Progress in cooling nanoelectronic devices to ultra-low temperatures
A. T. Jones, C. P. Scheller, J. R. Prance, Y. B. Kalyoncu, D. M. Zumbühl, R. P. Haley
PProgress in cooling nanoelectronic devices to ultra-low temperatures
A. T. Jones, C. P. Scheller, J. R. Prance, ∗ Y. B. Kalyoncu, D. M. Zumbühl, and R. P. Haley Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK Department of Physics, University of Basel, CH-4056 Basel, Switzerland
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 ofquantum technologies, sensors and metrological standards. The challenge is a longstanding one, withthe lowest reported on-chip electron temperature having remained around 4 mK for more than 15years. This is despite the fact that microkelvin temperatures have been accessible in bulk materialssince the mid 20th century. In this review we describe progress made in the last five years usingnew cooling techniques. Developments have been driven by improvements in the understanding ofnanoscale physics, material properties and heat flow in electronic devices at ultralow temperatures,and have involved collaboration between universities and institutes, physicists and engineers. Wehope that this review will serve as a summary of the current state-of-the-art, and provide a roadmapfor future developments. We focus on techniques that have shown, in experiment, the potential toreach sub-millikelvin electron temperatures. In particular, we focus on on-chip demagnetisationrefrigeration. Multiple groups have used this technique to reach temperatures around 1 mK, with acurrent lowest temperature below 0 . Keywords: Nanoelectronics; Ultra-low temperatures; Dilution refrigeration; Adiabatic nuclear demagnetiza-tion
1. Introduction
Millikelvin electronic measurements of mi-cro/nanoscale devices and materials are used in awide range of fields; from quantum technology, materialsscience and metrology to observational astrophysics anddark matter searches. In some cases, physical effectsemerge at low temperature that provide a new anduseful electronic behaviour, such as superconductivityor conductance quantisation. In other cases, low tem-peratures provide a “quiet” environment that can, forexample, improve the signal-to-noise ratio of sensitivedetectors or increase the coherence time of qubits.Regardless of the goal, or the refrigeration technologyused, it remains challenging to cool the conductionelectrons in a nanoscale device or material significantlybelow 10 mK. As the temperature drops, the thermalcoupling between conduction electrons and the hostlattice weakens and the heat capacity of the electronicsystem falls. This makes the electron temperature moresensitive to parasitic heating. In a nanoscale structure,where the physical size already limits the electronicheat capacity, it is very challenging to maintain lowelectron temperatures against the incoming heat fromelectromagnetic radiation, eddy-current heating, nearbyhot insulators, and the electronic connections neededfor measurement. This review outlines the currentprogress in cooling nanoelectronic systems below 10 mK,and the potential for new techniques to reach electrontemperatures deep in the microkelvin regime.The ability to access low-millikelvin or microkelvin ∗ [email protected] temperatures in nanoelectronic structures brings the ex-citing possibility of unexpected discoveries in a newregime. But there are also immediate goals that moti-vate much of the work we discuss here. Low electrontemperatures are needed to observe new predicted elec-tronic phases, including exotic quantum Hall states [1–5], topological insulators [6], collective electron-nuclearspin states [7–10], insulating ground states in 2D sys-tems [11, 12] and superconductivity in some materi-als [13]. In established applications, lower electron tem-peratures may improve coherence times of semiconductorand superconducting qubits [14–16] and hybrid Majoranadevices [17–19], as well as reducing error mechanisms inmetrological standards such as charge pumps [20, 21] andquantum Hall resistance standards [22].This review focuses on cooling techniques that we knowto have successfully produced on-chip electron tempera-tures significantly below 10 mK in experiment. We willnot discuss emerging refrigeration techniques, such asmicro/nanoscale electronic coolers, that may be able toreach ultralow temperatures but have not yet done so inexperiment. More information on micrometer-scale re-frigeration can be found in recent reviews such as [23, 24].We will also not discuss ultralow temperature thermom-etry in detail, although this is obviously an importantand relevant topic. Information about the current stateof metrology in ultralow temperature thermometry canbe found in [25]. More information about techniquesthat are particularly relevant to micro/nanoelectronic de-vices at ultralow temperatures can be found, for example,in [26–28] for noise thermometry, [29–32] for Coulombblockade thermometry and [33–35] for quantum dot-based thermometry. Almost all of the work discussedbelow makes use of one or more of these thermometrytechniques. a r X i v : . [ c ond - m a t . m e s - h a ll ] J un
2. Cooling techniques and heat flow innanoelectronic devices
When trying to cool micro/nanoelectronic devices toultralow temperatures, experimentalists are faced withseveral unfavourable physical scaling laws: the heat ca-pacity of the conduction electrons falls with temperature,as does their thermal coupling to other electronic systemsand to phonons in the host lattice. To achieve an elec-tron temperature T e that is close to the base temperatureof the surrounding environment, parasitic heat leaks intothe conduction electrons need to be carefully managed.What this means quantitatively depends on the detailsof each sample and how it is coupled to its local environ-ment; however, the scaling laws, discussed in more detailbelow, demonstrate the extent of the challenge of movingto lower temperatures. As a trivial example, consider adevice that has been well-thermalised to a refrigeratoroperating at 10 mK by cooling through bond wires andkeeping the total parasitic heat leak below 1 fW. Thesame device could require a total heat leak below 10 aWto stay similarly well-thermalised to a refrigerator oper-ating at 1 mK. In this section, we outline a general ther-mal model for an on-chip conductor at low temperaturesand use this model to illustrate the various cooling tech-niques that can be employed to reach on-chip electrontemperatures below 10 mK. The thermal model is outlined in Fig. 1. It shows sev-eral channels that are available to remove heat from theconduction electrons in an on-chip material. The ther-mal resistances of the channels are temperature depen-dent and so the optimal way to cool the electrons willalso change with temperature. The details of the ther-mal resistance for each cooling channel may be differentin different samples, but the illustration in Fig. 1 is oftena useful approximation and could apply to, for example,conduction electrons in a semiconductor nanostructureor in a thin-film metal circuit. In the following discus-sion we use the simple example of a metal conductor onthe surface of an insulating substrate.In the first instance, conduction electrons in an on-chipmaterial are coupled to phonons and spins in the samematerial. In many commonly-used metals and semicon-ductors, the low-temperature thermal coupling betweenconduction electrons at temperature T e and phonons attemperature T p is given by the heat flow˙ Q ep = Σ V (cid:0) T − T (cid:1) , (1)where Σ is a material-dependent coupling constant and V is the volume of the material. Note that the expo-nent of the temperatures in this equation is commonlyaccepted to be 5 in many materials [36–38], howeverin some systems, particularly those confined to fewer dimensions, it has been observed to take other values2 < n ≤ ∼ . × W m − K − in dopedsemiconductors to ∼ × W m − K − in metals [23].If the on-chip material contains spinful nuclei, heatwill flow between the nuclear spin bath and conductionelectrons through spin-lattice relaxation. In the limit ofsmall nuclear Zeeman splitting ( g n µ n B (cid:28) k B T , where g n is the g-factor, µ n is the nuclear magneton and B the magnetic field), the spin-lattice relaxation rate τ − isproportional to T e and characterised by the Korringa con-stant κ = τ T e [39]. Also in this limit, the heat capacityof the nuclei is above the Schottky anomaly and follows C n ∝ B /T . The thermal coupling between conductionelectrons and the nuclear spin bath at temperature T n isthen given by the heat flow [40, 41]˙ Q en = λ n nµ κ B T n ( T e − T n ) , (2)where λ n is the molar nuclear Curie constant of the mate-rial, n is the number of moles and µ is the permeabilityof free space. Equation 2 has been experimentally veri-fied in a broad range of metals and semiconductors [36]and we will assume that it is valid in the following discus-sion. However, it should be noted that deviations fromthe Korringa law, which invalidate Equ. 2, have been ob-served in some metallically-doped semiconductors below10 K in the disordered, interacting regime [42], semicon-ductors doped close to the metal-insulator transition [43]and Kondo metals [44].The thermal model in Fig. 1 shows an on-chip materialthat contains a thermal bath of nuclear spins. The samebasic model could also apply to a material that containsparamagnetic impurities. In this case, the nuclear spinbath is replaced by a bath of electron spins bound toimpurities or dopants. The thermal resistance betweenthese spins and the conduction electrons will be deter-mined by the spin relaxation time. The heat capacity ofthe spin bath is likely to include a Schottky anomaly inthe millikelvin temperature range [45].While the heat flows described by Equ. 1 and 2 re-distribute energy between the thermal subsystems of anon-chip material, the thermal resistances R WF and R K determine how well the material is coupled to the out-side environment. The thermal resistance R WF repre-sents electronic heat conduction through the electricalconnections to a device. It is related to the electrical re-sistance R of the connection via the Wiedemann–Franzlaw R WF = 3 π (cid:18) ek B (cid:19) R ( T e + T ) / , (3)where T is the temperature of the outside environment.In practice, the value of R can be chosen across a widerange. The resistance of a single gold bond wire, includ-ing contact resistance, can be less than 10 mΩ at lowtemperatures [46]. On the other hand, the electrical re- R en R ep R K R WF Phonons T p , C p Electrons T e , C e Nuclear spins T n , C n SubstrateOff-chipwiring ˙ Q par (a) (b) On-chip conductor
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 p , C e , C n andtemperatures T p , T e , T n respectively. Heat flow between the subsystems is determined by temperature differences and thethermal resistances R ep and R en . The conductor sits on an insulating substrate, which is assumed to be macroscopic and inthermal equilibrium with the base temperature of an external refrigerator. The thermal resistance between the conductor andthe substrate is the phonon boundary (Kapitza) resistance R K . The conductor is electrically connected to off-chip wiring, whichis also assumed to be well-thermalised with the external refrigerator. The thermal resistance R WF between on-chip electronsand electrons in the wiring is determined by the electrical resistance of the connection. (b) illustrates the location of eachcomponent in an optical image of a typical device on a low-temperature sample mount. sistance can easily be increased above 10 kΩ by includingon-chip thin-film resistors or tunnel junctions [47–50].The final component of the thermal model is thephonon boundary (Kapitza) resistance R K , which is typ-ically between the on-chip conductor and the substrate.The value of R K depends on the substrate material andthe on-chip material, as well as the microscopic propertiesof the interface [51]. The boundary resistance roughlyscales as R K ∝ T − with a prefactor that depends onthe acoustic mismatch between the two materials, thestrength of scattering at the interface and the area A of the interface. For common metals (including Al, Cu,Au, In) on insulating substrates (Sapphire, Quartz, Si)expected values are AR K T ∼ − K m W − [51]. Be-cause it is difficult to control the quality of interfaces inexperiment, a precise prediction of R K is rarely possible.However, at ultralow temperatures it is common to find R ep (cid:29) R K and therefore cooling of the conduction elec-trons through phonon channels is not limited by R K . Insome samples, for example a semiconductor 2D electrongas, the conduction electrons are inside the substrate ma-terial and couple directly to the substrate phonons. Inthis case, R K may be omitted from the thermal modelor it may be used to represent the boundary resistancebetween the substrate and its support.All of the thermal channels shown in Fig. 1 havetemperature-dependent thermal resistances. Figure 2shows predicted values of the corresponding thermal con-ductances for two example situations. In the first exam-ple, shown in Fig. 2(a), a thick ( ∼ µ m) on-chip copperfilm has a low-resistance electrical connection to someoff-chip wiring. Both the external wiring and substratechip are assumed to be macroscopic and well-thermalisedwith the refrigerator. Above ∼ (cid:28) Q par and the thermal resistance R WF . The second ex-ample, shown in Fig. 2(b), is a similar system but witha 0 . times smaller forthe high-impedance case to to reach the same electrontemperature as the low impedance case.The steady-state electron temperature in the thermalmodel is only determined by thermal resistances, theamount of parasitic heating and the temperature of thecold reservoir (refrigerator). However, the heat capaci-ties of the various subsystems are needed to understandany dynamic behaviour. Figure 3 shows how the heat ca-pacities of two example materials vary with temperaturebetween 3 K and 100 µ K. The heat capacity of the con-duction electrons falls linearly with temperature, mak-ing the instantaneous on-chip electron temperature moresensitive to intermittent sources of heat. In the case ofundoped silicon, shown in Fig. 3(b), its total heat capac- í í í í 7 . í í í í I : P . 5 í : ) N ȍ (cid:0) 5 H S 5 . (cid:1) í 5 í H Q 7 í í í í 7 . í í í í I : P . 5 í : ) P ȍ (cid:0) 5 H S 5 . (cid:1) í 5 í H Q P 7 (a) (b) Fig. 2. Predicted thermal conductances for the model shown in Fig. 1 in two example situations. In both, the on-chip conductoris a copper film of size 205 µ m × . µ m × µ m (similar to the device in [52]). Its substrate is a silicon chip, which is assumedto be well-thermalised to the external refrigerator at temperature T . In (a), the on-chip electrons are electrically connectedto well-thermalised external wiring through a low-resistance (10 mΩ) bond wire. This path provides the strongest thermalconnection to the electrons for T (cid:28) .
36 mT, the effective dipolar field in copper. In (b), the resistance of the electrical connection is 10 kΩ and a magnetic field of0 . T (cid:28) ity drops all the way down to 100 µ K, where it reachesa value ∼ times smaller than the room temperaturephonon heat capacity. The situation is different in cop-per, as shown in Fig. 3(a), because its total heat capacityis boosted below ∼ b = 0 .
36 mT [36]. The heat capacity of the nuclear spinbath can be used to stabilise the electron temperature[since R en (cid:28) R ep , as shown in Fig. 2(b)] and even tocool the electrons through demagnetisation refrigeration,as discussed in later sections. Eliminating parasitic heat leaks is one of the majorchallenges in cooling nano-electronic devices down to ul-tra low temperatures. Material heat release, microwaveradiation from higher temperature stages of the dilutionrefrigerator as well as RF and low frequency noise cou-pling to the sample though its electrical leads are wellknown sources of parasitic heating. The main counter-measures include installation of radiation shields, ther-mal anchoring of the sample leads at multiple temper-ature stages of the dilution refrigerator, elimination ofground loops and, in particular, intensive microwave fil-tering. Various different filtering approaches have beenproposed in the literature, a summary of which can befound in [53]. These designs include metal powder fil-ters [33, 54–56], micro-fabricated filters [57–60], thermo- coax cables [61, 62], copper ‘tape worm’ filters [63, 64],thin film filters [65] and lossy transmission lines [66].Depending on the application, specific filtering designsmay have advantages over others, for example the useof 50 Ω characteristic filters [55] when impedance match-ing is crucial, or dissipative cryogenic filters with zero dcresistance [63] for low impedance devices. Thermocoaxcables [61, 62] provide very strong attenuation in the mi-crowave and THz regime, and can be used as signal wiresfrom room temperature down to the mixing chamber.However, thermalisation of the inner conductor carryingthe measurement signal is rather challenging, and filter-ing in the MHz regime is not as effective. Therefore acombination of thermocoax cables for the high frequencyrange and, for example, low cut-off frequency silver-epoxymicrowave filters with improved thermalisation [33], asused for microkelvin experiments at the University ofBasel (see Fig. 4), ensures good filtering throughout therelevant frequency range and optimal thermal anchoring.
The combination of higher thermal resistances andlower electronic heat capacity makes it difficult to reachon-chip electron temperatures below 10 mK using stan-dard experimental techniques in a dilution refrigerator.The most common approach, which works well at highertemperatures, is to ensure that the substrate and externalwiring are well thermalised through solid contact with thecoldest stage of the refrigerator, and to reduce parasiticheating and dissipation in the device as much as possi-ble. Often, the latter requires careful filtering of electri-cal noise in the incoming wiring. For successful examples í í í í 7 . í í í í í í í í í í - . í P R O í & H & S í í í í 7 . í í í í í í í í í í - . í P R O í & H & S & Q P 7 (a) (b) Cu Si ( n = × cm − ) Fig. 3. Molar heat capacities of copper and silicon at low temperatures. (a) Total heat capacity (solid line) of copper, whichis the sum of contributions from conduction electrons C e , phonons C p and nuclear spins C n . (b) Total heat capacity (solidline) of undoped silicon in zero applied magnetic field with a free electron concentration of 1 × cm − . This could be, forexample, silicon in the channel of an accumulation-mode FET. In both materials, C p is insignificant for T (cid:28) C n for T (cid:28) C n growsas the square of applied magnetic field. -125-100-75-50-250 A tt enua t i on ( d B ) Frequency (Hz) thermo coax noisefloor Filters layered segmented π -Filters (+2*4.7 nF) layered segmented Fig. 4. Room temperature attenuation characteristics of a1 . . demonstrating 6 mK (cid:46) T e (cid:46)
10 mK, see [33, 34, 68–72].Reaching significantly lower on-chip electron tempera-tures requires a different approach to thermalising thesample and different refrigeration technology, since eventhe best dilution refrigerators are limited to temperaturesabove 1 mK [73].Demagnetisation cooling is, at present, the most widelyused technique for cooling bulk materials below the basetemperature of a dilution refrigerator. It is used in lowtemperature laboratories [74] and has been applied, al-though much less widely, to cool micro/nanoelectronicdevices. It is an application of the magnetocaloric effect, first discovered in iron in 1883 [75], whereby the tem-perature of a suitable material can be changed upon theapplication of a magnetic field. This occurs in materialsthat are paramagnetic by virtue of an electronic mag-netic moment or as a result of the nuclear spin. Nuclearparamagnets are most relevant for the temperature rangediscussed in this review, and the corresponding coolingtechnique is known as adiabatic nuclear demagnetisation.The principle of demagnetisation cooling is well estab-lished. For overviews, see for example [36, 41, 74]. Herewe provide a brief outline for those unfamiliar with thetopic to aid understanding of later sections. Nuclear de-magnetisation refrigeration operates by controlling theZeeman splitting of the nuclear spin energy levels in anapplied magnetic field. For small magnetic fields, theZeeman splitting is much less than the thermal energy k B T , leading to a random spin-orientation distributionthroughout the refrigerant. This gives an entropy contri-bution of S = R ln(2 I + 1), for R the ideal gas constantand I the nuclear spin. In suitable materials [76], wherethis is the dominant entropy contribution, a significantentropy reduction can be obtained by ordering the spinorientations in a large magnetic field. This can be usedas part of a cooling technique by first applying a mag-netic field of ∼
10 T and then waiting for the nuclearspins to thermalise to the base temperature of a dilutionrefrigerator (a process termed precooling). The refriger-ant is then thermally isolated from the mixing chamberof the dilution refrigerator, allowing it to remain at ap-proximately constant entropy, and the magnetic field isswept down, producing cooling.The molar nuclear spin entropy is approximately [77] S ≈ R ln(2 I + 1) − λ n µ (cid:18) BT n (cid:19) . (4)This shows that the entropy is entirely a function of B/T n , meaning that the minimum attainable final tem-perature is given by T f = T i B f /B i , where T i is the initialnuclear temperature, and B i & B f are the initial and finalmagnetic fields, respectively. Note that the total mag-netic field consists of the externally applied field B ext andthe effective nuclear internal field b , which arises from themagnetic dipole interactions in the nuclei. These fieldscombine to give the total field B = p B + b .Cooling by demagnetisation often uses elaborate refrig-eration stages [74] on state of the art, custom-built di-lution refrigerators [73], or vibration isolated systems oncommercial, cryogen-free dilution refrigerators [78, 79].While these systems can readily reach bulk electron tem-peratures ∼ µ K, it is not straightforward to use themto cool a nanoelectronic sample to similar temperatures.In the remainder of this review, we will discuss recentlydeveloped techniques that can be used to overcome someof the challenges and effectively apply demagnetisationrefrigeration to micro/nanoelectronic devices and sam-ples.
3. Immersion cooling
In the context of low temperature mi-cro/nanoelectronic devices, immersion cooling meansimmersing parts of the experiment, including the device,in liquid helium to improve thermal contact with thecoldest stage of a refrigerator. This coldest stage may bethe liquid helium refrigerant inside the mixing chamberof a dilution refrigerator or the solid refrigerant ofa demagnetisation refrigerator. The helium in theimmersion cell may be either He, He, or a mixture ofthe two and, depending on the working temperature,may be in either the normal state or the superfluidstate. Thermal contact to liquid in the immersion cellis often improved by the use of sintered metal-powderheat exchangers, which provide an extremely largesolid/liquid contact area to counteract the boundaryresistance at the solid/liquid interface [36, 80]. Forexample, a sintered silver heat exchanger with a volumeof a few cubic centimetres may have a contact surfacearea ∼
10 m [81]. In the context of the thermal modeldiscussed above, immersion cooling can be used toensure that the substrate, off-chip wiring and the sampleenvironment (which contributes to ˙ Q par ) are all wellthermalised at the base temperature of a refrigerator.Immersion cooling has been used to thermalise mi-cro/nanoelectronic devices to the base temperature ofdilution refrigerators [5, 12, 31, 35, 82, 83] and demag-netisation refrigerators [11, 46, 84, 85]. In all cases wherea separate immersion cell is used, sintered metal powderheat exchangers are used to make thermal contact be-tween the helium in the cell and the cold metal parts ofthe refrigerator. In many cases, sintered silver heat ex-changers in the immersion cell are also used to make goodthermal contact with the off-chip wiring [5, 11, 12, 31, 46,82, 84, 85]. The aim is to cool on-chip electrons through electronic heat conduction, exploiting the T − scaling ofthe electronic thermal resistance (Equ. 3) in preferenceto the T − scaling of the electron-phonon thermal resis-tance (Equ. 1). This approach is particularly effectivefor samples with a low electrical contact resistance, andoptimising sample fabrication for lower resistances canproduce lower base electron temperatures [46].Despite significant efforts, electron temperaturesreached with immersion cooling are rarely below ≈ He immersion cell cooled by a PrNi nucleardemagnetisation stage. Some of the same authors havealso reported temperature-dependent behaviour down to0 . . He immersion cell cooled by a coppernuclear demagnetisation stage has been used to reachan electron temperature below 2 mK in a 2D electrongas, as measured using current-noise-sensing thermome-try [46]. While successful, experiments of the type de-scribed above require custom-made or significantly cus-tomised refrigerators. Nicolí et al. [35] developed an im-mersion cell to reach an electron temperature of 6 .
4. Demagnetisation refrigeration of electricalcontacts
In traditional microkelvin experiments, the measure-ment wiring is typically thermalised at the lowest temper-atures on a single nuclear demagnetization stage by wrap-ping a long section of wiring but making thermal contactonly through a thin layer of electrical (and thus ther-mal) insulation, preventing undesired electrical shorting.At temperatures below 10 mK, or certainly below 1 mK,this becomes prohibitively inefficient. In this section wesummarize the results obtained using networks of demag-netisation stages, where each measurement lead passesthrough its own nuclear refrigerator (NR). This elimi-nates the weak cooling link through an electrical insulatorand replaces it with electronic Wiedemann–Franz cool-ing. The approach has been implemented in three suc-cessive versions at the University of Basel. We describethese experimental setups and review measurements ofmicro/nanoelectronic devices cooled through nuclear re-frigeration of their measurement leads. The first twogenerations of nuclear stages were developed for a Lei-den cryogenics MNK wet dilution refrigerator. The thirdgeneration was installed on a Bluefors LD dry dilutionrefrigerator.A full schematic of the latest (3rd generation) de-magnetisation setup, installed on a Bluefors LD refrig-erator, is shown in Fig. 5. With increasing generationof demagnetization stage, the lowest electron tempera-ture after demagnetization in the NRs was reduced from1 mK in [86] to 0 . .
15 mKin [88]. The improvements result mainly from increasingthe amount of copper per plate (0 .
57 mol / 1 mol / 2 mol)while optimizing the geometry for reduced eddy-currentheating and increasing the diameter of the silver wires(1 .
27 mm / 1 .
27 mm / 2 .
54 mm) connecting the NRs to sil-ver sintered heat exchanges residing inside the mixingchamber. Finally, the surface area of the silver sinteredheat exchangers, as determined from BET surface areaanalysis [89], was increased from 3 m in the first two gen-erations to 9 m in the third generation. An overview ofrelevant system parameters for the different generationsof demagnetization stage is given in table 1.Measurement setups on both dilution fridges (wet anddry) use ≈ . >
100 dB attenuation for frequenciesabove ≈
200 MHz are installed at the MC level in bothexperimental setups. Transmission spectra for the mi-crowave filters and a thermocoax cable for comparisonare shown in Fig. 4. The filters consist of ≈ . . Fig. 5. Schematic of a nuclear demagnetization stage mountedon a Bluefors LD dry dilution refrigerator. The measurementleads are thermalised with Ag powder sinters ( top right pic-ture , scale bar: 5 mm) in the mixing chamber (MC, blue) andpass through C-shaped Al heat switches (green) to the Cuplates. The gradiometer of a noise thermometer as well asthe (L)CMN thermometers are positioned in a region of can-celled magnetic field between the MC and the NR stage. Thegradiometer is double-shielded by a Nb tube and a outer NbTitube (red).
Middle right inset: photograph of the gradiome-ter pick-up coil made from insulated Nb wire with 100 µ mdiameter. The 2 ×
20 turns are wound non-inductively on ahigh-purity silver wire which is spot-welded to a NR. Scalebar: 2 mm.
Lower inset: schematic cross section through thenetwork of 16 parallel NRs. Each NR is 2 mol of Cu (99 .
99 %Cu, low-H content [90], RRR ∼ . × .
17 cm ×
12 cm. This figurewas taken from [88]. therefore reveal information solely about cooling devicesthrough their electrical contacts.The GaAs quantum dots were investigated in twomodes of operation, direct transport and charge sens-ing. In the first method, a small source drain bias of V SD = 70 µ V was applied to a single quantum dot andthe resulting DC current, shown in Fig. 6, was measuredas a function of plunger gate voltage V p used to shiftthe quantum dot level with respect to the source anddrain chemical potential. In the limit of small tunnellingrates, the temperature broadening of the resulting DCcurrent steps can be fit with a Fermi-Dirac distributionto obtain separately the electronic temperature of theadjacent source and drain leads. Strictly speaking, this I DC ( p A ) V P (mV) FD fits α = 30 µeV/mVT L = 11.0 ± 1.0 mKT R = 17.8 ± 1.4 mK V SD = -70 µV < T L > ( m K ) T MC (mK)
10 mK to 2 mK.This is not so surprising since Wiedemann–Franz coolingthrough the sample leads is expected to be effectiveonly for low impedance devices, as illustrated in Fig. 2.Presumably the small temperature reduction upondemagnetization results mainly from the sample holderbeing cooled by a nuclear refrigerator through a massive,99 . V , reflects the superconducting gap. The electron tem-perature T AN can then be directly extracted by perform-ing a linear fit [solid black lines in Fig. 8(a)] to the onsetof the quasiparticle current I in logarithmic scale, i.e. T AN = e/k B · dV /d (ln I ) where e and k B are the elemen-tary charge and the Boltzmann constant, respectively.Alternatively, a fit to the full bias profile can be applied[dashed red curves in Fig. 8(a)] to extract the electrontemperature of the normal metal as described in detailin [87].Due to the huge, mm-size macroscopic leads on theNIS device, one could hope for improved off-chip nu-clear demagnetization performance compared to the highimpedance arrays present in the metallic Coulomb block-ade thermometers. Indeed, the electron temperaturedrops by ≈
30 % from ≈
10 mK to ≈ ≈
15 % reduction in temperature inthe case of the CBTs in Fig. 7(b). The limiting factor inthis case is most likely the RMS voltage noise (cid:10) V (cid:11) in themeasurement leads which couples directly to the chemi-cal potential in the normal metal and translates into anelevated temperature reading if (cid:10) V (cid:11) (cid:29) k B T e /e . In ad-dition, residual perpendicular magnetic fields also leadto a drastic overestimation of the electronic temperature[87].Table 1 summarizes all the relevant system parame-ters such as sample mount, nuclear stage dimensions andmass, filtering, sinters and so forth for the three gener-ations of nuclear stage installed on a wet MNK systemfrom Leiden cryogenics (1st and 2nd generation) and ona dry LD system from Bluefors (3rd generation). In ad-dition, an overview is given of the electron temperaturemeasurements performed using quantum dots, NIS de-vices, and metallic CBTs. Details of the lowest tempera-ture results, which were reached using CBTs on the 3rdgeneration stage, can be found in section 5.2.The experiments discussed above show that low-millikelvin on-chip electron temperatures can be success-fully reached by magnetic refrigeration of external elec-trical connections. These experiments also demonstratethat the the base electron temperature is often limitedby the device being measured, not the external refriger-ator. In the case of quantum dots and NIS thermome-ters, intrinsic noise (charge fluctuations), extrinsic noise(voltage fluctuations) and residual perpendicular mag-netic field (for the NIS thermometer) likely limited thelowest T e that could be resolved. In the case of CBTs,their high impedance meant that cooling through electri-cal connections was less effective. In the following sec-tion, we discuss how on-chip magnetic refrigeration canbe used to overcome the latter challenge.0 g / g T -1 0 1V SD (mV)1.00.80.6 g / g T -1 0 1V SD (mV) fitsT Cu =31 mK10 mK2 mK67 k Ω V ac = 5 μ VE C = 16.6 mK 175 k Ω V ac = 5 μ VE C = 19 mK4.8 M Ω V ac = 40 μ VE C = 34.8 mKT CBT = 14.9 mK 12.2 mK11.4 mK 9.5 mK31.0 mK30.3 mK T C B T ( m K ) Cu (mK) Ω sensor67 k Ω sensorheat applied3 separate(T Sp + T Cup ) p = 4.9 ± 0.4, ~ 40 aWp = 2 T CBT = T Cu demagnetization runs Ω sensorp = 3.9± 0.4p = 2 (a) (b) Fig. 7. Thermometry using various metallic Coulomb blockade thermometers with differing resistance. Normalised differentialconductance g/g T as a function of applied DC bias is shown in panel (a) for various copper plate temperatures T Cu . Off-chipdemagnetization down to T Cu = 2 mK slightly reduces the electronic temperature for the 4 . . . T Cu . Open (closed) markers represent the67 kΩ (4 . (a) (b) Fig. 8. Normal metal-insulator-superconductor (NIS) tunnel junction thermometry. (a) Linear fits (solid black) to the onsetof 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].
5. On-chip demagnetisation refrigeration
On-chip demagnetisation refrigeration uses asmall quantity of refrigerant integrated onto a mi-cro/nanoelectronic device. The refrigerant is electricallyconnected to the device’s conduction electrons, provid-ing a thermal link to the nuclear spins via hyperfineinteractions between the nuclei and electrons [36, 95].This bypasses the electron-phonon coupling bottleneckassociated with cooling a sample through its electricallyinsulating substrate. It also bypasses the weak thermallink to off-chip wiring in high impedance devices.The earliest observations of on-chip magnetic coolingwere made where, instead of using a conventional nucleardemagnetisation refrigerant such as copper, the spin en-tropy was provided by electronic paramagnetism within the material of the device structure. In [96], which is aninvestigation into the anomalous Hall effect in a topolog-ical insulator, an unexpected variation in the Hall bar’sresistivity was found and ultimately identified as the re-sult of unexpected temperature changes. These temper-ature changes arose from a magnetocaloric effect in someunknown part of the device. During experiments, thedevice temperature was reduced to 25 mK from a mix-ing chamber temperature of 40 mK. This resulted in avery low longitudinal resistance and excellent Hall con-ductance quantisation. Unexpected cooling has also beenobserved in measurements of aluminium SETs [97]. Inthis work, the aluminium was doped with manganese inorder to suppress superconductivity, which was undesir-able for good device operation. The doping was found tohave the side effect of allowing demagnetisation refriger-1
Generation stage 1st generation [86] [29, 87] [88]Refrigerator model Leiden cryogenics Leiden cryogenics Bluefors(MNK) (MNK) (LD)Wet / dry system Wet Wet DrySample mount Sample stage Sample stage Nuclear stageDemagnetization Off-chip Off-chip On- & off-chip · · . · . · .
25 cm · (12 · . · .
17 cm )Cu NR mass 0 .
57 mol 1 mol 2 molSinter surface area 3 m · . Ag wire diameter 1 .
27 mm 1 .
27 mm 2 .
54 mmDiscrete filter @ MC None [29] RC 2-pole, 10 kHz BW RC 2-pole, 45 kHz BW820 Ω/22 nF, 1 . . · [2 kΩ/680 pF][87] RC 2-pole, 30 kHz BW1 . . . T e . . .
15 mKPower curves Power curves Noise thermometryLowest sample T e Not measured QD [91]: 10 . . . . ≈ . . surface area were installed for each measurementlead. The lowest electron temperatures for the nuclear stage and sample are indicated in the last two rows. ation of the SET to 140 mK, down from the 300 mK basetemperature of the He cryostat in which the sample wasmounted.For on-chip cooling to a few millikelvin, the most ef-fective approach to date uses relatively small blocks ofmetallic refrigerant in direct electrical connection withthe circuit elements of a device. Provided the connectionhas a low enough electrical resistance, the conductionelectrons in the device and the refrigerant are essentiallya single thermal bath, cooled by demagnetisation of therefrigerant’s nuclear spins. A number of demonstrationshave been made using CBTs to measure electron temper-ature during the cooling process, at Lancaster University[52], the University of Basel [94] and Delft University ofTechnology [98, 99]. The CBT is particularly well suitedto the demonstration of magnetic cooling since the oper-ation of the device itself is insensitive to the applied mag-netic field [100–102], and it can also be fabricated withconveniently sized metallic islands for the addition of re-frigerant, which can be electroplated up to a thickness of ∼ µ m [see Fig. 9(a) and (b)]. Electroplating is usedto avoid stress build-up in the thick metal film, which of-ten occurs with more conventional deposition techniques(e.g. sputtering or evaporation). On-chip nuclear refrigeration was first demonstratedusing 6 µ m thick copper refrigerant electroplated ontothe 32 ×
20 metal island array of a CBT device [52]. Thissample was pre-cooled to T e ≈ . / s, the CBT conductance was seen todrop as would be expected for a falling on-chip electrontemperature. Repeated experiments made with differentDC biasing of the CBT confirmed that the conductancechange was indeed due to a change in temperature, andnot the result of electromagnetic induction. The lowesttemperatures reached with such single-rate demagnetisa-tions were T e ≈ . T e m pe r a t u r e ( m K ) T e T n T p E l e c t r on T e m pe r a t u r e ( m K ) mm Fig. 9. Demonstration of on-chip demagnetisation refrigeration with copper refrigerant. The CBT device shown schematically in(a) features large (6 µ m thick) Cu refrigerant blocks applied to an array of metallic islands. A photograph of the 6 . × . ×
20 array of metal islands taking up the left 3 / . / s demagnetisation, to which the three subsystem model was fitted,allowing extraction of the phonon and nuclear spin temperatures. Panel (d) shows how the base temperature and hold timewere extended by using three different demagnetisation rates instead of one. Details of the demagnetisation profiles ‘Optimised1’ and ‘Optimised 2’ can be found in [52]. also confirming that the heat flow to the nuclear spinsgoes as B , as expected from Equ. 2. With the heatleak due to eddy-current heating going as ( dB/dt ) [36],it was expected that reducing the ramp rate as the de-magnetisation proceeded to lower fields would lead tolower base temperatures (see also [103, 104]). The re-sult of this optimisation is shown by the third (red) andfourth (black) traces in Fig. 9(d), in which the latter lineshows the benefit of having a larger nuclear heat capacityif the demagnetisation is completely stopped at a highermagnetic field. Optimisation of the demagnetisation pro-file resulted in a slightly lower base electron temperatureof 4 . B/T n by using larger magnetic fieldsand lower precooling temperatures. A similar CBT wastherefore cooled in a different, Lancaster-built dilutionrefrigerator with an 8 T superconducting solenoid anda base temperature of 2 . B/T n over the dry cryostat. The Lancaster-built cryostat features an openable plasticmixing chamber [106] and sintered silver heat exchang-ers were added to the mixing chamber to help precool theCBT, as shown in Fig. 10(a) and (b). This cryostat alsohas the inherent benefit of lower mechanical vibrationsbecause there is no pulse-tube cooler, from which [107]there can be a significant additional heat leak througheddy-current heating [79, 88] and additional electricalnoise [108]. The particular dilution refrigerator used forthe results shown in Fig. 11 also features extensive vi-bration isolation and is located within a shielded roomwhich further removes vibrations and electrical noise.In Fig. 11(a), we see that the transition to a colder dilu-tion refrigerator significantly improved the base electrontemperature from 4 . . . . B/T e , scaled such that its initial value is equal tounity at the start of the wet demagnetisations. As de-scribed in section 2, the entropy of the system is entirely a3 ConeJointPackageSinterLeadSintersShieldedSilverWires ScrewTerminalsCBTPackageMeasurementLeadsCopperWiresAralditeSupportMixingChamber(a) (b)
Fig. 10. The ‘coldfinger’ used for precooling a CBT sensor in its package on a dilution refrigerator in Lancaster. Panel (a)shows the coldfinger mated with the mixing chamber hence, when cooled, the sinters shown at the top of the diagram in panel(b) are immersed in the liquid He– He refrigerant of the dilution refrigerator. Cooling is provided through the silver wireconnected to the package and also the shielded silver wires attached to each measurement lead. function of
B/T n , and since the electron-phonon couplinghere is extremely weak compared to the electron-nuclearspin coupling, we can assume T e = T n . Figure 11(b)therefore shows deviation from the ideal case of constantentropy, which would be represented by a straight hori-zontal line. There is a clear initial benefit to the use of acryostat with a lower base temperature and higher fieldmagnet, since this leads to a larger initial B/T e valueand hence a larger entropy reduction during precooling.We also see that the optimised sweeps are able to avoidthe sudden entropy change as the nuclear heat capacityis exhausted at the end of the single-rate sweeps.Traditionally, the naturally abundant copper isotopes Cu and Cu, both with spin I = 3 /
2, have been usedfor large bulk demagnetisation stages capable of them-selves reaching electron temperatures of 12 µ K [109] andcooling liquid helium to 100 µ K [110]. Copper has beenwidely used both due to its thermodynamic benefits, suchas a relatively large nuclear magnetic moment for all iso-topes and low temperature of spontaneous magnetic or-dering, but also more practical considerations such as theease at which it can be machined into a desired shape andits good availability in high purity form [36, 74]. How-ever, there are other materials which have some bene-fits over copper, particularly in terms of the magnitudeof the Korringa constant which determines the thermalcoupling between the nuclear spins and conduction elec-trons.An alternative nuclear refrigerant is indium, whichhas spin I = 9 /
2, nuclear Curie constant λ n /µ =13 . µ JKT − mol − and Korringa constant κ = 0 .
09 Ks.Indium therefore seems promising when compared to cop-per, which has smaller λ n /µ = 3 . µ JKT − mol − , and hence a smaller nuclear heat capacity, and longer Kor-ringa constant κ = 1 . µ K [111], and has a super-conducting transition at 28 mT [112], limiting the lowesttemperatures that can be reached during demagnetisa-tions. This means indium has seldom been used for theconstruction of bulk demagnetisation stages. However,for on-chip cooling, where the refrigerant is applied byelectroplating, and the minimum temperatures obtainedare currently above 300 µ K, these limitations are not nec-essarily important.Yurttagül et al. at Delft University of Technologyhave demonstrated on-chip magnetic cooling using 25 µ mthick, on-chip indium refrigerant blocks [98]. Theseblocks were electroplated onto a CBT consisting of a35 ×
15 array of metallic islands. Precooling was per-formed using a ‘wet’ dilution refrigerator equipped witha 12 . . / s, a minimum electron temperature of 3 . T e ( m K ) (a) Time (s)0.000.250.500.751.001.25 B T / ( S c a l ed ) e (b) Dry SingleDry OptimisedWet SingleWet Optimised
Fig. 11. Comparison of demagnetisation cooling using the same cooling platform on wet and dry dilution refrigerators. Panel(a) shows a comparison of the electron temperatures achieved during single rate and optimised multi-rate demagnetisations onthe wet and dry dilution refrigerators. Panel (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. electron-phonon coupling and conduction through themeasurement leads, with no controllable heat switch tobreak this link during the demagnetisation. While thismakes for easy construction of the cooling platform, apenalty is paid in terms of the continuous heat input,particularly from phonons, when the CBT electrons arecooled significantly below the temperature of the fridge.Therefore, one approach for improving the minimum elec-tron temperatures is to thermally isolate the device usinga heat switch [113] and to cool the environment surround-ing the CBT chip. This has been performed by combin-ing the on-chip demagnetisations with demagnetisationof both the incoming measurement lines and the box thesample is mounted in, as described below.
Coulomb blockade thermometers with on-chip copperrefrigerant have been studied in Basel using the 3rd gen-eration magnetic refrigeration stage on a Bluefors LD di-lution refrigerator (see section 4 and table 1 for details).In this case, the heat-leak into the cold on-chip islands isreduced by ensuring that substrate phonons and the off-chip wiring are also cooled below the base temperatureof the dilution refrigerator.While dry dilution refrigerators, such as the Blue-fors LD, seem to be the future path of low temperaturephysics, with obvious advantages compared to wet sys-tems such as lower operating costs and independence ofthe worlds helium production, there are also disadvan-tages. Stronger magnets are available for wet systems dueto the more efficient cooling when immersing the mag- net directly into liquid helium. Furthermore, the pulsetube coolers used in dry systems introduce higher levelsof vibrations, which is detrimental for adiabatic nucleardemagnetization experiments due to vibration inducededdy current heating. It is these vibrations that resultin relatively high CBT precooling temperatures in theseexperiments, as shown in Figs. 12,13. This is the currentbottleneck for this setup.In contrast to previous experiments using the Baselrefrigeration stages, here the sample is placed inside asmall copper box which is mounted directly onto a nu-clear refrigerator while using two other NRs as sampleleads. This allows for direct on-chip demagnetization ofthe copper electroplated CBT islands in addition to off-chip demagnetization. The CBTs are operated in sec-ondary mode, i.e. recording only the zero bias conduc-tance during demagnetization. While this method re-quires high temperature calibration it comes with theadvantage that no DC current passes through the devicewhich otherwise would lead to Joule heating effects. Infact, a single bias trace after demagnetization is sufficientto destroy the nuclear polarization in the Cu-plated CBTislands that was built up during precooling at large mag-netic field. The Joule heating effect is already visibleat the lowest temperatures obtained in continuous modeoperation of the dilution refrigerator without demagne-tization, as demonstrated in [33, 94].The inset in Fig. 12(a) shows the relative conduc-tance dip size δg = 1 − g ( V SD ) /g T as a function of Cu-plate temperature, where g ( V SD ) is the differential con-ductance as a function of applied source-drain bias V SD and g T the temperature-independent high-bias differen-tial conductance. The relative conductance dip size canbe approximated by δg = u/ − u /
60 + u /
630 where5 Cu ( mK ) E C = 6.55 mKfit eq.(1)806040200 T C B T ( m K ) Cu ( mK )(a) T CBT T CBT =T cu δ g ( % ) Cu ( mK ) E C = 6.55 mKfit eq.(1) 86420 T C B T ( m K ) T C u ( m K ) warm up T CBT model T Cu (d) T ( m K ) demagnetization T CBT model T Cu (c) AND ξ = 1 (CBT) AND ξ = 1 (Cu) T ( m K ) electrons CBT phononsnuclei precooling T CBT model (b) C u p l a t e Q coup Q par • • T Cu T MC (mixing chamber) T Cu T CBT
Fig. 12. Nuclear adiabatic demagnetization of a metallic Coulomb blockade thermometer. (a) CBT temperature T CBT versuscopper plate temperature T Cu . The diagonal dashed line indicates ideal thermalisation T CBT = T Cu . The inset shows thenormalized zero bias conductance dip δg as a function of T Cu . A fit [30] (solid black curve) in the high temperature regime T Cu >
30 mK is used to extract the charging energy E C = 6 .
55 mK. The three steps of nuclear demagnetization, precooling,demagnetization, and warmup, are shown in panels (b), (c), and (d) respectively. Light blue, yellow, and red data indicate T CBT , T Cu , and T MC , respectively. Black dashed curves in (b-d) are predictions from a thermal model schematically indicatedin (b). A light blue line in (c) indicates ideal adiabatic demagnetization. This figure was adapted from [94]. u = E C / ( k B T CBT ) [30]. Therefore a fit to δg in the hightemperature regime where T CBT = T Cu allows one to ex-tract the charging energy E C as the only free fit parame-ter. Subsequently, any measured conductance dip can beconverted back to an electronic temperature T CBT usingthe previously determined charging energy. The CBTagrees very well with the Cu-plate temperature and onlystarts to deviate slightly at low temperatures, reaching T CBT = 9 . T Cu = 8 . T MC and Cu-platetemperature T Cu , which both saturate just below 10 mK.The high precooling temperature is limited by pulse tubevibrations leading to either eddy current induced heat-ing in the CBT islands and/or voltage fluctuations inthe measurement wires that are then dissipated through Joule heating in the sample. The pulse tube vibrationsare clearly visible in voltage noise measurements acrossthe device (see supplemental information in [94]) show-ing up as frequency combs with a 1 . . T CBT = 7 . E C = 6 . ± .
04 mK. Fixing the NRs with re-spect to MC and still shields, together with an increasedprecooling time of 140 h results in a CBT temperatureof 20 . . .
3, giving a final CBT tempera-ture of T CBT = 1 . µ K. The electron tempera-ture then remained below 700 µ K for some 85 hours, ow-ing to a small heat leak of 27 aW per island.
6. Conclusions and open questions
Techniques for cooling micro/nanoelectronic devices toultralow temperatures have progressed significantly inthe last five years, largely through the development ofnew experimental methods based on nuclear demagneti-sation refrigeration. By using multiple, macroscopic de- magnetisation refrigerators to cool the substrate and elec-trical contacts of a device, and by incorporating micro-scopic volumes of nuclear refrigerant into a device struc-ture, it is now possible to produce and measure low- andsub-millikelvin electron temperatures on-chip. The con-tinuing development of immersion cells cooled by demag-netisation refrigeration may also provide a solution, par-ticularly for very low impedance devices. Despite theseadvances, the sensitivity of on-chip electrons to parasiticheating and electrical noise mean that it is still exper-imentally challenging to get the electrons cold and toperform accurate thermometry. Coulomb blockade ther-mometers have proven to be an excellent testbed for newcooling techniques, as they provide both reliable ther-mometry and a degree of built-in protection against elec-trical noise. It is an open question how these new coolingtechniques can be applied effectively to other types of de-vice. Based on the work to-date, it seems unlikely thatone single approach to cooling will be effective for everytype of micro/nanoelectronic device or sample. It alsoseems inevitable that careful consideration and design ofthe on-chip thermal environment will be needed for anyexperiment where sub-millikelvin electron temperaturesare required.
Acknowledgements
This work was supported by the European Mi-crokelvin Platform (the European Union’s Horizon 2020research and innovation programme, grant agreementNo. 824109), the Swiss Nanoscience Institute, NCCRQSIT, Swiss NSF No. 179024 and an ERC start-ing grant (DMZ). The data used for Fig. 11 andFig. 13 are available at https://doi.org/10.5281/zenodo.3759633 . Data from the other figures was pre-viously published, please see the respective references. [1] S. Chesi and D. Loss, Phys. Rev. Lett. , 146803(2008).[2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, Rev. Mod. Phys. , 1083 (2008).[3] A. Stern, Nature , 187 (2010).[4] W. Pan, K. W. Baldwin, K. W. West, L. N. Pfeiffer,and D. C. Tsui, Phys. Rev. B , 041301 (2015).[5] N. Samkharadze, I. Arnold, L. N. Pfeiffer, K. W. West,and G. A. Csáthy, Phys. Rev. B , 081109 (2015).[6] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. ,3045 (2010).[7] P. Simon and D. Loss, Phys. Rev. Lett. , 156401(2007).[8] P. Simon, B. Braunecker, and D. Loss, Phys. Rev. B , 045108 (2008).[9] B. Braunecker and P. Simon, Phys. Rev. Lett. ,147202 (2013).[10] C. P. Scheller, T.-M. Liu, G. Barak, A. Yacoby, L. N.Pfeiffer, K. W. West, and D. M. Zumbühl, Phys. Rev. Lett. , 066801 (2014).[11] J. Huang, J. S. Xia, D. C. Tsui, L. N. Pfeiffer, andK. W. West, Phys. Rev. Lett. , 226801 (2007).[12] T. Knighton, Z. Wu, J. Huang, A. Serafin, J. S. Xia,L. N. Pfeiffer, and K. W. West, Phys. Rev. B ,085135 (2018).[13] E. Schuberth, M. Tippmann, L. Steinke, S. Lausberg,A. Steppke, M. Brando, C. Krellner, C. Geibel, R. Yu,Q. Si, and F. Steglich, Science , 485 (2016).[14] R. Hanson, L. P. Kouwenhoven, J. R. Petta,S. Tarucha, and L. M. K. Vandersypen, Rev. Mod.Phys. , 1217 (2007).[15] J. Clarke and F. K. Wilhelm, Nature , 1031 (2008).[16] M. H. Devoret and R. J. Schoelkopf, Science , 1169(2013).[17] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys.Rev. Lett. , 077001 (2010).[18] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev.Lett. , 177002 (2010). T C B T ( m K ) δ g ( % ) MC (mK) relative cond. dip polynomial fitE C = 6.72 ± 0.04 mK T CBT = 1.8 mK (b)(a)
Fig. 13. Nuclear adiabatic demagnetization of a metallic Coulomb blockade thermometer. (a) Schematic showing the demagne-tization stage being fixed rigidly with respect to mixing chamber shield. A second set of PEEK screws fixes the mixing chambershield with respect to the still radiation shield. (b) The main panel shows the extracted CBT temperature as a function ofmagnetic field during the demagnetization process. Calibration data are shown in the inset, where blue markers correspondto measurements of the relative conductance dip δg/g T and a fit to the data in the high temperature regime from 30 mK to65 mK is shown in solid red. The resulting charging energy is E C = 6 . ± .