Abstract

The Casimir force between electric and magnetic hyperbolic metamaterial slabs is investigated. Due to hyperbolic dispersion, the electromagnetic features of these metamaterials along the optical axis are different from those perpendicular to the optical axis; consequently, these features contribute differently to the Casimir effect. The repulsive Casimir force is formed between electric and magnetic hyperbolic metamaterial slabs; moreover, hyperbolic dispersion can enhance the repulsive effect. However, by utilizing the extremely anisotropic behavior of hyperbolic metamaterials and changing the separation distance between the two slabs, the restoring Casimir force emerges. Additionally, by considering the dispersion of both the permittivity and the permeability of hyperbolic metamaterials, the Casimir force reaches several equilibrium points at different separation distances. Furthermore, the Casimir force at room temperature is discussed. Although the temperature can weaken the effect of the restoring Casimir force, stable equilibria may remain upon choosing suitable filling factors. This work shows that hyperbolic metamaterials have potential applications in micro- and nanoelectromechanical systems, especially for maintaining stability and overcoming adhesion problems.

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

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    [Crossref]
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2018 (1)

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

2017 (1)

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

2016 (4)

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(1), 11329 (2016).
[Crossref] [PubMed]

M. S. Mirmoosa, S. Y. Kosulnikov, and C. R. Simovski, “Magnetic hyperbolic metamaterial of high-index nanowires,” Phys. Rev. B 94(7), 075138 (2016).
[Crossref]

V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
[Crossref] [PubMed]

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

2015 (1)

2014 (1)

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

2013 (5)

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Tuning the Casimir force via modification of interface properties of three-dimensional topological insulators,” J. Appl. Phys. 113(20), 204302 (2013).
[Crossref]

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

T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21(12), 14943–14955 (2013).
[Crossref] [PubMed]

2012 (3)

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20(14), 15100–15105 (2012).
[Crossref] [PubMed]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

2011 (5)

S. I. Maslovski and M. G. Silveirinha, “Mimicking Boyer’s Casimir repulsion with a nanowire material,” Phys. Rev. A 83(2), 022508 (2011).
[Crossref]

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
[Crossref]

A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
[Crossref] [PubMed]

R. Esquivel-Sirvent and G. C. Schatz, “Mixing rules and the Casimir force between composite systems,” Phys. Rev. A 83(4), 042512 (2011).
[Crossref]

2010 (4)

M. G. Silveirinha and S. I. Maslovski, “Physical restrictions on the Casimir interaction of metal-dielectric metamaterials: An effective-medium approach,” Phys. Rev. A 82(5), 052508 (2010).
[Crossref]

M. G. Silveirinha, “Casimir interaction between metal-dielectric metamaterial slabs: Attraction at all macroscopic distances,” Phys. Rev. B Condens. Matter Mater. Phys. 82(8), 085101 (2010).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

S. I. Maslovski and M. G. Silveirinha, “Ultralong-range Casimir-Lifshitz forces mediated by nanowire materials,” Phys. Rev. A 82(2), 022511 (2010).
[Crossref]

2009 (6)

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, “The Casimir force between real materials: Experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009).
[Crossref]

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

V. Yannopapas and N. V. Vitanov, “First-principles study of Casimir repulsion in metamaterials,” Phys. Rev. Lett. 103(12), 120401 (2009).
[Crossref] [PubMed]

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
[Crossref] [PubMed]

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

2008 (3)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir interactions for anisotropic magnetodielectric metamaterials,” Phys. Rev. A 78(3), 032117 (2008).
[Crossref]

2007 (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

2006 (2)

2003 (1)

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, “Measurement of the Casimir force between dissimilar metals,” Phys. Rev. Lett. 91(5), 050402 (2003).
[Crossref] [PubMed]

2002 (3)

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
[Crossref] [PubMed]

O. Kenneth, I. Klich, A. Mann, and M. Revzen, “Repulsive Casimir forces,” Phys. Rev. Lett. 89(3), 033001 (2002).
[Crossref] [PubMed]

2001 (2)

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
[Crossref]

1998 (2)

F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 μ m,” Phys. Rev. Lett. 81(21), 4549–4552 (1998).
[Crossref]

1956 (1)

E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2(2), 73–83 (1956).

1948 (1)

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Alekseyev, L. V.

Atkinson, J.

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Converg. 1(1), 14 (2014).
[Crossref] [PubMed]

Bandlow, B.

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
[Crossref]

Barnakov, Y. A.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[Crossref] [PubMed]

Belov, P.

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

Bi, M. H.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

Bordag, M.

M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rep. 353(1–3), 1–205 (2001).
[Crossref]

Bressi, G.

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

Buks, E.

E. Buks and M. L. Roukes, “Casimir force changes sign,” Nature 419(6903), 119–120 (2002).
[Crossref] [PubMed]

E. Buks and M. L. Roukes, “Stiction, adhesion energy, and the Casimir effect in micromechanical systems,” Phys. Rev. B Condens. Matter Mater. Phys. 63(3), 033402 (2001).
[Crossref]

Capasso, F.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

J. N. Munday, F. Capasso, and V. A. Parsegian, “Measured long-range repulsive Casimir-Lifshitz forces,” Nature 457(7226), 170–173 (2009).
[Crossref] [PubMed]

Carugno, G.

G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88(4), 041804 (2002).
[Crossref] [PubMed]

Casimir, H. B. G.

H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. K. Ned. Akad. Wet. 51(7), 793–795 (1948).

Chen, H.

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

Chen, L.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

Chen, X.

J. C. Martinez, X. Chen, and M. B. A. Jalil, “Casimir effect and graphene: Tunability, scalability, Casimir rotor,” AIP Adv. 8(1), 015330 (2018).
[Crossref]

Cortijo, A.

A. G. Grushin and A. Cortijo, “Tunable Casimir Repulsion with Three-Dimensional Topological Insulators,” Phys. Rev. Lett. 106(2), 020403 (2011).
[Crossref] [PubMed]

Dalvit, D. A. R.

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, “Casimir-Lifshitz theory and metamaterials,” Phys. Rev. Lett. 100(18), 183602 (2008).
[Crossref] [PubMed]

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F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
[Crossref]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Weiland, T.

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
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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(1), 11329 (2016).
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Woolf, D.

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
[Crossref]

Xu, J. P.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
[Crossref]

Yang, Y. P.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
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R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

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[Crossref] [PubMed]

Zeng, R.

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
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Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

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Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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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(1), 11329 (2016).
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J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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Zhao, R.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
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Zhou, J.

R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
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Zhu, C. J.

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
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Zhu, G.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
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G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
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R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
[Crossref]

W. J. Nie, R. Zeng, Y. H. Lan, and S. Y. Zhu, “Casimir force between topological insulator slabs,” Phys. Rev. B Condens. Matter Mater. Phys. 88(8), 085421 (2013).
[Crossref]

R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
[Crossref]

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Y. P. Yang, R. Zeng, H. Chen, S. Y. Zhu, and M. S. Zubairy, “Controlling the Casimir force via the electromagnetic properties of materials,” Phys. Rev. A 81(2), 022114 (2010).
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Appl. Phys. Lett. (3)

A. W. Rodriguez, D. Woolf, P.-C. Hui, E. Iwase, A. P. McCauley, F. Capasso, M. Loncar, and S. G. Johnson, “Designing evanescent optical interactions to control the expression of Casimir forces in optomechanical structures,” Appl. Phys. Lett. 98(19), 194105 (2011).
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M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
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IEEE Trans. Microw. Theory Tech. (1)

G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective Modeling of Double Negative Metamaterial Macrostructures,” IEEE Trans. Microw. Theory Tech. 57(5), 1136–1146 (2009).
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F. M. Serry, D. Walliser, and G. J. Maclay, “The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS),” J. Appl. Phys. 84(5), 2501–2506 (1998).
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Nat. Commun. (1)

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A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
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Nature (3)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
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Opt. Express (3)

Phys. Lett. A (1)

R. Zeng, L. Chen, W. J. Nie, M. H. Bi, Y. P. Yang, and S. Y. Zhu, “Enhancing Casimir repulsion via topological insulator multilayers,” Phys. Lett. A 380(36), 2861–2869 (2016).
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R. Zeng and Y. P. Yang, “Tunable polarity of the Casimir force based on saturated ferrites,” Phys. Rev. A 83(1), 012517 (2011).
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R. Zeng, Y. P. Yang, and S. Y. Zhu, “Casimir force between anisotropic single-negative metamaterials,” Phys. Rev. A 87(6), 063823 (2013).
[Crossref]

G. Song, J. P. Xu, C. J. Zhu, P. F. He, Y. P. Yang, and S. Y. Zhu, “Casimir force between hyperbolic metamaterials,” Phys. Rev. A 95(2), 023814 (2017).
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R. Zhao, J. Zhou, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Repulsive Casimir force in chiral metamaterials,” Phys. Rev. Lett. 103(10), 103602 (2009).
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Figures (6)

Fig. 1
Fig. 1 Scheme of the considered system. A and B are an ε-HMM and a μ-HMM, respectively.
Fig. 2
Fig. 2 Real parts of ε x x A (solid blue lines) and ε z z A (dotted red lines) as functions of ω with different filling factors: (a) f A = 0.2 , and (b) f A = 0.5 . Real parts of μ x x B (solid blue lines) and μ z z B (dotted red lines) with different filling factors: (c) f B = 0.2 , and (d) f B = 0.5 . Other parameters are mentioned in the text. Shaded areas indicate regions of hyperbolic dispersion.
Fig. 3
Fig. 3 The relative Casimir force F r = F C / F 0 between an ε-HMM and a μ-HMM as a function of the separation a with different filling factors. (a) The filling factor of the ε-HMM f A varies, but the filling factor of the μ-HMM f B is fixed as f B = 0.5 . The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when f B = 0.5 . (b) The filling factor of the μ-HMM f B varies, but the filling factor of the ε-HMM f A is fixed as f A = 0.5 . The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when f B = 0.2 . Other parameters are mentioned in the text.
Fig. 4
Fig. 4 The relative Casimir force F r between an ε-HMM and a μ-HMM as a function of the separation a with different f B . The restoring forces are shown in the curves for f B = 0.2 (dashed blue line), 0.5 (dotted magenta line), and 1 (dash-dotted red line). The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with f B = 0.2 .
Fig. 5
Fig. 5 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation a with different f B . Multiequilibrium appears when f B = 0.5 (dash-dotted magenta line). Circles and squares are the transition points of the force polarity. Squares indicate stable equilibria. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with f B = 0.5 . Other parameters are mentioned in the text.
Fig. 6
Fig. 6 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation a with different f B at room temperature T = 300 K . Circles and squares are the transition points of the force polarity. Squares indicate a stable equilibrium. The parameters are the same as those in Fig. 5.

Equations (19)

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ε A = ( ε x x A 0 0 0 ε y y A = ε x x A 0 0 0 ε z z A ) ,
ε x x A = f A ε m A + ( 1 f A ) ε d A ,
ε z z A = ε m A ε d A f A ε d A + ( 1 f A ) ε m A .
μ B = ( μ x x B 0 0 0 μ y y B = μ x x B 0 0 0 μ z z B ) ,
μ x x B = f B μ e f f B + ( 1 f B ) ,
μ z z B = μ e f f B f B + ( 1 f B ) μ e f f B .
F C = π Re 0 d ω d 2 k | | ( 2 π ) 2 ω 2 c 2 k | | 2 p = TE,TM r p A ( ω , k | | ) r p B ( ω , k | | ) e 2 i a ω 2 / c 2 k | | 2 1 r p A ( ω , k | | ) r p B ( ω , k | | ) e 2 i a ω 2 / c 2 k | | 2 ,
F C = 2 π 2 0 d ξ 0 k | | d k | | ξ 2 c 2 + k | | 2 p = TE,TM r p A ( i ξ , k | | ) r p B ( i ξ , k | | ) e 2 a ξ 2 / c 2 + k | | 2 1 r p A ( i ξ , k | | ) r p B ( i ξ , k | | ) e 2 a ξ 2 / c 2 + k | | 2 .
r TE A = μ A k 0 z k A z TE μ A k 0 z + k A z TE , r TM A = ε x x A k 0 z k A z TM ε x x A k 0 z + k A z TM .
k | | 2 + ( k A z TE ) 2 = K 2 ε y y A μ A ,
k | | 2 ε z z A + ( k A z TM ) 2 ε x x A = K 2 μ A .
r TE B = μ x x B k 0 z k B z TE μ x x B k 0 z + k B z TE , r TM B = ε B k 0 z k B z TM ε B k 0 z + k B z TM .
k | | 2 μ z z B + ( k B z TE ) 2 μ x x B = K 2 ε B ,
k | | 2 + ( k B z TM ) 2 = K 2 ε B μ y y B .
ε m A = 1 Ω m 2 ω 2 + i γ m ω ,
ε d A = 1 Ω d 2 ω 2 ω d 2 + i γ d ω .
μ e f f B = 1 Ω e f f 2 ω 2 ω e f f 2 + i γ e f f ω .
ε B = 1 Ω B 2 ω 2 + i γ B ω .
F C = k B T π m = 0 ( 1 δ m 0 2 ) 0 k | | d k | | ξ m 2 c 2 + k | | 2 p = TE,TM r p A ( i ξ m , k | | ) r p B ( i ξ m , k | | ) e 2 a ξ m 2 / c 2 + k | | 2 1 r p A ( i ξ m , k | | ) r p B ( i ξ m , k | | ) e 2 a ξ m 2 / c 2 + k | | 2 ,

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