Nanosecond thermometry with Josephson junction
NNanosecond thermometry with Josephson junction
M. Zgirski, M. Foltyn, A. Savin, M. Meschke, and J. Pekola Institute of Physics, Polish Academy of Sciences,Aleja Lotnikow 32/46, PL 02668 Warsaw, Poland Low Temperature Laboratory, Department of Applied Physics,Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland (Dated: April 18, 2017)We demonstrate a novel approach to thermometry at the nanoscale exploiting a superconductingweak link. Such a weak link probed with nanosecond current pulses serves as a temperature sensingelement and, due to the fast inherent dynamics, is capable of delivering unprecedented temporalresolution. We employ the thermometer to measure dynamic temperature of electrons in a longsuperconducting wire relaxing to the bath temperature after application of the heating pulse. Ourmeasurement delivers nanosecond resolution thus providing the proof-of-concept of the fastest-to-date all-solid-state thermometry. Our method improves the state-of-the-art temporal resolutionof mesoscopic thermometry by at least two orders of magnitude, extending temporal resolution ofexisting experiments and introducing new possibilities for ultra-sensitive calorimeters and radiationdetectors.
Investigations of thermal processes in mesoscopic sys-tems demand application of a fast thermometry thatcan be easily integrated with a structure . With re-duction of the volume of a thermodynamic system itsthermal inertia rapidly vanishes leaving often very shorttime interval for observation of a transient from whichall important thermodynamical quantities can be de-rived. Static methods employed normal metal-insulator-superconductor (N-I-S) tunnel junctions , SQUID noisethermometry or quantum dots to explore hot elec-tron effects , quantization of heat conductance , buildMaxwell‘s demons and microcoolers . Some dynam-ical thermal properties were measured in steady statese.g. relaxation time of an excess electron energy to aphonon bath τ e − p can be accessed by measuring electron-phonon thermal conductance G e − p and assuming theusual linear temperature dependence for the electronicheat capacity . However to get a complete understand-ing of thermodynamics at nanoscale one needs obviouslyto have a thermometer operating at time scales muchshorter than thermal relaxation times . Fast ther-mometry is prerequisite for time-resolved bolometers -detectors of electromagnetic radiation, especially in thefar-infrared and THz band, for health , security andastronomical applications . A typical thermal relax-ation time τ of a nanoisland is the ratio of its heat ca-pacity C to thermal conductance G providing a path tothermal reservoir for an excess energy. If such an island isused as a sensing element of a bolometer (e.g. absorbingsingle photons), to increase device sensitivity, it is highlydesirable to reduce its heat capacity to maximize temper-ature rise upon photon absorption (∆ T = hν/C ) and re-duce thermal conductance to reservoir. However, such anoptimization may lead to reduction of the relaxation timeof the nanoisland calling for the application of even fasterthermometers. Fast thermometry would also lend strong support to development of cryoelectronics and quantumcomputing devices making it possible to control temper-ature of different components of the devices and monitortheir thermal coupling to environment.One approach to boost the temporal resolution of athermometer is to embed a temperature sensor into amicrowave or RF resonator . A change in magni-tude and phase of transmitted or reflected signal bearsinformation about thermal dynamics of the system. Themethod circumvents the problem of unavoidable stray ca-bling capacitance offering a typical bandwidth of 10 MHz.The need to use a resonator increases the sensor complex-ity and inhibits a higher level of integration (microwaveon-chip resonators are mm-sized structures). In an effortto explore thermal processes at significantly faster rateswe have developed a completely different strategy: weemploy a hysteretic superconducting weak link probedwith fast current pulses for its switching threshold as atemperature sensing element. Our thermometer is ca-pable of measuring temperature transients with unprece-dented temporal resolution, being inherently limited onlyby plasma frequency of the Josephson junction (JJ), withresponse in the picoseconds range. Moreover, it also ex-hibits other valuable characteristics: (i) it is, to the bestof our knowledge, the smallest all-solid-state-based ther-mometer; (ii) it is very simple to fabricate e.g. the Dayemnanobridge is just a piece of a nanowire interrupting athicker wire; (iii) it can be easily integrated with dif-ferent nanostructures providing high spatial resolutionfor the temperature read-out; and (iv) it requires muchsimpler hardware configuration compared to existing RF-techniques. The ease of integration, true nanometer sizeand simplicity make our thermometer a candidate forultra-low energy calorimetry and bolometry applications.Switching thermometry, as described below, can proveto be very attractive in many physical experiments e.g. a r X i v : . [ c ond - m a t . s up r- c on ] A p r FIG. 1.
Switching current measurement. (a)
IV charac-teristics of a JJ biased through R B bias resistor (cf. Fig. 4).The JJ supports supercurrent only to a certain level. Oncrossing the threshold value i sw a finite voltage developsacross the JJ. Dots, revealing bias line with slope − /R B ,are measuring artifacts related to finite response of room tem-perature electronics. In reality switching process is instanta-neous. (b) An estimator for switching probability P at givencurrent amplitude i J is measured with a train of N pulses. (c) An S-curve: P ( i J ) dependence. in determination of heat capacity, thermal conductivity,studying mechanisms of heat exchange in nanostructuresor even in experiments detecting single photons, providedpossibility to launch them on demand synchronized to thepulses probing the JJ.Below we describe our approach to fast thermometryat nanoscale. First we show how the switching featureof any superconducting weak link i.e. its transition fromthe superconducting to the normal state can be utilizedto derive the weak link temperature. We validate the in-troduced probing protocol by studying dynamic temper-ature of our model system (Aluminum superconductingnanowire), with true nanosecond resolution, and compareour measurement with prediction of the heat flow equa-tion. Subsequently, prior to a summary, we outline thepowerful perspectives for future studies that our methodbrings about. Josephson junction as a temperature-sensitive switch
JJs are sometimes referred to as switches for their abil-ity to carry supercurrent only to a certain level and,above this level, they switch to a finite voltage state(Fig. 1). The methodology of the switching current mea-surement is known . A rectangular current pulse isapplied to the junction and the response of the junction ismeasured: it switches or remains in the superconductingstate. The switching process exhibits stochastic char-acter, for it involves thermal or quantum fluctuations i sw ( m A) for P = 0.5 T ( K ) - 1 0 - 5 0 5 1 0- 1 0 0- 5 005 01 0 0 T = 1 6 m K T = 1 . 2 K T = 1 . 3 2 K i ( m A) V ( m V ) R n = 7 0 W FIG. 2.
Calibration curve.
Temperature dependence of theswitching current for the superconducting weak link studiedin this work. Inset: IV curves collected at three temperatures:16.5 mK, 1.2 K, 1.32 K (from left to right, offset horizontally). driving JJ out of its metastable state . Sending apulse train allows to determine the switching probability P corresponding to a given pulse amplitude. Repeat-ing the same experiment for different current amplitudesgives what is called S-curve: current amplitude depen-dence of the switching probability (Fig. 1). The switchingexperiments on JJs have shed some light on the natureof Andreev bound states in the superconducting pointcontacts , allowed for magnetization measurementswith nanoSQUIDs , and have been statistically studiedproving to be useful for generating random numbers .The key observation in the current context is the de-pendency of the switching current threshold on tempera-ture (Fig. 2) , a feature required for a temperature sen-sor. The JJ thermometer is calibrated by measuring itsswitching current corresponding to P = 0 . T . To bring in the temporal resolutionof the thermometer we make use of a pump & probe idea,somewhat familiar from the laser physics. This is thekey ingredient for our new approach: a nanostructure inthermal contact with the JJ is heated with a pump pulseand then, say several dozen of nanoseconds later, the JJis tested with a probe pulse (Fig. 3). The probe pulseamplitude is adjusted to yield P = 0 . FIG. 3.
The principle of the pump and probe experiment ( pump & probe pulse definition). By applying the currentpulse larger than the switching threshold we force the junction to go to the normal state (1). Then we bring the junction and itssurroundings to thermal steady state (2). The probe sequence (4,5) is delayed by time τ (3) with respect to the pump sequence(1,2) and its testing part (4), if tuned to obtain P = 0 . A P , A H and A denotecurrent amplitudes for different parts of the pump&probe pulse. Model system for testing the proposed thermometry
The suggested thermometry scheme can be realizedbased on different types of JJs, such as tunneling JJs(a very thin oxide layer sandwiched between two super-conducting electrodes), proximity JJs (a piece of normalmetal interrupting superconductor) or superconductingbridges e.g. Dayem nanobridges, etc. . In the currentwork the Aluminum Dayem nanobridge has been utilizedto demonstrate applicability and reliability of the newthermometry and highlight its superior temporal sensi-tivity. The device is presented in Fig. 4. It consists ofa narrow superconducting bridge placed in the middle ofsuperconducting wire anchored at both ends to large areacontact pads serving as energy reservoirs. Such a struc-ture is easily obtained on a silicon substrate with con-ventional one-step e-beam lithography and, what is im-portant for benchmarking, its thermal dynamics is easyto simulate as thermal properties of Aluminum are wellknown. The device is placed in the dilution refrigeratorwith base temperature of 10 mK. First we measure itsswitching current dependence on temperature i sw ( T ) at well defined bath temperatures determined with a con-ventional calibrated RuO x thermometer (Fig. 2). Then,with application of pump&probe pulse train, we performswitching current relaxation measurements of the junc-tion after it switched first to a normal state, was thenbrought to a steady state and was finally left to cooldown. For each delay between pump pulse and probepulse, we find the switching current amplitude corre-sponding to P = 0 . N = 10 ,
000 pump&probe pulses to measure eachpoint. Period of 100 µ sec have ensured complete ther-mal relaxation after each pump&probe pulse. Switch-ing current temporal changes for the junction heatedabove a critical current and relaxing back to the bathtemperature of 300 mK are displayed in Fig. 5. To con-vert switching current into dynamic temperature we usethe calibration relationship i sw ( T ) (Fig. 2). Such an ap-proach imposes the maximum temporal uncertainty ofthe temperature determination equal to the testing pulseduration i.e. about 10 ns in the presented study (see theSupplementary Note 3), but allows for a straightforwarduse of the calibration curve i sw ( T ) for recalculating of FIG. 4.
Nanostructure and measurement setup usedto benchmark the new thermometry.
The wire of length75 µ m is interrupted in the middle with a Dayem nanobridge.The width of the wire is 600 nm and its thickness is 30 nm.Two voltage amplifiers depicted with triangles measure cur-rent i J flowing into nanobridge and voltage V J across it. the switching current into dynamic temperature. Wehighlight an outstanding temporal resolution using log-arithmic scale. It reveals nanosecond resolving capabil-ity of the thermometer. We can monitor temperature ofthe link immediately after it reenters to superconductingstate, about 20 ns after switching off the heating current(Fig. 5). This time is refered to as the dead time of ourexperiment. We use the same weak link both for heatingthe wire to an elevated temperature and for sensing thedynamic temperature during relaxation. For heating weneed to transfer the weak link to the normal state, forsensing it must be in the superconducting state. Transi-tion between these two regimes requires several dozen ofnanoseconds. It is possible to make these two function-alities independent by introducing a separate heater andeliminate dead time in experiments. Steady state temperature profile - Modeling
To get a temperature profile of the wire once the wireswitches to the normal state we solve the heat balanceequation in a steady state characterized by a constantelectric current dissipating the Joule heat in the wire.The equation reads: − ddx (cid:32) κ ( T e ) dT e dx (cid:33) = f ( T e ) (1)where left part of equation deals with hot electron dif-fusion ( κ ( T e ) is the electron thermal conductivity) and f ( T e ) = H ( T c ) · r · i b S − ˙ q ep ( T e ) is a source/drain term isw ( m A) for P = 0.5 C o o l i n g t i m e ( n s ) T b a t h = 3 0 0 m K T (K) C o o l i n g t i m e ( n s ) T (K) FIG. 5.
Nanowire thermal relaxation.
Switching currenttemporal changes recorded for the bridge undergoing thermalrelaxation to the bath temperature of 300 mK (circles) andthe corresponding electron temperature inferred with the aidof the i sw ( T ) curve (triangles). A graphical conversion ispresented in the Supplementary Note 4. The pump&probepulse used in the experiment is shown in the Fig. 3. Thetiming is: τ = 100 ns, τ = 5 µ s, τ = cooling time, τ =10 ns, τ = 3 µ s. Inset shows the same T vs. cooling timedata in the linear scale. accounting for the heat generation and absorption in aunit length of the wire with r - resistance per unit length,S - wire cross-section, i b - biasing current, H ( T c ) - theHeaviside step function and ˙ q ep ( T e ) - hot electron powerfed back to phonons. κ ( T e ) and ˙ q ep ( T e ) are numericaldata calculated according to the integrals found in ref. 26and 27. At T e > T c , ˙ q ep ( T e ) is equal to (cid:80) ( T e − T ph )with (cid:80) = 1 . × W/m / K being the electron-phononcoupling constant in Aluminum. The theoretical steadystate profile corresponding to our experimental realiza-tion is presented in the inset of Fig. 6. Since the pro-file is flat in the center of the wire the only heat trans-fer under consideration is heat flow from hot electronsto phonons ( f ( T e ) = 0), yielding for our experimentwith A H = 20 µ A (cf. Fig. 3) electron temperature of T e (cid:39) . Relaxation - Modeling
The cooling of the wire is governed by the temporalrelaxation equation taking into account two relaxationpaths for the excess electron energy: the electron-phononcoupling and the diffusion of hot electrons: − ddx (cid:32) κ ( T e ) dT e dx (cid:33) = c s ( T e ) · δT e δt + ˙ q ep ( T e ) (2) FIG. 6.
Numerical modeling of temperature relaxationin the superconducting nanowire. (a)
Temporal evolu-tion of temperature profile in the wire. Temperature vari-ations in the bridge and in the pads are distinguished withseparate curves (red and black respectively). (b)
Modeledbridge (red solid curve) and pad (black solid curve) tempera-tures (the same as distinguished curves from panel (a)) com-pared with experimental relaxation (black triangles, the samedata as in Fig. 5). Cooling time t = 0 is set to be the endof the heating pulse bringing the wire to a steady state with T (cid:39) . where c s ( T e ) is experimentally determined heat capacityof Aluminum found in ref. 28. The initial condition isassumed to be the steady state profile introduced aboveand displayed in the inset of Fig. 6. The calculated tem-perature relaxation in the superconducting wire and themeasurement results are presented in Fig. 6. The fast re-laxation within first microsecond involves processes lead-ing to equalizing the temperature in the whole wire withthe temperature of the pads. More detailed analysis (seethe Supplementary Note 6) shows that above (cid:39) (cid:39) Discussion
We have benchmarked the ultrafast thermometry pro-tocol employing an Aluminum Dayem nanobridge as atemperature-sensing element. However the method willwork for any kind of JJ which exhibits switching cur-rent dependence on temperature. Subject to the temper-ature interval of interest one may use a tunnel junctionor proximity junction e.g. superconductor-normal-metal-superconductor SNS junction. By selection of proper ma-terials and adjusting the length of the normal bridge (inthe latter case) one can engineer operational temperaturerange and magnitude of the switching current to matchexperimental requirements.Temperature is an equilibrium concept and its appli-cation to describe a state of rapidly heating or coolingsystem may rise some questions. In solid state physicstemperature determines the occupation of electron states(in equilibrium according to the Fermi-Dirac distribu-tion). There are situations when electron distributionis far from equilibrium, but without external excitationit locally quickly converges to the Fermi-Dirac functiondue to fast electron-electron interaction. It happens attime scales much shorter than relaxation times consid-ered in the presented experiment. We assume that elec-tron subsystem undergoes quasi-static evolution and isdescribed locally at each moment with equilibrium Fermi-Dirac distribution. Thus temperature of electron systemis well-defined all the time during relaxation. Our con-cept of temperature measurement can be also extendedto probe non-equilibrium systems i.e. where electrons arenot Fermi-Dirac distributed. In such a case the measuredtemperature is called effective temperature and corre-sponds to the equilibrium temperature if the same phys-ical effect is observed e.g. the same switching current.Variations of switching current in JJs are set by twofactors: (i) value of the superconducting gap (it is localproperty of the junction that sets the critical current)and (ii) strength of electromagnetic fluctuations inducedby environment (it is non-local property that may makejunction sensitive to temperature of remote impedance).In case of a Dayem nanobridge with critical current ofabout 100 µ A, as the one studied here, first factor domi-nates and fluctuations play a minor role. However, for atunnel junction with low critical current fluctuations willdominate and a sensor based on such a junction will besensitive to a non-local temperature of the environment.The presented thermometry is unique with respect topower dissipation. Pulses probing JJ thermometer havesteep rising slopes and with standard equipment can bemade shorter than 1 ns. Consequently, they cause negligi-ble heating of the sample and do not rise sample temper-ature. It is in contrast to familiar DC and RF techniqueswhere probing signals may alter temperature which onewishes to measure.
Outlook
The proposed thermometer can be considered as abase sensing element for bolometers operating in the far-infrared, THz and microwave bands. Owing to its verysmall size the JJ-based thermometer could be integratedwith very small absorber with all three dimensions fallinginto a few tens of nanometers providing a versatile plat-form for sub-aJ/K calorimetry. Small metal volume of10 -10 nm employed as the absorber yields a heat ca-pacity of (10 − ) k B offering high gains in sensitiv-ity and temporal resolution of incident radiation . Asingle microwave photon absorption (2-20 GHz respec-tively) would produce a detectable 10 mK temperaturespike, allowing to count arriving photons and investigatestatistics of heat transport in superconducting quantumcircuits. The proposed thermometers, defined in the var-ious places of the electronic chip, will offer fast and highspatial resolution for mapping the temperature across thechip. They will allow to analyze the effect of fast elec-trical pulses on the chip temperature distribution thushelping to develop cryoelectronics and quantum comput- ing devices. The unprecedented temporal resolution ofthe thermometer allows to ”see” heat propagating acrossdifferent nanostructures e.g. get access to real time ob-servation of hot electron diffusion and investigate mecha-nisms of heat dissipation via phonon or photon emissionchannels . The temporal resolution of thermometer canbe pushed down into sub-ns range with application ofstandard GHz-limited arbitrary waveform generators andsample design compatible with microwave propagation. Conclusion
We have presented implementation of the hystereticsuperconducting weak link for nanosecond thermometryapplications. Such a weak link exhibits a very fast in-trinsic dynamics falling into picoseconds range and isperfectly suited for sensing rapidly changing electrontemperature, with the temporal resolution of a singlenanosecond easily achievable. Successful implementationof our approach paves the way to cutting-edge experi-ments in the field of thermodynamics of low tempera-ture quantum circuits. It gives rise to development ofultra-sensitive low noise calorimeters and bolometers forhealth, security and astronomical applications.
METHODS
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The work is supported by Foundation for PolishScience (First TEAM/2016-1/10), the EAgLE project(FP7-REGPOT-2013-1, Project No. 316014) and Pol-ish Ministry of Science and Higher Education (Grant2819/7.PR/2013/2). We acknowledge the availability ofthe facilities and technical support provided by Otaniemiresearch infrastructure for Micro and Nanotechnologies(OtaNano).