Journal of Low Temperature Physics manuscript No. (will be inserted by the editor) Y. Takei L. Gottardi H.F.C. Hoevers P.A.J. de Korte J. van der Kuur M.L. Ridder M.P. Bruijn Characterization of a high-performance Ti/Au TES microcalorimeter with a central Cu absorber Received July 23, 2007, Accepted September 15, 2007 Keywords Transition edge sensor, microcalorimeter, X-ray Abstract We are developing X-ray microcalorimeters based on Ti/Au transitionedge sensors (TES). Among sensors we have fabricated, one with a Cu absorber at the center of the TES shows a particularly good X-ray energy resolution: 1.56 ev at 250 ev and 2.5 ev at 5.9 kev. In this paper, a detailed study of its impedance and noise is presented. The noise is not explained by a sum of known sources. The magnitude of unexplained noise is largest when the sensitivity of the TES on temperature (α) and on current (β) are the highest. The observed relation between the noise level and sensitivity suggests a source of thermal fluctuations inside the TES or between the TES and the absorber. We also found that β is linearly correlated to the product of α and current, which limits the effective sensitivity that is expressed as α/(1+β). PACS numbers: 07.20.Fw,07.85.Nc,85.25.Am,85.25.Oj,95.55.-n 1 Introduction An X-ray microcalorimeter is a cryogenic spectrometer that, in principle, can achieve 1 ev energy resolution with a sensitive thermometer operated at T 100 mk. The transition-edge sensor (TES) microcalorimeter, which uses a sharp resistance drop of a superconducting film as a thermometer, is considered the most promising detector for future X-ray astronomy missions. The theoretical limit of the energy resolution is written as E 2.35 4kT 2 C/α, where E is the fullwidth half maximum energy resolution, k the Boltzmann constant, C the heat capacity, and α = ln R/ ln T the dimensionless sensitivity of the thermometer on SRON Netherlands Institute for Space Research Sorbonnelaan 2, 3584 CA, Utrecht, The Netherlands Tel.: +31-30-2535556 Fax: +31-30-2540860 E-mail: y.takei@sron.nl
2 Fig. 1 (Color online) Left: picture of the sensor. Right: R-T profile taken with small current. temperature. No group has, however, achieved the theoretical limit, due to the presence of so-called unexplained noise of unknown origin. Understanding and handling of this noise is quite essential to further optimize the resolution of a TES microcalorimeter. 1 We are developing microcalorimeters based on a Ti/Au bilayer TES. Among sensors we have fabricated, one with a Cu absorber at the center of the TES shows a particularly good X-ray energy resolution of 1.56 ev at 250 ev and 2.5 ev at 5.9 kev. 2,3 In this paper we present a detailed study of the characteristics of the sensor. Description of measurement setup and analysis of energy resolution is founds in Gottardi et al. 2 2 Basic characteristics of the sensor The sensor consists of a Ti/Au bilayer TES and a central Cu absorber. The TES is thermally connected to the heat bath through a silicon-nitride membrane. The top-view picture of the pixel is shown in Fig. 1 (left). The dimensions of the Ti/Au bilayer and the Cu absorber is 146 µm 150 µm and 100 µm 100 µm, respectively, while the thickness is 20 (Ti) / 50 (Au) nm and 1000 nm, respectively. The transition temperature of the TES is T c = 100 mk, and the normal-state resistance is R n = 143 mω. Fig. 1 (right) shows R T profile of the sensor taken with small current. The 1 µm-thick membrane gives thermal conductance of G = 0.36 nw/k at T c between the TES and the bath. We regulated the bath temperature of our ADR cooler at 73 mk for the measurement presented. The TES was virtually voltage-biased with a R th = 10 mω (Thevenin equivalent) load resistance and kept in the transition by negative electrothermal feedback with a Joule power of P = 6.5 pw. The stability of bath temperature is 7 µk rms with a sampling interval of 7 s. We canceled a remaining magnetic field perpendicular to the TES using a superconducting coil beneath the TES. 3 Impedance measurement The thermal and electrical response of a TES is characterized through several parameters that vary with bias voltage: resistance R, sensitivity on temperature α, sensitivity on current β = ln R/ ln I, and heat capacity C. A measurement of
3 Fig. 2 Left: impedance curve at R = 0.5R n. Points show the measured data in f =10 Hz 30 khz range; each point corresponds to a frequency. Solid and dashed lines are models calculated with and without thermal decoupling, respectively. Right: parameters determined from the impedance analysis. From top to bottom, α, β, and C are plotted as a function of bias point resistance. impedance of a TES allows us to derive these parameters 4. We applied a bandwidthlimited white noise ( f < 30 khz) voltage to the TES in addition to the standard DC bias. The input and output signals were stored with an analog-to-digital converter in time domain. Then, the impedance was calculated from the ratio of the input and output signals in frequency domain, after the transfer function of the signal lines was corrected. Note that it is not trivial to completely correct the transfer function because any stray capacitance, inductance or the response of electronics would make it quite complicated. We did not model i.e., formalize the transfer function, but instead, we derived it experimentally. We took a ratio of the data of the normal and superconducting state to determine the inductance L in the TES SQUID circuit, in which the transfer function is canceled out. Then, we calculate the transfer function by dividing the data of superconducting state by (R th + iωl). The impedance Z of a TES would trace a semi-circle in a complex (Re-Im) plane as Z = Z inf +(Z inf Z 0 )/( 1+iωτ eff ), where Z 0 = R(1+β +L )/(L 1) and Z inf = R(1+β) are the impedance at low and high frequency limit, respectively, and τ eff = C/(G(L 1)) is the thermal time constant of the TES 4. Here L = Pα/GT is the loopgain of the electro-thermal feedback. We found that this simple model is not sufficient to fit our data in a whole frequency range as shown with a dashed line in Fig. 2 (left). This suggests presence of an additional heat capacity that is decoupled from the TES. 5 The impedance is well fitted after the decoupled heat capacity is taken into account 6 (also indicated in Fig. 2-left). We found that the decoupling time constant is 0.1 ms and the dangling heat capacity
4 Fig. 3 (Color online) Left: Noise spectrum at R = R n /2. The measured values are shown with a thick solid line. Two dotted lines represents the calculated phonon (at lower frequency) and Johnson (at higher frequency) noises, a dashed line the thermal fluctuation noise between the TES and the dangling heat capacity, and a dash-dotted line unexplained noise. The sum of the former three noise components are shown with a dash-dot-dot-dotted line, while the sum of all noise components are indicated with a thin solid line. Right: M vs. R/R n. The solid line indicates scaled α/(1 + β) dependence (see 5.2) is 0.04 pj. The dangling heat capacity might be the Si substrate, or other pixels on the chip. The origin is not clear yet. It cannot be the absorber, because we then should see strongly non-exponential X-ray pulses, which is not the case. The derived parameters, i.e., α, β and C, are shown in Fig. 2 (right) as a function of R of the operating point. Sensitivities α and β show a peak at the middle of the transition, while C is almost constant over the transition. Note C contains the TES and absorber heat capacity but not the dangling one. A clear correlation between α and β is seen. Note that accuracy of these parameters is hardly affected by the modeling of the decoupling. Even when we do not include the decoupling into the model, the derived α, β and C stay same within 5%. Hence we do not discuss further the origin of the dangling capacity. 4 Noise analysis We also studied the noise spectrum of the sensor. A noise spectrum at R = R n /2 is shown in Fig. 3 (left), along with models. The models are calculated from the parameters determined from the impedance, i.e., no adjustable parameters involved. There is a contribution of thermal fluctuation noise between the TES and the dangling heat capacity (dashed line) in addition to the phonon and Johnson noise (two dotted lines). The measured noise is well reproduced at below the thermal response frequency (dash-dot-dot-dotted line), while the measured noise is larger at higher frequency; unexplained noise exists in our device. It was reported by many groups that unexplained noise has a similar frequency dependence as the Johnson noise. We expressed unexplained noise in terms of M-times the Johnson noise, according to Ullom et al. 1 This assumption is found reasonable (thin solid line of Fig. 3-left). As shown in Fig. 3 (right), M varies along the bias points. The noise has a peak at the bias point where the sensitivity (α or β) is the highest.
5 Fig. 4 Relation between β and αi. Points show the measured data, while the solid line indicates a relation β = α I/1.7 ma. 5 Discussion We have studied the impedance and noise of a Ti/Au TES microcalorimeter that achieved a energy resolution of 1.56 ev at 250 ev. The measurement gave us detailed information on the sensor. We discuss what limits the energy resolution and how we could improve it. The effective sensitivity of the TES α eff d lnr/d lnt is a combination of α and β. It is calculated as α eff = α/(1 + β) under constant-voltage bias. It is essential to fabricate a high-α eff TES without increasing the noise amplitude. The apparent correlation between α, β and M suggests that we did not benefit much from high α. Since β is larger than 1 when α is large, α eff of our device is limited to 100. The existence of unexplained noise degrades the energy resolution, by a factor of (1+M 2 ) 1/4 at small signal limit. 1 This factor reaches 1.7 at a bias point where α is high, since M has also a large value (M 3). The device has a dangling heat capacity of unknown origin that introduces a thermal fluctuation noise. The noise degrades the energy resolution by 0.2 ev at 250 ev. 5.1 α β correlation Besides an apparent correlation between α and β, we found a linear correlation between αi and β as shown in Fig. 4. The solid line indicates β = αi/1.7 ma. The correlation is stronger than one between α and β, and hence looks originated from the nature of superconductivity. Indeed, Ginzburg-Landau theory predicts that β is represented with α and I, namely β = α(i/i C0 ) 2/3 where I C0 is a critical current at zero temperature. 7 Although our relation has a different exponent, this may be due to resistance dependence of I C0. There is an upper limit on α eff under the correlation, since α eff α/β (I/1.7 ma) 1 when α is large. This limit is also written as α eff < 250 R/R n, based on R n = 143 mω and P = 6.5 pw. We also investigated the relation for different bath temperature and for different perpendicular magnetic fields. The relation is still valid with sightly different coefficient (±20%). A sensor with a small coefficient would improve the energy resolution. If it is related to the critical current density of the TES, a thicker TES would be better for the same I, i.e., the same G and the same bath temperature.
6 5.2 Origin of unexplained noise The magnitude of unexplained noise strongly depends on the bias point; it is high when α is high. A correlation between the noise level and α or α eff is also seen in other groups; e.g., noise of NIST sensors has M α eff dependence 1. Our sensor seems to have stronger dependence, close to M α eff. A possible noise source that may explain the relation between M and α is internal thermal-fluctuation noise (ITFN) 8, which is thermal fluctuation inside the TES or between the TES and the absorber. ITFN is caused by finite thermal conductance inside the TES and its magnitude is expected to be proportional to α eff. The level of the ITFN, calculated assuming the thermal conductance of 270 nw/k, is indicated with a solid line in the right panel of Fig. 3. This line corresponds to M = 0.015α eff. The model reasonably matches to the data and reproduces the trend of M vs. R at R = 0.2 0.8R n. Although we arbitrary chose the thermal conductance value, the value is physically not so strange, because it is of the same order as the conductance expected by the Wiedemann-Franz law. The temperature fluctuation caused by the ITFN is 1.4 nk/ Hz. If the ITFN indeed is responsible for the noise, a thicker (small R n ) TES would be helpful to reduce unexplained noise. 6 Conclusion We studied the impedance and noise characteristics of a Ti/Au TES microcalorimeter that achieves a good energy resolution of 1.56 ev at 250 ev. There is an apparent correlation between α, β and M; all of them have the largest value at the same bias point. We found a strong linear correlation between αi and β. There is also a roughly linear correlation between α eff and M. These relation will be considered to understand the limiting factor of the performance, and to fabricate a TES with better performance. Acknowledgements This work is financially supported by the Dutch Organization for Scientific Research (NWO). References 1. Ullom, J.N., et al., Applied Physics Letters, 87, 194103 (2005) 2. Gottardi, L. et al., Journal of Low Temperature Physics in this volume 3. Hoevers, H.F.C et al., Journal of Low Temperature Physics in this volume 4. Lindeman, M.A., et al., Review of Scientific Instruments, 75,1283 (2004) 5. Zink, B.L., et al., Applied Physics Letters, 89, 123101 (2006) 6. Galeazzi, M., McCammon, D., Journal of Applied Physics, 93, 4856, (2003) 7. Tinkham, M., Introduction to Superconductivity 2nd ed., Dover Publications, inc., Mineola, Newyork (2004) 8. Hoevers, H.F.C., et al., Applied Physics Letters, 77, 4422 (2000)