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Berreman effect in bimetallic nanolayered metamaterials
Optical Materials 99 (2020) 109578
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Berreman effect in bimetallic nanolayered metamaterials S. Cortés-López a , S.L. Gastélum-Acuña b , F.J. Flores-Ruiz c , V. Garcia-Vazquez a , R. García-Llamas d , F. Pérez-Rodríguez a,d ,∗ a
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico CONACYT-Departamento de Investigación en Física, Universidad de Sonora, Apdo. Post. 5-88, Hermosillo, Son. 83000, Mexico CONACYT-Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico d Departamento de Investigación en Física, Universidad de Sonora, Apdo. Post. 5-88, Hermosillo, Son. 83000, Mexico b c
ARTICLE Keywords: Metamaterials Photonic crystals Reflectivity Metals Nanofilms
INFO
ABSTRACT The visible and ultraviolet responses of a binary regular stack, composed of alternating Al and Ag nano-thin layers, are both experimentally and theoretically studied. It is found that the 𝑠- and 𝑝-polarization reflectivity spectra (𝑅𝑠 and 𝑅𝑝 , respectively) of the inherently anisotropic bimetallic system are noticeably different due to the appearance of a prominent dip near the zero of the Ag permittivity. Such an effect is similar to the Berreman one, which is observed in thin polar-crystals films near the longitudinal-optical phonon frequency, as well as in thin metal films near plasma frequency. Using both the effective medium approach and a nonlocal homogenization formalism, being valid even beyond the long wavelength limit, it is demonstrated that the bimetallic multi-layer stack behaves as a uniaxial anisotropic metamaterial. As it is shown, the appearance of the dip in the 𝑝-polarization spectrum is associated to the zero of the longitudinal component of the effective permittivity (𝜖𝑧 ). The calculation of 45-degree reflectometry spectrum, namely the difference 𝑅𝑝 − 𝑅2𝑠 , from experimental and theoretical reflectivity data confirmed that the observed dip in 𝑅𝑝 is connected to a plasma resonance of the energy-loss function ℑ(−1∕𝜖𝑧 ).
1. Introduction Among photonic metamaterials, the inherently-anisotropic hyperbolic metamaterials (HMs) are distinguished by exhibiting striking phenomena such as negative refraction [1,2] and subwavelength imaging [3] without optical magnetism. The HMs can be either regular metal–dielectric layered structures or periodic arrays of metallic nanowires embedded in a dielectric host medium [4–8]. Besides, there exist natural single-phase HMs [9,10], to which graphite [11], high-temperature cuprate superconductors [9,12–14], magnesium diboride [9], tetradymites [15], hexagonal boron nitride [16,17] and others belong. The optical response of the HMs can be explained in terms of their permittivity tensor [4–8], which has one principal value with sign different from the others in certain frequency range. For a uniaxial medium (𝜖𝑥 = 𝜖𝑦 = 𝜖⟂ ≠ 𝜖𝑧 = 𝜖∥ , with the optical axis being parallel to the 𝑧-axis), the photonic dispersion relation for transverse magnetic (TM) modes (extraordinary waves) becomes hyperbolic [4–8]. In the case 𝜖⟂ > 0, 𝜖𝑧 < 0, the isofrequency surface of TM modes is a twofold hyperboloid; the HM is called type I. In contrast, in the case, 𝜖⟂ < 0, 𝜖𝑧 > 0, the TM isofrequency surface corresponds to a one-fold hyperboloid; the HM is a type-II medium.
Another interesting anisotropic optical material is the bimetallic superlattice (see, for example, Refs. [18–25] and references therein). As is shown in some theoretical works [18,20,22], by applying the local Drude model for describing the frequency dependence of the conducting layers in a semi-infinite bimetallic superlattice and considering a moderate electron relaxation rate, two broad reflection minima appear inside the frequency interval, 𝜔𝑝1 < 𝜔 < 𝜔𝑝2 , limited by the plasma frequencies of the metals (i.e. 𝜔𝑝1 and 𝜔𝑝2 ). The predicted minima are associated with two pass bands of the transverse-magnetic photonic modes, which have almost-real Bloch wave numbers and, therefore, propagate along the growth direction of the superlattice. These TM propagating normal modes are originated by surface plasmons at each interface, which are coupled by the tails of their evanescent electromagnetic fields [18] as it also occurs in metal–dielectric superlattices [26]. Besides, in a finite-size multilayer stack, the Bloch wave vector is quantized and the reflection spectra exhibit Fabry–Perot resonances in the two TM pass bands [25]. The principal values of the permittivity tensor for a bimetallic laminar periodic structure can be calculated by employing the effective medium approach (EMA) [27–30]. Using it and the local Drude model, the low- and high-frequency TM pass bands
∗ Corresponding author at: Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico. E-mail address: [email protected] (F. Pérez-Rodríguez).
https://doi.org/10.1016/j.optmat.2019.109578 Received 18 August 2019; Received in revised form 7 November 2019; Accepted 26 November 2019 Available online 5 December 2019 0925-3467/© 2019 Elsevier B.V. All rights reserved.
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3. Theoretical reflectivity spectra
in the interval (𝜔𝑝1 , 𝜔𝑝2 ), which were predicted in Refs. [18,20,22], correspond to the photonic pass bands where the bimetallic superlattice behaves as a HM of type II and I, respectively [25]. In the present work, the optical properties of Al/Ag nanolayered stacks are both experimentally (Section 2) and theoretically (Sections 3 and 4) studied. In particular, the measured visible and UV reflectivity spectra, 𝑅𝑠 and 𝑅𝑝 , for 𝑠 and 𝑝-polarized incident light, respectively, are critically compared with those theoretically obtained by applying the methods based on the transfer matrix (TMM) [18,20] and the expansion into bulk modes (MEBM) [21,22,31]. In the calculations, we shall consider realistic metal frequency-dependent permittivities, taken from the literature. The Al and Ag permittivities used here consider not only the optical response of the conduction electrons, but also the bound-electron contributions, as well as the noticeable electron relaxation rate of nano-thin metal layers. In order to verify whether or not the bimetallic nanolayered superlattice behaves as a HM, its photonic band structure and the principal values of the effective permittivity are calculated and analyzed. The effective parameters of the anisotropic bimetallic metamaterial are obtained by employing both EMA formulas [27–30], and a nonlocal homogenization approach (NHA) that is valid even beyond the long wavelength limit (Section 4). The longitudinal component (𝜖𝑧 ) of the effective permittivity tensor may have zeros close to the zeros of the metal (Al or Ag) permittivities. The existence of zeros in the longitudinal component 𝜖𝑧 can lead to resonances in the energy loss function ℑ(−1∕𝜖𝑧 ) and, then, to minima (dips) in the spectra of 𝑅𝑝 [32–34] and 45-degree reflectometry [35] for thin metal films. The latter spectrum is given by the difference 𝛥45 ≡ 𝑅𝑝 − 𝑅2𝑠 at 45◦ incidence angle and allows to detect both zeros and poles of the longitudinal and transverse components of the effective permittivity, correspondingly. In fact, the appearance of the dips in the UV spectra of 𝑅𝑝 and 𝛥45 is the Berreman effect, which was firstly observed in infrared spectra of polaritonic media [36,37]. In Section 5, the lower-frequency zero of 𝜖𝑧 , which is near the zero of the Ag permittivity, is accurately determined from experimental and theoretical spectra of the 45-degree reflectometry. Our results are discussed and compared with other works in Section 6. Finally, our conclusions are presented in Section 7, whereas the experimental details are given in Appendix.
In order to provide an initial explanation of the features of the reflectivity spectra presented in the previous section, we have numerically calculated the optical spectra for two regular bimetallic (Al/Ag) nanolayered stacks with three (panel (a) of Fig. 2) and six (panel (b) therein) bilayers overlying a glass substrate. The Al- and Ag-layer thicknesses used in reflectivity calculations are: 𝑑𝐴𝑙 = 12.2 (6.1) nm and 𝑑𝐴𝑔 = 16.2 (8.1) nm for the three (six)-bilayer stack. The calculations were carried out by employing realistic Al and Ag wavelengthdependent permittivities (Fig. 3), the standard transfer matrix method and verified with the method of expansion into bulk modes. As was commented in the Introduction, the frequency (or wavelength) dependence of the metal permittivity can be described with the local Drude model, but it may also have contributions associated with bound electrons, which are usually modeled with Lorentz oscillators. Besides, the relatively-large electron relaxation rate in metal nano-thin layers, in comparison with the electron-collision rate for thick metal films, should be taken into account. The permittivity values of the 16.2 nm-thick Ag-layers in SAMPLE A were taken from Palik handbook [38], which in fact correspond to the bulk Ag permittivity (see solid lines in panel (a) of Fig. 3). The use of such values is justified because the Ag permittivity of thin films becomes noticeably different from the bulk value once the Ag film thickness 𝑑𝐴𝑔 is smaller than 12 nm, as was demonstrated in Refs. [39,41]. On the other hand, for the Ag-layers of 8.1 nm, we use the permittivity values determined in Ref. [39] for an Ag film with the same thickness (𝑑𝐴𝑔 = 8.1 nm). As can be seen in panel (a) of Fig. 3, the permittivity of the thinner Ag layers has an imaginary part (𝜖 ′′ ) which is larger than the bulk Ag permittivity, whereas the absolute value of its real part (|𝜖 ′ |) turns out to be smaller than that for the Ag bulk in a wide wavelength range (compare solid lines with dotted curves of the same color). The changes in the permittivity can be attributed to granular structure (heterogeneity) of the fabricated nano-thin films [39,41]. As is shown in Ref. [40], the permittivity of Al films is very close to the bulk value when their thickness is larger than 10 nm. Therefore, in the reflectivity calculations for the Al layers with 12.2 nm thickness we used the bulk Al permittivity values reported in Palik database [38]. The permittivity for thinner Al-layers, namely those with 𝑑𝐴𝑙 = 6.1 nm, coincides with that obtained in the work [40] for an Al film of the same thickness. As in the case of Ag-layers, the difference between the permittivity of the Al nano-thin layers and the bulk Al permittivity may be due to the heterogeneity of the ultra-thin metal films [40]. The calculated reflectivity spectra of the Al/Ag stacks for both 𝑠and 𝑝-polarization geometry (Fig. 2) correspond to the same incidence angles of the experimental optical spectra (Fig. 1). Notice that the same features of the measured reflectivity spectra are practically reproduced by the numerically-calculated spectra. It is important to mention that the dip, appearing at 𝜆 ≈ 327 nm (𝜆 ≈ 333 nm) and being close to the zero of the permittivity for Ag layers in SAMPLE A (B) [see Fig. 3(a)], is clearly observed in the 𝑅𝑝 spectra of both bimetallic nanolayered stacks. Moreover, for SAMPLE A, both 𝑅𝑠 and 𝑅𝑝 [panel (a) of Fig. 2] have relatively-large (small) values at wavelengths above (below) 500 nm (300 nm) in good agreement with the experiment [panel (a) of Fig. 1]. The moderate values of the reflectivity for SAMPLE B, observed in panel (b) of Fig. 1 at wavelengths larger than 500 nm, is also reproduced by the theoretical spectra [panel (b), Fig. 2]. Nevertheless, the wide minimum at 𝜆 ∼ 559 nm turns out to be significantly broader in the calculated spectra (compare panels (b) of Figs. 1 and 2).
2. Optical characterization of Al/Ag nanolayered stacks Visible and UV reflectance spectra were measured on two samples deposited by different deposition techniques. SAMPLE A was grown by thermal evaporation. The stack structure was substrate/[Al(12 nm)/ Ag(16 nm)]3 . SAMPLE B was deposited by magnetron sputtering, with a stack structure of the form substrate/[Al(6 nm)/Ag(8 nm)]6 . The thickness of each Al/Ag bilayer in SAMPLE A (28 nm) is twice the size of that in SAMPLE B (14 nm). In both cases, the Al/Ag thickness ratio (3/4) is the same. The number of repetitions in each sample was chosen to produce the same total thickness of the whole stack (84 nm). Details of the fabrication of each sample are indicated in Appendix. The reflectivity spectra for 𝑠- and 𝑝-polarized light of SAMPLE A and SAMPLE B were measured by using a fluororeflectomer system in the wavelength interval 250–800 nm (see the Appendix for experimental details). In Fig. 1, the measured optical spectra for 30◦ , 45◦ , and 60◦ angles of incidence are presented. As can be observed in panel (a), the reflectivity for SAMPLE A is relatively-large (∼ 0.9) at wavelengths (𝜆) larger than 500 nm. At wavelengths 𝜆 ∼ 305 nm, both 𝑅𝑠 and 𝑅𝑝 have a broad minimum. Surprisingly, the 𝑝-polarization reflectivity spectra of SAMPLE A have an additional feature. A narrow dip is observed at 𝜆 ≈ 327 nm. Furthermore, the dip has a depth that systematically increases with the angle of incidence 𝜃. On the other hand, the reflectivities 𝑅𝑠 and 𝑅𝑝 for SAMPLE B, panel (b) in Fig. 1, are smaller than those for SAMPLE A at wavelengths 𝜆 above 500 nm [compare panels (a) and (b)]. The reflectivity spectra for SAMPLE B also show a broad minimum at 𝜆 ∼ 283 nm. Similarly to 𝑅𝑝 (𝜆) of SAMPLE A, the appearance of a relatively-narrow dip at 𝜆 ≈ 333 nm, whose depth increases with the incidence angle 𝜃 is clearly observed in the 𝑝-polarization reflectivity spectra of SAMPLE B [see panel (b)].
4. Bulk photonic modes and the effective permittivity tensor In order to explain the measured and calculated reflectivity spectra for the regular nanolayered stacks (Figs. 1 and 2) more deeply, we shall analyze the photonic band structure and the wavelength dependence of the effective permittivity tensor of a bimetallic (Al/Ag) superlattice 2
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Fig. 1. Measured 𝑠- and 𝑝-polarization reflectivity spectra at different angles of incidence. (a) Multilayered stack with 3 unit cells (SAMPLE A). (b) Multilayered stack with 6 unit cells (SAMPLE B).
Fig. 2. Calculated 𝑠- and 𝑝-polarization reflectivity spectra for a multilayered stack of the form [Al/Ag]𝑁 . (a) N = 3. (b) N = 6. The spectra were calculated with both TMM (Transfer Matrix Method) and MEBM (Method of Expansion into Bulk Modes) at different angles of incidence, both methods coincide as expected.
Fig. 3. Permittivity data taken from Refs. [38–40]. (a) Silver. (b) Aluminum.
(Fig. 4). The dispersion relations between the Bloch wave number (𝑘𝑧 according to the assumed geometry) and frequency for 𝑠- and 𝑝polarized modes in a binary superlattice are given by the respective well-known formulas [42,43],
−
1 2
(
𝑘𝑧1 𝑘𝑧2 + 𝑘𝑧2 𝑘𝑧1
) sin(𝑘𝑧1 𝑑1 ) sin(𝑘𝑧2 𝑑2 ),
(1)
and cos(𝑘𝑧 𝑎) = cos(𝑘𝑧1 𝑑1 ) cos(𝑘𝑧2 𝑑2 ) ( ) 1 𝜀2 𝑘𝑧1 𝜀1 𝑘𝑧2 − + sin(𝑘𝑧1 𝑑1 ) sin(𝑘𝑧2 𝑑2 ), 2 𝜀1 𝑘𝑧2 𝜀2 𝑘𝑧1
cos(𝑘𝑧 𝑎) = cos(𝑘𝑧1 𝑑1 ) cos(𝑘𝑧2 𝑑2 ) 3
(2)
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Fig. 4. (a) and (b) Photonic dispersion relations of 𝑠- and 𝑝-polarized modes in Al/Ag superlattices with the same unit cells of the bimetallic stacks presented in Figs. 2(a) and (b), respectively. (c) and (d) Effective permittivity principal values calculated within EMA (Effective Medium Approach) and NHA (Nonlocal Homogenization Approach) for the corresponding Al/Ag superlattices of (a) and (b).
√ √ 2 2 where 𝑘𝑧1 = 𝜀1 𝜔𝑐 2 − 𝑘2𝑥 and 𝑘𝑧2 = 𝜀2 𝜔𝑐 2 − 𝑘2𝑥 are the wave numbers of the electromagnetic waves inside the metals 1 (Ag) and 2 (Al) with thicknesses 𝑑1 and 𝑑2 , respectively; 𝜔 = 2𝜋∕𝜆 is the frequency, 𝑐 is the velocity of light in vacuum, 𝑘𝑥 is the conserved wave-vector component, parallel to the metal layers, 𝑎 is the superlattice period and 𝑘𝑧 is the Bloch wave number. Panels (a) and (b) of Fig. 4 present the dispersion relation 𝑘𝑧 (𝜆) for two Al/Ag superlattices, correspondingly, having the same Al- and Aglayers thicknesses and permittivities as those of the regular bimetallic stacks considered in the previous section. Notice that the Bloch wave number has a large imaginary part in comparison with its real part [|ℜ(𝑘𝑧 )| ≪ ℑ(𝑘𝑧 )], particularly for the case of the superlattice with thicker metal layers (panel (a)): 𝑑1 = 𝑑𝐴𝑔 = 16.2 nm and 𝑑2 = 𝑑𝐴𝑙 = 12.2 nm. This implies that such modes should rapidly decay from the sample surface, explaining why the reflectivity for SAMPLE A is, in general, larger than that of SAMPLE B. It is also interesting to analyze the effective permittivity tensor of both Al/Al superlattices and to find their correlation with the reflectivity spectra of the bimetallic nanolayered stacks. In panels (c) and (d) of Fig. 4, we show the wavelength dependences of the principal values of the effective permittivity tensor, which were calculated within EMA [27–30] for SAMPLE A and SAMPLE B, respectively. From their inspection, it follows that the dip of the 𝑝-polarization reflectivity of SAMPLE A (B) at 𝜆 ≈ 327 nm (𝜆 ≈ 333 nm) is very close to the zero of the 𝑧-component (𝜖𝑧𝐸𝑀𝐴 ) of the permittivity tensor, which is in fact determined by the zero of the Ag permittivity [panel (a) of Fig. 3]. It is also observed that the superlattice with thicker metal layers [see panel (c) of Fig. 4] is characterized by relatively large absolute values of the negative 𝑥- and 𝑦-components of the effective permittivity at wavelengths 𝜆 > 500 nm, leading to a strong decay of
the electromagnetic field and, hence, to high values of the reflectivity ∼ 0.9 [see Fig. 1(a) and 2(a)] even in a sample with a few bilayers. The used EMA formulas [27–30] can only provide a qualitative explanation of the optical spectra of layered stacks with metal components since they are valid when the electromagnetic field slightly varies inside the layers. This can occurs in ultra-thin metals with thickness 𝑑 much smaller than the skin depth 𝛿 (𝑑 ≪ 𝛿) and at rather small frequencies (very long wavelengths) [44]. The application of EMA to periodic metal–dielectric layered structures fails because of the skin effect, which leads to a drastic variation of the electromagnetic field inside the metal layer [45]. EMA can also breaks down for deeply subwavelength all-dielectric superlattices since their reflectivity spectra near the critical angle depends on both very small period variations and the termination of the layered structure [46–48]. Due to the limitations of the EMA, various homogenization approaches have been developed in the past few years (see, for example Refs. [49–54]). The proposed theories describe the HM as a nonlocal anisotropic homogeneous medium and provide explicit formulas for calculating the effective permittivity components. In the present study of bimetallic nanolayered stacks, we shall apply a recently-proposed nonlocal homogenization approach (NHA) [55], which is valid even beyond the long wavelength limit (i.e., at high frequencies too) and for any values of the Bloch wave number 𝑘𝑧 . According to the NHA [55], the nonlocal effective parameters are given by the spatial averaging over the unit cell of the periodic part of the Bloch normal modes as: ⟨ ⟩ 𝐷𝑖,𝑝 ⃗ 𝑖 = 𝑥, 𝑦, 𝑧, (3) 𝜀ef f,𝑖 (𝜔, 𝑘) = ⟨ ⟩, 𝐸𝑖,𝑝 where ⟨⋯⟩ indicates the spatial averaging over the unit cell, 𝐷𝑖,𝑝 and 𝐸𝑖,𝑝 are the periodic parts of the displacement vector and electric field, 4
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⃗ = (𝑘𝑥 , 0, 𝑘𝑧 ). The respectively, having the form of Bloch waves, and 𝑘 nonlocal effective permittivity-tensor principal values, Eq. (3), allow us to rewrite the dispersion relations (1) and (2) for 𝑠- and 𝑝-polarized normal photonic modes in the same form as those for a nonlocal homogeneous anisotropic dielectric material, respectively: 𝑘2𝑧 + 𝑘2𝑥 =
𝜔2 ⃗ 𝜀ef f,𝑦 (𝜔, 𝑘), 𝑐2
𝑘2𝑧 ⃗ 𝜀ef f ,𝑥 (𝜔, 𝑘)
+
𝑘2𝑥 ⃗ 𝜀ef f,𝑧 (𝜔, 𝑘)
Al/Ag stacks with thinner ‘‘nanolayers", grown with a rather small deposition time. It was found that their reflectivity spectra do not exhibit a dip at the frequency of the Ag-permittivity zero. Such samples behave more like an isotropic Al/Ag nanocomposite having an effective scalar permittivity with a zero at wavelengths smaller than that of the Ag-permittivity zero.
(4)
=
𝜔2 . 𝑐2
6. Discussion
(5)
The results presented in the previous sections demonstrate that Al/Ag nanolayered stacks behave as uniaxial anisotropic media. Both EMA and NHA predict a zero of the effective permittivity-tensor 𝑧component (corresponding to the growth direction), where the 𝑝polarization reflectivity 𝑅𝑝 and the 45-degree reflectometry spectrum have a narrow dip. However, the large imaginary parts of the effective permittivity components, which are attributed to surface-electron scattering in nano-thin layers, does not allow the propagation of photonic modes with hyperbolic dispersion relation at frequencies just above the zero of 𝜖𝑧 , coinciding with that of the Ag permittivity. In other words, the HM behavior of Al/Ag nanolayered stacks could not be confirmed.
Panels (c) and (d) of Fig. 4 show the calculated nonlocal effective permittivity principal values for the bimetallic superlattices considered above. Notice that the wavelength dependences of the permittivity components, predicted by the EMA and NHA, are indeed quantitatively different. Both approaches, however, predict almost the same zero for the 𝑧-component of the effective permittivity, which is associated to the dip in the 𝑅𝑝 spectra of both bimetallic nanolayered stacks [compare solid lines with dotted curves in panels (c) and (d)]. It should be mentioned that we have also applied the MEBM [21,22,31] to calculate the reflectivity spectra. Within MEBM, the microscopic electric and magnetic fields inside the finite-size bimetallic stack are expressed as a superposition of two Bloch normal waves (with wave numbers 𝑘𝑧 and −𝑘𝑧 , respectively). The amplitudes of such waves, as well as the amplitudes of the reflected and transmitted fields, are straightforwardly calculated by imposing the continuity of the tangential components of the electric and magnetic fields at the front and rear surfaces of the bimetallic nanolayered stack. As it was verified, the calculated reflectivity spectra within MEBM coincide with those shown in Fig. 2, which were obtained with the TMM.
The appearance of a dip in the 𝑝-polarization reflectivity of the bimetallic laminar stack is attributed to the Berreman effect [32–34]. Hence, the origin of the effect is a plasma resonance in the energy-loss function ℑ(−1∕𝜖𝑧 ), being close to the Ag plasma frequency. It should also be mentioned that the contributions of the bound electrons to the Ag permittivity shift its zero to lower frequencies (larger wavelengths). In fact, the models for the Ag permittivity commonly make use of a local Drude term with plasma frequency of the order of 9 eV (138 nm), together with Lorentz oscillators with resonance frequencies ∼ 4 eV (310 nm), leading to zeros of the permittivity for Ag nano-thin films at ∼ 327–333 nm [38,56].
5. 45-degree reflectometry To confirm that the dip in the 𝑅𝑝 spectra is connected to a zero of the 𝑧-component of the effective permittivity, we have calculated the 45-degree reflectometry spectra, namely 𝛥45 ≡ 𝑅𝑝 − 𝑅2𝑠 , from the experimental and theoretical reflectivities (curves for 𝜃 = 45◦ in Figs. 1 and 2, respectively). As was demonstrated in Ref. [35], the 𝛥45 spectra for a multilayer stack of thickness 𝑑 ≪ 𝜆 are given by the principal values of the effective permittivity tensor as: [ ( )] √ −1 𝑑 ℑ(𝜀⟂ ) − 𝜀2𝑠 ℑ 𝛥45 = 4 𝜋 2 𝑅0𝑝 |𝜃=45◦ , (6) 𝜀∥ 𝜆(𝜀𝑠 − 1)2
Another factor that might alter the optical spectra is the metal nonlocality [34]. According to Refs. [18,22], in the case of spatiallydispersive metals, the reflectivity spectrum of a bimetallic semi-infinite superlattice has an extra series of sharp peaks, which are originated by single-film resonance frequencies and superimposed to the two broad reflection minima appearing near the plasma frequency of the metals. In addition, the TM photonic pass bands, corresponding to those minima, are split in several bands as a result of the coupling of surface-plasmons with the confined modes in the metal layers with lower plasma frequency (𝜔𝑝1 ). Nevertheless, the large damping parameters used in the permittivities of Al and Ag nano-layers obstruct the observation of effects associated with the Ag nonlocality.
if the homogenized laminar structure behaves as a uniaxial crystal (𝜖⟂ = 𝜖𝑥 = 𝜖𝑦 ≠ 𝜖∥ = 𝜖𝑧 ). In Eq. (6), 𝑅0𝑝 |𝜃=45◦ denotes the 𝑝-polarization reflectivity of the substrate, having a permittivity 𝜖𝑠 , at 45-degree incidence angle. According to Eq. (6), transverse resonances manifest as prominent maxima in 𝛥45 spectra, while longitudinal modes, corresponding to zeros of the parallel component of the effective permittivity 𝜖∥ = 𝜖𝑧 , should clearly manifest themselves as narrow minima (dips). Fig. 5 shows the 𝛥45 ≡ 𝑅𝑝 − 𝑅2𝑠 spectra, which were calculated by using the experimental (Fig. 1) and theoretical (Fig. 2) 𝑅𝑠 and 𝑅𝑝 reflectivities for the bimetallic nanolayered stacks with 3 and 6 bilayers. Evidently the 𝛥45 spectrum for SAMPLE A (B) exhibits a well-discernible dip at 𝜆 ≈ 327 nm (𝜆 ≈ 333 nm), whereas at larger wavelengths 𝛥45 ≈ 0 for both samples. The agreement between theory and experiment is excellent. We also show in Fig. 5 the wavelength dependence of the energy-loss function ℑ(−1∕𝜖𝑧 ), which was calculated by using the NHA. As can be observed, the latter has a resonance precisely at the frequency where both 𝑅𝑝 and 𝛥45 spectra have a dip. It should be also noticed that the dips in 𝑅𝑝 and 𝛥45 [peak in ℑ(−1∕𝜖𝑧 )] are more prominent for SAMPLE A (compare panels (a) and (b) in Fig. 5). This can be due to the fact that the thinner layers have not well defined interfaces between layers because of their granularity and the diffusion of Al and Ag to adjacent layers. To confirm such a statement, we have also carried out reflectivity measurements on
In order to reduce the large imaginary parts in the effective permittivity components and enhance the propagation of photonic modes in Al/Ag layered stacks, some improvements during the preparation of the films could be implemented. For example, the use of singlecrystal substrates instead of the amorphous substrates here utilized could induce the growth of highly-oriented metallic layers. Atomically smooth metallic surfaces would reduce or avoid the intersperse effect, producing sharp Al/Ag interfaces. On the other hand, substrate heating during deposition could enhance the crystallinity of the individual layers. Other improvements during deposition could also be added. The use of higher sputtering rates, for example, can increase the kinetic energy of the atoms reaching the substrate for a better nucleation density in the metallic layers. Target-to-substrate distances could also be varied. We expect that the quality of the Al/Ag layers can be improved by an optimization process during film deposition. Furthermore, we expect that the dip found in the 𝑝-polarization reflectivity will be enhanced by an appropriate choose of the deposition parameters. Further experiments will be needed to prove this. 5
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Fig. 5. 45-degree reflectometry spectra calculated from the measured (Fig. 1) and the theoretical (Fig. 2) reflectivity curves for an Al/Ag multilayer. (a) Stack with 3 unit cells. (b) Stack with 6 unit cells. The green curve indicates the frequency dependence of the energy-loss function.
7. Conclusion
(Ag), respectively. During the deposition process, the high vacuum pressure was 2 × 10−5 mbar and the thickness of each metallic layer was controlled by an optical monitoring system. The fabricated multilayer stack has three bimetallic layers (or three unit cells), which were grown as follows: first, an aluminum (Al) layer with a thickness of 12 nm was deposited on glass substrate. Afterwards, above this layer, a silver (Ag) layer with a thickness of approximately 16 nm was deposited, composing the first bimetallic layer. The process is repeated until the multilayer nanometric material contains three bimetallic layers, namely, the structure: substrate/[Al(12 nm)/Ag(16 nm)]3 . Preparation of SAMPLE B. SAMPLE B was deposited at room temperature using a Kurt J. Lesker sputtering system with two parallel DC magnetron sources and cross-contamination shields. The metallic multilayer has a stack structure of the form substrate/[Al(6 nm)/Ag(8 nm)]6 . A Corning 2947B glass was used as substrate. Base pressure prior to deposition was 3 × 10−8 Torrs. Elemental Al (99.99% pure) and Ag (99.99% pure) 2" dia. targets were sputtered at a 5 W/cm2 power density under an Ar pressure of 3.75 mTorr. Each target-to-substrate distance was fixed at 6 inches. The deposition rate of each source was calibrated by perfilometry, atomic force microscopy, and micro interferometry. Fluororeflectomer Equipment: A home-made fluororeflectomer system [57] was used to obtain the spectral measurements of the samples. The fluroreflectometer is a noncommercial instrument used to measure the emission, excitation, reflection and transmission of samples. The instrument operates in luminescence and excitation spectra (XL-mode) of the samples and specular reflection and transmission spectra of thick or thin films (RT-mode). The RT-mode was used to measure the specular reflection (pol-𝑠 and pol-𝑝) of the bimetallic samples. The spectra of specular reflection of the bimetallic samples were measured from 250 to 800 nm for 𝛥𝜆 = 1 nm resolution and for different angles of incidence, 𝜃, from 10 to 70◦ , in steps of 𝛥𝜃 = 5◦ .
A study of the Berreman effect in the 𝑝-polarization reflectivity and 45-degree reflectometry spectra of Al/Ag nanolayered stacks in the 250–800 nm wavelength range has been carried out. The effect consists in the appearance of a narrow dip in such optical spectra at wavelengths ∼327–333 nm. The theoretical optical spectra, calculated by using both the transfer matrix method and the method of expansion into bulk modes with realistic Al and Ag permittivities, reproduced the main features of the experimental observations. To explain the phenomenon, we calculated the effective permittivity-tensor principal values for an Al/Ag nanolayered superlattice within the effective medium approach and a nonlocal homogenization one. The latter allows us to describe the homogenized bimetallic superlattice as a nonlocal anisotropic medium and to obtain the exact photonic band structure at any values of the frequency and the Bloch wave number. The calculated effective permittivity-tensor components demonstrated that the homogenized bimetallic laminar stack behaves as uniaxial anisotropic medium with considerably-large imaginary parts of the effective permittivity-tensor components. Therefore, the propagation of photonic modes having a hyperbolic dispersion relation is practically forbidden. As it was shown, the Berreman effect in the Al/Ag nanostructure is due to a resonance in the energy-loss function ℑ(−1∕𝜖𝑧 ) at the zero of the effective permittivity-tensor component corresponding to the stack-growth direction. Such a zero coincides with the zero of the Ag-nanolayer permittivity. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments
References
This work was partially supported by CONACYT (Mexico), VIEPBUAP, Mexico (grants: 100160855-VIEP2018 and 100312733-VIEP2019), PFCE, Mexico, and PRODEP-BUAP-CA-250, Mexico. SLGA and RGL acknowledgments to CONACYT, Mexico (Grant 1893). FJFR acknowledges to FOINS-CONACYT, Mexico (Grant No. 2016-01-2488). Appendix. Experimental details Preparation of SAMPLE A. This sample was grown by the thermal evaporation technique in an Edwards high vacuum chamber (1 × 10−6 mbar). Corning glass substrate was used. The bimetallic stack was fabricated by alternating two metallic layers, aluminum (Al) and silver 6
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Berreman effect in bimetallic nanolayered metamaterials S. Cortés-López a , S.L. Gastélum-Acuña b , F.J. Flores-Ruiz c , V. Garcia-Vazquez a , R. García-Llamas d , F. Pérez-Rodríguez a,d ,∗ a
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico CONACYT-Departamento de Investigación en Física, Universidad de Sonora, Apdo. Post. 5-88, Hermosillo, Son. 83000, Mexico CONACYT-Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico d Departamento de Investigación en Física, Universidad de Sonora, Apdo. Post. 5-88, Hermosillo, Son. 83000, Mexico b c
ARTICLE Keywords: Metamaterials Photonic crystals Reflectivity Metals Nanofilms
INFO
ABSTRACT The visible and ultraviolet responses of a binary regular stack, composed of alternating Al and Ag nano-thin layers, are both experimentally and theoretically studied. It is found that the 𝑠- and 𝑝-polarization reflectivity spectra (𝑅𝑠 and 𝑅𝑝 , respectively) of the inherently anisotropic bimetallic system are noticeably different due to the appearance of a prominent dip near the zero of the Ag permittivity. Such an effect is similar to the Berreman one, which is observed in thin polar-crystals films near the longitudinal-optical phonon frequency, as well as in thin metal films near plasma frequency. Using both the effective medium approach and a nonlocal homogenization formalism, being valid even beyond the long wavelength limit, it is demonstrated that the bimetallic multi-layer stack behaves as a uniaxial anisotropic metamaterial. As it is shown, the appearance of the dip in the 𝑝-polarization spectrum is associated to the zero of the longitudinal component of the effective permittivity (𝜖𝑧 ). The calculation of 45-degree reflectometry spectrum, namely the difference 𝑅𝑝 − 𝑅2𝑠 , from experimental and theoretical reflectivity data confirmed that the observed dip in 𝑅𝑝 is connected to a plasma resonance of the energy-loss function ℑ(−1∕𝜖𝑧 ).
1. Introduction Among photonic metamaterials, the inherently-anisotropic hyperbolic metamaterials (HMs) are distinguished by exhibiting striking phenomena such as negative refraction [1,2] and subwavelength imaging [3] without optical magnetism. The HMs can be either regular metal–dielectric layered structures or periodic arrays of metallic nanowires embedded in a dielectric host medium [4–8]. Besides, there exist natural single-phase HMs [9,10], to which graphite [11], high-temperature cuprate superconductors [9,12–14], magnesium diboride [9], tetradymites [15], hexagonal boron nitride [16,17] and others belong. The optical response of the HMs can be explained in terms of their permittivity tensor [4–8], which has one principal value with sign different from the others in certain frequency range. For a uniaxial medium (𝜖𝑥 = 𝜖𝑦 = 𝜖⟂ ≠ 𝜖𝑧 = 𝜖∥ , with the optical axis being parallel to the 𝑧-axis), the photonic dispersion relation for transverse magnetic (TM) modes (extraordinary waves) becomes hyperbolic [4–8]. In the case 𝜖⟂ > 0, 𝜖𝑧 < 0, the isofrequency surface of TM modes is a twofold hyperboloid; the HM is called type I. In contrast, in the case, 𝜖⟂ < 0, 𝜖𝑧 > 0, the TM isofrequency surface corresponds to a one-fold hyperboloid; the HM is a type-II medium.
Another interesting anisotropic optical material is the bimetallic superlattice (see, for example, Refs. [18–25] and references therein). As is shown in some theoretical works [18,20,22], by applying the local Drude model for describing the frequency dependence of the conducting layers in a semi-infinite bimetallic superlattice and considering a moderate electron relaxation rate, two broad reflection minima appear inside the frequency interval, 𝜔𝑝1 < 𝜔 < 𝜔𝑝2 , limited by the plasma frequencies of the metals (i.e. 𝜔𝑝1 and 𝜔𝑝2 ). The predicted minima are associated with two pass bands of the transverse-magnetic photonic modes, which have almost-real Bloch wave numbers and, therefore, propagate along the growth direction of the superlattice. These TM propagating normal modes are originated by surface plasmons at each interface, which are coupled by the tails of their evanescent electromagnetic fields [18] as it also occurs in metal–dielectric superlattices [26]. Besides, in a finite-size multilayer stack, the Bloch wave vector is quantized and the reflection spectra exhibit Fabry–Perot resonances in the two TM pass bands [25]. The principal values of the permittivity tensor for a bimetallic laminar periodic structure can be calculated by employing the effective medium approach (EMA) [27–30]. Using it and the local Drude model, the low- and high-frequency TM pass bands
∗ Corresponding author at: Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico. E-mail address: [email protected] (F. Pérez-Rodríguez).
https://doi.org/10.1016/j.optmat.2019.109578 Received 18 August 2019; Received in revised form 7 November 2019; Accepted 26 November 2019 Available online 5 December 2019 0925-3467/© 2019 Elsevier B.V. All rights reserved.
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3. Theoretical reflectivity spectra
in the interval (𝜔𝑝1 , 𝜔𝑝2 ), which were predicted in Refs. [18,20,22], correspond to the photonic pass bands where the bimetallic superlattice behaves as a HM of type II and I, respectively [25]. In the present work, the optical properties of Al/Ag nanolayered stacks are both experimentally (Section 2) and theoretically (Sections 3 and 4) studied. In particular, the measured visible and UV reflectivity spectra, 𝑅𝑠 and 𝑅𝑝 , for 𝑠 and 𝑝-polarized incident light, respectively, are critically compared with those theoretically obtained by applying the methods based on the transfer matrix (TMM) [18,20] and the expansion into bulk modes (MEBM) [21,22,31]. In the calculations, we shall consider realistic metal frequency-dependent permittivities, taken from the literature. The Al and Ag permittivities used here consider not only the optical response of the conduction electrons, but also the bound-electron contributions, as well as the noticeable electron relaxation rate of nano-thin metal layers. In order to verify whether or not the bimetallic nanolayered superlattice behaves as a HM, its photonic band structure and the principal values of the effective permittivity are calculated and analyzed. The effective parameters of the anisotropic bimetallic metamaterial are obtained by employing both EMA formulas [27–30], and a nonlocal homogenization approach (NHA) that is valid even beyond the long wavelength limit (Section 4). The longitudinal component (𝜖𝑧 ) of the effective permittivity tensor may have zeros close to the zeros of the metal (Al or Ag) permittivities. The existence of zeros in the longitudinal component 𝜖𝑧 can lead to resonances in the energy loss function ℑ(−1∕𝜖𝑧 ) and, then, to minima (dips) in the spectra of 𝑅𝑝 [32–34] and 45-degree reflectometry [35] for thin metal films. The latter spectrum is given by the difference 𝛥45 ≡ 𝑅𝑝 − 𝑅2𝑠 at 45◦ incidence angle and allows to detect both zeros and poles of the longitudinal and transverse components of the effective permittivity, correspondingly. In fact, the appearance of the dips in the UV spectra of 𝑅𝑝 and 𝛥45 is the Berreman effect, which was firstly observed in infrared spectra of polaritonic media [36,37]. In Section 5, the lower-frequency zero of 𝜖𝑧 , which is near the zero of the Ag permittivity, is accurately determined from experimental and theoretical spectra of the 45-degree reflectometry. Our results are discussed and compared with other works in Section 6. Finally, our conclusions are presented in Section 7, whereas the experimental details are given in Appendix.
In order to provide an initial explanation of the features of the reflectivity spectra presented in the previous section, we have numerically calculated the optical spectra for two regular bimetallic (Al/Ag) nanolayered stacks with three (panel (a) of Fig. 2) and six (panel (b) therein) bilayers overlying a glass substrate. The Al- and Ag-layer thicknesses used in reflectivity calculations are: 𝑑𝐴𝑙 = 12.2 (6.1) nm and 𝑑𝐴𝑔 = 16.2 (8.1) nm for the three (six)-bilayer stack. The calculations were carried out by employing realistic Al and Ag wavelengthdependent permittivities (Fig. 3), the standard transfer matrix method and verified with the method of expansion into bulk modes. As was commented in the Introduction, the frequency (or wavelength) dependence of the metal permittivity can be described with the local Drude model, but it may also have contributions associated with bound electrons, which are usually modeled with Lorentz oscillators. Besides, the relatively-large electron relaxation rate in metal nano-thin layers, in comparison with the electron-collision rate for thick metal films, should be taken into account. The permittivity values of the 16.2 nm-thick Ag-layers in SAMPLE A were taken from Palik handbook [38], which in fact correspond to the bulk Ag permittivity (see solid lines in panel (a) of Fig. 3). The use of such values is justified because the Ag permittivity of thin films becomes noticeably different from the bulk value once the Ag film thickness 𝑑𝐴𝑔 is smaller than 12 nm, as was demonstrated in Refs. [39,41]. On the other hand, for the Ag-layers of 8.1 nm, we use the permittivity values determined in Ref. [39] for an Ag film with the same thickness (𝑑𝐴𝑔 = 8.1 nm). As can be seen in panel (a) of Fig. 3, the permittivity of the thinner Ag layers has an imaginary part (𝜖 ′′ ) which is larger than the bulk Ag permittivity, whereas the absolute value of its real part (|𝜖 ′ |) turns out to be smaller than that for the Ag bulk in a wide wavelength range (compare solid lines with dotted curves of the same color). The changes in the permittivity can be attributed to granular structure (heterogeneity) of the fabricated nano-thin films [39,41]. As is shown in Ref. [40], the permittivity of Al films is very close to the bulk value when their thickness is larger than 10 nm. Therefore, in the reflectivity calculations for the Al layers with 12.2 nm thickness we used the bulk Al permittivity values reported in Palik database [38]. The permittivity for thinner Al-layers, namely those with 𝑑𝐴𝑙 = 6.1 nm, coincides with that obtained in the work [40] for an Al film of the same thickness. As in the case of Ag-layers, the difference between the permittivity of the Al nano-thin layers and the bulk Al permittivity may be due to the heterogeneity of the ultra-thin metal films [40]. The calculated reflectivity spectra of the Al/Ag stacks for both 𝑠and 𝑝-polarization geometry (Fig. 2) correspond to the same incidence angles of the experimental optical spectra (Fig. 1). Notice that the same features of the measured reflectivity spectra are practically reproduced by the numerically-calculated spectra. It is important to mention that the dip, appearing at 𝜆 ≈ 327 nm (𝜆 ≈ 333 nm) and being close to the zero of the permittivity for Ag layers in SAMPLE A (B) [see Fig. 3(a)], is clearly observed in the 𝑅𝑝 spectra of both bimetallic nanolayered stacks. Moreover, for SAMPLE A, both 𝑅𝑠 and 𝑅𝑝 [panel (a) of Fig. 2] have relatively-large (small) values at wavelengths above (below) 500 nm (300 nm) in good agreement with the experiment [panel (a) of Fig. 1]. The moderate values of the reflectivity for SAMPLE B, observed in panel (b) of Fig. 1 at wavelengths larger than 500 nm, is also reproduced by the theoretical spectra [panel (b), Fig. 2]. Nevertheless, the wide minimum at 𝜆 ∼ 559 nm turns out to be significantly broader in the calculated spectra (compare panels (b) of Figs. 1 and 2).
2. Optical characterization of Al/Ag nanolayered stacks Visible and UV reflectance spectra were measured on two samples deposited by different deposition techniques. SAMPLE A was grown by thermal evaporation. The stack structure was substrate/[Al(12 nm)/ Ag(16 nm)]3 . SAMPLE B was deposited by magnetron sputtering, with a stack structure of the form substrate/[Al(6 nm)/Ag(8 nm)]6 . The thickness of each Al/Ag bilayer in SAMPLE A (28 nm) is twice the size of that in SAMPLE B (14 nm). In both cases, the Al/Ag thickness ratio (3/4) is the same. The number of repetitions in each sample was chosen to produce the same total thickness of the whole stack (84 nm). Details of the fabrication of each sample are indicated in Appendix. The reflectivity spectra for 𝑠- and 𝑝-polarized light of SAMPLE A and SAMPLE B were measured by using a fluororeflectomer system in the wavelength interval 250–800 nm (see the Appendix for experimental details). In Fig. 1, the measured optical spectra for 30◦ , 45◦ , and 60◦ angles of incidence are presented. As can be observed in panel (a), the reflectivity for SAMPLE A is relatively-large (∼ 0.9) at wavelengths (𝜆) larger than 500 nm. At wavelengths 𝜆 ∼ 305 nm, both 𝑅𝑠 and 𝑅𝑝 have a broad minimum. Surprisingly, the 𝑝-polarization reflectivity spectra of SAMPLE A have an additional feature. A narrow dip is observed at 𝜆 ≈ 327 nm. Furthermore, the dip has a depth that systematically increases with the angle of incidence 𝜃. On the other hand, the reflectivities 𝑅𝑠 and 𝑅𝑝 for SAMPLE B, panel (b) in Fig. 1, are smaller than those for SAMPLE A at wavelengths 𝜆 above 500 nm [compare panels (a) and (b)]. The reflectivity spectra for SAMPLE B also show a broad minimum at 𝜆 ∼ 283 nm. Similarly to 𝑅𝑝 (𝜆) of SAMPLE A, the appearance of a relatively-narrow dip at 𝜆 ≈ 333 nm, whose depth increases with the incidence angle 𝜃 is clearly observed in the 𝑝-polarization reflectivity spectra of SAMPLE B [see panel (b)].
4. Bulk photonic modes and the effective permittivity tensor In order to explain the measured and calculated reflectivity spectra for the regular nanolayered stacks (Figs. 1 and 2) more deeply, we shall analyze the photonic band structure and the wavelength dependence of the effective permittivity tensor of a bimetallic (Al/Ag) superlattice 2
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Fig. 1. Measured 𝑠- and 𝑝-polarization reflectivity spectra at different angles of incidence. (a) Multilayered stack with 3 unit cells (SAMPLE A). (b) Multilayered stack with 6 unit cells (SAMPLE B).
Fig. 2. Calculated 𝑠- and 𝑝-polarization reflectivity spectra for a multilayered stack of the form [Al/Ag]𝑁 . (a) N = 3. (b) N = 6. The spectra were calculated with both TMM (Transfer Matrix Method) and MEBM (Method of Expansion into Bulk Modes) at different angles of incidence, both methods coincide as expected.
Fig. 3. Permittivity data taken from Refs. [38–40]. (a) Silver. (b) Aluminum.
(Fig. 4). The dispersion relations between the Bloch wave number (𝑘𝑧 according to the assumed geometry) and frequency for 𝑠- and 𝑝polarized modes in a binary superlattice are given by the respective well-known formulas [42,43],
−
1 2
(
𝑘𝑧1 𝑘𝑧2 + 𝑘𝑧2 𝑘𝑧1
) sin(𝑘𝑧1 𝑑1 ) sin(𝑘𝑧2 𝑑2 ),
(1)
and cos(𝑘𝑧 𝑎) = cos(𝑘𝑧1 𝑑1 ) cos(𝑘𝑧2 𝑑2 ) ( ) 1 𝜀2 𝑘𝑧1 𝜀1 𝑘𝑧2 − + sin(𝑘𝑧1 𝑑1 ) sin(𝑘𝑧2 𝑑2 ), 2 𝜀1 𝑘𝑧2 𝜀2 𝑘𝑧1
cos(𝑘𝑧 𝑎) = cos(𝑘𝑧1 𝑑1 ) cos(𝑘𝑧2 𝑑2 ) 3
(2)
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Fig. 4. (a) and (b) Photonic dispersion relations of 𝑠- and 𝑝-polarized modes in Al/Ag superlattices with the same unit cells of the bimetallic stacks presented in Figs. 2(a) and (b), respectively. (c) and (d) Effective permittivity principal values calculated within EMA (Effective Medium Approach) and NHA (Nonlocal Homogenization Approach) for the corresponding Al/Ag superlattices of (a) and (b).
√ √ 2 2 where 𝑘𝑧1 = 𝜀1 𝜔𝑐 2 − 𝑘2𝑥 and 𝑘𝑧2 = 𝜀2 𝜔𝑐 2 − 𝑘2𝑥 are the wave numbers of the electromagnetic waves inside the metals 1 (Ag) and 2 (Al) with thicknesses 𝑑1 and 𝑑2 , respectively; 𝜔 = 2𝜋∕𝜆 is the frequency, 𝑐 is the velocity of light in vacuum, 𝑘𝑥 is the conserved wave-vector component, parallel to the metal layers, 𝑎 is the superlattice period and 𝑘𝑧 is the Bloch wave number. Panels (a) and (b) of Fig. 4 present the dispersion relation 𝑘𝑧 (𝜆) for two Al/Ag superlattices, correspondingly, having the same Al- and Aglayers thicknesses and permittivities as those of the regular bimetallic stacks considered in the previous section. Notice that the Bloch wave number has a large imaginary part in comparison with its real part [|ℜ(𝑘𝑧 )| ≪ ℑ(𝑘𝑧 )], particularly for the case of the superlattice with thicker metal layers (panel (a)): 𝑑1 = 𝑑𝐴𝑔 = 16.2 nm and 𝑑2 = 𝑑𝐴𝑙 = 12.2 nm. This implies that such modes should rapidly decay from the sample surface, explaining why the reflectivity for SAMPLE A is, in general, larger than that of SAMPLE B. It is also interesting to analyze the effective permittivity tensor of both Al/Al superlattices and to find their correlation with the reflectivity spectra of the bimetallic nanolayered stacks. In panels (c) and (d) of Fig. 4, we show the wavelength dependences of the principal values of the effective permittivity tensor, which were calculated within EMA [27–30] for SAMPLE A and SAMPLE B, respectively. From their inspection, it follows that the dip of the 𝑝-polarization reflectivity of SAMPLE A (B) at 𝜆 ≈ 327 nm (𝜆 ≈ 333 nm) is very close to the zero of the 𝑧-component (𝜖𝑧𝐸𝑀𝐴 ) of the permittivity tensor, which is in fact determined by the zero of the Ag permittivity [panel (a) of Fig. 3]. It is also observed that the superlattice with thicker metal layers [see panel (c) of Fig. 4] is characterized by relatively large absolute values of the negative 𝑥- and 𝑦-components of the effective permittivity at wavelengths 𝜆 > 500 nm, leading to a strong decay of
the electromagnetic field and, hence, to high values of the reflectivity ∼ 0.9 [see Fig. 1(a) and 2(a)] even in a sample with a few bilayers. The used EMA formulas [27–30] can only provide a qualitative explanation of the optical spectra of layered stacks with metal components since they are valid when the electromagnetic field slightly varies inside the layers. This can occurs in ultra-thin metals with thickness 𝑑 much smaller than the skin depth 𝛿 (𝑑 ≪ 𝛿) and at rather small frequencies (very long wavelengths) [44]. The application of EMA to periodic metal–dielectric layered structures fails because of the skin effect, which leads to a drastic variation of the electromagnetic field inside the metal layer [45]. EMA can also breaks down for deeply subwavelength all-dielectric superlattices since their reflectivity spectra near the critical angle depends on both very small period variations and the termination of the layered structure [46–48]. Due to the limitations of the EMA, various homogenization approaches have been developed in the past few years (see, for example Refs. [49–54]). The proposed theories describe the HM as a nonlocal anisotropic homogeneous medium and provide explicit formulas for calculating the effective permittivity components. In the present study of bimetallic nanolayered stacks, we shall apply a recently-proposed nonlocal homogenization approach (NHA) [55], which is valid even beyond the long wavelength limit (i.e., at high frequencies too) and for any values of the Bloch wave number 𝑘𝑧 . According to the NHA [55], the nonlocal effective parameters are given by the spatial averaging over the unit cell of the periodic part of the Bloch normal modes as: ⟨ ⟩ 𝐷𝑖,𝑝 ⃗ 𝑖 = 𝑥, 𝑦, 𝑧, (3) 𝜀ef f,𝑖 (𝜔, 𝑘) = ⟨ ⟩, 𝐸𝑖,𝑝 where ⟨⋯⟩ indicates the spatial averaging over the unit cell, 𝐷𝑖,𝑝 and 𝐸𝑖,𝑝 are the periodic parts of the displacement vector and electric field, 4
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⃗ = (𝑘𝑥 , 0, 𝑘𝑧 ). The respectively, having the form of Bloch waves, and 𝑘 nonlocal effective permittivity-tensor principal values, Eq. (3), allow us to rewrite the dispersion relations (1) and (2) for 𝑠- and 𝑝-polarized normal photonic modes in the same form as those for a nonlocal homogeneous anisotropic dielectric material, respectively: 𝑘2𝑧 + 𝑘2𝑥 =
𝜔2 ⃗ 𝜀ef f,𝑦 (𝜔, 𝑘), 𝑐2
𝑘2𝑧 ⃗ 𝜀ef f ,𝑥 (𝜔, 𝑘)
+
𝑘2𝑥 ⃗ 𝜀ef f,𝑧 (𝜔, 𝑘)
Al/Ag stacks with thinner ‘‘nanolayers", grown with a rather small deposition time. It was found that their reflectivity spectra do not exhibit a dip at the frequency of the Ag-permittivity zero. Such samples behave more like an isotropic Al/Ag nanocomposite having an effective scalar permittivity with a zero at wavelengths smaller than that of the Ag-permittivity zero.
(4)
=
𝜔2 . 𝑐2
6. Discussion
(5)
The results presented in the previous sections demonstrate that Al/Ag nanolayered stacks behave as uniaxial anisotropic media. Both EMA and NHA predict a zero of the effective permittivity-tensor 𝑧component (corresponding to the growth direction), where the 𝑝polarization reflectivity 𝑅𝑝 and the 45-degree reflectometry spectrum have a narrow dip. However, the large imaginary parts of the effective permittivity components, which are attributed to surface-electron scattering in nano-thin layers, does not allow the propagation of photonic modes with hyperbolic dispersion relation at frequencies just above the zero of 𝜖𝑧 , coinciding with that of the Ag permittivity. In other words, the HM behavior of Al/Ag nanolayered stacks could not be confirmed.
Panels (c) and (d) of Fig. 4 show the calculated nonlocal effective permittivity principal values for the bimetallic superlattices considered above. Notice that the wavelength dependences of the permittivity components, predicted by the EMA and NHA, are indeed quantitatively different. Both approaches, however, predict almost the same zero for the 𝑧-component of the effective permittivity, which is associated to the dip in the 𝑅𝑝 spectra of both bimetallic nanolayered stacks [compare solid lines with dotted curves in panels (c) and (d)]. It should be mentioned that we have also applied the MEBM [21,22,31] to calculate the reflectivity spectra. Within MEBM, the microscopic electric and magnetic fields inside the finite-size bimetallic stack are expressed as a superposition of two Bloch normal waves (with wave numbers 𝑘𝑧 and −𝑘𝑧 , respectively). The amplitudes of such waves, as well as the amplitudes of the reflected and transmitted fields, are straightforwardly calculated by imposing the continuity of the tangential components of the electric and magnetic fields at the front and rear surfaces of the bimetallic nanolayered stack. As it was verified, the calculated reflectivity spectra within MEBM coincide with those shown in Fig. 2, which were obtained with the TMM.
The appearance of a dip in the 𝑝-polarization reflectivity of the bimetallic laminar stack is attributed to the Berreman effect [32–34]. Hence, the origin of the effect is a plasma resonance in the energy-loss function ℑ(−1∕𝜖𝑧 ), being close to the Ag plasma frequency. It should also be mentioned that the contributions of the bound electrons to the Ag permittivity shift its zero to lower frequencies (larger wavelengths). In fact, the models for the Ag permittivity commonly make use of a local Drude term with plasma frequency of the order of 9 eV (138 nm), together with Lorentz oscillators with resonance frequencies ∼ 4 eV (310 nm), leading to zeros of the permittivity for Ag nano-thin films at ∼ 327–333 nm [38,56].
5. 45-degree reflectometry To confirm that the dip in the 𝑅𝑝 spectra is connected to a zero of the 𝑧-component of the effective permittivity, we have calculated the 45-degree reflectometry spectra, namely 𝛥45 ≡ 𝑅𝑝 − 𝑅2𝑠 , from the experimental and theoretical reflectivities (curves for 𝜃 = 45◦ in Figs. 1 and 2, respectively). As was demonstrated in Ref. [35], the 𝛥45 spectra for a multilayer stack of thickness 𝑑 ≪ 𝜆 are given by the principal values of the effective permittivity tensor as: [ ( )] √ −1 𝑑 ℑ(𝜀⟂ ) − 𝜀2𝑠 ℑ 𝛥45 = 4 𝜋 2 𝑅0𝑝 |𝜃=45◦ , (6) 𝜀∥ 𝜆(𝜀𝑠 − 1)2
Another factor that might alter the optical spectra is the metal nonlocality [34]. According to Refs. [18,22], in the case of spatiallydispersive metals, the reflectivity spectrum of a bimetallic semi-infinite superlattice has an extra series of sharp peaks, which are originated by single-film resonance frequencies and superimposed to the two broad reflection minima appearing near the plasma frequency of the metals. In addition, the TM photonic pass bands, corresponding to those minima, are split in several bands as a result of the coupling of surface-plasmons with the confined modes in the metal layers with lower plasma frequency (𝜔𝑝1 ). Nevertheless, the large damping parameters used in the permittivities of Al and Ag nano-layers obstruct the observation of effects associated with the Ag nonlocality.
if the homogenized laminar structure behaves as a uniaxial crystal (𝜖⟂ = 𝜖𝑥 = 𝜖𝑦 ≠ 𝜖∥ = 𝜖𝑧 ). In Eq. (6), 𝑅0𝑝 |𝜃=45◦ denotes the 𝑝-polarization reflectivity of the substrate, having a permittivity 𝜖𝑠 , at 45-degree incidence angle. According to Eq. (6), transverse resonances manifest as prominent maxima in 𝛥45 spectra, while longitudinal modes, corresponding to zeros of the parallel component of the effective permittivity 𝜖∥ = 𝜖𝑧 , should clearly manifest themselves as narrow minima (dips). Fig. 5 shows the 𝛥45 ≡ 𝑅𝑝 − 𝑅2𝑠 spectra, which were calculated by using the experimental (Fig. 1) and theoretical (Fig. 2) 𝑅𝑠 and 𝑅𝑝 reflectivities for the bimetallic nanolayered stacks with 3 and 6 bilayers. Evidently the 𝛥45 spectrum for SAMPLE A (B) exhibits a well-discernible dip at 𝜆 ≈ 327 nm (𝜆 ≈ 333 nm), whereas at larger wavelengths 𝛥45 ≈ 0 for both samples. The agreement between theory and experiment is excellent. We also show in Fig. 5 the wavelength dependence of the energy-loss function ℑ(−1∕𝜖𝑧 ), which was calculated by using the NHA. As can be observed, the latter has a resonance precisely at the frequency where both 𝑅𝑝 and 𝛥45 spectra have a dip. It should be also noticed that the dips in 𝑅𝑝 and 𝛥45 [peak in ℑ(−1∕𝜖𝑧 )] are more prominent for SAMPLE A (compare panels (a) and (b) in Fig. 5). This can be due to the fact that the thinner layers have not well defined interfaces between layers because of their granularity and the diffusion of Al and Ag to adjacent layers. To confirm such a statement, we have also carried out reflectivity measurements on
In order to reduce the large imaginary parts in the effective permittivity components and enhance the propagation of photonic modes in Al/Ag layered stacks, some improvements during the preparation of the films could be implemented. For example, the use of singlecrystal substrates instead of the amorphous substrates here utilized could induce the growth of highly-oriented metallic layers. Atomically smooth metallic surfaces would reduce or avoid the intersperse effect, producing sharp Al/Ag interfaces. On the other hand, substrate heating during deposition could enhance the crystallinity of the individual layers. Other improvements during deposition could also be added. The use of higher sputtering rates, for example, can increase the kinetic energy of the atoms reaching the substrate for a better nucleation density in the metallic layers. Target-to-substrate distances could also be varied. We expect that the quality of the Al/Ag layers can be improved by an optimization process during film deposition. Furthermore, we expect that the dip found in the 𝑝-polarization reflectivity will be enhanced by an appropriate choose of the deposition parameters. Further experiments will be needed to prove this. 5
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Fig. 5. 45-degree reflectometry spectra calculated from the measured (Fig. 1) and the theoretical (Fig. 2) reflectivity curves for an Al/Ag multilayer. (a) Stack with 3 unit cells. (b) Stack with 6 unit cells. The green curve indicates the frequency dependence of the energy-loss function.
7. Conclusion
(Ag), respectively. During the deposition process, the high vacuum pressure was 2 × 10−5 mbar and the thickness of each metallic layer was controlled by an optical monitoring system. The fabricated multilayer stack has three bimetallic layers (or three unit cells), which were grown as follows: first, an aluminum (Al) layer with a thickness of 12 nm was deposited on glass substrate. Afterwards, above this layer, a silver (Ag) layer with a thickness of approximately 16 nm was deposited, composing the first bimetallic layer. The process is repeated until the multilayer nanometric material contains three bimetallic layers, namely, the structure: substrate/[Al(12 nm)/Ag(16 nm)]3 . Preparation of SAMPLE B. SAMPLE B was deposited at room temperature using a Kurt J. Lesker sputtering system with two parallel DC magnetron sources and cross-contamination shields. The metallic multilayer has a stack structure of the form substrate/[Al(6 nm)/Ag(8 nm)]6 . A Corning 2947B glass was used as substrate. Base pressure prior to deposition was 3 × 10−8 Torrs. Elemental Al (99.99% pure) and Ag (99.99% pure) 2" dia. targets were sputtered at a 5 W/cm2 power density under an Ar pressure of 3.75 mTorr. Each target-to-substrate distance was fixed at 6 inches. The deposition rate of each source was calibrated by perfilometry, atomic force microscopy, and micro interferometry. Fluororeflectomer Equipment: A home-made fluororeflectomer system [57] was used to obtain the spectral measurements of the samples. The fluroreflectometer is a noncommercial instrument used to measure the emission, excitation, reflection and transmission of samples. The instrument operates in luminescence and excitation spectra (XL-mode) of the samples and specular reflection and transmission spectra of thick or thin films (RT-mode). The RT-mode was used to measure the specular reflection (pol-𝑠 and pol-𝑝) of the bimetallic samples. The spectra of specular reflection of the bimetallic samples were measured from 250 to 800 nm for 𝛥𝜆 = 1 nm resolution and for different angles of incidence, 𝜃, from 10 to 70◦ , in steps of 𝛥𝜃 = 5◦ .
A study of the Berreman effect in the 𝑝-polarization reflectivity and 45-degree reflectometry spectra of Al/Ag nanolayered stacks in the 250–800 nm wavelength range has been carried out. The effect consists in the appearance of a narrow dip in such optical spectra at wavelengths ∼327–333 nm. The theoretical optical spectra, calculated by using both the transfer matrix method and the method of expansion into bulk modes with realistic Al and Ag permittivities, reproduced the main features of the experimental observations. To explain the phenomenon, we calculated the effective permittivity-tensor principal values for an Al/Ag nanolayered superlattice within the effective medium approach and a nonlocal homogenization one. The latter allows us to describe the homogenized bimetallic superlattice as a nonlocal anisotropic medium and to obtain the exact photonic band structure at any values of the frequency and the Bloch wave number. The calculated effective permittivity-tensor components demonstrated that the homogenized bimetallic laminar stack behaves as uniaxial anisotropic medium with considerably-large imaginary parts of the effective permittivity-tensor components. Therefore, the propagation of photonic modes having a hyperbolic dispersion relation is practically forbidden. As it was shown, the Berreman effect in the Al/Ag nanostructure is due to a resonance in the energy-loss function ℑ(−1∕𝜖𝑧 ) at the zero of the effective permittivity-tensor component corresponding to the stack-growth direction. Such a zero coincides with the zero of the Ag-nanolayer permittivity. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments
References
This work was partially supported by CONACYT (Mexico), VIEPBUAP, Mexico (grants: 100160855-VIEP2018 and 100312733-VIEP2019), PFCE, Mexico, and PRODEP-BUAP-CA-250, Mexico. SLGA and RGL acknowledgments to CONACYT, Mexico (Grant 1893). FJFR acknowledges to FOINS-CONACYT, Mexico (Grant No. 2016-01-2488). Appendix. Experimental details Preparation of SAMPLE A. This sample was grown by the thermal evaporation technique in an Edwards high vacuum chamber (1 × 10−6 mbar). Corning glass substrate was used. The bimetallic stack was fabricated by alternating two metallic layers, aluminum (Al) and silver 6
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