约瑟夫森$0$结、$\pi$结与$\varphi$结

基本概念

Josephson 结通常由两个超导体通过一个弱连接区域连接而成，例如

$$\mathrm{S/N/S},\qquad \mathrm{S/F/S}$$

其中 S 表示超导体，N 表示普通金属，F 表示铁磁体。左右两个超导体的序参量可以写成

$$\Delta_L=\Delta_0 e^{i\theta_L},\qquad \Delta_R=\Delta_0 e^{i\theta_R}.$$

Josephson 结的核心变量是左右超导体之间的相位差

$$\phi=\theta_R-\theta_L.$$

体系的自由能一般是相位差的周期函数

$$F(\phi)=F(\phi+2\pi).$$

Josephson 电流由自由能对相位差的导数给出

$$I(\phi)=\frac{2e}{\hbar}\frac{\partial F(\phi)}{\partial \phi}.$$

因此，Josephson 结的基态相位由自由能最低点决定。不同类型的 Josephson 结，本质区别就在于$F(\phi)$的最低点位于哪里。

$0$结

最普通的 Josephson 结是$0$结。其自由能最低点位于$\phi=0$，简单有效自由能可以写成

$$F(\phi)=-E_J\cos\phi,$$

其中$E_J>0$，对应的Josephson电流为

$$I(\phi)=I_c\sin\phi.$$

此时体系倾向于让两个超导体保持同相位

$$\theta_L=\theta_R.$$

所以$0$结的物理含义是：左右超导体之间的 Josephson 耦合为正，基态相位差为零。

$\pi$结

$\pi$结的自由能最低点位于$\phi=\pi$，它可以看成 Josephson 耦合常数变号。自由能可以写成

$$F(\phi)=+E_J\cos\phi,\qquad \text{或者}\qquad F(\phi)=-E_J\cos(\phi+\pi).$$

对应电流为

$$I(\phi)=-I_c\sin\phi =I_c\sin(\phi+\pi).$$

因此，$\pi$结中左右两个超导体在基态下更倾向于相差一个$\pi$相位

$$\theta_R-\theta_L=\pi.$$

$\pi$结常见于含有磁性中间层的 Josephson 结，例如 S/F/S 结。在铁磁层中，交换场会使 Cooper 对获得有限动量，从而导致配对振荡。当结长或交换场满足某些条件时，最低能量状态从$0$相转变为$\pi$相。

$\varphi$结

$\varphi$结是更一般的情况，其自由能最低点既不在$0$，也不在$\pi$，而是在

$$\phi=\pm\varphi_0, \qquad 0<\varphi_0<\pi.$$

也就是说，体系自发选择一个非平庸相位差。为了稳定这种相位，通常需要高阶 Josephson 谐波。例如自由能可以写成

$$F(\phi)=-J_1\cos\phi-J_2\cos 2\phi.$$

对应电流为

$$I(\phi)=J_1\sin\phi+2J_2\sin 2\phi.$$

平衡态满足$I(\phi)=0$，除了$\varphi=0,\pi$之外，还可能存在非平庸解

$$\cos\varphi_0=-\frac{J_1}{4J_2}.$$

当该解存在并且对应自由能极小值时，体系就是$\varphi$结。通常需要

$$J_2<0,\qquad |J_1|<4|J_2|.$$

此时基态相位为

$$\phi=\pm\varphi_0.$$

所以$\varphi$结的物理含义是：左右超导体既不倾向于同相，也不倾向于反相，而是自发形成一个介于$0$和$$pi之间的相位差。

$0-\pi$转变

在许多 Josephson 结中，改变中间区长度、交换场强度、化学势或温度等参数，会导致基态从$0$相变为$\pi$相。这个过程称为$0-\pi$转变。在有效模型中，可以理解为第一谐波系数 $J_1$变号

$$F(\phi)=-J_1\cos\phi.$$

当$J_1>0$，基态为$0$结；当$J_1<0$，基态为$\pi$结。在转变点附近$J_1\approx 0$，第一谐波被压低，此时第二谐波$J_2$可能变得重要，从而有机会形成$\varphi$结。

代码模拟

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

"""
Effective Josephson-junction model for 0, pi, and phi junctions.

Free energy:
    F(phi) = -J1 cos(phi) - J2 cos(2 phi)

Current-phase relation:
    I(phi) = dF/dphi = J1 sin(phi) + 2 J2 sin(2 phi)

The code saves:
    1. data/*.dat
    2. figures/free_energy_three_junctions.png/pdf
    3. figures/cpr_three_junctions.png/pdf
"""

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import MaxNLocator
from pathlib import Path


# =========================================================
# 0. Global switches
# =========================================================
DO_CALCULATE_AND_SAVE = True
DO_PLOT_FROM_DATA = True


# =========================================================
# 1. Output directories
# =========================================================
ROOT_DIR = Path("josephson_cpr_demo")
DATA_DIR = ROOT_DIR / "data"
FIG_DIR = ROOT_DIR / "figures"

DATA_DIR.mkdir(parents=True, exist_ok=True)
FIG_DIR.mkdir(parents=True, exist_ok=True)


# =========================================================
# 2. Plot style
# =========================================================
def set_plot_style():
    plt.rcParams.update({
        "figure.dpi": 120,
        "savefig.dpi": 300,
        "font.family": "Times New Roman",
        "mathtext.fontset": "stix",
        "font.size": 15,
        "axes.labelsize": 18,
        "axes.linewidth": 1.25,
        "xtick.labelsize": 15,
        "ytick.labelsize": 15,
        "legend.fontsize": 13,
        "lines.linewidth": 2.8,
        "xtick.direction": "in",
        "ytick.direction": "in",
        "xtick.top": True,
        "ytick.right": True,
        "xtick.major.size": 5,
        "ytick.major.size": 5,
        "xtick.major.width": 1.15,
        "ytick.major.width": 1.15,
        "figure.facecolor": "white",
        "axes.facecolor": "white",
    })


# =========================================================
# 3. Model definitions
# =========================================================
def free_energy(phi, J1, J2):
    """
    F(phi) = -J1 cos(phi) - J2 cos(2 phi)
    """
    return -J1 * np.cos(phi) - J2 * np.cos(2.0 * phi)


def josephson_current(phi, J1, J2):
    """
    I(phi) = dF/dphi = J1 sin(phi) + 2 J2 sin(2 phi)
    """
    return J1 * np.sin(phi) + 2.0 * J2 * np.sin(2.0 * phi)


def find_ground_state_phase(phi, F):
    """
    Return all numerically degenerate minima within a tolerance.
    """
    F_min = np.min(F)
    tol = 1e-4
    idx = np.where(np.abs(F - F_min) < tol)[0]

    # Collect separated minima only
    minima = []
    for i in idx:
        if not minima:
            minima.append(phi[i])
        elif abs(phi[i] - minima[-1]) > 0.02:
            minima.append(phi[i])

    return np.array(minima), F_min


# =========================================================
# 4. Parameters
# =========================================================
N_PHI = 2001
PHI = np.linspace(-np.pi, np.pi, N_PHI)

# Parameter sets:
# 0 junction:       minimum at phi = 0
# pi junction:      minimum at phi = pi
# phi junction:     minima at phi = ±phi0
#
# For the phi junction chosen below:
#   cos(phi0) = -J1 / (4 J2) = 1/2
#   phi0 = pi/3
CASES = {
    "0_junction": {
        "label": r"$0$ junction",
        "J1": 1.00,
        "J2": 0.10,
        "color": "#0072B2",
        "linestyle": "-",
        "marker": "o",
    },
    "pi_junction": {
        "label": r"$\pi$ junction",
        "J1": -1.00,
        "J2": 0.10,
        "color": "#D55E00",
        "linestyle": "--",
        "marker": "s",
    },
    "phi_junction": {
        "label": r"$\varphi$ junction",
        "J1": 0.40,
        "J2": -0.20,
        "color": "#CC79A7",
        "linestyle": "-.",
        "marker": "^",
    },
}


# =========================================================
# 5. Calculation and save data
# =========================================================
def calculate_and_save():
    summary_lines = []
    summary_lines.append("# case  J1  J2  phi_min/pi")

    for key, cfg in CASES.items():
        J1 = cfg["J1"]
        J2 = cfg["J2"]

        F = free_energy(PHI, J1, J2)
        I = josephson_current(PHI, J1, J2)
        F_shift = F - np.min(F)

        minima, F_min = find_ground_state_phase(PHI, F)

        data = np.column_stack([
            PHI,
            PHI / np.pi,
            F,
            F_shift,
            I,
        ])

        header = (
            "phi   phi_over_pi   F_phi   F_phi_minus_Fmin   I_phi\n"
            f"J1 = {J1:.12f}, J2 = {J2:.12f}"
        )

        np.savetxt(DATA_DIR / f"{key}.dat", data, header=header)

        for phimin in minima:
            summary_lines.append(
                f"{key:16s}  {J1:+.8f}  {J2:+.8f}  {phimin / np.pi:+.8f}"
            )

    with open(DATA_DIR / "summary.txt", "w", encoding="utf-8") as f:
        f.write("\n".join(summary_lines))

    print("Calculation finished.")
    print(f"Data saved to: {DATA_DIR}")


# =========================================================
# 6. Helper functions for plotting
# =========================================================
def set_three_ticks(ax, x=True, y=True):
    """
    Limit the number of major ticks to at most three.
    """
    if x:
        ax.set_xticks([-1.0, 0.0, 1.0])
        ax.set_xticklabels([r"$-\pi$", r"$0$", r"$\pi$"])

    if y:
        ax.yaxis.set_major_locator(MaxNLocator(nbins=3))


def load_case_data(key):
    path = DATA_DIR / f"{key}.dat"
    if not path.exists():
        raise FileNotFoundError(f"Missing data file: {path}")
    return np.loadtxt(path)


def mark_minima_on_energy(ax, phi_over_pi, F_shift, cfg):
    """
    Mark the free-energy minima.
    """
    min_val = np.min(F_shift)
    tol = 1e-4
    idx = np.where(np.abs(F_shift - min_val) < tol)[0]

    # Reduce nearby points to separated markers
    selected = []
    for i in idx:
        if not selected:
            selected.append(i)
        elif abs(phi_over_pi[i] - phi_over_pi[selected[-1]]) > 0.03:
            selected.append(i)

    ax.scatter(
        phi_over_pi[selected],
        F_shift[selected],
        s=45,
        marker=cfg["marker"],
        color=cfg["color"],
        edgecolor="black",
        linewidth=0.6,
        zorder=5,
    )


# =========================================================
# 7. Plot free energy
# =========================================================
def plot_free_energy():
    fig, ax = plt.subplots(figsize=(5.6, 4.2))

    for key, cfg in CASES.items():
        data = load_case_data(key)

        phi_over_pi = data[:, 1]
        F_shift = data[:, 3]

        ax.plot(
            phi_over_pi,
            F_shift,
            color=cfg["color"],
            linestyle=cfg["linestyle"],
            linewidth=3.0,
            label=cfg["label"],
        )

        mark_minima_on_energy(ax, phi_over_pi, F_shift, cfg)

    ax.set_xlabel(r"$\phi$")
    ax.set_ylabel(r"$F(\phi)-F_{\min}$")

    ax.set_xlim(-1.0, 1.0)
    ax.set_ylim(bottom=-0.03)

    set_three_ticks(ax, x=True, y=True)

    ax.legend(frameon=True, fancybox=False, edgecolor="black", loc="best")

    # ax.text(
    #     0.03,
    #     0.93,
    #     r"$F(\phi)=-J_1\cos\phi-J_2\cos2\phi$",
    #     transform=ax.transAxes,
    #     ha="left",
    #     va="top",
    #     fontsize=14,
    # )

    fig.tight_layout()
    fig.savefig(FIG_DIR / "free_energy_three_junctions.png", bbox_inches="tight")
    fig.savefig(FIG_DIR / "free_energy_three_junctions.pdf", bbox_inches="tight")
    plt.show()


# =========================================================
# 8. Plot current-phase relation
# =========================================================
def plot_cpr():
    fig, ax = plt.subplots(figsize=(5.6, 4.2))

    for key, cfg in CASES.items():
        data = load_case_data(key)

        phi_over_pi = data[:, 1]
        I = data[:, 4]

        ax.plot(
            phi_over_pi,
            I,
            color=cfg["color"],
            linestyle=cfg["linestyle"],
            linewidth=3.0,
            label=cfg["label"],
        )

    ax.axhline(0.0, color="black", linewidth=1.0, linestyle=":")

    ax.set_xlabel(r"$\phi$")
    ax.set_ylabel(r"$I(\phi)$")

    ax.set_xlim(-1.0, 1.0)

    set_three_ticks(ax, x=True, y=True)

    ax.legend(frameon=True, fancybox=False, edgecolor="black", loc="best")

    # ax.text(
    #     0.03,
    #     0.93,
    #     r"$I(\phi)=J_1\sin\phi+2J_2\sin2\phi$",
    #     transform=ax.transAxes,
    #     ha="left",
    #     va="top",
    #     fontsize=14,
    # )

    fig.tight_layout()
    fig.savefig(FIG_DIR / "cpr_three_junctions.png", bbox_inches="tight")
    fig.savefig(FIG_DIR / "cpr_three_junctions.pdf", bbox_inches="tight")
    plt.show()


# =========================================================
# 9. Print physical interpretation
# =========================================================
def print_summary():
    print("\nGround-state phase summary")
    print("=" * 72)

    for key, cfg in CASES.items():
        data = load_case_data(key)

        phi = data[:, 0]
        phi_over_pi = data[:, 1]
        F = data[:, 2]

        minima, _ = find_ground_state_phase(phi, F)

        J1 = cfg["J1"]
        J2 = cfg["J2"]

        print(f"{cfg['label']:18s}: J1 = {J1:+.3f}, J2 = {J2:+.3f}")

        if key == "0_junction":
            print("    Expected ground state: phi = 0")
        elif key == "pi_junction":
            print("    Expected ground state: phi = pi")
        elif key == "phi_junction":
            if J2 < 0 and abs(J1) < 4.0 * abs(J2):
                phi0 = np.arccos(-J1 / (4.0 * J2))
                print(f"    Expected ground states: phi = ±phi0, phi0/pi = {phi0 / np.pi:.4f}")

        print("    Numerical minima:")
        for phimin in minima:
            print(f"        phi/pi = {phimin / np.pi:+.4f}")

    print("=" * 72)
    print(f"Figures saved to: {FIG_DIR}")


# =========================================================
# 10. Main
# =========================================================
def main():
    set_plot_style()

    if DO_CALCULATE_AND_SAVE:
        calculate_and_save()

    if DO_PLOT_FROM_DATA:
        plot_free_energy()
        plot_cpr()
        print_summary()


if __name__ == "__main__":
    main()

参考文献

1. Disorder-Induced Phase Transitions in Altermagnetic Josephson Junctions

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