Abstract
Generation of electric voltage in a conductor by applying a temperature gradient is a fundamental phenomenon called the Seebeck effect. This effect and its inverse is widely exploited in diverse applications ranging from thermoelectric power generators to temperature sensing. Recently, a possibility of thermoelectricity arising from the interplay of the nonlocal Cooper pair splitting and the elastic cotunneling in the hybrid normal metalsuperconductornormal metal structures was predicted. Here, we report the observation of the nonlocal Seebeck effect in a graphenebased Cooper pair splitting device comprising two quantum dots connected to an aluminum superconductor and present a theoretical description of this phenomenon. The observed nonlocal Seebeck effect offers an efficient tool for producing entangled electrons.
Introduction
Mesoscopic thermoelectric effects have been investigated in a variety of condensed matter systems that, besides fundamental normal metal–superconductor–normal metal (NSN) systems^{1,2,3,4,5}, also include quantum dots^{6,7,8,9,10}, atomic point contacts^{11,12,13}, Andreev interferometers^{14,15}, superconducting rings^{16} and nanowire heat engines^{17}. Thermoelectric effects in the superconducting systems^{18,19,20,21,22}, in particular those dealing with nonlocal thermoelectric currents in superconductor–ferromagnet devices^{23,24,25} and in bulk nonmagnetic hybrid NSN structures^{26,27,28} have attracted special attention. The connection between thermoelectric effects and the Cooper pair splitting (CPS)^{1,2}, proposed in ref. ^{29}, established a mechanism for the coherent nonlocal thermoelectric effect in hybrid superconducting systems. This connection was further studied and explicitly described for a ballistic NSN structure^{4}. It was revealed analytically in ref. ^{4} that the electric transport in the NSN structures depends on the elastic cotunneling (EC) process on par with the CPS. Contrary to intuitive expectations, together these two processes may enable the transfer of heat through the superconductor^{3,4,5}. The EC and CPS probabilities can in turn be made energy dependent by placing quantum dots between each normal lead and the superconducting region^{1,2}.
Here we present the experimental observation of the nonlocal thermoelectric current generated by imposing thermal gradient across a quantum dot–superconductor–quantum dot (QDSQD) splitter. We find that both CPS and EC processes contribute to the nonlocal thermoelectric current and that their relative contributions can be tuned by the gate potentials. The ability to tune between the CPS and EC allows for testing of fundamental theoretical concepts relating entanglement and heat transport in the graphene CPS systems.
Results
Theoretical considerations
Let us consider an QDSQD device within the Landauer formalism. Taking that the nonlocal 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
Here τ_{L(R)}(E) is the transmission probability of the left (right) quantum dot renormalized by Coulomb interaction, which depends on the energy of an electron and on the side gate potentials applied to the dots V_{sg,L(R)} (τ_{L(R)}(E) is given by the sum of Lorentzian peaks or Fano resonances associated with discrete energy levels, see Supplementary Note 4, and τ_{S} is the effective transmission probability of the superconducting lead. The latter corresponds to the probability for an electron coming out of one dot so that, instead of escaping into the bulk of the superconducting electrode, it reaches the other dot. It becomes independent on the electron energy E if the dots are separated by a distance shorter than the superconducting coherence length. This condition is reasonably well fulfilled in our experiment. The nonlocal thermoelectric currents in the dots can, in turn, be expressed in terms of the EC and CPS contributions, \({{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}=({{\Delta }}{I}_{{\rm{EC}}}+{{\Delta }}{I}_{{\rm{CPS}}})/2\), \({{\Delta }}{I}_{{\rm{R}}}^{{\rm{nl}}}=({{\Delta }}{I}_{{\rm{EC}}}+{{\Delta }}{I}_{{\rm{CPS}}})/2\), where
and \({f}_{{\rm{L(R)}}}(E)=1/(1+{e}^{E/{k}_{{\rm{B}}}{T}_{{\rm{L(R)}}}})\) is the distribution function in the left (right) electrode having the temperature T_{L(R)}.
Experiment
Now, we turn to experimental realization of CPS and EC. Several material platforms have been employed in the experiments^{30,31,32,33,34,35,36,37}, 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.
The temperature difference ΔT = T_{L} − T_{R} between the leads of the twoterminal device induces the thermoelectric current I = αGΔT, where G is the conductance of the device and α is the Seebeck coefficient^{39}. 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 energydependent electron transmission probability^{42}, and in superconductor–ferromagnet tunnel junctions^{25} 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 (\({I}_{{\rm{L(R)}}}^{{\rm{loc}}}\)) and small nonlocal contributions (\({{\Delta }}{I}_{{\rm{L(R)}}}^{{\rm{nl}}}\)), \({I}_{{\rm{L(R)}}}={I}_{{\rm{L(R)}}}^{{\rm{loc}}}({V}_{{\rm{sg,L(R)}}})+{{\Delta }}{I}_{{\rm{L(R)}}}^{{\rm{nl}}}({V}_{{\rm{sg,L}}},{V}_{{\rm{sg,R}}})\), see Supplementary Note 4. To infer the nonlocal contribution from the measured current I_{L(R)}, we subtract off its slowly varying average local background, 〈I_{L(R)}〉 (see “Methods”):
We thus obtain nonlocal currents \({{\Delta }}{I}_{{\rm{L(R)}}}^{{\rm{nl}}}\), which have a magnitude of order of 5–10% of the total thermoelectric currents. Figure 2 displays the maps of the nonlocal thermoelectric current in left dot \({{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}({V}_{{\rm{sg,L}}},{V}_{{\rm{sg,R}}})\) measured in the vicinity of the two conductance peaks of the right dot for the heating voltage V_{h} = 5 mV. In Fig. 2b–d, g, g_{L(R)} = hG_{L(R)}/e^{2} is the dimensionless conductance of the left (right) quantum dot. We find that \({{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}\) is symmetric with respect to the centers of the conductance peaks of the right dot, but it changes sign at the maxima of conductance peaks of the left dot. Thus the nonlocal current \({{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}\) approximately follows the same pattern as the product [dg_{L}(V_{sg,L})/dV_{sg,L}]g_{R}(V_{sg,R}).
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 \({\varepsilon }_{j,n}^{\prime}\), via the hopping matrix element t_{j,n}. This results in the transmission probabilities of the dots \({\tau }_{j}={\sum }_{n}{\gamma }_{j,n}{{{\Gamma }}}_{j,n}/[{(E{\varepsilon }_{j,n} {t}_{j,n}{ }^{2}/(E{\varepsilon }_{j,n}^{\prime}))}^{2}+{({\gamma }_{j,n}+{{{\Gamma }}}_{j,n})}^{2}/4]\), see Supplementary Note 4. The conductances \({g}_{{\rm{L}}}({V}_{{\rm{sg}},j})=4{\tau }_{j}^{2}(0,{V}_{{\rm{sg}},j})/{[2{\tau }_{j}(0,{V}_{{\rm{sg}},j})]}^{2}\), as predicted by the theory of Andreev reflection^{43}, 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}.
Nonlocal thermoelectricity
Figure 3 displays the main result of our study. There we plot the nonlocal 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 nonlocal current, they can also reasonably fit the conductance peaks (see the caption of Fig. 2). In the experiment, the nonlocal current \({{\Delta }}{I}_{{\rm{R}}}^{{\rm{nl}}}\) 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 crosscheck for our description. One sees that not only the magnitudes of the currents \({{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}\) and \({{\Delta }}{I}_{{\rm{R}}}^{{\rm{nl}}}\) are in good agreement with the theory, but their symmetric and antisymmetric combinations \({{\Delta }}{I}_{{\rm{CPS}}}={{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}+{{\Delta }}{I}_{{\rm{R}}}^{{\rm{nl}}}\) and \({{\Delta }}{I}_{{\rm{EC}}}={{\Delta }}{I}_{{\rm{L}}}^{{\rm{nl}}}{{\Delta }}{I}_{{\rm{R}}}^{{\rm{nl}}}\) 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 nonlocal signals. The observed general agreement in Fig. 3 provides strong support of the nonlocal coherent thermoelectric effect in our device.
Local thermoelectricity
Since, as noted, the nonlocal currents are relatively small, one can treat the measured currents foremost as local, \({I}_{{\rm{L(R)}}}\simeq {I}_{{\rm{L(R)}}}^{{\rm{loc}}}\). 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}/e^{2} 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 \({I}_{{\rm{L}}}^{\max }\propto {P}^{1/3}\), which is consistent with G_{th} ∝ T^{3} 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 \({\alpha }_{\max }=\max \{{I}_{{\rm{L}}}/{G}_{{\rm{L}}}({T}_{{\rm{L}}}{T}_{{\rm{S}}})\}\approx 250\) 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 energydependent transmission probability^{44}; the same theory was earlier employed in deriving the nonlocal 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 Δ_{0} 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.
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 nonzero, 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 nonlocal 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 nonlocal thermoelectric effect is shown to develop from single peak to double peak structure when temperature is lowered^{5}. The calculated double peak structure is similar to our local thermoelectric signal observed at low temperatures. However, the resonance energy dependence of the nonlocal signal with Coulomb blockade agrees poorly with the experimental results in Fig. 3a in comparison with our coherent transport model.
Discussion
This work has demonstrated the use of thermal gradient as primus motor for generating entangled electrons in graphene Cooper pair splitter. As the quantum dots in the device can be tuned individually, we are able to tune the device operation between EC and CPS regimes, thereby accomplishing direct control of two streams of entangled electrons. This type of scheme is useful not only for enabling devices where electrical drive is neither possible nor desired but also as a platform for realizing quantum thermodynamical experiments.
Methods
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 280nmthick 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 ebeam evaporator. Normal contacts to the graphene quantum dots were made using etched graphene nanoribbons with a small number of conductance channels at the operating point^{45}.
The strong p^{++} doping and the interfacial scattering at the Si/SiO_{2} 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 lockin 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 10^{6} 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/e^{2}, 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 \(P={V}_{{\rm{h}}}^{2}/R\) 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 lockin 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: \(\sim\! \exp (T/{E}_{{\rm{Th}}})\). Here \({E}_{{\rm{Th}}}=\hbar D/{L}_{{\rm{SGS}}}^{2}\) stands for the Thouless energy given by the length of the SGS section L_{SGS} and the diffusion constant \(D \sim \frac{1}{2}{v}_{{\rm{F}}}\lambda\), where the Fermi velocity v_{F} = 8 × 10^{5} m/s and λ is the charge carrier mean free path of graphene. Using λ ≃ 20 nm for graphene on SiO_{2} 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 \({R}_{{\rm{d}}}^{\max }\) 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 \({R}_{{\rm{d}}}^{\max }\) and the heating power to the value of \({R}_{{\rm{d}}}^{\max }\) recorded when varying the cryostat temperature. As detailed in Supplementary Note 2, we obtain the relation \({T}_{{\rm{SGS,L}}}=9.1\times {V}_{{\rm{h}}}^{0.70}+90\) mK between the SGS_{L} temperature and the heating voltage (V_{h} in Volts). For the SGS_{R} thermometer, we obtained an estimate \({T}_{{\rm{SGS,R}}}\simeq 8.4\times {V}_{{\rm{h}}}^{0.70}+90\) 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, \({I}_{{\rm{L(R)}}}^{{\rm{loc}}}\), and nonlocal, \({{\Delta }}{I}_{{\rm{L(R)}}}^{{\rm{nl}}}\), contributions. Note that, in theory, while the nonlocal 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 \({I}_{{\rm{L(R)}}}^{{\rm{loc}}}\) is nothing but the total current I_{L(R)} averaged over the gate voltage on the opposite dot; \({{\Delta }}{I}_{{\rm{L(R)}}}^{{\rm{nl}}}\) can be thus obtained by subtracting off this average background from I_{L(R)}. On practice, however, the gate electrodes may be subject to crosstalk, which the described simple processing does not account for. For the most part, the crosstalk was eliminated by a small rotation of the data array. Bearing in mind the remaining residual crosstalk, we construct a slowly varying background \(\langle {I}_{{\rm{L(R)}}}({V}_{{\rm{sg,L}}},{V}_{{\rm{sg,R}}})\rangle\) in the following manner (for clarity, let us consider the case of the left dot): for any fixed \({V}_{{\rm{sg,L}}}={V}_{{\rm{sg,L}}}^{{\rm{F}}}\), \(\langle {I}_{{\rm{L}}}({V}_{{\rm{sg,L}}}^{{\rm{F}}},{V}_{{\rm{sg,R}}})\rangle\) is the linear fit of \({I}_{{\rm{L}}}({V}_{{\rm{sg,L}}}^{{\rm{F}}},{V}_{{\rm{sg,R}}})\) as function of V_{sg,R}. The nonlocal contribution is then obtained using the formula
Linear fits in the background construction were found sufficient for compensating the remaining residual tilt of the current maps on the {V_{sg,L}, V_{sg,R}} plane.
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 nonlocal current, we employ a standard crossed Andreev reflection formalism^{2,49,50,51}. 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 backaction effect owing to the environmental impedance caused by graphene ribbons. The inclusion of the backactioninduced voltage sources makes the incoherent calculation selfconsistent. For details of the calculations, we refer to Supplementary Notes 4 and 5.
Data availability
All data needed to evaluate the conclusions in the paper are covered by the paper and its Supplementary Information. Additional data related to this work are available from authors upon reasonable request.
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Acknowledgements
We are grateful to C. Flindt, P. Burset, and T. Heikkilä for discussions and to I. A. Sadovskyy for sharing his numerical codes. This work was supported by Aalto University School of Science Visiting Professor grant to G.B.L., as well as by Academy of Finland Projects No. 290346 (Z.B.T., AF post doc), No. 314448 (BOLOSE), and No. 312295 (CoE, Quantum Technology Finland). The work of A.L. was support by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. This work was also supported within the EU Horizon 2020 program by ERC (QuDeT, No. 670743), and in part by MarieCurie training network project (OMT, No. 722923), COST Action CA16218 (NANOCOHYBRI), and the European Microkelvin Platform (EMP, No. 824109). The work of N.S.K. and G.B.L. was supported by the Government of the Russian Federation (Agreement No. 05.Y09.21.0018), by the RFBR Grants No. 170200396A and No. 180200642A, Foundation for the Advancement of Theoretical Physics and Mathematics BASIS, the Ministry of Education and Science of the Russian Federation No. 16.7162.2017/8.9. The work of N.S.K and A.G. at the University of Chicago was supported by the NSF grant DMR1809188. The work of V.M.V. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
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This research, initiated by P.J.H., is an outgrowth of a longterm collaboration between G.B.L. and P.J.H. The experimental setting and the employed sample configuration were developed by Z.B.T. and P.J.H. The patterned graphene samples were manufactured by Z.B.T. using Aalto University OtaNano infrastructure. The experiments were carried out at OtaNano LTL infrastructure by Z.B.T. and A.L. who were also responsible for the data analysis. A.S. and M.H. were adjusting and operating the LTL infrastructure. Theory modeling for coherent transport was performed by D.S.G., N.S.K., and G.B.L. The theory for incoherent transport was foremost developed and analyzed by N.S.K., A.G., V.M.V., and G.B.L. The results and their interpretation were discussed among all the authors. The paper and its Supplementary Information were written by the authors together.
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Tan, Z.B., Laitinen, A., Kirsanov, N.S. et al. Thermoelectric current in a graphene Cooper pair splitter. Nat Commun 12, 138 (2021). https://doi.org/10.1038/s41467020204767
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DOI: https://doi.org/10.1038/s41467020204767
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