High-temperature superconductors are reviewed in light of the fact that their binding energy is ionic. The conducting electrons are dominated by the much larger energy scales coming from ligand Coulomb integrals, including the out-of-plane ones, which are responsible for the Fermi arcs. The historic reinterpretation of Hund's rule from an intraelectronic to a central mean-field effect is applied to compare the cuprates to the pnictides. It is argued that the cuprates conform to the now-standard central-field paradigm, while the generally abandoned intraelectronic mechanism is exceptionally applicable to the pnictides. A non-adiabatic Fermi liquid paradigm is inferred from the phenomenological evidence. Glueless superconductivity is interpreted as the limiting case of Cooper-pair scattering in cuprates when the Cu ion is perfectly rigid.
We report the fabrication and measurements of ionic liquid gated Hall bar devices prepared on thin Ca$_3$Ru$_2$O$_7$ flakes exfoliated from bulk single crystals that were grown by a floating zone method. Two types of devices with their electrical transport properties dominated by $c$-axis transport in Type A or that of the in-plane in Type B devices, were prepared. Bulk physical phenomena, including a magnetic transition near 56 K, a structural and metal-insulator transition at a slightly lower temperature, as well as the emergence of a highly unusual metallic state as the temperature is furth…
When electron doped, the layered transition metal nitrides ${\cal T}$NCl (${\cal T}$ = group IVB transition metal ion) become impressive superconductors with critical temperature T$_c$= 15-26K. Here we take the most studied member, ZrNCl, as a representative and calculate the dielectric response $ε(ω)$ versus frequency and concentration of doped electronic carriers. The static dielectric constant $ε_{\infty}$=5 is reproduced extremely well. We establish that the differences between rigid band modeling and virtual crystal treatment are small, and compare also with actual lithium doping using su…
Effective Ionic Valence and Local Magnetic Moment in Kagome Superconductors
Shanghai Jiao Tong University（上海交通大学）；University of Cambridge（剑桥大学）；University of California Santa Barbara（加州大学圣塔芭芭拉分校）；Leibniz Institute for Solid State and Materials Research（莱布尼茨固体与材料研究所）
We report the temperature variation of the $^{119}$Sn-Mössbauer spectra of the antiperovskite (inverse perovskite) oxide superconductor Sr$_{3-x}$SnO. Both superconductive (Sr-deficient) and non-superconductive (nearly stoichiometric) samples exhibit major $γ$-ray absorption with isomer shift similar to that of Mg$_2$Sn. This fact shows that Sr$_{3-x}$SnO contains the metallic anion Sn$^{4-}$, which is rare especially among oxides. In both samples, we observed another $γ$-ray absorption with a larger isomer shift, indicating that there is another ionic state of Sn with a higher oxidation numbe…
When the electron density of highly crystalline thin films is tuned by chemical doping or ionic liq- uid gating, interesting effects appear including unconventional superconductivity, sizeable spin-orbit coupling, competition with charge-density waves, and a debated low-temperature metallic state that seems to avoid the superconducting or insulating fate of standard two-dimensional electron systems. Some experiments also find a marked tendency to a negative electronic compressibility. We suggest that this indicates an inclination for electronic phase separation resulting in a nanoscopic inhomo…
The associations between emergent physical phenomena (e.g., superconductivity) and orbital, charge, and spin degrees of freedom of $3d$ electrons are intriguing in transition metal compounds. Here, we successfully manipulate the superconductivity of spinel oxide Li$_{1\pm x}$Ti$_2$O$_{4-δ}$ (LTO) by ionic liquid gating. A dome-shaped superconducting phase diagram is established, where two insulating phases are disclosed both in heavily electron-doping and hole-doping regions. The superconductor-insulator transition (SIT) in the hole-doping region can be attributed to the loss of Ti valence ele…
A controversial issue on whether the electron-phonon interaction (EPI) is crucial for high-temperature superconductivity or it is weak and inessential has remained one of the most challenging problems of contemporary condensed matter physics. We employ a continuum RPA approximation for the dielectric response function allowing for a selfconsistent semi-analytical evaluation of the EPI strength, electron-electron attractions, and the carrier mass renormalisation in layered high-temperature superconductors. We show that the Fröhlich EPI with high-frequency optical phonons in doped ionic lattices…
Ionic gating is a very popular tool to investigate and control the electric charge transport and electronic ground state in a wide variety of different materials. This is due to its capability to induce large modulations of the surface charge density by means of the electric-double-layer field-effect transistor (EDL-FET) architecture, and has been proven to be capable of tuning even the properties of metallic systems. In this short review, I summarize the main results which have been achieved so far in controlling the superconducting (SC) properties of thin films of conventional metallic super…
Applying the Helmholtz Decomposition theorem to linearized General Relativity leads to a gauge-invariant formulation where the transverse-traceless part of the metric perturbation describes gravitational waves in matter. Gravitational waves incident on a superconductor can be described by a linear London-like constituent equation characterized by a "gravitational shear modulus" and a corresponding plasma frequency and penetration depth. Electric-like and magnetic-like gravitational tensor fields are defined in terms of the strain field of a gravitational wave. It is shown that in the DC limit,…
Recent findings of superconductors that simultaneously exhibit multiple spontaneously broken symmetries, such as ferromagnetic order or lack of an inversion center and even combinations of such broken symmetries, have led to much theoretical and experimental research. We consider quantum transport in a junction consisting of a ferromagnetic metal and a non-unitary ferromagnetic superconductor. It is shown that the conductance spectra provides detailed information about the superconducting gaps, and is thus helpful in determining the pairing symmetry of the Cooper pairs in ferromagnetic superco…
It seems that the remarkable properties of the high temperature superconducting oxides, especially the Insulator-Metal Transition (IMT) and the Metal-Superconductor Transition (MST) both originate from the competition (interplay) between ionic versus in-plane covalence nature of bonds in these materials. As a result of this competition, the microscopic order parameter, that is firmly identified to be the local field estimated from the ionic polarization at the sub-unit level (one half of the unit cell), shows a strong temperature as well as chemical doping dependence. While the out-of-plane io…
Based on first-principles FLAPW-GGA calculations, we have investigated the electronic structure of newly synthesized novel superconductors LaONiBi and LaOCuBi, the first bismuth-containing compounds from the family of quaternary oxypnictides which attract now a great deal of interest in search for novel 26-55K superconductors. The band structure, density of states and Fermi surfaces are discussed. Our results indicate that the bonding inside of the (La-O) and (Ni(Cu)-Bi) layers is covalent whereas the bonding between the (La-O) and (Ni(Cu)- Bi) blocks is mostly ionic. For both oxybismuthides, …
Anomalous metallic behavior, marked by a saturating finite resistivity much lower than the Drude estimate, has been observed in a wide range of two-dimensional superconductors. Utilizing the electrostatically gated LaAlO3/SrTiO3 interface as a versatile platform for superconductor-metal quantum phase transitions, we probe variations in the gate, magnetic field, and temperature to construct a phase diagram crossing from superconductor, anomalous metal, vortex liquid, to Drude metal states, combining longitudinal and Hall resistivity measurements. We find that the anomalous metal phases induced …
We present a theoretical framework for understanding the behavior of the normal and superconducting states of overdoped cuprate high temperature superconductors in the vicinity of the doping-tuned quantum superconductor-to-metal transition. The key ingredients on which we focus are d-wave pairing, a flat antinodal dispersion, and disorder. Even for homogeneous disorder, these lead to effectively granular superconducting correlations and a superconducting transition temperature determined in large part by the superfluid stiffness rather than the pairing scale.
We describe the thermal and electrical conductivities of quasi-one dimensional wires, across a quantum phase transition from a superconductor to a metal induced by pairbreaking perturbations. Fluctuation corrections to BCS theory motivate a field theory for quantum criticality. We describe deviations in the Wiedemann-Franz ratio from the Lorenz number, which can act as sensitive tests of the theory. We also describe the crossovers out of the quantum critical region into the metallic and superconducting phases.
We report on mutual proximity effects in d-wave superconductor/half-metal heterostructures which correspond to systems composed of high-Tc cuprates and manganite materials. In our study, proximity effects are induced by the interplay of two separate interface effects: spin-mixing (or rotation) surface scattering and spin-flip scattering. The surface spin-mixing scattering introduces spin-triplet pairing correlations in superconducting side; as a result, Andreev bound states are formed at energies within the superconducting gap. The spin-flip scattering introduces not only long range equal-spin…
The response of a superconductor to a gravitational wave is shown to obey a London-like constituent equation. The Cooper pairs are described by the Ginzburg-Landau free energy density embedded in curved spacetime. The lattice ions are modeled by quantum harmonic oscillators characterized by quasi-energy eigenvalues. This formulation is shown to predict a dynamical Casimir effect since the zero-point energy of the ionic lattice phonons is modulated by the gravitational wave. It is also shown that the response to a gravitational wave is far less for the Cooper pair density than for the ionic lat…
Ionic-Bond-Driven Atom-Bridged Room-Temperature Cooper Pairing in Cuprates and Nickelates: a Theoretical Framework Supported by 32 Experimental Evidences
Peking University（北京大学）；Beijing Technology and Business University（北京工商大学）
Motivated by the existence of superconductivity in pyrite-structure CuS$_2$, we explore the possibility of ionic-liquid-gating-induced superconductivity in the proximal antiferromagnetic Mott insulator NiS$_2$. A clear gating-induced transition from a two-dimensional insulating state to a three-dimensional metallic state is observed at positive gate bias on single crystal surfaces. No evidence for superconductivity is observed down to the lowest measured temperature of 0.45 K, however. Based on transport, energy-dispersive X-ray spectroscopy, X-ray photoelectron spectroscopy, atomic force micr…
We consider a system of superconducting grains embedded in a normal metal. At zero temperature this system exhibits a quantum superconductor-normal metal phase transition. This transition can take place at arbitrarily large conductance of the normal metal.
We study the proximity effect in a superconductor (S)-normal metal (N) bilayer under in-plane magnetic field. A compensation between the Zeeman effect and the energy splitting between bonding and anti-bonding levels leads to a magnetic field induced superconducting phase in the (H,T) phase diagram of the S-N bilayer well above the standard paramagnetic limit. We demonstrate that the presence of the non-magnetic impurities shrink the region of field induced superconductivity existence in S-N and S-S bilayers.
We investigate the coupling between two singlet superconductors separated by a half-metallic magnet. The mechanism behind the coupling is provided by the rotation of the quasiparticle spin in the superconductor during reflection events at the interface with the half metal. Spin rotation induces triplet correlations in the superconductor which, in the presence of surface spin-flip scattering, result in an indirect Josephson effect between the superconductors. We present a theory appropriate for studying this phenomenon and calculate physical properties for a superconductor/half metal/supercondu…
We predict that the Josephson current in a clean s-wave superconductor-normal metal-d-wave superconductor junction is periodic in superconducting phase difference $ϕ$ with period $π$ instead of $2π$. The frequency of non-stationary Josephson effect is correspondingly $2ω_J = 4eV.$ The effect is due to coexistence in the normal layer of current carrying Andreev levels with phase differences $ϕ$ and $ϕ+π.$
The tunneling spectrum of an electron and a hole in metal-superconductor-metal junctions is computed using the Blonder-Tinkham-Klapwijk theory. The incident and the outgoing currents finally balance each other by an interface charge inside the superconductor and metal junction. The present computation shows a more abundant structure compared to that on a metal-superconductor junction, such as the resonance at bias voltages above the energy gap of the superconductor. The density of the interface charge shows a quantum-like oscillation.
It is shown that the application of sufficiently strong magnetic field to the odd-frequency paired Pair Density Wave state described in Phys. Rev. B 94, 165114 (2016) leads to formation of a low temperature metallic state with zero Hall response. Applications of these ideas to the recent experiments on stripe-ordered La_{1.875}Ba_{0.125}CuO_4 are discussed.
We study the transport properties of ultrathin disordered nanowires in the neighborhood of the superconductor-metal quantum phase transition. To this end we combine numerical calculations with analytical strong-disorder renormalization group results. The quantum critical conductivity at zero temperature diverges logarithmically as a function of frequency. In the metallic phase, it obeys activated scaling associated with an infinite-randomness quantum critical point. We extend the scaling theory to higher dimensions and discuss implications for experiments.
We found La2Sb with a layered structure composed of alternate stacking of La square nets and LaSb layers exhibits bulk superconductivity with a critical temperature of 5.3 K. This suggests that the presence of the square net with strong La-La metal bonding is essential for the emergence of superconductivity.
We have fabricated quasi-two-dimensional arrays of nano-scale Pb grains coupled by an overlayer of Ag grains. Their temperature dependent resistive transitions follow predictions for an array of mesoscopic superconductor-normal-superconductor junctions. The decrease of their transition temperatures with Ag overlayer thickness systematically deviates from the Cooper limit theory of the proximity effect as the Pb grain size decreases. The deviations occur when the estimated number of Cooper pairs per grain is less than or equal to 1 and suggest the approach to a superconductor to metal transitio…
We explore the possibility that the ionic electron polarizabilities of the oxygen ions in the cuprates and the bismutates and the polarizabilities of As and Se ions in the iron pnictides contribute to charge carrier pairing leading to high Tc superconductivity. Using the fact that the ionic polarization response to a change in the electric field is practically instantaneous we find, that the inter carrier electrostatic potential is attractive in a limited distance range. This potential is used to calculate quantum mechanically the cooper-like pairing energy and wavefunction and the gap energy …
The zero temperature, or quantum, metal-superconductor phase transition is studied in disordered systems in dimension greater than two. A effective local field theory is developed that keeps all soft modes or fluctuations explicitly. A simple renormalization group analysis is used to exactly determine the quantum critical behavior at this transition.
Manipulating the superconducting states of high-T_c cuprate superconductors in an efficient and reliable way is of great importance for their applications in next-generation electronics. Traditional methods are mostly based on a trial-and-error method that is difficult to implement and time consuming. Here, employing ionic liquid gating, a selective control of volatile and non-volatile superconductivity is achieved in pristine insulating Pr_2CuO_{4\pmδ} film, based on two distinct mechanisms: 1) with positive electric fields, the film can be reversibly switched between non-superconducting and …
We find a series of topological phase transitions in a half-metal/superconductor heterostructure, by tuning the direction of the magnetization of the half-metal film. These include transitions between a topological superconducting phase with a bulk gap and another phase without a bulk gap but has a ubiquitous local gap. At the same time, the edge states change from counter-propagating Majorana edge modes to unidirectional Majorana edge modes. In addition, we find transitions between the second phase and a nodal phase which turns out to be a two-dimensional Weyl superconductor with Fermi line e…
The metallic ground state in two-dimensional (2D) superconductors has attracted much attention but is still under intense scrutiny. Especially, the measurements in ultralow temperature region are challenging for 2D superconductors due to the sensitivity to external perturbations. In this work, the resistance saturation induced by external noise, named as "extrinsic anomalous metallic state", is observed in 2D transition metal dichalcogenide (TMD) superconductor 4Ha-TaSe2 nanodevices. However, with further decreasing temperature, credible evidence of intrinsic anomalous metallic state is obtain…
To investigate the transport properties of iron based superconductors, we prepared planar hybrid superconductor-normal metal-superconductor (SNS') thin film junctions with BaFe1.8Co0.2As2 as base electrode. As counter electrode we used a lead indium alloy, while the normal metal layer was formed by thin gold films. Temperature dependent measurements of the electrical conductivity were strongly influenced by the properties of the electrodes. We developed a junction structure that allows us to characterize the electrodes, too, including the behavior of their normal state resistance in order to c…
We develop a detailed microscopic theory describing dc Josephson effect and Andreev bound states in superconducting junctions with a half-metal. In such systems the supercurrent is caused by triplet pairing states emerging due to spin-flip scattering at the interfaces between superconducting electrodes and the half-metal. For sufficiently clean metals we provide a detailed non-perturbative description of the Josephson current at arbitrary transmissions and spin-flip scattering parameters for both interfaces. Our analysis demonstrates that the behavior of both the Josephson current and Andreev …
In mesoscopic superconductor/normal-metal/superconductor (SNS) heterostructures, it is known that the resistance of the normal metal between the superconductors has a strong temperature dependence. Based on this phenomenon, we have developed a new type of thermometer, which dramatically enhances our ability to measure the local electron temperature Te at low temperatures. Using this technique, we have been able to measure small temperature gradients across a micron-size sample, opening up the possibility of quantitatively measuring the thermal properties of mesoscopic devices.
We study the tunneling conductance of a clean normal metal/chiral p-wave superconductor junction using the extended Blonder-Tinkham-Klapwijk formalism. It is shown that the spin-orbit coupling of the Rashba type that is present near the interface causes the subgap conductance peaks associated with the Andreev surface bound states to shift to a nonzero bias. We also investigate the effect of the Fermi wavevector mismatch between the normal metal and the superconductor.
We considered a mesoscopic superconductor/normal metal (S/N) structure in which the N reservoirs are maintained at different temperatures. It is shown that in the absence of current between the N reservoirs a voltage difference $V_{T}$ arises between the superconducting and normal conductors. The voltage $V_{T}$ oscillates with increasing phase difference $ϕ$ between the superconductors, and its magnitude does not depend on the small parameter $(T/ε_{F}).$
We show that Zeeman field can induce a topological transition in two-dimensional spin-orbit coupled metals, and concomitantly, a first-order phase transition in the superconducting state involving a discontinuous change of Cooper pair momentum. Depending on the spin-orbit coupling strength, we find different phase diagrams of 2D superconductors under in-plane magnetic field.
The Meissner effect in diffusive normal metal / insulator / d-wave superconductor junctions is studied theoretically in the framework of the Usadel equation under the generalized boundary condition. The effect of midgap Andreev resonant states (MARS) formed at the interface of d-wave superconductor is taken into account. It is shown that the formation of MARS suppresses the susceptibility of the diffusive normal metal.
On the basis of the Keldysh method of non-equilibrium systems, we develop a theory of electron tunneling in normal-metal/superconductor junctions. By using the tunneling Hamiltonian model (being appropriate for the tight-binding systems), the tunneling current can be exactly obtained in terms of the equilibrium Green functions of the normal metal and the superconductor. We calculate the conductance of various junctions. The discrepancy between the present treatment and the well-known scheme by Blonder, Tinkham, and Klapwijk is found for some junctions of low interfacial potential barrier.
We combine in-situ electrical transport and Seebeck coefficient measurements with the ionic liquid gating technique to investigate superconductivity and the normal state of FeySe1-xTex (FST) films. We find that the pristine FST films feature a non-Fermi liquid temperature dependence of the Seebeck coefficient, i.e., S/T ~ AS lnT, and AS is strongly correlated with the superconducting transition temperature (Tc). Ionic liquid gating significantly raises Tc of FST films, for which the Seebeck coefficient displays a novel scaling behavior and retains the logarithmic temperature dependence. Moreov…
We have studied the effect of perpendicular magnetic fields and temperatures on the nonlinear electronic transport in amorphous Ta superconducting thin films. The films exhibit a magnetic field induced metallic behavior intervening the superconductor-insulator transition in the zero temperature limit. We show that the nonlinear transport in the superconducting and metallic phase is of non-thermal origin and accompanies an extraordinarily long voltage response time.
We develop a theory of the conductance of superconductor/normal metal/superconductor junctions in the case where the superconducting order parameter has d-wave symmetry. At low temperature the conductance is proportional to the square root of the inelastic electron relaxation time in the bulk of the superconductor. As a result it turns out to be much larger than the conductance of the normal part of the junction.
We present measurements of current noise and cross-correlations in three-terminal Superconductor-Normal metal-Superconductor (S-N-S) nanostructures that are potential solid-state entanglers thanks to Andreev reflections at the N-S interfaces. The noise correlation measurements spanned from the regime where electron-electron interactions are relevant to the regime of Incoherent Multiple Andreev Reflection (IMAR). In the latter regime, negative cross-correlations are observed in samples with closely-spaced junctions.
Density functional studies of 26K superconducting LaFeAs(O,F) are reported. We find a low carrier density, high density of states, $N(E_F)$ and modest phonon frequencies relative to $T_c$. The high $N(E_F)$ leads to proximity to itinerant magnetism, with competing ferromagnetic and antiferromagnetic fluctuations and the balance between these controlled by doping level. Thus LaFeAs(O,F) is in a unique class of high $T_c$ superconductors: high $N(E_F)$ ionic metals near magnetism.
We consider the chi_0 dependence of the density of states inside the normal metal of a superconductor - normal metal - superconductor (SNS) junction.Here chi_0 is the phase difference of two superconductors of the junction. It is shown that in the absence of electron-electron interaction the energy dependence of the density of states has a gap which decreases as chi_0 increases and closes at chi_0= pi. Both the analytical expressions for the chi_0 dependence of the density of states and the results of numerical simulations are presented.
We study the proximity coherence in a mesoscopic normal-metal film (N) in contact with a superconductor (S). Accounting for a repulsive interaction between the electrons in the normal metal, we find an enhanced local density of states close to the NS interface. The sharp peak in the density is pinned to the Fermi energy and leads to spontaneous paramagnetic interface currents. The induced orbital magnetic moments exhibit the characteristic features of paramagnetic reentrance observed in normal-metal coated superconducting cylinders [Phys. Rev. Lett. 65, 1514 (1990)].
We apply the scattering matrix approach to the triplet proximity effect in superconductor-half metal structures. We find that for junctions that do not mix different orbital modes, the zero bias Andreev conductance vanishes, while the zero bias Josephson current is nonzero. We illustrate this finding on a ballistic half-metal--superconductor (HS) and superconductor -- half-metal -- superconductor (SHS) junction with translation invariance along the interfaces, and on HS and SHS systems where transport through the half-metallic region takes place through a single conducting channel. Our calcula…
We consider a process of the Andreev reflection between a normal metal and the s-wave superconductor in the FFLO state. It is shown that the process takes place if the energy of the incoming electron is bound within the finite interval called the Andreev window. The position of the window determines the value of the non-zero total momentum of Cooper pairs and the value of the gap.
As a half-metal is spin-polarized at its Fermi level by definition, it was conventionally thought to have little proximity effect to an s-wave superconductor. Here we show that, with interface spin-orbit coupling, $p_x +ip_y$ superconductivity without spin degeneracy is induced on the half-metal, and we give an estimate of its bulk energy gap. Therefore a single-band half-metal can give us a topological superconductor with a single chiral Majorana edge state. Our band calculation shows that two atomic layers of VTe or CrO$_2$ is a single-band half-metal for a wide range ($\sim$0.1eV) of Fermi …
We consider tunnel microrefrigerators at low temperature. There is a number of experimental studies performed on microrefrigeration in tunneling superconductor--normal metal (SN) structures. Related to these experiments, only the electron subsystem has been considered theoretically. Independently, the phonon deficit effect has been studied a while ago in superconductor-superconductor tunnel junctions. It can be regarded as a possible prototype scheme for superconducting microrefrigerators. We try to provide the missing link between experiments on the SN tunnel junction refrigerators and the th…
We study the point-contact tunneling between normal metal and ferromagnetic superconductor. In the case of magnon-induced pairing the tunneling conductance is continuous and smooth function of the applied voltage. For small values of the applied voltage the Ohm law holds. We show that one can obtain the magnetization and the superconducting order parameter from the tunneling conduc- tance. In the case of paramagnon-induced superconductivity the tunneling does not depend on the magnetization. We argue that tunneling experiment can unambiguously determine the correct pairing mechanism in the fer…
We discuss the surface impedance (Z=R-iX) of a normal-metal/superconductor proximity structure taking into account the spin-dependent potential at the junction interface. Because of the spin mixing transport at the interface, odd-frequency spin-triplet s-wave Cooper pairs penetrate into the normal metal and cause the anomalous response to electromagnetic fields. At low temperature, the local impedance at a surface of the normal metal shows the nonmonotonic temperature dependence and the anomalous relation R>X. We also discuss a possibility of observing such anomalous impedance in experiments.
We consider a mesoscopic four-terminal superconductor/normal metal (S/N) structure in the presence of a temperature gradient along the N wire. A thermoemf arises in this system even in the absence of the thermoelectric quasiparticle current if the phase difference between the superconductors is not zero. We show that the thermoemf is not small in the case of a negligible Josephson coupling between two superconductors. It is also shown that the thermoelectric voltage has two maxima: one at a low temperature and another at a temperature close to the critical temperature. The obtained temperature…
The results of calculations of the spatially-resolved density of states (DoS) in an S(F/N) bilayer are presented (S is a superconductor, F is a metallic ferromagnet, N is a normal metal) within quasiclassical theory in the dirty limit. Analytical solutions are obtained in the case of thin F, N layers which demonstrate the peculiar features of DoS in this system. The dependencies of the minigap and the DoS peak positions on the exchange energy and parameters of the layers are studied numerically
We discuss the quasiparticle entropy and heat capacity of a dirty superconductor-normal metal-superconductor junction. In the case of short junctions, the inverse proximity effect extending in the superconducting banks plays a crucial role in determining the thermodynamic quantities. In this case, commonly used approximations can violate thermodynamic relations between supercurrent and quasiparticle entropy. We provide analytical and numerical results as a function of different geometrical parameters. Quantitative estimates for the heat capacity can be relevant for the design of caloritronic d…
