Abstract

Composite structures based on metal open rings and thin wires are well established, for obtaining efficient negative index materials (NIM), acting as metamaterials in the long wavelength regime. The main losses are due both to metal absorption and to the inner electric resistance of metals; to overcome this latter loss we propose a new metal-semiconductor structure dimensioned by direct synthesis method, which offers an almost perfect Drude-like effective magnetic permeability. The choice of particular semiconductor components allows to get a negative resistance for the current induced by the electromagnetic field, which cancels that of the metal but puts a limit to the spectral response of the metamaterial. We consider some parasite effects, such as bianisotropy and incorrect values of structural parameters, to see limitations and features of this new NIM technology.

© 2007 Optical Society of America

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References

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  1. V. G. Veselago, "The Electrodynamics of substances with simultaneously negative vaues of ? and ?," Sov. Phys. Usp. 10, 509 (1968)
    [CrossRef]
  2. S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Pro. Phys. 68 (2005) 449-521
    [CrossRef]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
    [CrossRef]
  4. A. J. Ward, J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43773-793 (1996)
    [CrossRef]
  5. J. B. Pendry, "Negatve Refraction Makes a Perfect Lens," Phys. Rev. Lett.3966-3969 (2000)
    [CrossRef] [PubMed]
  6. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
    [CrossRef]
  7. R. W. Ziolkowski, "Superlumina Transmision of Information through an Electromagnetic Metamaterial," Phys. Rev. E 63, 046604 (2001)
    [CrossRef]
  8. S. D. Gupta, R. Arun, and G. S. Agarwal, "Subluminal to Superluminal Propagation in aLeft-Handed Medium," Phys. Rev. B 69, 113104 (2004)
    [CrossRef]
  9. D. R. Smith and N. Kroll, Phys., "Negative Refractive Index in Left-Handed Materials," Rev. Lett. 85, 2933 (2000)
    [CrossRef]
  10. R. W. Ziolkowski and A. D. Kipple, "Causality and Double-Negative Metamateria," Phys. Rev. E 68, 026615 (2003)
    [CrossRef]
  11. R. W. Ziolkowski and E. Heyman, "Wave Propagation in Media Having Negative Permittivity and Permeability," Phys. Rev. E 64, 056625 (2001)
    [CrossRef]
  12. J. D. Wilson, Z. D. Schwarz, "Multifocal Flat Lens with Left-Handed Metamaterial," Appl. Phys. Lett. 86, 021113 (2005)
    [CrossRef]
  13. D. Pines, D. Bohm, "A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions," Phys. Rev. 85, 338 (1952)
    [CrossRef]
  14. D. Bohm, D. Pines, "A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas," Phys. Rev. 92, 609 (1953)
    [CrossRef]
  15. J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
    [CrossRef] [PubMed]
  16. A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic Plasmon Resonance," Phys. Rev. E 73, 036609 (2006)
    [CrossRef]
  17. M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
    [CrossRef]
  18. M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis and S. Linden, "Single-slit split-ring resonators at optical frequencies: Limits of size scaling," submitted to Opt. Lett. (2006)
    [CrossRef]
  19. I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, "Tunable Split-Ring Resonators for Nonlinear-Negative Index Metamaterials," Opt. Express 14,9344 (2006)
    [CrossRef] [PubMed]
  20. I. Gil, J. Bonache, J. García-García, and F. Martín, "Tunable Metamaterial Transmission Lines Based on Varactor-Loaded Split-Ring Resonators," IEEE Trans. Microwave Theory Technol. 0018-9480 (2006)
  21. S. M. Sze, Semiconductor Devices: Physics and Technology, (Wiley and Sons Inc., 1985) pgs. 237-249
  22. R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002)
    [CrossRef]
  23. Z. Chen, Finite Element Methods and Their Applications, (Springer-Verlag Berlin Heidelberg New York, 2005) pgs. 364-368

2006 (2)

2005 (3)

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

J. D. Wilson, Z. D. Schwarz, "Multifocal Flat Lens with Left-Handed Metamaterial," Appl. Phys. Lett. 86, 021113 (2005)
[CrossRef]

S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Pro. Phys. 68 (2005) 449-521
[CrossRef]

2004 (1)

S. D. Gupta, R. Arun, and G. S. Agarwal, "Subluminal to Superluminal Propagation in aLeft-Handed Medium," Phys. Rev. B 69, 113104 (2004)
[CrossRef]

2003 (1)

R. W. Ziolkowski and A. D. Kipple, "Causality and Double-Negative Metamateria," Phys. Rev. E 68, 026615 (2003)
[CrossRef]

2002 (1)

R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002)
[CrossRef]

2001 (2)

R. W. Ziolkowski and E. Heyman, "Wave Propagation in Media Having Negative Permittivity and Permeability," Phys. Rev. E 64, 056625 (2001)
[CrossRef]

R. W. Ziolkowski, "Superlumina Transmision of Information through an Electromagnetic Metamaterial," Phys. Rev. E 63, 046604 (2001)
[CrossRef]

2000 (2)

D. R. Smith and N. Kroll, Phys., "Negative Refractive Index in Left-Handed Materials," Rev. Lett. 85, 2933 (2000)
[CrossRef]

J. B. Pendry, "Negatve Refraction Makes a Perfect Lens," Phys. Rev. Lett.3966-3969 (2000)
[CrossRef] [PubMed]

1999 (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
[CrossRef]

1996 (2)

A. J. Ward, J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43773-793 (1996)
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

1968 (1)

V. G. Veselago, "The Electrodynamics of substances with simultaneously negative vaues of ? and ?," Sov. Phys. Usp. 10, 509 (1968)
[CrossRef]

1953 (1)

D. Bohm, D. Pines, "A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas," Phys. Rev. 92, 609 (1953)
[CrossRef]

1952 (1)

D. Pines, D. Bohm, "A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions," Phys. Rev. 85, 338 (1952)
[CrossRef]

Agarwal, G. S.

S. D. Gupta, R. Arun, and G. S. Agarwal, "Subluminal to Superluminal Propagation in aLeft-Handed Medium," Phys. Rev. B 69, 113104 (2004)
[CrossRef]

Arun, R.

S. D. Gupta, R. Arun, and G. S. Agarwal, "Subluminal to Superluminal Propagation in aLeft-Handed Medium," Phys. Rev. B 69, 113104 (2004)
[CrossRef]

Bohm, D.

D. Bohm, D. Pines, "A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas," Phys. Rev. 92, 609 (1953)
[CrossRef]

D. Pines, D. Bohm, "A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions," Phys. Rev. 85, 338 (1952)
[CrossRef]

Economou, E. N.

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Gundogdu, T. F.

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Gupta, S. D.

S. D. Gupta, R. Arun, and G. S. Agarwal, "Subluminal to Superluminal Propagation in aLeft-Handed Medium," Phys. Rev. B 69, 113104 (2004)
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, "Wave Propagation in Media Having Negative Permittivity and Permeability," Phys. Rev. E 64, 056625 (2001)
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

Kafesaki, M.

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Kipple, A. D.

R. W. Ziolkowski and A. D. Kipple, "Causality and Double-Negative Metamateria," Phys. Rev. E 68, 026615 (2003)
[CrossRef]

Kivshar, Y. S.

Koschny, Th.

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, Phys., "Negative Refractive Index in Left-Handed Materials," Rev. Lett. 85, 2933 (2000)
[CrossRef]

Marqués, R.

R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002)
[CrossRef]

Medina, F.

R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002)
[CrossRef]

Morrison, S. K.

Penciu, R. S.

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negatve Refraction Makes a Perfect Lens," Phys. Rev. Lett.3966-3969 (2000)
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

A. J. Ward, J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43773-793 (1996)
[CrossRef]

Pines, D.

D. Bohm, D. Pines, "A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas," Phys. Rev. 92, 609 (1953)
[CrossRef]

D. Pines, D. Bohm, "A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions," Phys. Rev. 85, 338 (1952)
[CrossRef]

Rafii-El-Idrissi, R.

R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002)
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Pro. Phys. 68 (2005) 449-521
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
[CrossRef]

Sarychev, A. K.

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic Plasmon Resonance," Phys. Rev. E 73, 036609 (2006)
[CrossRef]

Schwarz, Z. D.

J. D. Wilson, Z. D. Schwarz, "Multifocal Flat Lens with Left-Handed Metamaterial," Appl. Phys. Lett. 86, 021113 (2005)
[CrossRef]

Shadrivov, I. V.

Shalaev, V. M.

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic Plasmon Resonance," Phys. Rev. E 73, 036609 (2006)
[CrossRef]

Shvets, G.

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic Plasmon Resonance," Phys. Rev. E 73, 036609 (2006)
[CrossRef]

Smith, D. R.

D. R. Smith and N. Kroll, Phys., "Negative Refractive Index in Left-Handed Materials," Rev. Lett. 85, 2933 (2000)
[CrossRef]

Soukoulis, C. M.

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "The Electrodynamics of substances with simultaneously negative vaues of ? and ?," Sov. Phys. Usp. 10, 509 (1968)
[CrossRef]

Ward, A. J.

A. J. Ward, J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43773-793 (1996)
[CrossRef]

Wilson, J. D.

J. D. Wilson, Z. D. Schwarz, "Multifocal Flat Lens with Left-Handed Metamaterial," Appl. Phys. Lett. 86, 021113 (2005)
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski and A. D. Kipple, "Causality and Double-Negative Metamateria," Phys. Rev. E 68, 026615 (2003)
[CrossRef]

R. W. Ziolkowski and E. Heyman, "Wave Propagation in Media Having Negative Permittivity and Permeability," Phys. Rev. E 64, 056625 (2001)
[CrossRef]

R. W. Ziolkowski, "Superlumina Transmision of Information through an Electromagnetic Metamaterial," Phys. Rev. E 63, 046604 (2001)
[CrossRef]

Appl. Phys. Lett. (1)

J. D. Wilson, Z. D. Schwarz, "Multifocal Flat Lens with Left-Handed Metamaterial," Appl. Phys. Lett. 86, 021113 (2005)
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 472075-2084 (1999)
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 11 (1999)
[CrossRef]

J. Mod. Opt. (1)

A. J. Ward, J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Opt. 43773-793 (1996)
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12 (2005)
[CrossRef]

Opt. Express (1)

Phys. Rev. (2)

D. Pines, D. Bohm, "A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions," Phys. Rev. 85, 338 (1952)
[CrossRef]

D. Bohm, D. Pines, "A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas," Phys. Rev. 92, 609 (1953)
[CrossRef]

Phys. Rev. B (2)

S. D. Gupta, R. Arun, and G. S. Agarwal, "Subluminal to Superluminal Propagation in aLeft-Handed Medium," Phys. Rev. B 69, 113104 (2004)
[CrossRef]

R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev. B 65, 144440 (2002)
[CrossRef]

Phys. Rev. E (4)

R. W. Ziolkowski and A. D. Kipple, "Causality and Double-Negative Metamateria," Phys. Rev. E 68, 026615 (2003)
[CrossRef]

R. W. Ziolkowski and E. Heyman, "Wave Propagation in Media Having Negative Permittivity and Permeability," Phys. Rev. E 64, 056625 (2001)
[CrossRef]

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic Plasmon Resonance," Phys. Rev. E 73, 036609 (2006)
[CrossRef]

R. W. Ziolkowski, "Superlumina Transmision of Information through an Electromagnetic Metamaterial," Phys. Rev. E 63, 046604 (2001)
[CrossRef]

Phys. Rev. Lett. (2)

J. B. Pendry, "Negatve Refraction Makes a Perfect Lens," Phys. Rev. Lett.3966-3969 (2000)
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

Rep. Pro. Phys. (1)

S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Pro. Phys. 68 (2005) 449-521
[CrossRef]

Rev. Lett. (1)

D. R. Smith and N. Kroll, Phys., "Negative Refractive Index in Left-Handed Materials," Rev. Lett. 85, 2933 (2000)
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, "The Electrodynamics of substances with simultaneously negative vaues of ? and ?," Sov. Phys. Usp. 10, 509 (1968)
[CrossRef]

Other (4)

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis and S. Linden, "Single-slit split-ring resonators at optical frequencies: Limits of size scaling," submitted to Opt. Lett. (2006)
[CrossRef]

I. Gil, J. Bonache, J. García-García, and F. Martín, "Tunable Metamaterial Transmission Lines Based on Varactor-Loaded Split-Ring Resonators," IEEE Trans. Microwave Theory Technol. 0018-9480 (2006)

S. M. Sze, Semiconductor Devices: Physics and Technology, (Wiley and Sons Inc., 1985) pgs. 237-249

Z. Chen, Finite Element Methods and Their Applications, (Springer-Verlag Berlin Heidelberg New York, 2005) pgs. 364-368

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Figures (5)

Fig. 1.
Fig. 1.

Basic scheme of the metal (blue)-semiconductor (brown) composite ring with negative real magnetic permeability. To feed the NDRM semiconductor, a conductive stripe along x is implemented.

Fig. 2.
Fig. 2.

Block scheme of the time-domain impedance.

Fig. 3.
Fig. 3.

Magneto-electric coupling bringing bianisotropy; with θ it is shown the aperture angle related to the NDRM device (a); the usual way to suppress bianisotropy’s effects is to alternate a column of composite rings with a complementary inverted column of the same kind (b).

Fig. 4.
Fig. 4.

Real (a) and imaginary (b) part of μ. versus angular frequency for Zc (ω) with many values of Q bigger (red) and smaller (green) than the exact value; the required μ. function has an ω 0 of 2πx100 THz, with γ=ω 0 /100; contiguous graphs differ for a ∆Q/Q ratio of 0.01%; (c) a magnification of the μ imaginary part for ∆Q/Q ratio of -0.01% (lower) and -0.02%> (upper).

Fig. 5.
Fig. 5.

A typical high-frequency model for NDRM semiconductor devices.

Equations (34)

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

F ( ω ) = μ eff ( ω ) = 1 ω 0 2 ω ( ω j γ )
j ω μ 0 S η H int ( ω ) = Z c ( ω ) I ( ω ) μ 0 Z c ( ω ) = I ( ω ) H int ( ω )
H int ( ω ) H ext ( ω ) = I ( ω ) h 1 H ext ( ω ) H int ( ω ) = I ( ω ) H int ( ω ) h
F ( ω ) = μ eff ( ω ) = B eff ( ω ) μ 0 H eff ( ω ) = μ 0 H int ( ω ) f + μ 0 H ext ( ω ) ( 1 f ) μ 0 H ext ( ω ) =
= 1 f jωQ Z c ( ω ) + jωQ
Q = μ 0 S η h
Z c ( ω ) = ( F ( ω ) + f 1 ) ( 1 F ( ω ) ) jωQ = jωQ ( F ( ω ) + f 1 ) ( F ( ω ) 1 )
Z c ( ω ) = jωQ ( F ( ω ) 1 ) + f ( F ( ω ) 1 ) = jωQ [ 1 + f ( F ( ω ) 1 ) ] =
= jωQ jωQf ( F ( ω ) 1 ) = jωQ + Qf ω 0 2 ( 3 + γω 2 ) =
= jωQ + ( Qf γ ω 0 2 ) ω 2 + j ( Qf ω 0 2 ) ω 3
jωQ + ( Qf γ ω 0 2 ) ω 2 + j ( Qf ω 0 2 ) ω 3
Z c ( t ) = Q d dt + Q d dt ( f γ ω 0 2 d dt ) + Q d dt [ + f γ ω 0 2 d dt ( 1 γ d dt ) ]
Z m ( ω ) = 1 σ ( ω ) + j ω ε 0 ε r ( ω ) l m S m σ ( ω ) j ω ε 0 ε r ( ω ) σ ( ω ) 2 l m S m =
= 1 σ ( ω ) l m S m ε 0 ε r ( ω ) σ ( ω ) 2 l m S m =
= R m ( ω ) j ω ε 0 ε r ( ω ) S m l m R m ( ω ) 2
R m ( ω ) = 1 σ ( ω ) l m S m 1 ω ε 0 ε r ( ω ) l m S m
R m ( ω ) = R α ω 2 , R α ω 2 , ω B
R m ( ω ) 2 R 2 R α ω 2 ω B
Z m ( ω ) = R α ω 2 ( ε 0 ε r ( ω ) S m l m R 2 ) +
+ j ( ε 0 ε r ( ω ) S m l m 2 ) ω 3
{ ε 0 ε r ( ω ) S m l m R 2 = Q R = l m Q ε 0 ε r ( ω ) S m ε 0 ε r ( ω ) S m l m 2 R α = ( Qf ω 0 2 ) α = f 2 ω 0 2 R
R TED ( ω ) = R + β ω 2 , R β ω 2 ω B
Z c ( ω ) = ( β α ) ω 2 ( ε 0 ε r ( ω ) S m l m R 2 ) +
+ j ( ε 0 ε r ( ω ) S m l m 2 ) ω 3 =
= jωQ + ( Qf γ ω 0 2 ) ω 2 + j ( Qf ω 0 2 ) ω 3
{ R = Q l m ε 0 ε r ( ω 0 ) S m = μ 0 S η l m ε 0 ε r ( ω 0 ) hS m β = f ω 0 2 ( Q γ + R 2 )
ω max = min { 1 10 l m Q ε 0 ε r ( ω 0 ) S m , ω 0 10 2 f , 1 10 ω 0 f ( Q R γ + 1 2 ) , 2 πc 7 D n s } =
= min { 1 10 c hl m ε r ( ω 0 ) S m S η , 1 10 ω 0 f ( γ c ε r ( ω ) S m S η l m h + 1 2 ) , 2 πc 7 D n s }
I WIRES I NDRM 1 + ρ NDPM ρ Metal Ring S Metal Ring S NDRM θ 2 π θ = r ͂ ( θ )
{ D ¯ = ε 0 ( 1 + χ ̿ e ) E ¯ j k ̿ c H ¯ B ¯ = μ 0 ( 1 + χ ̿ m ) H ¯ + j k ̿ T c E ¯
μ eff ( ω ) = { 1 f Q [ ( Q'f ω 0 2 ) ω 2 Δ Q ] [ ( Q'f γ ω 0 2 ) ω ] 2 + [ ( Q'f ω 0 2 ) ω 2 Δ Q ] 2 } +
j { fQ ( Q'f γ ω 0 2 ) ω [ ( Q'f γ ω 0 2 ) ω ] 2 + [ ( Q'f ω 0 2 ) ω 2 Δ Q ] 2 }
Z NDRM ( ω ) R n 1 + ( ω C R n ) 2 C R n 2 1 + ( ω C R n ) 2
Z NDRM ( ω ) R n + R n 3 C 2 ω 2 C R n 2 + j ω 3 ( C 3 R n 4 )

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