Continued advances in superconducting qubit performance require more detailed understandings of the many sources of decoherence. Within these devices, two-level systems arise due to defects, interfaces, and grain boundaries and are thought to be a major source of qubit decoherence at millikelvin temperatures. In addition to Al, Nb is a commonly used metallization layer in superconducting qubits. Consequently, a significant effort is required to develop and qualify processes that mitigate defects in Nb films. As the fabrication of complete superconducting qubits and their characterization at millikelvin temperatures is a time and resource intensive process, it is desirable to have measurement tools that can rapidly characterize the properties of films and evaluate different treatments. Here, we show that measurements of the variation of the superconducting critical temperature Tc with an applied external magnetic field H (of the phase boundary ) performed with very high-resolution show features that are directly correlated with the structure of the Nb films. In combination with x-ray diffraction measurements, we show that one can even distinguish variations in the size and crystal orientation of the grains in a Nb film by small but reproducible changes in the measured superconducting phase boundary.
Two-level systems (TLSs) in superconducting devices form significant sources of qubit decoherence at millikelvin temperatures1 and are an important factor limiting qubit performance. TLSs may exist at the interfaces between Nb films and the substrate,2 in the oxide layers at the surface of Nb films,3,4 at grain boundaries in Nb films,1 and at defects in the Nb film itself.5 As common fabrication processes preclude the use of epitaxial single crystal Nb, a number of techniques are being explored to mitigate the effects of interfaces, grain boundaries, and native surface oxides as sources of TLSs using wafer scale processing.
The ultimate test of these mitigation strategies would be to fabricate superconducting qubits with differently processed Nb films and measure their coherence times at millikelvin temperatures. However, this is a time consuming and expensive process that severely limits the throughput of materials development. Alternatively, tools such as scanning transmission electron microscopy (STEM),6,7 secondary ion mass spectroscopy (SIMS),2 or atom probe tomography (APT) can be used to instead quantify the structural quality of the Nb films used pre-characterized qubits. While such probes give detailed information about the disorder of films, oxides/interfaces, and concentration of impurities, the information is usually obtained for microscopic cross sections of the device, and the physical relationships between these observations and the operational performance of the qubits are not immediately evident.
Conversely, macroscopic electrical transport measurements are well suited for evaluation of process changes to superconducting films as they inherently probe the impacts of these material parameters to the superconductivity. For example, simple temperature dependent resistivity measurements can determine Tc, and the low temperature sheet resistance and residual resistivity ratio (RRR) provide information about the amount of disorder in the film. More sophisticated measurements can provide important microscopic parameters of the superconductivity itself.
Here, we focus on one such technique that measures the superconducting phase boundary rapidly and with high precision. From the Ginzburg–Landau (GL) theory, Tc of a superconducting film with a small external field H relative to the upper critical field applied perpendicular to the plane of the film varies with H as
where ξ0 is the zero-temperature GL coherence length, is the zero-field transition temperature, and is the superconducting flux quantum.8 Thus, Tc is a linear function of H, and its slope combined with the measured value of directly gives a measure of ξ0. At larger fields/at lower Tc, the phase boundary is better defined by microscopic theory as saturates to its maximum value,9 while ξ0 remains an important length-scale for the superconductivity.
When applied to the study of arbitrary superconducting films, this analysis is frequently performed by tracing the resistive transition of the film at various fixed fields, defining Tc as T at a fixed fraction of the normal state resistance, and plotting this Tc as a function of H. This can alternatively be performed by directly fitting R(T) to microscopic theory10 or by other approximate methods. Nevertheless, this approach suffers from several drawbacks. First, in the cryogenic probes used for such measurements, there is always a thermal lag between the temperature of the thermometer and that of the sample. Thus, it is important to sweep slowly both up and down in temperature through the resistive transition, so that the hysteresis in the traces is minimized. This is time consuming and incurs additional cryogen and operational costs. Second, the discrete nature of the magnetic fields at which Tc is determined may result in features of the phase boundary being missed. To avoid these issues, we have used a feedback technique that allows for the measurement of Tc as a continuous function of H. Our results show non-linear deviations from the GL theory in the phase boundary that, via comparison with X-Ray Diffraction (XRD) measurements, appear correlated with the crystal grain structure and disorder in the films.
The Nb films examined in this work were deposited via High Power Impulse Magnetron Sputtering (HiPIMS) at Torr base pressure. To study the role of the film/substrate interface, 40 or 155 nm films were deposited on three different substrates: (1) intrinsically doped (001) oriented Si; (2) a-plane Al2O3 with (110) crystal orientation; and (3) c-plane Al2O3 with (006) orientation. These substrates were chosen to qualify and improve upon current fabrication techniques, as (001) oriented Si is the substrate used for commercial production of many superconducting qubits,11 while Al2O3 can be used to control the epitaxial growth and orientation of Nb films.12 X-ray studies show that Nb films grow as (110) oriented grains on all substrates used in this work and show partial epitaxial growth on Al2O3 substrates (see supplementary material S1).
In addition, separate 40 nm thick films deposited on a-Al2O3 substrates were ultra-high vacuum (UHV) annealed at 1000 °C for 30 min at Torr. This treatment resulted in enhanced epitaxy of the Nb, with the Nb plane oriented along the direction on the (110) Al2O3 surface. X-ray diffraction (XRD) rocking curve data show that this treatment results in the average horizontal grain size of the Nb increasing from 7 to 124 nm. Similarly, after annealing, we see columnar grains extending from the substrate to the surface oxide (see supplementary material S2). Figure 1 shows Atomic Force Microscopy (AFM) images of Nb films on Al2O3 before and after this annealing process, which show a clearly pronounced increase in the size of the grains at the surface of the film. After deposition, the films were patterned using standard photolithography techniques into 1.2 mm by 30 μm Hall bars and then defined using an Ar ion mill with the photoresist serving as an etch mask.
The devices were measured in a custom probe inserted into a He Dewar with a 3 T maximum field perpendicular to the plane of the films. In the probe, the devices were mounted in vacuum and thermally anchored to a variable temperature stage that is weakly coupled to the 4 K flange of the probe. By controlling the power through a heater mounted on this stage, the temperature of the stage could be varied with a short response time. The accuracy of the thermometer was verified by measuring the transition of a 99.99% pure Nb wire, which showed a Tc of 9.22 K. The resistance of the devices was measured by custom modified Adler-Jackson AC resistance bridges,13 with AC excitations of nA at frequencies . Table I summarizes details of the transitions of the films included in this study. The transition temperatures of the majority of the films were in the range of 8.5–9 K with residual resistance ratios (RRR, the ratio of the room temperature resistance to Rn, the resistance just above the superconducting transition) of ∼5.
|Substrate .||d (nm) .||(K) .||(mK) .||RRR .|
|Substrate .||d (nm) .||(K) .||(mK) .||RRR .|
Values reported along the direction, see Fig. 6.
To initially qualify the superconducting transition of the films, the power to the stage heater was ramped slowly under computer control. Figure 2 shows examples of a superconducting transitions measured on a 40 nm thick film on an a-Al2O3 substrate ramping both up and down in temperature at various magnetic fields. For most films studied, the transition widths were 20 mK. Figure 2 also illustrates the problem of using the resistive transition to accurately measure the phase boundary: even with a slow ramp rate of 5 K/h, the traces corresponding to sweeps up and down in temperature show hysteresis comparable to the transition width.
To accurately measure , we use a feedback technique to maintain the resistance of the sample at a specified value along the superconducting transition by adjusting the set-point resistance of the bridge to the desired value as an applied magnetic field is swept. The error signal from the bridge is then used as the input to a homemade analog proportional–integral–differential (PID) controller that controls the power to the stage heater to maintain the sample resistance. By fixing the bridge resistance at , the temperature as read from the thermometer directly gives Tc when the error signal is minimized by the PID under feedback. To define the transition width, one can additionally measure T – H at other values of resistance, such as 5% Rn and 95% Rn to measure the transition width (see Fig. 5). This technique has been used to measure coherence lengths and anomalous Little-Parks oscillations in mesoscopic Al structures;14 superconductivity in star shaped mesoscopic devices;15 and hysteretic superconducting phase boundaries arising from underlying magnetism in two-dimensional LaAlO3/SrTiO3 heterostructures,16 but has not previously been applied to the assessment of superconducting materials for quantum applications.
Figure 3 shows an example measurement for a 40 nm thick Nb film on Si (001), ramping in both directions in field. As can be seen, the curves corresponding to the different sweep directions cannot be distinguished, with the reproducibility being better than 1 mK except at the field extrema where the sweep direction changes (see supplementary material S3). As noted, the GL theory predicts Tc to be a linear function of H, with a slope determined by ξ0. The traces in Fig. 3 show a clear nonlinear structure, with the derivative largest near zero field and decreasing asymptotically to a constant value at larger fields. This positive curvature of the superconducting phase boundary is seen to an extent in all as-deposited films measured. Before we explore the origin of this curvature, we first discuss the overall differences between Nb films grown on different substrates which can be seen by this method.
Figure 4 shows for two Nb films grown on Si (001) substrates. The of the 40 nm film is lower than that of the corresponding 155 nm film, in line with existing results that show a sharp drop in 's and RRRs of Nb films below a thickness of 200 nm.17 The corresponding slopes of the curves for these films give equivalent coherence lengths 13 nm (see supplementary material S4), far less than the 30 nm seen in bulk Nb18 but comparable to reports for thin deposited films.17,19 Notably, the non-linearity is more evident in the thinner 40 nm film, appearing as a decrease in slope between 2 and 3 kOe. This transition implies substantial inhomogeneity throughout the film, possibly due to greater influence of the Nb/substrate interface.
Figure 5 shows equivalent data for 40 nm thick Nb films grown on a- and c-plane Al2O3 substrates, where we now additionally plot the T–H curves taken under feedback at 5% and 95% of the normal state resistance, to indicate the full width of the transition. These values are comparable to 40 nm Nb films grown on Si (001) (Fig. 4), but a clear Tc and ξ0 enhancement can be seen in the a-Al2O3 film. We attribute this to improved epitaxy between the Nb (110) termination and the termination a-Al2O3 of the substrate (discussed in supplementary material S1). The shape of the curves for the two substrate orientations is characteristically different, and the curves for the 5% Rn and 95% Rn diverge more at larger fields for the a-Al2O3 film than for the c-Al2O3 film. Thus, there appears to be a correspondence between greater transition width broadening, positive curvature of the phase boundary, higher , and crystalline epitaxy.
These correlated features allow us to consider origins for the unexpected nonlinearities in . As the effect appears most strongly in thinner films and varies in magnitude for films of similar thickness on different substrates, it is likely that film granularity and inhomogeneity at the Nb-substrate interface contribute substantially. Evidence for this can be seen in our measurements of the more epitaxial UHV annealed Nb films, when studied along different crystal orientations with respect to the electronic current direction.
As noted before, Nb films grow with a (110) termination on a-Al2O3, with the direction of the Nb films oriented along the in-plane direction on the a-Al2O3. XRD measurements (supplementary material S2) show that UHV annealing yields columnar grains reaching from substrate to surface with lateral sizes as large at ∼120 nm, much larger the ξ0. X-ray reflectivity (XRR) measurements (supplementary material S2) separately show that the interfacial roughness of the UHV annealed Nb is reduced, without a significant drop in film thickness that results in a change in film dimensionality. To explore the dependence of the superconducting properties as a function of the angle between the in-plane crystal direction and the current direction, Hall bars oriented with their lengths at specific angles to the Nb orientation were fabricated from the film. curves for these samples are shown in Fig. 6 and are significantly different from the samples discussed earlier. First, for these films is considerably reduced in comparison to unannealed Nb films on all substrates. However, this is consistent with the reduced RRR, indicative of greater disorder in spite of the improved crystallinity and reduced roughness, possibly due to voids and dislocations generated in the film after relaxation of the crystal or diffusion of surface/native oxides into the bulk. Second, appears to vary systematically with the direction of the current with respect to the crystalline axes, being a maximum when the current is aligned along the Nb direction, and minimum close to the Nb direction (Fig. 6). Third, and most striking in view of the data on Nb films on other substrates, Tc is almost a perfectly linear function of H except for a small field regime near H = 0 where one now sees a negative curvature. Normalizing these phase boundaries by , such that their slope is only defined by , collapses them to a single curve [Fig. 6(b)], implying that in spite of the variation of with angle, ξ0 is the same for all directions. Given the lack of correlation between and ξ0 after annealing, we believe that this can occur when the grain size is much larger than ξ0.
These results enable us to propose a simple model to explain the nonlinearities in observed in polycrystalline Nb films with small, randomly oriented grains, wherein the Nb grains have a distribution of and ξ0 with higher corresponding to longer ξ0. This is possible when the GL parameter κ, the ratio of the London penetration depth λ and ξ0, increases as a function of impurity concentration, as is widely reported in Nb.20–22 Near H = 0, the grains with the largest contribute to . As H increases, the Tc of these grains is more rapidly reduced due to their longer ξ0, at which point grains with smaller and shorter ξ0 dominate the transport, leading to a reduction in the slope of the curves. This result may be enhanced in the case of highly granular films, where only small populations of grains are required to carry the current to achieve a total resistance of , leading to larger variation in the phase boundary than expected of the transition width at zero field. For uniform epitaxial films with a single predominant crystal axis, one observes a linear behavior in as in Fig. 6. For films with grains with two distinct crystalline axes with respect to the current, one would observe two distinct slopes, as seen in Fig. 5. For films with more complicated crystal grain structures, one might observe multiple slopes or even a continual change in slope at low fields. By extension, increasing the grain size by annealing allows for more uniform transport and thus more linear .
In summary, we observe unexpected nonlinearities in the superconducting phase boundaries of thin polycrystalline Nb films fabricated on different substrates. These nonlinearities are due to the distributions of critical temperatures and coherence lengths of crystal grains in the films. Careful measurements of the superconducting phase boundary as demonstrated here thus enable a rapid means of determining the quality of Nb films used in the fabrication of superconducting qubits.
See the supplementary material for supporting XRD/XRR structural characterizations of deposited films before and after annealing; further experimental considerations related to phase boundary mapping under resistance feedback; and numerical derivatives/extracted coherence lengths taken from these phase-boundaries.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under Contract No. DEAC02-07CH11359. This work made use of the NUFAB facility of Northwestern University's NUANCE Center, which has received support from the SHyNE Resource (No. NSF ECCS-2025633), the IIN, and Northwestern's MRSEC program (No. NSF DMR-1720139). This work made use of the Jerome B. Cohen X-ray Diffraction Facility supported by the MRSEC program of the National Science Foundation (No. DMR-1720139) at the Materials Research Center of Northwestern University and the Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (No. NSF ECCS-2025633). Additional support and equipment were provided by the U.S. Office of Naval Research through a Defense University Research Instrumentation Program (DURIP) under Grant No. W911NF-20-1-0066.
Conflict of Interest
The authors have no conflicts to disclose.
Kevin Michael Ryan: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Mark C. Hersam: Conceptualization (supporting); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Venkat Chandrasekhar: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Investigation (supporting); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Carlos G. Torres-Castanedo: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Visualization (supporting); Writing – review & editing (equal). Dominic P. Goronzy: Conceptualization (supporting); Investigation (supporting); Writing – review & editing (equal). David A. Garcia Wetten: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting). Mark Field: Project administration (equal); Resources (equal). Cameron Kopas: Project administration (equal); Resources (equal); Writing – review & editing (equal). Jayss D. Marshall: Investigation (supporting); Resources (equal). Matthew Reagor: Project administration (equal); Resources (equal). Michael J. Bedzyk: Conceptualization (supporting); Funding acquisition (equal); Project administration (equal); Supervision (equal).
The data that support the findings of this study are available within the article and its supplementary material.