⇦Chinese Physics Letters, 2017, Vol. 34, No. 9, Article code 097701 Origin of Negative Imaginary Part of Effective Permittivity of Passive Materials * Kai-Lun Zhang(张凯伦)1, Zhi-Ling Hou(侯志灵)1**, Ling-Bao Kong(孔令宝)2, Hui-Min Fang(房慧敏)1, Ke-Tao Zhan(战可涛)1 Affiliations 1School of Science & Beijing Key Laboratory of Environmentally Harmful Chemicals Assessment, Beijing University of Chemical Technology, Beijing 100029 2School of Science, Beijing Technology and Business University, Beijing 100048 Received 10 April 2017 *Supported by the National Natural Science Foundation of China under Grant No 51102007, and the Fund for Discipline Construction of Beijing University of Chemical Technology under Grant No XK1702.
**Corresponding author. Email: zhilinghou@gmail.com Citation Text: Zhang K L, Hou Z L, Kong L B, Fang H M and Zhan K T 2017 Chin. Phys. Lett. 34 097701 Abstract The anti-resonant phenomenon of effective electromagnetic parameters of metamaterials has aroused controversy due to negative imaginary permittivity or permeability. It is experimentally found that the negative imaginary permittivity can occur for the natural passive materials near the Fabry–Perot resonances. We reveal the nature of negative imaginary permittivity, which is correlated with the magnetoelectric coupling. The anti-resonance of permittivity is a non-inherent feature for passive materials, while it can be inherent for devices or metamaterials. Our finding validates that the negative imaginary part of effective permittivity does not contradict the second law of thermodynamics for metamaterials owing to the magnetoelectric coupling. DOI:10.1088/0256-307X/34/9/097701 PACS:77.22.Ch, 77.22.-d, 78.67.Pt, 71.45.Gm © 2017 Chinese Physics Society Article Text Plasmonic metamaterials have attracted intense interest in the past few years due to the exotic optical properties and wide variety of potential applications such as invisibility cloaks,[1-3] high-resolution imaging,[4,5] quantum levitation,[6] polarization conversion,[7,8] small antennas,[9] ultrasmall cavity[10] and energy harvester.[11] Based on the assumption which treats the periodical collections of resonant inclusions as homogenized bulk materials, several homogenization models were proposed to macroscopically describe the unusual electromagnetic response of metamaterials using the effective permittivity and permeability.[12-16] One of the most common models, the $S$-parameter retrieval or Nicolson–Ross–Weir (NRW)[17,18] method describing the wave propagation in the natural material, is used to determine the constitutive parameters of metamaterials, when the size of unit cell is much smaller than the wavelength.[12-14] For natural materials, the NRW method provides straightforward analysis and characterization from the transmission and reflection parameters of a finite-thickness slab. However, it is still a challenging work to provide physical insight into the nature of the electromagnetic response of metamaterials. The left-handed or other anomalous behavior is usually accompanied by the presence of anti-resonances of permeability or permittivity. Although it is gradually acceptable in dozens of works, the retrieved effective constitutive parameter has still aroused serious doubts near the metamaterial resonances where most interesting behavior takes place. The focus of the argument is whether the imaginary part of both permittivity and permeability must be positive simultaneously for passive materials.[19-24] Recently, the antiresonance of permeability of metamaterials becomes acceptable,[25-28] since Markel[23] demonstrated that the imaginary part of permeability can be negative in naturally occurring diamagnetics or in artificial materials, which does not contradict the second law of thermodynamics or energy conservation. Nevertheless, the negative imaginary part of permittivity of metamaterials is usually regarded as an artifact that is obtained from the numerical errors or noise of $S$ parameters. Indeed, some small errors or noise near the slab resonances may cause the artifact, when the reflection coefficient tends to zero.[29] For high-loss material with large reflection coefficient, while the origin of negative imaginary permittivity is still unclear, here in our experiment, we report an unexpected antiresonance of permittivity of passive natural materials near the Fabry–Perot resonances (FPR) frequency with normal reflection coefficient. In the following we put forward the origin of the negative imaginary part of permittivity from the magnetoelectric coupling of multiferroic materials. This is a strong indication that the retrieval procedure is not bothered by the opposite sign of permittivity in metamaterials. The Er-substituted polycrystalline BiFeO$_{3}$ (BEFO) was synthesized by the sol-gel method. The starting materials are Bi(NO$_{3}$)$_{3}\cdot$5H$_{2}$O, Fe(NO$_{3}$)$_{3}$$\cdot$9H$_{2}$O and a solution of Er$_{2}$O$_{3}$ in HNO$_{3}$, with the stoichiometic proportion of 0.95:1:0.05. The ethylene glycol was used as a dispersing agent. The annealing temperature was 600$^\circ\!$C. The detailed procedures can be found in Ref. [30]. The as-prepared samples were compacted into a rectangular sheet (22.82 mm$\times$10.12 mm) to fit the geometrical dimensions of the waveguide sample holder. The thicknesses of samples 1 and 2 are 3.34 mm and 0.814 mm, respectively. The $S$ parameters were measured by a vector network analyzer (Anritsu37269D). All numerical simulations were carried out using the commercial available numerical package CST Microwave Studio 2014. The relative complex permittivity and magnetic permeability were determined from $S$ parameters by the NRW method[31] in the $X$ band.
cpl-34-9-097701-fig1.png
Fig. 1. The effective permittivity (a) and permeability (b) of BEFO near the Fabry–Perot resonances.
Figure 1 shows the experimental complex permittivity and magnetic permeability of the BEFO sample in thicknesses 3.34 mm. It is noteworthy that the negative imaginary part of effective permittivity ($\varepsilon''$) is observed from 9.5 to 12.4 GHz with a minimum value of $-$7.63 (see Fig. 1(a)). With the increasing frequency, $\varepsilon''$ decreases when the frequency is higher than 9.5 GHz, while the real part of effective permittivity ($\varepsilon '$) increases rapidly when the frequency is higher than 10.5 GHz. The corresponding magnetic permeability shows the resonance phenomenon in this frequency region (see Fig. 1(b)). In fact, the negative $\varepsilon''$ in Fig. 1 is not an intrinsic dielectric behavior for the BEFO material because the thickness of sample 1 is half the FPR wavelength shown in the inset of Fig. 1(b) (the FPR frequency is 12.358 GHz). The intrinsic dielectric properties are shown in Fig. 2. The value of $\varepsilon''$ is $\sim$1.7, and none of these parameters are less than 0 at all frequencies investigated. As is known, when the sample thickness is $\lambda _{\rm g}/2$, the concomitant FPR can manipulate the reflection of electromagnetic field.[31-33] Then, the retrieved values of permittivity and permeability could be effected near the FPR frequency. For low-loss materials or metamaterials, small errors or noise near the slab resonances may cause the artifact when the reflection coefficient is close to zero.[29] However, both $|S_{11}|$ and the Fresnel reflection coefficient are far from zero for the BEFO material (see Figs. 3(a) and 3(b)). Thus the negative $\varepsilon''$ is not produced by the small errors or noise. The absolute value of $\varepsilon''$ is so large that it should be a special response of BEFO rather than an artifact. To analyze the reasons behind the negative $\varepsilon''$, we investigate whether the thickness affects the dielectric properties of samples by simulating $S$ parameters for homogenous slabs of various thicknesses near the FPR. The variations of complex permeability and complex permittivity calculated from the $S$ parameters by the NRW method at various slab thicknesses are shown in Fig. 4. The negative $\varepsilon''$ is not observed for any thickness though the FPR occurs at the $X$ band for these thickness. The value of $\varepsilon''$ near the FPR produces some deviation from the norm, and the deviation at the FPR frequency is less than 8% for each thickness. Moreover, both complex permittivity and permeability from simulations show the inverse resonance characteristic as compared with the measured results. In detail, all the $\varepsilon''$ values in the $X$ band are positive, and show an upward peak for each thickness. The value of $\varepsilon '$ decreases with increasing frequency at all thickness. Nevertheless, $\mu''$ presents a downward peak for each thickness, which contrasts with the experimental results. Therefore, the inverse resonance between simulation and experiment suggests that the negative $\varepsilon''$ is a special response of BEFO rather than an artifact.
cpl-34-9-097701-fig2.png
Fig. 2. The intrinsic permittivity (a) and permeability (b) of BEFO.
cpl-34-9-097701-fig3.png
Fig. 3. Magnitude (a) and phase (b) of $S$ parameters and Fresnel reflection coefficient (c) of BEFO.
cpl-34-9-097701-fig4.png
Fig. 4. Thickness effects of complex permittivity (a) and complex permeability (b) retrieved by simulating $S$ parameters for homogenous slabs of various thicknesses near the FPR.
Not only $\varepsilon''$ but also $\mu''$ can be negative in passive natural materials near the Fabry–Perot resonance (for example, Fe$_{3}$O$_{4}$/SnO$_{2}$nanorods,[34] MWCNT/wax,[35] and hollow cobalt nanochain composites[36] show negative $\mu''$). We find that the negative $\varepsilon''$ is related to the magnetoelectric coupling. For magnetoelectric coupling material, ${\boldsymbol B}_{\rm eff}$ and ${\boldsymbol D}_{\rm eff}$ can be written as[37,38] $$\begin{align} {\boldsymbol D}_{\rm eff} =\,&\varepsilon \varepsilon _0 {\boldsymbol E}+\xi {\boldsymbol H},~~ \tag {1a}\\ {\boldsymbol B}_{\rm eff} =\,&\mu \mu _0 {\boldsymbol H}-\varsigma {\boldsymbol E},~~ \tag {1b} \end{align} $$ where $\xi$ and $\varsigma$ are magnetoelectric parameters, $\varepsilon$ and $\mu$ are complex permittivity and complex permeability, respectively. For an electromagnetic wave with a time dependent factor $e^{j\omega t}$, the source-free Maxwell equations with magnetoelectric coupling effect can be expressed as follows: $$\begin{align} \nabla {\rm \times }{\boldsymbol E}=-j\omega \mu \mu _0 {\boldsymbol H}-{\boldsymbol J}_{\rm EM},~~ \tag {2a}\\ \nabla \times {\boldsymbol H}=j\omega \varepsilon \varepsilon _0 {\boldsymbol E}+{\boldsymbol J}_{\rm ME},~~ \tag {2b} \end{align} $$ where ${\boldsymbol J}_{\rm ME}$ and ${\boldsymbol J}_{\rm EM}$ are the electric current density and magnetic current density resulting from the magnetoelectric coupling, respectively. They can be expressed as follows: $$\begin{alignat}{1} \!\!\!\!{\boldsymbol J}_{\rm ME}=\,&\varepsilon \varepsilon _0 \frac{\partial {\boldsymbol E}_{\rm ME}}{\partial t}=\varepsilon \varepsilon _0 \frac{\partial (-\varsigma {\boldsymbol E})}{\partial t}=-j\omega \varepsilon \varepsilon _0 \varsigma {\boldsymbol E},~~ \tag {3a}\\ \!\!\!\!{\boldsymbol J}_{\rm EM} =\,&\mu \mu _0 \frac{\partial {\boldsymbol H}_{\rm EM} }{\partial t}=j\omega \mu \mu _0 \xi {\boldsymbol H}.~~ \tag {3b} \end{alignat} $$ Thus the Maxwell Eqs. (2a) and (2b) can be rewritten as $$\begin{alignat}{1} \!\!\!\!\!\nabla {\rm \times }{\boldsymbol E}=\,&-j\omega \mu \mu _0 (1+\xi){\boldsymbol H}=-j\omega \mu _{\rm eff} \mu _0 {\boldsymbol H},~~ \tag {4a}\\ \!\!\!\!\!\nabla \times {\boldsymbol H}=\,&j\omega \varepsilon \varepsilon _0 (1-\varsigma){\boldsymbol E}=j\omega \varepsilon _{\rm eff} \varepsilon _0 {\boldsymbol E}.~~ \tag {4b} \end{alignat} $$ From Eqs. (4a) and (4b), we have $$\begin{align} \varepsilon _{\rm eff} =\,&\varepsilon (1-\varsigma),~~ \tag {5a}\\ \mu _{\rm eff} =\,&\mu (1+\xi),~~ \tag {5b} \end{align} $$ where $\varepsilon$, $\mu$, $\xi$ and $\varsigma$ satisfy the constitutive relations: $\varepsilon =\varepsilon'-j{\varepsilon }''$, $\mu =\mu'-j\mu''$, $\xi =\xi'-j\xi''$, and $\varsigma =\varsigma'-j\varsigma''$. According to Eqs. (5a) and (5b), we obtain $$\begin{alignat}{1} \!\!\!\!\!\varepsilon _{\rm eff} =\,&{\varepsilon }'(1-{\varsigma }')+{\varepsilon }''{\varsigma }''-j[{\varepsilon }''(1-{\varsigma }')-{\varepsilon }'{\varsigma }''],~~ \tag {6a}\\ \!\!\!\!\!\mu _{\rm eff} =\,&{\mu }'(1+{\xi }')-{\mu }''{\xi }''-j[{\mu }''(1+{\xi }')+{\mu }'{\xi }''].~~ \tag {6b} \end{alignat} $$ According to Eqs. (6a) and (6b), $\xi$ and $\varsigma$ are calculated as shown in Fig. 5. The values of $\xi$ and $\varsigma$ are strongly dependent on the resonant frequency, which present a large value in the vicinity of the resonance frequency and become very small away from the resonance frequency. The result suggests that the conversion between magnetic energy and electric energy becomes easy in the vicinity of resonance frequency.
cpl-34-9-097701-fig5.png
Fig. 5. Magnetoelectric coupling parameters $\varsigma$ (a) and electromagnetic coupling parameter $\xi$ (b) of BEFO near the Fabry–Perot resonances.
When $\xi =0$ and $\varsigma =0$, $\varepsilon''$ and $\mu''$ should be positive to meet the requirements of the second law of thermodynamics. When $\xi \ne 0$ and $\varsigma \ne 0$, while according to Eq. (6a), ${\varepsilon }''_{\rm eff}$ can be negative if ${\varepsilon }''(1-{\varsigma }') < {\varepsilon }'{\varsigma }''$. Considering the magnetoelectric coupling effect, the magnetic energy is transformed into the electric energy, and the total energy loss is still positive. Thus ${\varepsilon }''_{\rm eff}$ is negative, which does not contradict the second law of thermodynamics because of the electric loss. In particular, when the FPR or Mie resonance occurs, the transform between magnetic energy and electric energy is easy to conduct. Of course, the negative $\varepsilon''$ is not an intrinsic dielectric attribute for this material because it depends on the slab thickness. However, it may be classified as an intrinsic character when the sample is treated as a device. Owing to the fact that the magnetoelectric coupling can occur in metamaterials,[39,40] thus the retrieval procedure is not bothered by the opposite sign of permittivity in metamaterials. Moreover, the resonance of metamaterials associated with unit cell is not dependent on the slab thickness. Therefore, negative $\varepsilon''$ can be an intrinsic character of metamaterial. In summary, we have investigated the effective constitutive parameters retrieved from the scattering properties of the natural passive material BEFO, and found that ${\varepsilon }''_{\rm eff}$ is negative near the FPR. The negative ${\varepsilon }''_{\rm eff}$ is not the artifact caused by small unavoidable errors in numerical or experimental reflection and transmission data, due to the fact that the reflection coefficient does not tend to zero. We have shown that the negative ${\varepsilon }''_{\rm eff}$ may originate from the magnetoelectric coupling effect where a transform between magnetic energy and electric energy is produced. Although the negative $\varepsilon''$ is not an intrinsic dielectric behavior for this material due to the FPR, it is an intrinsic physical character for devices. For metamaterial, if the magnetoelectric coupling occurs, the negative $\varepsilon''$ can be classified as an intrinsic character rather than nonphysical one, as the resonance leading to negative $\varepsilon''$ will not be affected by variations in the slab thickness. References Metamaterial Electromagnetic Cloak at Microwave FrequenciesBroadband Ground-Plane CloakA full-parameter unidirectional metamaterial cloak for microwavesMetamaterial Apertures for Computational ImagingPlasmonic nanoresonators for high-resolution colour filtering and spectral imagingQuantum levitation by left-handed metamaterialsTerahertz Metamaterials for Linear Polarization Conversion and Anomalous RefractionPolarization switching and nonreciprocity in symmetric and asymmetric magnetophotonic multilayers with nonlinear defectMetamaterial-Based Efficient Electrically Small AntennasDeep subwavelength Fabry-Perot resonancesMetamaterial electromagnetic energy harvester with near unity efficiencyComposite Medium with Simultaneously Negative Permeability and PermittivityDetermination of effective permittivity and permeability of metamaterials from reflection and transmission coefficientsRobust method to retrieve the constitutive effective parameters of metamaterialsCausality relations in the homogenization of metamaterialsNonlinear interaction of two trapped-mode resonances in a bilayer fish-scale metamaterialMeasurement of the Intrinsic Properties of Materials by Time-Domain TechniquesAutomatic measurement of complex dielectric constant and permeability at microwave frequenciesComment I on “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”Comment II on “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”Resonant and antiresonant frequency dependence of the effective parameters of metamaterialsReply to Comments on “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”Can the imaginary part of permeability be negative?On the signs of the imaginary parts of the effective permittivity and permeability in metamaterialsNegative capacitor paves the way to ultra-broadband metamaterialsChiral metamaterials with negative refractive index based on four “U” split ring resonatorsRolled-up nanotechnology for the fabrication of three-dimensional fishnet-type GaAs-metal metamaterials with negative refractive index at near-infrared frequenciesA terahertz metamaterial with unnaturally high refractive indexCorrecting the Fabry-Perot artifacts in metamaterial retrieval proceduresEnhanced magnetization and improved leakage in Er-doped BiFeO 3 nanoparticlesMicrowave permittivity and permeability experiments in high-loss dielectrics: Caution with implicit Fabry-Pérot resonance for negative imaginary permeabilityEnhanced electromagnetic pressure in a sandwiched reflection gratingDeep subwavelength Fabry-Perot-like resonances in a sandwiched reflection gratingPorous Fe 3 O 4 /SnO 2 Core/Shell Nanorods: Synthesis and Electromagnetic PropertiesMicrowave absorbing performances of multiwalled carbon nanotube composites with negative permeabilityDual nonlinear dielectric resonance and nesting microwave absorption peaks of hollow cobalt nanochains composites with negative permeabilityDyadic Green's function and dipole radiation in chiral mediaOPTIMIZATION APPROACH TO THE RETRIEVAL OF THE CONSTITUTIVE PARAMETERS OF SLAB OF GENERAL BIANISOTROPIC MEDIUMRestoring the physical meaning of metamaterial constitutive parametersSubstrate-induced bianisotropy in metamaterials
[1] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F and Smith D R 2006 Science 314 977
[2] Liu R, Ji C, Mock J J, Chin J Y, Cui T J and Smith D R 2009 Science 323 366
[3] Landy N and Smith D R 2012 Nat. Mater. 12 25
[4] Hunt J, Driscoll T, Mrozack A, Lipworth G, Reynolds M, Brady D and Smith D R 2013 Science 339 310
[5] Xu T, Wu Y K, Luo X G and Guo L J 2010 Nat. Commun. 1 59
[6] Leonhardt U and Philbin T G 2007 New J. Phys. 9 254
[7] Grady N K, Heyes J E, Chowdhury D R, Zeng Y, Reiten A K, Taylor A J, Dalvit D A R and Chen H T 2013 Science 340 1304
[8] Tuz V R, Prosvirnin S L and Zhukovsky S V 2012 Phys. Rev. A 85 043822
[9] Ziolkowski R W and Erentok A 2006 IEEE Trans. Antennas Propag. 54 2113
[10] Huang C P and Chan C T 2014 EPJ Appl. Metamater. 1 2
[11] Almoneef T S and Ramahi O M 2015 Appl. Phys. Lett. 106 153902
[12] Smith D R, Padilla W J, Vier D C, Nemat-Nasser and Schultz S 2000 Phys. Rev. Lett. 84 4184
[13] Smith D R, Schultz S, Markos P and Soukoulis M 2002 Phys. Rev. B 65 195104
[14] Chen X D, Grzegorczyk T M, Wu B I, Pacheco J and Kong J A 2004 Phys. Rev. E 70 016608
[15] Alu A, Yaghjian A D, Shore R A and Silveirinha M G 2011 Phys. Rev. B 84 054305
[16] Tuz V R, Novitsky D V, Mladyonov P L, Prosvirnin S L and Novitsky A V 2014 J. Opt. Soc. Am. B 31 2095
[17] Nicolson A and Ross G 1970 IEEE Trans. Instrum. Meas. 19 377
[18] Weir W B 1974 Proc. IEEE 62 33
[19] Depine R A and Lakhtkia A 2004 Phys. Rev. E 70 048601
[20] Efros A L 2004 Phys. Rev. E 70 048602
[21] Koschny T, Markos P, Smith D R and Soukoulis C M 2003 Phys. Rev. E 68 065602
[22] Koschny T, Markos P, Smith D R and Soukoulis C M 2004 Phys. Rev. E 70 048603
[23] Markel V A 2008 Phys. Rev. E 78 026608
[24] Woodley J and Mojahedi M 2010 J. Opt. Soc. Am. B 27 1016
[25] Hrabar S, Krois I, Bonic I and Kiricenko A 2011 Appl. Phys. Lett. 99 254103
[26] Li Z, Zhao R, Koschny T, Kafesaki M, Alici K B, Colak E, Caglayan H, Ozbay E and Soukoulis C M 2010 Appl. Phys. Lett. 97 081901
[27] Rottler A, Harland M, Broell M, Schwaiger S, Stickler D, Stemmann A, Heyn C, Heitmann D and Mendach S 2012 Appl. Phys. Lett. 100 151104
[28] Choi M, Lee S H, Kim Y, Kang S B, Shin J, Kwak M H, Kang K Y, Lee Y H, Park N and Min B 2011 Nature 470 369
[29] Liu X X, Powell D A and Alu A 2011 Phys. Rev. B 84 235106
[30] Zhou H F, Hou Z L, Kong L B, Jin H B, Cao M S and Qi X 2013 Phys. Status Solidi A 210 809
[31] Hou Z L, Zhang M, Kong L B, Fang H M, Li Z J, Zhou H F, Jin H B and Cao M S 2013 Appl. Phys. Lett. 103 162905
[32] Huang C P, Wang S B, Yin X G, Zhang Y, Liu H, Zhu Y Y and Chan C T 2012 Phys. Rev. B 86 085446
[33] Huang C P, Yin X G, Zhang Y, Wang S B, Zhu Y Y, Liu H and Chan C T 2012 Phys. Rev. B 85 235410
[34] Chen Y J, Gao P, Wang R X, Zhu C L, Wang L J, Cao M S and Jin H B 2009 J. Phys. Chem. C 113 10061
[35] Deng L J and Han M G 2007 Appl. Phys. Lett. 91 023119
[36] Shi X L, Cao M S, Yuan J and Fang X Y 2009 Appl. Phys. Lett. 95 163108
[37] Eftimiu C and Pearson L W 1989 Radio Sci. 24 351
[38] Chen X, Grzegorczyk T M and Kong J A 2006 Prog. Electromagn. Res. 60 1
[39] Alu A 2011 Phys. Rev. B 83 081102
[40] Powell D A and Kivshar Y S 2010 Appl. Phys. Lett. 97 091106