Cascaded superconducting junction refrigerators: optimization and performance limits
A. Kemppinen, A. Ronzani, E. Mykkänen, J. Hätinen, J. S. Lehtinen, M. Prunnila
CCascaded superconducting junction refrigerators: optimization and performance limits
A. Kemppinen, ∗ A. Ronzani, E. Mykkänen, J. Hätinen, J. S. Lehtinen, and M. Prunnila
VTT Technical Research Centre of Finland Ltd (Dated: September 30, 2020)We demonstrate highly transparent vanadium–silicon and aluminium–silicon tunnel junctions, where siliconis doped to remain conducting even in cryogenic temperatures. We discuss using them in a cascaded electronicrefrigerator with two or more refrigeration stages, and where different superconducting gaps are needed fordifferent temperatures. The optimization of the whole cascade is a multidimensional problem, but we presentan approximative optimization criterion that can be used as a figure of merit for a single stage only.
Normal metal – insulator – superconductor (NIS) and semi-conductor – superconductor (Sm–S) tunnel junctions can beused for electrical refrigeration, since the superconducting en-ergy gap ∆ allows thermionic energy filtering of the tunnel-ing electrons [1–4]. Remarkable proof-of-concept demonstra-tions using suspended lateral assemblies with cold fingers in-clude the refrigeration of macroscopic objects [5–7], but de-spite the extensive efforts, miniature electrical refrigeratorshave not been able to replace more macroscopic techniquessuch as dilution refrigeration. Challenges for practical appli-cations include limited cooling power and complicated engi-neering of phonon and electron-phonon heat flows in cold fin-ger solutions [8, 9], and the limited temperature range for re-frigeration that depends on ∆ . The latter could in principlebe overcome by multi-stage refrigerators that utilize super-conductors with different ∆ [10–12], but still, practical multi-stage refrigerators have not been demonstrated.Recently, the electronic refrigeration of a macroscopic sili-con chip was demonstrated using aluminium–silicon junctionsas the electronic cooling element, mechanical support, andas a blockade for phonon heat transport [13]. Phonon trans-port was suppressed simply by the Kapitza resistance betweenAl and Si [14]. This approach avoids complex arrangementsof cold fingers, which should allow a simple multi-stage as-sembly, possibly even 3D integration, see Fig. 1(a). Refer-ence [13] concludes that refrigeration from above 1 K to be-low 100 mK is a realistic target, but improvements are stillneeded: (i) Since both cooling power and phonon heat leaksare proportional to the tunnel junction area A , it is benefi-cial to decrease the characteristic resistance R A = R T A to im-prove the ratio between cooling power and phonon heat leaks.Here, R T is the tunneling resistance of the junction. (ii) Whilethe Kapitza interface appeared to be a sufficient heat blockbelow about 500 mK, the suppression of phonon heat con-ductance using, e.g., nanowire constrictions, is necessary athigher temperatures. (iii)
A superconductor with larger gapthan aluminium, e.g., vanadium, is needed for refrigerationabove about 500 mK. In this letter, we demonstrate both Al–Si and V–Si tunnel junctions with small R A . We discuss theoptimization of a cascade cooler with multiple refrigerationstages (Fig. 1(a)) and in particular, we evaluate how the finitecooling efficiency affects the complete thermal balance.The cooling power of NIS or Sm–S tunnel junction is lim- ∗ antti.kemppinen@vtt.fi FIG. 1. (a) Schematic picture of a cascaded refrigrerator with F stages. A macroscopic refrigerator provides the base temperature T ,and each stage n has different total area A n of tunnel junctions (yel-low) and temperature T n . The thermal balance of stage n is definedby the cooling power P C , n , phonon heat leak ˙ Q ph , n from the previousstage n −
1, and the heating power P H , n + resulting from the refrig-eration of the next stage n +
1. (b-d) Optimisation between γ and R A at T S = . R A and γ for A eff = ,
30, and 60 nm from left to right, respectively. The dotted lines indicate the optimal γ and R A on each line. The only difference for panels (b-d) is thephonon heat conductance, which is 3900 W / m K , 390 W / m K ,and 39 W / m K for panels (b-d), respectively. ited by the leakage resistance R in the sub-gap regime, espe-cially at temperatures well below the critical temperature ofthe superconductor T (cid:28) T c . The sub-gap leakage can orig-inate from nonidealities of the superconductor or the tunneljunction, but the fundamental limit is defined by the Andreevreflection [15, 16]. According to theory for opaque NIS junc-tions [17], the sub-gap leakage parameter γ = R / R T due toAndreev reflection, is inversely proportional to the character-istic resistance γ = A ch R K / ( R A ) . Here, A ch is the area of oneconduction channel, R K = h / e is the quantum resistance, e isthe elementary charge and h is the Planck constant. We use an a r X i v : . [ c ond - m a t . s up r- c on ] S e p effective channel area A eff = γ R A R K (1)as a figure of merit for our tunnel junction, which describesthe trade-off between γ and R A . The observed channel areasof Andreev limited Al–AlO–Cu junctions have been of the or-der of 30 nm , which is about decade higher than theoreticalestimates [17]. This has been accounted for slight inhomo-geneities of the tunnel barrier, and exponentially dependenttunneling probabilities.Figures 1(b–d) demonstrate the tradeoff between γ and R A for relative cooling T N / T S with a V–Si junction at super-conductor temperature T S = . ∆ V = µ eV [18] ( T c = .
95 K). We as-sume that the phonon heat current is ˙ Q ph = g ph A ( T S − T N ) .Figure 1(b) shows results obtained with the prefactor g ph = / m K that is expected for the Kapitza interface [13].Figures 1(c–d) demonstrate cases where the phonon heat con-ductance has been suppressed by factors 10 and 100, respec-tively. The more transparent junctions, i.e., lower R A withcost of having higher γ , are favored when the phonon heatconductance is high. The suppression of the phonon heat con-ductance shifts the preference to higher R A and related lower γ but overall Figs. 1(b-d) motivate the fabrication of tunneljunctions with R A < Ω µ m .Figure 2(a) shows the voltage–current characteristics of Al–Si and an V–Si junctions fabricated with similar process thanin Ref. [19] but with aim of increased junction transparency.The Al–Si junctions have 500 nm thick Al as superconductorbut the V–Si junctions actually consists of a multilayer Al (25nm) – V (150 nm) – Al (400 nm). The vanadium junctionsrequire a thin layer of, e.g., Al, at the junction interface to al-low high-quality junctions [10], but also the same layer andthickness of the V, can be used to tune the effective supercon-ducting gap (Fig. 2(b)) between the gap of Al, ∆ Al = µ eV,and the gap of pure V, ∆ V = µ eV [18]. The thicker Allayer on top reliefs the multilayer film strain, but it also pro-vides efficient means for quasiparticle diffusion. The Al–Sijunction has the characteristic tunneling resistance of about R A ≈ Ω µ m , which is a factor 10 improvement comparedto the results of Ref.[13]. The sub-gap leakage parameter ofthe junction is γ = × − , which yields A eff =
62 nm . Thesimilar parameters for the V–Si junction are R A ≈ Ω µ m , γ = × − , and A eff =
65 nm . This is a significant im-provement to previous results with S–Si junctions [13, 19].Our results yield an encouraging upper limit of A ch (cid:46)
60 nm for the strength of Andreev reflection in our Sm–S junctions,which is only factor 2 higher than the Andreev limit observedfor Al–Cu junctions [17]. There is no reference value for A ch in Sm–S junctions and we have not yet observed any indica-tion that we would be at this fundamental limit.The optimisation in Fig. 1 and in existing literature onlyoptimises the cooling of a single stage without taking into ac-count cooling efficiency, i.e., the ratio between cooling powerone side of the refrigeration stage, P C , and related heating theother, P H . This approach is usually valid for the first stage ofthe cascade, if the macroscopic refrigerator provides the base FIG. 2. (a) Current–voltage characteristics of Al–Si (orange) and V–Si (blue) junctions with low leakage. Both junctions were measuredin series with a similar junction, but significantly larger area. Thismay have an effect on the observed superconducting gap, but not tothe measurement of γ . (b) The superconducting energy gap of twoV–Si junctions as a function temperature obtained from tunnellingrate fits to the IV(T)s (circles and squares). The lines are the bestmatching Bardeen-Cooper-Schrieffer theory superconducting energygaps corresponding to critical temperatures of 3.55 K and 3.8 K. temperature T with large cooling power P (cid:29) P H . However,cooling of stage n in Fig. 1(a) produces significant heatingpower for the subsequent stage n −
1. Numerical optimizationof the cascade refrigerator using the full heat balance is com-putationally demanding due to the large number of parametersthat affect the system. Neither does that provide the physicalinsight of the system. Therefore, we developed an approxima-tive method for optimising individual stages of the cascade.The cooling power of an Sm–S or NIS junction is P C ≈ . ∆ / ( eR T ) × ( k b T N / ∆ ) / [2]. The optimal cooling perfor-mance occurs roughly at a fixed ratio between T N and T c , since ∆ limits cooling at high temperatures, and γ at low tempera-tures. Ideally, for any target cooling temperature T N we use asuperconductor with T c that optimises the cooling power, i.e., T N / T c is fixed, which is possible, e.g., with the help of theproximity effect , see Fig. 2(b). Then we have T N ∝ T c ∝ ∆ ,which yields P C ∝ T N f ( T N / T c ) ∝ T N [13].The cooling and heating powers of each stage n are pro-portional to the total area of tunnel junctions of the stage,i.e. P C , n , P H , n ∝ A n . The finite cooling efficiency means that P H , n (cid:29) P C , n . If the heating of stage n is a dominant heatsource of stage n −
1, we have P H , n ≈ P C , n − ≈ ( A n − / A n ) × ( T n − / T n ) × P C , n , where T n and T n − are the temperatures ofstages n and n −
1, respectively. We then obtain the ratio re-quired for the areas of sequential stages a n = A n − A n ≈ P H , n T n P C , n T n − ≡ P H T N P C T S , (2)The right hand side is written as a function of normal metaland superconductor temperatures of the stage under consider-ation, T N , and T S , respectively. The structure of the cascaderefrigerator ensures that T N = T n , and T S = T n − . The heatload caused by the cooling of the higher stages, n = , . . . , F ,requires the area ratio between the first and the last stages A / A F ≡ a = ∏ Fn = a n .An optimal cascade refrigerator has small a , since the goalis to have a compact device that provides the maximum cool-ing power to the final stage. The constraint of this optimisa-tion problem is the target cooling ratio T / T F , where we as-sume that the first stage is optimized separately as in Fig. 1.To yield an approximative solution, we assume also that eachstage n = . . . F has the same a n , i.e. a = a F − n and thesame relative refrigeration performance T S / T N , i.e., T / T F =( T S / T N ) F − . Then we have F − = lg ( T / T F ) / lg ( T S / T N ) .This yields a = O lg ( T / T F ) , where O is our optimization pa-rameter O ≡ (cid:18) P H T N P C T S (cid:19) TS − lg TN . (3)An optimal cascade refrigerator thus has refrigeration stages n = . . . F that each have small O n . It is important to notethat O only depends of on the cooling and heating powersand the temperature difference of a single stage only. It al-lows us to optimize a single stage at a time, and to use it as afigure of merit for that stage. The optimization parameter O also does not explicitly depend on the experimental parame-ters such as γ , R A , or g ph , i.e., there may be several ways toobtain similar O . To gain intuitive insight to the magnitudeof O , let us consider a realistic refrigeration objective, to coolfrom T = T F =
100 mK. If all stages n = , . . . , F havethe same O = O n , the cascade then requires the total ratio ofareas A / A F = O lg ( T / T F ) = O .Below we will demonstrate the consequencies of O param-eter optimization for specific cooler stages and cascades. Wefirst demonstrate how O yields a different result than conven-tional cooler optimization. Then we study, how well Al–Siand V–Si junctions can perform as a higher n ≥ γ and R A using A eff =
15 nm , which is in agreement with thetargeted γ = − and R A = Ω µ m of Ref. [13].Figure 3(a-b) show the relative cooling T N / T S and O , re-spectively, for Al–Si stage at 0.3 K. Both are presented as thefunction of γ and relative bias voltage v = eV / ∆ where V isthe absolute bias voltage. The minimum of O is obtained atsmaller bias voltage than v opt = − . k B T N / ∆ , which is theapproximate voltage for the maximum cooling power [2]. Itshould be also noted that the optimisation of O yields smaller γ and higher R A than the optimisation of T N / T S . This meansthat for cascaded stages, it is of advantage to sacrifice some ofthe cooling power in the lower temperature stages to suppressthe heating of stages at higher temperature. For Figs. 3(c-d)we calculated O for Al-based ( ∆ = µ eV) and V-based( ∆ = µ eV) refrigeration stages as a function T S , g ph , γ ,and v . We show O ( T S , g ph ) where O has been optimized withrespect to γ and v as shown in Fig. 3(b) for all combinations of T S and g ph separately. Regimes with O (cid:46) indicate roughlythe operating regimes where Al- and V-based stages can beoperated as a higher stage n ≥ g ph in Figs. 3(c-d) is the expected g ph = / m K for aKapitza interface. This indicates that an improvement of thephonon thermal resistance is important when the Al–Si junc-tions are used in a cascade, even though the Kapitza interfacecan be sufficient for a single Al–Si stage [13]. Aluminium FIG. 3. (a-b) Al–Si refrigerator stage at T S = . g ph =
390 W / m K , and A eff =
15 nm . (a) Relative cooling T N / T S asfunction of γ and v = eV / ∆ . (b) Optimisation parameter O as a func-tion of v and γ . The minimum position of O is denoted by whitecircle in both (a) and (b). The expected optimum bias voltage v opt for T N of minimum O is shown by the dotted white line. (c-d) Theoptimal O with respect to v and γ for Al-based ( ∆ = µ eV) andV-based ( ∆ = µ eV) refrigeration stages, respectively. can be used for refrigeration below about 500 mK, but abovethat, a superconductor with higher ∆ is needed. However, asuperconductor with higher ∆ requires even lower g ph .Figures 4(a-d) show the parameters obtained from the op-timization of Figs. 3(c-d) as a function of T S for three valuesof g ph . These results demonstrate that the optimal γ variessignificantly as a function of temperature and phonon heatconductance, and that high temperature and phonon heat con-ductance yield high optimal γ , which is expected since thenlow R A maximizes cooling power. Optimization parameter O gets very high values for non-optimal refrigeration stages,which stringent requirements for a practical cascade refriger-ator. However, Fig. 4(c) shows that Al and V can be used in acascade below about 500 mK and 1 K, respectivelyFinally, we compare temperature optimizations based onoptimizing parameter O and on the full heat balance. Experi-mentally, a desired starting temperature would be T > . He. Thismotivates to consider a refrigerator that consists of 3 stageswith ∆ = µ eV, ∆ = µ eV, and ∆ = µ eV, whereboth ∆ and ∆ can be achieved with V–Si junctions. The firststage is optimised as in Fig. 1, and the other by minimizing O and O . The black symbols and lines of Fig. 4(d) illus-trate the refrigeration performance of stages 2 and 3, whenthe first stage yields temperature T = . FIG. 4. Results from the optimization of O with respect to v and γ forAl-based ( ∆ = µ eV, orange) and V-based ( ∆ = µ eV, blue)refrigeration stages. Data with g ph = / m K , 390 W / m K ,and 39 W / m K are shown with dashed, dash-dotted, and solid lines,respectively. (a) Optimal γ as a function T S . (b) The required areascaling a n as a function of T S . (c) O as a function of T S . Vana-dium stage with g ph = / m K has O > , which is be-yond the figure scale. (d) T N as a function of T S . The grey lineshows the threshold for cooling, i.e., T N = T S . The black lines andsymbols illustrate refrigeration performance in the case where a firststage cooler provides T = . ∆ = µ eV, blue), and third stage on Al ( ∆ = µ eV, orange).The cases g ph =
390 W / m K and 39 W / m K are denoted withopen squares and closed circles, respectively. (e) Refrigeration per-formance ( T and T are blue and orange, respectively) as a functionof T of a cascade with V-based ( ∆ = µ eV) second stage and Al-based third stage. (f) Parameters a , a , and a = a a (blue, orange,and grey lines, respectively) as a function of T for the cascade of (e). phonon heat conductance g ph ≈
39 W / m K allows to reach T =
100 mK. Figure 4(e) shows T and T as a function of T . Figure 4(f) shows the required area scalings a , a , and a = a a for the refrigeration results of Fig. 4(e). Our con-clusion from Figs. 4(e-f) is that it is very demanding to builda cascade refrigerator that cools a sample from above 1 K tobelow 100 mK, but if such a device is built, the area scalingcan be as low as a ∼
10, which enables a very compact de-vice. Table I shows a comparison between results obtainedfrom the O parameter approximative optimization and the fullheat balance for the 3-stage refrigerator. There is a relatively good agreement between the approximation and the full heatbalance, if the area scalings are multiplied with a small factorof 2. . . 3.To conclude, we have developed an optimization protocolfor cascaded electronic refrigerators based on NIS or Sm–Stunnel junctions, which takes into account the heating gener-ated in the cascaded device. We use it to show the requiredcriteria using a cascade of Al–Si and V–Si tunnel junctionsfor refrigeration from above 1 K down to about 100 mK. Ouranalysis shows that energy efficient refrigeration stages are re-quired to mitigate the heating of cascaded refrigeration stagesand demonstrates the importance of phonon engineering andoptimization of tunnel junctions. Furthermore, we have ex- TABLE I. Example of thermal balance with A ch =
15 nm . Here, T est are temperatures estimated from the optimisation of single stagesusing parameter O , and T m = k are the temperatures obtained from thethermal balance of the whole cascade, where m is an extra multiplierfor the areas. For m = a n . For m = k , the areas the second and the first stage is k × a , and k × a , respectively. Values for g ph , γ and v are obtainedfrom optimization of O . Same parameters are used in the simulationsof the full model. T = . n T est T m = T m = A n / A g ph γ v O n (K) (K) (K) (W / m K ) ( − ) v opt perimentally demonstrated transparent Al–Si and V–Si tunneljunctions, which set an upper limit for A ch (cid:46)
60 nm for thechannel area of Andreev reflection in both types of junctions,which is already close results on NIS junctions [17]. ACKNOWLEDGMENTS
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