A semiempirical model for two-level system noise in superconductingmicroresonatorsJiansong Gao,1,a兲Miguel Daal,2John M. Martinis,3Anastasios Vayonakis,1Jonas Zmuidzinas,1Bernard Sadoulet,2Benjamin A. Mazin,4Peter K. Day,4andHenry G. Leduc41Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California91125, USA2Physics Department, University of California at Berkeley, Berkeley, California 94720, USA3Department of Physics, University of California, Santa Barbara, California 93106, USA4Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA共Received 2 April 2008; accepted 9 May 2008; published online 29 May 2008兲We present measurements of the low-temperature excess frequency noise of four niobiumsuperconducting coplanar waveguide microresonators, with center strip widths srranging from3to20␮m. For a fixed internal power, we find that the frequency noise decreases rapidly withincreasing center strip width, scaling as 1 / sr1.6. We show that this geometrical scaling is readilyexplained by a simple semiempirical model which assumes a surface distribution of independenttwo-level system fluctuators. These results allow the resonator geometry to be optimized forminimum noise. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2937855兴Thin-film superconducting microresonators are of greatinterest for a number of applications 共see Refs. 1–4 and ref-erences therein兲. Excess frequency noise is universally ob-served in these resonators2,5,6and is very likely caused bytwo-level systems 共TLSs兲 in dielectric materials.3,7Indeed,the TLS hypothesis is supported by the observed dependenceof the noise on resonator internal power7,8and temperature.3In a recent paper4共paper A hereafter兲, we presented measure-ments of the TLS-induced low-temperature frequency shiftsof five niobium 共Tc=9.2 K 兲 coplanar waveguide 共CPW兲resonators with varying center strip widths sr. From the ob-served geometrical scaling of the frequency shifts 共⬃1/ sr兲,we showed that the TLS must be located in a thin 共few nan-ometer兲 layer on the surface of the CPW. In this letter, wepropose a semiempirical TLS noise model that assumes thissurface distribution, and we show that the model explainsour measurements of the geometrical scaling of the noise.The device used for the experiment in this paper is ex-actly the same device used in paper A. In brief, the chipcontains five CPW quarter-wavelength resonators 共Z0⬇50 ⍀ , fr⬇6 GHz兲 made by patterning a 120 nm thick Nbfilm deposited on a c-plane crystalline sapphire substrate.Each resonator is capacitively coupled to a common feedline,using a CPW coupler 共coupling quality factor Qc⬃50 000兲of length lc⬵200␮m and with a common center-strip widthof sc=3␮m. The coupler is then widened into the resonatorbody, with a center-strip width of sr=3, 5, 10, 20, or 50␮m,and a length of lr⬃5 mm. The noise was measured using astandard IQ homodyne technique;2,3both the measurementsetup and the analysis of the noise data are identical to ourprevious work.7The device is cooled in a dilution refrigerator to a basetemperature of 55 mK. The fractional frequency noise spec-tra S␦f共␯兲/ fr2of the five resonators were measured for micro-wave readout power P␮win the range −61 to − 73 dBm; the−65 dBm spectra are shown in Fig. 1共a兲. We clearly see thatthe noise has a common spectral shape but decreases as thecenter strip becomes wider. Unfortunately, the data for thelowest-noise 共50␮m兲 resonator are influenced by the noisefloor of our cryogenic microwave amplifier, so we excludethis resonator from further discussion. The noise levels at␯=2 kHz were retrieved from the noise spectra and areplotted as a function of resonator internal power Pint=2Qr2P␮w/␲Qcin Fig. 1共b兲. All resonators display a powerdependence close to S␦f/ fr2⬀ Pint−1/2as we have previouslyobserved.3,7,8In order to study the geometrical scaling of thenoise in more detail, we first fit the noise versus power datafor each resonator to a simple power law, and retrieve thevalues of the noise S␦f共2 kHz兲/ fr2at Pint=−25 dBm for eachgeometry. These results 共Fig. 2兲 again show that the noisedecreases with increasing sr, although not 共yet兲 as a simplepower law.To make further progress, we introduce a semiempiricalmodel for the TLS noise. We assume that the TLS have auniform spatial distribution within a volume of TLS-hostinga兲Electronic mail: [email protected]. 1. 共Color online兲 Fractional frequency noise spectra of the four CPWresonators measured at T=55 mK. 共a兲 Noise spectra at P␮w=−65 dBm.From top to bottom, the four curves correspond to CPW center strip widthsof sr=3, 5, 10, and 20␮m. The various spikes seen in the spectra are due topickup of stray signals by the electronics and cabling. 共b兲 Fractional fre-quency noise at␯=2 kHz as a function of Pint. The markers represent dif-ferent resonator geometries, as indicated by the values of srin the legend.The dashed lines indicate power law fits to the data of each geometry.APPLIED PHYSICS LETTERS 92, 212504 共2008兲0003-6951/2008/92共21兲/212504/3/$23.00 © 2008 American Institute of Physics92, 212504-1Downloaded 11 Jul 2008 to 128.111.8.176. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspmaterial Vhthat occupies some portion of the total resonatorvolume V. Consider a TLS labeled␣, located at a randomposition rជ␣苸 Vhand with an energy level separation E␣=共⌬␣2+⌬0,␣2兲1/2. Here ⌬␣and ⌬0,␣are the TLS asymmetryenergy and tunnel splitting, which are random and have ajoint distribution function f共⌬,⌬0兲= P/ ⌬0, where P is thetwo-level density of states introduced by Phillips.9The TLStransition dipole moment is given by dជ␣=nˆ␣d0⌬0,␣/ E␣,where d0is the maximum dipole moment for a TLS withenergy E␣and the dipole orientation unit vector nˆ␣is as-sumed to be random and isotropically distributed. In theweak-field, linear response limit, the TLS contribution to thedielectric tensor of the hosting medium is⌬⑀kl共␻,rជ兲 =−兺␣d␣,kd␣,l␦共rជ− rជ␣兲␹␣共␻兲␴z,␣, 共1兲where k , l represent Cartesian components,␹␣共␻兲=1/ 共E␣−ប␻+ j⌫␣兲+1/ 共E␣+ប␻− j⌫␣兲 is a damped single-pole re-sponse function for