Heat transport and electron cooling in ballistic normal-metal/spin-filter/superconductor junctions
Shiro Kawabata, Andrey S. Vasenko, Asier Ozaeta, F. Sebastian Bergeret, Frank W. J. Hekking
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Heat transport and electron cooling in ballisticnormal-metal/spin-filter/superconductor junctions
Shiro Kawabata a , Andrey S. Vasenko b , Asier Ozaeta c , Sebastian F. Bergeret c , d ,Frank W. J. Hekking b a Electronics and Photonics Research Institute (ESPRIT), National Institute of Advanced Industrial Science and Technology(AIST), Tsukuba, Ibaraki, 305-8568, Japan b LPMMC, Universit´e Joseph Fourier and CNRS, 25 Avenue des Martyrs, BP 166, 38042 Grenoble, France c Centro de F´ısica de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, Manuel de Lardizabal 5, E-20018 SanSebasti´an, Spain d Donostia International Physics Center (DIPC), Manuel de Lardizabal 5, E-20018 San Sebasti´an, Spain
Abstract
We investigate electron cooling based on a clean normal-metal/spin-filter/superconductor junction. Due to the sup-pression of the Andreev reflection by the spin-filter effect, the cooling power of the system is found to be extremelyhigher than that for conventional normal-metal/nonmagnetic-insulator/superconductor coolers. Therefore we can ex-tract large amount of heat from normal metals. Our results strongly indicate the practical usefulness of the spin-filtereffect for cooling detectors, sensors, and quantum bits.
Key words:
Electron cooling; Superconducting tunnel junction; Spin filter; Andreev reflection; Thermal transport
PACS:
1. Introduction
The quasiparticle transport across a normal-metal/insulator/superconductor (N/I/S) junctionis governed by single and Andreev processes. Whenthe energy E of quasiparticles is larger than thesuperconducting gap ∆, single quasiparticles cantunnel through the barrier I. This selective tun-neling of ”hot” quasiparticles gives rise to electroncooling of the normal metal in an N/I/S junc-tion [1–3]. Experimentally, the cooling of a normalmetal from 300mK down to below 100mK has beendemonstrated [1,4].On the other hand, an energy E below the gap( E < ∆), as a result of the Andreev reflection, twoquasiparticles can tunnel into S from N and form aCooper pair in the S electrodes. A limitation of the performance of N/I/S coolers is resulting from thesuch two-particle Andreev processes. The Andreevcurrent does not transfer heat through the N/I/Sinterface but rather generates so called the AndreevJoule heating [5–7]. At low temperature regimes, theAndreev Joule heating exceeds the single-particlecooling.A simple way to enhance the cooling power is toreduce the N/I/S junction transparency. However,small barrier transparency hinders ”hot” single-quasiparticle transport and leads to a serious limi-tation in the achievable cooling powers. In order toincrease the barrier transparency and to reduce theAndreev Joule heating, it was suggested to use fer-romagnetic metals (FM) as an interlayer [8–10]. Gi-azotto and co-workers have investigated the coolingof a clean N/FM/S junction theoretically and foundthe enhancement of the cooling power compared to
Preprint submitted to Elsevier 13 March 2018 onventional N/I/S junctions due to the suppressionof the Andreev Joule heating [8]. However in orderto realize such an efficient cooler, impractical FMswith extremely-high spin-polarization
P > .
94 likehalf metals [11] are needed.Recently, influences of the spin-filter effect inferromagnetic-semiconductors [12–14] on the prox-imity effect [15–21], the Josephson effect [22–32],and macroscopic quantum phenomena [33–36] havebeen investigated theoretically. Moreover, super-conducting tunnel junctions with spin-filters havebeen also realized experimentally [37–40] In thiswork we propose an novel electron-cooler based onclean N/spin-filter/S junctions [see Fig. 1(a)] andshow that the cooling power is drastically enhanceddue to the suppression of the Andreev reflection bythe spin-filter effect as described in Fig. 1(b). Pre-liminary result of this work has been reported in[41]. In this paper we will discuss about the theoret-ical derivation of the cooling power in more detail.
Fig. 1. (a) Schematic diagram of a nor-mal-metal/spin-filter/superconductor (N/SF/S) cooler and(b) the delta-function model of a SF barrier. In the SFinterface ( x = 0), the transmission probability of electronsor holes for one spin-channel is much larger than the otherone. This allows the suppression of the Andreev reflectionat the SF interface.
2. Theory
Let us first consider an one-dimensional ballisticN/SF/F junction as shown in Fig. 1(a). The spin-filtering barrier at x = 0 can be described by a spin-dependent delta-function potential [see Fig. 1(b)], i.e. , V σ ( x ) = ( V + ρ σ U ) δ ( x ) , (1)where V is a spin-independent part of the potential, U is the exchange-splitting, and ρ σ = +( − )1 for up(down) spins [22,42].The spin-filtering property of the barrier is quali-tatively characterized by the spin-filtering efficiency P = | t ↑ − t ↓ | t ↑ + t ↓ , (2)where t σ = 11 + ( Z + ρ σ S ) , (3)is the transmission probability of the spin-filteringbarrier for spin σ with m and k F being the massof electrons and the Fermi wave number. The nor-malized spin-independent and -dependent potentialbarrier-height are given by Z ≡ mV ~ k F , (4) S ≡ mU ~ k F . (5)For a perfect spin-filter with t ↑ > t ↓ = 0, weget P = 1. On the other hand, we have P = 0 forthe conventional non-magnetic barrier with U = 0( t ↑ = t ↓ ).The system can be described by the Bogoliubov-de Gennes (BdG) equation [22]: H − ρ σ U δ ( x ) ∆( x )∆ ∗ ( x ) − H + ρ σ U ( x ) δ ( x ) Φ σ ( x )= E Φ σ ( x ) , (6)where H is the spin-independent part of the single-particle Hamiltonian, i.e. , H = − ~ ∇ m + V δ ( x ) − µ F , (7)( µ F is the chemical potential),∆( x ) = ∆( T ) e iφ Θ( x ) (8)is a pair potential [ φ is the phase of the pair potentialand Θ( x ) is the Heaviside step function],2 ↑ ( x ) = u ↑ ( x ) v ↓ ( x ) , (9)Φ ↓ ( x ) = u ↓ ( x ) v ↑ ( x ) (10)are the eigenvectors, and the eigenenergy E is mea-sured from µ F .The wave function in N ( x <
0) and S ( x >
0) isgiven byΨ Nσ ( x ) = e ik + x + e − ik + x r eeσ + e ik − x r heσ , (11)Ψ Sσ ( x ) = u v e − iφ e iq + x t eeσ + v e iφ u e − iq − x t heσ , (12)where u = s (cid:18) E (cid:19) , (13) v = s (cid:18) − Ω E (cid:19) , (14) k ± = k F s ± Eµ F , (15) q ± = k F s ± Ω µ F , (16)withΩ = i p ∆( T ) − E . (17)The normal reflection coefficient r eeσ and the An-dreev reflection coefficient r heσ can be obtained bysolving the BdG equation with two boundary con-ditions at the spin-filtering barrier ( x = 0):Ψ Nσ (0) = Ψ Sσ (0) , (18) − ~ m (cid:18) ddx Ψ Sσ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 − ddx Ψ Nσ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 (cid:19) + V − ρ σ U V + ρ σ U Ψ Sσ (0) = 0 . (19) By assuming k ± ≈ q ± ≈ k F (20)based on the fact that E ∼ ∆( T ) ≪ µ F , we cananalytically obtain r eeσ and r heσ as follows: r eeσ ( E ) = − (cid:2) i (Ω Z − | E | ρ σ S ) + Ω( Z − S ) (cid:3) ( | E | + Ω) (1 − iρ σ S ) + 2Ω( Z − S ) , (21) r heσ ( E ) = ∆( T ) e − iφ ( | E | + Ω) (1 − iρ σ S ) + 2Ω( Z − S ) . (22)In the following calculations, we have determinedthe temperature T dependence of the superconduct-ing gap ∆( T ) by solving the BCS gap equation nu-merically.In order to check the suppression of the Andreevreflection by the spin-filter effect, firstly we studythe spin-dependent electron transport of the junc-tion. The voltage V dependence of the differentialconductance G of the system can be calculated fromthe Blonder-Tinkham-Klapwijk formula [43], G ( V ) = e h X σ = ↑ , ↓ [1 − B σ ( E = eV ) + A σ ( E = eV )] , (23)where B σ ( E ) ≡ | r eeσ ( E ) | ,A σ ( E ) ≡ (cid:12)(cid:12) r heσ ( E ) (cid:12)(cid:12) . (24)In Fig. 2 we plot the spin-filtering efficiency P de-pendence of the conductance G ( V ) /G N vs eV / ∆ for a junction with (a) the transparent ( t ↑ = 1 . t ↑ = 0 .
1) at zero tem-perature, where G N = 2 e h Z + S ( Z − S + 1) + 4 S , (25)stands for the conductance of an N/SF/N junc-tion and ∆ = ∆( T = 0K). If P is increased, thesub-gap conductance for | eV | ≤ ∆ is largely re-duced [41,44]. It is important to note that for thecase of the perfect spin-filter ( P = 1), the An-dreev reflection is completely inhibited, indicatingthat the spin-filter would suppress the unwantedAndreev Joule heating.In order to see the benefit of the spin-filtering bar-rier on the electron cooling, we numerically calculatethe cooling power by using the Bardas and Averinformula [1,5],3 ig. 2. The conductance G vs the bias voltage V of anormal-metal/spin-filter/superconductor (N/SF/S) cooler at T = 0K for (a) transparent ( t ↑ = 1) and (b) tunneling barrier( t ↑ = 0 . G N is the conductance of a N/SF/N junction, ∆ is the superconducting gap at T = 0K, and P is the spin-fil-tering efficiency, respectively. P = 0 . .
0) is correspondingto a nonmagnetic-insulating (a perfect SF) interlayer. By in-creasing P , the sub-gap conductance is reduced considerablydue to the suppression of the Andreev reflection. ˙ Q ( V ) = 2 eh X σ = ↑ , ↓ Z ∞−∞ dE [ E { − B σ ( E ) − A σ ( E ) }− eV { − B σ ( E ) + A σ ( E ) } ] × [ f ( E − eV ) − f ( E )] , (26)where f ( E ) stands for the Fermi-Dirac distributionfunction. In the case of ˙ Q >
0, we can realize coolingof N.The cooling power ˙ Q vs the bias voltage V for (a) t ↑ = 0 . t ↑ = 0 . T = 0 . T c , where T c isthe superconducting transition temperature. As willbe discussed later, the maximal cooling power can berealized for T ≈ . T c see Fig. 4(a). If we increase P ,the cooling power ˙ Q is enhanced drastically. Thesepeculiar results can be attributed to the suppressionof the Andreev reflection and equivalently the un-desirable Andreev Joule heating. This means that Fig. 3. The cooling power ˙ Q vs bias voltage V of an N/SF/Scooler with (a) t ↑ = 0 . T = 0 . T c for severalspin polarizations P . the spin-filter effect dramatically boosts the cool-ing power ˙ Q in comparison with conventional N/I/Scoolers.Next let us discuss about the optimization of thecooling power in terms of temperature T as wellas the spin-filtering efficiency P to design the high-performance cooler. In Fig. 4 we plot the coolingpower ˙ Q as a function of temperature T at the op-timal bias voltage V = V opt in which ˙ Q is maxi-mized as function of V . The theoretical upper-limitof the cooling power for conventional N/I/S coolers[ ˙ Q ( V opt ) ≈ . /h )] is realized for t ↑ = t ↓ ≈ .
05 and
T /T c ≈ . Q ( V opt ) is maximizedaround T ≈ . T c , decreasing at both higher andlower temperatures. From the view point of practi-cal applications, it is remarkable that if we increase P , both the maximum value of ˙ Q and the minimumtemperature in which ˙ Q ( V opt ) ≥ ig. 4. The optimal cooling power ˙ Q ( V opt ) of a N/SF/Scooler as a function of (a) temperature T and (b) the spin–filtering efficiency P . The dotted line is the theoretical upperlimit of the optimal cooling power ˙ Q max ( V opt ) ≈ . /h for conventional N/I/S coolers, which can be achieved in thecase of T/T c ≈ . t ↑ = t ↓ ≈ . the large cooling power ˙ Q is needed. We also plot thespin-filtering efficiency P dependence of the optimalcooling-power ˙ Q ( V opt ) for different values of t ↑ inFig. 4(b). The maximum cooling-power ˙ Q ( V opt ) forN/SF/S junctions can be achieved in the case of theperfect spin-filter ( P = 1) because of the completesuppression of the Andreev reflection. It is impor-tant to note that even in the small P value ( P ≪ Q ( V opt ) overcomes the theoretical upper-limit forconventional N/I/S coolers. More notably, for thecase of t ↑ = 0 .
3, ˙ Q ( V opt ) can be a factor of 15 largerthan the theoretical upper-limit for N/I/S coolers.Based on above results, we next discuss aboutthe advantage of SF-based coolers over FM-basedones (N/FM/S junctions) [8,10]. In order to realizepositive cooling-power for FM-based coolers, it wasfound that considerably high spin-polarization P > .
94 is needed. In this sense we have to use exotic andrecalcitrant FMs, like half-metals [11] in the FM in-terlayer. On the other hand, in N/SF/S cooler, muchsmaller value of P is enough for realizing the high performance cooler. This means that large numberof SF materials, e.g., Eu chalcogenides [13], rareearth nitrides, spinel ferrites [45–49], and mangan-ites [38,50–53] can be used for solid-state coolers.More importantly such junctions with large P havebeen already realized in an EuS/Al ( P ∼ .
9) [13],EuSe/Al( P ∼
1) [42], and GdN/NbN junction ( P ∼ .
8) [37]. Therefore we can conclude that SF-basedcooler is much more practical than the FM basedone. This is a crucial advantage of the SF-basedcooler.It is important to note that in spin-filter coolerswith large P , one of the spins (e.g., up-spin elec-trons) with E > ∆( T ) can tunnel through the SFbarrier, but opposite spins (e.g., down-spin elec-trons) with E > ∆( T ) can not be escaped from Nto S. This means that the only the up-spin elec-trons can contribute to the cooling. By using aS/SF1/N/SF2/S structure in which the magnetiza-tion direction of SF1 and SF2 layers is antiparallel,it is possible to effectively cool down both up- anddown-spin quasiparticles in N.
3. Summary
To summarize, we have proposed a novel electron-cooler based on ballistic N/SF/S junctions. Wefound that the cooling power ˙ Q is higher thanthe theoretical upper-limit of ˙ Q for conventionalN/I/S coolers, which results form the suppressionof the Andreev Joule heating. Our results open upa way to make efficient solid-based refrigerators forcooling several useful and practical devices, suchas superconducting X-ray detectors, single-photondetectors, magnetic sensors, NEMSs, and qubits. Acknowledgements
We would like to S. Nakamura for useful discus-sions. This work was supported by the TopologicalQuantum Phenomena (No.23103520) KAKENHIon Innovative Areas, a Grant-in-Aid for ScientificResearch (No. 25286046) from MEXT of Japan,the JSPS Institutional Program for Young Re-searcher Overseas Visits, the European UnionSeventh Framework Programme (FP7/2007-2013)under grant agreement ”INFERNOS” No. 308850,the Spanish Ministry of Economy and Competi-tiveness under Project FIS2011-28851-C02-02, andthe CSIC and the European Social Fund underJAE-Predoc program.5 eferences [1] F. Giazotto, T. T. Heikkil¨a, A. Luukanen, A. M. Savin,and J. P. Pekola, Rev. Mod. Phys. , 217 (2006).[2] P. Virtanen and T. T. Heikkil¨a, Appl. Phys. A , 625(2007).[3] J. T. Muhonen, M. Meschke, and J. P. Pekola, Rep.Prog. Phys. , 046501 (2012).[4] M. Nahum, T. M. Eiles, and J. M. Martinis, Appl. Phys.Lett , 3123 (1994).[5] A. Bardas and D. Averin, Phys. Rev. B , 12873 (1995).[6] S. Rajauria, P. Gandit, T. Fournier, F. W. J. Hekking,B. Pannetier, and H. Courtois, Phys. Rev. Lett. ,207002 (2008).[7] A. S. Vasenko, E. V. Bezuglyi, H. Courtois, and F. W.J. Hekking, Phys. Rev. B , 094513 (2010).[8] F. Giazotto, F. Taddei, R. Fazio, and F. Beltram, Appl.Phys. Lett. , 3784 (2002).[9] A. V. Burmistrova, I. A. Devyatov, M. Yu. Kupriyanov,and T. Yu. Karminskaya, JETP Letters , 203 (2011).[10] A. Ozaeta, A. S. Vasenko, F. W. J. Hekking, and F. S.Bergeret, Phys. Rev. B , 174518 (2012).[11] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.M. Daughton, S. von Moln´ar, M. L. Roukes, A. Y.Chtchelkanova, and D. M. Treger, Science , 1488(2001).[12] R. Meservey, and P. M. Tedrow, Phys. Rep. , 173(1994).[13] J. S. Moodera, T. S. Santos, and T. Nagahama, J. Phys.Cond. Mat. , 165202 (2007).[14] K. Inomata, N. Ikeda, N. Tezuka, R. Goto, S. Sugimoto,M Wojcik, and E Jedryka, Sci. Technol. Adv. Mater. ,014101 (2008).[15] T. Tokuyasu, J. A. Sauls, and D. Rainer, Phys. Rev. B , 8823 (1988).[16] J. C. Cuevas and M. Fogelstr¨om, Phys. Rev. B ,104502 (2001).[17] E. Zhao, T. L¨ofwander, and J. A. Sauls, Phys. Rev. B , 134510 (2004).[18] J. Linder, A. Sudbo, T. Yokoyama, R. Grein and M.Eschrig, Phys. Rev. B , 214501 (2010).[19] F. S. Bergeret, A. Verso, and A. F. Volkov, Phys. Rev.B , 214516 (2012).[20] P. Machon, M. Eschrig, and W. Belzig, Phys. Rev. Lett. , 047002 (2013).[21] F. Giazotto, and F. S. Bergeret, Appl. Phys. Lett. ,162406 (2013).[22] Y. Tanaka and S. Kashiwaya, Physica C , 357 (1997).[23] M. Fogelstr¨om, Phys. Rev. B , 11812 (2000).[24] M. Nishida, K. Murata, T. Fujii, N. Hatakenaka, Phys.Rev. Lett. , 207004 (2007).[25] S. Kawabata, Y. Asano, Y. Tanaka, A. A. Golubov, andS. Kashiwaya, Phys. Rev. Lett. , 117002 (2010).[26] S. Kawabata, Y. Asano, Y. Tanaka, and S. Kashiwaya,Physica E , 1010 (2010).[27] J. Linder and A. Sudbo, Phys. Rev. B , 020512(R)(2010).[28] S. Hikino, M. Mori, S. Takahashi, and S. Maekawa, J.Phys. Soc. Jpn. , 074707 (2011).[29] Y. H. Liao, M. Yang, C. Ma, and Y. B. Cao, Low. Temp.Phys. , 368 (2012). [30] F. S. Bergeret, A. Verso, and A. F. Volkov, Phys. Rev.B , 060506(R) (2012).[31] F. S. Bergeret, and F. Giazotto, Phys. Rev. B , 014515(2013).[32] S. Hikino, M. Mori, and S. Maekawa, J. Phys. Soc. Jpn. , 074704 (2014).[33] S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, andA. A. Golubov, Phys. Rev. B , 180502(R) (2006).[34] S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka,Physica C , 136 (2006).[35] S. Kawabata, and A. A. Golubov, Physica E , 386(2007).[36] S. Kawabata, Y. Asano, Y. Tanaka, S. Kashiwaya, andA. A. Golubov, Physica C , 701 (2007).[37] K. Senapati, M. G. Blamire, and Z. H. Barber, Nat.Mater. , 849 (2011).[38] T. Golod, A. Rydh, V. M. Krasnov, I. Marozau, M. A.Uribe-Laverde, D. K. Satapathy, Th. Wagner, and C.Bernhard, Phys. Rev. B , 134520 (2013).[39] A. Pal, Z. H. Barber, J. W. A. Robinson, and M. G.Blamire, Nat. Comm. , 3340 (2014).[40] P. K. Muduli, A. Pal, and M. G. Blamire, Phys. Rev. B , 094414 (2014).[41] S. Kawabata, A. Ozaeta, A. S. Vasenko, F.W. J.Hekking, and F. S. Bergeret, Appl. Phys. Lett. ,032602 (2013).[42] J. S. Moodera, X. Hao, G. A. Gibson, and R. Meservey,Phys. Rev. Lett. , 637 (1988).[43] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys.Rev. B , 4515 (1982).[44] S. Kashiwaya, Y. Tanaka, N. Yoshida, and M. R.Beasley, Phys. Rev. B , 3572 (1999).[45] U. L¨uders, M. Bibes, K. Bouzehouane, E. Jacquet, J.-P. Contour, S. Fusil, J. -F. Bobo, J. Fontcuberta, A.Barth´el´emy, and A. Fert, Appl. Phys. Lett. , 082505(2006).[46] Y. K. Takahashi, S. Kasai, T. Furubayashi, S. Mitani,K. Inomata, and K. Hono, Appl. Phys. Lett. , 072512(2010).[47] S. Matzen, J. B. Moussy, R. Mattana, K. Bouzehouane,C. Deranlot, and F. Petroff, Appl. Phys. Lett. ,042409 (2012).[48] N. M. Caffrey, D. Fritsch, T. Archer, S. Sanvito, and C.Ederer, Phys. Rev. B , 024419 (2013).[49] T. Nozaki, H. Kubota, A. Fukushima, and S. Yuasa,Applied Physics Express , 053005 (2013).[50] D. K. Satapathy, M. A. Uribe-Laverde, I. Marozau, V.K. Malik, S. Das, Th. Wagner, C. Marcelot, J. Stahn,S. Br¨uck, A. R¨uhm, S. Macke, T. Tietze, E. Goering,A. Fran´o, J.-H. Kim, M. Wu, E. Benckiser, B. Keimer,A. Devishvili, B. P. Toperverg, M. Merz, P. Nagel, S.Schuppler, and C. Bernhard, Phys. Rev. Lett. ,197201 (2012).[51] T. Harada, I. Ohkubo, M. Lippmaa, Y. Sakurai, Y.Matsumoto, S. Muto, H. Koinuma, and M. Oshima,Phys. Rev. Lett. , 076602 (2012).[52] Y. Liu, F. A. Cuellar, Z. Sefrioui, J. W. Freeland, M.R. Fitzsimmons, C. Leon, J. Santamaria, and S. G. E.te Velthuis, Phys. Rev. Lett. , 247203 (2013).[53] T. Harada, R. Takahashi, and M. Lippmaa, J. Phys.Soc. Jpn. , 014801 (2013)., 014801 (2013).