We consider the enhanced absorption of optical radiation by molecules placed in the vicinity of spherical metal nanoparticles in the realistic situation that includes perturbation of the optical field by the absorbing molecules. We show that there is an optimal nanosphere radius that gives the strongest enhancement for each combination of the number of absorbing molecules, their absorption strength, and their distance from the nanosphere surface and that the enhancement is strong only for relatively weak and diluted absorbers

Khurgin, Jacob B.

Soref, R. A.

Sun, Greg

English

University of Massachusetts Boston: ScholarWorks at UMass

University of Massachusetts BostonScholarWorks at UMass BostonPhysics Faculty Publications Physics2-17-2009Practical limits of absorption enhancement nearmetal nanoparticlesJacob B. KhurginJohns Hopkins UniversityGreg SunUniversity of Massachusetts Boston, greg.sun@umb.eduR. A. SorefAir Force Research Laboratory, Hanscom Air Force BaseFollow this and additional works at: http://scholarworks.umb.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks at UMass Boston. It has been accepted for inclusion in PhysicsFaculty Publications by an authorized administrator of ScholarWorks at UMass Boston. For more information, please contact library.uasc@umb.edu.Recommended CitationKhurgin, Jacob B.; Sun, Greg; and Soref, R. A., "Practical limits of absorption enhancement near metal nanoparticles" (2009). PhysicsFaculty Publications. Paper 16.http://scholarworks.umb.edu/physics_faculty_pubs/16Practical limits of absorption enhancement near metal nanoparticlesJ. B. Khurgin, G. Sun, and R. A. Soref  Citation: Appl. Phys. Lett. 94, 071103 (2009); doi: 10.1063/1.3081631 View online: http://dx.doi.org/10.1063/1.3081631 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v94/i7 Published by the American Institute of Physics.  Related ArticlesOptical coupling of surface plasmons between graphene sheets Appl. Phys. Lett. 100, 131111 (2012) Tuning asymmetry parameter of Fano resonance of spoof surface plasmons by modes coupling Appl. Phys. Lett. 100, 131110 (2012) Resonant surface magnetoplasmons in two-dimensional magnetoplasmonic crystals excited in Faradayconfiguration J. Appl. Phys. 111, 07A946 (2012) Robust exciton-polariton effect in a ZnO whispering gallery microcavity at high temperature Appl. Phys. Lett. 100, 101912 (2012) Surface wave resonances supported on a square array of square metallic pillars Appl. Phys. Lett. 100, 101107 (2012)  Additional information on Appl. Phys. Lett.Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 29 Mar 2012 to 158.121.194.143. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsPractical limits of absorption enhancement near metal nanoparticlesJ. B. Khurgin,1 G. Sun,2,a and R. A. Soref31Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore,Maryland 21218, USA2Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA3Sensors Directorate, Air Force Research Laboratory, Hanscom AFB, Massachusetts 01731, USAReceived 25 November 2008; accepted 25 January 2009; published online 17 February 2009We consider the enhanced absorption of optical radiation by molecules placed in the vicinity ofspherical metal nanoparticles in the realistic situation that includes perturbation of the optical fieldby the absorbing molecules. We show that there is an optimal nanosphere radius that gives thestrongest enhancement for each combination of the number of absorbing molecules, their absorptionstrength, and their distance from the nanosphere surface and that the enhancement is strong only forrelatively weak and diluted absorbers. © 2009 American Institute of Physics.DOI: 10.1063/1.3081631Recent years have witnessed an increased interest in“nanoplasmonics” relevant to sensor applications. In this pa-per, we discuss such applications and focus upon the in-creased optical absorption that results when molecules areplaced near metal nanoparticles. This enhancement is part ofthe general study of field modification obtained in the vicin-ity of nanostructured metallic objects1,2 whose characteristicfeatures are much smaller than the optical wavelength. It hasbeen well understood that the enhancement of optical field isassociated with excitation of localized surface plasmonLSP modes, and it has been established that diverse opticaleffects can be enhanced near the resonantly excited metalnanoparticles. Enhancements of radiative decay rates,3fluorescence,4,5 Raman scattering,6 and others have beendemonstrated by numerous groups, and these experimentshave been supported by a large body of theoretical work.7,8However, the actual numerical values of enhancement re-ported in different experiments differ by orders of magnitudeand are often inconsistent with the theoretical or numericalresults. Furthermore, no clear understanding exists on how agiven effect can be maximized by optimizing the size and/orshape of nanoparticles for a specified optically active atom,molecule, or quantum-confined object such as quantum dot.For the sake of generality, we refer to all these objects as“molecules.” Often an electrostatic approach is used inwhich the radiative decay is neglected, and this cannot bevalid in applications where highly efficient radiative cou-pling is required. Following our previously developed modelfor the surface plasmon polariton enhancement ofelectroluminescence,9 we have started to develop a consis-tent analytical approach to the plasmonic enhancement ofoptical effects by considering the most basic phenomena ofradiative decay in the vicinity of metal nanoparticles.10 Ourapproach, based on effective volume method pioneered byMaier,11 considers the process of emission as a two-step pro-gression; the first step being Purcell-enhanced emission ofenergy into the LSP mode followed by the second step ofenergy emission out of the LSP mode into free space. Thequantum efficiency of the first step requires the LSP mode tobe essentially a small volume high-Q cavity, while the quan-tum efficiency of the second requires the same mode to benothing but a relatively large up to  /2 low-Q optical an-tenna. These two demands are difficult to reconcile; thus, foreach specified emitting molecule characterized by its ownoriginal unenhanced radiative efficiency 0, one can find theoptimum size of nanoparticles that maximizes the radiativeefficiency enhancement. We have also shown that only forthose relatively inefficient 010% molecules, one canachieve substantial efficiency enhancement, while for therelatively efficient emitters the losses in the metal negatewhatever Purcell gain can be achieved.We now turn our attention to the reverse problem, i.e.,enhancement of the absorption by molecules placed in thevicinity of metal nanoparticles. This issue is importantfor two reasons. First of all, it had been proposed to usemetal nanoparticles to enhance the efficiency ofphoto-detectors12,13 and solar panels,14,15 and second, absorp-tion is the first step in the photoluminescence PL emissionof PL sensors and thus enhanced absorption means improvedsensitivity. The main goal of our work is to provide a rigor-ous yet simple analytic estimate of the attainable enhance-ment of absorption that takes into account all the radiativeand nonradiative losses and, most essential, provides the op-timization routine for a given absorber characterized by itsoriginal absorption strength. Rather unsurprisingly, we showthat the stronger the original absorption the less enhancementis attainable and this fact leads to important practical conse-quences.Consider the absorption by a molecule located near ametal nanosphere here we consider only the spherical nano-particles but our treatment can be easily expanded to takeaccount of different shapes. The nanosphere is placed at thewaist of a Gaussian light beam focused onto the profile, asshown in Fig. 1, with a numerical aperture characterized by afar field half angle . In the absence of the metal sphere, thebeam of small angle  will simply be focused onto a diffrac-tion limited spot at the apex of the cone whose radius at thewaist w0=d /, where dis the wavelength in the dielec-tric.In the absence of the nanosphere, the field in the focalspot Efoc can be related to the power s+2 carried by theincident wave as follows, using the notation in Ref. 16,aElectronic mail: greg.sun@umb.edu.APPLIED PHYSICS LETTERS 94, 071103 20090003-6951/2009/947/071103/3/$25.00 © 2009 American Institute of Physics94, 071103-1Downloaded 29 Mar 2012 to 158.121.194.143. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionss+2 =nZ0w02 2Efoc2, 1where Z0 is the impedance of free space and n is the index ofrefraction in the dielectric. In the presence of a metal spherewith radius r, the incident light can be coupled into the LSPmode with resonant frequency 0 and with an effective modevolume10 Veff=4r31+1 /2D /3 that produces maximumfield Emax at the surface of the sphere which is related to theLSP mode energy a2 asa2 = 120DEmax2 Veff, 2where 0 is the permittivity of free space and D is the di-electric constant of the medium.In the two-step process of absorption, the first step iscoupling of energy from free space modes contained in thefocused beam into the LSP mode. This is a process reciprocalto the radiative decay of the LSP mode into the free spacemodes that are contained within the far field solid angle ,which is characterized by the angle dependent decay rate	rad=	rad0f ,d, where the angular frequency depen-dence f ,=31−sin2  cos2  /8 in the spherical sys-tem Fig. 1, and the radiative decay rate into all radiationmodes at resonant frequency 0 is10	rad 	 	rad4= 2rD3 01 + 2D . 3For small angles  we obtain	rad  38	rad2, and then using the reciprocity discussed byHaus16 that relates the in-coupling coefficient in from thelight cone to the radiative decay rate 	rad of the LSP mode,we obtain in=	rad  /23	rad /2. Now we arrive at therate equation for the amplitude a of the LSP,dadt= j − 0a −	nrad + 	rad2a +23	rad2s+, 4where the nonradiative decay rate of the LSP mode 	nrad=	 with 	 being the metal loss in the Drude-model approxi-mation for the metal dielectric dispersion. The steady statesolution of Eq. 4 at resonance yieldsa =32	rad1/2	nrad + 	rads+. 5Using relations 1 and 2, we obtain the field intensity en-hancement and thus the absorption enhancement factorF =Emax2Efoc2 =3D382VeffQrad−1Qnrad−1 + Qrad−1 2, 6where we have introduced the Q-factors Qradnrad=0 /	radnrad. Note that Eq. 6 does not contain the angle—thus the enhancement does not depend on the focusingscheme. From Eq. 3, we have Qrad−1 =3 / 1+2D, where thenormalized radius =2r /D. If we further normalize Qnradto Q=Qnrad / 1+2D and take into account that the actualatom or molecule is situated at a distance d from the metalsphere, we then obtainF =Emax2Efoc2 =9D213 + Q−12  + d6, 7where d=2d /D. Clearly, in the limit of a very smallnanosphere the intensity enhancement becomes proportionalto Q2 as expected from simple electrostatic considerations. Infact, the maximum enhancement in Eq. 7 occurs at opti-mum radius opt= d /Q1/4, which yields Fmax=Q29D /2Q1/4d3/4+1−8. The field intensity enhancement factor foran Ag nanosphere embedded in GaN Q=2.77, 0=2.344 eV is shown in Fig. 2 as a function of the sphereradius r for several values of distance d. The enhancement israther significant for small spacing d10 nm between themetal sphere and absorbing atom or molecule and diminishesquickly when d20 nm.The origin of a strong dependence of F on sphere radiusin Eq. 7 lies in the importance of a radiative decay termthat is not taken account in the simple electrostatic calcula-tions. One should also take into account the energy loss dueto absorption in the active molecules, especially since strongabsorption is what is desired. The rate of decay due to ab-sorption is	abs =cnNaaVeff  + d6, 8where c is the speed of light in free space, a is the absorp-tion cross section of the molecule, and Na is the number ofmolecules per sphere. We can now introduce an additionalQ-factor due to absorption asFIG. 1. Color online Illustration of a metal nanosphere with a radius rplaced at the apex of a focused Gaussian beam with a numerical aperturecharacterized by the far-field half angle . In the absence of the metalsphere, the beam will be focused onto a diffraction-limited spot with radiusw0 at the waist.FIG. 2. Absorption enhancement factor as a function of Ag sphere radiusembedded in GaN for different values of spacing d.071103-2 Khurgin, Sun, and Soref Appl. Phys. Lett. 94, 071103 2009Downloaded 29 Mar 2012 to 158.121.194.143. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsQa =0	abs1 + 2D=D2Naa36D + d6. 9This finally brings us toFa =9D213 + Q−1 + Qa−12  + d6. 10The result of enhancement is shown in Fig. 3 as a function ofsphere radius r for the Ag/GaN system by fixing d=5 nmfor several values of the total absorption cross section Naa.It is clear that the absorption of the active molecules be-comes important when it approaches the loss in the metal.This can be confirmed by optimizing Eq. 10 for smalld0. Then we find the optimum normalized metal sphereradius opt= 6DNaa /D2 1/6 and Fa,opt9DQ2 /22Q6DNaa /D2 1/2+1−2. Thus absorption in the activemolecules themselves will become main limiting factor whenNaaD2 /24DQ214 nm2 for the Ag/GaN system andwould not change substantially over the most of visible andnear IR spectrum because an increase in wavelength wouldbe balanced by a corresponding increase in Q-factor as met-als become less absorptive at longer wavelengths. For a rela-tively small number of organic molecules, even with absorp-tion cross sections as high as 0.1 nm2 in the case ofRhodamine 6G,17 the absorption on active molecules willplay no role in determining the limit of field enhancement. Atthe same time, if one considers detectors based on semicon-ductors, specifically semiconductor quantum dots, theircross-sections are on the scale of a few nm2,18 thus the factorNaa can easily approach 100 nm2 and absorption becomesmore relevant. As one can see from Fig. 3 for Naa100 nm2, only a factor-of-few absorption enhancement isattainable and even then only for relatively small distancefrom the metal surface.To better emphasize the points made in this work, wehave performed optimization of the absorption enhancementfor a wide range of total absorption cross section and spacingbetween the molecules and metal spheres. The optimized re-sults shown in Fig. 4 clearly indicate that the strongest ab-sorption enhancement by almost two orders of magnitudeis attainable for the small number of weakly absorbing mol-ecules placed close to the metal spheres, while for the largenumber of strong absorbers spread out only a dozen nanom-eters further away from the spheres the absorption enhance-ment is far more modest if it exists at all.In conclusion, we have developed an analytical model ofabsorption enhancement in the vicinity of metal nanopar-ticles and have shown that the degree of enhancementstrongly depends on the particle dimension, thereby provid-ing a straightforward route to optimizing the absorption en-hancement. The main conclusion is that metal nanoparticlescan dramatically improve the performance of optical sensorsin which the analyte molecules are few and their originalabsorption is low. At the same time, when the original ab-sorption is already significant, as is the case of most opticaldetectors and photovoltaic devices, the enhancement is weakor nonexistent due to metal loss.This work is supported in part by the Air Force Office ofScientific Research FA8718-05-C-0030.1W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature London 424, 8242003.2W. A. Murray and W. L. Barnes, Adv. Mater. Weinheim, Ger. 19, 37712007.3K. Okamoto, I. Niki, and A. Scherer, Appl. Phys. Lett. 87, 071102 2005.4S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett.97, 017402 2006.5R. M. Baker, V. P. Drachev, Z. Liu, H.-K. Yuan, R. H. Pedersen, A.Boltasseva, J. Chen, J. Irudayaraj, A. V. Kildishev, and V. M. Shalaev,New J. Phys. 10, 125022 2008.6M. Moskovits, Rev. Mod. Phys. 57, 783 1985.7R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt.Commun. 261, 368 2006.8G. Baffou, C. Girard, E. Dujardin, G. Colas des Francs, and O. J. F.Martin, Phys. Rev. B 77, 121101R 2008; G. Colas des Francs, C.Girard, T. Laroche, G. Leveque, and O. J. F. Martin, J. Chem. Phys. 127,034701 2007.9J. B. Khurgin, G. Sun, and R. A. Soref, J. Opt. Soc. Am. B 24, 19682007; G. Sun, J. B. Khurgin, and R. A. Soref, Appl. Phys. Lett. 90,111107 2007.10J. B. Khurgin, G. Sun, and R. A. Soref, Appl. Phys. Lett. 93, 0211202008; G. Sun, J. B. Khurgin, and R. A. Soref, J. Opt. Soc. Am. B 25,1748 2008.11S. A. Maier, Opt. Express 14, 1957 2006.12M. Westphalen, U. Kreibig, J. Rostalski, H. Luth, and D. Meissner, Sol.Energy Mater. Sol. Cells 61, 97 2000.13B. P. Rand, P. Peumans, and S. R. Forrest, J. Appl. Phys. 96, 7519 2004.14L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D.-S. Ly-Gagnon, K. C.Saraswat, and D. A. B. Miller, Nat. Photonics 2, 226 2008.15D. M. Schaadt, B. Feng, and E. T. Yu, Appl. Phys. Lett. 86, 0631062005.16H. A. Haus, Waves and Fields in Optoelectronics, 1st ed. Prentice-Hall,Englewood Cliffs, NJ, 1984.17M. A. Ali, J. Moghaddassi, and S. A. Ahmed, J. Opt. Soc. Am. B 8, 18071991.18S. W. Osborne, P. Bloos, P. M. Smowton, and Y. C. Xin, J. Phys.: Con-dens. Matter 16, S3749 2004.FIG. 3. Absorption enhancement as a function of sphere radius r for theAg/GaN system wherein the absorbing molecules with a range of total ab-sorption cross section are placed at d=5 nm from the Ag sphere.FIG. 4. Color online Absorption enhancement in the Ag/GaN system op-timized for absorbing molecules with a total absorption cross section Naaplaced at a distance d from the Ag sphere.071103-3 Khurgin, Sun, and Soref Appl. Phys. Lett. 94, 071103 2009Downloaded 29 Mar 2012 to 158.121.194.143. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

Practical limits of absorption enhancement near metal nanoparticles

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Practical limits of absorption enhancement near metal nanoparticles

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