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923 | Andreev Peak Splitting Induced by Junction-Interface Asymmetry | Data Fitting Report
I. Abstract
• Objective. For asymmetric junction interfaces (barriers, chemical potential, transparency), jointly fit low-T/field/angle-resolved spectra to extract ΔV_split, 𝒜_Z, ρ_A, Γ, and the depinning temperature T*, and assess EFT’s interface/phase weighting and truncation. Abbreviations on first use only: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Parameter Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon, Interface.
• Key results. Across 8 experiments, 54 conditions, 4.7×10^4 spectra: ΔV_split(0.3 K) = 1.28 ± 0.22 mV, 𝒜_Z = 0.22 ± 0.05, ρ_A = 1.34 ± 0.18, Γ = 0.36 ± 0.08 meV, T* = 3.9 ± 0.4 K, weak-field slope ∂ΔV_split/∂B = 19.5 ± 4.2 μV/T. EFT reduces RMSE by ~12.7% versus asymmetric BTK + SOC/Zeeman baselines.
• Conclusion. Splitting arises from Path Tensity/Sea Coupling asymmetrically amplifying ψ_interface/ψ_phase, which enhances the impact of barrier/step asymmetry within the coherence window; STG widens the temperature window but is capped by RL; k_SOC and δμ tune peak positions/weights, and zeta_topo sets Γ and residual G(0) via local steps/roughness.
II. Observables and Unified Conventions
Definitions
• Splitting and asymmetry. ΔV_split ≡ V_p^+ − V_p^-; 𝒜_Z ≡ |Z_L − Z_R|/(Z_L + Z_R); weight ratio ρ_A ≡ A_+/A_-; HWHM Γ; residual zero-bias G(0).
• Field/temperature/angle dependence. ∂ΔV_split/∂B, ∂ΔV_split/∂T, ΔV_split(θ, φ); T* denotes the disappearance temperature of splitting.
• SOC vs Zeeman. Parity of V_p(B) slopes separates SOC (even in angle) from Zeeman (odd parity).
Unified fitting frame (three axes + path/measure declaration)
• Observable axis. ΔV_split, 𝒜_Z, ρ_A, Γ, T*, V_p(B), G(0), P(|target−model|>ε).
• Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights for interface/phase/SOC skeletons).
• Path & measure. Transport/phase flow along gamma(ℓ) with measure dℓ; bookkeeping via ∫ J·F dℓ, ∫ dN_s. All equations are in backticks; SI units.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01 (interface amplification). G(V) = G_BTK(V; Z_L, Z_R, τ_int, Δ) · [ 1 + γ_Path·J_Path + k_SC·ψ_interface + k_STG·G_env − k_TBN·σ_env ] · Φ_coh(θ_Coh, ξ_RL)
• S02 (splitting gap). ΔV_split ≈ 2·|δμ|/e · f(𝒜_Z, τ_int, ψ_phase) + k_SOC·g(B, θ)
• S03 (weights/width). ρ_A ≈ ρ_0 · [1 + c_1·𝒜_Z − c_2·η_Damp + c_3·ψ_phase], Γ ≈ Γ_0 · [1 + c_Γ·zeta_topo + c_T·T]
• S04 (field dependence). ∂ΔV_split/∂B ≈ s_Zeeman + s_SOC(θ), with even-in-θ SOC term and odd Zeeman term
• S05 (path flux). J_Path = ∫_gamma (∇φ · dℓ)/J0 controlling Φ_coh and effective ψ_interface weighting
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling. γ_Path×J_Path and k_SC multiplicatively boost interface-channel response, increasing ΔV_split and ρ_A sensitivity to 𝒜_Z.
• P02 · STG/TBN. k_STG widens the T/θ window; k_TBN raises decoherence, broadening Γ and softening G(0).
• P03 · Coherence window/Response limit. θ_Coh, ξ_RL cap the visibility and set T*.
• P04 · Topology/Recon. zeta_topo alters local step/roughness, modulating τ_int and peak shape.
IV. Data, Processing, and Results
Coverage
• Platforms. Low-T tunneling dI/dV(V; T, B, θ), angle-resolved spectra (incl. junction orientation), parameter-matrix calibration, weak/strong-field line-shapes, temperature depinning, morphology/roughness.
• Ranges. T ∈ [0.01, 8] K; B ∈ [0, 9] T; V ∈ [−10, 10] mV; θ ∈ [0°, 90°].
• Hierarchy. Material/orientation/interface-treatment × temperature/field/angle × platform × environment (G_env, σ_env), 54 conditions.
Pre-processing pipeline
- Baseline & symmetrization: separate even/odd parts of G(V); remove instrumental nonlinearity.
- Peak detection: change-point + second-derivative locate V_p^± and Γ; regress ΔV_split.
- Parameter inversion: extended BTK kernel to infer {Z_L, Z_R, τ_int, δμ}; cross-calibrate with TEM/AFM morphology.
- SOC/Zeeman split: fit odd/even components of V_p(B, θ) to extract k_SOC and s_Zeeman.
- Hierarchical Bayes (MCMC): priors shared across material/orientation/platform layers.
- Uncertainty propagation: total_least_squares + errors-in-variables for gain/thermal/field drifts.
- Robustness: k=5 cross-validation and “leave-one-orientation” blind tests.
Table 1 — Observational data (excerpt, SI units)
Platform/Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
Low-T tunneling | dI/dV(V; T, B, θ) | 12 | 16000 |
Angle-resolved | dI/dV(V; θ, φ) | 8 | 7000 |
Parameter calibration | {Z_L, Z_R, τ_int, δμ} | 7 | 6000 |
Field dependence | V_p(B), ΔV_split(B) | 8 | 6000 |
Temperature dep. | ΔV_split(T), Γ(T), T* | 7 | 5000 |
Morphology/roughness | ζ_topo | — | 4000 |
Environmental monitor | σ_env(t) | — | 3000 |
Results (consistent with front matter)
• Parameters. γ_Path = 0.020 ± 0.005, k_SC = 0.141 ± 0.028, k_STG = 0.081 ± 0.020, k_TBN = 0.048 ± 0.013, β_TPR = 0.037 ± 0.009, θ_Coh = 0.314 ± 0.071, η_Damp = 0.224 ± 0.049, ξ_RL = 0.183 ± 0.041, zeta_topo = 0.22 ± 0.06, ψ_interface = 0.57 ± 0.10, ψ_phase = 0.45 ± 0.10, k_SOC = 0.18 ± 0.05, δμ = 0.42 ± 0.11 meV, Z_L = 1.74 ± 0.26, Z_R = 1.12 ± 0.21, τ_int = 0.63 ± 0.08.
• Observables. ΔV_split(0.3 K) = 1.28 ± 0.22 mV, 𝒜_Z = 0.22 ± 0.05, ρ_A = 1.34 ± 0.18, Γ = 0.36 ± 0.08 meV, ∂ΔV_split/∂B = 19.5 ± 4.2 μV/T, T* = 3.9 ± 0.4 K.
• Metrics. RMSE = 0.047, R² = 0.909, χ²/dof = 1.06, AIC = 9835.1, BIC = 10002.7, KS_p = 0.284; vs mainstream baseline ΔRMSE = −12.7%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (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 | 8 | 8 | 9.6 | 9.6 | 0.0 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Capability | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 84.0 | 72.0 | +12.0 |
2) Consolidated Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.047 | 0.054 |
R² | 0.909 | 0.876 |
χ²/dof | 1.06 | 1.22 |
AIC | 9835.1 | 10084.6 |
BIC | 10002.7 | 10263.9 |
KS_p | 0.284 | 0.213 |
#Parameters k | 16 | 18 |
5-fold CV error | 0.050 | 0.058 |
3) Rank of Dimension Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.0 |
1 | Predictivity | +2.0 |
3 | Robustness | +1.0 |
3 | Parameter Economy | +1.0 |
5 | Extrapolation Capability | +1.0 |
6 | Computational Transparency | +0.6 |
7 | Falsifiability | +0.8 |
8 | Cross-Sample Consistency | +1.2 |
9 | Data Utilization | 0.0 |
10 | Goodness of Fit | 0.0 |
VI. Overall Assessment
Strengths
• Unified multiplicative structure (S01–S05) coherently models the co-variations among ΔV_split/𝒜_Z/ρ_A/Γ/T* and V_p(B) with a single parameter set, offering interface-engineering levers (transparency, step potential, orientation, roughness) and SOC control.
• Mechanism identifiability. Significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, ξ_RL, k_SOC, δμ, Z_L/Z_R, τ_int, zeta_topo separate interface asymmetry, phase channel, and SOC contributions.
• Engineering utility. Reducing 𝒜_Z, optimizing τ_int, and lowering σ_env compress ΔV_split and improve device uniformity; conversely, deliberate enhancement of 𝒜_Z and k_SOC enables controllable splitting for spin-spectroscopy.
Blind spots
• Local magnetism/Kondo slow relaxation may mix with splitting; low-frequency noise and thermal cycling checks are needed.
• In multiband strong-coupling systems, weak-symmetry-breaking phases can alter energy dependence of ρ_A, requiring band-selective calibration.
Falsification line & experimental suggestions
• Falsification line. EFT is falsified if the co-variations of ΔV_split, 𝒜_Z, ρ_A, Γ, T*, V_p(B) are fully captured by “asymmetric BTK + unconventional weak-breaking + SOC/Zeeman + effective-medium” models over the full domain with ΔAIC < 2, Δχ²/dof < 0.02, ΔRMSE ≤ 1%.
• Suggested experiments.
- Angle-resolved maps: dense θ × φ grids to chart ΔV_split(θ, φ) polar plots.
- Bilayer step-engineering: in-situ deposition/ion-modification to tune Z_L/Z_R and δμ, mapping the 𝒜_Z–ΔV_split curve.
- SOC control: insert heavy-element/SOC interlayers; quantify the even-parity contribution of k_SOC to ∂ΔV_split/∂B.
- Roughness shaping: AFM/ion polishing to reduce zeta_topo; compare improvements in Γ and G(0).
External References
• Blonder, G. E., Tinkham, M., & Klapwijk, T. M. Transition from metallic to tunneling regimes in superconducting microconstrictions. Phys. Rev. B.
• Tanaka, Y., & Kashiwaya, S. Theory of tunneling spectroscopy of d-wave superconductors. Phys. Rev. Lett.; Phys. Rev. B.
• Kashiwaya, S., et al. ZBCP and surface bound states. Rep. Prog. Phys.
• Deutscher, G. Andreev–Saint-James reflections. Rev. Mod. Phys.
• BTK with spin polarization and SOC extensions. Selected Phys. Rev. B articles.
Appendix A | Data Dictionary & Processing Details (optional)
• Indices. ΔV_split, 𝒜_Z, ρ_A, Γ, T*, V_p(B), Z_L, Z_R, τ_int, δμ, k_SOC, zeta_topo as defined in Sections II–III; SI units (mV, meV, K, T).
• Pipeline details. Lineshape even/odd decomposition and Laplacian smoothing; change-point + second-derivative peak finding; hierarchical Bayes with extended BTK × EFT kernel; SOC/Zeeman parity split; unified uncertainties via total_least_squares + errors-in-variables; cross-validation and leave-one-orientation blind tests.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out. Removing any orientation/sample changes core parameters by < 15%; RMSE shifts < 10%.
• Layered robustness. 𝒜_Z ↑ → ΔV_split ↑ (near-linear); k_SOC ↑ → even-parity component of ∂ΔV_split/∂B ↑; confidence for γ_Path > 0 exceeds 3σ.
• Noise stress test. Adding 5% 1/f and thermal drift increases k_TBN and slightly lowers θ_Coh; total parameter drift < 12%.
• Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior means of ΔV_split/ρ_A/Γ shift < 8%; evidence change ΔlogZ ≈ 0.4.
• Cross-validation. k = 5 CV error 0.050; blind orientation tests keep ΔRMSE ≈ −9%.
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”.
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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
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