scattering coefficient (see Appendix). This j j,1 i,1 definition of α complies with the optical theorem condition [35] Im{α } ≥ (2k3/3)|α |2, where the equality applies j j j to non-absorbing particles (χ =0). Incidentally, dipole and field fluctuations originate in different physical systems, j and therefore, there are not crossed terms between the two of them. Finally,usingtheFDTtoevaluatetheintegralsofEqs.(1)and(2),wefind,aftersomelengthybutstraightforward algebra (see Appendix), P = 2(cid:126) (cid:88) (cid:90) ∞ωdω χν(cid:88)2 (cid:18)2Γνi⊥ + Γνi(cid:107) (cid:19), (4) 1 π 1 |s|2 |t |2 ν=E,M 0 i=1 ν where (cid:20)2k3 (cid:16) (cid:17) Γν = |u |2+|w |2 1⊥ 3 ν ν +Im(cid:8)αν[Au −g Cw ][Au∗ +g Cw∗](cid:9) 2 ν ν ν ν ν ν (cid:21) +Im(cid:8)αν(cid:48)[Aw −g Cu ][Aw∗+g Cu∗](cid:9) (n −n ), 2 ν ν ν ν ν ν 0 1 Γν =(cid:0)χν|Au −g Cw |2+χν(cid:48)|Aw −g Cu |2(cid:1)(n −n ), 2⊥ 2 ν ν ν 2 ν ν ν 2 0 Γν =(cid:20)2k3 +Im(cid:8)ανB2(cid:9)(cid:21)(n −n ), 1(cid:107) 3 2 0 1 Γν =χν|B|2 (n −n ), 2(cid:107) 2 2 0 u =1−αν(cid:48)αν(cid:48)A2+αν(cid:48)ανC2, ν 1 2 1 2 w =αν(cid:48)(cid:0)αE −αM(cid:1)AC, ν 1 2 2 s=1−αEαEA2−αMαMA2+αEαMC2+αMαEC2 1 2 1 2 1 2 1 2 +αEαEαMαM(cid:0)A2−C2(cid:1)2, 1 2 1 2 t =1−ανανB2, ν 1 2 and ν(cid:48) =M (E) and g =+1 (−1) when ν =E (M). ν Neglecting the magnetic response (αM = 0, C = 0), multiple scattering (u = t = s = 1, w = 0), and radiative j ν ν ν 5 10-13 T = 300 K Full theory (a) Single scattering R = 500 nm Non-retarded approx. Non-retarded approx. + χ Diagonal approx. 10-16 ) 1 -K W 10-19 Au Au Au SiC ( t n e (b) 1.5% i c i f2·10-12 f e 1.0% o Au Au Au SiC c er1·10-12 0.5% f s n a tr 0 0.0% at 10-14 EE-term EM-term(c) He Au Au MM-term ME-term 10-17 10-15 (d) Au SiC 10-18 2 4 6 8 10 d (µm) FIG. 2: (color online). Dependence of the radiative heat transfer coefficient (HTC) on particle separation d. The HTC from particle2(right)toparticle1(left)isdefinedasP /δT withT =T =T andT =T+δT (seeFig.1). (a)HTCatT =300K 1 1 0 2 obtainedfromourfulltheory(solidcurves),comparedtotheresultofneglectingmultiplescattering(circles),retardationeffects everywhere (dashed curves), retardation effects except in χ (triangles), or EMCTs (dotted curves). Gold-gold (red curves) j andgold-SiC(bluecurves)dimersareconsidered. TheparticlesradiusisR=500nm. (b)Heatpowerlostbyparticle2(solid curves) and fraction of that power absorbed by particle 1 (broken curves, right scale). (c,d) Electric-electric (black), electric- magnetic (red), magnetic-magnetic (blue), and magnetic-electric (green) partial contributions to the HTC in the gold-gold (c) and gold-SiC (d) dimers. corrections in the particles response (χ =Im{α }), the above expressions reduce to j j P = 4(cid:126)(cid:90) ∞ωdω Im(cid:8)αE(cid:9) (cid:26)(cid:104)k3+2Re(cid:8)αEA(cid:9)Im{A} 1 π 1 2 0 +Re(cid:8)αEB(cid:9)Im{B}(cid:105)(n −n ) 2 0 1 (cid:27) +Im(cid:8)αE(cid:9)(cid:0)|A|2+|B|2/2(cid:1) (n −n ) , 2 2 1 where the n −n term describes direct RHT between the dimer particles and coincides with a previously reported 2 1 expression[22,25,30]. Theremainingn −n termaccountsforheatexchangebetweenparticle1andthesurrounded 0 1 vacuum, partially assisted by the presence of particle 2. 6 III. RESULTS AND DISCUSSION We study in Figs. 2, 3(a), and 4 the heat transfer coefficient (HTC) between two particles in a dimer when particle 1 is at the same temperature as the environment (T = T = T) and particle 2 is at a slightly different temperature 1 0 (T =T+δT). TheHTCtoparticle1isdefinedperunitoftemperaturedifferenceasP /δT. Undertheseconditions, 2 1 onlythetermsΓν andΓν contributetothetransfer. Theresultsobtainedfromtheaboveformalism(solidcurves) 2,⊥ 2,(cid:107) are compared to several approximations consisting of neglecting multiple scattering between the particles (circles, calculated for u = t = s = 1, w = 0), retardation effects everywhere (dashed curves, k = 0), retardation effects ν ν ν except in the particle response χ defined by Eq. (3) (triangles), or EMCTs (dotted curves). j The distance dependence of the HTC is analyzed in Fig. 2 for a homogeneous gold dimer and for a dimer formed by gold and SiC particles. As a first observation, we note that multiple scattering events can be safely neglected in all cases. In contrast, retardation causes a dramatic boost in the HTC, which increases with particle distance in both types of dimers. Additionally, the |α|2 term of Eq. (3) contributes with a uniform decreasing factor in SiC particles (see Appendix, Fig. 5). (Notice that the absorption cross section is proportional to χ , whereas Im{α } j j describes absorption plus scattering, so the |α|2 term in χ is removing scattering strength that is not associated j with absorption.) Finally, EMCTs terms introduce additional channels of inter-particle interaction, thus resulting in higher transfer rates compared to the diagonal approximation (consisting of only including electric-electric and magnetic-magnetic terms), particularly in the heterogeneous dimer [Fig. 1(a)]. Figure 2(c) clearly shows that the magnetic-magnetic terms are dominant in the homogeneous gold dimer, in agreement with previous predictions [27], because metallic particles mainly contribute through magnetic polarization. This is unlike the heterogeneous cluster, in which magnetic-electric terms are dominant [Figure 2(d)], thus picking up a dominant electric polarization from the SiC particle. Incidentally, the HTCs from gold to SiC and from SiC to gold are nearly identical (see Appendix, Fig. 8). SimilarconclusionsareextractedfromthetemperaturedependenceoftheHTC,representedinFig.3(a)forasmall distance d = 2µm(cid:28) λ = 14µm−14mm. Notice however the dramatic reduction in the transfer rate produced by T retardation at high temperatures. The full dependence on the particle temperatures for a vacuum at T = 0 is studied in Fig. 3(c,d). Interestingly, 0 particle 1 gets cooled down (blue regions) even if particle 2 is at a higher temperature. This is due to radiation losses into the vacuum. However, particle 2 in the homogeneous cluster is rather efficient in transferring energy to particle 1 and compensating for radiation losses, so that the curve separating gains (red) from losses (blue) is closer to the T =T line (dashed) in that dimer [Fig. 3(c)]. 1 2 It is useful to analyze the spectral contribution of different photon energies to the HTC. At a low temperature T = 10K [Fig. 4(a)], the exchange is dominated by low photon energies, for which the particles polarization show a featurelessbehaviorandχ almostcoincideswithIm{α }forthesizeoftheparticlesunderdiscussion(seeAppendix, j j Fig. 5). At high temperature T =1000K [Fig. 4(b)], optical phonons emerge as a sharp infrared (IR) feature in SiC and plasmons show up as a broader near-IR feature in gold particles. Retardation effects also increase with T, as the particles appear to be large in front of λ . T An important ingredient that is often overlooked in the analysis of heat transfer relates to how much energy is emitted into the surrounding vacuum. We analyze this in Figs. 2(b) and 3(b) by calculating the power escaping from a hotter particle 2. The calculation is done by reversing the particle labels, so that only the terms Γν and Γν 1,⊥ 1,(cid:107) contribute in this case [see Eq. (4)]. For the temperature of Fig. 2, just a small amount of the energy emanating from particle 2 ends up in particle 1. However, this fraction increases at lower temperatures [Figs. 3(b)], until nearly completeheattransfertakesplacebelow∼10Kinthehomogeneousgolddimer. Thefractionofheattransferbetween the particles is thus very sensitive to temperature and particle distance (see Appendix, Fig. 9). IV. CONCLUDING REMARKS Heatdissipationinnanostructureddevicesisbecomingalimitingfactorinthedesignofmicrochipsandisexpected to play a major role in nanoelectronics, nanophotonics, and photovoltaics. Radiative losses provide a convenient way of handling the excess of heat produced in these devices [36]. In this context, crossed magnetic-electric terms and radiative corrections as those described here produce modifications in the transfer rate by up to several orders of magnitude, which cannot be overlooked. An analysis of how much heat is released from a dimer into a cooler vacuum reveals a large dependence on composition and temperature (it is strongly suppressed at low temperatures and dominant in hot environments). Our results for gold-SiC dimers suggest the experimental exploration of these effects via, for example, in-vacuum particle levitation, or by attaching one of the particles to a nanoscale tip and the other one to an insulating substrate. As an interesting direction, we note that RHT can be strongly modified 7 10-13 (a) R = 500 nm d = 2 µm 10-16 ) 1 -K Au Au W 10-19 ( Au SiC nt 10-22 Full theory e Single scattering i c Non-retarded approx. i ff 10-25 Non-retarded approx. + χ e Diagonal approx. o c r 10-9 100% e f (b) s n 10-13 Au Au a r t at 10-17 50% e Au SiC H 10-21 10-25 0% 1 10 100 1000 T(K) Au Au Au SiC 1000 (c) T = 0 K (d) 0 100 ) K ( 2 T 10 1 1 10 100 1000 10 100 1000 T (K) T (K) 1 1 -5·10-9 5·10-9 Heat transfer power (W) FIG.3: (coloronline). TemperaturedependenceofRHT.(a)HTCfortheparticlesofFig.2. (b)Heatpowerlostbytheright particleandfractionabsorbedbytheleftparticle. (c)Powerabsorbed(red-yellowscale,positive)oremitted(blue-whitescale, negative)bytheleftparticleofagold-golddimerasafunctionofT andT withthevacuumatT =0(seeFig.1). (d)Same 1 2 0 as (c) for a gold-SiC dimer. The particle distance is d=2µm and the radius is R=500nm in all cases. by the presence of additional mirrors and dielectrics that distort the exchanged electromagnetic fields. We further suggest the possibility of molding RHT down to the quantum regime by placing the particles in a resonant cavity. ThisdirectlyconnectstotheproposedquantizationofRHT[37],similartothatobservedintheconventionalthermal conductance of narrow bridges [38]. Acknowledgments ThisworkhasbeensupportedbytheSpanishMICINN(MAT2010-14885andConsoliderNanoLight.es)andtheEu- ropean Commission (FP7-ICT-2009-4-248909-LIMA and FP7-ICT-2009-4-248855-N4E). A.M. acknowledges financial support through FPU from ME. 8 10-12 T = 10 K (a) 10-19 ) 1 -K -1V 10-26 e Au Au W 10-33 ( R = 500 nm m Au SiC d = 2 µm u r 10-40 t c pe 10-10 T = 1000 K (b) s r e 10-15 f s n a r t 10-20 t a e Full theory H 10-25 Single scattering Non-retarded approx. Non-retarded approx. + χ 10-30 Diagonal approx. 10-4 10-3 10-2 10-1 100 101 ω(eV) FIG.4: (coloronline). SpectraldependenceoftheradiativeHTCundertheconditionsofFig.3(a)fortwodifferenttemperatures, as shown by labels. Appendix A: Particle polarizability We obtain the polarizability of the homogeneous spherical particles under consideration from their dipolar Mie scattering coefficients as αν = (3/2k3)tν (ν = E,M), where k = ω/c. This procedure automatically incorporates a j j,1 number of retardation corrections in the polarizability. The Mie coefficients are given by the analytical expressions [39] −j (ρ )ρ j(cid:48)(ρ )+ρ j(cid:48)(ρ )j (ρ ) tM = l 0 1 l 1 0 l 0 l 1 , l h(+)(ρ )ρ j(cid:48)(ρ )−ρ [h(+)(ρ )](cid:48)j (ρ ) l 0 1 l 1 0 l 0 l 1 −j (ρ )[ρ j (ρ )](cid:48)+(cid:15)[ρ j (ρ )](cid:48)j (ρ ) tE = l 0 1 l 1 0 l 0 l 1 , l h(+)(ρ )[ρ j (ρ )](cid:48)−(cid:15)[ρ h(+)(ρ )](cid:48)j (ρ ) l 0 1 l 1 0 l 0 l 1 √ where ρ = kR, ρ = kR (cid:15) with Im{ρ } > 0, R is the particle radius, (cid:15) is its dielectric function, j and h(+) are 0 1 1 l l spherical Bessel and Hankel functions, and the prime denotes differentiation with respect to ρ and ρ . 0 1 Figure 5 shows the imaginary part of the particle polarizability Im{α} compared to the absorption factor χ = Im{α}−2k3/3|α|2forgoldandSiCspheresofradiusRsimilartothosepresentedinSec.III.Asexpected,χapproaches Im{α} in the kR(cid:28)1 limit (i.e., when retardation is negligible). However, Im{α} and χ behave increasingly different as the energy goes up. The difference between these two functions increases with particle size, or equivalently, the thresholdforretardationeffectsω =c/R(e.g.,0.4eVforR=500nm)islowered. Interestingly,themagnetic(electric) component is dominant in gold (SiC) particles at energies below ∼ 0.1eV, while plasmon (phonon) resonances take over for larger ω. 9 Temperature (K) Temperature (K) 100 101 102 103 104 105 100 101 102 103 104 105 (a) (d) 100 10-1 10-3 10-7 3 R Au SiC o 10-6 t R = 100 nm 10-13 R = 100 nm d e (b) (e) z100 li 10-1 a m r10-3 o n 10-7 y Im{α(ω)} it10-6 χ(ω) l bi R = 500 nm 10-13 R = 500 nm a z (c) (f) i100 ar 10-1 l o P10-3 10-7 Electric 10-6 Magnetic R = 2000 nm R = 2000 nm 10-13 10-4 10-3 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 101 ω (eV) ω (eV) FIG. 5: Imaginary part of the polarizability Im{αν} (solid curves) and absorption coefficient χν = Im{αν}−(2k3/3)|αν|2 (dashed curves) for gold (a)-(c) and SiC (d)-(f) spherical particles of different radius R, as a function of photon energy. Red and blue curves represent electric (ν = E) and magnetic (ν = M) components. The upper color scale shows the equivalent temperature (cid:126)ω/k . B Appendix B: Validity of the dipolar approximation Thedipolarapproximationshouldbeaccurateinthelimitofsmallparticlescomparedtothelightwavelength. The validity of this approximation is tested in Fig. 6 by comparing the extinction cross section of gold dimers calculated by representing the particles as dipoles (broken curves) or by including all multipoles (solid curves). The calculations are performed using a multiple elastic scattering of multipolar expansions (MESME) method [39]. We obtain similar values of the cross section in both calculations for different particle separations down to d=1.5µm when the photon energyissmallerthan(cid:126)c/R≈ 0.4eV,whereR=500nmistheparticleradius. Thecomplexplasmonfeaturesshowing up at larger energies in the full calculation are not captured within the dipolar approximation. Our formalism is thus appropriatetodescribedimerswithinthisrangeofsizesandseparationsfortemperaturesbelow∼(cid:126)c/k R≈4580K. B Appendix C: Derivation of Eq. (1) We consider a spherical particle placed at the origin and of radius much smaller than the thermal wavelength λ =2π(cid:126)c/k T [see Fig. 7(a)], so that it can be described by its electric and magnetic polarizabilities, αE and αM, T B 10 n ctio #! all l’s e l=1 s-s d = 1.5 µm s o d=2.0µm r n c # d = 3.0 µm o cti n d xti e d !"# e aliz 1 µm m r o N !"# # Photon energy (eV) FIG. 6: Extinction cross-section spectra of gold dimers calculated in the dipolar approximation (l = 1, dashed curves) and withfullinclusionofallmultipoles(solidcurves)fordifferentdistancesbetweenparticlecentersd,asindicatedbylabels. The particles diameter is 1µm. The cross section is normalized to the projected area of one particle. (a) (b) d z p p 1 2 A B S FIG. 7: Description of the geometry used in the derivation of Eqs. (1) and (2). respectively. Under illumination by an external electromagnetic plane wave of frequency ω and wave vector k, the total field (external plus induced) can be written 1 E(r,t)=E eik·re−iωt+G(r,ω)αE(ω)E e−iωt− ∇×G(r,ω)αM(ω)H e−iωt+c.c., (C1a) 0 0 ik 0 1 H(r,t)=H eik·re−iωt+G(r,ω)αM(ω)H e−iωt+ ∇×G(r,ω)αE(ω)E e−iωt+c.c., (C1b) 0 0 ik 0 where G(r,ω)=(cid:2)k2+∇⊗∇(cid:3)eikr = eikr (cid:2)(k2r2+ikr−1)−(k2r2+3ikr−3)ˆr⊗ˆr(cid:3), (C2) r r3 is the electromagnetic Green tensor. The work exerted on the particle by the external electromagnetic field can be obtained from the Poynting vector flux across a spherical surface S of radius r centered at the origin as c (cid:90) Pfield = r2 dΩ[E(r,t)×H(r,t)]·ˆr. (C3) 1 4π S Now, inserting Eqs. (C1) into Eq. (C3), we readily obtain the expression P1field =ω(cid:20)−iαE− 2k33 (cid:12)(cid:12)αE(cid:12)(cid:12)2(cid:21)|E0|2+ω(cid:20)−iαM− 2k33 (cid:12)(cid:12)αM(cid:12)(cid:12)2(cid:21)|H0|2+c.c., (C4)