Diffusive Spin-Triplet Josephson Junction Simulation

Overview

This package implements a comprehensive simulation framework for diffusive spin-triplet Josephson junctions with antiferromagnetically coupled perpendicular magnetization domains and spin-glass interface layers. The framework is designed to model the assymetrical phase shift under sweeping $\vec{B}$ field experimental observations in study of Nb/Fe/Cr multilayer Josephson junctions.

Key Features

1. Diffusive Transport (Usadel Equations)

• Appropriate for sputtered Nb with short mean free path (~5 nm)
• Distinguishes between singlet (short-range) and triplet (long-range) transport
• Computes coherence lengths: $\xi_s \approx 0.08$ nm (singlet), $\xi_T \approx 0.8-50$ nm (triplet)

2. Spin-Glass Interface Modeling

• Models F' layers (0.25 nm each) as frozen spin-glass with random exchange couplings
• Uses Heisenberg Random Exchange Hamiltonian for inter-domain frustration
• Implements Metropolis energy minimization for field-step evolution
• History-dependent magnetization that doesn't fully relax to external field

3. Long-Range Triplet Component (LRTC)

• LRTC amplitude: $\propto \sin(\theta_L) \sin(\theta_R)$ where $\theta_L$, $\theta_R$ are non-collinearity angles
• Maximum triplet current when magnetization layers are perpendicular to bulk Fe
• Decay: $\exp(-d_F/\xi_T)$ showing long-range persistence

4. Spectral Leakage

• Disorder in F' layers prevents complete suppression expected in clean synthetic antiferromagnets (SAF)
• Explains the observation: no suppression in upper field sweep despite antiferromagnetic configuration
• Quantified via leakage_ratio = std(sin theta) / |mean(sin theta)|

5. Magnetic Field Dependence & Hysteresis

• Full Fraunhofer pattern calculation with spatially non-uniform $j_c(x,y)$
• Hysteresis from spin-glass memory effects
• Phase shift $\delta$ extracted from residual magnetization

Project Structure

rcsj_sde/
├── rcsj_sde/
│   ├── diffusive_triplet.py          # Module 1: Geometry, materials, spin-glass
│   ├── fraunhofer_diffusive.py       # Module 3: Fraunhofer integration (Usadel kernel)
│   ├── diffusive_rcsj.py             # Module 4: RCSJ dynamics with B-dependent I_c
│   ├── validation_tools.py           # Module 5: Validation & parameter extraction
│   ├── junction.py                   # Base junction class (original)
│   └── ... (other original files)
├── examples/
│   ├── diffusive_complete_workflow.ipynb  # Full simulation tutorial
│   └── ... (other example notebooks)
├── test_simulation.py                # Validation test script
├── README.md                         # This file
└── requirements.txt                  # Python dependencies

Installation & Setup

1. Create Virtual Environment

cd rcsj_sde
python3 -m venv venv
source venv/bin/activate

2. Install Dependencies

pip install numpy scipy matplotlib scikit-optimize tqdm pandas numba

Or use:

pip install -r requirements.txt

3. Verify Installation

python3 test_simulation.py

Expected output: "[PASS] ALL TESTS PASSED"

Usage

Quick Start: Run Complete Workflow

source venv/bin/activate
jupyter notebook examples/diffusive_complete_workflow.ipynb

Programmatic Usage Example

from rcsj_sde.diffusive_triplet import (
    DiffusiveJunctionGeometry, MaterialParameters, MagneticConfiguration, UsadelKernel
)
from rcsj_sde.validation_tools import spectral_leakage_analysis

# Define junction geometry
geometry = DiffusiveJunctionGeometry(
    d_S=100.0,              # Nb superconductor (nm)
    d_F_prime=0.25,         # Spin-glass interface (nm)
    d_F=10.0,               # Bulk Fe (nm)
    junction_length=1000.0,
    junction_width=1000.0,
    n_grid_x=30
)

# Define material parameters
materials = MaterialParameters(
    Delta=1.5,      # Nb gap (meV)
    E_ex=100.0,     # Fe exchange (meV)
    T=2.0           # Temperature (K)
)

# Create magnetic configuration with spin-glass
mag_config = MagneticConfiguration(geometry, n_spins_per_interface=50)

# Evolve during field sweep
B_sweep = [0, 0.01, 0.02, 0.01, 0]
for B in B_sweep:
    mag_config.evolve_field_step(B, n_relax_steps=100, T_eff=0.05)

# Analyze spectral leakage (protects triplet from SAF suppression)
leakage = spectral_leakage_analysis(mag_config)
print(f"Disorder protected: {leakage['disorder_protected']}")

Physical Model

5-Layer Stack: S/F'/F/F'/S

Layer	Material	Thickness	Role
S (outer)	Nb	~100 nm	Superconductor (BCS gap = 1.5 meV)
F' (left)	Fe/Cr	0.25 nm	Spin-mixer with random exchange
F (bulk)	Fe	~10-15 nm	Ferromagnet with exchange splitting
F' (right)	Fe/Cr	0.25 nm	Spin-mixer with random exchange

Transport Kernel (Houzet-Buzdin)

Local supercurrent density from LRTC:

$$j_c(x,y) = j_0 \left[ \sin(\theta_L) \sin(\theta_R) \cdot e^{-d_F/\xi_T} + \cos(\theta_L) \cos(\theta_R) \cdot e^{-d_F/\xi_s} \right]$$

where:

• $\theta_L$, $\theta_R$ = angles between F' and bulk F magnetization
• $\xi_T$, $\xi_s$ = triplet and singlet coherence lengths
• Triplet term dominates due to much longer range

Fraunhofer Integral

Total critical current from spatial integration:

$$I_c(B) = \left| \int\int j_c(x,y) \cdot e^{i\phi_{flux}(x,y)} dx, dy \right|$$

where $\phi_{flux} = 2\pi \Phi / \Phi_0$ accounts for external and domain-induced magnetization.

Empirical Fit Function

$$I_c(\Phi) = I_{c0} \cdot e^{-d_{Nb}/\xi_T} \cdot \left| \frac{J_1\left(\pi(\Phi-\delta)/\Phi_0\right)}{\pi(\Phi-\delta)/\Phi_0} \right|$$

Extracted Parameters:

• $I_{c0}$: Maximum critical current
• $xi_T$: Triplet coherence length (meV*nm)
• $\delta$: Phase shift from residual F' magnetization (rad)

Key Physics Insights

1. "Missing Suppression" Anomaly

In a clean synthetic antiferromagnet (SAF), the two F' layers would create destructive interference, completely suppressing the supercurrent. However, Li observes no suppression in the upper field sweep.

Explanation from simulation:

• Spin-glass disorder creates broad distribution of magnetization angles
• Some regions have strong LRTC signal ($\sin\theta_L \sin\theta_R > 0$)
• Spatial averaging over disorder protects global triplet current: "spectral leakage"
• Quantified by leakage_ratio parameter

2. Hysteresis from Spin-Glass Memory

The F' layers freeze in history-dependent configurations:

• Up sweep: Spins partially aligned with $+B$
• Down sweep: Spins "lag" behind, partially aligned with previous $+B$
• Consequence: $I_c^{up}(B) \neq I_c^{down}(B)$

3. Phase Shift from Remanent Magnetization

After field sweep, F' layers retain net magnetization despite nominally antiparallel bulk Fe:

• Creates local magnetic flux contribution
• Shifts Fraunhofer pattern by phase $\delta$
• Directly observable in critical current vs. flux curve

Validation & Comparison

Metrics Computed

1. Fraunhofer Pattern

   • Period (in units of $\Phi_0$)
   • Contrast: $(I_{c,max} - I_{c,min}) / I_{c,max}$
   • Asymmetry: left-right asymmetry measure

2. Spectral Leakage

   • LRTC mean and std
   • Leakage ratio (disorder quantifier)
   • Boolean: disorder-protected or not

3. Hysteresis

   • Max difference between up/down sweeps
   • RMS difference
   • Hysteresis loss (area between curves)

4. Fit Quality

   • $\chi^2$
   • $R^2$ coefficient of determination
   • Extracted parameters: $I_{c0}$, $\xi_T$, $\delta$

Comparison with Experiment

The notebook includes code to compare simulation vs experimental data:

from rcsj_sde.validation_tools import compare_simulation_to_experiment

result = compare_simulation_to_experiment(
    B_range_sim, Ic_sim,
    B_range_exp, Ic_exp,
    junction_area
)

print(f"R^2 = {result['r_squared']:.4f}")
print(f"Extracted xi_T = {result['extracted_params']['xi_T']:.1f} nm")
print(f"Extracted delta = {result['extracted_params']['delta']:.4f} rad")

Computational Performance

Task	Time	Notes
Spin-glass init	~0.1 s	Random exchange matrix generation
Single field step	~0.5 s	100 Metropolis steps per point
Full field sweep (50 points)	~25 s	Up and down together
Fraunhofer integral	~2 s	Spatial integration over grid
RCSJ I-V single point	~1 s	5000 time steps
Full I-V map (50 fields, 50 currents)	~2500 s	~45 minutes

Optimization tips:

• Use coarser spatial grids (n_grid_x=10-20) for quick exploration
• Use fewer Metropolis steps (n_relax_steps=50) during development
• Parallelize I-V calculations over field values (not yet implemented)

References

Core Theory

1. Houzet & Buzdin (2007): "Long range triplet Josephson effect through a ferromagnetic trilayer", Phys. Rev. B 76, 060504(R)
2. Bergeret, Volkov, Efetov (2005): "Proximity effects in superconductor-ferromagnet heterostructures", Rev. Mod. Phys. 77, 1321
3. Eschrig (2011): "Spin-polarized supercurrents for spintronics", Physics Today 64 (1), 43

Experimental Context

4. Robinson et al. (2010): "Controlled Injection of Spin-Triplet Supercurrents", Science 329, 59
5. Khaire et al. (2010): "Observation of Spin-Triplet Superconductivity in Co-Based Josephson Junctions", PRL 104, 137002
6. Komori et al. (2021) : "Spin-orbit coupling suppression and singlet-state blocking of spin-triplet Cooper pairs" , Science sciadv.abe0128

Spin-Glass Physics

7. Spin-Glass Theory: Edwards-Anderson Model, Sherrington-Kirkpatrick Model
8. Random Exchange: Frustrated magnetic systems with disorder

Troubleshooting

Import Errors

# Ensure virtual environment is activated
source venv/bin/activate

# Reinstall packages if needed
pip install --upgrade numpy scipy numba

Numerical Issues

Problem: Saturated magnetic layers?

• Check: Are F' layers properly initialized with random orientations?
• Verify: Non-collinearity angles should be distributed across [0, pi]

Last Updated: January 2026
Framework Version: 1.0 (Diffusive Usadel + Spin-Glass)