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isolated metal nanoparticles and their coupled arrays. G. Sun,1,* J. B. Khurgin,2 and R. A. Soref 3. 1Department of Physics, University of Massachusetts Boston, ...
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Plasmonic light-emission enhancement with isolated metal nanoparticles and their coupled arrays G. Sun,1,* J. B. Khurgin,2 and R. A. Soref 3 1 Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA 3 Sensors Directorate, Air Force Research Laboratory, Hanscom Air Force Base, Massachusetts 01731, USA *Corresponding author: [email protected]

2

Received June 27, 2008; accepted July 2, 2008; posted July 16, 2008 (Doc. ID 97952); published September 26, 2008 We present a systematic study of the enhancement of radiative efficiency of light-emitting matter achieved by proximity to metal nanoparticles. Our goal is to ascertain the limits of the attainable enhancement. Two separate arrangements of metal nanoparticles are studied, namely isolated particles and an array of particles. The method of analysis is based on the effective mode volume theory. Using the example of an InGaN / GaN quantum-well active region positioned in close proximity to Ag nanospheres, we obtain optimal parameters for the nanoparticles for maximum attainable enhancement. Our results show that while the enhancement due to isolated metal nanoparticles is significant, only modest enhancement can be achieved with an ordered array. We further conclude that a random assembly of isolated particles holds an advantage over the ordered arrays for light-emitting devices of finite area. © 2008 Optical Society of America OCIS codes: 240.6680, 230.3670.

1. INTRODUCTION The promise of large efficiency enhancement of luminescence achieved by placing emitters in close proximity to metal nanoparticles has stimulated a great deal of research devoted to demonstrating such an enhancement in various media [1,2]. The experimental work has been supported by a large body of theoretical analysis [3–11], but two basic questions have not yet been answered: for an emitter with a radiative efficiency ␩rad what kind of enhancement can be achieved with a given metal, and what should be the optimal parameters of the nanoparticles? Most analyses performed in the literature have been heavily biased towards numerical calculations in which physical insight sometimes gets lost. On a more analytical basis, the nature of enhancement has been always well understood as the local enhancement of optical electric fields in the vicinity of metal–dielectric interfaces, but the quantitative analysis was usually imprecise, largely because quasi-electrostatic models had been used in which the effects of radiation and scattering losses had not been properly taken into account. In the quasi-electrostatic model, the variation of geometry on a subwavelength scale suggests that the time-dependent term in the wave equation can be eliminated. Eliminating that term also eliminates the radiation losses, which may be a perfectly legitimate approximation for those tasks in which these losses are indeed small. But if the whole purpose of introducing metal nanoparticles is to boost the emission of radiation, then in the successful structure the radiation losses become important enough to influence actual fields in the proximity of metal. In our previous work [12,13] we took a first step to0740-3224/08/101748-8/$15.00

wards creating an adequate theoretical description of the radiation enhancement when we considered the emission of excited matter owing to the presence of surfaceplasmon-polaritons (SPPs) at the metal–dielectric interface. In our approach the radiation was treated as a twostep process. The first step is the energy transfer from the matter to a SPP mode enhanced by the Purcell factor FP [14] due to the high mode density in the SPP. By quickly transferring the excitation into the SPP mode, the Purcell effect in essence reduces the loss due to the nonradiative recombination. The Purcell enhancement is more prominent in the tightly-confined SPPs that have large wavevectors and small group velocity. The second step consists of coupling the energy from the confined SPP modes to actual propagation modes with a certain coupling rate, and this process competes with the nonradiative loss owing to absorption in the metal. Typically, tightly confined large wavevector SPPs are more difficult to couple in the outside world, and they also suffer from large losses due to field penetration into the metal. Hence the overall radiation efficiency from the excited matter to the free-propagating wave has a complicated dependence on the SPP characteristics—and we have shown that for a given original radiative efficiency ␩rad there exists an optimum SPP mode that is sufficiently well-confined but can be still efficiently coupled into the radiation continuum using a surface grating. Thus for each original radiative efficiency ␩rad we have ascertained the exact value of SPP enhancement. Our results indicated that one can obtain large enhancement only if the original radiative efficiency of the emitter is very small. In this work, we shall employ our previously developed © 2008 Optical Society of America

Sun et al.

model for the SPP enhancement of electroluminescence (EL) [13] in conjunction with the effective mode volume approach recently developed for the nanocavities [15] to provide a simple analytical model that unambiguously answers the above questions. We apply our model to the extreme opposite case of the extended metal–dielectric interface that supports SPPs, namely the case of metal nanoparticles and their eigenmodes consisting of charge oscillations coupled to the dipole field. Two types of nanoparticle arrangements placed in the vicinity of emitters are studied—isolated metal nanoparticles and an ordered array of those particles. We treat the emission as a two-step process (Fig. 1). First comes the coupling of the material polarization into a closely confined surface plasmon (SP) eigenmode with −1 the radiative rate ␶rad enhanced by the Purcell factor FP [14]. This process competes with the nonradiative decay −1 . The second step is the in the emitter whose rate is ␶rad coupling of energy from the SP mode into the radiation continuum with the rate ␥rad. This coupling has to compete with the nonradiative loss with the rate ␥nrad due to absorption in the metal, and one can define the coupling efficiency as ␩pr = ␥rad / 共␥rad + ␥nrad兲. Usually the more tightly confined modes that have larger Purcell enhancement have relatively large nonradiative loss, and based upon this tradeoff, one can optimize the nanostructure parameters to achieve the best overall enhancement. At this point, we treat the nanoparticles in spherical form, but our analysis can be easily modified to other shapes. We augment the aforementioned two-step model with the effective mode volume approach recently developed for the nanocavities [15] and provide the unambiguous answers to the same basic questions which we repeat here– what is the attainable radiation efficiency enhancement for an emitter with a radiative efficiency ␩rad, and what are the optimal parameters of the nanoparticles?

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2. ENHANCEMENT DUE TO ISOLATED NANOPARTICLES The geometry of a single metal nanosphere being placed in the vicinity of an InGaN / GaN active quantum well (QW) region of a dielectric media is shown in Fig. 2. We first calculate the effective volume of the SP eigenmode that is supported by the metal sphere. In the spherical polar coordinate system with the z axis perpendicular to the sample plane as shown in Fig. 2, we obtain the dipole field inside and outside of the metal sphere with radius a [16]:

冦冉

3⑀D

E共r, ␪兲 =

⑀M + 2⑀D

E0

r⬍a

E0zˆ

⑀M − ⑀D



a3

⑀M + 2⑀D r

关2 cos ␪rˆ − sin ␪␪ˆ 兴 r ⬎ a 3



,

共1兲

and we use the Drude model approximation for the metal dispersion:

⑀M = 1 −

␻p2 ␻2 + j␻␥

,

共2兲

where ␻P is the plasmon frequency, ␥ is the metal loss, and ␧D is the dielectric constant of the surrounding media. Clearly, when the condition ␧M共␻o兲 + 2␧D = 0 is satisfied, the field inside the metal sphere remains even in the absence of the outside driving field Eo (polarized in the z direction)—an unambiguous indication of an eigenmode oscillating at the resonance frequency

␻o =

␻p 共1 + 2⑀D兲1/2

.

共3兲

It is easy to see that the maximum field occurs at the surface of the nanoparticle at r = a, where

Fig. 1. (Color online) Illustration of the two-step process, with Step 1 being the transfer of energy from emitter into SP modes and Step 2 being the radiative coupling of SP modes into radiation modes. Coupling rates associated with the various processes are shown.

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␥d = ␥nrad + ␥rad .

共10兲

The nonradiative decay rate ␥nrad can be estimated as the energy decay of the SP mode due to the absorption in metal:

冉 冊 冕 冋 册 冉 冊再 冋 册 dU

1

=

dt

r⬍a

nrad

4

=

3

2

⳵ 共 ␻ ⑀ M兲

⑀o Im 1

␲a3

2

␻ E 2d 3r

⳵␻

⳵ 共 ␻ ⑀ M兲

⑀o Im

⳵␻

␻E2



␥ = − U. 2

共11兲

We thus obtain Fig. 2. (Color online) Illustration of an isolated Ag nanosphere placed near an active InGaN QW in GaN.

Emax =

6 ⑀ DE o

⑀M + 2⑀D

共4兲

.

Assuming that the emission frequency is close to the SP resonance, i.e., ␧M共␻o兲 ⬇ −2␧D, we can calculate the effective volume Veff of the eigenmode via the total energy [15] U=



1

r⬍a

2

⑀o

⳵ 共 ␻ ⑀ M兲 ⳵␻

E 2d 3r +



1

r⬎a

2

␥ ␥nrad = , 2

共12兲

since essentially one half of the energy is contained in the metal. The radiative decay rate, however, can be found using the standard dipole radiation formula with the dipole itself given by [16] p = 4␲⑀o

⑀M − ⑀D 3⑀D

a3



3⑀D

⑀M + 2⑀D



Eo ⬇ − 2␲⑀oa3Emax . 共13兲

⑀ o⑀ DE 2d 3r

The radiating power at the SP resonance frequency is

1

2 = ⑀o⑀DEmax Veff 2

共5兲

冉 冊 dU

Prad = −

dt

= rad

␻4o n3 12␲⑀oc

3

p2 =

冉 冊 2␲a ␭D

3

␻o 1 + 2⑀D

U,

and obtain



共14兲



4 1 , Veff = ␲a3 1 + 3 2⑀D

共6兲

which is very close to the volume of the sphere itself. Now, for the dipole positioned at the distance d from the particle surface and oriented in the z direction normal to the surface, the effective density of the SP modes will be

␳SP =

L共␻兲 Veff

冉 冊 a

a+d

共7兲

,

Im关共⑀M共␻兲 + 2⑀D兲−1兴 兰 Im关共⑀M共␻兲 + 2⑀D兲−1兴d␻

.

共8兲

Using the Drude approximation Eq. (2) we obtain the standard Lorentzian shape L共␻兲 =

␥rad =

冉 冊 2␲a

␻o

3

␭D

1 + 2⑀D

共15兲

.

The radiative coupling efficiency of the SP mode is

6

where the normalized line shape of the dipole oscillation is L共␻兲 =

where c is the speed of light in free space, n is the index of refraction of the dielectric, and ␭D = ␭ / n is the emission wavelength in the dielectric. This equation is correct, of course, only as long as the size of the nanoparticle is still small relative to the wavelength. We thereby obtain

␥d/2␲ 共␻ − ␻o兲2 + ␥d2 /4

,

共9兲

where the SP resonance frequency ␻o is given in Eq. (3), and the dipole decay rate combines nonradiative and radiative components:

␩pr =

␥rad ␥nrad + ␥rad

=

Q␹3 1 + Q␹3

,

共16兲

where we have introduced the normalized sphere radius ␹ = 2␲a / ␭D and the effective Q-factor: Q=

2␻o 共1 + 2⑀D兲␥

.

共17兲

For the Ag/ GaN system, we have Q = 5.54 at SP resonance frequency ប␻o = 2.34 eV (for Au/ GaN, Q = 0.38 at ប␻o = 1.95 eV). Thus the radiative coupling efficiency increases with the nanoparticle volume, i.e., with the number of free electrons in the particle, which can be understood as the collective effect of all electrons in the metal emitting in phase. The Purcell factor can be estimated as a ratio of the effective density of the SP modes to that of the radiation

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continuum. Taking into account that the SP dipole is polarized in the z direction, we obtain the density of modes of the radiation continuum as

␳rad =

冉 冊 2␲

1

=

␳SP

3␲2 ␭D ␻o

␳rad



共18兲

.

冉 冊 册冋 冉 冊 册 冉 冊

−1 = Veff L共␻o兲

9 ⑀ DQ

6

a

a+d



␹3共1 + Q␹3兲 ␹ + ␹d

−1 −1 ␶rad + ␶rad Fp␩pr

共20兲

−1 −1 ␶nrad + ␶rad 共Fp + 1兲

to the original radiative efficiency 1

The Purcell factor can be then obtained at resonance

F p共 ␻ o兲 =

␩SP =

1751

2␲

1

3␲

2

␭D

3

1

␩rad =

−1 ␶rad −1 −1 ␶nrad + ␶rad

共21兲

.

We thus arrive at the expression for the enhancement factor due to a single metal nanoparticle

−1

Fsingle =

␻o

␩SP ␩rad

=

1 + Fp␩pr 1 + Fp␩rad

共22兲

,

6

,

共19兲

where ␹d = 2␲d / ␭D. This Purcell factor FP exhibits a strong dependence on the metal particle size as shown in Fig. 3 and diminishes very quickly with the separation d between the nanoparticle and emitter. The Purcell factor increases initially with the sphere size following the factor 共␹ / ␹ + ␹d兲6 in Eq. (19) and then decreases as the sphere volume 共␹3兲 increases. (For emitters placed very close to the surface of the metal sphere 共␹d Ⰶ ␹兲, the Purcell factor simply decreases with the sphere volume.) This decreasing trend of the Purcell factor with the sphere volume is nearly opposite that of the radiative coupling efficiency in Eq. (16). If one neglects the radiative decay rate of the SP mode in the denominator of the Lorentzian shape in Eq. (9) by making ␥d = ␥nrad = ␥ / 2, the two opposing trends in Eqs. (16) and (19) would cancel each other, thus making the overall efficiency enhancement independent of the nanoparticle size—the answer produced by the quasistatic models. Taking the radiative decay into consideration alters the picture and results in strong size dependence of the overall enhancement factor. This enhancement factor thus should be calculated as the ratio of the luminescence efficiency in the presence of SP modes

Fig. 3. Purcell factor of a single metal nanoparticle as a function of sphere radius.

which suggests that in order to have any enhancement at all, Fsingle ⬎ 1, the radiative coupling efficiency of the SP mode must be greater than the original radiative efficiency of the emitter. To analyze (22) we first consider the case of emitters with very low original radiative efficiency such that FP␩rad Ⰶ 1, Fsingle ⬇ 1 + FP␩pr = 1 +

9 ⑀ DQ 2

冉 冊 ␹

共1 + Q␹3兲2 ␹ + ␹d

6

.

共23兲

For small nanoparticles in which the nonradiative decay of SP mode dominates, Q␹3 Ⰶ 1, the result becomes essentially identical to the one obtained in the quasi-static approximation with a weak dependence of the enhancement factor on the nanoparticle size. In fact, when the emitter is placed very close to the surface of nanoparticle 共␹d ⬇ 0兲 the enhancement shows almost no dependence on the particle size. This observation is easy to interpret: with small original radiative efficiency ␩rad Ⰶ 1, the matter of most concern is the transfer of energy from the emitter into the SP mode before nonradiative decay takes place. This process clearly favors small nanoparticle size for its small effective volume of the SP mode, but the small volume also leads to small radiative coupling efficiency (15) of the SP mode. As a result, the increase of the enhancement factor with the reduction of the nanoparticle volume quickly saturates for very small original radiative efficiency ␩rad Ⰶ 1. Once we examine emitters with nonzero ␩rad, however, the situation changes, showing a strong nanoparticle size dependence. Using the example of an isolated Ag sphere in InGaN / GaN QW dielectric media shown in Fig. 2, we have calculated the enhancement factor, and the result for the QW positioned at a distance d = 10 nm from the bottom surface of the Ag sphere is shown in Fig. 4 for a range of the original radiative efficiency of the QW emitter. We can see that the enhancement factor exhibits strong dependence upon the nanosphere dimensions with the peak occurring when the radius is small enough to yield smaller effective mode volume for an enhanced Purcell factor, yet is still sufficiently large to assure strong radiative coupling of the SP mode. As expected, the higher the original radiative efficiency, the less critical becomes the concern of transferring energy from the emitter into the SP mode, and accordingly the more important grows the concern for the efficient energy transfer from the SP mode into free-space radiation modes. This situation favors larger nanoparticles that can

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for many applications where not only efficiency but also overall power is required, a single emitter simply cannot accomplish both. Therefore, to achieve practical enhancements for such devices as LEDs with a large area, one must consider arrays of spheres.

3. ENHANCEMENT DUE TO A 2-D ARRAY OF NANOPARTICLES

Fig. 4. Enhancement factor Fsingle due to a single isolated Ag nanosphere as a function of the sphere radius a for a range of the original radiative efficiency of the QW emitter ␩rad.

emit the SP energy photons into the free space before they get lost in the metal. With the Ag sphere size optimized for each QW position d and original radiative efficiency ␩rad, we can then obtain optimal enhancement factor Fsingle,opt as a function of the original radiative efficiency ␩rad shown in Fig. 5. It can be seen that the enhancement gets reduced rather quickly as the metal nanoparticle is placed farther away from the emitter. An important observation that can be made is that the overall efficiency enhancement is much stronger than the one provided by the interface of the dielectric with a metal layer considered in [13]. To understand the origin of this seeming discrepancy, we first note that the output coupling of the interface SPP is subject to the momentum conservation rules, while for the nanosphere SP these rules do not apply since the subwavelength dipole contains a wide range of momentum. This can be extremely valuable in detecting radiation from single molecules, but

Let us now consider an ordered 2-D array of nanospheres separated by a distance R that is placed on top of a QW emitter as shown Fig. 6. The coupling between the dipoles of the neighboring nanoparticles will result in propagating SP modes in the x y plane that are characterized by a 2-D dispersion—a relationship between the frequency and 2-D wavevectors of the SP modes. Such a dispersion can be obtained for simplicity by taking into account the interaction between only the nearest neighbors. The equation of motion for the m , n-th dipole pm,n can be written as

⳵2pm,n ⳵t

2

= − ␻2o pm,n − ␻2o pm−1,n − ␻2o pm,n−1

Eout共R, ␪兲

Eout共R, ␪兲 Ein

Ein

− ␻2o pm+1,n

− ␻2o pm,n+1

Eout共R, ␪兲

Eout共R, ␪兲 Ein

Ein ,

共24兲

where Ein and Eout are the magnitude of dipole field inside 共r ⬍ a兲 and outside 共r ⬎ a兲 of the sphere given by Eq. (1), respectively, and are related at near resonance with ␧M共␻o兲 ⬇ −2␧D by Eout共R, ␪兲 = Ein

a3 R3

关1 + 3 cos2 ␪兴.

共25兲

For the case where the dipoles are arranged in the TM mode, we should have ␪ = ␲ / 2 (Fig. 2): Eout共R, ␪兲 = Ein

a3 R3

共26兲

,

and we obtain

⳵2pm,n ⳵t2

=−

␻2o pm,n



␻2o

a3 R3

共pm−1,n + pm+1,n + pm,n−1 + pm,n+1兲. 共27兲

Substituting pm,n = poei共qxmR+qynR−␻qt兲

共28兲

into Eq. (27), we quickly obtain the dispersion relation in this tight-binding approximation:



␻q2 = ␻2o 1 + 2

Fig. 5. Optimal enhancement Fsingle,opt by an optimized single isolated Ag nanosphere as a function of the original radiative efficiency ␩rad of the QW emitter positioned at some distance d below the metal sphere.

a3 R3



关cos共qxR兲 + cos共qyR兲兴 ,

共29兲

which is a broad SP band in which each mode is characterized by a wavevector q = qxxˆ + qyyˆ as illustrated in Fig. 7. The luminescence gets efficiently coupled into the many modes inside the broadband:

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(Color online) 2-D ordered array of metal nanoparticles placed in the vicinity of the QW active region of a LED.

⌬␻SP ⬇ 4共a/R兲3␻o ,

共30兲

but only the modes with a small wavevector 兩q兩 ⬍ kD = 2␲ / ␭D within the narrower band can actually couple into the free propagating dielectric modes. These modes are confined within a narrow range of frequencies ⌬␻rad near the top of the band

␻o⬘ = 共1 + 4a3/R3兲1/2␻o .

共31兲

It is easy to see that these radiative modes make up only a fraction of all the SP modes,

−1 ␥radgrad

␩pr,array =

=

−1 ␥radgrad + ␥/2

Q␹3 grad + Q␹3

共33兲

.

For the densely packed spheres with small grad Ⰶ 1, the coupling efficiency can approach unity. These radiative modes with broadening 2 ␥d,array = 共␭D /␲R2兲␥rad + ␥/2

共34兲

and the normalized line shape Lrad共␻兲 =

␥d,array/2␲

共35兲

2 共␻ − ␻o⬘兲2 + ␥d,array /4

have a combined Purcell factor grad =

2 ␲kD

共2␲/R兲2

=

␲R2 2 ␭D

.

共32兲

(Obviously the grad = 1 for sparsely spaced metal arrays with R ⬎ ␭D / 冑␲.) Since each radiative SP mode within the radiative band ⌬␻rad is a collective nearly in-phase oscil−1 SP dipoles, the radiative decay time should lation of grad then be decreased by about the same factor grad. The radiative coupling efficiency should then be modified

FP,rad = grad

9 ␲ ⑀ D␥ Q 4␹3

¯ 共 ␻ ⬘兲 L rad o

a6 关共a + d兲2 + R2/6兴3

,

共36兲

where we have taken into account the average separation between the emitters and the 2-D metal spheres ¯d2 = 共a + d兲2 +

R2

共37兲

6

and the average of normalized line shape within the radiative band ⌬␻rad: k

¯ 共 ␻ ⬘兲 = L rad o

兰0DLrad共2␲q兲dq k

兰0D2␲qdq

共38兲

.

The remaining nonradiative modes with 兩q兩 ⬎ kD in the broad SP band ⌬␻SP have a broadening of ␥ / 2 and a normalized line shape of Lnrad共␻兲 =

␥/2␲ 共␻ − ␻o⬘兲2 + ␥2/4

共39兲

.

Electroluminescent coupling into these modes with a combined Purcell factor FP,nrad = 共1 − grad兲

9 ␲ ⑀ D␥ Q 4␹3

¯ L nrad共␻o⬘兲

a6 关共a + d兲2 + R2/6兴3 共40兲

Fig. 7. (Color online) Dispersion relationship of the SP modes of the 2-D array.

will produce no emission at all. The average of the nor¯ malized line shape L nrad共␻⬘ o 兲 within the nonradiative band

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⌬␻SP − ⌬␻rad is calculated similarly to Eq. (38) except the average is taken within the first Brillouin zone over the area where 兩q兩 ⬎ kD. The Purcell factors with which the luminescence couples into the radiative and nonradiative modes within the band ⌬␻rad and ⌬␻SP − ⌬␻rad (Fig. 7), respectively, are shown in Fig. 8 where the emitters are separated from the metal spheres by d = 10 nm, and the spacing between neighboring spheres is kept at R = 3a. In comparison with the single nanoparticle having equal separation between the emitter and metal sphere (Fig. 3), the Purcell factor of the 2-D array is at least an order of magnitude less. This significant reduction is due to the fact that the energy transfer from the emitter into the SP modes of the 2-D array can no longer be at resonance since the SP modes of a 2-D array spread into a band ⌬␻SP (Fig. 7) and that the average separation between the emitter and metal spheres is effectively increased [Eq. (37)]. It can be seen that the radiative and nonradiative Purcell factors are on the same order of magnitude, suggesting comparable coupling of luminescence into radiative as well as nonradiative modes inside of the broad SP band ⌬␻SP. We finally obtain the enhancement of EL by the 2-D array Farray =

1 + FP,rad␩pr,array 1 + 共FP,rad + FP,nrad兲␩rad

.

共41兲

Compared with Eq. (22) for a single nanometal sphere, this enhancement by the 2-D array is significantly reduced by the EL coupling into the nonradiative modes within the broad SP band through the Purcell factor FP,nrad. Obviously for the 2-D array of metal spheres, the enhancement depends not only on the radius of those spheres but also upon the spacing between them. For an InGaN / GaN QW with an original radiative efficiency ␩rad = 0.01 that is embedded d = 10 nm below the metal particles, the enhancement result due to a 2-D array of nanoparticles is shown in Fig. 9 where optimal values of sphere radius a and spacing R can be obtained for maxi-

Fig. 8. (Color online) Radiative 共FP,rad兲, nonradiative 共FP,nrad兲, and total Purcell factors 共FP,rad + FP,nrad兲 for a 2-D array as a function of metal sphere radius a with R = 3a and d = 10 nm.

Fig. 9. (Color online) Enhancement due to a 2-D array of Ag spheres that are separated 10 nm from InGaN / GaN QW emitters with an original radiative efficiency ␩rad = 0.01.

mum enhancement. The optimization result for the 2-D array of metal spheres is shown in Fig. 10 as a function of the original radiative efficiency for QW emitters embedded at d = 5, 10, and 20 nm below the metal array. The general trend is that the 2-D array of smaller metal spheres provides better enhancement results for inefficient emitters, while the array of larger spheres should be employed for improving those emitters that are already fairly efficient. This trend can be understood from the two competing factors in enhancement, namely the Purcell factor Fp that is larger for smaller metal particles and the out-coupling efficiency ␩pr array that is larger for larger particles. It can be seen from Eq. (41) that for smaller ␩rad, the Purcell factor plays more important role while the out-coupling efficiency tends to dominate for higher ␩rad. The center-to-center spacing Ropt between spheres is typically 3–4 times their radius. Taking these optimal particle sizes and spacing, we plot the result of optimal enhancement as a function of the original radiative efficiency in Fig. 11. In comparison with the result for the single isolated nanoparticle shown in Fig. 2, the enhancement is significantly reduced and ap-

Fig. 10. (Color online) Optimization of 2-D array of Ag spheres that are separated 5, 10, and 20 nm from InGaN / GaN QW emitters as a function of original radiative efficiency ␩rad.

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using local field enhancement near the metal–dielectric interface, but for the large-area emitters that are already reasonably efficient, very little additional enhancement can be obtained. The only way to increase the enhancement for emitters of finite area would be to introduce strong disorder and localization of SP modes, which tend to perform better than an ordered array of metal nanoparticles or an extended metal sheet with specially designed gratings. As future work we plan to expand our treatment to the possibility of enhancing photoluminescence and various nonlinear optical processes.

ACKNOWLEDGMENTS The authors thank the Air Force Office of Scientific Research for support of this work. Fig. 11. Optimal enhancement due to 2-D array of Ag spheres on InGaN / GaN QW emitter as a function of the original radiative efficiency for several separation values d.

proaches our previous result of metal–dielectric interface SPP enhancement [13]. It should be emphasized here that the array of metal spheres is a rather poor approximation for the continuous metal slab since the charges cannot flow between the spheres. Nevertheless, our model shows that the main reason for the dramatically poor efficiency of SPP on the metal dielectric interface in comparison with that of a nanoparticle plasmon is the momentum conservation rule that prevents interface SPPs from escaping. It also explains that the disordered interfaces in which the plasmon modes are localized tend to perform better than specially designed interfaces with gratings.

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4. SUMMARY In this work we have developed a simple yet rigorous twostep theory for emission enhancement in the vicinity of metal nanoparticles. Our theory properly takes into account the influence of both nonradiative loss and radiative coupling upon the Purcell factor—the correct treatment that had not been taken in most prior works. We have shown the interplay between the efficiency of energy coupling from the emitter into the nanoparticle SP mode and the subsequent energy transfer from this mode into the free space. It is due to this interplay, for every original radiative efficiency ␩rad, that one can optimize the nanoparticle size to maximize overall efficiency enhancement. The overall enhancement by a single nanoparticle was found to be substantially larger than the enhancement attainable with the assistance of SPP on the metal– dielectric interface. We have attributed this improvement to the violation of the momentum conservation rule by a single nanoparticle, and as proof we have considered the case of an ordered array of the metal nanoparticles in which the enhancement was found to be substantially lower than in a single nanoparticle and on a par with a metal–dielectric interface. Our main conclusion is that when dealing with a small number of very inefficient emitters, one can substantially enhance their emission by

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