Transport and excitations in a negative-U quantum dot at the LaAlO3/SrTiO3 interface

In a solid-state host, attractive electron–electron interactions can lead to the formation of local electron pairs which play an important role in the understanding of prominent phenomena such as high T c superconductivity and the pseudogap phase. Recently, evidence of a paired ground state without superconductivity was demonstrated at the level of single electrons in quantum dots at the interface of LaAlO3 and SrTiO3. Here, we present a detailed study of the excitation spectrum and transport processes of a gate-defined LaAlO3/SrTiO3 quantum dot exhibiting pairing at low temperatures. For weak tunneling, the spectrum agrees with calculations based on the Anderson model with a negative effective charging energy U, and exhibits an energy gap corresponding to the Zeeman energy of the magnetic pair-breaking field. In contrast, for strong coupling, low-bias conductance is enhanced with a characteristic dependence on temperature, magnetic field and chemical potential consistent with the charge Kondo effect.

Current-voltage characteristics are measured to find the critical temperature T c and critical magnetic field B z c of the source and drain leads. A current is applied using a 10 MΩ bias resistor, and the voltages V L , V R are measured across the side leads of the device, as indicated in Fig. 1 of the manuscript.
Supplementary Fig. 1 shows the current-voltage characteristics for varying temperatures, and 2 for varying perpendicular magnetic field. The red curve indicates a current sweep from a negative to a positive value, while the blue curve indicates a sweep in the opposite direction. At the critical current I c the device switches from a superconducting to a resistive state. I c is extracted from the data using a threshold detection. The data is fitted to the Ambegaokar-Baratoff relation, based on two weak links with different parameters for T c and R JJ . More details on the hysteretic nature of the IV curve and the theory fit to I c (T ) can be found in [1].
We find a critical temperature T c ≈ 270 mK and out-of-plane critical magnetic field B z c ≈ 90 mT, similar to previous studies of the room-temperature grown LAO/STO samples [1]. From measurements of samples grown in similar growth conditions and sheet resistance, we estimate the in-plane critical magnetic field B y c ≈ 2 T.  4 show the full-range bias spectroscopy for varying external magnetic fields for V g <-0.6 V. We find a negative charging energy U ≈ −160µeV (white arrows) and diamond height ∆E ≈ 600 µeV (red and orange arrows in Supplementary Fig. 3) which is associated with the level spacing ∆E and charging energy E C . Because of our split-gate device geometry, as dot occupation is increased, simultaneously the tunnel barriers are lowered and tunnel coupling between the dot and the reservoirs is increased. This becomes evident from the closing of the transport gap around zero bias as gate voltage is increased, shown in Supplementary Fig. 4.  In Supplementary Fig. 5 a zoom-in is shown for the first odd diamond at B = 6 T. The lever arm α g of the top gates is extracted from the height and width of the diamond, which are indicated in the figure by arrows, with the gate coupling coefficient α = C g /C Σ = ∆V sd ∆Vg ≈ 0.005. Measurements in the x and z-directions revealed that B x,y p > 1, T, beyond our current range for the 6,1,1 vector magnet. This shows that even though the critical fields of the superconducting state of the leads are much lower (by a factor ∼ 10) in the perpendicular direction, no such strong reduction occurs for B p (the details of the measurements will be reported elsewhere). Importantly, we have in general found no measurable consequences of superconductivity in the weak coupling regime supporting the use of normal metal reservoirs in the rate-equation model used to calculate the bias-spectroscopy in the weak coupling regime (Fig. 2 of main manuscript).
The magnetic field dependence of the conductance peaks is measured for the three axis directions B x , B y and B z using a 6-1-1 vector magnet. Supplementary figure 6 shows the full range for the B y dependence from which the conductance peak at V g = -0.92 V was studied in the manuscript. The B x and B z dependencies did not reveal any splitting of the conductance peaks.
The magnetic field dependence of the conductance peaks at zero bias voltage V sd for varying magnetic field. Each trace is taken at a finite magnetic field By from 0 to 6 T and offset vertically. The dependence of the first conductance peak on magnetic field in the y and z-direction, measured in two subsequent cool downs of the same device. Each trace is taken at a finite magnetic field By from 0 to 6 T and offset vertically. c,d The magnetic field dependence of the peak positions, each peak indicated by , , respectively, as indicated in panels (a, b). The points are then fit to two linear functions, indicated by red lines, which intersect at the pair-splitting field B y p and B z p . A more detailed fit is shown in Supplementary figure 9 In a subsequent cool down, the sample was rotated such that B x could be increased up to 6 T. The magnetic field dependence of the conductance peak at V g = -.92 V and extracted peak positions are shown in Supplementary Fig.  7. The pairing field at which the peaks split up is found using linear fitting of the data above B = 2 T. Regions in which the peaks strongly deviate from linearity in V g due to gate switching have been omitted from the fitting range. The values of the pairing field at which the linear fits cross are found to be B y p = 1.8 T and B z p = 2.06 T. It seems that in the closed regime of the dot, no strong angle dependence on magnetic field is present. However, the thermal cycle may have altered the disorder potential and important parameters of the sample that may alter U and/or g. A heat map representation of the data presented in Supplementary figure 7 (a, b). The color scale indicates the measured conductance with sweeping the gate voltage Vg on the x-axis and magnetic field By (left panel) and Bz (right panel) on the y-axis. All transport calculations shown in this work are based on the single-orbital Anderson model. This is in a sense the simplest possible model which can exhibit the desired physics and has only a single orbital which can be occupied with one electron of each spin projection, and with a double occupation associated with an additional charging energy U . This model has been used in many theoretical studies of electron transport in quantum dots, and the only difference here is that we take U < 0. The Hamiltonian of the quantum dot coupled to the leads is given by Here describes the quantum dot (QD), which has a single orbital that can be occupied by electrons with spin projection σ =↑, ↓ with associated occupation number n σ = d † σ d σ and energy ε σ , where ε ↑ − ε ↓ = gµ B B. U is the additional energy cost associated with double occupation of the dot orbital, which becomes negative in the case studied here. The source (r = s) and drain (r = d) are described by noninteracting electrons with n kσr = c † kσr c kσr . We assume that those states are always occupied according to a Fermi distribution, meaning that the source and drain are assumed to remain individually in equilibrium (at different chemical potentials) also when a source-drain voltage is applied. Finally, tunneling of electrons between the source/drain and the QD is described by the tunneling Hamiltonian For the transport calculations we use the real-time diagrammatic technique [2] to perform a perturbation expansion to order H 4 T r ∼ Γ 2 for the reduced density matrix of the QD, as well as the current through the dot [3,4]. This theory takes into account all coherent one-and two-electron tunneling processes, including, for example, singleelectron tunneling and cotunneling, as well as the electron pair tunneling terms investigated in [5]. We note that this transport theory takes the interaction U into account in a non-perturbative manner, which is crucial to capture the experimentally observed transport features.
As described in the manuscript, the experiment is compared to transport calculations [3,4]. Bias spectroscopy for varying magnetic field is calculated and shown in Supplementary figure 10 for negative U , for negative U and asymmetric tunnel coupling and for positive U . In the case of negative U, before a magnetic field is applied, the (0 → 1) transition is not allowed as accessing it requires a finite energy. This can either be achieved with a bias voltage V sd > U or a magnetic field such that E Z > U .
For asymmetric tunnel barriers, excited state tunneling originating from one lead is favored. This is evident from the simulations in Supplementary Fig. 10, indicated by "U < 0, asymmetric tunnel barriers" where Γ L = 10 × Γ R . In the case of positive U , the (0 → 1) transition dominates the (0 → 2) transition in terms of transport.

Supplementary Note 5. TEMPERATURE DEPENDENCE OF CONDUCTANCE PEAKS IN REGIMES OF ZERO AND FINITE MAGNETIC FIELD
To extract the temperature dependence of the peak height, we fit a multiple peak model to the data using the method of least squares. First, a peak finding algorithm is used to find the height and location V n of each peak n as an initial guess to the fitting procedure. The curves are then fitted to a multiple Gaussian model to provide the initial condition for a least squares fit to the following multiple peak model which is expected for a thermally broadened Coulomb Blockade peak in the sequential tunneling regime [6] (i.e. for B > B p ) and also agrees up to fourth order in with the theoretical lineshape ∝ δV g / sinh(δV g ) for pair tunneling [5]. The fit is excellent and the fitted peaks have amplitudes A n , positions V n and widths σ n for the n conductance peaks, α g = 0.0052 the gate voltage to energy conversion or lever arm. A and full width at half maximum (FWHM) are extracted from calculations and scaled to the data. For the sequential tunneling regime, the width relates to the electron temperature T el [6] (3.53k B T el = α g FWHM(V)) and for the B = 0 case where tunneling consists of both thermally excited sequential tunneling and pair tunneling [5] the FWHM scales linearly with T el [5]. Upon cooling the fitted FWHM stays constant until it saturates for T ≈ 100mK, which we attribute to a finite electron temperature in combination with lifetime broadening. In the case of U < 0, the charge Kondo affect can arise when the tunnel coupling Γ is sufficiently large such that the Kondo temperature k B T K = 1/2 √ ΓU exp[π 0 ( 0 + U )/ΓU ] > k B T sample . The charge Kondo effect manifests as a higher-order co-tunneling process that enhances conductance at low temperature at the degeneracy points of the even charge states.
In the main manuscript we presented detailed analysis of a single zero bias resonance. Not all resonances in the strong coupling regime show the conductance increase at low temperature. Interpreting the upturn as a consequence of the charge-Kondo model such behavior can be expected as the Kondo temperature depends on the coupling strength which naturally varies between resonances. In Supplementary Fig. 13 we present the corresponding data for a number of additional resonances as indicated in Supplementary Fig. 12. Supplementary Fig. 14 shows the details of the corresponding bias spectroscopy with the widths of the zero-bias resonance extracted both as a function of V sd and V g where the latter has been converted to energy using the gate-lever arm estimated from the individual bias spectra. In both cases the widths have been converted in to temperature to allow comparison with the characteristic temperature where the conductance starts increasing upon cooling. The theoretical treatment of the charge Kondo effect in negative-U quantum dots predicts that all three energy scales should be set by the Kondo temperature T K in reasonable agreement with our findings. We note, however, that the comparison with the theory is complicated by the finite magnetic field B z = 300 mT which is below the pairing field but required to avoid effect of superconductivity in the leads.  Supplementary  Fig. 13.  Supplementary Fig. 12. Panels a-d correspond to resonance A, e-h to resonance B, i-l to resonance C and m-p to resonance D. The first column (a, e, i, m) shows gate traces at V sd = 0 V for B = 0 (solid lines), B = 0.7 T (dashed lines), and B = 1.4 V (dotted lines). The second column (b, f, j, n) shows temperature dependence with the peak height extracted in third column (c, g, k, o). Column 4 (d, h, l, p) shows the conductance as a function of Vg and By  Supplementary Fig. 12. The characteristic detuning when the conductance has dropped by a factor of two is extracted as indicated after conversion into temperature by Boltzmanns constant. The gate-detuning width was converted into energy using the gate lever-arm determined from the bias spectroscopy in each case (first column: a, d, g, j).