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Journal of Physics Condensed Matter

The BCS–BE crossover phase diagram atT= 0 K for a d-wave superconductor: the importance of the Debye frequency and the tight binding band structure

José Jesús Rodríguez-Núñez, Alex A. Schmidt, Orlando Alvarez-Llamoza, Eligio Orozco

Published Jun 8, 2004• 0 citations
Publisher

Abstract

We consider the phase diagram of the BCS (Bardeen?Cooper?Schrieffer)?BE (Bose?Einstein) crossover in the ground state (T = 0?K) of a dx2?y2-wave superconductor, with a nearest neighbour tight binding structure, when we take into account the Debye (phononic) frequency around the chemical potential, ?. This approach is a continuation of the work of den Hertog (1999 Phys.?Rev.?B 60 559) and that of Soares et al (2002 Phys.?Rev.?B 65 174506). The latter authors considered the influence of the second-nearest neighbours, but neither set of authors took into account the effect of the Debye frequency, ?D, or the influence of the next nearest neighbour matrix hopping element. We have found the following results: (1)?there is not a metallic phase?that is, when , , , and , where n is the carrier density per site, V is the attractive interaction, t is the nearest neighbour hopping integral, and is the next nearest neighbour hopping ratio; (2)?the BCS?BE crossover line is strongly affected by the presence of ?D/4t and that of ?actually, the values of V/4t needed to achieve the Bose?Einstein regime become extremely large for small values of ?D/4t; and (3)?both ?/4t and ?/4t strongly depend on the values of ?D/4t and . The results?(1) are in agreement with the ones found by Perali et al (2003 Phys.?Rev.?B 68 066501 (Preprint cond-mat/0211132)) and Rodr?guez-N??ez et al (2003 Phys.?Rev.?B 68 066502), and in disagreement with those of den Hertog and Soares et al.

DOI
10.1088/0953-8984/16/25/009

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BCS to Bose-Einstein crossover phase diagram at zero temperature for a dx2−y2 order parameter superconductor: Dependence on the tight-binding structure

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We study the effect of the next-nearest-neighbor (NNN) hopping matrix element ${t}^{\ensuremath{'}}$ on the zero-temperature $(T=0\mathrm{K})$ crossover phase diagram, namely, from the BCS limit to the Bose-Einstein regime, for a model which presents a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor order parameter symmetry. This theoretical generalization is required by angle-resolved photoemission spectroscopy experiments. Our NNN hopping is characterized by the parameter $\ensuremath{\alpha}\ensuremath{\equiv}{t}^{\ensuremath{'}}/t\ensuremath{\ne}0,$ where t is the nearest neighbor hopping matrix element. For \ensuremath{\alpha}\ensuremath{\geqslant}0, there is a crossover of BCS binding energy type as described [B. C. den Hertog, Phys. Rev. B 60, 559 (1999)] by keeping the density fixed. However, for \ensuremath{\alpha}0, the system is always in the Bose-Einstein regime for $n~{n}_{c}\ensuremath{\approx}0.12.$ We conclude that the presence of the NNN hopping matrix element drastically influences the crossover diagram for a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor.

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Comment on “BCS to Bose-Einstein crossover phase diagram at zero temperature for a dx2−y2 order parameter superconductor: Dependence on the tight-binding structure”

A. Perali, P. Pieri, Giancarlo Calvanese Strinati

The work by Soares et al. [Phys. Rev. B 65, 174506 (2002)] investigates the BCS to Bose-Einstein crossover for d-wave pairing in the two-dimensional attractive Hubbard model. Contrary to their claims, we found that a nonpairing region does not exist in the density vs coupling phase diagram. The gap parameter at $T=0,$ as obtained by solving analytically as well as numerically the BCS equations, is in fact finite for any nonzero density and coupling, even in the weak-coupling regime.

arXiv:cond-mat/0211132•Physical review. B, Condensed matter•Aug 28, 2003•16 citations

Reply to “Comment on ‘BCS to Bose-Einstein crossover phase diagram at zero temperature for a dx2−y2 order parameter superconductor: Dependence on the tight-binding structure’ ”

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In a recent Comment, Perali, Pieri, and Strinati [Phys. Rev. B 68, 066501 (2003)] came forward with a correction to the work of Soares et al. [Phys. Rev. B 65, 174506 (2002)], in the sense that they correctly found no nonpairing region in the density vs coupling diagram of the two-dimensional attractive Hubbard model for a d-wave symmetry for the order parameter. But, the BCS-Bose-Einstein crossover diagram also changes, a point missed by Perali, Pieri, and Strinati in their Comment. Furthermore, they use an analytical treatment of the gap equation to find the order parameter as C function of interaction strength, namely, $\ensuremath{\Delta}/4t$ vs $V/4t.$ We point out that this treatment is not completely correct for the evaluation of the chemical potential, $\ensuremath{\mu}/4t$ vs $V/4t,$ when we include the Debye frequency, ${\ensuremath{\omega}}_{D}/4t,$ and their cutoff frequency, ${\ensuremath{\omega}}_{o}.$ As a consequence, the phase diagram, namely, n vs $V/4t,$ strongly depends on the value of ${\ensuremath{\omega}}_{D}/4t,$ i.e., for small values of ${\ensuremath{\omega}}_{D}/4t$ the phase diagram moves to higher values of $V/4t.$

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The role of random onsite and hopping disorder in inducing a Bardeen–Cooper–Schrieffer superconductor to Bose–Einstein condensate (BCS–BEC) crossover is investigated using the Bogoliubov–de Gennes (BdG) method on a negative- U Hubbard model in two dimensions. In the strong disorder limit, the BdG state becomes highly inhomogeneous with the formation of superconducting ' droplets ' which are characterized by large pairing amplitudes. The different physical properties calculated for the inhomogeneous state differ significantly from the pure system at all values of electron densities, while we show that a crossover to a BEC state, if at all, is possible at low electron densities. The random onsite disorder supports such a crossover scenario, which is confirmed by observing the chemical potential to slip below the noninteracting band (Leggett criterion). The hopping disorder, however, renders a delocalized phase which allows for overlapping between the pairs, thus preventing the formation of a 'local pair' phase. Also the increase in bandwidth induced by hopping disorder always accommodates the chemical potential inside the band, and hence rules out a crossover phenomenon. However, another variant of the off-diagonal inhomogeneity, i.e. an anisotropic hopping, offers a pro-crossover scenario in the large anisotropy limit.

Journal of Physics Condensed Matter•Oct 17, 2008•11 citations
We consider the phase diagram of the BCS (Bardeen?Cooper?Schrieffer)?BE (Bose?Einstein) crossover in the ground state (T = 0?K) of a dx2?y2-wave superconductor, with a nearest neighbour tight binding structure, when we take into account the Debye (phononic) frequency around the chemical potential, ?. This approach is a continuation of the work of den Hertog (1999 Phys.?Rev.?B 60 559) and that of Soares et al (2002 Phys.?Rev.?B 65 174506). The latter authors considered the influence of the second-nearest neighbours, but neither set of authors took into account the effect of the Debye frequency, ?D, or the influence of the next nearest neighbour matrix hopping element. We have found the following results: (1)?there is not a metallic phase?that is, when , , , and , where n is the carrier density per site, V is the attractive interaction, t is the nearest neighbour hopping integral, and is the next nearest neighbour hopping ratio; (2)?the BCS?BE crossover line is strongly affected by the presence of ?D/4t and that of ?actually, the values of V/4t needed to achieve the Bose?Einstein regime become extremely large for small values of ?D/4t; and (3)?both ?/4t and ?/4t strongly depend on the values of ?D/4t and . The results?(1) are in agreement with the ones found by Perali et al (2003 Phys.?Rev.?B 68 066501 (Preprint cond-mat/0211132)) and Rodr?guez-N??ez et al (2003 Phys.?Rev.?B 68 066502), and in disagreement with those of den Hertog and Soares et al.
We study the effect of the next-nearest-neighbor (NNN) hopping matrix element ${t}^{\ensuremath{'}}$ on the zero-temperature (T=0K)(T=0\mathrm{K})(T=0K) crossover phase diagram, namely, from the BCS limit to the Bose-Einstein regime, for a model which presents a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor order parameter symmetry. This theoretical generalization is required by angle-resolved photoemission spectroscopy experiments. Our NNN hopping is characterized by the parameter $\ensuremath{\alpha}\ensuremath{\equiv}{t}^{\ensuremath{'}}/t\ensuremath{\ne}0,$ where t is the nearest neighbor hopping matrix element. For \ensuremath{\alpha}\ensuremath{\geqslant}0, there is a crossover of BCS binding energy type as described [B. C. den Hertog, Phys. Rev. B 60, 559 (1999)] by keeping the density fixed. However, for \ensuremath{\alpha}0, the system is always in the Bose-Einstein regime for $n~{n}_{c}\ensuremath{\approx}0.12.$ We conclude that the presence of the NNN hopping matrix element drastically influences the crossover diagram for a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor.
The work by Soares et al. [Phys. Rev. B 65, 174506 (2002)] investigates the BCS to Bose-Einstein crossover for d-wave pairing in the two-dimensional attractive Hubbard model. Contrary to their claims, we found that a nonpairing region does not exist in the density vs coupling phase diagram. The gap parameter at T=0,T=0,T=0, as obtained by solving analytically as well as numerically the BCS equations, is in fact finite for any nonzero density and coupling, even in the weak-coupling regime.
In a recent Comment, Perali, Pieri, and Strinati [Phys. Rev. B 68, 066501 (2003)] came forward with a correction to the work of Soares et al. [Phys. Rev. B 65, 174506 (2002)], in the sense that they correctly found no nonpairing region in the density vs coupling diagram of the two-dimensional attractive Hubbard model for a d-wave symmetry for the order parameter. But, the BCS-Bose-Einstein crossover diagram also changes, a point missed by Perali, Pieri, and Strinati in their Comment. Furthermore, they use an analytical treatment of the gap equation to find the order parameter as C function of interaction strength, namely, $\ensuremath{\Delta}/4t$ vs V/4t.V/4t.V/4t. We point out that this treatment is not completely correct for the evaluation of the chemical potential, $\ensuremath{\mu}/4t$ vs V/4t,V/4t,V/4t, when we include the Debye frequency, ${\ensuremath{\omega}}_{D}/4t,$ and their cutoff frequency, ${\ensuremath{\omega}}_{o}.$ As a consequence, the phase diagram, namely, n vs V/4t,V/4t,V/4t, strongly depends on the value of ${\ensuremath{\omega}}_{D}/4t,$ i.e., for small values of ${\ensuremath{\omega}}_{D}/4t$ the phase diagram moves to higher values of V/4t.V/4t.V/4t.
We have analyzed the doping and temperature dependencies of the physical properties of a square-lattice narrow-band superconductor with a relevant for systems with local Hubbard repulsion nematic pairing due to effective nearest-neighbor attraction. Our focus was a possible crossover of the system from the Bardeen–Cooper–Schrieffer (BCS) superconductivity to the Bose–Einstein condensation (BEC) regimes with increasing temperature. We have found that in the d-channel the crossover does not take place at realistic values of doping and temperature, and even at a large attraction, i.e., the system is always in the BCS regime, unlike the “usual” d-wave superconductor. Contrary to non-nematic d-wave superconductors, the chemical potential μ always grows with temperature increase that might indicate an unexpectedly large size of Cooper pairs. At the same time, in the s-channel, we have found a BCS–BEC crossover with temperature for a wide range of temperatures even at rather high dopings, nf ≈ 0.2. This result is different from the “usual” s-case, where a narrow-band (i.e., at strong coupling) system is always in the BEC state. We have found for a given large value of attraction that there is a value of doping when for a wide range of temperature the system is in a “pre-crossover” regime with μ ≈ 0, i.e., in a regime of coexistence of local and extended pairs. Thus, in narrow-band systems a “usual” s-superconductor is in a BEC state, but in the nematic case one can expect a BCS–to–BEC crossover with temperature growth at realistic system parameters, i.e., the crossover might be measured experimentally.
We review aspects of the evolution from Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensation (BEC) in two dimensions, which have now become relevant in systems with low densities, such as gated superconductors Li xZrNCl, magic-angle twisted trilayer graphene, FeSe, FeSe1− xS x, and ultracold Fermi superfluids. We emphasize the important role played by chemical potentials in determining crossovers or topological quantum phase transitions during the BCS–BEC evolution in one-band and two-band superfluids and superconductors. We highlight that crossovers from BCS to BEC occur for pairing in nonnodal s-wave channels, whereas topological quantum phase transitions, in which the order parameter symmetry does not change, arise for pairing in any nodal higher angular momentum channels, such as d-wave. We conclude by discussing a few open questions regarding the BCS-to-BEC evolution in 2D, including modulus fluctuations of the order parameter, tighter upper bounds on critical temperatures, and the exploration of lattice effects in two-band superconductors and superfluids.
The role of random onsite and hopping disorder in inducing a Bardeen–Cooper–Schrieffer superconductor to Bose–Einstein condensate (BCS–BEC) crossover is investigated using the Bogoliubov–de Gennes (BdG) method on a negative- U Hubbard model in two dimensions. In the strong disorder limit, the BdG state becomes highly inhomogeneous with the formation of superconducting ' droplets ' which are characterized by large pairing amplitudes. The different physical properties calculated for the inhomogeneous state differ significantly from the pure system at all values of electron densities, while we show that a crossover to a BEC state, if at all, is possible at low electron densities. The random onsite disorder supports such a crossover scenario, which is confirmed by observing the chemical potential to slip below the noninteracting band (Leggett criterion). The hopping disorder, however, renders a delocalized phase which allows for overlapping between the pairs, thus preventing the formation of a 'local pair' phase. Also the increase in bandwidth induced by hopping disorder always accommodates the chemical potential inside the band, and hence rules out a crossover phenomenon. However, another variant of the off-diagonal inhomogeneity, i.e. an anisotropic hopping, offers a pro-crossover scenario in the large anisotropy limit.