Tuning the superconducting dome in granular aluminum thin films

Strongly disordered as well as granular superconductors can have very low superfluid density $n_{\textrm{s}}$, i.e. the density of Cooper pairs in the superconducting condensate can be much smaller than in crystalline superconductors [1]. Such superconductors with low $n_{\textrm{s}}$ can host various unconventional electronic states including Cooper pair insulators or pseudogap states [2]. These states often depend on the level of disorder in the material, which thus acts as a convenient tuning parameter during material growth. The corresponding phase diagrams, where the critical temperature $T_{\textrm{c}}$ of superconductivity typically is suppressed with increasing disorder, have been studied in great detail in the context of the superconductor-insulator transition [3]. Manifestations of low $n_{\textrm{s}}$ are long London penetration depth $\lambda_{\textrm{L}}$ (from a magnetic-field point of view) and high kinetic inductance $L_{\textrm{kin}}$ (from an electrodynamic point of view) [4]. Superconductors with high $L_{\textrm{kin}}$ play an important role as microwave circuit elements [5], or for parametric amplifiers [6, 7, 8] within the context of superconducting quantum computing [9]. Furthermore, strongly disordered superconductors are the core material for superconducting nanowire single photon detectors (SNSPDs) [10].

Granular aluminum plays a prominent role among the wider class of disordered superconductors due to several characteristics [11, 12]: (1) preparation of granular aluminum films is straightforward by deposition of aluminum vapor in an oxygen atmosphere (e.g. using thermal or electron-beam evaporation or sputtering) [13, 14, 15, 16]. (2) The normal-state resistivity and the superfluid density can be tuned over several orders of magnitude [13, 17]. (3) With increasing normal-state resistivity, $T_{\textrm{c}}$ first increases substantially before then decreasing, leading to a phase diagram with a so-called superconducting dome [13]. (4) These properties are advantageous in the field of quantum circuits and superconducting quantum computation, and therefore granular aluminum has been employed in numerous studies of concrete applications [18, 19, 20, 16].

Thus it might come as a surprise that the underlying mechanisms that control superconductivity in granular aluminum, including the shape of the superconducting dome, have been at the center of long-standing discussions. General considerations about material characteristics of granular aluminum are sketched in Fig. 1(a), and the previously established resulting phase diagram with superconducting dome [21, 13, 17, 22, 23] is shown schematically in Fig. 1(b).

Granular aluminum consists of nanometer-sized grains of pure aluminum within an aluminum oxide matrix. With overall properties being metallic, this means that neighbouring aluminum grains are strongly coupled via thin oxide barriers [24]. Microscopically, the superconducting state of granular aluminum thus can be considered a random three-dimensional array of superconducting islands coupled via Josephson junctions, while the macroscopic electronic properties of granular aluminum are similar to those of homogeneously disordered superconductors [25, 17, 26, 23]. The normal-state dc resistivity $\rho_{\textrm{dc}}$ is a commonly used material property to quantify grain decoupling in granular aluminum, which in the phase diagram (see Fig. 1(b)) is a parameter that can be tuned during growth. Previous studies have demonstrated that the initial increase of $T_{\textrm{c}}$ for increasing $\rho_{\textrm{dc}}$ is due to an increase of the superconducting energy gap $\Delta$ while the subsequent decrease of $T_{\textrm{c}}$ is caused by suppressed superfluid density $n_{\textrm{s}}$ [17, 27]. The size of the aluminum grains is another crucial parameter. Most experiments with superconducting granular aluminum employ thin films that were deposited on room-temperature substrates, where the diameter of the aluminum grains is around 3 nm, with maximum $T_{\textrm{c}}$ of the superconducting dome around 2.2 K [21]. If instead the films are deposited onto substrates cooled with liquid nitrogen, then the grain size is around 2 nm, and maximum $T_{\textrm{c}}$ is around 3.2 K [13, 28]. Superconductivity in such films grown at nitrogen temperatures was researched intensely by the Tel Aviv group [29, 23, 28, 11]. Earlier related investigations into superconducting aluminum films utilized even colder substrates [30, 31, 32], though with different scientific aims, with different fabrication, and with challenges concerning sample stability. Therefore, the goal of the present study is investigating the evolution of the superconducting dome of granular aluminum as a function of substrate temperature, from room temperature down to cooling with liquid helium.

The granular aluminum films presented in this work were prepared by thermal evaporation of high-purity aluminum from a tungsten boat onto sapphire (Al₂O₃) substrates mounted on a copper substrate holder held at a distance of 28 cm from the boat. The substrate holder plate is directly coupled to the cryogenic tank and has built-in heater and sensor for temperature control. The films were grown in presence of a constant oxygen flow in the chamber to achieve the granular structure. An adjustable and gradually increasing current was passed through the tungsten boat to control the rate of evaporation of the aluminum. The substrate was kept at a constant temperature with the help of the heater. The thickness of the film was calculated using a quartz crystal installed inside the evaporation chamber that acts as rate meter.

The pressure in the deposition chamber before evaporation, in absence of oxygen, ranged between $~{}6\times 10^{-7}\,\textrm{mbar}$ (for substrates at 300 K) and $~{}1.5-2\times 10^{-7}\,\textrm{mbar}$ (for substrates at 25 K). To reach cold substrate temperatures, liquid nitrogen (N₂; boiling temperature of 77 K) or liquid helium (He; boiling temperature of 4 K) was introduced in the cryostat above the sample holder.

Two main process variables that affect the resistivity of the films are oxygen flow in the chamber and rate of evaporation of aluminum. The lower the oxygen flow, the lower is the resistivity due to reduced aluminum oxide barriers between the aluminum grains. Similarly, the higher the evaporation rate, the more aluminum is deposited on the substrate with respect to the incorporated oxygen, hence decreasing the resistivity. The films were prepared with resistivities ranging from 4 to 7000 $\mu\Omega$cm. To cover this range, the oxygen flow in the chamber was varied between 0.1 and 1.0 SCCM (standard cubic centimeters per minute). Similarly, the rate of evaporation of aluminum was varied between 0.8 and 3.5 Å/s. The growth rate is set to a nearly constant level before the shutter in front of the substrate is opened.

For substrates cooled with liquid nitrogen, we investigated samples grown either at 150 K or 100 K substrate temperature. For helium-cooled substrates, the temperature was 25 K, which was the lowest temperature that we could routinely stabilize during the growth process of the films.

After film growth, the samples were taken out of the evaporator to ambient, and then contacted for temperature-dependent dc resistance measurements, which were performed in separate ⁴He-cooled bath cryostats. Most of the samples were grown on $10\times 10$ mm² substrates (suitable for THz measurements [33, 17, 23]) and studied using 4-point dc measurements in van der Pauw geometry (with contacts at the corners of the sample). A few samples were grown on $5\times 5$ mm² substrates and then contacted using gold Corbino contacts (suitable for GHz measurements [34]) deposited through a shadow mask in a separate thermal evaporator. Here dc resistance was measured in 2-point configuration, and the offset due to wiring was determined from a reference measurement. For all samples, the dc resistivity was obtained from the dc resistance, the known lateral geometry, and the thickness as determined during growth with the quartz rate meter. The normal-state resistivity at 5 K is used to relate the samples within the phase diagram below, and $T_{\textrm{c}}$ was defined as the temperature where the measured resistivity amounts to half of this normal-state value.

We have grown granular aluminum films at substrate temperatures ranging from room temperature down to 25 K. We find that the superconducting dome of $T_{\textrm{c}}$ becomes higher for decreasing substrate temperature, thus extending to the helium temperature range the trend previously observed down to nitrogen temperatures. This suggests that the reduction of grain size, which was previously characterized for room temperature and nitrogen temperature, continues into the helium temperature range. Clearly, structural characterization of films grown at helium temperatures is highly desired to confirm this evolution of grain size.

Often granular aluminum is considered to be a material system that is rather unsusceptible to the exact details of sample preparation, i.e. the type of disorder in the system generated by growth of aluminum grains with oxide barriers in between generally seems to be rather robust. Our study indicates that the story of superconductivity in granular aluminum might be even more intricate than already established. One aspect is the substantial thickness dependence. Another is the fact that the maximum of the superconducting domes that we observe for substrate temperatures of 300 K and 100 K is somewhat lower than previously reported in literature [21, 13, 17, 22, 23]. This might be important for quantitative optimization of devices that utilize the kinetic inductance of superconducting granular aluminum, since $T_{\textrm{c}}$ is not ‘universal’ even for room-temperature growth.

This makes the picture of superconductivity in granular aluminum more complex, and it asks for more theoretical investigations of this fascinating model system for superconductivity with mesoscopic disorder. It also means that additional tuning parameters during growth might be possible, which could be helpful during material optimization for specific applications. In this context, our observation that the transition width of granular aluminum grown on cryogenically cooled substrates, for the region of high resistivity and thus high kinetic inductance, is narrower than for room-temperature growth, might be particularly attractive. It also reinforces the general interest in unconventional behavior of superconductors with low superfluid density, e.g. including collective excitations [49, 50, 51, 52, 27], which now might become accessible with an even wider choice of tuning options.

We thank Gabriele Untereiner for support with sample preparation, Alessandro D’Arnese for initial work on the evaporator, and Nimrod Bachar, Ameya Nambisan, and Ioan Pop for helpful discussions. We acknowledge financial support by the Deutsche Forschungsgemeinschaft DFG (project SCHE 1580/6-1 / DR 228/57-1) and by the Baden-Württemberg Stiftung within project QT-10 (QEDHiNet).

We have prepared a set of films matching closely the top of the ‘150 K’-dome, with thickness 10 nm and $\rho_{\textrm{dc}}\approx 300\mu\Omega\textrm{cm}$, while changing the substrate temperature between 150 K and 290 K. The resulting values for $T_{\textrm{c}}$ are shown in Fig. 6. We find a continuous reduction of $T_{\textrm{c}}$ with increasing substrate temperature. This confirms the general conception that grain size decreases continuously for cooler substrates and that this induces higher $T_{\textrm{c}}$. Thus any cooling of the substrate, even if it does not reach the ‘nitrogen range’ around 77 K to 100 K, can significantly enhance $T_{\textrm{c}}$.