DEPARTMENT OF PHYSICS, UNIVERSITY OF JYVASKYLARESEARCH REPORT No. 2/2003

HIGH PERFORMANCE MICROBOLOMETERS AND

MICROCALORIMETERS: FROM 300 K TO 100 mK

BY

ARTTU LUUKANEN

Academic Dissertationfor the Degree of

Doctor of Philosophy

To be presented, by the permission of theFaculty of Mathematics and Science

of the University of Jyvaskyla,for public Examination in Auditorium FYS-1 of the

University of Jyvaskyla on May 9, 2003at 12 o’clock noon.

Jyvaskyla, FinlandMay 2003

Abstract

This thesis is a review of six publications which focus on the development of ther-mal detectors and on-chip coolers. The thermal detectors developed are antennacoupled microbolometers and transition-edge microcalorimeters. The bolometersare intended for the detection of submillimetre waves, a region in the electromag-netic spectrum located between the infrared and microwave frequencies, in whichdetection methods are least developed. A room temperature bolometer is pre-sented, which is suitable for applications where large levels of signal power arepresent, such as solar astronomy or imaging of concealed weapons under cloth-ing when combined with an illumination source, while the sensitivity of a novelsuperconducting bolometer, operated at 4.2 K, is more than sufficient for passiveimaging of terrestrial sources. The figure of merit for this detector is shown to beabout an order of magnitude better than that of existing 4.2 K bolometers.

Results obtained with X-ray transition-edge sensor (TES) microcalorimeters,the most sensitive type of X-ray detector in terms of energy resolution to date, arepresented. These devices are operated at temperatures below 100 mK, and canachieve energy resolutions of a few electron volts. The TES microcalorimeters willbe used in future X-ray science missions of the European Space Agency (ESA), theU.S. National Astronautics and Space Administration (NASA), and the NationalSpace Development Agency of Japan (NASDA). Until recently, the behaviour ofthese devices has been well understood, but there have been a number of reportsthat the performance of the detectors does not achieve that predicted by the theory.We have developed a Corbino disk geometry TES, which is used as a diagnostic toolto investigate the origins of the excess noise. We show that the excess noise can beaccurately modeled with a noise arising from fluctuational superconductivity fromregions of the TES which are close to the critical temperature Tc. We predict thatthis noise source will be of significance in TESs operated at large bias currentsand low operating point resistances. Ways to decrease this noise contribution arepresented.

The cooling of detectors to temperatures well below 1 K is a challenging task.

As a last part of this thesis we present the results obtained with a novel type of

cooler where the cooling effect is based on the evaporative cooling of electrons

from a normal metal island. These devices can cool efficiently the electron gas by

a factor of 3 from 0.3 K to 0.1 K. Related to this, it is shown that the cooling

of the electron gas alone does not improve the sensitivity of thermal detectors

sufficiently, and consequently that the cooling of the lattice is desirable. With the

lattice coolers presented, cooling of an isolated silicon nitride platform from 0.2

K to 0.1 K is demonstrated. Further increase of the cooling power is possible by

increasing the size of the tunnel junctions.

i

Preface

This work has been carried out in the Physics department of the University ofJyvaskyla, Finland.

First of all, I thank my supervisor Prof. Jukka Pekola for his dedicated guid-ance of my work, and his ability to see through the clutter of lesser issues to thehard core of a physical problem. It has been a great pleasure to work with him.Apart from Jukka, I am deepdly indebted to a large group of people. Especiallyimportant to my career has been Dr. Heikki Sipila, who hired me to work forMetorex already in 1994, and who I am to thank for choosing to work with ther-mal detectors in the first place. Moreover, the experience of working for a companyhas provided me with invaluable knowledge on how to conduct project orientedresearch.

From Jyvaskyla, I thank the departmental staff, and the entire applied physicsgroup for enjoyable comradeship, and especially Antti Nuottajarvi, whose exper-tise in the thin film processing combined with a resilient attitude was invaluablefor this work. The other irreplaceable ’clean room wizard’ is Tarmo Suppula,who produced the SINIS samples and with whom I had a number of importantdiscussions relating to the technological challenges in sample fabrication. Alsothe contribution of Kimmo Kinnunen is acknowledged for helping me with thecryogenic measurements over the years, and also for many discussions.

I gratefully acknowledge the comments, ideas, assistance, hardware and in-sights kindly and generously supplied by Konstantin Arutyunov, Marcel Bruijn,Bernard Collaudin, Erich Grossman, Klavs Hansen, Tero Heikkila, Panu Helisto,Juhani Huovelin, Vladimir Khaikin, Jani Kivioja, Mikko Kiviranta, Nikolai Kop-nin, Piet de Korte, Martin Kulawski, Jan van der Kuur, Leonid Kuzmin, MarkLindeman, Peter de Maagt, Ilari Maasilta, Juha Mallat, Didier Martin, Wim Mels,Seppo Nenonen, Antti Niskanen, Heikki Seppa, Jussi Tuovinen, Joel Ullom, andSeppo Vaijarvi, as well as contributions from all those I have forgotten to mention.

I had many interesting discussions with my co-authors, who I thank for delight-ful collaboration: Jouni Ahopelto, Dragos Anghel, Wouter Bergmann Tiest, HenkHoevers, Mikko Leivo, Antti Manninen, Mika Prunnila, and Alexander Savin.

Financial support from the Jenny and Antti Wihuri foundation, the Academyof Finland, and the European Space Agency are kindly acknowledged.

And finally, I thank my parents, and friends for their support. My deep-est gratitude is to Vilja, for putting up with me and being such a lovely companion.

Espoo, April, 2003

Arttu Luukanen

ii

List of Publications

This thesis is a review of the author’s work in the field of room temperaturebolometers, cryogenic X-ray radiation detectors, and tunnel junction refrig-erators. It consists of an overview and the following selection of the author’spublications in the field:

I. A. Luukanen, H. Sipila, K.M. Kinnunen, A.K. Nuottajarvi, and J.P.Pekola, Transition-edge microcalorimeters for X-ray space applications,Physica B 284-288, 2133-2134 (2000).

II. A. Luukanen, M.M. Leivo, J.K. Suoknuuti, A.J. Manninen and J.P.Pekola, On-chip refrigeration of hot electrons at sub-kelvin tempera-tures, J. Low Temp. Phys. 120, 281–290 (2000).

III. D.V. Anghel, A. Luukanen, and J.P. Pekola, Performance of cryogenicmicrobolometers and microcalorimeters with on-chip coolers, Appl.Phys. Lett. 78, 556–558 (2001).

IV. A. Luukanen, A.M. Savin, T.I. Suppula, J.P. Pekola, M. Prunnila, andJ. Ahopelto, Integrated SINIS refrigerators for efficient cooling of cryo-genic detectors, in LTD-9 AIP Conference Proceedings 605, 375–378(2002).

V. A. Luukanen and J.P. Pekola, A superconducting antenna-coupled hot-spot microbolometer, to appear in Appl. Phys. Lett. 82, (2003).

VI. A. Luukanen, K.M. Kinnunen, A.K. Nuottajarvi, H.F.C. Hoevers,W.M. Bergmann Tiest, and J.P. Pekola, Fluctuation superconductiv-ity limited noise in a transition-edge sensor, accepted for publicationin Phys. Rev. Lett. (2003).

iii

Author’s Contribution

The research reported in this thesis was carried out at the Physics Depart-ment of University of Jyvaskyla during the years 1998-2003. The author haswritten papers I, V and VI, and participated in writing papers II, III and IV.In paper III, the author contributed to the theoretical analysis. The authorcarried out a significant proportion of the sample fabrication, measurements,and data analysis in papers I and VI, and participated in the measurementsand data analysis in papers II and IV. In paper V, the author carried outall of the sample fabrication and measurements, and performed most of thedata analysis.

iv

Contents

Abstract i

Preface ii

List of Publications iii

Author’s Contribution iv

1 Introduction 1

1.1 The electromagnetic spectrum . . . . . . . . . . . . . . . . . . 11.2 Detection methods . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Figures of merit . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Thermal detectors 9

2.1 Operating principle . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Antenna-coupled microbolometers 15

3.1 Room temperature antenna-coupled microbolometers . . . . . 153.2 Superconducting antenna-coupled microbolometers . . . . . . 23

4 Transition-edge microcalorimeters 28

4.1 X-ray measurements with a square TES . . . . . . . . . . . . 294.2 TES in Corbino disk geometry . . . . . . . . . . . . . . . . . . 33

5 Excess noise in transition-edge microcalorimeters 42

6 On-chip cooling of microcalorimeters 51

6.1 Coolers with large junction area . . . . . . . . . . . . . . . . . 526.2 Results from electron coolers . . . . . . . . . . . . . . . . . . . 536.3 Lattice coolers . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Summary 58

References 66

Abstracts of publications I–VI 67

v

1 Introduction

1.1 The electromagnetic spectrum

Electromagnetic radiation envelopes us humans here on Earth, and in factperpetrates the entire Universe. All bodies emit a specific type of radiationwhose spectrum, or colour, is determined by the physical temperature of theobject. This radiation is commonly known as blackbody radiation. Our vi-sion is based on the interaction of radiation with a complex sensory system,the eye and the cortex in our brain. Due to the physical characteristics ofthe central star in our solar system, the Sun, most of the living entities onearth detect electromagnetic radiation in a fairly narrow wavelength range,which is commonly known as visible light. Determined by its surface tem-perature, the Sun emits most of its energy in this wavelength range, andthus evolution, as it tends to favour efficiency, has provided us with sensorsspecialized to visible light as there is an abundance of signal available. By nomeans is the visible region the only interesting region in the electromagneticspectrum. The spectrum, shown in Fig. 1, can be divided to subdivisionswith familiar names. Starting from long wavelengths, radio waves span theregion down to a wavelength of one centimeter, where the microwave regionstarts. Microwaves correspond to wavelengths between 10 cm down to about1 cm. Reducing the wavelength to a millimeter, the region is quite naturallycalled the millimeter wave region, which continues to the submillimeter waveregion down to a wavelength of about 100 micrometers (one micrometer cor-responds to one thousandth of a millimeter). These regions are of relevanceto this research and will be looked upon in more detail later. The regionbetween 100 micrometers and 1 micrometer is the infrared region, for whichthe evolution has kindly provided a sensor - the skin. Most of us will find itdifficult to roast a sausage in a large campfire if equipped with too short of astick: a burning sensation on your skin is due to the infrared radiation, mostof which is emitted at a wavelength λ given by Wien’s displacement law:λ = 2.898 × 10−3K m/Tcampfire = 2.2 µm. After the visible and ultravioletwavelengths, the radiation is commonly known as X-rays, a penetrating typeof radiation which also possesses some danger to living organisms due to itsionizing nature. The sun is also a strong emitter of X-rays, but these arefortunately blocked by the atmosphere. Also X-rays are under study withinthis thesis. The most energetic electromagnetic radiation is called gammarays, which interact rather weakly with matter. These photons are generatedat the very extremest of conditions in the Universe.

1

Figure 1: The electromagnetic spectrum. The curves correspond to theblackbody spectra of the cosmic microwave background (T = 3 K), a hu-man (T = 310 K), a campfire (T = 1200 K), the Sun (T = 6000 K), and anaccreting black hole (T=5 million K). The shaded regions are of relevance tothe work presented here.

2

1.2 Detection methods

The detection of radiation relies on the interaction of electromagnetic radia-tion with matter. The detection methods are commonly divided to two maincategories: Coherent detection and incoherent (or direct) detection. In theformer case, both the amplitude and the phase information of the incomingradiation is detected, while in the latter only the other quantity, usually theamplitude, is measured.

Most common type of incoherent detectors are photon detectors, whichrespond directly to absorbed photons. An absorbed photon generates boundcharge carriers through the photoelectric effect, and the resulting signal iseither a chemical change, modulation of the electrical current through thedevice, or the primary charge collected directly with a charge sensitive am-plifier. Photon detectors yield information on the energy and the absorptionrate of the incident photons. These devices are used throughout the gamma,X-ray, ultraviolet, visible, and infrared regions. A long wavelength cut-offin the far-infrared is imposed on photodetectors by the energy required tocreate primary excitations.

Another class of incoherent detectors are thermal detectors. In thermaldetectors, the operation is based on the absorption and thermalization ofphotons. Usually the resulting temperature change modulates an electricalproperty of the device, such as resistance. An attractive feature of thermaldetectors is that they are extremely broad band devices, as the energy of theprimary excitation, the phonon, is very small. Thermal detectors are thusused all the way from gamma rays to microwaves, and are the focus of thiswork.

In coherent detectors, the detection is based on the accelerating effect ofthe electric field of the incident radiation to the electrons in the detector. Atfrequencies where low noise amplifiers exist (. 200 GHz), the signal is ampli-fied prior to detection. At higher frequencies a method called heterodyning isused where the incoming electromagnetic field is mixed with a coherent localoscillator signal, and the resulting beat frequency, often called the interme-diate frequency (IF), is amplified while phase information of the photons ispreserved. This type of detectors are used primarily from the submillimeterto radio wavelengths, but also in the infrared. Notably, a coherent detectionsystem can incorporate incoherent detectors as mixing elements.

Comparison of the different detection techniques is difficult as it is im-possible to establish a single performance marker for the various methods.The application determines the best option: for instance, if high resolutionspectroscopy in the millimeter wave region is the application, incoherent de-tectors are typically not a good choice. Likewise, if one desires to do broad

3

band radiometry, incoherent detectors outperform coherent detectors. Simul-taneously, one has to bear in mind technological constraints: while coolingto 100 mK is considered acceptable for high resolution X-ray astrophysics, aportable X-ray fluorescence analyzer is unlikely going to house an adiabaticdemagnetization refrigerator. A brief summary of various detection meth-ods from visible to infrared is collected to table 1. Only one parameter isquoted as a figure of merit, which is the noise equivalent power (NEP), dis-cussed in more detail in Section 2. This parameter is relevant for radiometricapplications when detecting changes in the incident flux of photons.

1.3 Figures of merit

A perfect radiation detector would preserve all the information contained inthe incoming flux of photons. The important figures of merit are

i) Noise of the detector. In the ideal case, the internal noise of the detectorshould not hinder the accuracy of the measurement. The limiting accu-racy in the ideal detector is due to statistical fluctuations in the photonflux itself.

ii) Linearity. A linear detector produces an output signal which is propor-tional to the input power or energy.

iii) Quantum efficiency. This is given by the ratio of the number of absorbedphotons to those of the incident photons, η = Nabs/Ni. Ideally, all thephotons are absorbed, and quantum efficiency is unity. The quantumefficiency depends on the energy of the photons, and should remainrelatively unchanged over the range of energies to be detected.

iv) Dynamic range. A large dynamic range is required especially when de-tecting small variations in signal on top of a large (constant) backgroundsignal.

v) Speed. A fast detector is needed if rapid changes in the incident powerare to be detected.

vi) Scalability to large arrays. Imaging detectors usually use a large numberof pixels to form an image.

vii) Dissipation. Especially in cryogenic detectors the power consumption iscritical, as the available cooling power is limited at low temperatures.

4

The sensitivity of detectors in the visible and infrared spectral regionsis often expressed in terms of Noise Equivalent Power, or NEP, from theexpression for signal-to-noise ratio

S

N=ηPopt

NEP

√τint, (1)

where τint is the post detection integration time and Popt is the signal power.Another useful figure of merit for infrared detectors is the Noise EquivalentTemperature Difference, or NETD, which expresses the minimum detectablechange in the target blackbody temperature. A general form is given by

NETD =NEP

η∂Popt/∂T√τint

, (2)

where Popt is the blackbody power incident on the detector given by

Popt =

∫ λ2

λ1

LAΩ dλ =

∫ λ2

λ1

2hc

λ5(ehc/λkBT − 1)λ2 dλ, (3)

and diffraction limited throughput AΩ = λ2 of a single mode is assumed. Inthe Rayleigh-Jeans limit when hc/λ kBT , Eq. (3) simplifies to

PRJ = 2kBT∆ν, (4)

where ∆ν = c(λ2 − λ1)/(λ1λ2). Thus, the expression for NETD becomes

NETDRJ =NEP

2ηkB∆ν√τint

. (5)

In many situations, including comparisons with coherent detectors, NETDis a rather convenient method to compare detector sensitivity.

In coherent detection methods, such as heterodyne receivers or HEMTtuned RF receivers, the sensitivity is often expressed in terms of receivernoise temperature TN. The relation between TN and NEP is

NEP = kBTN

√∆ν, (6)

where ∆ν = 2∆fIF for double sideband receiver and ∆ν = ∆fIF for singlesideband receivers where ∆fIF is the bandwidth of the intermediate frequencyamplifier. The noise temperature of a receiver can be either limited by thenoise of the amplifier, the noise of the mixer, noise in the input power, or,ultimately, by the so called quantum noise. Quantum noise arises from theHeisenberg’s uncertainty relation according to which the accuracy with which

5

one can measure the energy and the arrival time of the incoming photon islimited to

∆E∆t & ~. (7)

For the measurement of power, the uncertainty becomes ∆P = hν/∆t, whichcorresponds to a noise temperature TN,QL = hν/kB.

This limit is not imposed on incoherent detectors which do not detect thephase of the incoming radiation. One way to compare the performance ofcoherent and incoherent detectors is to study the background limited noiselevel (often called as BLIP, from Background Limited Performance) as afunction of the ratio hν/kBT . The limiting NEP for a incoherent detectorobserving a single mode is given by

NEPBLIP,d = hν√

∆ν

√

n0(1 + ηdn0)

ηd

, (8)

where n0 = (ehν/kBT − 1)−1 is the background occupancy number and ηd isthe total quantum efficiency of the detector. For a coherent detector, thebackground limited performance is given by

NEPBLIP,c = hν√

∆ν(1 +mηcn0)

ηc

, (9)

where ηc is the quantum efficiency of the coherent detector. The constantm = 1 for single sideband mixer or an amplifier, and for double sidebandmixer m = 2 [1]. A comparison is shown in Fig. 2.

This thesis focuses on the development and optimization of thermal de-tectors for two wavelength ranges: for the X-ray region and for the submil-limeter to millimeter wavelength range, which are of particular interest forseveral reasons. X-ray astronomy is a blooming field of astrophysics and sev-eral ambitious space borne observatories, such as NASA’s Constellation X[2], and European Space Agency’s XEUS missions [3] which will study ener-getic phenomena at the far reaches of our Universe with an unprecedentedsensitivity. Ground-based applications of high resolution X-ray spectroscopyare in semiconductor industry where the detector technology benefits processcontrol, and also in materials analysis where the high resolution enables thedetection of chemical shifts in the fluorescence spectra of elements.

The submillimetre wave range is interesting as it contains a huge numberof vibrational and rotational spectra of molecules, implying a variety of spec-troscopy applications. Long range terrestrial applications are limited by thestrong attenuation of submillimetre waves in the lower atmosphere, a reasonwhy ground based submillimetre astronomy is carried out at mountain tops

6

10−3

10−2

10−1

100

101

102

10−2

10−1

100

101

hν/k B

T

NE

PB

LIP

,d/N

EP

BLI

P,c

Figure 2: A comparison of the background limited performance of a inco-herent detector and a coherent detector. Incoherent detectors outperformcoherent detectors when hν/kBT & 1, which corresponds to ν ≈ 2 THz fora 300 K blackbody and ν ≈ 60 GHz for a 3 K blackbody. The calculationassumes ηd = 0.1 and ηc = 0.5. Also an identical bandwidth ∆ν is assumed.

and from balloons. Cosmologists observe the echoes of the Big Band in thesubmillimetre waves, as its radiance peaks at submillimetre waves and theclutter from foreground objects, such as planetary and galactic dust, is low.An unprecedented view to the infant Universe will open within a few yearswith the commissioning of the Herschel-Planck mission [4].

Beyond Herschel-Planck, future science objectives require the develop-ment of much larger image forming incoherent detector arrays than are inuse today. The large array size, combined with stringent sensitivity require-ments pose a significant challenge for the experimentalist. Closer to earth,a number of applications are envisioned, one of which is the passive imag-ing of concealed weapons, explosives, or other contraband, under clothing.The ability of passing relatively unobscured through dielectric objects com-bined with the short wavelength allows the construction of cameras with areasonable aperture size.

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INCOHERENT Ref.

Detector operating λ range NEP Array Power

type temperature [W/Hz1/2] size /pixel

Photovoltaic ∼ 100 K 1 µm 10−17 > 106 ∼ 1 nW [5]∼ 2.5 µm

∼ 2 K 110 µm 10−17 100 [6]

Thermal ∼ 300 K 8 µm 10−13 105 ∼ 100 nW [7]- 12 µm

∼ 300 K 2 mm 10−11 101 ∼ 400 µW This work1

4.2 K 2 µm 10−13 101 [8]∼ 5 mm

4.2 K 150 µm- 10−14 100 20 nW This work1

660 µm∼ 370 mK 100 µm 10−17 101 10 pW [9]

5 mm

COHERENT

HEMT TRF ∼300 K > 1.5 mm 10−14 103 ∼1 W [10]20 K 10−16 101 10 mW [11]

Schottky ∼300 K ∼ 600 µm 10−15 101 1 mW [12]Schottky ∼300 K ∼ 120 µm 10−14 101 1 mW [12]SIS 4.2 K > 220 µm 10−16 101 ∼1 µW [12]HEB 4 K - 20 K > 120 µm 10−15 100 ∼100 nW [13, 14]

Table 1: A comparison of detection methods from the infrared to millimetre-wave spectral ranges. 1Electrical measurements.

8

Figure 3: A simplified diagram of the operating principle of a thermal de-tector. An incident photon causes a temperature rise in a thermally isolatedheat capacity (denoted by C). The heat escapes via a thermal link G to theheat sink in a time scale determined by the thermal time constant τ0 = C/G.

2 Thermal detectors

2.1 Operating principle

Thermal detectors measure energy. The absorbed energy results in a tem-perature rise, given by ∆T = E/C, where E is the input energy, and C is thetotal heat capacity of the absorbing system. In a detector, the temperaturerise ∆T is typically transformed to a suitable measurable electrical parame-ter, such as the dielectric constant, electrical resistivity, or inductance. Whenused to measure the energy of individual incident photons, thermal detectorsare called calorimeters (from latin calor=heat), and if changes in the incidentflux of photons (power) is measured, the devices are called bolometers (fromgreek bole=ray). At submillimeter wavelengths the rate of photons is verylarge compared to the typical time constant of a detector, and thus devicesare called bolometers, whereas in the X-rays, it is possible to use calorimetricmode as the rate of photons is small.

Thermal detectors are divided to several subgroups depending on the waythe signal (heat) is transformed to a measurable signal. The most commontypes are pyroelectric and resistive thermal detectors. In the former, thesignal is produced by the polarization of the detector material due to the

9

temperature change. In the latter, the electrical resistivity of the materialis modified due to a change in the temperature. The pulse response of athermal detector to a photon is shown in Fig. 3. The absorbed photoncauses a sharp rise in temperature with the rise time limited by the time ittakes for the photon energy to thermalize through the heat capacity C. Thepulse is followed by an exponential fall time to the steady state condition.

The behaviour of a calorimeter can be obtained from

C(T )dT

dt+ Pout = Pin (10)

where the first term on the left describes the time rate of change of heatstored to the heat capacity of the sensor, and the second term is the powerflowing out of the detector, while the term on the right side is the inputpower, which consists of contributions from the bias power, optical powerand the noise power. The power flowing out of the detector is given by [15]

Pout =

∫ Td

T0κ(T )dT

∫ xd

0[A(x)]−1dx

, (11)

where A is the cross section of the link, Td is the temperature of the detector(at x = xd), T0 is the temperature of the heat sink (at x = 0) and the thermalconductivity κ(T ) depends on the characteristics of the link. In a metallic(normal state) link at low temperatures (T ΘD) the heat transport ismainly by the electron gas in the metal, and thus the thermal conductivityis given by Wiedemann-Franz law, κ(T ) = LT/ρn where L = 2.45 · 10−8

V2/K2 is the Lorentz number and ρn is the resistivity of the metal. In thecase of a crystalline dielectric link the heat is transported by phonons in thelattice. The crystalline structure of the material has a strong influence onthe thermal conductivity, but typically κ(T ) = aT b with a a constant andb ∼ 2. In superconductors, the charge carriers, known as Cooper pairs, allsit in a low energy state of zero entropy, which is separated by an energy gap∆E = 1.76kBTc from the states of the single, unpaired electrons. Only brokenpairs (single electrons) can carry entropy (and thus heat), and their densityhas an exponential dependence on temperature. The thermal conductivity ofa superconducting link coincides with the thermal conductivity of a metalliclink when T → Tc, the critical temperature of the superconductor. Slightlybelow Tc, the electronic contribution κ(T ) = κ0e

−ΛTc/T with κ0 = LTeΛ/ρn,and Λ ≈ 1.76. In disordered superconductors (and at intermediate temper-atures of a few K), also the lattice conductivity has a significant effect onthe thermal transport. In this case the temperature dependence is ratherdifficult to estimate.

10

Bias circuit

G,G

L

RS I0

R

Figure 4: The electro-thermal circuit of a bolometer. The circuit is divided totwo major subparts: the electrical circuit (solid lines) and the thermal circuit(open lines). The interplay between the electrical and thermal circuits takesplace in the bolometer.

A typical bias circuit of a bolometer is shown in Fig. 4. It can bedivided to the thermal circuit, described by heat capacity C and thermalconductances G and G, defined below, and an electrical circuit consisting ofa bias supply V with a source impedance ZS(ω) = RS + iωL.

Considering a general case, where the power flow to the heat sink is givenby

Pout = K(T n − T n0 ), (12)

where K is a constant which depends on materials parameters and the ge-ometry of the link, the equation for a bolometer with a bias point resistanceof R = V/I absorbing a time-varying optical signal Popt = Poe

iωt becomes

Cd(δTeiωt)

dt+K(T n − T n

0 ) +GδT = Pbias + Poeiωt +

dPbias

dTδT, (13)

where δT is used to denote the temperature change due to the signal power.Equating the steady state components of the equation yields

Pbias = K(T n − T n0 ), (14)

from which one can obtain the result for the average operating temperatureof the bolometer, given by

T =Pbias

G+ T0 (15)

11

where an average thermal conductance G is defined by

G =K(T n − T n

0 )

T − T0

=Pbias

(

Pbias

K+ T n

0

)1/n − T0

. (16)

The change in input signal power modifies the bias dissipation, an effectdescribed by the last term in Eq. (13). This effect is commonly knownas electrothermal feedback (ETF). Taking a closer look at the temperaturechange we obtain

δT =Po

G + iωC − dPbias/dT(17)

where G = dP/dT ≈ nKT n−1 is the dynamic thermal conductance. Now,considering the electrothermal term

dPbias

dT=dPbias

dR

dR

dT= −I2

[

R− ZS(ω)

R+ ZS(ω)

]

αR

T= −Pbiasα

Tβ(ω), (18)

where we have used α ≡ d logR/d logT the dimensionless temperature coef-ficient of resistance for the thermometer, and β(ω) ≡ [R−ZS(ω)]/[R+ZS(ω)]the effect of the bias circuit on the ETF. Taking into account the thermalcut-off of the bolometer, the frequency-dependent loop gain is defined as

L(ω) ≡ Pbiasα

GT

β(ω)√

1 + ω2τ 20

= L0β(ω)

√

1 + ω2τ 20

, (19)

where τ0 = C/G is the intrinsic thermal time constant of the bolometer. Theelectro-thermal loop gain describes the effect of changing input power to thebias power of the detector. For positive bolometers with α > 0 the loopgain is positive for current bias (since Re[β(ω)] > 0) and negative for voltagebias (as Re[β(ω) < 0]). Negative bolometers will not be discussed within thescope of this thesis. For metallic bolometers operated at room temperatureα ∼ 1 and the loop gain is typically small (L0 .1) so that the role of ETF isnegligible. On the contrary, superconducting detectors with α ∼ 100 and Gsome three orders of magnitude smaller than for room temperature devicescan have large loop gain (L0 ∼ 50), so that ETF plays a significant role inthe detector characteristics.

For measurement purposes, it is worthwhile to notice that L can also beexpressed as

L(ω) =dR/R

dP/P=P

R

d(V/I)

d(V I)=Z(ω) −R

Z(ω) +Rβ(ω), (20)

12

where Z(ω) = dV/dI is the dynamic (electrical) impedance of the bolometerand as ω → ∞, Z = R since signals faster than the time constant do notaffect the operating point of the bolometer [16].

A major impact of the ETF is that the bolometer time constant is reducedfrom τ0 to τeff = τ0/[1 + β(0)L0].

A common way of biasing bolometers is to drive a constant current I0through a parallel combination of the bolometer and the load resistor RS.As long as RS R, the bolometer is effectively voltage biased. The respon-sivity of a voltage biased bolometer can be derived as follows: The currentresponsivity is defined as SI ≡ dI/dPo = (dI/dR)(dR/dT )(dT/dPo). Now,using Eqs. (17), (18) and (20) we can write

∣

∣

∣

∣

dT

dPo

∣

∣

∣

∣

=1

G

∣

∣

∣

∣

1

1 + iωτ0 + L0β

∣

∣

∣

∣

=1

G

1

1 + βL0

1√

1 + ω2τ 2eff

. (21)

Next, we note that (dR/dT ) = αR/T . To calculate the total derivativedI/dR we need to take into account the effect of the bias circuit on dI. Atzero frequency, the voltage across the bolometer is V = I0RSR/(RS + R),while the current is I = I0RS/(RS +R). Thus, dI/dR = −I0RS/(RS +R)2 =−I(β + 1)/(2R). Using these results, the current responsivity becomes

SI(ω) =IαR

GT

1 + β

2

1

1 + βL0

1√

1 + ω2τ 2eff

= − 1

V

(1 + β)L0

2(1 + βL0)

1√

1 + ω2τ 2eff

. (22)

In the limiting case at ω = 0 with β = 1 (perfect voltage bias) and L0 1,

SI(0) = − 1

V. (23)

A similar treatment can be carried out for a current biased bolometer,when RS is in series with the bolometer. This derivation yields

SV(ω) =1

I

(1 − β)L0

2(1 + βL0)

1√

1 + ω2τ 2eff

, (24)

which yields in the limit ω = 0, β = −1 and L0 1

SV(0) = −1

I. (25)

Next, we shall summarize the various sources of noise in thermal de-tectors. The limiting noise source in a thermal detector arises from the

13

spontaneous exchange of energy between the heat sink and the detector heatcapacity C across the thermal link G. It can be shown, that this energyexchange leads to a current noise at the output of the voltage biased detec-tor given by the so-called thermal fluctuation noise (TFN), also known asphonon noise [15]

δITFN =

√

4γkBT 2G

2V

(1 + β)L0

(1 + βL0)

1√

1 + ω2τ 2eff

(26)

where γ describes the effect of the temperature gradient across the thermallink with κ ∝ T b between the sensor and the heat sink [15] in the diffusivelimit

γ =(b+ 1)

(

Tcb+3 − Tc

−bT03+2 b

)

(3 + 2 b)Tc2(

Tcb+1 − T0

b+1) ≈ b+ 1

2b+ 3, (27)

where the approximation is valid when Tc T0.Another noise contribution is due to the Johnson noise, given by

δIJN =

√4kbTR−1

2

1 + β(ω)

1 + β(ω)L0

√

1 + ω2τ 20

√

1 + ω2τ 2eff

. (28)

It is often stated that the Johnson noise is suppressed by the ETF, whilemore appropriate would be to say that the Johnson noise is not amplified bythe electro-thermal gain of the sensor. Thus, in the output of the sensor,the relative contribution of Johnson noise compared to TFN is smaller atfrequencies below τ−1

0 .Additionally, there are contributions from the noise of the amplifier, as

well as from internal thermal fluctuations which arise from the distributedinternal thermal resistance of the detector [17, 18], and finally, as will beshown in Section 5, excess noise due to fluctuation superconductivity aboveTc.

14

3 Antenna-coupled microbolometers

3.1 Room temperature antenna-coupled microbolome-

ters

Antenna-coupled microbolometers can extend the wavelength limits ofbolometers from infrared to the millimetre-wave range [19, 20, 21, 22]. ThisSection briefly summarizes the results obtained with titanium air-bridge mi-crobolometers, coupled to a full-wave dipole antenna.

The main benefit of antenna coupling is the fact that the thermally activeelement can be made much smaller than the wavelength of the incomingradiation. This is of especial importance when millimetre-waves are detected,as the required size of the bolometer becomes prohibitingly large, as the areaof the absorbing element scales as λ2. Increased area results to a large heatcapacity, and thus slow time constant, as the thermal conductance G is set bythe sensitivity requirement. Since the thermal time constant of the antenna-coupled microbolometers is very short (of the order 1 µs), there is much roomto improve the thermal isolation. All of the power received by the antenna isdissipated to the bolometer bridge, given that the antenna and the bolometerimpedances are matched.

Our approach to improve the thermal isolation is the removal of the heatconductance to the substrate by fabricating the bolometer to a free standingbridge structure, and coupling the bolometer to an antenna with relativelyhigh input impedance Zant since G ∝ 1/Zant.

After a survey of lithographic antenna literature, we selected a full-wavedipole although dipoles on dielectric substrates do suffer from losses to backlobes. The most attractive property was the possibility of obtaining feedingpoint impedances Zant in excess of 200 Ω [23]. A more thorough study beyondthe scope of this work could utilize other antennae with better beam charac-teristics, combined with planar impedance transformers that could increasethe input impedance even further. The operating frequency was chosen tobe 160 GHz which is a trade-off between atmospheric transmission and 300K blackbody radiance, with the former decreasing with frequency while theradiance increases.

One important consideration in resistive bolometers is the 1/f noise. Al-though this noise source can be avoided by modulating the optical signalat a frequency above the knee frequency where 1/f noise equals the John-son noise, the knee frequency might be too high for practical modulationmethods. The 1/f noise in a resistor is given by

V 21/f (f) = V 2zf r = V 2αHneΩf

r, (29)

15

where r ' −1, V is the voltage across the detector, the 1/f noise parameteris z = αH/(neΩ), with αH the Hooge parameter [24], ne the mobile chargecarrier density, and Ω the volume of the bolometer. It is crucial for a resistiveantenna-coupled microbolometer to have a low 1/f noise since their volumeis much smaller than in typical surface-absorbing infrared microbolometers.We studied the 1/f spectra from three materials: platinum, niobium, andtitanium. Another important factor is the temperature coeffcient of resis-tance which should be as large as possible. The results from the materialscharacterisation are collected in Table 2. Based on these findings we choseTi as the bolometer material.

Material TCR (α/T ) [%/K] r z [Hz−(r+1)]Ti 0.31 -1.22 7·10−13

Nb 0.045 -1.13 13·10−13

Pt 0.19 -0.9 24·10−13

Table 2: Results for the temperature coefficient of resistance (TCR) and the1/f noise for Ti, Nb and Pt.

The bolometers were fabricated on a 525 µm thick nitridized high resis-tivity (10 kΩcm) silicon wafer. The fabrication details are very similar tothose used in the study of the superconducting Nb bolometers, described inpaper V. The main difference was the incorporation of a separate metalliza-tion layer for the full-wave dipole antenna. The antenna layer was patternedusing electron beam lithography and a double layer electron resist. The Auwas grown using ultra-high vacuum (UHV) electron gun evaporation to athickness of 200 nm, and a lift-off in acetone followed. A second double-layerelectron resist was spun on the wafer, and the bolometer bridge was pat-terned. After development, we used a short O2 plasma etch to remove resistresiduals from the antenna-bolometer contact region prior to the depositionof the Ti bolometer film. The Ti was evaporated to a thickness between 180

CF4 flow 7.35 sccmO2 flow 2.0 sccmRF power 245 mW/cm2

Pressure 35 mTorr - 70 mTorrTime ∼ 2 min

Table 3: Bridge release RIE parameters.

16

nm and 200 nm, followed by a lift-off. The release of the bridge was carriedout as with in the superconducting Nb bolometers using reactive ion etching(RIE) with oxygen and carbon tetrafluoride (CF4) at a relatively high pres-sure between 35 mTorr and 70 mTorr. The etch started at the lower pressurein order to break the surface. After ∼ 1 minute the pressure was raised to 75mTorr which results to an isotropic etch which removes the SiN underneaththe bridge. The etch parameters are tabulated in Table 3. The resultingbolometer structure is shown in Fig. 5.

750 mµ

15 mµ

a)

b)

Figure 5: A SEM image of the 15 µm × 1 µm × 180 nm Ti bridge bolometer,with a) an overview of the bolometer and the antenna. The widenings in thelow frequency measurement lines serve as λ/4 low-pass filters to prevent RFloss to the lines. b) A closeup of the antenna feed, showing the suspendedbridge.

17

0.0 0.5 1.0 1.5100

150

200

250

300

350

400

450

a) b)

Z,R

[Ω]

Current [mA]0.0 0.5 1.0 1.5

0

100

200

300

400

Res

pons

ivity

[V/W

]

Figure 6: a) The resistance R (circles) and differential resistance Z (squares)of the Ti bridge as a function of the bias current. A fit (line) to the Z(I)curve yields a value G = 3 µW/K for the thermal conductivity according toEq. (30). b) The measured electrical responsivity of the bolometer.

18

0.1 1 10

10

100

0.20 mA0.42 mA0.62 mA0.83 mA0.94 mA1.04 mA

a)

NE

P[p

W/H

z½]

Frequency [kHz]0.0 0.2 0.4 0.6 0.8 1.0 1.21

10

100b)

Current [mA]

Figure 7: a) The spectrum of NEP at various bias currents, indicated in thelegend of the figure. b) The NEP at 1 kHz (squares) and at 10 kHz (circles).The two solid lines represent the expected NEP with contributions fromthermal fluctuation noise, Johnson noise and 1/f noise with z = 7 · 10−13.The dashed line represents the thermal fluctuation noise limit, and the dash-dotted line is the contribution of the Johnson noise.

The electrical performance of the current biased bolometer was character-ized by measuring its I(V ) and differential resistance Z = dV/dI in vacuo.The differential resistance measurement was carried out using the lock-intechnique. The results are shown in Fig. 6. The voltage responsivity of thebolometer was determined using Eqs. (20) and (24). The Z(I) curve can beused to fit the thermal conductivity G using

Z(I) = R0

[

G2 −GI2αT−1R0 + 2I4α2T−2R20

(2I2αT−1R0 −G)2

]

, (30)

which is obtained from V (I) = R(I)I = IR0G/(G− αT−1I2R0) with whereR0 stands for the zero-bias resistance.

The noise of the bolometer was determined by measuring the noise voltageacross the bolometer using a pair of low noise preamplifiers with a gain of

19

1000 connected to the inputs of a correlation FFT spectrum analyzer. Whenused in the correlation mode, the noise contributions from the preamplifiersare eliminated, and the noise of the bolometer can be determined accurately.Combined with the results from the responsivity measurements, the electricalNEP of the bolometer is obtained. The results are shown in Fig. 7. The1/f noise is higher than expected, which might be due to the RIE processwhich can increase the density of impurities in the Ti. At 10 kHz, the deviceis almost limited by the Johnson noise with a NEP=15 pW/

√Hz. However,

at the more practical chopping frequency of 1 kHz the NEP is at best 42pW/

√Hz.

In the final stage of characterization we measured the optical response ofthe bolometer to a millimetre-wave signal. As a source, a Gunn oscillatorwith a frequency doubler was used. The source could be tuned from ∼150GHz up to about 180 GHz. The output of the source was connected to apyramidal horn antenna. A rotating chopper was placed in front of the hornto modulate the incident power on the bolometer at a chopping frequency of∼ 700 Hz. The bolometer was attached to the backside of a 10 mm diameterhemispherical silicon lens , and centred by visual inspection to an accuracy of∼50 µm. The bolometer/lens combination was placed 28.5 cm from the horn.The bolometer was again current biased, and the voltage across the bolometerwas connected to a low noise laboratory preamplifier with a voltage gain of10 000. Bandwidth of the preamplifier was limited from 30 Hz to 1000 Hz,and the output was connected to the signal input of a lock-in amplifier. Thereference signal was the 0-5 V TTL from the chopper, which was connectedto the reference input of the lock-in amplifier.

A frequency response measurement was performed in order to verify thatthe device indeed responds at the design frequency of 160 GHz. This measure-ment was carried out by placing the bolometer facing the horn antenna, andadjusting the oscillator to different frequency settings. The output powerfrom the waveguide was measured using a calibrated total power meter1.The results calibrated against the Dorado DPM-2 are shown in Fig. 8 (a).Clearly, there is a strong response at 160 GHz, as expected. More pointsbelow 150 GHz would be needed to get an accurate figure for the bandwidthof the device. If using a typical value for lithographic dipoles of about 30% bandwidth (3 dB) around the centre frequency, the response indicates acentre frequency of ∼144 GHz. This shift in the centre frequency can becaused by impedance mismatch between the antenna and the bolometer. Asthe final optical measurement we measured the response of the bolometerto the millimetre-wave signal as a function of the bias current. The source

1Dorado DPM-2 Total Power Meter

20

0.0 0.5 1.0 1.5 2.0 2.5 3.0245

250

255

260

265

270

275

280

285

290

150

160

170

180

190

200

0.0

0.2

0.4

0.6

0.8

1.0 b)

Abs

orbe

dop

tical

pow

er[n

W]

Bias current [mA]

a)

Nor

mal

ized

bolo

met

erre

spon

se

Frequency [GHz]

Figure 8: a) The optical response of the detector as a function of the RFfrequency. b) The absorbed optical power, determined using the electricalresponsivity from Pabs = Vsig/SV(I). The maximum response corresponds tothe bias where the bolometer resistance (207 Ω) matches the antenna inputresistance.

21

was tuned to 155 GHz, and the signal amplitude was recorded while thebolometer bias current was increased. The electrical responsivity SV(I) ofthe detector is known from the previous measurements. Figure 8 (b) showsthe optical response of the device, normalized with the electrical reponsivity.An interesting feature is that the optical response peaks at about 2.1 mA,after which it reduces. The bolometer resistance at this bias was 207 Ω. Weattribute the reducing optical response at larger currents to the increasingimpedance mismatch between the bolometer and the antenna. We considerthis as an interesting method of characterizing antenna input resistance witha direct measurement.

22

S S

N

ln/2 l/20

Tc

T0

T

xFigure 9: The hot-spot model for the Nb bridge.

3.2 Superconducting antenna-coupled microbolome-

ters

As was shown in Section 2.1, cooling of bolometers improves their NEP signif-icantly. Here we describe briefly the properties of superconducting hot-spotNb bolometers, discussed in detail in Paper V. Superconductors, when oper-ated at Tc are very sensitive thermometers making them an attractive choicefor bolometer thermal sensing elements. Transition-edge microcalorimetersalso utilize this property, and they shall be discussed in Section 4.

In order to make use of the large responsivity achievable with supercon-ductors, one has to optimize the noise performance such that the Johnsonnoise and the thermal fluctuation noise are comparable, i.e. that the thermalisolation of the bolometer is sufficient. This can be achieved by removing thesubstrate from below the thermal sensing element, as was discussed above inSection 3.1. The bridge is voltage biased, which introduces negative electro-thermal feedback which maintains a part of the bridge within Tc, while thebath temperature is maintained at 4.2 K. Such a bridge can be modeled witha simple hot-spot model [25, 26], where superconductivity enters the bridgefrom its ends, leaving a dissipative normal region to the centre portion of thebridge, as shown in Fig. 9. Incoming optical power Popt modulates the sizeof the normal region and thus the current through it. The model assumessteady state behaviour, which is justified by the fact that any typical signalis much slower than the estimated 1 µs thermal time constant of the bridgewhich is based on the bridge volume and specific heat of Nb in the normal

23

30 mµ

15 mµ

Figure 10: A SEM image of the Nb bridge bolometer, coupled to a logarithmicspiral antenna.

state. In the limit of small optical power, the solution for the I(V ) curve is

I(V )|Popt→0 =4κ(Tc − T0)

V l+V wt

ρNl, (31)

where κ is the thermal conductivity of the Nb, w, l, t are the width, length andthe thickness of the bridge, respectively, and ρN is the normal state resistivityof Nb. Our model assumes that the thermal conductivity is a constant, as thelattice thermal conductivity of the Nb is significant at 4.2 K. Moreover, theelectrical thermal conductivity is a non-monotonous function of temperaturedue to the fact that below Tc the thermal conductivity can actually increasesomewhat in disordered metals due to reduced electron-phonon scattering[27]. Thus, as it difficult to determine the exact dependence of κ on T , itis taken as a constant. The measurements were carried out in an evacuatedcan immersed in liquid He. First, the critical temperature of the bridge wasmeasured to be 6.8 K by measuring its resistance with a small bias currentagainst the bath temperature. From this measurement we also obtainedρN = 56 µΩcm. This is considerably higher than the tabulated value of 16µΩcm for Nb. The high normal state resistivity is likely caused by highimpurity content, which is supported also by the Tc, as the tabulated valuefor the critical temperature of Nb is 9.1 K. Next, the bridge, shown in Fig.10, was biased in parallel with a 1.2 Ω shunt resistor, and the current throughthe bridge was measured with a SQUID [28]. The I(V ) characteristics areshown in Fig. 12. The hot-spot model was fitted with the experimental data,

24

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

-2000

-1500

-1000

-500

0a)

Curr

entre

sponsi

vity

[A/W

]

Voltage [mV]

10 100 1000 10000 100000

1E-11

1E-10

1E-9 V=0.82 mVb)

nois

esp

ect

rald

ensi

ty[A

/Hz½

]

Frequency [Hz]

Figure 11: The electrical responsivity of the Nb bridge bolometer. The insetshows the current noise spectrum measured at V = 0.82 mV.

from which we obtained κ = 0.54 W/(Km), which is in agreement with thevalue predicted by the Wiedemann-Franz law [0.29 W/(Km)] and tabulatedvalue for NbTi [0.28 W/(Km)] [29]. Using Eqs. (20) and (22), the currentresponsivity of the bridge can be calculated, shown in Fig. 11. Finally, thecurrent noise spectra was measured at different bias points. A noise spectrummeasured at V = 0.82 mV is shown in the inset.

The noise equivalent power of the bolometer is shown in Fig. 13. We notethat the minimum NEP=14 fW/

√Hz, which is almost an order of magnitude

improvement over existing 4.2 K bolometers [8]. Our measurement was alsolimited by the noise of the SQUID, and better noise matching would enablea further improvement to NEP∼ 3 fW/

√Hz. Since the time constant is

short, there exists the possibility of reducing G even further to improve theNEP. The increase in the NEP at bias voltages below 0.82 mV are due toelectro-thermal oscillations which take place in the bias circuit when theeffective time constant of the bolometer becomes comparable to the LSQUID/Relectrical time constant of the bias circuit [30],[V]. Thanks to the simplefabrication process, the device is scalable to large imaging arrays, and thesearrays could be coupled to time or frequency multiplexed SQUID array read-outs, currently under intensive development for future 100 mK bolometerand microcalorimeter arrays [31, 32].

25

0.0 2.0 4.0 6.0 8.0 10.0

0

10

20

30

40

50

60

70

80

Curr

ent[m

A]

Voltage [mV]

Figure 12: The I(V ) characteristics of the Nb bridge. The experimental datais represented by the circles, the solid line is the fit with κ = 0.54 W/(Km),the dotted line represents the Ohmic asymptote with the bridge in fullynormal state, while the dashed line indicates the electro-thermal ∝ 1/V term.The smooth transition from ohmic to the partly superconducting region issmooth which is an indicative of a hot-spot type behaviour.

26

1.0 10.00.1

1.0

10.0

100.0

1000.0

NE

P[fW

/Hz½

]

Voltage [mV]

Figure 13: The electrical NEP of the Nb bridge bolometer. The experi-mental data are marked by triangles, while the the solid line represents thetotal modeled noise, consisting of contributions from the SQUID (dotted line,barely seen under the total noise), thermal fluctuation noise (dashed line),and Johnson noise (dash-dotted line).

27

4 Transition-edge microcalorimeters

When a transition from normal conductivity to superconductivity occurs ina thin film, a sharp drop to zero in the electrical resistance R of the filmcan be observed. Typical steepness of the transition α ≡ d logR/d logT isof the order of 1000, and thus the transition itself can be utilised as a verysensitive means of measuring changes in the temperature T of the film. Thisidea has been around for a long time [33, 34, 35], but experimenters hadto use external feedback to maintain the operating point of the bolometerwithin the transition. Later, the idea of using voltage bias to provide internalnegative feedback was introduced [36], and this, together with the existenceof SQUID ammeters, made the breakthrough for practical transition-edgesensors (TESs).

When voltage bias is used, the superconducting film is maintained withinthe transition by the electro-thermal feedback, discussed in Section 2.1. Asthe film is first switched to the normal state by either an external magneticfield or by a large current exceeding the critical current, the bias point can beapproached from the normal state by reducing the bias voltage. As transitionis approached, current increases with decreasing voltage, since the bias poweris maintained nearly constant, and thus I(V ) ≈ P/V where P is given bythe heat flow to the substrate at T0, Eq. (12). It is interesting to note thathad the bias source a truly zero impedance, the TES would switch to normalstate with all finite V . In practice however, TES switches instead to thesuperconducting state at V =

√RsP as the resistance of the film becomes

comparable to that of the internal resistance of the bias source Rs (the socalled shunt resistance).

Typically, when biased with small current bias and when the R(T ) curveis measured by sweeping the bath temperature, α has a maximum value ofabout 300. When biased with a voltage bias, the situation is quite different,and in many cases α can be smaller by an order of magnitude. Now, theinput joule power is much larger which has a two-fold effect on the transition:Firstly, the current flowing through the TES generates a magnetic field, whichitself effects the critical temperature across the film. Critical temperaturewill tend to be suppressed at the edges of the film where the perpendicularcomponent of the self field is at maximum. At the centre of the film, Tc isclose to zero-field case. Secondly, there is a temperature gradient across thefilm due to dissipation and the effective value for α is obtained as an averageover the temperature range in the film.

At this point it is relevant to discuss the magnetic properties of aTES film. The magnetic penetration depth of a titanium film is λ =√

me/µ0nee2 ≈ 30 nm where me is the electron mass and ne = 3.4 · 1028

28

m−3 is the free electron density for Ti. The type of the superconductorcan be determined from the ratio of the penetration depth to the coherencelength, κ = λ/ξ. At Tc and in the dirty limit κ = 0.715λ/l where the meanfree path is obtained from the normal state resistivity using

l =(rs/a0)

2

ρN× 92 · 10−18 Ωm2, (32)

with rs/a0 = 2.1 for Ti [37] where rs is the radius of the conduction electronoccupation volume and the Bohr radius a0 = 0.529 A. For a typical resistivityof ρN = 3.81 µΩcm for our films, l ≈ 110A. Thus, κ ≈ 2.7 which suggeststhat the film is a type II superconductor. If there is a type I superconductingground plane present, this will make the TES film also type I.

We can estimate the magnitude of the magnetic effect by comparing thecritical current Ic with the typical operating current. The Ginzburg-Landautheory, combined with BCS critical field and penetration depth [30], yields aresult for the critical current density Jc = Jc0(1 − T/Tc)

3/2 where

Jc0 ≈ 6.39(kBTc)3/2

√

N(0)

~ρN≈ 4 GA/m2, (33)

with N(0) = 4.25 · 1047 J−1m−3 is the density of states at Fermi surfacefor titanium [37]. For comparison, a typical current just before entering thetransition is of the order 10 µA which corresponds to a current density ≈ 1MA/m2 for a typical cross sectional area of 10 (µm)2. The current over therange of operating points is typically below 50 µA. From the Bean’s criticalstate model [38, 39], the current density distribution for a thin superconduct-ing film is given by

J(x) =2πJc

πarctan

(

√

w2 − a2

a2 − x2

)

|x| < a,

= Jc, a < |x| < w, (34)

where the majority of the current is carried at the edges where J = Jc whilethe width of the field-free region is determined by the penetration depthλ(T ) = λ(0)/

√

1 − (T/Tc)4 and is given by a = w√

1 − (I/Ic)2. Figure 14shows a finite-element calculation of the temperature and current distributionin a square TES using the critical state model.

4.1 X-ray measurements with a square TES

Initially, studies at our group focused on the development of square TESmicrocalorimeters based on titanium-gold proximity bilayers. Details of the

29

SiN membrane

Reduced tem

perature

Current

Figure 14: The temperature and current distribution across a 250 (µm)2 TESwith a normal state resistance of 0.3 Ω, calculated using a 2-dimensionalfinite-element model. The temperature gradient in reduced temperatureunits is indicated by the levels of gray, while contours indicate the constantcurrent density contrours. The bias voltage is 0.1 µV.

30

300 x 300 ( m) TESµ 2

SiN

a) b)

Si

2.5

mm

Figure 15: a) A SEM image of a square TES. The relevant dimensions aremarked to the picture. The thickness of the SiN membrane was 250 nm.b) The I(V ) curve of a square TES. The spikes below 0.9 µV are due toX-ray events absorbed to the detector during the voltage sweep. The dashedline corresponds to the normal state resistance of 0.15 Ω. The X-ray energydrives the sensor close to the normal state.

fabrication procedures can be found from Ref. [40]. A SEM image of the 300µm × 300 µm sensor is shown in Fig. 15 a). This device had a Tc = 150mK and a normal state resistance of 150 mΩ. The measured I(V ) curve isshown in Fig. 15 b). The detector was biased in parallel with a 7 mΩ shuntresistor, and the current through the TES was measured with a SQUID witha current noise of 10 pA/

√Hz. Bath temperature was maintained at ≈20

mK. The spikes in the curve are due to X-ray events absorbed during themeasurement. The pulses almost touch the line extrapolated from the normalstate, which implies that the photon energy is close to saturating the sensor.

Taking in to account the thermal fluctuation noise and Johnson noise inthe TES, the full-width at half maximum (FWHM) energy resolution of a

31

transition-edge sensor X-ray microcalorimeter is given by [36]

∆EFWHM = 2.36√

kBT 2c Cξetf (35)

with

ξetf = 2

[

γ

(

1

αL0

)

+

(

1

αL0

)2]1/4

(36)

describing the effect of the electro-thermal feedback to the energy resolutionof the sensor. If the detector noise is limited by the thermal fluctuation noise,ξetf < 1. For the SiN thermal link used, γ = 0.432.

We can make a crude estimate of the energy resolution for the detectorusing Eq. (35) with parameters obtained in the measurements. We estimatethe overall heat capacity of the sensor to be about 2.4 pJ/K. This number isobtained by calculating the contributions to the heat capacity from the 300µm × 300 µm bismuth absorber on top of the TES, and the Ti/Au TES at150 mK. The layer thicknesses were 3 µm for the Bi, 73 nm for the Au and26 nm for the Ti. The heat capacity is given by C = cVV for the layers withvolume V , where cV = γsρmTc/(amaNA) is the heat capacity per unit volumecalculated using the molar specific heat γs, mass density ρm and atomic massa for each material. NA and ma correspond to the Avogadro’s constant(NA = 6.022 · 1023 mol−1) and the atomic mass unit (ma = 1.661 · 10−27 kg),respectively. For the Ti layer, the heat capacity is multiplied by a factor2.43 due to the increase in the specific heat at Tc. The bias dissipation atthe operating range was 6 pW, which yields an average thermal conductivityG = Pb/(Tc−T0) ≈ 55 pW/K. The dynamic thermal conductance G in termsof the average thermal conductivity is obtained from [41]

G = nG1 − Tc/T0

1 − T nc /T

n0

, (37)

where n ≈ 3 for SiN, and using the values for this sensor G ≈ 120 pW/K,yielding an intrinsic time constant of τ0 = 20 ms. From measured pulses aneffective fall time of 260 µs was obtained, which implies an average loop gainof L0 ≈ τ0/τmeas ≈ 80. Next, we estimate α by noting that the characteristicX-rays at energy E = 5.89 keV almost saturate the sensor, which can beseen from the I(V ) curve with X-rays almost reaching the line correspondingto the normal state resistivity, ∆T = E/C ≈ 0.5 mK ≈ 0.8∆Tc. Thus,α ≈ Tc/∆Tc ≈ 300. Inserting the values to Eqs. (35) and (36), the resultingtheoretical resolution is 12.7 eV ·0.13 =1.7 eV.

The measured X-ray spectrum is shown in Fig. 16. A fit calculated usinga convolution of the gaussian detector response and the Lorenzian line shape

32

0.0 2.0 4.0 6.0 0

300

600

T c =155 mK

T 0 ~20 mK

Co

un

ts/B

in (

Bin

wid

th 1

.26

eV

)

Energy [keV]

5.85 5.86 5.87 5.88 5.89 5.90 5.91 5.92 5.93 5.94

Figure 16: The Mn K-line spectrum measured with the square TES. Thedetector resolution is (9.1 ± 0.1) eV. The insert shows an expaned view ofthe Kα line.

for the Mn Kα complex yields a detector resolution of (9.1±0.1) eV. Thus, theexperimental value for ξetf = 0.72. The possible explanations for the discrep-ancy are position dependent energy losses in the Bi absorber, non-optimumanalogue filtering, or alternatively excess noise, either due to internal energyfluctuations [17], or noise due to fluctuation superconductivity, which shallbe addressed in Section 5.

4.2 TES in Corbino disk geometry

Rectangular X-ray microcalorimeter TESs, such as the one described in Sec-tion 4.1, are rather vulnerable to imperfections and irregularities in the TESfilm. One common complication is imperfect contact between the supercon-ductor and the normal metal [42] or different relative thicknesses for the twometals due to different wetting properties of the metals. In either case, thecritical temperature at the edges is modified. In a typical situation, the Tc

is higher at the edges, which effectively decreases the steepness of the transi-

33

Figure 17: A diagram of the Corbino disk TES. The micrograph on top leftshows the lateral dimensions of the device. Film order and typical thicknessesare shown in the exploded view of the CorTES.

tion. Moreover, the self-field, discussed briefly above, spreads the transitionsignificantly when voltage biased. To counter these issues and eliminate theimperfections of the edges, we started investigating an edgeless geometry,the so-called Corbino disk [43]. A schematic of the geometry is shown inFig. 17. In this geometry, the bias current is fed to a circular inner con-tact, and collected at an outer contact at the perimeter of the annular TES.Thus, the current distribution is radial. Thanks to the current distributionand the ground plane, the magnetic field near the film is azimuthal, has noradial dependence. The magnetic field, calculated from the vector potential,is shown in Fig. 18. Not only does this geometry eliminate uncontrollablephenomena at the edges of the film, but it also makes the analytical modelingof the TES simpler, as the current density is determined by the geometry.The Corbino disk TESs are processed on a double-side nitridized 525 µmthick < 100 > silicon wafer. The low-stress Si3N4 is grown2 by low pressurechemical vapour deposition (LPCVD) technique to a thickness of 250 nm.

2The nitride is processed at the Microfabrication Laboratory, University of California,Berkeley, USA.

34

0 2 4 6 8 10 12 14 16

x 10−5

10−2

10−1

100

101

H [a

.u.]

z=400 nm

z=200 nm

z=1 µm

Radial coordinate [m]

Figure 18: The magnetic field of the transport current as a function of thedistance from the Corbino disk TES. The field is azimuthal, and constantover the film. The different curves correspond to different thicknesses of theinsulator, located between the TES and the superconducting ground plane.

First, a rectangular opening is patterned to Si3N4 at the back side by theuse of UV photolithography and reactive ion carbon tetrafluoride (CF4) andoxygen (O2) plasma etch (RIE). Next, an anisotropic KOH wet etch of thebulk Si takes place along the lattice boundaries, until a free-standing 750 µm× 750 µm nitride window is released.

Next, a double-layer polymer resist is spun on the wafer with thicknessesof about 300 nm - 400 nm for the two polymer layers. Patterning is done withelectron-beam lithography (EBL). Although the feature size of the structureswould allow the use of photolithography, we have decided to use EBL for itsversatility and better lift-off properties. The first metal layer to be depositedis the bottom superconducting contact, which makes a contact to the TES atthe circumference. The evaporation is done with a ultra-high vacuum (UHV)electron gun evaporator with a base pressure of about 10−8 mbar.

The niobium layer is deposited at a rate of 0.3 nm/s to a thickness of50 nm, followed by a contacting layer of gold with a thickness of 10 nm.Next, a second set of EBL resists are spun, and the annular TES shape ispatterned. Intially we had difficulties in controlling the critical temperaturesof our TES films when using proximity bilayers, see Fig. 19. The Ti layerwas deposited first, followed by the Au layer. This order was chosen due toadhesion, as Au tends to peel off the Si3N4 substrate. The problem arises asthe Ti layer getters impurities while the Au target being changed in place,

35

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0

100

200

300

400

500

600

700

JYU 1998

JYU 1999

JYU 2000

JYU 2000 (Trilayer)

CORTES (Trilayer)

Critical te

mpera

ture

[m

K]

Au/Ti thickness ratio

Figure 19: The dependence of Tc on the relative thicknesses of Au and Tiwith a thickness in the range 50 nm - 200 nm. The results from 1998 wereobtained after a new UHV evaporator was installed. The gradual contami-nation is clear from results from 1999, leading to the results of 2000 when thecontrol of Tc become impossible. A trilayer -TES was then developed whichavoided the contact contamination problems associated with bilayers. Alsothe fabrication of the CorTES proved to be difficult if the TES films weredeposited on the AlOx insulator, a generous source of impurities. After weswitched to evaporating the AlOx on the TES, the Tc become controllableagain. The dashed line represents a fit Tc = Tc0 −mdAu/dti with Tc0 = 642mK and m = −330 mK.

and generates an impurity layer which prevents a good electron-transparentinterface from forming with the gold as it is deposited. As a result, the criticaltemperatures measured seemed to saturate to about 200 mK, and increasingthe Au thickness had little or no effect on the Tc. To counter the problem, weresorted to trilayers, where the superconducting Ti is sandwiched betweentwo layers of Au. As passive Au is used as the first layer, the interface isclean for the Ti. The TES films are deposited with first a thin (10 nm) Tiadhesion layer, an Au layer of about 60 nm, followed by the Ti layer with athickness of about 40 nm. The last layer is another Au layer (10 nm thick)which serves as a contacting layer for the other Nb layer.

In order to avoid short circuiting the TES with the superconducting Nb

36

top electrode, an insulating layer is required. We have experimented withsilicon monoxide (SiO) and aluminium oxide (AlOx), of which the latter hasproven to provide more reliable results. The insulator is evaporated afterresist spin and EBL in a separate electron beam evaporator (base pressure∼ 10−5 mbar). To avoid step coverage issues, the sample is mounted at a 10

angle and the sample mount is rotated at about 60 r/min while the AlOx isevaporated to a thickness of 100 nm at a rate of 0.1 nm/s.

Finally, the top electrode which serves as a superconducting ground planeand the centre contact is fabricated using EBL and UHV evaporation to athickness of 40 nm. In addition, after deposition of the Au contacting layers,a short O2 plasma etch is performed to clean the metal interface.

Characterization of the CorTES was carried out by first measuring theR(T ) curve with a small current bias. The superconducting transition of aCorbino-disk TES is shown in Fig. 20. This measurement was performedusing an AC current bias (with a load impedance of 1 MΩ), and measur-ing the voltage across the TES using the lock-in technique while the bathtemperature is swept across Tc. In this method the bias dissipation is verysmall (. 1 pW) and the TES film is maintained isothermal and at the sametemperature as the substrate. Thus, Fig. 20 represents the intrinsic widthof the superconducting transition which is mainly due to nonuniformities inthe film.

Next, the critical current was measured by using a switching currentmeasurement, where the device was cooled to 30 mK, and short 1 µs currentpulses were injected with a 10 ppm duty-cycle to the device with increasingamplitude until a voltage pulse was detected across the CorTES. From themeasurement, we obtained a critical current density of 36 MA/m2. Thisdiscrepancy compared to the prediction given by Eq. (33) can be attributedto the Nb contacting layer. Further study of SEM images, for example Fig.21, showed that the top Nb ground plane has problems covering the openingin the SiO. This is due to the formation of a SiO eave as the SiO is depositedat an angle. We fabricated test samples with Nb wires (100 µm × 744 nm× 150 nm) crossing a 100 nm thick layer of silicon monoxide. The switchingcurrent measurement yielded a critical current density of 20 MA/m2, close tothe value measured for the CorTES. Although the step coverage has provento be problematic, TES operation is not compromised as long as the criticalcurrent of the wiring is larger than the maximum operating current given byImax =

√

P/Rs which is typically about 30 µA.The circular geometry allows for straightforward analytical modeling of

the TES. If we assume radial symmetry, the current density is given byJ(r) = I/2πrd, where r is the radial coordinate and d is the thickness of theTES. Approaching Tc from the normal state (which is the case when biasing

37

119 120 121 122 123 124 125 126 -0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Sa

mp

le r

esis

tan

ce

[O

hm

s]

Bath & film temperature [mK]

Figure 20: The constant-current superconducting transition of a Corbinodisk TES. The striped region corresponds to a typical (200 µK) temperaturegradient across the device when biased with voltage bias.

2 mµ

Nb+AlOx+Au/Ti/Au

Nb+Au/Ti/Au

Figure 21: Step coverage with evaporated Nb is problematic due to the for-mation of a SiO eave (marked with an arrow) during the insulator deposition.

38

0.04 0.06 0.08 0.1 0.12 0.1410

−6

10−5

10−4

10−3

10−2

10−1

Temperature [K]

The

rmal

con

duct

ivity

[W/K

m] 200 nm SiN

S, Λ=1.764kBT

cΛ=1.5k

BT

cΛ=1.2k

BT

cΛ=0.5k

BT

c

Figure 22: A comparison between the thermal conductivity of the supportingSiN membrane and the metal films on it. The solid line at the bottom is thethermal conductivity of a 200 nm thick SiN membrane. The various graphsat the top of the figure correspond to the thermal conducitivity of the TESfilms with various values of the parameter Λ.

TESs to the operating point), the edge of the CorTES at r1 reaches Tc first,and thus superconductivity begins to fill the disk from the edge. Assumingthat the resistivity is either zero or equals the normal state resistivity of thefilm, the resistance of the device is given by

R =ρN

2πdlnrbri, (38)

where ρN is the normal state resistivity of the TES film, d is the film thickness,ri is the radius of the inner contact, and rb is a time-averaged equilibriumnormal-superconductor boundary.

We note that the thermal conductivity of a corbino disk TES is completelydominated by the heat transport in the metal films. From experimentalresults of Ref. [44], the thermal conductivity of the 250 nm thick siliconnitride membrane at around 100 mK is κM = ATB with A = 14.5 · 10−3

W/(KB+1m) and B = 1.98, whereas the thermal conductivity of the TESfilms at temperatures below Tc is given by

κS(T ) ≈ κNe−Λ/kB(1/T−1/Tc), (39)

39

T

r

Pbias

T0

Tc

r r1 r

Nκ(T)∝ T

Sκ(T)∝ exp(-Λ/kBT)

SiNκ(T)≈14.5E-3T1.98

T1

T

r

Pbias

T0

Tc

rb r1 r0

Nκ(T)∝ T

Sκ(T)∝ exp(-Λ/kBT)

SiNκ(T)≈14.5E-3T1.98

T1

S

rb

r1

N

r0

r2

SiN

Si @ T0

S

rb

r1

N

ri

r0

SiN

Si @ T0

a) b)

rri

Figure 23: The thermal model of the CorTES. a) Different annular regionsof the CorTES depicted in a top view. The innermost circle (radius ri) is thesuperconducting central electrode, rb is the radius of the equilibrium phaseboundary. The superconducting region extends from rb to r1. The membraneis the region between r1 and r0. b) The temperature profile along the radialdirection in the CorTES. The normal state part (N) is treated as isothermal.In the superconducting part (S) the thermal conductivity is assumed to takean exponential dependence on the temperature.

where κN = LTc/ρ is the thermal conductivity of the TES films in nor-mal state as given by the Wiedemann-Franz law with L = 2.45 · 10−8V2/K2

the Lorentz number. The parameter Λ is of the order of the energy gap∆ ' 1.764kBTc of the superconductor. A comparison between the thermalconductivities are shown in Fig. 22. As the film thicknesses are compara-ble (dTES ≈ 100 nm), it is clear from the figure that the heat transport isdominated by the transport through the metal films. It is thus justified tosimplify the thermal model so that the effect of the nitride below the TESfilms can be neglected.

Now, the thermal model is depicted in Fig. 23. At this point we assumethat the annular normal state region (ri < r < rb) is isothermal with TN = Tc.

40

0 1 2 3 0

10

20

0.18 G

0.33 G

0.37 G

0.40 G

Cu

rre

nt

[ A

]

Voltage [ V]

Figure 24: The I(V ) curve of the voltage-biased CorTES. Different valuesof external magnetic field were applied, and the low critical current of thewiring is evident. The I(V ) curve of the TES itself is not affected, as theexternal field is screened by the ground plane.

Equating the input bias power with the radial heat flux we obtain

Q

2πd

∫ r1

rb

1

rdr = −

∫ T1

Tc

κS(T ) dT , S − region, (40)

Q

2πd

∫ r0

r1

1

rdr = −

∫ Tc

T0

κM(T ) dT ,membrane (41)

where Q = V 2/R = V 22πd/ρN ln(rb/ri) is the dissipated bias power andT1 is the film temperature at the outer edge of the CorTES. The radius ofthe membrane r0 is obtained by equating the circumference of a circularmembrane with the circumference of a square one, 2πr0 = 4w, where w isthe pitch of the square membrane. Eq. (41) gives a solution for T1

T1 =

[

V 2 ln r0

r1(B + 1)

ρN ln rb

riA + TB+1

0

]1

B+1

. (42)

Inserting this to Eq. (40), carrying out the integration and then solving nu-merically for rb, a solution for the I(V ) curve is obtained. The measuredI(V ) curve at T0 ≈ 10 mK is compared with the a fit in Fig. 24. Fittingparameter is the parameter Λ, i.e. the thermal conductance of the supercon-ducting film. Best fit yields Λ = 1.25kBTc, a reasonable value as typicallykBTc < Λ < 2kBTc [27]. The simplified model assumes zero width for the

41

0 100 200 300 400 50020

40

60

80

100

120

140

160

Radial coordinate [µm]

Tem

pera

ture

[mK

]

0 1 2 35

10

15

20

25

30

Voltage [µV]

Cur

rent

[µA

]

50 100 150

126

128

130

132

Figure 25: The temperature profile at various bias voltages, marked to theinset showing the measured I(V ) curve. The profile in the superconductingregion is shown expanded at top right.

transition which produces the sharp corner at V = 0.8 µV. The calculatedtemperature profile for a CorTES with ri=15 µm, r1=150 µm, r0 = 477 µm,Tc = 132 mK, T0 = 20 mK and Λ = 1.25 is shown in Fig. 25.

5 Excess noise in transition-edge micro-

calorimeters

Many groups working on TESs have reported that the observed noise can notbe accounted for by the combination of thermal fluctuation noise between theheat sink and the detector, Johnson noise and SQUID noise [45, 46, 47, 48,49]. Some resolution degradation can be accounted for by taking in accountposition-dependent energy losses of the incident photons which broadens thewidth of the spectral lines [50]. To add to the significance of the problem, firstreports concerning excess noise in the transition region are already from 1969[51]. To date, no conclusive explanation has come forth to explain this effect.In some cases, internal energy fluctuations across the thermal resistance ofthe TES can generate additional thermal fluctuation noise, which can bemodeled accurately [17, 18]. At this point it is interesting to point out thatthe very observation of this type of internal thermal fluctuation noise impliesthat there is a substantial variation in Tc or a temperature gradient presentinside a TES. This can be justified by a simple argument by taking two

42

resistors with resistances R1 and R2, with heat capacities C1 and C2, andassuming that energy fluctuations of δE take place in between them. NowR1 and R2 can be identified with two regions of a TES, say the two halvesof the device. We assume that the transition characteristics are identical,i.e. R1(T ) = R2(T ). Since δT1 = δE/C1 = −δT2, and thus δR1 = −δR2 thechange in the total resistance vanishes. Observation of ITFN thus requiresthat either R1 or R2 is zero (below Tc), or that α is vastly different betweenR1 and R2, i.e. that there is significant variation in Tc or a temperaturegradient, compared to the intrinsic transition width, across the device.

In all the cases where the performance is degraded by the increased noisearising from the detector, the excess noise current increases as the resistanceof the TES is decreased. This is highly undesirable, as the detectors arepreferably biased close to the bottom of the transition in order to keep theresponse time of the TES short while maximizing the temperature changethat results to detector saturation. If operated close to the normal state, theloop gain is smaller and the corresponding effective time constant is longer.

It is well known that superconductivity can exist in a metal even above Tc

[52, 53]. It has also been shown with the microscopic BCS theory that thesefluctuations generate current noise when an applied electric field is present[54]. Our approach is based on the Ginzburg-Landau (GL) theory, valid closeto Tc, and with some simplifications the calculation of fluctuation effects iseasy.

In the phenomenological GL theory, the superconducting phase is de-scribed by the complex order parameter ψ(r), and |ψ(r)|2 is attributed tothe local density of superconducting electrons, ns. In the following calcula-tions we assume that ψ(r) varies slowly across the film and thus ψ(r) = ψ.The GL free energy F is minimized at and above Tc when |ψ|2 = ns = 0.However, at and above Tc thermal fluctuations in the order parameter δψtake place and their probability is proportional to e−δF/kBT with δF ∼ kBTc

the fluctuation in the free energy of the condensate. Thus, close to Tc thesefluctuations are quite common in small volumes since δF ∝ Ω, where Ωis the volume associated with the fluctuations [55]. The coherence lengthdetermines the spatial extent of the fluctuations, given in the dirty limit by

ξ(T ) =0.855

√

ξ0l√

1 − Tc

T

(43)

where ξ0 is the BCS coherence length

ξ0 =0.18~vF

kBTc, (44)

43

10−4

10−1

100

101

Radial co−ordinate r [m]

ξ[T

(r)]

/r

Figure 26: The normalized coherence length ξ(T )/r plotted against the radialcoordinate r at various different bias settings. At small bias (V . 1 µV) thefluctuations are fully correlated with ξ(T )/r > 1. The divergence correspondsto the location of the equilibrium phase boundary at each bias setting.

and vF = 2·106 m/s is the Fermi velocity for Ti. Considering first the CorTEScase (paper VI), we associate a fluctuation energy δF ∼ kBTc with a fluctuat-ing volume δΩ within which the order parameter fluctuates between zero andδψ, and calculate the corresponding displacement of the phase boundary δr.Due to the radial temperature gradient in the CorTES, the coherence lengthξ diverges at the boundary, and the fluctuations are correlated. Figure 26shows the coherence length plotted against the radial coordinate using thetemperature distribution obtained from the fit to the I(V ) curve. Now

δF ' −〈α2GL〉

2βGL

δΩ =α2

0〈T/Tc − 1〉22βGL

δΩ, (45)

where α0 = 1.36~/(4meξ0l), me is the electron mass and βGL =0.108α2

0/[N(0)k2BT

2c ]. Next, we approximate 〈T/Tc − 1〉 ' kδr where

k = T−1c dT/dr|r=rb

, and set the fluctuation in volume to δΩ = 2πdrbδr,and solve for the displacement

δr =

[

0.108

πdk2rbN(0)kBTc

]1/3

. (46)

Deriving T (r) in the normal region from the diffusion equation yields aresult for the gradient at the boundary

k = − V 2

T 2c Lrb ln(rb/ri)

(47)

44

The spectral density of the phase boundary fluctuation is obtained bynoting that the lifetime is given by (see, e.g. [55])

τGL =π~

8kB(T − Tc)

≈ π~

8kBTckδr(48)

and assuming that the fluctuation spectrum is white upto τ−1GL, the equivalent

noise bandwidth is∫

∞

0(1 + ω2τ 2

GL)−1dω = π/(2τGL). The noise spectraldensity of the resistance fluctuations is

δRSD =ρNδr

2πrbd

√

2τGL

π= 0.12ρN

[

~3

π7N(0)k4BT

4c r

7bd

7k5

]1/6

, (49)

and the resulting current noise is given by

δIFSN =dI

dRδRSD =

I

RδRSD

1 + β

2(1 + βL0)

√

1 + ω2τ 20

√

1 + ω2τ 2eff

. (50)

A comparison between the model and the measured rms noise is shown inFig. 27. The measurement was carried out at a bath temperature of 20 mKand the rms noise was measured in a frequency band from 100 Hz to 20 kHzas a function of bias voltage. The modeled noise includes contributions fromthe thermal fluctuation noise (TFN), the Johnson noise (JN), SQUID noise(SQN), internal thermal fluctuation noise (ITFN), and the fluctuation super-conductivity noise (FSN). We used the Lorentz number as a fitting parameter,and a good agreement was obtained with 0.1L. The physical explanation ofthis reduced Lorentz number is likely the fact that the Wiedemann-Franzlaw is not obeyed when the film is near Tc, or alternatively simply due to thecrude definition of our model. Also the noise spectra at various bias pointswere measured. Figure 28 shows the measured spectra together with fits withthe model, and its components.

The discussion above assumed that the fluctuations correlate within aannular volume 2πrbdδr. It is possible that the fluctuations correlate withinthe entire disk [56]. In this case the fluctuations can be calculated by firstnoting that

δ2F

δψ2(δψ2) ≈ α0Ωkξeff(δψ2) ≈ kBTc, (51)

where ξeff = ξ0/(kξ0)1/3 is an effective coherence length in the presence of

the temperature gradient and Ω is the volume of the entire TES. The dis-placement of the phase boundary can be calculated by noting that 〈|δψ|2〉 =

45

1000

10

100

Measured

TFN+JN+SQN

TFN+JN+SQN+ITFN

TFN+JN+SQN+ITFN+FSN

TFN+JN+SQN+ITFN+FSN

(zero- dimensional)

rms c

urr

ent nois

e [pA

/Hz ½

]

Bias voltage [nV]

Figure 27: The measured rms current noise between 100 Hz and 20 kHz,marked by circles. The modeled noise, including the fluctuation supercon-ductivity noise is marked by the solid line in the partly correlated case, andby the dotted line in the zero- dimensional case. The irregularities in themodeled curve are due to the fact that measured values for the voltage wereused in the calculation of the noise. The dashed and dash-dotted lines rep-resents the modeled data in absence of FSN, and shows the effect of internalthermal fluctuation noise (ITFN), which peaks at the voltage where α is atmaximum.

46

0.01 0.1 1 10 100 1000 100001000001

10

100c)

Frequency [Hz]

1

10

100b)

Cur

rent

nois

esp

ectra

lden

sity

[A/H

z½]

1

10

100

ITFN

FSN

a)

TFNSQN

JN

Total modelednoise

Figure 28: The noise spectra of the CorTES, measured at bias voltages of a)0.75 µV, b) 0.62 µV, and c) 0.5 µV. The different contributions of thermalfluctuation noise (TFN), Johnson noise (JN), SQUID noise (SQN), internalthermal fluctuation noise (ITFN), and fluctuation superconductivity noise(FSN) are identified in the top graph.

47

|dψ/dx|2〈|δx0|2〉 ∼ (ψ0/ξeff)2〈|δx0|2〉. Using this result and Eq. (51), thefluctuation in the boundary radius is

δr ≈√

0.108

N(0)kBTcπr21dk

2. (52)

In this zero-dimensional case, the resistance fluctuations give

δRSD0 = 0.1ρN

[

~2

N(0)k3BT

3c π

3r21d

5k4

]1/4

. (53)

We note that the zero-dimensional case does not differ substantially from thepartly correlated case. The zero- dimensional result is shown also in Fig. 27,and a good fit is in this case obtained with 0.5L.

Next, we shall discuss shortly the possibility to optimize this noise inTES detectors. When approaching perfect voltage bias (β → 0), the currentnoise becomes δI = IδRSD/R. Taking the terms in highest powers out of thesquare brackets in Eq. (49), noting that in Eq. (47) ln(rb/ri) = 2πRd/ρN

and approximating rb ≈ ri(1 + 2πRd/ρN) the noise can be written as

δIFSN(0) =0.24ITcL

V 2(1 + L0)

[

~3T 2

c k

πN(0)k4Bdri(1 + 2πRdρ−1

N )

]1/6

≡ 0.24ITcL

V 2(1 + L0)Γ.

(54)Now the term in the square brackets denoted by Γ can be considered aconstant as it depends weakly on the parameters due to the 6th root. Re-markable is that very little is left to be optimized! In terms of NEP, the FSNcontribution is given by

NEPFSN =δIFSN

SI=

0.24LTc

RL0Γ√

1 + ω2τ 20 =

0.24LT 2cG

V 2α

√

1 + ω2τ 20 , (55)

i.e. that at low bias voltages this noise term will dominate over the othernoise contributions. The value of Γ is between 1.0·10−8 K/

√Hz and 1.4·10−8

K/√

Hz over the operating range in our CorTES. If the FSN term dominatesover the other noise sources, the energy resolution is [57]

∆EFWHM = 2.36

(∫

∞

0

4df

NEP2FSN

)−1/2

≈ 0.4LT 2

cG

V 2αΓ√τ0 (56)

This implies that the NEP contribution of FSN in transition-edge sensorscould be decreased if the normal state resistance is increased. However,one should be careful with this since it also increases the amplitude of the

48

internal thermal fluctuation noise, and is a potential problem in terms ofnoise matching to the SQUID.

To address the situation in a square TES sensor, similar treatment can beapplied with simplifying assumptions, mainly that the current distribution ishomogeneous and that we can treat the problem as one dimensional. We nowassume that the dissipative region in a square TES is a rectangular regionacross the device with length ln = Rwd/ρN where w is the width of thefilm. The solution for the temperature gradient in the normal region withboundary conditions T (±ln/2) = Tc and T ′(0) = 0 is

T (x) =1

2

(

V 2

L+ 4Tc −

4V 2x

Lln

)1/2

, (57)

from which the equivalent of Eq. (47) is k = −V 2ρN/(RwdLT2c ) =

−PbρN/(2wdLT2c ) where we have used notation ’’ to make a distinction

to the square geometry. Now

δIFSN, = 0.72ITcL

V 2(1 + L0)Γ, (58)

where Γ = [~3kT2c /(N(0)k4

Bwd)]1/6. Using typical parameters we obtain

Γ ≈ 1·10−8 K/√

Hz, and the expression for NEP is as in Eq. (55). Althoughthe treatment is somewhat oversimplified, Eq. (58) might be useful whenpredicting the excess noise in square TESs.

The fluctuation superconductivity noise contribution is larger in theCorTES geometry due to the fact that k increases faster towards the centreof the disk as the current density increases. A homogeneous current distribu-tion thus seems a favourable choice. In addition, the CorTES is particularlyvulnerable to FSN as the fluctuations are completely correlated. There re-mains a weak dependence on the temperature gradient, k1/6, which mightexplain the fact that some groups see decreased excess noise when the TESis equipped with thick normal metal bars on the edges, or a thick Cu ab-sorber in the centre, which effectively flatten out the temperature gradientacross the device. Moreover, the normal metal acts as a constant parallelresistance with the TES film, which reduces the amplitude of the resistancefluctuations. This naturally decreases also the responsivity of the TES, butmight allow the optimisation of the fluctuation superconductivity noise. An-other possibility to reduce the excess noise is by simply dividing a TES to nparallel TESs, since in this case the FSN decreases as n−1/3 since δr ∝ n1/6

and while δRSD ∝ n−1/2. In order to reduce FSN to the level of the thermalfluctuation noise,

n =

(

0.72T 2cGLΓ

V 2αNEPTFN

)3

(59)

49

parallel TESs are required. Using the parameters from our CorTES data,however, we would need to have n ∼ 102 parallel TESs, which renders thisapproach impractical.

It can be concluded that critical fluctuation effects are of fundamentalnature in transition-edge sensors, and will be the limiting noise source whenoperated at very small bias resistances, where these sensors would otherwiseexhibit optimum performance.

50

6 On-chip cooling of microcalorimeters

Cooling of detectors to temperatures below 100 mK is by all measures achallenging task. Especially instruments which have limited room availableand constraints regarding their operational lifetime, such as space-borne in-strumentation, pose a significant technological challenge in terms of theircryogenic systems. The refrigerators in use today involve complicated liq-uid or gas handling systems and large amounts of liquid refrigerants. Themost common cooling methods used well below 1 K are based either on theenthalpy of mixing of 3He and 4He or adiabatic demagnetization of param-agnetic salts, or nuclei in metals.

In this Section we present the results obtained with a solid state coolerwhere the effect is based on the evaporation of hot electrons of a metal. Thedevice consists of a pair of normal metal - insulator - superconductor (NIS)tunnel junctions, where the hot electrons of the normal metal are removed byplacing a voltage across the junctions which results to the tunnelling of thehottest electrons to the superconductor. This device is suitable for on-chipcooling of thermal detectors where the dissipation is of the order of picowatts.

The cooling effect in a NIS tunnel junction is based on the excistence ofan energy gap ∆ = 1.76kBTc in the superconductor. Biasing the normal sideat a voltage V . ∆/e the hottest electrons with an energy E above the Fermilevel EF can tunnel through the tunnel barrier to the superconducting side.As each electron carries out heat E − eV , the metal cools [58, 59, 60, 61].A combination of two sets of junctions leads to a SINIS structure, and thenet cooling power is doubled as the cooling is symmetric: On the oppositejunction, quasiparticles with energy below EF, i.e. ’cool’ quasiparticles tunnelto the normal metal. The net cooling power is obtained by integrating theenergy driven through the junction resistance RT over all the states in theelectrodes

Qc =1

e2RT

∫

∞

−∞

gS(E)[f(E − eV, TN) − f(E, TS)](E − eV ) dE, (60)

where gS(E) = gN(0)Re[E(E2 − ∆2)−1/2] is the density of states in the su-peconductor, and f(E, T ) = 1 + exp[(E − µ)/kBT ]−1 is the Fermi-Diracdistribution at T and µ ∼ EF. Subscripts N and S refer to the normalmetal and the superconductor, respectively. The maximum cooling power isreached when V . ∆/e [59, 60] when

Qcmax ≈ 0.6

√∆

e2RT(kBTN)3/2. (61)

The minimum achievable electron temperature in the normal metal is deter-mined by the heat flow from the electron system of the normal metal to the

51

phonon system, given by

TNmin =

(

T 50 − Qcmax

ΣΩ

)1/5

, (62)

where Σ ∼ ·109 WK−5m−3 is a material dependent constant and Ω is thevolume of the normal metal. A SINIS cooler can also be used to reduce thelattice temperature. This can be achieved by extending the normal electrodeas a cold finger to a thermally isolated platform. Lattice refrigeration isdesirable when cooling thermal detectors, as will be shown in Section 6.3.

6.1 Coolers with large junction area

The experimental results described in paper II were obtained with small areajunctions which were fabricated on nitridized silicon wafers using EBL andthin film shadow mask deposition. The superconductor was a 15 nm thicklayer of aluminium, which was oxidized in pure oxygen atmosphere prior tothe deposition of the 20 nm thick copper normal metal. This masking tech-nique allows fabrication of good quality junctions with a fairly small junctionarea of ∼ 1 µm2 and below. The junction quality sets a lower limit for thejunction barrier thickness: leakage currents degrade the cooling performancesignificantly in aluminium oxide junctions with a specific resistance (resis-tance × junction area) below 0.1 kΩµm2. Thus, in order to decrease RT

one has to fabricate junctions with larger area. With the EBL and shadowmasking technique this is possible only by connecting several small junctionsin parallel. A more desirable route is to increase the area of single junctions,as the parallel junction coolers tend to have excessive volume which increasesthe heat flow to the electron system from the phonons. This strategy waspursued in the work described in the first half of paper IV. In these exper-iments, the double-layer electron resist was replaced by a micromachinedshadow mask, which was fabricated using photolithography, anisotropic wetetching of Si, and reactive ion etching. The use of this type of mask enabledus to increase the junction area significantly, up to 100 (µm)2. However,with the large junction coolers it became evident that the thermalization ofthe superconductors becomes a crucial issue: the large tunnelling currentfrom these junctions leads to an excessive population of quasiparticles nearthe junctions which have a finite probability of back-tunnelling to the normalmetal, decreasing the cooling power drastically. The problem is circumventedby the use of so-called quasiparticle ’traps’ [62]. The traps can be of highconductivity normal metal, placed in the vicinity of the junctions, wherebythey transport the excess heat away from the junction area.

52

a) b)

Figure 29: a) A SEM image of an electron cooler with large junctions. Alsothe Cu quasiparticle ’traps’ are indicated with arrows. The two bars at thebottom of the figure are due to the shadows from the angle evaporation anddo not affect the performance of the cooler. b) A side view of the structureshowing the layer order.

6.2 Results from electron coolers

An electron cooler with 30 (µm2) junctions is shown in Fig. 29 a). Alsoshown in the figure are the normal metal traps in close vicinity of the junc-tions, while a pair of small junctions are used for thermometry. A side viewshowing the layer order is shown in Fig. 29 b). Although having the trapsdisplaced in the plane from the junctions gave satisfactory results, the con-trol of the distance was difficult due to the required accuracy required in theevaporation angle of the traps. A slight misalignment can short the junction,thus removing the cooling effect. The outcome of the investigations on theoptimum junction geometry and trap location indicated clearly that havingthe trap directly underneath the junction should provide the best coolingresults. The next task was to overcome the difficulties of having the thicknormal metal in contact with the Al, i.e. interdiffusion of the metals andproximity effect of the Cu on the Al. As a first trial, we tried depositing athin (< 10 nm) titanium layer in between the Al and the Cu trap as a dif-fusion barrier. In low temperature measurements, no cooling was observed.Since no additional trials were carried out, we are not sure whether the coolerfailure was caused by the Ti or having something else being wrong with thecooler. According to our analysis, a very thin oxide layer could be transpar-ent enough for the quasiparticles, while acting as a diffusion barrier for themetals. Also, the high probability of pin-holes through the thin oxide could

53

250 300 350 400 450 500 550 6000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

T e,m

in/T

bath

Tbath

(mK)

a)b)

50 75 100 250 50010

100

1000

Te,min

[mK]

Coo

ling

Pow

er[p

W]

Figure 30: The best cooling results obtained with the electron coolers. a)The minimum electron temperature vs. the bath temperature, and b) thecooling power of the refrigerator.

54

actually be beneficial for the trapping process. Sandwiched coolers with a 5nm thick AlOx between the superconducting Al electrode and the Cu trapwere fabricated. A SEM micrograph of one device and the obtained coolingresults from two devices are shown in Fig. 30. The trap appears to performvery well to the very lowest temperatures (80 mK), since no heating is seenin the cooling curves. Also, the cooling power, shown in the inset of thecooling curves is ∼20 pW at 80 mK, which is sufficient for most calorimeterapplications.

6.3 Lattice coolers

In optimised thermal detectors (such as bolometers and calorimeters), theperformance is limited by the random exchange of energy between the de-tector and its heat sink. This fluctuation is usually called ’phonon noise’,and can be reduced by increasing the thermal isolation of the detector, or bycooling the device. We have shown in paper III that one must cool both, thedetector and the heat sink in order to improve the sensitivity significantly.This is due to the fact that the phonon noise is in the non-equilibrium caseproportional to (T n+1 + T n+1

0 )1/2, where n is either 4 or 5 depending on thelimiting thermal conductance in the system [63].

Thus, in order to improve detector performance substantially with theSINIS, the actual heat sink of the device needs to be the cooled object. Thenatural way to start working towards this is to cool a thermally isolatedplatform, instead than just the electron gas in a metal. To do this, thenormal metal island can be extended as a cold finger across a thermally wellisolated bridge.

With the existing quality of materials, i.e. with non-superconductingmetal elements in this case, the choice of the geometry of the cold fingerextending between the heat bath and the membrane is a compromise. Thisarises from the competing effects of increased thermal conduction along thecold finger, when the cross sectional area is increased, and from the simulta-neous decrease of thermal isolation due to the increased volume, and this wayincreased electron phonon heat transport in the cold finger on the substrate,outside the membrane. Typically we use copper, electron-gun evaporated inUHV (<10−8 mbar), as the normal metal. RRR measurements of Cu filmshave been performed. The results are collected in Table 4. In thick (200nm) films, an RRR of about 10 can be achieved, resulting in a substantialimprovement in thermal conductance of the cold finger as compared to thatof 10 times thinner films with five times lower resistivity ratios.

We get important limitations for the structure and geometry of possibleSINIS refrigerators already from rather simple considerations. For exam-

55

w = 0.2 µm w = 1 µm w = 10 µmρ(300 K) RRR ρ(300 K) RRR ρ(300 K) RRR

d = 30 nm 2.9 1.9 3.2 2.1 3.3 2.2d = 75 nm 2.2 2.7 2.2 3.4 2.3 3.8d = 300 nm 1.8 5.3 1.8 8.7 1.7 10.3

Table 4: Resistivity ρ at 300 K (in µΩ cm) and RRR for Cu films withdifferent thicknesses d and widths w. The tabulated value for bulk Cu isρ(300K) = 1.67 µΩcm.

ple, the maximum thickness, d, of the normal-metal layer above the tunneljunction is determined by the competition between the cooling power andelectron-phonon coupling. The maximum cooling power of a single junctionwith resistance RT is given by Eq. 61 which is reached when V is slightlybelow ∆/e. Requiring that the electron-phonon heat flow in the junctionarea A, Qe−p = ΣΩ(T 5

p,n − T 5e,n), must be smaller than Qcmax, we get using

the equation for Qe−p and Eq. (61)

d <0.6

√∆

e2Σ

1

RTA

(kBTe,n)3/2

T 5p,n

, (63)

where we have taken into account that Te,n Tp,n in our case. If condition(63) is not valid, all cooling power is consumed by the electron-phonon heatflow in the junction area. Putting ∆ = 200 µeV, Σ = 4·109 WK−5m−3, andRTA = 2·10−10 Ωm2 which is the smallest specific resistance of the junctionwith which we have observed cooling, and requiring that Te,n = 0.1 K whenTp,n = 0.3 K, we get d < 110 nm. In practice d cannot be much above 30nm if we want to get a significant cooling power for a sensor on a dielectricmembrane.

The best lattice cooling result obtained is shown in Fig. 31. Contraryto the results obtained with the electron coolers, we have not been able tocool the lattice with the large junctions to 0.3Tbath. The reason is excessvolume of the normal metal on the bulk, which decreases the net coolingpower. In the case of the cooler in question with Ω ≈ 19 (µm)3, the powertransported from the electron system to the lattice is 22 pW. The maximumcooling power of the junctions with bath at 205 mK and electrons at 151 mKand a tunnel resistance of RT ≈ 8 Ω is from Eq. (61) 47 pW. The remaining∼ 20 pW is possibly lost due to the heating of the superconductor.

The importance of the result shown in Fig. 30 should not be underesti-mated: the fact that the trapping works well at the very lowest temperatures

56

100 150 200 250 300 350 400 450 5000.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05T m

in/T

bath

Tbath

(mK)

Figure 31: The best cooling result obtained with a lattice cooler with largearea junctions. Thermometry of the isolated SiN membrane and the bulk Siwas carried out with a pair of small SINIS junctions. The heating below 200mK is due to the excessive heating of the superconductor due to non-perfectquasiparticle traps.

opens up the possibility for much larger junction areas with cooling power inthe tens of nanowatts range.

57

7 Summary

The focus of this work has been the quest to pursue the sensitivity limitsof thermal radiation detectors. Even though bolometric detection has somedrawbacks, mainly the fact that it requires cooling to obtain competitivesensitivity, these detectors are unrivaled in terms of versatility and spectralrange. These virtues make thermal detectors a rewarding field for an ex-perimentalist such as myself to work in as there are so many applicationswhere these devices can be used. What would the American astronomerSamuel P. Langley say if told that the device he used in 1901 to detect a cowacross a field is nowadays used to detect nanokelvin variations in the Cosmicmicrowave background radiation!

In this thesis novel room temperature antenna-coupled microbolometerswere developed in which a simple, robust surface micromachining techniquewas used to fabricate free standing titanium bridges. Although sufferingfrom excessive 1/f noise, the results are encouraging and with some furthermaterials study could reach NEPs in the 10 pW/

√Hz range at modulation

frequencies below 1 kHz. There are immediate applications for these devices,two of which are the study of the millimetre-wave radiation from solar flareeruptions [64], and imaging of concealed weapons under clothing [65].

The superconducting Nb hot-spot bolometer developed within this the-sis demonstrated a significant performance increase in terms of speed, NEPand dynamic range over existing 4.2 K bolometers. The possibility of con-structing large integrated imaging arrays incorporating SQUID multiplexersopens one route to the realization of sensitive submillimetre-wave camerasfor passive imaging applications in the field of security, medical imaging, andremote sensing. As single pixel devices, these detectors could benefit fast-scan submillimetre fourier transform spectrometry in which fast, broadbanddetectors are needed, and where the detectors used today require cooling tobelow 1 K in order to reach the NEP level we demonstrated at 4.2 K.

Although the transition-edge sensor microbolometers and micro-calorimeters have been at the focus of a lot of attention in the recent years,we have shown that the picture hasn’t yet been complete. The introduc-tion of an additional source of noise has deepened our understanding of thetransition-edge sensor, and this knowledge can be used in optimizing novelsensors. We feel that this study is important as a lot of effort is put tothe development of hot-electron TES bolometers for the submillimetre-waverange [66, 67, 68]. Yet, the fluctuation superconductivity noise might nothave as severe impact on these devices as the bias power they are operatedat is much smaller than in the case of X-ray microcalorimeters.

In the end, the factor that will determine the prevalence of cryogenic de-

58

tectors is the efficiency, size, reliability and production cost of the coolingsystems available. The ultimate goal is to have temperatures well below 1 Kcompletely transparent to the end-user. One technology with such potentialis the SINIS tunnel junction cooler, on which the the final part of this thesiswas focused on. We showed that it is possible to increase the cooling power ofSINIS coolers to the level that they are applicable for typical microcalorime-ter dissipation levels. As the overheating of the superconducting electrodecan be avoided with quasiparticle trapping, a natural continuation of thiswork would be the further increase of the junction size in conjunction withthe use of novel substrate materials, such as porous silicon where the latticeheat transport is strongly suppressed. The physics have shown that the cool-ing method can be scaled to cooling stages, and the problems remaining aremostly technological.

59

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[53] R. Glover, “Ideal resistive transition of a superconductor,” Phys. Lett.A, vol. 25, pp. 542–544, Oct 1967.

[54] K. Nagaev, “Theory of excess noise in superconductors above Tc,” Phys-ica C, vol. 184, pp. 149–158, 1991.

[55] M. Tinkham, Introduction to superconductivity, ch. 8, pp. 296–315. Sin-gapore: McGraw-Hill, 1996.

[56] N. Kopnin. Private communication, 2002.

[57] D. M. S.H. Moseley, J.C. Mather, “Thermal detectors as x-ray spec-trometers,” J. Appl. Phys., vol. 56, pp. 1257–1262, September 1984.

[58] M. Nahum, T. M. Eiles, and J. M. Martinis, “Electronic microrefrigera-tor based on a normal-insulator-superconductor tunnel junction,” Appl.Phys. Lett, vol. 65, pp. 3123–3125, December 1994.

[59] M. M. Leivo, J. P. Pekola, and D. V. Averin, “Efficient peltier refrig-eration by a pair of normal metal/insulator/superconductor junctions,”Appl. Phys. Lett, vol. 68, pp. 1996–1998, April 1996.

[60] M. Leivo, A. Manninen, and J. Pekola, “Microrefrigeration by normal-metal/insulator/superconductor tunnel junctions,” Appl. Superc., vol. 5,pp. 227–233, 1998.

[61] P. A. Fisher, J. N. Ullom, and M. Nahum, “High-power on-chip mi-crorefrigerator based on a normal-metal/insulator/superconductor tun-nel junction,” Applied Physics Letters, vol. 74, pp. 2705–2707, May 1999.

65

[62] J. P. Pekola, D. V. Anghel, T. I. Suppula, J. K. Suoknuuti, A. J. Manni-nen, and M. Manninen, “Trapping of quasiparticles of a nonequilibriumsuperconductor,” Appl. Phys. Lett., vol. 76, pp. 2782–2784, May 2000.

[63] S. R. Golwala, J. Jochum, and B. Sadoulet, “Noise considerations in lowresistance NIS junctions,” in Proc. 7th Int. Workshop on Low Tempera-ture Detectors LTD-7 (S. Cooper, ed.), (Munich, Germany), pp. 64–65,MPI Physik, July 1997.

[64] V. Khaikin, S. Yakovlev, A. Kazarinov, I. E. anf A. Volkov, andE. Valtaoja, “Multi-beam solar radio telescope of tuorla observatory:test operations.,” in URSI/IEEE XXVII Convention on Radio Science(S. Tretyakov and J. Saily, eds.), (Espoo, Finland), pp. 84–87, HelsinkiUniversity of Technology, 2002.

[65] E. Grossman, A. Bhupathiraju, A. Miller, and C. Reintsema, “Concealedweapons detection using an uncooled millimeter-wave microbolome-ter system,” in Infrared and passive millimeter-wave imaging systems:Design, analysis, modeling and testing (R. Appleby, G. Holst, andD. Wikner, eds.), vol. 4719, (Bellingham, Washington 98227-0010 USA),pp. 364–369, SPIE, SPIE, 2002.

[66] A. Sergeev, V. Mitin, and B. Karasik, “Ultrasensitive hot-electronkinetic-inductance detectors operating well below the superconductingtransition,” Appl. Phys. Lett., vol. 80, pp. 817–819, Feb 2002.

[67] A. Goldin, J. Bock, C. Hunt, A. Lange, H. LeDuc, A. Vayonakis,and J. Zmuidzinas, “Samba: Superconducting antenna-coupled multi-frequency, bolometric array,” in Low Temperature Detectors 9 (LTD-9) (F. S. Porter, D. McCammon, M. Galeazzi, and C. Stahle, eds.),vol. 605 of AIP Conference Proceedings, (Melville, New York), pp. 251–254, American Institute of Physics, 2002.

[68] M. J. Myers, A. Lee, P. Richards, D. Schwan, J. Skidmore, A. Smith,H. Spieler, and J. Yoon, “Antenna-coupled arrays of voltage-biasedsuperconducting bolometers,” in Low Temperature Detectors 9 (LTD-9) (F. S. Porter, D. McCammon, M. Galeazzi, and C. Stahle, eds.),vol. 605 of AIP Conference Proceedings, (Melville, New York), pp. 247–250, American Institute of Physics, 202.

66

Abstracts of publications I–VI

I. In an European Space Agency funded research project, our goal is todevelop microbolometer technology for x-ray and far-infrared detectionfor ESA’s future scientific missions. We report results on the x-raycalorimeter, which is based on the superconducting transition of theTi/Au thermometer strip at about 200 mK. Incident x-rays heat up aBi absorber, deposited on top of the 400 µm × 400 µm thermometer.The temperature rise of the absorber is measured as a change of thethermometer current with a SQUID operating at 1 K.

II. Evaporation of hot electrons from a normal-metal into a superconductorcan be used for efficient Peltier type cooling in micrometer size tunneljunctions. We have cooled the electrons to one third, and, as the mainresult of the present paper, a separate silicon nitride membrane to aboutone half of its starting temperature; all results have been obtainedat temperatures well below 1 K, where the lattice is weakly coupledto electrons thermally. The micromachined membrane can serve as athermal bath for tiny samples, like bolometric radiation detectors inastronomy.

III. Astronomical observations of cosmic sources in the far-infrared and X-ray bands require extreme sensitivity. The most sensitive detectorsare cryogenic bolometers and calorimeters operating typically at about100 mK. The last stage of cooling (from 300 mK to 100 mK) oftenposes significant difficulties in space-borne experiments, both in systemcomplexity and reliability. We address the possibility of using refrig-eration based on normal metal/insulator/superconductor (NIS) tunneljunctions as the last stage cooler for cryogenic thermal detectors. Wecompare two possible schemes: the direct cooling of the electron gasof the detector with the aid of NIS tunnel junctions and the indirectcooling method, when the detector lattice is cooled by the refrigeratingsystem, while the electron gas temperature is decreased by electron-phonon interaction. The latter method is found to allow at least anorder of magnitude improvement in detector noise equivalent power,when compared to the direct electron cooling.

IV. In this paper we report recent results obtained with large areasuperconductor-insulator-normal metal-insulator-superconductor tun-nel junction coolers. With the devices we have successfully demon-strated electronic cooling from 260 mK to 80 mK with a cooling powerof 20 pW at 80 mK. At present, we are focusing on obtaining similar

67

performance in cooling cryogenic detectors. Additionally, we present re-cent results of successful operation of a metal-semiconductor structurewith a Schottky barrier acting as the tunnel barrier and the possibilityto use this kind of structures for on-chip cooling.

V. We report the electrical properties of an antenna-coupled niobiumvacuum-bridge bolometer, operated at a temperature of 4.2 K, in whichthe thermal isolation is maximized by the vacuum gap between thebridge and the underlying silicon substrate. The device is voltage-biased, which results in a formation of a normal state region in themiddle of the bridge. The device shows a current responsivity of -1430 A/W and an amplifier limited electrical noise equivalent power of1.4×10−14 W/

√Hz.

VI. In order to investigate the origin of the until now unaccounted ex-cess noise and to minimize the uncontrollable phenomena at the tran-sition in X-ray microcalorimeters we have developed superconductingtransition-edge sensors into an edgeless geometry, the so-called Corbinodisk (CorTES), with superconducting contacts in the centre and at theouter perimeter. The measured rms current noise and its spectral den-sity can be modeled as resistance noise resulting from fluctuations nearthe equilibrium superconductor-normal metal boundary.

68

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Journal of Low Temperature Physics, Vol. 120, Nos. 34, 2000

On-Chip Refrigeration by Evaporation ofHot Electrons at Sub-Kelvin Temperatures

A. Luukanen,1, 2, * M. M. Leivo,1 J. K. Suoknuuti,1 A. J. Manninen,1

and J. P. Pekola1

1 Department of Physics, University of Jyva skyla , P.O. Box 35, FIN-40351 Jyva skyla , Finland2 Metorex International Oy, Nihtisillankuja 5, P.O. Box 85, FIN-02631 Espoo, Finland

(Received January 18, 2000; accepted March 23, 2000)

Evaporation of hot electrons from a normal-metal into a superconductor canbe used for efficient Peltier type cooling in micrometer size tunnel junctions.We have cooled the electrons to one third,

1 and, as the main result of thepresent paper, a separate silicon nitride membrane to about one half of itsstarting temperature; all results have been obtained at temperatures wellbelow 1 K, where the lattice is weakly coupled to electrons thermally. Themicromachined membrane can serve as a thermal bath for tiny samples, likebolometric radiation detectors in astronomy.

Cryogenic refrigeration methods2, 3 developed over the past centurymake use of thermodynamic properties of liquids and gases, or of magnetismin solids. The common feature in all these techniques is that they involvecomplicated instrumentation and massive refrigerants. The most illustrativeand familiar method is the evaporative cooling of a liquid, usually 4Heor 3He. More advanced cooling methods rely on either enthalpy in mixingthe two different isotopes of helium liquid, or on adiabatic demagnetisationof either paramagnetic salts (electronic spins) or nuclei in metals. Here wedescribe recent results on a solid-state microcooler, variants of which havebeen suggested in the literature over the past few years.4, 5 We havedeveloped this method to allow construction of a prototype refrigerator.The device is fabricated by a combination of nanolithography and micro-machining. The principal idea is the evaporation of hot electrons from ametal to be cooled; this way we have already earlier achieved a reductionof electron temperature from 0.3 K down to 0.1 K,1 and now, by advancingthis method, we have refrigerated the lattice from 0.2 K to 0.1 K on amicromachined platform of silicon nitride.

* E-mail: arttu.luukanenmetorex.fi.

281

0022-2291000800-0281818.000 2000 Plenum Publishing Corporation

In an ordinary Peltier solid-state cooler near room temperature(thermoelectric cooler, TEC) two unequal conducting materials make acontact. Electric current is then driven through, whereby one side of thecontact cools down. Cooling powers of 10 W up to 100 W can be achievedwith a suitable choice of semiconductors. Single-stage coolers can reachtemperature differences up to 70 degrees starting from room temperature,and cascade structures about 130 degrees (centigrade) based on manufac-turers' specifications.

In a NIS- or SINIS-based refrigerator the cooling effect arises due tothe forbidden energy states within the superconducting energy gap 2

(N=normal-metal, I=insulator, S=superconductor). When bias voltageV2e is applied across the NIS junction, only electrons with energy Elarger than the Fermi energy, EF , can tunnel through the insulating barrier.Each tunneling electron carries out heat E&eV, and the net heat transferrate from the normal metal electrode through the NIS junction with a nor-mal-state resistance of RT can be derived by integrating the net energydriven through the junction over all states in the electrodes,

Q4 NIS=1

e2RT|

&gS(E )[[1& f (E, TS)] f (E&eV, Te)

& f (E, TS)[1& f (E&eV, Te)]](E&eV ) dE, (1)

where gS(E )rgN(0) |E |- E2&22 is the density of states in the super-conductor, and f (E, T )=(1+e(E&+)kBT )&1 is the FermiDirac distribu-tion at T, with +&EF . The temperature of the superconductor and normalmetal are denoted by TS and Te , respectively.

The maximum cooling power is obtained when V2e, and is givenby Refs. 1 and 6

Q4 maxr0.6212

e2RT(kBTe)

32, (2)

where Te is the electronic temperature of the normal-metal electrode. Equa-tion (2) is valid when Te<<TC , with TC the critical temperature of thesuperconductor. The cooling effect is symmetric in V,1 which allows us toconnect two NIS junctions in series as a SINIS structure which is thebuilding block of all our coolers. In this structure, the cooling power isdoubled, since heat is transferred out from the normal-metal island throughboth junctions even though the electric current is driven in one direction(see Fig. 1 for operational principle).

282 A. Luukanen et al.

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Fig. 1. Operational principle of a symmetric SINISstructure. Energy level diagrams of superconductorsand the normal-metal island are shown. When biasedat voltage V22e, the electrons above the Fermilevel, EF , are allowed to tunnel horizontally into theempty states of the left superconducting electrode andthe quasiparticles of the right superconducting elec-trode can tunnel into the empty states of the normal-metal below EF . The average temperature of theN-island tends to decrease due to the resulting coolingpower 2Q4 max . Tunnelling from above the gap at theright junction and to below the gap at the left junctionare taken in account in calculating the total coolingpower. The effect of these processes is negligible at lowtemperatures.

At temperatures T1 K the electron system of a normal-metaldecouples from the lattice thermally. This enables us to lower Te below thelattice temperature, T0 , if the volume of the normal-metal is sufficientlysmall and the cooling power is high enough. Heat flow from the lattice tothe cooled electrons,7, 8

Q4 0e=70(T 50&T 5

e), (3)

balances Q4 max and determines the minimum achievable electron temperature.Here, 7 is a material-dependent constant (of the order of 109 WK5 m3 formost metals) and 0 is the volume of the normal-metal electrode.

Microrefrigerators are fabricated using electron beam lithographyfollowed by thin film shadow mask deposition.9 Aluminium (t15 nm thick)is used as a superconductor and copper (t20 nm thick) as a normal-metal,and the tunnelling barriers are formed by oxidising the aluminium electrodein pure oxygen atmosphere prior to copper deposition. Silicon wafers

283On-Chip Refrigeration by Evaporation of Hot Electrons

(nitridised in the Microfabrication laboratory at UC Berkeley, USA) areused as substrates.

In our first experiment on electron cooling1 we applied the simplestconstruction with two identical NIS junctions in the symmetric SINISstructure (sample A in Fig. 2b). Though the cooling power was small, sub-stantial decrease in the electronic temperature of the small normal-metalisland, TminT0 &0.35, was achieved at T0&0.3 K. Here, Tmin is the elec-tronic temperature obtained at optimal bias voltage. Sample B shown inFig. 2a has ten SINIS coolers in parallel attached to the common normal-metal island. In sample B we obtained Tmin T0&0.45 at T0 &0.17 K. Thecooling power is, however, much larger than in sample A, since the volumeof the normal-metal island to be cooled in B is almost eight times larger.Thermometry was based on the temperature dependence of the current-voltage characteristics of two additional NIS junctions attached to thenormal-metal island.10 If a NIS junction is biased with a constant current, I,such that the voltage V across the junction is slightly below 2 (22 forSINIS junctions), the voltage depends strongly on the electronic tempera-ture Te in the normal-metal electrode. When kBTe<<2 and 0<<eV<2,the current through a NIS junction is given as

I(V )&I0e(eV&2)kB Te, (4)

where I0=2eRT - ?kBTe 22 is the characteristic current for an NIS junc-tion. The temperature sensitivity is

dVdTe

&kB

eln

II0

. (5)

Since the cooling power is inversely proportional to the normal-stateresistance of the junction (see Eq. (1)), one may conclude that the thinneris the barrier, the larger is the cooling power. Yet, one cannot increase thecooling power indefinitely this way. The practical limit for the specificresistance (resistance_area) of an Al-I-Cu junction on making a thinbarrier is of the order of 0.1 k0 +m2. Leakage currents degrade the coolingperformance especially at low temperatures due to ohmic losses. Also,highly transparent junctions inject a lot of power, which tends to heat upthe superconductor at the junction. Therefore, our samples are fabricatedin an ultra high vacuum (UHV) evaporation system, which ensures goodquality of the tunnel barriers and the electrodes.

In order to attain considerable cooling, we have to maximise the ratioof the total junction area to the size of the normal-metal electrode. Thisrequirement, combined with that of large cooling power, can be fulfilled

284 A. Luukanen et al.

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Fig. 2. Electronic SINIS cooler. (a) An electron micrograph of part of asample with 10+10 junctions for cooling and one additional SINIS struc-ture for thermometry. The electrical connections of the cooler and the ther-mometer are shown. Blue regions are made of aluminium (S), whereas thered color illustrates thin films made of copper (N). The 0.4 +m wide verti-cal line between the comb-like structures is the normal-metal island, whichis cooled down. (b) Performance of the electronic coolers A and B. On theright, typical cooling curves at few bath temperatures with correspondingmaximum cooling powers (at V& \22e) are represented. V is the biasvoltage across the SINIS cooler, and Te is the electronic temperature of theN electrode. On the left, Tmin is the minimum value of Te obtained at|V| &22e&0.4 mV for aluminium. On the left, we show Tmin scaled bythe bath temperature T0 , which is the the value of Te at V=0.

285On-Chip Refrigeration by Evaporation of Hot Electrons

either by increasing the number of junctions in parallel or by using largearea junctions. The latter option has been studied by us (unpublished) andalso in Ref. 12. Until now, the performance is inferior, especially at lowtemperatures. However, cooling powers of the order of a few tens of pWhave been observed in junctions of 20_20 +m2 surface area. Latticerefrigeration makes use of a thin dielectric membrane as a thermallyisolated platform (see Fig. 3a). We use low-stress silicon-rich nitride, whichis grown onto the (100)-oriented silicon by low pressure chemical vapordeposition (LPCVD). A rectangular opening through the silicon substrateis achieved by anisotropic etching of silicon, and the self-standing siliconnitride window is formed on the opposite side of the wafer. In thin mem-branes, typically 0.2 +m thick in our case, phonon propagation in themembrane is restricted into two dimensions. By further processing themembrane by reactive ion etching into a suspended bridge structure,propagation of phonons in the plane is also effectively suppressed bysurface scattering. By cutting the bridges t5 +m wide we are able tothermally isolate the membrane from the heat bath.13 The condensationof the phonon gas into lower dimensions in these membranes has beendiscussed in Refs. 14 and 15.

The normal-metal island of the SINIS cooler is extended from the bulkonto the membrane to serve as a cold finger.16 Figure 3a shows a workingrefrigerator with six 5 +m wide self-standing bridges. Three cold fingers areextended onto the membrane along the bridges. The volume of the normal-metal extension on the membrane is large to ensure effective thermalisa-tion. Temperature of the membrane is measured by a separate SINISstructure residing on the membrane with superconducting leads along twobridges on opposite sides.

As the main result of this work, we could cool the membrane fromabout 0.2 K down to 0.1 K (see Fig. 3b), which is the best refrigeration byNIS junctions reported anywhere to date. Around this operating tempera-ture the thermal time constant of the cooled membrane is of the order of100 ms, as was verified by AC calorimetric measurements.13 The thermalresistance between the bulk silicon (heat bath) and the membrane is of theorder of 0.1 KpW, and the heat capacity of the membrane is about 1 pJK.It should be straightforward to increase the cooling power from the presentpicowatt level to nanowatts by increasing either the number or the size ofthe junctions.

Despite the promising results at low temperatures, we have major con-cerns of applying this technique at higher temperatures. In contrast to theT 32 temperature dependence of the cooling power, thermal shunting ofthe normal-metal electrode to the lattice has much stronger temperaturedependence, B T 5, which disables us to apply the method straightforwardly

286 A. Luukanen et al.

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Fig. 3. SINIS microrefrigerator. (a) Scanning electron microscope image of a silicon nitridemembrane (in the center) with self-suspended bridges. Three normal-metal cold fingers (on theleft) extending onto the membrane conduct heat out from the membrane, but a heat load alongthe bridges compensates the refrigeration effect in the steady state. The SINIS cooler is on thebulk (far left) and the thermometer in the middle of the membrane. (b) Maximum decrease inlattice temperature of the membrane, Tmin T0 , as a function of T0 for two samples C and D. Bothsamples have coolers of the type shown in Fig. 2a; the specific resistance was 1.39 k0 +m2 and0.22 k0 +m2 for C and D, respectively. Typical cooling curves at few bath temperatures arepresented on the right like in Fig. 2b.

287On-Chip Refrigeration by Evaporation of Hot Electrons

above 1 K. Thermal coupling to lattice increases by a factor of 400 withbath temperature rise from 0.3 K to 1 K.

An important issue is the quasiparticle diffusion in a superconductor.``Hot'' quasiparticles relax by inelastic scattering with phonons, by recom-bination to form Cooper pairs with phonon emission,17 and by impurityscattering, which is the dominant relaxation process in such thin films atlow temperature. Heat removed by the superconducting electrodes from thecentral normal-metal can be conducted further away from the junction areaby depositing a separate thin film of normal-metal in contact with thesuperconducting electrodes. These ``quasiparticle traps'' near the junctionhelp the superconductor not to get heated excessively. In recent experi-ments we have obtained significant improvement in the cooling perfor-mance using these traps.18

Immediate applications of the microrefrigeration are in cooling ofultra-sensitive radiation detectors19 in space-borne applications. Super-conducting transition edge detectors20, 21 are typically used to measuresmall signals at low background levels. Since the total power dissipation inthese applications is typically of the order of a few picowatts, the coolingpowers obtained to date are already directly applicable to such devices atthe pixel level. These detectors are operated typically close to 0.1 K inorder to reach photon noise limited performance.

If operated at 0.3 K, a temperature readily achieved by for example a3He sorption refrigerator, the performance is severely degraded. In thefuture, SINIS cooler removes the need for a space based dilutionrefrigerator or an adiabatic demagnetisation refrigerator since the detectorassembly can be integrated to the 0.3 K stage, and the microrefrigerator isused as the final cooling stage from 0.3 to 0.1 K. One way of realising thestructure is illustrated in Fig. 4. The cold finger of the microrefrigerator isextended on the suspended bridge structure and provides the heat bath forthe calorimeter, located at the centre of the membrane. The silicon sub-strate is maintained at 0.3 K by a 3He sorption cooler.

In the discussion above, the cooling of the detector is performedindirectly, i.e., the temperature of the heat bath is lowered. The SINIScooler could also be used directly to cool the electrons in the detector filmitself,22 but recent results show that the indirect method yields better per-formance.23

As the conclusion, we have developed and demonstrated an on-chipmicrocooler based on NIS tunnelling. It can provide a temperaturedecrease by almost a factor of two at 0.2 K in a structure which can serveas a heat bath for, e.g., small radiation detectors for low-backgroundmeasurements. In the future, the cooler will be capable of cooling any smallsamples with a moderate (t1 nW) cooling power from 0.3 K to 0.1 K.

288 A. Luukanen et al.

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Fig. 4. A schematic illustration of a SINIS cooled microcalorimeter. Ifthe detector heat bath temperature is lowered from 0.2 K to 0.1 K by theSINIS cooler (Q4 cooler), energy resolution to the x-rays of cosmic sourcesis improved by a factor of two. The improvement in sensitivity is moreprominent in a far-infrared bolometer, where a factor of almost ten indetector noise equivalent power can be gained. Q4 rad is the power of theincident radiation.

ACKNOWLEDGMENTS

We thank Bernard Collaudin, Roberto Leoni, Dragos Anghel, andKonstantin Arutyunov for discussions. This work was supported by theEuropean Space Agency, the Technology Development Center of Finland,and the Academy of Finland. Correspondence and requests for materialshould be addressed to J.P. (e-mail: Jukka.Pekolaphys.jyu.fi).

REFERENCES

1. M. M. Leivo, J. P. Pekola, and D. V. Averin, Efficient Peltier refrigeration by a pair ofnormal metalinsulatorsuperconductor junctions, Appl. Phys. Lett. 68, 19961998 (1996).

2. O. V. Lounasmaa, Experimental Principles and Methods Below 1 K, Academic Press Ltd.,London (1974).

3. F. Pobell, Matter and Methods at Low Temperatures, Springer-Verlag, Germany (1996).4. M. Nahum, T. M. Eiles, and J. M. Martinis, Electronic microrefrigerator based on a

normal-insulator-superconductor tunnel junction, Appl. Phys. Lett. 65, 31233125 (1994).5. H. L. Edwards, Q. Niu, G. A. Georgakis, and A. L. de Lozanne, Cryogenic cooling using

tunneling structures with sharp energy features, Phys. Rev. B 52, 57145736 (1995). Street,Traverse Supercool AB, P.O. USA.

6. M. M. Leivo, A. J. Manninen, and J. P. Pekola, Microrefrigeration by normal-metalinsulatorsuperconductor tunnel junctions, Applied Superconductivity 5, 227233 (1998).

7. M. L. Roukes, M. R. Freeman, R. S. Germain, and R. C. Richardson, Hot electrons andenergy transport in metals at millikelvin temperatures, Phys. Rev. Lett. 55, 422425(1985).

8. F. C. Wellstood, C. Urbina, and J. Clarke, Hot-electron effects in metals, Phys. Rev. B 49,59425955 (1994).

9. G. J. Dolan, Offset masks for lift-off photoprocessing. Appl. Phys. Lett. 31, 337339(1977).

10. M. Nahum and J. M. Martinis, Ultrasensitive hot-electron microbolometer, Appl. Phys.Lett. 63, 30753077 (1993).

289On-Chip Refrigeration by Evaporation of Hot Electrons

11. R. G. Melton, J. L. Paterson, and S. B. Kaplan, Superconducting tunnel-junctionrefrigerator, Phys. Rev. B 21, 18581867 (1981) of a (1999).

12. P. A. Fisher, J. N. Ullom, and M. Nahum, High-power on-chip microrefrigerator basedon a normal-metalinsulatorsuperconductor tunnel junction, Appl. Phys. Lett. 74,27052707 (1999).

13. M. M. Leivo and J. P. Pekola, Thermal characteristics of silicon nitride membranes atsub-kelvin temperatures, Appl. Phys. Lett. 72, 13051307 (1998).

14. D. V. Anghel, J. P. Pekola, M. M. Leivo, J. K. Suoknuuti, and M. Manninen, Propertiesof the phonon gas in ultrathin membranes at low temperature, Phys. Rev. Lett. 81,29582961 (1998).

15. D. V. Anghel and M. Manninen, Behavior of the phonon gas in restricted geometries atlow temperatures, Phys. Rev. B 59, 98549857 (1999).

16. A. J. Manninen, M. M. Leivo, and J. P. Pekola, Refrigeration of a dielectric membraneby superconductorinsulatornormal-metalinsulatorsuperconductor tunneling, Appl.Phys. Lett. 70, 18851887 (1997).

17. S. B. Kaplan et al., Quasiparticle and phonon lifetimes in superconductors, Phys. Rev. B14, 48544873 (1976).

18. J. P. Pekola, D. V. Anghel, T. I. Suppula, J. K. Suoknuuti, A. J. Manninen, and M. Manninen,Trapping of quasiparticles of a non-equilibrium superconductor, submitted (1999).

19. P. L. Richards, Bolometers for infrared and millimeter waves, J. Appl. Phys. 76, 124(1994).

20. A. T. Lee, P. L. Richards, S. W. Nam, B. Cabrera, and K. D. Irwin, A superconductingbolometer with strong electrothermal feedback, Appl. Phys. Lett. 69, 18011803 (1996).

21. K. D. Irwin, An application of electrothermal feedback for high resolution cryogenicparticle detection, Appl. Phys. Lett 66, 19982000, (1995).

22. L. S. Kuzmin, D. S. Golubev, and I. A. Devyatov, Cold-electron bolometer with electronicmicrorefrigeration and general noise analysis, Proc. SPIE 3465, 193199 (1998).

23. A. Luukanen and J. P. Pekola, On-chip cooling of superconducting transition-edgemicrobolometers and calorimeters, in preparation (2000).

290 A. Luukanen et al.

APPLIED PHYSICS LETTERS VOLUME 78, NUMBER 4 22 JANUARY 2001

Performance of cryogenic microbolometers and calorimeterswith on-chip coolers

D. V. Anghel,a) A. Luukanen,b) and J. P. PekolaDepartment of Physics, University of Jyva¨skyla, P.O. Box 35, FIN-40351 Jyva¨skyla, Finland

~Received 11 May 2000; accepted for publication 9 November 2000!

Astronomical observations of cosmic sources in the far-infrared and x-ray bands require extremesensitivity. The most sensitive detectors are cryogenic bolometers and calorimeters operatingtypically at about 100 mK. The last stage of cooling~from 300 to 100 mK! often poses significantdifficulties in space-borne experiments, both in system complexity and in reliability. We address thepossibility of using refrigeration based on normal metal/insulator/superconductor~NIS! tunneljunctions as the last stage cooler for cryogenic thermal detectors. We compare two possibleschemes: direct cooling of the electron gas of the detector with the aid of NIS tunnel junctions andthe indirect cooling method, when the detector lattice is cooled by the refrigerating system, while theelectron gas temperature is decreased by electron–phonon interaction. The latter method is found toallow at least an order of magnitude improvement in detector noise equivalent power compared todirect electron cooling. ©2001 American Institute of Physics.@DOI: 10.1063/1.1339261#

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A thermal detector system, such as a bolometer ocalorimeter, consists of a thermal sensing element~TSE!which is connected to a heat sink. The TSE typically consof an absorber and a thermometer. The thermometer cantransition-edge sensor1–3 or a normal metal/insulatorsuperconductor~NIS! tunnel junction thermometer.4,5 In thisletter we address the fundamental question of whethershould cool the electrons of the detector directly, i.e., cthe TSE below the heat sink temperature, or cool the hbath of the TSE. The principle of the NIS cooler has beintroduced in Refs. 6 and 7.

The sensitivity of thermal detectors is strongly inflenced by the detector temperature. The figure of meritbolometric detectors is the noise equivalent power~NEP!. To

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calculate this quantity for thermal detectors coupled tochip coolers, we have to evaluate the fluctuations of the etron gas temperature due to power exchange with the latwith the superconductor of the cooler~in the case of directcooling!, and due to the bias of the thermometer~see Fig. 1!.We can then define the NEP as the optical input power~thepower to be detected! needed to produce a change in ttemperature of the electron gas equal to the square root omean square of these fluctuations. In the following calcutions the noise introduced by the bias powerQb is supposedto be very small.

By applying the prescription above and making usethe power balance equation for the TSE, schematicdrawn in Fig. 1, we arrive at the following expression for tNEP:8

NEP25^d2Qep,shot~v!&U11]Qep

]T1

1

ivCV11@]~Qep2QK!/]T1#U2

1^d2QK,shot~v!&S ]Qep

]T1D 2

1

v2CV12 1@]~Qep2QK!/]T1#2

1^d2QJ,shot~v!&11

v2 S ]eF

]N

]QJ

]E D 2

^d2NJ,shot~v!&, ~1!

onms

ol-

om-e of

whereE5eV (V is the voltage across the junction!, whiledQ(J/ep/K),shot(v) and dNJ,shot(v) are thev components ofthe Fourier transformedshot noise fluctuations~finite quan-tities transferred randomly at aconstantaverage rate! of thepower fluxesQ(J/ep/K) and particle fluxNJ ~through the NISjunctions!, respectively. The other notations are explainedFig. 1. We can now observe that in the case of indirect coing, the last two terms in Eq.~1! ~let us call them NEPJ)

a!NIPNE–HH, P.O. Box MG-6, R.O.-76900 Bucures¸ti–Magurele, Romania.b!Metorex International Oy, P.O. Box 85, FIN-02631 Espoo, Finland; el

tronic mail: arttu.luukanen@metorex.fi

nl-

disappear, since there are no NIS refrigerating junctionsthe thermometer. Moreover, if in three-dimensional systewe write Qep5SepV(T1

52Tep5 ),9 QJ5SKS(T2

42T14),8 then

^d2Qep,shot(v)&'5kBSKV(Te61T1

6) ~Ref. 10! ~the error ofthe approximation is within 2% for anyT1.Te) and^d2QK,shot(v)&58kB@z(5)/z(4)#SKS(T1

51T25), wherez(x)

is the Riemann function,V is the volume of the TSE,S is thecontact area between the TSE and the heat bath, whileSep

andSK are coupling constants. In the case of indirect coing, Te'T1'T2 , while for the direct coolingTe,T1,T2 ,the exact values depend on the coupling constants and geetry. Due to the high power in the temperature dependenc

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© 2001 American Institute of PhysicsIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp

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557Appl. Phys. Lett., Vol. 78, No. 4, 22 January 2001 Anghel, Luukanen, and Pekola

the noise terms, a slightly higher temperature of the latticeof the heat bath would have a big effect on the NEP.make this point more clear let us discuss two extreme caIn the first case suppose that the thermal resistance betwthe lattice and the heat bath~Kapitza resistance! is muchsmaller than the one between the electrons and phonu]QK /]T1u@u]Qep /]Teu, while in the second case the inequality is reversed. We define acritical frequency vc

[u]QK /]T1u/CV154tv/w, where t is the transmission coefficient for a phonon between the lattice and the heat ba11

v is the velocity of sound in the heat bath, andw is thethickness of the TSE. For typical devicesvc is of the orderof 1010s21. Therefore we can neglect in general thev de-pendence of the first two terms in Eq.~1!. Under such as-sumptions, incase1 the NEP reduces to

NEP2'^d2Qep,shot~v!&1^d2QK,shot~v!&

3S ]Qep

]T1D 2S ]QK

]T1D 22

1NEPJ2

'^d2Qep,shot~v!&1NEPJ2, ~2!

where the last approximation holds if (T2 /T1)5 is muchsmaller than the ratio between the electron–phonon andKapitza resistance@(T2 /T1)5(SVT1 /SKS)!1#, which iscertainly the case for indirect cooling. In the second casehave the approximation

FIG. 1. Schematic of the thermal sensing element. The two rectangboxes represent two distinct subsystems of this element: the electron syand the lattice. The electron system, the lattice, the heat bath, and thperconductor are at temperaturesTe , T1 , T2 , andTs , respectively. In thecase of direct cooling the superconductor is connected to the electronthrough NIS junctions, while in the case of indirect cooling it is used to cdown the thermal bath. The heat capacities of the electron gas and the lare CVe and CV1 , respectively. The power fluxes transmitted betweensystems are represented by long arrows and are denoted byQJ , Qep , andQK . Qb is the bias power, whileQo is the optical input power to be detected.

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NEP2'43^d2Qep,shot~v!&1NEPJ2. ~3!

Using Eqs.~2! and ~3!, we can write in general the ratiobetween the noise equivalent power in the case of dicooling (NEPd) and in the case of indirect cooling (NEPi):

NEPd

NEPi'ATe

61T16

2Te6 1

NEPJ2

NEPi2 . ~4!

If the working conditions requireTe'0.1 K and if the latticetemperature isT1'0.3 K ~Refs. 7, 12, and 13! in the case ofdirect cooling, we obtain, ignoring NEPJ , a ratio ofNEPd /NEPi'19, thus strongly favoring the indirect coolinmethod.

Unfortunately it is difficult to calculate the performancof NIS junctions as coolers, having the junctioparameters.12,14 This is due to the fact that the quasiparticenergy levels in the superconductor are populated duringcooling process and the superconductor has to be coitself using so callednormal traps for quasiparticles.12,13

In any case, if we suppose that the effective temperaturthe superconductor is an external parameter, at low temptures we can find analytical approximations forQJ . Keepingonly the highest order terms inD/kBTe and D/kBT1 ,whereD is the energy gap in the superconductor, the opmum cooling power isQopt,J'0.6(D2/e2RT)(kBTe /D)3/2

2A2pkBTsD3e2D/kBTs.7,13 Using this equation, supposin

that Ts'T1'T2'0.3 K and that the superconductor is Awith D'200meV, we can calculate the tunnel junction rsistanceRT from the power balance equation. To evaluaQep , which also enters into the power balance equation,assume that the TSE is made of copper (Sep'43109 W K25 m23) and V51 mm3. From these we findRT

'22V. Entering this value into the expression for NEPJ ,8,15

and since]QJ /]E50 at the optimum bias,7 we find NEPJ

51.2310217W/AHz. As a comparison, in the case of direcooling ^d2Qep,shot(v)&direct

1/2 '1.4310217W/AHz.In the case of direct cooling, there is another no

contribution, say NEPJ8 , due to the Johnson noise in the voage across the NIS junctions, that should be quadraticadded to NEPJ

2. If close to the optimum bias voltageVopt wewrite QJ'2g(V2Vopt)

2, then at low temperatureg'(0.33/RT)3(pD/2kBTe)

1/2.8 To evaluate the order omagnitude we write ^d2V(v)&54kBTeRT'1.2310223

V2/Hz and obtain NEPJ8'10223W/AHz. Therefore NEPJ8 ismany orders of magnitude smaller than NEPJ so we neglectit here, but in the future a more rigorous investigation of tvoltage fluctuations in such systems would be desirable.

In this letter we calculated the noise equivalent powermicrobolometers cooled by the direct~the refrigerator is con-nected to the thermal sensing element! and the indirect~therefrigerator cools the heat bath of the TSE! methods, respectively. It turned out that the NEPJ is more than an order omagnitude smaller~in our example it was 19 times smaller ithe situation in which the noise in the cooling power of tNIS junctions was not taken into account! if the TSE iscooled indirectly, a fact that would recommend this methfor applications such as the X-ray Evolving Universe Sptroscopy~XEUS! mission16 currently under study by the European Space Agency. There is also the possibility of cobining both methods in bolometers, which would allo

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558 Appl. Phys. Lett., Vol. 78, No. 4, 22 January 2001 Anghel, Luukanen, and Pekola

significant improvement in the dynamic range of far-infrarbolometers in terms of background saturation.

This work was funded by the European Space Ageunder Contract Nos. 13006/98/NL/PA~SC! and 12835/98/NL/SB and by the Academy of Finland under the FinniCenter of Excellence Program 2000-2005~Project No.44875, Nuclear and Condensed Matter Program at JYFL!.

1K. D. Irwin, G. C. Hilton, D. A. Wollman, and J. M. Martinis, Appl. PhysLett. 69, 1945~1996!.

2A. T. Lee, P. L. Richards, S. W. Nam, B. Cabrera, and K. D. Irwin, ApPhys. Lett.69, 1801~1996!.

3A. Luukanen, H. Sipila¨, K. Kinnunen, A. Nuottaja¨rvi, and J. P. Pekola,Physica B284, 2133~2000!.

4M. Nahum and J. M. Martinis, Appl. Phys. Lett.63, 3075~1993!.5M. Nahum and J. M. Martinis, Appl. Phys. Lett.66, 3203~1995!.6M. Nahum, T. M. Eiles, and J. M. Martinis, Appl. Phys. Lett.65, 3123~1994!.

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7M. M. Leivo, J. P. Pekola, and D. V. Averin, Appl. Phys. Lett.68, 1996~1996!.

8D. V. Anghel and J. P. Pekola, J. Low Temp. Phys.~submitted!.9F. C. Wellstood, C. Urbina, and J. Clarke, Phys. Rev. B49, 5942~1994!.

10S. R. Golwala, J. Jochum, and B. Sadoulet, inProceedings of the 7thInternational Workshop on Low Temperature Detectors LTD-7, edited byS. Cooper, 1997~MPI Physik, Munich, 1997!, pp. 64–65.

11F. Pobell,Matter and Methods at Low Temperatures, 2nd ed.~Springer,Berlin, 1996!.

12J. P. Pekola, D. V. Anghel, T. I. Suppula, J. K. Suoknuuti, A. J. Manninand M. Manninen, Appl. Phys. Lett.76, 2782~2000!.

13D. V. Anghel, Ph.D. thesis, University of Jyva¨skyla, 2000.14P. A. Fisher, J. N. Ullom, and M. Nahum, Appl. Phys. Lett.74, 2705

~1999!.15D. Golubev and L. Kuzmin, inFrom Andreev Reflection to the Interna

tional Space Station, edited by M. Tarasov~Chalmers University of Tech-nology, Gothenburg, Sweden, 1999!.

16M. Bavdaz, J. A. Bleeker, G. Hasinger, H. Inoue, G. G. Palumbo, APeacock, A. N. Parmar, M. J. Turner, J. Truemper, and J. SchiemProc. SPIE3766, 82 ~1999!.

IP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp

Integrated SINIS Refrigerators for Efficient Cooling Of Cryogenic Detectors

A. Luukanen+, A.M. Savin+, T.I. Suppula+ and J.P. Pekola+ M. Prunnila* and J. Ahopelto*

+Department of Physics, University of Jyväskylä, P.O.Box 35, FIN-40351 Jyväskylä, Finland *VTT Electronics, P.O. Box 1101, 02044 VTT, Finland

Abstract. In this paper we report recent results obtained with large area superconductor-insulator-normal metal-insulator-superconductor tunnel junction coolers. With the devices we have successfully demonstrated electronic cooling from 260 mK to 80 mK with a cooling power of 20 pW at 80 mK. At present, we are focusing on obtaining similar performance in cooling cryogenic detectors. Additionally, we present recent results of successful operation of a metal-semiconductor structure with a Schottky barrier acting as the tunnel barrier and the possibility to use this kind of structures for on-chip cooling.

Thermal detectors, such as bolometers and calorimeters, require cooling to low cryogenic temperatures in order to reduce the thermal noise present in the devices at higher temperatures. Typically, the cooling to 100 mK relies on either mixing two different isotopes of liquid helium in a dilution refrigerator, or on adiabatic demagnetization of either paramagnetic salts (electronic spins) or nuclei in metals. Superconductor – Insulator - Normal metal – Insulator - Superconductor (SINIS) tunnel junctions provide an attractive solid state alternative for these ‘classical’ methods, and they are ideal for miniaturized detectors, such as microcalorimeters [1]. For a detailed discussion on the operating principle of SINIS coolers, we refer the reader to references [2,3,4,6,7].

The SINIS coolers described in our earlier work [4] consisted of a large number of small junctions operated in parallel. Small junctions are easier to fabricate, but lack the cooling power required for detector applications. For this reason, we now fabricate the junctions using a micromechanical shadow mask, with which we can produce junctions with an area typically in the range of 30 to 100 (µm)2, and a tunnel resistance below 30 Ω. As the junction area is increased, trapping of the energetic quasiparticles becomes crucial, since excess quasiparticles in the superconductor will transport energy back to the normal metal [5]. The trapping can be efficiently carried out by using a normal metal electrode (quasiparticle ‘trap’) in close proximity to the cooling junction [6].

Figure 1 shows an SEM image of an electron cooler with a junction area of ~30 (µm)2. The Cu traps are located directly underneath the junction, and are separated from the superconducting electrode by a thin (5-7 nm) AlOx layer acting as a diffusion

barrier between Al and Cu. The best cooling results were obtained with the cooler with the thinnest oxide (5 nm) between the junction and the trap. The cooling curves prove that the cooling power can be increased by orders of magnitude, with close to the theoretically predicted performance, as long as the quasiparticles are efficiently trapped.

250 300 350 400 450 500 550 6000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 Cooler #1 Cooler #2

Tm

in/T

bath

Tbath (mK)

50 75 100 250 50010

100

1000

Te [mK]

Coo

ling

Pow

er [p

W]

FIGURE 1. (Left) An electronic SINIS cooler with quasiparticle traps placed underneath the junctions. (Right) Temperature drop and cooling power of two identical SINIS electron coolers with traps under the junctions.

Based on the successful electronic cooling results we are currently working on a

demonstration of cooling an isolated Si3N4 membrane from 0.3 K to 0.1 K. Figure 2 below shows an SEM image of a membrane cooler together with a measurement of the thermal transport in the cooler: this one had electronic thermometers close to the junctions on the bulk Si, and at the membrane end of the cold finger, as well as a phonon thermometer on the membrane. As shown in the figure, the temperatures are virtually identical. This proves that thermal equilibrium exists between the cooling junction and the membrane. The curve also shows that the trap layer does not function as it should, as indicated by the heating below ~400 mK. Our current efforts are concentrated on optimising the oxide thickness between the trap and the junction in this configuration.

200 250 300 350 400 450 500 550 6006500.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04 Electron temperature at cooling junctions Electron temperature on the membrane Lattice temperature on the membrane

Tm

in/T

bat

h

Tbath (mK)

FIGURE 2. (Left) An SEM image of a membrane cooler with a pair of SINIS refrigerators with large [90 (µm)2] junctions. (Right) Temperature measurement at different positions on this cooler.

Apart from the SINIS work discussed above, we suggest a new approach for developing low temperature micro-coolers: to use a heavily doped semiconductor instead of a normal metal in the cooler device. No additional insulating layer is required in this structure because the Schottky barrier in the semiconductor-superconductor (Sm-S) contact forms the tunnel barrier. The utilization of Schottky barrier is very promising because it allows fabrication of homogeneous tunnel junctions with large area. From practical and technological points of view, heavily doped silicon (n++silicon) is the material of choice because of its compatibility with a wide variety of other applications.

“S-Sm-S” structure is very similar to the NIS structure and it may be used both for thermometry and for microcooler applications. The operation of cooler based on S-Sm-S structure is illustrated in Fig. 3. As in SINIS coolers, at low bias voltage (V < ∆/e) the tunneling of electrons (with E > EF) from n++silicon into superconductor and the tunnelling of quasiparticles (with E < EF) from superconductor into n++silicon results in cooling of the electron system in n++silicon. The maximum cooling power for a S-Sm-S (or a SINIS) structure, which is obtained when V is slightly below 2∆/e, is given by [3,7] Pmax≈2⋅0.6∆½(kBTe)

3/2/(e2RT), where RT is the tunnel resistance for a single junction.

P P

eV

EF

2∆2∆

conduction band edge

Schottky barriers

I

S SSm

-0.4 -0.2 0.0 0.2 0.4

-80

-40

0

40

80

940 mK 1340 mK

100 mK 424 mK 631 mK

I / n

A

U / mV

FIGURE 3. (Left) Energy band diagram illustrating cooling in S-Sm-S structure. ∆ is the energy gap in the superconductor, EF - the Fermi energy of the semiconductor and P - heat flow out of the semiconductor. V is the applied voltage and I is the resulting current. The gray areas denote occupied (single particle) electron states. (Right) Current-voltage characteristics of an S-Sm-S thermometer with 6 × 6 (µm)2 Schottky junctions.

In our work we used thin silicon-on-insulator (SOI) film as the semiconductor and aluminum as the superconductor. The thickness of the SOI film (aluminum film) was 70 nm (300 nm) and n-type carrier concentration in n++silicon was 4.5*1019 cm-3. The area of the rectangular SOI film was 20 × 30 (µm)2, the size of the cooling contacts and thermometer contacts was 5 × 18 (µm)2 and 3 × 3 (µm)2 respectively. At 100 mK the sheet resistance of SOI film was 140 Ω/

and it did not noticeably change within

the experimental temperature range. Characteristic resistance of the aluminum- n++silicon junction was 70 kΩ(µm)2, which can be decreased by at least an order of magnitude by increasing the doping level of the Si.

The value of the electron–phonon coupling constant Σ in the cooled metal considerably affects the characteristics of the NIS cooler. The measured value of Σ in

n++silicon (Σ ≈ 1.0 × 108 W/K5m3) is more than one order of magnitude lower than in copper and few times lower than in aluminum [4].

-0.4 -0 .2 0.0 0 .2 0 .4

80

160

240

320

(b )(a )

T

/ m

K

V / m V100 200 300 400

0.3

0 .6

0 .9

Tm

in /

T0

T0 / m K FIGURE 4. (a) The electron temperature Te in n++silicon as a function of the voltage V across the

S-Sm-S structure at different substrate temperatures given by T at V=0. (b) Maximum decrease of electron temperature Te as a function of substrate temperature. Solid and dashed curves correspond to numerical solution of Pmax+Pel-ph = 0 with Σ = 1 × 108 W/K5m3 and Σ = 1.0 × 109 W/K5m3 respectively.

The electron temperature in n++silicon as a function of the voltage across the S-Sm-

S structure for few substrate temperatures is presented in Fig. 4a. The observed cooling effect is rather large: cooling of the electron system exceeds 60% at T0=150 mK (see Fig. 4b). The maximum cooling power of the n++silicon cooler (with volume Ω) can be estimated using equation for the heat flow from phonons (temperature T0) to electrons (Te = Tmin) [8] with Σ obtained in [9] for similar SOI films, given by Pel-ph=ΣΩ(Te

5-T05). The maximum cooling power of the device at T0 = 150 mK is

about 0.4 pW, which we expect to improve significantly with further work. ACKNOWLEDGMENTS

This work has been supported by the Academy of Finland under projects No. 46804 and No. 46805 , the Finnish Center of Excellence Program 2000-2005 (Project No. 44875, Nuclear and Condensed matter program at JYFL), and the European Space Agency (ESTEC Contract No. 3006/98/NL/PA(SC)).

REFERENCES

1. Anghel, D.V., Luukanen, A., Pekola, J.P., Appl. Phys. Lett. 78, 556-558, (2001). 2. Nahum, M., Eiles, T.M., and Martinis, J.M, Appl. Phys. Lett. 65, 3123-3125, (1994). 3. Leivo, M.M. , Pekola, J.P. , and Averin, D.V., Appl. Phys. Lett. 68, 1996-1998, (1996) 4. Luukanen, A., Leivo, M.M., Suoknuuti, J.K., Manninen, A.J., and Pekola, J.P., J. Low Temp. Phys., 120, 281-290, (2000). 5.Jochum, J., Mears, C., Golwala, S., Sadoulet, B. ,Castle, J.P., Cunningham, M.F.,Drury, O.B. ,Frank, M., Labov, S.E. ,Lipschultz, F.P. ,Netel, H. , and Neuhauser, B. , J. Appl. Phys. 83, 3217 (1998). 6. Pekola, J.P. ,Anghel, D.V., Suppula, T.I., Suoknuuti, J.K. , Manninen, A.J. ,and Manninen, M., Appl. Phys. Lett. 76,19 (2000). 7. Manninen, A.J., Leivo, M.M. and Pekola, J.P., Appl. Phys. Lett. 70, 1885-1887 (1997). 8. Wellstood, F. C., Urbina, C. and Clarke, John, Phys. Rev. B 49, 5942-5955 (1994). 9.Kivinen, P., Savin, A., Manninen, A., Pekola, J., Prunnila, M. and Ahopelto, J., in Physics, Chemistry and Application of Nanostructures, edited by V.E.Borisenko, S.V.Gaponenko and V.S.Gurin, Singapore: World Scientific, 2001, pp. 180-183.

A superconducting antenna-coupled hot-spot microbolometer

A. Luukanena),b),c) and J.P. Pekolad)

b)University of Jyvaskyla, Department of Physics

P.O. Box 35 (Y5), FIN-40014 University of Jyvaskyla, Finland

c)Metorex International Oy, P.O. Box 85, FIN-02631, Espoo, Finland

d)Low Temperature Laboratory, Helsinki University of Technology

P.O. Box 2200, FIN-02015 HUT, Finland

Abstract

We report the electrical properties of an antenna-coupled niobium vacuum-bridge bolometer,

operated at a temperature of 4.2 K, in which the thermal isolation is maximized by the vacuum

gap between the bridge and the underlying silicon substrate. The device is voltage-biased, which

results in a formation of a normal state region in the middle of the bridge. The device shows a

current responsivity of -1430 A/W and an amplifier limited electrical noise equivalent power of

1.4×10−14 W/√

Hz.

PACS numbers: 85.25.Pb,07.57.Kp

a)Electronic address: arttu.luukanen@vtt.fi,

present address: VTT Information Technology, Microsensing, P.O.Box 1207, FIN-02044 VTT, Finland

1

Antenna-coupling extends the detectable range of wavelengths of bolometers to millimeter

waves and beyond [1–4]. Antenna-coupled microbolometers consist of a lithographic antenna,

coupled electrically to a thermally sensitive element. Incident electromagnetic radiation

induces time-varying current in the antenna, which is dissipated in an impedance-matched

bolometer element acting as the antenna termination. Measurements have shown that nearly

perfect optical coupling is possible up to 30 THz [5]. One of the main benefits of this

technology is that the sensitivity is not limited by thermal time constant requirements, unlike

in the case of the state of the art infrared micromachined microbolometers [6], as well as in

several other bolometer experiments. The noise equivalent power (NEP) is (4kBT 2C/τ0)1/2

for phonon noise limited bolometer with a heat capacity C and a thermal time constant

τ0 = C/G, with G the thermal conductance between the bolometer and the heat sink. In

antenna-coupled devices, the thermal sensing element can be made much smaller than the

detected wavelength, resulting in a much smaller heat capacity, thus allowing for smaller

G and better NEP. Previously, antenna-coupled high-Tc vacuum-bridge microbolometers

showed excellent performance with a NEP=9 · 10−12 W/√

Hz at a bath temperature of

87.4 K [7]. However, the fabrication of such vacuum-bridges has proven to be difficult.

Additionally, high-Tc films usually require a buffer layer, such as yttrium stabilized zirconia,

and they often suffer from high levels of 1/f noise, and thus require the use of an optical

chopper.

If operated at or close to the temperature of liquid helium, conventional superconductors

such as Nb can be used. Besides the relatively simple processing, the thermal fluctuation

and Johnson noise of the bolometer are significantly lower at 4.2 K, as compared to 77 K.

A convenient way of biasing superconducting transition-edge sensors is by constant voltage,

as is done with X-ray microcalorimeters [8]. This results in a formation of a normal state

region which in the vacuum-bridge device is located in the centre portion of the bridge

[9, 10]. To model the performance of the vacuum-bridge, we first assume that the electron

and phonon populations are at equilibrium, i.e. Te = Tp. We also consider a steady state

treatment, since we estimate τ0 ∼ 1 µs, which is much faster than any typical signal to

be detected. The heat flow in the normal region of the bridge is in the steady state given

by −κd2T/dx2 = V 2/(ρln)2ρ + Popt/(wtln), with |x| < ln/2, while in the superconducting

part −κd2T/dx2 = Popt/(wtl) with |x| > ln/2. These equations include the assumption that

part of the bridge is in the normal state and that the thermal conductivity κ is same and

2

constant in both regions. We base this rather bold assumption on the fact that at this range

of temperatures the contribution of the lattice to the thermal conductivity is significant,

and can even increase below Tc in disordered metals due to the reduced electron-phonon

scattering as the number of quasiparticles decreases, so that the changes in the sum of the

lattice and the electronic contributions remain relatively small [11]. The bridge (length l,

width w, thickness t) is biased with voltage V , and has a normal state resistivity ρ. The

length of the normal state part of the wire is given by ln. Dissipation of both, the bias power,

Pb, and the optical power Popt( Pb) takes place in the normal part of the wire, whereas the

superconducting region of the bridge is assumed to dissipate only the RF. It should be noted

that also frequencies slightly below the gap frequency of Nb (f = 3.52kBTc/h = 500 GHz

with Tc = 6.8 K) are absorbed in the superconducting region as there is a large temperature

gradient present in the bridge.

We use boundary conditions T (0) = T (l) = T0, and dT (ln/2)/dx|S = dT (ln/2)/dx|N at

the superconducting-normal (S-N) interfaces. Additionally, we require the maximum of the

temperature to occur at the middle of the normal state part, i.e. that dT/dx|l/2 = 0.

In the limit of small optical power, Popt → 0, the solution for the I(V ) reduces to

I(V )0 =4κ(Tc − T0)wt

V l+

V wt

ρl. (1)

The first term on the right side of Eq.(1) is due to the electro-thermal feedback while the

second term on the right describes the ohmic behaviour of the bridge. When V is small, the

bias dissipation is constant and equal to 4κ(Tc − T0)wt/l. Saturation occurs when optical

power approaches the bias power.

The 20 µm × 1 µm × 100 nm Nb vacuum-bridge is fabricated on a nitridized high-

resistivity Si wafer. The nominal thickness of the nitride was 1 µm. For convenience, we

use electron beam lithography to pattern the structures, although optical lithography could

be used as well. The antenna and the bolometer bridge are patterned using electron beam

lithography on a double layer PMMA-MAA/MAA electron resist with a thickness of 350

nm for the bottom (PMMA/MAA) and 300 nm for the top (PMMA) layers. The antenna

is a logarithmic spiral antenna with a nominal band from 455 GHz to 2 THz determined by

the outer and inner radii of the spiral, respectively, and a real input impedance of 75 Ω on

Si (εr = 11.7). Following the patterning, a 100 nm thick Nb layer is evaporated at a rate

of 3 A/s. After lift-off, the sample is dry etched with a mixture of CF4 and O2 gases. This

3

dry etching step removes the Si3N4 from the sample. The etch is performed at a relatively

high pressure of 50 mTorr, which results in isotropic etch of the nitride, removing it also

underneath the narrow Nb bridge. By prolonging the etching step it is also possible to etch

the bulk Si, exposed after the Si3N4 has been removed. The resulting bridge is separated by

a ∼ 2 µm vacuum gap from the underlying substrate (see Fig. 1). Some under-etch (∼ 1

µm laterally) of the antenna also takes place, but we believe that this has a negligible effect

on the antenna properties. As the devices were intended for electrical measurements only,

the antenna does not incorporate thicker normal metal layer required to prevent losses in

the relatively thin Nb film.

The critical temperature and resistivity of the Nb was measured in an independent 4-wire

measurement against a CBT primary thermometer [12], yielding Tc = 6.8 K and ρ = 56

µΩcm. After this, the devices were characterised by measuring their current - voltage

characteristics with a SQUID current preamplifier [13]. The sample, together with the

SQUID encased in a superconducting Nb shield, were mounted to a vacuum can immersed

in liquid helium. A floating bias circuit consisted of a tunable 18 V battery, connected in

series with a 1.2 kΩ current limiting resistor at room temperature. The voltage bias for the

bridge was provided by a 1.2 Ω shunt resistor connected in parallel with the bridge.

Using Eq.(1), a fit was made to the I − V curve with κ as a fit parameter. All the

other parameters were fixed. For the thermal conductivity we obtained κ = 0.54 W/Km,

which is surprisingly close to the value predicted by Wiedemann-Franz law, L0Tc/ρ = 0.29

W/Km with L0 = 2.45×10−8 V2/K2, and to that reported for NbTi (0.26 W/Km) [14]. We

attribute this to the contribution of the phonons to the thermal conductivity below Tc.

For any resistive bolometer, the electrical responsivity can be calculated from the I − V

curve using the differential (Z = dV/dI) and bias point resistance (R = V/I) [15]. The

parameter describing the negative electro-thermal feedback (ETF) in the bolometer is the

loop gain, given by L = β(Z − R)/(Z + R), and can be calculated from Eq.(1), yielding

L = 4βκρ(Tc − T0)/V2. Here β = (R − Rs)/(R + Rs) describes the influence of the voltage

source impedance on the ETF. Again, we have omitted the frequency dependence of L as we

assume that the device response is much faster than any typical signal. A general treatment

of a voltage biased bolometer yields a current responsivity SI ≡ dI/dP = −V −1L/(L + 1)

which approaches a value −1/V when L is large. Figure 3 shows the current responsivity

determined from the I − V curve.

4

Noise measurements were performed by biasing the bridge at different points on the

I−V curve, and measuring the rms noise between 750 Hz and 25 kHz. The noise consists of

uncorrelated contributions from the fluctuations of heat between the bridge and the heat sink

(phonon noise), Johnson noise of the resistive part, and noise from the SQUID. The phonon

noise current is given by ip =√

γ4kBT 2c G|SI|, where γ = 0.67 describes the effect of the

temperature gradient in the bridge [16]. The Johnson current noise is not amplified by the

electrothermal gain, and is given at the sensor output by iJ =√

4kBTc/R(1 + β)/[2(1 +L)].

The current noise of the SQUID was measured separately yielding isq = 12 pA/√

Hz. The

total NEP of the detector is given by NEP2tot = (i2p + i2J + i2sq)/|SI|2.

The results of the noise measurement are shown in Fig. 4 with a comparison to the

theoretical prediction described above. The minimum NEP is reached at V = 0.82 mV

where NEP= 1.4×10−14 W/√

Hz, which is almost an order of magnitude improvement over

existing 4.2 K bolometers. Furthermore, improvement of the read out noise would improve

the NEP further down to the detector limited NEP of 2.6×10−15 W/√

Hz. Noteworthy

is also the excellent dynamic range of the device: The bias dissipation is 26 nW, yielding

dynamic range of 55 dB at a 30 Hz information bandwidth. The increase in the NEP seen

below 0.82 mV arises from oscillations of the electrical circuit. The intrinsic time constant is

τ0 ≈ 0.9 µs based on the heat capacity for the bridge in the normal state, and the measured

value for G, whereas the time constant of the bias circuit is τele = L/R where L ≈ 2 µH is

the combined inductance of the SQUID input coil and parasitic inductance from wiring. At

V =0.82 mV, R = 28 Ω, L = 4.5, and thus τeff/τele = τ0(1 +L)−1τ−1ele ≈ 2, while the stability

criterion requires this ratio to be more than 5.8 [17].

Some coupling mismatch is present due to the slightly larger resistance of the bridge (130

Ω versus the 75 Ω of the antenna), and to the inductance of the bridge. Taking in account the

resistance and the 10 pH self inductance of the bridge, we estimate the impedance mismatch

between the antenna and the bridge will reflect less than 20 % of the optical power at all

bias points if operated around 500 GHz.

In summary, we have modelled, fabricated and measured the electrical properties of a

novel type of a superconducting bolometer with an electrical NEP better by almost an order

of magnitude compared to the current state of the art bolometers operated at 4.2 K. The

simple fabrication process allows the construction of large imaging arrays while improving

the noise matching with the SQUID would allow for a further significant improvement of

5

NEP. The detector is compatible with novel SQUID multiplexing methods [18] which make

it an attractive choice for large imaging millimeter wave arrays.

6

Acknowledgements

The authors would like to thank K. Hansen, H. Sipila and T. Suppula for discussions.

This work has been funded by the Academy of Finland and the Finnish technology agency

TEKES within the framework of National ANTARES space research programmes High

Energy Astrophysics and Space Astronomy (HESA) consortium and by the Academy of

Finland under the Finnish Centre of Excellence Programme 2000-2005 (Project No. 44875,

Nuclear and Condensed Matter Programme at JYFL.

7

[1] T.-L. Hwang, S. Schwarz, and D. Rutledge, Appl. Phys. Lett. 34, 773 (1979).

[2] D. P. Neikirk and D. B. Rutledge, Appl. Phys. Lett. 44, 153 (1984).

[3] M. E. MacDonald and E. N. Grossman, IEEE Trans. Microwave Theory Tech. 43, 893 (1995).

[4] A. Luukanen and V.-P. Viitanen, Proc. SPIE 3378, 34 (1998).

[5] E. N. Grossman, J. E. Sauvageau, and D. G. McDonald, Appl. Phys. Lett. 59, 3225 (1991).

[6] R. Wood, Monolithic Silicon Microbolometer Arrays (Academic Press, 1997), vol. 47 of Semi-

conductors and Semimetals, chap. 3, pp. 43–122.

[7] J. Rice, E. Grossman, and D. Rudman, Appl. Phys. Lett. 65, 773 (1994).

[8] K. D. Irwin, Appl. Phys. Lett. 66, 1998 (1995).

[9] D. W. Floet, E. Miedema, and T. Klapwijk, Appl. Phys. Lett. 74, 433 (1999).

[10] H. F. Merkel, P. Khosropanah, S. Cherednichenko, K. Yngvesson, A. Adam, and E. L. Koll-

berg, IEEE Trans. Appl. Supercond. 11, 179 (2001).

[11] S. Wasim and N. Zebouni, Phys. Rev. 187, 539 (1969).

[12] Nanoway Oy, CBT primary thermometer, sensor CBT10, monitor model 400R.

[13] Conductus iMAG LTS SQUID system.

[14] F. Pobell, Matter and methods at low temperatures (Springer-Verlag, Berlin and New York,

1996), 2nd ed.

[15] R. C. Jones, J. Opt. Soc. Am. 43, 1 (1953).

[16] J. C. Mather, Appl. Opt. 21, 1125 (1982).

[17] K. D. Irwin, G. C. Hilton, D. A. Wollman, and J. M. Martinis, J. Appl. Phys. 83, 3978 (1998).

[18] M. Kiviranta, H. Seppa, J. van der Kuur, and P. Korte, in Low Temperature Detectors 9 (LTD-

9), edited by F. S. Porter, D. McCammon, M. Galeazzi, and C. Stahle (American Institute of

Physics, Melville, New York, 2002), vol. 605 of AIP Conference Proceedings, pp. 295–300.

8

Figure legends

1. A scanning electron micrograph of the antenna-coupled Nb bridge bolometer. The

inset at upper right shows a detailed image of the feed region taken at a steep angle

to show the separation between the bridge and the substrate. The diagram at lower

left shows the model used in the theoretical treatment with S indicating the supercon-

ducting regions, and the shaded area in the middle of the bridge marking the normal

region extending from −ln/2 to ln/2.

2. The I − V -curve of the parallel connection of the Nb vacuum-bridge and the shunt

resistor (circles). The linear part towards higher voltage values corresponds to the

ohmic behaviour of the bridge in its normal state. The fit with the theory is given by

the solid line with κ = 0.54 W/Km. As a reference, the asymptote corresponding to

the ohmic behaviour (∝ V ) is marked with a dash-dotted line. At voltages below about

1.8 mV the ends of the bridge turn superconducting, resulting in negative differential

resistance and dissipation which is independent of the bias voltage. The dashed line

represents the electro-thermal term which is proportional to 1/V . The noise present

in the experimental data arises from the bias source.

3. (a) The electrical responsivity of the bridge derived from the I − V curve (circles). A

30 -point adjacent averaging method was used to smooth the experimental data. The

dashed line is the responsivity calculated using the fit to the I −V characteristics. (b)

Noise spectrum of the detector at V = 0.82 mV. The cut-off at ∼ 50 kHz corresponds

to the bandwidth of the SQUID flux-locked loop electronics. The low frequency noise

seen below 10 Hz is attributed to the SQUID since it was not effected by the voltage.

4. The electrical NEP of the bolometer. The measured NEP is marked by triangles,

and the solid line corresponds to the modeled data which includes contributions from

thermal fluctuation noise (dashed line), Johnson noise (dash-dotted line), and SQUID

noise (dots). The NEP is limited by the noise of the SQUID until V = 0.82 mV. The

increase in the noise below this point is due to oscillations of the bias circuit as the

effective time constant τeff becomes comparable to the electrical time constant τele.

9

S S

20 mµ

Tc

xln/20

FIG. 1: A. Luukanen, Applied Physics Letters L030458

10

0.0 2.0 4.0 6.0 8.0 10.0

0

10

20

30

40

50

60

70

80

Cur

rent

[µA

]

Voltage [mV]

FIG. 2: A. Luukanen, Applied Physics Letters L030458

11

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

-2000

-1500

-1000

-500

0

(b)

(a)

Cur

rent

resp

onsi

vity

[A/W

]

Voltage [mV]

0.01 0.1 1 10 10010

100

1000 V=0.82 mV

nois

esp

ectra

lde

nsity

[pA

/Hz½

]

Frequency [kHz]

FIG. 3: A. Luukanen, Applied Physics Letters L030458

12

1.0 10.00.1

1.0

10.0

100.0

1000.0

NE

P[fW

/Hz½

]

Voltage [mV]

FIG. 4: A. Luukanen, Applied Physics Letters L030458

13

Fluctuation superconductivity limited noise in a transition-edge

sensor

A. Luukanen, K. M. Kinnunen, A. K. Nuottajarvi

Department of Physics, University of Jyvaskyla

P.O.Box 35 (YFL) FIN-40014 University of Jyvaskyla, Finland

H. F. C. Hoevers, W. M. Bergmann Tiest

SRON National Institute for Space Research

Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands

J. P. Pekola

Low Temperature Laboratory, Helsinki University of Technology

P.O. Box 2200, FIN-02015 HUT, Finland

Abstract

In order to investigate the origin of the until now unaccounted excess noise

and to minimize the uncontrollable phenomena at the transition in X-ray mi-

crocalorimeters we have developed superconducting transition-edge sensors

into an edgeless geometry, the so-called Corbino disk (CorTES), with super-

conducting contacts in the centre and at the outer perimeter. The measured

rms current noise and its spectral density can be modeled as resistance noise

resulting from fluctuations near the equilibrium superconductor-normal metal

boundary.

Typeset using REVTEX

1

At present, the most sensitive energy-dispersive X-ray detector is the transition-edge

sensor (TES) microcalorimeter, a thermal detector operated typically at a bath temperature

below 100 mK [1–3]. The device consists of an X-ray absorber (Bi, Au, Cu being the

most common materials), thermally coupled to a TES superconducting film with a critical

temperature Tc ≈ 100 mK. The TES film is typically a proximity-coupled bilayer, e.g.,

Ti/Au, Mo/Cu, or Mo/Au. A common detector geometry is a square TES film, covered

completely, or in some cases partially by the X-ray absorber film. The TES film - absorber

combination is located on a thermally isolating Si3N4 film, micromachined to a bulk Si

substrate which acts as the heat sink. Wires with a Tc much higher than that of the

TES film are used to connect the detector to the bias circuit. The device is connected to a

constant voltage bias, and the current through the sensor is measured with a superconducting

quantum interference device (SQUID). The theory of operation of these devices has been well

developed, but is not complete as the TES microcalorimeters consistently do not achieve the

energy resolution predicted by the models. Firstly, the TES microcalorimeters fail to reach

the expected energy resolution in calorimetry, especially when the deposited heat drives

the device through a large part of its superconducting transition. Secondly, most TESs

exhibit noise in excess of the sum of the commonly recognized noise components: thermal

fluctuation noise arising from the thermal link between the TES and the heat sink (TFN),

Johnson noise (JN), and SQUID (read-out) noise (SN). This letter presents a simple model

which explains this discrepancy in the detector subject to our study.

In the square devices, edges parallel to the current become crucial for the device perfor-

mance. Firstly thickness variations resulting from underetching or imperfect deposition of

the TES bilayer lead to spatial Tc variations. Secondly the edges have also proven to give

rise to flux creep noise with the higher concentration of trapping centres due to local defects

which can be observable at certain values of the bias voltage. A solution to overcome edge

effects is to deposit thick normal metal banks over the edges. The proximity effect of the

thick normal metal reduces the critical temperature of the edges well below that of bulk of

the TES film, resulting in well-defined edges [4].

2

Another way of removing the edges is to use a Corbino disk geometry, in which a current

source is placed in the apex of the annular TES film, and another superconducting contact is

placed to the outer circumference of the TES film. Here the current density is proportional

to 1/r, which results in a well defined phase boundary at certain distance rb from the centre

of the disk. A Corbino geometry TES, or the CorTES, is shown in the inset of Fig. 1. The

central contact is provided by a superconducting ground plane, covering the entire sensor.

By this we ensure a truly cylindrical symmetry and a homogeneous current distribution with

a concentric current return.

The devices are fabricated on a double-side nitridized, 525 µm thick Si wafer. Free

standing Si3N4 membranes with a thickness of 250 nm are fabricated by wet etching of the

Si. The CorTES layers are patterned by e-beam lithography, combined with UHV e-beam

evaporation and lift-off. The wiring layers consist of a circular Nb outer contact, and a Nb

ground plane, which contacts the TES film through an opening in an underlying insulator.

In contrast to a square TES, the phase boundary in the CorTES evolves controllably

from the centre of the disk and moves radially outwards with increasing current. This can be

modelled by a heat transfer model, similar to that used to describe suspended Nb microbridge

bolometers and hot electron mixers [5,6]. Assuming radial symmetry, the current density is

given by j(r) = I/(2πrt), where I is the current, r is the radial distance from the centre

of the disk, and t is the film thickness. Consequently, the resistance of the CorTES is

given by R = ρn/(2πt)∫ rb

r0

1/rdr = ρn/(2πt) ln(rb/r0), where r0 is the radius of the central

superconducting contact. The steady-state behaviour can be modeled by first noting that

the heat transport within the sensor is completely dominated by the metal films. The

thermal conductivity in the superconducting region (at radii r ≥ rb) is given approximately

by κS(T ) = κN exp[−Λ/kB(1/T − 1/Tc)] where Λ is of the order of the energy gap ∆ of the

superconductor, kB is the Boltzmann constant, κN = LTc/ρn is the normal state thermal

conductivity, ρn is the normal state electrical resistivity and L = 2.45 × 10−8 V2/K2 is

the Lorentz number [7]. Here we assume validity of Wiedemann-Franz law. The thermal

conductivity of the 250 nm thick SiN membrane is given by κM ' 14.5 × 10−3T 1.98 [8], and

3

at temperatures present in the system (20 mK - 150 mK) it is typically three orders of

magnitude smaller than κS. As the thicknesses of the SiN and the TES are comparable, the

problem reduces to two dimensions, and due to symmetry further to one dimension. At this

point we neglect the temperature gradient within the normal state part, as the gradient over

the superconducting annulus and especially over the surrounding membrane are much larger.

In the superconducting part we require a heat balance Q/(2πt)∫ r1

rb

1/r dr = −∫ T1

TcκS(T ) dT ,

where Q = V 22πt/ρn ln(rb/r0) is the dissipated bias power, with V the applied voltage across

the sensor and r1 the radius of the CorTES outer edge at temperature T (r1) = T1. A similar

equation can be written for the heat transport in the membrane but now the integration is

carried out from r1 to r0 = 2w/π, the radius of the ”equivalent” circular membrane to the

square one with a pitch of w, and in temperature from T1 to T0, the latter being the bath

temperature. This leads to a solution for T1, which can be inserted into the heat balance

equation of the superconducting region. This can then be numerically solved for rb(V ) from

which one obtains I(V ) = V/R = 2πtV/ρn ln[rb(V )/r0].

We first carried out an R − T0 measurement in a dilution refrigerator measuring the

resistance R of the CorTES using a 4-wire AC method with a current bias of 5 µA as a

function of T0. From this, Tc = 123 mK was obtained. Next, we measured a set current-

voltage [I(V )] curves using voltage bias with a source impedance Rs of 7 mΩ, and a SQUID

ammeter. The results are shown in Fig. 1, together with a fit using the model above. The

I(V ) curve is insensitive to the external magnetic field thanks to the Nb groundplane. The

fitting parameter is Λ in the the superconducting region. Best fit yields Λ = 1.25kBTc, which

is a reasonable value, somewhat smaller than the BCS gap, ∆ = 1.76kBTc. The sharp corner

present in the fit is due to the fact that the model assumes a step-wise transition with zero

width, whereas the actual transition is smooth, as seen in the inset of Fig. 2. Figure 2

shows the steepness of the transition, α = d lnR/d lnT , as a function of V measured at

different bath temperatures. In all the curves α has a maximum value of about 300. This

is one order of magnitude higher than in typical (square) microcalorimeters [2]. The high α

is attributed mainly to the self-screening property of the ground plane and the well defined

4

edge conditions.

The noise characteristics of the CorTES were measured as a function of the bias voltage.

Both noise spectra and the rms current noise between 100 Hz and 20 kHz were determined.

The intrinsic thermal time constant of the CorTES with heat capacity C and thermal con-

ductance to the bath of G, τ0 = C/G, was determined from pulse response to be about 1.2

ms. Thus, the rms measurements are mainly sensitive to noise which is not suppressed by

the electro-thermal feedback (frequencies above (2πτ0)−1=130 Hz). This method allows us

to investigate the current noise against the operating point. When biased in the operating

region (V . 1 µV), the noise in the CorTES can not be accounted for by assuming contri-

butions from TFN, JN and SN, as can be seen in Fig. 3. We argue that the discrepancy

can not be explained by including an internal TFN (ITFN) [9,10] arising from the finite

internal thermal impedance of the TES film. The ITFN is calculated as in Ref. [9] and it

is proportional to Iα exhibiting a peak at a bias corresponding to the maximum value of

α. However, the ITFN does not explain the noise at lower bias voltages where the noise is

more than a decade larger than what we would be expect just by assuming contributions

from the previously known terms.

According to the Ginzburg-Landau theory, the free energy difference between equilib-

rium superconducting and normal states of a volume Ω in the absence of any fields is

F = Ω(α|ψ|2 + 1/2β|ψ|4), where α = 1.36~2/(4mξ0l)(T/Tc − 1) ≡ α0(T/Tc − 1) and

β = 0.108/N(0)[α0/(kBTc)]2. Here m is the electron mass, and N(0) = 1.33×1034 cm−3eV−1

is the density of states at Fermi level for Ti, ξ0 ≈ 20 µm is the BCS coherence length and

l ≈ 100 A is the mean free path determined from the normal state resistivity. The tem-

perature gradient within the normal section can be solved by −∇(κN∇T ) = j(r)2ρn =

V 2L−1[r ln(rb/r0)]−2 using boundary conditions T (rb) = Tc and ∇T (r0) = 0. Fluctua-

tions of ψ with free energy variations δF . kBT are possible and they correspond to large

fluctuating volumes δΩ of the condensate near Tc. The temperature gradient restricts the

fluctuations to an annular volume at the outer perimeter of the normal state part, where

the coherence length ξ(T ) = 0.86√

ξ0l)/√

T/Tc − 1 diverges. We estimate the radial extent

5

of fluctuations, δr, assuming that order parameter fluctuates between zero and its equilib-

rium value at a distance δr away from the equilibrium phase boundary within a volume

δΩ = 2πrbtδr:

kBTc ' δF ' −〈α2〉

2βδΩ '

πα2

0tγ2rbβ

δr3. (1)

Here we have assumed that (T/Tc−1) can be approximated by δrγ, where γTc represents the

effective radial temperature gradient at the phase boundary. The solution for the tempera-

ture profile yields γ = −V 2/(T 2c aLrb ln(rb/r0)), where the numerical factor 0 ≤ a ≤ 1 is used

as the only fitting parameter which describes the reduction of the Lorentz number close to

the boundary due to the presence of Cooper pairs. Solving for δr and inserting the equations

for α0 and β, the fluctuation in boundary radius is given by δr = 0.48[πN(0)kBTcrbtγ2]−1/3.

In order to obtain the spectral density of the critical fluctuations, we note that the

relaxation time of a fluctuation is given by τGL = ~π[8kB(T−Tc)]−1 = ~π[8kBTcγδr]

−1. Thus,

the equivalent noise bandwidth is∫

∞

0(1+ω2τ 2

GL)−1dω = π/(2τGL), and the resulting spectral

density of the resistance fluctuations is given by δR = ρnδr(2πtrb)−1

√

2τGL/π. This can be

considered as a white noise source within the bandwidth of our measurement. The current

noise arising from resistance fluctuations is given by δI = dI/dR δR = I/(2R)(b + 1)δR

where R = V/I and b = (R−Rs)/(R+Rs) corrects for non ideal voltage bias. The resistance

fluctuations are suppressed by the ETF in a similar fashion as JN. More explicitly, the

fluctuation superconductivity noise (FSN) component is given by

δIFSN(ω) =IδR

R

1 + b

2(1 + bL0)

√

1 + ω2τ 20

√

1 + ω2τ 2

eff

, (2)

where bL0 = V Iα/(GT ) is the loop gain of the negative electrothermal feedback, and τeff =

τ0/(1 + bL0) is the effective time constant of the sensor. As Figs. 3 and 4 show, the noise

can be accurately modelled through out the transition and over a wide range in frequency

with only one fit parameter, a = 0.1. We should, however, keep in mind that the crude

definition of δr in our model may simply be compensated by this fitting parameter.

When FSN dominates over the other noise terms, the FWHM energy resolution of a

calorimeter can be estimated from the noise equivalent power NEPFSN = δIFSN/SI where SI

6

is the current responsivity of the sensor [11],

∆E ≈ 1.18

(∫

∞

0

df

NEPFSN2(f)

)

−1/2

= 1.18V I

L0RδR

√

2τ0π

= 1.18I2δR

L0

√

2τ0π. (3)

In the measured device, ∆EFSN has a minimum value of ∼0.1 eV at V = 0.55 µV (where

α is at maximum) and it increases to 29 eV at V = 0.3 µV. This implies that the energy

resolution of TES microcalorimeters degrades significantly if biased at a voltage below that

corresponding to maximum of α.

In summary, we have fabricated and analyzed an idealized transition-edge sensor in

which edge-effects are excluded. An analytical steady state model has been developed which

shows good agreement with the measured I(V ) curve. The CorTES is insensitive to external

magnetic fields due to a current carrying Nb ground plane. As a result, the α remains above

300 even when biased with constant voltage bias. We show that the previously unexplained

extra noise originates from thermal fluctuations of the phase boundary. The same noise

mechanism is present in all types of superconducting transition-edge sensors [12], but the

effect might not be observable in some cases depending on the way the phase boundaries

configure themselves.

ACKNOWLEDGMENTS

This work has been supported by the Academy of Finland under the Finnish Centre

of Excellence Programme 2000-2005 (Project No. 44875, Nuclear and Condensed Matter

Programme at JYFL), and by the Finnish ANTARES Space Research Programme under

the High energy Astrophysics and Space astronomy (HESA) consortium. The work of HFCH

and WBT is financially supported by the Dutch organisation for scientific research (NWO).

The authors gratefully acknowledge K. Hansen, N. Kopnin and H. Seppa for their comments.

7

REFERENCES

[1] K. D. Irwin, G. C. Hilton, D. A. Wollman, and J. M. Martinis, Appl. Phys. Lett. 69,

1945 (1996).

[2] W. B. Tiest et al., in Low Temperature Detectors 9 (LTD-9), Vol. 605 of AIP Confer-

ence Proceedings, edited by F. S. Porter, D. McCammon, M. Galeazzi, and C. Stahle

(American Institute of Physics, Melville, New York, 2002), pp. 199–202.

[3] A. Luukanen et al., Summary report, European Space Agency, ESA Contract

12835/98/NL/SB, ESA-ESTEC, ADM-A, Keplerlaan 1, NL-2200 AG Noordwijk.

[4] G. C. Hilton et al., IEEE Trans. Appl. Supercond. 11, 739 (2001).

[5] A. Luukanen and J. Pekola, to appear in Appl. Phys. Lett., (2003).

[6] D. W. Floet, E. Miedema, and T. Klapwijk, Appl. Phys. Lett. 74, 433 (1999).

[7] R. Berman, in Thermal conduction in solids (Oxford University Press, Oxford, 1976),

Chap. 12, pp. 164–168.

[8] M. M. Leivo and J. P. Pekola, Appl. Phys. Lett. 72, 1305 (1998).

[9] H. Hoevers et al., Appl. Phys. Lett. 77, (2000).

[10] J. M. Gildemeister, A. T. Lee, and P. L. Richards, Appl. Opt. 40, 6229 (2001).

[11] D. M. S.H. Moseley, J.C. Mather, J. Appl. Phys. 56, 1257 (1984).

[12] K. Nagaev, Physica C, 184,149 (1991).

8

FIGURE LEGENDS

1. a) The measured I(V ) characteristics of the CorTES at different external magnetic

fields denoted by the different symbols. The sensor itself is shielded by the Nb ground

plane, and the I(V ) curve is not affected by the field. The Nb bias lines, however, are

sensitive to the field and have a relatively small critical current due to difficulties in

step coverage. The solid line corresponds to the fit with Λ=1.25kBTc. b) An optical

micrograph of the sensor. The radii of the centre and outer contacts are 15 µm and

150 µm, respectively. c) A diagram showing the layer order.

2. Transition steepness, α, measured at different bath temperatures. A lower bath tem-

perature corresponds to a larger bias dissipation and correspondingly larger biasing

currents. However, the relative transition width remains almost unchanged. The inset

shows the corresponding R− T curves calculated from the I(V ) curve.

3. The average rms current noise in a frequency band from 100 Hz to 20 kHz (marked by

open circles) measured at a bath temperature of 20 mK. The total modeled noise is

the solid line, and it consists of TFN, JN, SN, ITFN, and FSN. The dash-dotted line

represents a model with only TFN, JN and SN included. The dashed line represents

a model with TFN, JN, SN, and ITFN.

4. Noise spectra measured at a bath temperature of 20 mK at bias voltages of a) 0.75 µV,

b) 0.62 µV, and c) 0.5 µV. The notation for the different modelled noise components

is identical to that of Fig. 3.

9

FIGURES

0.00 0.25 0.50 0.75 1.00

10

20

0.18 G0.33 G0.37 G0.40 G

Cur

rent

[µA

]

Voltage [µV]

Si N membrane3 4

Si N membrane3 4

Ti/Au TESAlOx AlOxNb

Centre contact

Out

er c

onta

ct

a) b)

c)

FIG. 1. A. Luukanen, H.F.C. Hoevers, K.M. Kinnunen, A.K. Nuottajarvi, J.P. Pekola, and

W.M. Bergmann Tiest

10

0.2 0.4 0.6 0.8 1.00

100

200

300

400

100 120

0

40

80

120

Resis

tance

[mW

]

Temperature [mK]

Bath temperature T0

50 mK69 mK77 mK87 mK102 mK

a

Bias voltage [mV]

FIG. 2. A. Luukanen, H.F.C. Hoevers, K.M. Kinnunen, A.K. Nuottajarvi, J.P. Pekola, and

W.M. Bergmann Tiest

11

1000

10

100

rms

curr

entn

oise

[pA

/Hz½

]

Bias voltage [nV]

FIG. 3. A. Luukanen, H.F.C. Hoevers, K.M. Kinnunen, A.K. Nuottajarvi, J.P. Pekola, and

W.M. Bergmann Tiest

12

100 101 102 103 104 10510

100 c)

Frequency [Hz]

10

100 b)

Cur

rent

nois

esp

ectra

lden

sity

[A/H

z½]

10

100 a)

FIG. 4. A. Luukanen, H.F.C. Hoevers, K.M. Kinnunen, A.K. Nuottajarvi, J.P. Pekola, and

W.M. Bergmann Tiest

13