## The ratio of the kinetic inductance to the geometric inductance: a key parameter for the frequency tuning of the THz semiconductor split-ring resonator |

Optics Express, Vol. 21, Issue 17, pp. 20363-20375 (2013)

http://dx.doi.org/10.1364/OE.21.020363

Acrobat PDF (2104 KB)

### Abstract

By introducing the frequency tuning sensitivity, an analytical model based on equivalent LC circuit is developed for the relative frequency tuning range of THz semiconductor split-ring resonator (SRR). And the model reveals that the relative tuning range is determined by the ratio of the kinetic inductance to the geometric inductance (RKG). The results show that under the same carrier density variation, a larger RKG results in a larger relative tuning range. Based on this model, a stacked SRR-dimer structure with larger RKG compared to the single SRR due to the inductive coupling is proposed, which improves the relative tuning range effectively. And the results obtained by the simple analytical model agree well with the numerical FDTD results. The presented analytical model is robust and can be used to analyze the relative frequency tuning of other tunable THz devices.

© 2013 OSA

## 1. Introduction

1. S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. **9**(5), 407–412 (2010). [CrossRef] [PubMed]

5. P. Dardano, M. Gagliardi, I. Rendina, S. Cabrini, and V. Mocella, “Ellipsometric determination of permittivity in a negative index photonic crystal metamaterial,” Light: Sci. Appl. **1**(12), e42 (2012). [CrossRef]

6. V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. **102**(13), 133902 (2009). [CrossRef] [PubMed]

7. V. Mocella, P. Dardano, I. Rendina, and S. Cabrini, “An extraordinary directive radiation based on optical antimatter at near infrared,” Opt. Express **18**(24), 25068–25074 (2010). [CrossRef] [PubMed]

8. C. H. Lin, R. L. Chern, and H. Y. Lin, “Polarization-independent broad-band nearly perfect absorbers in the visible regime,” Opt. Express **19**(2), 415–424 (2011). [CrossRef] [PubMed]

9. C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. **37**(3), 308–310 (2012). [CrossRef] [PubMed]

10. C. Ma and Z. Liu, “Focusing light into deep subwavelength using metamaterial immersion lenses,” Opt. Express **18**(5), 4838–4844 (2010). [CrossRef] [PubMed]

12. T. Roy, E. T. Rogers, and N. I. Zheludev, “Sub-wavelength focusing meta-lens,” Opt. Express **21**(6), 7577–7582 (2013). [CrossRef] [PubMed]

13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express **19**(2), 1563–1568 (2011). [CrossRef] [PubMed]

14. M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express **17**(20), 18330–18339 (2009). [CrossRef] [PubMed]

15. J. B. Wu, B. B. Jin, Y. H. Xue, C. H. Zhang, H. Dai, L. B. Zhang, C. H. Cao, L. Kang, W. W. Xu, J. Chen, and P. H. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express **19**(13), 12021–12026 (2011). [CrossRef] [PubMed]

16. H. T. Chen, H. Yang, R. Singh, J. F. O’Hara, A. K. Azad, S. A. Trugman, Q. X. Jia, and A. J. Taylor, “Tuning the resonance in high-temperature superconducting terahertz metamaterials,” Phys. Rev. Lett. **105**(24), 247402 (2010). [CrossRef] [PubMed]

17. H. T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics **2**(5), 295–298 (2008). [CrossRef]

23. D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express **19**(10), 9968–9975 (2011). [CrossRef] [PubMed]

24. J. Y. Ou, E. Plum, L. Jiang, and N. I. Zheludev, “Reconfigurable photonic metamaterials,” Nano Lett. **11**(5), 2142–2144 (2011). [CrossRef] [PubMed]

25. J. J. Li, C. M. Shah, W. Withayachumnankul, B. S.-Y. Ung, A. Mitchel, S. Sriram, M. Bhaskaran, S. J. Chang, and D. Abbott, “Mechanically tunable terahertz metamaterials,” Appl. Phys. Lett. **102**(12), 121101 (2013). [CrossRef]

13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express **19**(2), 1563–1568 (2011). [CrossRef] [PubMed]

14. M. J. Dicken, K. Aydin, I. M. Pryce, L. A. Sweatlock, E. M. Boyd, S. Walavalkar, J. Ma, and H. A. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express **17**(20), 18330–18339 (2009). [CrossRef] [PubMed]

21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. **107**(9), 093104 (2010). [CrossRef]

22. J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express **16**(19), 14390–14396 (2008). [CrossRef] [PubMed]

17. H. T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics **2**(5), 295–298 (2008). [CrossRef]

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. **1**(2), 128–132 (2013). [CrossRef]

13. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens, “Electrically controllable fishnet metamaterial based on nematic liquid crystal,” Opt. Express **19**(2), 1563–1568 (2011). [CrossRef] [PubMed]

23. D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express **19**(10), 9968–9975 (2011). [CrossRef] [PubMed]

## 2. Structure design and simulation model

*φ*= 0° or

*φ*= 180°, as shown in Figs. 1(a) and 1(b), respectively. And the material of SRRs is InSb, which can be grown on the GaAs or Si substrate by using molecular beam epitaxy (MBE) process [19

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. **1**(2), 128–132 (2013). [CrossRef]

26. G. Singh, E. Michel, C. Jelen, S. Slivken, J. Xu, P. Bove, I. Ferguson, and M. Razeghi, “Molecular-beam epitaxial growth of high quality InSb for p-i-n photodetectors,” J. Vac. Sci. Technol. B **13**(2), 782–785 (1995). [CrossRef]

19. L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. **1**(2), 128–132 (2013). [CrossRef]

20. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. **56**(4), 554–557 (2009). [CrossRef]

**1**(2), 128–132 (2013). [CrossRef]

27. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science **306**(5700), 1351–1353 (2004). [CrossRef] [PubMed]

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) **3**(2), 57–62 (2009). [CrossRef]

*C*is the capacitance and

*R*is the resistance of the SRR. Assume that the current has the form of

*I*=

*I*

_{0}exp(-

*βt*-

*iωt*), where

*β*accounts for the damping loss. Then the resonance frequency is obtained from Eq. (1) asDifferent from the commonly employed LC model [27

27. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science **306**(5700), 1351–1353 (2004). [CrossRef] [PubMed]

*C*(the sum of the gap capacitance

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) **3**(2), 57–62 (2009). [CrossRef]

*α*is a correction factor to the surface capacitance, since Eq. (4d) delivers a smaller result than the true value for a thin SRR [28

28. V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) **3**(2), 57–62 (2009). [CrossRef]

**1**(2), 128–132 (2013). [CrossRef]

21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. **107**(9), 093104 (2010). [CrossRef]

*N*), the electron charge (

*e*), the vacuum permittivity (

*ε*

_{0}) and the carrier effective mass (

*m**). The values of these parameters are taken as

*m** = 0.015

*m*

_{e}[20

20. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. **56**(4), 554–557 (2009). [CrossRef]

21. Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. **107**(9), 093104 (2010). [CrossRef]

30. X. Y. Dai, Y. J. Xiang, S. C. Wen, and H. Y. He, “Thermally tunable and omnidirectional terahertz photonic bandgap in the one-dimensional photonic crystals containing semiconductor InSb,” J. Appl. Phys. **109**(5), 053104 (2011). [CrossRef]

*m*

_{e}is the electron’s rest mass. These parameters allow for a good fitting of the InSb permittivity around 1 THz when the carrier density is below

*N*= 1 × 10

^{17}cm

^{−3}[31

31. S. C. Howells and L. A. Schlie, “Transient terahertz reflection spectroscopy of undoped InSb from 0.1 to 1.1 THz,” Appl. Phys. Lett. **69**(4), 550–552 (1996). [CrossRef]

22. J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express **16**(19), 14390–14396 (2008). [CrossRef] [PubMed]

## 3. Results and discussion

*N*is increased from 5 × 10

^{15}cm

^{−3}to 1 × 10

^{17}cm

^{−3}, as shown in Fig. 2(a). For

*N*= 5 × 10

^{15}cm

^{−3}and 1 × 10

^{16}cm

^{−3}, the higher order modes are also excited which will not be discussed in this work. Two features can be seen in Fig. 2. First, the transmission dip becomes pronounced with the increment of

*N*, which indicates that the resonance strength is gradually enhanced. This is because that InSb transits from an insulator to a conductor when considerable free carriers are thermally or optically excited. And a more conductive SRR is able to induce the stronger LC resonance [32

32. R. Singh, A. K. Azad, J. F. O’Hara, A. J. Taylor, and W. Zhang, “Effect of metal permittivity on resonant properties of terahertz metamaterials,” Opt. Lett. **33**(13), 1506–1508 (2008). [CrossRef] [PubMed]

*N*will reduce

*S*is introduced by us as:Based on Eq. (3) and Eq. (4), the tuning sensitivity is derived asFigure 2(b) shows that as

*N*increases, the tuning sensitivity is reduced (black line), indicating that the kinetic inductance and hence the resonance frequency is less sensitive to the increasing carrier density. Therefore, the resonance frequency undergoes a slow growth with

*N*and will finally saturate to the value

*h*) and radius (

*r*) of SRR at

*N*= 1 × 10

^{16}cm

^{−3}(Fig. 3(a)) and 1 × 10

^{17}cm

^{−3}(Fig. 3(b)). As shown from a comparison between Figs. 3(a) and 3(b), the dependence of RKG on the geometric sizes (

*h*and

*r*) is similar at the two carrier densities. But RKG is larger at the lower

*N*because the kinetic inductance is inversely proportional to the carrier density according to Eq. (4a). Figures 3(a) and 3(b) further show that for a fixed carrier density, RKG is reduced with an increasing

*h*. So under the same variation of

*N*, a lower relative tuning range is expected at a larger

*h*. This is verified in Fig. 3(c), where the relative tuning range (

*N*

_{1}= 1 × 10

^{16}cm

^{−3}and

*N*

_{2}= 1 × 10

^{17}cm

^{−3}) as a function of

*h*is obtained. The analytical result (red solid line) shows that the relative tuning range decreases from 170% to 130% as

*h*increases from 1 μm to 3 μm, which agrees well with the simulation results (blue dots). The decrease of RKG and hence the relative tuning range with

*h*can be explained by the LC model. An examination of Eqs. (4a) and (4b) reveals that the inductance

*h*is increased. Nonetheless,

*h*at the same time, as shown in Fig. 3(e). Although the capacitance

*h*, which is expected to reduce the resonance frequency according to Eq. (3), the simultaneous reduction of the inductance makes a more significantly competing effect, eventually increasing the resonance frequency.

*r*provides another means to adjust the RKG. Similar to the situation of

*h*, an increment of

*r*also results in reduced RKG, which then reduces the relative tuning range. However,

*r*than to the thickness

*h*, as shown in the contour plots. And another notable difference can be seen by comparing Fig. 3(e) with Fig. 3(f). Figure 3(e) shows that at a given

*N*the resonance frequency increases with the thickness, as mentioned above. While the resonance frequency is reduced with an increasing radius because both the inductance and capacitance are increased. In other words, by reducing

*h*, an increased relative tuning range is achieved and accompanied by the reduction of the resonance frequency. In contrast, by reducing

*r*, the relative tuning range Δ

*f*/

*f*(

*N*

_{1}) and the resonance frequencies

*f*(

*N*

_{1}) are both increased. Consequently, as

*r*is reduced the absolute resonance shift Δ

*f*is also increased. For example, as

*N*rises from 1 × 10

^{16}cm

^{−3}to 1 × 10

^{17}cm

^{−3}, the resonance frequency of SRRs with

*r*= 17 μm increases from 0.36 THz to 0.86 THz, with a relative tuning range of 139% and the absolute resonance shift Δ

*f*= 0.50 THz. While for

*r*= 9 μm, the resonance shifts from 0.58 THz to 1.47 THz, with a relative tuning range of 153% and Δ

*f*= 0.89 THz. So the adjustment of the radius of SRRs can result in the simultaneous increment of the relative tuning range and absolute resonance shift. In Fig. 3(f), there is a slight discrepancy between the analytical and simulated resonance frequency. The reason is that Eq. (4b) takes the expression of the geometric inductance of a closed ring, while the simulated SRR structure has a gap. Furthermore, the geometric inductance of a SRR can be approximated with Eq. (4b) only when the condition

**3**(2), 57–62 (2009). [CrossRef]

*g*, the larger the radius

*r*, the smaller discrepancy between the analytical and simulation results can be achieved, which is shown as in Fig. 3(f).

34. R. Singh, E. Smirnova, A. J. Taylor, J. F. O’Hara, and W. Zhang, “Optically thin terahertz metamaterials,” Opt. Express **16**(9), 6537–6543 (2008). [CrossRef] [PubMed]

*M*) arising from the inductive coupling between SRRs [35

35. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies,” Phys. Rev. B **76**(7), 073101 (2007). [CrossRef]

37. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics **3**(3), 157–162 (2009). [CrossRef]

*M*<0, the net geometric inductance is reduced from

*C*, which takes into account both the self capacitance and the mutual capacitance. Then the resonance frequency is obtained as:After a similar mathematical deduction to obtain Eq. (7), the relative tuning range taking mutual inductance into account is derived as Eq. (9). It takes the similar form with Eq. (7), but with

_{tot}*M*<0 and

*M*>0.Equation (9) reveals that the mutual inductance may affect the relative tuning range by altering RKG. While the mutual capacitance, no matter how large it is, has no effect on the relative tuning range. A comparison between Eq. (7) and (9) shows that under the same carrier density variation, an introduction of negative mutual inductance will increase RKG and hence improve the relative tuning range. On the contrary, a positive mutual inductance is expected to reduce the relative tuning range.

*s*and twist angle

*φ*. This so-called stereo-metamaterial has been investigated in terms of the plasmon coupling between the two stacked SRRs [37

37. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics **3**(3), 157–162 (2009). [CrossRef]

37. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics **3**(3), 157–162 (2009). [CrossRef]

_{z}at the anti-bonding mode for

*φ*= 180° (Fig. 4(a)) and the bonding mode for

*φ*= 0° (Fig. 4(b)), respectively. The magnetic fields induced by the two SRRs interferes destructively (constructively) at the anti-bonding (bonding) mode, which gives rise to the negative (positive) mutual inductance.

*φ*= 180°. Under the incident configuration as exhibited in Fig. 1, the bonding mode cannot be effectively excited and only the anti-bonding mode couples to the incident wave, which is manifested by the dip in the transmission spectrum. The simulated transmission spectra of the stacked SRR-dimers (solid lines) and single-layer SRRs (dash lines) are presented in Fig. 5. The carrier density ranges from 1 × 10

^{16}cm

^{−3}(black) to 1 × 10

^{17}cm

^{−3}(pink). For the same carrier density, the stacked SRR-dimer structure exhibits the higher resonance frequency than the single-layer SRRs. It can also be explained with the LC model. On one hand, a negative mutual inductance is induced in the stacked SRR-dimer MM. So the stacked SRRs have a lower geometric inductance

*N*is increased from 1 × 10

^{16}cm

^{−3}to 1 × 10

^{17}cm

^{−3}, the resonance frequency of single-layer SRR (stacked SRR-dimer) shifts from 0.39 THz (0.41 THz) to 0.97 THz (1.12 THz), with a relative tuning range of 149% (172%). So the improvement of relative tuning range due to the negative mutual inductance is observed, which meets the expectation from the analytical model as proposed above. Moreover, different from reducing the SRR thickness, the SRR-dimer structure effectively increases the relative tuning range with no expense of weakening the resonance strength.

35. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies,” Phys. Rev. B **76**(7), 073101 (2007). [CrossRef]

*s*. The mutual inductance is evaluated from Eq. (10), which is widely used to calculate the mutual inductance between two coaxially aligned circular coils [38]. Because the circumference of the SRR is much longer than the gap width, i.e.,

*E*and

*K*are the complete elliptic integrals of the first and second kind with the geometry-dependent parameter

*s*and shown in the red line of Fig. 6(a). Not surprisingly,

*M*decreases with increased

*s*due to weakened inductive coupling between SRRs. And the decreased mutual inductance leads to the decline of

*N*increases from 1 × 10

^{16}cm

^{−3}to 1 × 10

^{17}cm

^{−3}). And the improvement of the relative tuning range is especially pronounced at a smaller separation due to the stronger inductive coupling, which is responsible for a larger negative mutual inductance. As the separation

*s*grows, the relative tuning range of the stacked SRR-dimers is reduced as a result of the decreased mutual inductance. But even at a large separation of

*s*= 8 μm, the relative tuning range of stacked SRRs is still larger than that of the single-layer structure. The analytical results are well reproduced by the simulation data, indicating that the LC model effectively captures the main physics involved in the frequency tuning.

*φ*= 0°, rather than the structure with

*φ*= 180°.The reason is that the bonding mode is a dark mode for

*φ*= 180°, which weakly couples to the incident beam and so can hardly be discerned in the transmission spectrum [39

39. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. **4**(5), 899–903 (2004). [CrossRef]

*φ*= 0°, the bonding mode can be effectively excited by the incident THz wave. At the bonding resonance, the magnetic field along z axis (H

_{z}) induced by the two SRRs of the dimer oscillates in phase, giving rise to a positive mutual inductance and a net geometric inductance

*s*grows, the mutual inductance becomes smaller due to the gradually weakened inductive coupling. Consequently, RKG increases with

*s*until it approaches the value of the single-layer SRRs. Correspondingly, the relative tuning range increases with

*s*and asymptotically approaches the relative tuning range of the single-layer SRRs.

40. B. Sauivac, C. R. Simovski, and S. Tretyakov, “Double split-ring resonators: Analytical modeling and numerical simulations,” Electromagnetics **24**(5), 317–338 (2004). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

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19. | L. Y. Deng, J. H. Teng, H. W. Liu, Q. Y. Wu, J. Tang, X. H. Zhang, S. A. Maier, K. P. Lim, C. Y. Ngo, S. F. Yoon, and S. J. Chua, “Direct optical tuning of the terahertz plasmonic response of InSb subwavelength Gratings,” Adv. Opt. Mater. |

20. | J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. |

21. | Q. Bai, C. Liu, J. Chen, C. Cheng, M. Kang, and H. T. Wang, “Tunable slow light in semiconductor metamaterial in a broad terahertz regime,” J. Appl. Phys. |

22. | J. Han, A. Lakhtakia, and C. W. Qiu, “Terahertz metamaterials with semiconductor split-ring resonators for magnetostatic tunability,” Opt. Express |

23. | D. Shrekenhamer, S. Rout, A. C. Strikwerda, C. Bingham, R. D. Averitt, S. Sonkusale, and W. J. Padilla, “High speed terahertz modulation from metamaterials with embedded high electron mobility transistors,” Opt. Express |

24. | J. Y. Ou, E. Plum, L. Jiang, and N. I. Zheludev, “Reconfigurable photonic metamaterials,” Nano Lett. |

25. | J. J. Li, C. M. Shah, W. Withayachumnankul, B. S.-Y. Ung, A. Mitchel, S. Sriram, M. Bhaskaran, S. J. Chang, and D. Abbott, “Mechanically tunable terahertz metamaterials,” Appl. Phys. Lett. |

26. | G. Singh, E. Michel, C. Jelen, S. Slivken, J. Xu, P. Bove, I. Ferguson, and M. Razeghi, “Molecular-beam epitaxial growth of high quality InSb for p-i-n photodetectors,” J. Vac. Sci. Technol. B |

27. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science |

28. | V. Delgado, O. Sydoruk, E. Tatartschuk, R. Marqués, M. J. Freire, and L. Jelinek, “Analytical circuit model for split ring resonators in the far infrared and optical frequency range,” Metamaterials (Amst.) |

29. | A. Taflove and S. C. Hagness, |

30. | X. Y. Dai, Y. J. Xiang, S. C. Wen, and H. Y. He, “Thermally tunable and omnidirectional terahertz photonic bandgap in the one-dimensional photonic crystals containing semiconductor InSb,” J. Appl. Phys. |

31. | S. C. Howells and L. A. Schlie, “Transient terahertz reflection spectroscopy of undoped InSb from 0.1 to 1.1 THz,” Appl. Phys. Lett. |

32. | R. Singh, A. K. Azad, J. F. O’Hara, A. J. Taylor, and W. Zhang, “Effect of metal permittivity on resonant properties of terahertz metamaterials,” Opt. Lett. |

33. | R. S. Penciu, M. Kafesaki, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Magnetic response of nanoscale left-handed metamaterials,” Phys. Rev. B |

34. | R. Singh, E. Smirnova, A. J. Taylor, J. F. O’Hara, and W. Zhang, “Optically thin terahertz metamaterials,” Opt. Express |

35. | H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies,” Phys. Rev. B |

36. | N. T. Tung, D. T. Viet, B. S. Tung, N. V. Hieu, P. Lievens, and V. D. Lam, “Broadband Negative Permeability by Hybridized Cut-Wire Pair Metamaterials,” Appl. Phys. Express |

37. | N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics |

38. | L. D. Landau and E. M. Lifschitz, |

39. | P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. |

40. | B. Sauivac, C. R. Simovski, and S. Tretyakov, “Double split-ring resonators: Analytical modeling and numerical simulations,” Electromagnetics |

**OCIS Codes**

(130.5990) Integrated optics : Semiconductors

(260.5740) Physical optics : Resonance

(160.3918) Materials : Metamaterials

(250.5403) Optoelectronics : Plasmonics

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Metamaterials

**History**

Original Manuscript: June 17, 2013

Revised Manuscript: August 2, 2013

Manuscript Accepted: August 7, 2013

Published: August 22, 2013

**Citation**

Jiawei Cong, Binfeng Yun, and Yiping Cui, "The ratio of the kinetic inductance to the geometric inductance: a key parameter for the frequency tuning of the THz semiconductor split-ring resonator," Opt. Express **21**, 20363-20375 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-20363

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