FFLO-enhanced upper critical field in the extended saddle-point model: thermodynamic confirmation, quantitative cutoffs, and robustness of the nonuniform state remain undetermined

Abrikosov and Buzdin (Phys. Rev. B 63, 224506, 2001) showed that allowing Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) nonuniform superconductivity in the extended saddle-point model of high-temperature superconductors produces an upper critical field $H_{c2}(T)$ with positive curvature over the entire superconducting temperature range and no low-temperature saturation, formally removing the Pauli paramagnetic limit, with the ideal low-temperature divergence cut off only by impurity scattering or transverse dispersion. The paper itself stated limitations that define the open problem: its comparison with experiment remained qualitative, because, in the authors' words, the resistive results on overdoped $\mathrm{Tl}_2\mathrm{Ba}_2\mathrm{CuO}_6$ "were not confirmed by subsequent measurement of specific heat for the same substance"; its cutoff estimates are explicitly "simple" order-of-magnitude bounds that require material-specific impurity scattering times and transverse dispersion parameters; and its Gaussian variational ansatz is justified by the authors only near $T_c$ while being applied over the whole superconducting range, leaving open whether more general order-parameter profiles modify the FFLO region and the shape of $H_{c2}(T)$.

Background

The upper critical field $H_{c2}(T)$ is the magnetic field at which superconductivity is destroyed at temperature $T$. In a type-II superconductor two mechanisms suppress superconductivity: orbital pair breaking, in which the electronic motion couples to the field through the vector potential and organizes the flux into vortices, and the Pauli paramagnetic (Zeeman) effect, in which the spin splitting of the electron bands unbalances the opposite spins of a spin-singlet Cooper pair. The Zeeman effect alone bounds $H_{c2}$ by the Pauli paramagnetic limit (the Chandrasekhar-Clogston limit), the field at which the spin-polarization energy gain of the normal state exceeds the superconducting condensation energy. The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, predicted in the 1960s (Fulde and Ferrell, 1964), evades this limit in sufficiently clean systems: the Cooper pair acquires a finite momentum, matching partners between the spin-split Fermi surfaces, so the order parameter becomes spatially modulated with a modulation wavevector $q$. The state requires that orbital pair breaking be weak (for example, when the field lies parallel to the conducting planes of a layered material) and is rapidly destroyed by impurity scattering, so its experimental identification is delicate.

A separate thread concerns the electronic structure of the cuprate high-temperature superconductors. In several cuprates the dispersion has saddle points (van Hove singularities, i.e., points where the Fermi velocity vanishes and the density of states is enhanced) located near the Fermi level at the $(\pi,0)$ and $(0,\pi)$ points of the Brillouin zone. Abrikosov (1997) argued that these saddle points are extended: the dispersion is nearly flat along one momentum direction over a broad segment, so the Fermi surface contains quasi-one-dimensional (quasi-1D) regions of very high density of states. In his extended saddle-point model, with two inequivalent quasi-1D regions coupled by inter-region pairing, calculations with a spatially uniform order parameter reproduced the observed positive curvature of $H_{c2}(T)$ only in the immediate vicinity of the superconducting transition temperature $T_c$, while experiments on the overdoped cuprate Tl$_2$Ba$_2$CuO$_6$ (Mackenzie et al., 1993) reported resistive $H_{c2}$ with positive curvature persisting over the whole superconducting range.

Abrikosov and Buzdin (2001) asked whether the FFLO mechanism, known from earlier analyses of strictly one-dimensional superconductors to lift the paramagnetic limit, operates in this mixed geometry, where only parts of the Fermi surface are effectively one-dimensional. They derived a variational integral equation for $H_{c2}$ that simultaneously includes the FFLO modulation wavevector, the Zeeman splitting (measured by a dimensionless parameter $M$ giving the ratio of Zeeman to orbital pair breaking), the orbital effect, and the inter-region pairing (a dimensionless coupling $\eta$), using Gaussian trial profiles of width ratio $s$ for the order parameter. Numerically, for $\eta=1.2$ and $1.5$ with $M=0.01$ to $0.04$, the resulting $H_{c2}(T)$ curves have positive curvature over the whole range $0<T<T_c$ with no low-temperature saturation, and a finite-temperature onset of a low-temperature FFLO region. Analytically, in the clean, strictly 1D limit the critical Zeeman energy grows as $\beta H\propto T_c^2/T$ (with $\beta$ the Bohr magneton), so the ideal $H_{c2}$ diverges as $T\to0$ and the paramagnetic limit is removed. In real materials this divergence is expected to be cut off by two mechanisms: nonmagnetic impurity scattering, limiting the field to a scale set by the impurity scattering time $\tau_{\text{imp}}$ as $H_{pc}\approx\tau_{\text{imp}}T_c^2/\beta$, and weak dispersion transverse to the quasi-1D direction, which restores higher-dimensional kinematics and limits the field to $H_{pc}\approx(T_c/\beta)\sqrt{v/\delta v}$, where $v$ is the Fermi velocity along the quasi-1D direction and $\delta v$ its variation induced by the transverse dispersion. A closely related model study by Dias (2000) showed that a strong power-law van Hove density-of-states singularity alone (without FFLO physics) already produces positive $H_{c2}$ curvature and, for sufficiently strong singularities, a zero-temperature divergence of $H_{c2}$.

The limitations acknowledged in the 2001 paper itself, together with the scope caveat on its ansatz, define this problem: first, the paper states that its results "qualitatively agree[] with experiment," but that "the detailed comparison with experiment is complicated by the fact that the results of resistive measurements were not confirmed by subsequent measurement of specific heat for the same substance" (the specific-heat study of Tl$_2$Ba$_2$CuO$_6$ by Carrington et al. 1996, cited as Ref. 11 of the paper), so a direct thermodynamic confirmation of the FFLO-enhanced $H_{c2}$ scenario in those materials is lacking; second, the impurity- and dispersion-limited cutoff estimates are described by the authors as "simple estimate[s]" of order-of-magnitude character, and quantitative prediction for a given compound requires measured or microscopically computed $\tau_{\text{imp}}$ and $\delta v$, with the sensitivity of FFLO stability to realistic band structure unestablished; third, the Gaussian variational ansatz is justified by the authors only as "a very good approximation near $T_c$" yet is applied over the whole superconducting range, leaving open whether more general spatial profiles modify the location and extent of the FFLO region and the detailed shape of $H_{c2}(T)$.

Problem Statement

Determine whether the FFLO-stabilized upper critical field in the extended saddle-point model is robust and experimentally real. The source result (Abrikosov and Buzdin, 2001) is a weak-coupling, variational calculation with Gaussian order-parameter profiles showing, for inter-region coupling $\eta=1.2$ and $1.5$ and Zeeman-to-orbital ratio $M=0.01$ to $0.04$, a positively curved, non-saturating $H_{c2}(T)$ with a low-temperature FFLO region, a clean-limit low-temperature divergence $\beta H\propto T_c^2/T$, and order-of-magnitude cutoffs $H_{pc}\approx\tau_{\text{imp}}T_c^2/\beta$ (impurity scattering, with $\tau_{\text{imp}}$ the impurity scattering time) and $H_{pc}\approx(T_c/\beta)\sqrt{v/\delta v}$ (transverse dispersion, with $v$ the quasi-1D Fermi velocity and $\delta v$ its transverse variation). Three linked questions remain open, addressing whether the FFLO-enhanced $H_{c2}$ scenario is the correct explanation of the observed non-saturating, positively curved $H_{c2}(T)$ in overdoped $\mathrm{Tl}_2\mathrm{Ba}_2\mathrm{CuO}_6$-class cuprates:

(1) Robustness of the nonuniform state (theory): does the FFLO region, its onset temperature, its extent, and the detailed shape of $H_{c2}(T)$, survive when the Gaussian variational profiles — justified in the source only as "a very good approximation near $T_c$" while being applied over the whole superconducting range — are replaced by more general spatial forms of the order parameter?

(2) Material-specific limits (theory): how sensitive is the FFLO stability to realistic band structure beyond the idealized extended saddle points, and what are the quantitative (not order-of-magnitude) cutoff fields for a given compound, given measured or microscopically computed $\tau_{\text{imp}}$ and $\delta v$?

(3) Thermodynamic confirmation (experiment): does a bulk thermodynamic determination of $H_{c2}(T)$ (specific heat or equivalent) on overdoped $\mathrm{Tl}_2\mathrm{Ba}_2\mathrm{CuO}_6$-type samples, in the field and temperature regime where the resistive positive curvature was reported, corroborate the predicted persistent positive curvature, the absence of low-temperature saturation, and the finite-temperature onset of the FFLO state? The source paper itself notes that the resistive results were not confirmed by subsequent specific-heat measurement on the same substance.

The three questions concern the same scenario: components (1) and (2) address whether the theoretical prediction is stable under the relaxed assumptions that the original calculation could not treat, and component (3) concerns the experimental test of the same scenario. Component (3) is answerable from existing bulk thermodynamic measurements on overdoped $\mathrm{Tl}_2\mathrm{Ba}_2\mathrm{CuO}_6$-type samples and does not require new experimental data.

References

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