Magnetic field sensitivity of transition edge sensors

Here we describe measurements of the TES critical current $I_{\mathop{}\!\mathrm{c}}$ with magnetic field B_z normal to the plane of the TESs. For these measurements the bath temperature was regulated at a temperature close to $T_{\mathop{}\!\mathrm{c},i}$ to better than ± 0.1 mK. For clarity here and in what follows, we identify individual TES characteristics particularly $T_{\mathop{}\!\mathrm{c}}$ with an additional subscript e.g. $T_{\mathop{}\!\mathrm{c},i}$ to indicate that measurements apply to a specific device. Our measured $T_{\mathop{}\!\mathrm{c},i}$ varied with TES size. For each applied field, and each TES the bias voltage V_b was swept symmetrically about V_b = 0 V and $I_{\mathop{}\!\mathrm{c}}$ determined from the device current at the first onset of resistance for both positive and negative bias directions. The voltage sweep was carried out at a sufficiently low rate of 0.3 Hz to avoid thermal hysteresis, similar to Smith et al [19].

Assuming that the supercurrent density is uniform, a magnetic field B_z is expected to modulate the TES critical current such that [19]

$$\begin{equation} I_{\mathop{}\!\mathrm{c}} \left(\Phi_z\right) = I_{\mathop{}\!\mathrm{c}}\left(0\right) \left| \mathrm{sinc} \left(\pi \Phi_z/\Phi_0\right) \right|. \end{equation} \tag{ 1 }$$

Here $I_{\mathop{}\!\mathrm{c}}(0)$ is the critical current at zero field, $\Phi_z = B_z A_{\mathrm{eff}}$ is the magnetic flux through the plane of the TES and Φ₀ is the magnetic flux quantum. $A_{\mathrm{eff}}$ is an effective TES area defined further below. The variation of $I_{\mathop{}\!\mathrm{c}} (\Phi)$ is then the same as the Fraunhofer diffraction pattern. Equation (1) suggests that a smaller TES is less sensitive to a normal magnetic field B_z .

It is necessary to assume a value of A_eff to convert the known applied field to the flux $\Phi_z$. For a TES with order parameter modified by the proximity effect as shown in figure 2, it is not obvious that the effective area $A_{\mathrm{eff}}$ for the flux calculation is the same as the geometric bilayer area A, as indicated in figure 1(b). In particular, it might be expected that the contribution to A_eff from a particular area element will scale with the value of the superconducting order parameter at that point. The approach we take here is to instead determine A_eff for a given device as the area giving the best account of the central peak of the measured supercurrent oscillations. From the effective area we can define an effective side length $L_{\mathrm{eff}} = \sqrt{A_{\mathrm{eff}}}$.

Figure 4 shows the effective length calculated in this manner plotted as a function of true length for all of the devices tested in this study. For unpatterned devices with no bars the effective length is always smaller than, but comparable with, the actual length, with greater deviation seen for larger devices. The behaviour for devices with bars is more complicated and will be discussed in subsequent sections.

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3.1.1.  $I_{\mathop{}\!\mathrm{c}}(\Phi)$ for unpatterned devices.

Figure 5 shows the measured normalized critical currents $I_{\mathop{}\!\mathrm{c}}(\Phi)/I_{\mathop{}\!\mathrm{c}}(0)$ as a function of magnetic flux for a series of unpatterned TESs with different bilayer side lengths in the range 5–70 µm. The solid lines show the fitted dependence of $I_{\mathop{}\!\mathrm{c}}(\Phi)$ as given by equation (1). Measurements for each TES are plotted with a vertical offset of 0.5 between datasets where the coloured-matched dashed lines indicate the y-axis zero in each case. For the smaller devices, 20 µm side length and below, very good agreement is shown with the prediction, whilst the larger devices are less well described. In particular, the larger devices do not show well-defined minima in $I_{\mathop{}\!\mathrm{c}}$ when the flux is an integer of flux quanta.

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3.1.2. Effect of partial lateral bars on $I_{\mathop{}\!\mathrm{c}}(\Phi)$.

Figure 6 shows the measured the critical current as a function of magnetic flux for a series of TESs with 10 µm bilayers and increasing numbers of bars. Again the colour-matched horizontal dashed lines indicate the y-axis for the offset data sets. The currents have been normalised to $I_{\mathop{}\!\mathrm{c}}$ at zero flux and the magnetic flux has been calculated using the effective area that gave the best fit to the periodic structure in critical current. At first sight, figure 6 shows the expected Fraunhofer-like dependence of $I_{\mathop{}\!\mathrm{c}}(\Phi)$ with Φ for all of these 10 µm TESs. However, we find a clear difference between behaviour of $I_{\mathop{}\!\mathrm{c}}(\Phi)$ for between even and odd numbers of bars. For even numbers of bars $I_{\mathop{}\!\mathrm{c}}(m \Phi_0) \lt 0.02 I_{\mathop{}\!\mathrm{c}}(0)$ for integer m, as expected from equation (1). For odd numbers of bars we instead find $I_{\mathop{}\!\mathrm{c}}(m \Phi_0) \simeq 0.4 I_{\mathop{}\!\mathrm{c}}(0)$ for integer m, i.e. although they are local minima in the critical current, the points $\phi_z = m \phi_0$ for integer m are not the expected nulls.

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Empirically we found that the behaviour of the critical current for odd numbers of bars could be well described by modifying equation (1) with the addition of a quadratic term:

$$\begin{align} I_{\mathop{}\!\mathrm{c}} \left(\Phi\right) = I_{\mathop{}\!\mathrm{c}}\left(0\right) \left( a_1 \left| \mathrm{sinc} \left(f\left(\Phi\right)\right) \right| + \left(1-a_1\right) - a_2 \left(f\left(\Phi\right)\right)^2 \right), \end{align} \tag{ 2 }$$

where $f(\Phi) = \pi \Phi_z/\Phi_0$ and a_j are parameters of the model. The solid lines shown in figure 6(a) for 1 and 3 bars use a₁ = 0.7 and a₂ = 0.005. Despite the evidently different behaviour of $I_{\mathop{}\!\mathrm{c}} (\Phi)$ with bar number, all four TESs have similar effective lengths, as seen in figure 4.

The dependence of $I_{\mathop{}\!\mathrm{c}}(\Phi)$ on number of bars may arise from the geometry of the supercurrent flow in the different cases. Measurements on identical TESs with lateral bars that completely crossed the $\mathop{}\!\mathrm{S^{^{\prime}}}$ region have been carried out by Harwin et al [29]. These showed that the bar/bilayer structure itself remained in the normal state to temperatures at least to 80 mK, i.e. well below the temperatures of interest here, and with zero measurable supercurrent, despite the proximity of the bilayer. This implies negligible supercurrent current flow across the bars themselves, so partial bars should result in a meandering supercurrent pattern.

The self-magnetic field generated in the TES by a meandering current can be understood using a simplified Biot-Savart law calculation. Figure 7 shows B_z produced by a current of 5 µA following the path shown in red. Figure 7(a) shows the case with two normal metal bars and figure 7(b) the case with three normal metal bars. Critically, it can be seen that sign of the self-field alternates between adjacent current loops.

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The self-flux is the sum of the flux in each of these loops and adds to the applied flux, modifying the behaviour of I_c. With an even number of bars and therefore loops, we expect the sums of fluxes in adjacent pairs of loops to cancel; hence the self-field correction will be negligible. For an odd number of bars there is one unmatched loop, leading to a finite self-flux correction. Further, the scale of the correction should decrease as the applied B-field is increased and the self-field becomes less significant. This behaviour is consistent with the data.

Figure 8 shows the measured effect of adding three partial normal metal bars to a larger TES with a side length of 40 µm. The Fraunhofer pattern in $I_{\mathop{}\!\mathrm{c}}(\Phi)$ is evident for the device without bars, although the expected minima with Φ are not well defined. In the device with three bars the Fraunhofer pattern is absent and I_c is almost independent of $\Phi_z$ up to $3\Phi_0$. We explore this effect further in section 3.3.2.

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Figure 9(a) shows the critical currents of three of the smallest unpatterned TESs in zero field as a function of T_b . Figure 9(b) shows the same data and fits but expands the region around $T~\sim T_{\mathop{}\!\mathrm{c},i}$. The black lines show the expected temperature dependence of the current in a SS'S weak-link for $T\gt T_{\mathop{}\!\mathrm{c},i}$ given by both Golubov [30] and Sadleir [18]

$$\begin{equation} I_{\mathop{}\!\mathrm{c}}\left(T,L\right)\propto \frac{L}{\xi\left(T\right)} \exp\left( \frac{-L}{\xi\left(T\right)} \right), \end{equation} \tag{ 3 }$$

with $\xi(T) = \xi_i / \sqrt{T/T_{\mathop{}\!\mathrm{c},i} - 1}$. For the three TES lengths, we find a value ξ_i = 450 ± 20 nm and $T_{\mathop{}\!\mathrm{c}, i}$ = 197 ± 3 mK. This gives an excellent account of the measured supercurrents in this regime. Below $T_{\mathop{}\!\mathrm{c},i}$, I_c is well accounted for by the expected Ginzburg–Landau dependence $I_{\mathop{}\!\mathrm{c}}(T) \propto (1-T/T_{\mathop{}\!\mathrm{c},i})^{3/2}$. This dependence is shown by the dashed black line in figure 9(b) for the 20 µm TES. Using a free electron model with Fermi velocity $v_\textrm{F}$ = 1.39 × 10⁶ ms⁻¹, and the measured resistivity for our 120 nm thick Au films ρ = 1.33 × 10⁻⁸ Ωm, we calculate a normal-state electronic diffusion coefficient D = 2.80 × 10⁻² m²s⁻¹. Using $\xi_n = \sqrt{\hbar D/2\pi k_{\mathrm{B}}T_{\mathop{}\!\mathrm{c}}}$, we calculate ξ_n = 450 nm, in good agreement with the measurement. These observations are consistent with [18] and we identify $\xi_i\equiv \xi_n$.

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Sadleir et al [16, 18] investigated the variation of measured $T_{\mathop{}\!\mathrm{c}}$ with geometry and field for TESs at temperatures $T \gt T_{\mathop{}\!\mathrm{c}, i}$ where $T_{\mathop{}\!\mathrm{c}, i}$ is the temperature corresponding to the mid-point of the resistive transition, $R = R_{\mathop{}\!\mathrm{n}}/2$. Those measurements were performed on unreleased TESs, for which the thermal conductance to the heat bath is high and TES Joule power heating with current bias $P_\textrm{J} = I_{b}^2R$ is relatively unimportant for low bias currents I_b . Here we report $T_{\mathop{}\!\mathrm{c}, i}$ for TESs with thermal isolation from the substrate under voltage bias using the measured bias power at $R_0 = R_{\mathop{}\!\mathrm{n}}/2$ as a function of applied field in order to determine $T_{\mathop{}\!\mathrm{c}, i}(B_z)$. This approach explicitly explores the operating regime of a TES under voltage bias.

Figure 10 shows measurements of the TES Joule power determined at $R = R_{\mathop{}\!\mathrm{n}}/2$ for an unpatterned TES with 70 µm side length. These $P_\textrm{J}(T_b)$ curves were parametrized using $P_\textrm{J}(T_b) = K (T_{\mathop{}\!\mathrm{c},i}^{\,n} -T_b^{\,n})$, assuming that parameters K and n, that describe the heat flow to the heat bath [23] at temperature T_b , do not change with field for a particular TES. This procedure was used to determine $T_{\mathop{}\!\mathrm{c}}(B)$ for a range of TES lengths. The inset of figure 10 shows example $P_\textrm{J}(V_b)$ curves at three different applied magnetic fields for this TES, all measured at fixed T_b = 75 mK. The $P_\textrm{J}(V_b)$ plot at the maximum field shown here, B = 50 µT, would correspond to $\Phi/\Phi_0 \sim 15$, well-beyond the range of data shown in figure 5.

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Interestingly, although the field-dependent maxima of the supercurrents shown in figure 5 closely-follow the expected Fraunhofer-like dependence, the Joule power for the voltage-biased TES is changed only modestly (∼5%) with B. Additionally there is evidence of change in the overall $P_\textrm{J}(V)$ with B that would be consistent with reduced α with increasing field.

Figure 11 shows how $T_{\mathop{}\!\mathrm{c}}(B)$ changes with applied field B_z . Figure 11(a) shows $T_{\mathop{}\!\mathrm{c}, i}(B_z)$ for three unpatterned TESs with geometric length $\gt$ 20 µm. $T_{\mathop{}\!\mathrm{c},i}$ is reduced almost linearly with applied B_z . It is interesting to investigate this observation in terms of flux-flow resistance in a type II superconductor [31]. The applied field would then be identified as the critical field $B_{\mathop{}\!\mathrm{c},i}$ that gives the measured $T_{\mathop{}\!\mathrm{c},i}$ for each TES. The gradients of the linear fits to the data, $\partial B/\partial T |_{B\to0}$, with $B_{\mathop{}\!\mathrm{c}} = \phi_0/2\pi\xi^2$, [31] and $\xi (T) = \xi / \sqrt{1-T/T_{\mathop{}\!\mathrm{c}}}$ [32], give an estimate of ξ within this model. For the 40 µm and 70 µm TESs the correlation coefficients are high, $R^2\gt 0.99$, and we find that $\xi = 470 \pm 20$ nm would account for the field dependence of $T_{\mathop{}\!\mathrm{c},i}$. For the 20 µm TES the same calculation gives ξ = 520 ± 20 nm although the correlation is lower, $R^2\simeq 0.95$. At temperatures above the intrinsic TES transition temperature we might expect $\xi = \xi_n$, the normal-state correlation length, indeed the derived ξ_n would agree well with our earlier interpretation of the data shown in figure 9.

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Interpretation of the measurements of $T_{\mathop{}\!\mathrm{c}}(B)$ for smaller 5 µm and 10 µm TESs, shown in figure 11(b), in terms of flux-flow resistance is less compelling. This would be expected if $L \lt 3.49 \, \xi(T)$ [32]. The fluctuations in $T_{\mathop{}\!\mathrm{c},i}(B)$ appear more closely related to the Fraunhofer patterns of the underlying I_c as shown in the inset of figure 11(b). For the smaller TESs J_c is reduced to close to zero at the expected minima.

3.3.1. Effect of applied field on the TES bias current.

Figure 12 shows the relationship between the oscillations seen in transition temperature and bias current measured with increasing field. For the four 10 µm side length TESs studied, the change in $T_{\mathop{}\!\mathrm{c}}/T_{\mathop{}\!\mathrm{c}}(0)$ is plotted on the same axes as the normalised change in current $\mathop{}\!\Delta I_{\mathrm{bias}}/\mathop{}\!\Delta I_{\mathrm{bias}}(0)$. There is good agreement between the two sets of oscillations for all four TESs, suggesting that the same effects determine the period of oscillation of both critical current and bias current. For the TES with two partial bars, the broad minimum in $\mathop{}\!\Delta I_{\mathrm{bias}}$ at around 30–40 µT corresponds to a maximum in $T_{\mathop{}\!\mathrm{c}}$. The oscillations in $I_{\mathop{}\!\mathrm{c}}$ in figure 6 also show a maximum around this region, suggesting that another physical process may be suppressing the maximum in $\Delta I_\text{bias}$.

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3.3.2. Effect of magnetic field on TES responsivity.

In this section we consider the effect of magnetic fields on TES responsivity when used as a power detector. To do so it is first useful to consider how the calculation of the responsivity is modified in the presence of a magnetic field. Magnetic field affects the responsivity of a TES directly through its effect on the bias current (section 3.3.1).

TES are usually operated with voltage bias. However, the most-commonly used TES voltage-bias and readout circuit, as used here, is not an ideal voltage source [23]. In the standard circuit analysis, the combination of a low valued bias resistor R_b (typically ∼ 1 mΩ) and any stray resistance R_stray are represented by a load resistance R_L . This means that the bias voltage of the TES V_b changes if the TES current changes irrespective of details of the TES behaviour itself. For simplicity here we assume that the TES electrothermal parameters are near-ideal (α is large, β → 0) for all bias points. With these assumptions and for a TES used as a bolometer, the small-signal power-to-current responsivity of the TES in a small-signal analysis at an initial bias point is $s_{I,0} = 1/(I_0(R_0-R_L))$ [23]. In this analysis we identify $I_0\equiv I_{b}$ for consistency with the complete small-signal analysis of [23]. If for example a magnetic field changes the bias current so that $I_1 = I_0-\Delta I(B)$, the responsivity $s_{I,1}$ is also changed. Circuit analysis shows that

$$\begin{equation} \frac{s_{I,1}}{s_{I,0}} = \frac{I_0\left(R_0-R_L\right)}{I_0\left(R_0-R_L\right)-2\Delta I\left(B\right)R_L}. \end{equation} \tag{ 4 }$$

Note that for an ideal voltage bias $s_{I,1}/s_{I,0} = 1$ irrespective of $\Delta I(B)$. This responsivity is most-relevant for quantifying small-signal power detection in a bolometer, but similar considerations would apply for a calorimeter.

Figure 13(a) shows the responsivity as a function of magnetic field for a series of square TESs of 10 µm side length. The oscillations in responsivity are of similar magnitude for all of the TESs. Figure 13(b) shows the responsivity as a function of magnetic field for two 40 µm TESs. These larger devices display the smallest changes in responsivity with field, and the addition of partial normal metal bars further reduces s_I . Figure 13(b) also shows additional discontinuities near ± 3 µT and ± 15 µT that maybe indicative of phase-slip behaviour. These features were very reproducible as a function of applied field, with finer sampling with field, and even between cool-downs. The observation that larger TESs may display phase slip behaviour is consistent with the existing theoretical ideas [25, 30, 32]. These discontinuities in $I_b(B)$ were not observed in similar larger TESs with additional lateral bars.

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The existing theoretical work does not, however, predict the result that the smallest TESs display the largest changes in current with flux, as these TESs have less magnetic flux threading them, which would imply that any changes in current with flux would be smaller. This suggests that there may be non-uniformities in the larger TESs, such as vortices, which reduce magnetic flux penetration and hence magnetic field sensitivity.

3.3.3. Directional dependence of bias and critical current on applied field.

To investigate the directional magnetic field dependence of the bias current of the TESs at fixed V_b , the magnitude of the applied magnetic field was fixed and its direction varied. Using the full three-axis Helmholtz system, fixing $\vert \mathbf{B} \vert$ defines a sphere of fixed radius, which can be parametrised in terms of azimuthal and polar angles. In order to ensure efficient sampling, the increment of the azimuthal angle was fixed and the increment of the polar angle was varied to sample elements of equal area.

For the general field direction,

$$\begin{equation} \mathbf{B} = B_x \hat{\mathbf{x}}+B_y \hat{\mathbf{y}}+ B_z \hat{\mathbf{z}}, \end{equation} \tag{ 5 }$$

Equation (1) can be extended so that the critical current is [33]

$$\begin{equation} I_{\mathop{}\!\mathrm{c}}\left(\Phi\right) = I_{\mathop{}\!\mathrm{c}}\left(0\right) \prod_{i = x,y,z} \left| \mathrm{sinc} \left( \frac{\pi \Phi_i}{\Phi_0}\right) \right| . \end{equation} \tag{ 6 }$$

As B_x is parallel to the direction of current flow, the effect of $\Phi_x$ can be neglected. Figure 14 shows the model prediction of the change in critical current with the direction of a 50 µT applied field, for a 10 µm side length TES. The direction of the applied field is scaled by the resulting change in critical current, so for a TES that is equally sensitive to applied fields in all directions, the response surface would be spherical. The lobed response surface indicates that the thin film is most sensitive to magnetic fields applied perpendicular to the plane of the TES, as expected. The shading of the surface corresponds to the magnitude of the response. This model also predicts oscillations in the response, which can be seen both in the response surface and the cuts in the x − z and y − z planes shown underneath. The response is minimal when the applied field is in the x − y plane. The model suggests the general behaviour for I_b of a real TES as a function of field orientation, although experimentally (compare for example figures 5 and 12) although we have already shown that the correlation is imperfect.

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Figure 15 shows the measured directional dependence of the change in bias current for 10 µm and 40 µm TESs either plain or with three lateral bars. The scans were taken at a temperature of 90 mK and a bias point of $0.5\,R_{\mathop{}\!\mathrm{n}}$, with a field magnitude of 20 µT. In each plot, the change in bias current at a given angle has been normalised by the maximum value recorded for all angles so as to emphasise the directional dependence. The colour of the points corresponds to the normalised total magnitude of response. Device geometry is indicated by the cartoon in the top right-hand corner of each plot. TES size increases across the columns of the figure and the number of bars down the rows.

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The smaller unpatterned device shows the simplest directional behaviour: figure 15(a). This has a clean response surface, with a two-lobed form and with no visible oscillations (or ripples) with angle. The asymmetry in lobe sizes is most likely due to imperfect cancellation of the contribution from the earth's magnetic field and was seen for the other devices.

The larger unpatterned device, figure 15(b), displays more complicated behaviour. Ripples are visible in the response surface as the field angle is changed. This links to figure 13(a), which shows that for changes in magnetic field around and below 20 µT perpendicular to the plane of the thin film, S_I (hence I_b ) changes monotonically with field for an unpatterned TES but that a TES with three normal metal bars is operating close to a local maximum in S_I (minimum in I_b ). The relatively large oscillations in the response surface for this TES in an applied field of 20 µT arise from the local structure in $I_b(B_z)$ as the field direction is changed.

It can be seen by comparing the two rows of figure 15 that the addition of normal metal bars significantly reduces the directional magnetic field dependence of the TES about the z-axis. This is indicated by the broadening of the main response lobes, which are not described by the simple model in figure 14. In all cases the response in B_y and B_x is below our estimated experimental measurement error [29].

The key results of our measurements are summarised below, grouped by device parameter. For these Mo/Au TESs we observed different behaviour depending on whether the geometric side length was above or below ∼20 µm and we distinguish these two types of device as 'large' and 'small' respectively.

• 1.  
  Field and temperature dependence of the critical current, I_c : the behaviour of I_c with field and temperature is important because of its expected influence on TES operating parameters, such as current in bias I_b . All devices showed some degree of Fraunhofer-like fluctuations in I_c with applied field B_z . The small, unpatterned, devices showed behaviour closest to that of an ideal short Josephson junction demonstrating well-defined zeroes of I_c with field. Increasing the side-length and adding partial lateral bars were both found to reduce the modulation depth of the pattern. Very different behaviour was observed in small patterned devices with odd and even numbers of bars that we attribute to the effect of self-field. In all cases the effective area of the device as determined from the change in I_c with applied flux was smaller than the physical area of the TES. The temperature dependence of I_c above T_c was well-described by the description of a SS'S weak-link above T_c given in Golubov et al [30] that was derived from the Usadel model. We found a characteristic length ξ = 450 ± 20 nm that was consistent with the calculated normal-state correlation length of the S' bilayer. The same conclusion was also reached by Sadleir et al [18]. Below T_c , $I_c (T)$ followed the expected Ginzburg–Landau dependence for all TES sizes.Our measurements of I_c were carried out at fixed bath temperature with $T_{b}\simeq T_{c}$. We would expect differences in the magnitude of I_c under voltage-biased operation when $T_{b}\sim 0.5T_{c}$, and where Joule heating raises the operating temperature of the TES above T_b such that the actual TES operating temperature may indeed be lower than T_c . However, the functional dependence of the behaviour should be similar.
• 2.  
  Field dependence of the operating point of a voltage-biased TES with electrothermal feedback: all measurements were made on membrane suspended devices, so as to be representative of sensitive bolometric/calorimetric detectors for low incident powers or energies (and having low noise equivalent powers or high energy resolution respectively).Superconducting critical temperatures T_c were found to decrease with applied field by about 150 KT⁻¹ and 500 KT⁻¹ for large and small devices respectively. For larger TESs, the reduction in T_c could be modelled in terms of the effect of flux-flow resistance using a characteristic length ξ = 470 ± 20 nm, again consistent with ξ being the normal-state correlation length of the S' bilayer.The field dependence of I_b was found to be similar that of I_c , consistent with assumptions above. The Joule power in bias was found to be only weakly dependent on applied field, changing by ≈5% in a 50 µT field for a large TES.
• 3.  
  Dependence of the operating current, I_b on field orientation: we measured the dependence of the change in I_b on field direction and found significantly reduced sensitivity to in-plane fields, as expected. We found that smaller TESs were most sensitive to applied fields. The addition of lateral bars also reduced the angular dependence.
• 4.  
  TES power-to-current responsivity with applied field: TES power-to-current responsivity (change in current for a given change in input power) was shown to vary with field in all devices. Variations of order 10% with field were measured for small devices, with little reduction when bars were added. The variations were reduced in larger devices (∼ 2% for a 40 µm device) and the addition of three bars was found to suppress them nearly entirely. However, the larger device without lateral bars showed additional, reproducible, discontinuities in sensitivity (and bias current) not present in the small devices, which we attribute to the release or movement of pinned phase slip features.

The measurements suggest routes towards optimal TES designs for future applications with respect to the effect of magnetic field.

We will primarily use the behaviour of the bias current $I_b(B)$ hence the power-to-current responsivity S_I to inform our recommendations, as it is most directly relevant to detector operation. In addition, the effects of the magnetic field on other parameters, like critical temperature and current, are primarily felt through their effect on this $I_b(B)$. In this paper we have not attempted to model the behaviour of $I_b(B)$ in terms of, for example, the resistively shunted Josephson (RSJ) model [15, 19, 24] or a two-fluid model [34] but the key-point is that both require T_c and I_c as inputs in the modelling of I_b .

Based on the observations reported our overall recommendations are listed below:

• Zero field should be the goal in all applications. B and changes in B should be minimized. But this may not be a realistic solution.
• The smallest TESs are least sensitive to changes in field near $B_z = 0$. See equation (1). Our measurements confirm this.
• TESs are less sensitive to changes in B_y and B_x as expected: see equation (1). Our measurements confirm this. Routing of cabling harnesses or positioning of apertures, for example, should consider this observation.

For operation in non-zero fields with fluctuations (that may occur or be unavoidable in applications) the comments depend on the magnitude of the fields but generally:

• Smaller TESs are more sensitive to changes in applied field see $S_I(B)$ shown in figure 13.
• The larger TESs measured here were less susceptible to field (and variations).
• But for larger TESs we also observe discontinuities in I_b hence S_I with B_z that would seriously compromise detector operation within a varying stray field.
• The addition of lateral normal metal bars further reduces the magnetic field sensitivity in I_b hence S_I for larger TESs, and for these TESs no discontinuities in I_b were observed.

It is important to emphasise that these considerations do not represent an optimization for general TES design. Rather we expect the issues to be material, operating temperature and geometry dependent.