Temperature dependence of microwave losses in lumped-element resonators made from superconducting nanowires with high kinetic inductance

We study the response of several microwave resonators made from superconducting NbTiN thin-film meandering nanowires with large kinetic inductance, having different circuit topology and coupling to the transmission line. Reflection measurements reveal the parameters of the circuit and analysis of their temperature dependence in the range 1.7–6 K extract the superconducting energy gap and critical temperature. The lumped-element LC resonator, valid in our frequency range of interest, allows us to predict the quasiparticle (QP) contribution to internal loss, independent of circuit topology and characteristic impedance. Our analysis shows that the internal quality factor is limited not by thermal-equilibrium QP, but an additional temperature-dependent source of internal microwave loss.


I. INTRODUCTION
Superconducting nanowires with large kinetic inductance are interesting for applications in quantum technology due to their low loss and adjustable-strength nonlinearity.Their small size and low capacitance open the possibility of achieving high characteristic impedance, in some cases higher than the quantum resistance R Q = h/4e 2 = 6.45 kΩ [1][2][3][4].High kinetic inductance nanowires are fabricated from thin films with a single process step, resulting in compact, robust and reproducible resonant circuits in a microwave band, e.g.4-8 GHz.As lumped elements, nanowire inductors provide a versatile platform for the design of quantum devices, sensors and detectors.Here we explore the properties of such lumped elements, with a particular focus on understanding microwave losses and the role of thermal-equilibrium quasiparticles (QP).
Understanding microwave losses in high kinetic inductance materials requires consideration of the resonator design and coupling to the transmission line through which we measure its physical response.A lumpedelement model allows us to relate values of the individual circuit variables to the scattering parameters of a reflection measurement.We study the temperature depen- * haviland@kth.sedence of these variables in the range 1.7-6 K, fitting two models to the measured data: the traditional two-fluid model (TTF) [23] and the model based on the Mattis-Bardeen (MB) equations [24].The temperature dependence of the MB model accurately describes the behavior of NbTiN as a strong-coupling, disordered BCS superconductor, allowing us to determine the critical temperature T c and the energy gap ∆ 0 .We analyze resonators with both series and parallel circuit topologies, and with different couplings to the microwave transmission line.
Combining the lumped-element circuit description and two-fluid model, we are able to unambiguously quantify the contribution to microwave loss.As expected for an accurate lumped-element model, our analysis shows that the QP loss, at a given resonance frequency, is independent of the circuit topology and the values of the lumped elements.Comparing the data to the prediction of the model, we show that our resonators contain an additional temperature-dependent loss contribution which is not explained by thermal-equilibrium QPs.

II. DEVICES AND MODELS
We realize high-Q factor resonators in both series and parallel circuit topologies by combining high kinetic inductance superconducting nanowires and interdigital capacitors.Figure 1(a) shows a illustrative layout of the two circuit topologies and Fig. 1(b) the lumped-element model of the kinetic inductor.Figure 1(c)-(f) show scanning electron micrographs of the individual lumped elements.The circuits are fabricated from a single layer of NbTiN film with thickness t = 15 nm on a 600 nm-thick substrate of silicon nitride (SiN) on silicon (Si).The superconducting film was co-sputtered with argon (Ar) and nitrogen (N 2 ) gas from separate Nb and Ti targets.The patterns are defined with a combination of photolithography and e-beam lithography and realized with reactive ion etching.Details of design and fabrication of these types of devices can be found in [25].The resonators are either directly coupled to a transmission line or coupled via a reactive element (capacitor or inductor, depending on topology) to adjust the total quality factor and coupling coefficient η = κ ext /(κ ext + κ 0 ), where κ ext is the loss rate to the external transmission line, and κ 0 the loss rate internal to the resonator.
Our description of the kinetic inductor as a single lumped-element is valid for frequencies much lower than the first eigenmode of the meandering nanowire [25], ∼ 24 GHz for the design shown in Fig. 1 √ n s ≈ 250 nm [26,27].In the regime of small thickness t ≪ λ L the nanowire's kinetic inductance per unit length far exceeds its geometric inductance.The measurement setup is described in the Appendix.We measure the reflection coefficient, where Z R (ω) is impedance of the resonator and Z ext is the impedance seen by the resonator.For a single port high-Q resonator, Γ(ω) can be expressed as [28,29] where the resonance frequency ω 0 , internal loss rate κ 0 and external loss rate κ ext , are the key figures-of-merit that we use to compare different resonators.
The lumped-element model allows us to relate the resonator parameters to individual circuit variables.For a series topology, we have [28], and for parallel we have where R s (R p ) is an effective resistance describing losses.
Losses in the lumped-element nanowire kinetic inductor are well modeled by a temperature-dependent parallel resistance R(T ), as shown in Fig. 1

III. TWO-FLUID MODEL
Quasiparticle losses in the lumped-element kinetic inductor can be described by the two-fluid model of superconductivity [30].This model invokes a complex conductivity, with a real part describing dissipation due to QP current, and an imaginary part describing the inertia of the condensate.The QP resistance is inversely proportional to the normal carrier density R QP ∝ 1/n n , and the kinetic inductance inversely proportional to the density of pairs, L k ∝ 1/n s .The concentration of both normal and superconducting carriers changes with temperature, but the total density of charge carriers n = n s (T ) + n n (T ) is independent of temperature.With n = n s (0) = n n (T c ) we can explicitly write the temperature scaling of L k and R QP , and where ) is the normal state resistance of the kinetic inductor.We fit both the TTF and MB models to the measured data.The former predicts a quadratic dependence of the London penetration depth on temperature, leading to a power-four scaling of the kinetic inductance [30], and latter relates the complex conductivity to the BCS energy gap.In the low frequency limit ℏω ≪ ∆, where pair breaking from the driving field is negligible, the MB model predicts [31], In the fitting procedure we use an expression which is a good approximation to the temperature dependence of the superconducting energy gap of BCS superconductors [32]: IV. DATA ANALYSIS  Fig. 2(c)].We take the capacitor to be independent of temperature.For the directly coupled resonator we can analytically relate the resonator parameters to the lumped-element circuit variables, allowing us to express the temperature dependence of the resonance frequency We use Eqn.(10) to fit the TTF and MB models with Λ(T ) given by Eqn.(7) and Eqn.(8), respectively.The fits are shown in Fig. 2(a) where the MB model shows excellent agreement to the data, with critical temperature T c = 9.45 K and energy gap ratio ∆ 0 /k B T c = 1.85.
From this fit we conclude that NbTiN behaves as a strong coupling "dirty" BCS superconductor [33].
The temperature dependence of the external loss rate κ ext (T ) is well explained with the same Λ MB (T ) found in the fit of ω 0 (T ) by adjusting only one scaling factor κ ext (0) [see Fig. 2(d)], Here we point out that κ ext ≈ 30 MHz ≫ κ 0 over most of the temperature range studied.The low quality factor of this over-coupled resonator makes the determination of Γ(ω) sensitive to details of Z ext .For example, standing waves from small impedance mismatches in the transmission line give a frequency dependent Z ext (ω) which may vary significantly over the bandwidth of the resonator, increasing the uncertainty in the determination of the circuit variables.
To explain the temperature dependence of the internal loss rate κ 0 (T ) we first consider only the QP contribution, i.e.R = R QP (T ).The series topology gives One arrives at the same expression for the parallel topology, where κ QP 0 = (R QP (T )C) −1 .Combining Eqn.(5), Eqn.(6) and Eqn.(12) with the relation between the zero-temperature kinetic inductance and the normalstate resistance [30] we arrive at an expression for the internal loss rate due to QPs, Eqn. ( 14) tells us that, for a given resonance frequency, the QP loss rate is independent of the resonator's characteristic impedance Z c = L k /C and circuit topology.We plot Eqn.(14) in Fig. 2(d), where we see that the measured κ 0 is nearly an order of magnitude larger than κ QP 0 .This implies that there are additional sources of microwave dissipation, e.g.radiative losses (not into the transmission line) or temperature-dependent dielectric losses in the insulating layer [34].We model these as additional resistive channels in parallel with the kinetic inductor, The parameter R d0 describes residual losses at zero temperature, and the parameters R d and β describe a thermally activated loss mechanism.Figure 2(b) shows a best fit with the values of all variables given in Table I.
For the case of a series resonator directly connected to a transmission line, the value of L k = κ 0 (0)/ Re [Z ext ] in Table I is found by assuming Re [Z ext ] = 50 Ω.As previously mentioned, this assumption is not very reliable and furthermore the strongly over-coupled resonator leads to intrinsic uncertainty in the determination of κ 0 (T ) [29].More accurate analysis can be made closer to critical coupling where κ ext ≈ κ 0 .Critical coupling is difficult to achieve with direct connection because typically Re [Z R (ω 0 )] ≪ 50 Ω.Adding a reactive circuit element allows us to tune the external impedance closer to critical coupling while increasing total quality factor.

Reactively coupled resonators
The series resonator is reactively coupled to the transmission line with an inductive shunt L s to ground, realized by the short section of nanowire in the illustrative layout of Fig. 1(a), and shown schematically in the inset of Fig. 3(c).This circuit is the dual of the more common capacitively-coupled parallel resonator, which we discuss later.The shunt inductance L s modifies the external load impedance as with the approximation valid when ωL s ≪ 50 Ω, which is the case for the range of frequency and temperature studied here.With L s ≪ L k the resonator parameters become where R, L k and L s are temperature-dependent.Assuming L s and L k have the same temperature scaling, Eqn.(18) would predict κ ext to be independent of temperature.
The above expressions capture the qualitative behavior with temperature, but they are not accurate enough for fitting.Rather, we fit Eqn.(1) to the measured Γ(ω) at all temperatures to obtain the circuit variables, without the approximations and assumptions stated above.Here we fit only the MB model, as the fit does not converge with the TTF model.The results are given in Table I and the corresponding resonator parameters for this fit are shown by the solid lines in Fig. 3.As expected the fit shows a nearly constant κ ext ≈ 1 MHz.In the range of temperature studied κ 0 (T ) crosses κ ext , with the resonator going from over-coupled at low temperature to under-coupled at higher temperature.The analysis spans a range of coupling coefficient 0.25 < η < 0.9, where determination of the internal losses from the measured scattering parameters is most accurate.As for the directly coupled series resonator, we find the theoretically predicted κ QP 0 to be far lower than that determined from the measured scattering parameters.
We also studied the capacitively-coupled parallel LCresonator, with a layout illustrated in Fig. 1(a) and shown schematically in the inset of Fig. 4(c).The coupling capacitance C c transforms the external load admittance as (20) A high-Q factor requires a coupling capacitance such that near resonance ωC c ≪ 1/50 Ω.In this case, the external admittance becomes The real part reduces the external decay rate, increasing the quality factor, while the imaginary part adds to the resonator capacitance.For C c ≪ C there is a slight shift of the resonance frequency.The resonator parameters become with L k and R depending on temperature.
For the capacitively-coupled parallel LC-resonator, four circuit variables define the three resonator parameters.The fit requires prior knowledge of at least one circuit variable.The data in Fig. 4(c) shows the best fit of ω 0 (T ) using the TFF and MB models.Again the MB model fits better than the TFF model, and we find that the measured internal loss rate κ 0 (T ) exceeds the expected QP contribution by nearly an order of magnitude, as shown in Fig. 4(d).We also show the best fit of κ 0 (T ) to the loss model for R(T ) given in Eqn.(15) with best-fit values given in Table I.Here we note that the residual resistance R d0 for the parallel resonator is almost three orders of magnitude smaller that that for the series configuration, consistent with an increased contribution from dielectric losses due to the much larger resonator capacitance C.

V. CONCLUSION
In a temperature range where thermal-equilibrium quasiparticle (QP) contributions to the internal losses cannot be neglected, we analyzed the resonance frequency and loss rates of microwave resonators made from superconducting meandering nanowires with high kinetic inductance.We showed how microwave measurements of the resonator reflection coefficient in a relatively narrow temperature interval 1.7-6 K provide sufficient information to determine the critical temperature and superconducting energy gap.Quantifying these parameters is essential as they, for a given resonance frequency, uniquely determine the QP contribution to losses and therefore the maximum achievable quality factor at any given temperature, irrespective of the resonators characteristic impedance and circuit topology.
We studied three lumped-element circuit topologies with different coupling to the transmission line.For each we show that the Mattis-Bardeen model provides a consistently good description of the temperature dependence of the kinetic inductance derived from the shift in resonance frequency, with no topology-specific additional assumption.Combined with a proper characterization of the measurement setup, our analysis allows us to quantify the zero-temperature kinetic inductance and normalstate resistance of the nanowire.
Our analysis showed the presence of additional internal microwave losses, not explained by thermal-equilibrium QPs.A comparison between the measured and predicted QP losses for different circuit topologies provides information about the origin of these additional losses.Compared to the series case, the reduced residual resistance of a parallel topology points to the presence of substrate dielectric losses, consistent with the much larger area of the interdigital capacitor.
Our study provides tools for those interested in circuit design with high kinetic inductance, including alternative approaches to achieve a desired coupling to the transmission line.From our analysis we conclude that there is significant room for increasing the quality factor of microwave resonators realised with high kinetic inductance nanowires.Future work should analyze nanowires of different materials, on different substrates, or suspended in vacuum.We hope that the methods and analysis presented here will inspire such future studies.

FIG. 1 .
FIG. 1.(a) The device layout for a series resonator with inductive coupling and a parallel resonator with capacitive coupling.Scanning electron micrographs (SEM) of the components inside the colored boxes are shown in (c)-(f).(b) Lumped-element model of the meandering nanowire kinetic inductor consisting of two parallel channels: a temperature-dependent inductance L k (T ), and a temperature-dependent resistance R(T ).(c) An SEM image of the interdigital series capacitor.(d) An SEM image of the short nanowire shunting inductor, for inductive coupling of the series resonator.(e) An SEM image of an interdigital parallel capacitor for the parallel resonator.(f ) An SEM image of a meandering nanowire kinetic inductor used in both designs.
(f).Disordered superconductors such as NbTiN have low Cooper pair density n s and large London penetration depth λ L ∝ 1/

FIG. 2 .
FIG. 2. (a) Magnitude and (b) phase of the reflection coefficient Γ(ω) of a directly coupled series LC-resonator measured at multiple temperatures with an input power of -102 dBm.Black solid lines are best fit curves to the data.(c)Temperature dependence of the resonance frequency ω0 with fits using the Mattis-Bardeen (MB) model and the traditional two-fluid model (TTF).The inset displays the equivalent circuit diagram.(d) External κext and internal κ0 decay rates with fits using the MB model.The lowest solid green line traces the predicted internal losses due to the QP contribution in the two-fluid model κ QP 0 .

Figure 2
Figure2shows the temperature dependence of the parameters of a series resonator obtained by fitting Eqn.(2) to the measured Γ(ω) at each temperature [Fig.2(a,b)].The resonator consists of a single interdigital capacitor in series with a 100 nm-wide nanowire kinetic inductor, directly coupled to the transmission line [see inset of

FIG. 3 .
FIG. 3. (a) Magnitude and (b) phase of the reflection coefficient Γ(ω) of an inductively coupled series LC-resonator, measured at multiple temperatures with an input power of -122 dBm.Black solid lines are best fit curves to the data.(c)Temperature dependence of the resonance frequency ω0 with a fit using the Mattis-Bardeen (MB) model.The inset displays the equivalent circuit diagram.(d) External κext and internal κ0 decay rates with fits using the MB model.The lowest solid green line traces the predicted internal losses due to the QP contribution in the two-fluid model κ QP 0 .

FIG. 4 .
FIG. 4. (a) Magnitude and (b) phase of the reflection coefficient Γ(ω) of a parallel LC-resonator measured at multiple temperatures with an input power of -99.5 dBm.Black solid lines are best fit curves to the data.(c)Temperature dependence of the resonance frequency ω0 with fits using the Mattis-Bardeen (MB) model and the traditional two-fluid model (TTF).The inset displays the equivalent circuit diagram.(d) External κext and internal κ0 decay rates with fits using the MB model.The lowest solid green line traces the predicted internal losses due to the QP contribution in the two-fluid model κ QP 0 .

|S 31 |
FIG. A2.Scattering parameters of the directional coupler.The coupling corresponds to the scattering parameter S31 while the isolation is given by S41.The ratio D = S41/S31 yields the directivity of the directional coupler.