On the nature of the superconducting-to-normal transition in transition edge sensors

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Nuclear Instruments and Methods in Physics Research A 523 (2004) 234–245

On the nature of the superconducting-to-normal transition in transition edge sensors G.W. Fraser* Department of Physics and Astronomy, Space Research Centre, University of Leicester, Leicester LE1 7RH, UK Received 18 December 2003; accepted 23 December 2003

Abstract A transition edge sensor (TES) is an X-ray microcalorimeter based on a thin film superconductor voltage-biased within its resistive transition. This superconductor is usually modelled as a variable resistor, completely described by a single value of the logarithmic temperature coefficient, a: Recent measurements of excess noise (and poorer than expected X-ray energy resolution) demand, however, a more detailed physical model of the superconducting-to-normal transition. In this paper, we show, using data from Ir films and from Ti- , W- , Ir- , Mo/Cu- and Ti/Au-based devices, that the TES may be plausibly regarded as a two-dimensional superconductor exhibiting a Kosterlitz–Thouless– Berezinsky phase transition due to vortex dynamics. Adopting this model leads to : (i) expressions for the variation of both TES resistance and the a-parameter with temperature (or bias voltage) within the transition and (ii) a physical description of the excess noise source and its scaling laws, together with strategies for its reduction. r 2003 Elsevier B.V. All rights reserved. PACS: 07.85 Fv; 81.05 Cy; 95.55 Ka Keywords: X-rays; Superconductor; Transition edge sensor; Vortex dynamics

1. Introduction The transition edge sensor (TES), consisting of a normal metal absorber coupled to a superconducting thermistor, is a form of microcalorimeter with great promise for non-dispersive X-ray spectroscopy in the 0:1210 keV band [1–13]. X-ray energy is rapidly thermalised in the normal metal, leading to a prompt temperature rise in the superconducting film. This film is voltage biased within its superconducting-to-normal (S–N) transition so that

*Corresponding author. Tel.: +44-116-252-3542; fax: +44116-252-2464. E-mail address: [email protected] (G.W. Fraser).

the temperature rise produces a large change in film resistance which, finally, is sensed by a SQUID amplifier inductively coupled to the bias circuit. The superconducting film is usually represented [12] by a simple variable resistor fully characterised by the dimensionless temperature coefficient: a¼

d log R T dR ¼ : d log T R dT

ð1Þ

If C is the detector heat capacity and G the thermal conductance to the heat sink, negative electrothermal feedback (ETF) applied to the TES then results in [1,2]: (i) a detector time constant teff which is related to t ¼ ðC=GÞ; the thermal time constant in the

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absence of feedback, by a multiplicative factor ð1 þ a=nÞ1 ; where nðB325Þ is the logarithmic derivative of the bias power with respect to temperature T: (ii) an intrinsic energy resolution DE reduced below the thermodynamic limit 2:36ðkT 2 CÞ1=2 by a factor BOa: Since practical values of a can exceed 102 ; optimised TES arrays, working at temperatures B100 mK; have attracted considerable attention as focal plane detectors for future X-ray observatories [14,15] and as detectors for laboratory X-ray microanalysis [16]. FWHM energy resolutions of ð4:570:1Þ eV at 5:89 keV [4] and ð2:070:1Þ eV at 1:49 keV [5] have been reported in single pixel TESs. The sub-eV performance predicted by bolometer theory, however, remains elusive. Recent experiments have shown that the noise power spectrum of practical TESs, based on proximity bilayers such as Ti/Au and Mo/Au, is not completely described by the expected sum of: (a) Johnson noise, (b) SQUID noise and (c) phonon noise in the thermal link to the heat bath. The proposal of Hoevers et al. [7,8] that thermal fluctuation noise (TFN) [17] is wholly responsible for the observed excesses has now been questioned [9] by the SRON researchers themselves. Gildemeister et al. [18] have developed a multi-node thermal model of the microcalorimeter, which reproduces some features of the observed noise spectra in three specific test devices. Proper identification of this noise source is, however, equivalent to constructing a correct physical model of the S–N transition itself. To date, only Irwin et al. [3], using the two fluid model of superconductivity [19], and Ukibe et al. [20], adopting a modified Fermi function, have proposed a physical form for RðTÞ in the transition region. The function of Ref. [3], however, contains two free parameters and is not readily comparable to experimental transition data. The RðTÞ function of Ref. [20] has five free parameters. The modelling challenge is a significant one. Even in a perfectly uniform, impurity-free film, and neglecting edge effects, a must be expected to be a function of bias current, applied magnetic field and temperature (i.e. bias point) within the phase transition.

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Section 2 of this paper attempts to summarise, in the present experimental context, Kosterlitz– Thouless–Berezinsky (KTB) theory, a mathematical framework used to describe the transition in two-dimensional, current-carrying superconductors. In Section 3, we use this model to account for measured resistive transitions in both single element (Ir, Ti, W) films and in proximity bilayers (Ti/Au). We investigate the relationship between normal state sheet resistance and the transition width and set a limit to the maximum attainable a: We associate the excess noise source observed for TES bias points low down in the transition [7–9] with current-induced vortex unbinding in the superconducting film. We note that it may be possible to eliminate this excess noise by applying a magnetic field perpendicular to the plane of the superconducting film.

2. Theory Both TES detector physics and the detailed theory of two-dimensional superconductors— concentrating on the sheet conductance Gs ðo; TÞ Gs ¼ s1 ðo; TÞd  is2 ðo; TÞd

ð2Þ

are active areas of research which have hitherto remained largely distinct, possibly because of the mathematical difficulty of the latter endeavour, directed mainly at an understanding of high Tc superconducting films. In Eq. (2), d is the film thickness and s1 and s2 are, respectively, the real and imaginary parts of the complex conductivity. 2.1. Two-dimensionality In a practical TES, the superconducting film thickness is typically B100 nm: For any twodimensional theory to apply, d must obey the inequality [21]: doxðTÞ

ð3Þ

where x is the Ginzburg–Landau coherence length [19], found from the Pippard coherence length x0 E_nF =pDð0Þ by means of either the clean limit

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formula [22]: 1=2

xðTÞ ¼ 0:74x0 ½1  t

ð4aÞ

or the dirty limit expression xðTÞ ¼ 0:85½x0 l1=2 ½1  t1=2 :

ð4bÞ

nF is the Fermi velocity and Dð0Þ is the superconducting energy gap at absolute zero temperature. Eq. (4a) refers to the case where the electron mean free path l is much greater than x0 ; Eq. (4b) to the opposite extreme l5x0 : t denotes the ratio of the device temperature to ‘‘the transition temperature’’ (precisely defined in Section 2.2.3 below). For t sufficiently close to unity, the inequality doðx0 lÞ1=2 is a conservative measure of two-dimensionality. For proximity bilayers, the appropriate form of the effective coherence length is given by Nagel et al. [23]. 2.2. The KTB model 2.2.1. Outline In this picture, the S–N transition is accounted for by the creation and subsequent dynamics of vortex cores. Introductions to this model, originally developed to describe neutral superfluids such as He films, are given in Refs. [19,21,24–30]. The definitive account of the charged ‘‘two-dimensional Coulomb gas’’ is given by Minnhagen [31]. In a bulk Type II superconductor, resistive effects arise in an external field due to vortices associated with magnetic flux penetration. The simple view of a vortex (or vortex core) is as a cylinder of normal material whose radius is the Ginzburg–Landau coherence length xðTÞ defined in Eqs. (4a) and (b). The divergence of these equations as t-1; reflects the complete transformation to the normal state as vortices grow in size. Each vortex carries a flux quantised in multiples of the flux quantum Fo ¼ h=2e: In the presence of an electric field along the length L of the film, vortices move parallel to the width W : The fact that the vortices originate in Type II superconductor models while all the superconductors used to fabricate TESs are Type I, is not an impediment to the KTB model. Essentially, the distinction between Types I and II is not made in two dimensions. An ðH; TÞ phase diagram for a

two-dimensional superconductor is given by Doniach and Huberman [32]. In the absence of an external field, the free energy U to create a vortex– anti-vortex pair is of order kT; where k is Boltzmann’s constant. In the zero field, zero current superconducting state, therefore, thermally excited vortices occur in bound pairs of opposite sign and the transition to the resistive state begins at a precise temperature Tc ; above which a nonzero areal density of free vortices nf is present. In the presence of an ‘‘ionising’’ current (the bias current in the case of a TES), nf a0 even for ToTc : In this picture, there is, fundamentally, no true phase transition in a current carrying 2-d superconductor. Tc is the KTB transition temperature, otherwise known as the vortex unbinding temperature. This model of a 2-d superconductor is strictly only valid while a small fraction of the sample area consists of vortex cores [30]. We may therefore expect the model to break down, in the TES context, for bias points at the high temperature end of the S–N transition. 2.2.2. Finite size effects In applying the KTB model of the S–N transition to a practical device such as a TES [33–40], we must be aware of finite size effects [24,25,29], in addition to the film thickness criterion discussed in Section 2.1. The vortex free energy function UðrÞ depends on r; the separation of the centres of a vortex pair, either as r1 (just as for the neutral superfluid film) or as r2 ; according to whether r is less than or greater than the 2-d penetration length l2d : l2d ¼ 2l2 =d:

ð5Þ

Here, l is the London penetration length of the bulk material. Thus, provided the size of the film is small compared with l2d ; the dependence of U on r will be the same as in the superfluid case and the KTB theory should apply. According to Halperin and Nelson [24], l2d and Tc are simply related by l2d ðcmÞ ¼ 2=Tc ðKÞ:

ð6Þ

Beasley et al. [25] estimate the numerator on the right-hand side of Eq. (6) to be 0.98, rather than 2.0. For TESs designed to operate at temperatures B0:1 K; the relevant l2d values are of order

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10220 cm; compared to typical widths W B0:521 mm (Section 3). Thus, we can proceed with confidence, at least in terms of device geometry, to apply the KTB theory. 2.2.3. Transition width Experimentally, the width of the S–N transition is usually quoted as the temperature difference between 10% and 90% fractions of the normal state resistance. The width of the S–N transition in the KTB model, dTc ; is simply given by: dTc ¼ Tc0  Tc :

ð7Þ

Here, Tc0 ; the mean field or fluctuation-corrected BCS critical temperature, is the temperature at which the transition to the resistive state is essentially complete. The parameter t introduced in Section 2.1 is then seen to be t ¼ T=Tc0 :

ð8Þ

Combining BCS and KTB formalisms in the dirty limit, it follows [24,25,28] that the normalised transition width should vary with the normal state sheet resistance of the superconducting film RN according to dTc 0:17 RN e E ð9Þ Rc Tc0 provided RN is much less than Rc ; the characteristic resistance in two dimensions Rc ¼ _=e ¼ 4100 O 2

ð10Þ

e is a dimensionless parameter of order unity. Eq. (9) provides a challenge to the KTB theory, in the case of low-resistance TES devices—the predicted proportionality of transition width to normal state sheet resistance. TESs are normally fabricated with surface resistances RN B1 O=& and transition temperatures around 100 mK: For such low resistances, Eq. (9) predicts transition widths B4 mK; rather than the B1 mK typically observed experimentally. However, Eq. (9) does not take into account other established transitionbroadening effects such as (a) non-zero ambient magnetic field, (b) edge effects, (c) vortex pinning at impurity sites.

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The effect of small B-fields is to enhance the low-temperature ‘‘tail’’ or ‘‘step’’ of the S–N transition. Resistive tails in the S–N transition curves of high-quality Nb films have been observed for fields as low as B1 G [41]. The presence of impurities in the film promotes the possibility that vortex-antivortex pairs are trapped and then dissociate—adding to the effective resistance— rather than move within the film. Failure to observe a direct relationship between dTc and RN does not, therefore, rule out the KTB theory. Rather, Eq. (9) provides an upper limit to the ‘sharpness’ of the transition in practical TESs. 2.2.4. Transition RðTÞ and aðTÞ Adopting vortex dynamics as a plausible physical description of the S–N transition, while recognising (Section 2.2.3) the possible limitations of the KTB model, leads to a number of competing expressions for the variation of TES resistance with temperature within the S–N transition. According to Pierson et al. [29]:
1=2 RðTÞ ¼ RN expðz b=ðT=Tc  1Þ Þ ð11aÞ which is a generalisation of the Minnhagen form [31]:  1=2 ! bðTc0  Tc Þ RðTÞ ¼ RN exp 2 : ð11bÞ T  Tc The critical exponent z characterises the nature of the vortex dynamics; z ¼ 2 corresponds to simple diffusion. The parameter b is a fit parameter (usually in the range 1–10). The form of RðTÞ which we have used to make comparisons with measurements—with b set to unity for simplicity— is that suggested by Kadin et al. [28]:  0 1=2 ! Tc  T RðTÞ ¼ RN exp 2 : ð11cÞ T  Tc We obtain from Eqs. (1) and (11c) an analytical expression for the temperature coefficient a: dTc T : ð12aÞ aðTÞ ¼ 3=2 ðT  Tc Þ ðTc0  TÞ1=2 By inspection, a tends to infinity at both extrema of the S–N transition, T ¼ Tc and T ¼ Tc0 : The a value at the mid-point of the transition has the

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simple form:  0  T þ Tc am ¼ 2 c0 : Tc  Tc

ð12bÞ

In what follows, we use Eqs. (11c) and (12a) to analyse S–N transition data for both simple 2-d films and for actual TES devices. A combination of Eq. (9) and (12a) gives, for narrow S–N transitions, the result: am ¼ 23:5Rc =RN

ð12cÞ

providing for the first time an absolute (but not restrictive) upper limit to the a parameter (B1  105 for a TES with 1 O normal state resistance). 2.2.5. The I2V curve An important test of KTB theory is provided by the current–voltage characteristic of the superconducting film at temperatures both above and below the vortex unbinding temperature. The voltage V across the film is predicted to have a power-law dependence on the current I flowing through it. The critical exponent of the power law, aðI; TÞ; has been widely studied [26–28]. At the vortex unbinding temperature T ¼ Tc ; a ¼ 3: At T ¼ Tc0 ; a ¼ 1: A voltage response for small currents which is linear in current is the expected response for thermally excited free vortices [26]. TES I2V curves must always be measured in order to establish electrothermal feedback conditions, but their interpretation is complicated by the presence in parallel with the TES of a shunt resistor, whose resistance at some point low in the S–N transition will become large with respect to the resistance of the TES itself. For device operation, therefore, the lowest voltages are of

no real interest [11]; only a few authors [8,34] have reported I  V curves near V ¼ 0: In both these papers, however, the derived value of a is very close to unity, providing our initial evidence from the TES literature for the validity of the KTB model.

3. Measurements 3.1. Iridium films Ir thin film resistor meanders were produced by magnetron sputtering by Oxford Instruments Superconductivity (Cambridge, UK) under contract to the University of Leicester. These samples were fabricated prior to the production of actual TES detectors [6], in order to evaluate batch-tobatch reproducibility and the dependence of transition temperature on process parameters. Six Ir ‘‘chips’’ were produced, each with fifteen resistors. Transition measurements were made for both increasing and decreasing temperature. The film properties, controlled to first order by silicon nitride substrate temperature during sputtering, are summarised in Table 1 for five of the six chips. The sixth chip exhibited very broad, stepped superconducting transitions and was not considered further. The deposition process and subsequent low-temperature measurements are described by Trowell [33]. The transition temperatures are all significantly higher than the bulk iridium value of 112 mK; indicating, according to Fukuda et al. [34], the presence of an Irx Si1x interface layer.

Table 1 Measured [33] and modelled properties of iridium thin film resistors Figure no.

dð75 nmÞ

RRR ð70:01Þ

Fit Tc ðmKÞ

Fit Tc0 ðmKÞ

Fit RN ðOÞ

1(a) 1(b) 1(c) 1(d) 1(e)

100 100 200 200 100

1.74 1.74 1.93 1.94 1.71

146.5 153.5 130 138.5 152:2; 155:0

147.9 156.0 135.5 139.3 158:0; 159:0

1050 1095 450 3440 1005; 1100

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The quality of superconducting films is measured using the residual resistivity ratio:

RRR ¼

rð300 KÞ : rð4:2 KÞ

ð13Þ

The values given in Table 1 are less than those reported by Nagel et al. [23]. These authors’ Ir films ð120odo400 nmÞ were produced from 99.999% pure powder and exhibited RRR values in the range 1.3–23.4, with an average value over 26 samples of 4.48. Our RRR values, however, closely match those of Ref. [34]—1.71 for an iridium layer deposited on Si3 N4 and 1.50 for Ir on a silicon substrate. The iridium films of Table 1 were either 100 or 200 nm thick. For bulk iridium, x0 ¼ 4400 nm [23]. In the case of a bilayer system, the effective coherence length is less than x0 : Trowell [33] has used the De Gennes–Werthamer dirty limit bilayer model to estimate coherence length values greater than 328 nm for the present Ir films. Taking the increase in coherence length near transition expressed in Eqs. (4a) and (b) into account, it is reasonable to conclude that Eq. (3) is satisfied and that we can proceed to a trial of the vortex model. Figs. 1(a)–(e) compare measured RðTÞ curves for the Ir meanders with the expression given in Eq. (11c). For one of the Ir ‘‘chips’’, a single representative data set (Fig. 1a) is shown. For the remaining chips, two or more data sets are displayed—obtained from different test structures on the same chip and/or from measurements made by the heating up and cooling down of the same meander. There was no evidence for hysteresis in the temperature cycling of any of the Ir films [33]. Gratifyingly, the KTB model resistance described in Eq. (11c) gives reasonable fits to all the measured RðTÞ curves, in some cases accounting for two orders of magnitude change in resistance over only a few milli Kelvin. Figs. 1(b) and (e) shows evidence for resistive ‘‘tails’’ at the level of B1% of the normal resistance (see Section 2.2.3) of the resistive meander. We move now to the analysis of published S–N transitions in actual TES structures.

239

3.2. TES data Figs. 2(a)–(d) compare best-fit transition curves with measurements made on W [11], Ti [20], Ti/Au [40] and Ir-based [34] transition edge sensors. Expectations for the model should be constrained by two factors: (i) The strict validity of the KTB model only in the lower half of the S–N transition, where vortex densities are low. The discontinuity in RðTÞ and rapid increase in aðTÞ when T tends to Tc0 are the resultant limitations of the model. A physically correct model valid only in the lower half of the S–N transition, however, is still valuable, since it is in that regime that any detector must be biased for linear operation over the broadest energy range and where the excess noise is observed to be largest. (ii) The presence, at the low-temperature end of the S–N transition of secondary processes (Section 2.2.2) which form tails or otherwise broaden the transition. We note particularly the influence of TES current [37]; those transitions obtained with the lowest values of I (80 nA for Fig. 2(b) and B1 mA for Fig. 2(a)) do yield the better fits to the prediction of the KTB model, Eq. (11c). Nevertheless, the fits, taken together, are further evidence for the usefulness of the vortex model. Fig. 3 summarises an analysis of the S–N transition for the Ti/Au TES of Hoevers et al. [7,8]. Here, the ordinate is bias voltage, rather than explicit device temperature. Inverting the power balance equation [1]: V2 n ¼ KðT n  Tbath Þ R

ð14Þ

we transformed bias voltage values to equivalent device temperatures T; then fit the published aðV Þ curve using Eq. (12a) and published parameter values ðn ¼ 3:2; K ¼ 44 nW=K3:2 ; refrigerator base temperature Tbath ¼ 10 mK). The choice of Tc ¼ 107 mK and Tc0 ¼ 136 mK leads to a computed aðV Þ which reproduces rather

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Fig. 1. (a)–(e). Comparison of measured (individual symbols) and calculated resistance-versus-temperature curves for the Ir meanders described in Table 1. For T > Tc0 ; R ¼ RN in the theoretical model.

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241

Fig. 2. Comparison of measured (individual symbols) and calculated (full lines) TES resistance-versus-temperature curves for: (a) Tungsten TES of Cabrera et al. [11]. (b) Titanium TES of Ukibe et al. [20]. (c) Gold/Titanium TES of Ventura et al. [40]. (d) Iridium TES of Fukuda et al. [34].

4. Excess noise 4.1. Observations A survey of the literature [9,13,35–39] leads to the following description of pthe ffiffiffiffiffiffiffi observed ‘‘excess’’ noise current source in ðA= HzÞ:

Fig. 3. Comparison of measured (faint line [7,8]) and calculated (circles) aðV Þ curves.

well the minimum measured at 2 mV bias voltage, along with other features of the experimental counterpart curve.

(i) The excess noise originates inside the TES. (ii) The excess noise spectrum is white at frequencies above B100 Hz; with a 1=f component at lower frequencies. (iii) The magnitude of the excess noise increases as the TES bias-point resistance decreases i.e. in p1=R [36,37]. (iv) The magnitude of the noise is relatively device and material-independent at B100 pA= OHz: (v) The dependence of the excess noise on device size is unclear, but the smallest TES of the

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three Mo/Au devices described in Ref. [39] exhibited the least excess noise, at least at bias points low in the S–N transition. Other features of the excess noise—in particular, its dependence on the TES current I [9,35,39]—are the subjects of conflicting experimental reports. Tiest et al. [9] report that for a fixed resistance set point, the current noise decreased with increasing bath temperature Tbath ; approximately proportional to I: Stahle et al. [13] and Tan et al. [37] observed a small increase in noise current as the bias current increased but Lindeman et al. [39] claim independence of bias voltage, bias current and bath temperature. Interestingly, the last authors use their results to discount the noise source which emerges from the KTB picture of vortex dynamics [42,43]. This model is, nevertheless, developed in the next section. 4.2. Calculations The independent motion of individual vortices in a superconducting film gives rise to phase-slip shot noise [42,43] with a voltage noise spectral density:   Df Sv ðf Þ ¼ 2f0 V ð15Þ 2p for frequencies f below fmax ¼ t

1

spectral density becomes:   pffiffiffiffiffi 2f0 lv V 1=2 : in ¼ Sv =R ¼ R2 W

We note immediately that this current noise source is white (Section 4.1), up to a maximum frequency given by Eq. (16). We note also that, until the parameter-dependences of the mean-free path lv are expressed in full, the conclusion of Lindeman et al. [39], that phase-slip shot noise is not a candidate for the observed excess noise in TESs because it predicts only a direct dependence of Sv on V (Eq. (15)), is premature. According to Ref. [21]: lv ¼ nt

where t is the characteristic time associated with Df; a change in superconducting phase between the voltage contacts of the device. These contacts are assumed to extend the full-width of the TES, so rendering the vortex system one dimensional. When an individual vortex crosses the complete width of the superconducting film, a total phase slip of 2p occurs [42]. Whereas flux–flow noise [8] corresponds to the motion of many (B103 ) flux quanta in an applied magnetic field, we are dealing here with the motion of individual thermallyinduced vortices, travelling perhaps only small fractions of the width of the TES, in zero applied magnetic field. If lv is the vortex mean free path, then Df ¼ 2pðlv =W Þ; and the current noise

ð18Þ

where n; the vortex velocity, and the characteristic time t (now identified as the mean free time) are found from Jf0 If0 n¼ ¼ ð19aÞ Zc WdZc t¼

p Rc _ 7 RN kB T

ð19bÞ

where J is the current density in the superconducting film, Z is the vortex viscosity [44], c is the speed of light in vacuo, kB is Boltzmann’s constant. Working in the dirty limit, with the vortex core radius av written as: a2v ¼ ðp=3Þlx

ð16Þ

ð17Þ

ð20Þ

a straightforward formulation of the vortex viscosity (in units of Newton second=metre2 ) comes from Eqs. (1.5) and (2.14) of Bardeen and Stephen [44]. In the clean limit, the RHS of Eq. (20) is equal to ðp=2:718Þ x20 ¼ 1:16x20 : With our definition of the flux quantum (Section 2.2): Z ¼ F0 Hc2 ð0Þsn

ð21aÞ

1 2 Hc2 ð0Þ ¼ ð1=2pÞF0 a2 v ¼ ð3=2p ÞF0 ðlx0 Þ :

ð21bÞ

Here, sn is the normal state conductivity; x0 is the Pippard coherence length introduced in Section 2.1 and F0 is the magnetic flux quantum. Hc2 ð0Þ is the upper (or depairing) critical field of the 2-d film at zero temperature [32]. We resort to Eq. (21b) in order to establish a closed expression for in :

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Combining Eqs. (17)–(20), we find  2 h 1 1 VI x l in2 ¼ C 2 ðA2 =HzÞ: e kB T 0 A R2

The upper frequency limit fmax ; substituting RN ¼ 0:1 O and T ¼ 0:1 K into Eq. (19b), is of order 1 MHz—consistent with observations. Before carrying out any quantitative comparison of Eq. (22) with measurement, we must also account for (a) the frequency response of the SQUID amplifier, preamplifier and integrator chain and (b) suppression of noise sources internal to the TES by electro-thermal feedback. The SQUID response can be modelled as that of a low-pass filter with corner frequency [8]

ð22Þ

The prefactor C ¼ 21=ð2p2 Þ is only very slightly different from unity. Our model thus predicts that the excess current noise in should scale: (i) Within the transition, as the reciprocal of resistance, R: This agrees with the key observation of Fujimoto et al. [36] and Lindeman et al. [39]. (ii) With device size, as the reciprocal of device area ðA ¼ WLÞ to the power one-half. (iii) As the square root of the dissipated power, P ¼ IV : We note that, within the S–N transition, the power remains nearly constant (see Fig. 2 of Ref. [11]). Thus, the explicit voltage dependence of Eq. (15) has been reduced to a slow variation with the voltage–current product. (iv) Directly, as a material-dependent coherence length-electron mean free path product. Substituting for the dissipated power from Eq. (14), finally, brings out a dependence on the bath temperature: n in2 ¼ constðT n  Tbath Þ=T:

243

fSQUID ¼ R=ð2pLSQUID Þ

ð24Þ

where LSQUID is the input inductance of the SQUID readout circuit. The composite transfer function F has the form [8]   1þb 2 F¼ ðf0 =fe Þ2 2 " #" # 1 þ ðf =f0 Þ2 1  ð25Þ 1 þ ðf =fe Þ2 1 þ ðf =fSQUID Þ2 where, if Rs is the TES shunt resistance, b ¼ ðR  Rs Þ=ðR þ Rs Þ: Since some experimental reports on excess noise do not report the full parameter set ðf0 ; fe ; fSQUID ; Rs Þ; not all entries in Table 2— which compares measured and calculated current noise spectral densities—have had bandpass filtering fully taken into account.

ð23Þ

Increasing bath temperature, therefore, leads to a decrease in current noise, in keeping with (some of) the observations summarised in Section 4.1.

Table 2 Comparison of measured and calculated excess noise values at f ¼ 2:5 kHz for stated TES geometries and bias conditions Ref.

[7,8] [7,8] [9] [37] [39]

TES

Ti/Au Ti/Au Ti/Au Mo/Cu Mo/Cu

T ðKÞ

0.111 0.135 0.1 0.18 0.104

I ðmAÞ

70.5 29.0 17 20 60

V ðmVÞ

0.55 2.6 0.68 0.11 0.235

R ðmOÞ

7.8 93.0 40 5.5 3.9

L ðmmÞ

0.5 0.5 0.31 0.4 0.5

W ðmmÞ

0.25 0.25 0.31 0.4 0.5

d ðnmÞ

22+26 22+26 18+20 39+118 50+270

in ðpA=OHzÞ

Notes

Meas.

Calc.

9 15 30 27 190

64 14 14 40 192

a b c d e

In the column representing film thicknesses, the first figure refers to the superconducting layer and the second to the normal metal. Notes: (a) fSQUID ¼ 2:5 kHz; f0 ¼ 160 Hz; fe ¼ 195 Hz; Rs ¼ 7 mO: (b) fSQUID ¼ 29 kHz; f0 ¼ 160 Hz; fe ¼ 1:9 kHz; Rs ¼ 7 mO: (c) fSQUID ¼ 10 kHz; f0 ¼ 200 Hz; fe ¼ 1:1 kHz; Rs ðassumedÞ ¼ 7 mO: (d,e) Rs ¼ R; frequency-dependent terms on RHS of Eq. (25) set to unity.

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In predicting absolute values for the current noise spectral density, the only physically uncertain parameter is the effective coherence length i.e. the square root of x0 l: One difficulty of interpretation arises from the fact that almost all the TESs for which noise data have been reported are proximity bilayers. We can, however, produce estimates of in by recognising that the electron mean free path must be constrained by the superconductor film thickness, d: The results of computations, summarised in Table 2, based on an effective coherence length set exactly equal to d; show (perhaps surprisingly) respectable agreement with measurement.

5. Discussion A vortex dynamics model has been shown to account for most of the hitherto unexplained features of TES operation. The noise modelling presented here is essentially a rediscovery and extension of the work of Knoedler et al. [42,43] and at the very least provides a physical basis for prediction. One key prediction is that the current noise should be decreased by the application of an external magnetic field. At zero field [42] the neutral vortex plasma strongly screens the interaction of individual vortices. A magnetic field B, however, induces vortices of one sign with density nB ¼ B=F0 which at temperatures low down in the transition may exceed the density of thermally free vortices, nf : The vortices then interact via a longrange logarithmic potential that is observed to significantly reduce the shot (i.e. excess) noise [42]. Higher up in the S–N transition, the free vortex density increases, screening is re-established, and the noise approaches its zero-field value. Setting nB ¼ nf ; one can obtain an order-ofmagnitude estimate of the B-field required to suppress the excess noise. Recalling that the coherence length is a measure of the size of the vortex, the areal density of free vortices can be simply estimated using Eq. (4). In the case of thin film Iridium (Section 3.1), x0 ¼ 328 nm: Taking t ¼ 0:99 and working in the clean limit, we obtain BB1:2 G:

We note that Ullom et al. [45] have recently reported the empirical reduction of excess TES noise using external fields of magnitude 0:13– 0:21 G:

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