- Email: [email protected]
Stopping power and Cherenkov radiation in photonic crystals
Nuclear Instruments and Methods in Physics Research B 230 (2005) 24–30 www.elsevier.com/locate/nimb
Stopping power and Cherenkov radiation in photonic crystals N. Zabala a,b,*, F.J. Garcı´a de Abajo a,b, A. Rivacoba a,b,c, A.G. Pattantyus-Abraham d, M.O. Wolf d, L.A. Blanco a,b, P.M. Echenique
a,b
a
Unidad Fı´sica de Materiales, CSIC-UPV/EHU, Apartado 1072, 20080 San Sebastian, Spain Donostia International Physics Center (DIPC), Apartado 1072, 20080 San Sebastian, Spain Departamento de Fı´sica de Materiales, Facultad de Quı´mica, UPV/EHU, San Sebastian, Spain d Department of Chemistry, UBC, 2036 Main Mall, Vancouver, BC, Canada V6T 1Z1
b c
Available online 19 January 2005
Abstract Electron energy loss spectroscopy (EELS) induced by fast electrons in electron microscopes are used to probe photonic structures. Some of the loss features are shown to be associated to the excitation of radiative modes in the samples (Cherenkov radiation), from where information on photonic bands is extracted. The case of a 1D crystal is qualitatively discussed to explain the physics behind Cherenkov radiation in photonic crystals. For an electron beam collimated on one of the pores of 2D crystal consisting of a porous alumina film with 100-nm lattice constant, theory predicts a Cherenkov feature at around 6–9 eV, which is in excellent agreement with experiment. Finally, the features of the loss spectra are shown to be strongly correlated with the density of photonic states, suggesting the potential application of this technique to probe the quality and actual performance of photonic crystals. Ó 2004 Elsevier B.V. All rights reserved. PACS: 41.60.Bq; 79.20.m Keywords: Electron–solid interaction; Electron energy loss spectroscopy (EELS); Scanning transmission electron microscopy (STEM); Cherenkov effect; Photonic crystals; Porous solids
1. Introduction
*
Corresponding author. Address: Elektrika eta Elektronika Saila, Zientzia eta Teknologia Fakultatea, UPV/EHU, 644 P.K., 48080 Bilbao, Spain. Tel.: +34 94 6012538; fax: +34 94 6013071. E-mail address: [email protected] (N. Zabala).
When a charged particle moves inside a dielectric faster than light in that medium, electromagnetic radiation is emitted. This effect is customarily observed as a faint blue light in nuclear reactors and was first characterized experimentally by P.A. Cherenkov in 1934 and interpreted later by I.Y. Tamm and I.M. Frank in 1937. All
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.12.011
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
three were awarded the Nobel Prize in 1958 for these contributions. This Cherenkov effect has led to the development of the particle detector, used in experimental nuclear and particle physics. For a nice review on this effect and related phenomena see [1]. In the context of solid state physics, the use of this effect has been more restricted. It has been pointed out as a mechanism of energy loss in EELS measurements performed in scanning transmission electron microscopes (STEM) using energetic electron beams that satisfy the Cherenkov radiation (CR) condition, v > c/n, where v is the electron velocity and n is the refraction index of the sample. It is not necessary that the charge penetrates the material in order to observe this effect, since it also happens under aloof geometry conditions, whenever the above condition is fulfilled. Induced currents are necessarily subject to acceleration in inhomogeneous media, so that they give rise to radiative losses as well. The possibility of quasi-Cherenkov radiation by fast electrons in periodic structures of ordinary crystals, called as parametric X-ray radiation (PXR) was also discussed earlier in the literature [2]. This effect has been observed also in Josephson junctions [3]. More recently, the Cherenkov effect has received renewed attention because of its applicability to study nanostructures ranging from isolated and clustered particles [4–6] to photonic crystals [7,8]. In this work, we focus on the study of photonic crystals using this type of radiation. It has been recently shown that CR can be emitted by subluminal electrons moving in periodically structured materials and assisted by momentum transfer from the lattice (phonon creation) [9]. We propose a similar mechanism (momentum transfer from the lattice) to utilize the Cherenkov effect in EELS as a tool to probe the band structure of photonic crystals. We illustrate this by analyzing experimental EELS spectra obtained in aloof STEM along porous alumina and comparing them with the calculated photonic bands. The experiments show a sharp peak at 7–8 eV due to CR emission. The energy position of this peak depends both on the lattice parameters and on the electron velocity, and the latter is used to probe different regions of the photonic structure in energy–momentum space.
25
We present first a short tutorial on the stopping power due to CR when the charge moves in an homogeneous medium. Then, we discuss motion in a periodic 1D crystal, which illustrates photonic band folding and mixing in energy–momentum space. Finally we discuss 2D crystals formed in porous alumina films, where EELS experiments have been performed and the results explained with different models to calculate the stopping power.
2. Stopping power and Cherenkov effect in homogeneous media The electromagnetic field accompanying a charge moving in vacuum with constant velocity v is entirely evanescent. However, inside a transparent material, if v is larger than the speed of light in the medium, a propagating field is produced along the directions of the so-called Cherenkov cone, similar to Mach waves produced with sound. The threshold velocity and the aperture of the cone are given by c/n and coshC = c/vn, respectively, where n is the refractive index. This means that the velocity must be larger than the phase velocity of electromagnetic field in order to have Cherenkov radiation. The stopping power or energy loss per unit length is obtained from the electric field acting back on the charged particle along its trajectory, namely Z dt Eðr; tÞ ¼ Eðr; xÞeixt ; ð1Þ 2p and in frequency space a straightforward solution of Maxwells equations yields Z ix 1 d3 q 4peiqr Eðr; xÞ ¼ 2 2 v r 2 3 cv v ð2pÞ q2 xc2 dðx q vÞ; 2
ð2Þ
where = n is the dielectric function, which generally depends on frequency x. The function d(x q Æ v) appearing in this equation reveals the kinematical constraint relating frequency and momentum parallel to the velocity, which defines the frequency–momentum space region where excitations can be produced in the medium at the
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N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
expense of the moving charged particle. This is represented by shadowed regions in Fig. 1. Furthermore, the response of the medium is contained in the denominator of the above expression, and this defines the dispersion relation of light in the homogeneous medium, shown by a dotted straight line (this assumes no dependence of on x). After integrating over the momentum for a particle moving along the z axis, the above expression can be recast rffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ix 1 1 EðR; z; xÞ ¼ 2 2 2 v r K 0 xR 2 2 eixz=v ; cv v v c
ð3Þ
where R is the distance to the particle trajectory and K0 is the zeroth order modified Bessel func-
tion. When the Cherenkov condition is fulfilled, the argument inside this Bessel function becomes imaginary and it behaves as an oscillatory function that describes propagating radiation (this is the case depicted in Fig. 1). In that case, the number of photons emitted by a unit charge per unit of path length is given by the well-known formula [10] P bulk ¼
1 1 2 ; 2 c v
ð4Þ
which depends upon the photon wavelength exclusively via . There are several technologically interesting materials that satisfy the Cherenkov condition at the energies commonly used in STEM microscopes (about 100 or 200 keV), so that radiative energy losses play a role in the corresponding energy loss spectra. But when dielectric boundaries are present, the emission probability is no longer given by Eq. (2), and it exhibits an explicit dependence on the wavelength via reflections at the boundaries. Expressions for the energy loss probability have been obtained for different bounding geometries [11] and EELS with finite objects as spherical particles has revealed the need of relativistic expressions to account for these radiative losses.
3. Periodic systems: photonic crystals
Fig. 1. Cherenkov radiation (CR) cone (up) for a charged particle moving with velocity v in an homogeneous medium of dielectric constant , and representation of CR modes in a frequency–momentum space diagram (down). The light line in vacuum (dashed curve), the light line in the medium (dotted curve), and the x = qv kinematical constraint for a particle moving at velocity v larger that the velocity of light in the medium (continuous) are represented in this diagram. The shadowed area shows the momentum–energy region of excitations produced by the particle in the medium.
We study now the properties of CR effect in periodic structures, and in particular, we consider first 1D photonic crystals. As explained above, the Cherenkov radiation results from the coupling of the fast charged particle and the radiative modes of a dielectric medium in which light is slowed down at a velocity below the speed of light in vacuum. As we have noted above, the particle velocity must be larger than the speed of light in that medium if CR is to be emitted in a homogeneous medium. Otherwise, the coupling of the particle with the radiative modes is not possible, as shown in Fig. 2(a) and (b), a situation that corresponds to a real argument inside the Bessel function of Eq. (3), which leads to an exponentially decaying evanescent field. The situation is quite different in a photonic crystal, where the index of refraction is periodi-
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
27
condition of a Cherenkov threshold velocity, since virtually any particle, regardless how slow it moves, can couple to photonic bands; (2) band gaps will show up as dips in the Cherenkov losses of the particle, and also as frequency regions where no Cherenkov light is produced (notice that a full gap needs a 3D structure, so our 1D example gives just a qualitative explanation) and (3) sharp Cherenkov loss features are expected at frequencies where the x = qv kinematical constraint matches some bands. This will be illustrated in what follows by an actual example of Cherenkov losses in a 2D crystal formed by vacuum holes in porous alumina.
4. Results for porous alumina: EELS spectra Fig. 2. Brief explanation of the Cherenkov effect in photonic crystals. (a) and (b) show a charged particle moving in a homogeneous medium with velocity below the Cherenkov threshold, along with the corresponding frequency–momentum diagram illustrating the dispersion relation of light in the medium and the region where excitations can be transferred from the projectile (shaded area in (b)). (c) depicts the same projectile moving inside a one-dimentional photonic crystal. The dispersion relation of light is now modified by umklapp processes (absorption or emission of momentum from the lattice), as shown in (d). The periodicity of the problem suggests band folding onto the first Brillouin zone (e). Finally, light bands are mixed and directional band gaps show up (f). Now, the projectile can couple to crystal radiative modes and gives rise to Cherenkov radiation.
cally modulated, giving rise to more complicated dispersion relations for light. In particular, in a one-dimensional (1D) photonic crystal like that shown in Fig. 2(c), the 1D lattice of spacing d can produce momentum transfer to the light by multiples of 2p/d, so that the particle can couple to light by umklapp processes assisted by that transfer of momentum, as shown in Fig. 2(d). Band folding on the first Brillouin zone of the crystal (for an axis perpendicular to the interfaces), and mixing of the resulting bands leads to the actual photonic band structure of the crystal, where directional photonic band gaps are opened [see Fig. 2(e) and (f)]. Fig. 2(f) illustrates some general features that are expected to occur in the Cherenkov effect in photonic crystals: (1) band folding removes the
We focus our attention now on CR effects in the presence of cylindrical pores in alumina. EELS has been performed in porous alumina obtained by anodization [12]. This material has many applications from a technological point of view (e.g. as a mesoporous host for magnetic nanowires or to encapsulate electroluminiscent conjugate polymers with the aim of preparing light emitting devices). The samples are 1.4 lm thick and the pores, of about 58 nm in diameter and 90 nm of interpore distance, are oriented perpendicularly to the sample surfaces. The STEM beam is aligned along one pore and focused to fit inside only one pore. The energy loss of the emerging electrons is obtained for different impact parameters relative to the axis. The dominant spectral features for trajectories near the pore wall take place around 17 eV and originate in the excitation of surface plasmons, but additional loss features at 7 eV (8 eV) for 200 keV (120 keV) incident electrons are observed. We center our attention on the origin of this sharp peak, which we attribute to (aloof) Cherenkov effect modified by the sample structure [13,14], because it occurs in the region where the CR condition is satisfied in alumina. That peak remains almost at the same position regardless the impact parameter of the trajectory with respect to the axis of the pore, and it is the main feature for an axial trajectory. Different models have been studied to account for the behaviour of this peak.
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N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
The simplest one considers only the central channel along which the electron passes. For this case the energy loss probability per unit length (for the axial trajectory) is given by the following expression [15]: dP ðxÞ 2 1 ðxÞv2 =c2 ¼ 2 Imag F ðxÞ ; ð5Þ dz pv ðxÞ where F(x) depends on frequency via the dielectric function of the material and on the radius of the cylinder. The result provided by this expression for 200 keV electrons in a hole with diameter of 58 nm is the dashed line of Fig. 3. The loss probability increases continuously as x drops to zero, where it takes the value given by Eq. (2) corresponding to an unbounded medium. This behaviour for small x is general for all the impact parameters and it is a consequence of the long wavelength of the associated radiation. The radiation leaking from the pore extends to large distances as if there was no reflecting surface. So, this simple model is not enough to account for the experimental peak. The next model considered, slightly more complicated, is that of a cylindrical shell of a given thickness (61 nm in Fig. 3). The outer surface of the shell is qualitatively justified by the presence of other holes surrounding the central one on
Fig. 3. Calculated EELS probability per unit length for 200keV electrons moving along the axis of a pore drilled in alumina when the pore is isolated (dashed curve), surrounded by six other pores (solid curve), or inside an alumina shell (dashed– dotted curve). The radius of the pore is 29 nm.
which the electron is moving. The additional holes play the role of reflecting the radiation leaking out from the central one, so that the outer surface of the shell mimics that same role. In this case, there is a prominent peak at 7 eV, which agrees well with the experimentally obtained peak. Furthermore, no loss for x ! 0 is obtained. The expression for F(x) is evidently more complicated for this case, depending on the internal and external radii of the shell [14]. It is easy to prove that for this model the radiation is completely confined in the shell by total internal reflection, leading to the quantization of the momentum perpendicular to the pores q? = x(b2(x) 1)1/2/v in the shell. There are guided modes of CR that bounce back and forth between the inner and outer walls of the shell. For the thickness considered in the figure, there is only one peak but for thicker shells more and more modes are possible, giving rise to multiple peaks in the spectrum, which eventually converge to the continuous spectrum of the single hole considered above. This limit is recovered only for a thickness of several lms [14]. Finally, we consider a more realistic description consisting of seven pores with an hexagonal arrangement that mimics the samples that we have studied. In this case, the loss probability has been obtained from the induced electric field, which is in turn obtained from Maxwell s equations by means of the boundary element method [5]. Now the same peak at 7 eV is obtained but the probability goes to the unbounded value for x ! 0, as in the single pore, because the medium also extends to infinity and radiation is not completely confined by the hole-array structure, although it is certainly reflected in the surrounding pores with high efficiency around 7 eV. When more holes are considered [13,14] the loss spectra exhibit finer structures that converge eventually to the case of an infinite 2D crystal of triangular lattice [13] (the lattice unit is drawn in the inset of the figure). Furthermore, for all finite geometries the loss probability converges to the same value for small xs. This suggests that CR cannot resolve finite pore structures in the long-wavelength limit. For the infinite crystal, the calculations have been carried out within a 2D version of the Korriga–Kohn–Rostoker multiple-scattering method
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
[16] and the details of the calculation can be found in [17]. The so calculated energy loss spectrum for 200 keV electrons is shown in the right panel of Fig. 4. Note that there is a strong attenuation of Cherenkov losses in the long-wavelength limit (small x), which is consistent with the fact that lower energy losses require a larger number of cylinders to converge. Information about the band structures has been obtained from plane-wave expansion methods [17,18]. In the left panel of Fig. 4 we have plotted the density of states (DOS) of the same photonic crystal. The DOS intensity is represented as a function of the photon energy x and parallel momentum q. The kinematical constraint x = qv for the considered 200 keV electrons (v = 0.7c) allows us to sample regions of the (q, x) plane below the light cone in vacuum (dashed straight line) that are not easily accessible using external light. The features of the DOS along the x = qv line are given by the shadowed area in the right panel. Notice that, although the similarity is not complete, the main features of the DOS along that line are reflected in the loss spectrum.
29
When lower electron energies are considered the slope of the x = qv line decreases and consequently its intersection with the maxima of the DOS occurs at higher photon energies. That explains that with 120 keV electrons the Cherenkov peak moves from 7 eV to around 8 eV in the experiments. Fig. 4 is thus a prove that EELS can be used to sample local DOS.
5. Conclusions In conclusion we have shown that swift charged particles provide a localized source of evanescent electromagnetic field that couples to subwavelength nanostructures and permit probing their photonic properties by giving rise to the emission of Cherenkov radiation. In particular, we have proved that EELS measurements performed with fast electrons can be used to directly probe the band structures of 2D photonic crystals of porous alumina. Similarly, one could extend these studies to swift ions (e.g. protons), that will be advantageous in that they can move along larger distances in a medium without changing their velocity significantly, unlike what happens with fast electrons. This will relax the necessity for aloof geometry conditions.
Acknowledgements
Fig. 4. Sampling photonic density of states by EELS. The left panel shows the photonic density of states (DOS) of a 2D crystal of cylindrical air holes drilled in alumina as a function of photon frequency x and momentum component parallel to the cylinders, q. The crystal parameters are shown in the inset. The right panel shows the calculated EELS probability for 200-keV electrons (v = 0.7c) moving along the axis of one of the pores in an infinite crystal with the same pore diameter and lattice spacing as above, as compared to the DOS along the x = qv line, taken from the left panel.
The authors acknowledge support from the Basque Departamento de Educacio´n, Universidades e Investigacio´n, the University of the Basque Country UPV/EHU (contract no. 00206.21513639/2001), and Spanish Ministerio de Ciencia y Tecnologa (contract no. MAT2001-0946 and MAT2002-04087-C02-01).
References [1] V.L. Ginzburg, Physics-Uspekhi 39 (10) (1996) 973. [2] M.L. Ter-Mikaelian, High-Energy Electromagnetic Processes in Condensed Media, Wiley Interscience, New York, 1972;
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[3] [4] [5] [6] [7] [8] [9]
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30 P. Rullhusen, X. Artru, P. Dhez, Novel Radiation Sources Using Relativistic Electrons, Word Scientific, Singapore, 1998; J. Freudenberger, V.B. Gavrikov, M. Galemann, H. Genz, L. Groening, V.L. Morokhovskii, V.V. Morokhovskii, U. Nething, A. Richter, J.P.F. Sellschop, N.F. Shulga, Phys. Rev. Lett. 74 (1995) 2487, and references therein. E. Goldobin, A. Wallraff, N. Thyssen, A.V. Ustinov, Phys. Rev. Lett. 57 (1998) 130. S. Scholes, Discuss. Faraday Soc. 50 (1970) 222. F.J. Garcı´a de Abajo, A. Howie, Phys. Rev. Lett. 80 (1998) 5180. N. Yamamoto, K. Araya, F.J. Garcı´a de Abajo, Phys. Rev. B 64 (2001) 205419. F.J. Garcı´a de Abajo, Phys. Rev. Lett. 82 (1999) 2776. F.J. Garcı´a de Abajo, L.A. Blanco, Phys. Rev. B 67 (2003) 125108. T.E. Stevens, J.K. Wahlstrand, J. Kuhl, R. Merlin, Science 291 (2001) 627.
[10] J.D. Jackson, Classical Electrodynamics, third ed., Wiley, New York, 1999. [11] F.J. Garcı´a de Abajo, A. Rivacoba, N. Zabala, N. Yamamoto, Phys. Rev. B 69 (2004) 155420. [12] A.G. Pattantyus-Abraham, Doctoral Thesis, University of British Columbia, Vancouver, Canada, 2003. [13] F.J. Garcı´a de Abajo, N. Zabala, A. Rivacoba, A.G. Pattantyus-Abraham, M.O. Wolf, P.M. Echenique, Phys. Rev. Lett. 91 (2003) 143902. [14] N. Zabala, A.G. Pattantyus-Abraham, A. Rivacoba, F.J. Garcı´a de Abajo, M.O. Wolf, Phys. Rev. B 68 (2004) 245407. [15] N. Zabala, A. Rivacoba, P.M. Echenique, Surf. Sci. 209 (1989) 465. [16] K.M. Leung, Y. Qiu, Phys. Rev. B 48 (1993) 7767. [17] F.J. Garcı´a de Abajo, A. Rivacoba, N. Zabala, P.M. Echenique, Phys. Rev. B 68 (2003) 205105. [18] K.M. Ho, C.T. Chan, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152; K. Busch, S. John, Phys. Rev. E 58 (1998) 3896.
Stopping power and Cherenkov radiation in photonic crystals N. Zabala a,b,*, F.J. Garcı´a de Abajo a,b, A. Rivacoba a,b,c, A.G. Pattantyus-Abraham d, M.O. Wolf d, L.A. Blanco a,b, P.M. Echenique
a,b
a
Unidad Fı´sica de Materiales, CSIC-UPV/EHU, Apartado 1072, 20080 San Sebastian, Spain Donostia International Physics Center (DIPC), Apartado 1072, 20080 San Sebastian, Spain Departamento de Fı´sica de Materiales, Facultad de Quı´mica, UPV/EHU, San Sebastian, Spain d Department of Chemistry, UBC, 2036 Main Mall, Vancouver, BC, Canada V6T 1Z1
b c
Available online 19 January 2005
Abstract Electron energy loss spectroscopy (EELS) induced by fast electrons in electron microscopes are used to probe photonic structures. Some of the loss features are shown to be associated to the excitation of radiative modes in the samples (Cherenkov radiation), from where information on photonic bands is extracted. The case of a 1D crystal is qualitatively discussed to explain the physics behind Cherenkov radiation in photonic crystals. For an electron beam collimated on one of the pores of 2D crystal consisting of a porous alumina film with 100-nm lattice constant, theory predicts a Cherenkov feature at around 6–9 eV, which is in excellent agreement with experiment. Finally, the features of the loss spectra are shown to be strongly correlated with the density of photonic states, suggesting the potential application of this technique to probe the quality and actual performance of photonic crystals. Ó 2004 Elsevier B.V. All rights reserved. PACS: 41.60.Bq; 79.20.m Keywords: Electron–solid interaction; Electron energy loss spectroscopy (EELS); Scanning transmission electron microscopy (STEM); Cherenkov effect; Photonic crystals; Porous solids
1. Introduction
*
Corresponding author. Address: Elektrika eta Elektronika Saila, Zientzia eta Teknologia Fakultatea, UPV/EHU, 644 P.K., 48080 Bilbao, Spain. Tel.: +34 94 6012538; fax: +34 94 6013071. E-mail address: [email protected] (N. Zabala).
When a charged particle moves inside a dielectric faster than light in that medium, electromagnetic radiation is emitted. This effect is customarily observed as a faint blue light in nuclear reactors and was first characterized experimentally by P.A. Cherenkov in 1934 and interpreted later by I.Y. Tamm and I.M. Frank in 1937. All
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.12.011
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
three were awarded the Nobel Prize in 1958 for these contributions. This Cherenkov effect has led to the development of the particle detector, used in experimental nuclear and particle physics. For a nice review on this effect and related phenomena see [1]. In the context of solid state physics, the use of this effect has been more restricted. It has been pointed out as a mechanism of energy loss in EELS measurements performed in scanning transmission electron microscopes (STEM) using energetic electron beams that satisfy the Cherenkov radiation (CR) condition, v > c/n, where v is the electron velocity and n is the refraction index of the sample. It is not necessary that the charge penetrates the material in order to observe this effect, since it also happens under aloof geometry conditions, whenever the above condition is fulfilled. Induced currents are necessarily subject to acceleration in inhomogeneous media, so that they give rise to radiative losses as well. The possibility of quasi-Cherenkov radiation by fast electrons in periodic structures of ordinary crystals, called as parametric X-ray radiation (PXR) was also discussed earlier in the literature [2]. This effect has been observed also in Josephson junctions [3]. More recently, the Cherenkov effect has received renewed attention because of its applicability to study nanostructures ranging from isolated and clustered particles [4–6] to photonic crystals [7,8]. In this work, we focus on the study of photonic crystals using this type of radiation. It has been recently shown that CR can be emitted by subluminal electrons moving in periodically structured materials and assisted by momentum transfer from the lattice (phonon creation) [9]. We propose a similar mechanism (momentum transfer from the lattice) to utilize the Cherenkov effect in EELS as a tool to probe the band structure of photonic crystals. We illustrate this by analyzing experimental EELS spectra obtained in aloof STEM along porous alumina and comparing them with the calculated photonic bands. The experiments show a sharp peak at 7–8 eV due to CR emission. The energy position of this peak depends both on the lattice parameters and on the electron velocity, and the latter is used to probe different regions of the photonic structure in energy–momentum space.
25
We present first a short tutorial on the stopping power due to CR when the charge moves in an homogeneous medium. Then, we discuss motion in a periodic 1D crystal, which illustrates photonic band folding and mixing in energy–momentum space. Finally we discuss 2D crystals formed in porous alumina films, where EELS experiments have been performed and the results explained with different models to calculate the stopping power.
2. Stopping power and Cherenkov effect in homogeneous media The electromagnetic field accompanying a charge moving in vacuum with constant velocity v is entirely evanescent. However, inside a transparent material, if v is larger than the speed of light in the medium, a propagating field is produced along the directions of the so-called Cherenkov cone, similar to Mach waves produced with sound. The threshold velocity and the aperture of the cone are given by c/n and coshC = c/vn, respectively, where n is the refractive index. This means that the velocity must be larger than the phase velocity of electromagnetic field in order to have Cherenkov radiation. The stopping power or energy loss per unit length is obtained from the electric field acting back on the charged particle along its trajectory, namely Z dt Eðr; tÞ ¼ Eðr; xÞeixt ; ð1Þ 2p and in frequency space a straightforward solution of Maxwells equations yields Z ix 1 d3 q 4peiqr Eðr; xÞ ¼ 2 2 v r 2 3 cv v ð2pÞ q2 xc2 dðx q vÞ; 2
ð2Þ
where = n is the dielectric function, which generally depends on frequency x. The function d(x q Æ v) appearing in this equation reveals the kinematical constraint relating frequency and momentum parallel to the velocity, which defines the frequency–momentum space region where excitations can be produced in the medium at the
26
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
expense of the moving charged particle. This is represented by shadowed regions in Fig. 1. Furthermore, the response of the medium is contained in the denominator of the above expression, and this defines the dispersion relation of light in the homogeneous medium, shown by a dotted straight line (this assumes no dependence of on x). After integrating over the momentum for a particle moving along the z axis, the above expression can be recast rffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ix 1 1 EðR; z; xÞ ¼ 2 2 2 v r K 0 xR 2 2 eixz=v ; cv v v c
ð3Þ
where R is the distance to the particle trajectory and K0 is the zeroth order modified Bessel func-
tion. When the Cherenkov condition is fulfilled, the argument inside this Bessel function becomes imaginary and it behaves as an oscillatory function that describes propagating radiation (this is the case depicted in Fig. 1). In that case, the number of photons emitted by a unit charge per unit of path length is given by the well-known formula [10] P bulk ¼
1 1 2 ; 2 c v
ð4Þ
which depends upon the photon wavelength exclusively via . There are several technologically interesting materials that satisfy the Cherenkov condition at the energies commonly used in STEM microscopes (about 100 or 200 keV), so that radiative energy losses play a role in the corresponding energy loss spectra. But when dielectric boundaries are present, the emission probability is no longer given by Eq. (2), and it exhibits an explicit dependence on the wavelength via reflections at the boundaries. Expressions for the energy loss probability have been obtained for different bounding geometries [11] and EELS with finite objects as spherical particles has revealed the need of relativistic expressions to account for these radiative losses.
3. Periodic systems: photonic crystals
Fig. 1. Cherenkov radiation (CR) cone (up) for a charged particle moving with velocity v in an homogeneous medium of dielectric constant , and representation of CR modes in a frequency–momentum space diagram (down). The light line in vacuum (dashed curve), the light line in the medium (dotted curve), and the x = qv kinematical constraint for a particle moving at velocity v larger that the velocity of light in the medium (continuous) are represented in this diagram. The shadowed area shows the momentum–energy region of excitations produced by the particle in the medium.
We study now the properties of CR effect in periodic structures, and in particular, we consider first 1D photonic crystals. As explained above, the Cherenkov radiation results from the coupling of the fast charged particle and the radiative modes of a dielectric medium in which light is slowed down at a velocity below the speed of light in vacuum. As we have noted above, the particle velocity must be larger than the speed of light in that medium if CR is to be emitted in a homogeneous medium. Otherwise, the coupling of the particle with the radiative modes is not possible, as shown in Fig. 2(a) and (b), a situation that corresponds to a real argument inside the Bessel function of Eq. (3), which leads to an exponentially decaying evanescent field. The situation is quite different in a photonic crystal, where the index of refraction is periodi-
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
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condition of a Cherenkov threshold velocity, since virtually any particle, regardless how slow it moves, can couple to photonic bands; (2) band gaps will show up as dips in the Cherenkov losses of the particle, and also as frequency regions where no Cherenkov light is produced (notice that a full gap needs a 3D structure, so our 1D example gives just a qualitative explanation) and (3) sharp Cherenkov loss features are expected at frequencies where the x = qv kinematical constraint matches some bands. This will be illustrated in what follows by an actual example of Cherenkov losses in a 2D crystal formed by vacuum holes in porous alumina.
4. Results for porous alumina: EELS spectra Fig. 2. Brief explanation of the Cherenkov effect in photonic crystals. (a) and (b) show a charged particle moving in a homogeneous medium with velocity below the Cherenkov threshold, along with the corresponding frequency–momentum diagram illustrating the dispersion relation of light in the medium and the region where excitations can be transferred from the projectile (shaded area in (b)). (c) depicts the same projectile moving inside a one-dimentional photonic crystal. The dispersion relation of light is now modified by umklapp processes (absorption or emission of momentum from the lattice), as shown in (d). The periodicity of the problem suggests band folding onto the first Brillouin zone (e). Finally, light bands are mixed and directional band gaps show up (f). Now, the projectile can couple to crystal radiative modes and gives rise to Cherenkov radiation.
cally modulated, giving rise to more complicated dispersion relations for light. In particular, in a one-dimensional (1D) photonic crystal like that shown in Fig. 2(c), the 1D lattice of spacing d can produce momentum transfer to the light by multiples of 2p/d, so that the particle can couple to light by umklapp processes assisted by that transfer of momentum, as shown in Fig. 2(d). Band folding on the first Brillouin zone of the crystal (for an axis perpendicular to the interfaces), and mixing of the resulting bands leads to the actual photonic band structure of the crystal, where directional photonic band gaps are opened [see Fig. 2(e) and (f)]. Fig. 2(f) illustrates some general features that are expected to occur in the Cherenkov effect in photonic crystals: (1) band folding removes the
We focus our attention now on CR effects in the presence of cylindrical pores in alumina. EELS has been performed in porous alumina obtained by anodization [12]. This material has many applications from a technological point of view (e.g. as a mesoporous host for magnetic nanowires or to encapsulate electroluminiscent conjugate polymers with the aim of preparing light emitting devices). The samples are 1.4 lm thick and the pores, of about 58 nm in diameter and 90 nm of interpore distance, are oriented perpendicularly to the sample surfaces. The STEM beam is aligned along one pore and focused to fit inside only one pore. The energy loss of the emerging electrons is obtained for different impact parameters relative to the axis. The dominant spectral features for trajectories near the pore wall take place around 17 eV and originate in the excitation of surface plasmons, but additional loss features at 7 eV (8 eV) for 200 keV (120 keV) incident electrons are observed. We center our attention on the origin of this sharp peak, which we attribute to (aloof) Cherenkov effect modified by the sample structure [13,14], because it occurs in the region where the CR condition is satisfied in alumina. That peak remains almost at the same position regardless the impact parameter of the trajectory with respect to the axis of the pore, and it is the main feature for an axial trajectory. Different models have been studied to account for the behaviour of this peak.
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The simplest one considers only the central channel along which the electron passes. For this case the energy loss probability per unit length (for the axial trajectory) is given by the following expression [15]: dP ðxÞ 2 1 ðxÞv2 =c2 ¼ 2 Imag F ðxÞ ; ð5Þ dz pv ðxÞ where F(x) depends on frequency via the dielectric function of the material and on the radius of the cylinder. The result provided by this expression for 200 keV electrons in a hole with diameter of 58 nm is the dashed line of Fig. 3. The loss probability increases continuously as x drops to zero, where it takes the value given by Eq. (2) corresponding to an unbounded medium. This behaviour for small x is general for all the impact parameters and it is a consequence of the long wavelength of the associated radiation. The radiation leaking from the pore extends to large distances as if there was no reflecting surface. So, this simple model is not enough to account for the experimental peak. The next model considered, slightly more complicated, is that of a cylindrical shell of a given thickness (61 nm in Fig. 3). The outer surface of the shell is qualitatively justified by the presence of other holes surrounding the central one on
Fig. 3. Calculated EELS probability per unit length for 200keV electrons moving along the axis of a pore drilled in alumina when the pore is isolated (dashed curve), surrounded by six other pores (solid curve), or inside an alumina shell (dashed– dotted curve). The radius of the pore is 29 nm.
which the electron is moving. The additional holes play the role of reflecting the radiation leaking out from the central one, so that the outer surface of the shell mimics that same role. In this case, there is a prominent peak at 7 eV, which agrees well with the experimentally obtained peak. Furthermore, no loss for x ! 0 is obtained. The expression for F(x) is evidently more complicated for this case, depending on the internal and external radii of the shell [14]. It is easy to prove that for this model the radiation is completely confined in the shell by total internal reflection, leading to the quantization of the momentum perpendicular to the pores q? = x(b2(x) 1)1/2/v in the shell. There are guided modes of CR that bounce back and forth between the inner and outer walls of the shell. For the thickness considered in the figure, there is only one peak but for thicker shells more and more modes are possible, giving rise to multiple peaks in the spectrum, which eventually converge to the continuous spectrum of the single hole considered above. This limit is recovered only for a thickness of several lms [14]. Finally, we consider a more realistic description consisting of seven pores with an hexagonal arrangement that mimics the samples that we have studied. In this case, the loss probability has been obtained from the induced electric field, which is in turn obtained from Maxwell s equations by means of the boundary element method [5]. Now the same peak at 7 eV is obtained but the probability goes to the unbounded value for x ! 0, as in the single pore, because the medium also extends to infinity and radiation is not completely confined by the hole-array structure, although it is certainly reflected in the surrounding pores with high efficiency around 7 eV. When more holes are considered [13,14] the loss spectra exhibit finer structures that converge eventually to the case of an infinite 2D crystal of triangular lattice [13] (the lattice unit is drawn in the inset of the figure). Furthermore, for all finite geometries the loss probability converges to the same value for small xs. This suggests that CR cannot resolve finite pore structures in the long-wavelength limit. For the infinite crystal, the calculations have been carried out within a 2D version of the Korriga–Kohn–Rostoker multiple-scattering method
N. Zabala et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 24–30
[16] and the details of the calculation can be found in [17]. The so calculated energy loss spectrum for 200 keV electrons is shown in the right panel of Fig. 4. Note that there is a strong attenuation of Cherenkov losses in the long-wavelength limit (small x), which is consistent with the fact that lower energy losses require a larger number of cylinders to converge. Information about the band structures has been obtained from plane-wave expansion methods [17,18]. In the left panel of Fig. 4 we have plotted the density of states (DOS) of the same photonic crystal. The DOS intensity is represented as a function of the photon energy x and parallel momentum q. The kinematical constraint x = qv for the considered 200 keV electrons (v = 0.7c) allows us to sample regions of the (q, x) plane below the light cone in vacuum (dashed straight line) that are not easily accessible using external light. The features of the DOS along the x = qv line are given by the shadowed area in the right panel. Notice that, although the similarity is not complete, the main features of the DOS along that line are reflected in the loss spectrum.
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When lower electron energies are considered the slope of the x = qv line decreases and consequently its intersection with the maxima of the DOS occurs at higher photon energies. That explains that with 120 keV electrons the Cherenkov peak moves from 7 eV to around 8 eV in the experiments. Fig. 4 is thus a prove that EELS can be used to sample local DOS.
5. Conclusions In conclusion we have shown that swift charged particles provide a localized source of evanescent electromagnetic field that couples to subwavelength nanostructures and permit probing their photonic properties by giving rise to the emission of Cherenkov radiation. In particular, we have proved that EELS measurements performed with fast electrons can be used to directly probe the band structures of 2D photonic crystals of porous alumina. Similarly, one could extend these studies to swift ions (e.g. protons), that will be advantageous in that they can move along larger distances in a medium without changing their velocity significantly, unlike what happens with fast electrons. This will relax the necessity for aloof geometry conditions.
Acknowledgements
Fig. 4. Sampling photonic density of states by EELS. The left panel shows the photonic density of states (DOS) of a 2D crystal of cylindrical air holes drilled in alumina as a function of photon frequency x and momentum component parallel to the cylinders, q. The crystal parameters are shown in the inset. The right panel shows the calculated EELS probability for 200-keV electrons (v = 0.7c) moving along the axis of one of the pores in an infinite crystal with the same pore diameter and lattice spacing as above, as compared to the DOS along the x = qv line, taken from the left panel.
The authors acknowledge support from the Basque Departamento de Educacio´n, Universidades e Investigacio´n, the University of the Basque Country UPV/EHU (contract no. 00206.21513639/2001), and Spanish Ministerio de Ciencia y Tecnologa (contract no. MAT2001-0946 and MAT2002-04087-C02-01).
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