Effects of exchange correlation functional on optical permittivity of gold and electromagnetic responses In-Bai Lin, Tony Wen-Hann Sheu, and Jia-Han Li* Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan * [email protected]
Abstract: Understanding the optical properties of nanometer-scale noble metals is important for the nanoplasmonic devices. The bulk gold and thin film are calculated by density functional theory (DFT) with LDA, PBE, and GLLBSC functionals, respectively. The GLLBSC results for bulk gold are closer to the experimental data because the GLLBSC functional has better descriptions of transition energy. The Im(ε) of thin film calculated by LDA and PBE are overestimated. The effects of DFT-based optical properties are performed by conducting electromagnetic simulations. The transmission for the gold thin film by GLLBSC is blue-shifted. The gold grating structure with the GLLBSC-based optical permittivity has strong localized streamlines of Poynting vector in the corner edges at the resonance condition. ©2014 Optical Society of America OCIS codes: (250.5403) Plasmonics; (160.4760) Optical properties; (310.0310) Thin films.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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16. J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen, M. Dułak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H. A. Hansen, H. H. Kristoffersen, M. Kuisma, A. H. Larsen, L. Lehtovaara, M. Ljungberg, O. LopezAcevedo, P. G. Moses, J. Ojanen, T. Olsen, V. Petzold, N. A. Romero, J. Stausholm-Møller, M. Strange, G. A. Tritsaris, M. Vanin, M. Walter, B. Hammer, H. Häkkinen, G. K. Madsen, R. M. Nieminen, J. K. Nørskov, M. Puska, T. T. Rantala, J. Schiøtz, K. S. Thygesen, and K. W. Jacobsen, “Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method,” J. Phys. Condens. Matter 22(25), 253202 (2010). 17. J. Yan, J. J. Mortensen, K. W. Jacobsen, and K. S. Thygesen, “Linear density response function in the projector augmented wave method: Applications to solids, surfaces, and interfaces,” Phys. Rev. B 83(24), 245122 (2011). 18. J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77(18), 3865–3868 (1996). 19. M. M. Alvarez, J. T. Khoury, T. G. Schaaff, M. N. Shafigullin, I. Vezmar, and R. L. Whetten, “Optical Absorption Spectra of Nanocrystal Gold Molecules,” J. Phys. Chem. B 101(19), 3706–3712 (1997). 20. K. Leosson, A. S. Ingason, B. Agnarsson, A. Kossoy, S. Olafsson, and M. C. Gather, “Ultra-thin gold films on transparent polymers,” Nanophotonics 2(1), 3–11 (2013). 21. Lumerical Solutions, Inc., http://www.lumerical.com/tcad-products/fdtd/ 22. H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B Quantum Semiclassical Opt. 6(5), S404–S409 (2004). 23. Z. B. Wang, B. S. Luk’yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flows around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 032427 (2004).
1. Introduction The optical properties of noble metals and performances of their devices depend on their geometrical structures and material parameters. For most optical designs and applications, the optical properties of noble metals for bulk materials are taken from experimental results, such as the early measurements done by Johnson and Christy [1]. As the size scale of the devices is shrunk into few nanometers by state-of-art nanofabrication techniques, it is important to know the correct optical properties of the materials for designing new nanophotonic devices. With the success of the ab initio methods, it has been widely utilized to simulate many nanostructures such as graphene nanoribbons transistor [2], nanoparticle dimer [3], and nanoparticle chain [4]. However, analyzing the nanophotonic devices by using ab initio methods is not realistic because it needs to consume heavy computational cost. Therefore, using electromagnetic simulations with the optical permittivities calculated by density functional theory is one of the alternative and reliable strategies to investigate the nanophotonic devices [5, 6]. The time-dependent density functional theory (TDDFT) has been successfully applied to interpret the optical permittivity or dielectric function for various materials including the semiconductors and metals [7, 8]. The local density approximation (LDA) and the generalized gradient approximation (GGA) are widely applied to the exchange-correlation functionals. For the noble metal surface, a large screening emerges due to an incorrect calculation of the d band by LDA functional [9, 10]. Besides, the optical permittivity of bulk gold calculated by LDA or PBE functionals shows that the peak of the imaginary part of the dielectric constant at around 3 eV is different when comparing with the bulk gold experimental data from Johnson and Christy [5, 6, 8, 11]. Although the LDA and PBE functionals can capture most of the characteristic features of the optical permittivity, it is not sufficiently accurate in the visible light region which is important for nanoplasmonic applications. The many-body GW method can be used to describe d bands better in noble metals [12, 13]; however, it needs much more computational resource. This fact results in its less application usage in engineering practice. In this paper, the band structures and the optical permittivities of bulk gold are calculated by using three different exchange-correlation functionals, LDA, PBE, and GLLBSC. We found that the optical permittivity of bulk gold calculated by GLLBSC is closer to the experimental data of Johnson and Christy than the results calculated by LDA and PBE. The reason is due to a better description of the single-particle energies by GLLBSC compared to those of LDA and PBE [9, 10, 14]. The band structures and the optical permittivities of gold thin film by the three exchange-correlation functionals are also investigated. To show the effects of the calculated optical properties by these three functionals on plasmonic structures, the electromagnetic simulations for the gold thin film and gold grating atop the glass substrate are performed using the calculated DFT-based dielectric constants. Their transmissions and #223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30726
loss spectra for different exchange-correlation functionals are calculated and compared with the results calculated by the Johnson and Christy’s experimental dielectric function. The electric field intensities and Poynting vector for the gold grating are also simulated. 2. Methodology and simulation setup In this work, all first principles calculations were performed by the grid-based projector augmented wave (GPAW) method [15–17]. The Kohn-Sham wave functions and band energies were calculated for bulk gold and relaxed gold thin films by the LDA, PBE, and GLLBSC functionals. The LDA functional is derived from the homogeneous electron gas model. The PBE functional is a generalized gradient approximation reported by Perdew et al [18]. The GLLBSC functional, a revised Perdew-Burke-Ernzerhof functional for the solid and correlation, is an orbital-dependent functional based on the GLLB potential. It has successfully modified the unoccupied conduction band for semiconductor and insulator [14], and it can improve the description of the noble metal’s single-particle energies [9, 10]. The computational cost by using the GLLBSC potential is reasonable compared to the LDA and GGA functionals [9, 14]. In our calculation, the uniform mesh of the grid spacing is set at 0.02 nm for bulk gold and gold thin films. For the structure optimization of gold thin film, the atomic positions were relaxed by minimizing the forces based on the quasi-Newtonalgorithm. Also, the local density approximation (LDA) was elected to describe the exchange and correlation. For the k-point grid, a Monkhorst-Pack mesh of 8 × 8 × 1 was used for gold thin film in the (111) orientation. The gold thin films were modeled with 14 atomic layers (~3.63 nm) in a 1 × 1 unit cell. A supercell contains 2 nm vacuum layers in the directions perpendicular to the gold thin films. The wave functions and single particle energies for bulk gold and gold thin film were calculated by LDA, PBE, and GLLBSC. Then, they were used for the calculation of optical permittivities by the frequency-dependent density response function within the random phase approximation (RPA) framework [17]. The k-point grids corresponding to 40 × 40 × 40 (for bulk gold) and 40 × 40 × 1 (for gold thin film) were elected to construct a dense sampling of the Brillouin zone, which can describe the optical permittivity in the range between 400 nm and 1000 nm. A finite lifetime broadening of 0.02 eV was used. The dielectric function was calculated on a plane wave basis with the energy cut-off of 150 eV. 3. Bulk gold Figure 1 shows the real and imaginary parts of the optical permittivity calculated by the DFT method within different exchange correlation functionals and the experimental data reported by Johnson and Christy. The calculation is based on the linear response function of an interacting many-body system by solving Dyson’s equation from the ground state electronic structure [17]. To include all the transitions in our calculation of optical permittivity, a finite but small q, which is a wave vector restricted to the Brillouin Zone, is considered. The optical permittivities calculated by the GLLBSC functional are overall much closer to the experimental data than the calculations results by the LDA and PBE functionals. The minima of the Im(ε) calculated by LDA and PBE functionals occur at a longer wavelength compared to the experimental data. Because of the interband transition, the Im(ε) shall rise up at a higher energy. For noble metals, the minimum of the Im(ε) occurs when the interband transition has a greater contribution to Im(ε) than intraband transition. Thus, the minimum of Im(ε) occurring at a longer wavelength means the underestimation of the electron transition between the occupied d-bands and the unoccupied hybridized sp-bands above the Fermi level. According to [5, 8], the electronic transitions resulting from the d-bands to the p-bands close to the Fermi level is around the X and L points of Brillouin zones.
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Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30727
Fig. 1. The predicted dielectric constants for bulk gold using the LDA, PBE, and GLLBSC functionals. For comparison, the experimental data reported by Johnson and Christy (J&C) is also shown.
Figure 2 shows the band structures (left panel) and the corresponding density of states (right panel) from the Γ point to the X point. The energies of X point calculated by PBE (~1.2 eV) and LDA (~1.1 eV) functionals are smaller than that by the GLLBSC functional (~2.0 eV). Glantschnig and Ambrosch-Draxl reported that the orbitals at the X point in ~0 eV (Fermi level) were p-bands and most of the orbitals below 0 eV (Fermi level) were d-bands [8]. Figure 2 shows that some of the d-bands calculated by GLLBSC functional are higher than the d-bands calculated by LDA and PBE functionals. Compared to the band structures by GLLBSC functional, the band structures by LDA and PBE functional have the lower edge of the d-bands and have led to broadening d bands below Fermi level. It can be also found that the range of the occupied d-bands calculated by GLLBSC is narrower than the occupied dbands calculated by LDA and PBE in Fig. 2. At ~0 eV (Fermi level), the peak energy mainly consisting of p-bands by LDA and PBE functionals is smaller than the peak energy by GLLBSC functional. The interband transitions calculated by LDA and PBE functionals occur at the lower energy. Due to the underestimated p-bands energy at X points, indicating the electron transitions between the occupied d-bands and the unoccupied p-bands, the minimum (interband transition) calculated by LDA and PBE functionals occurs at a longer wavelength.
Fig. 2. Left panel: the predicted electronic band structures of bulk gold along the Γ-X direction for LDA, PBE, and GLLBSC; right panel: the corresponding density of states. The red dotted line at 0 eV indicates the Fermi-level.
To understand the interband contribution for the optical properties of bulk gold, we plot Fig. 3(a) to show the calculated Im(ε). It is noted that only the interband contribution is considered in Fig. 3(a). The experimental data by Johnson and Christy shown in Fig. 1 include the interband and intraband contributions. Alvarez et al. decomposed the dielectric #223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30728
functions of Johnson and Christy into the free-electron (Drude model) and the interband (5d to 6sp) parts [19]. Thus, for the comparison sake, the results of the calculated interband contribution of bulk gold by Alvarez et al. are also shown in Fig. 3(a). The curve of our calculated interband contribution by GLLBSC is close to the result reported by Alvarez et al. This means that the GLLBSC functional can have a better description of the interband contribution than those by the LDA and PBE functionals. Figure 3(b) shows the band structures of two possible paths of the interband transition. Also, the lowest energy transition is marked on the band structures in Fig. 3(b). The marked transition energy along Γ-X points of the Brillouin zone in Fig. 3(b) (left panel) calculated by LDA, PBE, and GLLBSC functionals are 1.663 eV, 1.683 eV, and 1.815 eV, respectively. The other path is along W-LΓ points of the Brillouin zone shown in Fig. 3(b) (right panel), and the marked transition energies calculated by LDA, PBE, and GLLBSC functional are 1.296 eV, 1.320 eV, and 1.489 eV, respectively. As the marked transition energy becomes smaller, the contribution to Im(ε) takes place from a smaller photon energy, i.e., a longer wavelength. These results are consistent with the plotted curves of Im(ε) for LDA, PBE, and GLLBSC functionals in Fig. 3(a), which shows that the curves for LDA and PBE are more red-shifted than GLLBSC and the results of Alvarez et al. The explanation is that the p-bands calculated by LDA and PBE functionals for the bulk gold are actually underestimated, thereby causing the discrepancy of the optical permittivity to occur compared to the experimental data.
Fig. 3. (a) The predicted imaginary parts of the dielectric constants of the bulk gold by LDA, PBE, and GLLBSC. Only the interband contributions are considered. For comparison, the interband contribution of bulk gold calculated by Alvarez et al. [15] is shown as the green line. (b) The calculated lowest energy transition of the bulk gold along the Γ-X (left panel) and W-LΓ (right panel) directions by LDA, PBE, and GLLBSC, where the green dotted lines at 0 eV indicate the Fermi-level.
4. Gold thin film: 14 atomic layers of Au(111) To understand the effects of the exchange correlation functionals on the predicted optical permittivities and band structures for gold thin films, the 14 atomic layers of gold (~3.63 nm) constitute the thin film structure which is considered in the (111) orientation in the GPAW simulations. Figure 4 shows the comparison of the results for the dielectric functions of the 14 atomic layers of gold and the bulk gold. The minimum of the Im(ε) calculated by LDA or PBE functional occurs in the case of a longer wavelength. Compared to the bulk gold calculations, the minima of the Im(ε) calculated by LDA or PBE functional are slightly redshifted and the minimum of the Im(ε) calculated by the GLLBSC is blue-shifted for the 14 atomic layers of gold. In comparison with the predicted Im(ε) of thin film and bulk gold, i.e. the dashed lines and solids lines in Fig. 4, it is found that the Im(ε) of thin film calculated by the GLLBSC is smaller than the Johnson and Christy bulk gold results in the entire wavelength range; however, the Im(ε) of thin film calculated by the LDA and PBE are larger than Johnson and Christy results in the wavelength range between 500 nm and 750 nm. It means that the predicted Im(ε) of thin film by LDA and PBE functionals are overestimated. It is also worthy to point out that the calculated Im(ε) for the thin film is smaller than Im(ε) for
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Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30729
the bulk gold in short wavelength range. The reason is that there are more electrons to transit from the occupied d bands to the unoccupied sp bands in shorter wavelength for the bulk gold. For Re(ε), it is a negative value but larger for the thin film compared to the bulk gold. It means that the optical property of the gold thin film is not quite like “metal” in view of electromagnetics compared to the bulk gold.
Fig. 4. Real parts (bottom panel) and imaginary parts (top panel) of the permittivity of Au(111) thin film (dotted line) predicted by the adopted DFT method in comparison with the data from bulk gold (solid line) and Johnson and Christy experimental data (green dotted line).
Figure 5 shows the band structures of the Au(111) thin film with 14 atomic layer thickness calculated by the LDA, PBE, and GLLBSC functionals. It should be noted that the band structures of the (111) surface of noble metals with 24 atomic layer thickness were already calculated by LDA and GLLBSC in [10]. In Fig. 5, the top of the Au d-bands calculated by PBE, LDA, and GLLBSC functionals are −0.930, −1.305 eV, and −2.087 eV, respectively. The difference at the top of the Au d-bands by LDA and GLLBSC is about 0.782 eV, which is similar to the results reported by Jane et al. [9, 10]. Before discussing in greater detail of the results in Fig. 5, it is worthy to read the band structures for bulk gold in Fig. 2 and the band structures for Au(111) thin film in Fig. 5. Figure 2 seems to indicate that most of the d-bands are pushed up by GLLBSC compared to LDA or PBE, while Fig. 5 shows the opposite trend, i.e. downshift of d-bands. However, it should be noticed that the results in Fig. 2 are Γ-X for bulk gold in face centered cubic Brillouin zone and the results in Fig. 5 are Γ-K-M-Γ for the Au(111) thin film in surface Brillouin zone. It is observed that the range of the occupied d-bands calculated by GLLBSC is narrower than the occupied d-bands calculated by LDA and PBE for Au(111) thin film in Fig. 5, which shows good agreement with [10]. This similar phenomenon can be also found for bulk gold in Fig. 2. Also, the upper d-bands calculated by GLLBSC are lower than LDA and PBE for Au(111) thin film in Fig. 5. It can be explained that the screening is overestimated compared to the results by GLLBSC functional in Fig. 5(a). The similar phenomena can be found in the predicted results of PBE and GLLBSC functionals shown in Fig. 5(b). For the GLLBSC functional, the energy of dbands close to Fermi level is lower than LDA and PBE functionals. The transition energy between the occupied d-bands and the unoccupied hybridized sp bands above the Fermi level predicted by GLLBSC is larger than LDA and PBE functionals. In accordance with the band structure of the bulk gold, the electronic transition from the occupied d band to the unoccupied p band around the X point and the L point of the Brillouin zone is related to the minimum value of the Im(ε). For the gold thin film, the relatively narrow energy gap between the d-bands and Fermi level taking place at the M points, which is projected from the X points and the L points of the Brillouin zone of face center cubic, causes the occurrence of minima of the Im(ε) at a longer wavelength in Fig. 4. For the case of a narrower energy gap between the d-bands and Fermi level, the lower energy can trigger the interband transition.
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Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30730
Fig. 5. Comparison of the predicted band structures of the Au(111) thin film by different functionals, where (a) shows LDA (gray) and GLLBSC (black) and (b) shows PBE (gray) and GLLBSC (black). The Fermi energy is set to be zero as shown in the red dashed lines. The red plus sign markers at the M points of the surface Brillouin zone indicate that top d-bands are −1.305 eV, −2.087 eV, and −0.930 eV for LDA, GLLBSC, and PBE, respectively.
5. Applications of gold thin film Gold-coated glass plate is one of the simple plasmonic devices. According to the recent research report [20], the simulated results of the gold thin films deposited on the glass based on experimental bulk gold dielectric functions cannot predict the experimental results very well as the thickness of gold thin film is only few nanometers. In this paper, two plasmonic structures are studied to show the effects of gold thin film thickness on plasmonic resonances. One structure is the gold thin film which has the thickness of h = 3.63 nm (14 atomic layers) atop the silicon dioxide substrate as shown in Fig. 6(a), and the other structure is the gold grating atop the silicon dioxide substrate in Fig. 6(b) with the thickness h = 3.63 nm, the period p = 50 nm, and the gap d = 10 nm. For simplicity in all calculations, the silicon dioxide substrate with infinitely large thickness is considered. The incident plane wave is x-polarized and propagates from top to down in + z direction. The Lumerical FDTD solutions package [21] is used to calculate the transmittance, reflection, and loss. The optical permittivities used in electromagnetic simulations are the predicted data of gold thin film in Fig. 4 by density functional theory with LDA, PBE and GLLBSC functionals, respectively. For the sake of comparison, we also show the simulation results for the structures using the experimental data from Johnson and Christy.
Fig. 6. Cross section view of (a) gold thin film and (b) gold grating, where the thickness of the thin film is h, the width of the gap is d, and the period is p.
For the gold thin film, Fig. 7 shows the high transmittance into the silicon dioxide substrate for LDA, PBE, GLLBSC and experimental data from Johnson and Christy. Compared to the transmittance adopted by Johnson and Christy, the calculated transmittance curves are red-shifted for the LDA and PBE functionals but blue-shifted for the GLLBSC functional. Based on [20], the results of the GLLBSC functional is closer to the experimental
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Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30731
transmission of gold thin film. Because the loss predicted by LDA and PBE functionals occurs at the longer wavelengths in Fig. 7, the transmittance curves are red-shifted. Also, more energies are reflected in longer wavelength. Generally speaking, the loss in Fig. 7 is proportional to the Im(ε) in Fig. 4.
Fig. 7. Loss (dashed line), reflection (solid line), and transmittance (cross line) for the gold thin films based on the GLLBSC (blue), PBE (red), LDA (black), and J&C (green) permittivities.
Figure 8 shows the transmittance, reflection, and loss for the gold grating structures. The simulation results by adopting the Johnson and Christy dielectric functions for the gold grating show that the maximum loss is at the wavelength 826.2 nm, i.e. the minimum of transmittance which is also close to the maximum of loss. The maximum losses are 882.8 nm, 959.2 nm and 968.5 nm for adopting the DFT-based optical permittivities by GLLBSC, PBE, and LDA, respectively. Because the Im(ε) is larger at the resonance for LDA and PBE functionals compared to GLLBSC in Fig. 4, the losses for the structures by LDA and PBE are slightly larger and the transmissions are slightly smaller. Besides, the maximum reflection calculated by GLLBSC functional is larger than the results predicted by LDA and PBE functionals. There are more energies which are reflected due to a smaller loss at the resonance for the results calculated by the GLLBSC functional.
Fig. 8. Loss (dashed line), reflection (solid line), and transmittance (cross line) for the gold grating based on the GLLBSC (blue), PBE (red), LDA (black), and J&C (green) permittivities.
Figure 9 shows the electric field distributions (in logarithmic scale) and Poynting vector streamlines for the gold grating structures at the respective maximum loss for the four cases in Fig. 8. The strong localized electric field exists at the corner edges with the dense streamlines of Poynting vector. It is interesting to see that the experimental data (J&C) and GLLBSC functional show fewer streamlines of Poynting vector through the grating holes in Figs. 9(a) and 9(b) compared to LDA and PBE functionals in Figs. 9(c) and 9(d). The GLLBSC results in Fig. 9(b) show that the dense streamlines of Poynting vector at the corner
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Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30732
edges have a great influence on the Poynting vector through the holes. The streamlines of Poynting vector are not straight lines around the center of the grating holes for the GLLBSC results compared to J&C’s results. Furthermore, the phenomenon of the dense streamlines of Poynting vector at the corner edge are related to the topological singularity [22, 23], and the DFT-based optical permittivity within different exchange-correlation functionals embody quite different results of the Poynting vector. For the LDA and PBE results, more Poynting vector lines straightly pass through the grating holes in Figs. 9(c) and 9(d). Campbell et al. reported that the Poynting vector streamlines with PBE-based optical permittivity dissipate as well at the corner edges [6]. Therefore, it can explain why the transmittances at the resonances calculated by LDA and PBE functionals are higher than the J&C’s and GLLBSC’s results in Fig. 8. We can also find that LDA and PBE functionals have greater loss of energies at the resonance in Fig. 8. However, the electric field magnitude distributions and streamlines of Poynting vector shown in Fig. 9 reveal that there are less localized energies occurring at the corner edge at the resonance. It is because that the Im(ε) of gold thin film predicted by LDA and PBE functionals are larger than GLLBSC functional and Johnson and Christy experimental data at the resonance. This can also explain that the streamlines of Poynting vector calculated by GLLBSC functional shown in Fig. 9 are densely distributed near the grating corner but the loss calculated by GLLBSC functional in Fig. 8 is much smaller at the resonance. The prediction of gold grating by GLLBSC is that the edge of the corner has the tendency of localizing power and causes less power passing through the hole between each gold strip.
Fig. 9. Electric filed magnitude distributions (in logarithmic scale) and streamlines of Poynting vector for the gold grating structures at the wavelengths of the maximal losses: 826.2 nm, 882.8nm, 968.5nm and 959.2 nm, respectively, for the (a) J&C; (b) GLLBSC; (c) LDA; (d) PBE optical permittivities. The white solid line is the boundary between the Au(111) thin film, silicon dioxide substrate and the vacuum.
6. Conclusion Having reduced the degree of underestimation of the transition energy, the predicted optical permittivities of bulk gold calculated by GLLBSC functional is much closer to experimental data in comparison with the LDA and PBE functionals from the visible to the near infrared region. For the gold thin film of 14 atomic layers, the energy gap between top d band and
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30733
Fermi level calculated by GLLBSC is larger than those by the LDA and PBE. The Im(ε) calculated by LDA or PBE functionals is slightly red-shifted and Im(ε) calculated by GLLBSC is blue-shifted. The Im(ε) of thin film calculated by the GLLBSC is smaller than the Johnson and Christy bulk gold results in the entire wavelength range, but the Im(ε) of thin film calculated by LDA and PBE are larger than the Johnson and Christy results in the wavelength range between 500 nm and 750 nm. It means that the predicted Im(ε) of thin film by LDA and PBE functionals are overestimated. To emphasize the discrepancy of the optical permittivity for gold thin film, we have demonstrated the transmittance, reflection and loss of gold thin film and gold grating structure based on the DFT-based optical permittivity and Johnson and Christy permittivity. For the gold thin film, the transmission curve of GLLBSC is blue-shifted which has a good consistency with [20]. For the gold grating structure, the transmittance predicted from the DFT-based optical permittivity shows the resonance taking place at different wavelengths in comparison with the experimental data of optical permittivity. All the resonances based on DFT-based optical permittivity are red-shifted. For the electric field distribution, GLLBSC-based optical permittivity predicts that the grating has a strong localized energy and localized streamlines of Poynting vector in the corner edges at the resonance. However, the loss based on the GLLBSC-based optical permittivity is the smallest at the resonance due to the smallest Im(ε). The effects of exchange correlation functional on optical permittivity of gold have been studied, and the electromagnetic responses of gold nanostructures by DFT-based optical permittivities have been investigated. As the sizes of the nanophotonic structures get smaller, our study of using the DFT-based optical properties of the noble metals in electromagnetic simulations may offer a more reliable strategy to design and to investigate some additional functional devices and to conduct more application studies. Acknowledgments This work was supported by the National Science Council of Taiwan (NSC 101-2221-E-002120 and NSC 102-2221-E-002-006) and NTU Project (103R891404 and 103R7816). We are grateful to the National Center for High-Performance Computing, Taiwan, for providing us with the computation time and facilities. The authors also thank the anonymous reviewers for their useful comments and constructive suggestions.
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30734
Abstract: Understanding the optical properties of nanometer-scale noble metals is important for the nanoplasmonic devices. The bulk gold and thin film are calculated by density functional theory (DFT) with LDA, PBE, and GLLBSC functionals, respectively. The GLLBSC results for bulk gold are closer to the experimental data because the GLLBSC functional has better descriptions of transition energy. The Im(ε) of thin film calculated by LDA and PBE are overestimated. The effects of DFT-based optical properties are performed by conducting electromagnetic simulations. The transmission for the gold thin film by GLLBSC is blue-shifted. The gold grating structure with the GLLBSC-based optical permittivity has strong localized streamlines of Poynting vector in the corner edges at the resonance condition. ©2014 Optical Society of America OCIS codes: (250.5403) Plasmonics; (160.4760) Optical properties; (310.0310) Thin films.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30725
16. J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen, M. Dułak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H. A. Hansen, H. H. Kristoffersen, M. Kuisma, A. H. Larsen, L. Lehtovaara, M. Ljungberg, O. LopezAcevedo, P. G. Moses, J. Ojanen, T. Olsen, V. Petzold, N. A. Romero, J. Stausholm-Møller, M. Strange, G. A. Tritsaris, M. Vanin, M. Walter, B. Hammer, H. Häkkinen, G. K. Madsen, R. M. Nieminen, J. K. Nørskov, M. Puska, T. T. Rantala, J. Schiøtz, K. S. Thygesen, and K. W. Jacobsen, “Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method,” J. Phys. Condens. Matter 22(25), 253202 (2010). 17. J. Yan, J. J. Mortensen, K. W. Jacobsen, and K. S. Thygesen, “Linear density response function in the projector augmented wave method: Applications to solids, surfaces, and interfaces,” Phys. Rev. B 83(24), 245122 (2011). 18. J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77(18), 3865–3868 (1996). 19. M. M. Alvarez, J. T. Khoury, T. G. Schaaff, M. N. Shafigullin, I. Vezmar, and R. L. Whetten, “Optical Absorption Spectra of Nanocrystal Gold Molecules,” J. Phys. Chem. B 101(19), 3706–3712 (1997). 20. K. Leosson, A. S. Ingason, B. Agnarsson, A. Kossoy, S. Olafsson, and M. C. Gather, “Ultra-thin gold films on transparent polymers,” Nanophotonics 2(1), 3–11 (2013). 21. Lumerical Solutions, Inc., http://www.lumerical.com/tcad-products/fdtd/ 22. H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B Quantum Semiclassical Opt. 6(5), S404–S409 (2004). 23. Z. B. Wang, B. S. Luk’yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flows around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 032427 (2004).
1. Introduction The optical properties of noble metals and performances of their devices depend on their geometrical structures and material parameters. For most optical designs and applications, the optical properties of noble metals for bulk materials are taken from experimental results, such as the early measurements done by Johnson and Christy [1]. As the size scale of the devices is shrunk into few nanometers by state-of-art nanofabrication techniques, it is important to know the correct optical properties of the materials for designing new nanophotonic devices. With the success of the ab initio methods, it has been widely utilized to simulate many nanostructures such as graphene nanoribbons transistor [2], nanoparticle dimer [3], and nanoparticle chain [4]. However, analyzing the nanophotonic devices by using ab initio methods is not realistic because it needs to consume heavy computational cost. Therefore, using electromagnetic simulations with the optical permittivities calculated by density functional theory is one of the alternative and reliable strategies to investigate the nanophotonic devices [5, 6]. The time-dependent density functional theory (TDDFT) has been successfully applied to interpret the optical permittivity or dielectric function for various materials including the semiconductors and metals [7, 8]. The local density approximation (LDA) and the generalized gradient approximation (GGA) are widely applied to the exchange-correlation functionals. For the noble metal surface, a large screening emerges due to an incorrect calculation of the d band by LDA functional [9, 10]. Besides, the optical permittivity of bulk gold calculated by LDA or PBE functionals shows that the peak of the imaginary part of the dielectric constant at around 3 eV is different when comparing with the bulk gold experimental data from Johnson and Christy [5, 6, 8, 11]. Although the LDA and PBE functionals can capture most of the characteristic features of the optical permittivity, it is not sufficiently accurate in the visible light region which is important for nanoplasmonic applications. The many-body GW method can be used to describe d bands better in noble metals [12, 13]; however, it needs much more computational resource. This fact results in its less application usage in engineering practice. In this paper, the band structures and the optical permittivities of bulk gold are calculated by using three different exchange-correlation functionals, LDA, PBE, and GLLBSC. We found that the optical permittivity of bulk gold calculated by GLLBSC is closer to the experimental data of Johnson and Christy than the results calculated by LDA and PBE. The reason is due to a better description of the single-particle energies by GLLBSC compared to those of LDA and PBE [9, 10, 14]. The band structures and the optical permittivities of gold thin film by the three exchange-correlation functionals are also investigated. To show the effects of the calculated optical properties by these three functionals on plasmonic structures, the electromagnetic simulations for the gold thin film and gold grating atop the glass substrate are performed using the calculated DFT-based dielectric constants. Their transmissions and #223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30726
loss spectra for different exchange-correlation functionals are calculated and compared with the results calculated by the Johnson and Christy’s experimental dielectric function. The electric field intensities and Poynting vector for the gold grating are also simulated. 2. Methodology and simulation setup In this work, all first principles calculations were performed by the grid-based projector augmented wave (GPAW) method [15–17]. The Kohn-Sham wave functions and band energies were calculated for bulk gold and relaxed gold thin films by the LDA, PBE, and GLLBSC functionals. The LDA functional is derived from the homogeneous electron gas model. The PBE functional is a generalized gradient approximation reported by Perdew et al [18]. The GLLBSC functional, a revised Perdew-Burke-Ernzerhof functional for the solid and correlation, is an orbital-dependent functional based on the GLLB potential. It has successfully modified the unoccupied conduction band for semiconductor and insulator [14], and it can improve the description of the noble metal’s single-particle energies [9, 10]. The computational cost by using the GLLBSC potential is reasonable compared to the LDA and GGA functionals [9, 14]. In our calculation, the uniform mesh of the grid spacing is set at 0.02 nm for bulk gold and gold thin films. For the structure optimization of gold thin film, the atomic positions were relaxed by minimizing the forces based on the quasi-Newtonalgorithm. Also, the local density approximation (LDA) was elected to describe the exchange and correlation. For the k-point grid, a Monkhorst-Pack mesh of 8 × 8 × 1 was used for gold thin film in the (111) orientation. The gold thin films were modeled with 14 atomic layers (~3.63 nm) in a 1 × 1 unit cell. A supercell contains 2 nm vacuum layers in the directions perpendicular to the gold thin films. The wave functions and single particle energies for bulk gold and gold thin film were calculated by LDA, PBE, and GLLBSC. Then, they were used for the calculation of optical permittivities by the frequency-dependent density response function within the random phase approximation (RPA) framework [17]. The k-point grids corresponding to 40 × 40 × 40 (for bulk gold) and 40 × 40 × 1 (for gold thin film) were elected to construct a dense sampling of the Brillouin zone, which can describe the optical permittivity in the range between 400 nm and 1000 nm. A finite lifetime broadening of 0.02 eV was used. The dielectric function was calculated on a plane wave basis with the energy cut-off of 150 eV. 3. Bulk gold Figure 1 shows the real and imaginary parts of the optical permittivity calculated by the DFT method within different exchange correlation functionals and the experimental data reported by Johnson and Christy. The calculation is based on the linear response function of an interacting many-body system by solving Dyson’s equation from the ground state electronic structure [17]. To include all the transitions in our calculation of optical permittivity, a finite but small q, which is a wave vector restricted to the Brillouin Zone, is considered. The optical permittivities calculated by the GLLBSC functional are overall much closer to the experimental data than the calculations results by the LDA and PBE functionals. The minima of the Im(ε) calculated by LDA and PBE functionals occur at a longer wavelength compared to the experimental data. Because of the interband transition, the Im(ε) shall rise up at a higher energy. For noble metals, the minimum of the Im(ε) occurs when the interband transition has a greater contribution to Im(ε) than intraband transition. Thus, the minimum of Im(ε) occurring at a longer wavelength means the underestimation of the electron transition between the occupied d-bands and the unoccupied hybridized sp-bands above the Fermi level. According to [5, 8], the electronic transitions resulting from the d-bands to the p-bands close to the Fermi level is around the X and L points of Brillouin zones.
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30727
Fig. 1. The predicted dielectric constants for bulk gold using the LDA, PBE, and GLLBSC functionals. For comparison, the experimental data reported by Johnson and Christy (J&C) is also shown.
Figure 2 shows the band structures (left panel) and the corresponding density of states (right panel) from the Γ point to the X point. The energies of X point calculated by PBE (~1.2 eV) and LDA (~1.1 eV) functionals are smaller than that by the GLLBSC functional (~2.0 eV). Glantschnig and Ambrosch-Draxl reported that the orbitals at the X point in ~0 eV (Fermi level) were p-bands and most of the orbitals below 0 eV (Fermi level) were d-bands [8]. Figure 2 shows that some of the d-bands calculated by GLLBSC functional are higher than the d-bands calculated by LDA and PBE functionals. Compared to the band structures by GLLBSC functional, the band structures by LDA and PBE functional have the lower edge of the d-bands and have led to broadening d bands below Fermi level. It can be also found that the range of the occupied d-bands calculated by GLLBSC is narrower than the occupied dbands calculated by LDA and PBE in Fig. 2. At ~0 eV (Fermi level), the peak energy mainly consisting of p-bands by LDA and PBE functionals is smaller than the peak energy by GLLBSC functional. The interband transitions calculated by LDA and PBE functionals occur at the lower energy. Due to the underestimated p-bands energy at X points, indicating the electron transitions between the occupied d-bands and the unoccupied p-bands, the minimum (interband transition) calculated by LDA and PBE functionals occurs at a longer wavelength.
Fig. 2. Left panel: the predicted electronic band structures of bulk gold along the Γ-X direction for LDA, PBE, and GLLBSC; right panel: the corresponding density of states. The red dotted line at 0 eV indicates the Fermi-level.
To understand the interband contribution for the optical properties of bulk gold, we plot Fig. 3(a) to show the calculated Im(ε). It is noted that only the interband contribution is considered in Fig. 3(a). The experimental data by Johnson and Christy shown in Fig. 1 include the interband and intraband contributions. Alvarez et al. decomposed the dielectric #223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30728
functions of Johnson and Christy into the free-electron (Drude model) and the interband (5d to 6sp) parts [19]. Thus, for the comparison sake, the results of the calculated interband contribution of bulk gold by Alvarez et al. are also shown in Fig. 3(a). The curve of our calculated interband contribution by GLLBSC is close to the result reported by Alvarez et al. This means that the GLLBSC functional can have a better description of the interband contribution than those by the LDA and PBE functionals. Figure 3(b) shows the band structures of two possible paths of the interband transition. Also, the lowest energy transition is marked on the band structures in Fig. 3(b). The marked transition energy along Γ-X points of the Brillouin zone in Fig. 3(b) (left panel) calculated by LDA, PBE, and GLLBSC functionals are 1.663 eV, 1.683 eV, and 1.815 eV, respectively. The other path is along W-LΓ points of the Brillouin zone shown in Fig. 3(b) (right panel), and the marked transition energies calculated by LDA, PBE, and GLLBSC functional are 1.296 eV, 1.320 eV, and 1.489 eV, respectively. As the marked transition energy becomes smaller, the contribution to Im(ε) takes place from a smaller photon energy, i.e., a longer wavelength. These results are consistent with the plotted curves of Im(ε) for LDA, PBE, and GLLBSC functionals in Fig. 3(a), which shows that the curves for LDA and PBE are more red-shifted than GLLBSC and the results of Alvarez et al. The explanation is that the p-bands calculated by LDA and PBE functionals for the bulk gold are actually underestimated, thereby causing the discrepancy of the optical permittivity to occur compared to the experimental data.
Fig. 3. (a) The predicted imaginary parts of the dielectric constants of the bulk gold by LDA, PBE, and GLLBSC. Only the interband contributions are considered. For comparison, the interband contribution of bulk gold calculated by Alvarez et al. [15] is shown as the green line. (b) The calculated lowest energy transition of the bulk gold along the Γ-X (left panel) and W-LΓ (right panel) directions by LDA, PBE, and GLLBSC, where the green dotted lines at 0 eV indicate the Fermi-level.
4. Gold thin film: 14 atomic layers of Au(111) To understand the effects of the exchange correlation functionals on the predicted optical permittivities and band structures for gold thin films, the 14 atomic layers of gold (~3.63 nm) constitute the thin film structure which is considered in the (111) orientation in the GPAW simulations. Figure 4 shows the comparison of the results for the dielectric functions of the 14 atomic layers of gold and the bulk gold. The minimum of the Im(ε) calculated by LDA or PBE functional occurs in the case of a longer wavelength. Compared to the bulk gold calculations, the minima of the Im(ε) calculated by LDA or PBE functional are slightly redshifted and the minimum of the Im(ε) calculated by the GLLBSC is blue-shifted for the 14 atomic layers of gold. In comparison with the predicted Im(ε) of thin film and bulk gold, i.e. the dashed lines and solids lines in Fig. 4, it is found that the Im(ε) of thin film calculated by the GLLBSC is smaller than the Johnson and Christy bulk gold results in the entire wavelength range; however, the Im(ε) of thin film calculated by the LDA and PBE are larger than Johnson and Christy results in the wavelength range between 500 nm and 750 nm. It means that the predicted Im(ε) of thin film by LDA and PBE functionals are overestimated. It is also worthy to point out that the calculated Im(ε) for the thin film is smaller than Im(ε) for
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30729
the bulk gold in short wavelength range. The reason is that there are more electrons to transit from the occupied d bands to the unoccupied sp bands in shorter wavelength for the bulk gold. For Re(ε), it is a negative value but larger for the thin film compared to the bulk gold. It means that the optical property of the gold thin film is not quite like “metal” in view of electromagnetics compared to the bulk gold.
Fig. 4. Real parts (bottom panel) and imaginary parts (top panel) of the permittivity of Au(111) thin film (dotted line) predicted by the adopted DFT method in comparison with the data from bulk gold (solid line) and Johnson and Christy experimental data (green dotted line).
Figure 5 shows the band structures of the Au(111) thin film with 14 atomic layer thickness calculated by the LDA, PBE, and GLLBSC functionals. It should be noted that the band structures of the (111) surface of noble metals with 24 atomic layer thickness were already calculated by LDA and GLLBSC in [10]. In Fig. 5, the top of the Au d-bands calculated by PBE, LDA, and GLLBSC functionals are −0.930, −1.305 eV, and −2.087 eV, respectively. The difference at the top of the Au d-bands by LDA and GLLBSC is about 0.782 eV, which is similar to the results reported by Jane et al. [9, 10]. Before discussing in greater detail of the results in Fig. 5, it is worthy to read the band structures for bulk gold in Fig. 2 and the band structures for Au(111) thin film in Fig. 5. Figure 2 seems to indicate that most of the d-bands are pushed up by GLLBSC compared to LDA or PBE, while Fig. 5 shows the opposite trend, i.e. downshift of d-bands. However, it should be noticed that the results in Fig. 2 are Γ-X for bulk gold in face centered cubic Brillouin zone and the results in Fig. 5 are Γ-K-M-Γ for the Au(111) thin film in surface Brillouin zone. It is observed that the range of the occupied d-bands calculated by GLLBSC is narrower than the occupied d-bands calculated by LDA and PBE for Au(111) thin film in Fig. 5, which shows good agreement with [10]. This similar phenomenon can be also found for bulk gold in Fig. 2. Also, the upper d-bands calculated by GLLBSC are lower than LDA and PBE for Au(111) thin film in Fig. 5. It can be explained that the screening is overestimated compared to the results by GLLBSC functional in Fig. 5(a). The similar phenomena can be found in the predicted results of PBE and GLLBSC functionals shown in Fig. 5(b). For the GLLBSC functional, the energy of dbands close to Fermi level is lower than LDA and PBE functionals. The transition energy between the occupied d-bands and the unoccupied hybridized sp bands above the Fermi level predicted by GLLBSC is larger than LDA and PBE functionals. In accordance with the band structure of the bulk gold, the electronic transition from the occupied d band to the unoccupied p band around the X point and the L point of the Brillouin zone is related to the minimum value of the Im(ε). For the gold thin film, the relatively narrow energy gap between the d-bands and Fermi level taking place at the M points, which is projected from the X points and the L points of the Brillouin zone of face center cubic, causes the occurrence of minima of the Im(ε) at a longer wavelength in Fig. 4. For the case of a narrower energy gap between the d-bands and Fermi level, the lower energy can trigger the interband transition.
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30730
Fig. 5. Comparison of the predicted band structures of the Au(111) thin film by different functionals, where (a) shows LDA (gray) and GLLBSC (black) and (b) shows PBE (gray) and GLLBSC (black). The Fermi energy is set to be zero as shown in the red dashed lines. The red plus sign markers at the M points of the surface Brillouin zone indicate that top d-bands are −1.305 eV, −2.087 eV, and −0.930 eV for LDA, GLLBSC, and PBE, respectively.
5. Applications of gold thin film Gold-coated glass plate is one of the simple plasmonic devices. According to the recent research report [20], the simulated results of the gold thin films deposited on the glass based on experimental bulk gold dielectric functions cannot predict the experimental results very well as the thickness of gold thin film is only few nanometers. In this paper, two plasmonic structures are studied to show the effects of gold thin film thickness on plasmonic resonances. One structure is the gold thin film which has the thickness of h = 3.63 nm (14 atomic layers) atop the silicon dioxide substrate as shown in Fig. 6(a), and the other structure is the gold grating atop the silicon dioxide substrate in Fig. 6(b) with the thickness h = 3.63 nm, the period p = 50 nm, and the gap d = 10 nm. For simplicity in all calculations, the silicon dioxide substrate with infinitely large thickness is considered. The incident plane wave is x-polarized and propagates from top to down in + z direction. The Lumerical FDTD solutions package [21] is used to calculate the transmittance, reflection, and loss. The optical permittivities used in electromagnetic simulations are the predicted data of gold thin film in Fig. 4 by density functional theory with LDA, PBE and GLLBSC functionals, respectively. For the sake of comparison, we also show the simulation results for the structures using the experimental data from Johnson and Christy.
Fig. 6. Cross section view of (a) gold thin film and (b) gold grating, where the thickness of the thin film is h, the width of the gap is d, and the period is p.
For the gold thin film, Fig. 7 shows the high transmittance into the silicon dioxide substrate for LDA, PBE, GLLBSC and experimental data from Johnson and Christy. Compared to the transmittance adopted by Johnson and Christy, the calculated transmittance curves are red-shifted for the LDA and PBE functionals but blue-shifted for the GLLBSC functional. Based on [20], the results of the GLLBSC functional is closer to the experimental
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30731
transmission of gold thin film. Because the loss predicted by LDA and PBE functionals occurs at the longer wavelengths in Fig. 7, the transmittance curves are red-shifted. Also, more energies are reflected in longer wavelength. Generally speaking, the loss in Fig. 7 is proportional to the Im(ε) in Fig. 4.
Fig. 7. Loss (dashed line), reflection (solid line), and transmittance (cross line) for the gold thin films based on the GLLBSC (blue), PBE (red), LDA (black), and J&C (green) permittivities.
Figure 8 shows the transmittance, reflection, and loss for the gold grating structures. The simulation results by adopting the Johnson and Christy dielectric functions for the gold grating show that the maximum loss is at the wavelength 826.2 nm, i.e. the minimum of transmittance which is also close to the maximum of loss. The maximum losses are 882.8 nm, 959.2 nm and 968.5 nm for adopting the DFT-based optical permittivities by GLLBSC, PBE, and LDA, respectively. Because the Im(ε) is larger at the resonance for LDA and PBE functionals compared to GLLBSC in Fig. 4, the losses for the structures by LDA and PBE are slightly larger and the transmissions are slightly smaller. Besides, the maximum reflection calculated by GLLBSC functional is larger than the results predicted by LDA and PBE functionals. There are more energies which are reflected due to a smaller loss at the resonance for the results calculated by the GLLBSC functional.
Fig. 8. Loss (dashed line), reflection (solid line), and transmittance (cross line) for the gold grating based on the GLLBSC (blue), PBE (red), LDA (black), and J&C (green) permittivities.
Figure 9 shows the electric field distributions (in logarithmic scale) and Poynting vector streamlines for the gold grating structures at the respective maximum loss for the four cases in Fig. 8. The strong localized electric field exists at the corner edges with the dense streamlines of Poynting vector. It is interesting to see that the experimental data (J&C) and GLLBSC functional show fewer streamlines of Poynting vector through the grating holes in Figs. 9(a) and 9(b) compared to LDA and PBE functionals in Figs. 9(c) and 9(d). The GLLBSC results in Fig. 9(b) show that the dense streamlines of Poynting vector at the corner
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30732
edges have a great influence on the Poynting vector through the holes. The streamlines of Poynting vector are not straight lines around the center of the grating holes for the GLLBSC results compared to J&C’s results. Furthermore, the phenomenon of the dense streamlines of Poynting vector at the corner edge are related to the topological singularity [22, 23], and the DFT-based optical permittivity within different exchange-correlation functionals embody quite different results of the Poynting vector. For the LDA and PBE results, more Poynting vector lines straightly pass through the grating holes in Figs. 9(c) and 9(d). Campbell et al. reported that the Poynting vector streamlines with PBE-based optical permittivity dissipate as well at the corner edges [6]. Therefore, it can explain why the transmittances at the resonances calculated by LDA and PBE functionals are higher than the J&C’s and GLLBSC’s results in Fig. 8. We can also find that LDA and PBE functionals have greater loss of energies at the resonance in Fig. 8. However, the electric field magnitude distributions and streamlines of Poynting vector shown in Fig. 9 reveal that there are less localized energies occurring at the corner edge at the resonance. It is because that the Im(ε) of gold thin film predicted by LDA and PBE functionals are larger than GLLBSC functional and Johnson and Christy experimental data at the resonance. This can also explain that the streamlines of Poynting vector calculated by GLLBSC functional shown in Fig. 9 are densely distributed near the grating corner but the loss calculated by GLLBSC functional in Fig. 8 is much smaller at the resonance. The prediction of gold grating by GLLBSC is that the edge of the corner has the tendency of localizing power and causes less power passing through the hole between each gold strip.
Fig. 9. Electric filed magnitude distributions (in logarithmic scale) and streamlines of Poynting vector for the gold grating structures at the wavelengths of the maximal losses: 826.2 nm, 882.8nm, 968.5nm and 959.2 nm, respectively, for the (a) J&C; (b) GLLBSC; (c) LDA; (d) PBE optical permittivities. The white solid line is the boundary between the Au(111) thin film, silicon dioxide substrate and the vacuum.
6. Conclusion Having reduced the degree of underestimation of the transition energy, the predicted optical permittivities of bulk gold calculated by GLLBSC functional is much closer to experimental data in comparison with the LDA and PBE functionals from the visible to the near infrared region. For the gold thin film of 14 atomic layers, the energy gap between top d band and
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30733
Fermi level calculated by GLLBSC is larger than those by the LDA and PBE. The Im(ε) calculated by LDA or PBE functionals is slightly red-shifted and Im(ε) calculated by GLLBSC is blue-shifted. The Im(ε) of thin film calculated by the GLLBSC is smaller than the Johnson and Christy bulk gold results in the entire wavelength range, but the Im(ε) of thin film calculated by LDA and PBE are larger than the Johnson and Christy results in the wavelength range between 500 nm and 750 nm. It means that the predicted Im(ε) of thin film by LDA and PBE functionals are overestimated. To emphasize the discrepancy of the optical permittivity for gold thin film, we have demonstrated the transmittance, reflection and loss of gold thin film and gold grating structure based on the DFT-based optical permittivity and Johnson and Christy permittivity. For the gold thin film, the transmission curve of GLLBSC is blue-shifted which has a good consistency with [20]. For the gold grating structure, the transmittance predicted from the DFT-based optical permittivity shows the resonance taking place at different wavelengths in comparison with the experimental data of optical permittivity. All the resonances based on DFT-based optical permittivity are red-shifted. For the electric field distribution, GLLBSC-based optical permittivity predicts that the grating has a strong localized energy and localized streamlines of Poynting vector in the corner edges at the resonance. However, the loss based on the GLLBSC-based optical permittivity is the smallest at the resonance due to the smallest Im(ε). The effects of exchange correlation functional on optical permittivity of gold have been studied, and the electromagnetic responses of gold nanostructures by DFT-based optical permittivities have been investigated. As the sizes of the nanophotonic structures get smaller, our study of using the DFT-based optical properties of the noble metals in electromagnetic simulations may offer a more reliable strategy to design and to investigate some additional functional devices and to conduct more application studies. Acknowledgments This work was supported by the National Science Council of Taiwan (NSC 101-2221-E-002120 and NSC 102-2221-E-002-006) and NTU Project (103R891404 and 103R7816). We are grateful to the National Center for High-Performance Computing, Taiwan, for providing us with the computation time and facilities. The authors also thank the anonymous reviewers for their useful comments and constructive suggestions.
#223929 - $15.00 USD (C) 2014 OSA
Received 29 Sep 2014; revised 14 Nov 2014; accepted 24 Nov 2014; published 3 Dec 2014 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.030725 | OPTICS EXPRESS 30734
