Terahertz detection using mechanical resonators based on 2D materials
Juha Hassel, Mika Oksanen, Teemu Elo, Heikki Seppä, Pertti J. Hakonen
TTerahertz detection using mechanical resonators based on 2Dmaterials
Juha Hassel , Mika Oksanen , Teemu Elo , Heikki Sepp¨a and Pertti J. Hakonen Low Temperature Laboratory, Department of Applied Physics,Aalto University School of Science, FI-00076, Finland and VTT Technical Research Centre of Finland,PO BOX 1000, FI-02044 VTT, Finland (Dated: July 16, 2018)
Abstract
We have investigated a THz detection scheme based on mixing of electrical signals in a voltage-dependent capacitance made out of suspended graphene. We have analyzed both coherent andincoherent detection regimes and compared their performance with the state of the art. Usinga high-amplitude local oscillator, we anticipate potential for quantum limited detection in thecoherent mode. The sensitivity stems from the extraordinary mechanical and electrical propertiesof atomically thin graphene or graphene-related 2D materials. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b . INTRODUCTION Interest in THz detection and imaging technologies is traditionally motivated by astron-omy and more recently also by a growing demand for new solutions for enhancing publicsecurity. Passive THz imaging using cryogenic sensor arrays has been successful in fulfillingthis demand. At present, incoherent detectors based on transition edge sensors and coher-ent detectors typically based on SIS mixers have applications in astronomical imaging [1, 2].Security applications using a few different approaches have been demonstrated as well [3–5].New solutions with enhanced sensitivity, increased operating temperature, or an increasedlevel of integration are being developed constantly [6]. Additional possibilities to such questare provided by THz detectors based on micro- (MEMS) or nanoelectromechanical (NEMS)systems, especially those employing novel 2D materials [7] such as graphene. Due to theextraordinary mechanical properties, atomically thin NEMS might yield a significant sensi-tivity improvement in the operation of mechanical radiation detection devices.THz detection using graphene has aroused considerable interest [8–13]. Previous workhas taken advantage of graphene’s linear band structure and the low heat capacity of single-layer graphene. THz detection has been done via a plasmonic mechanism [11], by bolometricdetection [10], and by noise thermometry [13]. A recent experiment [12] with graphene FETwith dissimilar contact metals reached noise equivalent power (NEP) around 20 pW/Hz / operating at room temperature. Svintsov et. al. [14] have proposed a scheme with suspendedgraphene FET, where they take advantage of the plasma resonance that naturally occursat THz frequencies for short graphene devices. The results so far have remained inferiorto the current state-of-the-art bolometers based on superconducting detectors, which reachnoise equivalent powers (NEP) around 10 fW/Hz / in the 0.2 – 1.0 THz band at T = 4 .
2K [15] and below 1 aW/Hz / at 20 mK [16]. For coherent detectors the relevant figureof merit is the receiver noise temperature T n . For SIS mixer receiver noise temperatures afew times above the quantum limit hf / k B (with h and k B denoting the Planck constantand the Boltzmann constant, respectively) have been reported [2]. Ultra-low-noise. coherentreceiver operation of graphene at THz is largely unexplored, albeit graphene based integratedsubharmonic mixer circuits have been demonstrated at 200-300 GHz frequencies [17, 18].Here we propose an original scheme of detecting THz radiation using antenna-coupledmechanical resonators based on atomically thin two-dimensional materials. As the perfor-2 ant C C1 C L RC G Z ant λ/2 = l + l G C G Antenna Coupling Detector Z in FIG. 1. (color on line) a) Schematic view of the analyzed MEMS THz detection system. Anatomically thin, suspended membrane, e.g. graphene (brown color), forms a capacitor located atthe end of an electrical cavity resonator line (yellow color), approximately 50 micron long. Antennais coupled to the MEMS detector via a coupling capacitor formed by the gap in the transmissionline. b) Schematic circuit of the radiation detector. The length of the detector unit correspondsto a half-wavelength resonator: λ/ (cid:96) + (cid:96) G with (cid:96) G denoting the length of the graphene sensor.c) Equivalent electrical circuit for sensitivity analysis. The equivalent capacitance of the strip line(or coplanar line) electrical resonator is denoted by C while the sensor and coupling capacitancesare given by C G and C c , respectively. R denotes electrical losses, mostly caused by the resistanceof the two-dimensional material. mance of this scheme relies heavily on the properties of the mechanical detector element, wehave chosen to employ graphene in our device. Graphene shows great promise for superiorsensitivity owing to its high Young’s modulus E ∼ I. POWER RESOLUTION
We analyze a system where a mechanical resonator is coupled to a THz regime antenna.The signal picked up by the antenna drives the mechanical resonator, and the resultingmechanical vibrations are detected. In our analysis, we don’t consider the detection ofmechanical motion - we just assume that it can be done down to the oscillation amplitudelimit set by thermal excitation. Demonstration of detection of such thermally driven motiondown to 50 mK has been presented, for example, in Ref. [19] in a bilayer graphene NEMSresonator using cavity optomechanics.Dipole or log spiral antennas can be designed with real impedance of several tens of ohms,the value of which can be matched to a transmission line resonator with a capacitance C c by requiring Z in = Z ant (Fig. 1). In the transmission line case, the input impedance is givenby Z in,d = π Q e Z ( ωC c ) , (1)where Q e and Z are the quality factor of the resonator and the characteristic impedance,respectively, and ω is the angular frequency of the electromagnetic radiation. In Fig. 1a thegeometry is such that the mechanical resonator covers only a fraction of the cavity lengthat the end of the microwave (THz) resonator. However, it is also possible in principle thatthe well-conducting mechanical resonator material forms the electrical cavity by itself. Sucha geometry would be optimal for detection sensitivity (see below).The antenna can also be matched to a mechanical resonator whose dimensions are smallerthan the wavelength, in which case the matching is done by a separate, lumped element LCstructure with equivalent performance. In the lumped element case, the matching is doneby setting Z in,l = Q e √ L/C ( ωC c ) equal to the antenna impedance, i.e. (cid:112) L/C acts as thecharacteristic impedance Z of a corresponding distributed resonator.The voltage amplitude of the standing wave induced by a signal can be expressed as V = Q e (cid:114) LC P sig , (2)where the capacitance C = C G + C includes the sensor capacitance C G and the equivalentcapacitance C of the strip line (or coplanar line) of the electrical resonator. The mechanicalresonator then feels an eletrostatic force according to F = C G V d , where d is the vacuum gapbetween the graphene element and its counterelectrode (ground). Using the two equations4bove, the force responsivity ∂ F/ ∂ P sig becomes ∂F∂P sig = Q e √ LC d (cid:114) C G C + C G . (3)In the ideal case, when electrical cavity is formed by the mechanical resonator, the capacitiveshunting factor (cid:113) C G C + C G reducing the sensitivity becomes equal to one, while for the settingin Fig. 1a it equals to ∼ . C G (cid:39) . C ).The power resolution of the detector is ultimately limited by mechanical thermal noise.The mechanical noise can be expressed using the fluctuation-dissipation theorem as S / F = (cid:113) π k B T γ m where T is the temperature of the detector element (phonon temperature), and γ m denotes the damping rate of the mechanical resonator; at low temperatures electronictemperature may deviate substantially from phonon temperature due to weak electron-phonon coupling [20]. By replacing the decay rate γ m / π = ω m mQ m , the force fluctuations canbe written as S / F = (cid:113) k B T ω m mQ m , where ω m is the mechanical angular frequency, m denotesthe resonator mass [21], and Q m is the mechanical quality factor. Noise equivalent power isthen given by N EP = S / F ∂F/∂P sig , which leads to N EP = 2 dωQ e (cid:115) k B T ω m mQ m C + C G C G . (4)The above equation works generally for any antenna matched MEMS system, regardlessof the design details. Thus, it can be used to estimate performance of the system, if thecharacteristics of the electrical and mechanical resonators are known. The only requirementis that the resonator motion can be measured at the thermal motion level. Although the de-tector can work at room temperature, lowering the temperature directly improves sensitivityby reducing thermal noise as shown in Eq. (4), and typically the Q m values of mechanicalgraphene resonators are significantly larger at cryogenic temperatures.A substantial amount of experiments have been performed on graphene mechanical res-onators [19, 22–26] and hence we can reliably estimate the expected performance. For arealistic case study, we adopt the experimental parameters for a bilayer graphene resonatorfrom Ref. 19, i.e. ω m /2 π = 24 MHz, vacuum gap d = 70 nm, and Q m = 15 000, and m = 10 − kg. Operated at T = 25 mK, the NEP goes down to ∼ / , providedthat the matching circuit can reach Q e C G C + C G = 100 at 500 GHz operating frequency. Ifthe temperature is increased to 3 K, which can be reached with a pulsed cryocooler, the5EP increases to ∼
14 fW/Hz / . Here we assume that the practical Q m for a graphenemechanical resonance at 3 K is not altered significantly from 15 000. The bandwidth of thedevice is only around ∆ ν = 1 /Q e i.e. a few gigahertz, which should be contrasted to thebroadband detection ∆ ν = 500 GHz offered by bolometric detectors. Hence, no improve-ment compared with present techniques is achieved by our mechanical detection scheme.This result is quite expected as the high impedance of the small sensor element allows onlya narrow-band detection to be utilized. III. COHERENT DETECTION
The device analyzed above can also be used as a mixer, in which case its operation canbe brought down to the ultimate sensitivity limit governed by the Heisenberg uncertaintyprinciple. For a couple of decades, mixers based on superconductor-insulator-superconductor(SIS) junctions or superconducting hot electron bolometers (SHEB) have formed the mainstate-of-the-art at frequencies up to few THz. Reliable quantum-limited mixers can beachieved at frequencies below 680 GHz [27], a limit which is set by the superconducting gap ofNb. At frequencies f > /h , where 2∆ is the Cooper pair breaking energy, significant lossesare introduced in the superconductor. While this does not prohibit heterodyne mixing atlarger frequencies, the noise temperature is then typically limited to few times the quantumlimit T q [28–31].A local oscillator (LO) signal V LO applied over the MEMS structure, is summed withthe measured signal V , so that the RMS force acting on the resonator at frequency ω − ω LO is given by F ( ω − ω LO ) = C G √ d V V LO . (5)Voltage noise can be written as S / V = 1 ∂F/∂V S / F = √ dC G V LO (cid:115) k b T ω m mQ m (cid:18) C + C G C G (cid:19) . (6)By using Eq. (2), the noise power can be referred to the signal power by S P = S V / (cid:2) Q e ( L/ ( C + C G ) / (cid:3) .By defining the noise temperature as T n = S P /k b , T n = 2 d ωk b CV LO Q e (cid:18) k b T ω m mQ m (cid:19) . (7)6e further replace V LO with P LO in Eq. (2), and rewrite T n = 2 d ω P LO Q e (cid:18) T ω m mQ m (cid:19) (cid:18) C + C G C G (cid:19) . (8)We emphasize that the quoted noise temperature contains only the noise fluctuations stem-ming from the mechanical dissipation. There is another noise contribution due to electricalfluctuations which are limited to zero point fluctuations in the signal band since (cid:126) ω (cid:29) k B T .The noise power per unit band for quantum fluctuations in a cavity mode at frequency ω/ π corresponds to an energy of half of a photon: (cid:126) ω . Under impedance matching con-ditions for signal frequency, this corresponds to the single-side-band noise temperature of T n = (cid:126) ω/ (2 k B ), i.e. the standard quantum limit. In practice, however, T n = (cid:126) ω/ ( k B )since separation of the image frequency is problematic in the THz regime, which leads toadditional quantum noise of (cid:126) ω .In order to reach the quantum limit in detection sensitivity, the contribution of the me-chanical fluctuations needs to be brought below that of the quantum noise. Eq. (8) suggeststhat this is accomplished by increasing P LO sufficiently. A fundamental limitation for P LO arises from the linearity requirement for the mechanical resonator motion. Furthermore,practical limitations include that the LO power dissipated in a dilution refrigerator at 20mK has to be limited to about 20 µ W. Using Eq. (8), and the experimental graphene res-onator and matching circuit parameters quoted in Section II, we find that the LO powerof about 100 nW, well in line with typical cryostat operation, is sufficient for reaching thequantum limit of T n ≈
12 K at ω/ π = 500 GHz. We have also checked that the excitationat f m via mixing (cf. drive e.g. in Ref. 24) and at f LO ( (cid:29) f m ) remain well in the linearregime of graphene mechanical motion. It is worth to note that, according to Eq. (8), thenoise temperature scales down with the resonator mass provided that other parameters canbe kept unchanged. Hence, narrow ribbons are expected to yield the optimum, but the op-timal width will depend strongly on other constraints like the capacitive participation ratio C G C + C G .There are two critical issues concerning practical applications: 1) Dissipation in electricalcavity at THz frequency, i.e. the value of Q e , and 2) Participation ratio of the graphenesensor capacitance. High Q e -values have been reported to whispering gallery modes at THzfrequencies [32]. If large-gap superconducting materials can be employed for on lithographicchip circuits, there are no principal obstacles in obtaining Q e ∼ up to 1 THz or 2.57Hz using NbTiN or MgB , respectively [33]. Such a Q e -factor combined with a grapheneparticipation ratio a few per cent would bring the thermal noise contribution well below thequantum noise. The performance would improve even further if R (cid:3) <
10 Ω can be employedas targeted for graphene touch screen displays [34].Our analysis has not discussed practical problems in the detection of the vibration of thegraphene membrane. As the participation ratio of the graphene capacitance becomes larger,heating by Joule dissipation becomes stronger. However, the electron-phonon coupling atmK temperatures is weak and the environmental temperature of the first fundamental modegrows up moderately with heating of the electrons, provided the Kapitza resistance betweengraphene and its support structure is optimized [19]. Hence, operation at the necessary highdrive powers is feasible in our detector configuration without losing sensitivity to imposedradiation.
ACKNOWLEDGEMENTS
This work was supported by the Academy of Finland (grant no. 286098 and 284594, LTQCoE), and FP7 FET OPEN project iQUOEMS. This research made use of the OtaNanoLow Temperature Laboratory infrastructure of Aalto University, that is part of the EuropeanMicrokelvin Platform. [1] W. S. Holland, D. Bintley, E. L. Chapin, A. Chrysostomou, G. R. Davis, J. T. Dempsey,W. D. Duncan, M. Fich, P. Friberg, M. Halpern, K. D. Irwin, T. Jenness, B. D. Kelly, M. J.MacIntosh, E. I. Robson, D. Scott, P. A. R. Ade, E. Atad-Ettedgui, D. S. Berry, S. C. Craig,X. Gao, A. G. Gibb, G. C. Hilton, M. I. Hollister, J. B. Kycia, D. W. Lunney, H. McGregor,D. Montgomery, W. Parkes, R. P. J. Tilanus, J. N. Ullom, C. A. Walther, A. J. Walton,A. L. Woodcraft, M. Amiri, D. Atkinson, B. Burger, T. Chuter, I. M. Coulson, W. B. Doriese,C. Dunare, F. Economou, M. D. Niemack, H. A. L. Parsons, C. D. Reintsema, B. Sibthorpe,I. Smail, R. Sudiwala, and H. S. Thomas, Mon. Not. R. Astron. Soc. , 2513 (2013).[2] A. M. Baryshev, R. Hesper, F. P. Mena, T. M. Klapwijk, T. A. van Kempen, M. R. Hoger-heijde, B. D. Jackson, J. Adema, G. J. Gerlofsma, M. E. Bekema, J. Barkhof, L. H. R. e Haan-Stijkel, M. van den Bemt, A. Koops, K. Keizer, C. Pieters, J. Koops van het Jagt,H. H. A. Schaeffer, T. Zijlstra, M. Kroug, C. F. J. Lodewijk, K. Wielinga, W. Boland, E. F.van Dishoeck, H. Jager, and W. Wild, Astron. Astrophys. , 1 (2015).[3] E. Grossman, C. Dietlein, J. Ala-Laurinaho, M. Leivo, L. Gronberg, M. Gronholm, P. Lap-palainen, A. Rautiainen, A. Tamminen, and A. Luukanen, Appl. Opt. , E106 (2010).[4] E. Heinz, T. May, D. Born, G. Zieger, G. Thorwirth, S. Anders, V. Zakosarenko, T. Krause,A. Kr¨uger, M. Schulz, H.-G. Meyer, M. Schubert, and M. Starkloff, Opt. Eng. , 113204(2011).[5] A. Luukanen, M. Gr¨onholm, M. M. Leivo, H. Toivanen, A. Rautiainen, and J. Varis, Proc.SPIE , 836209 (2012).[6] A. Timofeev, J. Luomahaara, L. Gr¨onberg, A. M¨ayr¨a, H. Sipola, M. Aikio, M. Metso,V. Vesterinen, K. Tappura, J. Ala-Laurinaho, A. Luukanen, and J. Hassel, IEEE Trans.Terahertz Sci. Technol. PP , 1 (2017).[7] A. Ferrari, F. Bonaccorso, V. Fal’ko, K. Novoselov, S. Roche, P. Bøggild, S. Borini, F. Koppens,V. Palermo, N. Pugno, J. Garrido, R. Sordan, A. Bianco, L. Ballerini, M. Prato, E. Lidorikis,J. Kivioja, C. Marinelli, T. Ryh¨anen, A. Morpurgo, J. Coleman, V. Nicolosi, L. Colombo,A. Fert, M. Garcia-Hernandez, A. Bachtold, G. Schneider, F. Guinea, C. Dekker, M. Barbone,Z. Sun, C. Galiotis, A. Grigorenko, G. Konstantatos, A. Kis, M. Katsnelson, L. Vandersypen,A. Loiseau, V. Morandi, D. Neumaier, E. Treossi, V. Pellegrini, M. Polini, A. Tredicucci,G. Williams, B. Hee Hong, J.-H. Ahn, J. Min Kim, H. Zirath, B. Van Wees, H. Van DerZant, L. Occhipinti, A. Di Matteo, I. Kinloch, T. Seyller, E. Quesnel, X. Feng, K. Teo,N. Rupesinghe, P. Hakonen, S. Neil, Q. Tannock, T. L¨ofwander, and J. Kinaret, Nanoscale , 4598 (2015).[8] L. Vicarelli, M. S. Vitiello, D. Coquillat, A. Lombardo, A. C. Ferrari, W. Knap, M. Polini,V. Pellegrini, and A. Tredicucci, Nat. Mater. , 865 (2012).[9] B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu,and H. G. Xing, Nat. Commun. , 780 (2012).[10] M. Mittendorff, S. Winnerl, J. Kamann, J. Eroms, D. Weiss, H. Schneider, and M. Helm,Appl. Phys. Lett. , 21113 (2013).[11] A. V. Muraviev, S. L. Rumyantsev, G. Liu, A. A. Balandin, W. Knap, and M. S. Shur, Appl.Phys. Lett. , 181114 (2013).
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