Electron pairing without superconductivity

Abstract

Strontium titanate (SrTiO3) is the first and best known superconducting semiconductor1. It exhibits an extremely low carrier density threshold for superconductivity2, and possesses a phase diagram similar to that of high-temperature superconductors3,4—two factors that suggest an unconventional pairing mechanism. Despite sustained interest for 50 years, direct experimental insight into the nature of electron pairing in SrTiO3 has remained elusive. Here we perform transport experiments with nanowire-based single-electron transistors at the interface between SrTiO3 and a thin layer of lanthanum aluminate, LaAlO3. Electrostatic gating reveals a series of two-electron conductance resonances—paired electron states—that bifurcate above a critical pairing field Bp of about 1–4 tesla, an order of magnitude larger than the superconducting critical magnetic field. For magnetic fields below Bp, these resonances are insensitive to the applied magnetic field; for fields in excess of Bp, the resonances exhibit a linear Zeeman-like energy splitting. Electron pairing is stable at temperatures as high as 900 millikelvin, well above the superconducting transition temperature (about 300 millikelvin). These experiments demonstrate the existence of a robust electronic phase in which electrons pair without forming a superconducting state. Key experimental signatures are captured by a model involving an attractive Hubbard interaction that describes real-space electron pairing as a precursor to superconductivity.

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Figure 1: Device schematic and transport characteristics.
Figure 2: Out-of-plane magnetic field dependence of device A transport characteristics at T = 50 mK.
Figure 3: Temperature dependence of Bp.
Figure 4: Comparison between experiment and attractive Hubbard model.

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Acknowledgements

We thank A. Akhmerov, A. Annadi, S. Frolov, R. Lutchyn, C. Nayak and D. Pekker for discussions. This work was supported by ARO MURI W911NF-08-1-0317 (J.L.), AFOSR MURI FA9550-10-1-0524 (C.-B.E., J.L.) and FA9550-12-1-0342 (C.-B.E.), grants from the National Science Foundation DMR-1104191 (J.L.), DMR-1124131 (C.-B.E., J.L.) and DMR-1234096 (C.-B.E.), and the Office of Naval Research through the Naval Research Laboratory’s Basic Research Program (C.S.H.).

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Authors

Contributions

G.C. and M.T. did most of the design and fabrication of the devices, performed the experiments and wrote the manuscript. S.L. and J.P.V. contributed to measurements. M.H. patterned the interface electrodes. P.I. contributed to manuscript writing and measurement set-up. S.R. and H.L. grew the samples. C.-B.E. supervised sample growth and reviewed the manuscript. C.S.H. performed theoretical calculations and co-wrote the manuscript. J.L. supervised all the related experiment procedures and co-wrote the manuscript.

Corresponding author

Correspondence to Jeremy Levy.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Nanoscale potential barrier engineering and low-temperature transport characteristics.

a, Single-barrier device schematic. It has the same structure as device A except that only one barrier is integrated in the design. b, Resistance change during barrier cutting (Methods); t is time. c, The differential conductance dI/dV as a function of Vsg at T = 75 mK. The wire can be pinched off by Vsg at the barrier site, as the wire conductance becomes negligibly small in lower Vsg values.

Extended Data Figure 2 Transport properties of three LQD = 500 nm SET devices of different single barrier heights at T = 50 mK.

Plots show colour-coded dI/dV as a function of Vsg and voltage across the QDs in devices F (V34F), G (V34G) and H (V34H). a, Device F (ΔR/2 = 20 kΩ) requires a back gate voltage Vbg applied on the substrate of −5.6 V to pinch off the device since Vsg has limited tunability due to leakage at high absolute values. b, Device G (ΔR/2 = 110 kΩ) shows similar properties to device A. c, Device H (ΔR/2 = 305 kΩ) shows no conductance diamonds.

Extended Data Figure 3 Transport characteristics due to barrier confinement of device A.

a, Zero-bias line cut of the dI/dV map in b; filled black symbols show positions of line cuts displayed in ce. c–e, Full suppression of transport in device A at Vsg = −25 mV (c), resonant tunnelling transport at Vsg = −19 mV (d), and fully superconducting transport at Vsg = 0 mV (e).

Extended Data Figure 4 Global shift correction of data from device A.

a, Original data for Figs 2f and 4a in the main text; B, applied magnetic field. The global shift is illustrated by the red trace, δVsg(B). b, The same data shown in Figs 2f and 4a in the main text that are corrected by Vsg(B) = Vsg0(B) − δVsg(B).

Extended Data Figure 5 Three-dimensional ‘waterfall’ plot of the magnetic-field dependence of ZBPs (same data as shown in Fig. 2f).

Plot shows lock-in dI/dV data at small (100 μV) bias as a function of Vsg, taken as the magnetic field is swept from −9 T to 9 T (additional right-axis).

Extended Data Figure 6 Transport characteristics of devices B, C, D and E, which are all of the same geometry as device A.

Device letter is shown at lower right-hand corner of all plots. a, Device B dI/dV (colour coded) as a function of Vsg and V34 at Vbg = −0.7 V and T = 100 mK. A small gap (4Δ) close to zero-bias in the diamonds is due to the absence of normal carriers in the superconducting source/drain leads. b, Device B ZBP splitting in an out-of-plane magnetic field B, which is the same data as Fig. 3a in the main text. c, Device C dI/dV as a function of Vsg and V34 at Vbg = −4.4 V and T = 100 mK. d, Device C ZBP splitting in an out-of-plane magnetic field. e, Device D dI/dV as a function of Vsg and V34 at Vbg = −1.4 V and T = 100 mK. f, Device D ZBP splitting in an out-of-plane magnetic field. g, Device E dI/dV dependent on Vsg and V34 at Vbg = −2.2 V and T = 100 mK. h, Device E ZBP splitting in an out-of-plane magnetic field.

Extended Data Figure 7 Parity effect.

a, Energies Ei of the Hubbard model (equation (1) in main text) of a one-dimensional 16-site chain with open boundary conditions, t = 1 meV, U = −0.8 meV, and B = 0 for fillings Ne ≤ 16. The slope of each line is proportional to −Ne; red (blue) lines have even (odd) Ne. For all chemical potentials µ, the ground state has even Ne. b, Energies of the Hubbard model for the same parameters as a shifted by a quadratic function of µ, Ei*(µ) = Ei(µ) + 2, where c is arbitrary. The lowest energy for each value of µ is easier to discern. The ground state always has even Ne.

Extended Data Figure 8 Phase diagram of the Hubbard model on a one-dimensional 16-site chain with t = 1 meV and U = −0.8 meV.

The total number of electrons Ne and total spin S are labelled for some of the larger phases as Ne(S). The quantum numbers of the other phases can be deduced from their neighbours.

Extended Data Table 1 Parameters of eight devices

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Cheng, G., Tomczyk, M., Lu, S. et al. Electron pairing without superconductivity. Nature 521, 196–199 (2015). https://doi.org/10.1038/nature14398

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