van der Waals interactions at the nanoscale: The … der Waals interactions at the nanoscale: The...

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van der Waals interactions at the nanoscale: The effects of nonlocality Yu Luo 1 , Rongkuo Zhao 1 , and John B. Pendry 2 The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2013. Contributed by John B. Pendry, November 12, 2014 (sent for review September 4, 2014; reviewed by Che Ting Chan and Peter Nordlander) Calculated using classical electromagnetism, the van der Waals force increases without limit as two surfaces approach. In reality, the force saturates because the electrons cannot respond to fields of very short wavelength: polarization charges are always smeared out to some degree and in consequence the response is nonlocal. Nonlocality also plays an important role in the optical spectrum and distribution of the modes but introduces complexity into calcu- lations, hindering an analytical solution for interactions at the nanometer scale. Here, taking as an example the case of two touching nanospheres, we show for the first time, to our knowl- edge, that nonlocality in 3D plasmonic systems can be accurately analyzed using the transformation optics approach. The effects of nonlocality are found to dramatically weaken the field enhance- ment between the spheres and hence the van der Waals interaction and to modify the spectral shifts of plasmon modes. van der Waals | nonlocality | transformation optics | plasmonics T he van der Waals force is an electromagnetic interaction between correlated fluctuating charges on two electrically neutral surfaces (14). As the surfaces approach more closely, the force increases as fluctuations of shorter and shorter length scale come into play, but ultimately the force will saturate when the surfaces are so close that the even shortest wavelength charge fluctuations are included. It is this saturation with which we are concerned in this paper and to treat it, we need to go beyond the conventional description of a solid by a permittivity, «ðωÞ, that depends only on the frequency, ω. Here we recognize that the response of a solid depends on the length scale of the fluctuations and introduce a formalism using a generalized nonlocal permit- tivity, «ðω; kÞ (513), that also depends on the wave vector, k, and hence takes into account the saturation. Neglect of nonlocality leads to an unphysical diverging van der Waals force at short distances. We apply the technique of transformation optics (1416) to solve the difficult problem of including nonlocal effects when two nanoscale bodies interact and illustrate our theory with calculations for two closely spaced nanospheres. Ultimately at a few tenths of a nanometer, just before the surfaces touch, direct contact of the charges will come into play through electron tunneling (1725); in other words, chemical bonding will dominate the final approach. Here we are not con- cerned with chemical bonding, which is extensively treated else- where in the literature. The van der Waals force is weaker than chemical bonding effects, but plays an important role in a broad range of areas, such as surface and colloidal science, nanoelectromechanical systems, and nanotechnology (2, 3, 2628). In the classical electrodynamics picture where nonlocality is neglected, the fluctuating surface charges are located precisely on the surface and these infinitely compressed charges result in the unphysically divergent van der Waals force (1, 29). In a more realistic framework, considering the inherent quantum nature of electrons, the surface charges are intrinsically smeared across the boundary in a subnanometer layer (7, 2022), which dramatically alters the van der Waals energy in the small gap limit (3033). Due to the complexity introduced by nonlocality, its influence on the van der Waals force in 3D geometries has never been described appropriately. Esquivel-Sirvent and Schatz investigated the influence of nonlocality on the van der Waals force between two spherical gold nanoparticles (32). However, they used Hamakers method (1), which has been widely recognized as an inaccurate method for such nanoparticle systems where the size and the separation are comparable. We developed a theoretical approach to treat nonlocal prob- lems, which is as accurate as other semiclassical nonlocal methods (such as the hydrodynamic model) but much more efficient in terms of both simulation time and memory consumption when implemented in numerical simulations (34). In this paper, we show that 3D nonlocal systems can be studied analytically by combining our nonlocal model with the technique of transformation optics. Results and Discussion We start by considering a dimer of nanospheres, a prototypical plasmonic system. Other structures such as a sharp metal tip (3537) and two overlapping spheres (3840) can be studied in a similar manner. Widely separated nanoparticles (of sizes larger than 10 nm) can be treated by neglecting nonlocality, and trans- formation optics provides an elegant analytical solution for the electromagnetic (EM) field (41). To calculate the van der Waals force, an analytic form of the permittivity is needed: metals can be described by a local permittivity, « M ðωÞ = « b ω 2 p =½ωðω + iγÞ, where « b is usually expressed as a summation of several Lorentz models (see SI Text for details). As the particle separation is re- duced to a nanometric scale, the nonlocal smearing of metallic electrons plays an important role, and the implementation of a spatially dispersive metal permittivity for longitudinal EM fields, « M ðω; kÞ = « b ω 2 p =½ωðω + iγÞ β 2 jkj 2 , is required. The β factor describes the degree of nonlocality and is proportional to the Fermi velocity. Significance The van der Waals interaction is a ubiquitous but subtle force between particles mediated by quantum fluctuations of charge. It is the most long-range force acting between particles and influences a range of phenomena such as surface adhesion, friction, and colloid stability. Calculations of the force between parallel surfaces >10 nm apart is a simple task, but when the geometry is more complex, e.g., a pair of nanospheres <5 nm apart, the task is more difficult. Furthermore a macroscopic description of the dielectric properties no longer suffices, and we must consider the diffuse nonlocal nature of the electron polarization cloud. In this paper, we propose a simple analytic treatment of the problem. Author contributions: Y.L., R.Z., and J.B.P. designed research; Y.L. and R.Z. performed research; and Y.L., R.Z., and J.B.P. wrote the paper. Reviewers: C.T.C., Hongkong Univ. of Science and Technology; and P.N., Rice University. The authors declare no conflict of interest. 1 Y.L. and R.Z. contributed equally to this work. 2 To whom correspondence should be addressed. Email: [email protected]. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1420551111/-/DCSupplemental. 1842218427 | PNAS | December 30, 2014 | vol. 111 | no. 52 www.pnas.org/cgi/doi/10.1073/pnas.1420551111

Transcript of van der Waals interactions at the nanoscale: The … der Waals interactions at the nanoscale: The...

van der Waals interactions at the nanoscale: Theeffects of nonlocalityYu Luo1, Rongkuo Zhao1, and John B. Pendry2

The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2013.

Contributed by John B. Pendry, November 12, 2014 (sent for review September 4, 2014; reviewed by Che Ting Chan and Peter Nordlander)

Calculated using classical electromagnetism, the van der Waalsforce increases without limit as two surfaces approach. In reality,the force saturates because the electrons cannot respond to fieldsof very short wavelength: polarization charges are always smearedout to some degree and in consequence the response is nonlocal.Nonlocality also plays an important role in the optical spectrum anddistribution of the modes but introduces complexity into calcu-lations, hindering an analytical solution for interactions at thenanometer scale. Here, taking as an example the case of twotouching nanospheres, we show for the first time, to our knowl-edge, that nonlocality in 3D plasmonic systems can be accuratelyanalyzed using the transformation optics approach. The effects ofnonlocality are found to dramatically weaken the field enhance-ment between the spheres and hence the van der Waals interactionand to modify the spectral shifts of plasmon modes.

van der Waals | nonlocality | transformation optics | plasmonics

The van der Waals force is an electromagnetic interactionbetween correlated fluctuating charges on two electrically

neutral surfaces (1–4). As the surfaces approach more closely,the force increases as fluctuations of shorter and shorter lengthscale come into play, but ultimately the force will saturate whenthe surfaces are so close that the even shortest wavelength chargefluctuations are included. It is this saturation with which we areconcerned in this paper and to treat it, we need to go beyond theconventional description of a solid by a permittivity, «ðωÞ, thatdepends only on the frequency, ω. Here we recognize that theresponse of a solid depends on the length scale of the fluctuationsand introduce a formalism using a generalized nonlocal permit-tivity, «ðω; kÞ (5–13), that also depends on the wave vector, k, andhence takes into account the saturation. Neglect of nonlocalityleads to an unphysical diverging van der Waals force at shortdistances. We apply the technique of transformation optics (14–16) to solve the difficult problem of including nonlocal effectswhen two nanoscale bodies interact and illustrate our theory withcalculations for two closely spaced nanospheres.Ultimately at a few tenths of a nanometer, just before the

surfaces touch, direct contact of the charges will come into playthrough electron tunneling (17–25); in other words, chemicalbonding will dominate the final approach. Here we are not con-cerned with chemical bonding, which is extensively treated else-where in the literature.The van der Waals force is weaker than chemical bonding

effects, but plays an important role in a broad range of areas, suchas surface and colloidal science, nanoelectromechanical systems,and nanotechnology (2, 3, 26–28). In the classical electrodynamicspicture where nonlocality is neglected, the fluctuating surfacecharges are located precisely on the surface and these infinitelycompressed charges result in the unphysically divergent van derWaals force (1, 29). In a more realistic framework, consideringthe inherent quantum nature of electrons, the surface charges areintrinsically smeared across the boundary in a subnanometer layer(7, 20–22), which dramatically alters the van der Waals energy inthe small gap limit (30–33). Due to the complexity introduced bynonlocality, its influence on the van derWaals force in 3D geometries

has never been described appropriately. Esquivel-Sirvent andSchatz investigated the influence of nonlocality on the van derWaals force between two spherical gold nanoparticles (32).However, they used Hamaker’s method (1), which has beenwidely recognized as an inaccurate method for such nanoparticlesystems where the size and the separation are comparable.We developed a theoretical approach to treat nonlocal prob-

lems, which is as accurate as other semiclassical nonlocal methods(such as the hydrodynamic model) but much more efficient interms of both simulation time and memory consumption whenimplemented in numerical simulations (34). In this paper, we showthat 3D nonlocal systems can be studied analytically by combiningour nonlocal model with the technique of transformation optics.

Results and DiscussionWe start by considering a dimer of nanospheres, a prototypicalplasmonic system. Other structures such as a sharp metal tip (35–37) and two overlapping spheres (38–40) can be studied in asimilar manner. Widely separated nanoparticles (of sizes largerthan 10 nm) can be treated by neglecting nonlocality, and trans-formation optics provides an elegant analytical solution for theelectromagnetic (EM) field (41). To calculate the van der Waalsforce, an analytic form of the permittivity is needed: metals can bedescribed by a local permittivity, «MðωÞ= «b −ω2

p=½ωðω+ iγÞ�,where «b is usually expressed as a summation of several Lorentzmodels (see SI Text for details). As the particle separation is re-duced to a nanometric scale, the nonlocal smearing of metallicelectrons plays an important role, and the implementation ofa spatially dispersive metal permittivity for longitudinal EMfields, «Mðω; kÞ= «b −ω2

p=½ωðω+ iγÞ− β2jkj2�, is required. The βfactor describes the degree of nonlocality and is proportional tothe Fermi velocity.

Significance

The van der Waals interaction is a ubiquitous but subtle forcebetween particles mediated by quantum fluctuations of charge.It is the most long-range force acting between particles andinfluences a range of phenomena such as surface adhesion,friction, and colloid stability. Calculations of the force betweenparallel surfaces >10 nm apart is a simple task, but when thegeometry is more complex, e.g., a pair of nanospheres <5 nmapart, the task is more difficult. Furthermore a macroscopicdescription of the dielectric properties no longer suffices, andwe must consider the diffuse nonlocal nature of the electronpolarization cloud. In this paper, we propose a simple analytictreatment of the problem.

Author contributions: Y.L., R.Z., and J.B.P. designed research; Y.L. and R.Z. performedresearch; and Y.L., R.Z., and J.B.P. wrote the paper.

Reviewers: C.T.C., Hongkong Univ. of Science and Technology; and P.N., Rice University.

The authors declare no conflict of interest.1Y.L. and R.Z. contributed equally to this work.2To whom correspondence should be addressed. Email: [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1420551111/-/DCSupplemental.

18422–18427 | PNAS | December 30, 2014 | vol. 111 | no. 52 www.pnas.org/cgi/doi/10.1073/pnas.1420551111

Our recently proposed nonlocal model simplifies the theoret-ical treatment of this problem by avoiding the implementation ofa k-dependent permittivity and replacing the spatial nonlocalitywith a thin dielectric layer decorating the metal surface (34). Thiseffective local treatment of a nonlocal problem has been shown tothe highly accurate. Using this model, the dimer of nanospherescan be represented by a pair of core-shell nanoparticles, as shownin Fig. 1A. The dielectric constant of the core is given by thenormal metal permittivity «C = «MðωÞ, whereas the dielectricfunction «S of the shell is approximately expressed by

«SΔd′

=«MðωÞ+ 1«MðωÞλLðωÞ; [1]

where Δd′ is the thickness of the shell; and λLðωÞ denotes thepenetration depth of the surface charges. It can be calculated as (5)

λL =

ffiffiffi35

rvFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ω2p

.«Mðω→∞Þ−ωðω+ iγÞ

r ; [2]

where vF and ωp represent the Fermi velocity and the bulkplasma frequency of the metal, respectively. Eq. 1 only definesthe ratio of the permittivity to the thickness of the dielectricshell. It indicates that we have a freedom to choose variouscombinations of «S and Δd′, corresponding to different config-urations of the system. Here, two approaches are considered.

Shifting the Metal Boundary. The simplest interpretation of thenonlocal effects is to describe the smearing of the induced po-larization charge densities across the boundary as a shifting ofthe metal boundary (42). The amount of boundary shifting canbe obtained by letting «S = 1 in Eq. 1;

Δd′ðωÞ=�

«MðωÞ«MðωÞ+ 1

�λLðωÞ; [3]

which is frequency dependent. Applying this shifting boundaryapproximation to the nanosphere dimer gives an effective geometry

of spheres with a slightly increased separation and reduced radii ofthe two spheres, as shown in Fig. 1B. This effective local system canbe analytically investigated using the transformation optics ap-proach (41, 43).

Asymmetric Shell. The shifting boundary approach is valid only atlow frequencies, where Δd′ is much smaller than the penetrationdepth of surface plasmons (34). Keeping Δd′ to a sufficientlysmall value gives a more rigorous approach, which can also de-scribe the optical responses of the structure in the higher fre-quency range. However, a dimer of core-shell nanoparticles withuniform shell thicknesses is difficult to study analytically. Alter-natively, we can assume a nonuniform Δd′ðr′Þ, which is pro-portional to r′2 (r′ is the distance to the origin in Fig. 1C). Aninverse transformation, r′=R2

T=ðr−R0Þ, maps the asymmetriccore-shell dimer structure in Fig. 1C to a concentric annulus asshown by Fig. 1D. Transformation optics theory requires the per-mittivity of the materials in the annulus geometry as

«BðrÞ= «MR2T

�jr−R0j2; [4]

in the two blue regions

«WðrÞ=R2T

�jr−R0j2; [5]

in the white region, and

«YðrÞ= «MΔdð«M − 1ÞλL

�RT

jr−R0j�4

; [6]

in the yellow regions.We can solve for the electrostatic potential ϕðrÞ in this annulus

geometry by expanding it in spherical harmonics and solvinga penta-diagonal scattering matrix. This procedure will in turngives ϕðr′Þ in the asymmetric core-shell dimer in Fig. 1C. De-tailed derivations are provided in Materials and Methods.To verify our approach, we first apply our methodologies to

calculating the optical spectra of two spherical dimers. Fig. 2 Aand B displays the absorption efficiency (defined as the absorptioncross section over the dimer physical size) contour plot for gold

R’ Δd’~δ’

R0

δ

R2

Δd

ε’ ~r

ε’ ~ 0θ

1|r−R0|2

|r−R0|2Δd

{

R’

δ’

Nonlocal problem

Δd’= constant

R’-Δd’

δ+2Δd’

Shifting boundary

Asymmetric core-shell Transformed annulus

z

x

z’

x’

Inversion point

R1

Inversion point

ε ~rε ~ 0θ

1|r−R0|4{

A B

C D

Fig. 1. The schematic of the problem. A dimer of separated nanospheres where the surface charge smearing is described by an effective cover layer of (A)a constant thickness Δd′, (B) a constant permittivity, «S = 1 (shifting the metal boundary by Δd′), or (C) a variable thickness Δd′ and spatially dependentpermittivity. Under an inverse transformation, the asymmetric core-shell structure in C can be mapped to a dielectric annulus (shown in D) defined bya dielectric-coated metal sphere and a dielectric-coated hollow sphere.

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nanosphere dimers of 5 and 30 nm radii vs. the frequency and thegap size. As the gap narrows, the resonance shows a red shift,which saturates at the gap size δ≈ 0:2 nm irrespective of the size ofnanoparticles. However, the dimer of nanospheres with largerradius shows a more pronounced red shift (larger shiftingfrequency range) and therefore a broader plasmonic resonanceband at small separations where nonlocal effects dominate. Asimilar result has been observed in the 2D case of nanowire di-mer (8, 10). Fig. 2 C and D shows the numerical simulationresults (using COMSOL Multiphysics) for the light absorption byspherical dimers of different sizes with (green dots) and without(cyan dashed curves) inclusion of the nonlocal effects. The redsolid curves correspond to the absorption spectrums obtainedwith our analytical calculations, showing excellent agreementwith the nonlocal numerical simulations.Recent experiments have shown that nonlocal effects set an

upper limit for maximum field enhancement achievable withplasmonic nanoparticles (44, 45). Here, we investigate the fieldenhancement spectra at the center of the gap between twonanospheres. The separation between nanoparticles is 0.4 nm,above the quantum tunneling regime (17, 18). We consider firsta dimer of two identical silver spheres (R= 30 nm; the JohnsonChristy data for silver are used) (46) and compare the classicalcalculation (black dashed curve) with our theoretical nonlocalcalculations. Both the nonuniform coating assumption (red solidcurve) and shifting boundary assumption (blue dots) are con-sidered. In the nonlocal description, the spreading of the chargeeffectively smoothes the boundary at the gap, resulting in a re-duction of the field enhancement with the peaks shifting towardsmaller wavelengths compared with the local description. How-ever, as depicted in Fig. 3A, considerable field enhancement canstill be achieved with a maximum value larger than 1,000. Theshifting-boundary calculation predicts the resonance shift and

field enhancement in good agreement with the exact asymmetric-shell approach at frequencies below the bulk plasma frequency.We next replace one silver sphere with a silicon sphere of the

equal size and keep the separation between particles at 0.4 nm.The Palik data for silicon (47) are used. We neglect nonlocalityin the silicon. At such a small separation, the fundamental modein the silicon sphere hybridizes with the surface plasmon polar-itons in the silver surface, allowing for a subwavelength con-finement and large field enhancement. In this metal-dielectrichybrid system, more energy resides in the dielectric sphere,resulting in less energy confinement at the gap and smaller fieldenhancement compared with the silver sphere dimer system.However, as shown in Fig. 3B, the maximum field enhancementcan still reach up to more than 500, and the frequency of the fieldenhancement peak is less sensitive to the nonlocal effect. Resultsfor the sphere-plane configuration given in SI Text also confirmthat extremely large field enhancements are possible even in therealistic nonlocal case.It is worth pointing out that the field enhancement in the

junction of two spheres also gives an indirect measurement of thevan de Waals force. On one hand, the van de Waals force orig-inates from the zero point energy of the modes, whose change ismeasured by the resonance shifts. On the other hand, the fieldenhancement is directly related to the coupling between the twoparticles, which is also measured by the amount of resonanceshifts of the modes (44). Therefore, the reduction of the fieldenhancement in the nonlocal case indicates that the change ofzero point energy, and hence the van de Waals force, is smallerthan those in the local case.To illustrate this point, we first plot in Fig. 4 the shifts of res-

onances in terms of the gap size. The plasmonic system consists oftwo identical gold spheres of 5 nm radius. Both local (Fig. 4A) andnonlocal (Fig. 4B) cases are considered. Different from the studyin Figs. 2 and 3, where we apply the plane wave incidence to excite

−2

−1

0

110

10

10

10 −2

−1

0

110

10

10

10

δ (n

m)

δ (n

m)

δ

R = 5 nm

δR = 30 nm

0.5 1.0 1.5 2.0 2.5

10−2

10−1

100

101

frequency (eV)0.5 1.0 1.5 2.0 2.5

10−2

10−1

100

101

frequency (eV)

0.5 1.0 1.5 2.0 2.5frequency (eV)

0.5 1.0 1.5 2.0 2.5frequency (eV)

0.2 nm

5 nm

R = 30 nm

0.2 nm

Local (simulation)Nonlocal (simulation)Nonlocal (theory)

Local (simulation)Nonlocal (simulation)Nonlocal (theory)A

bsor

ptio

n cr

oss-

sect

ion

Abs

orpt

ion

cros

s-se

ctio

n

C D

A B

Fig. 2. The absorption spectrum for a dimer of spherical particles. The contour plot of the absorption cross section vs. the frequency and the separation fora pair of gold nanospheres with equal radii of (A) 5 and (B) 30 nm. Comparison of our analytical calculations with local and nonlocal numerical simulations fortwo closely separated (δ= 0:2 nm) gold spheres with equal radii of (C) 5 and (D) 30 nm.

18424 | www.pnas.org/cgi/doi/10.1073/pnas.1420551111 Luo et al.

only the bright modes, here we investigate all of the plasmonicmodes (including the dark ones) by solving the characteristicequation (41, 43). For a simple test, the dielectric constant ofgold is described by the Drude model, «MðωÞ= 1−ω2

p=ðω+ iγÞ2,with a bulk plasma frequency ωp = 3:3  eV and a damping fre-quency γ = 0. The nonlocality is described by the parameterβ= 0:0036c. Here the loss is ignored to give a more critical testof our theory. In both local and nonlocal calculations, the oddmodes tend to redshift and the even modes show a blue shift asthe two spheres approach (see the inset of Fig. 4 for the defi-nition of the odd and even modes). However, different from thelocal picture, all of the frequency shifts in the nonlocal case saturateat certain frequencies for small separations. As pointed out before,the van der Waals force results from the change of zero pointmotion of these modes, and saturation of the shifts indicates thatthe van der Waals force would be saturated in the touching limit.Nonlocality also results in the odd modes shifting above the

surface plasmon frequency toward the bulk plasmon frequencyωp. It is worth noting that the bulk longitudinal plasmon modes(black dots/green circles in Fig. 4B) also emerge in the nonlocalcase. However, the shifts of these modes are considerably smallerthan the shifts of the surface modes. Therefore, the longitudinalplasmon modes have relatively small contribution to the van deWaals force.

To study the van de Waals interaction quantitatively, wefit the experimental data for gold using one Drude and tenLorentz terms

«ðωÞ= 1−ω20

ωðω+ iγ0Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}free electrons

−X10j=1

fjω2j

ω2 −ω2j + iγjω|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

core electrons

: [7]

Generally speaking, both free electrons and core electrons con-tribute to the dielectric response of metal. The Drude term in Eq.7 corresponds to the free electrons, which show a divergent polar-izability at the zero frequency (48). The core electrons, on theother hand, are bonded by the nuclei. They exhibit the strongestresponses at finite resonance frequencies, and their contributioncan be approximately described by Lorentizians. The fitting param-eters in Eq. 7 are given in SI Text.Using the aforementioned shifting boundary method, the van

der Waals energy can be directly calculated within the local picturewith an effective separation δ+ 2Δd and an effective diameterD− 2Δd, as shown by the inset sketch in Fig. 5A. Using our pre-vious procedure in (43), the van der Waals energy, taking intoaccount the nonlocal effect, can easily be obtained. It is shown by

R’ = 30nm

δ’ = 0.4 nm

Fiel

d E

nhan

cem

ent

Fiel

d E

nhan

cem

ent

Silver / Silver Silver / Silicon

R’ = 30nm

δ’ = 0.4 nm

300 400 500 600 700 800 9000

500

1000

1500

2000

2500

wavelength (nm)300 400 500 600 700 800 900

0

200

400

600

800

wavelength (nm)

Local Nonlocal (asymmetric cover) Nonlocal (shifting boundary)

A B

Fig. 3. The field enhancements at the center of the gap between two spherical particles (A) between two silver spheres of equal radius 30 nm, separated bya 0.4-nm gap and (B) between a silver and a silicon spheres of equal radius 30 nm, separated by a 0.4-nm gap.

10−2 10−1 100 1011.5

2.0

2.5

3.0

3.5

Gap (nm)

Freq

uenc

y (e

V)

odd (nonlocal)even (nonlocal)

odd (longitu.)

even (longitu.)

ε = 0

10−2 10−1 100 101

Gap (nm)

odd (local)even (local)

ε = −1

Longitudinal modes

++

+

-

-

-

-

- ----

++

+

-

-

-

-

---- -

---

+

+

+

+

+++

++++

+

-

-

-

-

---- -

even (antibonding) mode

odd (bonding) mode

A B

Fig. 4. Plots of plasmon resonance modes supported by a pair of 5-nm radius gold spheres vs. the separation. The permittivity of gold is «gold =1− 3:32=ω2. (A) In the local description, as the gap size decreases, the odd modes (red dashed lines) tend to zero frequency and the even modes (bluedashed lines) fall into two groups: one group of modes tends to the bulk plasma frequency ωp = 3:3  eV («= 0) and another to discretized values below thesurface plasmon frequency ωsp = 2:33  eV («=−1). (B) In the nonlocal description with β= 0:0036c, only one group of even modes is observed. Moreover,the longitudinal plasmon modes (black dots for odd modes and green circles for even modes) emerge above ωp, and they remain nearly constant for variousparticle separations. The inset on the right plots the surface charge distributions for the odd (bonding) and even (antibonding) dipolar modes (49).

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the blue dashed line in Fig. 5A. Although the calculation in thelocal picture is exact, the shifting boundary approach itself is in-trinsically an approximate procedure. To check its accuracy, weused the multipole expansion method (50) (see Materials andMethods for details). In the multipole expansion calculation,the nonlocal effect is exactly considered as long as enoughterms in the series are included. The results become moreaccurate if more terms are used. As shown in Fig. 5A, we trun-cated the series at l= lmax − jmj for each m(jmj≤ lmax), wherelmax = 50 (cyan), 100 (magenta), and 150 (red). We found thatthe multipole expansion results converge to the shiftingboundary one when increasing the truncating number lmax. Thisphenomenon indicates that the shifting boundary method con-verges more rapidly than the multipole expansion method. It isalso considerably more rapidly calculated.For comparison, the van der Waals energy calculated neglect-

ing the nonlocal effect is also shown in Fig. 5A by the black dottedline. It diverges as −1=d in the small gap limit. However, thenonlocal results are eventually saturated at the touching limit.The magnitude of the nonlocal results is thus much smallerthan the local one at subnanometer separations. We also studythe van der Waals energy considering the nonlocal effect butneglecting the Lorentz terms (keeping the Drude term only) inthe fitted permittivity formula. As shown by the black shortdashed line, the magnitude of the van der Waals energy calcu-lated neglecting the Lorentz terms is much smaller than the re-alistic nonlocal one. This result indicates that the core electronsmake a significant contribution to the van der Waals forces atsuch a scale.In Fig. 5B, the same method is used to study the van der Waals

energy between gold and silicon nanospheres. The silicon as a di-electric is described in the local picture. Because the permittivityof silicon at imaginary frequencies is smaller than that of gold inthe whole spectrum range (Fig. 5B, Inset), the van der Waalsenergy between gold and silicon nanospheres in the local picture issmaller than that between two gold spheres, as shown by the blackdotted line.In summary, we presented an analytical approach based on

transformation optics to study the nonlocal effects in 3D plas-monic structures. As an example, nanosphere dimers with variousseparations are considered, and solutions for the optical spec-trum, distribution of the modes, and van de Waals energy havebeen obtained. Our methodology not only makes feasible theanalytical investigation of 3D nonlocal problems but also sheds

insight into the understanding of nonlocal effects in plasmonicnanostructures.

MethodsSolution to the Concentric Annulus in Fig. 1D. The core-shell structure in thephysical space in Fig. 1C has an inhomogeneous thickness, which acquires anr-dependence

Δd′1ð2Þ =R2TR1ð2Þ −R0

2 Δd: [8]

Then, the tangential and normal permittivity components of the dielectricmaterial can be obtained as

«′θ,φ = 0,  «91ð2Þr =«b«MqLi′1

qLR′1ð2Þ

�ð«M − «bÞi1

qLR′1ð2Þ

�Δd′1ð2Þ =C1ð2ÞΔdR1ð2Þ −R0

2, [9]

where we introduced

C1ð2Þ =«b«MqLR2

Ti′1qLR′1ð2Þ

�ð«M − «bÞi1

qLR′1ð2Þ

� , [10]

and the longitudinal plasmon normal wave vector qL =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2p=«b −ωðω+ iγÞ

q=β.

ilð · Þ and i9l ð · Þ are the modified spherical Bessel function and its derivative. Inthe transformed annulus frame, the permittivity of virtual dielectric materialcan be calculated as

«θ,φ = 0,  «1ð2Þr =R2TR1ð2Þ −R0

2 «91ð2Þr =C1ð2ÞR2TΔdR1ð2Þ −R0

4: [11]

The electrostatic potential in an anisotropic medium characterized by Eq. 10must satisfy the following partial differential equation

∇ðe ·∇ϕÞ= 0⇒1r2

∂∂r

r2R1ð2Þ −R0

4 ∂∂r

ϕ

!= 0: [12]

The general solution to Eq. 12 can be found as

ϕ=−b1ðθ,φÞ24 R2

1ð2ÞR20

+ 1

!2

−4R1ð2ÞR0

R21ð2ÞR20

+1

!cos θ+

4R21ð2Þ cos

2 θ

R20

35R0

r+b2ðθ,φÞ:

[13]

For the spherical dimer problem, we can write the electrostatic potential inthe anisotropic dielectric region as

10−2 10−1 100 101−10

−8

−6

−4

−2

0

δ (nm)

E (e

V)

lmax=50

150

100

Drude

shift

bou

ndar

y local

−100

−10−1

−10−2

10 nmΔd

0.8

0.9

1.1

EAu

- Si /

EAu

- Au

1.0

10−2 10−1 100 101

local

nonlocal

10 nm

Au Si

ε ( i

ξ)

Frequency (eV)102101

10010-1100

101

102

103

104

Au

Si

δ (nm)

δδ

A B

Fig. 5. van der Waals energy between two 10-nm-diameter nanospheres as a function of the separation. (A) Between two gold nanoparticles. For the bluedashed line, the nonlocal effect is considered by using the shifting boundary method as shown by the inset sketch, where the separation becomes δ+ 2Δd andthe diameter of each sphere decreases to D−2Δd. For the solid lines, lmax = 50 (cyan), 100 (magenta), and 150 (red), the nonlocal van der Waals energies arecalculated using the multipole expansion method, truncating the series at l= lmax − jmj for each m. The black dotted line shows the van der Waals energycalculated neglecting the nonlocal effect. The energy considering the nonlocal effect but neglecting the Lorentz terms in the permittivity is shown by theshort dashed line. (Inset) Log-log plot in the separation range between 1 and 10 nm. (B) Between a gold and a silicon nanoparticle, normalized to thatbetween two gold nanoparticles of the same size. Solid and dotted lines are nonlocal and local results, respectively. (Inset) Permittivities of gold and silicon atimaginary frequencies, fitted from the experimental data (SI Text).

18426 | www.pnas.org/cgi/doi/10.1073/pnas.1420551111 Luo et al.

ϕ=Xlm

8>>>>>>><>>>>>>>:−b+

lm

266666664

R21ð2ÞR20

+1

!2

+4R2

1ð2ÞR20

cos2θ

−4R1ð2ÞR0

R21ð2ÞR20

+1

!cos θ

377777775R0

r+b−

lm

9>>>>>>>=>>>>>>>;R1ð2Þ −R0

Ylmðθ,φÞ

=Xlm

R21ð2ÞR0r

8>>><>>>:

−�R1ð2Þ

�R0 +R0

�R1ð2Þ

�2 +4Am2

l +Am2l+1

� b+lm

+4R1ð2Þ

�R0 +R0

�R1ð2Þ

�b+l+1mA

ml+1 +b+

l−1mAml

�−4b+l+2mA

ml+2A

ml+1 +b+

l−2mAml−1A

ml

�+b−

lmR0r�R21ð2Þ

9>>>=>>>;R1ð2Þ −R0

Ylmðθ,φÞ,

[14]

where we introduced Aml =

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl2 −m2Þ=ð4l2 − 1Þ

pif jmj< l,

0 if jmj≥ l,and

R1ð2Þ −

R0= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R21ð2Þ +R2

0 − 2R1ð2ÞR0 cos θq

:

Matching the boundary conditions (see SI Text for details), we can see thatthe presence of spatial nonlocality results in penta-diagonal matrices com-pared with the tridiagonal matrices (41) in the local case.

Multipole Expansion Method. The total van der Waals energy between twospheres can be calculated in terms of the reflection amplitude of each sphere,cooperating with translation matrices that transfer a solution expressed ina basis appropriate to one sphere to a basis appropriate to the other

U=Z

4πi

X+∞m=0

Z+∞−∞

ln det    ð1−M ·R1 ·N ·R2Þdω, [15]

where M and N are full matrices given by

Mll′ = ð−1Þl′+m ðl+ l′Þ!ðl−mÞ!ðl′+mÞ!, Nll’ = ð−1Þjl−l’jMll’::

R1 and R2 are diagonal matrices given by

R1,ll′ = δll′

�−«M + 1+A1

«M + ð1+A1Þð1+ lÞ=l��

R′1R′1 +R′2 + δ

�2l+1

,

R2,ll′ = δll′

�−«M + 1+A2

«M + ð1+A2Þð1+ lÞ=l��

R′2R′1 +R′2 + δ

�2l+1

,

where R′1 and R′2 are the radii of each sphere in the physical space; δ is theseparation between them; and A1 and A2 are the parameters where thenonlocal effect enters and given by

A1 =lð«M − 1ÞqLR1

ilðqLR1Þi9l ðqLR1Þ

, A2 =lð«M − 1ÞqLR2

ilðqLR2Þi9l ðqLR2Þ

, [16]

where «M is the permittivity of each sphere. If qL →∞, A1,2 = 0, which goes tothe local case.

ACKNOWLEDGMENTS. Helpful discussions with Stefan A. Maier andA. I. Fernandez-Dominguez are gratefully acknowledged. We thank thefollowing for support: the Gordon and Betty Moore Foundation (J.B.P.), theRoyal Commission for the Exhibition of 1851 (R.Z.), and the LeverhulmeTrust (Y.L. and J.B.P.).

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Luo et al. PNAS | December 30, 2014 | vol. 111 | no. 52 | 18427

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