Quantized critical supercurrent in SrTiO₃-based quantum point contacts

A superconducting quantum constriction is realized by locally screening ionic liquid gating in SrTiO₃.

Abstract

Superconductivity in SrTiO₃ occurs at remarkably low carrier densities and therefore, unlike conventional superconductors, can be controlled by electrostatic gates. Here, we demonstrate nanoscale weak links connecting superconducting leads, all within a single material, SrTiO₃. Ionic liquid gating accumulates carriers in the leads, and local electrostatic gates are tuned to open the weak link. These devices behave as superconducting quantum point contacts with a quantized critical supercurrent. This is a milestone toward establishing SrTiO₃ as a single-material platform for mesoscopic superconducting transport experiments that also intrinsically contains the necessary ingredients to engineer topological superconductivity.

INTRODUCTION

Conductance quantization in ballistic quantum point contacts (QPCs) is a notable example of departure from the classical Drude picture of electrical conductivity set by the rate of charge carrier scattering (1). When a constriction between two electron reservoirs is sufficiently narrow and disorder free, its conductance becomes quantized according to the number of occupied modes: discrete transverse momenta allowed within the constriction’s confinement potential. Each mode contributes a conductance quantum δG = 2e²/h (spin-degenerate case), a value that does not depend on the exact geometry of the device.

A related phenomenon is expected to arise in a constriction between two superconducting reservoirs (2, 3), i.e., a superconducting QPC (SQPC). Again, the transverse momentum spectrum becomes discretized under the constriction confinement potential. The supercurrent carried by each mode is determined by the Andreev bound state (ABS) spectrum, which is typically a function of constriction geometry. SQPCs are thus characterized by quantized critical supercurrent I_c with a nonuniversal step height δI_c. However, in the limit of a short junction length, only one ABS per ballistic mode remains, and the current carried by each mode can reach a maximum value δI_c = eΔ/ℏ. This ideal step height is again geometry independent and scales only with the superconducting gap Δ.

The widespread route for fabricating gate-tunable superconducting weak links has been to combine two optimal components in a hybrid system: a clean semiconductor (typically a III-V semiconductor or Ge) and metallic superconducting leads (e.g., Nb and Al). These hybrid systems have been successfully used to demonstrate quantized critical supercurrent, but with quantization step heights far below eΔ/ℏ (4–9). The two major challenges for reaching the universal limit for quantized supercurrent are the geometric requirement that the distance between superconducting leads be much less than the superconducting coherence length ξ and the need for near-perfect semiconductor/superconductor contact transparency (3). Achieving the latter in hybrid semiconductor-superconductor systems has been a major materials science challenge that has required deployment of in situ heteroepitaxial growth techniques (10).

An alternate route taken in this work is to form both leads and constriction in a single electrostatically tunable superconducting material, such as SrTiO₃ (STO). Working within a single-material platform is attractive for fabricating SQPCs, as the superconductor/normal metal (SN) boundary can be purely electronic (no structural discontinuity) and thus potentially highly transparent.

One of STO’s remarkable aspects is superconductivity in the extremely dilute charge carrier density limit (11, 12). In two-dimensional (2D) electron systems (2DESs) at the surface of STO, such as LaAlO₃ (LAO)/STO, LaTiO₃/STO, and ionic liquid–gated STO, superconductivity occurs in the range of 0.01 electrons per unit cell (13, 14). Consequently, one can electrostatically control the transition between superconductor, normal metal, and insulator in this material. On the macroscopic scale, this control is well established using back gating through the STO substrate, top gating through a dielectric layer, and ionic liquid gating (13–18).

More recently, several approaches have emerged for nanoscale patterning of conduction in LAO/STO, leading to demonstration of quantization effects in normal state, but, so far, not in superconducting transport. Realization of a conventional split-gate QPC geometry in LAO/STO is challenging as it involves depleting and/or accumulating charge densities of at least ≈10¹³ cm^–2, close to the limit of conventional dielectrics. Spatial inhomogeneity and relatively short mean free paths in these 2DESs present another challenge, leading patterned constrictions to often be dominated by tunneling through accidental quantum dots (19–21). A QPC with normal state but not superconducting conductance quantization has recently been demonstrated in underdoped, nonsuperconducting LAO/STO (22). In (21), a constriction defined by split gates with normal state conductance about half of a single spin–degenerate ballistic mode was estimated to have corresponding partially transmitting single-mode supercurrent, although it did not show direct effects of quantization. A different technique is to write conductive channels on LAO/STO with voltage-biased atomic force microscope (AFM) tips. This method enabled demonstration of quantum wires and dots coupled by tunnel barriers to superconducting leads, with quantized normal-state transport and indirect signatures of electron pairing (23–26) but not superconductivity.

In this work, we demonstrate quantized supercurrent in QPCs in a split-gate geometry based on ionic liquid–gated STO. We observe a discretized step structure in the critical current, with tuning from zero to three ballistic modes. Step height per mode δI_c is only three to five times smaller than the canonical value eΔ/ℏ, as close to ideal as achieved in any hybrid system (6). The fabrication process of our devices is enabled by the fine patterning of local electrostatic gates using liftoff of metal and atomic layer–deposited Hafnia (HfO₂) with a feature size close to 40 nm. This is distinct from the approaches taken in previous works on LAO/STO weak links (20–22, 27–29). Notably, we avoid an epitaxial growth step at high temperature, which complicates the workflow for patterning and potentially introduces disorder [see, e.g., (30, 31)]. We thus consider this fabrication technique an attractive alternative for further development of STO as a platform for mesoscale superconducting devices.

RESULTS

Our devices are 20-μm-wide Hall bars covered by ionic liquid, which is polarized to accumulate a 2D carrier density at any exposed STO surface. The coarse contours of the Hall bar are defined by patterning an insulating SiO₂ layer, which separates the surface of undoped STO from the ionic liquid (Fig. 1, A and B); underneath the SiO₂, the STO surface remains insulating, while the carrier density in the Hall bar region is tuned into the superconducting regime. Split gates with thin, self-aligned HfO₂ dielectrics define 40-nm-wide constrictions (Fig. 1C) between neighboring superconducting reservoirs. The design includes five or six ohmic contacts on each side of the split gates (Fig. 1B) to enable four-terminal measurements of both the constriction and the adjacent superconducting leads.

Fig. 1. Electrostatically defined constriction in superconducting SrTiO₃.

(A) Schematic cross section of the device and illustration of the gate voltage definitions. (B) Confocal laser microscope image of the Hall bar region of the device and illustration of the measurement scheme. The dashed arrow indicates the location of the cross section in (A). (C) Scanning electron microscope image of the constriction region on a reference device. (D) Superconducting transition in the constriction and lead resistance. “Right” and “Left” refer to measurement of V_lead on both sides of the constriction. (E) Constriction conductance map with temperature and split gate voltage. (F) Constriction conductance map with magnetic field and local gate voltage. Symbols in (F) indicate the selected gate voltage values for which line cuts in field are shown in (G). Lead resistance at extremes of V_G12 is also shown in (G) to illustrate the independence of local gate voltage. The top axis shows the mapping from critical field B_c (red circles) to the coherence length. The estimated ξ is shown in (C) for comparison with device dimensions, along with the mean free path from Hall measurements in the leads (see section S4). In (D) to (G), V_GIL = 3 V and V_BG = 50 V.

The carrier density profile is electrostatically defined by voltages on four gates, as illustrated in Fig. 1A: a large coplanar gate that controls the polarization of the ionic liquid (V_GIL), a back gate (V_BG) and two split gates (V_G1 and V_G2, denoted as V_G12 for the case V_{G1 = VG2}). V_GIL and V_BG are “global” gates that tune the carrier density and vertical confinement in both the 2DES leads and the constriction. V_G12 is a “local” gate that only tunes carrier density and lateral confinement in the constriction. V_GIL is set when the device is near room temperature and maintained as the sample is cooled below the freezing temperature of the ionic liquid (220 K). V_GIL is used to polarize a drop of ionic liquid that covers both the coplanar gate electrode and the device. At lower temperatures, the polarization of the ionic liquid is frozen in. V_GIL is the primary control knob for the carrier density in the leads, which can be tuned from ≈5 × 10¹² to 10¹⁴ cm⁻² (32, 33). The superconducting transition temperature as a function of density has a maximum near 3 × 10¹³ cm⁻² (see section S3). The main results presented here will focus on this nearly optimally doped state obtained by cooling the device under V_GIL = +3 V. For additional data on the second constriction on the right side of the Hall bar in Fig. 1B, different devices, and cooldowns with carrier density tuned across a larger range, see sections S2 to S6.

The voltage V_BG on a back gate contacting the bottom of the STO crystal provides additional global tuning of the 2DES at base temperature, primarily by modulating the depth of the 2DES. For most experiments on this device, we set V_BG = +50 V to pull the electron density farther away from surface disorder [see (34) and section S4].

Figure 1D shows the superconducting transition T_c measured by sourcing a small AC excitation through a constriction at V_G12 = +3 V and V_BG = +50 V. In the following, constriction resistance and conductance will be denoted as R = dV_QPC/dI_AC and G = 1/R and the resistances of the leads as R_lead = dV_lead/dI_AC (see Fig. 1B and Materials and Methods for more details). On both sides of the constriction, R_lead shows a sharp transition near 350 mK. This is near the optimal T_c value for 2D STO (14, 17). The measured Hall density of 3.05 × 10¹³ cm⁻² and the slight increase of T_c by 20 mK upon removing the back-gate voltage suggest that this device state is slightly on the overdoped side of the superconducting dome (see section S4).

The constriction resistance R also starts decreasing near the lead T_c, but its transition to zero resistance (within accuracy of our measurement) is significantly broader than that of the leads. Decreasing V_G12 suppresses both the zero resistance state and the normal state conductance and eventually pinches off the weak link (Fig. 1, E and F). At base temperature, superconductivity can also be suppressed by a perpendicular magnetic field (Fig. 1F). Using ξ² = Φ₀/(2πB_C) (35), with Φ₀ = h/2e being the flux quantum, the critical field B_c = 130 to 140 mT in the constriction yields an estimated coherence length of ξ = 50 nm (43 nm in the leads). This estimate is consistent with the dirty-limit Bardeen-Cooper-Schrieffer superconductor picture (36, 37) in which the coherence length is set by the mean free path L_MFP. From Hall measurements on the leads, we extract a Hall mobility μ = 600 cm²/Vs and L_MFP = 55 nm.

The shortness of these length scales illustrates the challenge of fabricating QPCs and SQPCs in STO (see Fig. 1C). Observing ballistic transport requires junction length L < L_MFP. Achieving a single-ABS junction with critical current quantization also requires short junction length: L < ξ. Although the junction length is not well defined in a split-gate geometry, we fabricated the gates with very narrow lateral spacing (40 nm) and sharp tips to strive for the ballistic (or quasi-ballistic) regime.

The ballistic nature of the SQPC is most apparent in differential resistance at finite DC current. Filling of states in the constriction with V_G12 results in a staircase shape of the critical supercurrent I_c(V_G12) (Fig. 2). Adopting a definition of I_c as the current at which the resistance R is halved with respect to the high-V_DC normal state, plateaus at both positive and negative integer multiples of δI_c = 2.48 nA are seen in the V_{G12 − IDC} map of constriction resistance normalized to its normal state value (Fig. 2B).

Fig. 2. Critical current quantization.

(A) DC current dependence of constriction and lead resistances at V_G12 = 3V. (B) Constriction resistance, normalized to normal state resistance at V_DC = 100 μV. The solid red line indicates the critical current I_c. The dashed lines indicate 1, 2, and 3 integer multiples of δI_c = 2.48 nA. (C) V_G12 dependence of I_c normalized to δI_c and (D) normal state conductance G_N at V_DC = 100 μV, with a series resistance of 800 Ω subtracted from the raw data. The shaded connection between (C) and (D) emphasizes the numerical correspondence in the observed number of ballistic modes n. The dashed line in (C) is a fit to the saddle potential QPC model (see section S1). (E) Split- and back-gate voltage dependence of zero-bias conductance above T_c. G has been corrected for a variable series resistance gradually increasing from 1.15 to 2.1 kilohm. Short plateaus can be seen at integer multiples of 2e²/h (n = 1, 2, and hints at higher multiples). Unintentional Coulomb blockade levels can be seen near 0.2e²/h, e²/h, and 2.5e²/h.

In the ballistic SQPC picture, I_c/δI_c corresponds to n, which is the number of ballistic modes below the Fermi energy in the constriction (Fig. 2C). The first mode plateau is intermittent as a function of gate voltage due to resonant transmission through the weak link, correlated with the charging levels of an accidental Coulomb blockade observed near pinch-off at low V_G12 (see section S7), whereas the second and third plateaus are more stable. An alternative way to estimate the number of modes is from normal state conductance G_N, where each fully transmitting spin-degenerate mode is expected to contribute a conductance δG = 2e²/h. The number of modes inferred by dividing G_N by this increment matches that extracted from the sequence of steps in supercurrent. We also see hints of plateaus in normal state conductance at fixed V_DC = 100 μV near n = 1 and 2 (Fig. 2D). Features suggestive of normal state conductance quantization are more clearly apparent above T_c (Fig. 2E), where one does not need to apply a DC bias to suppress the supercurrent, and disorder-induced fluctuations are reduced. The plateau structure persists as a function of back gate voltage (detailed further in section S5).

Ideally, the magnitude of steps in I_c through a constriction should scale only with the superconducting gap as

$$\mathrm{\delta }{\mathit{I}}_{\mathrm{c}}=\frac{\mathit{e}\mathrm{\Delta }}{\mathrm{\hslash }}$$ (1)

This scaling is expected to hold for a short junction (L ≪ ξ) with perfectly transparent SN contacts (2, 3).

For most experimental realizations of SQPCs in hybrid metal superconductor/semiconductor devices, neither of these requirements is fully satisfied, and δI_c is generally suppressed by at least an order of magnitude (4, 5, 7–9). One work on Si/Ge nanowires with Nb contacts reported suppression by only a factor of 2.9 (6). In our case, data in Fig. 2 suggest a comparable factor of 3 to 5. The uncertainty comes from the choice of method to extract Δ (see section S8): from T_c of the constriction [δI_c/(eΔ/ℏ) = 2.9], from T_c of the leads [δI_c/(eΔ/ℏ) = 4.1], or from the temperature dependence of the excess current [δI_c/(eΔ/ℏ) = 4.8].

Analysis of the excess current I_exc allows separating the role of imperfect SN contact transparency τ_SN from that of finite junction length. We define I_exc as the zero-bias intercept of the normal-state resistance extrapolated from high V_DC (Fig. 3A). Both I_c and I_exc are expected to scale with G_N (see section S1) and, thus, approximately track each other with V_G12. However, these two quantities encode different physics: the shape of the ABS spectrum for I_c (2) and the balance between ordinary and Andreev reflections for I_exc (38). The quantity eI_excR_N/Δ can be nonlinearly mapped onto τ_SN following the treatment of Andreev reflections in an superconductor/normal metal/superconductor (SNS) junction in (39, 40). Over the gate voltage range with a well-defined and quantized supercurrent (1.5 < V_G12 < 2.5), we thereby extract ${\mathrm{\tau }}_{\text{SN}}=0.75_{-0.08}^{+0.12}$.

Fig. 3. SN transparency and junction length.

(A) The DC current–voltage curve of the constriction at V_G12 = 3 V and the definition of the excess current I_exc. (B) Split-gate voltage dependence of the excess and critical currents. (C) eI_excR_N/Δ, the input quantity of the SNS model in (39, 40), and its mapping onto SN boundary transparency τ_SN. (D) δI_c suppression by finite transparency and finite junction length. Comparison to the ballistic short-limit model (2) (solid black line), full calculation at L/ξ = 0.56 (3) (blue squares), and the approximate correction for arbitrary L/ξ from (41) (dashed lines). The shaded region reflects in (C) and error bars in (D) reflect the uncertainty on the gap (see section S8).

In the short junction limit L ≪ ξ (2), we can predict the suppression of δI_c as a function of τ_SN (Fig. 3D). The experimentally measured δI_c is only slightly below the theoretical curve, and its full suppression can be accounted for by multiplying it by an additional factor α= 0.7. This additional suppression can be explained by considering the finite length of the junction. An approximate theoretical description obtained in (41) is α = 1/(1 + L/2ξ), which is in good agreement with calculations for the case in (3), where L = 0.56ξ. In this work, assuming that α = 0.7 yields L = 0.85ξ = 42 nm, which is close to the 40-nm lithographic width of our QPC.

DISCUSSION

Transparency is likely to be the main driver for the reduction in δI_c from its ideal value despite being competitive with the hybrid III-V/superconductor systems, where τ_SN is typically estimated below 0.85 (7–9, 42) except for pristine epitaxial interfaces (10). An advantage of our single-material system is that the SN contact interface is electrostatically defined and presumably does not have a structural discontinuity. In our present realization, transparency is likely limited by the smooth gate-induced density variation, which, in turn, entails a gradually varying order parameter. We anticipate that τ_SN can be further improved by manipulating the SN boundary with additional local gates near the weak link.

Furthermore, we anticipate improvements by increasing the mean free path. In ionic liquid–gated STO and LAO/STO, L_MFP is typically less than 100 nm. However, improvements to μ > 10⁴ cm²/Vs and L_MFP> 1 μm have been demonstrated by separating the ionic liquid from the channel by an ultrathin spacer layer (16), band engineering with spacer layers in LAO/STO (43), or forming the channel from high-quality molecular beam epitaxy–grown STO in the 3D case (44). These demonstrations have so far only been accomplished in unpatterned or coarsely patterned films, but we anticipate that these approaches can be compatible with the nanopatterning technique developed here. The fabrication route used in this work is relatively simple—based on commercially available STO crystals, avoiding epitaxial growth steps—so complex patterning, device, or heterostructure design refinements could be added without rendering it unwieldy.

Using ionic liquid–gated STO as a platform, we have realized SQPCs with quantized critical supercurrent, tunable between zero and three ballistic modes by split gates. This is a first realization of a quantized gate-tunable SQPC in a single material system, enabling highly transparent SN contacts without structural discontinuity at the boundary. This work establishes spatially patterned screening of ionic liquid from an STO surface as a promising alternative to existing methods for nanoscale patterning of conduction and superconductivity in STO: patterning LAO/STO with pregrowth templates (19, 20, 28, 29), electrostatic depletion by patterned gates (21, 22, 27), or conductive channel writing by voltage-biased AFM tips (23–26, 45). Our method appears particularly suited for realizations of ballistic superconducting transport, which require maintaining high carrier densities within nanopatterned constrictions. Naturally occurring depletion near the edges of an STO-based conducting channel (45, 46) can be counteracted with local gates as we have shown.

Our approach may also be especially attractive for exploring topological superconductivity in several contexts. Combining ballistic transport with superconductivity, strong spin-orbit coupling, and tunable dimensionality offers hope for engineering extrinsic topological superconductivity in 1D nanostructures (47–49). Even an unpatterned STO 2DES may host intrinsic topological superconductivity in certain conditions because of interplay between its multiorbital band structure, spin-orbit coupling, and ferroelectricity (50–52). A ballistic point contact similar to the SQPC demonstrated here could serve as the tunnel probe central to many detection schemes for the resulting Majorana bound states (53–55). The single-mode ballistic Josephson junction regime demonstrated here is also a requisite ingredient of theoretical proposals for realizing topological ABS spectra in multiterminal junctions (56, 57). Last, this work is an important step toward realizing controlled negative U quantum dots (20, 23) in the classic geometry of an “island” coupled to two QPCs (58).

MATERIALS AND METHODS

Fabrication is based on commercially (001) oriented STO single-crystal substrates, purchased from MTI. To obtain a Ti-terminated surface with terrace-step morphology, these substrates were soaked in heated deionized water for 20 min and annealed at 1000°C for 2 hours in flowing Ar and O₂ in a tube furnace.

All subsequent patterning was performed with liftoff processes using e-beam–patterned poly(methyl methacrylate) (PMMA) 950K, 4% in anisole for the first step and 8% for all subsequent steps. The first step is the local split-gate pattern, written on a 100-kV e-beam write system. Atomic layer deposition was used to deposit 15 nm of HfO₂ (100 cycles of Hf precursor and water.) The deposition stage temperature was 85°C. We note the importance of loading the sample and starting the deposition quickly to avoid PMMA pattern reflow. The 5-nm Ti/50-nm Au gate contact was then deposited by e-beam evaporation. Liftoff of both HfO₂ and Ti/Au layers was then performed by soaking in heated N-Methyl-2-pyrrolidone (NMP), followed by ultrasonication in acetone.

The remaining patterning was performed with a 30-kV e-beam write system. The second step is the gate contact using liftoff of 40-nm Ti/100-nm Au in acetone. The third step is the ohmic contact deposition. It requires exposing the pattern to Ar⁺ ion milling before e-beam evaporation of 10-nm Ti/80-nm Au followed by liftoff in acetone. The fourth patterning step is the mesa insulation, deposited by magnetron sputtering 70 nm of SiO₂, followed by liftoff in acetone. The measured devices were imaged with a conventional optical microscope and with a Keyence VK-X confocal laser microscope. Scanning electron microscope imaging was performed on reference patterns written on the same chips.

Finished devices were annealed for 20 min at 150°C in air. The back gate contact to a gold pad on an alumina ceramic chip carrier was made with silver paste. Immediately after depositing a drop of ionic liquid diethylmethyl(2-methoxyethyl)ammonium bis(trifluoromethylsulfonyl)imide (DEME-TFSI) to cover both the device and the surrounding side gate, the samples were loaded into the dilution refrigerator system then vacuum-pumped overnight to minimize contamination of the ionic liquid by water from exposure to air.

The ionic liquid gate voltage V_GIL was slowly ramped up to desired value at room temperature, followed by several minutes of stabilization and then rapid-cooling the measurement probe below the freezing point of DEME-TFSI (220 K).Typical measured 2-terminal resistance per successful ohmic contact was 3 to 10 kilohm, which includes a 2- to 3-kilohm contribution from the measurement lines and built-in radio frequency filters in the probe. All measurements presented in the main text and in the Supplementary Materials were performed in a four-probe configuration (with the exception of fig. S9). Measurements were performed by voltage-sourcing nominal AC and DC excitations (${\mathit{V}}_{\text{AC}}^{*}$ and ${\mathit{V}}_{\text{DC}}^{*}$) through an adder circuit and measuring the drained current (I_AC and I_DC). V_QPC and V_DC refer to the AC and DC components of the voltage drop across the weak link measured at the adjacent voltage probes (as shown in Fig. 1B). The constriction resistance is R = 1/G = dV_AC/dI_AC. V_lead is the AC voltage drop between the next adjacent pair of voltage probes. The resistance of the unpatterned 2DES is R_lead = V_lead/I_AC.

Measured R contains two contributions: the resistance of the constriction itself (tuned by V_G12) and a series resistance R_S. Unless specified, R and G are shown as measured. In selected plots of G in the normal state, a specified V_G12-independent value of R_S was subtracted from the measured R. As discussed in more detail in section S5, R_S was chosen to match the plateau structure to integer multiples of e²/h, with R_S/R_lead ≈ 2 to 3. On the basis of the device geometry, R_S is expected to be approximately equal or larger than R_lead.

The Supplementary Materials to this report present extensive additional discussion and characterization of a total of six devices fabricated on three different STO chips. The contents of each supplementary section are briefly summarized below to facilitate their navigation. Section S1: theoretical framework for the ballistic SNS constriction model, I_C dependence on τ_NS, L, and V_G12, I_EXC dependence on τ_NS. Section S2: description of additional devices and their fabrication. Section S3: tuning of carrier density and superconductivity with V_GIL. Section S4: V_G12 and V_BG effect on R, R_lead, B_c, T_c, I_c, and Hall density. Additional characterization of transport through the constriction in the normal state (section S5), in the superconducting state (section S6), and in the tunneling and accidental Coulomb blockade regimes (section S7). Section S8: discussion of the different methods to extract the superconducting gap Δ and the associated uncertainty.

Supplementary Materials

This PDF file includes:

Supplementary Text

Sections S1 to S8

Figs. S1 to S40

References