Kinetic planar resonators from strongly disordered ultra-thin MoC superconducting films investigated by transmission line spectroscopy
KKinetic planar resonators from strongly disordered ultra-thin MoCsuperconducting films investigated by transmission line spectroscopy
M. Baránek, P. Neilinger, a) D. Manca, and M. Grajcar b) Department of Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia (Dated: 1 March 2021)
The non-contact broadband transmission line flip-chip spectroscopy technique is utilized to probe resonances of mm-sized square kinetic planar resonators made from strongly disordered molybdenum carbide films, in the GHz frequencyrange. The temperature dependence of the resonances was analyzed by the complex conductivity of disordered super-conductor, as proposed in Ref. 1, which involves the Dynes superconducting density of states. The obtained Dynesbroadening parameters relate reasonably to the ones estimated from scanning tunneling spectroscopy measurements.The eigenmodes of the kinetic planar 2D resonator were visualized by EM model in Sonnet software. The properunderstanding of the nature of these resonances can help to eliminate them, or utilize them e.g. as filters.Thin disordered superconducting films find a variety ofapplications in advanced technologies, such as supercon-ducting single-photon detectors, parametric amplifiers, superinductors, and superconducting quantum bits. Thesedevices make use of the distinctively high sheet resistanceR s of these films and thus the high kinetic sheet induc-tance, which is approximately related to R s and the su-perconducting energy gap ∆ by L k = ¯ hR s / π ∆ at temper-atures well below the superconducting critical temperatureT c . These devices typically operate in the GHz frequencyrange, and the knowledge of their complex conductivity isof crucial importance for the device design and properties.Moreover, the electrodynamic response of these films pro-vides important insight into the fundamental topics of su-perconductivity, such as the superconductor-insulator quan-tum phase transition and the Berezinskii-Kosterlitz-Thoulesstransition. To probe the complex conductivity of thin films,either as surface impedance or through penetration depth mea-surements, several microwave spectroscopy techniques are ex-ploited. These are either narrowband techniques, such ascavity resonators, the parallel plate resonator technique(which is commonly employed to study high-T c superconduc-tive films), planar transmission line resonators, or broad-band spectroscopy techniques like the transmission line orCorbino spectroscopy. The complex conductivity of su-perconductors is often studied in the vicinity of the super-conducting transition.
In several cases, resonances in thespectra were reported for metallic and superconducting filmson dielectric substrates.
In general, the presence ofthese resonances is undesirable as they can strongly limit theupper frequency range of the spectroscopy. The origin ofthese resonances in thin metallic films on a dielectric sam-ple was unveiled by simulations and experiments as substrateand cavity resonances and analyzed in detail by neglectingthe metallic films.In this letter, we report on resonances present in thetransmission spectra of disordered superconducting films of a) Also at Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta,Bratislava, Slovakia.; Electronic mail: [email protected] b) Also at Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta,Bratislava, Slovakia. molybdenum carbide (MoC) measured by broadband trans-mission line spectroscopy in temperature range from T c downto ≈
500 mK and frequency range from 1 GHz to 16 GHz.The measurements were carried out in flip-chip geometry. The5 nm thin films with different R s were sputtered on sapphiresubstrate and placed on top of the transmission line facingthe center conductor. The present resonances are explainedby referring them to eigenmodes of a kinetic planar 2D res-onator. Low resonance frequencies of the resonator are theresult of the high kinetic inductance of the strongly disor-dered MoC films. Moreover, in contrast to the aforemen-tioned works, we utilize these resonances to determine theDynes broadening parameter Γ of these films, which is usu-ally determined by scanning tunneling spectroscopy (STS) attemperatures well below T c . The superconducting density ofstates (SDOS) of disordered superconductors is characterizedby the smearing of coherence peaks and the presence of in-gapstates. Their STS spectra are generally analyzed in terms ofthe Dynes SDOS N ( E ) = Re { ( E + i Γ ) / [( E + i Γ ) − ∆ ] / } .A thorough study of the STS spectra of highly disorderedMoC films is presented in Ref. 27. As a result of disorder,the temperature and frequency dependent complex conduc-tivity σ = σ − i σ deviates from the well known Mattis-Bardeen conductivity, in contrast to clean BCS super-conductors. These deviations were directly related to thecomplex conductivity of disordered superconductors utiliz-ing Nam’s theory. This approach was furthered to the opticalconductivity of disordered superconductors. A series of four MoC films was sputtered by means of re-active magnetron deposition from a molybdenum target, inargon-acetylene atmosphere on top of 5 × s of the MoC film was subsequentlyincreased by measuring the carbon content, controlled via theargon-acetylene flow in the chamber during the deposition. The increased disorder results in increased R s and the sup-pression of T c . Eventually, at critical R s , these films undergoa superconductor-insulator transition. The properties of thesamples are listed in the Table I. R s is measured by Van derPauw method at room temperature. As the transition of highlydisordered superconductors, measured by the DC transport, isbroadened, the maximum of the resistivity temperature deriva-tive T DCc is listed.Next, the samples were measured by broadband non- a r X i v : . [ c ond - m a t . s up r- c on ] F e b TABLE I: Sample parameters: R s is sheet resistance at roomtemperature, T DCc is superconducting critical temperaturefrom transport measurement, T specc , Γ spec , and Γ STS arevalues estimated from spectroscopic measurement in the GHzfrequency range and from STS measurements, respectively.
Sample R s ( Ω ) T DCc (K) T specc (K) Γ spec / ∆ Γ STS / ∆ A 120 7.89 - - 0B 212 7.04 7.02 0 0.03C 565 4.85 5.11 0.2 0.20D 974 2.44 2.52 0.5 0.44 contact flip-chip transmission line spectroscopic technique,where the sample is placed on top of the transmission linefacing the center conductor. The model and the experimentaltransmission line are shown in Fig. 1.FIG. 1: (Color online) (a) The 3D model of the coplanarwaveguide created in SONNET software. Thesuperconducting film is at the bottom of the sapphiresubstrate. (b) Photo of a rectangular film sample suspendedabove the transmission line.The coplanar waveguide was fabricated on a RogersRO4003 PCB. The tapering in the middle of the transmis-sion line ensures impedance matching upon placing a bare5 × sapphire substrate with thickness 460 µ m abovethe tapering. The bandwidth of the transmission line is upto 18 GHz. The transmission line was fixed to a copper boxby silver epoxy and soldered to a SMA Through Hole con-nector. To ensure electrical insulation, the sample was placedon a thin cigarette paper and fixed by Ge-varnish. The tem-perature dependence measurements were carried out in a Herefrigerator and the transmission spectrum was measured bya vector network analyzer, see Fig 2a. Whereas the transmis-sion spectra of sample A with the lowest R s show the expectedstep-like increase in the transmission below T c , the tempera-ture dependence of the spectra for sample B with R s =212 Ω isslightly disturbed by unexpected resonances close to T c . Thefrequency of these resonances increases with decreasing tem-perature, indicating their dependence on the superconduct-ing properties of the film. Further, these resonances gain instrength and shift to lower frequencies with the increase ofR s , following the L k = ¯ hR s / π ∆ dependence, suggesting theirorigin in the resonances of the superconducting film.To further prove this assumption, samples C and D wereshaped by optical lithography and dry etching process ontoa square with 2.25 mm sidelength. Indeed, this resulted inthe increase of the frequencies co-responding to the geomet- FIG. 2: (a) The scheme of the experimental set-up fortransmission measurement. (b) Temperature dependence ofthe normalized transmission spectra from top to bottom, at 3,6 and 9 GHz of sample C. The curves are offset by -5dB.FIG. 3: (Color online) Normalized transmission spectra ofthe sample D, showing temperature-dependent resonances; fitof the resonances ( + ); temperature dependence of R s ( ◦ );The T DCc from DC measurement is denoted be dash-dottedline and the grey dashed line is the fit of DC transitionfollowing . For presentation, Savitzky–Golay filter was usedto suppress the background.rical factor and suppressed T c . Furthermore, the well-definedshape of the samples increased the quality of the resonances,showing the importance of the edges, typical for resonators.The temperature dependence of the transmission spectrum forsample D is shown in Fig. 3.The spectrum is normalized to the normal state transmis-sion. As it is visible, several resonances emerge from ω −→ c and with decreasing temperature, they shift toward higherfrequencies. To analyze the resonances, we model them assimple capacitively-coupled resonance modes. Their temper-ature dependent frequency can be approximated, assuming R −→
0, with the LC circuit resonance equation ( S ( ω n ) = ω n ( T ) ≈ (cid:112) (( C r + C c ) L r ) n + ( C r + C c ) n L k ( ω n , T ) , (1)where C r and L r are geometric parameters defining the res-onance mode. Assuming these parameters are constant forthe n th resonance mode in the measured narrow temperaturerange, the temperature dependence can be expressed by thesimple equation: ω n ( T ) ≈ (cid:112) A n + B n L k ( ω n , T ) , (2)where A n , B n are fitting parameters, specific for each reso-nance mode.The kinetic sheet inductance of a thin superconducting filmwith thickness much smaller than the London penetrationdepth is given as L k ( ω , T ) = σ ( ω n , T ) / ωσ ( ω n , T ) . Thecomplex conductivity of the films with finite Γ are numeri-cally calculated for a set of Γ , and the full superconductinggap ∆ = ∆ ( T = ) parameters and according to Eq. 2, thetemperature dependence of the resonances in the transmissionspectra are fitted, resulting in the corresponding values of Γ and ∆ . For MoC films, the relation ∆ = . T c is used andthe temperature dependence of ∆ ( T ) follows the BCS relation.The fitted curves for sample D are shown in Fig. 3. The esti-mated values of T c and Γ resulting in the best fit, together withthe expected values of Γ estimated from STS spectra arelisted in Table I. The values of Γ obtained from spectroscopycorrespond reasonably to the ones obtained from STS. Theestimated T c is slightly above the values estimated from DCtransport measurement, but in the broadened superconduct-ing transition. The complex conductivity of Dynes supercon-ductors reproduces the experimentally observed temperaturedependence in the low-temperature limit, where MB conduc-tivity model ( Γ −→
0) is already saturated. As a result, the MBconductivity fit underestimates the T c and fits the experimen-tal dependence with several times the error of the model withfinite Γ .To further investigate the nature of the resonances, the mea-surement was modeled in EM software Sonnet (Fig. 1a). Thecalculated complex conductivity of the film was inserted intothe software. The model reproduced the measurement qual-itatively, showing resonant absorption dips in the transmis-sion spectra. The resonances below 10 GHz are governedby the thin film properties. Above 10 GHz, the metal boxresonances and the dielectric resonances of the substrate arepresent, too. The latter resonances were recognized and an-alyzed in Ref. 25 for experiment with thin metallic films ondielectric substrates. The kinetic inductance and geometricinductance of the metallic films are negligible in comparisonto the disordered superconductors, and their resonances wouldbe present at much higher frequencies. The high L k of MoCfilms results in planar resonances present at frequencies as lowas 3 GHz, despite the small dimensions of the resonator. Thesimulated power loss and the corresponding resonances areshown in Fig. 4. The dominant current component in the su-perconducting film for the first 5 resonant modes illustratesthe nature of the resonances (Fig. 4). The frequencies of theresonances do not fit the experiment perfectly, partly due tothe lack of experimental control in sample position limitingthe accuracy of the model, and partly due to the difficulty ofthe simulations. However, they exhibit the same temperaturedependence as in the experiment.To conclude, we utilized a broadband transmission lineflip-chip spectroscopy technique to study the superconducting FIG. 4: (Color online) Simulated power loss in thetrasmission line and the corresponding eigenmodes of theplanar resonator. The relative amplitude of the j y , j x , j x , j y , j x current densities in superconducting layer are color coded.transition of thin, strongly disordered MoC films sputtered onsapphire substrates. The spectra contained unexpected reso-nances, which were identified as planar resonances of the su-perconducting film. These resonances have low resonance fre-quency due to the high L k of disordered superconductors andthey correspond to the eigenmodes of the planar 2D resonator.In contrast to previous works , the dynamics of the reso-nances is given by the complex conductivity of the disorderedfilms and dimensions of the films. This is further supportedby the EM model in Sonnet software. The resonances wereobserved on rectangular ≈ × films set by the sub-strate solely, and on optically lithographed 2.25 × squares with different R s . The temperature dependence ofthe resonances was fitted by numerically calculated complexconductivity of disordered superconductor with finite Γ . Theobtained Γ parameters relate reasonably to the ones expectedfrom STS measurements, showing that this theory could beutilized to describe the electrodynamic response of highlydisordered superconductors, and vice versa: this non-contactmethod could be used to estimate the Γ and T c of highly disor-dered films, avoiding the nontrivial calibration procedure re-quired in similar spectroscopic techniques , or the require-ment of structuring special resonators on the films . More-over, the understanding of the nature of these resonances canhelp eliminate them in the required bandwidth by smart de-sign, as for example in Ref. 38, or to utilize them as filters insensors and QED circuitry design. Their possible advantagecould be a higher power handling capability than their quasi1D on-chip microwave counterparts and their disjunctionfrom the circuitry design. Finally, the bare awareness of theexistence of these resonances in highly disordered films is im-portant, as they can affect spectroscopic experiments, wherestrongly disordered films are studied and delicate deviationsin the complex conductivity close to T c are analyzed . ACKNOWLEDGMENTS
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