Supercurrent-induced phonon angular momentum

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crystal lattice not as a rigid, static grid of atoms, but as a bustling dance floor. Usually, when these atoms vibrate (which we call phonons), they just jiggle back and forth in straight lines, like people doing the "running man."

But in certain special crystals, the atoms do something more exciting: they spin in circles or ellipses as they vibrate. Think of them as dancers doing a perfect pirouette. These spinning vibrations are called chiral phonons. Because they are spinning, they carry a tiny bit of angular momentum—essentially, a microscopic amount of "spin" or rotation.

This paper, written by physicist Takehito Yokoyama, proposes a fascinating new way to make these atoms spin faster or change their spin direction, not by pushing them with a magnet or heating them up, but by pushing an electric current through a superconductor.

Here is the breakdown of the idea using simple analogies:

1. The Setup: The Super-Dance Floor

The author looks at two types of special superconductors (materials that conduct electricity with zero resistance):

Mixed Parity Superconductors: These are materials where the "dance rules" are a bit chaotic and asymmetric.
S-wave Superconductors with Spin-Orbit Coupling: These are materials where the electrons' movement is tightly linked to their internal "spin" (like a top spinning while moving).

In both cases, the crystal structure is "chiral," meaning it has a handedness (like a left-handed or right-handed screw). This structure allows the atoms to naturally perform those circular dance moves (chiral phonons).

2. The Trigger: The Supercurrent

Usually, if you push a current through a normal wire, the electrons just flow straight. But in these special superconductors, the author suggests that the flowing current acts like a conductor on a dance floor.

When you send a supercurrent (a frictionless flow of electrons) through the material, it doesn't just move the electrons; it creates a subtle "wind" or a magnetic field that interacts with the spinning atoms.

3. The Effect: The Phonon "Edelstein" Effect

The paper draws a parallel to a famous effect in electronics called the Edelstein effect.

The Analogy: Imagine a crowd of people (electrons) walking in a circle. If the floor is tilted or the room is spinning, the people might start leaning or spinning themselves. In electronics, a current can make electrons spin up (creating spin polarization).
The New Discovery: Yokoyama proposes the reverse happens here. The flowing supercurrent doesn't just spin the electrons; it transfers that "spin energy" to the atoms themselves, making the chiral phonons (the spinning atoms) rotate with more intensity or in a specific direction.

It's like a river (the supercurrent) flowing through a turbine (the crystal). The river doesn't just push the turbine blades; it actually makes the blades spin in a specific, coordinated way that they wouldn't do on their own.

4. How It Works (The Physics Simplified)

The author uses math to show that this happens because of a connection between the atoms' rotation and the electrons' spin:

The Current: You apply a supercurrent.
The Electron Spin: Because of the material's special properties, this current forces the electrons to align their spins in a specific direction (like a crowd all looking North).
The Magnetic Whisper: These aligned electrons create a tiny, effective magnetic field.
The Atomic Spin: The spinning atoms (phonons) feel this magnetic field. Just like a compass needle aligns with a magnet, the atoms adjust their rotation. The result is a net angular momentum in the phonons.

5. Why Should We Care?

This isn't just a theoretical curiosity; it opens the door to a new kind of technology called Phononics (using sound/vibrations instead of electricity to process information).

The "Switch": If you can control the spin of atoms using a supercurrent, you could potentially build devices that use sound waves to store data or control magnets, without needing huge external magnets.
The Test: The author suggests we could prove this exists by shining light on the material. If the supercurrent is on, the light (Raman scattering) should bounce off the spinning atoms differently than when the current is off, revealing the hidden "spin" of the atoms.

Summary

In short, this paper predicts that flowing electricity in a superconductor can make the atoms inside it spin like tops. It's a bridge between the world of electricity (current) and the world of mechanics (spinning atoms), suggesting that we can "spin up" the very fabric of a material just by running a current through it. This could lead to new, ultra-efficient ways to control magnetism and information in future computers.

1. Problem Statement

The paper addresses the generation of phonon angular momentum (chiral phonons) in superconducting systems. While it is well-established that external stimuli like temperature gradients or electric fields can induce phonon angular momentum (analogous to the Edelstein effect in electronic spin systems), the mechanism for inducing this momentum via a supercurrent in the absence of an external magnetic field has not been fully explored.

The author investigates whether a supercurrent flowing through a superconductor with specific symmetry properties (lacking inversion symmetry) can generate a net angular momentum in the lattice vibrations (phonons). This is crucial for understanding the interplay between superconductivity, lattice chirality, and spin dynamics, potentially opening new avenues for phonon-mediated spin control in spintronic devices.

2. Methodology

The author employs a perturbative theoretical framework based on many-body Green's function techniques to derive analytical expressions for the induced phonon angular momentum ( $\langle L_z \rangle$ ).

Model Systems: Two distinct classes of superconductors are analyzed:
1. Mixed Parity Superconductors: Materials lacking inversion symmetry where the superconducting gap function contains both singlet ( $\Delta_s$ ) and triplet ( $\mathbf{d} \cdot \boldsymbol{\sigma}$ ) components.
2. s-wave Superconductors with Spin-Orbit Coupling (SOC): Conventional s-wave superconductors where inversion symmetry is broken by a Weyl-type spin-orbit coupling term ( $\alpha \mathbf{k} \cdot \boldsymbol{\sigma}$ ).
Hamiltonian Construction:
- Phonons: Described by a Hamiltonian $H_p$ with chiral modes (circularly polarized).
- Electrons: Described by BCS-like Hamiltonians incorporating the specific gap structures and SOC.
- Coupling: The interaction between chiral phonons and electrons is modeled via a spin-orbit coupling term $H_{ep} = \lambda L_z \sigma_z$ . This arises from the effective magnetic field generated by the rotating atoms (Zeeman effect).
- Supercurrent: Introduced via a vector potential $A_z$ (along the z-axis) using minimal substitution, treating the current operator $j_z$ as a perturbation.
Calculation:
- The expectation value of the phonon angular momentum $\langle L_z \rangle$ is calculated to the first order in the electron-phonon coupling ( $\lambda$ ) and the vector potential ( $A_z$ ).
- Diagrammatic expansions of the phonon Green's function are used, involving traces over phononic and electronic degrees of freedom (particle-hole, spin, and Matsubara frequencies).
- The London equation ( $j = \rho A$ ) is utilized to express the final results in terms of the supercurrent density $j_z$ rather than the vector potential, ensuring gauge invariance.

3. Key Contributions

Proposal of a New Mechanism: The paper proposes and theoretically validates a mechanism where a supercurrent induces phonon angular momentum in both mixed-parity and SOC-induced s-wave superconductors. This is identified as a superconducting analog of the "phonon Edelstein effect."
Analytical Derivations: The author derives closed-form analytical expressions for $\langle L_z \rangle$ as a function of temperature, gap parameters, and supercurrent density for both model systems.
Physical Interpretation: The effect is interpreted as an orbital Zeeman effect. The supercurrent induces a spin polarization (via the Edelstein effect) in the electronic system. Through the electron-phonon coupling ( $\lambda L_z \sigma_z$ ), this electronic spin polarization acts as an effective magnetic field that aligns the phonon angular momentum.
Experimental Proposal: The paper suggests a concrete experimental signature: detecting the difference in circularly polarized Raman scattering spectra with and without an applied supercurrent. It specifically proposes Nb/Te/Nb Josephson junctions as a candidate system, given that Tellurium (Te) is a chiral crystal with observed chiral phonons.

4. Key Results

A. Mixed Parity Superconductors

For superconductors with both singlet ( $\Delta_s$ ) and triplet ( $\Delta_t$ ) pairing near the critical temperature ( $T_c$ ):

The induced angular momentum is proportional to the product of the singlet and triplet gaps ( $\Delta_s \Delta_t$ ).
Result: $\langle L_z \rangle \propto -\frac{\lambda \Delta_s \Delta_t}{T^3} j_z$ .
Magnitude Estimate: For typical parameters ( $T \sim 1$ meV, $j_z \sim 10^6$ A/cm $^2$ ), the induced angular momentum density is estimated to be $\sim 2 \times 10^{-3} \hbar \text{\AA}^{-3}$ .

B. s-wave Superconductors with Spin-Orbit Coupling

For s-wave superconductors with Weyl-type SOC ( $\alpha$ ):

The effect arises at the third order in the spin-orbit coupling strength ( $\alpha^3$ ). The first-order terms vanish due to symmetry.
Result: $\langle L_z \rangle \propto -\frac{\alpha^3 \lambda \Delta^2}{T^3} j_z$ .
Magnitude Estimate: Under similar conditions, the magnitude is estimated to be $\sim 9 \times 10^{-6} \hbar \text{\AA}^{-3}$ .

C. General Findings

The response is linear with respect to the supercurrent density $j_z$ .
The effect is strictly dependent on the lack of inversion symmetry (gyrotropic crystals) to allow the coupling between the vector potential and the phonon angular momentum.
The derived expressions are gauge-invariant when expressed in terms of the supercurrent density.

5. Significance

Fundamental Physics: This work bridges the gap between superconductivity and chiral phononics, demonstrating that supercurrents can act as a control knob for lattice chirality, similar to how electric currents control electron spin.
Spintronics Applications: The mechanism offers a new route for phonon-mediated spin control. Since chiral phonons can generate effective magnetic fields and influence spin dynamics, supercurrents could be used to manipulate magnetization or spin currents without external magnetic fields.
Experimental Feasibility: By identifying specific material systems (like Tellurium-based Josephson junctions) and detection methods (circularly polarized Raman scattering), the paper provides a clear roadmap for experimental verification.
Theoretical Framework: The perturbative approach developed here can be extended to study other non-equilibrium phenomena in superconductors with broken inversion symmetry.

In summary, Yokoyama establishes that supercurrents can induce a net angular momentum in the crystal lattice of specific superconductors, driven by the interplay of spin-orbit coupling, superconducting pairing, and lattice chirality. This "supercurrent-induced phonon angular momentum" represents a novel transport phenomenon with potential implications for next-generation quantum and spintronic technologies.