IbIs Curves as a useful sensor diagnostic

IbIs Curves as a useful sensor diagnostic Tarek Saab November 7, 999 Abstract The IbIs plot is a very useful diagnostic for understanding the the behaviour and parameters of a TES as well as extracting its R vs. T dependence. This note will briefly discuss the information that can be extracted from the IbIs curves and its application towards understanding the behaviour of the CDMS ZIP detectors. Introduction Recent detector fabrication runs have produced W films with some unexpected and undesired properties, such as high T c and T c variations. For optimal performance, it is desired that the ZIP detectors have spatially homogeneous T c s in the range of 6 7 mk. Recent tests [] with Fe ion implantation has demonstrated the ability to suppress the overall T c of detector by tens of mk. This, combined with T c measurements of the detectors tells us an idea of how much the T c of a given sensor needs to be suppressed. However, these measurements do not provide information on any gradient of T c within the sensor since such measurements will be dominated by the portion of the sensor with the highest T c. IbIs measurements can be useful in characterizing a TES s T c, the width of the transition, or any variation in the critical temperature, as well as in predicting optimal bias points based on optimal signal to noise performance. In the next section I will describe some of the theory behind IbIs measurements, then in sections I ll describe how they can be used to characterize the T c profile of a detector and to back out R vs. T curves for the T- ESs. Then, in section I ll show some IbIs data from recently fabricated ZIPS. IbIs Curves. Ideal Curves An IbIs curve is basically the I-V curve of a TES. The circuit shown in figure illustrates how TESs are operated. For a sensor with R T ES mω, most of the bias current will flow through the bias resistor R bias (= mω) effectively voltage biasing the TES. There R para R TES ias Figure : Schematic of the circuit used in operating and reading out TESs. Shown are the bias current, the bias resistor R bias, and the inductor coupling the signal into the SQUID. R bias are three states that the TES can be in depending on the value of the voltage bias (or ):. Normal : If the current through the TES (I s ) is sufficiently large the W will revert to its normal

state with a resistance of Ω. Since the normal state resistance is much larger than R bias the TES will be voltage biased and I s is given by the linear relation I s = (R bias /R T ES ) (). Biased : Once drops sufficiently, the Joule heating generated in the W TES becomes equal to the power lost to the cold substrate (through electron phonon interactions). It is in this regime that electrothermal feedback occurs and the sensors can be operated at an arbitrary point in their superconducting transition. The temperature dependence of the power dissipated by the TES into the substrate is given by P = κ(t elec T phon) () where κ is the product of the electron-phonon coupling constant Σ and V, the volume of the TES. Since the superconducting transition is only a few mk wide it is a good approximation that the power dissipation is constant throughout the transition. Therefore, from the relationships P = V I = V /R () we see that the resistance of the TES will decrease quadratically with bias voltage while I s will vary inversely.. Superconducting : Once drops below a certain value such that R T ES becomes comparable with R bias the TES will cease to be voltage biased and begin to become current biased. In such a case, the Joule heating will decrease with the bias current and the TES will quickly cool (snap) and become fully superconducting. In this state one would expect I s to be equal to, however, due to small, few mω, parasitic resistances it will be slightly smaller, but linearly dependent on. For all three cases, the exact dependence of I s on is given by the following equation I s = R bias (R T ES + R para + R bias ) () The three states of the TES are shown in figure. The top plot shows I s as a function of ; the two linear regimes, corresponding to the superconducting and normal can be easily distinguished from the biased regime ( < µa) with its characteristic inverse relationship. Similarly, the second plot shows the constant resistances, with R T ES = R N and R T ES =, in the normal and superconducting regions respectively. The bias region shows a quadratic dependence of R T ES on. Finally, the third plot shows the P vs. curve. The I R[Ω] P[pW].... 8 6 IbIs plots W Bias Current Figure : Plots of IbIs data. The top plot is that of I s as a function of. The second and third plots are of R T ES vs., and P vs. respectively. In the bias region, < µa, the characteristic dependence on I s, R T ES, and P on can be seen.

bias regions shows the characteristic constant power dissipation, while in the normal region it is quadratically dependent on.. Deviation from the ideal The characteristics of the IbIs curves described in the previous section are qualitatively correct, and quantitatively accurate to first order. The diagnostic power of IbIs curves, however, lies in the strong dependence of the curves on any deviations from the ideal case... Finite Transition Width W films produced by the Balzers, both patterned and bulk, have had superconducting transition widths mk. This means that that the power dissipated by the TES when in the bias regime is not actually a constant. If we consider equation for T c 6 mk, T phon mk, and a transition width of mk then there will be a % decrease in the power dissipated at the lower end of the superconducting transition with respect to the higher end. The slope of the P vs. plot in the bias regime is then a measure of the width of the TES s superconducting transition. Figure shows the P vs. plot for TESs with a T c of 6 mk, referenced to the center of the transition, with widths of and mk. p [pw] Power dissipation in the bias region T c = 6 mk, w = mk T c = 6 mk, w = mk Bias Region I b Figure : IbIs curves of two TESs with T c = 6mK and widths of, and mk are shown for comparison. The effect of a superconducting transition with a finite width is to give the P vs. curve a non zero slope in the bias region. Normal Region.. Phase separation Another phenomenon that can affect the shape of the IbIs curve, in addition to impacting the performance of the TES, is phase separation. This refers to a superconducting - normal phase separation in which a portion of the TES is normal (resistive) while the remainder is fully superconducting. In comparison, everything mentioned so far has assumed that the entire length of the TES is at the same temperature and in the same phase. The mechanism behind phase separation in a TES is due to the balance of heat flow along the TES with that of heat flow into the substrate. The general idea is that the equilibrium power balance between the Joule heating (at a given, or V b ) and a T c can be satisfied by having a fraction of the TES being normal, with T T c. Equation gives the necessary criterion for phase stability and is a consequence of the solution of the heat flow equations along the TES []. g wf g ep α /n π () g ep is the thermal conductivity into the substrate due to electron-phonon interactions, g wf is the Wiedeman- Franz thermal conductivity along the TES, and, α is d ln R d ln T, and n =. The equilibrium point (in the limit of an infinitely sharp transition, i.e. α = ) is given by the solution to the following two equations []: K N d T dx + α d (T T ph) = ( ) I ρ (6) W d K S d T dx + α d (T T ph) = (7) where equation 6 refers to the normal portion of the TES and equation 7 refers to the superconducting portion. K N, K S are the thermal conductivities along the length of the TES for the normal and superconducting phases respectively. α is the heat transfer coefficient (per unit area) into the substrate, and d is the thickness of the TES W film. The solutions of equations 6 and 7 yield the current and voltage across the TES as a function of the size of the normal region (x ). From that, IbIs curves can be calculated. Figure compares the power curves of two TESs with the same T c = mk for the phase separated and non-phase separated cases. Since phase separation is dependent on heat flow from the TES into the substrate, raising the temperature of the substrate and hence decreasing the heat flow, makes the TES less likely phase separate.

Equations 6 and 7 refer to a TES with an infinitely sharp transition. As a result, phase separation is expected to occur for all TESs, independently of their lengths. However, using equation the phase separation stability criterion can be rewritten in terms of the various parameters of a TES as follows []: l max = π L Lor ασt c ρ n (8) where l max is the maximum length for a stable TES, L Lor the Lorentz constant nw Ω/K, α is d ln R d ln T and is for the TESs we produce and ρ n is the normal state bulk resistivity and is. Ω µm. With the aforementioned constants, and for a T c of 8 mk, we calculate a maximum TES length of µm for phase stable operation. Empirically, we see phase stable operation for TES lengths of µm for T c 8 mk... Non Uniform T c Recent films produced by the Balzers appear to have non-uniform T c across the wafer surface. This T c nonuniformity will affect the TES performance as well as the shape of the IbIs curve. Figure shows the power curve for two TESs of equal area and resistance, but with T c s of 6 and 8 mk, connected in parallel. It can be seen that if the TESs are biased with a sufficiently low bias current,, their net behaviour will be equivalent to a TES with a single T c of T c eqv = ( ( T c + Tc) ) (9) such that the power dissipation is equal to the average of the two different TESs. If three TESs with different temperatures are connected in parallel, one sees that, although, the general behaviour of the power curve is similar, it has acquired an extra kink, each corresponding to the point where a TES enters its bias region. As the T c distribution becomes continuous the kinks themselves merge into a smooth curve.the flat portion of the power curve, nonetheless, is still determined by the TES with the lowest T c, while the first deviation from a normal resistance power curve is determined by the highest T c element. IbIs Simulation. Calculating IbIs Curves A simulation has been developed, based on the principles described in the previous sections, that calculates the theoretical IbIs curves as well as signal to noise ratios for a given configuration of TESs. The parameters required for the IbIs calculations are listed below For a non-phase separated calculation all that is needed is : T c : The T c is obtained experimentally by measuring a bulk W sample or a TES. ρ n : The resistance per square for our W films has historically been Ω/. This corresponds to a normal state bulk resistivity of. Ω µm. κ: The electron-phonon coupling for a TES is given by the product of the electron-phonon coupling constant (Σ. 9 W/K µm ) and the vol- Power dissipation in the bias region Power dissipation in the bias region Non phase separated Phase separated Normal resistor 6 mk 8 mk 6/8 mk 6/7/8 mk 8 p [pw] 6 Bias region Normal region All TES s Biased 6 7 I b All TES s Normal 6 8 Figure : IbIs curves of two TESs with T c = mk under the phase separated and non-phase separated conditions. Figure : IbIs curves for TESs with different Tc distributions. The power dissipation curves for TESs with a range of T c s lie within the region defined by the curve of TES with uniform T c s at the extrema of the T c range

ume of the TES (V = 888 (. µm )). Σ has been determined empirically using equation and a measured value of T c. For the ZIP TESs we need to multiply the ΣV product by an additional factor of 7 to get the correct value of κ. This is done to take into account the additional volume of W that connect the quasiparticle traps on the Al fins to the W meander. w: The width of the superconducting transition has been measured to be mk. For a phase separated calculation the following additional parameters are required: K N : The normal thermal conductivity along the TES is given by K N = L Lor T c ρ n () K S : The superconducting thermal conductivity is equal to K N /. α: The heat transfer coefficient into the substrate is calculated from the electron-phonon coupling constant. ( g ep T α = (T c T ph) = Σ c T ph ) () (T c T ph). Calculating Signal to Noise The signal height, to first order, is inversely proportional to the TES s bias voltage (or ). Since the signal being measured corresponds to the current through the TES, while the power being dissipated by the TES is independent of bias voltage, in the approximation of a sharp transition, sensor current will vary inversely with bias voltage. Sensor noise, on the other hand has contributions from the TES s phonon noise as well as the Johnson noise of the TES, bias, and parasitic resistances. The Johnson noise contribution for the TES is given by [] i noise = k ( ) bt T ES R T ES () R T ES R T ES + R par + R bias The bias and parasitic resistance have similar contributions. The phonon noise contribution is given by ( ) n α + ω τ eff i noise(ω) = k bt T ES R T ES + ω τ eff n + + ω τeff () where n =, and τ eff is the electrothermal feedback time. Figures 6 shows the signal to noise as a function bias for TES with T c s of 8, 7, and 6 mk. S/N (arb. units).9.8.7.6..... Bias region Signal to Noise in the bias region Normal region 8 mk 7 mk 6 mk 6 7 8 Figure 6: Signal to Noise ratios for TES with different T c. All the superconducting transition widths were taken to be mk and the sensors were assumed to be non-phase separated. R vs. T Curves Assuming that no phase separation occurs, a TES s R vs. T curve can be calculated from the IbIs power curve. This simply makes use of equation to calculate the temperature associated with with every data point based on the power being dissipated. The TES resistance can be calculated from the I s vs. curve simply by inverting equation. We then have ( ) Ib R T ES = R bias R para () I s Consequently, we can obtain R vs. T curves as is shown in figure 7. The usefulness of this technique lies in the ability of the TES to temperature regulate itself, while in electrothermal feedback, rather than having to temperature regulate a dilution refrigerator to the precision necessary for measuring R vs. T in a more conventional manner. IbIs Curves : Device performance diagnostic Figure 8 is an example of ideal TES behaviour, exhibiting a flat P vs. region. The calculated curve was

based on a T c of 8 mk and a transition width of mk. A good example of a phase separated TES is shown in figure 9. The calculation (red line) is in good agreement with the data (blue dots). The parameters used to produce the calculated IbIs curve were T c = 9 mk, Σ =.77 nw/kµm, and the assumption of phase separation. The measured T c for this device, in the low excitation current limit, was 9mK. Recent W films coming out of the Balzers have exhibited a gradient in T c. The IbIs curve of one such TES is shown in figure. The red and cyan line are the P vs. curves of TES with T c = 7, 9 mk corresponding to the limits of the T c range. The blue dots and magenta line correspond to the IbIs data and the calculated IbIs respectively. The calculation was based on a T c gradient ranging from 7 9 mk with the assumption of phase separation. 6 Making the measurement One idiosyncrasy that should be taken into account with IbIs curves is the measurement technique. Since we re really measuring the output of the feedback amplifier there will be some overall DC offset to the measured current I s. Moreover, as the TES moves out of the superconducting regime to the biased regime there (i.e. the data Tc=9mK w/ ph. sep. Sample W TES R vs. T. 8. 6 R [Ω].. 6 7 8 9 6 7 8 9 T [mk] Figure 7: R vs. T curve of a TES reconstructed from IbIs data. Figure 9: P vs. curve of a TES with a uniform T c, exhibiting phase separation. The red curve is a the result of a calculation based on phase separation and a T c of 9 mk, compared to a measured T c of 9 mk.. data Tc=8mK w=mk Sample W TES 99 GS A D data normal Tc=9mK Tc=7mK Tc=7 9mK w. ph. sep. 8. 6.. 6 8 6 8 6 Figure 8: P vs. curve of a TES with a uniform T c, that is not exhibiting phase separation. The red curve is a the result of a calculation based on a T c of 8 mk and a transition width of mk, compared to a measured T c of 86 mk. Figure : P vs. curve of a TES exhibiting a gradient in T c, as well as phase separation. The magenta curve is a calculation based on a T c range of 7 9 mk, while the blue dots are the data points. The red and cyan curves correspond to TES with uniform T c s corresponding the T c range endpoints. 6

TES snaps normal) I s will undergo a large change within a very short period. The feedback amplifier might not be able to keep and will lose lock. The consequence of losing lock is that the feedback amplifier may find a different, stable, lock point. This introduces an additional shift in the measured value of I s that may be different for the superconducting and normal parts of the curve. These magnitude of these shifts will be equal to an integer multiple of the squid s Φ and can be easily subtracted over the appropriate range. The overall DC shift is taken out by fitting the normal and superconducting regions to straight lines and subtracting out any non-zero y-intercepts. 7 Summary I have briefly discussed the theory behind IbIs curves and showed how it can be applied to TES data. The usefulness of IbIs as a diagnostic tool lies in its ability to ability to characterize the bias region of a TES in more detail than a T c measurement. Parameters such as T c gradient, transition width, and phase separation, all of which impact device performance can be simulated and understood with IbIs data. References [] B.A Young, T. Saab, B. Cabrera, J.J. Cross, R.M. Clarke, and R.A. Abusaidi. Measurement of T c supression in tungsten using magnetic impurities. Journal of Applied Physics, 999. [] S.W. Nam. Development of Phonon-mediated Cryogenic Particle Detectors with Electron and Nuclear Recoil Discrimination. PhD thesis, Stanford University, 998. [] W. Skopol, M. Beasely, and M. Tinkham. Selfheating hotspots in superconducting thin-film microbridges. Journal of Applied Physics, (9): 66, 97. 7