ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 620 (2010) 294–298

Contents lists available at ScienceDirect

Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/nima

Analytical expressions for transition edge sensor excess noise models

Daniel Brandt �, George W. Fraser

Space Research Centre, Michael Atiyah Building, Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK

a r t i c l e i n f o

Article history:

Received 2 February 2010

Accepted 28 February 2010Available online 6 March 2010

Keywords:

Transition edge sensor

Excess noise

Percolation

Vortex dynamics

02/$ - see front matter & 2010 Elsevier B.V. A

016/j.nima.2010.02.271

esponding author. Tel.: +44 116 223 10 52.

ail address: Daniel.Brandt@star.le.ac.uk (D. Br

a b s t r a c t

Transition edge sensors (TESs) are high-sensitivity thermometers used in cryogenic microcalorimeters

which exploit the steep gradient in resistivity with temperature during the superconducting phase

transition. Practical TES devices tend to exhibit a white noise of uncertain origin, arising inside the

device. We discuss two candidate models for this excess noise, phase slip shot noise (PSSN) and

percolation noise. We extend the existing PSSN model to include a magnetic field dependence and

derive a basic analytical model for percolation noise. We compare the predicted functional forms of the

noise current vs. resistivity curves of both models with experimental data and provide a set of equations

for both models to facilitate future experimental efforts to clearly identify the source of excess noise.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

A transition edge sensor (TES) is an extremely sensitivethermometer, utilizing the rapid variation of resistivity withtemperature at the superconducting-normal phase transition(sn-transition). It has been proposed that using TESs asthermometers in microcalorimeters might yield a photoncounting spectrometer capable of energy resolution 1 eV forX-ray energies in the range of 0.3–10 keV [1]. Using practical TESsit has so far been impossible to reach the energy resolutionpredicted by calorimeter theory [1,2]. The reason for this is a noisesource of uncertain origin intrinsic to the device [1]. The noiseappears to be white at frequencies above 100 Hz and inverselyproportional in magnitude to the TES bias point resistance [3]. Anumber of theories have been suggested to explain the origin ofexcess noise [3–5], but so far no closed, self contained model hasbeen produced which is in agreement with all of the availableexperimental data. Recently, good progress has been madepredicting TES noise behavior by employing accurate thermalmodels of the TES thin films and absorbers [6]. However, not allfeatures of the excess noise phenomenon reported can beexplained by thermal models, and the origin of the excess noisephenomenon remains unclear [7].

The excess noise observed depends on a range of experimentalparameters, such as bias point resistance R, bias current I, biasvoltage V, operating temperature T and applied magnetic field ~B

[8]. Since it is generally impossible to vary one of theseparameters without influencing the remaining parameters, anexpression predicting the level of excess noise as a function of R, I,

ll rights reserved.

andt).

V, T, B is required if a meaningful comparison of a proposed noisemodel with experimental data is to be attempted.

In this paper we aim to provide complete analytical expres-sions for the dependence of the noise current expected from thecompeting excess noise models of phase slip shot noise (PSSN) [3]and percolation noise [5]. We begin with an overview of themathematical form of excess noise in Section 2. We then extendthe PSSN model derived by one of us [3] to include a dependenceon the applied magnetic field B (Section 3). In Section 4 wepresent the model of percolation noise and derive the firstanalytical description of percolation noise in TESs. Finally, wecompare the predictions of both models to each other and toexperimental results reported by Takei [9].

2. Mathematical form of excess noise

A number of experimenters (e.g. [3,4,9]) have reported theexcess noise phenomenon. Excess noise cannot be explained bythe sum of Johnson noise, thermal fluctuation noise and read-out(amplifier) noise (Fig. 1). Its two most dominant and mostconsistently reported features are its inverse proportionality tothe TES bias point resistance, leading to the name ‘‘ 1

R�noise’’ [9]and the fact that it appears to occur only above a given thresholdbias current. The variation of Johnson noise spectral currentdensity and empirical excess noise spectral current density withbias point resistance is shown in Fig. 1.

Mathematically, excess noise takes the form of a constantvoltage noise source. It has been reported by Takei et al. [9] that inthe case of their Ti/Au bilayer TES the best fit to the experimen-tally observed excess noise voltage spectral density is given by

ve ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffi4kBTc

pð1Þ

ARTICLE IN PRESS

Fig. 1. Noise spectral current density originating inside a TES with normal state

resistance Rn ¼ 100 mO. The solid line is the total noise observed, made up from

the quadrature sum of an analytical fit to experimental excess noise according to

Eq. (1) (dashed black line) and Johnson noise (dashed gray line).

Fig. 2. Variation of excess noise spectral current density with device resistivity.

The solid gray line is the best fit for experimental excess noise according to [9]. The

black line shows the noise predicted by the PSSN model (2) for the same device.

Any discrepancy between the two models may result from the fact that the

proximity effect of the Au absorber was not taken into account when determining

device thickness.

D. Brandt, G.W. Fraser / Nuclear Instruments and Methods in Physics Research A 620 (2010) 294–298 295

where Tc is the critical temperature of the TES and for a typicalsetup ve � 225 pV Hz�1=2 [9]. This result was obtained using a500mm square bridge-type Ti/Au bilayer TES, operating between63 and 77 mK. Thus, within this temperature range, any successfulnoise model must be in agreement with expression (1) whenapplied to a 500mm square bridge-type Ti/Au bilayer TES.

3. Phase slip shot noise

The theory of phase slip shot noise is based on the dynamics ofvortices in thin film superconductors [3,10].

A current flowing in the superconductor will exert a drivingforce on any magnetic vortices present, in a direction normal tothe current flow [11]. If a current causes a vortex to move acrossthe superconductor it causes the phase difference of the orderparameter at two points along the direction of the super currentto change with time, causing the occurrence of voltage shot noisein the direction normal to the vortex motion [10,12]. If vortexmotion in the sample is random, the summed voltage signal froma series of vortex movements can appear as phase slip shot noise(PSSN) [3,10]. Considering vortex dynamics in a thin super-conducting film, one of us [3] derived the following expression forthe noise spectral current density in terms of experimentalvariables:

i2nðV ,I,T ,A,RÞ ¼ C

h2

e2

� �1

kBTx0l

VI

R2Að2Þ

where in is the noise spectral current density, l is the electronicmean free path, x0 is the Pippard coherence length, A is the area offilm, R is the film resistance, V is the voltage across the film, I isthe current through the film and C ¼ p=21 is a constant ofproportionality [3]. Since both the electronic mean free path l andthe coherence length x0 are limited by the device thickness d it ispossible to approximate the normalized effective coherencelength

ffiffiffiffiffiffiffix0l

p� d [3]. It can be seen from Fig. 2 that the

predictions of the excess noise model according to Eq. (2) are ingood agreement with the best-fit function Eq. (1). For a betterquality fit, precise values for the coherence length x and electronicmean free path l are required. Due to the proximity effect inbilayer TESs, theoretical determination of these parameters is adifficult task.

3.1. Magnetic field dependence of phase slip shot noise

In this section we investigate the effect of applying a magneticfield B to a TES at bias resistance Rb, with the field directionperpendicular to the plane of the TES. The applied magnetic fieldintroduces a number nF of additional free vortices into the TESthin film, given by

nF ¼ B=f0 ð3Þ

where f0 ¼ h=2e is the magnetic flux quantum. Those freevortices created by the field B introduce a flux-flow resistanceinto the TES, given by [13]

Rff ¼ 2pnFx2RN ð4Þ

where x is the Ginzburg–Landau (GL) coherence length in the filmand RN is the film’s normal state resistance.

In the dirty limit, the GL-coherence length x can be found fromthe Pippard coherence length x0 and the electronic mean free path l:

x¼ 0:85x0l

1�t

� �1=2

ð5Þ

where t (B)¼T/Tc(B) is the reduced temperature and Tc(B) is the TEScritical temperature in the presence of an applied field B. The criticaltemperature at non-zero applied field is then calculated asTcðBÞ ¼ T0

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðBc�BÞ=Bc

p, where Bc is the critical applied field [12]

and Tc0 is the critical temperature at zero applied field. The Pippard

coherence length x0 and electron mean free path l can be found fromthe microscopic parameters of the TES material as [12,14]

x0 ¼ 0:18‘vf

kbT0c

ð6Þ

l¼mevf

nee2rð7Þ

where vf is the Fermi velocity, me, e are the electronic mass andcharge, respectively, ne is the number density of electrons and r is thematerial normal state resistivity. The electron number microscopicparameters required to evaluate (6) and (7) can be found in [1].

The flux flow resistance Rff added by the applied magnetic fieldcan then be found by evaluating Eqs. (3)–(7) and the predicted

ARTICLE IN PRESS

Fig. 3. Noise suppression factor as a function of applied magnetic field for a MoCu

TES with coherence length x¼ 8mm and mean free path l as predicted by the PSSN

model (solid black line) and as reported experimentally (circles) [8]. The dashed

line is provided as a guide to the eye.

D. Brandt, G.W. Fraser / Nuclear Instruments and Methods in Physics Research A 620 (2010) 294–298296

phase slip shot noise becomes

inðV ,I,T,A,RÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC

h2

e2

� �1

kBTx0l

VI

A

s1

RbþRffð8Þ

where according to Eqs. (3)–(5) Rff is given by

Rff ¼ 1:445p B

f0

x0l

1�tðBÞ

� �RN ð9Þ

It can be seen from (8) and (9) that we expect an applied magneticfield to suppress PSSN and that the magnitude of this suppressiondepends strongly on the coherence length x and electron meanfree path l in the TES film. For the case of bilayer TESs thosequantities are governed by the proximity effect, and determiningthem accurately is not a trivial task. In order to be able to comparethe magnetic PSSN model derived above with experimental datafor a MoCu bilayer TES acquired by [8], we use Eqs. (6) and (7) toestimate the Pippard coherence length of the MoCu TES asx0 ¼ 8mm, and the average electron mean free path l as 20 nm. Ascan be seen from Fig. 3, the PSSN model provides a reasonableestimate of the relative noise suppression introduced by anapplied magnetic field.

4. Percolation noise

While PSSN appears like a very promising candidate model forexcess noise, the model according to Eq. (2) predicts a depen-dency of spectral noise current density on device area A, whichhas not hitherto been observed. Other geometric effects have beenobserved which are also difficult to explain in the framework ofPSSN, such as the noise suppression by deposition of ‘‘zebrastripes’’—normal material stripes deposited onto the TES perpen-dicular to the direction of the bias current, forcing the current tozig-zag across the device [8]. These apparent deficiencies of thePSSN model when it comes to taking account of device geometryhave prompted the search for alternative, geometric, sources ofexcess noise [5].

Percolation theory is often used to describe a variety of phasetransitions. The theory of percolation noise applied to TESs isbased on the behavior of random resistor-superconductor net-works. The entire system is modeled as a grid of domains of area(coherence length)2. Any domain can either be in the normal orsuperconducting state.

The average size of superconducting clusters depends on thefraction of sites in the superconducting state. In the case of

infinite networks the average cluster size goes to infinitydiscontinuously as the superconducting fraction reaches a criticalvalue rc . In the case of two-dimensional site percolation rc can bedetermined numerically to be 0.593 [15].

Percolation noise assumes that in the sn-transition region thefluctuation of superconducting domains allows superconductingchannels to form and collapse spontaneously, giving rise to strongfluctuations in the system’s resistivity and was first proposed as asource of excess noise by Lindemann et al. [5]. One of us hasdeveloped a semi-analytical treatment of percolation noise andused Monte Carlo methods to show that the functional form of thenoise originating in random resistor-superconductor networks isin good agreement with the excess noise observed in TESs [16]. Todate, no analytical expression is available describing the depen-dence of percolation noise on experimental parameters.

4.1. Temperature dependence

In order to find a set of analytical expressions governing thepercolation behavior of superconducting domains during the sn-transition we start from the basic expressions given by Kiss andSvedlindh [17] for the behavior of resistivity and noise in randomresistor-superconductor networks:

Rrspðrr�r0crÞ

sð10Þ

i2npR�lrsrs ð11Þ

for supercritical networks (i.e. rr 4r0cr). Here Rrs is the networkresistivity, rr is the density of normal state sites, r0cr is theconjugate critical fraction defined as r0cr ¼ 1�rcs, rcs is the criticalfraction of superconducting domains (� 0:593), in is the currentnoise power spectral density of the network and lrs¼0.86 ands¼1.297 are geometrical scaling constants [17]. Those equations,and consequently all following derivations, are valid only in thepercolation region, where 0orr orcr . By substituting (10) into(11) it is possible to obtain an expression for in in terms of rr . Thenumber of electrons in the superconducting state can be foundfrom the Gorter–Casimir two fluid model [12]:

n0s ðTÞ ¼ n0

s ð0ÞT0

c�T

T0c

� �4

ð12Þ

where ns0 is the density of superconducting electrons and Tc

0 is thecritical superconductor temperature and superscript 0 indicateszero applied field.

Trying to build a percolation picture of the sn-transition, weassume that the fraction of domains in the superconducting staters is proportional to the fraction of charge carriers in thesuperconducting state ns

0(T)/ns0(0), yielding

rspT0

c�T

T0c

� �4

ð13Þ

Using rr ¼ 1�rs (i.e. all sites that are not superconducting arein the normal state) it is now possible to obtain an expression forthe noise current spectral density in terms of temperature. Webegin by substituting (10) into (11), using r0cr ¼ 1�rcs:

i2npðrcs�rsÞ�lrss

ð14Þ

We now substitute rs according to (13) into Eqs. (10) and (14)to obtain an expression for Rrs in terms of T:

Rrs ¼ k1 rcs�k2T0

c�T

T0c

� �4 !s

ð15Þ

i2n ¼ k23 rcs�k2

T0c�T

T0c

� �4 !�lrss

pR�lrsrs ð16Þ

ARTICLE IN PRESS

Fig. 4. Left: Variation of percolation noise spectral current density (black) and experimental excess noise spectral current density according to Eq. (1) with device

resistance. Right: Variation of percolation noise spectral current density with applied magnetic field for a TES with Hc¼100 G and operating point T¼0.9Tc. The constants of

proportionality used were k3 ¼ffiffiffiffiffiffi80p

pA and k2¼105.

D. Brandt, G.W. Fraser / Nuclear Instruments and Methods in Physics Research A 620 (2010) 294–298 297

where k1, k2 and k3 are constants of proportionality. To the bestof the authors’ knowledge this is the first analytical expressionderived for the dependence of percolation noise on experimentalparameters (i.e. critical temperature Tc and temperature T). Inorder to use the percolation model to make quantitativepredictions of the spectral noise current density, the constantsof proportionality have to be determined by experiment.

We now have a model of the resistive transition of aquasi-two-dimensional superconducting film as a function oftemperature in terms of a random resistor-superconductor sitepercolation network in the form of Eq. (15). The noise spectralcurrent density is given as a function of the network resistanceEq. (16).

Our assumption that the fraction of domains in the super-conducting state rs should be proportional to the fraction ofcharge carriers in the superconducting state is a rather naiveview of the situation. It is well known that the coherence length x,which governs the size of the superconducting domains, varieswith temperature. Since the number density of carriers inthe superconducting domains also varies with temperature, alinear relationship between rs and ns

0(T) is likely to be anoversimplification. Any physically based theory describing thevariation of the fraction of superconducting domains withtemperature will likely find k1 to be a function of T. Furthermore,the site percolation model assumes that the bias current doesnot exceed the local critical current, so that the resistivity ofany site is isotropic. Thus, in order to find the correct value forrcs applicable to current biased thin film superconductors, alongwith physically derived values for k1, k2, k3, more theoretical workis needed.

4.2. Magnetic field dependence

The magnetic field dependence of the percolation noise modelis introduced via the variation of the critical temperature Tc withthe applied magnetic field H. In non-zero applied field, allinstances of Tc

0 in Eqs. (15) and (16) are replaced by the fielddependent critical temperature Tc given by [12]

Tc ¼ T0c

Hc�H

Hc

� �1=2

ð17Þ

where a superscript 0 indicates zero applied field. Substituting(17) into (16) we now find an equation for the percolation noise

spectral current density in terms of the temperature T and appliedfield H:

i2n ¼ k2

3 rcs�k 1�T

T0c 1�

H

Hc

� �1=2

26664

37775

40BBB@

1CCCA�lrss

ð18Þ

The predicted noise spectral current density according to (18)has been plotted as a function of temperature and magneticfield in Fig. 4. While the approximate magnitude of the predictednoise can be set via the constants of proportionality, it appearsthat the functional form of the noise current vs. device resistancecurve is not of the correct shape to describe TES excess noise(Eq. (1)). The reduction of excess noise by applied magnetic fieldfor a film at T¼0.9Tc is predicted to be of order six parts in 105

over an applied field range of 0–15 G. Experimentally, a reductionof almost 60% can be expected over a range of � 130 mG [8]. Thisdiscrepancy is in itself an important result: It indicates that thechange in carrier concentration as a result of the applied field is initself insufficient to allow for the noise supression effect ofapplied magnetic fields to be described in the framework of thepercolation model. We suggest that taking into account the non-local magnetic interactions between domains might solve thisdiscrepancy.

5. Conclusions

We have investigated the two competing TES excess noisemodels of phase slip shot noise (PSSN) and percolation noise. Ourmain goal was to present analytical models of the two noisesources in order to enable quantitative discussion of the modelpredictions with regard to experimental results available.

From our investigation of the PSSN model the predicted noisecurrent of Eq. (2) and our extension to include the magnetic field(Eqs. (8) and (9)) are in order-of-magnitude agreement with best-fit functions to experimental data (Eq. (1)). The predictions of thePSSN model differ from the experimental data by a constantmultiplicative factor. Part of this inaccuracy may stem from thedependence of in on the electron mean free path l and super-conducting coherence length x, two parameters which can bedifficult to determine precisely for the sort of bi-layer structurestypical for TES films. The level of excess noise suppression

ARTICLE IN PRESS

D. Brandt, G.W. Fraser / Nuclear Instruments and Methods in Physics Research A 620 (2010) 294–298298

afforded by magnetic fields and the dependence of in on theinverse square root of the device area are testable predictions.Both, the mean free path l and device area A can be varied duringdeposition for constant composition bi-layer structures.

The analytical percolation noise model developed in Section 4does not yield the required pR�1 functional dependency. This isbelieved to be due to the fact that Eqs. (10) and (11) describe theresistance encountered by a current flowing from a point-likesource to a point-like sink [17]. In a practical TES device thesuperconducting contacts serving as the current source and sinkare usually deposited along an entire edge of the TES film. One ofus [16] has successfully used numerical methods to show that forthis case percolation noise does indeed yield the required R�1

dependence. This difference in behavior of percolation networksfor line-like and point-like current sources/sinks is an importantresult since it provides a testable prediction: For otherwiseidentical TES devices with varying contact geometries thefunctional form of the excess noise should change if percolationnoise is the source of the phenomenon. No such change ispredicted in the PSSN model.

The failure of the analytical percolation noise model tocorrectly predict noise suppression by applied magnetic fields isnot overly troubling, since all our derivations have assumedconstant domain size and non-interacting domains. It is wellknown that in the presence of an external field the vortices in thinfilms interact, forming Abrikosov lattices [12]. Any such interac-tion between magnetic vortices will have to be taken into accountexplicitly in any analytical theory of percolation noise in thinsuperconducting films.

References

[1] C. Enss, Cryogenic Particle Detection, Springer-Verlag GmbH, Heidelberg,Germany, 2005.

[2] V. Polushkin, Nuclear Electronics, John Wiley and Sons Ltd, Chichester,England, 2004.

[3] G.W. Fraser, Nuclear Instruments and Methods in Physics Research A 523(2003) 234.

[4] H. Hoevers, M. Bruijn, B. Dirks, L. Gottardi, P. de Korte, J. van der Kuur, A.Popescu, M. Ridder, Y. Takei, D. Takken, Journal of Low Temperature Physics151 (2008) 94.

[5] M. Lindemana, M. Anderson, S. Bandler, N. Bilgria, J. Chervenak, S. GwynneCrowder, S. Fallows, E. Figueroa-Feliciano, F. Finkbeinerb, N. Iyomoto, R.Kelley, C. Kilbourne, T. Lai, J. Man, D. McCammon, K. Nelms, F. Porter, L. Rocks,T. Saab, J. Sadleir, G. Vidugiris, Nuclear Instruments and Methods in PhysicsResearch A 559 (2006) 715.

[6] D. Goldie, M. Audley, D. Glowacka, V. Tsaneva, S. Withington, ournal ofApplied Physics 105 (2009) 074512-1–074512-7.

[7] B. Cabrera, Journal of Low Temperature Physics 151 (2008) 82.[8] J.N. Ullom, W.B. Doriese, G.C. Hilton, J.A. Beall, S. Deiker, W.D. Duncan, L.

Ferreira, K.D. Irwin, C.D. Reintsema, L.R. Vale, Applied Physics Letters 84(2004) 4206.

[9] Y. Takeia, K. Tanaka, R. Fujimotoand, Y. Ishisaki, U. Morita, T. Morooka, T.Oshima, K. Futamoto, T. Hiroike, T. Koga, K. Mitsuda, N.Y.T. Ohashi, N.Iyomoto, T. Ichitsubo, K. Sato, T. Fujimori, K. Shinozaki, S. Nakayama, K.Chinone, Nuclear Instruments and Methods in Physics Research A 523 (2004)134.

[10] R.F. Voss, C.M. Knoedler, P.M. Horn, Physical Review Letters 45 (1980) 1523.[11] O. Narayan, Journal of Physics A: Mathematical and General 36 (2003)

372.[12] D. Tilley, J. Tilley, Superfluidity and Superconductivity, third ed., IOP

Publishing Ltd, Bristol, England, 1990.[13] P. Minnhagen, Physical Review B 23 (1981) 5745.[14] B. Tanner, Introduction to the Physics of Electrons in Solids, Cambridge

University Press, University Printing House, Shaftesbury Road, Cambridge,UK, 1995.

[15] Wolfram research, /http://mathworld.wolfram.com/PercolationThreshold.htmlS.

[16] D. Brandt, Analytical and numerical treatment of percolation noise intransition edge sensors, in: Proceedings of the 13th International Workshopon Low Temperature Detectors-LTD13, AIP Conference Proceedings,vol. 1185, 2009, pp. 52–55.

[17] L.B. Kiss, P. Svedlindh, Physical Review Letters 71 (1993) 2817.