Nanoelectronic primary thermometry below 4 mK

Cooling nanoelectronic structures to millikelvin temperatures presents extreme challenges in maintaining thermal contact between the electrons in the device and an external cold bath. It is typically found that when nanoscale devices are cooled to ∼10 mK the electrons are significantly overheated. Here we report the cooling of electrons in nanoelectronic Coulomb blockade thermometers below 4 mK. The low operating temperature is attributed to an optimized design that incorporates cooling fins with a high electron–phonon coupling and on-chip electronic filters, combined with low-noise electronic measurements. By immersing a Coulomb blockade thermometer in the 3He/4He refrigerant of a dilution refrigerator, we measure a lowest electron temperature of 3.7 mK and a trend to a saturated electron temperature approaching 3 mK. This work demonstrates how nanoelectronic samples can be cooled further into the low-millikelvin range.

U nderstanding how to obtain and measure electron temperatures approaching 1 mK has the potential to open a new regime for studying nanoelectronics and pave the way towards pioneering sub-millikelvin techniques 1 . This would benefit numerous areas of activity; for example, investigations of the fractional quantum Hall effect in two-dimensional electron gases 2,3 and solid-state quantum technologies including superconducting and semiconducting qubits 4,5 . To access these temperatures one must minimize parasitic heating and internal Joule heating, and maximize the coupling to cold contact wires and phonons in the host lattice, all the while overcoming the decrease in electron-phonon coupling and electrical heat conduction as temperatures drop 6 .
Here we study Coulomb blockade thermometers (CBTs) that have been designed to operate significantly below 10 mK and demonstrate cooling of electrons in a nanoelectronic device to below 4 mK. A CBT consists of an array of Coulomb-blockaded metallic islands connected by tunnel junctions. The conductance of the array is temperature dependent, due to the balance between thermal excitations and an electrostatic barrier to single electron tunnelling across the islands 7,8 . CBTs typically function over a decade of temperature and have previously been demonstrated to work at temperatures as low as 7.5 mK (refs 9,10). Perhaps most importantly, they can be viewed as a primary thermometer of their internal electron temperature. We have used CBTs as a diagnostic tool to quantify and optimize the thermal environment of a nanoelectronic device and to unambiguously determine the local electron temperature. In the future, similar CBTs could be used to probe the electron temperature of nearby samples with thermal contact provided by direct electrical connections. Furthermore, the techniques that are used to cool the CBTs can be used to cool other nanoelectronic samples.

Results
Structure of the thermometer devices. The structure of the CBTs studied here is shown in Fig. 1. Devices are fabricated using an ex situ tunnel junction process 11 , which provides excellent tunnel junction uniformity 12 , and has also been used to fabricate superconducting qubits 13 (see Methods for details). Efficient thermal coupling between electrons and phonons in the metallic islands of the CBT is critical for reaching low electron temperatures 14,15 . The electron-phonon heat flow P ep is described by the material-dependent electron-phonon coupling constant S and the volume of the metallic island O, where T e is the electron temperature and T p is the phonon temperature 16 . To minimize T e , the island volume should be large and the material chosen to maximize S. We use electroplated Au on top of the CBT islands to increase their volume to nominal 5 Â 205 Â 38.5 mm 3 (see Methods for details). The effective electron-phonon coupling in these islands, with a relatively large volume and a high coupling constant 17 in Au S ¼ 2.4 Â 10 9 WK À 5 m À 3 , is estimated to be more than two orders of magnitude larger than in previous CBTs fabricated using the ex situ junction process 18 . In addition to efficient thermalization of the CBT itself, it is important to cool the incoming leads through robust thermal anchoring and heavy electromagnetic filtering 19 . We improve the chain of thermalization and filtering by including on-chip resistive meander structures in line with all electrical contacts. These form a distributed resistive-capacitive chain with a cutoff frequency E40 MHz. Similar filters based on a large area capacitor and tunnel junctions in series have previously been incorporated in a CBT 18 .
CBT characteristics above 7 mK. Figure 2 shows the behaviour of a CBT fabricated using the process described above, focusing on temperatures between 7 and 80 mK. The sensor is measured in both a commercial cryogen-free dilution refrigerator (Bluefors Cryogenics LD250) with a base temperature E7 mK and in a custom dilution refrigerator manufactured at Lancaster University 20 with a base temperature E2.5 mK. The conductance of the CBT is measured in a current-driven four-wire configuration, with the drive current and voltage amplification provided by an Aivon PA10 amplifier. A small AC excitation (typically 5 pArI AC r100 pA) is added to the DC bias I DC , allowing the differential conductance G to be measured with a lock-in amplifier.
In both refrigerators, the CBT is in a vacuum and housed in a gold-plated copper package (Aivon SH-1) that is attached to the mixing chamber plate. The package includes RC filters with a cutoff frequency E300 kHz. Electrical contacts to the CBT are thermalized in additional cold RC filters potted in Eccosorb CR 124 (Aivon Therma), which are also attached to the mixing chamber plate. The filters have a cutoff frequency E15 kHz.
As shown in Fig. 2a, the CBT conductance dips around zero bias and the dip becomes deeper and narrower at lower temperatures. Its full-width at half maximum is related to temperature by V 1/2 E5.439Nk B T/e, where N is the number of tunnel junctions in series 7 . This result does not account for self-heating in the sensor and so is only applicable when T e ¼ T p T. A more practical parameter to determine temperature is the zero-bias conductance G 0 , which has an approximate analytic relation to temperature 21 where u N E C /k B T is the dimensionless inverse temperature, E C [(N À 1)/N]e 2 /C S is the charging energy of the system, C S is the total capacitance of each island and G T is the asymptotic conductance. When u N o2.5, the temperature measurement error is o2.5% (ref. 15). Thus, if C S and G T are known, it is possible to determine T by measuring only G 0 . The most complete method to determine temperature is using a full tunnelling model to calculate G(V DC ) numerically 7 . We use this last approach to find C S and G T for the device, and these parameters are then used to find the temperature from subsequent measurements. Numerical calculations of conductance are made using an algorithm derived from the free, open-source library pyCBT (see Methods for further details). In addition, we account for overheating in the sensor by predicting the electron temperature T e in the islands using a model for the heat flow P into each island, where the first term is Joule heating at tunnel junctions of resistance R T , the second term is heat flow to phonons and P 0 accounts for parasitic heating. For a given phonon temperature, the minimum electron temperature T min e is found at Figure 2a shows the result of fitting the calculated G(V DC ) simultaneously to three measurements made between 20 and 60 mK. The fit parameters are R T , C S and T P for each measurement. The fit is found to be insensitive to the value of P 0 and hence parasitic heating is assumed to be negligible at these temperatures. Having calibrated the CBT, the fitted C S is used to fit further measurements with T p and R T as the free parameters. An example is given in Fig. 2a, where the electron temperature is found to be 7.2 ± 0.1 mK at a refrigerator temperature of 7.2 mK. Figure 2b shows that the electron temperature measured by the CBT is in close agreement with the refrigerator temperature T mxc in both refrigerators between 7 and 80 mK. Details of how T mxc was measured in each fridge can be found in the Methods section.
CBT characteristics below 7 mK. We investigate the behaviour of the CBT below 7 mK in the custom dilution refrigerator. Figure 3a shows the electron temperature of the sensor when in vacuum gradually cooling below 4 mK, while the refrigerator was held at T mxc u2.8 mK. Here, T e is determined from the value of G 0 , which is found by measuring G over a small range of V DC (E30 mV) close to V DC ¼ 0. We observe an extremely long equilibration time (over 3 days) but a rapid cooling of the CBT following a heating event (a refill of the liquid helium bath that briefly heats the CBT above 5.5 mK and the fridge above 3.5 mK). This suggests that the thermal contact between the CBT and the refrigerator is relatively strong, and that its slow cooling is probably due to heat leak from an external warm object.
A second CBT was immersed in the 3 He/ 4 He refrigerant of the custom dilution refrigerator to improve thermal coupling and better isolate external sources of heating. A schematic of the immersion cell is shown in the Fig. 3c. Sintered silver blocks increase the thermal contact between the refrigerant and the incoming contact wires 2 . Several sinters are attached to the sensor package, to each of the four measurement wires and to a grounding wire for the package and the RC filters. The immersed CBT is found to equilibrate much faster, as shown in Fig. 3a, reaching T e E3.8 mK at T mxc E2.7 mK. It is important to note that even when T e is elevated above T p and T mxc , the CBT remains a primary thermometer of its internal electron temperature.
To study the CBT at j V DC j 40, the time needed for the sensor to reach thermal equilibrium after a change of Joule heating needs to be known. Figure 3b shows the relaxation of T e after the CBT has been heated by a large drive current for long enough to reach thermal equilibrium (430 min). The subsequent value of T e is measured by scanning close to V DC ¼ 0, where Joule heating should be negligible. The relaxation of T e is found to have a time constant of 570 s. Figure 4a shows the calibration of the immersed sensor. The three warmest measurements are fitted simultaneously to determine C S ¼ 209.5 fF and R T ¼ 23.21 kO. The fitted temperatures agree with T mxc to within 6%. Given the agreement between the fitted T min e and T mxc , we can assume that parasitic heating is still negligible down to 10 mK.
The coldest measurement in Fig. 4a is fitted using the above values, yielding a minimum electron temperature of 3.86±0.01 mK. This measurement was made over a period of 7 h, to ensure that the CBT was in thermal equilibrium at each value of V DC . At these temperatures, the parasitic heating of the CBT is now significant and T min e does not match the refrigerator temperature of T mxc ¼ 2.7 mK. To fit this conductance dip, the thermal model, equation 3, is used with T p ¼ T mxc , and with the parasitic heating P 0 and the electron-phonon coupling SO as free parameters. Figure 4b shows how the CBT electron temperature diverges from the refrigerator temperature below E7 mK. Here the value of T e is found by measuring G 0 close to V DC ¼ 0 and so Joule heating can be neglected. The lowest temperature reported by the  CBT is below 3.7 mK when operating the fridge in single-shot mode (see Fig. 4c).

Discussion
The overheating of the sensor at V DC ¼ 0 constrains the value of P 0 (SO) À 1 in the fit to the coldest measurement in Fig. 4a. However, the parasitic heating is not large enough to reliably separate the values of P 0 and SO in the fitting. Qualitatively, the fits suggest that P 0 Z300 aW per island and SO is at least four times larger than expected from the nominal size of the thermalization blocks and the literature value of S for Au 17 . It is not possible to determine an upper bound on P 0 without constraining SO. It is worth noting that the power required to measure the CBT conductance (B1 aW per island due to Joule heating from I AC ) is much lower than our estimate of P 0 . As such, we believe that CBTs of this type can be operated at still lower temperatures by further reducing the parasitic heating.
The functional form of T e versus T mxc , as shown in Fig. 4b, should have the same temperature dependence as the dominant thermalization mechanism, that is, T 5 for electron-phonon coupling. However, other power laws have been observed 9 .
Here we find that the best fit of T x e ¼ T x mxc þ c gives x ¼ 2.7 and a saturated T e of c 1/x ¼ 3.4 mK. The fitted exponent x cannot be confirmed by fits to the conductance dips in Fig. 4a, because the overheating is still relatively weak, even at the lowest temperatures, and there is little effect on the shape of the dip. We find that a thermal model with a T 3 thermalization term fits the conductance dips equally well as a model using T 5 .
The saturation of the measured CBT temperature below 7 mK could be caused by parasitic heating of the islands or excess voltage noise across the tunnel junctions. It is possible that the operating temperature could be lowered by reducing parasitic heating through better shielding and by lowering the voltage noise in the measurement circuit. To understand the cause of saturation in more detail, or to test an improved measurement environment, this sensor would need to be cooled closer to 1 mK.
In conclusion, the CBTs described here have been shown to operate as reliable primary thermometers of electron temperature down to 3.7 mK. The large thermalization blocks incorporated in the device and a relatively low level of parasitic heating ensure that the electron subsystem in the sensor is well coupled to the phonon subsystem down to E7 mK. An immersion cell is shown to improve thermal coupling between a CBT and a dilution refrigerator. This allows the onset of overheating to be observed below 7 mK, and although the presence of overheating can be seen, the effect is sufficiently weak that the sensor will need to be cooled further to fully characterize the thermalization mechanisms.

Methods
Device fabrication. The CBT devices are fabricated using an ex situ tunnel junction process 11 . The Al films that define the CBT circuit have a thickness of 250 nm. Tunnel junctions between sections of Al are formed by an insulating layer of 250 nm SiO x deposited by plasma-enhanced chemical vapour deposition (PECVD). The junctions have a nominal diameter 0.8 mm and a resistivity B10 kO mm 2 . The substrate is undoped Si with 300 nm thermal oxide on the surface.
The island thickness achievable with the ex situ tunnel junction process or other deposition techniques used for tunnel junction devices is typically up to 1 mm. Thicker films suffer from stress build-up, causing poor adhesion between the film and the substrate. This is a severe problem at mK temperatures where poor adhesion can lead to poor thermalization and even mechanical failure due to thermal motion during cool down. We avoid these problems by using a combination of the ex situ process followed by masked electroplating of Au on top of the CBT islands 22 , which we refer to as thermalization blocks. Electroplating can produce B10 mm-thick, low-stress films and here we choose a nominal thickness of 5 mm for the thermalization blocks.
Refrigerator thermometry. Two different dilution refrigerators are used in the experiments described above. In the commercial refrigerator, the mixing chamber temperature T mxc is measured by a calibrated RuO 2 resistor (Sensor model RU-1000-BF0.007 supplied and calibrated by Bluefors Cryogenics) in contact with the mixing chamber plate. In the custom refrigerator, T mxc is measured using a conventional vibrating wire resonator viscometer immersed in the saturated dilute phase of the 3 He/ 4 He refrigerant in the mixing chamber 23,24 . The vibrating wire resonator is validated by comparison with a calibrated RuO 2 resistor (calibrated to 20 mK and supplied by Lake Shore Cryogenics). This resistor is thermally connected to the refrigerant via an immersed pad of sintered silver.
Data and software availability. All data used in this paper are available at http://dx.doi.org/10.17635/lancaster/researchdata/31, including descriptions of the data sets. The python-based pyCBT software library is freely available from Aivon Oy at https://github.com/AivonOy/pyCBT.