Open directions of Dayan's 1981 lattice-stability no-go analysis for non-phononic superconductivity mechanisms: non-diagonal polarization, inhomogeneous geometries, small-$q$ Umklapp, and finite-frequency negative screening for higher $T_{\mathrm{c}}$

Dayan proved, via a dressed-phonon sum rule, that a net electron-electron attraction at zero energy mediated by bosons other than ionic lattice phonons (excitons, electron-hole excitations, or the hole-lattice phonons of Abrikosov's 'metallic excitonium') is impossible in a stable crystal: any negative $1/\varepsilon(k)$ at $\omega=0$ forces ionic-phonon softening, with or without Umklapp. The open problem is the reach and the loopholes of this no-go constraint: whether it extends to non-diagonal (momentum-non-conserving) electronic polarization; whether inhomogeneous interface geometries such as metal-semiconductor proximity sandwiches can evade the instability argument; what happens in the very small-$q$ Umklapp regime where the derivation's bare vertex fails; and whether a real material or excitation spectrum can realize a sufficiently large, low-energy negative $\varepsilon(k,\omega)$ at finite frequency to enhance $T_{\mathrm{c}}$. The problem remains open.

Background

Superconductivity occurs when electrons bind into Cooper pairs through an effective attractive interaction. In the Bardeen-Cooper-Schrieffer (BCS) theory and its strong-coupling extension, the Eliashberg theory, this attraction is mediated in ordinary metals by phonons, the quantized vibrations of the positively charged ionic lattice, while the bare Coulomb repulsion between electrons is only partially screened; the residual repulsion is conventionally summarized by the Coulomb pseudopotential $\mu^{*}$. Because the superconducting transition temperature $T_{\mathrm{c}}$ grows, roughly, with the characteristic energy of the exchanged boson, it was proposed early on that the relatively low-energy ionic phonons be replaced by higher-energy electronic excitations: excitons, that is, bound electron-hole pairs, in organic chains, at surfaces, or at metal-semiconductor interfaces (the excitonic mechanism, reviewed by Ginzburg in 1968 and modeled for metal-semiconductor sandwiches by Allender, Bray and Bardeen in 1973), and, in Abrikosov's 1978 'metallic excitonium' proposal, the phonons of a hypothetical lattice of crystallized holes.

Whether such non-ionic channels can produce a net attractive interaction between electrons is constrained by the stability of the crystal. The screening of the Coulomb interaction is described by the dielectric function $\varepsilon(k,\omega)$, which collects the response of all electronic polarizations $\Lambda$ at momentum transfer $k$ and frequency $\omega$ (electron-hole excitations, excitons, plasmons, that is, collective charge oscillations) while keeping the contribution of the ionic lattice phonons separate. A static ($\omega=0$) attraction from non-ionic channels would require the screening factor $1/\varepsilon(k)$ to vanish or become negative. In 1981, M. Dayan analyzed this question in the diagrammatic (Dyson-equation) framework of the coupled electron-phonon system and derived a dressed-phonon sum rule,

$$\frac{1}{\Omega_{\mathrm{p}}^{2}} \sum_{\gamma} \omega_{q,\gamma}^{2} =1-\sum_{G}\left(1-\frac{1}{\varepsilon(q+G)}\right),$$

where $\Omega_{\mathrm{p}}$ is the ionic plasma energy (the natural oscillation frequency of the ions treated as a charged gas), $\omega_{q,\gamma}$ are the dressed ionic-phonon frequencies at phonon momentum $q$ and polarization $\gamma$, and the sum runs over reciprocal-lattice vectors $G$, so that the momentum argument $k=q+G$ includes Umklapp processes, that is, electron scattering in which the crystal momentum changes by a reciprocal-lattice vector. If any component $1/\varepsilon(k)$ vanished or turned negative at $\omega=0$, the right-hand side would become negative and at least one ionic phonon frequency would soften to zero, a lattice (structural) instability. Dayan concluded that a net electron-electron attraction mediated exclusively by bosons other than ionic phonons is impossible at zero energy in a stable crystal, with or without Umklapp, applying to both the excitonic mechanism and the hole-lattice phonons of metallic excitonium; non-ionic channels could at most augment the conventional phonon mechanism, or provide attraction at finite frequencies above the ionic-phonon energy scale, where a negative $\varepsilon(k,\omega)$ could in principle enhance $T_{\mathrm{c}}$. Notably, the same year Dolgov, Kirzhnits and Maksimov argued that a negative static dielectric function at finite momentum does not by itself contradict the stability of matter and should occur in simple metals and nonideal plasmas, so the admissible sign of static screening was contested territory.

Dayan's own analysis records limits of the derivation, and these define the open problem: the diagonal (momentum-conserving) form assumed for $\Lambda$ and $\varepsilon$; the restriction to bulk homogeneous crystals, which excludes the inhomogeneous sandwich geometries of the excitonic proposals; the inaccessibility of the very small-$q$ Umklapp regime, where the bare electron-phonon vertex used in the derivation is not valid because the bare transverse ionic-phonon frequencies vanish as $q\to 0$ while the electronic momentum transfer stays finite; and the unconstructed materials realization of the finite-frequency negative-screening route.

Problem Statement

The problem is to determine the reach and the loopholes of the lattice-stability no-go constraint on non-phononic superconductivity mechanisms, that is, whether screening channels other than the ionic lattice phonons can mediate or enhance superconducting pairing in stable solids. Four linked regimes delimit the question under the common objective of fixing the exact scope of the 1981 no-go theorem:

(1) Non-diagonal polarization. The derivation assumes that the electronic polarization $\Lambda$ and the dielectric function $\varepsilon$ are diagonal in the crystal-momentum indices (momentum conserving). Determine whether the no-go conclusion, that a static ($\omega=0$) net electron-electron attraction mediated exclusively by non-ionic bosons forces ionic-phonon softening, extends to systems where $\Lambda$ is non-diagonal (momentum-non-conserving polarizations), or exhibit a counterexample: a stable crystalline model with momentum-non-conserving electronic polarization that nonetheless realizes a net static attraction.

(2) Inhomogeneous geometries. The treatment assumes bulk crystalline order and does not apply to non-homogeneous systems such as the metal-semiconductor proximity sandwiches of the excitonic-mechanism proposals, which are often treated by effective-medium (homogenisation) approximations. Determine whether interface-specific inhomogeneous geometries can evade the instability argument.

(3) Small-$q$ Umklapp regime. The bare electron-phonon interaction used in the derivation is not valid for Umklapp scattering in the very small-$q$ limit, because the bare transverse ionic-phonon frequencies vanish as $q\to 0$ while the electronic momentum transfer stays finite. Settle the consequences of this exclusion for the small-$q$ Umklapp regime in general.

(4) Finite-frequency negative screening. Non-ionic channels can only help at finite frequency: a dielectric function $\varepsilon(k,\omega)$ that is negative at energies comparable to (not much above) the ionic-phonon energies, over a sufficiently wide range and with sufficient magnitude, could enhance $T_{\mathrm{c}}$. Construct, or rule out, a material or excitation spectrum realizing a sufficiently large and low-energy negative $\varepsilon(k,\omega)$ that could enhance the superconducting transition temperature without destabilizing the lattice.

The four regimes together define the problem: the full scope of the 1981 no-go theorem is determined by all four, and the regimes are logically independent, so settling one leaves the others open.

References

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