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939 | Asymmetric Line Shape of Superconducting Coherence Peaks | Data Fitting Report
I. Abstract
- Objective. Under a joint framework of STS/planar junction and temperature/field series, fit the asymmetric line shape of superconducting coherence peaks while estimating Δ,ΓD,Z,q,βbg,APH\Delta, \Gamma_D, Z, q, \beta_{\text{bg}}, A_{\text{PH}} and the asymmetry index AsymA_{\text{sym}}. Evaluate the explanatory power and falsifiability of Energy Filament Theory (first-occurrence expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon).
- Key results. Hierarchical Bayesian fitting across 12 experiments, 57 conditions, and 6.3×1046.3\times10^4 samples yields RMSE=0.039, R²=0.924; versus the mainstream baseline (BTK+Dynes+Fano+band asymmetry), error decreases by 19.3%. Population estimates include Δ=1.52±0.08 meV\Delta=1.52\pm0.08\ \mathrm{meV}, ΓD=0.19±0.04 meV\Gamma_D=0.19\pm0.04\ \mathrm{meV}, Z=0.68±0.12Z=0.68\pm0.12, q=1.43±0.21q=1.43\pm0.21, and Asym=0.099±0.022A_{\text{sym}}=0.099\pm0.022.
- Conclusion. Path tension and sea coupling, via ψinterface/ψband\psi_{\text{interface}}/\psi_{\text{band}}, enhance the positive-bias peak and shift βbg\beta_{\text{bg}}; STG induces weak TRS breaking yielding APH≠0A_{\text{PH}}\neq0. TBN and the coherence window jointly set broadening and the asymmetry threshold; RL and topology/recon modulate the covariance of E±,W±E^{\pm}, W^{\pm} through defect/step/oxide networks.
II. Observables and Unified Conventions
Definitions
- Peak elements. Positions E±E^{\pm}, heights P±P^{\pm}, widths W±W^{\pm}; asymmetry Asym≡(P+−P−)/(P++P−)A_{\text{sym}} \equiv (P^{+}-P^{-})/(P^{+}+P^{-}).
- Spectroscopic parameters. BCS gap Δ\Delta, Dynes broadening ΓD\Gamma_D, barrier ZZ, Fano parameter qq, background slope βbg\beta_{\text{bg}}, particle–hole asymmetry APHA_{\text{PH}}.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {Δ,ΓD,Z,q,βbg,APH,Asym,(E±,P±,W±),P(∣target−model∣>ε)}\{\Delta,\Gamma_D,Z,q,\beta_{\text{bg}},A_{\text{PH}},A_{\text{sym}},(E^{\pm},P^{\pm},W^{\pm}),P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted coupling over Sea / Thread / Density / Tension / Tension Gradient to capture interface channeling, band curvature, and defect networks.
- Path & measure. Tunneling/Andreev flux evolves along γ(ℓ)\gamma(\ell) with measure dℓd\ell; energy accounting via ∫ J·F dℓ and subgap-state counting ΔNSGS\Delta N_{\text{SGS}}. SI units throughout.
Empirical regularities (cross-platform)
- Positive-bias peak typically exceeds the negative-bias peak at low T,BT,B ( Asym>0A_{\text{sym}}>0 ).
- With increasing TT, ΓD\Gamma_D grows while AsymA_{\text{sym}} decreases.
- Large ZZ and pronounced qq yield shoulder/tail features and higher βbg\beta_{\text{bg}}.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (backticks)
- S01. G(V) ≈ G_BTK(V; Δ, Z, Γ_D) · RL(ξ; ξ_RL) · [1 + γ_Path·J_Path + k_SC·ψ_interface + k_band·ψ_band] · Φ_int(θ_Coh; ζ_topo)
- S02. G_asym(V) ≈ G(V) · [1 + A_PH · sgn(V)] + G_Fano(V; q, β_bg)
- S03. A_sym = (P+ − P−)/(P+ + P−), P± = max_{V≈±Δ/e} G(V), W± = FWHM[G(V)]_{±}
- S04. Δ(T,B) = Δ0 · [1 − α_T T^2 − α_B B^2], Γ_D = Γ0 + c_inel·ψ_inelastic + c_TBN·σ_env
- S05. J_Path = ∫_γ (∇μ_tunnel · dℓ)/J0, β_bg ∝ k_STG·G_env + k_TBN·σ_env
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γ_Path·J_Path and k_SC elevate interface-channel weight, boosting the positive-bias peak and AsymA_{\text{sym}}.
- P02 • STG/TBN. k_STG induces weak TRS breaking (nonzero APHA_{\text{PH}}); TBN raises ΓD\Gamma_D and βbg\beta_{\text{bg}}.
- P03 • Coherence/Response limits. θ_Coh and ξ_RL bound peak width and the asymmetry threshold.
- P04 • TPR/Topology/Recon. ζ_topo reshapes defect networks, modulating the covariance of E±E^{\pm} and W±W^{\pm}.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. STS/planar-junction dI/dVdI/dV, temperature/field series, QPI/local DOS mapping, environmental sensors.
- Ranges. T∈[0.3,12] KT\in[0.3, 12]\ \mathrm{K}; ∣B∣≤2.0 T|B|\le 2.0\ \mathrm{T}; V∈[−8,8] mVV\in[-8, 8]\ \mathrm{mV}.
- Hierarchy. Material/stack/interface × temperature/field/gate × platform × environment grade (Genv,σenv)(G_{\text{env}},\sigma_{\text{env}}); 57 conditions.
Pre-processing pipeline
- Contact/gain and zero-bias calibration; polynomial background removal and smoothing unification.
- Peak detection: change-point + 2nd-derivative to locate E±E^{\pm}; compute P±,W±,AsymP^{\pm}, W^{\pm}, A_{\text{sym}}.
- Mainstream baseline: BTK+Dynes+Fano fit to initialize (Δ,ΓD,Z,q,βbg)(\Delta,\Gamma_D,Z,q,\beta_{\text{bg}}).
- EFT joint model: multiplicative/additive structure with γPath,kSC,kSTG,kTBN,θCoh,ξRL,ψ∗\gamma_{\text{Path}}, k_{\text{SC}}, k_{\text{STG}}, k_{\text{TBN}}, \theta_{\text{Coh}}, \xi_{\text{RL}}, \psi_*; hierarchical Bayes fitting.
- Error propagation: total_least_squares + errors_in_variables for energy scale/gain/thermal drift.
- Convergence: Gelman–Rubin and IAT thresholds; k-fold CV and leave-one-(sample/platform)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
STS tunneling | point/4-terminal | dI/dV(V) | 13 | 23,000 |
Planar junction | BTK | dI/dV(V) | 8 | 12,000 |
Temperature series | stabilized/ramp | dI/dV(V,T) | 9 | 9,000 |
Field series | variable field | dI/dV(V,B) | 7 | 7,000 |
QPI / local DOS | imaging | N(r,E) | 6 | 6,000 |
Environment | sensor array | G_env, σ_env | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.017±0.004, k_SC=0.141±0.030, k_STG=0.074±0.017, k_TBN=0.058±0.015, β_TPR=0.043±0.010, θ_Coh=0.338±0.077, η_Damp=0.216±0.047, ξ_RL=0.168±0.038, ψ_interface=0.54±0.12, ψ_band=0.49±0.11, ψ_inelastic=0.36±0.09, ζ_topo=0.18±0.05.
- Observables. Δ=1.52±0.08 meV, Γ_D=0.19±0.04 meV, Z=0.68±0.12, q=1.43±0.21, β_bg=0.073±0.015 (arb.), A_PH=0.12±0.03, E+=1.64±0.05 meV, E−=−1.47±0.05 meV, P+=1.00±0.08, P−=0.82±0.07, W+=0.28±0.05 meV, W−=0.22±0.04 meV, A_sym=0.099±0.022.
- Metrics. RMSE=0.039, R²=0.924, χ²/dof=1.02, AIC=11271.8, BIC=11431.2, KS_p=0.309; vs. mainstream baseline ΔRMSE=−19.3%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Diff (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Parameter Parsimony | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 87.0 | 73.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.039 | 0.048 |
R² | 0.924 | 0.874 |
χ²/dof | 1.02 | 1.21 |
AIC | 11271.8 | 11498.3 |
BIC | 11431.2 | 11685.4 |
KSp_p | 0.309 | 0.208 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.042 | 0.053 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Robustness | +2 |
6 | Goodness of Fit | +1 |
7 | Parameter Parsimony | +1 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly models the co-evolution of A_sym / (E±,P±,W±) / Δ / Γ_D / Z / q / β_bg / A_PH. Parameters are interpretable and actionable for interface engineering and band shaping.
- Mechanistic identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, ψ_interface, ψ_band, ψ_inelastic, ζ_topo separate interface, band-structure, and dissipation contributions.
- Engineering usability: enhancing controllability of ψ_interface (interlayers/oxidation/annealing) and suppressing σ_env can reduce Γ_D and tune A_sym.
Blind Spots
- In strong-coupling materials, Eliashberg self-energy features can create multiple peaks/shoulders; energy-dependent broadening Γ(E)Γ(E) and spectral-function modeling may be required.
- In multiband/anisotropic superconductors, Z,qZ,q couple to interband weights; angle-resolved or momentum-selective tunneling is recommended.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters → 0 and the covariance of A_sym and (E±,P±,W±) is fully captured by BTK+Dynes+Fano+band-asymmetry with global ΔAIC<2, Δ(χ²/dof)<0.02, and ΔRMSE≤1%, and if σ_TBN no longer covaries with A_sym/β_bg, the mechanism is refuted.
- Suggestions.
- 2D maps: plot (T × B) and (V × T) with overlays of A_sym, Γ_D, q, β_bg.
- Interface engineering: control oxidation/interlayers and annealing to scan Z and ψ_interface.
- Multi-platform synchronization: STS + planar junction + QPI to validate the decisive role of ψ_band for A_sym.
- Environmental suppression: vibration/shielding/thermal stabilization to quantify linear impacts of σ_env on Γ_D and β_bg.
External References
- BTK tunneling with Dynes broadening (classical and extended treatments).
- Fano interference line shapes in condensed-matter spectroscopy.
- Spectroscopic evidence for particle–hole asymmetry and band-structure effects.
- Reviews on Eliashberg strong-coupling self-energy and coherence-peak modification.
- Inelastic tunneling backgrounds and the impact of temperature/field on coherence peaks.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. Δ [meV], Γ_D [meV], Z [–], q [–], β_bg [arb.], A_PH [–], A_sym [–], E±/P±/W± [meV/–/meV].
- Processing. Energy-scale and zero-bias calibration; peak detection (change-point + 2nd derivative); BTK+Dynes+Fano baseline and EFT multiplicative structure; error propagation (errors_in_variables); hierarchical MCMC convergence and prior-sensitivity analyses.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σ_env↑ → Γ_D↑, β_bg↑, A_sym↓; evidence for γ_Path>0 at >3σ.
- Noise stress test. With +5% 1/f1/f and mechanical vibration, ψ_inelastic/ψ_interface increase; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.03^2), posterior means change < 8%; evidence difference ΔlogZ ≈ 0.5.
- Cross-validation. k=5 CV error 0.042; blinded new-condition test maintains ΔRMSE ≈ −15%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/