Large Cavity-Optomechanical Coupling with Graphene at Infrared and

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Large Cavity-Optomechanical Coupling with Graphene at Infrared and Terahertz Frequencies Ian A.D. Williamson, S. Hossein Mousavi, and Zheng Wang ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00553 • Publication Date (Web): 31 Oct 2016 Downloaded from http://pubs.acs.org on November 16, 2016

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Large Cavity-Optomechanical Coupling with Graphene at Infrared and Terahertz Frequencies Ian A. D. Williamson, 1 S. Hossein Mousavi, 1 Zheng Wang *,1 1

Microelectronics Research Center, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78758 US

Abstract: Graphene exhibits many unusual elastic properties, making it an intriguing material for mechanical measurement and actuation at the quantum limit. We theoretically examine the viability of graphene for cavity optomechanics from near-infrared to terahertz wavelengths, fully taking into account its large optical absorption and dispersion. A large optomechanical coupling coefficient, on the same order of that observed in state-of-the-art optomechanical materials, can be realized in the mid-infrared spectrum with highly doped graphene, a high optical quality factor, and optimal positioning of graphene. Around 100 THz, the dispersive coupling coefficient reaches 180 MHz/nm and 500 MHz/nm in the resolved and unresolved sideband regimes, respectively. We find that predominantly dispersive coupling requires a high graphene Fermi level and mid-infrared excitation, while predominantly dissipative coupling favors a moderate graphene Fermi level and near-infrared excitation. Keywords: optomechanics, graphene, optical resonators, plasmonics, terahertz, infrared

TOC GRAPHIC

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Cavity optomechanics can benefit significantly from the unique mechanical properties of singleand few-layer graphene membranes. Notably, graphene has an exceptionally large in-plane Young’s modulus (~1 TPa) and a low mass ( ρ ~ 10 −19 kg/µm2) with large yield strength1–4. These

features

have

made

suspended

graphene

membranes

very

attractive

in

microelectromechanical (MEMS) and nanoelectromechanical systems (NEMS) for mass sensing5, displacement detection6, and transduction7,8. State-of-the-art graphene mechanical resonators9–14 have resonance frequencies on the order of 100 MHz and quality factors above 10 4 . Emerging fabrication techniques15–17 may open up the possibility of reaching resonance frequencies in the 100’s of MHz and approaching 1 GHz2. Simultaneously, graphene’s low mass leads to extremely large zero-point fluctuation amplitudes1 x zpf . Recent graphene NEMS implementations have achieved values of x zpf on the order of a picometer, which is several orders of magnitude larger than what is possible with higher dimensional materials such as silicon nitride. Such a low mass also allows graphene resonators to support a reduced noise floor for optomechanical force readout, which allows for high-sensitivity detection of nanoscale forces and displacements18–20. Although trapped nanoparticle optomechanical systems are known for their low masses and exceptional readout sensitivities21,22, graphene membranes support guided phonon modes with large quality factors that allow for the realization of coherent photon-phonon interfaces23. Despite these advantages, graphene has yet to be considered as a promising material at optical frequencies for cavity optomechanics which would allow for the manipulation and detection of quantum-limit mechanical motion with light. The fundamental challenge of using graphene at optical frequencies in cavity optomechanics is its large absorption, which is thought to set a limit on the optical quality factor that is too low for most optomechanical phenomena. For this reason, graphene-based cavity optomechanical 2 ACS Paragon Plus Environment

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experiments have been conducted mostly at microwave frequencies, where graphene is simply used as one of the conducting electrodes as part of a mechanically variable LC resonator9,24,25. At optical frequencies, graphene has been used for quantum-limited position and force readouts via evanescent coupling to quantum emitters26,27 and microspheres28, where the impact of radiation pressure on graphene is not exploited. The strong dispersion of graphene makes it more challenging to choose the ideal combination of operational frequency and material parameters that maximize optomechanical coupling. This challenge is exemplified by recent work in near-IR standing wave cavities29, where dissipative coupling was observed with a lower overall magnitude than those of conventional optomechanical materials. Considering that highly reflective interfaces are typically needed for large optomechanical coupling, the fact that graphene is essentially transparent with a constant 2.3% absorption in the near-IR and visible hardly make it an ideal material for cavity optomechanics. Even photothermal interactions, which deliberately utilize the optical absorption of graphene, have only realized a moderate readout sensitivities30 due to their low overall optical quality factor. However, in the THz and mid-IR spectra, recent works have shown that the plasmonic response of graphene can strongly modify optical resonances through near-field coupling31–34. Also in this spectral range, graphene has been shown to generate significant optical forces that are on the same order of magnitude as those arising from perfect reflectors in non-resonant structures35. These results suggest a potentially strong optomechanical coupling in graphene in the THz and mid-IR, approaching that of integrated cavity optomechanical systems, such as micro-ring resonators and photonic crystal cavities36–41. In this article, we examine the optomechanical coupling in a graphene membrane-in-cavity system throughout the THz and mid-IR spectra, to obtain a comprehensive understanding of the 3 ACS Paragon Plus Environment

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influence of the optical cavity design, as well as the intrinsic absorption and dispersion of graphene. We first present a closed-form optical response of a high-Q optical cavity loaded with graphene. We then identify the conditions that maximize the optomechanical coupling coefficients and the resulting predominantly dispersive or dissipative coupling regimes at THz and mid-IR wavelengths. Taking into account the inherent optical absorption of graphene, we proceed to refine the cavity configuration to support resolved-sideband optomechanics and analyze the impact of graphene’s optical absorption. Together, these results point towards the optimal combinations of cavity design and graphene properties that enable various cavity optomechanical phenomena.

RESULTS Optical cavity response We consider a single-layer graphene membrane suspended within a high-Q Fabry-Pérot cavity (Figure 1a), i.e. a membrane-in-cavity system42 that has been extensively utilized in cavity optomechanics, for optomechanical cooling, displacement sensing, and dynamical back-action, as well as other effects such as giant Faraday rotation43,44. An asymmetric Fabry-Pérot cavity is formed between two mirrors with complex reflection coefficients r1e jθ1 and r2 e jθ 2 . Each mirror is a Bragg reflector with alternating layers of dielectric with ε r1 = 21.16 and ε r 2 = 2.56 and thickness of t1 = 0.9 µm and t2 = 1.8 µm (Figure 1a). The bottom mirror has a reflection coefficient magnitude of nearly unity ( r2 ≅ 1 ), making the cavity a Gires–Tournois etalon45 that can be characterized as a one-port system in scattering-matrix formalism. Without loss of generality, we focus on the interactions between the second-order longitudinal optical mode (inset of Figure 1a) and suspended graphene membrane. The second-order longitudinal mode is


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the lowest order mode that provides a stable optomechanical equilibrium position for graphene in the center of the cavity (Figure 1b), allowing the graphene to be suspended a sufficient distance away from the mirrors so as to avoid the effect of van der Waals forces46. The optical response of graphene is modeled as an infinitesimally-thin conducting surface with a surface conductivity47 given by σ (ω ) = σ d (ω ) + σ i (ω ) , where the Drude conductivity is given by,

σ d (ω ) = − j

 EF  e 2 k BT 1 + 2 ln ( e − EF / kBT + 1)  ,  2 π h (ω − jγ )  k BT 

(1)

and the interband conductivity is given by,

σ i (ω ) =

je 2  2 EF − (ω − jγ ) h  ln   . 4π h  2 EF + (ω − jγ ) h 

(2)

EF is the Fermi level, determined by chemical doping or electrostatic gating, and γ is the phenomenological free-carrier scattering rate which parameterizes the losses. To quantify the optomechanical response of graphene, we employ the transfer matrix method48,49 (see Methods section), which analytically relates the optical response of the cavity to the movement of graphene. Throughout this paper, we refer to the optical cavity as being loaded when the suspended graphene is present or as being unloaded when the cavity is empty (equivalent to setting σ = 0 ). The linewidth of the unloaded cavity is given by50 −1

 ∂ψ  1 − r1r2 Γ= ,  r1r2  ∂ω 

(3)


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where ψ = 2η1 + 2η 2 is the round-trip phase delay between the mirrors and the graphene with,

2η1 = 2k0 ( L − h ) − θ1 and 2η2 = 2k0 h − θ 2 where k0 = ω / c ; θ1 and θ2 are the phase angles of the reflection coefficients of the top and bottom mirror, respectively. No approximation has been made for the unloaded cavity linewidth given by Equation 3. Graphene’s influence on the optical cavity is apparent in the reflection spectrum, where both the resonance frequency and linewidth of the cavity resonance follow sinusoidal shifts (Figure 1b) with respect to the graphene suspension height h. For a high-Q cavity (see Methods section), the cavity resonance frequency and linewidth are given by

ω0′ ( h ) = ω0 −

2Γ Im (σ ) cos 2 (η2 ( h ) ) 1 − r1r2 Y0

(4)

and

  2 Re (σ ) Γ′ ( h ) = Γ 1 + cos 2 (η2 ( h ) )  .  1 − r1r2 Y0 

(5)

Y0 is the free space wave admittance. Equation 4 reveals that when graphene is suspended at exactly the maximum of the optical mode’s electric field, the cavity experiences maximal ′ = −2Γ Im (σ ) / Y0 (1 − r1r2 )  . Due to the change in sign of the frequency detuning ∆ω0,max

imaginary part of graphene’s conductivity, the induced detuning corresponds to a blue shift at THz/mid-IR frequencies and a red shift above the interband transition frequency ωinter = 2 EF / h , which is consistent with predictions for the coupling between graphene and other types of optical resonances51.


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The maximal linewidth broadening also occurs when graphene is suspended at the maximum of the

electric

field

(Figure

1c),

and

is

dominated

by

the

graphene

absorption

Γ′max ≅ c Re (σ ) / (Y0 L ) . In contrast, when graphene is suspended exactly within the electric field null of the cavity mode, the total cavity quality factor is the same as that of the unloaded cavity, no matter how large the scattering rate of the graphene. The locations of the electric field nulls also correspond to the stable equilibrium positions of the graphene membrane in the 1D optomechanical system (Figure 1b).


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Figure 1. (a) Schematic of Fabry-Pérot resonator loaded with graphene and the associated second-order cavity mode magnitude ( E ). (b) Optomechanical potential and (c) power reflection coefficient of graphene membrane-in-cavity system as a function of suspension height h and excitation frequency ω . (d) Dispersive optomechanical coupling coefficient gom and (e) dissipative optomechanical coupling coefficient gi as various suspension height h. The cavity length is L = 7 µm, defined by two mirrors with N1 = 4, and N 2 = 20. Graphene is characterized at room temperature by a scattering rate γ = 3 THz and Fermi level of EF = 0.3 eV.

Optomechanical coupling regimes In this section, we identify the spectral range and graphene material constants that lead to predominantly dispersive or dissipative optomechanical coupling. Other constraints, such as maintaining a high optical quality factor, which is relevant for operating in the resolved sideband regime, will be considered in the next section. Following directly from Equation 4 and Equation 5, the dispersive and internal dissipative optomechanical coupling coefficients (Figure 1d,e), and their maximum magnitudes within the cavity are given by

g om =

∂ω0′ 2k0 Γ Im (σ ) = sin ( 2η 2 ) ∂h 1 − r1r2 Y0

max ( g om

(6)

2k Γ Im (σ ) ) = 1 − r0 r Y 1 2 0

and

gi =

2k Γ Re (σ ) ∂Γ ' =− 0 sin ( 2η 2 ) 1 − r1r2 Y0 ∂h

2k0 Γ Re (σ ) max ( gi ) = 1 − r1r2 Y0

.

(7)

All quantities are frequency-dependent through σ and the free-space wave number k 0 . g om and g i also have the same dependence on the graphene height h (shown in Figure 1d,e), implicitly through η 2 and the optical cavity properties. Both coupling coefficients are inversely proportional to the cavity length and increasing the optical quality factor of a high-Q cavity (


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r1,2 ≈ 1 ) does not change max ( g i

)

or max ( g om

)

because the unloaded linewidth Γ scales

proportionally with the term (1 − r1r2 ) (Figure S4). g om and g i differ only by the imaginary part and the real part of σ . Note that the expressions for the coupling coefficients in Equation 6 and Equation 7 are similar to those describing an optical cavity coupled to a subwavelength nanomechanical bar49, except that in our derivation graphene is accurately modeled as an infinitesimally thin conducting surface. The maximum values of g om and g i for a given cavity, as a function of unloaded cavity resonance frequencies, which is very close to the operating frequency, as well as the Fermi level and scattering rate of graphene are shown in Figure 2 and Figure 3.

Dispersive coupling For a fixed graphene scattering rate of 3 THz (Figure 2a), two distinct regions of large dispersive optomechanical coupling emerge: a narrowband peak at the interband-transition threshold frequency ωinter (white dotted line in Figure 2a), and a broadband peak with its maxima approximately 50% lower in frequency (white dashed line in Figure 2a). Except for very low doping levels ( EF < 0.1 eV), an increased doping, i.e. a higher EF , translates to an increased dispersive optomechanical coupling, especially between 10 and 100 THz where graphene’s plasmonic response becomes dominant (Figure 2a). For EF = 0.5 eV, the dispersive coupling peaks at approximately 2 GHz/nm for the high-frequency narrowband region (inset of Figure 2f). In the lower frequency broadband region, above the scattering rate but below the interband transition frequency ( γ γ ), g om becomes largely independent of γ (vertical dark blue contours of Figure 3a), and the high-frequency boundary of the predominantly dispersive coupling is linearly proportional to the Fermi level, which agrees with Equation 8. Thus, both the bandwidth and the peak coupling strength of this region benefit from simultaneously minimizing

γ and maximizing EF . Interestingly, a similar “sweet spot” in the THz spectrum has been predicted for transmission lines with broadband frequency-flat attenuation34. At operating frequencies below γ , g om drops off quickly and g i dominates (Figure 3), which qualitatively agrees with graphene behaving as a lossy conductor in the microwave regime. Here, graphene would be more suited as an electrode in electrostatic devices

9,24,25

rather than as a reflective

membrane within an optical cavity. Near the interband transition frequency, although the dispersive coupling reaches its highest strength across all frequencies, predominantly dispersive coupling is much more challenging to realize, and is reliant on a Fermi level of at least 0.375 eV and scattering rates below 1.25 THz. As previously mentioned, this spectral range coincides with a strong absorption edge that results in an optical cavity linewidth nearly 10x greater than that of the low-frequency broadband dispersive coupling region (Figure S1). More problematic is a significant increase in g i which reduces the contrast between dispersive and dissipative coupling. For relatively low doping ( EF = 0.1 eV), g om never outpaces the rising g i (Figure 2d); only for sufficiently high EF > 0.25 eV, does the ratio g om / gi exceed unity (Figure 2e,f). In general, the ratio g om / gi in this narrow interband peak is highly sensitive to the scattering rate. For γ = 3 THz, the narrow interband peak has g om / gi ≈ 1 throughout the range of Fermi levels that we consider in Figure 2c. For higher quality graphene ( γ ~ 1 THz), a 10x contrast between g om and g i can be achieved, 11 ACS Paragon Plus Environment

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corresponding to the narrow dark purple region in bottom right of Figure 3e. Alternatively, the simultaneously large dispersive and dissipative coupling in this region can be combined to maximize the overall displacement readout sensitivity, since both types of coupling contribute to the total optomechanical gain of the system18. Again, maximal sensitivity relies on achieving a large Fermi level and a small scattering rate in graphene. Overall, these optomechanical coupling peaks are of practical interest given that high-power and low-noise coherent sources are available in the mid-IR region.

Dissipative coupling At operating frequencies above the interband absorption edge (to the right of the dotted line in Figure 2b), dissipative coupling dominates, making graphene-loaded cavities good candidates for phenomena such as unresolved sideband cooling, photothermal backaction30,52, and optomechanical squeezing. Here, gi increases linearly with frequency and is independent of the Fermi level. The interband absorption leads to values of g i exceeding 1 GHz/nm, while the ratio of | g i / g om | rapidly reaches more than an order of magnitude (with gi >> g om in the bottom right of Figure 2c). Note that this ratio is largely independent of the scattering rate at operating frequencies above the interband transition frequency (Figure 3c,d,e). In particular, at moderate doping levels (i.e. EF = 0.1 eV) the ratio of dissipative coupling to dispersive coupling can reach two orders of magnitude in the near-IR spectrum (Figure 2c), while simultaneously providing g i on the order of many GHz/nm (Figure 2d). In contrast, for frequencies below the interband transition threshold (dotted line in Figure 2c), predominantly dissipative coupling is difficult to realize, and high Fermi levels are required ( EF ~ 0.5 eV) to reach just 0.1 GHz/nm of dissipative coupling.


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Figure 2. (a) Maximum g om and (b) gi for a range of optical cavities characterized by the unloaded resonance of the second-order mode ω0 and for various Fermi levels EF of graphene. (c) Ratio of gom / gi indicating the dispersive coupling regime ( g om / gi >> 1 ) and the dissipative coupling regime ( g om / gi > 1 ) and dissipative coupling regime( g om / gi 1 ), is shown in Figure 5b, again for the graphene cavity discussed for Figure 4a and Figure 4b. A Qm f m on the order of 1013 , i.e. Qm = 105 , is sufficient for enabling coherent coupling, which is nearly within reach of current

graphene mechanical resonators (Figure S5).


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Figure 5. (a) Single-photon coupling rate versus mechanical sideband resolution f m / Γ′ for graphene cavity optomechanical system operating in the broadband dispersive coupling regime, compared with photonic crystal36–38, micro-ring39, and microwave optomechanical systems that lie out of range to the lower right. (b) Single-photon cooperativity for the graphene optomechanical system with Qm f m of 1011, 1012, 1013.

DISCUSSION Our analysis reveals that despite its significant optical absorption, graphene can provide large optomechanical coupling in a membrane-in-cavity system that is comparable to state-of-the-art cavity optomechanical systems36–39. For quantum-limited force and displacement readout, predominantly dispersive coupling occurs in the mid-IR frequency range, between the scattering rate and the interband transition frequency of graphene, with a strength of up to 0.5 GHz/nm in graphene with the Fermi level above 0.45 eV. Generally speaking, short cavity lengths and large


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Fermi levels are essential to realizing the largest dispersive coupling, while small scattering rates suppress dissipative coupling and optical heating. Surprisingly, even for graphene with a scattering rate as large as 6 THz, dispersive optomechanical coupling can dominate the dissipative coupling by more than 100-fold. On the other hand, a graphene Fermi level around 0.1 eV leads to a predominantly dissipative coupling region in the near-IR above the interband transition threshold. The contrast ratio exceeds 100-fold regardless of the scattering rate of graphene, making the graphene an excellent candidate for exploring dissipative optomechanical cooling. Additionally, near the interband transition frequency, simultaneously large dispersive and dissipative optomechanical coupling are useful for maximizing sensitivity in displacement and force readout, at the expense of significant optical heating. For optomechanics requiring resolved sideband operation and an associated narrow optical cavity linewidth, the optical loss of graphene can be managed by suspension near the electricalfield nulls of a cavity with sufficiently high unloaded optical quality factor. Dispersive optomechanical coupling around 180 MHz/nm at 100 THz is realizable with graphene mechanical resonators with resonant frequencies above 100 MHz and with a graphene scattering rate of several THz. For graphene with higher scattering rates, advances in raising graphene mechanical resonance frequencies into the few-GHz range would be necessary. On the other hand, coherent photon-phonon coupling is within reach of existing graphene resonators with mechanical quality factors on the order of 105 . Even though we have focused on membrane-incavity systems, our general conclusions regarding the viability of graphene cavity optomechanics in the mid-IR regime and the trade-off between coupling strength and cavity linewidth are applicable to other cavity optomechanical systems that couple graphene with quantum dots, single-particle emitters26,27, or traveling-wave resonators28. 20 ACS Paragon Plus Environment

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METHODS Transfer matrix modeling The goal of the transfer matrix analysis is to obtain an expression for the fields inside the cavity, the denominator of which, will be mapped into a complex pole whose imaginary and real components define the cavity resonance frequency and linewidth. The transfer matrix formulation of the membrane-in-cavity system is T  1   = Tm2 ⋅ T2 ⋅ Tgr ⋅ T1 ⋅ Tm1   , 0 R

(9)

where R and T represent the reflection and transmission coefficients. The mirror transfer matrix is given by  r2 + t2 − t Tm =  r   − t 

(

r  t 1  t

(10)

)

where r = r1e jθ1 r2 e jθ 2 and t = t1 (t2 ) for the front (back) mirror. Note that we don’t explicitly define the phase angle of the mirror transmission coefficients ( t1 and t 2 ) since they do not appear in the final expression for the denominator of the cavity field amplitude. Also note that the mirrors of the cavity are symmetric, meaning that they have N1 ( N 2 ) periods with a single additional layer of thickness t1 . The transfer matrices for propagation between the graphene and the front mirror and the back mirror are given by,


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 e − jk0 ( L − h ) 0  T1 =   jk L − h  0 e 0 ( )    e− jk0 h 0  T2 =   e jk0 h   0

(11)

and the transfer matrix for the graphene membrane is given by

σ  1 + 2Y 0 Tgr =   σ  − 2Y  0

σ

 2Y0  . σ  1− 2Y0 

(12)

The forward ( a3 ) and backward ( b3 ) propagating field-amplitudes at the inside interface of the front mirror are given by

 a3  1   = Tm1   ,  R  b3 

(13)

which can be evaluated by first solving Equation 9 for R . The denominator of the expressions for a3 and b3 is given by,

1 − r1r2 e

−2 j (η1 +η2 )

+

σ 2Y0

(1 + r e 1

−2 jη1

)(1 + r e 2

−2 jη2

)

(14)

which can be mapped into a complex pole of the form, j (ω − ω0′ ) + Γ′ where ω 0′ and Γ′ correspond to the loaded cavity resonance frequency and linewidth. The unloaded cavity resonance frequency, ω 0 satisfies the condition, η1 + η 2 = mπ , which causes the field amplitude within the cavity to diverge for perfectly reflecting mirrors ( r1,2 = 1 ). If we assume that the


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second term of Equation 14 is nearly constant in the vicinity of the unloaded resonance frequency, ω 0 then Equation 14 can be rewritten as

r1r2

δψ  j (ω − ω0 ) + Γ  + R + jI , δω 

(15)

where R and I are the real and imaginary parts of the third term of Equation 14. The expression in Equation 15 leads to Equations 4 and 5 which are valid if ω 0 is near the center of the photonic band gap of the Bragg mirrors and if the quality factor of the cavity is high ( r1,2 ≈ 1 ).

ASSOCIATED CONTENT Supporting Information Available: Additional considerations for the optical cavity including the optical cavity finesse, quality factor, and linewidth as a function of operating frequency and graphene parameters; the optomechanical coupling’s dependence on the unloaded quality factor; and the experimental properties of graphene mechanical resonators. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.XXXXXXX.

AUTHOR INFORMATION Corresponding Author *

Email: [email protected].

Notes The authors declare no competing financial interest.


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ACKNOWLEDGEMENTS This work was supported in part by the Packard Fellowships for Science and Engineering, the Alfred P. Sloan Research Fellowship, and ONR program Grant N00014-16-1-2687 under program manager Dr. Dan Green.

REFERENCES (1) (2) (3) (4)

(5)

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(10)

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