Abstract

Most of hyperbolic metamaterials (HMMs) investigated to date are based on isotropic materials resulting in uniaxial HMMs in which dielectric permittivities perpendicular to the propagation direction are the same. Using an anisotropic material constituent to form a HMM is a promising research direction providing opportunities to control the dielectric permittivity in all three directions independently. Herein, we propose and theoretically demonstrate novel biaxial HMMs composed of multilayer stacks of few-layer black phosphorus (BP) and Au thin films. Black phosphorus is an anisotropic material exhibiting crystal axis-dependent dielectric permittivity due to its puckered crystal structure. The proposed HMM provides previously unattained hyperbolic dispersion relations in which the dielectric permittivity in Z-direction of the structure shows opposite sign from that in X- and Y-directions in the most wavelengths from 400~900nm. Furthermore, we calculated the Purcell factor of the proposed biaxial HMMs using full-field electromagnetic simulations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Metamaterials have drawn widespread attention in many aspects because of their astonishing freedom in controlling electromagnetic properties by arranging the components manually in either ordered or disordered way [1–5]. Among all kinds of metamaterials that have been designed and fabricated since its first announcement in 1960s, hyperbolic metamaterials (HMMs) [6–8] are distinguished due to their highly anisotropic features enabling hyperbolic iso-frequency dispersion curve and redefine the directional nature of light propagation [9,10].

Hyperbolic metamaterials (HMMs) [11] usually have negative permittivity in at least one direction. As a matter of fact, hyperbolic metamaterials offer great flexibility in tailoring the near-field distribution [12], which facilities prospective application in spontaneous emission enhancement [13,14], sub-wavelength image [15,16], super lenses [17], bio-sensing [18], hot-electric devices [19], broadband acoustic [20], broadband absorption [21], and super-planckian thermal emission [6].

One of the most promising applications of hyperbolic metamaterials is engineering the spontaneous emission, which plays a crucial role in single-photonics [22,23], light-emission devices [24,25], plasmonic lasers [26], etc. Besides, Purcell factor is an efficient parameter to characterize the spontaneous emission properties, which describes the modification of the spontaneous emission lifetime [27,28]. Specifically, Purcell factor could be estimated by the ratio of radiative rate in a particular electromagnetic structure to that in vacuum, proportional to the imaginary part of Green’s function in a medium [7]. Although the Purcell factor of hyperbolic metamaterials stays finite due to the discreteness of actual structures, it is reported to be sensitive to structure geometry, layer thickness and dipole orientation [22,29].

2. Theory

As originated from optical crystals, the effective permittivity tensors of hyperbolic material (ε) can be described in the following form [30]:

ε=(εxx000εyy000εzz)

In general, for isotropic materials the permittivity tensors in all directions are equal, which means

εxx=εyy=εzz 

However anisotropic materials have at least one of their permittivity tensor different. Conventionally, HMMs are engineered anisotropic materials that exhibit uniaxial hyperbolicity, in which

εxx=εyyεzz
corresponding to a hyperboloid iso-frequency surface [7]. While, for the circumstance of
εxxεyyεzz
and at least one of the tensors is negative, it’s considered to be an biaxial HMM with an asymmetric hyperboloid, in which four degenerate points could be observed [31]. In contrast, abnormal surface wave with elliptically polarized state exists at the interface of biaxial HMM, which leads to more freedom for promising application in quantum nanophotonics [32,33]. Such a material could be realized by alternative dielectric-metal multilayers, or by placing metal wires in a dielectric matrix [34]. Here, we propose and demonstrate a biaxial HMM based on few-layer black phosphorus and gold multilayer thin film stacks.

Black phosphorus is an exciting material system exhibiting 2D hyperbolic behavior at IR wavelengths [35]. As one of the emerging 2D materials with typically anisotropic electronic and optical properties, BP has drawn attention to be candidate in potential fields such as photovoltaic devices and field effect transitions [36,37]. Few-layer BP, as a layered material, is a semiconductor with a direct band gap(~0.3eV), with lattice distance az = 10.7Å in the out of plane direction [38]. Actually, the effective distance between layers is about half of the lattice [39]. While the band structure of few-layer BP has a direct energy gap at 𝛤 point other than at the Z point for bulk BP. In particular, few-layer BP would show anisotropic properties along X-, Y-, Z-directions [40], while metals such as gold would show negative permittivity over all the optical range. Thus, few-layer BP and gold multilayer thin films are potential candidates to form a biaxial HMM.

In this paper, we report the dipole emission of biaxial HMMs, which is constructed by alternating few-layer black phosphors (BP) and gold (Au) multilayers, as shown in Fig. 1(a), as the side view of few-layer BP in armchair and zigzag direction is illustrated in Fig. 1(b) and 1(c).

 

Fig. 1 (a) Schematics of layered few-layer BP-Au multilayer structure; Illustration the XZ plane vision of few-layer BP (b), which corresponding to the arm-chair property in X-axe and the YZ plane vision of few-layer BP (c), which corresponding to the zigzag property in Y-axe;

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3. Results and discussion

A biaxial HMM has been demonstrated, simulated, and analyzed in this paper. In the numerical simulation, we adopted a commercial software from Lumerical, in which a finite difference time domain (FDTD) method had been applied to solve the Maxwell equations of the whole structure. The dielectric function of few-layer BP is extracted from [41].

In our simulation, permittivity tensors in X, Y, Z directions separately and the thickness of few-layer BP can be expressed as 𝜀BPx, 𝜀BPy, 𝜀BPz, dBP (dBP = n*az/2, n is the layer number of BP), and the permittivity and thickness of Au could be given by εAu  and dAu. For the structure formulated by alternating few-layer BP and Au, when the thickness of each layer is much thinner than the incidence wavelength, it is an efficient methodology by adopting effective permittivity method to treat the whole system as a single anisotropic medium. Hence, the components of the effective permittivity tensors along the X, Y, Z-direction(𝜀xx, 𝜀yy, 𝜀zz) in an orthogonal system could be given by [7,42]:

εxx=εAudAu+εBPxεBPxdAu+dBP=εAudAudBP+εBPxdAudBP+1=εAup+εBPxp+1
εyy=εAudAu+εBPxεBPydAu+dBP=εAudAudBP+εBPydAudBP+1=εAup+εBPyp+1
εzz=dAu+dBPzdAuεAu+dBPεBPz=dAudBP+1dAudBP1εAu+1εBPz=p+1pεAu+1εBPz
where here p (p=dAudBP) is the thickness ratio of Au and BP layers. Obviously, the permittivity tensors of the whole structure are related to their original permittivity properties and thickness ratio (p). It should be noted here that the calculated permittivity parallel to every direction of an orthogonal system is only valid based on the premise of long wavelength limit.

As shown from the calculated results (Fig. 2(a)), a transition of the effective permittivity could be observed in 400~900 nm when the thickness of few-layer BP and gold equals to each other (p=1). In most of the discussed wavelength range, the relationship εzzεxx<0 or εzzεxx<0 would exist, which indicates hyperbolic characteristics. In details, the permittivity of the few-layer BP-Au multilayer structured could be divided into five different regions owing to separate relationship with each other and zero in the orthogonal system, as displayed in Fig. 2(b). In addition, the total absorption of structures formulated by BP-Au layers can’t be neglected in optical range, as the imaginary parts of permittivity suggest.

 

Fig. 2 (a) The effective permittivity in the three directions of the orthogonal system for the alternative BP-Au multilayer structure, as solid lines and dash lines represent the real part and imaginary part of permittivity respectively. (b) The magnification the permittivity near the zero point. It should be noted here that the span of εzz keeps from −250~250 for a better vision. Actually, the three curves would meet, not exactly, but almost at the same point (~623.3 nm) when magnifies that specific region.

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In region I from 400~623 nm, 𝜀zz<0<𝜀xx<𝜀yy. Actually, the isotropic surface of biaxial HMM would have for conical intersection points, which means the hyperboloid and the ellipsoid would intersect at four degenerate points [31]. Actually, 𝜀zz increases sharply crossing the 𝜀zz = 0 point (called ENZ region), and it meets with 𝜀xx and 𝜀yy at 623.3 nm, which means from 623~623.3 nm,0<𝜀zz<𝜀xx<𝜀yy, regarded as region II. After that, the real part of εzz keeps increasing until the highest point, then it would decrease. While 𝜀xx and 𝜀yy decreasing as before, only 𝜀xx is smaller than 𝜀xx from now on. Region III would come about until 𝜀yy meets zero at 648.9 nm, thus from 623.3 nm to 648.9 nm, the Eq. (0)<𝜀yy<𝜀xx<𝜀zz is valid. Then permittivity keeps decreasing, 𝜀xx would come to zero point at 653.1nm as well, thus 𝜀yy<0<𝜀xx<𝜀zz is valid in region IV exactly from 648.9~653.1 nm. In the end, during region V from 653.1 nm to 900 nm, both 𝜀xx and 𝜀yy are smaller than zero, while the value εzz remains above zero, ie, 𝜀yy<𝜀xx<0<𝜀zz. Even though the value of 𝜀xx is similar with 𝜀yy as the permittivity if armchair and zigzag direction is close to each other in long wavelength, but they are not equal exactly. On the other hand, the imaginary part of permittivity is always higher than zero, with the imaginary part of 𝜀xx and 𝜀yy descend when the wavelength increases. Only the imaginary part of 𝜀zz exhibits a sharp peak at ~623.3 nm.

According to the equations above, the permittivity of the alternating few-layer BP and Au could be adjusted by their ratio thickness (p). As shown in Fig. 3(a), the real part of 𝜀xx and 𝜀yy decrease when more Au exists in the structure. While the changing tendency of imaginary part would be opposite for 𝜀xx and 𝜀yy, as the imaginary part of 𝜀xx decrease but 𝜀yy increase when there is more gold compared to BP. On the other hand, both the transition points of real and imaginary part of εzz shift to longer wavelengths with more gold, as depicted in Fig. 3(b), (c), (d). Besides, the highest value of real and imaginary part of εzz increase.

 

Fig. 3 (a) Dependence of 𝜀xx and 𝜀yy on thickness ratio p (p = dAu/dBP), while the black and blue lines correspond to the permittivity in X and Y-direction respective. (b) Dependence of εzz on thickness ratio p from 0.8 to 1.4 over wavelength (400~900 nm). The solid and dash lines correspond to real and imaginary parts of the permittivity respectively. The permittivity map of real (c) and imaginary (d) part of 𝜀zz with respect to thickness ratio p sweep.

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The Purcell factor of an 8-layer hyperbolic metamaterial composing by four BP-Au units is further investigated, which could be extracted from the ratio of power radiated in surrounded environment and source power radiating in a homogeneous medium [28]. As illustrated in Fig. 4(a), the 8-layer structure formulates a peak at a same wavelength (~669 nm) for dipoles polarized along X-, Y-, Z-direction respectively, although the Purcell factor is sensitive to the orientation of electric dipole. Obviously, the peak of the Purcell factor locates at region V, during which the component of effective permittivity in Z-direction is much higher than that in other two polarizations, originating from resonance hybridization associated to BP-Au interfaces. Obviously, Purcell factor originated from Z-direction polarized dipole is much higher than the others. The oscillation of Purcell factor for dipoles polarized along Z-direction tends to be more prominent than other two directions due to the remarkable field distribution diversion along the stacked direction. In spite of the fact that variation in unit number of HMMs would not affect the effective permittivity according to the effective medium theory (EMT), the scattering parameters would make a difference when the total thickness of the structure changes. As a result, the spatial dispersion would lead to a deviation in the Purcell factor. When the periodic unit of BP-Au stacked in perpendicular direction decreases from 4 to 2, as shown in Fig. 4(b), the peak position of Purcell factor would move slightly to longer wavelengths, and the peak intensity would become more profound as well, which could be attributed to thinner thickness of the whole structure.

 

Fig. 4 (a) Purcell factor for 8-layer structure constructed by alternating few-layer BP (5nm) and Au (5nm) HMM, Fpx, Fpy, Fpz, corresponding to the Purcell factor activated by the electrical dipole at 10nm above HMM with orientation along X-, Y-, Z-direction respectively and Fpiso corresponding to isotropic Purcell factor for the whole multilayer structure. (b) Comparison of isotropic Purcell factor over frequency for the HMMs composed by 2, 3, 4 units’ structures along vertical direction. 1 unit composes by one-layer BP and one-layer Au. Thickness ratio p=1.2, dBP=5nm is adopted in this model.

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The Purcell factor could be analyzed in transverse electric (TE) and transverse magnetic (TM) polarizations through electrical and magnetic dipoles, but only the electrical dipole is discussed here. An isotropic Purcell factor (Fpiso) [24]could be calculated through an average over all Purcell factors under the electrical dipoles polarized along X-, Y-, and Z- directions, as shown in Fig. 1(a), characterized by Fpx, Fpy, and Fpz respectively.

Fpiso=Fpx+Fpy+Fpz3

Apparently, Purcell factor relates to the distance between dipole and hyperbolic metamaterial, here we fix the dipole 10 nm above the structure [24] which is much smaller than the incident wavelength so that the whole structure could be treated as effective medium.

Furthermore, the thickness ratio and specific layer’s thickness would significantly affect the Purcell factor. As illustrated in Fig. 5(a), the thickness ratio of few-layer BP and Au influences both the peak positions and maximum values of Purcell factor. For the 8-layer BP-Au structure, the Purcell factor at p=1.2 is higher than others, and the peak generally moves to shorter wavelengths while decreasing the BP percentage. It should be noted here that the impact of thickness ratio (p) on peak position is rather different from that on effective permittivity, which would shift to a reverse direction. Besides, if the thickness ratio keeps constant, varying the thickness of few-layer BP could definitely lead to variation in Purcell factor. Actually, when changing BP thickness from 3 nm to 6 nm, as shown in Fig. 5(b), the wavelength corresponding to highest Purcell factor would not have a distinguish vibration, concentrating around 645 nm. Only the peak intensity decreases when BP thickness increases, due to reduced coupling between successive layers. Besides, the oscillation of Purcell factor could be suppressed when increasing the thickness of each layer, which weakened originating from the coupling between thin layers. Overall, Purcell enhancement in BP-Au structure is comparable to that in general metal-dielectric multilayer structures [28,30], but the biaxial HMM here shows more flexible tunability owning to anisotropically intrinsic property of BP.

 

Fig. 5 (a) Dependence of isotropic Purcell Factor (Fpiso) over wavelength (400~900nm) for varying thickness ratio p (p=dAu/dBP) from 0.8 to 1.4 while the thickness of BP remains unchanged (5nm). (b) Dependence of isotropic Purcell Factor (Fpiso) over wavelength (400~900nm) for varying BP thickness (dBP) from 3 nm to 6 nm while the thickness ratio remains unchanged (p=1.2).

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4. Conclusions

In summary, a biaxial hyperbolic metamaterial is proposed and discussed in present work, which is composed of alternating layers of few-layer BP and gold. In the majority range from 400~900 nm, permittivity in Z-direction has opposite sign with that in X- or Y-direction. Besides, it has been shown that the increase of thickness ratio of gold and few-layer BP increase would lead to a shift of longer wavelength of transition point of εzz, while the Purcell factor would move to an opposite wavelength. Although, making this kind of structure into practice is still challenging at present, we believe that experimental feasibility is expected in the near future once breakthrough had been made in the fabrication of high quality BP. We hope the current work would provide inspiration and guidance for future research and improvement of biaxial hyperbolic metamaterials.

Funding

Office of Naval Research Young Investigator Program (ONR-YIP) Award (N-00014-17-1-2425); National Natural Science Foundation of China (NSFC) (61505111, 11604216); Science and Technology Planning Project of Guangdong Province (2016B050501005); Educational Commission of Guangdong Province (2016KCXTD006).

Acknowledgments

K.A. acknowledges support from the Office of Naval Research Young Investigator Program (ONR-YIP) Award (N00014-17-1-2425). The program manager is Brian Bennett.

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References

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  1. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [Crossref] [PubMed]
  2. C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
    [PubMed]
  3. C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley-IEEE Press, 2006).
  4. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
    [Crossref] [PubMed]
  5. N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
    [Crossref] [PubMed]
  6. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 1–9 (2012).
    [Crossref]
  7. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
    [Crossref]
  8. T. Galfsky, J. Gu, E. E. Narimanov, and V. M. Menon, “Photonic hypercrystals for control of light-matter interactions,” Proc. Natl. Acad. Sci. U.S.A. 114(20), 5125–5129 (2017).
    [Crossref] [PubMed]
  9. J. S. Gomez-Diaz and A. Alu, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
    [Crossref]
  10. S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7, 11329 (2016).
    [Crossref] [PubMed]
  11. A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
    [Crossref] [PubMed]
  12. K. Shi, F. Bao, and S. He, “Enhanced near-field thermal radiation based on multilayer graphene-hBN heterostructures,” ACS Photonics 4(4), 971–978 (2017).
    [Crossref]
  13. D. J. Roth, A. V. Krasavin, A. Wade, W. Dickson, A. Murphy, S. Kéna-Cohen, R. Pollard, G. A. Wurtz, D. Richards, S. A. Maier, and A. V. Zayats, “Spontaneous emission inside a hyperbolic metamaterial waveguide,” ACS Photonics 4(10), 2513–2521 (2017).
    [Crossref]
  14. M. Cuevas, “Surface plasmon enhancement of spontaneous emission in graphene waveguides,” J. Opt. 18(10), 105003 (2016).
    [Crossref]
  15. M. Kim, S. So, K. Yao, Y. Liu, and J. Rho, “Deep sub-wavelength nanofocusing of UV-visible light by hyperbolic metamaterials,” Sci. Rep. 6(1), 38645 (2016).
    [Crossref] [PubMed]
  16. P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
    [Crossref] [PubMed]
  17. J. Sun, M. I. Shalaev, and N. M. Litchinitser, “Experimental demonstration of a non-resonant hyperlens in the visible spectral range,” Nat. Commun. 6(1), 7201 (2015).
    [Crossref] [PubMed]
  18. K. V. Sreekanth, M. ElKabbash, Y. Alapan, E. I. Ilker, M. Hinczewski, U. A. Gurkan, and G. Strangi, “Hyperbolic metamaterials-based plasmonic biosensor for fluid biopsy with single molecule sensitivity,” EPJ Appl. Metamat. 4, 1 (2017).
    [Crossref]
  19. M. Sakhdari, M. Hajizadegan, M. Farhat, and P. Y. Chen, “Efficient, broadband and wide-angle hot-electron transduction using metal-semiconductor hyperbolic metamaterials,” Nano Energy 26, 371–381 (2016).
    [Crossref]
  20. C. Shen, Y. Xie, N. Sui, W. Wang, S. A. Cummer, and Y. Jing, “Broadband acoustic hyperbolic metamaterial,” Phys. Rev. Lett. 115(25), 254301 (2015).
    [Crossref] [PubMed]
  21. K. V. Sreekanth, M. ElKabbash, Y. Alapan, A. R. Rashed, U. A. Gurkan, and G. Strangi, “A multiband perfect absorber based on hyperbolic metamaterials,” Sci. Rep. 6(1), 26272 (2016).
    [Crossref] [PubMed]
  22. S. Axelrod, M. K. Dezfouli, H. M. K. Wong, A. S. Helmy, and S. Hughes, “Hyperbolic metamaterial nanoresonators make poor single-photon sources,” Phys. Rev. B 95(15), 155424 (2017).
    [Crossref]
  23. M. Y. Shalaginov, V. V. Vorobyov, J. Liu, M. Ferrera, A. V. Akimov, A. Lagutchev, A. N. Smolyaninov, V. V. Klimov, J. Irudayaraj, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Enhancement of single-photon emission from nitrogen-vacancy centers with TiN/ (Al, Sc) N hyperbolic metamaterial,” Laser Photonics Rev. 9(1), 120–127 (2015).
    [Crossref]
  24. D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
    [Crossref] [PubMed]
  25. J. S. T. Smalley, F. Vallini, S. A. Montoya, L. Ferrari, S. Shahin, C. T. Riley, B. Kanté, E. E. Fullerton, Z. Liu, and Y. Fainman, “Luminescent hyperbolic metasurfaces,” Nat. Commun. 8, 13793 (2017).
    [Crossref] [PubMed]
  26. T. Galfsky, Z. Sun, C. R. Considine, C. T. Chou, W. C. Ko, Y. H. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16(8), 4940–4945 (2016).
    [Crossref] [PubMed]
  27. M. N. Gjerding, R. Petersen, T. G. Pedersen, N. A. Mortensen, and K. S. Thygesen, “Layered van der Waals crystals with hyperbolic light dispersion,” Nat. Commun. 8(1), 320 (2017).
    [Crossref] [PubMed]
  28. A. P. Slobozhanyuk, P. Ginzburg, D. A. Powell, I. Iorsh, A. S. Shalin, P. Segovia, A. V. Krasavin, G. A. Wurtz, V. A. Podolskiy, P. A. Belov, and A. V. Zayats, “Purcell effect in hyperbolic metamaterial resonators,” Phys. Rev. B 92(19), 195217 (2015).
    [Crossref]
  29. L. Li, W. Wang, T. S. Luk, X. Yang, and J. Gao, “Enhanced quantum dot spontaneous emission with multilayer metamaterial nanostructures,” ACS Photonics 4(3), 501–508 (2017).
    [Crossref]
  30. L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2014).
    [Crossref]
  31. K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “Conical diffraction and the dispersion surface of hyperbolic metamaterials,” Phys. Rev. A 90(1), 013803 (2014).
    [Crossref]
  32. J. Sun, J. Zeng, and N. M. Litchinitser, “Twisting light with hyperbolic metamaterials,” Opt. Express 21(12), 14975–14981 (2013).
    [Crossref] [PubMed]
  33. W. Gao, F. Fang, Y. Liu, and S. Zhang, “Chiral surface waves supported by biaxial hyperbolic metamaterials,” Light Sci. Appl. 4(9), e328 (2015).
    [Crossref]
  34. J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
    [Crossref]
  35. A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
    [Crossref] [PubMed]
  36. S. Yu, X. Wu, Y. Wang, X. Guo, and L. Tong, “2D materials for optical modulation: challenges and opportunities,” Adv. Mater. 29(14), 1606128 (2017).
    [Crossref] [PubMed]
  37. Z. Liu and K. Aydin, “Localized surface plasmons in nanostructured monolayer black phosphorus,” Nano Lett. 16(6), 3457–3462 (2016).
    [Crossref] [PubMed]
  38. S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
    [Crossref] [PubMed]
  39. T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
    [Crossref] [PubMed]
  40. T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
    [Crossref]
  41. A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Solids Surf. 39(4), 227–242 (1986).
    [Crossref]
  42. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
    [Crossref]

2017 (9)

T. Galfsky, J. Gu, E. E. Narimanov, and V. M. Menon, “Photonic hypercrystals for control of light-matter interactions,” Proc. Natl. Acad. Sci. U.S.A. 114(20), 5125–5129 (2017).
[Crossref] [PubMed]

K. Shi, F. Bao, and S. He, “Enhanced near-field thermal radiation based on multilayer graphene-hBN heterostructures,” ACS Photonics 4(4), 971–978 (2017).
[Crossref]

D. J. Roth, A. V. Krasavin, A. Wade, W. Dickson, A. Murphy, S. Kéna-Cohen, R. Pollard, G. A. Wurtz, D. Richards, S. A. Maier, and A. V. Zayats, “Spontaneous emission inside a hyperbolic metamaterial waveguide,” ACS Photonics 4(10), 2513–2521 (2017).
[Crossref]

K. V. Sreekanth, M. ElKabbash, Y. Alapan, E. I. Ilker, M. Hinczewski, U. A. Gurkan, and G. Strangi, “Hyperbolic metamaterials-based plasmonic biosensor for fluid biopsy with single molecule sensitivity,” EPJ Appl. Metamat. 4, 1 (2017).
[Crossref]

S. Axelrod, M. K. Dezfouli, H. M. K. Wong, A. S. Helmy, and S. Hughes, “Hyperbolic metamaterial nanoresonators make poor single-photon sources,” Phys. Rev. B 95(15), 155424 (2017).
[Crossref]

J. S. T. Smalley, F. Vallini, S. A. Montoya, L. Ferrari, S. Shahin, C. T. Riley, B. Kanté, E. E. Fullerton, Z. Liu, and Y. Fainman, “Luminescent hyperbolic metasurfaces,” Nat. Commun. 8, 13793 (2017).
[Crossref] [PubMed]

L. Li, W. Wang, T. S. Luk, X. Yang, and J. Gao, “Enhanced quantum dot spontaneous emission with multilayer metamaterial nanostructures,” ACS Photonics 4(3), 501–508 (2017).
[Crossref]

M. N. Gjerding, R. Petersen, T. G. Pedersen, N. A. Mortensen, and K. S. Thygesen, “Layered van der Waals crystals with hyperbolic light dispersion,” Nat. Commun. 8(1), 320 (2017).
[Crossref] [PubMed]

S. Yu, X. Wu, Y. Wang, X. Guo, and L. Tong, “2D materials for optical modulation: challenges and opportunities,” Adv. Mater. 29(14), 1606128 (2017).
[Crossref] [PubMed]

2016 (9)

Z. Liu and K. Aydin, “Localized surface plasmons in nanostructured monolayer black phosphorus,” Nano Lett. 16(6), 3457–3462 (2016).
[Crossref] [PubMed]

T. Galfsky, Z. Sun, C. R. Considine, C. T. Chou, W. C. Ko, Y. H. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16(8), 4940–4945 (2016).
[Crossref] [PubMed]

M. Sakhdari, M. Hajizadegan, M. Farhat, and P. Y. Chen, “Efficient, broadband and wide-angle hot-electron transduction using metal-semiconductor hyperbolic metamaterials,” Nano Energy 26, 371–381 (2016).
[Crossref]

M. Cuevas, “Surface plasmon enhancement of spontaneous emission in graphene waveguides,” J. Opt. 18(10), 105003 (2016).
[Crossref]

M. Kim, S. So, K. Yao, Y. Liu, and J. Rho, “Deep sub-wavelength nanofocusing of UV-visible light by hyperbolic metamaterials,” Sci. Rep. 6(1), 38645 (2016).
[Crossref] [PubMed]

J. S. Gomez-Diaz and A. Alu, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
[Crossref]

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7, 11329 (2016).
[Crossref] [PubMed]

K. V. Sreekanth, M. ElKabbash, Y. Alapan, A. R. Rashed, U. A. Gurkan, and G. Strangi, “A multiband perfect absorber based on hyperbolic metamaterials,” Sci. Rep. 6(1), 26272 (2016).
[Crossref] [PubMed]

A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
[Crossref] [PubMed]

2015 (7)

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

C. Shen, Y. Xie, N. Sui, W. Wang, S. A. Cummer, and Y. Jing, “Broadband acoustic hyperbolic metamaterial,” Phys. Rev. Lett. 115(25), 254301 (2015).
[Crossref] [PubMed]

J. Sun, M. I. Shalaev, and N. M. Litchinitser, “Experimental demonstration of a non-resonant hyperlens in the visible spectral range,” Nat. Commun. 6(1), 7201 (2015).
[Crossref] [PubMed]

M. Y. Shalaginov, V. V. Vorobyov, J. Liu, M. Ferrera, A. V. Akimov, A. Lagutchev, A. N. Smolyaninov, V. V. Klimov, J. Irudayaraj, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Enhancement of single-photon emission from nitrogen-vacancy centers with TiN/ (Al, Sc) N hyperbolic metamaterial,” Laser Photonics Rev. 9(1), 120–127 (2015).
[Crossref]

S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
[Crossref] [PubMed]

A. P. Slobozhanyuk, P. Ginzburg, D. A. Powell, I. Iorsh, A. S. Shalin, P. Segovia, A. V. Krasavin, G. A. Wurtz, V. A. Podolskiy, P. A. Belov, and A. V. Zayats, “Purcell effect in hyperbolic metamaterial resonators,” Phys. Rev. B 92(19), 195217 (2015).
[Crossref]

W. Gao, F. Fang, Y. Liu, and S. Zhang, “Chiral surface waves supported by biaxial hyperbolic metamaterials,” Light Sci. Appl. 4(9), e328 (2015).
[Crossref]

2014 (6)

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2014).
[Crossref]

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “Conical diffraction and the dispersion surface of hyperbolic metamaterials,” Phys. Rev. A 90(1), 013803 (2014).
[Crossref]

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref] [PubMed]

2013 (2)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

J. Sun, J. Zeng, and N. M. Litchinitser, “Twisting light with hyperbolic metamaterials,” Opt. Express 21(12), 14975–14981 (2013).
[Crossref] [PubMed]

2012 (2)

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 1–9 (2012).
[Crossref]

2011 (1)

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

2007 (1)

N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007).
[Crossref] [PubMed]

2006 (3)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[Crossref]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

1986 (1)

A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Solids Surf. 39(4), 227–242 (1986).
[Crossref]

Akimov, A. V.

M. Y. Shalaginov, V. V. Vorobyov, J. Liu, M. Ferrera, A. V. Akimov, A. Lagutchev, A. N. Smolyaninov, V. V. Klimov, J. Irudayaraj, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Enhancement of single-photon emission from nitrogen-vacancy centers with TiN/ (Al, Sc) N hyperbolic metamaterial,” Laser Photonics Rev. 9(1), 120–127 (2015).
[Crossref]

Alapan, Y.

K. V. Sreekanth, M. ElKabbash, Y. Alapan, E. I. Ilker, M. Hinczewski, U. A. Gurkan, and G. Strangi, “Hyperbolic metamaterials-based plasmonic biosensor for fluid biopsy with single molecule sensitivity,” EPJ Appl. Metamat. 4, 1 (2017).
[Crossref]

K. V. Sreekanth, M. ElKabbash, Y. Alapan, A. R. Rashed, U. A. Gurkan, and G. Strangi, “A multiband perfect absorber based on hyperbolic metamaterials,” Sci. Rep. 6(1), 26272 (2016).
[Crossref] [PubMed]

Alu, A.

J. S. Gomez-Diaz and A. Alu, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
[Crossref]

Atwater, H. A.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

Avouris, P.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

Axelrod, S.

S. Axelrod, M. K. Dezfouli, H. M. K. Wong, A. S. Helmy, and S. Hughes, “Hyperbolic metamaterial nanoresonators make poor single-photon sources,” Phys. Rev. B 95(15), 155424 (2017).
[Crossref]

Aydin, K.

Z. Liu and K. Aydin, “Localized surface plasmons in nanostructured monolayer black phosphorus,” Nano Lett. 16(6), 3457–3462 (2016).
[Crossref] [PubMed]

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

Ballantine, K. E.

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “Conical diffraction and the dispersion surface of hyperbolic metamaterials,” Phys. Rev. A 90(1), 013803 (2014).
[Crossref]

Bao, F.

K. Shi, F. Bao, and S. He, “Enhanced near-field thermal radiation based on multilayer graphene-hBN heterostructures,” ACS Photonics 4(4), 971–978 (2017).
[Crossref]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Belov, P. A.

A. P. Slobozhanyuk, P. Ginzburg, D. A. Powell, I. Iorsh, A. S. Shalin, P. Segovia, A. V. Krasavin, G. A. Wurtz, V. A. Podolskiy, P. A. Belov, and A. V. Zayats, “Purcell effect in hyperbolic metamaterial resonators,” Phys. Rev. B 92(19), 195217 (2015).
[Crossref]

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref] [PubMed]

Boltasseva, A.

M. Y. Shalaginov, V. V. Vorobyov, J. Liu, M. Ferrera, A. V. Akimov, A. Lagutchev, A. N. Smolyaninov, V. V. Klimov, J. Irudayaraj, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Enhancement of single-photon emission from nitrogen-vacancy centers with TiN/ (Al, Sc) N hyperbolic metamaterial,” Laser Photonics Rev. 9(1), 120–127 (2015).
[Crossref]

Briggs, R. M.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

Carvalho, A.

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Castro Neto, A. H.

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Chen, P. Y.

M. Sakhdari, M. Hajizadegan, M. Farhat, and P. Y. Chen, “Efficient, broadband and wide-angle hot-electron transduction using metal-semiconductor hyperbolic metamaterials,” Nano Energy 26, 371–381 (2016).
[Crossref]

Chen, Y.

S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
[Crossref] [PubMed]

Chou, C. T.

T. Galfsky, Z. Sun, C. R. Considine, C. T. Chou, W. C. Ko, Y. H. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16(8), 4940–4945 (2016).
[Crossref] [PubMed]

Considine, C. R.

T. Galfsky, Z. Sun, C. R. Considine, C. T. Chou, W. C. Ko, Y. H. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16(8), 4940–4945 (2016).
[Crossref] [PubMed]

Cortes, C. L.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 1–9 (2012).
[Crossref]

Cuevas, M.

M. Cuevas, “Surface plasmon enhancement of spontaneous emission in graphene waveguides,” J. Opt. 18(10), 105003 (2016).
[Crossref]

Cummer, S. A.

C. Shen, Y. Xie, N. Sui, W. Wang, S. A. Cummer, and Y. Jing, “Broadband acoustic hyperbolic metamaterial,” Phys. Rev. Lett. 115(25), 254301 (2015).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

de Leon, N. P.

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B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
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T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
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T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Wang, W.

L. Li, W. Wang, T. S. Luk, X. Yang, and J. Gao, “Enhanced quantum dot spontaneous emission with multilayer metamaterial nanostructures,” ACS Photonics 4(3), 501–508 (2017).
[Crossref]

C. Shen, Y. Xie, N. Sui, W. Wang, S. A. Cummer, and Y. Jing, “Broadband acoustic hyperbolic metamaterial,” Phys. Rev. Lett. 115(25), 254301 (2015).
[Crossref] [PubMed]

Wang, Y.

S. Yu, X. Wu, Y. Wang, X. Guo, and L. Tong, “2D materials for optical modulation: challenges and opportunities,” Adv. Mater. 29(14), 1606128 (2017).
[Crossref] [PubMed]

Wangberg, R.

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[Crossref]

Watts, C. M.

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

Wild, D. S.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Wong, H. M. K.

S. Axelrod, M. K. Dezfouli, H. M. K. Wong, A. S. Helmy, and S. Hughes, “Hyperbolic metamaterial nanoresonators make poor single-photon sources,” Phys. Rev. B 95(15), 155424 (2017).
[Crossref]

Wong, Z. J.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7, 11329 (2016).
[Crossref] [PubMed]

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Wu, C.

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2014).
[Crossref]

Wu, J.

S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
[Crossref] [PubMed]

Wu, X.

S. Yu, X. Wu, Y. Wang, X. Guo, and L. Tong, “2D materials for optical modulation: challenges and opportunities,” Adv. Mater. 29(14), 1606128 (2017).
[Crossref] [PubMed]

Wurtz, G. A.

D. J. Roth, A. V. Krasavin, A. Wade, W. Dickson, A. Murphy, S. Kéna-Cohen, R. Pollard, G. A. Wurtz, D. Richards, S. A. Maier, and A. V. Zayats, “Spontaneous emission inside a hyperbolic metamaterial waveguide,” ACS Photonics 4(10), 2513–2521 (2017).
[Crossref]

A. P. Slobozhanyuk, P. Ginzburg, D. A. Powell, I. Iorsh, A. S. Shalin, P. Segovia, A. V. Krasavin, G. A. Wurtz, V. A. Podolskiy, P. A. Belov, and A. V. Zayats, “Purcell effect in hyperbolic metamaterial resonators,” Phys. Rev. B 92(19), 195217 (2015).
[Crossref]

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref] [PubMed]

Xia, F.

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Xie, Y.

C. Shen, Y. Xie, N. Sui, W. Wang, S. A. Cummer, and Y. Jing, “Broadband acoustic hyperbolic metamaterial,” Phys. Rev. Lett. 115(25), 254301 (2015).
[Crossref] [PubMed]

Yang, F.

S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
[Crossref] [PubMed]

Yang, S.

S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
[Crossref] [PubMed]

Yang, X.

L. Li, W. Wang, T. S. Luk, X. Yang, and J. Gao, “Enhanced quantum dot spontaneous emission with multilayer metamaterial nanostructures,” ACS Photonics 4(3), 501–508 (2017).
[Crossref]

Yao, K.

M. Kim, S. So, K. Yao, Y. Liu, and J. Rho, “Deep sub-wavelength nanofocusing of UV-visible light by hyperbolic metamaterials,” Sci. Rep. 6(1), 38645 (2016).
[Crossref] [PubMed]

Yu, S.

S. Yu, X. Wu, Y. Wang, X. Guo, and L. Tong, “2D materials for optical modulation: challenges and opportunities,” Adv. Mater. 29(14), 1606128 (2017).
[Crossref] [PubMed]

Zayats, A. V.

D. J. Roth, A. V. Krasavin, A. Wade, W. Dickson, A. Murphy, S. Kéna-Cohen, R. Pollard, G. A. Wurtz, D. Richards, S. A. Maier, and A. V. Zayats, “Spontaneous emission inside a hyperbolic metamaterial waveguide,” ACS Photonics 4(10), 2513–2521 (2017).
[Crossref]

A. P. Slobozhanyuk, P. Ginzburg, D. A. Powell, I. Iorsh, A. S. Shalin, P. Segovia, A. V. Krasavin, G. A. Wurtz, V. A. Podolskiy, P. A. Belov, and A. V. Zayats, “Purcell effect in hyperbolic metamaterial resonators,” Phys. Rev. B 92(19), 195217 (2015).
[Crossref]

P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref] [PubMed]

Zeng, J.

Zhang, S.

W. Gao, F. Fang, Y. Liu, and S. Zhang, “Chiral surface waves supported by biaxial hyperbolic metamaterials,” Light Sci. Appl. 4(9), e328 (2015).
[Crossref]

Zhang, X.

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7, 11329 (2016).
[Crossref] [PubMed]

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2014).
[Crossref]

ACS Photonics (4)

J. S. Gomez-Diaz and A. Alu, “Flatland optics with hyperbolic metasurfaces,” ACS Photonics 3(12), 2211–2224 (2016).
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K. Shi, F. Bao, and S. He, “Enhanced near-field thermal radiation based on multilayer graphene-hBN heterostructures,” ACS Photonics 4(4), 971–978 (2017).
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D. J. Roth, A. V. Krasavin, A. Wade, W. Dickson, A. Murphy, S. Kéna-Cohen, R. Pollard, G. A. Wurtz, D. Richards, S. A. Maier, and A. V. Zayats, “Spontaneous emission inside a hyperbolic metamaterial waveguide,” ACS Photonics 4(10), 2513–2521 (2017).
[Crossref]

L. Li, W. Wang, T. S. Luk, X. Yang, and J. Gao, “Enhanced quantum dot spontaneous emission with multilayer metamaterial nanostructures,” ACS Photonics 4(3), 501–508 (2017).
[Crossref]

Adv. Mater. (2)

C. M. Watts, X. Liu, and W. J. Padilla, “Metamaterial electromagnetic wave absorbers,” Adv. Mater. 24(23), OP98–OP120 (2012).
[PubMed]

S. Yu, X. Wu, Y. Wang, X. Guo, and L. Tong, “2D materials for optical modulation: challenges and opportunities,” Adv. Mater. 29(14), 1606128 (2017).
[Crossref] [PubMed]

Adv. Optoelectron. (1)

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 1–9 (2012).
[Crossref]

Appl. Phys. Lett. (1)

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006).
[Crossref]

Appl. Phys., A Solids Surf. (1)

A. Morita, “Semiconducting black phosphorus,” Appl. Phys., A Solids Surf. 39(4), 227–242 (1986).
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EPJ Appl. Metamat. (1)

K. V. Sreekanth, M. ElKabbash, Y. Alapan, E. I. Ilker, M. Hinczewski, U. A. Gurkan, and G. Strangi, “Hyperbolic metamaterials-based plasmonic biosensor for fluid biopsy with single molecule sensitivity,” EPJ Appl. Metamat. 4, 1 (2017).
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J. Opt. (1)

M. Cuevas, “Surface plasmon enhancement of spontaneous emission in graphene waveguides,” J. Opt. 18(10), 105003 (2016).
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Laser Photonics Rev. (1)

M. Y. Shalaginov, V. V. Vorobyov, J. Liu, M. Ferrera, A. V. Akimov, A. Lagutchev, A. N. Smolyaninov, V. V. Klimov, J. Irudayaraj, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Enhancement of single-photon emission from nitrogen-vacancy centers with TiN/ (Al, Sc) N hyperbolic metamaterial,” Laser Photonics Rev. 9(1), 120–127 (2015).
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Light Sci. Appl. (1)

W. Gao, F. Fang, Y. Liu, and S. Zhang, “Chiral surface waves supported by biaxial hyperbolic metamaterials,” Light Sci. Appl. 4(9), e328 (2015).
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Nano Energy (1)

M. Sakhdari, M. Hajizadegan, M. Farhat, and P. Y. Chen, “Efficient, broadband and wide-angle hot-electron transduction using metal-semiconductor hyperbolic metamaterials,” Nano Energy 26, 371–381 (2016).
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Nano Lett. (2)

T. Galfsky, Z. Sun, C. R. Considine, C. T. Chou, W. C. Ko, Y. H. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16(8), 4940–4945 (2016).
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Z. Liu and K. Aydin, “Localized surface plasmons in nanostructured monolayer black phosphorus,” Nano Lett. 16(6), 3457–3462 (2016).
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Nat. Commun. (7)

S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay, and J. Wu, “Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K,” Nat. Commun. 6(1), 8573 (2015).
[Crossref] [PubMed]

M. N. Gjerding, R. Petersen, T. G. Pedersen, N. A. Mortensen, and K. S. Thygesen, “Layered van der Waals crystals with hyperbolic light dispersion,” Nat. Commun. 8(1), 320 (2017).
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P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5, 3226 (2014).
[Crossref] [PubMed]

J. Sun, M. I. Shalaev, and N. M. Litchinitser, “Experimental demonstration of a non-resonant hyperlens in the visible spectral range,” Nat. Commun. 6(1), 7201 (2015).
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J. S. T. Smalley, F. Vallini, S. A. Montoya, L. Ferrari, S. Shahin, C. T. Riley, B. Kanté, E. E. Fullerton, Z. Liu, and Y. Fainman, “Luminescent hyperbolic metasurfaces,” Nat. Commun. 8, 13793 (2017).
[Crossref] [PubMed]

S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Y. S. Kivshar, and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nat. Commun. 7, 11329 (2016).
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K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
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Nat. Nanotechnol. (1)

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
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Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Nature (1)

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
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Opt. Express (1)

Phys. Rev. A (1)

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “Conical diffraction and the dispersion surface of hyperbolic metamaterials,” Phys. Rev. A 90(1), 013803 (2014).
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Phys. Rev. B (4)

A. P. Slobozhanyuk, P. Ginzburg, D. A. Powell, I. Iorsh, A. S. Shalin, P. Segovia, A. V. Krasavin, G. A. Wurtz, V. A. Podolskiy, P. A. Belov, and A. V. Zayats, “Purcell effect in hyperbolic metamaterial resonators,” Phys. Rev. B 92(19), 195217 (2015).
[Crossref]

S. Axelrod, M. K. Dezfouli, H. M. K. Wong, A. S. Helmy, and S. Hughes, “Hyperbolic metamaterial nanoresonators make poor single-photon sources,” Phys. Rev. B 95(15), 155424 (2017).
[Crossref]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

Phys. Rev. Lett. (3)

T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
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A. Nemilentsau, T. Low, and G. Hanson, “Anisotropic 2D materials for tunable hyperbolic plasmonics,” Phys. Rev. Lett. 116(6), 066804 (2016).
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C. Shen, Y. Xie, N. Sui, W. Wang, S. A. Cummer, and Y. Jing, “Broadband acoustic hyperbolic metamaterial,” Phys. Rev. Lett. 115(25), 254301 (2015).
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Proc. Natl. Acad. Sci. U.S.A. (1)

T. Galfsky, J. Gu, E. E. Narimanov, and V. M. Menon, “Photonic hypercrystals for control of light-matter interactions,” Proc. Natl. Acad. Sci. U.S.A. 114(20), 5125–5129 (2017).
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Prog. Quantum Electron. (1)

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2014).
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Sci. Rep. (2)

K. V. Sreekanth, M. ElKabbash, Y. Alapan, A. R. Rashed, U. A. Gurkan, and G. Strangi, “A multiband perfect absorber based on hyperbolic metamaterials,” Sci. Rep. 6(1), 26272 (2016).
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M. Kim, S. So, K. Yao, Y. Liu, and J. Rho, “Deep sub-wavelength nanofocusing of UV-visible light by hyperbolic metamaterials,” Sci. Rep. 6(1), 38645 (2016).
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Science (2)

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Other (1)

C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley-IEEE Press, 2006).

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Figures (5)

Fig. 1
Fig. 1 (a) Schematics of layered few-layer BP-Au multilayer structure; Illustration the XZ plane vision of few-layer BP (b), which corresponding to the arm-chair property in X-axe and the YZ plane vision of few-layer BP (c), which corresponding to the zigzag property in Y-axe;
Fig. 2
Fig. 2 (a) The effective permittivity in the three directions of the orthogonal system for the alternative BP-Au multilayer structure, as solid lines and dash lines represent the real part and imaginary part of permittivity respectively. (b) The magnification the permittivity near the zero point. It should be noted here that the span of ε zz keeps from −250~250 for a better vision. Actually, the three curves would meet, not exactly, but almost at the same point (~623.3 nm) when magnifies that specific region.
Fig. 3
Fig. 3 (a) Dependence of 𝜀xx and 𝜀yy on thickness ratio p (p = dAu/dBP), while the black and blue lines correspond to the permittivity in X and Y-direction respective. (b) Dependence of ε zz on thickness ratio p from 0.8 to 1.4 over wavelength (400~900 nm). The solid and dash lines correspond to real and imaginary parts of the permittivity respectively. The permittivity map of real (c) and imaginary (d) part of 𝜀zz with respect to thickness ratio p sweep.
Fig. 4
Fig. 4 (a) Purcell factor for 8-layer structure constructed by alternating few-layer BP (5nm) and Au (5nm) HMM, Fpx, Fpy, Fpz, corresponding to the Purcell factor activated by the electrical dipole at 10nm above HMM with orientation along X-, Y-, Z-direction respectively and Fpiso corresponding to isotropic Purcell factor for the whole multilayer structure. (b) Comparison of isotropic Purcell factor over frequency for the HMMs composed by 2, 3, 4 units’ structures along vertical direction. 1 unit composes by one-layer BP and one-layer Au. Thickness ratio p = 1.2 , d B P = 5 n m is adopted in this model.
Fig. 5
Fig. 5 (a) Dependence of isotropic Purcell Factor ( F p i s o ) over wavelength (400~900nm) for varying thickness ratio p ( p = d A u / d B P ) from 0.8 to 1.4 while the thickness of BP remains unchanged (5nm). (b) Dependence of isotropic Purcell Factor ( F p i s o ) over wavelength (400~900nm) for varying BP thickness ( d B P ) from 3 nm to 6 nm while the thickness ratio remains unchanged ( p = 1.2 ).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

ε = ( ε x x 0 0 0 ε y y 0 0 0 ε z z )
ε x x = ε y y = ε z z  
ε x x = ε y y ε z z
ε x x ε y y ε z z
ε x x = ε A u d A u + ε B P x ε B P x d A u + d B P = ε A u d A u d B P + ε B P x d A u d B P + 1 = ε A u p + ε B P x p + 1
ε y y = ε A u d A u + ε B P x ε B P y d A u + d B P = ε A u d A u d B P + ε B P y d A u d B P + 1 = ε A u p + ε B P y p + 1
ε z z = d A u + d B P z d A u ε A u + d B P ε B P z = d A u d B P + 1 d A u d B P 1 ε A u + 1 ε B P z = p + 1 p ε A u + 1 ε B P z
F p i s o = F p x + F p y + F p z 3

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