## Terahertz meta-atoms coupled to a quantum well intersubband transition |

Optics Express, Vol. 19, Issue 14, pp. 13700-13706 (2011)

http://dx.doi.org/10.1364/OE.19.013700

Acrobat PDF (888 KB)

### Abstract

We present a method of coupling free-space terahertz radiation to intersubband transitions in semiconductor quantum wells using an array of meta-atoms. Owing to the resonant nature of the interaction between metamaterial and incident light and the field enhancement in the vicinity of the metal structure, the coupling efficiency of this method is very high and the energy conversion ratio from in-plane to *z* field reaches values on the order of 50%. To identify the role of different aspects of this coupling, we have used a custom-made finite-difference time-domain code. The simulation results are supplemented by transmission measurements on modulation-doped GaAs/AlGaAs parabolic quantum wells which demonstrate efficient strong light-matter coupling between meta-atoms and intersubband transitions for normal incident electromagnetic waves.

© 2011 OSA

## 1. Introduction

1. Z. Schlesinger, J. C. M. Hwang, and J. S. J. Allen, “Subband-Landau-level coupling in a two-dimensional electron gas,” Phys. Rev. Lett. **50**, 2098–2101 (1983). [CrossRef]

2. J. S. Smith, L. C. Chiu, S. Margalit, A. Yariv, and A. Y. Cho, “A new infrared detector using electron emission from multiple quantum wells,” J. Vac. Sci. Technol. B **1**, 376–378 (1983). [CrossRef]

3. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science **22**, 553–556 (1994). [CrossRef]

*z*direction, can induce ISBs [4]. Additionally, free-space radiation has to be efficiently coupled into the device. Common ways to achieve both requirements are, for example, incidence under Brewster’s angle [5

5. L. C. West and S. J. Eglash, “First observation of an extremely large-dipole infrared transition within the conduction band of a GaAs quantum well,” Appl. Phys. Lett. **46**, 1156–1158 (1985). [CrossRef]

6. B. F. Levine, R. J. Malik, J. Walker, K. K. Choi, C. G. Bethea, D. A. Kleinman, and J. M. Vandenberg, “Strong 8.2*μ*m infrared intersubband absorption in doped GaAs/AlAs quantum well waveguides,” Appl. Phys. Lett. **50**, 273–275 (1987). [CrossRef]

7. D. Heitmann, J. P. Kotthaus, and E. G. Mohr, “Plasmon dispersion and intersubband resonance at high wavevectors in Si(100) inversion layers,” Solid State Commun. **44**, 715–718 (1982). [CrossRef]

8. 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**, 2075–2084 (1999). [CrossRef]

9. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics **2**, 351–354 (2008). [CrossRef]

10. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express **18**, 24140–24151 (2010). [CrossRef] [PubMed]

*z*axis. Thus, it is necessary that the distance between the QW and the meta-atom is smaller than the decay length in the substrate, which is typically on the order of several microns in the THz region.

*et al.*[11]. Thereby, a multi-QW structure has been sandwiched between two gold layers, providing sub-wavelength confinement of the electromagnetic radiation. Geiser

*et al.*[12] have been able to demonstrate strong coupling of a LC resonator to THz ISBs in a parabolic quantum well (PQW) even at room temperature (RT). The strong confinement is in this case also achieved by a vertical double-metal structure which has the drawback of a large capacitance per unit area. In consequence, such structures have to be small and provide only a very limited cross-section for the interaction with free space electromagnetic waves.

## 2. Model

*x*axis. The QW is located beneath the meta-atom with its growth direction parallel to the

*z*axis. In this geometry, the incident THz light cannot directly couple to the ISBs due to the selection rules, but it can couple to the meta-atom. Figure 1b shows a cross-section of the meta-atom on resonance along the dashed line in Fig. 1a. The electric field of this resonant mode, illustrated by the red field lines, extends both above and below the meta-atom, thereby possessing a non-zero

*z*component. Hence, the meta-atom couples the incident in-plane field to the

*z*field that is necessary to induce ISBs. Owing to the resonant nature of the interaction and the field enhancement known from such meta-atoms, the conversion efficiency between in-plane and

*z*field easily reaches values on the order of several ten percent.

*z*axis. The computational space is divided in two parts, air or vacuum on one side and substrate on the other side. The substrate is taken into account as frequency independent and lossless dielectric with permittivity

*ɛ*. The extension to a more realistic, dispersive medium is straightforward, but for the frequency range and substrate material we are interested in, not necessary (the refractive index of semi-insulating GaAs is well approximated by

*ɛ*= 12.96 up to 5 THz). The THz radiation is taken to be normally incident. Therefore, we can use simple periodic boundary conditions at the four side walls,

**F**(

*i, j,k*) =

**F**(

*i*+

*N*,

_{x}*j, k*) and

**F**(

*i, j, k*) =

**F**(

*i, j*+

*N*), where

_{y}, k**F**represents any field variable. At the top and bottom plane, we use first-order Engquist-Majda absorbing boundary conditions (ABC) [15

15. B. Engquist and A. Majda, “Absorbing boundary conditions for numerical simulation of waves,” Proc. Natl. Acad. Sci. USA **74**, 1765–1766 (1977). [CrossRef] [PubMed]

16. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. **EMC-23**, 377–382 (1981). [CrossRef]

16. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. **EMC-23**, 377–382 (1981). [CrossRef]

17. K. Umashankar and A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. **24**, 397–405 (1982). [CrossRef]

*E*

_{0}is the peak field amplitude,

*t*

_{0}a time offset and

*τ*the FWHM of the pulse.

*x*–

*y*plane at the interface between air and substrate. We simulate the metalized parts of the meta-atom as perfect electric conductor by fixing the tangential components of the electric field to zero. This is a very good approximation in the THz frequency range, where Ohmic losses in the metal can be neglected, provided the thickness of the metal is larger than the skin depth. To minimize the error introduced by approximating the smooth contour of the meta-atom by rectangular grid cells, we use the diagonal split-cell method for approximating round shapes [14].

*x*and

*y*direction, respectively. From Figs. 2b and 2c, it is evident that the resonant mode for polarization along the

*x*direction is a LC mode using the combined capacitance of the splits. The resonance along the

*y*direction is the fundamental dipole mode of the closed ring. The corresponding transmission spectra are shown in Fig. 2d. In the following, we are only interested in the LC resonance along the

*x*direction. This resonance shows a strong enhancement of the electric field at the gap and the two side bars. At a distance of Δ

*r*/2 = 250nm to the metal, corresponding to the center of the PQW, the

*z*peak field is about ten times stronger than the incident field. From a least squares fit of a Lorentzian to the simulated frequency domain data for this resonance, shown in Fig. 2d, we determined the resonance frequency and FWHM to be

*ω*= 2

_{c}*π*× 2.30THz and Δ

*ω*= 2

_{c}*π*× 0.27THz, respectively, giving a Q-factor of 8.5. The energy conversion efficiency calculated at the

*z*coordinate of the QW, is in this case

*η*= 54%.

*x*(

_{c}*t*) and

*x*(

*t*) are the electron displacement from equilibrium,

*δ*the damping coefficient and

_{c}*ω*the resonance frequency of the meta-atom, and Ω the coupling coefficient.

_{c}*E*. The fitted values

_{x}*ω*

_{12}= 2

*π*× 2.3THz and

*δ*= 0.4 × 10

^{12}rad/s correspond to the simulation parameters. The coupling coefficient Ω = 5.4 × 10

^{12}rad/s is larger than the damping of the QW,

*δ*, which is a clear indication of strong light-matter coupling. The results suggest that the coupling efficiency of the meta-atom to the ISB is already good enough that no further confinement of the electric field, for example in a microcavity [12], is necessary. The transmission coefficient predicted by the Maxwell-Bloch model with an incident field strength of 50V/cm (dark green curve in Fig. 3a) is also in excellent agreement to the simple Lorentz model.

## 3. Experiments

_{0.3}Ga

_{0.7}As material system. The distance of the well to the surface is 185 nm. We have used both modulation doped and undoped structures for reference [20

20. R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett. **25**, 272–274 (2000). [CrossRef]

21. R. Bratschitsch, T. Müller, R. Kersting, G. Strasser, and K. Unterrainer, “Coherent terahertz emission from opticall pumped intersubband plasmons in parabolic quantum wells,” Appl. Phys. Lett. **76**, 3501–3503 (2000). [CrossRef]

*L*= 140nm leading to a theoretical transition frequency of

*ω*

_{12}= 2

*π*× 2.2THz. The effective sheet density of the doped sample has been determined by Hall measurements to

*n*= 5×10

_{s}^{11}cm

^{–2}[22

22. J. Ulrich, R. Zobl, K. Unterrainer, G. Strasser, E. Gornik, K. D. Maranowski, and A. C. Gossard, “Temperature dependence of far-infrared electroluminescence in parabolic quantum wells,” Appl. Phys. Lett. **74**, 3158–3160 (1999). [CrossRef]

*T*

_{2}= 2.5ps [20

20. R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett. **25**, 272–274 (2000). [CrossRef]

*μ*

_{12}= 8.6nm, leading to an oscillator strength close to one. For coupling the QW to the free-space radiation, we have used an array of meta-atoms consisting of rectangular double-split rings, as shown in Fig. 2a. The rings consist of 150 nm Ag and 10 nm Au and have been fabricated using standard lithography and e-beam evaporation. The spacing between adjacent rings is 40

*μ*m × 40

*μ*m. The measured frequency response of the bare meta-atoms is in excellent agreement with the simulation (Fig. 2d). These measurements were done with a standard TDS setup on the undoped sample with a frequency resolution of 56GHz. From a Lorentzian fit to the experimental data, we extracted

*ω*= 2

_{c}*π*× 2.28THz and Δ

*ω*= 2

_{c}*π*× 0.46THz, giving a Q-factor of 4.9, which is lower than expected. We attribute this broadening to geometrical variations between different meta-atoms caused by imperfections in the fabrication process.

*δ*= 0.5 × 10

^{12}rad/s < Ω = 4.7 × 10

^{12}rad/s. The values have been extracted from a least squares fit of the coupled oscillator model, Eqs. (7) and (8), with the polarization averaging taken into account (shown as gray lines). The meta-atom transition frequency is determined to

*ω*

_{12}= 2

*π*× 2.3THz. All these values are in excellent agreement with our simulations.

## 4. Summary

## Acknowledgments

## References and links

1. | Z. Schlesinger, J. C. M. Hwang, and J. S. J. Allen, “Subband-Landau-level coupling in a two-dimensional electron gas,” Phys. Rev. Lett. |

2. | J. S. Smith, L. C. Chiu, S. Margalit, A. Yariv, and A. Y. Cho, “A new infrared detector using electron emission from multiple quantum wells,” J. Vac. Sci. Technol. B |

3. | J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science |

4. | M. Helm, The Basic Physics of Intersubband Transitions, vol. 62 of |

5. | L. C. West and S. J. Eglash, “First observation of an extremely large-dipole infrared transition within the conduction band of a GaAs quantum well,” Appl. Phys. Lett. |

6. | B. F. Levine, R. J. Malik, J. Walker, K. K. Choi, C. G. Bethea, D. A. Kleinman, and J. M. Vandenberg, “Strong 8.2 |

7. | D. Heitmann, J. P. Kotthaus, and E. G. Mohr, “Plasmon dispersion and intersubband resonance at high wavevectors in Si(100) inversion layers,” Solid State Commun. |

8. | 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. |

9. | N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics |

10. | N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express |

11. | Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. |

12. | M. Geiser, C. Walther, G. Scalari, M. Beck, M. Fischer, L. Nevou, and J. Faist, “Strong light-matter coupling at terahertz frequencies at room temperature in electronic LC resonators,” Appl. Phys. Lett. |

13. | K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

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

15. | B. Engquist and A. Majda, “Absorbing boundary conditions for numerical simulation of waves,” Proc. Natl. Acad. Sci. USA |

16. | G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. |

17. | K. Umashankar and A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. |

18. | R. W. Ziolkowski, “The incorporation of microscopical material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. |

19. | B. Bidégaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Methods Partial Differ. Eq. |

20. | R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett. |

21. | R. Bratschitsch, T. Müller, R. Kersting, G. Strasser, and K. Unterrainer, “Coherent terahertz emission from opticall pumped intersubband plasmons in parabolic quantum wells,” Appl. Phys. Lett. |

22. | J. Ulrich, R. Zobl, K. Unterrainer, G. Strasser, E. Gornik, K. D. Maranowski, and A. C. Gossard, “Temperature dependence of far-infrared electroluminescence in parabolic quantum wells,” Appl. Phys. Lett. |

**OCIS Codes**

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(160.3918) Materials : Metamaterials

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 29, 2011

Revised Manuscript: June 7, 2011

Manuscript Accepted: June 7, 2011

Published: June 30, 2011

**Citation**

D. Dietze, A. Benz, G. Strasser, K. Unterrainer, and J. Darmo, "Terahertz meta-atoms coupled to a quantum well intersubband transition," Opt. Express **19**, 13700-13706 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13700

Sort: Year | Journal | Reset

### References

- Z. Schlesinger, J. C. M. Hwang, and J. S. J. Allen, “Subband-Landau-level coupling in a two-dimensional electron gas,” Phys. Rev. Lett. 50, 2098–2101 (1983). [CrossRef]
- J. S. Smith, L. C. Chiu, S. Margalit, A. Yariv, and A. Y. Cho, “A new infrared detector using electron emission from multiple quantum wells,” J. Vac. Sci. Technol. B 1, 376–378 (1983). [CrossRef]
- J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 22, 553–556 (1994). [CrossRef]
- M. Helm, The Basic Physics of Intersubband Transitions , vol. 62 of Semiconductors and Semimetals (Academic Press, 2000).
- L. C. West and S. J. Eglash, “First observation of an extremely large-dipole infrared transition within the conduction band of a GaAs quantum well,” Appl. Phys. Lett. 46, 1156–1158 (1985). [CrossRef]
- B. F. Levine, R. J. Malik, J. Walker, K. K. Choi, C. G. Bethea, D. A. Kleinman, and J. M. Vandenberg, “Strong 8.2μm infrared intersubband absorption in doped GaAs/AlAs quantum well waveguides,” Appl. Phys. Lett. 50, 273–275 (1987). [CrossRef]
- D. Heitmann, J. P. Kotthaus, and E. G. Mohr, “Plasmon dispersion and intersubband resonance at high wavevectors in Si(100) inversion layers,” Solid State Commun. 44, 715–718 (1982). [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, 2075–2084 (1999). [CrossRef]
- N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2, 351–354 (2008). [CrossRef]
- N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18, 24140–24151 (2010). [CrossRef] [PubMed]
- Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102, 186402 (2009).
- M. Geiser, C. Walther, G. Scalari, M. Beck, M. Fischer, L. Nevou, and J. Faist, “Strong light-matter coupling at terahertz frequencies at room temperature in electronic LC resonators,” Appl. Phys. Lett. 97, 191107 (2010).
- K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method , 2nd ed. (Artech House, 2000).
- B. Engquist and A. Majda, “Absorbing boundary conditions for numerical simulation of waves,” Proc. Natl. Acad. Sci. USA 74, 1765–1766 (1977). [CrossRef] [PubMed]
- G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981). [CrossRef]
- K. Umashankar and A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. 24, 397–405 (1982). [CrossRef]
- R. W. Ziolkowski, “The incorporation of microscopical material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997). [CrossRef]
- B. Bidégaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Methods Partial Differ. Eq. 19, 284–300 (2003). [CrossRef]
- R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett. 25, 272–274 (2000). [CrossRef]
- R. Bratschitsch, T. Müller, R. Kersting, G. Strasser, and K. Unterrainer, “Coherent terahertz emission from opticall pumped intersubband plasmons in parabolic quantum wells,” Appl. Phys. Lett. 76, 3501–3503 (2000). [CrossRef]
- J. Ulrich, R. Zobl, K. Unterrainer, G. Strasser, E. Gornik, K. D. Maranowski, and A. C. Gossard, “Temperature dependence of far-infrared electroluminescence in parabolic quantum wells,” Appl. Phys. Lett. 74, 3158–3160 (1999). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.