Transition-Edge Sensors
K.D.Irwin and G.C.Hilton
National Institute of Standards and Technology,Boulder,CO80305-3328,USA irvin@v
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Abstract.In recent years,superconducting transition-edge sensors(TES)have emerged as powerful,energy-resolving detectors of single photons from the near infrared through gamma rays and sensitive detectors of photonfluxes out to mil-limeter wavelengths.The TES is a thermal sensor that measures an energy de-position by the increase of resistance of a superconductingfilm biased within the superconducting-to-normal transition.Small arrays of TES sensors have been demonstrated,and kilopixel arrays are under development.In this Chapter,we de-scribe the theory of the superconducting phase transition,derive the TES calorime-ter response and noise theory,discuss the state of understanding of excess noise, and describe practical implementation issues including materials choice,pixel de-sign,array fabrication,and cryogenic SQUID multiplexing.
1Introduction
In1911,Kammerlingh Onnes cooled a sample of mercury in liquid he-lium,and made the dramatic dis
covery that its electrical resistance drops abruptly to zero as it cools through its superconducting transition temper-ature,T c=4.2K[1].A large number of materials have since been found to have phase transitions into a zero-resistance state at various transition tem-peratures.The superconducting phase transition can be extremely sharp, suggesting its use as a sensitive thermometer(Fig.1).In fact,the logarith-mic sensitivity(the Chapter“Thermal Equilibrium Calorimeters–An In-troduction”by McCammon in this book)of a superconducting transition,α=d log R/d log T,can be two orders of magnitude more sensitive than that of the semiconductor thermistor thermometer that has been used so success-fully in cryogenic calorimeters(the Chapter“Semiconductor Thermistors”also by McCammon).
A superconducting transition-edge sensor(TES),also called a supercon-ducting phase-transition thermometer(SPT),consists of a superconducting film operated in the narrow temperature region between the normal and su-perconducting state,where the electrical resistance varies between zero and its normal value.A TES thermometer can be used in a bolometer(to measure power)or in a calorimeter(to measure a pulse of energy).The sensitivity of a TES makes it possible in principle to develop thermal detectors with faster Chr.Enss(Ed.):Cryogenic Particle Detection,Topics Appl.Phys.99,63–152(2005)
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64K.Irwin and G.Hilton
Fig.1.The transition of a superconductingfilm(a Mo/Cu proximity bilayer)from the normal to the superconducting state near96mK.The sharp phase transition suggests its use as a sensitive thermometer
response,larger heat capacity,and smaller detectable energy input than ther-mal detectors made using conventional semiconductor thermistors.However, the sharp transition leads to a greater tendency for instability and lower saturation energy,so that careful design is required.
In1941,Andrews et al.applied a current to afine tantalum wire operating in its superconducting transition region at3.2K and measured the change in resistance caused by an infrared signal[2].This was thefirst demonstra-tion of a TES bolometer.In1949,the same researcher applied a current to a niobium nitride strip within its superconducting transition at15K and measured the voltage pulses when it was bombarded by alpha particles[3]–thefirst reported demonstration of a TES calorimeter.Th
is work followed on earlier suggestions by Andrews himself in1938[4]and Goetz in1939[5].
During thefirst half century after their invention,TES detectors were seldom used in practical applications.One of the principal barriers to their adoption was the difficulty of matching their noise to FET amplifiers(the TES normal resistance is typically a few ohms or less).In order to noise match,the TES was sometimes read out using a cross-correlation circuit to cancel noise[6],ac biased in conjunction with a step-up transformer[7],or fabricated in long meander lines with high normal resistance[8,9].In recent years,this problem has been largely eliminated by the use of superconduct-ing quantum interference device(SQUID)current amplifiers[10],which are easily impedance-matched to low-resistance TES detectors[11,12].In addi-tion to their many other advantages,SQUID amplifiers make it possible to multiplex the readout of TES detectors(Sect.4.2),so that large arrays of detectors can be instrumented with a manageable number of wires to room temperature.Large arrays of TES detectors are now being deployed for a number of different applications.
Transition-Edge Sensors65 Another barrier to the practical use of TES detectors has been that it is difficult to operate them within the extremely narrow superconducting tran-sition region.When they are current-biased,Joule heating of the TES by the current can lead to thermal runaway,and smallfluctuations in bath tem-perature significantly degrade performance.Furthermore,variations in th
e transition temperature between multiple devices in an array of TES detec-tors can make it impossible to bias them all at the same bath temperature. As will be explained in Sect.2.5,when the TES is instead voltage-biased and read out with a current amplifier,the devices can easily be stably biased and they self-regulate in temperature within the transition with much less sensitivity tofluctuations in the bath temperature[13].The introduction of voltage-biased operation with SQUID current readout has led to an explosive growth in the development of TES detectors in the past decade.
The potential of TES detectors is now being realized.TES detectors are being developed for measurements across the electromagnetic spectrum from millimeter[14,15,16]through gamma rays[17,18]as well as with weakly interacting particles[19]and biomolecules[20,21,22].They have contributed to the study of dark matter and supersymmetry[23,24],the chemical compo-sition of materials[25],and the newfield of quantum information[26].They have extended the usefulness of the single-photon calorimeter all the way to the near infrared[27],with possible extension to the far infrared.They are being used in thefirst multiplexed submillimeter,millimeter-wave,and X-ray detectors for spectroscopy and astronomical imaging[15,16,28,29,30].
2Superconducting Transition-Edge Sensor Theory
We now describe the theory of a superconducting transition-edge sensor.We describe the physics of the superconducting transition(Sect.2.1),summa-rize the equations for TES small-signal theory(Sect.2.2),and analyze the bias circuit for a TES and its electrical and thermal response(Sect.2.3),the conditions for the stability of a TES(Sect.2.4),the consequences of neg-ative electrothermal feedback(Sect.2.5),thermodynamic noise(Sect.2.6), unexplained noise(Sect.2.7),and the effects of operation outside of the small-signal limit(Sect.2.8).Particular implementations of both TES single pixels and arrays,including performance results,will be described in Sect.3and Sect.4.
2.1The Superconducting Transition
In this work,we discuss sensors based on traditional“low-T c”superconduc-tors(often those with transition temperatures below1K).Other classes of su-perconductors,including the cuprates such as yttrium-barium-copper-oxide, are also used in thermal detectors.Transition-edge sensors based on these
66K.Irwin and G.Hilton
“high-T c ”materials have much lower sensitivity and much higher saturation levels than those that are discussed here.
In low-T c materials,the phenomenon of superconductivity has been fairly well understood since the 1950s,when detailed microscopic and macroscopic theories were developed.Superconductivity in low-T c materials occurs when two electrons are bound together in “Cooper”pairs,acting as one particle.The energy binding Cooper pairs prevents them from scattering,allowing them to flow without resistance.Bardeen,Cooper,and Schrieffer first ex-plained the formation of Cooper pairs in 1957in the landmark microscopic BCS theory [31].The energy binding the two electrons in a Cooper pair is due to inter-actions with positive ions in the lattice mediated by phonons (quantized lattice
vibrations).When a negatively charged electron flows in a supercon-ductor,positive ions in the lattice are drawn towards it,creating a cloud of positive charge.A second electron is attracted to this cloud.The energy binding the two electrons is referred to as the “superconducting energy gap”of the material.In the BCS theory,the size of the Cooper pair wave function is determined by the temperature-dependent coherence length ξ(T ),which has the zero-temperature value ξ0≡ξ(0)≈0.18v F /(k B T c ).Here v F is the Fermi velocity of the material,k B is the Boltzmann constant,and T c is the superconducting transition temperature.At temperatures above the transi-tion temperature,thermal energies of order k B T spontaneously break Cooper pairs and superconductivity vanishes.In a BCS superconductor,the transi-tion temperature T c is related to the
superconducting energy gap E gap of the material by E gap =≈3.5k B T c .In addition to perfect dc conductivity below T c ,a second hallmark of superconductivity is the Meissner effect:the free energy of the is minimized when an external magnetic field is excluded from the interior of a superconducting sample.An applied mag-netic field is exponentially screened by an induced Cooper-pair supercurrent with an effective temperature-dependent penetration depth,λeff(T ).The ap-proximate zero-temperature value of the penetration depth is the London penetration depth,λL (0).
Near the transition temperature,the physics of a superconductor is well described by the macroscopic Ginzburg–Landau theory [32],which was de-rived by a Taylor expansion of a phenomenological order parameter Ψ.Ψwas later shown to be proportional to the density of superconducting pairs [33].One result of the Ginzburg–Landau theory is that the characteristics of a superconductor with penetrating magnetic flux (such as a superconductor on its transition)are strongly dependent on its dimensionless Ginzburg–Landau parameter,κ≡λeff(T )/ξ(T ).If κ<1/√2,the superconductor is of Type I,and the free energy is minimized when magnetic flux that has penetrated the material clumps together.If κ>1/√2,the is of Type II,and magnetic flux that has penetrated the material separates into individual flux quanta that repel each other.The flux quantum is 本页已使用福昕阅读器进行编辑。福昕软件（Ｃ）２００５－２００７，版权所有，仅供试用。
Transition-Edge Sensors 67Φ0=h/2e =2.0678×10−15Wb.Whether a film is of Type I or II influences the physics of the transition,its noise behavior,current-carrying capability,and sensitivity to magnetic field.Transition-edge sensors with T c <1K can be either Type I or II.
The superconducting films considered in this section are usually in the dirty limit
(the coherence length is typically >1µm for T c <1K,and mean free paths are usually a few tens or hundreds of nanometers.)See Table 2in Sect.3for a list of coherence lengths of typical TES superconductors.A film in the dirty limit at T c has an approximate Ginzburg–Landau Parameter ([34]p.120),
κ≈0.715λL (0)/ (d ),(1)where  (d )is the electron mean free path,which is a function of the film thickness d .See Table 2for a list of London penetration depths,and Sect.3.4for a discussion of the electron mean free path.As can be shown by (1),TES detectors with a high mean free path,such as many TES X-ray detectors,have a low κand are typically Type I.TES detectors with a shorter mean fr
ee path,including optical TES detectors fabricated using thin tungsten films,have a higher κand are typically Type II.
The physics of a BCS superconductor well below T c is largely understood.However,the situation is more complicated in the transition region.In a typical TES,the measured transition width in the presence of a very small bias current (e.g.the current from a sensitive resistance bridge)is 0.1mK to ∼1mK.In the presence of typical operational bias currents,the transition width is usually a few mK.The variation of resistance with temperature can be caused by nonuniformities in T c ,by an external field,by transport current densities approaching the critical current density,by magnetic fields induced by transport current,or by variations in temperature within the TES due to Joule heating or other sources of power.The transition is strongly influenced by the geometry and by imperfections in the boundaries of the film and in the film itself.However,the transition width is finite even for a uniform film with near-zero applied current and no external field,which is the case that we consider first.
In a Type II superconductor,a current exerts a Lorentz force on flux quanta at pinning sites in the film.When the current is near zero,the Lorentz force is insufficient to overcome the pinning force,and the superconductor does not exhibit dc electrical resistance.However,as the temperature ap-proaches the transition temperature,thermal energy of order k B T allow flux lines to jump between pi
nning sites,creating a voltage and a finite transition width.The number of vortices present is a function of the magnetic field.Even at zero field,vortex–antivortex pairs can be thermally generated in the interior of the film.At the Kosterlitz–Thouless transition temperature of the film,T KT [35,36],thermal energies are sufficient to generate and unbind vortex–antivortex pairs,creating a thermally excited distribution of vortices 本页已使用福昕阅读器进行编辑。福昕软件（Ｃ）２００５－２００７，版权所有，仅供试用。

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