Pairing Excitations in Quantum Systems

• Pairing excitations are energy- and momentum-dependent responses that characterize the formation and disruption of paired states in quantum systems, including superconductors and semiconductors.
• They reveal spatial and temperature inhomogeneities in the pairing gaps, indicating that local pairing can persist above Tc and is strongly linked to underlying electronic correlations.
• Advanced techniques such as atomic-scale tunneling spectroscopy and complex gap modeling are used to disentangle electron–electron and electron–boson interactions governing high-Tc superconductivity.

Pairing excitations refer to the fundamental energy- and momentum-dependent responses associated with the formation, dynamics, and disruption of paired states in quantum systems—most prominently, Cooper pairs in superconductors, correlated electron–hole pairs in semiconductors, and nucleonic pairs in atomic nuclei. These excitations govern not only the emergence and robustness of the paired (superfluid or superconducting) ground state but also the spectrum and spatial structure of low-energy and collective modes, single-particle excitation continua, and the interplay with competing or coexisting electronic, lattice, and spin degrees of freedom. Contemporary research addresses pairing excitations through local probes, dynamical spectroscopy, and a combination of mean-field, beyond-mean-field, and fully microscopic theoretical frameworks, aiming to disentangle the competing mechanisms underpinning high-temperature superconductivity and strongly correlated phenomena.

1. Temperature Dependence and Spatial Inhomogeneity of Pairing Energy Gaps

High-resolution, lattice-tracking tunneling spectroscopy performed on Bi₂Sr₂CaCu₂O₈₊δ samples (0804.2623) demonstrates spatially inhomogeneous pairing gaps Δ(r, T) that decrease monotonically with increasing temperature. Critically, even above the superconducting transition temperature Tc (the loss of long-range phase coherence), a finite, spatially resolved pairing gap persists up to locally defined pairing temperatures Tₚ(r) > Tc. This observation reveals an extended regime of preformed pairs or local pairing, a result inconsistent with conventional mean-field BCS theory, where gap closure is synchronized with the loss of superconductivity.

The measured temperature dependence is quantitatively described using the thermally broadened density of states for a d-wave superconductor:

$$N_S(r, V, T) ∝ \int dE\, [−∂f(E, T)/∂E]\, \mathrm{Re} \left[ \frac{|E|}{\sqrt{E^2 - Δ(r, T)^2}} \right]$$

where $f(E, T)$ is the Fermi function and Δ(r, T) the local gap. The fits also yield the quasiparticle inverse lifetime Γ(r, T), revealing a broadening of the spectral features as T approaches and exceeds Tc, reflecting rapidly decohering superconducting quasiparticles.

Spatial inhomogeneity in gap magnitude is prominent even at the atomic scale; local regions with larger Δ(r, T) sustain a gap to higher temperature, whereas those with smaller Δ(r, T) become gapless just above Tc, indicating nanoscale variation in the local pairing strength.

2. Normal-State Electronic Excitations and Correlated Electron Physics

Well above Tₚ(r), local tunneling spectra revert to a gapless form but retain significant nontrivial energy dependencies and spatial inhomogeneities uncharacteristic of a conventional Fermi liquid metal. Conductance maps display marked electron–hole asymmetry and broad “hump” features centered around −150 to −300 meV, with spatial variation on the scale of about 20 Å.

These high-energy spectral features are tightly correlated with the later development of strong superconducting pairing gaps: areas with more pronounced normal-state anomalies correspond to regions with higher low-temperature Δ(r). Such excitations are characteristic of a doped Mott insulating state with strong electron–electron correlations, not accounted for in standard weak-coupling BCS/Eliashberg paradigms.

This inhomogeneity and persistence of correlated normal-state excitations even above Tc point to a scenario where pre-existing electronic correlations in the normal state “seed” the pairing interaction—an interpretation reinforced by the strong anticorrelation (correlation coefficient ≈ −0.75) between the zero-bias normal-state conductance and low temperature Δ(r).

3. Role of Electron–Boson Coupling in Pairing Excitations

The local tunneling conductance ratio $R(r, V, T)$, defined as the ratio of superconducting to normal-state spectra, exhibits systematic deviations from the simple d-wave thermal broadening model for energies above the pairing gap. Specifically, a pronounced dip appears in the range 50–80 meV, consistent with established tunneling signatures of electron–boson coupling.

Analysis of these deviations across regions with different local Δ suggests that the characteristic boson energy is approximately 35 ± 3 meV. Yet, crucially, the spatial variation in the bosonic coupling strength (quantified via the root-mean-square deviation from the pure d-wave model) does not correlate with local gap magnitude. The energy-dependent gap function is modeled as:

$Δ(r, ω) = Δ_R(r, ω) + i Δ_I(r, ω)$

with the bosonic coupling modifying primarily Δ_I (the imaginary part linked to quasiparticle scattering), but not controlling the essence of the pairing strength encoded in Δ_R(r, ω). The local pairing mechanism is therefore not dictated by changes in electron–boson interaction at the atomic scale.

4. Pairing Excitation Mechanism: Dominance of Correlated Electronic Excitations

The convergence of experimental and analytical evidence supports the conclusion that unusual, high-energy electronic excitations—emergent from strong electron–electron correlations in the normal state—determine the local pairing interaction in Bi₂Sr₂CaCu₂O₈₊δ (0804.2623). This stands in contrast to models invoking variation in low-energy (< 0.1 V) electron–boson coupling as the primary driver for superconductivity.

The spatial and spectral mapping reveals that local suppression of normal-state electronic conductance at the Fermi level (i.e., signatures of stronger correlations) predicts the nucleation of larger pairing gaps at low temperature. Moreover, the distinct way in which pairing gaps close—locally, and not sharply at Tc—implies that the intrinsic properties of highly correlated, spatially inhomogeneous normal-state excitations “precondition” regions for subsequent strong pairing. This points to a mechanism fundamentally tied to the electronic structure arising from proximity to a Mott insulating state.

5. Quantitative and Modeling Aspects: Signature Formulas and Analysis

The main equations employed for quantitative fitting and mechanistic analysis include:

Quantity	Formula/Definition	Context/Purpose
Thermally broadened LDOS	$$N_S(r, V, T) ∝ \int dE [ −∂f(E,T)/∂E ] \mathrm{Re} \left[ |E|/\sqrt{E^2 - Δ(r, T)^2} \right]$$	Fit local STM spectra to extract Δ(r, T), Γ(r, T)
Conductance Ratio (bosonic dip)	$R(r, V, T)$ = SC/NS conductance ratio	Highlights deviation from BCS theory, diagnosis of coupling
Complex, energy-dependent gap	$Δ(r, ω) = Δ_R(r, ω) + i Δ_I(r, ω)$	Incorporates effect of bosonic and high-energy electronic modes
Correlation coefficient Δ/(NS cond)	$r \approx -0.75$	Quantifies anticorrelation between normal-state conductance and gap size
Average boson energy	$\langle E_{boson} \rangle = 35\pm3~\mathrm{meV}$	Local bosonic mode energy influencing R(r, V, T) dip location

The modeling shows that while a complex gap function can capture bosonic renormalizations, the inhomogeneity and strength of pairing are primarily controlled by the underlying high-energy electronic structure.

6. Implications for High-Tc Pairing Theories

These findings provide stringent constraints on microscopic theories of high-Tc pairing. The absence of spatial correlation between bosonic coupling strength and Δ(r) disfavours models that attribute large-scale gap inhomogeneity to phonon or low-energy bosonic mode variations. Instead, a scenario emerges wherein strong correlations in the normal state—manifested as momentum- and energy-dependent spectral features in gapless regions—directly seed and govern the emergent inhomogeneous pairing interaction.

A plausible implication is that attempts to enhance Tc by manipulating lattice or bosonic degrees of freedom are unlikely to surpass the limitations set by the electronic structure inherent to the cuprate family. Instead, understanding and controlling the degree of electronic inhomogeneity and correlation effects at the nanoscale may be key to optimizing high-Tc superconductivity.

7. Outlook and Future Research Directions

The use of atomic-scale, temperature-dependent STM spectroscopy as in (0804.2623) uncovers intrinsic links between normal-state electronic inhomogeneity and the spatial landscape of pairing excitations. Future research may expand these measurements to wider sets of dopings, other cuprate and non-cuprate high-Tc families, and to time-resolved probes testing the dynamical response of pairing modes.

The evidence for pairing excitations rooted in unusual normal-state electronic structure rather than bosonic coupling aligns with theories invoking dominant strong correlation physics—such as RVB models, doped Mott insulator approaches, or emergent Mottness concepts. This paradigm guides a research direction focused on unraveling the microscopic organization and dynamics of correlated electrons as the foundation of high-temperature superconductivity.