Theoretical considerations

Let us consider an QD-S-QD device within the Landauer formalism. Taking that the non-local transport is primarily coherent and that the electron energies are smaller than the superconducting gap, ∣E∣ < Δ, we find, see Supplementary Note 4, that the EC, τ_EC(E), and CPS, τ_CPS(E), probabilities are given by the expressions

Experiment

Now, we turn to experimental realization of CPS and EC. Several material platforms have been employed in the experiments^30–37, and particularly promising results have been obtained in carbon nanotube, graphene, and nanowire settings, where the splitting efficiencies approaching 90% have been observed. Our present device, depicted in Fig. 1, consists of an Al superconducting injector in contact with two graphene quantum dots. Two side gate electrodes allow us to tune the resonance levels of the dots independently. In order to perform thermoelectric measurements, our device additionally contains two thermometers and a resistive heater, fabricated from a graphene monolayer. The thermometers are superconductor–graphene–superconductor (SGS) Josephson junctions that reveal local temperature through the temperature dependence of the switching current, I_sw(T)³⁸. The resistive heater comprises the graphene nanoribbon and two attached aluminum leads. The heater is distinctly apart and electrically isolated from the rest of the device, the heat to the Cooper pair splitter being transmitted through the substrate.

Fig. 1

False color SEM image of the device.

Green indicates graphene, blue corresponds to metallic Al/Ti sandwich leads, and the silicon substrate with 280-nm-thick silicon dioxide on top is colored in gray. The Joule heated region is indicated by red color. The superconducting graphene junctions are located between the leads marked by SGS_L and SGS_R. The left and right graphene quantum dots QD_L and QD_R, respectively, have an area 200 × 150 nm², foremost located under the Al injector and thus invisible in the image. Side gates with voltages V_sg,L and V_sg,R are also carved out of graphene. The inset at lower left corner illustrates the graphene quantum dots before overlaying the metallic Cooper pair injector. In the thermoelectric measurements, the quantum dot currents are tracked by current preamplifiers connected to leads 1 and 3 (virtual ground 20 Ω), while the Cooper pair injector, lead 2, is grounded.

The temperature difference ΔT = T_L − T_R between the leads of the two-terminal device induces the thermoelectric current I = αGΔT, where G is the conductance of the device and α is the Seebeck coefficient³⁹. For typical metals, such as aluminum or copper, the Seebeck coefficient is quite small, α ~ 3−7 μV/K. For graphene, α is inversely proportional to the square root of charge density, and it can reach much higher values close to the charge neutrality point^40,41. In quantum dots with energy-dependent electron transmission probability⁴², and in superconductor–ferromagnet tunnel junctions²⁵ large α up to a few k_B/e ~ 100 μV/K can be achieved. In our experiment, we observe similar values of the Seebeck coefficient in graphene quantum dots. We operate the graphene heater at frequency f = 2.1 Hz and record thermoelectric currents through both quantum dots at the double frequency 2f (see “Methods”). Thermal gradient induced by the heater is measured by SGS thermometers, which were calibrated separately as discussed in Supplementary Note 2.

The thermoelectric current induced by the heater in the left (right) quantum dot is given by the sum of dominating local ($$ I_{\mathrm{L(R)}}^{\mathrm{loc}}$$) and small non-local contributions ($$ \mathrm{\Delta }{I}_{\mathrm{L(R)}}^{\mathrm{nl}}$$), $$ I_{\mathrm{L(R)}}=I_{\mathrm{L(R)}}^{\mathrm{loc}}\left(V_{\mathrm{sg,L(R)}}\right)+\mathrm{\Delta }{I}_{\mathrm{L(R)}}^{\mathrm{nl}}\left(V_{\mathrm{sg,L}},V_{\mathrm{sg,R}}\right)$$, see Supplementary Note 4. To infer the non-local contribution from the measured current I_L(R), we subtract off its slowly varying average local background, 〈I_L(R)〉 (see “Methods”):

Fig. 2

Local and non-local contributions to the thermoelectric current.

a Thermally generated current at 2f in the left and right dot measured as a function of V_sg,L and V_sg,R; the data include both local and non-local thermoelectric contributions. b, c Zero bias conductance of the right quantum dot vs. V_sg,R in two intervals: −1.35 V < V_sg,R < −1.15 V, and −0.95 V < V_sg,R < −0.6 V; green arrow in the b points to the minor peak in the vicinity of the main conductance peak, and the red curve is the fit by the Fano resonance model with the parameters Γ_R = 20 μeV, γ_R = 252 μeV, $$ {\epsilon }_{\mathrm{R}}-{\epsilon }_{\mathrm{R}}^{\prime }=120$$ μeV, t_R = 55 μeV; the additional Fano peak in the fit is intentionally made stronger than the experimentally observed one in order to better reproduce the behavior of the non-local thermoelectric current in Fig. 3d. d Zero bias conductance of the left quantum dot in the interval −5 V < V_sg,L < −4.6 V. e, f Experimental non-local contribution to the thermal current in the left quantum dot, $$ \mathrm{\Delta }{I}_{\mathrm{L}}^{\mathrm{nl}}$$, in two selected regions of (V_sg,R, V_sg,L) plane. g Zero bias conductance of the left quantum dot; data as in d, but the red curve displays the fit of one of the peaks with the Fano resonance model with the parameters Γ_L = 6 μeV, γ_L = 98 μeV, $$ {\epsilon }_{\mathrm{L}}-{\epsilon }_{\mathrm{L}}^{\prime }=24$$ μeV, and t_L = 10 μeV. h, i Theoretically predicted non-local contribution $$ \mathrm{\Delta }{I}_{\mathrm{L}}^{\mathrm{nl}}$$ in the same regions as in e, f.

Before proceeding to our main result, note that some conductance peaks are split into two closely located peaks (see Fig. 2b, c). The splitting is explained by the Fano resonant effect, see Supplementary Note 4. Namely, we introduce the coupling rates Γ_j,n and γ_j,n (here j = L, R enumerates the dots) between the nth energy level of the dot (with energy ε_j,n) and, respectively, normal and superconducting leads; we also assume that the nth level is coupled to a dark energy level, having the energy $$ {\epsilon }_{j,n}^{\prime }$$, via the hopping matrix element t_j,n. This results in the transmission probabilities of the dots $$ {\tau }_j={\sum }_n{\gamma }_{j,n}{\mathrm{\Gamma }}_{j,n}/\left[{\left(E-{\epsilon }_{j,n}-\mid t_{j,n}{\mid }^2/\left(E-{\epsilon }_{j,n}^{\prime }\right)\right)}^2+{\left({\gamma }_{j,n}+{\mathrm{\Gamma }}_{j,n}\right)}^2/4\right]$$, see Supplementary Note 4. The conductances $$ g_{\mathrm{L}}\left(V_{\mathrm{sg},j}\right)=4{\tau }_j^2\left(0,V_{\mathrm{sg},j}\right)/{\left[2-{\tau }_j\left(0,V_{\mathrm{sg},j}\right)\right]}^2$$, as predicted by the theory of Andreev reflection⁴³, exhibit splitted peaks for t_j,n ≠ 0. In our model, we ignore Coulomb interaction because the charging energies of the dots are relatively small, E_C,j ≲ γ_j,n + Γ_j,n.

Non-local thermoelectricity

Figure 3 displays the main result of our study. There we plot the non-local thermal currents for both quantum dots together with the theory predictions based on Eqs. (1) and (2). The involved model parameters are chosen in such a way that, besides accounting well for the non-local current, they can also reasonably fit the conductance peaks (see the caption of Fig. 2). In the experiment, the non-local current $$ \mathrm{\Delta }{I}_{\mathrm{R}}^{\mathrm{nl}}$$ changes its sign three times in the vicinity of the conductance peak of the right dot located at V_sg,R = −1.24 V. Although in order to reproduce this behavior we had to take hopping amplitude, t_R, larger than required by the perfect fit to the conductance peak (see Fig. 2b), this offers a fair cross-check for our description. One sees that not only the magnitudes of the currents $$ \mathrm{\Delta }{I}_{\mathrm{L}}^{\mathrm{nl}}$$ and $$ \mathrm{\Delta }{I}_{\mathrm{R}}^{\mathrm{nl}}$$ are in good agreement with the theory, but their symmetric and anti-symmetric combinations $$ \mathrm{\Delta }{I}_{\mathrm{CPS}}=\mathrm{\Delta }{I}_{\mathrm{L}}^{\mathrm{nl}}+\mathrm{\Delta }{I}_{\mathrm{R}}^{\mathrm{nl}}$$ and $$ \mathrm{\Delta }{I}_{\mathrm{EC}}=\mathrm{\Delta }{I}_{\mathrm{L}}^{\mathrm{nl}}-\mathrm{\Delta }{I}_{\mathrm{R}}^{\mathrm{nl}}$$ exhibit the expected gate voltage dependence, although the comparison for I_EC is hampered by large noise as the EC current is a difference between two small non-local signals. The observed general agreement in Fig. 3 provides strong support of the non-local coherent thermoelectric effect in our device.

Fig. 3

Interplay of non-local thermoelectric currents in the left and right dots.

Non-local contributions to the thermal currents of the right and left dots (a–d) and the corresponding CPS and EC currents (e–h). The left column (graphs a, c, e, g) shows the experiment and the right one (graphs b, d, f, h)—theory. Horizontal dotted lines in the plots indicate the positions of the conductance peak maxima of the left dot.

Local thermoelectricity

Since, as noted, the non-local currents are relatively small, one can treat the measured currents foremost as local, $$ I_{\mathrm{L(R)}}\simeq I_{\mathrm{L(R)}}^{\mathrm{loc}}$$. The measured thermoelectric current of the left quantum dot is shown in Fig. 4a. The lowest curve in this panel shows the dimensionless conductance of the left quantum dot g_L = hG_L/e² as a function of the side gate voltage V_sg,L. Thermoelectric current I_L, depicted by the upper curves of Fig. 4a, varies with the same period as the conductance. Its magnitude grows with the increasing heating power P as $$ I_{\mathrm{L}}^{\max }\propto P^{1/3}$$, which is consistent with G_th ∝ T³ for the thermal conductance between electrons in graphene and phonons in the substrate. The maximum thermal power of the left quantum dot reaches a value of $$ {\alpha }_{\max }=\max \left\{I_{\mathrm{L}}/G_{\mathrm{L}}\left(T_{\mathrm{L}}-T_{\mathrm{S}}\right)\right\}\approx 250$$ μV/K, which is close to the values reported in ref. ⁴². Since we cannot reliably measure the temperature of the superconductor T_S, we set T_S = (T_L + T_R)/2 in evaluating $$ {\alpha }_{\max }$$ and in our theory modeling. In Fig. 4a, we also show the local thermoelectric current predicted by the theory of Andreev reflection with energy-dependent transmission probability⁴⁴; the same theory was earlier employed in deriving the non-local contributions using Eq. (2). In the local case, only those quasiparticles with energies above the superconducting gap, ∣E∣ > Δ, contribute. The zero temperature value of the gap Δ₀ is set by the transition temperature T_c = 1.0 K of the Al/Ti leads, and the transmission probability of the dot τ_L(E, V_sg,L) is inferred from the experimentally measured conductance g_L(V_sg,L), as explained in Supplementary Note 5. We find rather good agreement between theory and experiment except for the lowest values of the heating voltage. This agreement provides further confirmation for our model.

Fig. 4

Drive dependence of thermoelectric current.

a Upper curves: thermoelectric current in the left quantum dot I_L,exp vs. gate voltage V_sg,L measured at heating voltages V_h = [5, 9, 19, 25, 29] mV, where V_h = 5 mV is the blue curve and 29 mV is the red curve. We estimate the induced temperature difference between the left and right quantum dots to be T_L − T_R ≃ 17 mK for V_h = 5 mV, and T_L − T_R ≃ 59 mK for V_h = 29 mV. Middle curves: theory predictions based on the coherent model (see Supplementary Note 4) for the thermoelectric current I_L,theory plotted in the same manner as the upper curves for V_h = [5, 10, 20, 25, 30] mV. The gap of Al/Ti leads is set to Δ₀ = 150 μeV at T = 0 while the BCS gap formula Δ(T_S) with T_S = (T_L + T_R)/2 defines the T dependence. Lowest curve: experimental conductance of the left quantum dot. b Incoherent modeling for the low temperature regime: theoretical fits (dashed) to the measured thermoelectric currents I_L and conductance g_L (solid) at V_h = 7 and 11 mV, blue and red curves, respectively. The model is based on the assumption that the system may be split into the coherent subsystems, which, in turn, are joined incoherently into a circuit. Details of the model and fitting parameters are given in Supplementary Note 5.

In the low temperature regime, the coherent model predicts very small current due to lack of quasiparticles, while the experimental thermoelectric current remains significant and exhibits additional sign changes in the vicinity of some of the conductance peaks. These features can originate from non-zero, thermally induced voltages across the dots. To capture these effects, we propose that electrons may undergo quick inelastic relaxation, see Supplementary Note 5. This introduces incoherent effects that facilitate description of quantum dots and NS interfaces as independent conductor elements connected in series. The results of such an inelastic model are shown in Fig. 4b. The incoherent description accurately predicts the character of the local thermoelectricity at small temperatures. Incidentally, although at odds with the effect of local thermoelectricity, the non-local currents are dominantly determined by coherent electrical transport.

Alternatively, the additional peak in the local thermoelectricity could originate from Coulomb blockade^3,5 as the non-local thermoelectric effect is shown to develop from single peak to double peak structure when temperature is lowered⁵. The calculated double peak structure is similar to our local thermoelectric signal observed at low temperatures. However, the resonance energy dependence of the non-local signal with Coulomb blockade agrees poorly with the experimental results in Fig. 3a in comparison with our coherent transport model.

Samples and fabrication

Our graphene films were manufactured using mechanical exfoliation of graphite (Graphenium, NGS Naturgraphit GmbH) and placed on a highly p⁺⁺ doped silicon wafer, coated by 280-nm-thick thermal silicon dioxide. The conducting substrate was employed as a backgate for coarse tuning of the graphene quantum dots, while fine tuning was performed by adjusting the side gates. Electron beam lithography (EBL) on PMMA resist was used to pattern a mask for plasma etching of the graphene structures. A second EBL step was carried out to expose the pattern for electrode structures, followed by deposition of Ti/Al (5/50 nm, superconducting T_c = 1.0 K) leads using an e-beam evaporator. Normal contacts to the graphene quantum dots were made using etched graphene nanoribbons with a small number of conductance channels at the operating point⁴⁵.

The strong p⁺⁺ doping and the interfacial scattering at the Si/SiO₂ interface reduce the phonon mean free path in the substrate to one micron range, which facilitates the use of the heat diffusion equation for estimating thermal gradients along the substrate near the graphene ribbon heater and the splitter. Heat transport analysis was done separately for each component involved in the operation of the CPS, as well as a COMSOL simulation, see Supplementary Note 1.

Measurement scheme

Our conductance and thermoelectric current measurements employed regular lock-in techniques at low frequencies. In the thermoelectric experiments, we had one DL1211 current preamplifier connected to each quantum dot, while the superconductor was grounded on top of the cryostat. The current gain of DL1211 amplifier was set to 10⁶ V/A, which provides a virtual ground of 20 Ω. The lead resistance including filters was approximately 100 Ω, which is much less than the quantum dot resistance ~h/e², the quantum resistance. The galvanically separated heater was driven at f = 2.1 Hz, with an ac voltage amplitude ranging between 1 and 40 mV (for data without galvanic separation, see Supplementary Note 3). Because the resistance R of the graphene ribbon heater was independent of temperature in its regime of operation, the heating power $$ P=V_{\mathrm{h}}^2/R$$ was fully governed by the voltage V_h. The heating power oscillated at frequency 2f = 4.2 Hz, which resulted in thermoelectric currents at 4.2 Hz, recorded using a lock-in time constant of 1 s. The thermal response time of our device appears to be well below 1 ms, i.e., much less than a measurement period, so that the thermal response is not suppressed. The use of such a low frequency for the experiments was dictated by the need to eliminate the capacitive coupling between the wires in the measurements.

The local temperature was monitored using two SGS junctions. At low temperature, because of the proximity effect, graphene becomes superconducting, with a supercurrent exponentially proportional to temperature: $$ ~\exp \left(-T/E_{\mathrm{Th}}\right)$$. Here $$ E_{\mathrm{Th}}=ħD/L_{\mathrm{SGS}}^2$$ stands for the Thouless energy given by the length of the SGS section L_SGS and the diffusion constant $$ D~\frac{1}{2}{v}_{\mathrm{F}}\lambda $$, where the Fermi velocity v_F = 8 × 10⁵ m/s and λ is the charge carrier mean free path of graphene. Using λ ≃ 20 nm for graphene on SiO₂ and L_SGS = 200 nm, we estimate E_Th ≃ 130 μeV for our SGS thermometers. This kind of SGS junctions in the intermediate length regime (E_Th ≃ Δ) were experimentally found to provide good thermometers over the relevant range of temperatures in our work.

We have used the amplitude of the differential resistance peak $$ R_{\mathrm{d}}^{\max }$$ vs. T to infer the effective local temperature within the graphene sample. The SGS temperature under the voltage bias V_h in the graphene ribbon heater was obtained by direct comparison between $$ R_{\mathrm{d}}^{\max }$$ and the heating power to the value of $$ R_{\mathrm{d}}^{\max }$$ recorded when varying the cryostat temperature. As detailed in Supplementary Note 2, we obtain the relation $$ T_{\mathrm{SGS,L}}=9.1\times V_{\mathrm{h}}^{0.70}+90$$ mK between the SGS_L temperature and the heating voltage (V_h in Volts). For the SGS_R thermometer, we obtained an estimate $$ T_{\mathrm{SGS,R}}\simeq 8.4\times V_{\mathrm{h}}^{0.70}+90$$ mK.

Background subtraction

The direct experimental measurement allows to obtain the total electric current through the left (right) dot, I_L(R), constituting the sum of local, $$ I_{\mathrm{L(R)}}^{\mathrm{loc}}$$, and non-local, $$ \mathrm{\Delta }{I}_{\mathrm{L(R)}}^{\mathrm{nl}}$$, contributions. Note that, in theory, while the non-local contribution depends on both gate voltages, the local one is determined only by the gate voltage on the corresponding dot. This suggests that the local current through the left (right) dot $$ I_{\mathrm{L(R)}}^{\mathrm{loc}}$$ is nothing but the total current I_L(R) averaged over the gate voltage on the opposite dot; $$ \mathrm{\Delta }{I}_{\mathrm{L(R)}}^{\mathrm{nl}}$$ can be thus obtained by subtracting off this average background from I_L(R). On practice, however, the gate electrodes may be subject to cross-talk, which the described simple processing does not account for. For the most part, the cross-talk was eliminated by a small rotation of the data array. Bearing in mind the remaining residual cross-talk, we construct a slowly varying background $$ ⟨I_{\mathrm{L(R)}}\left(V_{\mathrm{sg,L}},V_{\mathrm{sg,R}}\right)⟩$$ in the following manner (for clarity, let us consider the case of the left dot): for any fixed $$ V_{\mathrm{sg,L}}=V_{\mathrm{sg,L}}^{\mathrm{F}}$$, $$ ⟨I_{\mathrm{L}}\left(V_{\mathrm{sg,L}}^{\mathrm{F}},V_{\mathrm{sg,R}}\right)⟩$$ is the linear fit of $$ I_{\mathrm{L}}\left(V_{\mathrm{sg,L}}^{\mathrm{F}},V_{\mathrm{sg,R}}\right)$$ as function of V_sg,R. The non-local contribution is then obtained using the formula

Theoretical modeling

Our theoretical calculations are based on both coherent and incoherent modeling of transport. In coherent modeling, we employ the Landauer approach with Andreev reflection^43,46 for calculating the local thermoelectric current; Lorentzian resonance line shapes are employed for transport in the quantum dots^35,44,47,48. For the non-local current, we employ a standard crossed Andreev reflection formalism^2,49–51. In our incoherent theory, based on scattering matrix formalism^1,4,52, we also include the influence of internal thermally generated current sources and their back-action effect owing to the environmental impedance caused by graphene ribbons. The inclusion of the back-action-induced voltage sources makes the incoherent calculation self-consistent. For details of the calculations, we refer to Supplementary Notes 4 and 5.