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

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 CuO2 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.

For resistive TES bolometers, a key figure for performance is the so-called "temperature coefficient of resistance" (TCR), given by [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]: High bolometric sensitivity requires a large value of TCR. For instance, structures of vanadium oxides V x O y , 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, T c . This is the case mainly when using conventional low-temperature superconductors with T c ≤ 1 K (the so-called low-T c TES bolometers), that achieve TCR ∼ 1000 K −1 or even more [17][18][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][18][19]. Note that for these measurements the very low temperature required to operate the low-T c 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-T c TES in other applications.
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 ∆T up 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 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 T c . 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 5 to 7 for 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.

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][2][3][4]6,[8][9][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][31][32][33][34][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.

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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Contrarily to the most common case of low-T c 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 constant with I) in all the operational T-range (see also below; this is also in contrast to low-T c 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 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 will be henceforth called the operational temperature interval. The other important parameter is P max , the maximum power measurable without saturation. In a I-bias resistive TES, it is possible to obtain P max at good aproximation [18,19] by just applying the heat flow equilibrium condition at saturation: where I 2 R(T + ) is the heat rate due to the Joule effect, G∆T is the power dissipated towards the cryobath, and G is 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.

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 µm) 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 µA, 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].

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][2][3][4][5][6][8][9][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", S V , an important parameter defined by: where ε is the absorbance of the sample, I is the bias current, and 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 L 0 should be smaller than 1, the value L 0 = 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 I 2 R contribution to L 0 in Equation (6) (Let us also note here that for I-bias the L 0 may remain constant, and L 0 = 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 µm × 100 nm as previously for µm-sensors. We then choose the meander length for each sensing material so that always L 0 ∼ 0.3. We also consider the same bias current and current density, I = 6 µm and j = 10 3 A/cm 2 , than for µm-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 µmand 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 = 6 µA and j = 10 3 A/cm 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 T c (p) below which, the superconductivity transition occurs, and for the R(T) magnitude and T-dependence in the normal state T > T c (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 T c (p), the material presents a normal-state background electrical resistivity, ρ b (T, p), that is linear on T above a certain so-called pseudogap temperature T * , and is pseudoparabolic semiconducting-like [38] for T c < T < T * . In YBCO, it is T * (K) ≈ 270 − 3000(p − 0.1) [38], so that for p 0.16, it is T * < T c 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 T c (p), T * (p), and ρ b (T, p) given in reference [38] for YBCO.
Near T c (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 T c (p). This rounding is known to have two contributions: critical fluctuations and doping inhomogeneities, that we describe in the following subsections.

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][42][43][44][45][46][47][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 where ρ b is the normal-state background resistivity (see previous subsection). Because ∆σ follows critical-divergence laws near the transition, its effect far from T c (for T 1.7T c ) is totally negligible.

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 T c -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 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 T c (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 T c = T c (p). The corresponding full width at half maximum (FWHM) for such intrinsic T c 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 T c (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 T c (p) is maximum and dT c /dp ∼ 0). However, for other dopings the situation may become very different and the T c 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].

Obtainment of the R(T) Curve of Nonpatterned HTS TES Using Finite-Element Computations
To calculate the resistance transition curves, R(T), of the T c -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 p i , and thus its own T ci and resistivity curve. We have used Equations (7)-(9) for the ρ i (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-εpsilon, that comprises about 12,000 floating-point units and is described in [56]). It was 100% allocated to run our software during several weeks. Electrical resistivity ρ versus temperature T obtained for YBa 2 Cu 3 O δ (YBCO) films with a single, uniform value for the nominal doping level p, including the case with negligible T c nanostructuration (or maximum-T c doping, p = 0.155, in which T c saturates near its maximum value and the T c disorder is negligible) and various cases in which the p value corresponds to significant T c nanostructuration (see Section 4 for 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 T c inhomogeneity in YBCO. [32,54,55] Therefore, the surface of the simulated HTS film is going to be (6 µm) 2 , in agreement with the microsensor HTS TES device implementation of Section 2.1.1.
In the case of the nonpatterned HTS considered in this section, the only spatial variation of doping and T c is the unavoidable intrinsic dopant ion structuration and, thus, we assign the local p i and T ci 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 p i are transformed to T ci values (and corresponding ρ i (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).

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 T c 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 σ of each region with random doping inhomogeneities [31,[33][34][35]: Here, p, p and ω(p, p) retain the same meaning as in Equation (10), and σ p 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]).

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][22][23][24][25][26][27][28][29][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 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) + p random (x, y), with p random (x, y) consistent with Equation (10) evaluated using the local p(x).

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 p i corresponding to the pattern to be simulated. Then we randomly calculate the local doping p i following the Gaussian distribution given by Equation (10), evaluated with the nominal doping p i of each node. Finally, to each node we assign the T ci and ρ i (T) corresponding to their local p i as 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 p i 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 T c -monodomain in YBCO (see Section 2.4 and [32,54]), the whole 200 × 200 sample corresponds to (6 µm) 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.

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: 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): For a discrete distribution (stepwise function p(x)), the above equation becomes instead a summation: where N is the number of discrete domains, each with its own nominal doping p i and length L i . Note that Equations (14) to (16) do not explicitly take into account the transverse currents when associating the different domains p i and L i . Non-longitudinal transport inside each domain is built-in by using the EM approach for each σ(p i , 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.

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 a uniform nominal doping p. As already mentioned, the doping level p 0.155 (corresponding to the stoichiometry YBa 2 Cu 3 O 6.93 ) is the one that has been used up to now to experimentally produce HTS TES, and, due to the saturation of T c near such doping level, it corresponds to a HTS film without a T c -nanostructure. However, the cases with uniform p < 0.155 correspond to films with random T c -nanostructuration.

Results for the R(T) Profile and Operational Parameters for Resistive TES Case
In Figure 1 we show the R(T) resulting from our finite-element computations for the case of YBCO films with nominal dopings p = 0.135, 0.140, 0.147 and 0.155, represented as circles, squares, diamonds, and triangles, respectively. As previously mentioned, to these nominal dopings, a random intrinsic structuration has to be added (following a Gaussian distribution as per Equation (10)) in order to obtain the local p(x, y) values; this is illustrated by the zoom in the pictured drawn in Figure 1.
As it is clearly shown in that figure, the reduction of p induces not only a shift of the transitions towards somewhat lower temperatures (as expected from the T c -vs-p phase diagram of HTS), but also a widening of the T-width of the transition region. The gray areas in Figure 1 are the T-range, ∆T, in which the R(T) transition occurs and R-vs-T is linear, i.e., the operational interval ∆T of Equation (3). This ∆T increases as p decreases, as may be noticed both in Figure 1 and also in Table 1. In this table we summarize the bolometric operational parameters obtained by using such R(T) results and the methods in Section 2 for three example resistive HTS TES devices designs (described in Sections 2.1.1 and 2.1.2).
It is clear that the improvement that the T c -nanostructure may provide for such bolometric characteristics: Notably, the increase of ∆T (about five times higher for the more nanostructured case than in the nonstructured one) is translated in an enhancement of P max of the bolometer device. This means the ability to receive a higher amount of radiation without saturating the sensing material. Note that a larger ∆T also implies a less demanding cryogenic setup in terms of required stability.
Therefore, our studies suggest that this first simple T c nanostructuration already improves some of the bolometric operational parameters with respect to the conventional, nonstructured HTS materials proposed until now. Table 1. Summary of the main operation parameters for resistive high-T c cuprate superconductors (HTS) transition-edge bolometer devices (TES) devices using, as sensing materials, YBCO superconductors with the various doping nanostructurations and patterns explored in this work. These include the usual nonstructured YBCO (p = 0.155) and the novel options considered by us: structured nonpatterned YBCO (p ≤ 0.155, Section 4) and structured patterned YBCO (Sections 5 to 7). The ∆T and TCR follow from the ρ(T) curves presented in Figures 1 and 3-5. The P max values were calculated for the three example device implementations described in Sections 2.1.1 and 2.1.2 (microbolometers with direct cooling and SrTiO 3 (STO) or silicon/Yttria-stabilized zirconia (YSZ)/Zirconia (CMOS)-type substrates, plus millimeter-wave sensors using a meander geometry).  Figure 1 also shows (as continuous lines) the results obtained by applying the EM approach, i.e., Equation (11), to the same parameter values and doping levels as used in the previous subsection. It can be seen in that figure that the coincidence between the finite-element computation and the EM approximation is excellent, even in the R → 0 + tails (a log-log zoom of such tails evidences moderate deviations in relative values, negligible in the absolute scale of Figure 1, as coherent with the expectation that the EM approximation is less accurate when percolative current paths appear [31,57]; see also Figure 2 for better evidence of these R ∼ 0 + deviations. This comparison gives, then, a first argument supporting the validity, at least concerning the main features, of our calculation methods for the doping structuring effects.

Verification Using the Analytical EM Approximation and Against Existing Measurements
A second argument supporting such validity is the comparison with actual measurements for R(T). In the case of structured nonpatterned HTS films considered in this section, measurements valid for such a comparison do exist. In particular, we have plotted in Figure 2 the data measured in reference [48] in high-quality YBCO (YBa 2 Cu 3 O δ ) films comprised by a single zone of nominal oxygen stoichiometries O 6.78 and O 6.85 (i.e., p 0.140 and p 0.156 respectively; to get the relations between oxygen ratio and doping we have interpolated the experimental data described in reference [58]). We can see in that figure the good accuracy of the theoretical (EM approach) and computational (finite-element) methods used in this article to reproduce the experimental resistance curves of nonpatterned YBCO films in the studied doping range.

Results for Structured HTS Materials Patterned with a Linear p(x) Variation
Let us now study different instances of doping-patterned HTS, seeking to progressively identify pattern designs optimizing the performance as bolometric sensor materials. We start by considering in this section, a simple linear variation of p along the longitudinal direction (the direction of the overall externally-applied electrical current, see the scheme in Figure 3): where p 0 and p L are the p-values at the opposite ends of the film x = 0 and x = L. In terms of the λ(p) function of Equation (12), this linear-in-x p-pattern simply becomes the constant value For our computations we chose the rather typical values p 0 = 0.135 and p L = 0.161.

Results for the R(T) Profile and Operational Parameters for Resistive TES Use
The results of our numerical finite-element evaluation for this p-pattern are displayed in Figure 3 (see also Table 1 for a comparative summary). As evidenced in the Figure 3, this type of structuring of the HTS film significantly broadens the R(T) transition (compare, e.g., with Figure 1 that represents, in the same T-scale, the results for HTS with comparable, but uniform, p-values). However, this structuring does not lead to a linear dependence of R vs. T in that transition region. This may pose a difficulty in resistive TES applications, that ideally require a R(T) variation both large (i.e., large TCR) and linear (i.e., constant dR/dT). The range ∆T in which both conditions are met is merely about 1.4 K for this type of p-pattern, already suggesting that further structuring optimizations would be desirable (see next section). In Table 1, we summarize the operational parameters obtained for the linear p(x) resistive bolometer. We can conclude that they are of the same order or worse than the parameters obtained for the typical already existing resistive HTS TES (the case p 0.155). This even includes the TCR, that is lower due to the increase of the operational temperature T − and then of R(T − ). The maximum energy and power may be somewhat higher due to the small increase of the width of the linear regime. In the upper row, we illustrate a YBCO film patterned following the linear variation of the nominal doping p(x) studied by us in Section 5 (see Equations (17) and (18)). 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 lineal 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 the analytical EM approximation, see Equation (19)). Note that the transition widens considerably with respect to Figure 1, but the operational ∆T range (shaded gray region) is small due to the nonlinearity of ρ with T in most of the transition.
We can conclude that this first patterning does not effectively optimize the operational parameters of the resistive HTS TES, mainly because it broadens the R(T) transition but does not achieve R(T) linearity in it.

Verification Using the Extended-EM Analytical Approximation
To check the validity of our results for the linear p(x) variation, we also have used the formulae described in Section 3.2. For that, our first step has been to combine Equation (15) with the λ(p) formula for this type of pattern (Equation (18)). This leads us to the new equation: where σ(p, T) results from Equation (11). The result of this analytical estimate is displayed in Figure 3 as a continuous line. As in the case described for constant nominal doping, this estimate achieves good agreement with the finite-element computation, confirming the basic accuracy of our results.

Results for Structured HTS Materials Patterned with a Continuous Exponential-Like Doping Variation
Seeking to find a p(x) profile producing a R(T) transition that improves the bolometric operational characteristics, we have explored numerous p(x) options beyond the simple linear function discussed above. In the present section we present the results that we obtained with the continuous p(x) functionality that led us to better bolometric performance (and a step-like, noncontinuous variation will be later discussed, in Section 7). This continuous p(x) profile is more intuitively described by means of the length weight function λ(p). In particular, we consider p-profiles leading to the following exponential λ(p) function: where δp and A are constants, the latter being easy to obtain by normalization considerations as In these equations, p 0 and p L are, as in the previous sections, the nominal doping at x = 0 and x = L respectively, being L the size of the film. Note that, by applying Equation (12), this corresponds to: For the case of YBCO films considered in this article, and for p 0 = 0.135 and p L = 0.161 as in the previous section, we found that the δp value that best optimizes the bolometric characteristics (most notably ∆T) is δp = 0.007. We also employed in our evaluations the same common parameter values as described in Sections 2.3 to 2.5.
In the upper row of Figure 4, the corresponding doping profile is pictured, both as a p(x) representation and as a 2D color density plot. It may be noticed that at the qualitative level the p(x) function itself is not too dissimilar to an exponential (however a purely exponential dependence of p with x would produce less optimized bolometric performance).

Results for the R(T) Profile and Operational Parameters for Resistive TES Use
The results of our numerical finite-element evaluation for this p-pattern are displayed in the second raw of Figure 4 (see also Table 1). As evidenced in that Figure 4, not only the transition is significantly broadened with respect to nonpatterned HTS but also (unlike what happened in the case of a linear p(x) variation) ∆T is highly increased. In particular, the ∆T region is increased up to 8.3 K, almost 10 times more than for nonstructured HTS.
As may be seen in Table 1, the improvements also occur in the P max parameter, that increase about one order of magnitude with respect to nonstructured HTS. However, note that the TCR value is one order of magnitude worse than in the case of such conventional, nonstructured HTS. This shortcoming and other improvements will be addressed in Section 7 with an evolved p-pattern design.

Verification Using the Extended-EM Analytical Approximation
We have checked our results also using an analytical estimate, by adapting to this pattern the effective-medium approach described in Section 3.2. For that, we have combined Equation (15) with the Equation (22) defining this p-pattern, to obtain the new formula: where σ(p, T) results from the Equation (11). The result of this analytical estimate is displayed in Figure 4 as a continuous line. Again it slightly overestimates R(T) in the tail of the transition, but basically confirms the finite-element results. As already mentioned, the overestimation is expected to be linked to precursor percolation effects.  In the upper row, we illustrate a YBCO film patterned following the exponential-like variation of nominal doping p(x) studied by us in Section 6 (see Equations (20) to (22)). 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 exponential-like 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 the analytical EM approximation, adapted in this work to this p(x) case, see Equation (23)).

Results for Structured HTS Materials Patterned with a Four-Step Exponential-Like Doping Variation
While the p(x)-pattern design considered in the previous section produced notable improvements of the bolometric features, at least two concerns may be expressed about it: First, any current structuration experimental technique [21][22][23][24][25][26][27][28][29][30] may have difficulties producing such a smooth and exponential-like variation of p with x (Equations (20) to (22)). Instead, it would be preferable a simpler and, mainly, discrete-pattern, i.e., one comprised of a few zones, each with a single p. This would ease fabrication, e.g., by means of several stages of deoxygenation of YBCO films using different cover masks in each stage. Secondly, the continuous-pattern of the previous section presents somewhat worsened TCR value with respect to some of the nonpatterned resistive HTS.
To address both issues, we consider now the discrete p(x) pattern described in the upper row of Figure 5. This pattern defines four zones, each with a single uniform p chosen to optimize the linear region of the transition. These values of p follow a discretized version of the exponential pattern: where L i is the length of the zone of nominal doping p i , and B is a constant so that ∑ i L i = L. We tested the bolometric performance for various doping levels p i of the four zones. We obtained the best results with the set p i = {0.136, 0.141, 0.145, 0.160}, that we describe next.

Results for the R(T) Profile and Operational Parameters for Resistive TES Use
The results for our numerical finite-element evaluation for this p-pattern are displayed in Figure 5 (see also Table 1). As evidenced there, the R(T) transition becomes significantly broad and linear, with such linear region conveniently starting at T − = 76.6 K (so that the HTS TES could be operated with the simplest liquid-nitrogen bath, at 77 K). The corresponding ∆T is now almost 13 K, the largest obtained in this paper. Also the TCR value > 5 K −1 is the largest obtained in this work, being almost double than for conventional nonstructured (i.e., maximum T c ) YBCO. The P max values (see Table 1) also reflect these improvements, being again the best among the HTS options considered in this work and more than one order of magnitude larger than for the nonstructured HTS.
To sum up, these finite-element computations reveal that this relatively simple-to-fabricate p-pattern produces order-of-magnitude improvements over nonstructured HTS materials in ∆T and P max , and also a 66% improvement in TCR.

Verification Using the Extended-EM Analytical Approximation
We have checked our results for this four-step p(x) pattern also using an analytical estimate, by adapting to this pattern the approach described in Section 3.2. In this case we used their discretized version , given by Equations (16). By combining it with the p(x)-pattern given by Equation (24) we now obtain: where the p i are the nominal dopings of each of the i zones, and σ(p i , T) results from Equation (11). This analytical estimate is displayed in Figure 5 as a continuous line. It fully confirms the main features obtained by the finite-element computations. Similarly to the case of the other p-patterns considered in our work, the estimate is expected to be less reliable in the lower part of the transition. 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).

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 1 for 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 T c -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 P max (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 5 and consists of just four zones of different sizes and doping levels (related by the exponential-like Equation (24) evaluated at p i = 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 P max values are also optimized about one order of magnitude.