Critical Current Density in <i>d</i>-Wave Hubbard Superconductors
In this work, the Generalized Hubbard Model on a square lattice is applied to evaluate the electrical current density of high critical temperature <i>d</i>-wave superconductors with a set of Hamiltonian parameters allowing them to reach critical temperatures close to 100 K. The appropria...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Materials |
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| Online Access: | https://www.mdpi.com/1996-1944/15/24/8969 |
| _version_ | 1797456510316445696 |
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| author | José Samuel Millán Jorge Millán Luis A. Pérez Harold S. Ruiz |
| author_facet | José Samuel Millán Jorge Millán Luis A. Pérez Harold S. Ruiz |
| author_sort | José Samuel Millán |
| collection | DOAJ |
| description | In this work, the Generalized Hubbard Model on a square lattice is applied to evaluate the electrical current density of high critical temperature <i>d</i>-wave superconductors with a set of Hamiltonian parameters allowing them to reach critical temperatures close to 100 K. The appropriate set of Hamiltonian parameters permits us to apply our model to real materials, finding a good quantitative fit with important macroscopic superconducting properties such as the critical superconducting temperature (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula>) and the critical current density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><msub><mi>J</mi><mi>c</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We propose that much as in a dispersive medium, in which the velocity of electrons can be estimated by the gradient of the dispersion relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the electron velocity is proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><mi>E</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in the superconducting state (where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow><mo>−</mo><mi>μ</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mo>Δ</mo><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></msqrt></mrow></semantics></math></inline-formula> is the dispersion relation of the quasiparticles, and <b><i>k</i></b> is the electron wave vector). This considers the change of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with respect to the chemical potential (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>) and the formation of pairs that gives rise to an excitation energy gap <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in the electron density of states across the Fermi level. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi></mrow></semantics></math></inline-formula> 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 <i>d</i>-wave symmetry of the pairing interaction leads to a maximum critical current density in the vicinity of the antinodal <i>k</i>-space direction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>π</mi><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of approximately <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.407236</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></semantics></math></inline-formula> A/cm<sup>2</sup>, with a much greater current density along the nodal direction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>,</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.214702</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow></semantics></math></inline-formula> A/cm<sup>2</sup>. These results allow for the establishment of a maximum limit for the critical current density that could be attained by a <i>d</i>-wave superconductor. |
| first_indexed | 2024-03-09T16:08:51Z |
| format | Article |
| id | doaj.art-37c4ea035f6d4ec2a4906a287f613772 |
| institution | Directory Open Access Journal |
| issn | 1996-1944 |
| language | English |
| last_indexed | 2024-03-09T16:08:51Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
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| series | Materials |
| spelling | doaj.art-37c4ea035f6d4ec2a4906a287f6137722023-11-24T16:24:51ZengMDPI AGMaterials1996-19442022-12-011524896910.3390/ma15248969Critical Current Density in <i>d</i>-Wave Hubbard SuperconductorsJosé Samuel Millán0Jorge Millán1Luis A. Pérez2Harold S. Ruiz3Facultad de Ingeniería, Universidad Autónoma del Carmen, Cd. del Carmen C.P. 24180, Campeche, MexicoFacultad de Ingeniería, Universidad Autónoma del Carmen, Cd. del Carmen C.P. 24180, Campeche, MexicoInstituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-360, Ciudad de Mexico C.P. 04510, CDMX, MexicoSchool of Engineering and Space Park Leicester, University of Leicester, University Rd., Leicester LE1 7RH, UKIn this work, the Generalized Hubbard Model on a square lattice is applied to evaluate the electrical current density of high critical temperature <i>d</i>-wave superconductors with a set of Hamiltonian parameters allowing them to reach critical temperatures close to 100 K. The appropriate set of Hamiltonian parameters permits us to apply our model to real materials, finding a good quantitative fit with important macroscopic superconducting properties such as the critical superconducting temperature (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula>) and the critical current density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><msub><mi>J</mi><mi>c</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We propose that much as in a dispersive medium, in which the velocity of electrons can be estimated by the gradient of the dispersion relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the electron velocity is proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><mi>E</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in the superconducting state (where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow><mo>−</mo><mi>μ</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mo>Δ</mo><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></msqrt></mrow></semantics></math></inline-formula> is the dispersion relation of the quasiparticles, and <b><i>k</i></b> is the electron wave vector). This considers the change of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with respect to the chemical potential (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>) and the formation of pairs that gives rise to an excitation energy gap <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in the electron density of states across the Fermi level. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">k</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi></mrow></semantics></math></inline-formula> 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 <i>d</i>-wave symmetry of the pairing interaction leads to a maximum critical current density in the vicinity of the antinodal <i>k</i>-space direction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>π</mi><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of approximately <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.407236</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></semantics></math></inline-formula> A/cm<sup>2</sup>, with a much greater current density along the nodal direction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>,</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.214702</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow></semantics></math></inline-formula> A/cm<sup>2</sup>. These results allow for the establishment of a maximum limit for the critical current density that could be attained by a <i>d</i>-wave superconductor.https://www.mdpi.com/1996-1944/15/24/8969critical current density<i>d</i>-wave superconductorsHubbard model |
| spellingShingle | José Samuel Millán Jorge Millán Luis A. Pérez Harold S. Ruiz Critical Current Density in <i>d</i>-Wave Hubbard Superconductors Materials critical current density <i>d</i>-wave superconductors Hubbard model |
| title | Critical Current Density in <i>d</i>-Wave Hubbard Superconductors |
| title_full | Critical Current Density in <i>d</i>-Wave Hubbard Superconductors |
| title_fullStr | Critical Current Density in <i>d</i>-Wave Hubbard Superconductors |
| title_full_unstemmed | Critical Current Density in <i>d</i>-Wave Hubbard Superconductors |
| title_short | Critical Current Density in <i>d</i>-Wave Hubbard Superconductors |
| title_sort | critical current density in i d i wave hubbard superconductors |
| topic | critical current density <i>d</i>-wave superconductors Hubbard model |
| url | https://www.mdpi.com/1996-1944/15/24/8969 |
| work_keys_str_mv | AT josesamuelmillan criticalcurrentdensityinidiwavehubbardsuperconductors AT jorgemillan criticalcurrentdensityinidiwavehubbardsuperconductors AT luisaperez criticalcurrentdensityinidiwavehubbardsuperconductors AT haroldsruiz criticalcurrentdensityinidiwavehubbardsuperconductors |