## Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies

Optics Express, Vol. 18, Issue 3, pp. 2797-2807 (2010)

http://dx.doi.org/10.1364/OE.18.002797

Acrobat PDF (646 KB)

### Abstract

Terahertz plasmonic resonances in semiconductor (indium antimonide, InSb) dimer antennas are investigated theoretically. The antennas are formed by two rods separated by a small gap. We demonstrate that, with an appropriate choice of the shape and dimension of the semiconductor antennas, it is possible to obtain large electromagnetic field enhancement inside the gap. Unlike metallic antennas, the enhancement around the semiconductor plasmonics antenna can be easily adjusted by varying the concentration of free carriers, which can be achieved by optical or thermal excitation of carriers or electrical carrier injection. Such active plasmonic antennas are interesting structures for THz applications such as modulators and sensors.

© 2010 OSA

## 1. Introduction

1. K. H. Yang, P. L. Richards, and Y. R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO_{3},” Appl. Phys. Lett. **19**(9), 320–323 (1971). [CrossRef]

4. J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolivar, and H. Kurz, “Time domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B **69**(15), 155427 (2004). [CrossRef]

9. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics **3**(3), 152–156 (2009). [CrossRef]

8. A. Rusina, M. Durach, K. A. Nelson, and M. I. Stockman, “Nanoconcentration of terahertz radiation in plasmonic waveguides,” Opt. Express **16**(23), 18576–18589 (2008). [CrossRef]

9. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics **3**(3), 152–156 (2009). [CrossRef]

12. R. Ulrich and M. Tacke, “Submillimeter waveguiding on periodic metal structure,” Appl. Phys. Lett. **22**(5), 251–253 (1973). [CrossRef]

18. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics **2**(3), 175–179 (2008). [CrossRef]

19. J. G. Rivas, M. Kuttge, P. H. Bolivar, H. Kurz, and J. A. Sánchez-Gil, “Propagation of surface plasmon polaritons on semiconductor gratings,” Phys. Rev. Lett. **93**(25), 256804 (2004). [CrossRef]

20. R. Parthasarathy, A. Bykhovski, B. Gelmont, T. Globus, N. Swami, and D. Woolard, “Enhanced Coupling of Subterahertz Radiation with Semiconductor Periodic Slot Arrays,” Phys. Rev. Lett. **98**(15), 153906 (2007). [CrossRef] [PubMed]

21. J. G. Rivas, P. H. Bolivar, and H. Kurz, “Thermal switching of the enhanced transmission of terahertz radiation through sub-wavelength apertures,” Opt. Lett. **29**(14), 1680–1682 (2004). [CrossRef] [PubMed]

26. E. Hendry, F. J. Garcia-Vidal, L. Martin-Moreno, J. Gómez Rivas, M. Bonn, A. P. Hibbins, and M. J. Lockyear, “Optical control over surface plasmon polariton-assisted THz transmission through a slit aperture,” Phys. Rev. Lett. **100**(12), 123901 (2008). [CrossRef] [PubMed]

27. J. Gómez Rivas, M. Kuttge, H. Kurz, P. Haring Bolivar, and J. A. Sánchez-Gil, “Low-frequency active surface plasmon optics on semiconductors,” Appl. Phys. Lett. **88**(8), 082106 (2006). [CrossRef]

29. J. Gómez Rivas, J. A. Sánchez-Gil, M. Kuttge, P. Haring Bolívar, and H. Kurz, “Optically switchable mirrors for surface plasmon polaritons propagating on semiconductor surfaces,” Phys. Rev. B **74**(24), 245324 (2006). [CrossRef]

30. H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature **444**(7119), 597–600 (2006). [CrossRef] [PubMed]

31. H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics **3**(3), 148–151 (2009). [CrossRef]

32. V. Giannini, J. A. Sánchez-Gil, J. V. García-Ramos, and E. R. Méndez, “Collective electromagnetic emission from molecular layers on metal nanostructures mediated by surface plasmon,” Phys. Rev. B **75**(23), 235447 (2007). [CrossRef]

33. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science **308**(5728), 1607–1609 (2005). [CrossRef] [PubMed]

## 2. Theoretical model

34. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires with arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A **24**(9), 2822–2830 (2007). [CrossRef]

*y*-direction are placed close to each other at a distance Δ.

*z*axis, its electric field being in the (

*x, z*) plane (

*p*-polarization). The wires are defined by an isotropic and homogeneous frequency-dependent permittivity ε(ω). The restriction to wires with translational symmetry has the advantage that it notably simplifies the formulation; in fact, we can reduce the initial three-dimensional vectorial problem to a two-dimensional scalar one, where the EM field is entirely described in the case of

*p*-polarization by the

*y*component of the magnetic field. Applying Green's integral theorem, and proceeding as in Ref [34

34. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires with arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A **24**(9), 2822–2830 (2007). [CrossRef]

**r**is outside Γ

_{j}andif

**r**is inside Γ

_{j.}

*j*can take the values 1 or 2 and indicates the

*j*-th wire, Γ

_{j}represents the parametric curve that describes the

*j*-th wire and

**R**

*(s) is a point on the parametric curve that delimitates wire. The superscripts*

_{j}*(out)*denotes that the magnetic field

*H*(

^{(out)}**r**) is evaluated outside the wires.

*H*(

^{(i)}**r**) is the incident field and

*G*(

^{(out)}**r, R**) is the Green function of the system, and ∂/∂

*N*is the normal derivative along the parametric curve. The function

*G*(

^{(out)}**r, R**) is the Green function of free space and it is obtained from Helmholtz equations describing the magnetic field outside the scatterers. If translational symmetry is considered, the Green function is given by the zeroth-order Hankel function of the first kind. Similar expressions are obtained employing the Green’s integral theorem inside the wires [34

34. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires with arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A **24**(9), 2822–2830 (2007). [CrossRef]

35. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express **15**(26), 17736–17746 (2007). [CrossRef] [PubMed]

*N*= 5.9⋅10

_{Au}^{22}cm

^{−3}). Due to the similar permittivities of InSb at THz frequencies and gold in the visible, it is reasonable to expect localized plasmon resonances at THz frequencies on InSb antennas with scaled dimensions. In the next section we have investigated the scattering cross section and the THz near-field distribution of InSb dimer plasmonic antennas.

## 3. Plasmonics resonances in InSb antennas

### 3.1 Scattering cross sections

*L*and height

*h*separated by a subwavelength gap Δ. We consider the interaction of a plane wave at THz frequencies impinging on the antenna at θ=0°. The incident field is

*p*-polarized, i.e., the electric field is in the plane of the figure along the

*x*-direction, i.e., parallel to the long axis of the antenna. From the far field scattered by the antennas we have calculated the scattering cross section (SCS) [34

**24**(9), 2822–2830 (2007). [CrossRef]

*L*. This normalization provides an intuitive characterization of the strength of the interaction of the incident THz wave with the antennas.. For scattering cross sections bigger than the geometrical cross section, the energy of the incident field around the antenna is concentrated on the antenna, leading to a local enhancement of the electromagnetic field. In Fig. 3(a) the scattering cross section normalized to the antenna length

*L*, i.e., the scattering efficiency (Q

_{eff}), is shown for different lengths ranging from 30 µm to 70 µm. The gap width is 2 µm and the height of each dimer is 5 µm. As expected, the SCSs exhibit a resonance when the real part of the permittivity of InSb is negative and the imaginary part is relatively small, i.e., around 1.5 THz. These resonances are the lowest order plasmonic resonances, i.e., the semiconductor structure acts as half wavelength (λ/2) antenna. As in the case of metal nanoantennas [34

**24**(9), 2822–2830 (2007). [CrossRef]

35. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express **15**(26), 17736–17746 (2007). [CrossRef] [PubMed]

*L*=70µm (green triangles in Fig. 3(a)) exhibits a small shoulder around 2.2 THz due to a weak excitation of a higher order resonance, i.e., the 3λ/2 resonance.. Narrower resonance widths are found for shorter lengths due to a reduced radiative damping.

*L+*Δ and Δ <<

*L*, this resonant condition can be written as

*L≈mλ/*4

*n*, with

*n*the refractive index of the surrounding medium and

*m*a positive integer which indicates the resonance order. Taking into account the symmetry of the antenna, when the light impinges on the top (see Fig. 1) only odd order resonances can be excited [34

**24**(9), 2822–2830 (2007). [CrossRef]

*m*order given by the classical antenna theory.

### 3.2 Near-fields

*L*=50µm, Δ=2µm and

*h*=5µm. The near-fields have been calculated at their respective resonance frequencies, i.e., 1.62 THz for the InSb antenna and at 1.73 THz for the Au antenna. The near-field intensities of Fig. 4 have been normalized to the incident field and plotted in logarithmic scale to distinguish the highest enhancements in the gap. Figures 4(c) and (d) represent the near-field enhancement along a line through the middle of the InSb (Fig. 4(c)) and Au (Fig. 4(d)) antenna parallel to its long axis. The near-fields present several similarities: there are maxima in the enhancement at the edges of the antenna and in the antenna gap. These maxima are caused by the coexistence of two effects: the lightning rod effect at the sharp corners and by the localized plasmon resonance. The gap enhancements are almost uniformly distributed inside the gap. This can be better appreciated in the zooms around the gap displayed in Figs. 4(e) for InSb and Fig. 4(f) Au. Quantitatively, the enhancement factors at the center of the gap are larger for InSb (~95) than those for Au (~75), highlighting the relevance of LSPRs in semiconductor antennas. For a gap size of 2 μm as displayed on Fig. 4, the field enhancement is increased by a factor of 1.25 when choosing semiconductors. The main difference in the near-field is found inside the antennas themselves. Due to the large value of the permittivity of gold, the electric field does not penetrate inside the antenna and the field vanishes in contrast to the finite field in the InSb antenna. The enhancements in Fig. 4 and 5 are calculated at the resonant frequency obtained from the SCS spectrum. It should be pointed out that when retardation effects are important the largest near-field enhancement is at a frequency slightly shifted with respect to the far field resonance [39

39. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local-fields at the surface of noble metal microspheres,” Phys. Rev. B **24**(2), 649–657 (1981). [CrossRef]

41. G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the Plasmon Resonances of Metallic Nanoantennas,” Nano Lett. **8**(2), 631–636 (2008). [CrossRef] [PubMed]

42. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. **94**(1), 017402 (2005). [CrossRef] [PubMed]

*z*=0 are shown in Fig. 5(b). For this particular antenna we can achieve enhancements factors larger than 10

^{3}in the gap.

### 3.3 Tuneability of semiconductor THz antennas

*L*=40µm and

*h*=5µm (see Fig. 5), we have explored the tuneable properties of this resonance upon variation of the carrier concentration. Figure 7(a) shows the near-field enhancement at the central point of dimer gap as a function of the free-electron concentration. The calculations are done at room temperature, using a carrier and temperature dependent mobility [37], and for different dimensions of the gap Δ = 0.1

*μ*m (down blue triangles), 0.2

*μ*m (up green triangles), 0.5

*μ*m (red circles), and 1.0

*μ*m (black squares). A minimum in the intensity enhancement appears for

*N*= 6⋅10

^{15}cm

^{−3}. Below this value modest enhancements are obtained. A rapid increase in the enhancement is observed for values of the carrier concentration above 6⋅10

^{15}cm

^{−3}, with a local maximum around

*N*= 2⋅10

^{16}cm

^{−3}and a decrease of the enhancement for larger

*N*. This behavior can be understood by looking to the value of the permittivity of InSb at 1.5 THz as a function of the carrier concentration. This value is represented in Fig. 7(b), where the blue triangles are –Re(ε) and the red circle are Im(ε). The real component of the permittivity is negative, i.e., InSb behaves as a conductor, only above

*N*~6⋅10

^{15}cm

^{−3}. For this carrier concentration the plasma frequency is 1.5 THz. The permittivity of InSb has positive real component and a small imaginary component for

*N*< 6⋅10

^{15}cm

^{−3}, i.e., InSb behaves as a dielectric and no large enhancements are expected. Above

*N*~ 6⋅10

^{15}cm

^{−3}, as the imaginary component of the permittivity increases, the excitation of surface plasmon resonances is less efficient due to larger Ohmic losses. Therefore, the most favorable carrier concentrations are below

*N*~ 10

^{17}cm

^{−3}. If the free carrier concentration is further increased, InSb approaches the perfect conductor limit behaving as a metal at THz frequencies. In this limit, the enhancements are lower and due to the lightning rod effect at the corners of the antenna. The dependence of the enhancement with carrier concentration does not change qualitatively when the dimension of the gap is varied (see Fig. 7 (a)). Only the enhancement increases inversely proportional the size of the gap. Note that the strong dependence of the field enhancement with carrier concentration, which in turn can be easily controlled in semiconductors, represents an important step towards active plasmonics functionalities.

*L*= 40 µm,

*h*= 5 µm, and gap width Δ = 1 µm. The dimensions of the antenna are chosen to optimize the field enhancement in the gap at 1.5 THz. A

*p*-polarized electromagnetic wave field impinges on top. The LSPRs for a low carrier concentration evolves into a broader resonance as the carrier concentration is increased. Also the resonant frequency shifts to higher frequencies, from ~1 THz to ~2 THz, as the carrier concentration increases from 10

^{16}to 10

^{17}cm

^{−3}. We can understand this behavior by considering the dependence of the permittivity and the plasma frequency on the carrier concentration. By increasing the carrier concentration the plasma frequency shifts to higher values. This, in turn, increases the frequency of the LSPR [43

43. L. Novotny, “Effective Wavelength Scaling for Optical Antennas,” Phys. Rev. Lett. **98**(26), 266802 (2007). [CrossRef] [PubMed]

## 4. Conclusion

^{2}-10

^{3}times the incident field). The achieved field enhancement is larger than that obtained with gold dimer antennas with identical dimensions due to the enhanced plasmonics character of semiconductors at THz frequencies. These antennas are the analogue at THz frequencies of metal plasmonic nanoantennas in the visible. Moreover, semiconductor based THz antennas are more versatile than their metallic counterparts since the dielectric properties of semiconductors depend on the free-carrier concentration and mobility. These parameters can be controlled in several ways, e.g., by changing temperature, doping concentration, applying an electric potential or by optical pumping. The control of the permittivity allows active tuning of plasmonic resonances and enhancement factors. Therefore, semiconductor dimer antennas are excellent candidates for applications in which large THz field enhancement and active control of these enhancements are required. These applications can be found in THz spectroscopy and sensing of low concentrations and in THz modulators.

## Acknowledgements

^{o}FP7-224189 (ULTRA project) and is part of the research program Stichting voor Fundamenteel Onderzoek der Materie (FOM), financially supported by the Nederlandse Organisatie voor Wetenschappelik Onderzoek (NWO). J. A. Sánchez-Gil acknowledges support from the Spain Ministerio de Ciencia e Innovación through the Consolider-Ingenio project EMET (CSD2008-00066) and NANOPLAS (FIS2009 11264). SM acknowledges support by the US Air Force Office of Scientific Research (AFOSR).

## References and links

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**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(290.4210) Scattering : Multiple scattering

(040.2235) Detectors : Far infrared or terahertz

(250.5403) Optoelectronics : Plasmonics

(300.6495) Spectroscopy : Spectroscopy, teraherz

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 13, 2009

Revised Manuscript: December 21, 2009

Manuscript Accepted: December 21, 2009

Published: January 26, 2010

**Citation**

Vincenzo Giannini, Audrey Berrier, Stefan A. Maier, José Antonio Sánchez-Gil, and Jaime Gómez Rivas, "Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies," Opt. Express **18**, 2797-2807 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2797

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### References

- K. H. Yang, P. L. Richards, and Y. R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19(9), 320–323 (1971). [CrossRef]
- D. H. Auston, A. M. Glass, and A. A. Ballman, “Optical Rectification by Impurities in Polar Crystals,” Phys. Rev. Lett. 28(14), 897–900 (1972). [CrossRef]
- Terahertz Spectroscopy, Principles and Applications, S.L. Dexheimer (ed.) (CRC Press, 2008).
- J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolivar, and H. Kurz, “Time domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004). [CrossRef]
- D. Qu, D. Grischkowsky, and W. Zhang, “Terahertz transmission properties of thin, subwavelength metallic hole arrays,” Opt. Lett. 29(8), 896–898 (2004). [CrossRef] [PubMed]
- K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [CrossRef] [PubMed]
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