Subscriber access provided by GEORGIAN COURT UNIVERSITY
Communication
Control of strong light-matter coupling using the capacitance of metamaterial nanocavities Alexander Benz, Salvatore Campione, John F Klem, Michael B. Sinclair, and Igal Brener Nano Lett., Just Accepted Manuscript • Publication Date (Web): 27 Jan 2015 Downloaded from http://pubs.acs.org on January 27, 2015
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Control of strong light-matter coupling using the capacitance of metamaterial nanocavities Alexander Benz,†,‡ Salvatore Campione,†,‡ John F. Klem,‡ Michael B. Sinclair,‡ and Igal Brener†,‡,* †
Center for Integrated Nanotechnologies (CINT), Sandia National Laboratories, P.O. Box 5800,
Albuquerque, NM 87185, USA ‡
Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185, USA
ACS Paragon Plus Environment
1
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 2 of 28
ABSTRACT Metallic nanocavities with deep subwavelength mode volumes can lead to dramatic changes in the behavior of emitters placed in their vicinity. This collocation and interaction often leads to strong coupling. Here, we present for the first time experimental evidence that the Rabi splitting is directly proportional to the electrostatic capacitance associated with the metallic nanocavity. The system analyzed consists of different metamaterial geometries with the same resonance wavelength coupled to intersubband transitions in quantum wells.
KEYWORDS metamaterial; nanocavity; strong light-matter interaction; intersubband transitions; mid-infrared; plasmonics.
ACS Paragon Plus Environment
2
Page 3 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Strong coupling between an optical cavity and a two-level system can lead to new and unusual coupled light-matter states (or quasi-particles) called polaritons1, 2. In this regime the cavity and the two-level system exchange energy coherently at a characteristic rate called the vacuum Rabi (angular) frequency Ω which is dominant with respect to all loss mechanisms present in the system. Experimentally, the energy exchange can be observed by measuring the optical reflectance or transmittance spectrum of the coupled system and/or a temporal beating in the time domain. Due to the strong light-matter interaction, the single bare cavity resonance splits into two resonances in the frequency domain forming an upper and a lower polariton branch. When the bare cavity and the two-level system are brought exactly into resonance, the two polaritons show an energy difference of 2 Ω . A larger Rabi frequency corresponds to a faster energy exchange or a more efficient coupling. This fundamental energy exchange between an optical cavity and a two-level system is the basis for a variety of fascinating phenomena such as polariton electroluminescence3-6, polariton lasing7-9, parametric scattering10, 11, polariton superfluidity12-14, and Bose-Einstein condensation of polaritons12-15. Recently, studying strong light-matter coupling in extremely small interaction volumes using plasmonic cavities has become a very active field of research16-32. Particularly, planar metamaterials seem ideally suited as nanocavities due to the possibility to engineer their electric and magnetic resonances by controlling the geometry of the individual subwavelength constituents. Strong coupling of such plasmonic metamaterial nanocavities has been achieved with cyclotron resonances19, optical phonons in polar crystals33, ultra-thin doped plasma layers34 or intersubband transitions (ISTs) in semiconductor quantum-wells (QWs)20-23, 35. A variety of applications can be realized using a combination of metamaterials and optical dipole transitions in semiconductors, such as voltage tunable optical filters36, intersubband-based light emitting
ACS Paragon Plus Environment
3
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 28
diodes37, and efficient non-linear devices38, 39. These types of actively tunable filters rely on controlling the resonance energy of ISTs using the quantum-confined Stark effect40, 41 and thereby the spectral response of the strongly coupled system. Furthermore, the coupled entity of a metamaterial resonator and an IST can form a polariton emitter which can be used for incoherent light generation. ISTs on their own cannot be used directly to realize efficient spontaneous emitters due to the extremely fast non-radiative scattering rates42. New classes of higher order non-linear media can be engineered to exploit the large non-linearities that can be designed using ISTs43. Recent investigations show that the non-linear conversion efficiency in strongly coupled metamaterial-IST systems can be enhanced by eight orders of magnitude compared to conventional non-linear crystals38, 39. All of these applications require an extremely efficient energy exchange between a metamaterial nanocavity and a two-level system or, equivalently, a large Rabi frequency. Previous attempts at increasing the light-matter coupling focused mainly on the properties of the two-level system while ignoring the effect of the metamaterial geometry20-22. Here, we will focus on the influence of the metamaterial nanocavity design on the type of strong light-matter interaction discussed above. We demonstrate experimentally that the strong light-matter coupling can be described by a simple equivalent circuit model and that the electrostatic capacitance of the single metamaterial nanocavity is the crucial parameter determining the coupling. We model the bare metamaterial (bare cavity) as a simple resistorinductor-capacitor (RLC) resonant circuit44. The strong light-matter coupling is added without free fitting parameters in the form of a dispersive capacitor45; a detailed schematic will be shown later in this work. This model simplifies the complex three dimensional problem of a metamaterial nanocavity coupled to an IST to a simple equivalent electrical circuit based on
ACS Paragon Plus Environment
4
Page 5 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
passive components. More importantly, the model shows that the electrostatic capacitance of the metamaterial resonator is the key that controls the light-matter coupling (or Rabi frequency)45. We verify the model experimentally using four metamaterial nanocavity geometries with identical bare cavity resonances but different values for the capacitive, inductive, and ohmic circuit components. We fabricate all four metamaterials atop the same wafer to exclude variations in the two-level system resonance frequency. Furthermore, and contrary to other strongly coupled systems, our experiments also show that the quality-factor (or damping) of the metamaterial does not affect the Rabi frequency noticeably. The basic geometry of our strongly coupled system is depicted in Figure 1(a). The metamaterial is fabricated directly atop the two-level system which is realized as an IST in a semiconductor heterostructure. Thin layers of different semiconductor crystals are grown on top of each other forming potential wells, so-called quantum-wells, for carriers. Thereby, the free motion of the electrons is inhibited along one dimension resulting in discrete energy states with atomic-like optical transitions. The energies of ISTs depend primarily on the width of the individual barriers and wells. Choosing appropriate host crystals, the transition energy can be tuned from the terahertz46 to the near-infrared47, 48 spectral range. The metamaterial resonators efficiently convert z-propagating normal incidence radiation to an electric field polarized along the z-axis and thereby makes coupling of free-space radiation to ISTs possible49. This combination of metamaterial and ISTs gives us a highly flexible model system to study strong light-matter interaction between a nanocavity and an artificial two-level system. However, we want to stress here that the exact details of the two-level system are less important in this work. Our findings can be applied to all other two-level systems used in similar experiments such as cyclotron resonances, optical phonons, excitons, or ultra-thin doped plasmonic layers.
ACS Paragon Plus Environment
5
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 28
In our experiments we use 12.5nm-thick In0.53Ga0.47As QWs, homogenously doped to 1018 cm3
and separated by 20nm-thick Al0.48In0.52As barriers. The samples are grown by molecular beam
epitaxy. This leads to a transition energy for the lowest bound-to-bound electronic transition (ground state to first excited state inside the same QW) of 100 meV or a transition frequency of 24.2 THz. The calculated50 energy levels for one QW-barrier pair are shown in Figure 1(b). Uncoupled, rectangular QWs show an oscillator strength49 of ~1 for this ground to first excited state transition, essentially restricting the entire interaction with light to this one single transition. We repeat the basic QW-barrier pair sequence 20 times to achieve a total thickness of 650 nm for the QW-stack. The QW stack growth is capped with a 30 nm thin Al0.48In0.52As layer. The thick QW-stack in combination with the thin cap layer is essential to ensure an almost perfect overlap of the metamaterial cavity mode with the QWs responsible for ISTs for all metamaterial geometries. The penetration depth of the cavity mode into the semiconductor is on the order of a few hundred nanometers in the mid-infrared spectral range49. Here, we will be using four basic metamaterial geometries labeled: dogbone, dumbbell, Jerusalem Cross and circular split-ring resonator (SRR). A schematic representation of all four geometries including their typical dimensions is illustrated in Figure 1(c). It should be stressed here that none of our metamaterial geometries requires a metallic ground plane to achieve the necessary mode confinement or the necessary z-polarized electric field in its vicinity22. We first model the interaction between a metamaterial nanocavity and the IST using finitedifference time-domain (FDTD) simulations51. This offers a fast and accurate way to determine the bare cavity resonance of the metamaterial (when the IST is turned off) and to calculate the upper and lower polariton branches (including the IST) for a given metamaterial geometry. We model one unit cell and apply periodic boundary conditions to capture the effect of the two-
ACS Paragon Plus Environment
6
Page 7 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
dimensional metamaterial array in the fabricated sample. The permittivity for the gold layer is extracted from spectral ellipsometry measurements performed on a separate 100 nm thin gold film prepared under identical conditions as the metamaterial samples. The ISTs are modeled as anisotropic harmonic oscillators49. Only the electric field component that is polarized along the growth direction of the QW-stack interacts with the ISTs, so the permittivity can be written as: 0 0 0
0 0
(1) ,
0
Δ ∗
where
2
is the background permittivity of the QW, Δ
difference between ground and excited state, oscillator strength,
∗
(2)
,
is the carrier concentration
is the effective electron mass,
is the intersubband transition frequency,
the vacuum permittivity and
is the
is the IST broadening,
is
is the elementary electron charge. The center frequency of the IST
is set to 24.2 THz, while the full-width-at-half-maximum is set to 2.4 THz. Both values were confirmed by intersubband absorption measurements in a multi-pass waveguide geometry using an unprocessed piece of the same wafer. The FDTD simulation results for a dogbone metamaterial nanocavity interacting with the ISTs are presented in Figure 2(a). We control the bare metamaterial resonance by geometrically stretching its dimensions thus allowing a step-wise sweep of the bare cavity frequency across the IST transition frequency. Due to the strong light-matter coupling we observe a clear anti-crossing behavior of the two polariton branches. Using a coupled oscillator model52 we can extract a corresponding Rabi frequency of 2.48 THz. The FDTD simulations in combination with the
ACS Paragon Plus Environment
7
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 8 of 28
analytic expression for the intersubband permittivity (Eq. (2)) already point at a few parameters that can be optimized to increase the Rabi frequency such as the population difference between ground and excited state (Δ
) or the oscillator strength for this transition (
). Increasing
either one of these parameters increases the influence of the IST on the QW permittivity and, as a consequence, the interaction with the light field. These parameters have also been identified in the quantum-mechanical modeling1, 20-22 to be essential for the determination of the Rabi frequency. It should be noted here that also in the classical picture the Rabi frequency increases with the number of oscillators coupled to the cavity53. In our case this corresponds to the carrier concentration difference between ground and first excited state, Δ
. Furthermore, the spatial
overlap between a cavity field and the volume where the two-level system resides shows a strong effect on the polariton splitting54. The thin cap layer (30 nm Al0.48In0.52As) in combination with the thick QW-stack (650 nm) yields an overlap close to 100% between the metamaterial cavity mode and the heterostructure regardless of the exact cavity field profile. This allows us to focus our study on the metamaterial geometry dependence and minimize the influence of the two-level system. This initial analysis also reveals one of the major weaknesses of solely relying on FDTD simulations: it is easy to analyze a given metamaterial geometry but very difficult to derive a new one that is optimized for light-matter coupling. The results of FDTD simulations are the full electromagnetic fields inside the computational volume. However, these results do not provide a direct (or intuitive) relation between the simulated geometry and a complex physical behavior such as light-matter coupling. The typical design flow of choosing a metamaterial geometry, calculating the reflectance or transmittance, evaluating the Rabi frequency, varying the geometry
ACS Paragon Plus Environment
8
Page 9 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
and then recalculating the field profiles offers no possibility to estimate the final results without running the full simulation, making trial-and-error the only optimization strategy available. The use of a recently published equivalent electrical circuit model to describe the light-matter interaction in these systems can solve some of the shortcomings mentioned before45. Modeling the bare metamaterial nanocavity using an RLC resonant circuit44 has obvious advantages compared to a mechanical analogue. The electrical circuit components can be related closely to the electromagnetic properties of the nanocavity. Furthermore, we can obtain the parameters for the equivalent circuit by splitting the metamaterial resonator into basic geometric shapes and describe the system as a combination of lumped circuit elements55. Analytical expressions for the self-capacitance of free-standing metal disks56 or rectangular metal patches57 have been reported. Even more complex geometries such as split-rings58 or Jerusalem Cross resonators59 can be modeled analytically with high accuracy. The inductive part for all metamaterial geometries used in this work can be found in reference books, e.g. 60. These two quantities set the bare resonance frequency of the metamaterial (
1
√
) and can be used to estimate the Rabi frequency.
Extracting the ohmic component from this kind of model is more difficult. The resistor in the equivalent circuit captures the ohmic losses of the metal and the radiative damping of the cavity, but the latter is difficult to model analytically. However, the RLC parameters can be extracted by fitting the reflectance or transmittance spectra of the equivalent one-dimensional transmission line model to the spectra obtained by FDTD simulations as explained in45. The equivalent parameters for the four different geometries, using this procedure are given in Table 1 for a bare cavity frequency of approximately 24.2 THz (resonant with the IST). These parameters also motivate our choices for the four different metamaterial geometries used to test the circuit model: we can cover high capacitance with different geometric shapes (dogbone and dumbbell), low
ACS Paragon Plus Environment
9
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
metamaterial damping (
Page 10 of 28
) resulting in a higher quality-factor (Jerusalem Cross) and high
inductance (circular SRR). The full equivalent circuit including the influence of the substrate is illustrated in Figure 1(a).45 The interaction between metamaterial and IST can be included by adding one dispersive capacitor to the circuit, which can be modeled as:
1 where
and
,
are the same permittivities that are used for the FDTD simulations and
are described in Eq. (1) and Eq. (2); and
is the capacitance used to model the metamaterial
bare cavity. Increasing the Rabi frequency corresponds to increasing the value for which can be achieved in two ways. First, modified to increase nanocavity
(3)
in Eq. (3),
(which represents the strength of the IST) can be
as dictated by Eq. (2). Second, the capacitance of the metamaterial
can be used to increase
linearly (we remind the reader that we examine this
second condition experimentally in this paper). It should be stressed here that the parameters for the dispersive capacitor in Eq. (3) are not free fitting parameters but rather given by the IST design. After we determined the equivalent circuit parameters for the bare metamaterial nanocavity, we can calculate the reflectance (or transmittance) spectra including the strong light-matter coupling by adding the dispersive capacitor defined in Eq. (3). The excellent agreement between the equivalent circuit model and FDTD simulations in the entire frequency range between 10 and 45 THz is presented in Figure 2(b). As an example we show three different bare cavity resonance
ACS Paragon Plus Environment
10
Page 11 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
frequencies for each of the metamaterial geometries where the metamaterial is blue-detuned from the IST (blue line), on resonance (black line) and red-detuned (red line). From these reflectance spectra we can extract the Rabi frequencies using a coupled oscillator model52. The Rabi frequencies reported in Table 1 confirm the expected trend: the larger the capacitance of the nanoresonator, the larger the Rabi frequency. The dogbone, the dumbbell and the Jerusalem Cross have all approximately the same capacitance leading to a similar Rabi frequency. Neither the exact geometric shape of the capacitor (dogbone versus dumbbell) nor the metamaterial quality-factor (the Jerusalem Cross shows a significantly lower damping factor
compared to
dogbone and dumbbell) cause any noticeable effect on the Rabi splitting. On the other hand, the circular SRR, which shows the lowest capacitance of all analyzed metamaterial geometries, also shows the lowest Rabi frequency. To confirm these theoretical predictions and the validity of the equivalent circuit model, we fabricated all four resonator geometries on top of the same semiconductor wafer which allows us to eliminate the effects of different wafer growths (variations in intersubband resonance frequency or different doping concentration) and focus on the interaction between IST and metamaterial. The metamaterial nanocavities are defined by electron-beam lithography followed by a Ti/Au (5/100 nm) evaporation and lift-off process. Scanning electron micrograph images showing all four fabricated metamaterial geometries are presented in Figure 3. We measure the transmittance at room temperature using a Bruker IFS66v for all samples. Similar to the FDTD simulations, we geometrically scale the metamaterial dimensions to sweep the bare cavity resonance across the IST and map out the two polariton branches. As an example, we present the experimental transmittance for the dogbone metamaterial as a function of bare cavity frequency in Figure 4(a). The avoided polariton crossing is most clearly visible in the frequency region
ACS Paragon Plus Environment
11
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 12 of 28
where the bare cavity frequency and the IST are similar in energy (~100 meV or ~24 THz). We have also added the polariton eigenfrequencies obtained using a coupled oscillator model to Figure 4(a) to emphasize the anti-crossing behavior. To extract the Rabi frequencies from the measurements we use a coupled oscillator model52 that allows us to fit the entire transmittance spectrum for each metamaterial scaling factor rather than using only two isolated data-points per bare cavity frequency (the upper and the lower polariton eigenfrequencies). We present the fitting results for the dogbone metamaterial and three different bare cavity frequencies (equivalent to three different geometric scaling factors) in Figure 4(b). These three fits yield the same IST frequency (always using the same wafer) and the same Rabi frequency (the coupling strength does not depend on the unperturbed resonance frequency of either the metamaterial or the IST in first approximation) but different bare cavity frequencies (the metamaterial is geometrically stretched). To determine the Rabi frequency for a given metamaterial geometry, we include all experimental transmittance spectra where the bare cavity resonances are within the frequency range: IST ±25%. The Rabi frequencies reported in Table 1 for all four metamaterial geometries are the arithmetic means within the defined frequency range (the standard deviation is given in Table 1 in brackets). This approach allows us to minimize the influence of individual measurements and increase the confidence for the extracted Rabi frequencies. Our experiments verify the prediction from the equivalent RLC circuit model: increasing the capacitance of the metamaterial resonator increases the Rabi frequency. The comparison between experimental data and simulations presented in Figure 4(c) shows that our measured Rabi frequency is approximately 0.2 THz smaller than the simulated one. However, the general trend is not affected by this discrepancy. In addition, we mention that no clear dependence between the
ACS Paragon Plus Environment
12
Page 13 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Rabi frequency and metamaterial damping is observed, neither in the experiments nor in the corresponding simulations. It should be noted here that quantum-mechanical models20-22 often used to describe strongly coupled system consisting of metamaterial resonators and ISTs neglect the exact geometry of the resonators and only include the spatial overlap between the cavity mode and the volume where the two-level system resides. Due to the thick QW-stacks (650 nm) and the thin Al0.48In0.52As cap layer (30 nm) in our samples we achieve a ~100% overlap with the QW-stack, resulting in a 38.5% overlap of the cavity mode and the ISTs (the remaining part of the cavity mode is within the Al0.48In0.52As barriers). Using such a quantum-mechanical model would lead to a Rabi frequency of 1.75 THz for all metamaterial samples regardless of their exact metamaterial geometry, which underestimates the experimental results and cannot capture the measured trend. In conclusion, we confirmed experimentally that the equivalent electrostatic capacitance of a metamaterial nanocavity is the crucial parameter defining the strong light-matter coupling. We fabricated four different metamaterial geometries with varying values for the capacitance, inductance, and ohmic resistance but with the same bare cavity resonance wavelength (~12 m) on top of the same semiconductor wafer. The experimental data are in good agreement with the theoretical predictions confirming the key role of the capacitance for strong light-matter coupling: geometries with a large capacitance such as dogbone or dumbbell metamaterial resonator show a larger Rabi frequency compared to conventional SRRs. Other metamaterial cavity parameters such as damping do not play a major role for strong coupling. This results in very simple metamaterial resonator design rules to optimize light-matter interaction using semiconductor heterostructures and potentially other dipolar excitations. Phenomena such as polariton electroluminescence, polariton lasing, parametric scattering, polariton superfluidity,
ACS Paragon Plus Environment
13
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 14 of 28
and Bose-Einstein condensation of polaritons can greatly benefit from an optimized light-matter coupling. Furthermore, enhancing the energy exchange between the nanocavity and IST is essential for applications such as voltage tunable optical filters, intersubband-based light emitting diodes, or efficient non-linear crystals.
ACS Paragon Plus Environment
14
Page 15 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
FIGURES Figure 1. Metamaterial structures and intersubband level scheme. (a) The metamaterial resonator is situated directly above the quantum-well stack which is based on an In0.53Ga0.47As / Al0.48In0.52As heterostructure and lattice matched to the InP substrate. The RLC series circuit describes the fundamental resonance of the metamaterial. The parameter
captures the
reflection at the air-semiconductor optical interface. The strong coupling to the intersubband transitions is modeled by the dispersive capacitor
. (b) The individual In0.53Ga0.47As quantum
wells are 12.5 nm wide and separated by 20 nm Al.48In.52As barriers. The homogenous doping level of 1018 cm-3 inside the quantum-well sets the Fermi level between the ground (black line) and the first excited state (red line). (c) Schematic representation of the four metamaterial geometries used in this work. The given dimensions (scale bar in the bottom right corner corresponds to 1 m) lead to an approximate resonance frequency of 24.2 THz for all geometries. Figure 2. Reflectivity results from the equivalent circuit model and from FDTD simulation. (a) Calculated reflectance spectra for the dogbone metamaterial resonator as a function of the bare cavity frequency. The white dashed lines represent the predicted polariton eigenfrequencies using a coupled oscillator model. (b) Comparison between the equivalent circuit model and finitedifference time domain simulations. The dashed lines represent finite-difference time-domain results for all four metamaterial geometries; the bare cavity frequencies are given for each individual line. The solid lines show the calculated reflectance using the equivalent electrical circuit model and including the effect of the strong coupling. The individual lines are offset vertically for clarity.
ACS Paragon Plus Environment
15
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 16 of 28
Figure 3. Scanning electron micrograph images showing fabricated metamaterial resonators. The metamaterial pattern is defined using electron-beam lithography followed by metal evaporation (Ti/Au: 5/100 nm) and a lift-off process. The white scale bar corresponds to 1 m in all four figures. Figure 4. Experimental transmittance spectra and analyzed Rabi frequencies. (a) Measured transmittance spectra for the dogbone metamaterial as a function of bare cavity frequency. The polariton anti-crossing is most clearly visible when the intersubband and the bare cavity frequency become similar (~24 THz). The white lines show the polariton eigenfrequencies obtained using a coupled oscillator mode. (b) Comparison between measured transmittance (dashed lines) and predicted transmittance (solid lines) using a coupled oscillator model for a dogbone metamaterial. The increased noise for frequencies below 20 THz is caused by the reduced detector sensitivity. The position of the individual lines in the contour plot in sub-panel (a) is indicated by the vertical, dashed lines. The coupled oscillator model predicts a Rabi frequency of 2.27 THz and an intersubband frequency of 24 THz. The bare cavity frequency is given next to each line. (c) Rabi frequency as a function of bare metamaterial capacitance (solid symbols are from experiment and open symbols from simulation). As predicted by the equivalent circuit model, the Rabi frequency increases with an increasing metamaterial resonator capacitance. The capacitance values are calculated for a bare cavity resonance of approximately 24.2 THz for all four geometries (Table 1).
ACS Paragon Plus Environment
16
Page 17 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
TABLES Dogbone
Dumbbell
Jerusalem cross
Circular SRR
Bare cavity frequency (THz)
24.6
25.0
24.2
24.9
R ()
20.7
17.2
14.6
36.2
L (pH)
2.84
2.71
2.87
4.22
C (aF)
14.8
15.1
15.5
9.7
Rsub ()
105.6
103.6
112.7
113.7
R/2 – simulation
2.48 (0.07)
2.51 (0.09)
2.51 (0.04)
2.28 (0.08)
R/2 – experiment
2.27 (0.21)
2.30 (0.18)
2.25 (0.18)
2.12 (0.17)
Table 1. Metamaterial and coupling circuit parameters. The RLC values for the equivalent electrical circuit are presented for a bare metamaterial cavity that is almost in resonance with the IST (see reflectance spectra in Figure 2(b)). The values for the Rabi frequencies are calculated as the arithmetic mean including all cavities that show a bare cavity frequency within the range IST ±25%. The values in brackets give the standard deviations.
ACS Paragon Plus Environment
17
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 18 of 28
AUTHOR INFORMATION Corresponding Author *E-mail: (I.B.) [email protected]. ASSOCIATED CONTENT Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources Any funds used to support the research of the manuscript should be placed here (per journal style). Notes The authors declare no competing financial interest. ACKNOWLEDGMENT Parts of this work were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering and performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
ACS Paragon Plus Environment
18
Page 19 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
ABBREVIATIONS QW, quantum-wells; IST, intersubband transition; FDTD, finite-difference time-domain; MIR, mid-infrared; QCL, quantum-cascade laser.
ACS Paragon Plus Environment
19
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 20 of 28
REFERENCES 1. Ciuti, C.; Bastard, G.; Carusotto, I. Phys. Rev. B 2005, 72, 115303. 2. Liberato, S. D.; Ciuti, C. Phys. Rev. Lett. 2007, 98, 103602. 3. Khalifa, A. A.; Love, A. P. D.; Krizhanovskii, D. N.; Skolnick, M. S.; Roberts, J. S. Appl. Phys. Lett. 2008, 92, 061107. 4. Bajoni, D.; Semenova, E.; Lemaître, A.; Bouchoule, S.; Wertz, E.; Senellart, P.; Bloch, J. Phys. Rev. B 2008, 77, 113303. 5. Tsintzos, S. I.; Pelekanos, N. T.; Konstantinidis, G.; Hatzopoulos, Z.; Savvidis, P. G. Nature 2008, 453, 372-375. 6. Christogiannis, N.; Somaschi, N.; Michetti, P.; Coles, D. M.; Savvidis, P. G.; Lagoudakis, P. G.; Lidzey, D. G. Adv. Optical Mater. 2013, 1, 503-509. 7. Dang, L. S.; Heger, D.; André, R.; Bœuf, F.; Romestain, R. Phys. Rev. Lett. 1998, 81, 3920-3923. 8. Senellart, P.; Bloch, J. Phys. Rev. Lett. 1999, 82, 1233-1236. 9. Christopoulos, S.; von Högersthal, G. B. H.; Grundy, A. J. D.; Lagoudakis, P. G.; Kavokin, A. V.; Baumberg, J. J.; Christmann, G.; Butté, R.; Feltin, E.; Carlin, J. F.; Grandjean, N. Phys. Rev. Lett. 2007, 98, 126405. 10. Savvidis, P. G.; Baumberg, J. J.; Stevenson, R. M.; Skolnick, M. S.; Whittaker, D. M.; Roberts, J. S. Phys. Rev. Lett. 2000, 84, 1547-1550. 11. Baumberg, J. J.; Savvidis, P. G.; Stevenson, R. M.; Tartakovskii, A. I.; Skolnick, M. S.; Whittaker, D. M.; Roberts, J. S. Phys. Rev. B 2000, 62, R16247-R16250. 12. Lagoudakis, K. G.; Wouters, M.; Richard, M.; Baas, A.; Carusotto, I.; Andre, R.; Dang, L. S.; Deveaud-Pledran, B. Nat Phys 2008, 4, 706-710. 13. Amo, A.; Sanvitto, D.; Laussy, F. P.; Ballarini, D.; Valle, E. d.; Martin, M. D.; Lemaitre, A.; Bloch, J.; Krizhanovskii, D. N.; Skolnick, M. S.; Tejedor, C.; Vina, L. Nature 2009, 457, 291-295. 14. Amo, A.; Lefrere, J.; Pigeon, S.; Adrados, C.; Ciuti, C.; Carusotto, I.; Houdre, R.; Giacobino, E.; Bramati, A. Nat Phys 2009, 5, 805-810. 15. Kasprzak, J.; Richard, M.; Kundermann, S.; Baas, A.; Jeambrun, P.; Keeling, J. M. J.; Marchetti, F. M.; Szymanska, M. H.; Andre, R.; Staehli, J. L.; Savona, V.; Littlewood, P. B.; Deveaud, B.; Dang, L. S. Nature 2006, 443, 409-414. 16. Torma, P.; Barnes, W. L. arXiv:1405.1661 2014. 17. Coles, D. M.; Somaschi, N.; Michetti, P.; Clark, C.; Lagoudakis, P. G.; Savvidis, P. G.; Lidzey, D. G. Nat Mater 2014, 13, 712-719. 18. Dintinger, J.; Klein, S.; Bustos, F.; Barnes, W. L.; Ebbesen, T. W. Phys. Rev. B 2005, 71, 035424. 19. Scalari, G.; Maissen, C.; Turcinkova, D.; Hagenmüller, D.; Liberato, S. D.; Ciuti, C.; Reichl, C.; Schuh, D.; Wegscheider, W.; Beck, M.; Faist, J. Science 2012, 335, 1323-1326. 20. Todorov, Y.; Andrews, A. M.; Colombelli, R.; Liberato, S. D.; Ciuti, C.; Klang, P.; Strasser, G.; Sirtori, C. Phys. Rev. Lett. 2010, 105, 196402. 21. Geiser, M.; Castellano, F.; Scalari, G.; Beck, M.; Nevou, L.; Faist, J. Phys. Rev. Lett. 2012, 108, 106402. 22. Benz, A.; Campione, S.; Liu, S.; Montano, I.; Klem, J. F.; Allerman, A.; Wendt, J. R.; Sinclair, M. B.; Capolino, F.; Brener, I. Nat. Comm. 2013, 4, 2882-2882.
ACS Paragon Plus Environment
20
Page 21 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
23. Dietze, D.; Benz, A.; Strasser, G.; Unterrainer, K.; Darmo, J. Opt. Express 2011, 19, 13700-13706. 24. Benz, A.; Campione, S.; Moseley, M. W.; Jonathan J. Wierer, J.; Allerman, A. A.; Wendt, J. R.; Brener, I. ACS Photonics 2014, 254-260. 25. Todorov, Y.; Sirtori, C. Phys. Rev. X 2014, 4, 041031. 26. Bellessa, J.; Bonnand, C.; Plenet, J. C.; Mugnier, J. Phys. Rev. Lett. 2004, 93, 036404. 27. Wang, W.; Vasa, P.; Pomraenke, R.; Vogelgesang, R.; De Sio, A.; Sommer, E.; Maiuri, M.; Manzoni, C.; Cerullo, G.; Lienau, C. ACS Nano 2013, 8, 1056-1064. 28. Baieva, S.; Hakala, T.; Toppari, J. Nanoscale Res Lett 2012, 7, 1-8. 29. Fofang, N. T.; Park, T.-H.; Neumann, O.; Mirin, N. A.; Nordlander, P.; Halas, N. J. Nano Lett. 2008, 8, 3481-3487. 30. Vasa, P.; Pomraenke, R.; Schwieger, S.; Mazur, Y.; Kunets, V.; Srinivasan, P.; Johnson, E.; Kihm, J.; Kim, D.; Runge, E.; Salamo, G.; Lienau, C. Phys. Rev. Lett. 2008, 101, 116801. 31. Neubrech, F.; Pucci, A.; Cornelius, T.; Karim, S.; García-Etxarri, A.; Aizpurua, J. Phys. Rev. Lett. 2008, 101, 157403. 32. Zengin, G.; Johansson, G.; Johansson, P.; Antosiewicz, T. J.; Käll, M.; Shegai, T. Sci. Rep. 2013, 3, 3074. 33. Shelton, D. J.; Brener, I.; Ginn, J. C.; Sinclair, M. B.; Peters, D. W.; Coffey, K. R.; Boreman, G. D. Nano Lett. 2011, 11, 2104-2108. 34. Jun, Y. C.; Reno, J.; Ribaudo, T.; Shaner, E.; Greffet, J.-J.; Vassant, S.; Marquier, F.; Sinclair, M.; Brener, I. Nano Lett. 2013, 13, 5391-5396. 35. Larsen, K.; Austin, D.; Sandall, I. C.; Davies, D. G.; Revin, D. G.; Cockburn, J. W.; Adawi, A. M.; Airey, R. J.; Fry, P. W.; Hopkinson, M.; Wilson, L. R. Appl. Phys. Lett. 2012, 101, 251109-251109. 36. Benz, A.; Montano, I.; Klem, J. F.; Brener, I. Appl. Phys. Lett. 2013, 103, 263116. 37. Geiser, M.; Scalari, G.; Castellano, F.; Beck, M.; Faist, J. Appl. Phys. Lett. 2012, 101, 141118. 38. Campione, S.; Benz, A.; Sinclair, M. B.; Capolino, F.; Brener, I. Appl. Phys. Lett. 2014, 104, 131104. 39. Lee, J.; Tymchenko, M.; Argyropoulos, C.; Chen, P.-Y.; Lu, F.; Demmerle, F.; Boehm, G.; Amann, M.-C.; Alu, A.; Belkin, M. A. Nature 2014, 511, 65-69. 40. Bastard, G.; Mendez, E. E.; Chang, L. L.; Esaki, L. Phys. Rev. B 1983, 28, 3241-3245. 41. Miller, D. A. B.; Weiner, J. S.; Chemla, D. IEEE J. Quantum Elec. 1986, 22, 1816-1830. 42. Faist, J.; Capasso, F.; Sirtori, C.; Sivco, D. L.; Hutchinson, A. L.; Chu, S. N. G.; Cho, A. Y. Appl. Phys. Lett. 1993, 63, 1354-1356. 43. Capasso, F.; Sirtori, C.; Cho, A. Y. IEEE J. Quantum Elec. 1994, 30, 1313-1326. 44. Jun, S. Y.; Sarabandi, K. IEEE Antennas and Wireless Propagation Letters 2012, 11, 1366-1369. 45. Campione, S.; Benz, A.; Klem, J. F.; Sinclair, M. B.; Brener, I.; Capolino, F. Phys. Rev. B 2014, 89, 165133. 46. Liu, H. C.; Capasso, F., Intersubband Transitions in Quantum Wells: Physics and Device Applications I. Academic Press: San Diego, 2000. 47. Nevou, L.; Julien, F. H.; Colombelli, R.; Guillot, F.; Monroy, E. Electron. Lett. 2006, 42, 1308-1309. 48. Driscoll, K.; Liao, Y.; Bhattacharyya, A.; Zhou, L.; Smith, D. J.; Moustakas, T. D.; Paiella, R. Appl. Phys. Lett. 2009, 94, 081120.
ACS Paragon Plus Environment
21
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 22 of 28
49. Gabbay, A.; Brener, I. Opt. Express 2012, 20, 6584-6597. 50. NextNano http://www.nextnano.com/nextnano3/ (accessed Jan 26, 2015). 51. Lumerical http://www.lumerical.com (accessed Jan 26, 2015). 52. Klein, M. W.; Tritschler, T.; Wegener, M.; Linden, S. Phys. Rev. B 2005, 72, 115113115113. 53. Rudin, S.; Reinecke, T. L. Phys. Rev. B 1999, 59, 10227. 54. Benz, A.; Campione, S.; Liu, S.; Montano, I.; Klem, J. F.; Sinclair, M. B.; Capolino, F.; Brener, I. Opt. Express 2013, 21, 32572. 55. Das, A.; Dhar, S.; Gupta, B. In Lumped Circuit Model Analysis of Meander Line Antennas. Proceedings of the 11th Mediterranean Microwave Symposium (MMS), Hammamet, Tunisia. Nov 8-10, 2011, Hammamet: pp 21-24. 56. Nishiyama, H.; Nakamura, M. IEEE Transactions on Components, Hybrids and Manufacturing Technology 1993, 16, 360-366. 57. Reitan, D. K.; Higgins, T. J. IEEE Transactions on Communications. 1957, 75, 761-766. 58. Sydoruk, O.; Tatartschuk, E.; Shamonina, E.; Solymar, L. J. Appl. Phys. 2009, 105, 014903. 59. Hosseinipanah, M.; Wu, Q. Radioengineering 2009, 18, 544-550. 60. Grover, F. W., Inductance Calculations: Working Formulas and Tables. Dover Publications, Inc.: Mineola, N.Y., 2004.
ACS Paragon Plus Environment
22
Page 23 of 28
1 µm Jerusalem Cross
Dumbbell
1 µm Circular SRR
32 Dogbone 28
24
Reflectance
Dogbone
Frequency (THz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Nano Letters
20
20 24 28 32 Bare cavity frequency (THz) 1 µm
1 µm
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
ACS Paragon Plus Environment
Page 24 of 28
Page 25 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Nano Letters
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
ACS Paragon Plus Environment
Page 26 of 28
(a)
(b)
Page 27 of 28
20.9 THz
Transmittance
1.5
24.6 THz 1.0
27.6 THz 0.5
20.0
24.0
28.0
Frequency (THz)
(c) Dogbone Dumbbell Jerusalem C. Circular SRR
2.5
OPEN: theory SOLID: experiments
2.4
Ω R /2π (THz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
2.3
2.2
2.1
6
8
10
12
14
16
Capacitance (aF)
ACS Paragon Plus Environment
18
32.0
36.0
Nano Letters
1 µm Jerusalem Cross
Dumbbell
1 µm Circular SRR
32 Dogbone 28
24
Reflectance
Dogbone
Frequency (THz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Page 28 of 28
20
20 24 28 32 Bare cavity frequency (THz) 1 µm
1 µm
ACS Paragon Plus Environment
Communication
Control of strong light-matter coupling using the capacitance of metamaterial nanocavities Alexander Benz, Salvatore Campione, John F Klem, Michael B. Sinclair, and Igal Brener Nano Lett., Just Accepted Manuscript • Publication Date (Web): 27 Jan 2015 Downloaded from http://pubs.acs.org on January 27, 2015
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Control of strong light-matter coupling using the capacitance of metamaterial nanocavities Alexander Benz,†,‡ Salvatore Campione,†,‡ John F. Klem,‡ Michael B. Sinclair,‡ and Igal Brener†,‡,* †
Center for Integrated Nanotechnologies (CINT), Sandia National Laboratories, P.O. Box 5800,
Albuquerque, NM 87185, USA ‡
Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185, USA
ACS Paragon Plus Environment
1
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 2 of 28
ABSTRACT Metallic nanocavities with deep subwavelength mode volumes can lead to dramatic changes in the behavior of emitters placed in their vicinity. This collocation and interaction often leads to strong coupling. Here, we present for the first time experimental evidence that the Rabi splitting is directly proportional to the electrostatic capacitance associated with the metallic nanocavity. The system analyzed consists of different metamaterial geometries with the same resonance wavelength coupled to intersubband transitions in quantum wells.
KEYWORDS metamaterial; nanocavity; strong light-matter interaction; intersubband transitions; mid-infrared; plasmonics.
ACS Paragon Plus Environment
2
Page 3 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Strong coupling between an optical cavity and a two-level system can lead to new and unusual coupled light-matter states (or quasi-particles) called polaritons1, 2. In this regime the cavity and the two-level system exchange energy coherently at a characteristic rate called the vacuum Rabi (angular) frequency Ω which is dominant with respect to all loss mechanisms present in the system. Experimentally, the energy exchange can be observed by measuring the optical reflectance or transmittance spectrum of the coupled system and/or a temporal beating in the time domain. Due to the strong light-matter interaction, the single bare cavity resonance splits into two resonances in the frequency domain forming an upper and a lower polariton branch. When the bare cavity and the two-level system are brought exactly into resonance, the two polaritons show an energy difference of 2 Ω . A larger Rabi frequency corresponds to a faster energy exchange or a more efficient coupling. This fundamental energy exchange between an optical cavity and a two-level system is the basis for a variety of fascinating phenomena such as polariton electroluminescence3-6, polariton lasing7-9, parametric scattering10, 11, polariton superfluidity12-14, and Bose-Einstein condensation of polaritons12-15. Recently, studying strong light-matter coupling in extremely small interaction volumes using plasmonic cavities has become a very active field of research16-32. Particularly, planar metamaterials seem ideally suited as nanocavities due to the possibility to engineer their electric and magnetic resonances by controlling the geometry of the individual subwavelength constituents. Strong coupling of such plasmonic metamaterial nanocavities has been achieved with cyclotron resonances19, optical phonons in polar crystals33, ultra-thin doped plasma layers34 or intersubband transitions (ISTs) in semiconductor quantum-wells (QWs)20-23, 35. A variety of applications can be realized using a combination of metamaterials and optical dipole transitions in semiconductors, such as voltage tunable optical filters36, intersubband-based light emitting
ACS Paragon Plus Environment
3
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 28
diodes37, and efficient non-linear devices38, 39. These types of actively tunable filters rely on controlling the resonance energy of ISTs using the quantum-confined Stark effect40, 41 and thereby the spectral response of the strongly coupled system. Furthermore, the coupled entity of a metamaterial resonator and an IST can form a polariton emitter which can be used for incoherent light generation. ISTs on their own cannot be used directly to realize efficient spontaneous emitters due to the extremely fast non-radiative scattering rates42. New classes of higher order non-linear media can be engineered to exploit the large non-linearities that can be designed using ISTs43. Recent investigations show that the non-linear conversion efficiency in strongly coupled metamaterial-IST systems can be enhanced by eight orders of magnitude compared to conventional non-linear crystals38, 39. All of these applications require an extremely efficient energy exchange between a metamaterial nanocavity and a two-level system or, equivalently, a large Rabi frequency. Previous attempts at increasing the light-matter coupling focused mainly on the properties of the two-level system while ignoring the effect of the metamaterial geometry20-22. Here, we will focus on the influence of the metamaterial nanocavity design on the type of strong light-matter interaction discussed above. We demonstrate experimentally that the strong light-matter coupling can be described by a simple equivalent circuit model and that the electrostatic capacitance of the single metamaterial nanocavity is the crucial parameter determining the coupling. We model the bare metamaterial (bare cavity) as a simple resistorinductor-capacitor (RLC) resonant circuit44. The strong light-matter coupling is added without free fitting parameters in the form of a dispersive capacitor45; a detailed schematic will be shown later in this work. This model simplifies the complex three dimensional problem of a metamaterial nanocavity coupled to an IST to a simple equivalent electrical circuit based on
ACS Paragon Plus Environment
4
Page 5 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
passive components. More importantly, the model shows that the electrostatic capacitance of the metamaterial resonator is the key that controls the light-matter coupling (or Rabi frequency)45. We verify the model experimentally using four metamaterial nanocavity geometries with identical bare cavity resonances but different values for the capacitive, inductive, and ohmic circuit components. We fabricate all four metamaterials atop the same wafer to exclude variations in the two-level system resonance frequency. Furthermore, and contrary to other strongly coupled systems, our experiments also show that the quality-factor (or damping) of the metamaterial does not affect the Rabi frequency noticeably. The basic geometry of our strongly coupled system is depicted in Figure 1(a). The metamaterial is fabricated directly atop the two-level system which is realized as an IST in a semiconductor heterostructure. Thin layers of different semiconductor crystals are grown on top of each other forming potential wells, so-called quantum-wells, for carriers. Thereby, the free motion of the electrons is inhibited along one dimension resulting in discrete energy states with atomic-like optical transitions. The energies of ISTs depend primarily on the width of the individual barriers and wells. Choosing appropriate host crystals, the transition energy can be tuned from the terahertz46 to the near-infrared47, 48 spectral range. The metamaterial resonators efficiently convert z-propagating normal incidence radiation to an electric field polarized along the z-axis and thereby makes coupling of free-space radiation to ISTs possible49. This combination of metamaterial and ISTs gives us a highly flexible model system to study strong light-matter interaction between a nanocavity and an artificial two-level system. However, we want to stress here that the exact details of the two-level system are less important in this work. Our findings can be applied to all other two-level systems used in similar experiments such as cyclotron resonances, optical phonons, excitons, or ultra-thin doped plasmonic layers.
ACS Paragon Plus Environment
5
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 28
In our experiments we use 12.5nm-thick In0.53Ga0.47As QWs, homogenously doped to 1018 cm3
and separated by 20nm-thick Al0.48In0.52As barriers. The samples are grown by molecular beam
epitaxy. This leads to a transition energy for the lowest bound-to-bound electronic transition (ground state to first excited state inside the same QW) of 100 meV or a transition frequency of 24.2 THz. The calculated50 energy levels for one QW-barrier pair are shown in Figure 1(b). Uncoupled, rectangular QWs show an oscillator strength49 of ~1 for this ground to first excited state transition, essentially restricting the entire interaction with light to this one single transition. We repeat the basic QW-barrier pair sequence 20 times to achieve a total thickness of 650 nm for the QW-stack. The QW stack growth is capped with a 30 nm thin Al0.48In0.52As layer. The thick QW-stack in combination with the thin cap layer is essential to ensure an almost perfect overlap of the metamaterial cavity mode with the QWs responsible for ISTs for all metamaterial geometries. The penetration depth of the cavity mode into the semiconductor is on the order of a few hundred nanometers in the mid-infrared spectral range49. Here, we will be using four basic metamaterial geometries labeled: dogbone, dumbbell, Jerusalem Cross and circular split-ring resonator (SRR). A schematic representation of all four geometries including their typical dimensions is illustrated in Figure 1(c). It should be stressed here that none of our metamaterial geometries requires a metallic ground plane to achieve the necessary mode confinement or the necessary z-polarized electric field in its vicinity22. We first model the interaction between a metamaterial nanocavity and the IST using finitedifference time-domain (FDTD) simulations51. This offers a fast and accurate way to determine the bare cavity resonance of the metamaterial (when the IST is turned off) and to calculate the upper and lower polariton branches (including the IST) for a given metamaterial geometry. We model one unit cell and apply periodic boundary conditions to capture the effect of the two-
ACS Paragon Plus Environment
6
Page 7 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
dimensional metamaterial array in the fabricated sample. The permittivity for the gold layer is extracted from spectral ellipsometry measurements performed on a separate 100 nm thin gold film prepared under identical conditions as the metamaterial samples. The ISTs are modeled as anisotropic harmonic oscillators49. Only the electric field component that is polarized along the growth direction of the QW-stack interacts with the ISTs, so the permittivity can be written as: 0 0 0
0 0
(1) ,
0
Δ ∗
where
2
is the background permittivity of the QW, Δ
difference between ground and excited state, oscillator strength,
∗
(2)
,
is the carrier concentration
is the effective electron mass,
is the intersubband transition frequency,
the vacuum permittivity and
is the
is the IST broadening,
is
is the elementary electron charge. The center frequency of the IST
is set to 24.2 THz, while the full-width-at-half-maximum is set to 2.4 THz. Both values were confirmed by intersubband absorption measurements in a multi-pass waveguide geometry using an unprocessed piece of the same wafer. The FDTD simulation results for a dogbone metamaterial nanocavity interacting with the ISTs are presented in Figure 2(a). We control the bare metamaterial resonance by geometrically stretching its dimensions thus allowing a step-wise sweep of the bare cavity frequency across the IST transition frequency. Due to the strong light-matter coupling we observe a clear anti-crossing behavior of the two polariton branches. Using a coupled oscillator model52 we can extract a corresponding Rabi frequency of 2.48 THz. The FDTD simulations in combination with the
ACS Paragon Plus Environment
7
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 8 of 28
analytic expression for the intersubband permittivity (Eq. (2)) already point at a few parameters that can be optimized to increase the Rabi frequency such as the population difference between ground and excited state (Δ
) or the oscillator strength for this transition (
). Increasing
either one of these parameters increases the influence of the IST on the QW permittivity and, as a consequence, the interaction with the light field. These parameters have also been identified in the quantum-mechanical modeling1, 20-22 to be essential for the determination of the Rabi frequency. It should be noted here that also in the classical picture the Rabi frequency increases with the number of oscillators coupled to the cavity53. In our case this corresponds to the carrier concentration difference between ground and first excited state, Δ
. Furthermore, the spatial
overlap between a cavity field and the volume where the two-level system resides shows a strong effect on the polariton splitting54. The thin cap layer (30 nm Al0.48In0.52As) in combination with the thick QW-stack (650 nm) yields an overlap close to 100% between the metamaterial cavity mode and the heterostructure regardless of the exact cavity field profile. This allows us to focus our study on the metamaterial geometry dependence and minimize the influence of the two-level system. This initial analysis also reveals one of the major weaknesses of solely relying on FDTD simulations: it is easy to analyze a given metamaterial geometry but very difficult to derive a new one that is optimized for light-matter coupling. The results of FDTD simulations are the full electromagnetic fields inside the computational volume. However, these results do not provide a direct (or intuitive) relation between the simulated geometry and a complex physical behavior such as light-matter coupling. The typical design flow of choosing a metamaterial geometry, calculating the reflectance or transmittance, evaluating the Rabi frequency, varying the geometry
ACS Paragon Plus Environment
8
Page 9 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
and then recalculating the field profiles offers no possibility to estimate the final results without running the full simulation, making trial-and-error the only optimization strategy available. The use of a recently published equivalent electrical circuit model to describe the light-matter interaction in these systems can solve some of the shortcomings mentioned before45. Modeling the bare metamaterial nanocavity using an RLC resonant circuit44 has obvious advantages compared to a mechanical analogue. The electrical circuit components can be related closely to the electromagnetic properties of the nanocavity. Furthermore, we can obtain the parameters for the equivalent circuit by splitting the metamaterial resonator into basic geometric shapes and describe the system as a combination of lumped circuit elements55. Analytical expressions for the self-capacitance of free-standing metal disks56 or rectangular metal patches57 have been reported. Even more complex geometries such as split-rings58 or Jerusalem Cross resonators59 can be modeled analytically with high accuracy. The inductive part for all metamaterial geometries used in this work can be found in reference books, e.g. 60. These two quantities set the bare resonance frequency of the metamaterial (
1
√
) and can be used to estimate the Rabi frequency.
Extracting the ohmic component from this kind of model is more difficult. The resistor in the equivalent circuit captures the ohmic losses of the metal and the radiative damping of the cavity, but the latter is difficult to model analytically. However, the RLC parameters can be extracted by fitting the reflectance or transmittance spectra of the equivalent one-dimensional transmission line model to the spectra obtained by FDTD simulations as explained in45. The equivalent parameters for the four different geometries, using this procedure are given in Table 1 for a bare cavity frequency of approximately 24.2 THz (resonant with the IST). These parameters also motivate our choices for the four different metamaterial geometries used to test the circuit model: we can cover high capacitance with different geometric shapes (dogbone and dumbbell), low
ACS Paragon Plus Environment
9
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
metamaterial damping (
Page 10 of 28
) resulting in a higher quality-factor (Jerusalem Cross) and high
inductance (circular SRR). The full equivalent circuit including the influence of the substrate is illustrated in Figure 1(a).45 The interaction between metamaterial and IST can be included by adding one dispersive capacitor to the circuit, which can be modeled as:
1 where
and
,
are the same permittivities that are used for the FDTD simulations and
are described in Eq. (1) and Eq. (2); and
is the capacitance used to model the metamaterial
bare cavity. Increasing the Rabi frequency corresponds to increasing the value for which can be achieved in two ways. First, modified to increase nanocavity
(3)
in Eq. (3),
(which represents the strength of the IST) can be
as dictated by Eq. (2). Second, the capacitance of the metamaterial
can be used to increase
linearly (we remind the reader that we examine this
second condition experimentally in this paper). It should be stressed here that the parameters for the dispersive capacitor in Eq. (3) are not free fitting parameters but rather given by the IST design. After we determined the equivalent circuit parameters for the bare metamaterial nanocavity, we can calculate the reflectance (or transmittance) spectra including the strong light-matter coupling by adding the dispersive capacitor defined in Eq. (3). The excellent agreement between the equivalent circuit model and FDTD simulations in the entire frequency range between 10 and 45 THz is presented in Figure 2(b). As an example we show three different bare cavity resonance
ACS Paragon Plus Environment
10
Page 11 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
frequencies for each of the metamaterial geometries where the metamaterial is blue-detuned from the IST (blue line), on resonance (black line) and red-detuned (red line). From these reflectance spectra we can extract the Rabi frequencies using a coupled oscillator model52. The Rabi frequencies reported in Table 1 confirm the expected trend: the larger the capacitance of the nanoresonator, the larger the Rabi frequency. The dogbone, the dumbbell and the Jerusalem Cross have all approximately the same capacitance leading to a similar Rabi frequency. Neither the exact geometric shape of the capacitor (dogbone versus dumbbell) nor the metamaterial quality-factor (the Jerusalem Cross shows a significantly lower damping factor
compared to
dogbone and dumbbell) cause any noticeable effect on the Rabi splitting. On the other hand, the circular SRR, which shows the lowest capacitance of all analyzed metamaterial geometries, also shows the lowest Rabi frequency. To confirm these theoretical predictions and the validity of the equivalent circuit model, we fabricated all four resonator geometries on top of the same semiconductor wafer which allows us to eliminate the effects of different wafer growths (variations in intersubband resonance frequency or different doping concentration) and focus on the interaction between IST and metamaterial. The metamaterial nanocavities are defined by electron-beam lithography followed by a Ti/Au (5/100 nm) evaporation and lift-off process. Scanning electron micrograph images showing all four fabricated metamaterial geometries are presented in Figure 3. We measure the transmittance at room temperature using a Bruker IFS66v for all samples. Similar to the FDTD simulations, we geometrically scale the metamaterial dimensions to sweep the bare cavity resonance across the IST and map out the two polariton branches. As an example, we present the experimental transmittance for the dogbone metamaterial as a function of bare cavity frequency in Figure 4(a). The avoided polariton crossing is most clearly visible in the frequency region
ACS Paragon Plus Environment
11
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 12 of 28
where the bare cavity frequency and the IST are similar in energy (~100 meV or ~24 THz). We have also added the polariton eigenfrequencies obtained using a coupled oscillator model to Figure 4(a) to emphasize the anti-crossing behavior. To extract the Rabi frequencies from the measurements we use a coupled oscillator model52 that allows us to fit the entire transmittance spectrum for each metamaterial scaling factor rather than using only two isolated data-points per bare cavity frequency (the upper and the lower polariton eigenfrequencies). We present the fitting results for the dogbone metamaterial and three different bare cavity frequencies (equivalent to three different geometric scaling factors) in Figure 4(b). These three fits yield the same IST frequency (always using the same wafer) and the same Rabi frequency (the coupling strength does not depend on the unperturbed resonance frequency of either the metamaterial or the IST in first approximation) but different bare cavity frequencies (the metamaterial is geometrically stretched). To determine the Rabi frequency for a given metamaterial geometry, we include all experimental transmittance spectra where the bare cavity resonances are within the frequency range: IST ±25%. The Rabi frequencies reported in Table 1 for all four metamaterial geometries are the arithmetic means within the defined frequency range (the standard deviation is given in Table 1 in brackets). This approach allows us to minimize the influence of individual measurements and increase the confidence for the extracted Rabi frequencies. Our experiments verify the prediction from the equivalent RLC circuit model: increasing the capacitance of the metamaterial resonator increases the Rabi frequency. The comparison between experimental data and simulations presented in Figure 4(c) shows that our measured Rabi frequency is approximately 0.2 THz smaller than the simulated one. However, the general trend is not affected by this discrepancy. In addition, we mention that no clear dependence between the
ACS Paragon Plus Environment
12
Page 13 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
Rabi frequency and metamaterial damping is observed, neither in the experiments nor in the corresponding simulations. It should be noted here that quantum-mechanical models20-22 often used to describe strongly coupled system consisting of metamaterial resonators and ISTs neglect the exact geometry of the resonators and only include the spatial overlap between the cavity mode and the volume where the two-level system resides. Due to the thick QW-stacks (650 nm) and the thin Al0.48In0.52As cap layer (30 nm) in our samples we achieve a ~100% overlap with the QW-stack, resulting in a 38.5% overlap of the cavity mode and the ISTs (the remaining part of the cavity mode is within the Al0.48In0.52As barriers). Using such a quantum-mechanical model would lead to a Rabi frequency of 1.75 THz for all metamaterial samples regardless of their exact metamaterial geometry, which underestimates the experimental results and cannot capture the measured trend. In conclusion, we confirmed experimentally that the equivalent electrostatic capacitance of a metamaterial nanocavity is the crucial parameter defining the strong light-matter coupling. We fabricated four different metamaterial geometries with varying values for the capacitance, inductance, and ohmic resistance but with the same bare cavity resonance wavelength (~12 m) on top of the same semiconductor wafer. The experimental data are in good agreement with the theoretical predictions confirming the key role of the capacitance for strong light-matter coupling: geometries with a large capacitance such as dogbone or dumbbell metamaterial resonator show a larger Rabi frequency compared to conventional SRRs. Other metamaterial cavity parameters such as damping do not play a major role for strong coupling. This results in very simple metamaterial resonator design rules to optimize light-matter interaction using semiconductor heterostructures and potentially other dipolar excitations. Phenomena such as polariton electroluminescence, polariton lasing, parametric scattering, polariton superfluidity,
ACS Paragon Plus Environment
13
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 14 of 28
and Bose-Einstein condensation of polaritons can greatly benefit from an optimized light-matter coupling. Furthermore, enhancing the energy exchange between the nanocavity and IST is essential for applications such as voltage tunable optical filters, intersubband-based light emitting diodes, or efficient non-linear crystals.
ACS Paragon Plus Environment
14
Page 15 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
FIGURES Figure 1. Metamaterial structures and intersubband level scheme. (a) The metamaterial resonator is situated directly above the quantum-well stack which is based on an In0.53Ga0.47As / Al0.48In0.52As heterostructure and lattice matched to the InP substrate. The RLC series circuit describes the fundamental resonance of the metamaterial. The parameter
captures the
reflection at the air-semiconductor optical interface. The strong coupling to the intersubband transitions is modeled by the dispersive capacitor
. (b) The individual In0.53Ga0.47As quantum
wells are 12.5 nm wide and separated by 20 nm Al.48In.52As barriers. The homogenous doping level of 1018 cm-3 inside the quantum-well sets the Fermi level between the ground (black line) and the first excited state (red line). (c) Schematic representation of the four metamaterial geometries used in this work. The given dimensions (scale bar in the bottom right corner corresponds to 1 m) lead to an approximate resonance frequency of 24.2 THz for all geometries. Figure 2. Reflectivity results from the equivalent circuit model and from FDTD simulation. (a) Calculated reflectance spectra for the dogbone metamaterial resonator as a function of the bare cavity frequency. The white dashed lines represent the predicted polariton eigenfrequencies using a coupled oscillator model. (b) Comparison between the equivalent circuit model and finitedifference time domain simulations. The dashed lines represent finite-difference time-domain results for all four metamaterial geometries; the bare cavity frequencies are given for each individual line. The solid lines show the calculated reflectance using the equivalent electrical circuit model and including the effect of the strong coupling. The individual lines are offset vertically for clarity.
ACS Paragon Plus Environment
15
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 16 of 28
Figure 3. Scanning electron micrograph images showing fabricated metamaterial resonators. The metamaterial pattern is defined using electron-beam lithography followed by metal evaporation (Ti/Au: 5/100 nm) and a lift-off process. The white scale bar corresponds to 1 m in all four figures. Figure 4. Experimental transmittance spectra and analyzed Rabi frequencies. (a) Measured transmittance spectra for the dogbone metamaterial as a function of bare cavity frequency. The polariton anti-crossing is most clearly visible when the intersubband and the bare cavity frequency become similar (~24 THz). The white lines show the polariton eigenfrequencies obtained using a coupled oscillator mode. (b) Comparison between measured transmittance (dashed lines) and predicted transmittance (solid lines) using a coupled oscillator model for a dogbone metamaterial. The increased noise for frequencies below 20 THz is caused by the reduced detector sensitivity. The position of the individual lines in the contour plot in sub-panel (a) is indicated by the vertical, dashed lines. The coupled oscillator model predicts a Rabi frequency of 2.27 THz and an intersubband frequency of 24 THz. The bare cavity frequency is given next to each line. (c) Rabi frequency as a function of bare metamaterial capacitance (solid symbols are from experiment and open symbols from simulation). As predicted by the equivalent circuit model, the Rabi frequency increases with an increasing metamaterial resonator capacitance. The capacitance values are calculated for a bare cavity resonance of approximately 24.2 THz for all four geometries (Table 1).
ACS Paragon Plus Environment
16
Page 17 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
TABLES Dogbone
Dumbbell
Jerusalem cross
Circular SRR
Bare cavity frequency (THz)
24.6
25.0
24.2
24.9
R ()
20.7
17.2
14.6
36.2
L (pH)
2.84
2.71
2.87
4.22
C (aF)
14.8
15.1
15.5
9.7
Rsub ()
105.6
103.6
112.7
113.7
R/2 – simulation
2.48 (0.07)
2.51 (0.09)
2.51 (0.04)
2.28 (0.08)
R/2 – experiment
2.27 (0.21)
2.30 (0.18)
2.25 (0.18)
2.12 (0.17)
Table 1. Metamaterial and coupling circuit parameters. The RLC values for the equivalent electrical circuit are presented for a bare metamaterial cavity that is almost in resonance with the IST (see reflectance spectra in Figure 2(b)). The values for the Rabi frequencies are calculated as the arithmetic mean including all cavities that show a bare cavity frequency within the range IST ±25%. The values in brackets give the standard deviations.
ACS Paragon Plus Environment
17
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 18 of 28
AUTHOR INFORMATION Corresponding Author *E-mail: (I.B.) [email protected]. ASSOCIATED CONTENT Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources Any funds used to support the research of the manuscript should be placed here (per journal style). Notes The authors declare no competing financial interest. ACKNOWLEDGMENT Parts of this work were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering and performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
ACS Paragon Plus Environment
18
Page 19 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
ABBREVIATIONS QW, quantum-wells; IST, intersubband transition; FDTD, finite-difference time-domain; MIR, mid-infrared; QCL, quantum-cascade laser.
ACS Paragon Plus Environment
19
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 20 of 28
REFERENCES 1. Ciuti, C.; Bastard, G.; Carusotto, I. Phys. Rev. B 2005, 72, 115303. 2. Liberato, S. D.; Ciuti, C. Phys. Rev. Lett. 2007, 98, 103602. 3. Khalifa, A. A.; Love, A. P. D.; Krizhanovskii, D. N.; Skolnick, M. S.; Roberts, J. S. Appl. Phys. Lett. 2008, 92, 061107. 4. Bajoni, D.; Semenova, E.; Lemaître, A.; Bouchoule, S.; Wertz, E.; Senellart, P.; Bloch, J. Phys. Rev. B 2008, 77, 113303. 5. Tsintzos, S. I.; Pelekanos, N. T.; Konstantinidis, G.; Hatzopoulos, Z.; Savvidis, P. G. Nature 2008, 453, 372-375. 6. Christogiannis, N.; Somaschi, N.; Michetti, P.; Coles, D. M.; Savvidis, P. G.; Lagoudakis, P. G.; Lidzey, D. G. Adv. Optical Mater. 2013, 1, 503-509. 7. Dang, L. S.; Heger, D.; André, R.; Bœuf, F.; Romestain, R. Phys. Rev. Lett. 1998, 81, 3920-3923. 8. Senellart, P.; Bloch, J. Phys. Rev. Lett. 1999, 82, 1233-1236. 9. Christopoulos, S.; von Högersthal, G. B. H.; Grundy, A. J. D.; Lagoudakis, P. G.; Kavokin, A. V.; Baumberg, J. J.; Christmann, G.; Butté, R.; Feltin, E.; Carlin, J. F.; Grandjean, N. Phys. Rev. Lett. 2007, 98, 126405. 10. Savvidis, P. G.; Baumberg, J. J.; Stevenson, R. M.; Skolnick, M. S.; Whittaker, D. M.; Roberts, J. S. Phys. Rev. Lett. 2000, 84, 1547-1550. 11. Baumberg, J. J.; Savvidis, P. G.; Stevenson, R. M.; Tartakovskii, A. I.; Skolnick, M. S.; Whittaker, D. M.; Roberts, J. S. Phys. Rev. B 2000, 62, R16247-R16250. 12. Lagoudakis, K. G.; Wouters, M.; Richard, M.; Baas, A.; Carusotto, I.; Andre, R.; Dang, L. S.; Deveaud-Pledran, B. Nat Phys 2008, 4, 706-710. 13. Amo, A.; Sanvitto, D.; Laussy, F. P.; Ballarini, D.; Valle, E. d.; Martin, M. D.; Lemaitre, A.; Bloch, J.; Krizhanovskii, D. N.; Skolnick, M. S.; Tejedor, C.; Vina, L. Nature 2009, 457, 291-295. 14. Amo, A.; Lefrere, J.; Pigeon, S.; Adrados, C.; Ciuti, C.; Carusotto, I.; Houdre, R.; Giacobino, E.; Bramati, A. Nat Phys 2009, 5, 805-810. 15. Kasprzak, J.; Richard, M.; Kundermann, S.; Baas, A.; Jeambrun, P.; Keeling, J. M. J.; Marchetti, F. M.; Szymanska, M. H.; Andre, R.; Staehli, J. L.; Savona, V.; Littlewood, P. B.; Deveaud, B.; Dang, L. S. Nature 2006, 443, 409-414. 16. Torma, P.; Barnes, W. L. arXiv:1405.1661 2014. 17. Coles, D. M.; Somaschi, N.; Michetti, P.; Clark, C.; Lagoudakis, P. G.; Savvidis, P. G.; Lidzey, D. G. Nat Mater 2014, 13, 712-719. 18. Dintinger, J.; Klein, S.; Bustos, F.; Barnes, W. L.; Ebbesen, T. W. Phys. Rev. B 2005, 71, 035424. 19. Scalari, G.; Maissen, C.; Turcinkova, D.; Hagenmüller, D.; Liberato, S. D.; Ciuti, C.; Reichl, C.; Schuh, D.; Wegscheider, W.; Beck, M.; Faist, J. Science 2012, 335, 1323-1326. 20. Todorov, Y.; Andrews, A. M.; Colombelli, R.; Liberato, S. D.; Ciuti, C.; Klang, P.; Strasser, G.; Sirtori, C. Phys. Rev. Lett. 2010, 105, 196402. 21. Geiser, M.; Castellano, F.; Scalari, G.; Beck, M.; Nevou, L.; Faist, J. Phys. Rev. Lett. 2012, 108, 106402. 22. Benz, A.; Campione, S.; Liu, S.; Montano, I.; Klem, J. F.; Allerman, A.; Wendt, J. R.; Sinclair, M. B.; Capolino, F.; Brener, I. Nat. Comm. 2013, 4, 2882-2882.
ACS Paragon Plus Environment
20
Page 21 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
23. Dietze, D.; Benz, A.; Strasser, G.; Unterrainer, K.; Darmo, J. Opt. Express 2011, 19, 13700-13706. 24. Benz, A.; Campione, S.; Moseley, M. W.; Jonathan J. Wierer, J.; Allerman, A. A.; Wendt, J. R.; Brener, I. ACS Photonics 2014, 254-260. 25. Todorov, Y.; Sirtori, C. Phys. Rev. X 2014, 4, 041031. 26. Bellessa, J.; Bonnand, C.; Plenet, J. C.; Mugnier, J. Phys. Rev. Lett. 2004, 93, 036404. 27. Wang, W.; Vasa, P.; Pomraenke, R.; Vogelgesang, R.; De Sio, A.; Sommer, E.; Maiuri, M.; Manzoni, C.; Cerullo, G.; Lienau, C. ACS Nano 2013, 8, 1056-1064. 28. Baieva, S.; Hakala, T.; Toppari, J. Nanoscale Res Lett 2012, 7, 1-8. 29. Fofang, N. T.; Park, T.-H.; Neumann, O.; Mirin, N. A.; Nordlander, P.; Halas, N. J. Nano Lett. 2008, 8, 3481-3487. 30. Vasa, P.; Pomraenke, R.; Schwieger, S.; Mazur, Y.; Kunets, V.; Srinivasan, P.; Johnson, E.; Kihm, J.; Kim, D.; Runge, E.; Salamo, G.; Lienau, C. Phys. Rev. Lett. 2008, 101, 116801. 31. Neubrech, F.; Pucci, A.; Cornelius, T.; Karim, S.; García-Etxarri, A.; Aizpurua, J. Phys. Rev. Lett. 2008, 101, 157403. 32. Zengin, G.; Johansson, G.; Johansson, P.; Antosiewicz, T. J.; Käll, M.; Shegai, T. Sci. Rep. 2013, 3, 3074. 33. Shelton, D. J.; Brener, I.; Ginn, J. C.; Sinclair, M. B.; Peters, D. W.; Coffey, K. R.; Boreman, G. D. Nano Lett. 2011, 11, 2104-2108. 34. Jun, Y. C.; Reno, J.; Ribaudo, T.; Shaner, E.; Greffet, J.-J.; Vassant, S.; Marquier, F.; Sinclair, M.; Brener, I. Nano Lett. 2013, 13, 5391-5396. 35. Larsen, K.; Austin, D.; Sandall, I. C.; Davies, D. G.; Revin, D. G.; Cockburn, J. W.; Adawi, A. M.; Airey, R. J.; Fry, P. W.; Hopkinson, M.; Wilson, L. R. Appl. Phys. Lett. 2012, 101, 251109-251109. 36. Benz, A.; Montano, I.; Klem, J. F.; Brener, I. Appl. Phys. Lett. 2013, 103, 263116. 37. Geiser, M.; Scalari, G.; Castellano, F.; Beck, M.; Faist, J. Appl. Phys. Lett. 2012, 101, 141118. 38. Campione, S.; Benz, A.; Sinclair, M. B.; Capolino, F.; Brener, I. Appl. Phys. Lett. 2014, 104, 131104. 39. Lee, J.; Tymchenko, M.; Argyropoulos, C.; Chen, P.-Y.; Lu, F.; Demmerle, F.; Boehm, G.; Amann, M.-C.; Alu, A.; Belkin, M. A. Nature 2014, 511, 65-69. 40. Bastard, G.; Mendez, E. E.; Chang, L. L.; Esaki, L. Phys. Rev. B 1983, 28, 3241-3245. 41. Miller, D. A. B.; Weiner, J. S.; Chemla, D. IEEE J. Quantum Elec. 1986, 22, 1816-1830. 42. Faist, J.; Capasso, F.; Sirtori, C.; Sivco, D. L.; Hutchinson, A. L.; Chu, S. N. G.; Cho, A. Y. Appl. Phys. Lett. 1993, 63, 1354-1356. 43. Capasso, F.; Sirtori, C.; Cho, A. Y. IEEE J. Quantum Elec. 1994, 30, 1313-1326. 44. Jun, S. Y.; Sarabandi, K. IEEE Antennas and Wireless Propagation Letters 2012, 11, 1366-1369. 45. Campione, S.; Benz, A.; Klem, J. F.; Sinclair, M. B.; Brener, I.; Capolino, F. Phys. Rev. B 2014, 89, 165133. 46. Liu, H. C.; Capasso, F., Intersubband Transitions in Quantum Wells: Physics and Device Applications I. Academic Press: San Diego, 2000. 47. Nevou, L.; Julien, F. H.; Colombelli, R.; Guillot, F.; Monroy, E. Electron. Lett. 2006, 42, 1308-1309. 48. Driscoll, K.; Liao, Y.; Bhattacharyya, A.; Zhou, L.; Smith, D. J.; Moustakas, T. D.; Paiella, R. Appl. Phys. Lett. 2009, 94, 081120.
ACS Paragon Plus Environment
21
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 22 of 28
49. Gabbay, A.; Brener, I. Opt. Express 2012, 20, 6584-6597. 50. NextNano http://www.nextnano.com/nextnano3/ (accessed Jan 26, 2015). 51. Lumerical http://www.lumerical.com (accessed Jan 26, 2015). 52. Klein, M. W.; Tritschler, T.; Wegener, M.; Linden, S. Phys. Rev. B 2005, 72, 115113115113. 53. Rudin, S.; Reinecke, T. L. Phys. Rev. B 1999, 59, 10227. 54. Benz, A.; Campione, S.; Liu, S.; Montano, I.; Klem, J. F.; Sinclair, M. B.; Capolino, F.; Brener, I. Opt. Express 2013, 21, 32572. 55. Das, A.; Dhar, S.; Gupta, B. In Lumped Circuit Model Analysis of Meander Line Antennas. Proceedings of the 11th Mediterranean Microwave Symposium (MMS), Hammamet, Tunisia. Nov 8-10, 2011, Hammamet: pp 21-24. 56. Nishiyama, H.; Nakamura, M. IEEE Transactions on Components, Hybrids and Manufacturing Technology 1993, 16, 360-366. 57. Reitan, D. K.; Higgins, T. J. IEEE Transactions on Communications. 1957, 75, 761-766. 58. Sydoruk, O.; Tatartschuk, E.; Shamonina, E.; Solymar, L. J. Appl. Phys. 2009, 105, 014903. 59. Hosseinipanah, M.; Wu, Q. Radioengineering 2009, 18, 544-550. 60. Grover, F. W., Inductance Calculations: Working Formulas and Tables. Dover Publications, Inc.: Mineola, N.Y., 2004.
ACS Paragon Plus Environment
22
Page 23 of 28
1 µm Jerusalem Cross
Dumbbell
1 µm Circular SRR
32 Dogbone 28
24
Reflectance
Dogbone
Frequency (THz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Nano Letters
20
20 24 28 32 Bare cavity frequency (THz) 1 µm
1 µm
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
ACS Paragon Plus Environment
Page 24 of 28
Page 25 of 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Nano Letters
ACS Paragon Plus Environment
Nano Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
ACS Paragon Plus Environment
Page 26 of 28
(a)
(b)
Page 27 of 28
20.9 THz
Transmittance
1.5
24.6 THz 1.0
27.6 THz 0.5
20.0
24.0
28.0
Frequency (THz)
(c) Dogbone Dumbbell Jerusalem C. Circular SRR
2.5
OPEN: theory SOLID: experiments
2.4
Ω R /2π (THz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Nano Letters
2.3
2.2
2.1
6
8
10
12
14
16
Capacitance (aF)
ACS Paragon Plus Environment
18
32.0
36.0
Nano Letters
1 µm Jerusalem Cross
Dumbbell
1 µm Circular SRR
32 Dogbone 28
24
Reflectance
Dogbone
Frequency (THz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Page 28 of 28
20
20 24 28 32 Bare cavity frequency (THz) 1 µm
1 µm
ACS Paragon Plus Environment
