# Critical Current Density in d-Wave Hubbard Superconductors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

**k**is the electron wave vector). This considers the change of $\epsilon \left(\mathit{k}\right)$ with respect to the chemical potential ($\mu $) and the formation of pairs that gives rise to an excitation energy gap $\Delta \left(\mathit{k}\right)$ in the electron density of states across the Fermi level. When $\epsilon \left(\mathit{k}\right)=\mu $ at the Fermi surface (FS), only the term for the energy gap remains, whose magnitude reflects the strength of the pairing interaction. Under these conditions, we have found that the d-wave symmetry of the pairing interaction leads to a maximum critical current density in the vicinity of the antinodal k-space direction $\left(\pi ,0\right)$ of approximately $1.407236\times {10}^{8}$ A/cm

^{2}, with a much greater current density along the nodal direction $\left(\frac{\pi}{2},\frac{\pi}{2}\right)$ of $2.214702\times {10}^{9}$ A/cm

^{2}. These results allow for the establishment of a maximum limit for the critical current density that could be attained by a d-wave superconductor.

## 1. Introduction

_{2}[3] and the copper-free oxide superconductors of the family BKBO [4]. However, there are exemptions such as the Bi and Li [5] and, more exceptionally, most practical superconducting compounds within the families of cuprates [6,7], iron pnictides, and chalcogenides [8], where the BCS coupling of electrons by the weak attraction caused by phonons does not explain all the above phenomena. Therefore, although the BCS theory explains how the density of states is changed on entering the superconducting state, which for electronically isotropic materials means that there are no electronic states anymore at the Fermi level, in the case of anisotropic electronic materials such as the LSCO, BSCCO, and YBCO cuprates [9], it is necessary to move beyond the BCS theory.

## 2. The Hubbard Model Approach and Related Considerations

_{2}planes of the superconducting cuprates, the formalism of the three-band Hubbard model is well-known [35]. Under this framework, the electronic states are close to the Fermi energy (${E}_{F}$) and can be described reasonably well by a single-band tight-binding model on a square lattice with second-neighbor hoppings [36]. This supports the idea that, on the one hand, the current density in the superconducting state of HTS materials can be incorporated via a single-band Hubbard model on a square lattice where the second-neighbor charge-bond interaction (correlated hopping) leads to the formation of Cooper pairs with d-wave symmetry. On the other hand, low-temperature superconductors showing an isotropic gap in all directions can be treated by an s-wave pairing function, where a generalized Hubbard model that includes the nearest-neighbor hopping (t) and the so-called nearest-neighbor correlated hopping interaction (∆t) can explain superconducting states with an extended s-symmetry gap [37]. Thus, for d-wave superconductors, our model contains the nearest-neighbor ($\mathsf{\Delta}t$) and second-nearest-neighbor ($\mathsf{\Delta}{t}_{3}$) correlated hopping interactions, in addition to the on-site ($U$) and inter-site ($V$) Coulombian repulsions [38].

- The electrons in the superconductor state travel across the crystal at a finite velocity (v) of less than c. Otherwise, there would not be a finite critical current.
- In a dispersive medium, the velocity of electrons can be estimated by the gradient of the relation of dispersion ε(
**k**), but in the case of superconductors, it needs to be estimated from the quasiparticle’s relation corresponding to Cooper pairs. - The electronic states that mainly participate in the formation of Cooper pairs are those near the Fermi; therefore, the higher velocity corresponds to that on the FS. Thus, for a given direction of $\mathit{k}$
**,**the group velocity involves the states $\mathit{k}$ such that $\left|\mathit{k}\right|<\left|{\mathit{k}}_{F}\right|$ and $\left|v\left(\mathit{k}\right)\right|\le \left|v\left({\mathit{k}}_{F}\right)\right|$. - The Cooper pairs are formed by electrons with the wave vectors
**k**and −**k**, whereby an electron travels in an opposite direction from the other. Analogously to the Mott-insulator transition, hole doping is considered for the carriers’ density from when it is half-filled [39]. - In anisotropic superconductivity, the electrons with wave vectors close to the nodes have a weak superconducting gap and require very low temperatures to form the Cooper pairs; therefore, the anti-nodal states play a more dynamic role in carrying the superconducting current [40].

^{2}is approximately ${10}^{22}/{\mathrm{cm}}^{3}$ [41], the carrier charge is $q=2e$, with $e$ the bare charge of the electron. The group velocity (${v}_{g}$) is given by

_{2}lattice parameters found for several cuprates [42,43].

## 3. Critical Current Density

^{3}, this is in good agreement with the recent observations reported in Ref. [41].

**k**on the FS, say $a{\mathit{k}}_{F-1}=\left(1.321354,1.321774\right)$ and $a{\mathit{k}}_{F-2}=\left(\pi ,0.087194\right)$

**,**represented by stars and circles in Figure 2), it is possible to notice that, as $T$ increases, the gap amplitude ${\Delta}_{d}$ decreases with a very rapid drop in the current density near ${T}_{c}$. In fact, even for temperatures lower than that of liquid nitrogen (77 K), the microscopic current density does not change substantially (or at least not within the logarithmic scale of Figure 2). It is clear that when assuming 77 K as a reference, $J$ slightly increases along with ${\Delta}_{d}$. When the temperature decreases (i.e., within the same order of magnitude), it can rapidly decrease with a parabolic tendency toward higher temperatures. This result is in good agreement with the experimental observations for ${J}_{c}$ along diverse YBCO samples [48,49], which means that the current density $J$ calculated from microscopic principles is independent of macroscopic factors, such as geometry (bulk, films, crystals, etc.), as well as of mesoscopic or vortex dynamics parameters influenced by materials deposition, fabrication techniques, and composite pinning properties. $J$ is to be understood as the most fundamental and possibly maximum critical current density that can be exhibited by the superconductor.

^{2}for the analyzed case, while for wave vectors close to the antinodal direction, it is $J\approx 1.407236\times {10}^{8}$ A/cm

^{2}(see Figure 2). This result indicates the existence of a maximum threshold value for ${J}_{c}$ in d-wave superconductors of about 140 MA/cm

^{2}because even though a higher current density can exist along the nodal direction due to its augmented group velocity, in practical applications, a transport current density higher than this value would destroy the superconductivity along ${\mathit{k}}_{F-2}$, creating an instantaneous avalanche of vortices moving from the superconducting state to the normal one. Thus, putting our findings into the context of the most recent measurements of the critical current density in PLD-deposited YBCO thin films [34,41], it should be noticed that, indeed, d-wave superconductors such as YBCO can reach ultra-high critical current densities of approximately 90 MA/cm

^{2}for oxygen overdoped samples, denoting significant room for improvement. Therefore, by having shown that from microscopic principles the maximum current density that a cuprate superconductor may stand is around 140 MA/cm

^{2}, we reaffirm the hypothesis at Ref. [41], which suggests that overdoping strategies with oxygen post-processing treatments and nanoengineering pinning can truly offer powerful prospects to increase the limits of dissipation-free current transport in cuprate superconductors and coated conductors for practical applications. Nevertheless, regardless of how the ${J}_{c}$ properties of the superconductor are enhanced, either by the inclusion of pinning elements (defects, dopants, inclusions, etc.) or the increment of charge carriers by the hole doping of the superconducting CuO

_{2}planes, the ${J}_{c}$ of the cuprates cannot exceed the microscopic limits imposed by the Cooper pairs formation.

## 4. Conclusions

_{2}planes (otherwise a stable flux pinning event cannot occur). Thus, our results support the idea that the further nanoengineering of superconducting thin films based on the deposition of rare-earth barium copper oxygen (REBCO) compounds and the increment of the charge density by the p-doping of superconducting CuO

_{2}planes are both likely to lead to critical current densities even higher than the record of $90{\mathrm{MA}/\mathrm{cm}}^{2}$ in overdoped YBCO films. Nevertheless, although these are already ultrahigh critical current densities, the capacity of the superconductors for transporting current is not unlimited, but on the contrary, from purely microscopic principles, we predicted that critical current densities higher than $140{\mathrm{MA}/\mathrm{cm}}^{2}$ in d-wave superconductors are unlikely to appear. Still, to fully ratify this conclusion, further research for the 3D case of tetragonal lattices is needed, such that families of superconductors with more Cooper oxygen planes and higher critical temperatures could be studied.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Theory of Superconductivity. Phys. Rev.
**1957**, 108, 1175–1204. [Google Scholar] [CrossRef] [Green Version] - Roberts, B.W. Survey of superconductive materials and critical evaluation of selected properties. J. Phys. Chem. Ref. Data
**1976**, 5, 581–822. [Google Scholar] [CrossRef] - Dadiel, J.L.; Naik, S.P.K.; Pęczkowski, P.; Sugiyama, J.; Ogino, H.; Sakai, N.; Kazuya, Y.; Warski, T.; Wojcik, A.; Oka, T.; et al. Synthesis of Dense MgB
_{2}Superconductor via In Situ and Ex Situ Spark Plasma Sintering Method. Materials**2021**, 14, 7395. [Google Scholar] [CrossRef] - Pęczkowski, P.; Łuszczek, M.; Szostak, E.; Muniraju, N.K.C.; Krztoń-Maziopa, A.; Gondek, Ł. Superconductivity and appearance of negative magnetocaloric effect in Ba
_{1–x}K_{x}BiO_{3}perovskites, doped by Y, La and Pr. Acta Mater.**2021**, 222, 117437. [Google Scholar] [CrossRef] - Huang, G.Q.; Xing, Z.W.; Xing, D.Y. Prediction of superconductivity in Li-intercalated bilayer phosphorene. Appl. Phys. Lett.
**2015**, 106, 113107. [Google Scholar] [CrossRef] [Green Version] - Ruiz, H.S.; Badía-Majós, A. Nature of the nodal kink in angle-resolved photoemission spectra of cuprate superconductors. Phys. Rev. B
**2009**, 79, 054528. [Google Scholar] [CrossRef] [Green Version] - Božović, I.; Bollinger, A.T.; Wu, J.; He, X. Can high-T
_{c}superconductivity in cuprates be explained by the conventional BCS theory? Low Temp. Phys.**2018**, 44, 519–527. [Google Scholar] [CrossRef] - Fernandes, R.M.; Coldea, A.I.; Ding, H.; Fisher, I.R.; Hirschfeld, P.J.; Kotliar, G. Iron pnictides and chalcogenides: A new paradigm for superconductivity. Nature
**2022**, 601, 35–44. [Google Scholar] [CrossRef] - Ruiz, H.; Badía-Majós, A. Strength of the phonon-coupling mode in La
_{2−x}Sr_{x}CuO_{4}, Bi_{2}Sr_{2}CaCu_{2}O_{8+x}and YBa_{2}Cu_{3}O_{6+x}composites along the nodal direction. Curr. Appl. Phys.**2012**, 12, 550–564. [Google Scholar] [CrossRef] - Bednorz, J.G. Possible highTc superconductivity in the Ba-La-Cu-O system. Eur. Phys. J. B
**1986**, 64, 189–193. [Google Scholar] [CrossRef] - Tang, C.Y.; Lin, Z.F.; Zhang, J.X.; Guo, X.C.; Zhong, Y.G.; Guan, J.Y.; Gao, S.Y.; Rao, Z.C.; Zhao, J.; Huang, Y.B.; et al. Antinodal kink in the band dispersion of electron-doped cuprate La
_{2−x}Ce_{x}CuO_{4±δ}. npj Quantum Mater.**2022**, 7, 1–6. [Google Scholar] [CrossRef] - Sigrist, M.; Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys.
**1991**, 63, 239–311. [Google Scholar] [CrossRef] - Bean, C.P. Magnetization of Hard Superconductors. Phys. Rev. Lett.
**1962**, 8, 250–253. [Google Scholar] [CrossRef] - Badía-Majós, A.; López, C.; Ruiz, H.S. General critical states in type-II superconductors. Phys. Rev. B
**2009**, 80, 144509. [Google Scholar] [CrossRef] [Green Version] - Ruiz, H.S.; Badía-Majós, A.; Rondan, H.S.R. Smooth double critical state theory for type-II superconductors. Supercond. Sci. Technol.
**2010**, 23. [Google Scholar] [CrossRef] - Robert, B.C.; Fareed, M.U.; Ruiz, H.S. How to Choose the Superconducting Material Law for the Modelling of 2G-HTS Coils. Materials
**2019**, 12, 2679. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Huang, Z.; Ruiz, H.S.; Zhai, Y.; Geng, J.; Shen, B.; Coombs, T.A. Study of the Pulsed Field Magnetization Strategy for the Superconducting Rotor. IEEE Trans. Appl. Supercond.
**2016**, 26, 1–5. [Google Scholar] [CrossRef] - Ruiz, H.S.; Zhang, X.; Coombs, T.A. Resistive-Type Superconducting Fault Current Limiters: Concepts, Materials, and Numerical Modeling. IEEE Trans. Appl. Supercond.
**2014**, 25, 1–5. [Google Scholar] [CrossRef] [Green Version] - Baghdadi, M.; Ruiz, H.S.; Coombs, T.A. Crossed-magnetic-field experiments on stacked second generation superconducting tapes: Reduction of the demagnetization effects. Appl. Phys. Lett.
**2014**, 104, 232602. [Google Scholar] [CrossRef] - Fareed, M.U.; Kapolka, M.; Robert, B.C.; Clegg, M.; Ruiz, H.S. 3D FEM Modeling of CORC Commercial Cables with Bean’s Like Magnetization Currents and Its AC-Losses Behavior. IEEE Trans. Appl. Supercond.
**2022**, 32, 1–5. [Google Scholar] [CrossRef] - Kapolka, M.; Ruiz, H.S. Maximum reduction of energy losses in multicore MgB
_{2}wires by metastructured soft-ferromagnetic coatings. Sci. Rep.**2022**, 12, 1–11. [Google Scholar] [CrossRef] - Hubbard, J. Electron correlations in narrow energy bands. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1963**, 276, 238–257. [Google Scholar] [CrossRef] - Spalek, J.; Oleá, A.M.; Chao, K.A. Magnetic Phases of Strongly Correlated Electrons in a Nearly Half-Filled Narrow Band. Phys. Status solidi (b)
**1981**, 108, 329–340. [Google Scholar] [CrossRef] - Ogata, M.; Himeda, A. Superconductivity and Antiferromagnetism in an Extended Gutzwiller Approximation fort–JModel: Effect of Double-Occupancy Exclusion. J. Phys. Soc. Jpn.
**2003**, 72, 374–391. [Google Scholar] [CrossRef] - Scalapino, D.J. A common thread: The pairing interaction for unconventional superconductors. Rev. Mod. Phys.
**2012**, 84, 1383–1417. [Google Scholar] [CrossRef] [Green Version] - Kordyuk, A.; Zabolotnyy, V.; Evtushinsky, D.; Inosov, D.; Kim, T.; Büchner, B.; Borisenko, S. An ARPES view on the high-T c problem: Phonons vs. spin-fluctuations. Eur. Phys. J. Spéc. Top.
**2010**, 188, 153–162. [Google Scholar] [CrossRef] - Spałek, J. Fifty years of Hubbard and Anderson lattice models: From magnetism to unconventional superconductivity—A brief overview. Philos. Mag.
**2014**, 95, 661–681. [Google Scholar] [CrossRef] [Green Version] - Pickett, W.E. Single Spin Superconductivity. Phys. Rev. Lett.
**1996**, 77, 3185–3188. [Google Scholar] [CrossRef] - Yang, J.; Luo, J.; Yi, C.; Shi, Y.; Zhou, Y.; Zheng, G.-Q. Spin-triplet superconductivity in K
_{2}Cr_{3}As_{3}. Sci. Adv.**2021**, 7. [Google Scholar] [CrossRef] - Steffens, P.; Sidis, Y.; Kulda, J.; Mao, Z.Q.; Maeno, Y.; Mazin, I.I.; Braden, M. Spin Fluctuations in Sr
_{2}RuO_{4}from Polarized Neutron Scattering: Implications for Superconductivity. Phys. Rev. Lett.**2019**, 122, 047004. [Google Scholar] [CrossRef] - Pérez, L.A.; Wang, C. dx
^{2}–y^{2}pairing in the generalized Hubbard square-lattice model. Solid State Commun.**2001**, 118, 589–593. [Google Scholar] [CrossRef] - Zhang, X.; Zhong, Z.; Ruiz, H.S.; Geng, J.; A Coombs, T. General approach for the determination of the magneto-angular dependence of the critical current of YBCO coated conductors. Supercond. Sci. Technol.
**2016**, 30. [Google Scholar] [CrossRef] - Zhang, X.; Zhong, Z.; Geng, J.; Shen, B.; Ma, J.; Li, C.; Zhang, H.; Dong, Q.; Coombs, T.A. Study of Critical Current and n-Values of 2G HTS Tapes: Their Magnetic Field-Angular Dependence. J. Supercond. Nov. Magn.
**2018**, 31, 3847–3854. [Google Scholar] [CrossRef] [Green Version] - Osipov, M.; Starikovskii, A.; Anishenko, I.; Pokrovskii, S.; Abin, D.; Rudnev, I. The influence of temperature on levitation properties of CC-tape stacks. Supercond. Sci. Technol.
**2021**, 34, 045003. [Google Scholar] [CrossRef] - Schüttler, H.-B.; Fedro, A.J. Copper-oxygen charge excitations and the effective-single-band theory of cuprate superconductors. Phys. Rev. B
**1992**, 45, 7588–7591. [Google Scholar] [CrossRef] - Mazin, I.I.; Singh, D.J. Ferromagnetic Spin Fluctuation Induced Superconductivity inSr
_{2}RuO_{4}. Phys. Rev. Lett.**1997**, 79, 733–736. [Google Scholar] [CrossRef] [Green Version] - Millán, B.; Hernández-Hernández, I.J.; Pérez, L.A.; Millán, J.S. A comparison of optimal doping behaviors between d- and s*-wave superconducting ground states. Rev. Mex. De Física
**2021**, 67, 312–317. [Google Scholar] [CrossRef] - Pérez, L.A.; Millán, J.S.; Wang, C. Spin singlet and triplet superconductivity induced by correlated hopping interactions. Int. J. Mod. Phys. B
**2010**, 24, 5229–5239. [Google Scholar] [CrossRef] - Imada, M.; Fujimori, A.; Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys.
**1998**, 70, 1039–1263. [Google Scholar] [CrossRef] [Green Version] - Perez, L.; Millán, J.; Domínguez, B.; Wang, C. Electronic specific heat of anisotropic superconductors and its doping dependence. J. Magn. Magn. Mater.
**2007**, 310, e129–e131. [Google Scholar] [CrossRef] - Stangl, A.; Palau, A.; Deutscher, G.; Obradors, X.; Puig, T. Ultra-high critical current densities of superconducting YBa
_{2}Cu_{3}O_{7-δ}thin films in the overdoped state. Sci. Rep.**2021**, 11, 1–12. [Google Scholar] [CrossRef] - Varshney, D.; Yogi, A.; Dodiya, N.; Mansuri, I. Alkaline Earth (Ca) and Transition Metal (Ni) Doping on The Transport Properties of Y
_{1-X}Ca_{x}Ba_{2}(Cu_{1-Y}Ni_{y})_{3}O_{7-δ}Superconductors. J. Mod. Phys.**2011**, 02, 922–927. [Google Scholar] [CrossRef] [Green Version] - Beyers, R.; Shaw, T. The Structure of Y1Ba2Cu3O7-δ and its Derivatives. Solid State Phys.
**1989**, 42, 135–212. [Google Scholar] [CrossRef] - Marder, M.P. Condensed Matter Physics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2010; pp. 455–456. [Google Scholar]
- Millan, B.; Perez, L.A.; Millan, J.S. Optimal doping for d-wave superconducting ground states within the generalized Hubbard model. Rev. Mex. Física
**2018**, 64, 233–239. [Google Scholar] [CrossRef] [Green Version] - Hossain, M.A.; Mottershead, J.D.F.; Fournier, D.; Bostwick, A.; McChesney, J.; Rotenberg, E.; Liang, R.; Hardy, W.N.; Sawatzky, G.A.; Elfimov, I.S.; et al. In situ doping control of the surface of high-temperature superconductors. Nat. Phys.
**2008**, 4, 527–531. [Google Scholar] [CrossRef] - Voo, K.-K.; Chen, H.-Y.; Wu, W.C. Defect and anisotropic gap-induced quasi-one-dimensional modulation of the local density of states ofYBa
_{2}Cu_{3}O_{7−δ}. Phys. Rev. B**2003**, 68, 012505. [Google Scholar] [CrossRef] [Green Version] - I Kosse, A.; Prokhorov, A.Y.; A Khokhlov, V.; Levchenko, G.; Semenov, A.; Kovalchuk, D.G.; Chernomorets, M.P.; Mikheenko, P.N. Measurements of the magnetic field and temperature dependences of the critical current in YBCO films and procedures for an appropriate theoretical model selection. Supercond. Sci. Technol.
**2008**, 21. [Google Scholar] [CrossRef] - Koblischka-Veneva, A.; Koblischka, M.R.; Berger, K.; Nouailhetas, Q.; Douine, B.; Muralidhar, M.; Murakami, M. Comparison of Temperature and Field Dependencies of the Critical Current Densities of Bulk YBCO, MgB
_{2}, and Iron-Based Superconductors. IEEE Trans. Appl. Supercond.**2019**, 29, 1–5. [Google Scholar] [CrossRef] - Badía-Majós, A.; López, C. Modelling current voltage characteristics of practical superconductors. Supercond. Sci. Technol.
**2014**, 28. [Google Scholar] [CrossRef]

**Figure 1.**Critical current density (${J}_{c}$) (blue line) as a function of

**k**on the Fermi surface for $T=0$ K for a set of Hamiltonian parameters with $-{t}^{\prime}/t=0.06$, $\mathsf{\Delta}t=0.5$ eV, $\mathsf{\Delta}{t}_{3}=0.05$ eV, and $n=0.805$. The corresponding Fermi surface is depicted on the k

_{x}-k

_{y}plane by a black line.

**Figure 2.**Critical current density (${J}_{c}$) as a function of temperature and gap amplitude for the same system as that which Figure 1 represents. The green stars correspond to the vector

**k**

_{F-1}on the FS where ${J}_{c}\left(\mathit{k}\right)$ is maximum and the red circles correspond to the vector

**k**

_{F-2}where ${J}_{c}\left(\mathit{k}\right)$ is minimal. The temperature dependence of the amplitude of the d-wave superconducting gap (${\Delta}_{d}\left(T\right)$ ) is depicted on the kx-ky plane (black solid circles).

**Figure 3.**Group velocity (${v}_{g}$) versus temperature and superconducting gap amplitude, evaluated at the points $a{\mathit{k}}_{F-1}=\left(1.321354,1.321774\right)$ (blue open squares) and $a{\mathit{k}}_{F-2}=\left(\pi ,0.087194\right)$ (red open circles) of the FS shown in Figure 2. The temperature dependence of the amplitude of the d-wave superconducting gap (${\Delta}_{d}\left(T\right)$) is depicted on the horizontal plane (black solid circles).

**Table 1.**Expressions for the Hubbard model parameters with $u\left(\mathit{r}\right)$ as the lattice periodic potential and $v\left(\mathit{r}-{\mathit{r}}^{\prime}\right)$ as the interaction potential between two electrons in the lattice.

Single-particle parameters |

${t}_{i,j}={\displaystyle \int}{d}^{3}\mathit{r}{\phi}^{*}\left(\mathit{r}-{\mathit{R}}_{i}\right)|-\frac{{\hslash}^{2}{\nabla}^{2}}{2m}+u\left(\mathit{r}\right)|\phi \left(\mathit{r}-{\mathit{R}}_{j}\right)$ |

$t={t}_{i,j}$ with $<i,j>$ ${t}^{\prime}={t}_{i,j}$ with $\ll i,j\gg $ |

Electron-electron interaction parameters |

${U}_{ij}^{\mathit{k}l}={\displaystyle \int}{d}^{3}\mathit{r}{d}^{3}{\mathit{r}}^{\prime}{\phi}^{*}\left(\mathit{r}-{\mathit{R}}_{i}\right){\phi}^{*}\left({\mathit{r}}^{\prime}-{\mathit{R}}_{j}\right)v\left(\mathit{r}-{\mathit{r}}^{\prime}\right)\phi \left(\mathit{r}-{\mathit{R}}_{\mathit{k}}\right)\phi \left({\mathit{r}}^{\prime}-{\mathit{R}}_{l}\right)$ $U={U}_{ii}^{ii}$ $V={U}_{ij}^{ij}$ ${U}_{ii}^{ij}=\mathsf{\Delta}t$, with $<i,j>$ |

${U}_{il}^{lj}=\mathsf{\Delta}{t}_{3}$, with $<i,l>$, $<j,l>$, and $\ll i,j\gg $ |

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**MDPI and ACS Style**

Millán, J.S.; Millán, J.; Pérez, L.A.; Ruiz, H.S.
Critical Current Density in *d*-Wave Hubbard Superconductors. *Materials* **2022**, *15*, 8969.
https://doi.org/10.3390/ma15248969

**AMA Style**

Millán JS, Millán J, Pérez LA, Ruiz HS.
Critical Current Density in *d*-Wave Hubbard Superconductors. *Materials*. 2022; 15(24):8969.
https://doi.org/10.3390/ma15248969

**Chicago/Turabian Style**

Millán, José Samuel, Jorge Millán, Luis A. Pérez, and Harold S. Ruiz.
2022. "Critical Current Density in *d*-Wave Hubbard Superconductors" *Materials* 15, no. 24: 8969.
https://doi.org/10.3390/ma15248969