Calculations of Some Doping Nanostructurations and Patterns Improving the Functionality of High-Temperature Superconductors for Bolometer Device Applications

nanomaterials Article Calculations of Some Doping Nanostructurations and Patterns Improving the Functionality of High-Temperature Superconductors for Bolometer Device Applications Jose C. Verde 1, Alberto S. Viz 1,2, Martín M. Botana 1,2, Carlos Montero-Orille 2,3 and Manuel V. Ramallo 1,2,* 1 Quantum Materials and Photonics Research Group, Department of Particle Physics, University of Santiago de Compostela, ES-15782 Santiago de Compostela, Spain; [email protected] (J.C.V.); [email protected] (A.S.V.); [email protected] (M.M.B.) 2Strategic Grouping in Materials AeMAT, University of Santiago de Compostela, ES-15782 Santiago de Compostela, Spain; [email protected] 3 Quantum Materials and Photonics Research Group, Department of Applied Physics, University of Santiago de Compostela, ES-15782 Santiago de Compostela, Spain *Correspondence: [email protected] or mv[email protected]; Tel.: +34-881813965 Received: 26 November 2019; Accepted: 30 December 2019; Published: 3 January 2020   Abstract: We calculate the effects of doping nanostructuration and the patterning of thin films of high-temperature superconductors (HTS) with the aim of optimizing their functionality as sensing materials for resistive transition-edge bolometer devices (TES). We focus, in particular, on spatial variations of the carrier doping into the CuO 2 layers due to oxygen off-stoichiometry, (that induce, in turn, critical temperature variations) and explore following two major cases of such structurations: First, the random nanoscale disorder intrinsically associated to doping levels that do not maximize the superconducting critical temperature; our studies suggest that this first simple structuration already improves some of the bolometric operational parameters with respect to the conventional, nonstructured HTS materials used until now. Secondly, we consider the imposition of regular arrangements of zones with different nominal doping levels (patterning); we find that such regular patterns may improve the bolometer performance even further. We find one design that improves, with respect to nonstructured HTS materials, both the saturation power and the operating temperature width by more than one order of magnitude. It also almost doubles the response of the sensor to radiation. Keywords: superconducting devices; photodetectors; nanostructured materials, nanostructured and microstructured superconductors; high temperature superconductors; bolometers 1. Introduction Bolometers are radiation sensors that detect incident energy via the increase of the temperature T caused by the absorption of incoming photons [ 1 – 16 ]. Bolometers are often used, e.g., for thermal infrared cameras (see, e.g., [ 1 – 10 ]), mm-wave sensing [ 1 – 6 , 8 – 13 ], space-based [ 12 , 13 ], laboratory far-infrared spectroscopy [ 13 ], etc. [ 1 – 16 ]. Superconductors are among the best candidate materials for bolometers, due to their extreme sensitivity to T near the superconducting transition, measurable for instance through the sharp variations of the electrical resistance R (resistive transition-edge bolometer—TES). Nanomaterials 2020,10, 97; doi:10.3390/nano10010097 www.mdpi.com/journal/nanomaterials Nanomaterials 2020,10, 97 2 of 20 For resistive TES bolometers, a key figure for performance is the so-called “temperature coefficient of resistance” (TCR), given by [1–19]: TCR = 1 R dR dT . (1) High bolometric sensitivity requires a large value of TCR. For instance, structures of vanadium oxides VxOy , commonly used in semiconductor-based bolometers, present TCR ∼ 0.025 K−1 [ 20 ]. Much larger TCR may be achieved with superconductor materials kept at base temperatures coincident with their normal–superconducting transition, Tc . This is the case mainly when using conventional low-temperature superconductors with Tc≤ 1 K (the so-called low- Tc TES bolometers), that achieve TCR ∼ 1000 K −1 or even more [ 17 – 19 ], making them a technology of choice for detecting the most faint radiations, as the cosmic infrared background or in quantum entanglement and cryptography applications [ 17 – 19 ]. Note that for these measurements the very low temperature required to operate the low- Tc TES is often not seen as a major problem, because cryogenizing the sensor below a few Kelvin is required anyway in order to minimize the thermal noise coming from the bolometer itself. However, the requirement of a highly-stabilized liquid-helium-based cryogenics is a serious difficulty for adoption of low-TcTES in other applications. After the discovery of high- Tc cuprate superconductors (HTS), various authors have explored their use for resistive bolometers with simpler liquid-nitrogen-based cryogenics (the so-called resistive HTS TES bolometers [ 1 – 16 ]). The compound YBa2Cu3Oδ (YBCO) is the HTS material usually considered for this application, usually with maximum- Tc doping, i.e., stoichiometry δ≃ 6.93. Such YBCO thin films provide TCR ∼ 1.5 K−1 , low noise at an operational temperature Tc∼ 90K, and also favorable values for the rest of parameters contributing to good bolometric performance (thermal conductivity, infrared absorbance, response time, etc.) [1–16]. Besides the cryogenics, the other difference with respect to low- Tc TES is that in actual implementations [ 1 – 16 ] the resistive HTS TES operate under current bias and usually in the ohmic regime (instead of non-ohmic resistance and the voltage bias employed in low- Tc TES to avoid thermal runaways [17–19]). However, the HTS TES until now proposed still share some of the significant shortcomings of low- Tc TES: First, thermal stability of the cryogenic bath is still challenging (liquid-nitrogen systems are simpler but tend to thermally oscillate more than those based on liquid helium). Secondly, both types of TES have useful TCR only at the superconducting transition, corresponding to operational temperature intervals ∆T of just ∼ 0.1K or less for low- Tc TES, and ∼ 1K for the resistive HTS TES proposed until now [1–19]. The HTS TES systems proposed until today are homogeneous in nominal composition and critical temperature [ 1 – 16 ]. However, in the recent years different novel techniques have been developed to impose regular patterns on HTS thin films, creating custom designs, down to the micro- and the nano-scales [ 21 – 30 ]. This allows custom-engineering regular variations of the critical temperature over the film surface. Realization of these regular and controlled patterning has been experimentally achieved using, e.g., local ferroelectric field-effect [ 21 ], nanodeposition [ 22 ], focused ion beam [ 23 ], etc. In fact, the nanostructuring of HTS has become the specific subject of recent conferences [ 24 – 27 ] and networks [28,29] funded by the European Union. However, the use of nanostructured films for optimizing HTS TES has been considered only very marginally up to now, the only precedent to our knowledge being Reference [6] by Oktem et al., who considered films with random distributions of nonsuperconducting incrustations producing limited increases of ∆Tup to only ∼2 K (and also small, and not always favorable, TCR variations). Our aim in the present work is to propose that certain custom nanostructurating and patterning of HTS materials may improve their functionality for resistive HTS TES sensors. In particular, we calculate the case of nanostructuring and patterning of the local carrier doping level p (the number of carriers per CuO 2 unit cell) in the prototypical HTS compound YBCO via local variations of oxygen stoichiometry (as realizable, e.g., via local desoxigenation, ion bombardement with different masks, etc.). Our main Nanomaterials 2020,10, 97 3 of 20 objective will be to obtain an increase of the operational temperature interval, ∆T , in which (i) the TCR is large and (ii) R is linear with T (i.e., dR/dT constant with T , that is another desirable feature that simplifies both the electronic control of the bolometer and the required stability of the cryogenic setup). Accompanying this ∆T increase we will also obtain improvements of other bolometric characteristics, such as the saturation power and in some cases the TCR itself. We organize our studies of the structured HTS materials in two parts: First we study the simplest case of carrier doping nanostructuring, namely the random nanoscale structuration that appears by just using oxygen stoichiometries that do not maximize Tc . We present our methods for those randomly structured HTS in Section 2; these consists of finite-element computations (and also, to confirm their validity, analytical estimates using effective-medium approximations [ 31 ]) that we apply to calculate the performance of the material in various example bolometer device implementations. The results following these methods in random nanoscale structurations are presented in Section 4. These results indicate that this first simple structuration may already improve some of the bolometric parameters with respect to conventional, nonstructured HTS materials The rest of the paper considers structurations that include not only the unavoidable random disorder but also the additional imposition of custom regular arrangements of zones with different nominal doping levels (patterning), studying different examples aimed to progressively improve the bolometric performance. The additional methods needed to calculate this added patterning are presented in Section 3, and the results are discussed in Sections 5to 7for various custom pattern designs, each of them improving the previous one. The most optimized pattern design (Section 7) is a four-step discretized exponential-like dependence of nominal doping with the longitudinal position. This arrangement should be also the easier one to fabricate. With respect to conventional nonstructured HTS TES materials, it improves by more than one order of magnitude the ∆T and the saturation power, and it also doubles the TCR sensitivity. 2. Methods for Structured Nonpatterned Resistive HTS TES Our methodology consists of computing the electrical resistance R versus temperature T of each of the structured HTS materials considered by us, and then using such R(T) to calculate the corresponding performance for bolometric operation. For completeness we will, in fact, consider various example bolometer-device designs, including simple square-shaped sensors such as those in [ 7 , 16 ] (that may be a micrometric size as appropriate for building megapixel cameras; we shall also consider two different substrates for completeness) and also larger sensors for millimeter-wavelength sensing using a meander geometry (as those built, always with nonstructured HTS, in [ 1 – 4 , 6 , 8 – 10 , 12 , 13 ]). While naturally we could not calculate in this work the whole range of possible device designs for a bolometer, our results in these example implementations show that our proposed nano optimizations of the materials should lead to improvements in at least some popular types of resistive HTS TES device designs. Also, for our R(T) calculations we shall use two alternative calculations, so as to be confident about the validity of the results: Finite-element computations first, and then effective-medium formulae (both paths have been successful in other studies of structured HTS [ 30 – 35 ], and we shall also include some confirming example comparisons with real data). Let us provide the details of all such procedures in the following sections. 2.1. Main Operational Parameters for Resistive HTS TES Devices We consider in this work, a HTS TES of resistive type, i.e., the temperature increase is sensed through the measurement of the electrical resistance, as in [ 1 – 16 ]. Contrarily to the most common case of low- Tc TES, the measurement is in current-bias ( I -bias) configuration in all experimentally implemented resistive HTS TES published to our knowledge (the experimental difficulties for voltage-bias ( V -bias) sensing in HTS TES were explained, e.g., by Khrebtov et al. [ 15 ]). Also the I value usually employed [1,7,9,10] is sufficiently small as to correspond to the ohmic regime ( R=V/I Nanomaterials 2020,10, 97 4 of 20 constant with I ) in all the operational T -range (see also below; this is also in contrast to low- Tc TES). As already mentioned in the Introduction, for ohmic resistive TES a main parameter of merit is the TCR, that may be also expressed as TCR =R(T+)−R(T−) ∆T R(T−), (2) where T− is the base operation temperature (the one in absence of radiation), T+ is the maximum temperature up to which the ohmic R(T) maintains the strong and constant slope with temperature, and ∆T=T+−T−(3) will be henceforth called the operational temperature interval. The other important parameter is Pmax , the maximum power measurable without saturation. In a I -bias resistive TES, it is possible to obtain Pmax at good aproximation [ 18 , 19 ] by just applying the heat flow equilibrium condition at saturation: Pmax +I2R(T+) = G∆T, (4) where I2R(T+) is the heat rate due to the Joule effect, G∆T is the power dissipated towards the cryobath, and Gis the thermal conductance between the film and the bath. We shall consider in this work three example resistive HTS TES device designs, to probe the effects of our proposed material optimizations in them. In particular, we consider two cases of microsensor bolometers, plus one case adapted to millimeter-wavelength sensing that we specify in the following section. 2.1.1. Microsensor Device Design The first resistive HTS TES device design consists of depositing a thin layer of YBCO HTS material over a substrate, the area of the HTS and the substrate being micrometric. In particular, we consider the convenient area (6 µ m) 2 , that makes possible building a 1 megapixel array of sensors in ∼ 1 cm 2 . Each substrate is in direct contact with a cryogenic liquid-nitrogen bath and we shall consider two possible substrate compositions: SrTiO 3 (STO, most appropriate to grow HTS films) and a CMOS-type substrate of interest for technological integration (note that HTS TES over CMOS substrates, in particular silicon/Yttria-stabilized zirconia (YSZ)/zirconia, have been already fabricated [ 10 ], using nonstructured YBCO). The experimental value of G for both types of substrate can be obtained from [ 10 , 36 ]. We consider YBCO thickness 100 nm and substrate thickness 1 mm. Also, we consider a bias current of I= 6 µ A, that corresponds to a current density j= 10 3 A/cm 2 , a value used in [ 13 , 15 ] and that corresponds to the ohmic range in all the T-range of operation [13,15,37]. 2.1.2. Millimeter-Wave Sensor Device Design The second device design we shall consider corresponds to the one most employed by experimentalists having produced resistive HTS TES [ 1 – 6 , 8 – 10 , 12 , 13 ]. It corresponds to a larger design using, as substrate, a suspended membrane of millimetric surface and CMOS-type composition; on top of such thin (micrometric thickness) membrane substrate a single meander of YBCO material is deposited. The larger area precludes building small megapixel sensors, but this is not important, e.g., for sensing millimeter wavelengths (that could not be constrained in smaller pixel sizes anyway, and that are among the main applications of bolometers). The meander geometry allows instead the ability to optimize the so-called ”static voltage responsivity“, SV , an important parameter defined by: SV=εL0 I(1−L0), (5) Nanomaterials 2020,10, 97 5 of 20 where εis the absorbance of the sample, Iis the bias current, and L0=TCR I2R G(6) is the so-called loop gain coefficient, which is a relative measure of the positive electrothermal feedback of the device [ 15 ] and G is again the thermal conductance towards the bath (whose experimental value may be found, e.g., in [ 13 ]); we consider (3 mm) 2 membranes for that evaluation. For a stable operation, the loop gain coefficient L0 should be smaller than 1, the value L0= 0.3 being usually taken as optimal. Therefore, the maximum static responsivity is obtained by tuning the geometry of the meander so to tune the I2R contribution to L0 in Equation (6) (Let us also note here that for I -bias the L0 may remain constant, and L0= 0.3, only if R is linear with T , so that maximizing ∆T is also interesting in this respect). In our calculations, we will use for the meander section the same size 6 µ m × 100 nm as previously for µ m-sensors. We then choose the meander length for each sensing material so that always L0∼ 0.3. We also consider the same bias current and current density, I = 6 µ m and j= 10 3 A/cm 2 , than for µ m-sensors (again corresponding to the ohmic range [ 13 , 15 , 37 ] and comparable to values used in experimental meander resistive HTS TES [ 13 , 15 ]). These choices not only are realistic but also they will allow us to use the same computer calculations of R(T) for both µ m- and mm-device designs (as only a geometric correction prefactor is needed to change R from one design to the other) and the same pair of bias current and current density, I=6µA and j=103A/cm2. 2.2. R(T)in the Normal State of Nonstructured HTS In HTS materials, the ohmic resistance versus temperature, R(T) , markedly varies with the doping level p (number of carriers per CuO 2 unit cell that for instance in YBCO may be changed through the oxygen content). This is true both for the value of the critical temperature Tc(p) below which, the superconductivity transition occurs, and for the R(T) magnitude and T -dependence in the normal state T>Tc(p) . The R(T) -versus- p phase diagram has been extensively studied in many works such as the review [ 38 ] (see also, e.g., [ 39 , 40 ]). Here, let us recall that the superconducting critical temperature is maximum at p∼ 0.155 (separating the so-called underdoped p< 0.155 and overdoped p> 0.155 compositions). Above Tc(p) , the material presents a normal-state background electrical resistivity, ρb(T , p) , that is linear on T above a certain so-called pseudogap temperature T∗ , and is pseudoparabolic semiconducting-like [ 38 ] for Tc<T<T∗ . In YBCO, it is T∗(K)≈ 270 − 3000 (p− 0.1 ) [ 38 ], so that for p& 0.16, it is T∗<Tc and the semiconducting-like region disappears. Instead of these rapid crude approximations, we will use in our analysis, all through the present work, the detailed quantitative results for Tc(p),T∗(p), and ρb(T,p)given in reference [38] for YBCO. Near Tc(p) , obviously R(T) undergoes the superconducting transition towards R(T) = 0. This transition is not fully sharp, instead, a sizable rounding of R(T) occurs in the vicinity of Tc(p) . This rounding is known to have two contributions: critical fluctuations and doping inhomogeneities, that we describe in the following subsections. 2.3. Rounding of R(T)Near the Superconducting Transition Due to Critical Phenomena The critical fluctuations around the transition play an important role in HTS and have been studied in detail, e.g., in [ 32 , 41 – 48 ]. The effects of critical fluctuations in the resistance curves are commonly summarized via the so-called paraconductivity, ∆σ , defined as the additional contribution to the electrical conductivity due to fluctuations: In particular, the total conductivity σ(T) near the transition becomes σ=∆σ+1/ρb, (7) where ρb is the normal-state background resistivity (see previous subsection). Because ∆σ follows critical-divergence laws near the transition, its effect far from Tc (for T& 1.7 Tc ) is totally negligible. Nanomaterials 2020,10, 97 6 of 20 Closer to Tc , however, ∆σ becomes progressively important and two T ranges may be distinguished. For 1.01 Tc.T. 1.7 Tc , i.e., the so-called Gaussian fluctuations region, ∆σ is well described by the Lawrence–Doniach paraconductivity equation for layered superconductors: [41–45,49] ∆σ=e2 16¯hd (1 t1+BLD t−1/2 −2 c+t+BLD/2 c2), (8) where e is the electron charge, ¯h is the reduced Planck constant, t=ln(T/Tc) is the reduced temperature, BLD = ( 2 ξc( 0 )/d)2 is the Lawrence–Doniach [ 49 ] layering parameter, ξc( 0 ) is the Ginzburg–Landau coherence length amplitude in the out-of-plane direction ( ≃ 1.1 Å in YBCO [ 44 , 45 ]), d is the superconducting layer periodicity length ( ≃ 5.85 Å in YBCO [ 44 , 45 ]) and c is a high-temperature cutoff constant ≃0.7 [43,44,50,51]. Closer to Tc , for TBKT .T. 1.01 Tc we find the strong phase fluctuation regime, dominated by the Berezinskii–Kosterkitz–Thouless (BKT) transition temperature TBKT ( ∼Tc− 2 K in YBCO) [ 42 , 52 ]. In this regime, the paraconductivity can be obtained using the equation: [33–35,47,53] ∆σ=ABKT exp"4sTc−TBKT T−TBKT #, (9) where ABKT is a constant, obtainable by requiring continuity of Equations (8) and (9) at the intersection of the Gaussian and BKT regimes, i.e., at T=1.01Tc. 2.4. R(T)Transition Rounding Due to Intrinsic Structuration of the Carrier Doping Level; Nominal vs. Local Doping As it has been explicitly demonstrated in various relatively recent experimental and theoretical works [ 32 , 46 , 47 , 54 , 55 ], an additional (and crucial for some doping levels) ingredient to understand the phenomenology of the resistive transition in HTS is to take into account the random Tc -inhomogeneities associated with the intrinsic disorder of the doping level. This intrinsic structuration is due to the fact that HTS compounds are non-stoichiometric, and therefore each dopant ion has at its disposal various lattice positions to occupy. For concreteness, we focus our present article in the case of the YBa 2 Cu 3 O δ superconductor with oxygen as a dopant ion. Experimental measurement indicates that a typical size of each local inhomogeneity is about (30 nm) 2 for HTS [ 32 , 54 , 55 ]. This produces, therefore, a certain randomness in the doping at the local scale, unavoidably present in even the more carefully grown HTS samples. A relatively easy geometrical calculation [ 32 , 54 , 55 ] reveals that this intrinsic structuration shall produce a Gaussian distribution of local dopant levels, as ω(p,p) = 2√ln2 √π∆pexp − p−p ∆p/(2√ln2)!2 , (10) where ω(p , p) is the fraction distribution of local doping levels, p , for a film with average doping level p (henceforth called nominal doping), and ∆p is the FWHM of the Gaussian distribution. This ∆p may be obtained, in turn, on the grounds of coarse-graining averages (see, e.g., Equation (6) of References [ 32 , 54 ]) and for YBCO it is ∆p∼ 0.006 (with a small dependence on p that may be considered in excellent approximation linear ∆p=0.0032 +0.0189p) [32,54]. Due to the Tc(p) dependence in HTS, the above distribution of local p values leads, in turn, to a corresponding distribution of local critical temperatures around the nominal value Tc=Tc(p) . The corresponding full width at half maximum (FWHM) for such intrinsic Tc structuration has been also considered, e.g., in [ 32 , 54 ]. Not surprisingly, it becomes quite negligible ( ∼ 1 K in YBCO) for the nominal dopings p≃ 0.155 that maximize Tc (and that has been used up to now for HTS TES; p≃ 0.155 corresponds to YBa 2 Cu 3 O 6.93 stoichiometry at which Tc(p) is maximum and dTc/dp ∼ 0). However, Nanomaterials 2020,10, 97 7 of 20 for other dopings the situation may become very different and the Tc distribution can reach FWHM values as large as, e.g., ∼5 K for p=0.13, significantly influencing the R(T)roundings [32,54,55]. 2.5. Obtainment of the R(T)Curve of Nonpatterned HTS TES Using Finite-Element Computations To calculate the resistance transition curves, R(T) , of the Tc -structured HTS material, we have used software (TOSERIS, available by request to authors) that numerically solves the electrical mesh-current matrix equations of a film modeled as a 200 × 200 square lattice of monodomains, where each domain i may have its own doping pi , and thus its own Tci and resistivity curve. We have used Equations (7)–(9) for the ρi(T , p) functionality of each monodomain i . The model also includes an I-bias power source and a voltmeter connected with zero-resistance contacts to opposite edges of the sample (see, e.g., scheme in Figure 1) and the R(T) of the film will be obtained as the external V/I . Calculating the circuit requires to numerically invert, for each temperature, the sparse matrix with dimensions 40,001 × 40,001 that defines the mesh-current equations. This is a parallelizable computation for which we employed a 31 Tflops supercomputer (LBTS- ε psilon, that comprises about 12,000 floating-point units and is described in [56]). It was 100% allocated to run our software during several weeks. uniform p(x) local p(x,y) 0.14 0.16 p= EM approximation (p=0.140 pictured) y x ✁ (10-7 Ωm) Figure 1. Electrical resistivity ρ versus temperature T obtained for YBa2Cu3Oδ (YBCO) films with a single, uniform value for the nominal doping level p , including the case with negligible Tc nanostructuration (or maximum- Tc doping, p= 0.155, in which Tc saturates near its maximum value and the Tc disorder is negligible) and various cases in which the p value corresponds to significant Tc nanostructuration (see Section 4for details). The data points correspond to the finite-element computations, and the continuous lines to the analytical effective-medium (EM) equations.The shaded gray region signals the operational temperature range ∆T (in which ρ is strongly dependent and linear in T ). In the upper drawing we illustrate the simulated sample and setup, also including a zoom at smaller length scales, imaging the spatial variation of the local doping level p(x , y) (each p(x , y) monodomain has typical size (30 nm) 2 and the distribution is Gaussian around the average p (x), see main text for details). We have performed our calculations with numerical values representative of the HTS compound YBCO and therefore for the area of a finite-element monodomain i we used (30 nm) 2 , that is expected to correspond to the size of a doping Tc inhomogeneity in YBCO. [ 32 , 54 , 55 ] Therefore, the surface of the simulated HTS film is going to be (6 µ m) 2 , in agreement with the microsensor HTS TES device implementation of Section 2.1.1. Nanomaterials 2020,10, 97 8 of 20 In the case of the nonpatterned HTS considered in this section, the only spatial variation of doping and Tc is the unavoidable intrinsic dopant ion structuration and, thus, we assign the local pi and Tci value to each of our 200 × 200 monodomains i as follows: We first build a set of 40,000 values of dopings following the Gaussian distribution given by Equation (10). We then assign each of those p -values to each node i randomly. Finally, those pi are transformed to Tci values (and corresponding ρi(T) functions) following the quantitative results of [ 38 ]. A scheme of an example of the resulting spatial distribution is provided in Figure 1(note the random nanostructuration in the zoomed area). 2.6. Analytical Estimates Using an Effective-Medium Approximation To additionally probe the consistency of our computations, we will use, as a useful test, semi-analytical results that we calculate using the so-called effective-medium equations (EM approximation). The EM approximation was first introduced by Bruggeman [ 57 ] for general random inhomogeneous materials, and then adapted, e.g., by Maza and coworkers [ 31 ] for HTS with Gaussian random Tc distributions. As shown in those early works, the EM approximation is a coarse-averaging model that may be considered accurate for temperatures not too close to the R→0 point (at which percolation effects may be expected to invalidate the approximation). In the case of our 2D media, the EM equations may be summarized as the following implicit condition for the conductivity σof each region with random doping inhomogeneities [31,33–35]: Z∞ 0 σp−σ σp+σω(p,p)dp=0. (11) Here, p , p and ω(p , p) retain the same meaning as in Equation (10), and σp is the electrical conductivity corresponding to doping level p . The above equation has to be numerically solved to obtain σ ; however, the computational weight is much lower than the finite-element computation method (seconds versus hours or even days in our parallel computer [56]). 3. Additional Methods for Structured and Patterned Resistive HTS TES We now describe the additional methods needed to obtain the R(T) curve of HTS films in which, additional to the random nanostructuration considered in previous section, also a regular pattern of nominal doping levels is imposed, with the aim to obtain designs that optimize the bolometric functionality. In these films, a regular spatial variation of the nominal doping level p is created by the samples’ grower by using any of the different methods for micro- and nanostructuration developed in the recent years by experimentalists in HTS films (see, e.g., [ 21 – 30 ]; for instance, for YBCO, this is possible by local deoxygenation using cover masks, ion bombardment, etc.). In particular, all the specific example patterns considered in this work will be expressible as functions p(x) , where x is the coordinate in the direction parallel to the external bias current (see scheme in Figure 1) and thus it will be useful to introduce the corresponding function λ(p) , or relative length weight of each nominal p value in the film, defined as λ(p) = 1 L dx dp, (12) where L is the total length of the film in the x -direction. Crucial for our studies, one has still to add to these nominal p(x) variations the unavoidable nanometric-scale randomness of the doping level (considered in the previous sections), i.e., p(x,y) = p(x) + prandom(x,y), (13) with prandom(x,y)consistent with Equation (10) evaluated using the local p(x). Nanomaterials 2020,10, 97 9 of 20 3.1. Obtainment of the R(T)Curve of Patterned HTS Using Finite-Element Computations To obtain the R(T) curves of patterned resistive HTS TES, we use the finite-element software TOSERIS already described in Section 2.5. We again use a 200 × 200 simulation mesh and now we assign to each of those finite elements a local doping as follows: First, we associate to each element i a nominal doping pi corresponding to the pattern to be simulated. Then we randomly calculate the local doping pi following the Gaussian distribution given by Equation (10), evaluated with the nominal doping piof each node. Finally, to each node we assign the Tci and ρi(T)corresponding to their local pias per the quantitative results of [38] for the HTS material YBCO (see Sections 2.2 and 2.3). We also tested that the sets of nodes sharing the same pi value follow the Gaussian distribution, and each R(T) simulation was repeated for several so-generated samples to verify their reproducibility. These checks indicate that our choice of a 200 × 200 node mesh provides enough statistical size. If we attribute to each node the size ( 30 nm )2 corresponding to each Tc -monodomain in YBCO (see Section 2.4 and [ 32 , 54 ]), the whole 200 × 200 sample corresponds to ( 6 µ m )2 , that is realistic for a microbolometric pixel. Unless stated otherwise, we will again use in our calculations the numerical values in Sections 2.2 to 2.5 for the common material characteristics, such as, e.g., a film thickness of 100 nm or values for the critical-fluctuation parameters as per Section 2.3. 3.2. Analytical Estimates Using an Extended-EM Approximation Besides performing finite-element computations, we will test our results against estimates based on the EM approach. For that purpose, we must suitably extend this approximation to account for the 1D gradient of nominal dopings corresponding to each example pattern to be considered in this work. For that, we consider the film as an association in a series of domains, each one with its own resistance and nominal doping p, so that: R(T) = Zx=L x=0 dp σ(p(x),T)S, (14) where L is again the total length of the superconductor, S is its transversal surface, and σ(p(x) , T) is the ohmic conductivity obtained using the monodomain-EM approach, i.e., using Equation (11) for each doping p (x). Equation (14) can be also written as the following integration over nominal doping, with the help of the λ(p)function defined in Equation (12): R(T) = LZpL p0 λ(p)dx σ(p(x),T)S, (15) For a discrete distribution (stepwise function p(x) ), the above equation becomes instead a summation: R(T) = N ∑ i=1 Li σ(pi,T)S, (16) where Nis the number of discrete domains, each with its own nominal doping piand length Li. Note that Equations (14) to (16) do not explicitly take into account the transverse currents when associating the different domains pi and Li . Non-longitudinal transport inside each domain is built-in by using the EM approach for each σ(pi , T) . However, this approximation may be expected to fail when it has to describe percolations (because both the sum in a series of domains and the EM approximation do not take them into account). Therefore it could be expected to overestimate the value of R(T) in the very close proximity to the fully superconducting R(T)→0+state. 4. Results for Structured Nonpatterned HTS Materials In the reminder of this article, we describe the results of applying our methods to different structured HTS materials. We consider first the case of nonpatterned HTS materials, i.e., those with Nanomaterials 2020,10, 97 16 of 20 y 0,14 0,16 0 6 x (  m) local p(x,y) p(x) p=0.14 0.16 p= 0 4 8 12 70 80 90 100 110 4-step exponential-like p(x) T(K)  T = 12.7 K EM approximation 4-step exponential p(x) variation . . x ρ(10-7 Ωm) Figure 5. In the upper row, we illustrate a YBCO film patterned following the discretized four-step exponential-like variation of the nominal doping p(x) studied by us in Section 7. We also illustrate doping across the film as a 2D color map, taking into account that the local doping level p(x , y) (zoom in the picture) results from accumulating the four-step exponential p(x) and the random Gaussian disorder at smaller length scales of about (30 nm) 2 (see main text for details). In the lower row, we plot the resistivity vs. temperature ρ(T) that we obtain for such film (data points for finite-element computations, continuous line for analytical EM approximation, see Equation (25)). This is the most optimized design found in this work for the use in resistive HTS TES devices (see also Table 1). 8. Conclusions To sum up, we considered the advantages of structuring and patterning of the doping level (and hence of the critical temperature) in high-temperature superconductors with respect to their operational characteristics as resistive bolometric sensors (resistive HTS TES) of electromagnetic radiation. In particular we studied some chosen examples of spatial variations of the carrier doping into the CuO 2 superconducting layers due to oxygen off-stoichiometry. Our main results are (see also Table 1for a quantitative account): (i) Non-patterned structured HTS materials (i.e., those with a nominal doping level uniform in space but that does not maximize the critical temperature, thus having random Tc -nanostructuring) may already provide some benefit for bolometric use with respect to the nonstructured HTS materials used up to now for those devices. In particular, they present a widened transition leading to a larger operational temperature interval ∆T and also larger Pmax (corresponding to the larger maximum detectable radiation power before sensor saturation). However, these improvements come at the expense of a certain reduction of the sensibility of the sensor as measured by the TCR value. (ii) The bolometric performance may be significantly more optimized with the use of HTS materials including an additional regular dependence on the position of the nominal doping level, p(x) (doping-level patterning). In that case, ad-hoc pattern designs may be found by progressively seeking widened and linear R(T) transitions. Our more optimized design is shown in Figure 5and consists of just four zones of different sizes and doping levels (related by the exponential-like Equation (24) evaluated at pi= 0.136, 0.141, 0.145, and 0.160). With this design the operational temperature is conveniently located at T−= 76.6 K, the operational temperature interval ∆T is almost 13 K (more than one order of magnitude larger than for the conventional nonstructured YBa 2 Cu 3 O 6.93 ), the TCR value is > 5 K −1 (almost double than for the nonstructured case), and the Pmax values are also optimized about one order of magnitude. Nanomaterials 2020,10, 97 17 of 20 Author Contributions: J.C.V. and M.V.R. derived the EM equations. J.C.V., A.S.V., M.M.B., and M.V.R. wrote the manuscript. J.C.V., A.S.V., C.M.-O., and M.V.R. contributed to the design of the studied patterns. All authors contributed to the computations and analyses of the results. Acknowledgments: This work was supported by projects FIS2016-79109-P (AEI/FEDER, UE) and AYA2016-78773-C2-2-P(AEI/FEDER,UE), by the Xunta de Galicia under grants ED431D 2017/06 and ED431C 2018/11, the Consellería de Educación Program for Development of a Strategic Grouping in Materials AeMAT under Grant No. ED431 2018/08, Xunta de Galicia, and by the CA16218 nanocohybri COST Action. JCV thanks the Spanish Ministry of Education for grant FPU14/00838. Conflicts of Interest: “The authors declare no conflict of interest.” Abbreviations The following abbreviations are used in this manuscript: TCR Temperature coefficient resistance YBCO YBa2Cu3Oδ STO SrTiO3 CMOS silicon/YSZ/Zirconia YSZ Yttria-stabilized zirconia TES Transition-edge sensor HTS High-temperature superconductor FWHM Full width at half maximum EM Effective medium BKT Berezinskii–Kosterlitz–Thouless References 1. 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