Transition-edge sensors (TESs) are two-dimensional superconducting films utilized as highly sensitive detectors of energy or power. These detectors are voltage biased in the superconducting-normal transition where the film resistance is both finite and a strong function of temperature. Unfortunately, the amount of electrical noise observed in TESs exceeds the predictions of existing noise theories. We describe a possible mechanism for the unexplained excess noise, which we term “mixed-down noise.” The source is Johnson noise, which is mixed down to low frequencies by Josephson oscillations in devices with a nonlinear current–voltage relationship. We derive an expression for the power spectral density of this noise and show that its predictions agree with measured data.

Transition-edge sensors (TESs) are versatile detectors of energy or power with applications in cosmology,^{1,2} quantum information,^{3,4} neutrino physics,^{5} x-ray spectroscopy,^{6–9} and nuclear security.^{10} Large satellite missions based on TESs are currently under development for cosmology^{11} and x-ray astrophysics.^{12} A TES consists of a two-dimensional superconducting film that is voltage-biased in the phase transition between the normal and superconducting states where the resistance is a strong function of temperature. A TES can be used to very precisely measure small changes in temperature caused by the absorption of single photons and particles or by an average power deposited by many photons. For example, the energy of single x rays can be determined with resolving powers (E/ΔE) exceeding 10^{3}. As demanding applications such as soft x-ray spectroscopy at synchrotrons and free electron lasers push transition-edge sensors to even higher resolving powers,^{13} it is critical to form a reliable noise theory dependent on parameters that can be measured experimentally and altered in the design process. Noise that is commonly observed in TESs in excess of the noise predicted by known mechanisms slows the development process and limits the scope of applications. In this Letter, we propose a mechanism for unexplained excess noise, derive an expression for its magnitude, and compare the predictions of the expression to measured noise data.

As shown in Fig. 1(a), the TES is a rectangular thin-film structure with typical lateral dimensions of tens to hundreds of micrometers. The critical temperature of the TES film can vary from a few millikelvins to a few kelvins depending on the application, and the normal state resistance of the film can vary from a few milliohms to a few ohms. Current enters and exits opposite sides of a TES from superconducting leads that have a higher critical current and temperature than the TES. Thermal isolation between the TES and the heat bath is achieved by micromachining the underlying substrate or from the intrinsic electron–phonon decoupling within the film itself.

A TES consists of a superconducting film operated near the transition to the normal state. The property most usually associated with the superconducting state is zero resistance for direct currents. However, a unique property of the superconducting state is that it can also be resistive. In this case, it signals the time-dependent evolution of the phase-difference between the two points used to measure the voltage-difference.^{14,15} An important consequence is that the observed voltage is dependent on both the temperature of the device and the current through the superconducting film, as is observed in TESs. The specific model applicable for this phase-difference generated voltage in our superconducting system is determined by the specific geometry and boundary conditions, but this is not essential to the arguments presented in this manuscript. The mechanism for mixed down noise described later in this manuscript only requires an oscillating current, which is implied by the voltage drop across the device, and a nonlinear current–voltage relationship.

Noise sources in a TES can be placed into one of three categories: (1) noise sources internal to the TES, (2) noise sources from the circuit in which the TES is embedded such as Johnson noise in the bias resistor and the noise contribution of the readout amplifier, and (3) noise from the external environment such as RF-pickup, stray photon arrivals, and fluctuations in the temperature bath. The second and third categories of noise from this list are either well understood or not fundamental and are not the source of the excess noise discussed here.

The two primary internal noise sources that have been identified in TESs are the voltage fluctuations due to Johnson noise from the resistance of the TES^{16} and the thermal fluctuation noise (TFN) between the TES and the heat bath. The power fluctuations due to thermal fluctuation noise (TFN) are expressed as $SPTFN=4kBT2G(T)FL(Tb,T)$, where *T* is the temperature of the TES, *T _{b}* is the temperature of the bath,

*G*(

*T*) is the thermal conductance to the bath, and

*F*is a dimensionless function, which scales the equilibrium thermal fluctuation noise, so that it is appropriate for the nonequilibrium operating conditions of a TES.

_{L}^{17}

The voltage fluctuations due to the Johnson noise of the TES are expressed as $SV=4kBTR$, where *R* is the resistance of the TES at its operating point. The current in a TES is usually measured by a SQUID ammeter, and it is, therefore, convenient to convert all types of noise into current noise as measured by the SQUID. The conversion of power and voltage fluctuations to current fluctuations via the power-to-current responsivity and the admittance of the TES circuit reveals a signature frequency dependence for each type of noise.

An additional internal noise source is sometimes detected due to internal thermal fluctuations between distributed heat capacities inside the TES.^{18,19} The frequency dependence of internal thermal fluctuation noise (ITFN) can sometimes mimic that of the Johnson noise. However, ITFN is well understood, such that when the TES parameters are carefully measured, it can usually be definitively separated from the unexplained Johnson noise.^{20} Also, ITFN can usually be suppressed below all other noise sources by using low resistance TESs like the ones used in this study. For more information on separating ITFN from excess Johnson noise, see the supplementary material.

Figure 2(a) shows the measured and predicted current noise based on the noise theory discussed above. The measured data show noise in excess of the noise predictions with the same frequency dependence as the TES Johnson noise. This excess noise, often referred to as “unexplained noise,” is the focus of this Letter.

Excess noise has been consistently observed in TESs^{21–23} since the first comparisons of detector data with noise theories for bolometers^{17} and microcalorimeters.^{24} For a more detailed discussion on excess noise, see reviews by Irwin and Hilton^{16} and Galeazzi.^{25} A number of explanations for the excess Johnson noise have been proposed including fluctuations due to vortex dynamics,^{26} fluctuations in the superconducting phase boundary,^{27} fluctuations in the superconducting order parameter,^{28} and by percolation models.^{29} However, none of these mechanisms give quantitative predictions consistent with the measured dependencies of the excess electrical noise.

In 2006, Irwin^{30} predicted an enhancement of the Johnson noise based on an analysis of a simple nonlinear resistive bolometer operated near equilibrium. This analysis predicted that the Johnson noise in a TES would be

where *β _{I}* is the logarithmic current sensitivity of the TES defined as

The parameters *R*, *β _{I}*, and

*α*, the logarithmic temperature sensitivity defined as

_{I}are commonly used to describe the *R*(*I*, *T*) surface in the small signal limit.^{16}

In some scenarios, the $(1+2\beta I)$ term has reasonably predicted measured noise. In many other scenarios, especially low in the transition and for devices with high *α _{I}* and

*β*, the $(1+2\beta I)$ expression dramatically underpredicts the measured noise.

_{I}^{31}The factor of $(1+2\beta I)$ is included in the predicted Johnson noise in Fig. 2. To derive Eq. (1), Irwin used a first order Taylor expansion of resistance as a function of temperature and current in place of a specific transition model, which may be insufficient to describe the regime that TESs are typically operated within.

^{36}

To predict electrical noise in a nonlinear device due to dissipation, a current–voltage (I–V) relationship is needed. As discussed above, this relationship can be calculated given the specific geometry and boundary conditions of the system. Unfortunately, the size and complexity of these devices makes these calculations intractable. As a result, the TES community typically compares its devices to one of two limiting cases: the resistively shunted junction (RSJ) model in the case of small weak-link like devices^{32,33} and the two-fluid model in larger devices.^{34,35} In the first case, at temperatures below the critical temperature of the leads, the superconducting order parameter from the leads extends into the film due to the longitudinal proximity effect, and the device can form an SS′S or SNS (superconducting-normal-superconducting) weak link, which can be modeled as a resistively shunted junction. In Fig. 1(a), the sketch of the superconducting order parameter is shown for the superconducting and normal states of a TES. In the second case, that of the two-fluid model, the resistance mechanism is the appearance of phase-slip lines in the film as the detector goes through the superconducting transition.

Regardless of what specific I–V relationship describes any given device, the drop in voltage across the superconducting film indicates that there will be an induced oscillating current at the Josephson frequency ($\omega J=2eV\xaf/\u210f$, where $V\xaf$ is the time averaged voltage). The presence of high-frequency oscillations and a nonlinear current–voltage relationship suggest that the noise could be mixed down from higher frequencies as is commonly seen in dc-SQUIDs and has been studied extensively in weak-link Josephson junctions.

The magnitude of the Johnson noise mixed down to low frequencies in weak-link junctions was calculated by Likharev and Semenov^{37} assuming the RSJ model. In this formulation, the primary intrinsic source of fluctuations is the normal current passing through the shunt resistor. When the constant total current bias exceeds the critical current, the supercurrent begins to oscillate at angular frequency *ω _{J}*, and the normal current traversing the junction is forced to oscillate at this frequency as well. With the normal current subjected to random fluctuations as well as a nonlinear current phase relationship, the resulting voltage fluctuations will consist of a component proportional to the current fluctuations along with components produced by the nonlinearity of the device. This means that the spectral density of the voltage noise at frequency

*ω*is determined by both the voltage noise at that frequency and the noise mixed-down to

*ω*by noise near the Josephson frequency (

*ω*) and its harmonics. They were able to derive the power spectral density of the voltage noise across the weak link at $\omega \u226a\omega J$. The result is

_{J}where *I* is the total current in the junction, *I _{c}* is the critical current of the junction, and

*R*is the dynamic resistance of the junction defined as $Rd=dV\xaf/dI|T$. In a resistively shunted junction, $SI(0)=4kBT/Rn$.

_{d}Inspired by the weak-link effects shown to be present in TESs, Kozorezov *et al.*^{36} calculated the spectral density of noise in TESs using the method of Coffey *et al.*^{38} However, in order to reach a calculable expression, Kozorezov *et al.* used a current–resistance relationship derived from the RSJ model, and the resulting spectral density was equivalent to Eq. (4).

Since the transition shape in most TESs is not reliably predicted by the RSJ model, we would like a theory for the mixed-down noise that is independent of how closely the V(I) relationship of the device follows this model. The current-phase relationship (CPR) of different types of junctions varies depending on the physics of the junction and the surrounding circuit. The classic RSJ model assumes a CPR of $Is=Ic\u2009sin\u2009\varphi $, but weak links and phase slip centers are known to display a variety of different periodic CPRs. The CPR alters the form of the I–V curve in general. Kogan and Nagaev^{39} derived a more general form for the mixed down noise expression using Nagaev's method,^{40,41} for an arbitrary CPR in the case $I>Ic$. The expression is

In the context of a TES, $dRd/dI$ is assumed to be evaluated at a constant temperature.

Lhotel *et al.*^{42} showed that measurements of long diffusive junctions at low voltage bias agreed with this theory, with the caveat that $SI(0)=4kBT/(V/I)$ instead of $SI(0)=4kBT/Rn$. This substitution of *R* for *R _{n}* seems to improve the prediction of noise in TESs as well and will be compared to data later in this paper.

In more familiar terms, Eq. (5) for a TES is

For comparison with data, we also write down the results of evaluating Eq. (6) for the RSJ model

and for the two-fluid model

The result from the two-fluid model is simply the first term of Eq. (6), and it aligns with the predicted Johnson noise for a current biased nonlinear resistor given by Stratonovich^{43} and applied to TESs by Irwin and Hilton.^{44} This expression is analogous to Eq. (1) and is based on the same physical principles but assumes a different circuit model. Low in the transition, where elevated levels of excess noise have been measured,^{45} the voltage bias of the TES is less stiff, and the expected Johnson noise may not be precisely Eq. (1). The expected thermodynamic noise from a TES with no mixed down noise effects could, therefore, be greater than Eq. (1) but not more than Eq. (8). The second term of Eq. (6) represents the magnitude of Johnson noise, which is mixed down from high frequencies.

Ideally, Eq. (6) could be applied to data directly by measuring the parameter $dRd/dI|T$. However, this is not a standard TES characterization parameter and is not easily measurable in a DC biased TES because of the electrothermal nature of the device. Neighboring points on the I–V curve are at slightly different temperatures, and the device is more sensitive to these small changes in temperature than it is to the small deviations in current between voltage bias points. Extracting the derivative by taking the slope of the *R _{d}* vs

*I*data along the I–V curve of the TES is, therefore, not a reliable way to calculate this term. It may be possible to measure this term accurately by using a larger amplitude input signal and measuring at frequencies well above the thermal roll-off frequency (

*G*/

*C*), so that the TES is held at a constant temperature. This type of measurement proves difficult to do with DC biased TESs like the ones discussed in this paper because there is an electrical roll-off at $f=R/L$ that inhibits measurement at high frequencies. Since a reliable method for measuring this derivative has not yet been developed, we simply compare the equations given by the RSJ and two fluid models to data and show that this is a viable theory which may be applied more directly to detectors in the future.

To compare Eqs. (4)–(8) to data, we extracted the relevant parameters from measurements on two TESs composed of MoCu bilayers with transition temperatures near 75 mK and normal state resistances around 20 mΩ. The input parameters for the noise theory were measured using standard techniques with the thermal conductance parameters extracted from power law fits to curves of bias power vs bath temperature, and the heat capacity and *α _{I}* and

*β*extracted from fits to the measured complex impedance vs frequency. Additional details are given in the supplementary material.

_{I}The extracted Johnson noise data divided by the Johnson noise of a linear resistor $(4kBTR)$ is shown in Fig. 3 as a function of bias point. The two TESs discussed here are excellent examples of the types of devices where the $(1+2\beta I)$ term fails to predict the measured noise. The two versions of Eq. (6) compare nicely with noise measurements. Though Eq. (8) may overestimate the true magnitude of the baseline nonlinear Johnson noise [previously defined as Eq. (1)] as discussed above, when the total excess Johnson noise exceeds Eq. (8), the noise is certainly in the regime of mixed down noise. Although Eqs. (8) and (7) do not perfectly predict the measured excess noise, the agreement is impressive. One would expect better agreement by using Eq. (6) with an accurate value for $dRd/dI|T$. The important role of $dRd/dI|T$ in the noise expression could explain measurements where dramatically different levels of excess noise are obtained for the same *β _{I}*,

^{31}since the amount of noise mixed down from higher frequencies depends not just on

*β*but also on how

_{I}*β*is changing across the transition.

_{I}The theory described above will help identify TESs with better noise characteristics, and therefore better energy resolution. It may also explain trends that we see in different detector designs. Detectors that act like weak links and have an *R _{d}* that changes continuously throughout the transition should display mixed down noise changing predictably over different bias points. Detectors displaying two-fluid behavior and evidence of phase slip lines will, in theory, have regions where $dRd/dI|T$ is zero, and therefore mixed down noise is zero. The trade-off will be more complex noise behavior from other sources around regions where phase slips are forming as certain bias points show instability. Here,

*R*may be difficult to measure or nondifferentiable, making this theory difficult to apply. If the spikes in noise are due to smoothly changing dynamic resistance, this theory ought to make an accurate prediction of excess noise due to mixed down Johnson noise.

_{d}In summary, we have introduced a model for excess noise in TESs based on high-frequency Johnson noise that is mixed down into the signal band by Josephson oscillations due to a nonlinear voltage–current relationship. The model is most useful in describing excess noise in detectors with $\beta I>1$ and significant $dRd/dI|T$. This model compares well with data assuming a specific transition model; however, the ideal test of this theory's validity will be using a correctly measured $dRd/dI|T$ to directly predict the magnitude of the excess Johnson noise. Our results give insight into a long-standing mystery of TES behavior and are likely to lead to improved TES designs for applications such as soft x-ray spectroscopy.

See the supplementary material for information on analysis of the data in this Letter and the method used to account for potential ITFN.

This work was supported by the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, and by the DOE NEUP and NASA APRA programs.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.