Design considerations for the optimization of {\lambda}-SQUIDs

Cryogenic microcalorimeters are key tools for high-resolution X-ray spectroscopy due to their excellent energy resolution and quantum efficiency close to 100%. Multiple types of microcalorimeters exist, some of which have already proven outstanding performance. Nevertheless, they can't yet compete with cutting-edge grating or crystal spectrometers. For this reason, novel microcalorimeter concepts are continuously developed. One such concept is based on the strong temperature dependence of the magnetic penetration depth of a superconductor operated close to its transition temperature. This so-called ${\lambda}$-SQUID provides an in-situ tunable gain and promises to reach sub-eV energy resolution. Here, we present some design considerations with respect to the optimization of such a detector that are derived by analytic means. We particularly show that for this detector concept the heat capacity of the sensor should match the heat capacity of the absorber.


λ-coil I in
Fig. 1 Simplified equivalent circuit diagram of a λ-SQUID.Inductors depicted in black are made from a superconductor with critical temperature Tc.The λ-coil displayed in blue is made from a different superconductor with critical temperature T c,λ ≪ Tc, and is in strong thermal contact with the absorber.The device is operated at temperature T 0 ⪅ T c,λ .A constant current I in is injected into an input coil with inductance L in and causes a temperature dependent magnetic flux within the SQUID loop via the a temperature dependent mutual inductance M in (T ).

Introduction
Cryogenic microcalorimeters such as superconducting transition-edge sensors (TESs) [1,2] or metallic magnetic calorimeters (MMCs) [3,4] have proven to be outstanding tools for measuring the energy of X-ray photons with unprecedented precision.They rely on sensing the change in temperature of an X-ray absorber upon photon absorption using an extremely sensitive thermometer that is based either on a superconducting (TES) or paramagnetic (MMC) sensor material.Due to their unique combination of excellent energy resolution and quantum efficiency close to 100 %, they offer significant advantages as compared to state-of-the-art grating or crystal X-ray spectrometers [5].Their quantum efficiency significantly relaxes the requirements on X-ray beam intensity, which especially benefits measurements on strongly diluted or radiation sensitive samples [6,7].Moreover, they cover the entire tender X-ray range [5] that is hardly accessible with both, grating and crystal spectrometers.State-of-the-art TES-and MMC-based X-ray detectors achieve an energy resolution (FWHM) of 0.72 eV for 1.5 keV photons [8] and of 1.25 eV for 5.9 keV photons [9] at near-unity quantum efficiency.However, despite of various ongoing developments, all targeting to improve the performance of these microcalorimeters, an energy resolution as low as of 100 meV, required for investigating vibrations or d-d-excitations in soft X-ray spectroscopy or resonant inelastic X-ray scattering, has yet to be demonstrated.Against this background, we have recently proposed a novel type of microcalorimeter, called λ-SQUID, that promises to provide the required energy resolution [10].
Fig. 1 shows a simplified equivalent circuit diagram of a λ-SQUID.Similar to a conventional dc-SQUID, it consists of a superconducting loop interrupted by two Josephson tunnel junctions, each with critical current I c , capacitance C JJ and normal state resistance R. The superconducting loop is divided into two parts, i.e. a section L sfb that couples to a flux biasing coil and a section with inductance L λ , denoted as λ-coil, which couples to an external input coil.Most parts of the device, including both, the input coil and flux biasing coil, the section L sfb , the Josephson junction electrodes, and the junction wiring, are made from a superconductor with a transition temperature T c much larger than the device operating temperature T 0 , i.e.
T c ≫ T 0 .In contrast, the λ-coil is made from a different superconductor with a transition temperature T c,λ that barely exceeds the operating temperature, i.e.T c,λ ⪆ T 0 .As a consequence, the magnetic penetration depth λ(T ) of the λ-coil and hence the current distribution within the cross-section of the λ-coil shows a strong temperature dependence that affects both, the inductance L λ (T ) of the λ-coil as well as the mutual inductance M in (T ) between the λ-coil and the input coil.Assuming that a constant current I in is injected into the input coil, the flux induced into the λ-SQUID then also becomes temperature sensitive: Φ(T ) = I in M in (T ).As in a typical dc-SQUID, this change in flux can be measured as a change in voltage across or current through the λ-SQUID, depending on the mode of operation.By bringing a suitable particle absorber with specific heat C abs into close thermal contact with the λ-coil, the temperature rise upon particle absorption can be accurately detected and measured.

Design considerations of the λ-coil
As the central sensing element, the λ-coil has an enormous influence on the performance of the λ-SQUID.Assuming a sophisticated readout chain in which the noise of subsequent amplifiers do not affect the noise performance of the λ-SQUID, e.g. by using an N -dc-SQUID series array as a first-stage low-temperature amplifier, the energy resolution ∆E FWHM of a λ-SQUID is set by two noise contributions, i.e. thermodynamic energy fluctuations S E,TD caused by random energy fluctuations among the absorber, sensor and heat bath, and the noise contribution S E,SQ from the λ-SQUID itself.In the following, we will derive the conditions for the λ-coil which minimize the λ-SQUID noise contribution S E,SQ .
We assume a hypothetical λ-SQUID as schematically depicted in Fig. 2. For simplicity, the flux bias coil is neglected as it does not affect the temperature sensitivity.We consider a λ-coil entirely separable into small, identical unit elements, each with inductance L and volume Ṽ .The loop comprises N parallel rows of M unit elements in series each, resulting in a total of N × M unit elements in total.Each unit element couples to a segment of input coil with inductance Lin via a mutual inductance Min = κ L Lin .Here, κ is the geometric coupling factor.We can thus conclude the following relations for the total loop inductance L and total volume V of the λ-coil and the combined mutual inductance M in between the λ-SQUID and the input coil: With only two free parameters to choose, we see immediately that we can not choose all three quantities independently.To quantify the remaining dependence, we have introduced the two parameters ν = M/N and µ = M N that fully define a specific layout of the λ-coil.
We use the geometric parameters ν and µ to express other relevant parameters: First, we assume that our λ-SQUID is optimized in terms of noise performance [11], In each expression, the term in brackets is independent of the specific design of the λ-SQUID, and depends only on fabrication parameters and the design of a unit element of the λ-coil.
The noise contribution S E,SQ caused by the λ-SQUID, expressed in fluctuations of the energy content, is given by the expression Here, S Φ denotes the magnetic flux noise of the SQUID.Moreover, ∂Φ ∂T and ∂T ∂E = 1/C tot are the temperature-to-flux transfer coefficient of the λ-SQUID and the inverse total heat capacity of the detector, respectively.For an optimized dc-SQUID, the flux noise can be approximated by S Φ ≈ 18k B T L 2 /R at the operating temperature T [12].The total heat capacity of the detector comprises both, the absorber and the λ-coil, i.e.C tot = C abs +µ Ṽ c, with c the specific heat per unit volume of the λ-coil.
The temperature-to-flux transfer coefficient ∂Φ ∂T of the λ-SQUID is given by where, again, the term in brackets does not depend on the design of the λ-coil as a whole, but rather on the unit element.Using Eq. 4 and Eq. 5 we obtain for the noise contribution Here, we have introduced the substitutions We see that S E,SQ neatly separates into a term η which depends only on the unit element, but is independent of the specific layout of the λ-coil, and a function g (µ), which contains the dependence of the λ-SQUID noise S E,SQ on the arrangement of unit elements that comprises the λ-coil.It is interesting to note that only µ appears here, and that the parameter ν has dropped out.This can be understood as follows: An increase in ν causes a proportional increase in total inductance L and a rise of M in ∝ √ ν.While the latter results in an increase of detector signal, the former affects the flux noise negatively.Ultimately, the signal-to-noise of the λ-SQUID remains unaltered by a change of the inductance L by varying the parameter ν as introduced above.
To minimize the noise level S E,SQ , we only have to consider the design parameter µ.Since the prefactor η has no influence on this optimization, we restrict our efforts to the function g (µ) and find for its first and second derivative.In this way, we find that the function g (µ), and thus the λ-SQUID energy noise contribution S E,SQ , has a minimum if the condition µ = µ opt is satisfied, with Thus, we can conclude that the layout of the λ-coil should be chosen such that the total specific heat of the λ-coil exactly equals the specific heat of the absorber C sens = C abs .This resembles a well-known result for cryogenic microcalorimeters [13].

Conclusion
The recently proposed λ-SQUID is a superconducting microcalorimeter with in-situ tunable gain and, with a proper choice of absorber and sensor material, promises to reach sub-eV energy resolution.In preparation for such a demonstration, we have presented theoretical design considerations related to the optimization of the layout of the λ-coil, which is the fundamental sensing element of a λ-SQUID.By sub-dividing the λ-coil into a large number of small unit elements, we could abstract the layout of any possible design of the λ-coil for a given fabrication method, and describe the remaining degrees of freedom in the design process by two parameters µ and ν.When considering the influence of these parameters on the energy noise contribution S E,SQ of the λ-SQUID, we draw two conclusions.First, the total inductance L of the λ-coil has no effect on the resulting energy noise, and can thus be chosen freely.Second, the noise contribution S E,SQ is minimized if the total volume of the λ-coil is chosen such that its specific heat directly equals that of the X-ray absorber: C sens = C abs .

Fig. 2
Fig.2Schematic circuit diagram of a hypothetical λ-SQUID.The SQUID loop consists of N identical segments connected in parallel, with each segment itself comprising M unit elements connected in series.One such unit element is framed in red.Each unit element has an inductance L with volume Ṽ , and couples to a unit of input loop with inductance Lin with a mutual inductance Min .The total arrangement is equivalent to a single effective λ-coil with inductance L, volume V and mutual inductance M in to an input coil with inductance L in .