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A terahertz photomixer based on plasmonic nanoantennas coupled to a graphene emitter
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 Nanotechnology 24 455202 (http://iopscience.iop.org/0957-4484/24/45/455202) View the table of contents for this issue, or go to the journal homepage for more
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IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 24 (2013) 455202 (7pp)
doi:10.1088/0957-4484/24/45/455202
A terahertz photomixer based on plasmonic nanoantennas coupled to a graphene emitter Pai-Yen Chen1 and Andrea Alu` Department of Electrical and Computer Engineering, The University of Texas at Austin, 1 University Station C0803, Austin, TX 78712, USA E-mail: [email protected] and [email protected]
Received 10 April 2013, in final form 19 September 2013 Published 15 October 2013 Online at stacks.iop.org/Nano/24/455202 Abstract We propose the concept of a graphene-based nanoantenna-enhanced photomixer to realize wideband-tunable terahertz (THz) frequency generation. When two laser beams are focused on the graphene nanoemitter of a planar field-emission diode, THz current oscillations can be created at the emitter tip through the optical heterodyne. Graphene’s optical transparency allows suitably designed plasmonic nanoantennas to boost the mixing of laser radiation at the emitter tip, significantly increasing the overall produced photomixing current. The THz wave generated at the graphene emitter is then coupled to a loading circuit, thanks to the THz wave confinement in the graphene nanostructures. Our design is ideally suited for THz sources that may be tuned from DC to 10 THz by simply shifting the frequency offset of two pumping lasers. (Some figures may appear in colour only in the online journal)
Graphene is a monatomic layer of carbon atoms arranged in a honeycomb lattice, whose interesting electrical and optical properties have been discovered quite recently [1]. Since then, graphene has attracted a great deal of interest because of its ultrahigh charge carrier mobility, high Fermi velocity (νF = 108 cm s−1 ) at low temperature, quantum Hall effect, unexpectedly high opacity for a single atomic layer, and gate-tunable responses [1–4]. In addition, its remarkable mechanical strength and thermal stability make graphene an extremely promising material in a wide range of electronic and optical applications. In the past few years, graphene nanodevices have become a subject of intensive research because of their exciting potential for ultrahigh-speed radio-frequency (RF) field effect transistors [5, 6], ultrafast and broadband optical modulators and photodetectors [7–10] and transparent electrodes for flexible flat-panels displays and solar panels [11], among many others. Graphene plasmonics is another emerging field of exciting applications for this nanomaterial platform, because
of the fact that graphene can provide a flat ultrathin substrate to tailor strong light–matter interactions in the terahertz (THz) and infrared range [12–16]. In addition to its extreme thinness, graphene is appealing for its gate-tunable plasmon resonance of massless Dirac fermions, enabling wideband tunability in photonic and optoelectronic nanodevices, such as plasmonic oscillators [17], flatland transformation optics and tunable metamaterials [18, 19], cloaking devices [20], terahertz wave switches [21], modulators [22] and phase shifters [23]. The realization of these functionalities at the nanoscale may pave the way to graphene-enabled wireless communication systems [23–28], in which graphene patches may serve as frequency-reconfigurable, electrically small THz antennas connecting all the other nanodevices mentioned above. These may have exciting applications in data transmission and nano-communication [25], biomedicine, sensing and actuation [23–28]. One of the main technological hurdles in THz systems is represented by the lack of efficient sources providing large power generation and good bandwidth of operation. Photomixers based on field-emission devices have been only
1 Author to whom any correspondence should be addressed.
0957-4484/13/455202+07$33.00
1
c 2013 IOP Publishing Ltd Printed in the UK & the USA
Nanotechnology 24 (2013) 455202
P-Y Chen and A Al`u
Figure 1. Schematics of (a) the proposed graphene-based THz photomixer based on field-emission (vacuum) nanodiodes and (b) its device cross-sectional view.
recently introduced for the generation of THz radiation with potentially increased output power and very large bandwidths, ideally spanning from DC to 100 THz [29–32]. As an additional advantage, electron field-emission, with a ballistic transport in vacuum, is insensitive to the environment temperature and ionizing radiation. For these reasons, field-emission nanodevices with very fast switching time and high current densities are widely used in vacuum micro/nanoelectronics [33–38]. Recently, there have been some efforts to explore the electron field emissions from freestanding graphene monolayers [39–42], which is expected to be largely enhanced due to the large geometric (electrostatic field) enhancement factor of graphene (103 –104 ). Numerical results provide a simple fitting formula for the geometric enhancement factor of a graphene patch [39]: βemitter = 2.07(h/t)0.75 × (1 + 0.09h/w), where h, w, t are respectively its height, width and thickness of graphene patch. As a result, a graphene emitter can have very low turn-on voltage due to its atomically sharp edge and extreme thinness (t ∼ 2 nm). In addition, well-defined graphene nanoemitters can be produced using lithographic patterning [5–8, 19], with significant fabrication advantages over carbon nanotubes or other nanomaterial emitters. In this paper we explore the concept and provide a complete design of a THz photomixer based on graphene field-emission nanodiodes [41], as shown in figure 1. The current density produced by a graphene emitter can be described by the well-known Fowler–Nordheim equation [33–38, 40–42]: J(E) = AE2 e−B/E (A m−2 ), 10−6 /8
Figure 2. (a) Equivalent circuit model for a graphene-based photomixer, as in figure 1. (b) Lumped-element transmission-line model for a graphene parallel-plate waveguide.
If the magnitude of E1 and E2 are much smaller than E0 , an explicit expression for the perturbed current density can be J = J0 + JD + Jω1 −ω2 + J2ω1 + J2ω2 + Jω1 +ω2 + · · · ! " 2 # J0 B B2 E1 2 E2 = J0 + 1+ + 2 + 2 E0 E E 2E0 0 0 ! 2 B B E1 E2 + J0 1 + + 2 E0 2E0 E0 E0
(1)
× cos[(ω1 − ω2 )t] + · · · .
where A = 1.4 × (A eV), B = 6.83 × 109 81.5 (V eV−3/2 m−1 ), 8 is the electron work function (8 = 4.5 eV for graphene) and E is the electric field at the emitter surface. If two sinusoidal (optical) fields E1 and E2 are superimposed to the electrostatic field E0 , the total electric field is E = E0 + E1 cos(ω1 t) + E2 cos(ω2 t), where ωi are the pumping oscillation frequencies. The electrostatic field at the emitter surface E0 = βemitter Vc /dvac depends on the emitter geometry, collector voltage Vc and vacuum gap distance dvac . V−2
(2)
At the apex of the graphene emitter, THz photomixing currents will be generated at the offset frequency ω1 − ω2 of two lasers. For small vacuum gaps, due to the large parasitic collector-to-emitter capacitance CAE , the THz wave is not effectively coupled to the collector (C) load resistor RA , due to the low-pass filtering effects associated with the shunt capacitance [30] (see figure 2(a)). On the other hand, a graphene emitter can support THz 2
Nanotechnology 24 (2013) 455202
P-Y Chen and A Al`u
surface plasmon-polariton waves [16–18, 14] that may be efficiently coupled to the load resistor at the source (S) terminal RL . Moreover, if another graphene sheet is placed below the graphene emitter with proper separation using an insulating oxide, a parallel-plate waveguide transmission-line [23, 43] can be formed to guide the strongly confined, non-dispersive quasi-transverse electromagnetic (quasi-TEM) THz wave, interconnecting the generated signal to different on-chip ports and processors [23, 43]. In equation (2), JD is the rectified current density, and all terms with frequencies larger than the mixing frequency ω1 − ω2 (i.e. 2ω1 , 2ω2 , and ω1 + ω2 ) can be neglected, as they are expected to be significantly attenuated along the graphene sheet. As observed from (2), the THz photomixing current depends on the optical field intensity E1 and E2 at the emitter tip. Recently, optical nanoantennas have been proposed to enhance the intrinsically weak spontaneous emission, nonlinear optical effects [44–47], and photomixing for THz generation [48, 49]. When subwavelength nanoantennas are placed around the emitter tip in our geometry (figure 1), the optical field enhancement associated with their plasmonic features can largely enhance the photomixing efficiency, as evident in (2). The high optical opacity of the graphene monolayer may allow us to readily integrate the graphene emitter to the optical nanoantennas. Since graphene is largely transparent to visible light (∼97.7% transmission in the near-infrared and visible range (see appendix))2 , light can pass through the grapheme monolayer in figure 1 and excite the optical antennas enhancing the optical near-field intensity [11, 47]. Figure 3 shows the predicted average local field enhancement versus frequency for an array of silver nanoantennas with width wa = 25 nm, length la = 60 nm, height ha = 25 nm and separation distance s = 150 nm. These parameters have not been optimized, but simply tailored to provide a good field enhancement around the frequency range of interest. In figure 3, we have assumed that the light polarization is parallel to the gaps and the field distributions have been calculated using full-wave simulations [50] using an empirically fitted Drude-type dispersion for silver [51]. The average field enhancement is defined by calculating the line integral of the local electric field normalized to the incident field, weighted by the total length. The observation line is 15 nm from the array and shows a single, rather broad resonance, resulting in a broadband enhancement of the THz photomixing process, provided that the emitter tip is the close proximity of optical antennas. The model in figure 2(a) ensures that the THz waves created by the current oscillations at the apex of the graphene emitter may be coupled to the load resistor at the source terminal ZL , which may represent a graphene patch antenna [24–28] or arbitrary matched THz circuit load. Here we assume the perfect transition between graphene waveguide
Figure 3. Averaged optical near-field enhancement versus frequency. The observation line is parallel and 15 nm distant from the array of silver nanoantennas. The right inset shows the cross-sectional view of the electric field distribution along the antenna axis (yellow dashed line in the left inset).
and load, ignoring possible parasitics. Due to the significantly shortened wavelength in this graphene waveguide (compared to the wavelength in free space) [23, 43], the graphene channel should be modeled as a segment of transmission-line, and retardation effects within its length should be carefully taken into consideration. The input impedance at the emitter tip is given by Zin = Zg
RL + Zg tanh γ l , Zg + RL tanh γ l
(3)
where γ = α + jβ and Zg are the propagation constant and characteristic impedance of the graphene transmission-line, respectively. The output power generated at the emitter tip is given by Pout = 21 Iω2 1 −ω2 Re[Zin ],
(4)
where the photomixing current Iω1 −ω2 = gp Jω1 −ω2 wt, and gp is the current gain taking into account the resonant photon-assisted field-emission process [29–31]. For graphene with a work function of 4.5 eV [40–42], a gain of about 40 dB is obtained from rigorous quantum mechanical simulations [30]. Although an elongated graphene sheet results in a large geometric (electrostatic) field enhancement factor proportional to (h/t)0.75 [39], the non-negligible attenuation constant α fundamentally may limit the maximum available length for the graphene layer. Also, the emitter length is important to properly phase the THz signal such that the maximum power can be delivered to the load. This happens when the phase shift coincides with a Fabry–Perot resonance of the transmission-line segment (βl = mπ with m an integer). For a subwavelength waveguide (i.e., d 2π c/(ω1 − ω2 )), a quasi-static circuit model in figure 2(b) may be used to calculate the propagation constant, phase velocity and characteristic impedance of the transmission-line [4]. The relevant lumped elements include the kinetic inductance LK (which is caused by the electron-inertia effect typically observed in high carrier mobility conductors like graphene at
2 The graphene monolayer is highly transparent in the near-infrared and
visible regions, showing a frequency-independent 97.7% optical transparency that is consistent with graphene’s minimum interband conductivity σmin = q2 /4h¯ . 3
Nanotechnology 24 (2013) 455202
P-Y Chen and A Al`u
very high frequencies), sheet resistance Rs (which is mainly caused by the carrier scattering by defects and impurities in graphene), magnetic inductance Lm , and electrostatic capacitance Ces per unit length: 2 Im[1/σs ] 2 Re[1/σs ] , Rs = , ωw w d Lm = µ0 , w w π Ces = εox ε0 + εox ε0 , d ln[6(d/w + 1)] LK =
(5)
where εox = 3.9 is the relative permittivity of silicon dioxide (SiO2 ) filling the waveguide, ε0 and µ0 are respectively the vacuum permittivity and permeability, σs is the complex surface conductivity derived from the Kubo formula [14]; see appendix for details. The first term in Ces represents the parallel-plate capacitance between the two graphene sheets and the second term is the fringing capacitance [52], negligible when w d. In the THz range, σs follows a Drude-type dispersion σs = −j(q2 EF /π h¯ 2 )/(ω − j), where is the carrier intraband scattering rate ( = 0.065 meV for high-quality graphene [13]) and the plasmon spectrum is tunable by Fermi energy EF . From the transmission-line theory, the propagation constant and characteristic impedance may then be calculated using p γ = α + jβ = jωCes [R + jω(LK + Lm )] s p R ∼ (6a) = jω LK Ces 1 + jωLK s s s R + jω(LK + Lm ) ∼ LK R Zg = 1+ . (6b) = jωCes Ces jωLK
Figure 4. Phase constant versus signal frequency for a graphene parallel-plate waveguide with different Fermi energy. The solid lines and dots are results obtained from our analytical transmission-line model (figure 2(b)) and the rigorous eigenmodal solution, respectively.
At THz frequencies and below the interband transition threshold, LK = Rs / ' π(h¯ /q)2 /EF is valid. Typically, LK is larger than Lm by several orders of magnitude, and value of LK is carrier-density-dependent, providing interesting possibilities to enhance the phase velocity and enable tunable and reconfigurable THz components [23]. Figure 4 shows the calculated phase constant β versus frequency for intrinsic (∼ EF = 250 meV) [53] and doped (EF = 500 meV) graphene parallel-plate waveguides with d = 50 nm and w = 30 µm; here the solid and dotted lines are respectively calculated using the transmission-line model introduced here and a rigorous eigenmodal solution in [23, 43]. It is evident in figure 3 that our model can accurately predict the THz wave propagation properties. It is also important to highlight that at THz frequencies ω R/LK = , the group (signal) velocity is essentially independent of frequency (vg = ∂β/∂ω = constant), typically desirable for signal communications. For an ideally lossless transmission-line, the output power delivered to the load Pout = Iω2 1 −ω2 Re[Zin ]/2. However, due to intrinsic losses in graphene, we expect that the real power delivered to the load PL = ηPout , where the efficiency η = (1 − |0|2 )/[(1 − |0e−γ l |2 )e2αl ] and the signal reflection caused by the impedance mismatch 0 = (ZL − Zg )/(ZL − Zg ). The output power is proportional to the load impedance when ZL is small, but it varies inversely with it
Figure 5. Contours of output power, varying the flux intensity of two lasers (P1 = P2 = P0 ) and the electrostatic field strength at the emitter surface (E0 ).
when ZL becomes large. Therefore, there is an optimal value of ZL satisfying the condition ∂PL (ZL )/∂ZL = 0. Figure 5 shows the contours of Pout calculated using equations (3)–(5) at the mixing frequency (ω1 − ω2 )/2π = 1 THz (here ω1 /2π = 393 THz and ω2 /2π = 392 THz), varying the electrostatic field strength E0 and the laser flux intensities P1 = P2 = P0 . Here the width w and length l of the graphene sheets are 30 µm and 2.3 µm respectively, and the optimized load resistance RL = 390 . The nanoantennas are placed 15 nm behind the emitter tip and the optical field enhancement may be obtained from figure 3. Figure 5 shows that the total generated power Pout increases with the static field and laser flux intensity. In order to protect the nanoemitter from vacuum arcing, it is necessary to limit the dc current density J0 to 109 A m−2 in steady state or 1012 A m−2 when operating with microsecond pulses [29–31]. In figure 4(a), Pout = 1 µW and J0 = 1.37 × 109 A m−2 are obtained at E0 = 7 V nm−1 and P0 = 1 MW cm−2 . Figures 6(a) and (b) show the contours of total generated THz power Pout and real power delivered to the load PL for the 4
Nanotechnology 24 (2013) 455202
P-Y Chen and A Al`u
Figure 6. Contours of: (a) output power at the emitter tip, (b) real power delivered to the load, varying the length and width of graphene sheets; the inset of (b) shows the contours of power delivery efficiency.
same photomixer design as in figure 5, with E0 = 7.5 V nm−1 and P0 = 1 MW cm−2 , varying the graphene width w and length l. The maximum output power sensibly depends on the graphene length, which is associated with the total phase shift experienced by the THz wave traveling along the transmission-line. Inspecting the figure, we note that, although a long emitter can enhance the geometric field enhancement factor βemitter and therefore reduce the required dc bias voltage Vc , after considering the losses in graphene, the first Fabry–Perot resonance mode is the preferred operation to achieve maximized PL and therefore optimum grapheme length loptimum = 2π/β = λg /2. The inset of figure 6(b) shows the contours of power efficiency η, showing that the efficiency rapidly deteriorates with increasing width and length, due to increased losses. Comparing figures 6(a) and (b), it is clear that the real power delivered to the load is attenuated because of propagation losses (1−η)Pout , especially for long graphene sheets. Figure 7 shows the real power PL delivered to the load versus the synthesized THz frequency for the same photomixer as in figure 5, with E0 = 7.5 V nm−1 and P0 = 1 MW cm−2 . For comparison, we also show the available power without the presence of the nanoantennas. In this plot we consider ω1 /2π = 393 THz, while varying ω2 /2π from 392 to 382 THz, in order to generate THz signals over a wide frequency range. For each mixing frequency, the optimal values of l and ZL are obtained with the physics-driven transmission-line and circuit models presented in figure 2. It is evident that the concept of designing nanoantennas coupled to an optimal graphene transmission-line emitter can greatly increase the output power over the whole dc 10 THz frequency range. For instance, at the mixing frequency (ω1 − ω2 )/2π = 1 THz, the maximum delivered power increases from 6.6 nW to 1.44 µW, thanks to the effect of the plasmonic nanoantennas. Typically, field-emission arrays can consist of 104 emitters per centimeter square [33–38] (i.e., flat-panel displays), so arrays of graphene-based, vacuum-nanoelectronic photomixers as the ones presented here can potentially realize mW power level.
Figure 7. Real power delivered to the load versus the synthesized THz frequency for the optimized graphene-based photomixer with and without nanoantennas.
In realistic designs, the physical length of the graphene emitter and of the waveguide cannot be changed. However, a tunable phase shift in the graphene channel is possible with a double-gate structure, as recently proposed in [23], which tunes the phase to β(EF )l = mπ . Also, a wideband-tunable impedance ZL can be realized by the gate-tunable graphene antenna, waveguide, and transformer [18, 23]. Figure 8 presents the output power for a nanoantenna-enhanced graphene photomixer with emitter length l = 2.3 µm and load impedance ZL = 420 , varying the Fermi energy. It is seen that the synthesized THz wave exhibits multiple peaks, significantly shiftable with Fermi energy. This feature is of great interest for applications in narrow-spectral-linewidth, tunable THz sources. To conclude, we have proposed here the concept of a graphene-based, vacuum-nanoelectronic photomixer with large tuning range covering the whole terahertz band (0.3–10 THz). The graphene layer has a triple functionality in our design. First, its high optical transparency may allow exciting plasmonic nanoantennas around its tip in order to boost the intensity of pumped visible light. Second, its extreme thinness is ideal to enhance the generation 5
Nanotechnology 24 (2013) 455202
P-Y Chen and A Al`u
over σinter , and the sheet conductivity of graphene can be explicitly expressed as [14]: σintra = −j
q2 KB T
π h¯ 2 (ω − j) EF EF × + 2 ln 1 + exp − . KB T KB T
(A.2)
For doped graphene with |EF | kB T (A.2), further reduces to the Drude-like dispersion: σintra ≈ −j
q2 EF π h¯ (ω − j) 2
.
(A.3)
In general, σinter must be evaluated numerically, however, for hω, ¯ |EF | kB T, it can be approximated as [14]: q2 2|EF | − h¯ (ω − j) σinter = −j ln . (A.4) 4π h¯ 2|EF | + h¯ (ω − j)
Figure 8. Power output versus synthesized THz frequency for a nanoantenna-enhanced, graphene-based photomixer, with emitter length l = 2.3 µm, load impedance RL = 390 , and Fermi energy swept from 250 to 400 meV. The gray line is the maximum achievable power of this photomixer.
For frequencies below the THz regime, the interband conductivity is almost negligible when compared with the intraband contributions. In the near-infrared and visible spectrum, the interband contribution may dominate over the intraband contribution, and σinter = q2 /4h¯ is a real constant. This is in consistent with graphene’s high optical opacity.
of THz signals. Finally, the graphene emitter can directly couple the THz wave to the loading circuit or emitting antenna, enabling all-graphene nano-transmitters and nanosensors [23–28]. This concept may have a great potential to realize mW terahertz sources by graphene field-emission arrays, offering much greater bandwidth and increased output power compared to currently available solutions.
References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S Y, Grigorieva I V and Firsov A A 2004 Science 306 666 [2] Geim A K and Novoselov K S 2007 Nature Mater. 6 183 [3] Grigorenko A N, Polini M and Novoselov K S 2012 Nature Photon. 6 749 [4] Wong H S P and Akinwande D 2011 Carbon Nanotube and Graphene Device Physics (Cambridge: Cambridge University Press) [5] Lin Y M et al 2011 Science 332 1294 [6] Lee J H, Parrish K N, Chowdhury Sk F, Ha T J, Hao Y, Tao L, Dodabalapur A, Ruoff R S and Akinwande D 2012 IEEE Int. Electronic Device Mtg. (San Francisco, CA, Dec. 2012) [7] Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, Wang F and Zhang X 2011 Nature 474 64 [8] Liu M, Yin X and Zhang X 2012 Nano Lett. 12 1482 [9] Xia F, Mueller T, Lin Y, Valdes-Garcia A and Avouris Ph 2009 Nature Nanotechnol. 4 839 [10] Fang Z, Liu Z, Wang Y, Ajayan P M, Norlander P and Halas N J 2012 Nano Lett. 12 3808 [11] Kim K S, Zhao Y, Jang H, Lee S Y, Kin J M, Kim K S, Ahn J H, Kim P, Choi J Y and Hong B H 2009 Nature 457 706 [12] Jablan M, Buljan H and Sojaˇci´c M 2009 Phys. Rev. B 80 245435 [13] Bouzianas G D, Kantartzis N V and Tsiboukis T D 2011 General Assembly and Scientific Symposium, 2011 30th URSI (Aug. 2011) [14] Hanson G W 2008 J. Appl. Phys. 103 06430 [15] Koppens F H L, Chang D E and Garcia de Abajo F J 2011 Nano Lett. 11 3370 [16] Chen J et al 2012 Nature 487 77 [17] Rana F 2008 IEEE Trans. Nanotechnol. 7 91 [18] Vakil A and Engheta N 2011 Science 332 1291 [19] Ju L et al 2011 Nature Nanotechnol. 6 630 [20] Chen P Y and Al`u A 2011 ACS Nano 5 5855 [21] Lee S H et al 2012 Nature Mater. 11 936
Acknowledgments The authors would like to thank Dr Deji Akinwande for fruitful discussions. This work was supported by the NSF CAREER award No. ECCS-0953311, the AFOSR YIP award No. FA9550-11-1-0009 and the ARO grant No. W911NF-11-1-0447.
Appendix Graphene’s sheet conductivity includes both semi-classical intraband conductivity σintra and quantum-dynamical interband conductivity σinter [14]: σs (ω, EF , , T) = σintra + σinter Z +∞ jq2 (ω − j) |ε| =− 2 (ω − j)2 π h¯ −∞ ∂fd (ε) dE × ∂ε Z +∞ ∂fd (−ε) − ∂fd (ε) − dE (A.1) (ω − j)2 − 4(ε/h) ¯ 2 0 where fd = 1/(1 + exp[(ε − EF )/(kB T)]) is the Fermi–Dirac distribution, ε is the energy, EF is the Fermi energy, T is the temperature, q is the electron charge, h¯ is the reduced Planck’s constant, and is the carrier scattering rate. The first and second terms in (A.1) account for the intraband and interband contributions, respectively. In the THz region and below the interband transition threshold (hω ¯ < 2|EF |), σintra dominates 6
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[37] Guerrera S A, Velasquez-Garcia L F and Akinwande A I 2012 IEEE Trans. Electron Devices 59 2524 [38] Chen P Y, Huang H, Akinwande D and Al`u A 2012 IEEE Trans. Nanotechnol. 11 1201 [39] Watcharotone S, Ruoff R S and Read F H 2008 Phys. Procedia 1 71 [40] Eda G, Unalan H E, Rupesinghe N, Amaratunga G A J and Chhowalla M 2008 Appl. Phys. Lett. 93 233502 [41] Lee S, Lee S S and Yang E H 2009 Nanoscale Res. Lett. 4 1218 [42] Wu Z S, Pei S, Ren W, Tang D, Gao L, Liu B, Li F, Liu C and Cheng H M 2009 Adv. Mater. 21 1756 [43] Hanson G W 2008 J. Appl. Phys. 104 084314 [44] Agio M and Alu A 2013 Optical Antennas (Cambridge, UK: Cambridge University Press) [45] Novotny L and van Hulst N 2011 Nature Photon. 5 83 [46] Chen P Y and Al`u A 2011 Nano Lett. 11 5514 [47] Argyropoulos C, Chen P Y, Monticone F, D’Aguanno G and Al`u A 2012 Phys. Rev. Lett. 108 263905 [48] Berry C W, Wang N, Hashemi M R and Jarrahi M 2013 Proc. Int. Conf. and Exhibition on Lasers, Optics & Photonic (San Antonio, TX, Oct. 2013) [49] Berry C W and Jarrahi M 2012 New J. Phys. 14 105029 [50] CST Microwave Studio: www.cst.com [51] Johnson P B and Christy R W 1972 Phys. Rev. B 6 4370 [52] Liao A D, Wu J Z, Wang X R, Tahy K, Jena D, Dai J J and Pop E 2011 Phys. Rev. Lett. 106 256801 [53] Berciaud S, Ryu S, Brus L E and Heinz T F 2009 Nano Lett. 9 346
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A terahertz photomixer based on plasmonic nanoantennas coupled to a graphene emitter
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 Nanotechnology 24 455202 (http://iopscience.iop.org/0957-4484/24/45/455202) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 194.27.18.18 This content was downloaded on 15/08/2014 at 15:31
Please note that terms and conditions apply.
IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 24 (2013) 455202 (7pp)
doi:10.1088/0957-4484/24/45/455202
A terahertz photomixer based on plasmonic nanoantennas coupled to a graphene emitter Pai-Yen Chen1 and Andrea Alu` Department of Electrical and Computer Engineering, The University of Texas at Austin, 1 University Station C0803, Austin, TX 78712, USA E-mail: [email protected] and [email protected]
Received 10 April 2013, in final form 19 September 2013 Published 15 October 2013 Online at stacks.iop.org/Nano/24/455202 Abstract We propose the concept of a graphene-based nanoantenna-enhanced photomixer to realize wideband-tunable terahertz (THz) frequency generation. When two laser beams are focused on the graphene nanoemitter of a planar field-emission diode, THz current oscillations can be created at the emitter tip through the optical heterodyne. Graphene’s optical transparency allows suitably designed plasmonic nanoantennas to boost the mixing of laser radiation at the emitter tip, significantly increasing the overall produced photomixing current. The THz wave generated at the graphene emitter is then coupled to a loading circuit, thanks to the THz wave confinement in the graphene nanostructures. Our design is ideally suited for THz sources that may be tuned from DC to 10 THz by simply shifting the frequency offset of two pumping lasers. (Some figures may appear in colour only in the online journal)
Graphene is a monatomic layer of carbon atoms arranged in a honeycomb lattice, whose interesting electrical and optical properties have been discovered quite recently [1]. Since then, graphene has attracted a great deal of interest because of its ultrahigh charge carrier mobility, high Fermi velocity (νF = 108 cm s−1 ) at low temperature, quantum Hall effect, unexpectedly high opacity for a single atomic layer, and gate-tunable responses [1–4]. In addition, its remarkable mechanical strength and thermal stability make graphene an extremely promising material in a wide range of electronic and optical applications. In the past few years, graphene nanodevices have become a subject of intensive research because of their exciting potential for ultrahigh-speed radio-frequency (RF) field effect transistors [5, 6], ultrafast and broadband optical modulators and photodetectors [7–10] and transparent electrodes for flexible flat-panels displays and solar panels [11], among many others. Graphene plasmonics is another emerging field of exciting applications for this nanomaterial platform, because
of the fact that graphene can provide a flat ultrathin substrate to tailor strong light–matter interactions in the terahertz (THz) and infrared range [12–16]. In addition to its extreme thinness, graphene is appealing for its gate-tunable plasmon resonance of massless Dirac fermions, enabling wideband tunability in photonic and optoelectronic nanodevices, such as plasmonic oscillators [17], flatland transformation optics and tunable metamaterials [18, 19], cloaking devices [20], terahertz wave switches [21], modulators [22] and phase shifters [23]. The realization of these functionalities at the nanoscale may pave the way to graphene-enabled wireless communication systems [23–28], in which graphene patches may serve as frequency-reconfigurable, electrically small THz antennas connecting all the other nanodevices mentioned above. These may have exciting applications in data transmission and nano-communication [25], biomedicine, sensing and actuation [23–28]. One of the main technological hurdles in THz systems is represented by the lack of efficient sources providing large power generation and good bandwidth of operation. Photomixers based on field-emission devices have been only
1 Author to whom any correspondence should be addressed.
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Figure 1. Schematics of (a) the proposed graphene-based THz photomixer based on field-emission (vacuum) nanodiodes and (b) its device cross-sectional view.
recently introduced for the generation of THz radiation with potentially increased output power and very large bandwidths, ideally spanning from DC to 100 THz [29–32]. As an additional advantage, electron field-emission, with a ballistic transport in vacuum, is insensitive to the environment temperature and ionizing radiation. For these reasons, field-emission nanodevices with very fast switching time and high current densities are widely used in vacuum micro/nanoelectronics [33–38]. Recently, there have been some efforts to explore the electron field emissions from freestanding graphene monolayers [39–42], which is expected to be largely enhanced due to the large geometric (electrostatic field) enhancement factor of graphene (103 –104 ). Numerical results provide a simple fitting formula for the geometric enhancement factor of a graphene patch [39]: βemitter = 2.07(h/t)0.75 × (1 + 0.09h/w), where h, w, t are respectively its height, width and thickness of graphene patch. As a result, a graphene emitter can have very low turn-on voltage due to its atomically sharp edge and extreme thinness (t ∼ 2 nm). In addition, well-defined graphene nanoemitters can be produced using lithographic patterning [5–8, 19], with significant fabrication advantages over carbon nanotubes or other nanomaterial emitters. In this paper we explore the concept and provide a complete design of a THz photomixer based on graphene field-emission nanodiodes [41], as shown in figure 1. The current density produced by a graphene emitter can be described by the well-known Fowler–Nordheim equation [33–38, 40–42]: J(E) = AE2 e−B/E (A m−2 ), 10−6 /8
Figure 2. (a) Equivalent circuit model for a graphene-based photomixer, as in figure 1. (b) Lumped-element transmission-line model for a graphene parallel-plate waveguide.
If the magnitude of E1 and E2 are much smaller than E0 , an explicit expression for the perturbed current density can be J = J0 + JD + Jω1 −ω2 + J2ω1 + J2ω2 + Jω1 +ω2 + · · · ! " 2 # J0 B B2 E1 2 E2 = J0 + 1+ + 2 + 2 E0 E E 2E0 0 0 ! 2 B B E1 E2 + J0 1 + + 2 E0 2E0 E0 E0
(1)
× cos[(ω1 − ω2 )t] + · · · .
where A = 1.4 × (A eV), B = 6.83 × 109 81.5 (V eV−3/2 m−1 ), 8 is the electron work function (8 = 4.5 eV for graphene) and E is the electric field at the emitter surface. If two sinusoidal (optical) fields E1 and E2 are superimposed to the electrostatic field E0 , the total electric field is E = E0 + E1 cos(ω1 t) + E2 cos(ω2 t), where ωi are the pumping oscillation frequencies. The electrostatic field at the emitter surface E0 = βemitter Vc /dvac depends on the emitter geometry, collector voltage Vc and vacuum gap distance dvac . V−2
(2)
At the apex of the graphene emitter, THz photomixing currents will be generated at the offset frequency ω1 − ω2 of two lasers. For small vacuum gaps, due to the large parasitic collector-to-emitter capacitance CAE , the THz wave is not effectively coupled to the collector (C) load resistor RA , due to the low-pass filtering effects associated with the shunt capacitance [30] (see figure 2(a)). On the other hand, a graphene emitter can support THz 2
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surface plasmon-polariton waves [16–18, 14] that may be efficiently coupled to the load resistor at the source (S) terminal RL . Moreover, if another graphene sheet is placed below the graphene emitter with proper separation using an insulating oxide, a parallel-plate waveguide transmission-line [23, 43] can be formed to guide the strongly confined, non-dispersive quasi-transverse electromagnetic (quasi-TEM) THz wave, interconnecting the generated signal to different on-chip ports and processors [23, 43]. In equation (2), JD is the rectified current density, and all terms with frequencies larger than the mixing frequency ω1 − ω2 (i.e. 2ω1 , 2ω2 , and ω1 + ω2 ) can be neglected, as they are expected to be significantly attenuated along the graphene sheet. As observed from (2), the THz photomixing current depends on the optical field intensity E1 and E2 at the emitter tip. Recently, optical nanoantennas have been proposed to enhance the intrinsically weak spontaneous emission, nonlinear optical effects [44–47], and photomixing for THz generation [48, 49]. When subwavelength nanoantennas are placed around the emitter tip in our geometry (figure 1), the optical field enhancement associated with their plasmonic features can largely enhance the photomixing efficiency, as evident in (2). The high optical opacity of the graphene monolayer may allow us to readily integrate the graphene emitter to the optical nanoantennas. Since graphene is largely transparent to visible light (∼97.7% transmission in the near-infrared and visible range (see appendix))2 , light can pass through the grapheme monolayer in figure 1 and excite the optical antennas enhancing the optical near-field intensity [11, 47]. Figure 3 shows the predicted average local field enhancement versus frequency for an array of silver nanoantennas with width wa = 25 nm, length la = 60 nm, height ha = 25 nm and separation distance s = 150 nm. These parameters have not been optimized, but simply tailored to provide a good field enhancement around the frequency range of interest. In figure 3, we have assumed that the light polarization is parallel to the gaps and the field distributions have been calculated using full-wave simulations [50] using an empirically fitted Drude-type dispersion for silver [51]. The average field enhancement is defined by calculating the line integral of the local electric field normalized to the incident field, weighted by the total length. The observation line is 15 nm from the array and shows a single, rather broad resonance, resulting in a broadband enhancement of the THz photomixing process, provided that the emitter tip is the close proximity of optical antennas. The model in figure 2(a) ensures that the THz waves created by the current oscillations at the apex of the graphene emitter may be coupled to the load resistor at the source terminal ZL , which may represent a graphene patch antenna [24–28] or arbitrary matched THz circuit load. Here we assume the perfect transition between graphene waveguide
Figure 3. Averaged optical near-field enhancement versus frequency. The observation line is parallel and 15 nm distant from the array of silver nanoantennas. The right inset shows the cross-sectional view of the electric field distribution along the antenna axis (yellow dashed line in the left inset).
and load, ignoring possible parasitics. Due to the significantly shortened wavelength in this graphene waveguide (compared to the wavelength in free space) [23, 43], the graphene channel should be modeled as a segment of transmission-line, and retardation effects within its length should be carefully taken into consideration. The input impedance at the emitter tip is given by Zin = Zg
RL + Zg tanh γ l , Zg + RL tanh γ l
(3)
where γ = α + jβ and Zg are the propagation constant and characteristic impedance of the graphene transmission-line, respectively. The output power generated at the emitter tip is given by Pout = 21 Iω2 1 −ω2 Re[Zin ],
(4)
where the photomixing current Iω1 −ω2 = gp Jω1 −ω2 wt, and gp is the current gain taking into account the resonant photon-assisted field-emission process [29–31]. For graphene with a work function of 4.5 eV [40–42], a gain of about 40 dB is obtained from rigorous quantum mechanical simulations [30]. Although an elongated graphene sheet results in a large geometric (electrostatic) field enhancement factor proportional to (h/t)0.75 [39], the non-negligible attenuation constant α fundamentally may limit the maximum available length for the graphene layer. Also, the emitter length is important to properly phase the THz signal such that the maximum power can be delivered to the load. This happens when the phase shift coincides with a Fabry–Perot resonance of the transmission-line segment (βl = mπ with m an integer). For a subwavelength waveguide (i.e., d 2π c/(ω1 − ω2 )), a quasi-static circuit model in figure 2(b) may be used to calculate the propagation constant, phase velocity and characteristic impedance of the transmission-line [4]. The relevant lumped elements include the kinetic inductance LK (which is caused by the electron-inertia effect typically observed in high carrier mobility conductors like graphene at
2 The graphene monolayer is highly transparent in the near-infrared and
visible regions, showing a frequency-independent 97.7% optical transparency that is consistent with graphene’s minimum interband conductivity σmin = q2 /4h¯ . 3
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very high frequencies), sheet resistance Rs (which is mainly caused by the carrier scattering by defects and impurities in graphene), magnetic inductance Lm , and electrostatic capacitance Ces per unit length: 2 Im[1/σs ] 2 Re[1/σs ] , Rs = , ωw w d Lm = µ0 , w w π Ces = εox ε0 + εox ε0 , d ln[6(d/w + 1)] LK =
(5)
where εox = 3.9 is the relative permittivity of silicon dioxide (SiO2 ) filling the waveguide, ε0 and µ0 are respectively the vacuum permittivity and permeability, σs is the complex surface conductivity derived from the Kubo formula [14]; see appendix for details. The first term in Ces represents the parallel-plate capacitance between the two graphene sheets and the second term is the fringing capacitance [52], negligible when w d. In the THz range, σs follows a Drude-type dispersion σs = −j(q2 EF /π h¯ 2 )/(ω − j), where is the carrier intraband scattering rate ( = 0.065 meV for high-quality graphene [13]) and the plasmon spectrum is tunable by Fermi energy EF . From the transmission-line theory, the propagation constant and characteristic impedance may then be calculated using p γ = α + jβ = jωCes [R + jω(LK + Lm )] s p R ∼ (6a) = jω LK Ces 1 + jωLK s s s R + jω(LK + Lm ) ∼ LK R Zg = 1+ . (6b) = jωCes Ces jωLK
Figure 4. Phase constant versus signal frequency for a graphene parallel-plate waveguide with different Fermi energy. The solid lines and dots are results obtained from our analytical transmission-line model (figure 2(b)) and the rigorous eigenmodal solution, respectively.
At THz frequencies and below the interband transition threshold, LK = Rs / ' π(h¯ /q)2 /EF is valid. Typically, LK is larger than Lm by several orders of magnitude, and value of LK is carrier-density-dependent, providing interesting possibilities to enhance the phase velocity and enable tunable and reconfigurable THz components [23]. Figure 4 shows the calculated phase constant β versus frequency for intrinsic (∼ EF = 250 meV) [53] and doped (EF = 500 meV) graphene parallel-plate waveguides with d = 50 nm and w = 30 µm; here the solid and dotted lines are respectively calculated using the transmission-line model introduced here and a rigorous eigenmodal solution in [23, 43]. It is evident in figure 3 that our model can accurately predict the THz wave propagation properties. It is also important to highlight that at THz frequencies ω R/LK = , the group (signal) velocity is essentially independent of frequency (vg = ∂β/∂ω = constant), typically desirable for signal communications. For an ideally lossless transmission-line, the output power delivered to the load Pout = Iω2 1 −ω2 Re[Zin ]/2. However, due to intrinsic losses in graphene, we expect that the real power delivered to the load PL = ηPout , where the efficiency η = (1 − |0|2 )/[(1 − |0e−γ l |2 )e2αl ] and the signal reflection caused by the impedance mismatch 0 = (ZL − Zg )/(ZL − Zg ). The output power is proportional to the load impedance when ZL is small, but it varies inversely with it
Figure 5. Contours of output power, varying the flux intensity of two lasers (P1 = P2 = P0 ) and the electrostatic field strength at the emitter surface (E0 ).
when ZL becomes large. Therefore, there is an optimal value of ZL satisfying the condition ∂PL (ZL )/∂ZL = 0. Figure 5 shows the contours of Pout calculated using equations (3)–(5) at the mixing frequency (ω1 − ω2 )/2π = 1 THz (here ω1 /2π = 393 THz and ω2 /2π = 392 THz), varying the electrostatic field strength E0 and the laser flux intensities P1 = P2 = P0 . Here the width w and length l of the graphene sheets are 30 µm and 2.3 µm respectively, and the optimized load resistance RL = 390 . The nanoantennas are placed 15 nm behind the emitter tip and the optical field enhancement may be obtained from figure 3. Figure 5 shows that the total generated power Pout increases with the static field and laser flux intensity. In order to protect the nanoemitter from vacuum arcing, it is necessary to limit the dc current density J0 to 109 A m−2 in steady state or 1012 A m−2 when operating with microsecond pulses [29–31]. In figure 4(a), Pout = 1 µW and J0 = 1.37 × 109 A m−2 are obtained at E0 = 7 V nm−1 and P0 = 1 MW cm−2 . Figures 6(a) and (b) show the contours of total generated THz power Pout and real power delivered to the load PL for the 4
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Figure 6. Contours of: (a) output power at the emitter tip, (b) real power delivered to the load, varying the length and width of graphene sheets; the inset of (b) shows the contours of power delivery efficiency.
same photomixer design as in figure 5, with E0 = 7.5 V nm−1 and P0 = 1 MW cm−2 , varying the graphene width w and length l. The maximum output power sensibly depends on the graphene length, which is associated with the total phase shift experienced by the THz wave traveling along the transmission-line. Inspecting the figure, we note that, although a long emitter can enhance the geometric field enhancement factor βemitter and therefore reduce the required dc bias voltage Vc , after considering the losses in graphene, the first Fabry–Perot resonance mode is the preferred operation to achieve maximized PL and therefore optimum grapheme length loptimum = 2π/β = λg /2. The inset of figure 6(b) shows the contours of power efficiency η, showing that the efficiency rapidly deteriorates with increasing width and length, due to increased losses. Comparing figures 6(a) and (b), it is clear that the real power delivered to the load is attenuated because of propagation losses (1−η)Pout , especially for long graphene sheets. Figure 7 shows the real power PL delivered to the load versus the synthesized THz frequency for the same photomixer as in figure 5, with E0 = 7.5 V nm−1 and P0 = 1 MW cm−2 . For comparison, we also show the available power without the presence of the nanoantennas. In this plot we consider ω1 /2π = 393 THz, while varying ω2 /2π from 392 to 382 THz, in order to generate THz signals over a wide frequency range. For each mixing frequency, the optimal values of l and ZL are obtained with the physics-driven transmission-line and circuit models presented in figure 2. It is evident that the concept of designing nanoantennas coupled to an optimal graphene transmission-line emitter can greatly increase the output power over the whole dc 10 THz frequency range. For instance, at the mixing frequency (ω1 − ω2 )/2π = 1 THz, the maximum delivered power increases from 6.6 nW to 1.44 µW, thanks to the effect of the plasmonic nanoantennas. Typically, field-emission arrays can consist of 104 emitters per centimeter square [33–38] (i.e., flat-panel displays), so arrays of graphene-based, vacuum-nanoelectronic photomixers as the ones presented here can potentially realize mW power level.
Figure 7. Real power delivered to the load versus the synthesized THz frequency for the optimized graphene-based photomixer with and without nanoantennas.
In realistic designs, the physical length of the graphene emitter and of the waveguide cannot be changed. However, a tunable phase shift in the graphene channel is possible with a double-gate structure, as recently proposed in [23], which tunes the phase to β(EF )l = mπ . Also, a wideband-tunable impedance ZL can be realized by the gate-tunable graphene antenna, waveguide, and transformer [18, 23]. Figure 8 presents the output power for a nanoantenna-enhanced graphene photomixer with emitter length l = 2.3 µm and load impedance ZL = 420 , varying the Fermi energy. It is seen that the synthesized THz wave exhibits multiple peaks, significantly shiftable with Fermi energy. This feature is of great interest for applications in narrow-spectral-linewidth, tunable THz sources. To conclude, we have proposed here the concept of a graphene-based, vacuum-nanoelectronic photomixer with large tuning range covering the whole terahertz band (0.3–10 THz). The graphene layer has a triple functionality in our design. First, its high optical transparency may allow exciting plasmonic nanoantennas around its tip in order to boost the intensity of pumped visible light. Second, its extreme thinness is ideal to enhance the generation 5
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over σinter , and the sheet conductivity of graphene can be explicitly expressed as [14]: σintra = −j
q2 KB T
π h¯ 2 (ω − j) EF EF × + 2 ln 1 + exp − . KB T KB T
(A.2)
For doped graphene with |EF | kB T (A.2), further reduces to the Drude-like dispersion: σintra ≈ −j
q2 EF π h¯ (ω − j) 2
.
(A.3)
In general, σinter must be evaluated numerically, however, for hω, ¯ |EF | kB T, it can be approximated as [14]: q2 2|EF | − h¯ (ω − j) σinter = −j ln . (A.4) 4π h¯ 2|EF | + h¯ (ω − j)
Figure 8. Power output versus synthesized THz frequency for a nanoantenna-enhanced, graphene-based photomixer, with emitter length l = 2.3 µm, load impedance RL = 390 , and Fermi energy swept from 250 to 400 meV. The gray line is the maximum achievable power of this photomixer.
For frequencies below the THz regime, the interband conductivity is almost negligible when compared with the intraband contributions. In the near-infrared and visible spectrum, the interband contribution may dominate over the intraband contribution, and σinter = q2 /4h¯ is a real constant. This is in consistent with graphene’s high optical opacity.
of THz signals. Finally, the graphene emitter can directly couple the THz wave to the loading circuit or emitting antenna, enabling all-graphene nano-transmitters and nanosensors [23–28]. This concept may have a great potential to realize mW terahertz sources by graphene field-emission arrays, offering much greater bandwidth and increased output power compared to currently available solutions.
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Acknowledgments The authors would like to thank Dr Deji Akinwande for fruitful discussions. This work was supported by the NSF CAREER award No. ECCS-0953311, the AFOSR YIP award No. FA9550-11-1-0009 and the ARO grant No. W911NF-11-1-0447.
Appendix Graphene’s sheet conductivity includes both semi-classical intraband conductivity σintra and quantum-dynamical interband conductivity σinter [14]: σs (ω, EF , , T) = σintra + σinter Z +∞ jq2 (ω − j) |ε| =− 2 (ω − j)2 π h¯ −∞ ∂fd (ε) dE × ∂ε Z +∞ ∂fd (−ε) − ∂fd (ε) − dE (A.1) (ω − j)2 − 4(ε/h) ¯ 2 0 where fd = 1/(1 + exp[(ε − EF )/(kB T)]) is the Fermi–Dirac distribution, ε is the energy, EF is the Fermi energy, T is the temperature, q is the electron charge, h¯ is the reduced Planck’s constant, and is the carrier scattering rate. The first and second terms in (A.1) account for the intraband and interband contributions, respectively. In the THz region and below the interband transition threshold (hω ¯ < 2|EF |), σintra dominates 6
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