Progress in Cooling Nanoelectronic Devices to Ultra-Low Temperatures

Abstract

Here we review recent progress in cooling micro-/nanoelectronic devices significantly below 10 mK. A number of groups worldwide are working to produce sub-millikelvin on-chip electron temperatures, motivated by the possibility of observing new physical effects and improving the performance of quantum technologies, sensors and metrological standards. The challenge is a longstanding one, with the lowest reported on-chip electron temperature having remained around 4 mK for more than 15 years. This is despite the fact that microkelvin temperatures have been accessible in bulk materials since the mid-twentieth century. In this review, we describe progress made in the last 5 years using new cooling techniques. Developments have been driven by improvements in the understanding of nanoscale physics, material properties and heat flow in electronic devices at ultralow temperatures and have involved collaboration between universities and institutes, physicists and engineers. We hope that this review will serve as a summary of the current state of the art and provide a roadmap for future developments. We focus on techniques that have shown, in experiment, the potential to reach sub-millikelvin electron temperatures. In particular, we focus on on-chip demagnetisation refrigeration. Multiple groups have used this technique to reach temperatures around 1 mK, with a current lowest temperature below 0.5 mK.

Keywords: Adiabatic nuclear demagnetisation; Dilution refrigeration; Nanoelectronics; Ultra-low temperatures.

Fig. 1

Thermal model of an on-chip conducting material at low temperature. The on-chip conductor (dashed box in a) contains three thermal subsystems: phonons, conduction electrons and nuclear spins, with heat capacities $C_{\mathrm{p}}$, $C_{\mathrm{e}}$, $C_{\mathrm{n}}$ and temperatures $T_{\mathrm{p}}$, $T_{\mathrm{e}}$, $T_{\mathrm{n}}$ respectively. Heat flow between the subsystems is determined by temperature differences and the thermal resistances $R_{\mathrm{ep}}$ and $R_{\mathrm{en}}$. The conductor sits on an insulating substrate, which is assumed to be macroscopic and in thermal equilibrium with the base temperature of an external refrigerator. The thermal resistance between the conductor and the substrate is the phonon boundary (Kapitza) resistance $R_{\mathrm{K}}$. The conductor is electrically connected to off-chip wiring, which is also assumed to be well-thermalised with the external refrigerator. The thermal resistance $R_{\mathrm{WF}}$ between on-chip electrons and electrons in the wiring is determined by the electrical resistance of the connection. b Illustrates the location of each component in an optical image of a typical device on a low-temperature sample mount (Color figure online)

Fig. 2

Predicted thermal conductances for the model shown in Fig. 1 in two example situations. In both, the on-chip conductor is a copper film of size $205\mathrm{\mu }\mathrm{m}\times 38.5\mathrm{\mu }\mathrm{m}\times 5\mathrm{\mu }\mathrm{m}$ (similar to the device in [52]). Its substrate is a silicon chip, which is assumed to be well-thermalised to the external refrigerator at temperature T. a The on-chip electrons are electrically connected to well-thermalised external wiring through a low-resistance ($10\mathrm{m}\mathrm{\Omega }$) bond wire. This path provides the strongest thermal connection to the electrons for $T\ll 1\mathrm{K}$. No external magnetic field is applied and the internal magnetic field is assumed to be 0.36 mT, the effective dipolar field in copper. b The resistance of the electrical connection is $10\mathrm{k}\mathrm{\Omega }$ and a magnetic field of 0.1 T is applied. As a result, coupling between the on-chip electrons and the external refrigerator is much weaker for $T\ll 1\mathrm{K}$ and, below a few millikelvin, the nuclear spin bath in copper becomes strongly coupled to the electrons (Color figure online)

Fig. 3

Molar heat capacities of copper and silicon at low temperatures. a Total heat capacity (solid line) of copper, which is the sum of contributions from conduction electrons $C_{\mathrm{e}}$, phonons $C_{\mathrm{p}}$ and nuclear spins $C_{\mathrm{n}}$. b Total heat capacity (solid line) of undoped silicon in zero applied magnetic field with a free electron concentration of $1\times {10}^{17}{\mathrm{cm}}^{-3}$. This could be, for example, silicon in the channel of an accumulation-mode FET. In both materials, $C_{\mathrm{p}}$ is insignificant for $T\ll 1\mathrm{K}$. Even in zero applied magnetic field, the total heat capacity of copper is dominated by $C_{\mathrm{n}}$ for $T\ll 1\mathrm{mK}$. The contribution from $C_{\mathrm{n}}$ grows as the square of applied magnetic field (Color figure online)

Fig. 4

Room temperature attenuation characteristics of a 1.5-m-long thermocoax cable (green) and different silver epoxy microwave filters. Blue and red represent layered and segmented filters, respectively, where the segmented filters have reduced parasitic capacitance. For the dashed characteristics, a 4.7 nF discoidal capacitor from Pacific Aerospace [67] was added to both filter ends. A picture of a silver epoxy microwave filter and centimetre scale bar is shown in the inset. This figure was taken from [33] (Color figure online)

Fig. 5

Schematic of a nuclear demagnetisation stage mounted on a Bluefors LD dry dilution refrigerator. The measurement leads are thermalised with Ag powder sinters (top right picture, scale bar: 5 mm) in the mixing chamber (MC, blue) and pass through C-shaped Al heat switches (green) to the Cu plates. The gradiometer of a noise thermometer as well as the (L)CMN thermometers are positioned in a region of cancelled magnetic field between the MC and the NR stage. The gradiometer is double-shielded by a Nb tube and a outer NbTi tube (red). Middle right inset: photograph of the gradiometer pick-up coil made from insulated Nb wire with $100\mathrm{\mu }\mathrm{m}$ diameter. The $2\times 20$ turns are wound non-inductively on a high-purity silver wire which is spot-welded to a NR. Scale bar: 2 mm. Lower inset: schematic cross section through the network of 16 parallel NRs. Each NR is 2 mol of Cu (99.99% Cu, low-${\mathrm{H}}_2$ content [90], $\mathrm{RRR}\sim 500$) and consists of two half-plates, spot-welded together at the top and bottom. Each half-plate is of dimension $3.4\mathrm{cm}\times 0.17\mathrm{cm}\times 12\mathrm{cm}$. This figure was taken from [88] (Color figure online)

Fig. 6

Electron temperature extracted from direct DC transport (a, b) and charge sensing (c, d) of quantum dots formed at the heterointerface of GaAs/AlGaAs structures. a Measured DC current (red dots) along with Fermi–Dirac curve fits (solid black curves). b Extracted electron temperature $T_L$ from measurements as shown in a as a function of mixing chamber temperature $T_{\mathrm{MC}}$. The inset shows a SEM image of a similar device. c Charge stability diagram of a double quantum dot similar to the one shown in the inset. d Electron temperatures extracted by fitting a Fermi–Dirac distribution to the charge sensing signal at base temperature and $T_{\mathrm{MC}}=132\mathrm{mK}$ are shown in the main panel and inset, respectively. This figure was adapted from [91] (Color figure online)

Fig. 7

Thermometry using various metallic Coulomb blockade thermometers with differing resistance. Normalised differential conductance $g/g_T$ as a function of applied DC bias is shown in a for various copper plate temperatures $T_{\mathrm{Cu}}$. Off-chip demagnetisation down to $T_{\mathrm{Cu}}=2\mathrm{mK}$ slightly reduces the electronic temperature for the $4.8\mathrm{M}\mathrm{\Omega }$ device from 11.4 mK (light blue) to 9.5 mK (dark blue). b Extracted electron temperatures as a function of $T_{\mathrm{Cu}}$. Open (closed) markers represent the $67\mathrm{k}\mathrm{\Omega }$ ($4.8\mathrm{M}\mathrm{\Omega }$) device. Data for the red and blue markers were collected during regular dilution refrigerator operation and adiabatic nuclear demagnetisation, respectively. This figure was adapted from [29] (Color figure online)

Fig. 8

Normal metal–insulator–superconductor (NIS) tunnel junction thermometry. a Linear fits (solid black) to the onset of the measured quasiparticle current (blue dots) in an NIS device. Fits to the full current profile are shown in dashed red. The inset shows a close-up for mixing chamber (bath) temperatures of 10 mK and 7 mK on the left and right, respectively. b Extracted electronic temperatures from a for the full curve fit and the linear fit are shown as red squares and black triangles, respectively. This figure was adapted from [87] (Color figure online)

Fig. 9

Demonstration of on-chip demagnetisation refrigeration with copper refrigerant. The CBT device shown schematically in a features large ($6\mathrm{\mu }\mathrm{m}$ thick) Cu refrigerant blocks applied to an array of metallic islands. A photograph of the $6.5\mathrm{mm}\times 2.3\mathrm{mm}$ chip is shown in b, with the $32\times 20$ array of metal islands taking up the left 3/4 of the device. The black crosses in c show the measured electron temperature during a 2.5 mT/s demagnetisation, to which the three subsystem model was fitted, allowing extraction of the phonon and nuclear spin temperatures. d Shows how the base temperature and hold time were extended by using three different demagnetisation rates instead of one. Details of the demagnetisation profiles ‘Optimised 1’ and ‘Optimised 2’ can be found in [52] (Color figure online)

Fig. 10

The ‘coldfinger’ used for precooling a CBT sensor in its package on a dilution refrigerator in Lancaster. a shows the coldfinger mated with the mixing chamber hence, when cooled, the sinters shown at the top of the diagram in b are immersed in the liquid ${}^3\mathrm{He}$–${}^4\mathrm{He}$ refrigerant of the dilution refrigerator. Cooling is provided through the silver wire connected to the package and also the shielded silver wires attached to each measurement lead (Color figure online)

Fig. 11

Comparison of demagnetisation cooling using the same cooling platform on wet and dry dilution refrigerators. a Shows a comparison of the electron temperatures achieved during single rate and optimised multi-rate demagnetisations on the wet and dry dilution refrigerators. b Shows a quantity related to the entropy change during the demagnetisations, and therefore shows the amount of deviation from the ideal case of constant entropy (Color figure online)

Fig. 12

Nuclear adiabatic demagnetisation of a metallic Coulomb blockade thermometer. a CBT temperature $T_{\mathrm{CBT}}$ versus copper plate temperature $T_{\mathrm{Cu}}$. The diagonal dashed line indicates ideal thermalisation $T_{\mathrm{CBT}}=T_{\mathrm{Cu}}$. The inset shows the normalised zero bias conductance dip $\delta g$ as a function of $T_{\mathrm{Cu}}$. A fit [30] (solid black curve) in the high temperature regime $T_{\mathrm{Cu}}>30\text{mK}$ is used to extract the charging energy $E_{\mathrm{C}}=6.55\text{mK}$. The three steps of nuclear demagnetisation, precooling, demagnetisation, and warmup, are shown in b, c and d, respectively. Light blue, yellow and red data indicate $T_{\mathrm{CBT}}$, $T_{\mathrm{Cu}}$, and $T_{\mathrm{MC}}$, respectively. Black dashed curves in b–d are predictions from a thermal model schematically indicated in b. A light blue line in c indicates ideal adiabatic demagnetisation. This figure was adapted from [94] (Color figure online)

Fig. 13

Nuclear adiabatic demagnetisation of a metallic Coulomb blockade thermometer. a Schematic showing the demagnetisation stage being fixed rigidly with respect to mixing chamber shield. A second set of PEEK screws fixes the mixing chamber shield with respect to the still radiation shield. b The main panel shows the extracted CBT temperature as a function of magnetic field during the demagnetisation process. Calibration data are shown in the inset, where blue markers correspond to measurements of the relative conductance dip $\delta g/g_T$ and a fit to the data in the high temperature regime from 30 to 65 mK is shown in solid red. The resulting charging energy is $E_C=6.72\pm 0.04\text{mK}$ (Color figure online)