Recombination limited energy relaxation in a BCS superconductor
A.V. Timofeev, C. Pascual Garcia, N.B. Kopnin, A.M. Savin, M. Meschke, F. Giazotto, J.P.Pekola
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Recombination limited energy relaxation in a BCS superconductor
A.V. Timofeev,
1, 2
C. Pascual Garc´ıa, N.B. Kopnin,
1, 4
A.M. Savin, M. Meschke, F. Giazotto, and J.P. Pekola Low Temperature Laboratory, Helsinki University of Technology, P.O. Box 3500, 02015 TKK, Finland Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, 142432 Russia NEST CNR-INFM & Scuola Normale Superiore, I-56126 Pisa, Italy L. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia
We study quasiparticle energy relaxation at sub-kelvin temperatures by injecting hot electronsinto an aluminium island and measuring the energy flux from electrons into phonons both in thesuperconducting and in the normal state. The data show strong reduction of the flux at low tem-peratures in the superconducting state, in qualitative agreement with the presented quasiclassicaltheory for clean superconductors. Quantitatively, the energy flux exceeds that from the theoryboth in the superconducting and in the normal state, possibly suggesting an enhanced or additionalrelaxation process.
PACS numbers:
Superconducting nanostructures attract lots of atten-tion currently, partly because of their potential appli-cations, for instance, in single Cooper pair and single-electron devices, in quantum information processing, andin detection of radiation. Although the operation ofmany of these devices is based on charge transport, theenergy relaxation in them is also of importance to war-rant proper functioning either under driven conditions, orwhen subjected to environment fluctuations. Thermal-ization of the electron system with the surrounding bathis a serious concern at sub-kelvin temperatures for non-superconducting structures, but securing proper thermal-ization of a superconductor is an even greater challenge.In particular, recombination of hot quasiparticles (QP:s)into Cooper pairs slows down exponentially towards lowtemperatures. QP scattering rates in usual BCS super-conductors have been assessed theoretically already sev-eral decades ago [1, 2], and there have been measure-ments of them, both soon after the first predictions (forreview, see [1]), and recently also at very low tempera-tures [3, 4, 5]. However, although a well recognized issuein normal systems, the most relevant property of relax-ation, the associated heat flux, has not been addressedin the past. This is the topic of the present letter. Wepresent both experimental and theoretical results whichdemonstrate the importance of slow thermal relaxationin superconducting nanostructures.Energy relaxation in normal metals has been investi-gated thoroughly in experiment for a long time [6, 7, 8, 9].The central results can be summarized as follows. Inthree dimensional systems electron-phonon (e-p) heatflux P ep is P ep = Σ V ( T − T ) . (1)Here, Σ is a material constant [10], V is the volume of theelectronic system, and T e and T p are the temperaturesof electrons and phonons, respectively. Deviations fromthis behaviour towards the fourth power of temperaturehave been seen for lower temperatures (see, for example, I inj I prb V prb I inj I prb V prb FIG. 1: A typical sample (Sample C) for measuring energyrelaxation in an aluminium superconducting bar. The circuitson the right indicate injection of hot QP:s and probing theisland temperature.
Ref. [9] and also more recent Ref. [11]), and are usu-ally explained by the impurity effects [12, 13, 14] whenthe wave length of a thermal phonon becomes of the or-der of the electron mean free path or of the sample size.Nevertheless, Eq. (1) gives a good account of the heatflux observed in most experiments at sub-kelvin temper-atures. Under the same conditions electron-electron (e-e) relaxation is typically much faster; most experimentsdemonstrate the so-called quasi-equilibrium, where elec-trons have a well-defined temperature, usually differentfrom that of the phonons. Deviations from this simplepicture have been observed, e.g., in voltage biased diffu-sive wires [15].Relaxation processes in superconductors have alsobeen studied, see, e.g., Ref. [1]. The most obvious fea-tures different from the normal state are: (i) The QP:sneed to emit or absorb an energy in excess of the energygap ∆ to be recombined or excited, respectively. Thisleads to exponentially slow e-p relaxation rates at lowtemperatures. (ii) The number of QP:s is very small wellbelow T C , leading to slow e-e relaxation as well. Thefocus has been in relaxation times, with no attempts toobtain energy flux in the spirit of Eq. (1). The relaxationtime was addressed recently, e.g., in experiments on su-perconducting photon detectors [3, 4, 5]; these measure-ments suggest to confirm the recombination limited rate τ − ∝ p T /T C e − ∆ /k B T down to T /T C ≃ .
2. At lower T the relaxation time saturates due to presently poorlyknown reasons.The e-p processes in clean superconductors can becharacterized by the rate τ − k , k − q of a QP with wavevector k to emit a phonon with wave vector q , τ − k , k − q = (2 π/ ~ ) |M k , k − q | δ ( E k − E k − q − ǫ q ) N ( E k − q )[1 − f ( E k − q )][ n p ( q , T p ) + 1]. Here, M k , k − q is the ma-trix element of electron-phonon coupling in a supercon-ductor, f ( E ) is the distribution function of electrons;it is the Fermi function f ( E, T e ) = (1 + e E/k B T e ) − if electrons are in equilibrium with a temperature T e .Phonons are assumed to be in equilibrium with oc-cupation n p ( q , T p ) = ( e ǫ q /k B T p − − . Comparedto the normal state [8], we have inserted the normal-ized density of states (DOS) N ( E ). For a supercon-ductor with energy gap ∆( T ), the DOS also dependson temperature N ( E, T ) = | E | / p E − ∆( T ) Θ( E − ∆( T ) ), where Θ( x ) is the Heaviside step func-tion. Electrons emit energy to phonons at the rate P e = N ( E F ) R dE k N ( E k ) f ( E k , T e ) R d qD p ( q ) ǫ q τ − k , k − q . D p ( q ) is the phonon DOS and N ( E F ) is the normal-stateDOS at the Fermi level. Writing a similar expression forabsorption of phonons P a , we obtain the net heat flux, P ep = P e − P a . Calculating the rate and the matrixelements from the quasiclassical theory for clean super-conductors [16] we find the e-p heat flux P ep = − Σ V ζ (5) k B Z ∞−∞ dE E Z ∞−∞ dǫ ǫ sign( ǫ ) M E,E + ǫ × (cid:2) coth( ǫ k B T p )( f (1) E − f (1) E + ǫ ) − f (1) E f (1) E + ǫ + 1 (cid:3) . (2)Here, f (1) E = f ( − E ) − f ( E ); for equilibrium, f (1) E =1 − f ( E, T e ) = tanh( E k B T e ), and M E,E + ǫ is givenby M E,E ′ = N ( E ) N ( E ′ )[1 − ∆ ( T e ) EE ′ ]. In the regime T p ≪ T e ≪ ∆ /k B we obtain P ep ≃ ζ (5) Σ V T e − ∆ /k B T e ,which is by a factor 0 . e − ∆ /k B T e smaller than the resultfor the normal state [Eq. (1) with T p ≪ T e ].Figure 1 shows a typical configuration of our experi-ments. The samples were made by electron beam lithog-raphy and shadow evaporation. The parameters of thestructures are given in Table I. The aluminium block inthe centre of Fig. 1 is the volume in which energy re-laxation is investigated. Two small and two large tunneljunctions connect the island to aluminium leads. The hotQP:s are injected via one of the small tunnel junctionsin series with a large one. Because of the large asym-metry of junction parameters, essentially all the power isinjected by the small junction. The steady-state distri-bution on the island is deduced from the current-voltagecurves (IVs) of the opposite pair of junctions. We observe TABLE I: Sample dimensions and junction resistances.Sample volume ( µ m ) R , R , R , R (kΩ)A 21 · · · · · · the QP current of only the small junction; the large junc-tion remains in the supercurrent state. Measurements ina configuration with two small junctions in series as in-jectors and the two large junctions in series as probeswere also made with essentially identical results.If the island is at temperature T e and the leads at T ext , the QP current I is given by eR T I = R dEN ( E − eV, T e ) N ( E, T ext )[ f ( E − eV, T e ) − f ( E, T ext )]. Here R T is the normal state resistance of the junction. For equalisland and lead temperatures, T e = T ext , the calculated(a) and measured (b) IVs for various T e /T C are shownin Fig. 2. Wide plateaus in the regime 0 < eV < eV = ∆ is shown in (c). The agreement between ex-periment and theory is good down to T e /T C ≃ . γ ≡ Γ / ∆which yields a smeared density of states, N ( E, T ) = | Re( E + i Γ) / p ( E + i Γ) − ∆( T e ) | . In the figure weshow lines with γ = 10 − and γ = 10 − . Since at highertemperatures no fit parameter is needed, we focus ouranalysis to the range 0 . < T e /T C < T e is ele-vated, and the leads remain at T ext = 0 . T C . This is theexpected behaviour under power injection, provided thee-e relaxation is strong and that the junctions are opaqueenough not to conduct heat from the island into the leads.A peak in the IVs arises at eV = ∆( T ext ) − ∆( T e ). In Fig.2 (e) we show the corresponding measured curves at var-ious levels of injected power. The resemblance between(d) and (e) is obvious but the features in experimentalcurves are broadened in comparison to those from thetheory, which is common for small junctions [17]. In thedata analysis we next find the minimum current in theplateau-like regime at bias voltages between the ”match-ing” peak and the strong onset of QP current. This cur-rent is converted into temperature by comparing it to the T e dependent minimum current of the theoretical IVs.The power ˙ Q ( V ) deposited on the island by a biasedjunction is given by e R T ˙ Q ( V ) = R ( E − eV ) N ( E − eV, T e ) N ( E, T ext )[ f ( E, T ext ) − f ( E − eV, T e )] dE . Thisequation allows us to determine the injected power, aswell as the heat flux through all the junctions. There aretwo features to note: (i) Since typical injection voltagesin the experiment are V ≫ ∆ /e , it is sufficient to assumethat the power injected into the island equals IV /
2, i.e.,it is divided evenly between the two sides of the junc- -0.20.00.2 -2 -1 0 1 2-0.20.00.2 -11 -10 -9 -2 -1 -4 -3 -2 -1 (f)(c) (e)(d)(b)(a) T ext / T C T ext / T C eV / T ext =50 mK T e / T C e I R T / e I R T / T ext
910 mK800 mK670 mK560 mK465 mK385 mK44 mK
Q (W)-2 -1 0 1 2 T ext =50 mK >1 T e / T C T ext / T C FIG. 2: Tunnel currents under equilibrium and quasi-equilibrium conditions for a superconductor. Theoretical (a)and experimental (b) IVs of a junction at several bath temper-atures. (c) Theoretical and experimental currents at eV = ∆.The two theory lines correspond to two different realistic pairbreaking parameters γ = 10 − (upper curve) and γ = 10 − (lower curve). (d) Calculated IVs when the two leads of thejunction have different temperatures. (e) The measured IVsunder a few injection conditions. (f) The current in SampleA on the plateau between the initial peak and the rise of thecurrent at the conduction threshold around 2∆ /e . tion: the junction behaves essentially as a normal junc-tion, where this statement is true always. (ii) The heatconductance of the (probing) junction is almost constantover a wide range of voltages within the gap region. Itsvalue is low and can be neglected under most experimen-tal conditions. Yet, to test this, we varied the resistancesof the large tunnel junctions by a factor of five betweensamples A and C, without a significant effect on the re-sults. Figure 2 (f) shows the current on the plateau,as described in the previous paragraph, as a function ofpower injected, at various bath temperatures. In a widerange, from 30 mK up to 380 mK, the behaviour is al- most identical: only the higher temperature among T e and T p plays a role, in consistence with the theoreticaldiscussion. Therefore we compare the experimental re-sults at the base phonon temperature (of about 50 mK)to the theoretical results for T p ≪ T e in what follows.We studied P ep in the normal state as well by apply-ing a magnetic field of about 120 mT to suppress thesuperconductivity and measuring the partial Coulombblockade (CB) signal [18]. Like in the superconductingstate, two regimes are possible. In equilibrium the re-sults of Ref. [18] apply. Under injection, the typicalsituation is such that T ext ≪ T e , which we discuss nowin more detail. The tunnelling rates in a state with anextra charge n for adding (+) or removing ( − ) an elec-tron to the normal island with electrostatic energy change∆ F ± ( n ) = ± E C ( n ± / ∓ eV / ± ( n ) = 1 e R T Z ∞−∞ dE f ( E )[1 − f ( E − ∆ F ± ( n ))] . (3)Here E C = e / C Σ is the charging energy of the islandwith the total capacitance C Σ , and f and f are thedistributions on the source and target electrodes. Forequilibrium distribution f i ( E ) = (1 + e E/k B T i ) − with T = T , Eq. (3) yields the result of Ref. [18]. Here wehave the opposite limit of low bath temperature, T ext = T ≪ T = T e . For T = 0, f ( E ) = 1 − Θ( E ), yieldingΓ ± ( n ) = ( k B T /e R T ) ln(1 + e − ∆ F ± ( n ) /k B T e ). The cur-rent into the island is I = e P ∞ n = −∞ σ ( n )[Γ + ( n ) − Γ − ( n )]where σ ( n ) is the probability of having n extra electronson the island. Since P ∞ n = −∞ nσ ( n ) = 0 by symmetry,and P ∞ n = −∞ σ ( n ) = 1, we find for the differential con-ductance up to the first order in E C /k B T e G neq G T = 1 − E C k B T e ( eV / k B T e ) . (4)The depth of the conductance minimum at V = 0 is∆ G/G T = E C / k B T e which is 50% larger than thatin the equal temperature case. To find the width athalf minimum we need to solve cosh ( eV ± / k B T e ) = 2.The full width is V neq1 / = | V + − V − | or V neq1 / = 4 ln(3 +2 √ k B T e /e . This is about 65% of the equal-temperaturevalue, V eq1 / ≃ . k B T e /e [18].Figure 3 is a collection of the data at the base temper-ature ( ≃
50 mK), in form of island temperature T e /T C asa function of injected power. The superconducting statewas measured for the three samples. The power has beennormalized by that at T C , to present data from differentsamples on the same footing. For samples A, B and C, P ( T C ) = 14 nW, 3 nW, and 3 nW, respectively. Thedata on the three samples are mutually consistent. Thequasiclassical result for a superconductor is shown by asolid line. The normal state data were taken for SampleC which is ideal for a measurement of the island temper-ature via partial CB: it has E C /k B ≃
20 mK yielding anapproximately 8% deep Coulomb dip in conductance at (cid:1)(cid:0)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:8)(cid:9)(cid:10)(cid:11) (cid:12)(cid:13)(cid:14)(cid:15) (cid:16)(cid:17)(cid:18)(cid:19) (cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !"
FIG. 3: (color online) Energy relaxation from theory and ex-periment. The data in the superconducting state are fromSample A (squares), B (diamonds), and C (circles). The opentriangles are from Sample C in the normal state. The solidline is the result of Eq. (2) in the superconducting state. Thedotted line indicates
P/P ( T C ) = ( T /T C ) , and the dashedline P/P ( T C ) = ( T /T C ) . The inset shows three Coulombpeaks measured in the normal state under different levels ofpower injection: the solid lines are theoretical fits to them. zero bias at 90 mK (see the inset of Fig. 3). The twolarge junctions were used for probing and the small onesfor power injection. We first checked that the value V eq1 / yields a good quantitative agreement with the equilib-rium temperature data over the whole range of the ex-periment. Next we measured the quasi-equilibrium elec-tronic temperature under injection. The low base tem-perature permits the use of the expression of V neq1 / aboveto extract T e in the range displayed in Fig. 3. Power lawtype behavior can be observed over the whole tempera-ture range 0 . T C < T e . T C . The data approach those ofthe superconducting state near T C ≃ .
45 K, as expected.The power law for P ep is, however, better approximatedby T (dashed line) instead of T (dotted line) of Eq.(3), yielding a deviation of the same sign with respect tothe basic theories as in the superconducting state.The experimental data demonstrate that e-p couplingin a superconductor is weaker than in the normal state,by two orders of magnitude at T e /T C = 0 .
3. But, likein the relaxation time experiments in a superconductor[3, 4, 5], the energy flux is larger than that from the qua-siclassical theory [1, 16]. This observation could suggestthat the electron relaxation rate both in the supercon-ducting and in the normal state might be sensitive tothe microscopic quality and the impurity content of theparticular film [14]. The impurity effects on the e-p re-laxation are controlled by the parameter qℓ where ℓ is the electronic mean free path and q = k B T e / ~ u is the wavevector of an emitted phonon with energy of the orderof the electronic temperature. With the speed of sound u ∼ ℓ ∼
20 nm in our samples, we have qℓ ∼ . − T e . Thus the impurity effects can becomeessential below 1 K. More theoretical studies are thusneeded of the impurity effects on the e-p interaction inthe superconducting state.Our experiments were performed on three samples withvery different lengths and junction parameters but theyyielded essentially identical results when normalized bythe island volume. Therefore we believe that issues likethermal gradients, slow electron-electron relaxation, andthe presence of tunnel contacts have only a minor influ-ence on the results. The data thus yield the intrinsic en-ergy relaxation of QP:s in the superconducting and in thenormal state. In summary, the experiment follows quali-tatively the theoretical model that we presented. On thequantitative level there is a substantial discrepancy espe-cially for superconductors, which would imply that oneneeds to invoke an extra relaxation channel to accountfor. Solution of this quantitative disagreement and ex-periments at still lower temperatures remain as topics offuture work.We thank M. Gershenson, H. Courtois, F. Hekking, A.Niskanen, T. Heikkil¨a, and Yu. Galperin for useful dis-cussions. This work was supported by the NanoSciERAproject ”NanoFridge”, by Russian Foundation for BasicResearch grant 06-02-16002, and by the Academy of Fin-land. [1] S.B. Kaplan et al. , Phys. Rev. B , 4854 (1976).[2] M.Yu. Reizer, Phys. Rev. B , 5411 (1989).[3] P.K. Day et al. , Nature , 817 (2003).[4] A.G. Kozorezov et al. , Appl. Phys. Lett. , 3654 (2001).[5] R. Barends et al. , arXiv:0802.0640.[6] V.F. Gantmakher, Rep. Prog. Phys. , 317 (1974).[7] M.L. Roukes et al. , Phys. Rev. Lett. , 422 (1985).[8] F.C. Wellstood, C. Urbina, and J. Clarke, Phys. Rev. B , 5942 (1994).[9] E. Chow, H.P. Wei, S.M. Girvin, and M. Shayegan, Phys.Rev. Lett. , 1143 (1996).[10] F. Giazotto et al. , Rev. Mod. Phys. , 217 (2006).[11] J.T. Karvonen and I.J. Maasilta, Phys. Rev. Lett. ,145503 (2007).[12] B.L. Altshuler, Zh. Eksp. Teor. Fiz. , 1330 (1978) [Sov.Phys. JETP , 670 (1978)].[13] M.Yu. Reizer and A.V. Sergeev, Zh. Eksp. Teor. Fiz. ,1056 (1986) [Sov. Phys. JETP , 616 (1986)].[14] A. Sergeev and V. Mitin, Phys. Rev. B , 6041 (2000).[15] H. Pothier et al. , Phys. Rev. Lett. , 3490 (1997).[16] N.B. Kopnin, Theory of Nonequilibrium Superconductiv-ity (Clarendon Press, Oxford, 2001).[17] A. Steinbach et al. , Phys. Rev. Lett. , 137003 (2001).[18] J.P. Pekola et al. , Phys. Rev. Lett.73