Josephson phase shift and diode effect due to the inverse spin Hall effect

This paper theoretically demonstrates that a superconductor-normal metal-superconductor junction with inversion-symmetric spin-orbit coupling can exhibit a Josephson phase shift and a superconducting diode effect via the inverse spin Hall effect induced by a spatially inhomogeneous magnetic field, without requiring broken structural inversion symmetry.

The Gist

The Big Picture: A Superconducting "One-Way Street"

Imagine a superconductor as a highway where cars (electrons) can drive forever without ever slowing down or burning fuel. This is the superconducting state. Usually, if you drive this highway in the morning (one direction) or at night (the other direction), the rules are exactly the same. It's perfectly symmetrical.

However, scientists are trying to build a Superconducting Diode. Think of a diode as a traffic gate that lets cars flow easily in one direction but blocks them or makes them work much harder in the other. If you can build this with superconductors, you could create ultra-fast, zero-energy computers and memory devices.

For a long time, scientists thought you needed a very specific, "broken" structure (like a lopsided road) to make this one-way street. This paper says: "No, you don't need a broken road. You just need a clever magnetic trick."

---

The Cast of Characters

1. The Junction (The Bridge): Imagine a bridge made of normal metal connecting two islands of superconductors. This is called an SNS Junction (Superconductor-Normal-Superconductor).
2. The Spin Hall Effect (The Spin-Sorting Machine): In normal metals, if you push electrons through a pipe, some spin left and some spin right, piling up at the edges. This paper shows that even in a superconducting bridge, if you push a "super-current," it sorts the spins just like a bouncer at a club, sending left-spinners to one side and right-spinners to the other.
3. The Inverse Spin Hall Effect (The Magnetic Push): This is the reverse. If you create a magnetic field that changes strength across the bridge (stronger on the left, weaker on the right), it pushes the electrons to create a current.

The Magic Trick: The "Phase Shift"

In the world of superconductors, the "flow" of electricity isn't just about how many cars are moving; it's about their synchronization, called the phase. Think of the supercurrent as a marching band.

• Normal March: The band marches in perfect lockstep.
• The Anomalous Phase Shift: The paper shows that if you apply a specific magnetic gradient (a magnetic field that gets stronger as you move across the bridge), it acts like a conductor waving a baton. It tells the band to start marching a tiny bit earlier or later than usual.

This shift is the key. It's like telling the band, "Start marching when the clock says 12:01 instead of 12:00."

Why This Creates a "Diode" (The One-Way Street)

Here is where the magic happens. The authors realized that this "phase shift" alone isn't enough to make a one-way street. It just shifts the timing.

However, in real life, bridges aren't perfect. There are bumps, turns, and reflections (called higher harmonics). When you combine the magnetic phase shift with these bumps, the road becomes lopsided.

• Going Forward: The magnetic push helps the band march over the bumps. It's easy.
• Going Backward: The magnetic push fights against the bumps. It's hard.

Suddenly, the "critical current" (the maximum speed the band can go before they trip and stop) is different depending on which way they are marching. You have created a Superconducting Diode.

The Big Breakthrough: No "Broken Symmetry" Needed

Usually, to make a one-way street, you have to build the road asymmetrically (e.g., using materials that are inherently "left-handed" or "right-handed"). This is called breaking inversion symmetry.

This paper's discovery is revolutionary because:

• The Road is Perfect: The metal bridge is perfectly symmetrical. It looks the same from the left or the right.
• The Trick is External: The "one-way" nature comes entirely from the magnetic field gradient (the magnetic gradient acts like a wind blowing harder on one side).
• The Result: The magnetic field breaks the symmetry for the electrons, even though the road itself is perfectly straight.

Real-World Analogy: The Treadmill and the Fan

Imagine a treadmill (the superconductor) that is perfectly balanced.

1. Normal Mode: If you walk forward or backward, it feels the same.
2. The Spin Hall Effect: If you run fast, the air pressure builds up on one side of the room (spin accumulation).
3. The Inverse Effect: Now, imagine you have a giant fan blowing air across the treadmill, but the fan is stronger on the left side than the right.
4. The Diode Effect: If you walk with the wind gradient, the air helps you push off the belt. If you walk against it, the air pushes you back. Even though the treadmill belt is perfectly symmetrical, the wind gradient makes it much easier to go one way than the other.

Why Does This Matter?

1. New Materials: We don't need to find weird, exotic materials with broken structures. We can use standard, clean materials and just apply a magnetic field.
2. Better Control: We can turn the "diode" on and off just by turning a magnet on or off, or moving it closer.
3. Future Tech: This opens the door to "Spintronic" computers that use the spin of electrons (like a tiny compass needle) instead of just their charge. These computers would be faster, use less energy, and could store data without needing power (non-volatile).

In a nutshell: The authors found a way to make superconductors act like one-way streets using a magnetic gradient, without needing to build a weird, lopsided structure. It's like making a perfectly straight road act like a one-way street just by blowing a specific wind pattern across it.

Technical Summary

1. Problem Statement

The paper addresses the mechanism behind the superconducting diode effect (non-reciprocal critical current) in Superconductor–Normal metal–Superconductor (SNS) Josephson junctions.

• Context: While the diode effect is well-known in systems with broken structural inversion symmetry (e.g., Rashba spin-orbit coupling systems) via the inverse Rashba-Edelstein effect, the authors investigate whether this effect can arise in systems that preserve spatial inversion symmetry.
• Gap: Conventional theories often rely on structural asymmetry. The authors aim to demonstrate that a spatially inhomogeneous magnetic field, which generates a steady spin current, can induce a diode effect even in inversion-symmetric systems by breaking inversion symmetry dynamically through the spin current itself.

2. Methodology

The authors employ a theoretical framework based on Keldysh Green's function formalism (path-ordered) to analyze an SNS junction.

• System Model:
  • A normal metal (N) region with random impurities possessing spin-orbit (SO) interaction that preserves both inversion and time-reversal symmetries.
  • Coupled to two conventional singlet superconductors (L and R) via tunneling.
• Theoretical Approach:
  • Direct Spin Hall Effect (SHE): Calculated the spin accumulation induced by a supercurrent (SC) using the response function between spin density and electric current. They utilize a diagrammatic expansion involving impurity scattering and Cooperon propagation (diffusive regime).
  • Inverse Spin Hall Effect (ISHE): Calculated the supercurrent induced by a spatially inhomogeneous static magnetic field ($B_s$) coupling to electron spins.
  • Harmonic Analysis: To derive the diode effect, the authors go beyond the linear response. They calculate higher-order contributions (specifically the fourth-order in self-energy) to capture higher harmonics in the current-phase relation, which are essential for breaking the symmetry of the phase potential.
• Validation: The analytical results were cross-verified using numerical simulations on a ring superconductor with a normal metal gap containing random SO interaction.

3. Key Contributions

1. ISHE in Inversion-Symmetric SNS Junctions: The paper establishes that the Inverse Spin Hall Effect can induce a supercurrent phase shift in SNS junctions even when the crystal structure possesses inversion symmetry.
2. Mechanism of Symmetry Breaking: The authors clarify that while the material is inversion-symmetric, the steady spin current generated by a spatially inhomogeneous magnetic field breaks both inversion and time-reversal symmetries, which is the necessary condition for the diode effect.
3. Role of Higher Harmonics: The paper rigorously demonstrates that a static anomalous phase shift alone does not create a diode effect (as it merely shifts the potential minimum). The diode effect emerges only when higher harmonics (e.g., $\sin(2\phi)$) arising from multiple interface scattering are included, creating an asymmetric potential barrier for positive and negative currents.
4. Distinction from Rashba Systems: Unlike the Rashba-Edelstein effect which requires structural asymmetry and reacts to uniform fields, the ISHE mechanism relies on the gradient of the magnetic field (spatial asymmetry of the field).

4. Key Results

A. Supercurrent-Induced Spin Hall Effect

• A supercurrent flowing through the SNS junction induces a static spin accumulation at the edges of the normal metal with opposite polarizations.
• The spin density $s(r)$ is proportional to the curl of the Josephson current density:
  $$s(r) = \lambda_{sh} (\nabla \times j_J)$$
  where $\lambda_{sh}$ is the spin Hall coefficient for the superconductor.

B. Inverse Spin Hall Effect and Phase Shift

• An inhomogeneous static magnetic field $B_s$ induces a spin current, which couples to the superconducting channel, generating an anomalous phase shift $\delta\phi$.
• The induced current is given by:
  $$j_{ish} = \gamma \lambda_{ish} (\nabla \times B_s) \cos \phi$$
• This results in a total current-phase relation:
  $$j = j_{c2} \sin(\phi + \delta\phi)$$
  where $\delta\phi \propto \nabla B_s$. This confirms that the superconductor adjusts its phase to absorb the inverse spin current.

C. The Diode Effect

• Including the fourth-order contribution (higher harmonics), the total current becomes:
  $$j = j_{c2} \sin(\phi + \delta\phi) + j_{c4} \sin(2\phi + \delta\phi^{(4)})$$
• This creates an asymmetric potential energy landscape $E(\phi)$. When a current is applied, the critical current for positive ($I_{c+}$) and negative ($I_{c-}$) directions differs.
• Quantitative Estimates:
  • In the tunneling regime, the simulation showed a diode effect of approximately 1%.
  • In the metallic regime (where higher harmonics are more prominent), the diode effect was enhanced to approximately 18% for specific parameter sets ($B_s/t_0 = 0.16$).

5. Significance

• New Control Mechanism: The work proposes a novel method for controlling Josephson junctions using equilibrium spin currents generated by proximate magnetic textures (e.g., magnetic domain walls, magnetic dots) or field gradients, without requiring structurally asymmetric interfaces.
• Fundamental Physics: It resolves the ambiguity regarding the necessity of structural inversion symmetry for the superconducting diode effect, showing that dynamic symmetry breaking via spin currents is sufficient.
• Applications: The findings open pathways for:
  • Dissipationless Memory and Logic: Creating non-volatile superconducting diodes.
  • Spin Current Detection: Providing a new route to detect both equilibrium and non-equilibrium spin currents (e.g., via spin pumping) by measuring the diode effect in SNS junctions.
• Experimental Feasibility: The authors suggest practical setups, such as placing a magnetic dot near the edge of the normal metal or utilizing domain walls, to generate the necessary field gradients to observe this effect.

In summary, this paper theoretically establishes that the Inverse Spin Hall Effect, driven by magnetic field gradients, is a robust mechanism for inducing the superconducting diode effect in inversion-symmetric SNS junctions, offering a versatile tool for future spintronic and superconducting devices.