Gate-tunable superconducting weak link behavior in top-gated LaAlO₃-SrTiO₃ Free

The two dimensional electron gas (2DEG) at the interface of the insulating oxides LaAlO₃ (LAO) and SrTiO₃ (STO) has attracted a lot of attention since its discovery in 2004.^1–3 It was found that the 2DEG undergoes a superconducting transition below about 300 mK,⁴ and that it can be tuned through a superconductor-insulator transition (SIT) by the application of a back-gate² or top-gate voltage.^5,6 The strength of superconductivity is a nonmonotonic function of the gate voltage. This offers a handy way to define tunable barriers in this system, with fine spatial control, using a combination of local top-gates and a global back-gate. We expect that by applying appropriate top-gate voltages, we can tune the area underneath the top-gate from a superconductor, through a normal metal, to finally an insulator, thereby enabling us to create various device structures in the system, without additional lithography and ion-milling steps that may potentially damage the interface. In this work, we use this technique to define top-gate tunable weak links in the 2DEG at the LAO-STO interface.

The nature of superconductivity at the LAO-STO interface is a subject of intense inquiry, as it occurs in two dimensions, in an environment of strong spin-orbit coupling,⁷ and in coexistence with magnetism.^8–11 Various theories have been proposed to explain superconductivity in this system, including a Fulde-Ferrel-Larkin-Ovchinnikov state,¹² a mixed parity state,¹³ and a conventional superconducting state involving electrons from d_xy orbitals in a few TiO₂ layers of STO next to the interface.¹⁴ Penetration depth experiments¹⁵ point to doubly gapped s-wave superconductivity with strong coupling, whereas out-of-plane tunneling measurements do not rule out other symmetries.¹⁶ Thus, a clear consensus is still lacking. By creating weak links in the LAO-STO system as described below and measuring their current-phase relations, we can garner important information about the order parameter symmetry of the system.^17,18

We measured two devices fabricated on the same substrate, both of which showed similar behavior, corresponding to short, overdamped weak links. In the remainder of the paper, we focus on data on one of these devices. Figure 1 shows a scanning electron micrograph (SEM) of the device, which was fabricated on 10 unit cells of LAO grown epitaxially by pulsed laser deposition (PLD) on TiO₂ terminated (001) STO. The details of the PLD synthesis are discussed in earlier papers.^19,20 Using photolithography and Ar ion milling, Hall bars were patterned onto the sample. Our earlier measurements showed that deposition of the top-gate electrode directly on top of LAO can lead to shorting of the top-gate and the 2DEG after just a few sweeps of the top-gate voltage, V_tg, even though LAO itself is an insulator. Hence, we used e-beam lithography and e-beam evaporation of 70 nm of SiO₂ in an oxygen atmosphere to define an additional insulating layer, covering one of the sections of the Hall bar between two pairs of voltage probes. This was done in order to have a more reliable barrier between the top-gate and 2DEG, so that we could use a larger range of values of V_tg and hence go deeper into the insulating regime of the 2DEG. Finally, the same lithography techniques were used to deposit Ti/Au for the top-gate on top of the SiO₂. The top-gated region in its narrowest section was a strip of length 6 μm and width 180 nm. We expect that for certain values of V_tg and temperature, T, the area underneath the top-gate should be non-superconducting, while the other regions remain superconducting. In this case, the resistance is a parallel combination of resistances of the two broad top-gated sections and the narrowest, 180 nm wide top-gated section. If this narrow section can support superconducting correlations across its width, then there would be a supercurrent through this narrow section, while the broad top-gated sections would carry no current. In this regime, the narrowest top-gated section would behave like a weak link between the superconducting banks on either side of the top-gated region.

Measurements were performed using a standard lockin amplifier technique with a Stanford designed SR124 lock-in amplifier and a home built current source, with an ac excitation frequency of 107 Hz and amplitude ∼10 nA. An Agilent synthesizer and a home-built summer were used to apply a dc current bias. The sample was measured in an Oxford MX100 dilution refrigerator with a base temperature of 25 mK. For all top-gate voltages studied, the leakage current from top-gate electrode to the 2DEG was less than 14 nA, which is a negligible dc offset as compared to critical current values of the device.

We measured dV/dI vs. I_dc characteristics of the device at V_tg = 0 V and V_bg = 0 V, before and after sweeping V_tg in the following sequence: 0V → +40 V → −40 V → 0 V, with V_bg = 0 V. The data are shown in Fig. 2(a). Before the V_tg sweep, dV/dI was found to be hysteretic in I_dc, with normal state resistance R_N = 4 kΩ, as shown by the red curve. During the V_tg sweep, we found that dV/dI values increased drastically as we swept to high negative values of V_tg. dV/dI vs. I_dc characteristics after this V_tg sweep, at V_tg = 0 V and V_bg = 0 V, were very different from those before the V_tg sweep, and are shown by the black curve. We note four characteristic features in the black curve, which are different from those in the red curve. These features are evident in all further measurements, although modulated by V_tg and T, which illustrates the fact that after the V_tg sweep, the system is fundamentally altered. First, R_N increased to 11.8 kΩ, which is much higher than its value before sweeping V_tg. Second, there are two distinct current scales in the data after the V_tg sweep, shown by the black curve in Fig. 2(a). One, which we denote by I_ci, is where the system transitions out of the zero resistance state, and the other, denoted by I_co, corresponds to the current beyond which the superconducting banks themselves go normal. The I_co current scale after the V_tg sweep coincides well with the current at which the zero resistance state is lost in data before the V_tg sweep, as seen in the figure. We define I_ci as the current at which dV/dI rises beyond 500 Ω, and I_co as the position of the most prominent peak in dV/dI. Third, additional peaks appear in dV/dI, characteristic of the particular V_tg and reproducible in different I_dc sweeps. Finally, the dV/dI vs. I_dc plot has negligible hysteresis near I_ci, although some hysteresis remains near I_co. We now discuss the implications of these features in detail, and how they together suggest weak link like behavior of our device.

Critical current measurements showed that superconductivity for our sample is strongest at V_bg = 90 V. In order to ensure that the banks are strongly superconducting, we set V_bg = 90 V for all further measurements. Transport characteristics of the device were found to be hysteretic in V_tg, hence all data shown were taken going from positive to negative values of V_tg.

Figure 2(b) shows how dV/dI varies with I_dc for various values of V_tg, at T = 30 mK. We observe that R_N increases monotonically with the decrease in V_tg. Since R_N is a sum of the normal state resistance of the banks, which is independent of V_tg, and the normal state resistance of the top-gated part, which is a function of V_tg, the increase in R_N after the V_tg sweep is attributed only to the top-gated region. Before the V_tg sweep, at V_bg = 90 V, R_N of the 600 μm long and 100 μm wide section of the Hall bar was 3 kΩ, which gives a sheet resistance of the banks, $RNBs$⁠, of 500 Ω. The top-gated section can be considered to be a parallel combination of the resistances of the two broader top-gated parts (20 μm long and 40 μm wide), which correspond to 0.5 squares each, and the narrowest top-gated part, which corresponds to 0.03 squares. Knowing $RNBs$ at V_bg = 90 V and assuming that the sheet resistance is uniform throughout the top-gated part, we can get rough estimates for the normal state sheet resistance $RNGs$ of the top-gated part. $RNGs$ increases from 244 kΩ to 314 kΩ in going from V_tg = 20 V to −15 V. These values of $RNGs$ are two orders of magnitude greater than the maximum values obtained by back-gating.⁸ This striking increase in sheet resistance due to application of large negative top-gate voltages has been observed earlier.⁶ At these large values of sheet resistance, we expect the top-gated section to be non-superconducting on the basis of previous experiments.^2–4,8 However, the nature of this non-superconducting state with such a high $RNGs$ has not yet been completely clarified.¹¹ The very high values of $RNGs$ obtained demonstrate the efficiency of the top-gate, and indicate that the area underneath it is much more depleted/disordered after the V_tg sweep.

We now analyze the structure in dV/dI vs. I_dc and the origin of the two current scales, I_ci and I_co. From our discussion above, we expect that the top-gated region is most likely non-superconducting. The superconducting correlations of the banks cannot be supported across the width of the broad top-gated part (20 μm). The 180 nm wide section of the top-gated region, however, may support superconducting correlations across its width, hence the supercurrent would be confined to flow only through this section. This would then define a weak link between the non-top-gated superconducting banks. The zero resistance state would disappear when I_dc exceeds the critical current of the narrowest top-gated section, which corresponds to I_ci. As discussed earlier, the top-gated section is highly disordered, and it is possible that only a narrow section of the narrowest top-gated region behaves as a weak link. Once I_dc exceeds the critical current of the superconducting banks, the final high-bias resistance, which we call R_N, is reached. From Fig. 2(b), we see that the I_co peaks for various V_tg fall on top of each other, supporting the claim that they are related to the superconducting banks going normal. The I_ci values on the other hand vary strongly with V_tg. The peaks in dV/dI between I_ci and I_co could appear as different parts of the narrow top-gated region exceed their critical currents. Since the degree of disorder varies with V_tg, these features are characteristic of the specific value of V_tg. From this discussion, we conclude that we have a weak link tunable by the top-gate.

We measured I_ci as a function of perpendicular magnetic field $B⊥$⁠, in order to confirm this hypothesis. These measurements were performed by using a PID (proportional-integral-differential controller) to bias I_dc at the steepest part of the dV/dI vs. I_dc curve as the system transitions out of the zero resistance state and then sweeping $B⊥$⁠. From Fig. 3, we see that I_ci does oscillate with $B⊥$⁠, as expected for a Josephson junction. The period of oscillations varies with V_tg. However, there are notable differences between our data and the Fraunhofer patterns expected from a standard Josephson junction. One is that I_ci never goes down completely to zero, which highlights the inhomogeneous nature of current flow through the junction. Second is that the period of oscillations is about 20 mT, which is about an order of magnitude larger than what we would expect from a junction of size 6 μm × 180 nm. This supports our claim that it is very likely that only a narrow strip of the 180 nm wide top-gated section acts as our weak link. As V_tg becomes more negative, the period decreases, which implies an increase in junction area, and the amplitude of oscillations reduces. At the most negative values of V_tg, the oscillations seem to disappear. Finally, the most striking feature of this data is the minimum in I_ci seen at zero field values for the more positive values of V_tg, which becomes shallower and finally reverses sign for negative values of V_tg. As has been seen earlier,⁸ our sample also shows evidence of coexistence between superconductivity and magnetism, demonstrated by the hysteresis in our magnetoresistance data in the inset of Fig. 3. Hence, it is plausible that the peculiar shape of the I_ci vs. $B⊥$ curves results from a combination of 0 and π coupling between the banks caused by the ferromagnetism in the sample.²¹ 

Near the transition temperature, we expect that the current-phase relation of Josephson junctions becomes approximately sinusoidal,¹⁸ and hence, the junction dynamics can be described by the well known tilted washboard model.²² Josephson junctions can be classified as either underdamped or overdamped based on junction dynamics, irrespective of the kind of barrier between the banks. The I–V characteristics of our device are not hysteretic, suggesting an overdamped junction.²² In order to estimate the Josephson coupling energy, we measured dV/dI as a function of I_dc at different temperatures for various V_tg and fit the zero bias resistance, R₀, to the Ambegaokar-Halperin thermally activated phase slip model for overdamped junctions.^23,24 According to this model, $R0∝(1/T) exp(−2EJ/kBT)$⁠, where E_J is the Josephson coupling energy. The proportionality holds if $EJ≫kBT$⁠. In our device, the estimated E_Js from the fits are about an order of magnitude larger than k_BT, ranging from 1.4 eV for V_tg = −15 V to 3.98 eV for V_tg = 10 V. We see from Fig. 4(a) that the $ln(R0T)$ data do vary linearly with $1/T$ for various V_tg, thus validating our model of overdamped weak links. From the slopes of the linear fits, we extract E_J, which are plotted as a function of V_tg in Fig. 4(b), again showing a non-monotonic dependence on V_tg. On the same plot, we also show E_J obtained from the measured I_ci at 30 mK, as calculated from $EJ=ℏIci/2e$⁠. We see that it too shows the expected non-monotonic dependence on V_tg, although the E_J values obtained in this way are larger than those obtained from the fits. The fact that the transport parameters vary systematically with V_tg suggests that V_tg provides a very good handle on the weak link properties.

Figure 4(c) shows the variation of I_ci and I_co with T for five different values of V_tg. The behavior of the two data sets is distinctly different. The variation of I_co with T seems to be almost independent of V_tg, which reinforces our claim that it is related to the critical current of the banks. However, the I_ci vs. T plots are very different for different V_tg. For a long junction, the critical current decays exponentially with temperature. No exponential decay of I_ci is evident from Fig. 4(b), at least for the values of V_tg tested. Hence, we conclude that we are in the short junction limit, consistent with our earlier analysis.

We note that after the submission of our manuscript, a similar study using top-gating to create a weak link in bare STO has been published.²⁵ Also, a shadow hard mask technique has been used to define LaAlO₃-SrTiO₃ nanowires,²⁶ and the authors have discussed the possibility of inhomogeneous ferromagnetism in the sample to explain their data.

In conclusion, we have created nanoscale devices showing evidence of top-gate tunable weak links. However, due to the geometry of our device and the inhomogeneous nature of the 2DEG, we cannot get good estimates of the critical current-normal state resistance products for our weak links. Nevertheless, the critical current shows very interesting modulation by perpendicular magnetic field, hinting at inhomogeneous ferromagnetism coexisting with superconductivity. The weak link fits well to an overdamped junction model. The strength of the superconducting order parameter is a nonmonotonic function of the top-gate voltage, whereas the normal state resistance is a monotonic function of the top-gate voltage, similar to what has been reported for back-gate tuning. However, the large increase in the resistance of the top-gated section is different from what is observed with back-gating, and merits further study. We believe that the method discussed in this paper provides an important way to further our understanding of the LAO-STO system.

The U.S. Department of Energy, Office of Basic Energy Sciences supported the work at Northwestern University through Grant No. DE-FG02-06ER46346. Research at UW-Madison (Design and synthesis of thin film heterostructures used in this work) was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award number DE-FG02-06ER46327.