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Quantum topological transition in hyperbolic metamaterials based on high Tc superconductors

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2014 J. Phys.: Condens. Matter 26 305701

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Recent advances in electromagnetic metamaterials have revealed such fascinating optical effects as negative refraction and electromagnetic cloaking. Even more consequential for condensed matter physics appears to be the recent observation that hyperbolic metamaterials may exhibit a drastic transition in the topology of the photon iso-frequency surface, from a closed ellipsoid to an open hyperboloid [1]. This novel topo-logical transition results in a considerable increase of the pho-tonic density of states, which deeply affects both optical and electronic properties of the material. This topological transi-tion may also be described as a change of metric signature of the effective ‘optical space’, via its metric signature tran-sition [2]. Indeed, propagation of monochromatic extraordi-nary light inside a hyperbolic metamaterial is described by a wave equation, which exhibits 2 + 1 dimensional Lorentz symmetry. The role of time in the corresponding effective 3D Minkowski spacetime is played by the spatial coordinate, which is oriented along the optical axis of the metamaterial [2, 3]. Since the best description of a physical system must

take into account its topology and symmetries, the language of ‘effective Minkowski spacetime’ appears to be quite con-venient. In addition, this language establishes a connec-tion between hyperbolic metamaterials and other condensed matter analogue models of gravity (see the recent extensive review [4] and the references therein). In some sense, hyper-bolic metamaterials offer an interesting experimental window into the physics of Minkowski spacetimes [5–8]. However, because of the different natures of metamaterial and cosmo-logical systems, there are no direct implications of metamate-rial research on gravitational physics.

Here we analyze electromagnetic properties of high Tc superconductors and demonstrate that they may exhibit hyperbolic metamaterial behavior in the far infrared and THz frequency ranges. Moreover, according to the calculations pre-sented below, in the THz range the hyperbolic behavior occurs only in the normal state, while no propagating photon modes exist in the superconducting state. Therefore, we predict that a novel quantum topological transition may be observed for

Journal of Physics: Condensed Matter

Quantum topological transition in hyperbolic metamaterials based on high Tc superconductors

Igor I Smolyaninov

Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA

E-mail: smoly@umd.edu

Received 11 March 2014, revised 18 May 2014Accepted for publication 9 June 2014Published 8 July 2014

AbstractHyperbolic metamaterials are known to exhibit a transition in the topology of the photon iso-frequency surface from a closed ellipsoid to an open hyperboloid, resulting in a considerable increase of the photonic density of states. This topological transition may also be described as a change of metric signature of the effective optical space. Here we demonstrate that high Tc superconductors exhibit hyperbolic metamaterial behavior in the far infrared and THz frequency ranges. In the THz range the hyperbolic behavior occurs only in the normal state, while no propagating photon modes exist in the superconducting state. Thus, a quantum topological transition may be observed for THz photons at zero temperature as a function of the external magnetic field, in which the effective Minkowski spacetime arises in the mixed state of the superconductor at some critical value of the external magnetic field. Nucleation of effective Minkowski spacetime occurs via the formation of quantized Abrikosov vortices.

Keywords: superconductivity, metamaterial, topological transition

(Some figures may appear in colour only in the online journal)

I I Smolyaninov

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THz photons in high Tc superconductors at zero temperature as a function of the external magnetic field. As a result of this quantum topological transition an effective Minkowski space-time arises in the mixed state of the superconductor at some critical value of the external magnetic field. Nucleation of this effective Minkowski spacetime occurs via formation of quan-tized Abrikosov vortices. These results appear to be even more interesting given the fact that the actual physical Minkowski vacuum behaves as a somewhat similar superconducting hyperbolic metamaterial when subjected to very strong mag-netic field [9, 10].

As a first step, let us demonstrate that the wave equa-tion describing propagation of monochromatic extraordinary light inside a hyperbolic metamaterial does indeed exhibit 2 + 1 dimensional Lorentz symmetry. A detailed derivation of this result can be found in [2, 3]. We assume that the meta-material in question is uniaxial and non-magnetic (µ = 1), so that the electromagnetic field inside the metamaterial may be separated into ordinary and extraordinary waves. Vector ⃗E of the extraordinary light wave is parallel to the plane defined by the k–vector of the wave and the optical axis of the metamate-rial. Since hyperbolic metamaterials typically exhibit strong temporal dispersion, we will work in the frequency domain and assume that in some frequency bands around ω = ω0 the metamaterial may be described by anisotropic dielectric tensor, having the diagonal components εxx = εyy = ε1 > 0 and εzz = ε2 < 0. In the linear optics approximation all the non-diagonal components are assumed to be zero. Propagation of extraordinary light in such a metamaterial may be described by a coordinate-dependent wave function Ψω = Ez obeying the following wave equation [2, 3]:

⎛

⎝⎜

⎞

⎠⎟ω ψ

ψε ε

ψ ψ− =

∂∂

+∂∂

+∂∂ω

ω ω ω

c z x y

12

2

2

12

2

2

2

2

2 (1)

(note that the ordinary portion of the electromagnetic field does not contribute to Ψω. This wave equation coincides with the Klein–Gordon equation for a massive scalar field Ψω in 3D Minkowski spacetime:

⎛

⎝⎜

⎞

⎠⎟

ψε ε

ψ ψ ω ψ ψ−∂

∂+

−∂∂

+∂∂

= = *ℏ

ω ω ωω ωz x y c

m c1

( )

2

12

2

2

2

2

20

2

2

2 2

2 (2)

in which spatial coordinate z = τ behaves as a ‘timelike’ vari-able. For example, it is easy to check that equation (2) remains invariant under the effective Lorentz coordinate transformation

ββ′ =

−−

εε−

z z x1

1( )

( )1

2

(3)

⎛⎝⎜

⎞⎠⎟

ββ ε

ε′ =

−−

−εε−

x x z1

1 ( )( )

1

21

2

where β is the effective Lorentz boost. Moreover, similar to our own Minkowski spacetime, together with translations and rotations in the xy plane these effective Lorentz transforma-tions form the Poincare group. Thus, equation  (2) describes world lines of massive particles which propagate in a flat 2 +

1 dimensional Minkowski spacetime [2, 3]. Note that the com-ponents of metamaterial dielectric tensor define the effective metric gik of this spacetime: g00 = − ε1 and g11 = g22 = − ε2. The metric signature has Minkowski character if ε1 and ε2 have opposite signs, and Euclidean character if the signs of ε1 and ε2 are the same. It is also clear that the extraordinary photon dispersion law

ωε ε

= ++

c

k k kz x y2

2

2

1

2 2

2 (4)

experiences a transition in the topology of the photon iso-fre-quency surface, from a closed ellipsoid (when the signs of both ε1 and ε2 are positive) to an open hyperboloid (when ε1 and ε2 have opposite signs). This observation establishes a clear cor-respondence between the topological transitions studied in [1] and the metric signature transitions studied in [2].

As a second step, let us demonstrate that copper oxide-based high Tc superconductors exhibit hyperbolic metamate-rial behavior in the far infrared and THz frequency ranges. Hyperbolic metamaterials are typically composed of multi-layer metal-dielectric or metal wire array structures. However, a few natural materials, such as sapphire and bismuth [11] also exhibit hyperbolic behavior in a limited frequency range. Let us demonstrate that typical high Tc superconduc-tors also belong to this class of ‘natural’ hyperbolic materials. Comparison of the crystallographic unit cell of a BSCCO high Tc superconductor and geometry of a layered hyperbolic metamaterial (shown in figure 1) strongly indicates potential hyperbolic behavior of the high Tc superconductors in an extended spectral range. Indeed, DC conductivity of typical high Tc compounds is known to come from charge carriers (electrons or holes), which mainly reside on the copper oxide planes (figure 1(a)), while conductivity perpendicular to the copper oxide planes is strongly suppressed [12]. In BSCCO the anisotropy of DC conductivity may reach 104 for the ratio of in plane to out of plane conductivity in high quality single crystal samples. Polarization-dependent AC reflec-tance spectra measured in the THz and far-infrared frequency ranges [12] also indicate extreme anisotropy. In the normal state of high Tc superconductors the in-plane AC conduc-tivity exhibits Drude-like behavior with a plasma edge close to 10000 cm − 1, while AC conductivity perpendicular to the copper oxide planes is nearly insulating. Extreme anisotropy is also observed in the superconducting state. The typical values of measured in-plane and out of plane condensate plasma frequencies in high Tc superconductors are ωs,ab = 4000–10 000 cm  −  1, and ωs,c = 1–1000 cm  −  1, respectively [12]. Once again, the measured anisotropy is the strongest in the BSCCO superconductors. Thus, experimental meas-urements strongly support the qualitative picture of BSCCO structure as a layered hyperbolic metamaterial (figure 1(b)) in which the copper oxide layers may be represented as metallic layers, while the SrO and BiO layers may be represented as the layers of dielectric. This qualitative picture is supported by the following quantitative analysis.

Let us calculate the diagonal components of the permit-tivity tensor of a typical high Tc superconductor in the THz

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and far-infrared ranges using Maxwell-Garnett approxima-tion. These components can be calculated similarly to [13], as follows:

ε αε α ε ε ε εα ε αε

= + − =− +

(1 ) ,(1 )

m dm d

m d1 2 (5)

where α is the volume fraction of superconducting phase, and εm  <  0 and εd  >  0 are the dielectric permittivities of the superconductor and dielectric, respectively. The validity of the Maxwell-Garnett approximation has been clearly demonstrated in [13]. Analytical calculations, based on the Maxwell-Garnett approximation and performed for the peri-odic array of metal nanolayers, were confronted with exact numerical solutions of the Maxwell equations. An excellent agreement between numerical simulations and analytical results was demonstrated. Note that calculations of permit-tivity tensor components in [13] has been performed for anisotropic nanoplasmonic composites in the limit of nano-meter scale individual layer thicknesses. Therefore, the theo-retical description developed in [13] is also applicable in our case. However, as has been shown by Larkin and Stockman [14], permittivity of silver and gold may deviate from the pre-dictions of the Drude model when particle size (or layer size) becomes smaller than 5 nm. While these corrections may need to be calculated from the first principles, it is reason-able to assume that they will affect the motion of electrons perpendicular to layers stronger than they affect the motion of electrons along the layers.

Our calculations will be performed for T < Tc and centred around the spectral range ωs,c < ω < ωs,ab, so that AC conduc-tivity perpendicular to the copper oxide layers may indeed be considered dielectric. For the in-plane permittivity we assume

Drude-like behavior supported by AC measurements (see figure 5 from [12]):

ε εωω

= −∞ ,m ms ab,2

2 (6)

while the out of plane permittivity will be approximated as

ε εωω

= −∞ ,d ds c,2

2 (7)

where ε ~∞ 4m is the dielectric permittivity of copper oxide layers above the plasma edge, and ε ∞d is the dielectric per-mittivity of the undoped insulating parent copper oxide com-pound. The parent perovskite compounds typically exhibit rather large values of ε ∞d , which may be estimated from their out of plane reflectivity. Based on figure 5 from [12] ε ~∞ 25d may be assumed. The calculated diagonal components of the permittivity tensor for a high Tc superconductor having ωs,ab = 10000 cm − 1 and ωs,c = 1000 cm − 1 are presented in figure 2. Based on the crystallographic unit cell of BSCCO α = 0.36 has been assumed in these calculations. The hyperbolic behavior appears in the 200 cm − 1  < ω <  1200 cm − 1 and 2400 cm  −  1  <  ω  <  4800 cm  −  1 spectral ranges. The appear-ance of hyperbolic bands is quite generic, independent of a particular choice of ωs,ab and ωs,c, as long as strong anisot-ropy of ωs is maintained. Within these spectral bands ε1 and ε2 have opposite signs, so that the effective spacetime described by equation (2) has Minkowski character and the topology of the photon iso-frequency surface is an open hyperboloid. The appearance of such an effective Minkowski spacetime has clear experimental signatures. If a dipole source, oscillating at a frequency ω0 and located within the hyperbolic band, is placed inside the high Tc superconductor, its radiation pattern

Figure 1. Comparison of the crystallographic unit cell of a BSCCO high Tc superconductor (a) and geometry of a layered hyperbolic metamaterial (b).

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looks like a light cone in Minkowski spacetime[5] (see inset in figure 2).

As a side note, we should point out that the demonstrated hyperbolic character of photon dispersion in high Tc super-conductors may have important consequences for the electron pairing interaction. The importance of plasmon modes for electron pairing in layered high Tc superconductors has been emphasized by many authors in the past (see, for example [15]). However, the fact that hyperbolic character of photon dispersion leads to appearance of broadband divergences in the photonic (plasmonic) density of states has been real-ized only very recently [2, 16]. This broadband divergence (which is cut off at very large scale k ~ 3 nm-1 corresponding to distance between copper oxide layers) should lead to con-siderable enhancement of the plasmon-mediated pairing of electrons proposed in [15]. Another potentially important consequence of hyperbolic behavior of high Tc superconduc-tors is that these materials may be used in hyperlens-based superresolution imaging [17, 18] in the THz and far-infrared frequency ranges. Such novel hyperlenses will have very interesting switchable properties as a function of temperature, current and the external magnetic field. We should also point out that a similar conclusion has been reached in [19–21].

Let us now investigate if a metric signature/topology transition [1, 2] is likely to occur upon transition from the superconducting to the normal state of a high Tc supercon-ductor. Based on experimental data reported in figure  5 from [12] it appears that the far-infrared hyperbolic band at 2400 cm − 1  < ω < 4800 cm − 1 is not affected by the super-conducting transition. This band is primarily defined by the plasma edge of the in-plane AC conductivity located close to 10 000 cm  −  1. This plasma edge is mostly unaffected by

the superconducting transition. On the other hand, as dem-onstrated in figure  3, the lower frequency hyperbolic band at ω < 1200 cm − 1 is strongly affected by the transition from superconducting to normal state. Calculations performed using the same parameters as in figure 2 indicate that a metric signa-ture transition is observed for ω <  180 cm − 1. This transition is caused by the disappearance of the ωs,c term in equation (7) in the normal state, which is observed in the experiment [12]. The same conclusion may also be reached based on results of [19], since the zeros of the permittivity components in equa-tion (5) are related to the Josephson plasma frequency, which depends on temperature. Thus, in the superconducting state photons cannot propagate below ω <  180 cm − 1 since ε is neg-ative in all directions. Upon transition into the normal state, an effective 3D Minkowski spacetime appears for ω <  180 cm − 1 extraordinary photons. Such a transition may occur at T = 0 as a function of the external magnetic field. Therefore, the transi-tion belongs to a class of quantum phase transitions. A some-what similar quantum metric signature/topology transition may occur at T = 0 in physical vacuum subjected to a much stronger magnetic field [9, 10]. Note that the measured room temperature in-plane reflectivity of the typical high Tc super-conductors below ω <  180 cm − 1 exceeds 94% (see figure 5 from [12]). Therefore, Im(ε1)/Re(ε1), which is dominated by

Figure 2. Diagonal components of the permittivity tensor of a high Tc superconductor calculated as a function of frequency assuming ωs,ab = 10000 cm − 1, ωs,c = 1000 cm − 1, ε =∞ 4m and ε =∞ 25d . The hyperbolic bands appear at 200 cm − 1  < ω <  1200 cm − 1 and 2400 cm − 1  < ω <  4800 cm − 1. The inset shows the radiation pattern of a dipole source placed inside a hyperbolic metamaterial calculated using COMSOL Multiphysics 4.2 solver. Radiation pattern looks like a light cone in a 2 + 1 dimensional Minkowski spacetime in which spatial z coordinate plays the role of a ‘timelike’ coordinate.

Figure 3. Comparison of diagonal components of the permittivity tensor of a high Tc superconductor in superconducting and normal states in the ω <  1200 cm − 1 frequency range. Calculations were performed using the same parameters as in figure 2. The metric signature transition is observed for ω <  180 cm − 1.

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Im(εm)/Re(εm), does not exceed 0.06, even at room tempera-ture, and further decreases on cooling, making high Tc super-conductors an excellent choice for studying the emergence of effective Minkowski spacetimes in hyperbolic metamaterials.

Let us consider the detailed mechanism of this transi-tion from the point of view of extraordinary photons in the ω <  180 cm − 1 spectral range. At H < Hc1 magnetic field does not penetrate into the superconductor, and no propa-gating electromagnetic modes exist in the superconducting state. On the other hand, as demonstrated by figure  3, at H > Hc2 the high Tc superconductor will be in the normal ‘hyperbolic metamaterial’ state. As described earlier, extraordinary photons in the ω <  180 cm − 1 spectral range perceive this state as an effective 3D Minkowski space-time. ‘Nucleation’ of this effective Minkowski spacetime from superconductive ‘nothingness’ occurs via the mixed state, in which the external magnetic field gradually pen-etrates into the superconductor via formation of quantized Abrikosov vortices [22]. These vortices have normal cores with dimensions approximately  equal to the supercon-ducting coherence length ξ. Therefore, they may be consid-ered to be elementary ‘Minkowski spacetime quanta’. Even though ξ is much smaller than the free space wavelength λ in the ω <  180 cm − 1 spectral range, the macroscopic elec-trodynamics treatment of vortices as small pieces of hyper-bolic metamaterial makes sense due to the open hyperboloid (divergent) nature of photon dispersion law in a hyperbolic metamaterial (see equation  (4)). This divergence (due to opposite signs of ε1 and ε2) is cut off only when the photon k vector components are comparable to the inverse crystal-lographic cell size [2, 16]. When the external magnetic field

is gradually increased above Hc1, the individual Abrikosov vortices form a crystal lattice. This lattice of ‘Minkowski spacetime quanta’ may be considered to be a homogeneous medium once the Abrikosov lattice periodicity becomes much smaller than λ. The volume fraction of normal phase in the mixed state may be obtained as [22]

β ξΦ

= ≈H H

Hc

2

0 2 (8)

where Φ0 is the magnetic flux quantum. Using equations (5–8) we may calculate the averaged diagonal components of the permittivity tensor in the mixed state as a function of the external magnetic field. These calculations are performed by taking into account the increase in volume fraction of the normal phase, given by equation (8), which reduces ωs,ab in equation  (5). Our use of the Maxwell-Garnett theory in the case of 2 D periodic structure (such as wire array metamate-rials and the Abrikosov vortex lattice) is based on the results of [13]. In this paper analytical calculations, based on the Maxwell-Garnett approximation and performed for peri-odic wire array nanostructures, were confronted with exact numerical solutions of Maxwell equations. Excellent agree-ment between numerical simulations and analytical results was demonstrated. Results of our calculations performed at ω = 150 cm − 1 are presented in figure 4. These calculations were performed using the same parameters as in figure  2. Macroscopic effective Minkowski spacetime emerges from the Abrikosov vortex lattice at the critical field HM = 0.47Hc2. Emergence of macroscopic Minkowski domains within the Abrikosov lattice state also has clear experimental signatures. As demonstrated in the inset in figure 4, the electromagnetic field at the Minkowski domain boundaries diverges in the zero loss limit. As has been demonstrated in [6, 23], from the electro-magnetic standpoint these boundaries are somewhat similar to the black hole and cosmological event horizons. Metamaterial losses, which are unavoidable in hyperbolic metamaterials tame these divergences. However, considerable field enhance-ment is still observed at Im(ε1)/Re(ε1) ~ 0.06, used in our simulations. Thus, the described system may indeed be used as an experimental model of hypothesized quantum nucleation of Minkowski spacetime from ‘nothing’. An interesting addi-tional feature of this model is that the transition from quantum to continuous effective Minkowski spacetime may be traced as a function of the external magnetic field.

We should also note that even though superconductors behave as perfect diamagnetics in the zero frequency limit, at frequencies above the superconducting gap the magnetic permeability of superconductors and normal metals is about the same, so that µ = 1 value may be assumed at high frequen-cies. Our calculations presented in figure 4 are performed at 150 cm − 1, which corresponds to 190 K, well above the super-conducting gap of BSCCO. Therefore, at the frequencies of interest, treatment of the superconducting phase as non-mag-netic material is well justified.

In conclusion, we have demonstrated that high Tc super-conductors exhibit hyperbolic metamaterial behavior in the far infrared and THz frequency ranges. Such superconductor-based hyperbolic metamaterials exhibit a quantum topological

Figure 4. Diagonal components of the permittivity tensor of a high Tc superconductor calculated at T = 0 as a function of the external magnetic field at ω= 150 cm − 1. Calculations were performed using the same parameters as in figure 2. Macroscopic effective Minkowski spacetime emerge from the Abrikosov vortex latice at HM = 0.47Hc2. The inset shows an example of electromagnetic field distribution in a random configuration of Minkowski domains calculated using COMSOL Multiphysics 4.2 solver. In the limit of zero losses electromagnetic field diverges at the Minkowski domain boundaries (the shown simulation assumes Im(ε1)/Re(ε1) ~ 0.06).

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phase transition at T = 0, in which the effective Minkowski spacetime arises in the mixed state of the superconductor at some critical value of the external magnetic field. Nucleation of Minkowski spacetime occurs via formation of quantized Abrikosov vortices, so that these vortices play the role of effective Minkowski spacetime quanta.

Acknowledgement

I acknowledge helpful conversations with Vera Smolyaninova.

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