PHYSICAL REVIEW LETTERS week ending PRL 100, 106804 (2008) 14 MARCH 2008
Probing the Photonic Local Density of States with Electron Energy Loss Spectroscopy
F. J. Garcı´a de Abajo1,* and M. Kociak2 1Instituto de O´ ptica—CSIC, Serrano 121, 28006 Madrid, Spain 2Laboratoire de Physique des Solides, CNRS, UMR8502, Universite´ Paris Sud XI, F91405 Orsay, France (Received 29 October 2007; revised manuscript received 24 January 2008; published 12 March 2008) Electron energy loss spectroscopy performed in transmission electron microscopes is shown to directly render the photonic local density of states with unprecedented spatial resolution, currently below the nanometer. Two special cases are discussed in detail: (i) 2D photonic structures with the electrons moving along the translational axis of symmetry and (ii) quasiplanar plasmonic structures under normal incidence. Nanophotonics in general and plasmonics, in particular, should benefit from these results connecting the unmatched spatial resolution of electron energy loss spectroscopy with its ability to probe basic optical properties such as the photonic local density of states.
DOI: 10.1103/PhysRevLett.100.106804 PACS numbers: 73.20.Mf, 78.20.Bh, 79.20.Uv
While a plethora of nanophotonic structures is currently derivation of this statement is offered, illustrated by nu- being devised for diverse applications like achieving single merical examples for both translationally invariant geome- molecule sensitivity in biosensing [1] or molding the flow tries and planar structures. Our results allow directly of light over nanoscale distances for signal processing [2], interpreting EELS data in terms of local photonic proper- no optical characterization technique exists that can render ties that encompass the full display of optical phenomena spectroscopic details with truly nanometer spatial resolu- exhibited by nanostructures, ranging from localized and tion. The need for that kind of technique is particularly propagating plasmons in patterned metallic surfaces [6]to acute in nanometric plasmonic designs that benefit from band gaps in dielectric photonic crystals [11]. sharp edges and metallic surfaces in close proximity to Green tensor and LDOS.—The optical response of a yield large enhancements of the electromagnetic field. nanostructure is fully captured in its electric Green tensor Scanning transmission electron microscopes (STEM) and its LDOS [12]. In particular, the electric field produced can plausibly cover this gap, as they perform electron by an external current density j r;! in an inhomogeneous energy loss spectroscopy (EELS) with increasingly im- medium of permittivity r;! can be written in frequency proved energy resolution that is quickly approaching the space ! as width of plasmon excitations in noble metals [3] and with Z 0 0 0 spatial resolution well below the nanometer [4]. The con- E r;! 4 i! dr G r; r ;! j r ;! (1) nection between EELS and photonics can be readily estab- lished when low-energy losses in the sub-eV to a few eV in terms of G, the electric Green tensor of Maxwell’s range are considered, compatible with typical photon en- equations in Gaussian units, satisfying
ergies in photonic devices. A formidable amount of infor- 0 mation is available in the literature for this so-called r r G r; r ;! valence EELS, including for instance studies of single 1 !2=c2 r;! G r; r0;! r r0 (2) nanoparticles of various shapes [3,5], interacting nanopar- c2 ticles [6], thin films [7], composite metamaterials [8], and carbon nanostructures [9]. Many of these reports are rele- and vanishing far away from the sources. vant to current nanophotonics research, in which the opti- We then define the LDOS projected along a unit vector n^ cal response of nanoparticles, nanoparticle assemblies, and as [13] patterned nanostructures plays a central role. In this con- 2!
r;! Imfn^ G r; r;! n^ g: (3) text, EELS has been recently demonstrated to image plas- n^ mon modes with spatial resolution better than a hundredth of the wavelength in triangular nanoprisms [3]. However, In free space, the uniform LDOS is known from blackbody despite significant progress from the theoretical side [10], theory (with a factor of 1=3 arising from projection over a no synthetic and universal picture has emerged to explain specific Cartesian direction n^ ): the spatial modulation of EELS measurements on arbitrary 0 2 2 3 r;! ! =3 c : (4) nanostructures. n^ In this Letter, we show that EELS provides direct infor- Similar to its electron counterpart in solid state physics, the mation on the photonic local density of states (LDOS), and photonic LDOS equals the combined local intensity of all thus it constitutes a suitable tool for truly nanometric eigenmodes of the system under consideration, provided characterization of photonic nanostructures. A rigorous they are well defined (e.g., in the absence of lossy media)
0031-9007=08=100(10)=106804(4) 106804-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 106804 (2008) 14 MARCH 2008
[14]. An alternative interpretation, which holds even in the becomes presence of lossy materials [15], comes from the realiza- 2 tion that 4 2!D2=@ is the decay rate for an excitation 4e ind R0;! ImfGzz R0; R0;q; q; ! g dipole strength D [16,17]. Finally, we point out that a @ complete definition of the LDOS should include a mag- 2 e2 R ;q;! ; (11) netic part [18], which is however uncoupled to our fast @! z^ 0 electrons. Energy loss probability.—The energy loss suffered by a where q !=v and we have defined fast electron passing near an inhomogeneous sample and 2! moving with constant velocity v along a straight line R;q;! Imfn^ G R; R;q; q; ! n^ g (12) n^ trajectory r re t can be related to the force exerted by the induced electric field Eind acting back on the electron as as a generalized density of states that is local in real space [19] along the R directions and local in momentum space along Z Z 1 the remaining z direction, parallel to the electron velocity ind @ E e dt v E re t ;t !d! ! ; (5) vector. 0 The value of q !=v reflects conservation of energy where the e electron charge has been included (i.e., and momentum in the transfer of excitations of frequency E>0) and ! and momentum @q from the electron to the sample. It is Z interesting to note that for an electron moving in an infinite e i!t ind ! dt Refe v E r t ;! g (6) vacuum one has @! e ! 2 2 2 is the loss probability. The Fourier transform z^ R;q;! L 2 1 q c =! !=c q ; (13) Z 2 c ind d! i!t ind E r;t e E r;! (7) which is always zero for subluminal electrons moving with 2 velocity v R ;! 0 @ along z, the Green tensor G r; r0;! depends on z and z0 Z 0 0 only via z z , and thus one can write dtdt0 Imfei! t t Gind r t ; r t0 ;! g; zz e e Z dq 0 0 ~ 0 iq z z G r; r ;! G R; R ;q;! e ; (14) (10) 2 where Gzz z^ G z^, the dependence of the loss proba- 0 0 ind where G~ R;R ;q;! 1=L G R;R ;q; q;! . Accord- bility on R0 is shown explicitly, and G denotes the G ingly, the local density of states can be decomposed into induced Green tensor obtained from by subtracting the @ free-space Green tensor. momenta components q along the z axis, Relation between EELS and LDOS.—Noticing that Z dq r;! ~ R;q;! ; (15) ze t vt, the time integrals of Eq. (10) yield the n^ 2 n^ Fourier transform of the induced Green tensor with respect 0 ind 0 0 to z and z , Gzz R; R ;q; q ;! , in terms of which where 106804-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 106804 (2008) 14 MARCH 2008 2! ~ ABC ~ R;q;! Imfn^ G R; R;q;! n^ g n^ 1 R;q;! : (16) L n^ Finally, combining Eqs. (11) and (16), one finds a relation 500 nm between the loss probability per unit of path length and the LDOS, DEFG 2 R0;! 2 e ~z^ R0;q;! ; (17) L @! with q !=v. 0>0.1 Equation (17) provides a solid link between LDOS and Loss probability EELS in a wide class of geometries that includes aloof −1 nm−1) trajectories in semi-infinite surfaces and thin films. F Interestingly, only q > !=c values are probed, lying out- 0.10 ) 1 side the light cone, and therefore difficult to study via − optical techniques. Trapped modes such as surface- nm 1 0.08 G plasmon polaritons lie in that region and are a natural − target for application of our results. Besides, the present E study can be directly applied to cathodoluminescence (CL) 0.06 B in all-dielectric structures, in which energy loss and CL D emission probabilities are identical. 0.04 A connection between the photonic density of states in A C momentum space and EELS has been previously reported 0.02 for electrons moving parallel to pores in 2D self-assembled alumina photonic crystals [11]. However, the above deri- Loss probability (% eV 0.00 vation is the first proof to our knowledge that a formal 860 880 900 relation exists between LDOS and EELS. Wavelength (nm) For illustration, we offer in Fig. 1 the EELS probability for electrons moving inside a finite hexagonal 2D crystal of FIG. 1 (color online). EELS probability for 200-keV electrons aligned Si nanowires, calculated with the boundary ele- moving parallel to an array of 13 Si nanowires of 300 nm in ment method (BEM) [20]. The photon wavelength range diameter (solid curve), as compared to an isolated nanowire under consideration includes two Mie modes of the iso- (broken curve). The nanowires are arranged in a hexagonal lated wire (broken curve), and significant hybridization lattice with a period of 350 nm. The contour-plot insets show between neighboring wires takes place in the array. The the impact-parameter dependence of the EELS probability for selected energy losses (labeled A–G), with the position of the loss probability varies relatively smoothly with impact electron beam considered in the curves indicated by an open parameter, a behavior which was expected in the LDOS symbol and the cylinder contours shown by dashed curves. for photon wavelengths relatively large compared to the diameter of the cylinders. Interestingly, the loss probability takes significant values in the interstitial regions, several tures are routinely obtained using scanning near-field opti- tens of nanometers away from the Si. Finite structures like cal microscopy, although the lateral resolution of this that considered in Fig. 1 present a colorful evolution of technique can hardly reach 50 nm. Here again EELS modes, the analysis of which can be useful for instance in renders much higher lateral resolution (down to the nano- the design of microlaser cavities. In infinite crystals, the meter) and permits obtaining information directly related loss probability exhibits van Hove singularities [21], which to the LDOS. Besides the formal relation between EELS follow rigorously those of the LDOS in translationally and the momentum-resolved LDOS expressed in Eq. (11), invariant systems according to Eq. (17). we offer in Fig. 2 a more detailed comparison between Planar geometries.—As microchip features continue to EELS and r;! (local in all spatial directions) for an Ag shrink, lithographically patterned metal structures are be- disk calculated with BEM [20]. This figure proves how the coming natural candidates to replace current electronic EELS probability can mimic quite closely z r;! , with z microcircuits. The new structures will operate at frequen- perpendicular to the planar structure and parallel to the cies above the terahertz rather than gigahertz, and will electron velocity. This resemblance holds for different carry electric signals strongly mixed with the electromag- accelerating voltages, making the interpretation robust netic fields that they generate in what is known as surface with respect to experimental details. Figure 2 provides a plasmons. The optical properties of metallic planar struc- solid example supporting the use of EELS to measure 106804-3 PHYSICAL REVIEW LETTERS week ending PRL 100, 106804 (2008) 14 MARCH 2008 5 electron 160 4 30 nm R LDOS FIG. 2 (color online). Relation be- 10 nm 3 normalized tween EELS and LDOS in planar ge- z to vacuum ometries. (a) We consider an Ag disk 2 of height 10 nm and radius 30 nm. The Photon energy (eV) ρ (r,ω) ρ(r,ω) 0 (a) (b)z (c) electrons move along z, perpendicular to 5 the disk. The EELS probability for 200- 2.5% keV (d) and 100-keV (e) electrons mim- 4 Probability ics closely the z-projected LDOS in a 3 per electron plane 10 nm above the disk (b), and less per eV of closely the unprojected LDOS ( x energy range 2 EELS EELS CL y z) (c). The CL emission only picks Energy loss (eV) (d)200 keV (e)100 keV (f) 100 keV 0 up part of the inelastic signal (f). 1 0 10203040 0 10203040 0 1020304050 Impact parameter R (nm) plasmon intensities with unprecedented lateral resolution. [5] A. Howie and R. H. Milne, Ultramicroscopy 18, 427 A first demonstration of such types of measurements has (1985). been recently reported [3]. [6] D. Ugarte, C. Colliex, and P. Trebbia, Phys. Rev. B 45, We thus conclude that the energy loss probability is 4332 (1992). directly related to the local density of states in arbitrary [7] C. H. Chen and J. Silcox, Phys. Rev. Lett. 35, 390 (1975). [8] D. W. McComb and A. Howie, Nucl. Instrum. Methods systems, where we understand locality in real space for the Phys. Res., Sect. B 96, 569 (1995). directions perpendicular to the electron trajectory and in [9] O. Ste´phan et al., Phys. Rev. B 66, 155422 (2002). momentum space along the direction of the electron ve- [10] A. Rivacoba, N. Zabala, and J. Aizpurua, Prog. Surf. Sci. locity vector. In 2D systems and for electrons moving 65, 1 (2000). along the direction of translational symmetry, the loss [11] F. J. Garcı´a de Abajo et al., Phys. Rev. Lett. 91, 143902 probability is exactly proportional to the photonic local (2003). density of states projected on the trajectory and decom- [12] There is a univocal correspondence between the spatially posed into parallel momentum transfers @q. Numerical dependent local dielectric function and the Green tensor examples have been presented showing a similar relation that works frequency by frequency, as can be clearly between LDOS outside planar metallic disks and EELS deduced upon inspection of Eq. (2). Furthermore, the spectra for electrons traversing them perpendicularly. Our photonic LDOS is directly obtained from the Green tensor following Eq. (3). The inverse relation is less direct, results provide a solid foundation for the use of EELS although one can extend the equivalent of Hohenberg- performed in STEMs to directly probe photonic properties Konh theorem [see P. Hohenberg and W. Kohn, Phys. 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