A first-principles study and some applied researches of high-temperature superconductors and other low-dimensional functional materials

TESE DE DOUTORAMENTO A FIRST-PRINCIPLES STUDY AND SOME APPLIED RESEARCHES OF HIGH-TEMPERATURE SUPERCONDUCTORS AND OTHER LOW-DIMENSIONAL FUNCTIONAL MATERIALS. Jose Lorenzo Castaño Verde ESCOLA DE DOUTORAMENTO INTERNACIONAL PROGRAMA DE DOUTORAMENTO EN CIENCIA DE MATERIAIS SANTIAGO DE COMPOSTELA ANO 2019 DECLARACIÓN DO AUTOR/A DA TESE A FIRST-PRINCIPLES STUDY AND SOME APPLIED RESEARCHES ON HIGH-TEMPERATURE SUPERCONDUCTORS AND OTHER LOW-DIMENSIONAL FUNCTIONAL MATERIALS D. Jose Lorenzo Castaño Verde Presento a miña tese, seguindo o procedemento axeitado ao Regulamento, e declaro que: 1) A tese abarca os resultados da elaboración do meu traballo. 2) De selo caso, na tese faise referencia ás colaboracións que tivo este traballo. 3) A tese é a versión definitiva presentada para a súa defensa e coincide coa versión enviada en formato electrónico. 4) Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. En Santiago de Compostela, 7 de Maio de 2019 Asdo. Jose Lorenzo Castaño Verde AUTORIZACIÓN DO DIRECTOR / TITOR DA TESE A FIRST-PRINCIPLES STUDY AND SOME APPLIED RESEARCHES ON HIGH-TEMPERATURE SUPERCONDUCTORS AND OTHER LOW-DIMENSIONAL FUNCTIONAL MATERIALS D. Manuel Vázquez Ramallo INFORMA/N: Que a presente tese, correspóndese co traballo realizado por D/Dna. Jose Lorenzo Castaño Verde , baixo a miña dirección, e autorizo a súa presentación, considerando que reúne os requisitos esixidos no Regulamento de Estudos de Doutoramento da USC, e que como director desta non incorre nas causas de abstención establecidas na Lei 40/2015. En Santiago de Compostela, 12 de Maio de 2019 Manuel Vázquez Ramallo Asdo Aos meus pais, ´a mi˜na irm´a e aos meus av´os. Acknowledgments Almost at the end of this stage, I would like to thank some people and institutions for their support and help. Without them this PhD thesis would have not been possible. Firstly, I want to acknowledge Manuel V. Ramallo for the supervision of this thesis and his guidance in all the steps of this path. His scientific knowledge and his experience were crucial at every time to take decisions. His invaluable help and orientation have been pivotal to reach all the aims of this work. I performed a very important part of this thesis in the University of Illinois at Urbana-Champaign (USA) in 2016 and 2017. I would like to express my gratitude to Professor Anthony James Leggett for all the advices related to the research work performed in the first part of the thesis. I also want to remark the kindness and the treatment that I have received from him during my two visits to Urbana. They were a key to help me to have a much more comfortable stay. I would also like to acknowledge the rest of our research group for their support and help with experimental procedures. First of all, I would like to mention specially the help received by Jesus Mosqueira, that has been crucial to carry out the SQUID measurements, the evaporations and the deposition of the contacts in the samples. His advices have also been pivotal to perform the resistance measurements, and to analyze the SQUID measurements. I want also to thank Antonio Veira for the growth of the samples and for their experimental advices, that also have been a key contribution to finish this work. The other tenures of the group, Carolina, Carlos, and Jes´us were also always available when I needed their help. I want also to thank Magdalena and Benito, the technicians of our group, for their inestimable help in all the experimental details of my work. Without their support, this work wouldn’t have been possible. 16 Contents II: Some applied researches of microand nano-structured superconductors and other materials IIA: Superconducting devices 4 Design optimization of high-temperature superconducting bolometers HTS TES using doping nanostructuration and 1D patterning 67 4.1 Introduction............................... 69 4.2 Methods for structured non-patterned HTS TES . . . . . . . . . . 71 4.3 Results for structured non-patterned HTS TES . . . . . . . . . . . 78 4.4 Methods for patterned HTS TES . . . . . . . . . . . . . . . . . . . 83 4.5 Results for HTS TES structured with a linear p(x) variation . . . . 85 4.6 Results for HTS TES with a continuous exponential-like doping patterning................................ 88 4.7 Results for HTS TES with a 4-step exponential-like doping patterning.................................. 90 4.8 Conclusions............................... 93 5 Modification of the resistive transition width of high-temperature superconducting films using 2D Tc-dot patterns 97 5.1 Introduction............................... 99 5.2 Methods.................................100 5.3 Broadening the R(T) transition with patterns of enhanced-Tcdots 102 5.4 Sharpening the R(T) transition with patterns of depressed-Tcdots 103 5.5 Conclusions...............................105 6 Hybrid piezoelectric + superconducting devices 107 6.1 Introduction...............................109 6.2 Samplepreparation...........................110 6.3 Characterization under no piezostress . . . . . . . . . . . . . . . . 111 6.4 XRD measurements of the strain under di↵erent piezostresses . . . 115 6.5 Electrical resistivity measurements under di↵erent piezostresses . . 118 6.6 Conclusions...............................122 Contents 17 IIB: Electrostatic nanostructured microporous media for nanofluid filtration 7 Electrostatic nanostructured microporous (ENM) media: Overview of theory and computational methods 127 7.1 Introduction...............................129 7.2 Modelequations ............................130 8 Results for the influence of the microgeometry of ENM on the nanofiltration characteristics 137 8.1 Overview ................................139 8.2 Initial, clean state performance of the nanofunctionalized microporous media . . . . . . . . . . . . . . . . 139 8.3 Initial, clean state energy consumption rate for filtration . . . . . . 141 8.4 Time-evolution of the nanofiltration . . . . . . . . . . . . . . . . . 144 8.5 Conclusions...............................152 Final conclusions 155 Resumen en castellano 157 Bibliography 167 List of publications by J.C. Verde 180 Chapter 1 Introduction This thesis presents research done with funding grants from Xunta de Galicia1 and the Spain’s Ministery of Education2. Both of these funding agencies required the research to include not only fundamental aspects but also applied studies. Correspondingly, this doctoral work comprises two parts: Part I presents novel results using a first-principle study of the high-temperature cuprate superconductor (HTS) materials, performed in the framework of a collaboration with Professor A.J. Leggett, in whose laboratory I performed di↵erent stays3. The study presents an extension, to the case of HTS, of equations originally derived for simple superconductors by G.V. Chester. They link the energy saved by the superconductivity with the structural properties of the materials. Then, we apply these extended equations to various prototypical HTS. Part II is devoted to various applied studies in di↵erent micro and nanostructured functional materials and devices. We divided these studies in two groups: In part IIA, we focus again on HTS and propose structuration designs that optimize their functionality as bolometric sensors of electromagnetic radiation (a development that led to our Spain’s patent application coded P201930020) and we also study other structured HTS systems (2D dot-pattern structured films; hybrid superconductor+piezoelectric films). In part IIB, we study instead electrostatic nanostructured micropore (ENM) media of the type recently introduced by industry in various forms (among other, 1predoctoral grant from Xunta de Galicia during 6 months. 2predoctoral FPU grant during 3.5 years. 3I performed a 1-month stay in 2016 and a second 3-month stay in 2017, both of them in the Institute of Condensed Matter Theory (ICMT) of the University of Illinois at UrbanaChampaign, USA, under the direct supervision of Professor A.J. Leggett. 19 20 Chapter 1. Introduction nanoalumina-coated microporous membranes) for filtration of nanofluids (liquids carrying nanoentities in suspension). By employing model equations previously developed in Santiago de Compostela, we find microgeometry configurations able to optimize the energy efficiency and performance of filters based on such media. We now present a brief but more detailed introduction to these di↵erent fundamental and applied researches: •Part I: First-principles study of the interplays between structural and superconducting energies in cuprate superconductors In contrast to the classic, BCS superconductors in which a phononic mechanism for Cooper pairing is well accepted and is thoroughly understood, [1] the high-temperature superconducting cuprates (and other complexly stronglycorrelated superconductors) are still a fully open puzzle in what concerns the microscopic mechanism for superconducting pairing. [2, 3] Putting light, even if only partially, on such pairing mechanism remains of maximum interest. [4] Probably due to the past success of the BCS theory, most part of the community has focused their attempts to explain this non-conventional materials trying to introduce some form of e↵ective Hamiltonian, describing the system in terms of a few degrees of freedom (often di↵erent in each approach: phononic, di↵erent magnetic models, etc.) that, hopefully, could dominate the wide variety of phenomenology measured for this type of superconductivity. [5–8] However, until this date, no theory has arisen with a good level of acceptance in the research community able to explain all the relevant properties of these compounds. Moreover, most of these superconductors are complexly correlated electron systems, what means, among other features, that it could be expected to be hard, or even impossible, to reduce one of such e↵ective Hamiltonians (for instance, taking into account the long-range part of the Coulomb interaction, believed to play an important role in HTS). [2] All of this raises the issue of whether this is the only way to study this problem or if we should also try some other methodology. Following this line of ideas, a di↵erent approach to the problem is to start from a first-principles general Hamiltonian and try to obtain the properties of the superconductor from it. The exact solution of all the statistical mechanics properties of the system is of course impossible to reach with today’s computing power. However, it is possible to obtain directly from these approaches some general, but very important properties. [3,9] One promising way to seek information from the general Hamiltonian is focusing on the evolution of the general thermodynamic energy averages when changing from the normal to the superconducting state. [9–12] The basic thermodynamic energy to be studied is the thermal-averaged Hamiltonian, defined as the internal 21 energy U. When one compound becomes superconducting, its internal energy should decrease, as this new state minimizes U. Furthermore, the total Hamiltonian is a sum of di↵erent types of energies, with the easiest way to describe it being a sum of the kinetic energy of all the particles plus the Coulomb interaction energy of all of them. It would be also very informative to know how each of these thermal averaged energy contributions evolve from the normal to the superconducting state, so to more precisely pinpoint the energy contribution saved in the transition from the normal to the superconducting state. In fact, if we were able to know where in this sum is exactly the energy saved by the superconductivity, we could be able to discard any microscopic theories that do not lead to energy savings in the right contributions, so we can focus instead on new theories taking this important information into account. For HTS, this approach is not entirely unheard, and at least one partially related new precedent is the discussion by di↵erent groups of some theory models [10–15] that explored the interlayer interactions as possible dominant drivers of the HTS superconducting transition. In particular, P.W. Anderson and coworkers [13–15] famously proposed the so-called interlayer-tunneling (ILT) model in which condensation reduces the out-of-plane kinetic energy of carriers. This lead to several experiments and theory comparisons [16–18] that mainly obtained that the interlayer kinetic energy saving in the conduction electrons is not strong enough to explain the condensation energy of these materials, indicating then that the ILT mechanism is not likely to be the dominant one for the superconducting condensation. In related e↵orts, various groups have focused in the measurement of the energy savings from the normal to the superconducting state in HTS. [16,19–22] For example, there were attempts to measure the total kinetic energy saving for the conduction electrons of the cuprates using the “single band” sum rule, in connection with tight binding models. [22, 23] These measurements showed varied behavior in the phase diagram of cuprates: for the very overdoped part superconductivity increased such kinetic energy; in the other part of the phase diagram, condensation was accompanied by savings in the kinetic energy of the conduction electrons strong enough to explain the condensation energy. [22] A particularly relevant previous reference for our own e↵orts in this thesis is the MIR scenario introduced by A.J. Leggett [10–12] build over a first-principle Hamiltonian to which some general properties thought to be probably common to all the HTS are applied. [10–12] Among these assumptions, Leggett introduces the possibility of Coulomb energy saving in the region of the mid-infrared frequencies for HTS. In spite of relevant experimental e↵orts, [19–21] the scenario still remains experimentally untested. Still ongoing e↵orts could possibly be discriminative enough as to finally probe the relevance of the MIR scenario. However, the Hamiltonian introduced by A.J. Leggett although being com- 22 Chapter 1. Introduction paratively quite general, is still not an exact Hamiltonian. As an example, it does not take into account the kinetic energy of the nucleus in the system, that gives information about the role of the phonons in the superconducting mechanism. It has become quite frequent to discard this term in the Hamiltonian of HTS because of the general idea that the phonons play a residual role in the mechanism of superconductivity. This would be coherent with the common conjecture that the reason why the isotope e↵ect seems to be negligible for the highest-Tccuprate composition close to optimal doping, but appreciable elsewhere, is simply that a BCS (i.e., phonon-induced) interaction contributes a doping-independent amount to the condensation energy. Then the BCS mechanism, and the isotope e↵ect, would become negligible when other more relevant mechanisms produce high-Tc. However, beyond these rough qualitative arguments, a quantitative probe of the consistency of the phonon mechanism is still somewhat lacking. In this thesis, we are going to study an even more general method to obtain information for cuprate superconductors, using the total Hamiltonian of the system. Our analysis is based on the method first introduced by G.V. Chester in 1956, before the appearance of the BCS theory to study the general evolution of the thermal averages of the kinetic energy of all the electrons of the system (conduction and non-conduction electrons), the kinetic energy of the nucleus in the system, and the total Coulomb energy (which includes the three interactions that play a role in the material, i.e., the interaction between electrons, the repulsion between nucleus, and the attraction between nucleus and electrons). As common in 1956, Chester assumed in his work a monoatomic and type-I superconductor. [9] In our study, we will adapt the Chester equations for the energy di↵erences so to extend their validity to generic type-II polyatomic superconductors. Consequently, we employ a Hamiltonian for polyatomic materials that is a general sum of the three terms, i.e., the kinetic energy of all the electrons of the system, the kinetic energy of the nucleus in the system, and the total Coulomb energy. Then we extend for this polyatomic case the three theorems employed in the original Chester formalism. Moreover, we express these equations in terms of the condensation energy of the system, so that the so-extended Chester-like equations are then valid also for type-II superconductors. These analytical calculations are presented in Chapter 2, while in Chapter 3 we express them in terms of quantities for which measurements exist in some HTS. This leads to singularize, for instance, the kinetic energy of the oxygen nucleus, or to identify terms that seem to be negligible for HTS. Also in Chapter 3, we quantify the energetic balances using experimental data available for three typical cuprate superconductors. [24–28] Our results indicate that the phononic contribution is not doping-independent, but rather evidence three di↵erent regions in the phase diagram: In the underdoped side, we find that the kinetic energy savings of the oxygen nucleus favors the superconductivity. In contrast, in the slightly overdoped side the movement of the oxygen nucleus 23 opposes to the superconductivity and, finally, in the highly overdoped side the nucleus contribution is almost negligible. •Part IIA: Some applied researches on microand nano-structured superconductors As commented before, this doctoral work also comprises studies (mainly computational and/or experimental) devoted to the optimization of functional materials as used on devices or other applications. The first half of those studies focus again on HTS, while the second half will address di↵erent materials. Concerning our applied studies on HTS systems (part IIA of this thesis), they have addressed the optimization of HTS bolometric radiation sensing devices (what led us to the application of the Spain Patent No. P201930020). We also performed two additional complementary studies on 2D dot-patterned HTS and piezoelectric+HTS hybrid devices. We briefly introduce now these three contributions: IIA.1Design optimization of high-temperature superconducting bolometers using doping nanostructuration and 1D patterning Bolometers are electromagnetic-radiation sensors that detect incident energy via the increase of the temperature Tcaused by the absorption of incoming photons. Sensors formed by an array of microsized bolometers (microbolometers) are often used, e.g., for infrared cameras, detectors in quantum cryptography, astrophysics, satellites, aviation, etc, manily when high-sensitivity requirements are more important than raw cost. [29–34] Superconductors operating slightly above their transition temperature Tcare among the best candidate materials for bolometers, for their extreme sensitivity to Tin an easy-to-measure observables such as the electrical resistance R(resistive transition-edge sensors, TES), and also for their wide spectral range, speed of response, and other competitive operational features. A key figure influencing the efficiency is the so-called “temperature coefficient of resistance” or TCR, given by: [29–34] TCR =  1 R dR dT .(1.1) High bolometric sensitivity requires a large value of TCR. Resistive TES fabricated with low-Tcsuperconductors (the so-called low-TcTES) achieve huge TCR⇠1000 K1or even more [29,35] (for comparison, TCR of VxOy, the most common, and non superconducting material used for inexpensive bolometers, achieve TCR ⇠0.025 K1[34]). This makes low-TcTES a technology of choice for detecting the most faint radiations, as the cosmic microwave background. [29–31] However, the need of liquid-helium cryogenics evidently limits widespread adoption of low-TcTES. 24 Chapter 1. Introduction After the discovery of HTS, various authors have explored their use for bolometer sensors (the so-called HTS TES) with simpler liquid-nitrogen-based cryogenics. [36–41] YBa2Cu3O(YBCO) is the HTS compound usually considered for this application, usually with maximum-Tcdoping, i.e., stoichiometry ⇠6.9. As shown in References [36–41], YBCO thin films provide a viable competition to VxOy. Not only they provide much larger TCR ⇠1.5K 1and low noise at an operational temperature ⇠90 K, but also they have a faster thermal response than VxOy. Moreover, the rest of parameters contributing to a large sensitivity (thermal conductivity, infrared absorbance, etc.) are also favorable or at least competitive with semiconducting bolometers. [36–41] However, the HTS TES until now proposed still share some of the significant shortcomings of low-TcTES: 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 Tof just ⇠0.1 K or less for low-TcTES, and ⇠1 K for the HTS TES proposed until now. [36–41] This makes them easy to saturate (low dynamic range). The HTS TES systems proposed until today are homogeneous in nominal composition and critical temperature. [36–40] However, in the recent years di↵erent novel techniques have been developed to impose regular patterns to HTS thin films, creating custom designs, down to the microand the nano-scales. [42–47] This type of micro and nanostructuration of HTS has even become the main subject of two successive European COST actions (NanoSC and nanocohybri, [46]). However, to our knowledge no author has up to now considered the possibility of designing a nanostructure pattern that could improve the performance as bolometer of a YBCO film. In chapter 4 we address that task, using mainly computational methods. For that, we propose improvements of HTS TES ohmic-resistive microbolometers by considering the use of nanostructuring and patterning of the local doping level p (the number of carriers per unit cell). In particular, in our calculations we will consider the prototypical HTS compound YBCO, in which pmay be easily varied by just changing the oxygen stoichiometry (e.g., via local deoxygenation, ion bombardment with di↵erent masks, etc.) and we focus on 1D patternings (i.e., p, or , varying along one direction). We concentrate in obtaining an increase of the operational temperature interval, T,inwhichi) the TCR is large and ii) Ris linear with T(i.e.,dR/dTconstant with T, that is another desirable feature that simplifies both the electronic control of the microbolometer and the required stability of the cryogenic setup). We propose several pattern designs that successively improve such Tand TCR (and also other bolometric parameters). Our most optimized design achieves a very competitive TCR⇠5K 1and T⇠ 13 K (and led to our Spain patent application code P201930020). 25 IIA.2Modification of the resistive transition width of high-temperature superconducting films using 2D Tc-dot pattterns The above applied studies on 1D-patterned HTS film devices are supplemented in this thesis with two complementary studies. The first is devoted to show that the shape of the resistive transition of a HTS film may be also customtailored by using a 2D-pattern design (instead of the simpler, though e↵ective, 1D-pattern used in the previous sections). We again use mainly computational approaches, and in this case we focus on achieving the opposite e↵ect over the transition than the one achieved by our previous 1D-pattern: Now we seek the possibility of sharpening the transition (lowering T) instead of broadening it. This may be of interest for some applied devices, like super-thermometers (sensitive inside a constrained window of temperatures T) [48] or resistive fault current limiters (as Tit may influence both the occurrence and speed of thermal avalanches and thus their ability to respond to ultrafast current strikes and their recovery time once the fault is cleared). [49,50] Our results in this study indicate that the resistive-transition sharpening may be achieved with certain 2D-dot patterns of Tcvariation. Our most optimized designs lower Tby about 75% versus the non-structured YBCO films (and presumably would shorten in a similar amount the response time of a resistive fault current limiter). IIA.3 Hybrid piezoelectric + high temperature superconducting devices Our second complementary study of HTS viewed as functional materials is an experimental work. We have performed various measurements and characterizations of hybrid piezoelectric+HTS devices. These systems, fabricated in our research group in Santiago de Compostela, consist on a YBCO layer of about 100 nm thickness grown on top of a piezoelectric monocrystal (Pb(Mg1/3Nb2/3)0.72Ti0.28O3). We have deposited electrical contacts on both sides of such hybrids, enabling us to exert custom externally-imposed voltage over the piezoelectric, that produces then custom mechanical strain over the HTS. Because strain induces a change of Tcin these superconductors, this opens a way to study (and/or functionally use) the variation of Tcwith the mere application of an external voltage. We have focused on the characterization of the strain that can be induced by applying a transversal (piezo)voltage, on the so-induced shift of the critical temperature, and on the thermal fluctuation e↵ects (paraconductivity) in such devices. •Part IIB: Some applied studies of functional nanofluid filters based on electrostatic nanostructured media 32 Chapter 2. Extended Chester equations for polyatomic superconductors ergy saving in the region of the mid-infrarred frequencies for HTS, that should be closely related with the superconducting mechanism. [10–12, 19] The partial Coulomb energy restricted to the region where the exchanged wave length verifies q!0 has been recently measured using ellipsometry experiments. [19] They have found that in this region there is an energy saving only in the overdoped part of the cuprate phase diagram that is not strong enough to explain the condensation energy. Conversely, the underdoped side of the phase diagram shows an increase in the Coulomb energy. Very recent Electron Loss Espectroscopy experiments (EELS) have been performed to obtain the partial Coulomb energy saving at higher q, showing a similar tendency in comparison to the ellipsometry measurements. [20, 21] To find where is the region where the Coulomb energy is saved, further EELS experiments at even higher qor at di↵erent frequencies should be performed.1 There where other attempts to measure the total kinetic energy saving within the tight binding model for the conduction electrons of the cuprates using a well known sum rule. [22] These measurements showed opposite behavior in some parts of the phase diagram of cuprates. For the very overdoped part of the phase diagram, there is a waste in kinetic energy. In the other part of the phase diagram, there is a saving in the kinetic energy of the conduction electrons that is strong enough to explain the condensation energy. [22] In this chapter we are going to focus on a method that Chester developed before the appearance of the BCS theory to study the general evolution of the thermal averages of the Kinetic energy of all the electrons of the system (conduction and non-conduction electrons), the kinetic energy of the nucleus in the system, and the total Coulomb energy (which includes the three interactions that play a role in the material, i.e., the interaction between electrons, the repulsion between nucleus, and the attraction between nucleus and electrons). [9] He developed these equations for a type I monoatomic material, and our aim is to extend the formalism for type I and type II polyatomic superconductors. In this chapter, in Section 2.2, we introduce the Chester equations for the energy balances of a general monoatomic type I superconductor. In Subsection 2.2.1 we introduce the Chester Hamiltonian for a monoatomic superconductor, then in subsection 2.2.2 we write the three theorems used in the formalism. After that we introduce in subsection 2.2.3 the Chester equations for a monoatomic superconductor, that will be complemented in subsection 2.2.4 with the introduction of the thermodynamic equation for the Gibbs free energy variation from the nor1There are also some theories based in e↵ective Hamiltonians that try to predict where is the energy saved in cuprate superconductors. The Interlayer Tunneling model (ILT) for HTS, developed by Anderson and coworkers, [13–15] introduces the hypothesis where there is a kinetic energy saving in the conduction electrons kinetic energy due to a Josephson coupling of the electrons between the superconducting CuO2planes. Subsequent measurements showed that this kinetic energy saving is not strong enough to explain the condensation energy of these materials. [17] 2.2. Chester equations for type I monoatomic superconductors 33 mal to the superconducting state for a generic type I superconductor. Then, in Section 2.3 we obtain the extend Chester equations for a general polyatomic superconductor at T= 0 K. To implement this we first introduce in subsection 2.3.1 a new thermodynamic quantity that is valid also for type II superconductors. Then we follow the same procedure, introducing the new Hamiltonian for polyatomic materials in subsection 2.3.2, introducing the theorems for the extended method in subsection 2.3.3, and obtaining the extended Chester equations in subsection 2.3.4. Finally we summarize our conclusions in section 2.4 2.2 A summary of the Chester equations for type I monoatomic superconductors G.V. Chester proposed a general approach based on the study of the energy balances at temperatures below the superconducting transition. [9] He obtained within his calculations exact equations for the di↵erences between the normal and superconducting states of the thermal averaged total Coulomb energy, the total thermal averaged kinetic energy of the electrons, and the total thermal averaged kinetic energy of all the nucleus of the material. In this section we briefly summarize these original calculations, so to put ourselves in position to generalize them later in Section 2.3. As a general rule, and to make our presentation easier to follow, here we will state the main steps of such Chester equations. A full demonstration of such steps, we refer to either Chester original’s work [9] or to our Section 2.3, in which we demonstrate the formalism in a generalized scenario. 2.2.1 Hamiltonian G.V. Chester develops all the equations starting from a first-principles Hamiltonian Hfor a monoatomic superconductor, given by: H=KM+Km+,(2.1) where KM= N X i=1 P2 i 2M,(2.2) Km= ZN X i=1 p2 i 2m,(2.3) 34 Chapter 2. Extended Chester equations for polyatomic superconductors = N X i<j N X j Z2e2 |RiRj| N X i=1 ZN X j=1 Ze2 |Rirj|+ ZN X i<j ZN X j e2 |rirj|.(2.4) In these equations, KMis the total kinetic energy of the nucleus of the monoatomic material with mass Mand atomic number Z,Kmis the total kinetic energy of the electrons with mass mand charge e, and is the total Coulomb energy represented by the sum of the interaction energy between the electrons, the interaction energy between the nucleus and the interaction energy between the nucleus and the electrons of the material. Note that capital letters Pand Rare used to refer to the momentum and positions of the nucleus, respectively. Conversely, pand rare used to describe the momentum and the positions of the electrons, respectively. It is important to remark that this Hamiltonian should describe the phenomena of the superconductivity, because it take into account all the basic interactions that take place in the material at low energies. 2.2.2 Three intermediate theorems Chester begins by demonstrating that this Hamiltonian verifies three generic theorems in quantum mechanics. •Contributions to H The first one comes just from the contributions to the Hamiltonian and is directly derived from Equation 2.1. H=U=KM+Km+,(Hcontributions) (2.5) where the bar stands for the thermal average calculated quantity. •Virial Theorem for H The second basic theorem is the virial theorem, that in this form is valid when the interparticle interaction is Coulomb-type (i.e., each term of the sum is proportional to the inverse of the distance between articles), and for hydrostatic pressure. PV =2 3(KM+Km)+1 3,(virial theorem) (2.6) where Pis the pressure, and Vis again the volume of the material. •Equation derived from Hellman Feynman theorem applied to H Finally, he uses a general result from statistical mechanics that comes almost directly from the application of the Hellman Feynman (HM) theorem to this 2.2. Chester equations for type I monoatomic superconductors 35 Hamiltonian. It introduces a relationship between the Gibbs free energy Gand the total kinetic energy of the nucleus of the system, and is given by: M@G @MT,P =KM.(derivation from HM theorem) (2.7) 2.2.3 Chester equations Therefore, making the di↵erence of all these quantities between the normal and superconducting state (represented by ), and expressing the internal energy and the volume in terms of the Gibbs Free energy, we obtain the Chester equations: First Chester equation (electronic kinetic energy saving) Km=M@G @MT,P +T2@G/T @TP,M +4P@G @PT,M (Chester 1) (2.8) Second Chester equation (nucleus kinetic energy saving) KM=M@G @MT,P (Chester 2) (2.9) Third Chester equation (Coulomb energy saving) =2T2@G/T @TP,M 5P@G @PT,M (Chester 2) (2.10) 2.2.4 Thermodynamic equation The three Chester equations are expressed in function of the di↵erence between the normal and superconducting value of the Gibbs free energy G. Therefore we need a way to measure this quantity for monoatomic superconductors. There is a well known thermodynamic equation for type I superconductors, that links this quantity with the critical magnetic field required to destroy the superconducting state Hc: G=VH2 c 2,(2.11) where Vis the volume of the sample. Due to the fact that Hcis relatively easy to measure in type-I conventional low-Tc superconductors, it is convenient to calculate the energy balances with available experimental data of Hc,V, and 36 Chapter 2. Extended Chester equations for polyatomic superconductors some derivatives of them. Taking into account that all these magnitudes are measurable, we can obtain all these energy di↵erences for a generic monoatomic type I superconductor and their dependence with Temperature. In fact, the results found within this approach are far more generic than the results of the BCS theory, because they are obtained from the real Hamiltonian of the system, instead of an e↵ective one. Chester achieves that the energy balance can be summarized in a Coulomb energy saving, an increase in the kinetic energy of the electrons, and also a saving in the kinetic energy of the nucleus. In fact, at 0K, the Coulomb energy saving perfectly balance the increase of the kinetic energy of the electrons. The final energy saving that leads to the condensation energy is explained by the kinetic energy saving in the movement of the nucleus, which is half of the value of the Coulomb saving at that temperature. This result remarks the importance of the phonons in the BCS superconducting mechanism. 2.3 Extended Chester equations for polyatomic type I or II superconductors The question that is raised in this chapter is the possibility to adapt these equations to extract generic results for other types of superconductors, whose superconducting mechanisms are still unknown. There have been many attempts to obtain the energy balances in non conventional superconductors, [19,22] but these method has never been applied for these type of materials. 2.3.1 Thermodynamic equation for type I or II superconductors The first step is to realize where are the assumptions in the Chester formalism that limit the equations for a type I monoatomic superconductor. First of all, it is hard to use the equation 2.11 for a type II superconductor, because the critical field Hcthat destroys the superconductivity is not well defined due to the appearance of the vortex, that implies the definition of two critical fields Hc1 and Hc2, that are not directly related with the Gibbs free energy saving without using any established theory or model. Therefore, is more natural to rewrite the equations in terms of a quantity that can be measured avoiding the problems related with the definition of Hcwithin a given theory. The most natural way is to choose the condensation energy as reference, because it can be obtained in an unambiguous way through specific heat measurements, [24–26, 64, 65] and is given by: [63] Econd =Un(T=0K)U(T=0K)= 2.3. Extended Chester equations for polyatomic type I or II superconductors 37 Z1 0 [Cn(T)C(T)] dT =Z1 0 [Sn(T)S(T)] dT, (2.12) where Cis the measured electronic specific heat of the sample, Sis the entropy, and nstands for the value in the normal state. However, the first conclusion that is raised from our thermodynamic equation is that to express our equations in terms of the condensation energy, we are going to fix all our energy balances at T= 0 K, limiting the scope of our extended formalism. 2.3.2 Hamiltonian for polyatomic superconductors Another problem to use the Chester arguments is that the Chester Hamiltonian described by equation 2.1 is only valid for a monoatomic materials, and unconventional superconductivity is usually manifested in polyatomic materials. Therefore, we will rewrite the Hamiltonian like in equation 2.1 as a sum of three di↵erent energies, as: H=KM+Km+,(2.13) but these energies are now given by: KM= Nz X k=1 Nk X i=1 P2 ik 2Mk ,(2.14) Km= Ne X i=1 p2 i 2m,(2.15) = Nz X k=1 Nz X k0k Nk X i=1 Nk0 X j=1 ZkZk0e2 |Rik Rjk0| Nz X k=1 Nk X i=1 Ne X j=1 Zke2 |Rik rj|+ Ne X i=1 Ne X i<j e2 |rirj|,(2.16) where when k=k0it is verified that i<j,Nzis the number of nucleus of di↵erent atomic number Z, and Nkis the number of nucleus of each type k.Mkis the mass of the nucleus of type k,Neis the total number of electrons in the system, and Zkis the atomic number of the nucleus of type k. In the last equation, the first summand is the nucleus-nucleus Coulomb energy, the second is the electronnucleus Coulomb energy, and the third is the electron-electron Coulomb energy, in analogy with Equation 2.4. 38 Chapter 2. Extended Chester equations for polyatomic superconductors 2.3.3 Three intermediate theorems •Contributions to H From now, this is going to be the Hamiltonian that we are going to use in our calculations. First of all, it is direct to establish that the Equation 2.5 for the internal energy is still valid with this Hamiltonian because our new Hamiltonian can be represented also as a sum of three terms. H=U=KM+Km+,(Hcontributions) (2.17) •Virial Theorem for H The generic expression for the virial theorem is given by: 1 2 Ntot X k=1 <Fk·rk>=KM+Km,(2.18) where Ntot =PNz kNk+Neis the sum of the total number of nucleus and electrons in the material, Fkis the total force over the particle k, the notation <> stands also for the thermal averaged quantities and rkis the position of the particle k. If we divide the external and internal forces (due to Coulomb repulsion or attraction), we obtain that: 1 2 Ntot X k=1 <Fext k·rk>=KM+Km+⌅,(2.19) where Fext kis the external force applied to the particle kand ⌅is the virial of interparticle forces for this Hamiltonian, given by: ⌅= Nz X k=1 Nz X k0k Nk X i=1j Nk0 X j=1 <(Rik Rjk0)·rRikRjk0 ZkZk0e2 |Rik Rjk0|>  Nz X k=1 Nk X i=1 Ne X j=1 <(Rik rj)·rRikrj Zke2 |Rik rj|>+ Ne X i=1 Ne X i<j <(rirj)·rrirj e2 |rirj|>, (2.20) 2.3. Extended Chester equations for polyatomic type I or II superconductors 39 where when k=k0it is verified that i<j. Due to the fact that the internal interaction forces are still of Coulomb type, it is easy to demonstrate that ⌅=, so this theorem is not a↵ected by the fact that His polyatomic, and we can write that: 1 2 Ntot X k=1 <Fext k·rk>=KM+Km+.(2.21) Here we have to deal with the type of external force that is being applied to the system. We are going to restrict by now to the hydrostatic pressure. Note that this virial theorem depends highly on the type of pressure, and here the anisotropy of the system plays an important role. Non hydrostatic are interesting cases to explore in further works. De di↵erential external forces dFion the i-direction can be interpreted in terms of the stress tensor ik as follows: dFi=ikdSk,(2.22) where dSkis the di↵erential surface vector in the k-direction, and in the hydrostatic case the stress tensor is given by: ik =2 4 P00 0P0 00P3 5,(2.23) where Pis the pressure in each of the faces of the material. So we can write the sum of equation 2.21: 1 2 Ntot X k=1 <Fext k·rk>=I3 X i,k=1 riikdSk,(2.24) where the integral goes through all the surface of the material. Taking into account the Gauss theorem: I3 X i,k=1 riikdSk=Zr·rdV =3PV. (2.25) Therefore we can conclude that we can use the same expression 2.6 for the virial theorem that Chester is using in his article for our more generic Hamiltonian: PV =2 3(KM+Km)+1 3,(virial theorem) (2.26) 40 Chapter 2. Extended Chester equations for polyatomic superconductors •Equation derived from Hellman Feynman theorem applied to H Finally we have to check that the equation for the third theorem is valid or if it has changed. The derivation is exactly the same, but now we have to take into account that we have di↵erent masses Mkfor our di↵erent types of nucleus. We thus start from the partition function for the Gibbs Free energy: Q=X VX l exp[(El(V)+PV)],(2.27) where Elis the lth energy level of the system for a volume V, =1/kT.The equation that correlates it with the Gibbs Free energy: G(T,P)=kT ln Q, (2.28) where kis the Boltzman’s constant. The energy levels, El(V), are the eigenvalues of our Hamiltonian. If lare the eigenfunctions of this Hamiltonian, then: Hl=Ell.(2.29) If we apply the Hellmann-Feynman theorem [66] to the derivative of this energy eigenvalues respect to the nucleus mass Mkof an ion of type k, we get: @El @Mk =ZV l⇤✓@H @Mk◆ldV. (2.30) Consequently we can write that: Mk @El @Mk =(KMk)ll,(2.31) where KMkis the partial kinetic energy of the nucleus of type k. Combining this relationship with equations 2.27, 2.28 and 2.31, we obtain that: Mk✓@G @Mk◆T,P =X VX l (KMk)ll exp[(El(V)+PV)]/Q =KMk,(2.32) and we can obtain the total kinetic energy of the nucleus.  Nz X k Mk✓@G @Mk◆T,P = Nz X k KMk=KM.(derivation from HM theorem) (2.33) Thus we have re-derived the third theorem for our Hamiltonian needed to apply the Chester formalism. 2.3. Extended Chester equations for polyatomic type I or II superconductors 41 2.3.4 Extended Chester equations The Final step is to extend these equations to 0K, introduce the di↵erence between the normal and superconducting state, and express the Gibbs free energy in terms of the condensation energy. Gibbs free energy is related with internal energy as given at T=0K: G=UPV. (2.34) Fixing the temperature to 0K, di↵erentiating between the normal and superconducting state, and substituting definition 2.34 this into the equation 2.33 we have: Econd =U=KM+Km+,(2.35) 3PV= 2(KM+Km)+,(2.36) Nz X k"MkP✓@V @Mk◆T,P Mk✓@Econd @Mk◆T,P#=KM.(2.37) Therefore, to know the energy balances for each doping, we need to know the value of the condensation energy Econd, the variation of the volume Vbetween the normal and superconducting state at T= 0 K, the variation of the condensation energy with the isotopic mass of each type of nucleus k, and the variation with the isotopic mass of the di↵erence of the volume Vbetween the normal and superconducting state at T= 0 K. Solving these equations we obtain the extended Chester equations for any type of superconductor: First extended Chester equation (electronic kinetic energy saving) Km=3PVEcond Nz X k"MkP✓@V @Mk◆T,P Mk✓@Econd @Mk◆T,P#,(Chester 1) (2.38) Second extended Chester equation (nucleus kinetic energy saving) KM= Nz X k"MkP✓@V @Mk◆T,P Mk✓@Econd @Mk◆T,P#,(Chester 2) (2.39) Third extended Chester equation (Coulomb energy saving) =2Econd 3PV. (Chester 3) (2.40) Note that these are exact equations to get the energy balance for any polyatomic superconducting material. 48 Chapter 3. Application of extended Chester equations to cuprate superconductors for each element in cuprates. Moreover, if we take into account the di↵erence between the normal and superconducting states of these quantities ,thedi↵erence should be even lower than the number estimated in the Equation 3.1. If we compare this with the typical order of magnitude of the condensation energy, we realize that we can discard the terms MkP(@V/@Mk)T,P in the first and second extended-Chester equations. The next term that we are going to estimate is PV, the one that corresponds to the term P@G/@Pof the original Chester equations. Chester himself in his pioneering work [9] remarks that this term is of the order of 106cal/mol in BCS superconductors while the condensation energy is of the order of 103cal/mol, so it can be discarded. Our strategy here is to obtain an upper limit for this value in cuprates. For that purpose, we are going to use the data of Pasler et al. [79], where they measure the thermal expansion coefficients ↵i=(1/Li)dLi/dT for each lattice parameter Liin the YBCO HTS. We are going to obtain the Li=a, band clattice parameters of YBCO depending on temperature Tby integrating the thermal expansion coefficients as follows: Li(T)=Lref exp ✓ZT Tref ↵i(T)dT◆,(3.2) where Lref and Tref are the reference lattice parameter and the reference temperature. In this case we have chosen for our approximation the temperature Tref = 190 K and the lattice parameters for that temperature a=3.82 ˚ A, b=3.89 ˚ A, and c= 11.68 ˚ A, that are the approximated values at room temperature. Note that we are going to calculate di↵erences of products of these parameters, so to obtain the order of magnitude that we are trying to achieve it does not matter if we choose as a starting point an approximate value of the exact lattice parameter at that temperature. With this approximation we have plotted in Figure 3.1 the evolution of the three lattice parameters of optimally doped YBCO with temperature, in the same scale, extracted from the data of Reference [79]. Reviewing the literature, one can find that the evolution of lattice parameters of both superconductors [79, 82] and non superconducting [80, 81] materials has the same shape, and can be described fitting the data to a second-order polynomial. In first approximation we have fitted the normal state of the lattice parameters to this kind of function to obtain a possible value of the lattice parameters in the normal state at 0 K (named as an,bnand cn). Due to the fact that the higher ↵ivalue is the one related to the c lattice parameter, this is the one that have a bigger change with temperature, as we can see in the upper part of the Figure 3.1. We have calculated the volume di↵erence anbncnabc, to get a possible order of magnitude of the amount V. With those values, we have obtained: 3.2. Terms that may be estimated as quantitatively negligible for HTS 49 PV⇡0.0055 J/mol,(3.3) where P= 105Pa is the atmospheric pressure. This is a value 3 orders of magnitude lower than the typical value obtained for the condensation energy in cuprates. [24–26,64,65] Figure 3.2: Product of the lattice parameters obtained integrating the data from Reference [79] are plotted versus the temperature. The red line represents the parabolic fit to normal state volume T>94 K and the blue dashed line represents the linear fit to the normal state volume. We obtain an estimation of the value of Vwith this extrapolations to 0 K. We have used another approach to get the maximum order of magnitude of the volume change that is possible to extract from this data. In this case, we have obtained firstly the volume as a function of the temperature, as is pictured in Figure 3.2, and then we have done another fit to a parabolic polynomial to extract the value of the volume in the normal Vnstate at 0 K. With this approach the di↵erence PV⇡0.0057 J/mol, very near from the last estimate. However, we would like also to estimate the highest possible change in volume from the normal to the superconducting state from these data. To do that we have plotted the extreme case in which the normal state fits a linear behavior, being the quantity Vthe highest. In this case we have obtained a value PV⇡0.0193 J/mol, which is still at least two orders of magnitude lower than the typical value for the condensation energy in cuprates. Note that we have studied the extreme cases to get the maximum order of magnitude to demonstrate that we can discard this term from our equations. A realistic analysis should discard the volume change from normal to superconducting state in the c direction, because there is no evidence of change in the behavior of this lattice parameter with temperature. On the other hand, it is possible to 50 Chapter 3. Application of extended Chester equations to cuprate superconductors see directly some jumps of opposite sign in the behavior of lattice parameters aand bin the critical temperature, that leads to a change of volume at least one order of magnitude lower than the one that we have obtained in our lasts approaches.1[79] •Extended-Chester equations without terms estimated negligible in cuprates Now that we have discarded the term associated to the derivative of the volume respect to the isotopic mass, and the term related to the external forces in the virial theorem we can rewrite the extended Chester Equations for HTS like: KM= Nz X k Mk✓@Econd @Mk◆T,P ,(3.4) KM=Econd + Nz X k Mk✓@Econd @Mk◆T,P ,(3.5) =2Econd.(3.6) There is some direct information that we can infer from these equations. First of all, there is a Coulomb energy saving through all the phase diagram that is always twice the condensation energy Econd. This is a well known result that is in agreement with more specific scenarios, like the MIR proposed by A.J. Leggett. [10–12] It is also important to remark that, although the Hamiltonian introduced by A.J. Leggett is quite general, it is not the exact Hamiltonian that should be used to describe the exact HTS. For instance, it does not take into account the kinetic energy of the nucleus in the system, that give us information about the role of the phonons in the superconducting mechanism of these materials. The Hamiltonian that we are using is more general, and it let us to obtain information about the role of the phonons in the superconducting mechanism of non conventional superconductors. 1We also want to remark that if the quantity Vdoes not change very much with pressure, all the energy balances can change with the right order of magnitude in hydrostatic pressure, introducing new phenomenology that may be of interest for HTS such as H2S. 3.3. Singularizing the oxygen-nucleus contributions KMO51 3.3 Singularizing the oxygen-nucleus contributions KMO For continuing our evaluation of the extended-Chester equations in cuprates, the ideal situation would be to count on the experimental measurements of the quantities Mk(@Econd/@Mk)T,P (see Equations 3.4 to 3.6). However, to the best of our knowledge, there are no measurements of the specific heat with temperature for cuprates with di↵erent isotopes. This fact lead us to make some assumptions to try to obtain this derivative. We are going to suppose that the condensation energy depends on the isotopic mass of the element kthrough di↵erent intermediate variables, U(Mk)=U(A(Mk),B(Mk),...), where A,B, etc., are intermediate properties that depend on the isotopic mass. In this context, we can write that: ✓@Econd @Mk◆T,P =✓@Econd @A◆T,P ✓@A @Mk◆T,P +✓@Econd @B◆T,P ✓@B @Mk◆T,P +... (3.7) The problem now is identifying what those intermediate variables could be. For instance, in the BCS analysis, Chester [9] used the critical magnetic field as an intermediate variable to obtain all the derivatives, taking into account that Hc=H0h(t), where t=T/Tcis the reduced temperature, and his an unknown function. Knowing the derivatives @H0/@Mand @Tc/@Mit was possible to obtain the energy balances with temperature. Our problem is more complicated, because we do not know the general equation for the condensation energy through all the phase diagram of the cuprates. Here, we could assume that the two major candidates for these intermediate variables are the critical temperature Tcand the pseudogap energy Eg.There are several experimental data that correlates the evolution of Econd,Tc, and Eg through the phase diagram of some cuprates. [24,25,64,65] However, as we have mentioned before, to obtain the condensation energy we have to know what is the value of the entropy or the specific heat of the sample in the normal state at temperatures below the critical temperature. To obtain these quantities, in principle three options would be: calculating it through some theory or model, using a high magnetic field to kill the superconducting state, or extrapolating it by taking into account the behavior at higher temperatures. The first option is only useful if we already know the microscopic mechanism that takes place in HTS. The second option is impossible to reach because the upper critical magnetic fields are too high for today’s experimental resources. Nevertheless, there have been attempts to introduce magnetic dopants like Zn to suppress the superconductivity and get the value in the normal state. [64] This method could be rather informative, but nowadays it is believed that the normal state of HTS is a competing phase between di↵erent states, and even a small 52 Chapter 3. Application of extended Chester equations to cuprate superconductors amount of dopants can influence this competition, changing the normal state entropy. [63] Last option is the most used, but it has the problem of whether it is necessary to take into account the pseudogap region as a normal state region, or take only the region above it. D. van der Marel et al. have proposed some generic constraints for the functional behavior of the normal state entropy based only in general thermodynamic principles. [63] For example, one of them is that the normal state entropy at 0 K should be equal to the measured one at that temperature. J.W. Loram et al. [64] have tried to extrapolate the normal state entropy above the pseudogap temperature in underdoped cuprates and they found that this extrapolation does not accomplish some of these constraints. Conversely, extrapolating the entropy associated to the pseudogap regime, it is possible to fulfill all the constraints, and thus obtain an unambiguous value of the condensation energy. Moreover, this extrapolation fits also with the magnetic dopant method. [64] Therefore, we can assume that the pseudogap is a normal state property, and thus it does not take part directly in the calculation of Econd,because above the superconducting fluctuation regime, in the pseudogap region, Sn(T)=S(T). Conversely, it is clear that Econd depends highly on the Tcvalue, as we can see directly from the specific heat measurements. [24, 25, 64, 65] In fact, BCS compounds verify that Econd /T2 c. Therefore, to estimate the energy balances, we are going to assume that the highest dependence of the condensation energy on the isotope masses is through the critical temperature Tc. Note that this approximation have been assumed in the past by A.J. Leggett for a similar estimation. [2] Within this scenario, it is reasonable to make the following approximation: KMk=Mk✓@Econd @Mk◆T,P ⇡Mk Econd Tc Tc Mk =Econd Tc ↵kTc,(3.8) where ↵k=(dln Tc/d ln Mk) is the isotope coefficient that corresponds to the element k, and means the di↵erence of the value between two dopings. Within this approximation, knowing the phenomenology of the phase diagram of cuprates in relation to the critical temperature, the isotope coefficient, and the condensation energy, we can extract some qualitative results for the value of KMk. The most measured isotope coefficient in all the phase diagram of cuprates is the isotope exponent for the oxygen ↵O. [27, 28, 67, 68, 70–75]Therefore, we can calculate the value of KMOfor the whole phase diagram. On the other hand, there are some experiments where they study the e↵ect of the copper isotope mass, [69, 75–77] and some measurements for the isotope coefficient for Ba in YBa2Cu3O7(YBCO) at optimal doping (with an almost negligible e↵ect). [69] For the copper coefficient of LSCO, there are measurements 3.3. Singularizing the oxygen-nucleus contributions KMO53 in two points of the phase diagram (p=0.15 and p=0.125), so there is not enough data use them for our calculations. [68,71,75] However, these data reflect the same trend for Cu than for oxygen (positive coefficient, and enhanced values for the underdoped region, concretely in the region where p=1/8). For the copper coefficient of YBCO, there are data measured by J.P. Frank et al. [77] for a wide part of the phase diagram. But these data contrast with those obtained by other authors [69,75] where the copper isotope coefficient shows the opposite sign, with a behavior similar to the case of LSCO (making these data more reliable). Therefore, while there is not enough amount of data to obtain accurately the KMCu through all the phase diagram with precision, with the data available we can induce that the trend is similar than for the case of oxygen. The measurement of the other isotope e↵ects for the other elements would rise the opportunity to estimate the behavior of KMk. Due to the fact that we are only able to reliably obtain the kinetic energy difference of the oxygen nucleus, we can rewrite the extended Chester equations 3.4 to 3.6 taking into account this fact and the new approximations, as follows: KMO⇡Econd Tc ↵OTc,(3.9) Kmix =KM+Krest =Econd KMO(3.10) =2Econd,(3.11) where Krest is the sum of the kinetic energy di↵erences between the normal and superconducting state of the nucleus of the cuprate except the oxygen and Kmix is defined as the sum of KMand Krest. With these equations we will be able to estimate the value of the three energy di↵erences for the cuprates LSCO, Bi2212, and YCBCO through the phase diagram with the available experimental data. 3.3.1 Three regions in the Econd vs doping, Econd(p), and O-isotope coefficient vs doping, ↵O(p), phase diagrams of cuprate HTS First of all, it is remarkable that the isotope coefficient for oxygen has a similar behavior in the phase diagram for most cuprates. [27] The only remarkable difference in this trend is in the point p⇠1/8 in LSCO, where there is a large jump in the isotope coefficient. [28, 71, 72, 75] The rest of the cuprates verify that the isotope coefficient is always positive, ⇠0 at optimal doping and it increases at both sides of the phase diagram, achieving values up to 0.8 in the underdoped side, and up to ⇠0.1 in the overdoped side. [27,28,67,69,70,73,74,76] The phenomenology of the condensation energy and the critical temperature phase diagrams are described in Figure 3.3. The condensation energy general 54 Chapter 3. Application of extended Chester equations to cuprate superconductors trend is that it increases continuously from small dopings until some point p⇠ 0.19, where it peaks, and then it starts to fall until maximum doping. [24–26,63– 65] Conversely, the Tcphase diagram show for all the cuprates a superconducting dome that follows an approximate inverse parabol that peaks at p⇠0.16. [24– 26,63–65] Figure 3.3: Scheme of the qualitative behavior of the critical temperature Tcand the condensation energy Econd through the phase diagram in cuprate superconductors. Within the scenario where most part of the dependence of Econd with the isotope masses of the material is through the critical temperature, we distinguish three regimes A, B and C. These regimes are characterized by a specific sign of the derivative of Econd respect to the Tc, and for di↵erent values of the isotope coefficient ↵. Due to the fact that the condensation energy and the oxygen isotope coefficient have the same qualitative behavior through the phase diagram of all the cuprates, we can infer some qualitative information about the sign of the kinetic energy variation of the oxygen nucleus through the phase diagram of these materials. Due to the fact that ↵Oand Tcare always positive in cuprates, the sign of KMOis going to be given by the quantity Econd/Tc. In this context we distinguish in Figure 3.3 three regimes. In regime A, both Econd and Tchave the same sign, so we expect that in this region KMO>0. Then, when we reach the maximum of the dome, we achieve a point in which Tc!0. This would lead us to a divergence of the quantity 3.4. Quantitative analysis of KMOin the phase diagram 55 KMO. However, we have to remark that at this point, ↵O⇠0 for all HTS, so we have an indetermination that is going to make hard to get what are exactly the value of KMOat this point. However, in region B we have that Econd and Tc have opposite sign, so KMO<0 here. Therefore, we have uncertainties to know what happens exactly in the maximum of the superconducting dome, but we can ensure that KMOchanges its sign. Finally, in the boundary between regions B and C the condensation energy peaks, and thus at some point Econd !0. This means that KMO= 0 in that boundary. Finally, in the C region, we reach again the same behavior where KMO>0 because Econd and Tchave again the same sign. Note that with this qualitative analysis we can infer the evolution of the sign of KMOthrough the phase diagram, but not its relative value respect to the condensation energy, an important information to know to what extent the movement of the oxygen nucleus play a role in the superconducting mechanism of HTS. This feature will be obtained by our calculations from experimental data that we have collected from the bibliography to plot the energy balances through all the phase diagram of some generic HTS. 3.4 Quantitative analysis of KMOin each of the regions of the phase diagram of cuprates In this subsection we will obtain our most important result from our estimates of the energy balances in cuprate HTS, that reveal three di↵erent regions in the phase diagram versus doping level in these compounds (roughly corresponding to regions A-C in Figure 3.3) in which KMOfavours, opposes or is negligible for superconductivity (roughly KMO>0 in A, KMO<0 in B, and KMO⇠0 in C). First of all, there are measurements of the condensation energy phase diagram and critical temperature phase diagram for the superconductors LSCO, [26] Bi2212, [25, 64] and YCBCO. [24, 65] We have also found data for the isotopic coefficient of LSCO [28,68,71,72,75] and Bi2212 [27,28,67,73]. We also found one point in the phase diagram for the YCBCO isotope coefficient, [27] so we decided to use the isotope coefficients of generic YBCO, [27, 28, 67, 69–71, 74, 76] taking into account that the trend of the function of the isotope exponent through all the phase diagram is similar for all the cuprates. [27] With all these data, we have approximated the value of KMOevaluated in the mean point (p1+p2)/2 of each pair of the Econd data obtained experimentally by: KMO(p=p2+p1 2)⇡Econd(p=p2)Econd(p=p1) Tc(p=p2)Tc(p=p1)↵O(p=p2+p1 2)Tc(p=p2+p1 2),(3.12) 56 Chapter 3. Application of extended Chester equations to cuprate superconductors where p2and p1are the dopings where the Econd have been measured. The values of Econd, the isotope coefficient and the critical temperature at these dopings have been interpolated from the data of the references [24–28]. We have plotted in Figure 3.4 the adimensional fraction of KMO/Econd for the phase diagram of the three cuprates. This is an important amount, that tells us to what extent the phonons play a role in the mechanism of HTS in di↵erent parts of the phase diagram. 3.4.1 Region A of the phase diagram: KMOfavors superconductivity The Figure 3.4 shows clearly the three regimes described in Section 3.3. In the region A, there is a kinetic energy saving in the oxygen kinetic energy, that near optimal doping is not negligible, i.e., the quantity KMO/Econd is in the order of the unity. This means that in this region of the phase diagram, the movement of the oxygen nucleus favors superconductivity, with a non negligible contribution to the condensation energy. This is due in part to the fact that in this region of the phase diagram, the isotope coefficient ↵Oreaches very high values, and the value of the condensation energy derivative respect to the critical temperature is also high. It is also remarkable that in this region, the behavior of KMO/Econd in LSCO is quite di↵erent, because the kinetic energy di↵erence increases at low dopings. This is due to the fact that in this region this compound has a very di↵erent isotope coefficient, extremely high (↵O⇡0.8). Moreover, LSCO has a quite di↵erent phase diagram in the region around p⇡1/8, which also contributes to enhance this value. The other two compounds show a di↵erent behavior, showing a lower value for lower dopings and the highest values near the optimal doping. 3.4.2 Indeterminate crossover between regions A and B As we discussed in Section 3.4, between the region A and B we have an indeterminate region due to the fact that there the value of Tc⇡0 and the value of the isotope coefficient is also negligible. This leads to an indetermination, that can only be solved measuring directly the isotope e↵ect in the condensation energy at optimal doping. However, due to the universal shape of the phase diagram of the cuprates, we can ensure that the sign of the kinetic energy change of the oxygen nucleus is di↵erent at both sides of the optimal doping. 3.4. Quantitative analysis of KMOin the phase diagram 57 Figure 3.4: Fraction of the kinetic energy variation of the oxygen nucleus in relation to the condensation energy in the phase diagram for three di↵erent cuprates obtained by Equation 3.12. In the upper part of the Figure is represented the calculations for LSCO, in the lower part the calculations for YCBCO are plotted, and in the lower part the calculations for Bi2212 are presented. We distinguish three di↵erent regions (A,B and C) common to all the cuprates. 3.6. Conclusions 65 IIA: Superconducting devices The contentsin this part are based on our Spain’s patent application coded P201930020 and an article by J.C. Verde et al. submitted for publication in Nanomaterials (Chapter 4), and in articles [47,88] and book [89] by J.C. Verde et al. (Chapter 5). Chapter 4 Design optimization of high-temperature superconducting bolometers HTS TES using doping nanostructuration and 1D patterning 67 68 Chapter 4. Design optimization of high-temperature superconducting bolometers Summary of the chapter: We consider the possibility of improving by means of micro and nanostructuration of the doping level the operational characteristics of cuprate superconductors (HTS) when used as thin film active elements in bolometric radiation-sensor devices (HTS TES). We develop for that task both numerical computations and e↵ective-medium theoretical formulas. The results of both of these two independent methods suggest that doping structuration may significantly improve the operational characteristics of these devices. For instance, the most optimized structuration design found by our analysis (a 1D discrete 4-step patterning of the doping level) with respect to conventional, non-structured HTS TES doubles the sensitivity and improves by more than one order of magnitude the saturation power and operating temperature range. 4.1. Introduction 69 4.1 Introduction Bolometers are radiation sensors that detect incident energy via the temperature increase caused by the absorption of incoming photons. Sensors formed by an array of microsized bolometers (microbolometers) are often used, e.g., for infrared cameras, detectors in quantum cryptography, astrophysics, satellites, aviation, etc. [29–34] A key figure influencing the efficiency of resistive bolometers (that measure variations of temperature Tthrough the corresponding changes of the electrical resistance R) is the so-called “temperature coefficient of resistance” or TCR, given by: [29–34] TCR =  1 R dR dT .(4.1) Highly sensitive bolometers require a large value of TCR. For instance, structures of vanadium oxides VxOy, commonly used in commercial microbolometers, present TCR ⇠0.025 K1. [34] Much larger TCR may be achieved with superconductors kept at base temperatures coincident with their normal-superconducting transition, Tc.Thisis the case mainly when using conventional low-temperature superconductors with Tc1K (the so-called low-Tctransition edge sensors, or low-TcTES), that achieve TCR ⇠1000 K1or even more. [29,35] Such high values are possible because low-Tcsuperconductors have a particularly narrow transition, due both to the achievable high purity and to the fact that their critical e↵ects near Tcare negligible and produce almost unobservable rounding of the normal-superconductor transition (also when operated in a voltage-biased configuration). The very high TCR values of low-TcTES make them a technology of choice for detecting the most faint radiation incidences, as for cosmic microwave background measurements or for other new applications where single-photon detection is important, as in quantum entanglement and cryptography. [29–31] Note that for these applications the very low temperature required to operate the low-TcTES 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 (that would otherwise mask the faint radiation to be measured). However, the requirement of a highly-stabilized liquid-helium-based cryogenic setup is a serious difficulty for adoption of low-TcTES in other applications. For these scenarios, thermal self-noise problems may be avoided already at much higher temperatures, feasible using less problematic coolers (the general rule is to operate the bolometer below the radiant temperature of the source to be measured) [39,40]. After the discovery of high-Tccuprate superconductors (HTS), various authors have explored their use for bolometer sensors (the so-called HTS TES) with simpler liquid-nitrogen-based cryogenics. [36–41] The compound YBa2Cu3O(YBCO) 70 Chapter 4. Design optimization of high-temperature superconducting bolometers is the HTS usually considered for this application, usually with optimal nominal doping. As shown in [36–41], YBa2Cu3O7thin films provide in some aspects a viable competition to VxOy. These YBa2Cu3O7films not only provide TCR ⇠1.5K 1and low noise at an operational temperature ⇠90 K, but also a faster thermal response than VxOy. Moreover, the rest of parameters contributing to a large sensitivity (thermal conductivity, infrared absorbance, etc.) are also favorable or at least competitive with semiconducting bolometers. [36–41] However, the HTS TES until now proposed still share some of the significant shortcomings of low-TcTES: 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 Tof just ⇠0.1 K or less for low-TcTES, and ⇠1 K for the HTS TES proposed until now. [36–41] In contrast, T⇠25 K is common for VxOy. [34] Importantly, this also means that small quantities of incoming radiation suffice to saturate existing HTS TES, making them appropriate only for “very poorly lit” imaging (low dynamic range). The HTS TES systems proposed until today use superconductors homogeneous in nominal composition and critical temperature. [36–40] However, in the recent years di↵erent novel techniques have been developed to impose regular patterns to HTS thin films, creating custom designs, down to the microand the nano-scales. [42–46] 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-e↵ect, [42] nanodeposition, [43] focused ion beam, [44] etc. In fact, nanostructuring of HTS has become the specific subject of recent conferences [45] and networks [46] funded by the European Union. Most growers of structured HTS films have focused up to now on increasing their critical current and achieving collective vortex pinning at certain matching magnetic fields. [42–45] Also, in a recent work [47] some of us preliminary showed that some simple ad hoc patterns may also moderately modify the shape of the R(T) curves near the transition (e.g., sharpening it). 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 [90] by Boktem et al., who consider 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 improvements of HTS TES microbolometers by considering the use of YBa2Cu3O7thin films including doping nanostructuring and doping patterns. In particular, our main objective will be to obtain an increase of the operational temperature interval, T,inwhich i) the TCR is large and ii) Ris linear with T(i.e.,dR/dTconstant with T, that is another desirable feature that simplifies both the electronic control of the 4.2. Methods for structured non-patterned HTS TES 71 microbolometer and the required stability of the cryogenic setup). Accompanying this Tincrease we will also obtain notable improvements of other bolometric characteristics, as the saturation energy and power, and in some cases the TCR itself. To fulfill that objective, we will study the operational characteristics of dopingnanostructured HTS TES . We will consider two major types of structuration: First, the random spatial disorder intrinsically associated to non-optimal doping levels. Secondly, we consider the imposition of regular arrangements of zones with di↵erent nominal doping levels each (patterning). For the first type of structuration (structured non-patterned HTS TES, presented in Sections 4.2 and 8.5), we implement as methods finite-element parallelsuperconducting calculations and also, as a way to confirm the results, analytical estimates using an e↵ective-medium approximation (based on Reference [91]). These results are also verified against existing measurements in equivalent YBCO films by other authors [92] (who never implemented them in a HTS TES but measured the R(T) curve in the relevant temperature interval). Our results indicate that simple type of structuration already improves some of the bolometric parameters with respect to this conventional non-structured HTS TES proposed until now (see, e.g., Table 4.1 for a rapid summary). Concerning the HTS TES structured by means of ad-hoc pattern designs of di↵erent zones, with di↵erent nominal doping levels (Sections 4.4 to 4.7) we have adapted our methodology to include the possibility of such zones (also extending the corresponding analytical formulae of Reference [92] both to the general case and to each example pattern considered). We will explicitly discuss three specific patterns that we found of special interest, each successively presenting better bolometer performance (see Table 4.1). Our more optimized design is a 4-step discretized exponential-like dependence of nominal doping with the longitudinal position. This pattern design should be the one easier to fabricate among those considered by us, and also the one presenting the most improved bolometric characteristics: with respect to conventional non-structured HTS TES, it improves by more than one order of magnitude Tand the saturation measurable power, and it also almost doubles the TCR sensitivity. 4.2 Methods for structured non-patterned HTS TES A prerequisite for our study is to have a valid description method for superconductors comprised by a single zone of uniform doping. Such single-doping domains will be later the building blocks which associations as a mesh circuit will form the doping-structured HTS film. In this section, we first briefly review the basic equations for the main operational bolometric characteristics of HTS TES sensors, and next we collect, from the existing bibliography, phenomenological formulae describing for non-patterned HTS films the electrical resistance vs 72 Chapter 4. Design optimization of high-temperature superconducting bolometers temperature curves, R(T), crucial to determine most of such bolometric characteristics. 4.2.1 Main operational parameters for HTS TES sensors Operation of HTS TES is based on the increase of the electrical resistance Rwith temperature Tas radiation heats the sensor: Therefore, a fundamental feature is the range of temperatures over which the increase of Ris large. In particular, we define the operational temperature interval Tas the di↵erence T=T+T,(4.2) where Tis the base thermal bath temperature (i.e., the one that occurs in absence of radiation) and T+is the maximum temperature up to which Rmaintains the strong and constant slope with T. With these definitions, the TCR may be obtained as: TCR = R(T+)R(T) TR(T).(4.3) Another important operational parameter that follows from the R(T) transition is the saturation energy Emax, defined as the maximum energy that the bolometer is able to measure: Emax =CT, (4.4) where Cis the heat capacity of the ensemble formed by the superconducting film and its substrate. Note that the mass of the later is always much larger than the one of the superconducting layer (typical thicknesses are 100 nm for the HTS and 1 mm for the substrate). Thus, Cis dominated by the substrate and is essentially independent of the doping of the superconductor. However, the Cof the substrate is temperature-dependent. Therefore, the operational temperature Tcorresponding to each doping must be taken into account in the calculation of C. The most commonly used substrate for YBCO superconductors is single-crystalline SrTiO3(STO) and henceforth we will evaluate Cusing the experimental data of Reference [93] for the heat capacity of STO at cryogenic temperatures. More important than Emax is often Pmax, the maximum power that can be measured without saturation: Pmax =GT, (4.5) where Gis the thermal conductance between the sensor and the thermal bath. For concreteness, we will consider a microbolometer in which the substrate is 4.2. Methods for structured non-patterned HTS TES 73 directly in contact with the heat sink. Therefore, to evaluate Gwe use the experimental data of Reference [94] for the thermal conductivity of single-crystalline STO, interpolating them at the operation temperature of each considered superconductor. We emphasize here that the Cand Gdependence on Tis moderate, with variations below ⇠20% between 75 K and 90 K. Therefore, for optimizing Emax and Pmax via doping structuring, the main factor to consider will be the corresponding increase of Tin Equations (4.4) and (4.5). Another important parameter to describe the bolometer operation is the time constant ⌧given by: ⌧=C/G. (4.6) This parameter estimates the time necessary for the bolometer to return to the temperature of the sink when the radiation is turned o↵[95]. Note that this quantity does not depend on the width T, and it is not going to change deeply with doping structuring. 4.2.2 R(T)in the normal state of non-structured HTS The temperature versus doping, T-vs-p, phase diagram of HTS displays various regions with markedly distinct R(T) behavior. This phase diagram has been, in fact, extensively studied by di↵erent authors, and today its account is fairly complete at the quantitative phenomenological level, as recently reviewed in detail, e.g., in Reference [96] (independently of the fact that many of the fundamental microscopic reasons for the occurrence of such phase diagram are still open to considerable controversy at the highest-profile academic forums [97,98]). This HTS phase diagram is convenient summarized in the Figure 4 of Reference [96]. Here, for the sake of brevity and clarity, let us just recall that the superconducting critical temperature is maximum at p⇠0.155 (optimal doping, separating the underdoped p<0.155 and overdoped p>0.155 compositions). Above Tc(p), the material presents a normal-state background electrical resistivity, ⇢b(T,p), that is linear on Tabove a certain so-called pseudogap temperature T⇤, and is pseudoparabolic semiconducting-like [96] for Tc<T<T⇤. In YBCO, it is T⇤(K)⇡270 3000(p0.1), [96] so that for p&0.16, it is T⇤<T cand 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 ⇢b(T,p) given in Reference [96] 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. 80 Chapter 4. Design optimization of high-temperature superconducting bolometers crobolometers considered in this Section because it mainly depends on the operational temperature Tvia the heat capacity and the thermal conductance and, in this region, the ratio between these parameters is almost constant with temperature. [93,94] Figure 4.1: Resistance R versus temperature T obtained for YBCO thin films with a single, uniform value for the nominal doping level p, including the case with negligible Tcnanostructuration (or optimal doping p=0.155 in which Tc saturates near its maximum value and Tcdisorder is negligible) and various cases in which the pvalue corresponds to significant Tcnanostructuration (see Sections 4.2 and 8.5 for details). The data points correspond to the finite-element computations, and the continuous lines to the analytical EM approximation (see Equation (4.12)). The shaded gray region signals the operational temperature range T(in which Ris strongly dependent and linear in T). In the upper drawing we image the simulated sample and setup, including also a zoom at smaller length scales illustrating the spatial variation of the local doping level p(x, y) (each p(x, y) monodomain has typical size (30 nm)2and the distribution is Gaussian around the average p(x), see main text for details). On the other hand, with the help of Table 4.1 it is also possible to compare these bolometric parameters with the typical values of classical Vanadium oxide bolometers. It is remarkable that the TCR values of the HTS TES considered 4.3. Results for structured non-patterned HTS TES 81 in this Section are 2 orders of magnitude higher than the value for the V2O5 bolometer. This means that this HTS TES have higher sensitivity to measure the radiation. It is also interesting that the time constant is lower in HTS TES, which means that they have a faster thermal response. However, the Tin V2O5bolometers is one order of magnitude higher than the value of the type of HTS TES considered in this section. Therefore, this type of HTS TES saturates much before than V2O5bolometers (as also shown, in fact, by the Emax and Pmax values in Table 4.1). Thus, our next goal in this work will be to increase Tto improve the amount of radiation that the HTS TES are able to receive without achieving the saturation point. This will be done by considering doping structurations beyond the intrinsic random disorder of the doping level. 4.3.2 Verification using the analytical EM approximation Figure 4.1 also shows, as continuous lines, the results obtained by applying the EM approach, i.e., Equation (4.12), 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 4.1, as coherent with the expectation that the EM approximation is less accurate when percolative current paths appear, [91,110] what happens when R⇠0+(see also Figures 4.2 and 4.3 that better evidence these R⇠0+deviations). [102] This comparison gives, therefore, a first argument supporting the validity, at least concerning the main features, of our calculation methods for the doping structuring e↵ects in HTS TES. 4.3.3 Verification against existent measurements To test the accuracy of our finite-element calculations (and also those using the EM approximation) we have plotted in Figures 4.2 and 4.3 the data measured in Reference [92] in high quality YBCO (YBa2Cu3O) films comprised by a single zone of nominal oxygen stoichiometries O6.78 and O6.85 (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 [113]). We can see in these figures the good accuracy of the theoretical (EM approach) and computational (finite-element) methods used in this article to reproduce the experimental resistance curves of non-patterned YBCO films in the studied doping range. Note that these existing non-optimal YBCO fils have never been used to implement HTS TES. Our present considerations obviously indicate that such implementation would be of interest. 82 Chapter 4. Design optimization of high-temperature superconducting bolometers Figure 4.3: Comparison between the resistance vs temperature resulting from our methodology (open symbols for the finite-element computations and solid line for the EM analytical approximation) and the measurements of Reference [92] (solid symbols), for YBCO films with uniform nominal oxygen stoichiometry O6.78 (corresponding to p=0.140, i.e.,a Tc-structured non-patterned HTS). Figure 4.2: Comparison between the resistance vs temperature resulting from our methodology (open symbols for the finite-element computations and solid line for the EM analytical approximation) and the measurements of Reference [92] (solid symbols), for YBCO films with uniform nominal oxygen stoichiometry corresponding to p=0.156, i.e., very near from the optimal doping level. Note that the normalization of the axes significantly zoom the transition region with respect to the Figure 4.1. 4.4. Methods for patterned HTS TES 83 4.4 Methods for patterned HTS TES Let us now study the R(T) curve of HTS films in which a regular pattern of nominal doping levels is imposed, with the aim to obtain pattern designs that optimize the bolometric functionality. In these films, a regular spatial variation of the nominal doping level pis created by the samples’ grower by using any of the di↵erent methods developed in the recent years by experimentalists in HTS films (see, e.g., References [42,43, 47]). In what follows, all the specific example patterns considered in this work will be expressible as functions p(x), where x is the coordinate in the direction perpendicular to the experimental current (see scheme in Figure 4.1). It will be also useful for our subsequent descriptions to introduce the corresponding function (p), or relative length weight of each nominal pvalue in the film, defined as (p)= 1 L dx dp; (4.13) where Lis the 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 microscopic-scale randomness of the doping level (considered in the previous sections), i.e., p(x, y)=p(x)+prandom(x, y),(4.14) with prandom(x, y) consistent with Equation (4.10) evaluated using the local p(x). Naturally this double, two-scale spatial variation considerably difficult the task of calculating the resistive transition of these films, for what we have implemented the methodology that we describe next. 4.4.1 Obtainment of the R(T)curve of patterned HTS TES using finite-element computations To obtain the R(T) curves of patterned HTS TES, we use the finite-element software TOSERIS already described in Subsection 4.2.5. 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 ia nominal doping pi corresponding to the pattern to be simulated. Then we randomly calculate the local doping pifollowing the Gaussian distribution given by Equation (4.10), evaluated with the nominal doping piof each node. Finally, to each node we assign the Tci and ⇢i(T) corresponding to their local pias per the quantitative results of Reference [96] for the HTS material YBCO (see Subsections 4.2.2 and 4.2.3). We also tested that the sets of nodes sharing the same pivalue to follow the Gaussian distribution, and each R(T) simulation was repeated for several 84 Chapter 4. Design optimization of high-temperature superconducting bolometers so-generated samples to verify their reproducibility. These tests indicate that our choice of a 200 ⇥200 node mesh provides enough statistical size. If we attribute to each node the size of (30 nm)2corresponding to each Tc-monodomain in YBCO (see Section 4.2.4 and References [103,108]), the whole 200⇥200 sample corresponds to (6 µm)2, that is realistic for a bolometer pixel. Unless stated otherwise, we will use in our contributions the numerical values in Sections 4.2.2 to 4.2.5 for the common material characteristics, such as, e.g., a film thickness of 100 nm or the di↵erent critical fluctuation parameters as per Section 4.2.3. 4.4.2 Analytical estimates using an extended-EM approximation Besides performing finite-element computations, we are going to test our results against estimates based on the EM approach, for which purpose we must first 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 series of domains, each one with its own resistance and nominal doping p. Thus the total resistance is then the sum of the resistance of each domain, i.e.: R(T)=Zx=L x=0 dx (p(x),T)S(4.15) where Lis again the total length of the bolometer, Sis the transversal surface, and (p(x),T) is the conductivity obtained using the monodomain EM approach, i.e., using Equation (4.12) for each doping p(x). Equation (4.15) can be also written as the following over nominal doping, with the help of the (p) function defined in Equation (4.13): R(T)=LZpL p0 (p)dx (p(x),T)S.(4.16) For a discrete distribution (stepwise function p(x)), the above equations become discrete summations: Where piis the nominal doping of the bolometer in at the beginning of the bolometer (x= 0), pfis the nominal doping in the end of the bolometer (x=L) and (p)=(1/L)dx/dp is the relative weight of each domain of doping pin the sample. The value of (p)is given by equation 4.19 for this pattern. R(T)= N X i=1 Li (pi,T)S,(4.17) 4.5. Results for HTS TES structured with a linear p(x) variation 85 where Nis the number of discrete domains, each with its own nominal doping pi and length Li. Note that Equations (4.15) to (4.17) do not explicitly take into account the transverse currents when associating the di↵erent domains piand Li. Nonlongitudinal transport inside each domain is built-in by using the EM approach for each (pi,T). However, this approximation may be expected to fail when it has to describe percolations (because both the sum in series between domains and the EM approximation do not take this into account). This can make this approximation overestimate the value of R(T) in the very close proximity to the superconducting state, where percolations are expected to be important. 4.5 Results for HTS TES structured with a linear p(x) variation In the remainder of this article, we shall apply our methodology to di↵erent instances of doping-patterned HTS TES, seeking to progressively identify pattern design optimizing the bolometric sensor performances. In this section, in particular, we consider a simple linear variation of palong the longitudinal direction (the direction of the overall externally-applied electrical current, see scheme in Figure 4.4): p(x)=p0+✓pLp0 L◆x, (4.18) where p0and pLare the p-values at the opposite ends of the film x= 0 and x=L. In terms of the (p) function of Equation (4.13), this linear-in-x p-pattern simply becomes the constant value (p)= 1 pLp0 .(4.19) More specifically, we have chosen for our computations the rather typical values p0=0.135 and pL=0.161. 4.5.1 Results for the R(T) profile and bolometric operational parameters The results of our numerical finite-element evaluation for this p-pattern are displayed in Figure 4.4 (see also Table 4.1 for a comparative summary). The spatial pattern imposed to the HTS TES bolometer is drawn in the upper row of the Figure 4.4. 86 Chapter 4. Design optimization of high-temperature superconducting bolometers Figure 4.4: In the upper row, we illustrate a YBCO film patterned following the linear variation of the nominal doping p(x) studied in Section 4.5 (see Equations (4.18) and (4.19)). We also illustrates doping over 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 resistance vs temperature R(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 (4.20)). Note that the transition widens considerably with respect to Figure 4.1, but the operational Trange (shaded gray region) is small due to the nonlinearity of R with Tin most of the transition. As evidenced in that Figure 4.4, this type of structuring of the HTS film significantly broadens the R(T) transition (compare, e.g., with Figure 4.1 that represents, in the same T-scale, the results for HTS TES with comparable but uniform p-values). However, this structuring does not lead to a linear dependence of Rvs Tin that transition region. This may pose a difficulty in TES applications, that ideally require a R(T) variation both large (i.e., large TCR) and linear (i.e., constant dR/dT). The range Tin which both conditions are met is merely about 1.4 K for this type of p-pattern, already suggesting that further design op- 4.5. Results for HTS TES structured with a linear p(x) variation 87 timizations would be desirable (see next Section). In Table 4.1 we summarize the operational parameters obtained for the linear p(x) bolometer. We can conclude that they are of the same order or worse than the parameters obtained for an HTS TES bolometer at optimal doping without p-patterning. 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 are somewhat higher due to the small increase of the width of the linear regime (but they are below the values for non-patterned but underdoped YBCO). On the other hand, the time constant remains basically the same, because the ratio between heat capacity and thermal conductance remains almost unchanged. We can conclude that this patterning does not e↵ectively optimize the operational parameters of the HTS TES, mainly because it broadens the R(T) transition but does not achieve R(T) linearity in it. Better tuned p(x) variations need to be found (See Sections 4.6 and 4.7) to neatly surpass the bolometric characteristics of underdoped non-patterned HTS-TES. 4.5.2 Verification using the extended-EM analytical approximation To check the validity of our results dor the linear p(x) variation, we also have used the formulae described in Subsection 4.4.2. For that, our first step has been to combine Equation (4.16) with the (p) formula for this type of pattern (Equation (4.19)). This leads us to the new equation: R(T)= L S(pLp0)ZpL p0 dp (p, T)(linear p(x) pattern),(4.20) where (p, T) results from Equation (4.12). The result of this analytical estimate is displayed in Figure 4.4 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. Note that this agreement is somewhat worsened (but still very good at the qualitative level) for the lower tail of the R(T) transition. As already commented in Subsection 4.2.6 and 4.4.2 this slight overestimation of R(T) as R!0+is expected in any EM-type approach and is due to the progressive occurrence of percolation e↵ects as more regions of the film become fully superconductor. 88 Chapter 4. Design optimization of high-temperature superconducting bolometers 4.6 Results for HTS TES with a continuous exponentiallike doping patterning Seeking to find a p(x) profile producing a R(T) transition leading to improved 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, non continuous variation will be later discussed, in Section 4.7). This continuous p(x) profile is more intuitively described by means of the auxiliary length-weight function (p). In particular, we consider p-profiles leading to the following exponential (p)function: (p)=Aexp ✓p0p p◆,(4.21) where pand A are constants, the latter being easy to obtain by normalization considerations as A=1 p 1 1exp ⇣p0pL p⌘.(4.22) In these equations p0and pLare, as in the previous sections, the nominal doping at x= 0 and x=Lrespectively, being Lthe size of the film. Note that, by applying Equation (4.13), this corresponds to the following p(x) profile: p(x)=p0pln ⇢1x L1exp ✓p0pL p◆ (4.23) For the case of YBCO films considered in this article, and for p0=0.135 and pL=0.161 as already used in the previous section, we found that the p value that best optimizes the bolometric characteristics (most notably T)is p=0.007. Besides these p0and pLvalues, we employed in our evaluations the same common parameter values as described in our Sections 4.2.3 to 4.2.5 about methodology. In the upper row of Figure 4.5 the corresponding doping profile is drawn, 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 pwith xwould produce however less optimized bolometric performance). 4.6. Results for HTS TES with a continuous exponential-like doping patterning 89 Figure 4.5: In the upper row, we illustrate a YBCO film patterned following the exponential-like variation of nominal doping p(x) studied in Section 4.6 (see Equations (4.21) to (4.23)). We also illustrates doping over 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 resistance vs temperature R(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 (4.26)). 4.6.1 Results for the R(T) profile and bolometric operational parameters The results of our numerical finite-element evaluation for this p-pattern are displayed in the second raw of Figure 4.5 (see also Table 4.1 for a comparative summary). As evidenced in that Figure 4.5, not only the transition is significantly broadened with respect to the one of non-patterned HTS TES but also (unlike what happened in the case of a linear p(x) variation) the region Tof the transition with linear R(T) dependence is highly increased. In particular, the Tregion is increased up to 8.3 K, almost 10 times more than in the case of Chapter 5 Modification of the resistive transition width of high-temperature superconducting films using 2D Tc-dot patterns 97 98 Chapter 5. Resistive transition modification of HTS Films Summary of the chapter: We extend our studies of HTS TES films with custom micro and nanostructuration of the doping level. In particular, now we consider also 2D microstructure patterns. We show that with custom 2D designs the resistive transition may be not only widened (as achieved in the previous chapter with 1D designs) but also it may be sharpened (as interesting, e.g.,toimprove the speed of HTS resistive fault-current limiter devices). 5.1. Introduction 99 5.1 Introduction Among the applications of superconductors, an example is device components that change their resistance Rby orders of magnitude in a delimited temperature range, the vicinity of the critical temperature Tc. [114,115] These may be part of, e.g., bolometers, [116] of super-thermometers sensitive inside a constrained window of temperatures T, [48] etc. The sharpness versus Tof the R(T) transition is also of interest in resistive fault current limiters, as it may influence both the occurrence and speed of thermal avalanches and thus their ability to respond to ultrafast current strikes and their recovery time once the fault is cleared. [49,50] Figure 5.1: Our finite-element representation of a superconducting film with a regular array of Tc-domains, that also considers a Gaussian Tc-distribution inside each domain of the regular pattern as shown in the zooming insets. The example pattern used for this illustration corresponds to the simulation in Fig. 2 with interdomain separation d= 3. In typical cuprates, each subcell in the insets is of approximate area (30 nm)2, so that each cell in the main figure is (0.3µm)2and the whole film (6 µm)2. See main text for details. In these and other superconducting components, the range of temperature over which the change of resistance occurs is not easily custom-engineered, as it is constrained by the Tcvalue of the superconductor itself and by the T-width 100 Chapter 5. Resistive transition modification of HTS Films of its R(T) transition. This width is well below 1 K in most low-Tcsuperconductors. [107,117] In contrast, in the the high-Tccuprates the width of the R(T) transition rounding is of about 10K or larger, even in the best samples due to the unavoidable existence of critical phenomena around Tc. [117–119] The latter follow specific power laws with respect to TTcand, thus, custom-engineering the detailed shape of the rounding is also challenging. In this paper, by means of computer simulations we explore the possibility of custom-broadening and custom-sharpening the resistive transition of high-Tc superconducting films by means of microor nano-structuration causing a regular pattern of domains with di↵erent Tcvalues over the film. Realization of this type of regular and controlled patterning has become experimentally possible in the recent years, thanks to advances in di↵erent techniques such as, e.g., local ferroelectric field-e↵ect, [42] nanodeposition, [43] focused ion bean, [44] etc. In fact, it has become the specific subject of recent conferences [45] and networks funded by the European Union. [46] While most growers of those structured superconductors have focused up to now on increasing their critical current and achieving collective vortex pinning at certain matching fields, [42–45] our present work proposes that these techniques may also open a way to produce superconducting films designed to have tailored shapes and widths of their R(T)curves. In section 5.2 we briefly describe our computational methods and parameter values. In section 5.3 we show that broadening the R(T) transition is possible with square array patterns of enhanced-Tcdots. In section 5.4 we show that the opposite e↵ect, i.e., sharpening the R(T) transition, is possible with certain regular patterns of depressed-Tcdomains. In section 5.5 we summarize our conclusions. 5.2 Methods 5.2.1 Critical fluctuations To achieve a realistic simulation of actual patterned high-Tcsuperconductors, a first necessary ingredient is a precise knowledge of the resistivity ⇢(T) near the transition of a single-Tcdomain in a cuprate. In that respect, in this paper we shall restrict ourselves to a manageable case, the ohmic regime with no external applied magnetic field. For that case, it was experimentally shown in cuprates of various dopings and with particularly well characterized inhomogeneity levels that the intrinsic ⇢(T) agrees with remarkable accuracy with a single set of equations comprising the di↵erent regimes of critical phenomena around Tc; [106,119–121] a convenient and explicit summary of such equation set may be found in. [106] This achievement reflects, in fact, notable advances in the understanding of the critical phenomena in cuprates (including the so-called Gaussian, uncertaintyprinciple, and strong phase-fluctuation regimes in layered type-II superconductors 5.2. Methods 101 [106, 120, 121]), although for all practical purposes in the present paper they simply provide us a valid empirical approximation to the ⇢(T) transition of a single-Tcdomain in a cuprate. 5.2.2 The role of random inhomogeneities As explicitly demonstrated in various recent experimental and theoretical works, [101–103,108,109] a second crucial ingredient to understand the rounding of the transition in cuprates is to correctly account for the random Tc-inhomogeneities associated with doping. This disorder exist even in the best cuprate samples and is due to the nonstoichiometry of the dopant ions, that leads to randomness in their placement in the crystal lattice. As shown in [103, 108, 109], the Tcdomains resulting from the intrinsic disorder lead to Gaussian distributions of Tcwith dispersions about Tc'2 K, and have a typical planar size of about 30 nm. This Tc-distribution overimposes a further rounding of the transition, additional to the critical fluctuations. Therefore it becomes necessary, in order to realistically study the R(T) transition of a high-Tcfilm with a regular pattern of Tc-domains produced by micro or nanostructuration, to also consider that inside each of these patterned domains an additional random Gaussian distribution of Tc-values exists. Naturally, this significantly increases the computational weight of the corresponding finite-element simulations. 5.2.3 Finite-element computations To calculate the R(T) transition curves of each of the superconducting films discussed in this work, we wrote ad-hoc software (toseris, available by request to authors) that numerically solves the mesh-current matrix equations of a film modeled, as illustrated in Fig. 1, as a 200 ⇥200 square lattice of domains where each domain imay have its own Tci and thus its own resistivity vs. temperature curve. As already mentioned, for the ⇢(T,Tci) functionality we use the equations explicited in Reference [106]. As Fig. 1 also illustrates, the model includes a power source and a voltmeter connected with zero-resistance contacts to opposite edges of the sample; the R(T) of the film will be obtained as the external V/I. Calculating the circuit requires to solve, for each temperature, a sparse linear system of 40001 variables, the mesh currents. This is a parallelizable computation for which we employed the supercomputer lbts-"psilon, which comprises about 12000 floating-point units, is able of 31 Tflops and is described in [89]. It was 100% allocated to run our software during several days. To assign a Tci to each domain iin our simulations, we employ the following two-step procedure: First, we divide the sample into a 20⇥20 grid, and associate to each of these cells an average critical temperature Tcaccording to the regular pattern that we want to simulate. Then, each of these cells is in turn subdivided 102 Chapter 5. Resistive transition modification of HTS Films into a 10⇥10 grid and to each of the resulting subcells a Tci is randomly assigned, following a Gaussian distribution that has mean value the Tcof their parent domain in the regular pattern and full-width at half-maximum Tc. Figure 1 illustrates an example. 5.2.4 Parameter values common to all simulations We perform our simulations with the following numerical values, representative of typical cuprates: For the area of a finite-element domain i, we use (30 nm)2, which as mentioned in subsection 5.2.2 is expected to correspond to the size of a doping Tc-inhomogeneity [103, 108,109]. Therefore, a patterned dot in the main Fig. 1 has area (0.3µm)2, a size well feasible with current microand nanostructuration techniques, [42–44] and the whole simulated film is (6 µm)2. For the film thickness we use 100 nm. For the full-width at half-maximum Tcof the random doping Tc-inhomogeneities, we take 2 K (see subsection 5.2.2 and [103,108,109]). For the physical parameters entering the critical fluctuation expressions summarized in Ref. [106], we use a parameter set similar to those in our previous works [106] and [102]: in-plane coherence length amplitude ⇠ab(0) = 1 nm, outof-plane coherence length much smaller than the typical distance between superconducting CuO2planes s⇠1 nm (so that fluctuations are in the 2D limit), Ginzburg parameter "LG =0.01, uncertainty-principle high-temperature fluctuation cuto↵"c=0.55, and vortex-antivortex fluctuation parameters BKT = 2K and b0= 4. Finally, we take (108⌦m/K) Tfor the normal-state background, i.e., the resistivity at temperatures well above the transition, where fluctuation and Tc-inhomogeneity e↵ects are no longer expected to be appreciable. 5.3 Broadening the R(T)transition with patterns of enhanced-Tcdots We start by considering the simple case of a square lattice of dots having critical temperature Tdot cconstant and higher than the one of the background areas Tback c. This corresponds to the example configuration illustrated in Fig. 1. We summarize in Fig. 2 the R(T) obtained from the corresponding simulations, for patterns with di↵erent dot packing densities, i.e.,di↵erent values of the dot interdistance d. These results indicate a two-step-like transition at Tdot cand Tback c, as maybe it could have been na¨ıvely expected, at the qualitative level, in advance. It is noticeable that increasing the dot packing density, i.e., decreasing d, increases the widening of the transition. This not only happens through an increase of the height of the Rjump at the larger critical temperature, Tdot c, but also by the less trivial e↵ect (and more relevant for some applications) of diminishing the slope dR/dTas R!0+: In particular, for the 25% dots configuration, i.e.,d= 1, that slope is almost halved with respect to its value for the no-dot configuration, 5.4. Sharpening the R(T) transition with patterns of depressed-Tcdots 103 Figure 5.2: Our results for the resistance vs. temperature of a film with a square pattern of dot domains having critical temperature Tdot cconstant and higher than the background one, Tback c(see Fig. 1). Note that the step-like broadening of the transition achieved with this patterning is intensified by increasing the dot packing density, i.e., by decreasing the interdomain distance d, given here in units of the number of cells in the main Fig. 1. See main text for details. being 66% of the value of the no-dot configuration when evaluated at a threshold of R'0.5⌦,i.e.,⇢'5⇥108⌦m. To widen the transition but avoiding the step-like R(T) shape, one possible option is to produce the same dotted patterns, but allowing now a di↵erent Tdot c for each column of the array, in particular producing a discretized gradient of Tdot cin the direction from one voltage contact to the other. Our results for that configuration, shown in Fig. 3, no longer present strong step-like features. This is in spite of the discreteness of the Tdot cvariations; the reason may be traced back to the rounding due to critical fluctuations and inhomogeneities. As also visible in Fig. 3, increasing the dot packing density, i.e., decreasing d, increases the widening of the transition. It also diminishes the slope dR/dTas R!0+, this last aspect in a way quantitatively similar to what is achieved without the gradient: The slope dR/dTevaluated at R'0.5⌦for the 25% dots configuration, i.e.,d= 1, is 66% of the value for the no-dot configuration. 5.4 Sharpening the R(T)transition with patterns of depressed-Tcdots We now consider regular arrays of dots with a Tdot clower than Tback c. As it will be shown in this section, this case will produce the interesting situation that the overall R(T) transition of the film will become sharper, instead of broader. 104 Chapter 5. Resistive transition modification of HTS Films Figure 5.3: Our results for the resistance vs. temperature of a film with a square pattern of dot domains with an enhanced Tcwhich value decreases according to a constant horizontal gradient in Fig. 1. Note that R(T) is no longer step-like as in Fig. 2, and that the transition gets broader by increasing the dot packing density, i.e., decreasing the interdomain distance d, given here in units of the number of cells in the main Fig. 1. See main text for details. This contrasts not only with the results of the previous section, but also with what would be obtained if the spatial pattern was fully random, in which case simulations and analytical approximations as, e.g.,e↵ective medium equations, predict widened transitions [102, 122] in agreement also with measurements in randomly granular cuprates. [122] Figure 4 shows our simulation results for dotted patterns in which Tdot c< Tback c, corresponding to the example in Fig. 1 but inverting its color scale, and hence the Tdot cand Tback cvalues. Note the modest sharpening of the transition, that becomes more pronounced as the dot packing density increases, i.e., as ddecreases. This moderate sharpening happens by i) a displacement of the inflection temperature (dR/dT= max) towards a lower part of the transition and ii) an increase of the R(T) slope as R!0+. In particular, dR/dTevaluated at R'0.5⌦for the 25% dots configuration, i.e.,d= 1, is 120% of the value for the no-dot configuration. A valid question is whether the above moderate sharpening of the R(T) transition could be made stronger by increasing the packing dot density. Given that Fig. 4 already includes the limit d= 1 that represents the minimum dot interdistance in our simulation procedures, we explored the e↵ects of increasing the dot packing density by means of elongated dots, i.e., regions of critical temperature Tdot cthat would span in Fig. 1 horizontally only one column but vertically a number (henceforth called elongation) of consecutive rows. For the distance between adjacent elongated dots we take always d= 1 (in both planar direc- 5.5. Conclusions 105 Figure 5.4: Our results for the resistance vs. temperature of a film with a square pattern of dot domains with depressed and constant Tdot c.TheR(T) transition becomes sharper as the dot packing density grows by decreasing the interdomain distance d, given here in units of the number of cells in the main Fig. 1. See main text for details. tions). We show in Fig. 5 our results for those elongated dots. The R(T) of the no-dot configuration is also presented, to ease comparison with Fig. 4. Note that increasing the dot packing density using elongated dots further sharpen the transition significantly. The slope dR/dTat R'0.5⌦also markedly increases, and for the 45% dot configuration, i.e., elongation 4, is almost the double (190%) of the value for the no-dot configuration. A crude qualitative interpretation of the above R(T) transition sharpening is that the R(T)!0 temperature is determined mainly by the high-Tcbackground current paths, while when R(T) begins to be finite it becomes also influenced by the lower-Tcphase, what decreases the rounding due to the large distance to Tdot c. 5.5 Conclusions We performed numerical simulations of the R(T) curves under zero external magnetic field, in a high-Tcsuperconducting film with a regular pattern of local Tcvariations induced, e.g., by micro or nanofunctionalization. Our simulations also include the e↵ects of the transition rounding due both to critical fluctuations and to the intrinsic inhomogeneity associated to the nonstoichiometry of dopant ions. Our results indicate that a regular square lattice of domains with enhanced critical temperature Tdot cmay broaden the transition. Also that when the regularly patterned domains have depressed critical temperatures the R(T) transition 112 Chapter 6. Hybrid piezoelectric + superconducting devices Figure 6.1: In this figure we show the XRD di↵ractogram of one of our YBCO+PMN-PT hybrids. The substrate (PMN-PT) peaks are marked with an S, whereas the YBCO ones are marked with only the corresponding crystallographic indexes. 6.3.2 Susceptibility measurements The macroscopic superconducting critical temperature and the distribution of critical temperatures in our samples, without any transversal piezovoltage, can be obtained from the magnetic susceptibility of the superconducting films, following standard procedures our group used and described in detail, e.g.,in References [103, 135]. We have measured the susceptibility using a Quantum Design’s MPMS-XL magnetometer, that achieves enough sensitivity as to perform this kind of measurements in thin films. The procedure is based on the measurement of the magnetic moment of the sample when it is under a dc magnetic field applied in the out of plane direction following a zero-field-cooled procedure. The nominal magnetic fields used to measure the magnetic moment of di↵erent samples are between 3 and 3.3 Oe. Because the substrate value of the magnetic susceptibility is so much larger in the superconducting state (where '1) than in the normal state (where ||⇠105), and being an extensive quantity, the value |(T)|directly reports the fraction of the sample that has become superconducting at that temperature T. Therefore, the d/dT near the transition is an excellent approximation to the distribution of local Tcvalues in the sample, !(Tc). [103,135,136] Susceptibility measurements in two examples of our YBCO+PMN-PT hybrids 6.3. Characterization under no piezostress 113 are plotted in Figure 6.2 and its insets. The criteria that we have used to fix the critical temperature of each sample is taking the maximum of the Gaussian fit to the solid lines in d/dT peak at the transition. These derivatives and the fits are plotted in Figure 6.2. Figure 6.2: In this figure we show two SQUID measurements of the susceptibility for two samples of our YBCO+PMN-PT hybrid. The main figures show the derivative of the susceptibility, that may be identified to the Tcdistribution, !(Tc), of the film. We have also plotted, in the insets, the value of the susceptibility (in SI units) in function of the temperature for these samples. The data have been fitted by a Gaussian function, which maximum may be interpreted as the average of the critical temperature of the film Tc, and the Full Width at Half Maximum (FWHM, Tc) as a measure of the inhomogeneity. These d/dT peaks indicate that the quality of the films grown in our laboratory is optimal in terms of the transition width and in terms of the homogeneity of Tc, as they shows transition widths (Tc, taken as the FWHM of the Gaussian 114 Chapter 6. Hybrid piezoelectric + superconducting devices Figure 6.3: In this Figure resistance measurements obtained for a YBCO thin film are plotted with the normal state background. The black rhombs represent the data obtained for E= 0 kVcm and the red line is the background Resistance Rbackground. A zoom near the transition may be seen in Figure 6.8 (rhomb data again). peaks) between 0.5 and 1.3 K. Moreover, the critical temperatures shows that our samples are near the optimal doping, where Tc⇠90 K. 6.3.3 Resistivity measurements Another way to study the quality of the superconducting YBCO films is to analyze the resistance curves versus temperature in a wide range of temperatures. We have measured the electrical resistance of our films with the four-probe method, with a current bias of 0.9 mA. To achieve the cryogenic temperatures we have used a cryostat to measure with a very low temperature rate of change, measuring the resistance applying opposite currents at each temperature, to calculate the mean value with the aim of eliminating possible thermoelectric e↵ects. In Figure 6.3 we show an example of resistance curve measured in one of our hybrid devices, taken without any polarization of the piezosubstrate (no voltage between any of the upper contacts and the one at the bottom of the piezosubstrate). The curve shows an inflexion-point critical temperature of about 88.9 K, a value near the one expected for optimal doping in YBCO films of similar thickness in SrTiO3substrates. The transition width is T⇠1 K, which stands for a very good film quality. The normal-state resistance Rbackground follows a polynomial function, as is also plotted in the Figure 6.3. 6.4. XRD measurements of the strain under di↵erent piezostresses 115 6.4 XRD measurements of the strain under di↵erent piezostresses As it has commented in the introduction, the inverse piezoelectric e↵ect enables us to change the lattice parameters of the sample by applying an electric field. To study this change, we have measured the strain "defined as: "c=c(E)c(E= 0) c(E= 0) ,(6.1) where Eis the applied electric field applied transversally (along the out of plane or c-direction) to the piezosubstrate and cis the out-of plane lattice parameter. For these measurements we have used a Rigaku Miniflex II di↵ractometer. The radiation used for the measurement has a wavelength (K↵1)=1.5406˚ A obtained from a sealed Cu tube. In this case we don’t have the monocromator, so that the radiation has also contributions (K↵2)=1.5443 ˚ A, and (K)= 1.3920 ˚ Arespectively, with an intensity of 0.5Iand 0.05 I,whereIis the intensity of the K↵1radiation. Because of the proximity of (K↵1) and (K↵2), the peaks they produce overlap each other; we remove this K↵2contribution from our data using the commercial software PDXL which uses the Rachinger method. The data treatment also includes an smoothing (that is performed optimizing the B-spline function to the measured data using the least-squares method) and a background substraction (obtained using the Sonneveld-Visser method). The angular region used for the measurement fixed a step of 0.02 s1and accumulating a time of 2 s per step. 6.4.1 Comparison between the strains in the piezosubstrate and in the superconductor In the Figure 6.4 we show the movement of the (003) peak of a PMN-PT and the (005) peak of YBCO for an applied electric field of 10 kV/cm at room temperature. Note that (as we describe in more detail in the next section) the values of the properties of related to the hybrid compound (strain, resistance, magnetization, etc.) depend highly on the history of the sample, because of the hysteresis of the inverse piezoelectric e↵ect. [137–143] Here we have plotted the peaks obtained after polarizing the sample with 10 kV/cm, taking the first measurement at 0 kV/cm and then increasing the electric field up to 10 kV/cm. By applying Bragg’s law to the location othe maximums of these peaks, we obtain the corresponding "cby applying Equation 6.1. What we can infer from the measurements is that only a about 10% of the strain is transmitted to the YBCO film in the c-direction. It is important to remark that the direct transmission of strain is in the ab-plane, and this strain induces an strain in the c-direction, which 116 Chapter 6. Hybrid piezoelectric + superconducting devices is always lower than the strain in the ab-plane, due to the Poisson coefficient of the material. Figure 6.4: The top figure shows the (003) Bragg peak of the PMN-PT XRD spectra in one of our piezodevices, when two di↵erent piezovoltages are applied to the sample corresponding to transversal electrical fields E= 0 and 10 kV/cm. The bottom figure shows the (005) peak of YBCO for the same electric fields and sample. All these measurements have been taken at room temperature. In both cases the measurements at E= 0 kV/cm were obtained after poling the sample during 30 minutes at 10 kV/cm, and the other values after the direct increase of the electric field after the first measurement. 6.4.2 Temperature and hysteresis e↵ects An interesting point to be made is the fact that the inverse piezoelectric e↵ect in our devices, and thus the piezostress, displays at room temperature hysteresis behavior, while at cryogenic temperature the e↵ect is instead reversible and 6.4. XRD measurements of the strain under di↵erent piezostresses 117 almost linear. This behavior could be in fact expected already from earlier measurements as a function of temperature of the coercive field of PMN-PT. [137] We measured such behavior by means of XRD measurements under di↵erent piezovoltages and under di↵erent temperatures, including cryogenic ones. For that, we built a cryogenic module that we incorporated to the same Rigaku Miniflex II device described before. The module also allows application of the piezovoltage. Teh so-obtained "cstrains (as given by Equation 6.1 using the displacement of the (003) PMN-PT peak) are plotted in Figures 6.5 and 6.6, for room temperature and 162 K respectively. These figures evidence the markedly di↵erent behavior at room and cryogenic temperatures. Figure 6.5: The hysteresis loop represented in this plot was obtained by measuring the (003) PMN-PT Bragg peak (in the same sample as in Figure 6.4) at room temperature. We start the measurement after poling the sample during 30 minutes, and then we measure the value at 0 kV/cm. Then we start to increase the transversal electric field, showing the data obtained using the black upwards triangles. We rise then the electric field up to 15 kV/cm and then we start to decrease it up to 15 kV/cm (red downwards triangles). Finally we increase again the electric field up to 0 kV/cm (green diamonds). To calculate the value of the strain "c, we have taken as a reference the last value of the lattice parameter that has been measured in this hysteresis loop at 0 kV/cm. First of all, at cryogenic temperatures, as shown in Figure 6.6, the strain of the substrate depends highly on the history of the sample. This is due to the fact that at room temperature, the coercive field of the PMN-PT only about 1.5 kV/cm [137]. Below the coercive field, the response of the PMN-PT is linear and reversible, which means that there is no hysteresis. This is due to the fact that the electric field a↵ects directly to the polarization of the unit cells, and does not change the orientation of the polarization domains of the sample. [142,143] When this domains change their orientation, the substrate response is no longer 118 Chapter 6. Hybrid piezoelectric + superconducting devices Figure 6.6: E↵ect of the transversal electric field over the "of the PMN-PT at a temperature of 162 K. This measurement demonstrates that the coercive field is increased at cryogenic temperatures, and thus the linear region of the system is also increased. This lead us to obtain expansion or compression in the sample applying positive or negative electric fields, respectively. Also, hysteresis is not seen during these measurements, that are reproducible when increasing or decreasing Ewithin this range (in contrast to the situation at room temperatures, see Figure 6.5). reversible and depends on the history of the system. As shown in Figure 6.5 we have measured the strain of the c-axis parameter of the PMN-PT at room temperature for applied electric fields between 15 to 15 kV/cm starting from 0 kV/cm to 15, achieving the so-called butterfly piezoloop, as studied in general, e.g., in References [138,143]. In contrast, at cryogenic temepreatures the coercive field is well higher. [137] Therefore, we can apply high electric fields (at least to 10 kV/cm) without entering in the hysteresis loop. This lead us to obtain more reproducible measurements, and to obtain expansion and compression for positive and negative applied electric fields, respectively. Our results in Figure 6.5 also show that the e↵ect is not exactly symmetric. This is, we can obtain higher strain with positive electric fields than with negative ones. This is a similar result to the one obtained in Reference [137] at 90 K for PMN-PT monocrystals. 6.5 Electrical resistivity measurements under di↵erent piezostresses To measure the resistance of the YBCO layer under di↵erent stresses exerted by the piezosubstrate, we have used the same experimental setup as we described in 6.5. Electrical resistivity measurements under di↵erent piezostresses 119 Figure 6.7: In this graph it is shown the experimental data obtained for the resistance curves for the three applied electric fields plus the background polynomial curve obtained from the normal state data. The black diamonds represent the data obtained for E= 0 kVcm, the red triangles represent the data for E= 10 kV/cm, the blue circles are the data obtained for E=10 kV/cm, and the red line is the background Resistance Rbackground. Subsection 6.3.3, i.e., we measure in a 4-wire configuration the in-plane resistance of the YBCO layer. The piezovoltage is applied between the bottom contact in the piezoelectric and one of the voltage contacts in the upper side of the YBCO film. The protocol for the application of the piezovoltage is as follows: First of all, we polarize the system at room temperature during 30 minutes with a piezovoltage of 300 V. After this, we decrease the temperature until 77 K, where we apply the desired piezovoltage for the measurement during another 30 minutes at stable temperature. Finally we increase the temperature slowly up to room temperature with the applied piezovoltage to measure the resistivity. 6.5.1 Shift of the resistive transition with piezostress Our measurements for the YBCO resistivity measured under di↵erent electrical fields Eapplied to the piezoelectric are plotted in Figures 6.7 and 6.8. The first Figure shows an overview with respect to temperature and also the same normal-state background Rbackground already used previously in Section 6.3.3 and Figure 6.3 and that, as shown by Figure 6.7, is valid for the three electrical fields E. In Figure 6.8, we zoom the T-region near the superconducting transition. This figure clearly evidences that, contrarily to the normal-state background, the critical temperature is significantly shifted by the piezostress. In particular, a first account of this shift may be quantified through the inflexion temperature in the middle of the transition, that for no piezovoltage is Tci = 88.93 K, while for 120 Chapter 6. Hybrid piezoelectric + superconducting devices Figure 6.8: Zoom of the resistance transition of the hybrid compound YBCO+PMN-PT for di↵erent electric fields. The black diamonds represent the data obtained for E= 0 kVcm, the red triangles represent the data for E= 10 kV/cm, and the blue circles are the data obtained for E=10 kV/cm. E= 10 kV/cm is Tci = 89.07 K and for E=10 kV/cm is Tci = 88.83 K. The positive electric fields corresponds to the case in which the c lattice parameter of the PMN-PT is expanded, so the in plane lattice parameter of both the PMNPT and the YBCO is compressed, and thus the c-axis parameter of the YBCO is expanded. This leads to an increase of the critical temperature. Conversely, with negative electric field, the c-axis lattice parameter of the YBCO is decreased, and the critical temperature is also diminished. When combined with our measurements of the piezostress as a function of the piezovoltage, i.e., our data of Figure 6.6, our resistivity measurements provide a value of the variation of Tci with respect to the c-axis of dTci/d"c=2.6K. 6.5.2 Paraconductivity The shift of the critical temperature with piezostress indicates a change of the fundamental-state energies that determine the appearance of superconductivity. It would be interesting, however, to test whether also the excitations above that groundstate are a↵ected by piezostress. A common related experimental probe is the so-called paraconductivity or fluctuation-induced electrical conductivity, i.e., the rounding of the resistivity transition, or the di↵erence, in conductivity units, between the measured resistance near but above Tcand Rbackground.InSection 4.2.3 we have written down a definition formula for the paraconductivity  (Equation 4.7) and we also detailed di↵erent equation predictions (Equation 4.8) for as a function of the reduced temperature ln(T/Tc) obtained on the grounds of Ginzburg-Landau approaches with Gaussian fluctuation thermodynamic theories. The comparison of our data with such predictions also open a way to probe 6.5. Electrical resistivity measurements under di↵erent piezostresses 121 Figure 6.9: In the upper part of the Figure it is shown the representation of the data obtained for the paraconductivity for the three resistance transitions measured for three di↵erent electric fields. The fit to the paraconductivity equation 4.8 is also plotted as a black line. In the lower part of the graph, it is shown the experimental measurements in terms of T/TFL c,beingTFL cthe value of the critical temperature obtained for each fit to the paraconductivity equation. In both parts of the graph, the black diamonds represent the data obtained for E= 0 kVcm, the red triangles represent the data for E= 10 kV/cm, and the blue circles are the data obtained for E=10 kV/cm. 128 Chapter 7. Electrostatic nanostructured microporous media Summary of the chapter: We present and adapt the main theory tools that we shall use in our studies of the influence of microgeometry over the nanofiltration characteristics of electrostatic-nanostructured microporous (ENM) media. First, the governing equations for the process of successive impurity trapping in ENM media are presented, based on previous work of our research group in Santiago (e↵ective-charge model). Then, we adapt these model equations to the case of interest in our present work, i.e., we update these equations to account for possible variations of the radius of the microconduits along a finite length. We also bring them into a finite-di↵erence form, so to allow their computational implementation. Other simulation details, including also typical parameter values in actual ENM media introduced by academia and industry, are also provided. 7.1. Introduction 129 7.1 Introduction One of the most innovative and promising techniques for water filtration is the application of nanotechnology. For example, in recent years both academia and industry [51–60] have become progressively interested in filtering media and membranes that achieve nanofiltration (i.e., removal of impurities of size .20 nm) by means of nanofunctionalization of the inner walls of pores of diameters in the micrometric ⇠0.2–2 µm (such micrometric pores are somewhat counterintuitively called macropores in IUPAC terminology [144]). In these investigations [51–60] it has been revealed the surprisingly good capability of such media to filter out nanosized impurities from an aqueous solution, principally when the signs of the ⇣-potentials of impurities and functionalized pore walls are opposite and electrostatic e↵ects enhance the filtration. This is in spite of the relatively large diameter of the pores (note that for a pressure-driven flow passing through a cylindrical conduit with diameter 1 µm, only about 0.04% of the fluid will transit closer than 10 nm from the walls). A key practical advantage of these novel ENM is that they are not a↵ected by the very large hydrodynamic resistance of the media composed by pores of nanometric diameter. In fact, industry has already commercialized some of those nanoenhanced filters, [58–60] not only for drinking water purification but also for other applications (clinical analyses, industrial e✏uent treatment, etc.). It is currently possible to custom-engineer the geometry profiles of micrometricdiameter pores of filtering media and membranes (see, e.g., References [145–149] for a description of di↵erent ad-hoc geometries and the experimental techniques used to obtain them). These feasible designs include, among others, tubular conduits with cylindrical shapes having di↵erent diameters, [146] corrugated conduits, [145] or pores with cone-like shapes. [147–149] Investigating conduits with di↵erent nominal geometries may also be relevant for random media, in which interconnections and throats between pores may be appropriately studied in terms of conduits with sinus-like corrugated shapes. [150–153] However, no investigation seems to exist up to now of the influence over the nanofiltration characteristics of the nominal conduit geometry of ENM. For that purpose, in part IIB of the present thesis we will present numerical simulations (and also some analytical results) of the nanofiltration performance and operational lifetime of nanoenhanced micropores with di↵erent geometries. The governing equations used for our numerical simulations will be taken from a recent model [61] describing the trapping of nanoimpurities by analogous electrostatic inner walls in a short cylindrical conduit. We rewrite these equations into a finite-di↵erence form, enabling them to be used in the computational assessment of conduits of finite length and arbitrary variations of radius (and therefore allow to study conduits of di↵erent microgeometry). The model in [61] may be also related to the so-called lubrication model independently developed in a di↵erent context (electrophoresis) by Ghosal in [154–156] (see also Section 7.2.1). 130 Chapter 7. Electrostatic nanostructured microporous media The present chapter is organized as follows: in Sections 7.2.1 to 7.2.3 we introduce the model and governing equations of [61] in a finite-di↵erence version appropriate for numerical treatment. Sections 7.2.4 and 7.2.5 describe our computational methods and the parameter values used in the simulations. The application of these methods to di↵erent specific conduit geometries is deferred to Chapter 8. 7.2 Model equations 7.2.1 Initial considerations We consider a tubular conduit of micrometric diameter whose inner walls have a (initially homogeneous) nanotexture. The tube has a nominal diameter d(x) that may, in general, vary along the axial coordinate x(with 0 xL,where Lis the tube length). A fluid passes through, carrying a certain concentration of nanoimpurities to be filtered out. All of the impurities are taken to be equal and have average radius ⇢0. Our task is to obtain the evolution with space x and time tof the impurity concentration Cimp(x, t) in the fluid, and of the areal density of impurities n(x, t) trapped in the tubes (throughout this article, by “areal density” we refer to quantities normalized using the nominal area of the tube inner wall). Initially (t= 0) the system is in the “clean state”, in which n(x, 0) = 0 and the diameter available to the flow is d(x). As time and flow passes, impurities get trapped and progressively cover the tube inner walls. Eventually, a “saturated state” is reached in which the nanotexture of the walls no longer electrostatically attracts further impurities. In the saturated state, nis nsat and the diameter available to the flow decreases down to a value d(xi)sat. We will consider in this article that after this saturated state the impurity trapping capability of the walls (and hence the filtration performance) is zero, so that we focus only on the e↵ects of nanostructuring and neglect conventional filtration mechanisms. In order to evaluate the e↵ects of variations of the diameter along the tube’s axial coordinate, and also to allow computation of successive spacial and temporal finite-di↵erence iterations, we discretize the problem as follows: We divide the tube into i=0,1,...N parallel slices of coordinates xi=(L/N)iand also discretize time as tj+1 =tj+tjwith j=0,1,... (the reason to allow the timestep to vary with jwill be to optimize our computational algorithms, see Section 7.2.4). Let us now briefly describe the key ingredients of the model introduced in [61] for the impurity-trapping dynamics of the nanostructured conduits, focusing mainly on its adaptation to the finite-di↵erence formalism needed for our present purposes (see [61] itself for a more comprehensive discussion of the underlying 7.2. Model equations 131 basic aspects of the model): i) The starting point of model [61] is to consider the e↵ects of electrostatic attraction on the probability of collision between the impurities circulating in the liquid flow and the conduit walls. For that, it introduces an “e↵ective charge” of the walls (proportional to their ⇣-potential) that will decay as impurities cover them and screen out the charges exposed in the nanotexturing. It is then calculated the “collision distance” ⇢e, or typical impurity-wall separation below which an impurity in the flow will acquire course of inevitable collision with the wall. From simple electrostatic escape-distance estimates, it may be expected that ⇢e should grow roughly linearly with ze. [61] When also taking into account the effects of screening due to the Debye length of the fluid D, a more precise law is obtained: [61] ⇢e(xi,t j)=⇢0+DW✓⇢e0 ⇢0 D✓1    n(xi,t j) nsat    ◆exp ✓⇢e0 ⇢0 D◆◆,(7.1) where ⇢e0 is the collision distance in the clean state, W(x) is the the principal Lambert function and the notation kxkstands for min(1,x). ii) Secondly, the model [61] assigns a impurity-wall binding probability for the impurities that do collide with the walls. This has to linearly decrease with the number of binding centers in the nanotextured walls, i.e.,withn(x, t). Therefore, it is maximum in the clean state. It is possibly noteworthy to note here that, when adapting the equations in [61] to a finite-di↵erence treatment, the survival probabilities for x-slices of finite thickness accumulate over length multiplicatively, not linearly as considered for di↵erential x-lengths in [61]. So, the binding probability for the clean state ⌦0in [61] has to be replaced by ⌦finite 0=(N/L)[1 (1 ⇢0⌦0)L/⇢0N]. As expected, for N!1this recovers probabilities linear with length. In all the simulations performed in our present work, Nis as large as 106and the di↵erences between ⌦finite 0and ⌦0remain below 0.05%. However, for smaller values of Nthe di↵erence may be relevant. iii) Finally, model [61] assumes flow conditions realistic for most water filtering applications. This includes moderate flow velocities, for which it is valid to neglect turbulence and the e↵ects of axial advection on the transversal dynamics (see Section 7.2.5 for a more comprehensive discussion of this latter aspect and the implications over the parameter values chosen for the simulations). The model also considers a pressure-driven flow (in contrast to electrical-driven-like in the models proposed by Ghosal in [154–156] for electrophoresis applications). As discussed in detail for instance in [157,158], if the radiuses of the conduits remain &10 nm and with moderate variations in space, the aqueous flows driven by hydrostatic pressure are in the Poiseuille regime. The corresponding velocity profile may be thus used to obtain the fraction feof impurities that travel within 132 Chapter 7. Electrostatic nanostructured microporous media the collision distance ⇢efrom the walls: fe(xi,t j)= ✓    2⇢e(xi,t j) d(xi)n(xi,t j)sat/nsat    1◆2 1!2 .(7.2) where ⇢eis again given by Equation (7.1). Therefore, of the impurities that flow through a given slice of the conduit, the fraction ftrapped of them that become trapped by the conduit walls is, in the clean state, the multiplication of fe(xi,t j) by ⌦finite 0and by the length of the finite-di↵erence slice L/N. As further flow passes and impurities progressively cover the nanotexture of the walls, the fraction will decrease linearly with the increase of n(xi,t i) and become zero when n=nsat. Therefore its expression becomes ftrapped(xi,t j)= L Nfe(xi,t j)⌦finite 0✓1    n(xi,t i) nsat    ◆.(7.3) This allows now to calculate the finite-di↵erence equations for the full filtration dynamics of the nanostructured filter, that are summarized in the following Subsection. 7.2.2 Recursive finite-di↵erence equations for n(xi,t j),Cimp(xi,t j) and (tj) By using the trapping probabilities discussed in the previous Subsection, finitedi↵erence-iterative expressions may be now obtained for the areal density of trapped impurities n(xi,t j) in each xislice of the tube at the time tj, for the concentration of flowing impurities in the liquid flow Cimp(xi,t j), and for the liquid flow (tj) passing through the tube. For the areal density of trapped impurities we have: n(xi,t j+1)=n(xi,t j)+tjCimp(xi,t j)(tj)⌦finite 0 ⇡d(xi)✓1    n(xi,t j) nsat    ◆fe(xi,t j). (7.4) The finite-di↵erence expression (but now with the iteration happening in the spatial variable) for the concentration of impurities in the fluid may be also obtained: Cimp(xi+1,t j)=Cimp(xi,t j)⇡Ld(xi) Ntj(tj)⇣n(xi,t j)n(xi,t j1)⌘.(7.5) For the fluid flow (tj), as already mentioned above we will assume in this article that the liquid is driven by a constant hydrostatic pressure di↵erence P 7.2. Model equations 133 between opposite ends of the tubular conduits. We thus use a Poiseuille relationship: [61,154,157] 1(tj)=128⌘L ⇡PN X i✓d(xi)n(xi,t j)sat nsat ◆4 .(7.6) Note that the flow becomes t-dependent even with constant P, due to the progressive narrowing of the conduit as the filter becomes dirtier. Eventually, if d(xi)n(xi,t j)sat/nsat = 0 at some xi, the conduit is clogged and = 0. 7.2.3 Logarithmic removal value LRV and operational lifetimes As customary, [51–53, 55–59,159] we will characterize the filtration performance through the so-called “logarithmic removal value”: LRV(t)=log10 Cimp(x=L, t) Cimp(x=0,t).(7.7) For instance, 99.9% impurity removal corresponds to LRV=3. We will use the notation LRV0for LRV(t= 0). Also, to characterize the endurance of the filter, we define the following two characteristic times: The so-called “LRV5 lifetime” and the “LRV1 lifetime”, defined as the t-values at which the LRV becomes 5-log and 1-log, respectively. In practice, which of these endurance criteria (5-log or 1-log) is more relevant will depend on the specific application intended for the filter. 7.2.4 Computational procedures Equations (7.4) to (7.6) form a set that may be iterated to compute the xand t-variation of the characteristics of the filter. For that purpose, the parallel computer lbts-"psilon was allocated in full to that task. This computer is composed of about 9000 math coprocessors working in parallel and has 20 Tflops peak computing speed at 32-bit precision (for a more detailed description, see http://lbts.usc.es/epsilon/version2014). The simulations were implemented writing ad-hoc software that accepts arbitrary user-defined geometries for the nominal diameters d(x). It iterates in both the jand iindexes (parallelizing the later), continuously calculating self-adaptively the timesteps tjby requiring that successive instants vary n(xi,t j) less than 0.01% (for all xivalues). This results in about 104time-intervals for each simulation. For the spacial discretization, we divided the tubular conduits into N= 106finite-di↵erence slices (this leads in our simulations to a trapping probability for each impurity colliding with the walls always below 104per finite-di↵erence x-slice). 134 Chapter 7. Electrostatic nanostructured microporous media For the boundary conditions of our simulations we take as initial state n(x, 0) = 0(i.e., clean state for t= 0) and Cimp(x6=0,0) = 0. For the spatial boundary condition we take Cimp(0,t)=C0(i.e., a constant impurity concentration for the fluid at the entry of the tubular conduit). 7.2.5 Typical parameter values For the various parameters describing the characteristics of the nanofunctionalized macropore conduits in Equations (7.4) to (7.6) we will use values of the same order of magnitude as the ones that were found in [61] to produce good agreement with the available data in various real systems (in particular with those in [51, 52, 58, 59]): We will use a nanoimpurity radius ⇢0= 10 nm, an e↵ective collision distance in the clean state ⇢e0 = 30 nm, a binding probability in the clean state ⌦0= 104/nm, a saturation impurity layer thickness sat = 40 nm and a saturation impurity areal density nsat = 102nm2. For the Debye length, we will use D= 10 nm. We will also use an incoming impurity concentration C0= 1010/m3, the viscosity typical of water ⌘= 103Pa s, and a pressure di↵erence P= 105Pa. These parameter values are also of the same order as the experimental conditions imposed in [51,52] and in [58, 59]. We have checked, however, that the results of our simulations are invariant with respect to the last three parameters except for a multiplicative rescaling of the time values; as already described in detail in [61], this is due to the symmetry of the starting formulas. Consequently, in the present article we shall preferentially express the times resulting from all our simulations normalized with respect to a common reference value tref . In particular, for this tref we will choose (as described in detail in Section 8.4.1) the half-saturation time of the cylindrical cylinder with diameter 300 nm. Increases in ⌘would merely increase tref linearly, while increases in C0and Pwould induce an inversely proportional change in tref . Concerning the size of the tubular conduits, we will always use a length L= 1 mm. For cylinders (Section 8.4.1) we will use diameters from 250 nm to 400 nm. For tube geometries with variable diameter (Sections 8.4.2 to 8.4.4) we will use minimum, average and maximum diameters of, respectively, 200 nm, 300 nm and 400 nm. Let us also briefly comment here on the fact that the above typical parameter values (especially pressure) correspond to moderate flow velocities, in particular small enough to fulfill the assumption made by the model equations that axial advection e↵ects do not complicate the recovery, after each impurity is removed from the liquid flow, of transversal equilibrium in the impurity concentration profile. [61] This is because the typical time it takes di↵usion to rebuild transversal equilibrium is orders of magnitude smaller than the typical time each impurity travels within the collision distance from the walls before being removed from 7.2. Model equations 135 the liquid: The first of those times may be estimated as 2/D, [160] where D is the di↵usion coefficient (⇠109m2s) and the typical transversal distance over which equilibrium has to be reestablished (in our case this may be taken somewhere between Dand ⇢e,i.e., 10–30 nm). This leads to times of the order of 106s. In contrast, the typical time that each impurity travels within the collision distance from the walls before being removed from the liquid may be estimated to be orders of magnitude larger, of the order of 4 ⇥102s (by simply combining an average Poiseuille velocity of Pd2/(32L⌘), a typical diameter d⇠300 nm, and a typical distance traveled before being trapped of 1/⌦0= 105m). Chapter 8 Results for the influence of the microgeometry of ENM on the nanofiltration characteristics 137 144 Chapter 8. Influence of the microgeometry on the nanofiltration characteristics However, when the filter starts to get saturated, the amount ⌦0f0 eL⇡0, and the energy rate increases. For a non-cylindrical shape, it is possible to integrate numerically equation 8.10 to obtain the value at initial-clean state, taking into account that Cimp(xi,t= 0) is given by equation 8.11. We have also checked that these values are the same as calculated by the numerical integration of equations 7.4 and 7.5, confirming the validity of our simulations. That main result in this case is that in the limit that we are analyzing, the initial Edoes not depend on the shape of the conduit if it has the same mean diameter and length. 8.4 Time-evolution of the nanofiltration 8.4.1 Simulation results for cylindrical conduits We first explore the filtration characteristics of cylinders as a function of their diameter. For that purpose, let us consider four cylinders, of diameters 250, 300, 350 and 400 nm (the rest of the parameters entering Equations (7.4) to (7.6) being, for all the cylinders, equal to those detailed in Section 7.2.5). We present in Table 8.1 the numerical results obtained for the initial logarithmic removal value LRV0and the operational lifetime (for the LRV5, LRV2 and LRV 1 criterions) of these cylindrical nanofunctionalized macropore conduits. Also, Figure 8.1 illustrates the results of our simulations for two example diameters (300 and 400 nm). In particular, the left-hand column of Figure 8.1 (i.e., panels 8.1(a) and 8.1(c)) shows the internal profile of the trapped-impurity layer in each tube at di↵erent instants of the time evolution, i.e., their true internal radius as the conduits become narrower due to the accumulation of trapped impurities. These profiles are shown for the following instants of time: For t= 0, or clean state (when the radius coincides with its nominal value d(x)/2); for t=tsat,i.e., when all of the conduit has reached its saturated state (and the internal radius equals (d(x)sat)/2); and also for two intermediate times, t0.15 and t1/2, defined by the conditions ¯n(t=t0.15)=0.15 and ¯n(t=t1/2)=0.5, where ¯nis the average areal density of trapped impurities in the whole tube. A first result of our simulations, already easily visible in these Figures 8.1(a) and 8.1(c), is that the impurities accumulate first at the beginning of the cylinders, instead of uniformly. The right-hand column of Figure 8.1 (i.e., panels 8.1(b) and (d)) shows the time-evolution of the areal density of trapped impurities at opposite ends of the cylindrical tube, n(x=0,t) and n(x=L, t). It also shows the average areal density ¯n(t) used to calculate the times t0.15 and t1/2(that are marked as solid points in these panels). The timescale in this Figure is normalized as t/tref ,where tref is chosen as the t1/2value of the cylinder with diameter 300 nm. The same 8.4. Time-evolution of the nanofiltration 145 Figure 8.1: Results of our numerical simulations for nanofunctionalized macropore conduits with the geometries shown in the insets of the left column (cylinders of nominal diameter 300 and 400 nm). Sections 7.2.4, 7.2.5 and 8.4.1 of the main text detail the parameter values and the procedural details used to obtain these results. Figures (a) and (c) show the internal profiles of the trapped-impurity layer as time advances and impurities accumulate on the walls. At t=0the internal radius equates the nominal value. The time tsat corresponds to the fullysaturated state. The intermediate times t0.15 and t1/2(solid points in Figures (b) and (d)) are defined as the instants when the areal density of trapped impurities averaged over all of the tube, ¯n(t), becomes respectively 0.15 and 1/2 times the saturation value nsat. Figures (b) and (d) show the time-evolution of the areal density of trapped impurities evaluated at both the entrance and the exit of the conduit, and also the average ¯n(t). Note that the time axis is logarithmic and normalized by a reference value, tref , common for all our simulations and geometries and defined as the value of t1/2of the cylinder of diameter 300 nm (see Figure (a)). See also Table 8.1 for further numerical information on these simulations. 146 Chapter 8. Influence of the microgeometry on the nanofiltration characteristics common tref will be used in all of the Figures and Tables of the present work. These representations evidence that for cylindrical conduits the growth with time of n(x, t) happens earlier when closer to the start of the tube (x= 0) than when closer to the exit point (x=L), by at least one order of magnitude in time value (note that the time axis in Figure 8.1 is logarithmic). As it could be expected, the behavior of ¯n(t) is intermediate between the ones at both edges of the conduit. The results of our simulations also serve to study the influence of the diameter of the cylinders over the filtration performance and operational lifetime, as given by the logarithmic removal value LRV. In Table 8.1 we list for each diameter the LRV value computed at the initial time, LRV0, that characterizes the maximum filtration capability of the cylinder (achieved in the initial, clean state). The results in Table 8.1 indicate a better LRV0for the smaller diameters, as it could be probably expected by arguing that for smaller diameters a larger percentage of the fluid moves closer than ⇢e0 from the walls. Table 8.1 also lists the times at which the LRV reaches the thresholds LRV= 5, LRV= 2 and LRV= 1, i.e., the operational lifetimes for three di↵erent filtration requirements. Logically, the LRV 5 lifetime is null for the cases in which LRV0<5. The results in this Table show that the lifetimes become significantly larger when the diameter becomes smaller. This trend may be already noticed in Figures 8.1(b) and (d), where a similar change may be seen in the times at which the n(t) curves reach the saturation value (in fact, a similar evolution may be also noticed for t0.15 and t1/2). This is likely also influenced by the fact that the flow is slower when the diameter decreases, making the absorption slower too (note that our simulations are performed at constant P, rather than at constant ). Note that initial hydrodynamic energy rate is going to be constant for all the diameters that we are analyzing, because we are in the regime where ⌦0f0 eL 1, so Eis given by Equation (8.15). The limit when the exponential factor is about 0.01, and this correction starts to be significant in the calculation of the hydrodynamic rate, is d= 530 nm. However, in this regime, the value of the hydrodynamic energy rate is going to change di↵erently for di↵erent diameters with time, as we can see from Equation 8.10. Thus, this special case only happens for the initial value. 8.4.2 Simulation results for increasing-conical tubular conduits. Let us now show that other conduit geometries may improve the filtration performance and/or operational lifetimes with respect to those obtained for cylinders. We begin that investigation by studying, in this Subsection, conical tubular conduits with diameter increasing in the direction of the fluid flow. Note that these are indeed feasible macropore geometries: there exist experimental studies of 8.4. Time-evolution of the nanofiltration 147 membranes with conical (increasing and decreasing) pore shapes [147–149] for di↵erent applications, such as the study of resistive-pulse biosensors [147] or capillary electrophoresis [149]. However, none of these researches have studied the nanofunctionalization of these conical-shape pores for water filtration applications. In Figure 8.2 we present the results obtained by performing our simulations with the d(x) function corresponding to the increasing-conical geometry. In particular, the diameter increases linearly from the value 200 nm at the entrance to the tube up to the value 400 nm at the exit point (this is to be compared therefore with the 300 nm cylinder). Again it is visible that impurities accumulate sooner at the entrance of the tubes than at their exit. But it can be noticed in the Figures that the di↵erences are now even larger than in the cylindrical case. As a consequence, the ¯n(t) evolution also spreads over a wider t-range. Figure 8.2: Results of our numerical simulations for a nanofunctionalized macropore conduit of the increasing-conical geometry pictured in the inset of (a) (with a nominal diameter value 200 nm for the narrower section and 400 nm for the wider section, and therefore 300 nm as average diameter). Sections 7.2.4, 7.2.5 and 8.4.2 of the main text provide parameter values and further details of these simulations. Figure (a) shows the internal profile of the trapped-impurity layer as time advances and nanoimpurities accumulate in the walls. The times tref , tsat,t0.15 and t1/2are defined as in Figure 8.1 and its caption (in Figure (b) t0.15 and t1/2are signaled as solid points). Figure (b) shows the time-evolution of the areal density nof trapped impurities evaluated at both the entrance and the exit of the conduit, and also the average ¯n(t). See also Table 8.2 and Figure 8.5 for further information on these simulations. As it may be noticed from the values listed in Table 8.2, the LRV0value achieved by the increasing cone improves the one of the equivalent cylinder (6.2log versus 5.6-log). More significantly, Table 8.2 also indicates that the increasing-cone geometry 148 Chapter 8. Influence of the microgeometry on the nanofiltration characteristics increases the operating lifetimes by about 24% (for the LRV5 criterion), a 40% (for the LRV2 criterion) or by about 71% (for the LRV1 criterion) with respect to cylinders with the same average diameter. These di↵erences are also illustrated in Figure 8.5, that compares the LRV(t) curves of conduits with di↵erent shapes (but with equal average diameter along the axial direction). We note again that the time axis in all our Figures is expressed normalized by a common reference value (tref defined as t1/2of the d= 300 nm cylinder) and in logarithmic scale. These results indicate, therefore, that a possible path to greatly increase the operational lifetime of nanoenhanced macropore filters is to employ a supporting media containing pores with variable diameter. In the following Sections, this conclusion will be extended to other example conduit shapes. Figure 8.3: Results of our numerical simulations for a nanofunctionalized macropore conduit of the decreasing-conical geometry pictured in the inset of (a) (with a nominal diameter value 200 nm for the narrower section and 400 nm for the wider section, and therefore 300 nm as average diameter). Sections 7.2.4, 7.2.5 and 8.4.3 of the main text provide parameter values and further details of these simulations. Figure (a) shows the internal profile of the trapped-impurity layer as time advances and nanoimpurities accumulate in the walls. The times tref , tsat,t0.15 and t1/2are defined as in Figure 8.1 and its caption (in Figure (b) t0.15 and t1/2are signaled as solid points). Figure (b) shows the time-evolution of the areal density nof trapped impurities evaluated at both the entrance and the exit of the conduit, and also the average ¯n(t). See also Table 8.2 and Figure 8.5 for further information on these simulations. In relation to the hydrodynamic energy rate, Figure 8.6 shows taht the initial Eis the same for all the channels with same mean diameter. However, the evolution with time is quite di↵erent depending on the geometry. This figure and its inset shows that trapping impurities costs more hydrodynamic energy for the increasing-cone geometry than for the cylinder after the time given by the 8.4. Time-evolution of the nanofiltration 149 LRV2 criterion. To sum up, this geometry improves the initial LRV and the lifetime in relation to the cylinder, but it needs a higher amount of hydrodynamic energy to filter. 8.4.3 Simulation results for decreasing-conical tubular conduits. We consider now a decreasing-conical geometry, i.e., a linearly decreasing diameter d(x) function. Again we use in our simulations 200 and 400 nm for the extreme diameter values, and the mean diameter along the axial direction remains 300 nm. Figure 8.3 illustrates the results of our simulations for that geometry. A significant qualitative di↵erence with respect to the previous increasing-cone conduit is that now the plots versus time of n(x=0,t) and n(x=L, t) are significantly closer to each other. In fact, near the saturation state both lines even cross each other. This means that no longer is always true that the beginning of the conduit, x= 0, accumulates impurities faster than its end, x=L. Instead, the smaller section at the end of the cone is able to overcome the rhythm of impurity accumulation of the first slices of the cone before these get saturated. This more uniform filling by impurities results in a di↵erent evolution with time of the filtration performance. As revealed by the values listed in Table 8.2 (see also Figure 8.5), the initial LRV values of the increasing and decreasing cones are coincident, but the LRV5 lifetime of the decreasing cone is even longer than the one of the increasing cone (about 3.5 times longer than in the increasing cone, and also ⇠335% longer than in the cylinder of equal mean diameter). In contrast, the LRV1 lifetime of decreasing cones is shorter that the one of increasing cones (14% shorter, though still 48% longer than for equivalent cylinders). On the other hand, the LRV2 criterion reveals a tendency located in the middle of the other two criterions, getting a higher lifetime in comparison both with the cylinder (73% higher) and the increasing-cone (24% higher). This can be understood by observing in Figure 8.5 that the LRV(t) curves of both types of conical shapes cross each other near LRV⇠1.5. In relation to the hydrodynamic energy, Figure 8.6 reveals that below the point where n/nsat ⇠0.95, the decreasing cone is the geometry that optimizes better the hydrodynamic energy of the system. Above this amount, the inset shows that is the cylinder the type of channel that improves better the energy efficient. Therefore, to sum up, the decreasing-cone has better lifetime than the cylinder and the increasing-cone (in the higher LRV criterions), it has better capability for filtration in the most part of the time-region, and it is the most efficient energetically speaking when the filter is no. 150 Chapter 8. Influence of the microgeometry on the nanofiltration characteristics 8.4.4 Simulation results for sinusoidally corrugated tubular conduits. Let us now consider tubular macropores of diameter increasing and decreasing successively along their axis. Again this macropore geometry is experimentally realizable, and studies exist on macropore networks composed by pores with spherical shape and interconnections (throats) of small radius forming sinus-like shapes. For an example see [145]. Also relevant is the fact that some theoretical studies have analyzed these porous media as a network of pores with sinus-like shapes. [150,151] Figure 8.4: Results of our numerical simulations for a nanofunctionalized macropore conduit of the sinusoidally corrugated geometry pictured in the inset of (a) (with a nominal diameter value 200 nm for the narrower section and 400 nm for the wider section, and therefore 300 nm as average diameter). Sections 7.2.4, 7.2.5 and 8.4.4 of the main text provide parameter values and further details of these simulations. Figure (a) shows the internal profile of the trapped-impurity layer as time advances and nanoimpurities accumulate in the walls. The times tref ,tsat,t0.15 and t1/2are defined as in Figure 8.1 and its caption (in figure (b) t0.15 and t1/2are signaled as solid points). Figure (b) shows the time-evolution of the areal density nof trapped impurities evaluated at both the entrance and the exit of the conduit, and also the average ¯n(t). See also Table 8.2 and Figure 8.5 for further information on these simulations. A sin(x)-like diameter functionality is possibly one of the simplest of the corrugated geometries, and could a priory be expected to serve as a fair first approximation to, at least, the most essential features of real corrugated pores. [145, 150, 151] In Figure 8.4 we present the results for our simulations using for the diameter profile the function d(x) = (300 nm) + (100 nm) sin(6⇡x/L), i.e., the maximum, minimum and mean diameter are the same as in the previous two Subsections, and d(x) presents three oscillations over the conduit’s length (we 8.4. Time-evolution of the nanofiltration 151 diameter of cylindrical conduit LRV0LRV5 lifetime LRV2 lifetime LRV1 lifetime 250 nm 7.7 1.16 tref 1.28 tref 3.56 tref 300 nm 5.6 0.18 tref 1.79 tref 1.77 tref 350 nm 4.3 - 2.22 tref 0.95 tref 400 nm 3.3 - 2.66 tref 0.53 tref Table 8.1: Summary of the main filtration performance figures for nanofunctionalized macropore conduits of cylindrical geometry and di↵erent diameters, obtained by using the numerical simulation procedures and parameter values described in Sections 7.2.4, 7.2.5 and 8.4.1 (see also Figures 8.1 and 8.5 for plots of the detailed time evolution of the simulation results). The reference time tref is the same as defined in Figure 8.1 and its caption. The initial logarithmic removal values, LRV0, obtained from these simulations fully coincide with the direct application of Equation (8.4). The operational lifetime using the LRV5 criterion is null for the cylinders in which LRV0<5, and is not shown. have checked that increasing the number of oscillations would not qualitatively a↵ect our results; note also that according to Equation (8.6) LRVsinus 0does not depend on the number of oscillations). The time-evolution of the internal profiles shown in Figure 8.4(a) evidences that in the corrugated conduits the impurities accumulate first in the initial oscillation periods, and within that tendency they accumulate first in their narrower portions. This result is coherent with a naive vision of crudely considering the corrugated conduits as a combination in series of alternately increasing and decreasing cones. The time-evolution of the average ¯n(t) and of the filtration performance LRV(t) also resembles somewhat the results obtained for cones. However, it must be noted that the initial LRV performance is better than the one of the cylindrical and conical shapes (see Table 8.2 and Figure 8.5). The di↵erence is almost of one logarithmic unit with respect to the cylinder. Concerning the operational lifetime, with the LRV5 criterion it is almost as long as in the decreasing cone (14% shorter) and still much longer than for cylinders (276% longer). This is achieved without penalizing the LRV1 lifetime: on the contrary, with the LRV1 criterion the operational lifetime is the longest in Table 8.2 (being in particular 108% longer than for the equivalent cylinder). Moreover, the LRV2 criterion neither penalizes the lifetime, because is also 108% longer than the cylinder. In terms of the hydrodynamic energy rate, it can be seen in Figure 8.6 that for n/nsat <0.9, the sinus geometry mimics quite well the behavior of its equivalent cylinder. However, for higher values, the sinus becomes energetically less efficient, and starts to become more similar to the increasing cone, which is the one less efficient. To sum up, the sinus geometry optimizes the lifetimes and the LRV, 152 Chapter 8. Influence of the microgeometry on the nanofiltration characteristics but is less efficient than the decreasing cone in terms of hydrodynamic energy. conduit geometry (average diameter 300 nm) LRV0LRV5 lifetime LRV2 lifetime LRV1 lifetime cylinder 5.6 0.17 tref 1.28 tref 1.71 tref increasing cone 6.2 0.21tref 1.79 tref 2.93 tref decreasing cone 6.2 0.74 tref 2.22 tref 2.53 tref sinusoidally corrugated 6.4 0.64 tref 2.66 tref 3.56 tref Table 8.2: Summary of the main filtration performance figures for conduits with di↵erent geometries but equal average diameter 300 nm, obtained by using the numerical simulation procedures and parameter values described in Sections 7.2.4 and 7.2.5. The reference time tref is the same as defined in Figure 8.1 and its caption. The LRV0values obtained from these simulations fully coincide with the direct application of Equations (8.4) to (8.6). 8.5 Conclusions To sum up, we have studied the performance of macropores (conduits of micrometric diameter) with nanostructured inner walls, with regards to their performance and operational lifetime when used as filters of nanoparticles in suspension in aqueous liquids (nanofluids). We have considered di↵erent geometries, including cylinders of di↵erent diameters and tubes with diameter varying along the direction of the fluid flow (increasing and decreasing conical tubes and sinuscorrugated tubes). We focused mainly on the influence of the nominal geometry over the initial logarithmic removal value LRV0, the operational lifetimes, energetic balances, and the spatial profile of the trapped-impurity layer in the inner walls of the conduits. We first studied the impact of the diameter size for cylindrical macropore conduits. We found that decreasing the diameter improves the LRV0and operational lifetimes and maintains the hydrodynamic energy rate E. Then we tested tubular macropores with cross-sectional diameter varying along the flow direction, and found that these variations allow to improve the filtration characteristics without decreasing the average diameter of the conduit. In particular i) increasing-conical conduits have, with respect to cylinders with the same average diameter, moderately better LRV0significantly longer operational times (about 70% longer) and worse hydrodynamic energetic rates; ii) decreasingconical conduits equally improve LRV0and extend even further the operational times for high-LRV filtration requirements (more than 300% longer than equivalent cylinders) while the low-LRV operational lifetime is extended less than for the previous case. Moreover, is the geometry that improves better the energetic balances; and iii) sinus-corrugated tubular conduits improve the LRV0values 8.5. Conclusions 153 by almost one logarithmic unit with respect to equivalent cylinders, and present extended operational lifetimes for both highand low-LRV requirements (about 275% and 100% longer than equivalent cylinders, respectively). Furthermore, for the region far from the saturated regime, it keeps an energetic balance similar to the case of the cylinder. Figure 8.5: Logarithmic removal value LRV as a function of time resulting from our simulations of nanofunctionalized macropore conduits with di↵erent geometries and the same average nominal diameter 300 nm. Note that the cylinder leads to the lowest LRV0(defined as LRV(t= 0)) and the lowest operational lifetimes (defined as the times at which the filtration performance reaches a certain LRV threshold; see also Table 8.2). Note also that the time axis is logarithmic and normalized by the same common reference time tref as in Figures 8.1 to 8.4 and Tables 8.1 and 8.2. Our results also reveal that the nanoenhanced conduits get covered by trapped nanoimpurities at a faster pace in the regions with either smaller diameters or nearer to the entrance of the flow in the conduit. When both types of regions do not coincide in space, their competition may significantly increase some operational lifetimes. For example, when comparing increasing and decreasing conical macropores, in spite of having equal LRV0, their time evolution is significantly di↵erent from each other (note, e.g., their LRV5 lifetimes in Table 8.2).