## Distributed gain in plasmonic reflectors and its use for terahertz generation |

Optics Express, Vol. 20, Issue 18, pp. 19618-19627 (2012)

http://dx.doi.org/10.1364/OE.20.019618

Acrobat PDF (1100 KB)

### Abstract

Semiconductor plasmons have potential for terahertz generation. Because practical device formats may be quasi-optical, we studied theoretically distributed plasmonic reflectors that comprise multiple interfaces between cascaded two-dimensional electron channels. Employing a mode-matching technique, we show that transmission through and reflection from a single interface depend on the magnitude and direction of a dc current flowing in the channels. As a result, plasmons can be amplified at an interface, and the cumulative effect of multiple interfaces increases the total gain, leading to plasmonic reflection coefficients exceeding unity. Reversing the current direction in a distributed reflector, however, has the opposite effect of plasmonic deamplification. Consequently, we propose structurally asymmetric resonators comprising two different distributed reflectors and predict that they are capable of terahertz oscillations at low threshold currents.

© 2012 OSA

## 1. Introduction

1. J. Gómez 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**, 256804 (2004). [CrossRef]

3. 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**, 123901 (2008). [CrossRef] [PubMed]

4. S. A. Mikhailov, “Plasma instability and amplification of electromagnetic waves in low-dimensional electron systems,” Phys. Rev. B **58**, 1517–1532 (1998). [CrossRef]

5. S. Riyopoulos, “THz instability by streaming carriers in high mobility solid-state plasmas,” Phys. Plasmas **12**, 070704 (2005). [CrossRef]

7. O. Sydoruk, V. Kalinin, and L. Solymar, “Terahertz instability of optical phonons interacting with plasmons in two-dimensional electron channels,” Appl. Phys. Lett. **97**, 062107 (2010). [CrossRef]

8. M. Dyakonov and M. Shur, “Shallow water analogy for a ballistic field-effect transistor: new mechanism of plasma wave generation by a dc current,” Phys. Rev. Lett. **71**, 2465–2468 (1993). [CrossRef] [PubMed]

9. F. J. Crowne, “Contact boundary conditions and the Dyakonov–Shur instability in high electron mobility transistors,” J. Appl. Phys. **82**, 1242–1254 (1997). [CrossRef]

12. O. Sydoruk, R. R. A. Syms, and L. Solymar, “Plasma oscillations and terahertz instability in field-effect transistors with Corbino geometry,” Appl. Phys. Lett. **97**, 263504 (2010). [CrossRef]

13. J. Lusakowski, W. Knap, N. Dyakonova, L. Varani, J. Mateos, T. Gonzalez, Y. Roelens, S. Bollaert, A. Cappy, and K. Karpierz, “Voltage tuneable terahertz emission from a ballistic nanometer InGaAs/InAlAs transistor,” J. Appl. Phys. **97**, 064307 (2005). [CrossRef]

15. A. El Fatimy, N. Dyakonova, Y. Meziani, T. Otsuji, W. Knap, S. Vandenbrouk, K. Madjour, D. Theron, C. Gaquiere, M. A. Poisson, S. Delage, P. Prystawko, and C. Skierbiszewski, “AlGaN/GaN high electron mobility transistors as a voltage-tunable room temperature terahertz sources,” J. Appl. Phys. **107**, 024504 (2010). [CrossRef]

16. T. Otsuji, T. Watanabe, A. El Moutaouakil, H. Karasawa, T. Komori, A. Satou, T. Suemitsu, M. Suemitsu, E. Sano, W. Knap, and V. Ryzhii, “Emission of terahertz radiation from two-dimensional electron systems in semiconductor nano- and heterostructures,” J. Infrared Milli. Terahz. Waves **32**, 629–645 (2011). [CrossRef]

## 2. Single plasmonic reflectors

*n*

_{0}surrounded by a dielectric with a relative permittivity

*ε*

_{d}and sandwiched between two perfectly conducting planes placed symmetrically at a distance

*w*from the channel. The geometry is as shown in Fig. 1(a) but with a single homogenous channel instead of a set of cascaded sections. The role of the metallic boundaries is to control the mode spectrum. We assumed TM waves with electric field components

*E*and

_{x}*E*, magnetic field component

_{z}*H*, angular frequency

_{y}*ω*, and wavenumbers

*k*and

_{x}*k*. The relationship between the wavenumbers is

_{z}*c*is the light velocity. The boundary conditions are

*E*|

_{z}_{x=±w}= 0 at the metal and at the channel. Here

*e*is the electron charge;

*ε*

_{0}is the vacuum permittivity;

*n*and

*J*are, respectively, the harmonically varying electron and current densities. Solution of Maxwell’s equations in the dielectric yields the field components; for example, the amplitude of the magnetic field above the channel is where

*A*is a constant.

4. S. A. Mikhailov, “Plasma instability and amplification of electromagnetic waves in low-dimensional electron systems,” Phys. Rev. B **58**, 1517–1532 (1998). [CrossRef]

12. O. Sydoruk, R. R. A. Syms, and L. Solymar, “Plasma oscillations and terahertz instability in field-effect transistors with Corbino geometry,” Appl. Phys. Lett. **97**, 263504 (2010). [CrossRef]

*n*

_{tot}, and velocity,

*v*

_{tot}, have the form where

*v*

_{0}and

*v*are the dc and the ac electron velocities, respectively. As is usually done, we then assumed that the amplitudes of the dc components are much smaller than their dc counterparts and ignored the products of the small quantities

*n*×

*v*and

*v*×

*v*. Hence, for the total current density we had The first term of the right-hand side in Eq. (4) is the dc component of the total current density,

*J*

_{0}=

*en*

_{0}

*v*

_{0}, and the sum of the last two terms is the ac component of the total current density,

*J*=

*en*

_{0}

*v*+

*env*

_{0}. Note that the dc current density is independent of the ac components, but the ac current density is affected by the dc components. The latter indicates coupling between the plasmons and drifting electrons, which is responsible for the plasmonic gain discussed below. If the amplitudes of the ac and dc components of the electron density and velocity have comparable values, the term

*n*×

*v*in Eq. (4) can no longer be neglected. It would, in particular, lead to a dc current due to the ac fields. We were, however, concerned with the onset of oscillations when the ac terms are small and Eq. (4) is valid.

*m*is the electron effective mass. Substituting Eqs. (2), (4), and (5) into Eq. (1) leads to the dispersion relation in the form where

*v*

_{0}. Due to the Doppler term

*k*

_{z}v_{0}, waves propagating at the same frequency in opposite directions have different values of

*k*and

_{x}*k*. Figure 1(b) shows this for the dispersion curves of the plasmons propagating in opposite directions at low wavenumbers. These are the solutions of the dispersion relation (6) yielding real values of

_{z}*k*and imaginary values of

_{z}*k*. The parameters used are two different electron densities

_{x}*ε*

_{d}= 12.8,

*m*= 0.067

*m*

_{0}(corresponding to GaAs),

*w*= 100 nm, and the same dc current density

17. R. E. Collin, *Foundations for Microwave Engineering* (Wiley-IEEE Press, Hoboken, New Jersey, 2001). [CrossRef]

*μ*m for the channel with the dc electron density

*w*chosen, the dispersion relation yields infinite number of solutions with complex-valued

*k*and

_{z}*k*. These represent evanescent modes that decay along both the

_{x}*x*- and

*z*-directions. The fields in the channel are given, in general, by a superposition of all modes, low- and high-wavenumber plasmons and evanescent waves, as discussed next.

*J*

_{0}flowing through it. A plasmon incident on the interface partially reflects and transmits, as shown schematically in Fig. 2 where the step represents the change in the dc electron density.

*z*= 0 should be known. Our approach to the boundary conditions differed markedly from the one usually used to describe reflection of drifted plasmons [8

8. M. Dyakonov and M. Shur, “Shallow water analogy for a ballistic field-effect transistor: new mechanism of plasma wave generation by a dc current,” Phys. Rev. Lett. **71**, 2465–2468 (1993). [CrossRef] [PubMed]

12. O. Sydoruk, R. R. A. Syms, and L. Solymar, “Plasma oscillations and terahertz instability in field-effect transistors with Corbino geometry,” Appl. Phys. Lett. **97**, 263504 (2010). [CrossRef]

*z*= 0,

*x*= 0 of the junction between the channels, we consider them at the whole interface

*z*= 0,

*x*∈ [−

*w*,

*w*]. Everywhere except the junction, the interface is a continuous dielectric, and the standard field boundary conditions apply To solve the problem in the presence of dc current, the boundary conditions for the ac current and velocity at the junction

*z*= 0,

*x*= 0 should also be known. We discuss first the ac current density. Due to the symmetry of the structure, Eq. (1) gives so that the boundary condition for the current should follow from the conditions for the magnetic field. To obtain these, we considered a cylinder of a small radius

*δ*with the axis at the junction as shown in Fig. 3. The surface of the cylinder is in the dielectric, where the magnetic field is a continuous function. Hence, as the radius of the cylinder decreases,

*δ*→ 0, one gets and because

*J*

^{(1)}|

_{z=−δ}→

*J*

^{(2)}|

_{z=δ}. As a point discontinuity of the current is implausible, the latter finally leads to Hence, the ac current density is continuous at the junction. As then follows from the continuity equation, no line charge can accumulate at the junction. The same arguments are valid for the

*E*component, and hence, for the ac charge density at the interface, leading to Finally, from Eqs. (10) and (11), the boundary condition for the electron velocity is

_{x}*α*denotes the mode number (the low-wavenumber plasmons have

*α*= 1). The amplitude of the incident plasmon is unity, and

*R*are the mode reflection coefficients. Analogously, for the transmitted magnetic field, we took

_{α}*T*are the transmission coefficients, and the summation is for low- and high-wavenumber plasmons and the evanescent waves. The boundary conditions (7) take then the form Combining Eqs. (13) with the analogous expressions for the electron velocity and current density, which follow from Eqs. (10) and (12), and using the orthogonality relation [18

_{α}18. A. D. Bresler, G. H. Joshi, and N. Marcuvitz, “Orthogonality properties for modes in passive and active uniform wave guides,” J. Appl. Phys. **29**, 794–799 (1958). [CrossRef]

19. R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implications for nanoscale cavities,” Phys. Rev. B **76**, 035,408 (2007). [CrossRef]

20. S. Thongrattanasiri, J. Elser, and V. A. Podolskiy, “Quasi-planar optics: computing light propagation and scattering in planar waveguide arrays,” J. Opt. Soc. Am. B **26**, B102–B110 (2009). [CrossRef]

*z*-direction from the channel with the density

## 3. Distributed plasmonic reflectors and oscillators

21. U. Mackens, D. Heitmann, L. Prager, J. P. Kotthaus, and W. Beinvogl, “Minigaps in the plasmon dispersion of a two-dimensional electron gas with spatially modulated charge density,” Phys. Rev. Lett. **53**, 1485–1488 (1984). [CrossRef]

*z*-direction; we now denote these as

*t*

^{(+)}and

*r*

^{(+)}, respectively. Then, we reversed the incidence direction and obtained another pair of reflection and transmission coefficients,

*t*

^{(−)}and

*r*

^{(−)}. If the lengths of the channels in a reflector are large enough, the evanescent waves excited at one interface do not influence the field distributions at the neighboring interfaces. As numerical calculations showed, evanescent waves in our examples can be neglected if the lengths exceed 200 nm. In addition, we ignored in the following the high-wavenumber plasmons. Because of the large wavenumber difference, scattering into these modes is ineffective, and their amplitudes are low. Under these assumptions, the four coefficients,

*i*-th interface in a distributed reflector. The transmission and reflection coefficients for a reflector comprising

*N*interfaces,

*N*− 1)-th channel that has the length

*L*

_{N}_{−1}.

*z*-direction). In the absence of dc current, the frequency variation of the reflection and transmission coefficients has the expected Bragg-like form (see Fig. 4(b)). In a number of frequency bands, the reflection coefficient is close to unity, and the transmission coefficient is close to zero. The lack of periodicity in the curves is due to plasmon dispersion (see Fig. 1(b)).

*R*

^{(−)}| = 1.09 at 1 THz for

*J*

_{0}= −0.05 A/cm. It is, however, less than unity at the same frequencies for the positive dc current, Fig. 4(c). At 1 THz and

*J*

_{0}= 0.05 A/cm, |

*R*

^{(+)}| = 0.91.

*G*| = |

*R*

^{(−)}

*R*

^{(+)}| = 1.09 × 0.91 ≈ 1.

*J*

_{0}= 0.029 A/cm, when a high and narrow peak appears at 1 THz (see Fig. 5(c) and (d)).

*R*| rises rapidly near the threshold current.

*k*′ − j

*k*″, where

*k*″ is the attenuation constant. Assuming that the attenuation constants are equal for all channels, we studied the effect of loss for the distributed reflector of Fig. 4(a). At 1 THz, the reflection coefficients decreased linearly with small

*k*″. The decrease of |

*R*| of 5%, from 1.09 to about 1.04, was reached for

*k*″ ≈ 2 rad/

*μ*m. For this value, we estimated

*ωτ*≈ 50, where

*τ*is the collision time. The corresponding mobilities are of the order of 10

^{5}cm

^{2}/(Vs), achievable in GaAs channels at low temperatures.

## 4. Conclusions

^{6}cm/s and it could be further decreased by increasing the number of sections in the distributed reflectors. The results suggest a new way of realizing THz oscillators.

## References and links

1. | J. Gómez Rivas, M. Kuttge, P. H. Bolivar, H. Kurz, and J. A. Sánchez-Gil, “Propagation of surface plasmon polaritons on semiconductor gratings,” Phys. Rev. Lett. |

2. | D. Veksler, F. Teppe, A. P. Dmitriev, V. Y. Kachorovskii, W. Knap, and M. S. Shur, “Detection of terahertz radiation in gated two-dimensional structures governed by dc current,” Phys. Rev. B |

3. | 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. |

4. | S. A. Mikhailov, “Plasma instability and amplification of electromagnetic waves in low-dimensional electron systems,” Phys. Rev. B |

5. | S. Riyopoulos, “THz instability by streaming carriers in high mobility solid-state plasmas,” Phys. Plasmas |

6. | S. M. Kukhtaruk, “High-frequency properties of systems with drifting electrons and polar optical phonons,” Sem. Phys. Quant. Electr. & Optoelectr. |

7. | O. Sydoruk, V. Kalinin, and L. Solymar, “Terahertz instability of optical phonons interacting with plasmons in two-dimensional electron channels,” Appl. Phys. Lett. |

8. | M. Dyakonov and M. Shur, “Shallow water analogy for a ballistic field-effect transistor: new mechanism of plasma wave generation by a dc current,” Phys. Rev. Lett. |

9. | F. J. Crowne, “Contact boundary conditions and the Dyakonov–Shur instability in high electron mobility transistors,” J. Appl. Phys. |

10. | M. V. Cheremisin and G. G. Samsonidze, “D’yakonov–Shur instability in a ballistic field-effect transistor with a spatially nonuniform channel,” Semiconductors |

11. | M. Dyakonov and M. Shur, “Current instability and plasma waves generation in ungated two-dimensional electron layers,” Appl. Phys. Lett. |

12. | O. Sydoruk, R. R. A. Syms, and L. Solymar, “Plasma oscillations and terahertz instability in field-effect transistors with Corbino geometry,” Appl. Phys. Lett. |

13. | J. Lusakowski, W. Knap, N. Dyakonova, L. Varani, J. Mateos, T. Gonzalez, Y. Roelens, S. Bollaert, A. Cappy, and K. Karpierz, “Voltage tuneable terahertz emission from a ballistic nanometer InGaAs/InAlAs transistor,” J. Appl. Phys. |

14. | Y. Tsuda, T. Komori, A. El Fatimy, K. Horiike, T. Suemitsu, and T. Otsuji, “Application of plasmon-resonant microchip emitters to broadband terahertz spectroscopic measurement,” J. Opt. Soc. Am. B |

15. | A. El Fatimy, N. Dyakonova, Y. Meziani, T. Otsuji, W. Knap, S. Vandenbrouk, K. Madjour, D. Theron, C. Gaquiere, M. A. Poisson, S. Delage, P. Prystawko, and C. Skierbiszewski, “AlGaN/GaN high electron mobility transistors as a voltage-tunable room temperature terahertz sources,” J. Appl. Phys. |

16. | T. Otsuji, T. Watanabe, A. El Moutaouakil, H. Karasawa, T. Komori, A. Satou, T. Suemitsu, M. Suemitsu, E. Sano, W. Knap, and V. Ryzhii, “Emission of terahertz radiation from two-dimensional electron systems in semiconductor nano- and heterostructures,” J. Infrared Milli. Terahz. Waves |

17. | R. E. Collin, |

18. | A. D. Bresler, G. H. Joshi, and N. Marcuvitz, “Orthogonality properties for modes in passive and active uniform wave guides,” J. Appl. Phys. |

19. | R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implications for nanoscale cavities,” Phys. Rev. B |

20. | S. Thongrattanasiri, J. Elser, and V. A. Podolskiy, “Quasi-planar optics: computing light propagation and scattering in planar waveguide arrays,” J. Opt. Soc. Am. B |

21. | U. Mackens, D. Heitmann, L. Prager, J. P. Kotthaus, and W. Beinvogl, “Minigaps in the plasmon dispersion of a two-dimensional electron gas with spatially modulated charge density,” Phys. Rev. Lett. |

22. | Z. Knittl, |

**OCIS Codes**

(230.4910) Optical devices : Oscillators

(240.6680) Optics at surfaces : Surface plasmons

(260.3090) Physical optics : Infrared, far

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: May 23, 2012

Revised Manuscript: June 29, 2012

Manuscript Accepted: July 2, 2012

Published: August 13, 2012

**Citation**

O. Sydoruk, R. R. A. Syms, and L. Solymar, "Distributed gain in plasmonic reflectors and its use for terahertz generation," Opt. Express **20**, 19618-19627 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19618

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### References

- J. Gómez 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, 256804 (2004). [CrossRef]
- D. Veksler, F. Teppe, A. P. Dmitriev, V. Y. Kachorovskii, W. Knap, and M. S. Shur, “Detection of terahertz radiation in gated two-dimensional structures governed by dc current,” Phys. Rev. B73, 125328 (2006). [CrossRef]
- 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, 123901 (2008). [CrossRef] [PubMed]
- S. A. Mikhailov, “Plasma instability and amplification of electromagnetic waves in low-dimensional electron systems,” Phys. Rev. B58, 1517–1532 (1998). [CrossRef]
- S. Riyopoulos, “THz instability by streaming carriers in high mobility solid-state plasmas,” Phys. Plasmas12, 070704 (2005). [CrossRef]
- S. M. Kukhtaruk, “High-frequency properties of systems with drifting electrons and polar optical phonons,” Sem. Phys. Quant. Electr. & Optoelectr.11, 43–49 (2008).
- O. Sydoruk, V. Kalinin, and L. Solymar, “Terahertz instability of optical phonons interacting with plasmons in two-dimensional electron channels,” Appl. Phys. Lett.97, 062107 (2010). [CrossRef]
- M. Dyakonov and M. Shur, “Shallow water analogy for a ballistic field-effect transistor: new mechanism of plasma wave generation by a dc current,” Phys. Rev. Lett.71, 2465–2468 (1993). [CrossRef] [PubMed]
- F. J. Crowne, “Contact boundary conditions and the Dyakonov–Shur instability in high electron mobility transistors,” J. Appl. Phys.82, 1242–1254 (1997). [CrossRef]
- M. V. Cheremisin and G. G. Samsonidze, “D’yakonov–Shur instability in a ballistic field-effect transistor with a spatially nonuniform channel,” Semiconductors33, 578–585 (1999). [CrossRef]
- M. Dyakonov and M. Shur, “Current instability and plasma waves generation in ungated two-dimensional electron layers,” Appl. Phys. Lett.87(11), 111501 (2005). [CrossRef]
- O. Sydoruk, R. R. A. Syms, and L. Solymar, “Plasma oscillations and terahertz instability in field-effect transistors with Corbino geometry,” Appl. Phys. Lett.97, 263504 (2010). [CrossRef]
- J. Lusakowski, W. Knap, N. Dyakonova, L. Varani, J. Mateos, T. Gonzalez, Y. Roelens, S. Bollaert, A. Cappy, and K. Karpierz, “Voltage tuneable terahertz emission from a ballistic nanometer InGaAs/InAlAs transistor,” J. Appl. Phys.97, 064307 (2005). [CrossRef]
- Y. Tsuda, T. Komori, A. El Fatimy, K. Horiike, T. Suemitsu, and T. Otsuji, “Application of plasmon-resonant microchip emitters to broadband terahertz spectroscopic measurement,” J. Opt. Soc. Am. B26, A52–A57 (2009). [CrossRef]
- A. El Fatimy, N. Dyakonova, Y. Meziani, T. Otsuji, W. Knap, S. Vandenbrouk, K. Madjour, D. Theron, C. Gaquiere, M. A. Poisson, S. Delage, P. Prystawko, and C. Skierbiszewski, “AlGaN/GaN high electron mobility transistors as a voltage-tunable room temperature terahertz sources,” J. Appl. Phys.107, 024504 (2010). [CrossRef]
- T. Otsuji, T. Watanabe, A. El Moutaouakil, H. Karasawa, T. Komori, A. Satou, T. Suemitsu, M. Suemitsu, E. Sano, W. Knap, and V. Ryzhii, “Emission of terahertz radiation from two-dimensional electron systems in semiconductor nano- and heterostructures,” J. Infrared Milli. Terahz. Waves32, 629–645 (2011). [CrossRef]
- R. E. Collin, Foundations for Microwave Engineering (Wiley-IEEE Press, Hoboken, New Jersey, 2001). [CrossRef]
- A. D. Bresler, G. H. Joshi, and N. Marcuvitz, “Orthogonality properties for modes in passive and active uniform wave guides,” J. Appl. Phys.29, 794–799 (1958). [CrossRef]
- R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implications for nanoscale cavities,” Phys. Rev. B76, 035,408 (2007). [CrossRef]
- S. Thongrattanasiri, J. Elser, and V. A. Podolskiy, “Quasi-planar optics: computing light propagation and scattering in planar waveguide arrays,” J. Opt. Soc. Am. B26, B102–B110 (2009). [CrossRef]
- U. Mackens, D. Heitmann, L. Prager, J. P. Kotthaus, and W. Beinvogl, “Minigaps in the plasmon dispersion of a two-dimensional electron gas with spatially modulated charge density,” Phys. Rev. Lett.53, 1485–1488 (1984). [CrossRef]
- Z. Knittl, Optics of Thin Films (Wiley, London, 1981).

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