Tunable Electron-Electron Interactions in ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ Nanostructures

The interface between the two complex oxides ${\mathrm{LaAlO}}_3$ and ${\mathrm{SrTiO}}_3$ has remarkable properties that can be locally reconfigured between conducting and insulating states using a conductive atomic force microscope. Prior investigations of “sketched” quantum dot devices revealed a phase in which electrons form pairs, implying a strongly attractive electron-electron interaction. Here, we show that these devices with strong electron-electron interactions can exhibit a gate-tunable transition from a pair-tunneling regime to a single-electron (Andreev bound state) tunneling regime where the interactions become repulsive. The electron-electron interaction sign change is associated with a Lifshitz transition where the $d_{xz}$ and $d_{yz}$ bands start to become occupied. This electronically tunable electron-electron interaction, combined with the nanoscale reconfigurability of this system, provides an interesting starting point towards solid-state quantum simulation.

Popular Summary

Superconductors are materials in which electrons overcome their natural tendency to repel one another. This mutual attraction among electrons leads to their bunching into pairs, and the coordinated motion of these pairs results in the formation of a superconducting state in which electrons carry current without resistance. The theory of superconductivity, originally formulated by Bardeen, Cooper, and Schrieffer (BCS), describes how very weak attractive interactions due to electron-phonon coupling can lead to a superconducting state. However, so-called “high-temperature” superconductors do not behave according to the BCS paradigm; their electron pairing is exceptionally strong, with a physical origin that has not yet been determined. Electron pairing is known to be exceptionally strong in strontium titanate (${\mathrm{SrTiO}}_3$), a material that exhibits superconductivity at record-low electron densities. For ${\mathrm{SrTiO}}_3$, it is known (and suspected for high-temperature cuprate superconductors) that electrons can pair at temperatures and magnetic fields far outside the superconducting regime. Here, we show that the electron-electron interactions can be tuned from attractive to repulsive in a superconducting single-electron transistor fabricated at a ${\mathrm{SrTiO}}_3$-based interface.

${\mathrm{SrTiO}}_3$-based heterostructures and nanostructures exhibit superconducting critical temperatures of roughly $T_c\sim 0.3\mathrm{K}$. In this regime, we observe electron-pair tunneling for low electron densities through a quantum dot, and Andreev bound states that are a clear manifestation of repulsive electron-electron interaction at high densities. We attribute the sign change of the electron-electron interaction to a Lifshitz transition at which electron subbands with a different electron orbital character emerge.

Our work, which shows that control can be exerted over electron-electron interactions in a reconfigurable, two-dimensional interface, provides new insights into the mechanism of superconductivity in ${\mathrm{SrTiO}}_3$ and helps pave the way for the development of a solid-state quantum simulator with controllable electron-electron interactions.

Figure 1

Superconducting single electron transistor (SSET). (a) The excitation spectra of a QD depends on the sign of the interaction strength $U$. When $U<0$ (top two panels), the two-electron ground state (top left-hand panel) is lower than the one-electron ground state. When $U>0$ (bottom two panels), the one-electron ground state is lowest (bottom right-hand panel). (b) Electron-electron interactions are probed by a SSET fabricated by c-AFM lithography. The nanowire QD is defined by two barriers between leads 3 and 4 separated by $1\mu \mathrm{m}$. A side gate tunes the chemical potential of the QD.

Figure 2

Transport characteristics. At $T=50\mathrm{mK}$, $dI/dV$ is measured as a function of $V_{34}$ and $V_{\mathrm{sg}}$ at (a) $B=0\mathrm{T}$ and (b) $B=1\mathrm{T}$. The dashed line in (a) is a guide to the eye showing how the diamonds are offset. The fact that the diamonds can be connected by a straight line indicates that one lead has a gap while the other is not gapped. The red arrow indicates the location of zero-bias peak. (c) Zero-bias line cuts at $B=0–4\mathrm{T}$ in low $V_{\mathrm{sg}}$ range ($-60<V_{\mathrm{sg}}<-35\mathrm{mV}$). The ZBPs bifurcate above $B_c$ ($1–2\mathrm{T}$), signifying pair tunneling. Curves are shifted by $1.16\mu \mathrm{S}$ starting from $B=4\mathrm{T}$ data for clarity. (d) Zero-bias line cuts at $B=0–4\mathrm{T}$ in high $V_{\mathrm{sg}}$ range ($-30<V_{\mathrm{sg}}<-10\mathrm{mV}$). The ZBPs do not bifurcate, signifying single-electron tunneling. Curves are shifted by $7.75\mu \mathrm{S}$ starting from $B=4\mathrm{T}$ data for clarity.

Figure 3

Simulation of pair conductance diamonds on varying gapped excitations in the leads. (a) When both source and drain leads only have gapped excitations, the diamonds are offset away from the gapless excitations indicated by the dashed lines. An insulating gap of $4({\mathrm{\Delta }}_s+{\mathrm{\Delta }}_d)/e$ appears between the tips of diamonds, where ${\mathrm{\Delta }}_s$ and ${\mathrm{\Delta }}_d$ are the pairing gaps of source and drain leads. (b) When the drain lead has gapless excitations, one side of the diamonds stays connected by a straight line. Note electron pairs can still tunnel through the device when $|V_{34}|<2{\mathrm{\Delta }}_s/e$, as shown in the conductance peak at zero-bias in the bottom panel. $\alpha$ on the $x$-axes is the lever arm ratio converting $V_{\mathrm{sg}}$ to energy.

Figure 4

Theoretical calculation of DOS spectra in a single-level QD in the presence of (a) attractive ($U=-4\mathrm{\Delta }$) and (b) repulsive ($U=2\mathrm{\Delta }$) electron-electron interaction. For the case (a) of strong attractive interactions, the two-electron “$\mathsf{X}$”-shaped resonances are dominant, whereas for case (b) of strong repulsion, the dominant subgap “loop” features are one-electron resonances with Andreev bound states.

Figure 5

Comparison between data and calculation. (a) Magnified data plot in $-33<V_{\mathrm{sg}}<-19\mathrm{mV}$. (b) Calculation of the DOS on the QD in the same $V_{\mathrm{sg}}$ range. The QD is restricted to 4 levels, with negative (positive) interaction for the bottom (upper) 2 levels in band 1 (2).

Figure 6

Low-field dependence of ABS. (a)–(h) ABS loops at $B=0$, 0.06, and 0.18–0.78 T in steps of 0.12 T. (i) Average vertical line cuts (averaged in $-14<V_{\mathrm{sg}}<-11\mathrm{mV}$). Curves are shifted for clarity. (j) Extracted pairing gap size as function of $B$.

Figure 7

RCSJ model fitting. (a) Schematic. (b) RCSJ fitting of $I\text{−}V$ curve at $V_{\mathrm{sg}}=0\mathrm{mV}$ yielding $I_c=2.8\mathrm{nA}$, $R_J=40.4\mathrm{k}\mathrm{\Omega }$, and $R_L=5.0\mathrm{k}\mathrm{\Omega }$.