Implementation of SNS thermometers into molecular devices for cryogenic thermoelectric experiments

TL;DR

This paper presents a novel device integrating SNS thermometers into molecular junctions, enabling precise thermoelectric measurements at millikelvin temperatures to study quantum properties and energy conversion efficiency.

Contribution

The work introduces a new device architecture that combines SNS thermometers with single-molecule devices for accurate temperature measurement at cryogenic levels.

Findings

Thermometer accurately measures electronic temperature from 100 mK to 1.6 K.

Device enables high-precision thermoelectric experiments on single molecules.

Potential to improve understanding of quantum thermoelectric phenomena.

Abstract

Thermocurrent flowing through a single-molecule device contains valuable information about the quantum properties of the molecular structure and, in particular, on its electronic and phononic excitation spectra, and entropy. Furthermore, accessing the thermoelectric heat-to-charge conversion efficiency experimentally can help to select suitable molecules for future energy conversion devices, which - predicted by theoretical studies - could reach unprecedented efficiencies. However, one of the major challenges in quantifying thermocurrents in nanoscale devices is to determine the exact temperature bias applied to the junction. In this work, we have incorporated a superconductor-normal metal-superconductor (SNS) Josephson junction thermometer into a single-molecule device. The critical current of the Josephson junction depends accurately on minute changes of the electronic temperature in a wide temperature range from 100 mK to 1.6 K. Thus, we present a device architecture which can enable thermoelectric experiments on single molecules down to millikelvin temperatures with high precision.

Full text

††preprint: AIP/123-QED

Thermoelectric effects, i.e., the conversion between heat and charge currents, have received renewed interest from the nanoelectronics community Heremans et al. (2013); Gemma and Gotsmann (2021) and in particular in the field of single-molecule electronics Rincón-García et al. (2016); Wang, Meyhofer, and Reddy (2019). Recent studies Josefsson et al. (2018); Kleeorin et al. (2019); Dutta et al. (2019); Pyurbeeva et al. (2021); Hsu et al. (2022) show that thermoelectric measurements at cryogenic temperatures contain important information about the physical and quantum-thermodynamic properties of nanoscale and molecular systems, which are hard to access with conventional transport experiments. Furthermore, according to theoretical predictions, molecular devices could achieve a very high dimensionless figure of merit Paulsson and Datta (2003) $ZT$ – a quantity that is a measure for the heat-to-charge conversion efficiency. While the highest observed $ZT$ of inorganic materials is currently only Hinterleitner et al. (2019) about 5-6, the predicted $ZT$ of molecular heat engines could reach values of $\approx 100$, which would result in efficiencies close to the Carnot efficiency limit Finch, García-Suárez, and Lambert (2009). This efficiency could be even further enhanced at cryogenic temperature Harzheim et al. (2020). However, to quantify thermoelectric effects it is primordial to know the exact temperature drop across a molecular junction. This is in particular challenging at cryogenic temperature because most thermometers used so far are metal-based resistive sensors with low sensitivity at $T<20$ K. This work fills this gap by developing a superconducting thermometer, which is sensitive down to mK temperature, and by implementing this thermometer into an electromigrated break junction (EMBJ) device.

Different types of low-temperature thermometry approaches have been explored in the literature, including Johnson Noise thermometry Casey et al. (2003), Coulomb Blockade thermometry Pekola et al. (1994), and Hybrid Tunnel Junction thermometry Rowell and Tsui (1976). In this article, we focus on superconductor-normal metal-superconductor (SNS) thermometers because of their excellent properties: low impedance, high sensitivity at low temperatures, and a negligible access resistance Dutta (2018). Furthermore, they can be easily implemented into molecular junctions: To this end, we further improve our single-molecule thermoelectric junctions Gehring et al. (2019), by contacting the source and drain gold contacts with two superconductors, forming a local SNS junction. By measuring the critical current over this SNS junction the temperature can be extracted. In this manner, the contacts can be simultaneously used for thermometry and transport measurements. Here, we use a molybdenum rhenium (MoRe) superconductor, for its high critical temperature of 8.7 K (with a zero-temperature superconducting gap of about $\Delta\approx 1.8k_{\rm B}T_{\rm C}\approx$ 1.3 meV).

The fabrication procedure of the devices on Si wafers with a 285 nm layer of SiO2 oxide is illustrated in Figures 1(a)-(d). First (Figure 1(a)), the local gate was made by depositing a 1 nm thick adhesion layer of titanium (Ti) and a 7 nm thick layer of palladium (Pd) by standard electron-beam lithography and metal evaporation. This gate thickness is chosen to decrease the heat transport from source to drain Gehring et al. (2019). Then, the heater was fabricated by depositing 3 nm of Ti and 27 nm of Pd (Figure 1(b)). Subsequently, 10 nm Al2O3 was deposited by atomic layer deposition. The aluminium oxide forms the insulating layer between the heater and the electrical contacts, and acts as the dielectric of the local gate. Afterwards, the 13 nm thick Au bridge was made (see Figure 1(c)). To get a high-quality gold bridge, the deposition rate during evaporation was kept low ($0.5$ A/s) and a high vacuum around $10^{-8}$ mbar was used. In the last step, the 100 nm thick MoRe contacts were created by electron-beam lithography, metal sputtering and lift-off (Figure 1(d)).

False-colored scanning electron microscope (SEM) images of the final device together with a zoom-in of an individual Josephson junction are shown in Figures 1(e)-(f). The spacing between the superconducting contacts varies, and is 253 nm on one side of the gold bridge (thermometer A) and 247 nm on the other side (thermometer B), in the device shown here.

To calibrate the thermometers, devices were cooled down to 100 mK in a dilution refrigerator. A four-point measurement scheme was applied, where a DC current ($I_{\mathrm{JJ}}$) was biased over the junction and the voltage response ($V_{\mathrm{JJ}}$) was simultaneously recorded. A typical DC current-voltage characteristic at 100 mK for thermometer A is shown in Figure 2(a). In the low-current regime, the gold bridge between the two superconductors is proximitized and the Cooper pairs can move from one superconductor to the other without dissipation, forming an Andreev bound state Sauls (2018). At a certain current level – i.e., at the switching current ($I_{\rm SW}$) – the gold weak link changes its state from superconducting to normal, resulting in a voltage increase in the current-voltage ($I-V$) characteristic. From the slope of the $I-V$ characteristic at $I>I_{\rm SW}$ the normal resistance of the gold weak link is calculated and the diffusion coefficient of the electrons in this region is estimated. When ramping the current back, the gold weak link becomes superconducting again at the retrapping current ($I_{\rm R}$). This current value typically differs from $I_{\rm SW}$, which has been explained by capacitive effects in the junction Stewart (1968) or by heating of the electrons in the normal conducting junction during current sweeping Courtois et al. (2008). Since the geometrical capacitance of the junctions used in this work is not sufficiently large to explain the existence of the retrapping current, we attribute the observed hysteresis to heating.

Importantly, the switching to the superconducting state with increasing current bias is a stochastic process Angers et al. (2008), so several $I-V$ curves need to be recorded and a mean value of $I_{\mathrm{SW}}$ is calculated. To this end, the critical current was measured with an AC technique Dutta et al. (2020) (see Figure 2b). In a typical measurement the junction is biased with a 300 Hz triangular AC current from -20 $\upmu$A to 87 $\upmu$A. At these settings, gold stays in the normal state only for a short time, preventing the system from heating. Also, the offset is chosen to ensure that the junction stays proximitized in the negative current part and can relax to the base temperature. Using this approach, in a typical experiment we recorded 3000 measurements of switching events with a current resolution of about 0.07 $\upmu A$.

Using the AC measurement technique, it is possible to measure $I_{\rm SW}$ at different base temperatures and calibrate the thermometers in the device. The $I_{\rm SW}$ histograms for thermometer A are shown in Figure 2(c). It can be seen that the switching current is very sensitive to the sample temperature and decreases from 80 $\upmu$A at 150 mK to 61 $\upmu$A at 750 mK. We also note an asymmetry in the switching current distributions. Such asymmetry has been observed before Spahr et al. (2020) and attributed to thermal fluctuations. The stochasticity of switching current is dominated by quantum noise at temperatures below the Thouless energy, while thermal fluctuations become the dominating mechanism at higher temperatures (see discussion in SI III). From the fits to the current histograms we extract the average switching current $<I_{\mathrm{sw}}>$ as a function of temperature for the two thermometers (see Figure 3). The sensitivity of the thermometers, defined as $\frac{dI_{\rm SW}}{dT}$, in the temperature range from 100 mK to 750 mK is 31.5 $\mu$A/K and 42 $\mu$A/K for thermometer A and B, respectively (see SI II. for data of other thermometers and SI V. for a comparison with other Au-based SNS thermometers). These values are typical for long SNS junctions and similar to those reported for Nb-Cu-Nb SNS junctions Dubos et al. (2001a). Furthermore, both thermometers have a temperature resolution better than 15 mK within the investigated temperature range (see SI IV. for more details).

In the following, we discuss the temperature dependence of $<I_{\mathrm{sw}}>$ (see Figure 3). In Josephson junctions this dependence is very sensitive to the interplay between the energy scales of the proximity effect, the Thouless energy, $E_{\rm TH}=\frac{\hbar D}{L^{2}}$, and the superconducting gap $\Delta$. Here, $D=\frac{v_{\rm F}l_{\rm e}}{3}$ is the diffusion coefficient in the normal metal, where $v_{\rm F},~{}l_{\rm e}$ are the Fermi velocity and the elastic mean free path of electrons, respectively, and $L$ is the distance between the superconducting electrodes. In the long junction limit $\Delta>E_{\rm TH}$, the temperature dependence of $I_{\rm SW}$ can be described theoretically by using a quasi-classical approach based on Green’s functions in imaginary space Belzig et al. (1999); Dubos et al. (2001b); it requires solving the Usadel equation at all energies Dubos et al. (2001b).

We used the usadel1 package Virtanen and Heikkilä (2007) to fit the experimental data. We assumed that the interfaces between the superconductor and the normal metal are perfectly transparent and that the phase difference between the superconductors is fixed at $\phi=1.27\pi/2$, similar to Dubos et al. Dubos et al. (2001b). Furthermore, we fixed the normal resistance, $R_{\rm N}$, of the junction and $\Delta$ of the superconductor to values obtained from transport measurements. The solid lines in Figure 3 show the resulting fits where the Thouless energy and the suppression coefficient, $\alpha$, which accounts for the non-ideality of the normal metal-superconductor contact, were used as fitting parameters. Details about the fitting and the resulting parameters can be found in the supporting information. We find a suppression coefficient $\alpha\approx 0.5$, similar to the value obtained by Courtois et.al. Courtois et al. (2008). The Thouless energies, $E^{\rm FIT}_{\rm TH}=49.4-87.2\mu$eV, obtained from the fits in Figure 3 are close to the values determined from the transport measurements, $E^{\rm EXP}_{\rm TH}\approx 40\upmu$eV, which can be calculated as Janssen (2001) $E^{\rm EXP}_{\rm TH}=\frac{1}{VN(E_{\rm F})}\frac{R_{\rm Q}}{2\pi R_{\rm N}}$, where $V=L\cdot w\cdot t$ is the volume of the gold film, $N(E_{\rm F})$ is the volumetric electronic density of states of gold at the Fermi level and $R_{\rm Q}=\frac{h}{2e^{2}}$ is the resistance quantum. $N(E_{\rm F})$ can be obtained from the electronic specific heatStewart (1983) $\gamma=0.69\cdot 10^{-3}$ $\rm J\ mol^{-1}\ K^{-2}$ and the molar volumeSingman (1984) $v=1.021\cdot 10^{-5}$ $\rm m^{3}\ mol^{-1}$ asBeck and Claus (1970) $N(E_{\rm F})=\frac{3\gamma}{\pi^{2}k_{\rm B}^{2}v}=1.73\cdot 10^{28}$ $\rm eV^{-1}\ m^{-3}$.

In summary, we implemented an SNS superconducting thermometer in a molecular thermoelectric device. MoRe is used as the superconductor, which allows to perform thermometry in a temperature range from 100 mK to 1.6 K with a high sensitivity of up to $40~{}\upmu$A/K. Other than previously used resistance thermometers, the SNS thermometers developed in this work directly measure the temperature of the electronic system in the immediate proximity to the molecule. Therefore, our devices will allow to e.g. extract the absolute Seebeck coefficient from thermopower measurements on a molecule and will thus pave the way for precise investigations of molecular heat engines.

supplementary materials

See supplementary materials SI I. for theoretical fit to thermometer A with two parameters; SI II. for results of test thermometers with different L– spacing; SI III. for discussion of $I_{\rm SW}$ fluctuations; SI IV. for temperature resolution of thermometers; and SI V. for comparison with other Au-based SNS junctions.

Acknowledgements.

The authors acknowledge financial support from the F.R.S.-FNRS of Belgium (FNRS-CQ-1.C044.21-SMARD, FNRS-CDR-J.0068.21-SMARD, FNRS-MIS-F.4523.22-TopoBrain), from the Federation Wallonie-Bruxelles through the ARC Grant No. 21/26-116 and from the EU (ERC-StG-10104144-MOUNTAIN, FET-767187-QuIET). This project (40007563-CONNECT) has received funding from the FWO and F.R.S.-FNRS under the Excellence of Science (EOS) programme. This work was supported by the Netherlands Organisation for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program and Natuurkunde Vrije Programma’s: 680.90.18.01.

Author Declarations

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Serhii Volosheniuk: Investigation (lead); Resources (lead); Methodology (lead); Visualization (lead) Software (equal); Validation (equal); Writing - original draft (lead); Writing review & editing (equal). Damian Bouwmeester: Validation (equal); Software (equal); Investigation (support). Chunwei Hsu: Validation (equal); Methodology (support); Resources (support). H.S.J. van der Zant: Supervision (equal); Writing - review & editing (equal) Pascal Gehring: Project administration (lead); Supervision (equal); Writing -review & editing (equal)

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon request.