Long-range spin-polarized Josephson effect in ballistic S/F/S junctions with precessing magnetization

This paper presents a theory for ballistic S/F/S junctions with precessing magnetization, demonstrating that the resulting non-equilibrium distribution of Andreev bound states generates long-range equal-spin superconducting correlations and a non-sinusoidal Josephson current, enabling a fully polarized half-metal junction to switch from an insulating "off" state to a conducting "on" state.

The Gist

The Big Picture: A Superconducting "Traffic Jam" That Needs a Spin

Imagine a superconductor as a super-highway where cars (electrons) can drive without any friction or traffic jams. Usually, these cars travel in pairs (called Cooper pairs), holding hands and moving in perfect sync.

Now, imagine you put a magnet in the middle of this highway. In a normal magnet, the cars are forced to drive in opposite lanes based on their "spin" (a quantum property like a tiny compass needle). Because the magnet pushes them apart, the pairs break up, and the frictionless traffic stops. This is why you usually can't send a supercurrent through a magnet.

The Twist: This paper proposes a clever trick. What if we make the magnet wobble (precess)? Imagine the magnet isn't a solid wall, but a spinning top that is tilted and wobbling as it spins.

The authors show that this wobbling acts like a magic bridge. It allows the electron pairs to cross the magnet without breaking up, creating a "long-range" supercurrent where none should exist.

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The Key Concepts (Translated)

1. The "Half-Metal" Switch (The Off/On Button)

The paper focuses heavily on a special type of magnet called a Half-Metal.

• The Analogy: Imagine a toll booth that only lets in Red Cars. Blue cars are strictly forbidden.
• The Problem: In a normal superconductor, cars come in Red/Blue pairs. If you try to send them through a "Red-Only" toll booth, the Blue car gets stuck, the pair breaks, and the traffic stops. The junction is OFF.
• The Solution: When the magnet wobbles (precesses), it acts like a shapeshifting toll booth. For a split second, it lets a Blue car in, then instantly turns it into a Red car (or vice versa) so it can pass.
• The Result: The junction flips from OFF (no current) to ON (current flows). This is a switch controlled by the wobbling of the magnet.

2. The "Dance Floor" vs. The "Stadium" (Bound States vs. Continuum)

In quantum physics, electrons can exist in specific "parking spots" (bound states) or float freely in a "sea" (continuum).

• Normal Junction: The parking spots are full, and the sea is empty below a certain energy level.
• Wobbling Magnet Junction: The wobble shakes the ground. Some parking spots disappear, and the "sea" rises up to fill the gap.
• The Analogy: Imagine a dance floor. Usually, people only dance in specific spots. But if the DJ (the magnet) starts spinning the room (precession), the floor tilts. Now, people can dance in the aisles (the continuum) and the spots.
• Why it matters: This creates a very strange, non-smooth relationship between the current and the phase (the timing of the electron waves). It's not a smooth sine wave anymore; it's jagged and full of sudden jumps, like a rollercoaster with sharp drops.

3. The "Resonance" Effect (The Swing)

The paper mentions that if you wiggle the magnet at just the right speed, the effect gets huge.

• The Analogy: Think of pushing a child on a swing. If you push at random times, nothing happens. But if you push exactly when the swing is at the peak (resonance), the swing goes super high with very little effort.
• The Paper's Finding: If the wobbling frequency matches the magnet's natural "swing" frequency (ferromagnetic resonance), the supercurrent spikes dramatically. This suggests a way to control superconductors using microwave signals.

4. Long-Range vs. Short-Range (The "Ghost" vs. The "Fog")

• Normal Magnet: The supercurrent dies out very quickly as the magnet gets thicker, like a fog that clears up after a few meters.
• Wobbling Magnet: The "wobble-induced" current is like a ghost. It can travel through a very thick magnet without fading away.
• Why? The "ghost" current is made of "equal-spin" pairs (Red/Red or Blue/Blue). These pairs don't care about the magnet's direction as much as the normal pairs do. The wobble creates these special pairs, allowing the current to travel long distances.

The "So What?" (Why should we care?)

1. Super-Spintronics: We are entering an era of "spintronics," where computers use electron spin instead of just charge. This research shows how to build switches that are controlled by microwaves. You could turn a superconducting wire on and off just by wiggling a magnet with a radio signal.
2. Quantum Computers: These junctions are potential building blocks for quantum computers. The ability to switch a supercurrent on and off with high precision is crucial for qubits (quantum bits).
3. Energy Efficiency: Because the current flows without resistance, these devices could be incredibly energy-efficient, provided we can manage the tiny amount of heat generated by the wobbling magnet (which the paper calculates is negligible).

Summary in One Sentence

By making a magnet wobble, the authors discovered a way to turn a "dead" magnetic barrier into a "super-highway" for electricity, creating a switchable, long-range supercurrent that could revolutionize future quantum electronics.

Technical Summary

1. Problem Statement

The paper addresses the generation of long-range, spin-polarized supercurrents in ballistic Josephson junctions containing a ferromagnetic (F) or half-metallic (HM) layer.

• Context: In conventional Superconductor-Ferromagnet-Superconductor (S/F/S) junctions, supercurrents are carried by spin-singlet Cooper pairs. Due to the exchange field in the ferromagnet, these pairs oscillate and decay rapidly over a short length scale (typically the Fermi wavelength), limiting the range of the proximity effect.
• The Gap: While "equal-spin" (triplet) Cooper pairs can propagate long distances, they typically require static, non-collinear magnetic textures (e.g., domain walls or spirals) to be generated.
• The Question: Can a uniformly precessing magnetization (dynamic non-collinearity) generate equal-spin superconducting correlations in a ballistic junction, and what are the specific transport characteristics (current-phase relationship, conductance) in the limit of zero temperature and ballistic transport?

2. Methodology

The authors employ a rigorous theoretical framework based on the scattering approach within the Bogoliubov-de Gennes (BdG) formalism.

• Model System: One-dimensional ballistic junctions with a central magnetic region (length $L$) sandwiched between:
  • One Normal metal and one Superconductor (N/F/S or N/HM/S).
  • Two Superconductors (S/F/S or S/HM/S) with a phase difference $\phi$.
• Hamiltonian: The system is described by a time-dependent Hamiltonian due to the precessing magnetization $\mathbf{m}(t)$ with frequency $\Omega$ and tilt angle $\theta$.
• Rotating Frame Transformation: To handle the time-dependence, the authors transform the problem into a rotating reference frame. This converts the time-dependent Hamiltonian into a time-independent one but introduces an effective Zeeman field term ($\pm \hbar\Omega/2$) along the $z$-axis. This creates a non-collinear exchange field configuration essential for converting singlet pairs into triplet pairs.
• Spectrum Analysis:
  • Discrete Spectrum: Andreev Bound States (ABS) exist in the subgap region $0 \le E < |\Delta| - \hbar|\Omega|/2$.
  • Continuous Spectrum: For energies $E > |\Delta| - \hbar|\Omega|/2$, the spectrum becomes continuous.
• Non-Equilibrium Distribution: A key theoretical innovation is the treatment of the occupation numbers of the ABS. Since the precession drives the system out of equilibrium, the authors derive the steady-state occupation of bound states based on their coupling to a fictitious normal reservoir, leading to a non-Fermi-Dirac distribution.
• Dimensionality: Results are first derived for 1D junctions and then generalized to 2D and 3D by integrating over transverse momenta ($k_\parallel$).

3. Key Contributions

1. Theory of Ballistic Precessing Junctions: Unlike previous works focusing on diffusive limits or tunnel barriers, this paper provides a detailed theory for ballistic junctions, revealing distinct features like sharp resonances and non-sinusoidal current-phase relations.
2. Mechanism for Long-Range Transport in Half-Metals: The authors demonstrate that in a half-metal (where only one spin species exists), precession acts as a switch. Without precession, the junction is an insulator (no subgap current). With precession, it becomes a conductor with finite Andreev conductance and Josephson current.
3. Non-Sinusoidal Current-Phase Relationship (CPR): For large tilt angles $\theta$, the CPR deviates strongly from the standard sinusoidal form ($I \propto \sin \phi$), exhibiting sharp jumps and complex structures due to the interplay between discrete bound states and the continuous spectrum.
4. Resonant Enhancement: The theory predicts a resonant enhancement of the critical current and conductance when the precession frequency $\Omega$ matches the ferromagnetic resonance frequency $\Omega_{res}$, driven by the dependence of the tilt angle $\theta$ on the driving field.

4. Key Results

A. N/HM/S and S/HM/S Junctions (Fully Polarized)

• Switching Effect: In a half-metal, the zero-bias conductance $G$ vanishes without precession ($\theta=0$). With precession, $G$ becomes finite and scales as $G \propto \theta^2 \Omega^2$ for small angles.
• Current-Phase Relation: For small $\theta$, the Josephson current $I \propto \theta^2 \sin \phi$. For large $\theta$, the relation becomes highly non-sinusoidal.
• Spectral Features: The precession reduces the energy window for bound states. As the junction length $L$ varies, bound states can merge with the continuum or cross the chemical potential, causing abrupt jumps in the current (due to changes in occupation numbers).
• Long-Range Nature: In 2D/3D, the current averaged over transverse momenta remains non-zero even for large $L$, confirming the long-range nature of the precession-induced triplet supercurrent.

B. N/F/S and S/F/S Junctions (Partially Polarized)

• Coexistence: Both opposite-spin (short-range, oscillatory) and equal-spin (long-range, non-oscillatory) proximity effects coexist.
• Averaging: The opposite-spin contribution oscillates rapidly with junction length $L$ and averages to zero in higher dimensions. The precession-induced equal-spin contribution does not oscillate and survives averaging, dominating the long-range transport.
• Scaling: The critical current scales as $\langle I_c \rangle \propto (\hbar\Omega/|\Delta|)^2 \theta^2$.

C. Experimental Feasibility

• Frequency Range: The required precession frequencies ($\Omega$) are in the GHz to THz range, compatible with ferromagnetic resonance frequencies.
• Thermal Management: The authors estimate the heating due to Gilbert damping. Even in vertical junctions with poor thermal conductivity, the temperature rise is negligible ($\sim 30$ mK), suggesting experimental realization is feasible.
• Materials: Half-metals like $CrO_2$ or $Co_2MnGe$ are ideal candidates due to their low Gilbert damping ($\alpha$) and lack of competing singlet supercurrents.

5. Significance

• Fundamental Physics: The work establishes that time-dependent non-collinearity is a robust mechanism for generating equal-spin triplet superconductivity, complementing the well-known spatial non-collinearity.
• Spintronics Applications: It proposes a mechanism for microwave-controlled switching of supercurrents. By tuning the frequency of an AC magnetic field to the ferromagnetic resonance, one can switch a junction from an "off" state (zero current) to an "on" state (finite current).
• Device Potential: The ability to generate long-range supercurrents in half-metals without static magnetic textures opens new avenues for superconducting spintronics, potentially enabling low-dissipation spin-logic devices and novel qubits.
• Theoretical Advancement: The detailed treatment of non-equilibrium occupation in ballistic junctions provides a more accurate description of the Josephson effect in dynamic magnetic environments than previous diffusive models.

In summary, this paper theoretically validates that precessing magnetization in ballistic ferromagnetic junctions can induce a robust, long-range, spin-polarized Josephson effect, offering a pathway to dynamically control supercurrents for future quantum and spintronic technologies.