Progress in Cooling Nanoelectronic Devices to Ultra-Low Temperatures.

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.

Edge-State Wave Functions from Momentum-Conserving Tunneling Spectroscopy.

We perform momentum-conserving tunneling spectroscopy using a GaAs cleaved-edge overgrowth quantum wire to investigate adjacent quantum Hall edge states. We use the lowest five wire modes with their distinct wave functions to probe each edge state and apply magnetic fields to modify the wave functions and their overlap. This reveals an intricate and rich tunneling conductance fan structure which is succinctly different for each of the wire modes. We self-consistently solve the Poisson-Schrödinger equations to simulate the spectroscopy, reproducing the striking fans in great detail, thus, confirming the calculations. Further, the model predicts hybridization between wire states and Landau levels, which is also confirmed experimentally. This establishes momentum-conserving tunneling spectroscopy as a powerful technique to probe edge state wave functions.

Machine learning enables completely automatic tuning of a quantum device faster than human experts.

Variability is a problem for the scalability of semiconductor quantum devices. The parameter space is large, and the operating range is small. Our statistical tuning algorithm searches for specific electron transport features in gate-defined quantum dot devices with a gate voltage space of up to eight dimensions. Starting from the full range of each gate voltage, our machine learning algorithm can tune each device to optimal performance in a median time of under 70 minutes. This performance surpassed our best human benchmark (although both human and machine performance can be improved). The algorithm is approximately 180 times faster than an automated random search of the parameter space, and is suitable for different material systems and device architectures. Our results yield a quantitative measurement of device variability, from one device to another and after thermal cycling. Our machine learning algorithm can be extended to higher dimensions and other technologies.

Evolution of the quantum Hall bulk spectrum into chiral edge states.

One of the most intriguing and fundamental properties of topological systems is the correspondence between the conducting edge states and the gapped bulk spectrum. Here, we use a GaAs cleaved edge quantum wire to perform momentum-resolved spectroscopy of the quantum Hall edge states in a tunnel-coupled 2D electron gas. This reveals the momentum and position of the edge states with unprecedented precision and shows the evolution from very low magnetic fields all the way to high fields where depopulation occurs. We present consistent analytical and numerical models, inferring the edge states from the well-known bulk spectrum, finding excellent agreement with the experiment-thus providing direct evidence for the bulk to edge correspondence. In addition, we observe various features beyond the single-particle picture, such as Fermi level pinning, exchange-enhanced spin splitting and signatures of edge-state reconstruction.

Magnetic cooling for microkelvin nanoelectronics on a cryofree platform.

We present a parallel network of 16 demagnetization refrigerators mounted on a cryofree dilution refrigerator aimed to cool nanoelectronic devices to sub-millikelvin temperatures. To measure the refrigerator temperature, the thermal motion of electrons in a Ag wire-thermalized by a spot-weld to one of the Cu nuclear refrigerators-is inductively picked-up by a superconducting gradiometer and amplified by a SQUID mounted at 4 K. The noise thermometer as well as other thermometers are used to characterize the performance of the system, finding magnetic field independent heat-leaks of a few nW/mol, cold times of several days below 1 mK, and a lowest temperature of 150 µK of one of the nuclear stages in a final field of 80 mT, close to the intrinsic SQUID noise of about 100 µK. A simple thermal model of the system capturing the nuclear refrigerator, heat leaks, and thermal and Korringa links describes the main features very well, including rather high refrigerator efficiencies typically above 80%.

Intrinsic Metastabilities in the Charge Configuration of a Double Quantum Dot.

We report a thermally activated metastability in a GaAs double quantum dot exhibiting real-time charge switching in diamond shaped regions of the charge stability diagram. Accidental charge traps and sensor backaction are excluded as the origin of the switching. We present an extension of the canonical double dot theory based on an intrinsic, thermal electron exchange process through the reservoirs, giving excellent agreement with the experiment. The electron spin is randomized by the exchange process, thus facilitating fast, gate-controlled spin initialization. At the same time, this process sets an intrinsic upper limit to the spin relaxation time.

Possible evidence for helical nuclear spin order in GaAs quantum wires.

We present transport measurements of cleaved edge overgrowth GaAs quantum wires. The conductance of the first mode reaches 2e(2)/h at high temperatures T≳10 K, as expected. As T is lowered, the conductance is gradually reduced to 1e(2)/h, becoming T independent at T≲0.1 K, while the device cools far below 0.1 K. This behavior is seen in several wires, is independent of density, and not altered by moderate magnetic fields B. The conductance reduction by a factor of 2 suggests lifting of the electron spin degeneracy in the absence of B. Our results are consistent with theoretical predictions for helical nuclear magnetism in the Luttinger liquid regime.

Metallic Coulomb blockade thermometry down to 10 mK and below.

We present an improved nuclear refrigerator reaching 0.3 mK, aimed at microkelvin nanoelectronic experiments, and use it to investigate metallic Coulomb blockade thermometers (CBTs) with various resistances R. The high-R devices cool to slightly lower T, consistent with better isolation from the noise environment, and exhibit electron-phonon cooling [proportional] T(5) and a residual heat-leak of 40 aW. In contrast, the low-R CBTs display cooling with a clearly weaker T-dependence, deviating from the electron-phonon mechanism. The CBTs agree excellently with the refrigerator temperature above 20 mK and reach a minimum-T of 7.5 ± 0.2 mK.

Method for cooling nanostructures to microkelvin temperatures.

We propose a new scheme aimed at cooling nanostructures to microkelvin temperature based on the well established technique of adiabatic nuclear demagnetization: we attach each device measurement lead to an individual nuclear refrigerator, allowing efficient thermal contact to a microkelvin bath. On a prototype consisting of a parallel network of nuclear refrigerators, temperatures of ∼1 mK simultaneously on ten measurement leads have been reached upon demagnetization, thus completing the first steps toward ultracold nanostructures.

Electrical control of spin relaxation in a quantum dot.

We demonstrate electrical control of the spin relaxation time T1 between Zeeman-split spin states of a single electron in a lateral quantum dot. We find that relaxation is mediated by the spin-orbit interaction, and by manipulating the orbital states of the dot using gate voltages we vary the relaxation rate W identical withT1(-1) by over an order of magnitude. The dependence of W on orbital confinement agrees with theoretical predictions, and from these data we extract the spin-orbit length. We also measure the dependence of W on the magnetic field and demonstrate that spin-orbit mediated coupling to phonons is the dominant relaxation mechanism down to 1 T, where T1 exceeds 1 s.

Energy-dependent tunneling in a quantum dot.

We present measurements of the rates for an electron to tunnel on and off a quantum dot, obtained using a quantum point contact charge sensor. The tunnel rates show exponential dependence on drain-source bias and plunger gate voltages. The tunneling process is shown to be elastic, and a model describing tunneling in terms of the dot energy relative to the height of the tunnel barrier quantitatively describes the measurements.

Asymmetry of nonlinear transport and electron interactions in quantum dots.

The symmetry properties of transport beyond the linear regime in chaotic quantum dots are investigated experimentally. A component of differential conductance that is antisymmetric in both applied source-drain bias V and magnetic field B, absent in linear transport, is found to exhibit mesoscopic fluctuations around a zero average. Typical values of this component allow a measurement of the electron interaction strength.

Cotunneling spectroscopy in few-electron quantum dots.

Few-electron quantum dots are investigated in the regime of strong tunneling to the leads. Inelastic cotunneling is used to measure the two-electron singlet-triplet splitting above and below a magnetic field driven singlet-triplet transition. Evidence for a nonequilibrium two-electron singlet-triplet Kondo effect is presented. Cotunneling allows orbital correlations and parameters characterizing entanglement of the two-electron singlet ground state to be extracted from dc transport.

Gate-controlled spin-orbit quantum interference effects in lateral transport.

In situ control of spin-orbit coupling in coherent transport using a clean GaAs/AlGaAs two-dimensional electron gas is realized, leading to a gate-tunable crossover from weak localization to antilocalization. The necessary theory of 2D magnetotransport in the presence of spin-orbit coupling beyond the diffusive approximation is developed and used to analyze experimental data. With this theory the Rashba contribution and linear and cubic Dresselhaus contributions to spin-orbit coupling are separately estimated, allowing the angular dependence of spin-orbit precession to be extracted at various gate voltages.

Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots.

We investigate antilocalization due to spin-orbit coupling in ballistic GaAs quantum dots. Antilocalization that is prominent in large dots is suppressed in small dots, as anticipated theoretically. Parallel magnetic fields suppress both antilocalization and also, at larger fields, weak localization, consistent with random matrix theory results once orbital coupling of the parallel field is included. In situ control of spin-orbit coupling in dots is demonstrated as a gate-controlled crossover from weak localization to antilocalization.