Pumping of spin supercurrent in unitary triplet superconductors

This paper proposes a general principle for generating dissipationless spin supercurrents in unitary triplet superconductors by utilizing magnetization dynamics in a proximity magnetic nanostructure to convert particle spin into Cooper pair spin, thereby extending conventional spin pumping mechanisms through emergent particle-hole symmetry.

The Gist

The Big Picture: Moving "Spin" Without Friction

Imagine you have a highway where cars (electrons) usually drive. Sometimes, these cars carry a special cargo called spin (a tiny magnetic property, like a spinning top). In normal materials, moving this cargo is messy; the cars crash into things, lose energy, and the cargo gets scattered. This is like driving on a bumpy dirt road.

However, in a special type of material called a superconductor, the cars can link up in pairs (Cooper pairs) and drive on a frictionless, magical highway. They move without losing any energy. This is called a supercurrent.

Usually, we know how to make a charge supercurrent (moving the cars themselves). But scientists have been trying to figure out how to move just the spin cargo (the spinning tops) without moving the cars, and without friction. This is called a spin supercurrent.

The Problem: The "Empty" Pairs

In most superconductors, the linked pairs of cars are "neutral." They spin in opposite directions, canceling each other out. It's like two dancers holding hands, one spinning clockwise and the other counter-clockwise. The net spin is zero.

The paper asks: If the pairs have zero net spin, how can we pump a flow of spin through them?

The Solution: The "Spin Pump" Analogy

The authors propose a clever trick based on a concept called Andreev Reflection.

The Charge Analogy (The Old Way):
Imagine a normal metal (a busy parking lot) connected to a superconductor (the magical highway). When a single car (electron) tries to enter the highway, it can't drive alone. So, it hits the gate, bounces back as a "hole" (an empty space), and leaves behind a new pair of cars that join the highway flow.

• Result: The charge is conserved. The car didn't disappear; it just transformed into a pair. This creates a supercurrent.

The Spin Analogy (The New Way):
The authors realized we can do the same thing with spin, but we need a different kind of superconductor: a Triplet Superconductor.

• In these materials, the pairs are "unitary," meaning they are balanced (net spin = 0) when sitting still.
• However, the authors discovered that if you wiggle a magnet right next to the superconductor, you can create a Spin Torque.

The Mechanism: The "Shaking Magnet"

Imagine the superconductor is a calm lake. The "Cooper pairs" are ducks swimming in perfect synchronization.

1. The Setup: You place a tiny, vibrating magnet (a magnetic nanostructure) right next to the lake.
2. The Action: The magnet spins and wiggles (this is called magnetization dynamics). This creates a "wind" or a "push" on the ducks (the electrons) in the water.
3. The Magic: Because of the special rules of this triplet superconductor, this "wind" doesn't just push the ducks; it forces them to swap their spinning directions.
   • Normally, if a duck spins one way, it stays that way.
   • Here, the interaction with the magnet forces the duck to flip its spin.
   • The Conservation Law: Since the total spin of the system must be conserved, if the duck flips its spin, the "Cooper pair" it belongs to must absorb that change.
4. The Result: The Cooper pairs, which were previously neutral, now carry a net flow of spin. They start moving the "spin cargo" down the frictionless highway.

Why This is a Big Deal

1. It's "Unitary" Magic:
Usually, to get a spin supercurrent, you need a material that is already "unbalanced" (non-unitary). This paper shows you can generate a spin supercurrent even in materials that are perfectly balanced at rest. It's like making a river flow without needing a waterfall to start it; you just need to shake the ground.

2. Beyond the "Standard Shake":
In normal physics, if you shake a magnet, the spin flow is usually limited to a specific formula (proportional to how fast the magnet spins crossed with its direction).
The authors found that in these triplet superconductors, the rules are different. The "spin pump" is much more efficient and complex. It's like discovering that a specific type of windmill can generate power not just from the wind's speed, but from the wind's shape and direction in ways we didn't think possible.

3. No Friction, No Waste:
Because this is a supercurrent, the spin moves without losing energy to heat. This is the "Holy Grail" for future electronics (spintronics). Imagine computers that process information using spin instead of electricity, running cooler and faster, with zero energy loss.

Summary in One Sentence

The paper proposes a new way to generate a frictionless flow of magnetic "spin" through special superconductors by wiggling a nearby magnet, effectively turning a balanced, neutral system into a powerful engine for spin transport, much like how a bouncing ball can create a flow of water in a pipe.

Technical Summary

1. Problem Statement

The paper addresses a fundamental challenge in spintronics and superconductivity: how to generate and manipulate dissipationless spin supercurrents in unitary triplet superconductors.

• Context: Spin supercurrents (flow of spin angular momentum without dissipation) are highly desirable for low-power spintronic devices. While non-unitary triplet superconductors (where Cooper pairs have intrinsic spin polarization) can transport spin via center-of-mass momentum, unitary triplet superconductors (where the condensate spin polarization $\text{Tr}(\Delta^\dagger \sigma \Delta) = 0$) are theoretically difficult to drive into a spin-supercurrent state using conventional methods like temperature gradients or charge currents.
• Gap: Existing theories of "spin pumping" (generating spin currents via magnetization dynamics) typically apply to normal metals or non-unitary superconductors. It was unclear if a unitary triplet superconductor, which has no net equilibrium spin polarization, could support a spin supercurrent driven by external dynamics.
• Core Question: Can a unitary triplet superconductor support a spin supercurrent, and if so, what is the mechanism to drive it?

2. Methodology

The authors employ a combination of theoretical frameworks to derive a general principle and calculate specific outcomes:

• Analogy with Andreev Reflection: The authors draw a parallel between charge conservation in conventional $s$-wave superconductors and spin conservation in triplet superconductors.
  • In conventional superconductors, the singlet order parameter acts as a "charge sink," converting quasiparticle charge into Cooper pair charge (Andreev reflection).
  • The authors propose that in triplet superconductors, the triplet order parameter acts as a "spin sink" (or spin torque), converting quasiparticle spin into Cooper pair spin.
• Hamiltonian Formulation: They model a prototypical unitary $p$-wave superconductor using a Nambu space Hamiltonian. They define spin density, spin-current density, and spin-torque density operators.
• Scattering Theory: To analyze the system under dynamic conditions, they develop a nonlinear scattering theory. They treat the time-dependent exchange field from a proximate magnetic nanostructure as a scattering potential.
• Perturbation Theory: Using time-dependent perturbation theory (up to second order), they calculate the wave functions of quasiparticles interacting with an AC exchange field generated by ferromagnetic resonance (FMR).
• Continuity Equation: They utilize the spin continuity equation to relate the divergence of the spin supercurrent to the spin torque density: $\nabla \cdot \mathbf{J}_{sc} = -\mathbf{T}_s$.

3. Key Contributions

• General Principle of Spin Conservation: The paper establishes that even in unitary triplet superconductors (where equilibrium Cooper pairs carry no spin), the triplet order parameter $\Delta_p$ provides a mechanism to convert injected quasiparticle spin into the spin of Cooper pairs. This restores global spin conservation, allowing for the generation of a dissipationless spin supercurrent.
• Beyond Conventional Spin Pumping: The authors demonstrate that the generated spin supercurrent is not limited to the conventional form proportional to $\mathbf{dM}/dt \times \mathbf{M}$ (which characterizes spin pumping in normal metals). Due to the emergent particle-hole symmetry and the specific structure of the triplet order parameter, new combinations of magnetization components contribute to the current.
• Mechanism Proposal: They propose a specific implementation: using the magnetization dynamics $\mathbf{M}(t)$ of a proximate magnetic nanowire (driven by FMR) to pump spin into the superconductor via interfacial exchange coupling.
• Theoretical Framework: They derive explicit analytical expressions for the DC spin-current density and DC spin-torque density in terms of the scattering matrix elements and the triplet order parameter.

4. Key Results

The authors applied their theory to a model system: a LaAlO$_3$/KTaO$_3$ superconducting interface (a candidate for unitary $p$-wave superconductivity) covered by a CoFeB magnetic nanowire.

• Spin Torque and Current Generation:
  • The AC exchange field from the precessing magnetization induces a spin torque on quasiparticles.
  • This torque acts as a source for Cooper pairs, generating a DC spin supercurrent flowing away from the pumping region.
  • The spin current is polarized along the saturation magnetization direction ($\hat{z}$).
• Temperature Dependence (Crucial Finding):
  • Spin Current ($J_s$): The quasiparticle-carrying component of the spin current is suppressed at $T \to 0$ (due to lack of thermal quasiparticles) and increases with temperature, peaking near $T_c$.
  • Spin Torque ($T_s$): The spin torque density is proportional to the order parameter $\Delta_p$. It increases with temperature initially (due to more quasiparticles) but decreases as $T \to T_c$ because $\Delta_p \to 0$.
  • Spin Supercurrent ($J_{sc}$): By integrating the spin torque, the authors show the resulting dissipationless spin supercurrent exhibits a unique behavior: it vanishes at $T \to 0$, rises to a maximum at $T \lesssim T_c$, and vanishes again at $T_c$. This distinguishes it from the purely quasiparticle current.
• Efficiency: The calculated spin current magnitude is comparable to spin Hall currents generated by electric fields of 1 kV/cm, suggesting the effect is experimentally measurable.
• Selection Rules: Table I in the paper outlines "selection rules" showing how different combinations of magnetization vectors ($\mathbf{M}, \tilde{\mathbf{M}}, \bar{\mathbf{M}}$) contribute to different spin current/torque components, offering a rich landscape for control beyond simple $\mathbf{dM}/dt \times \mathbf{M}$.

5. Significance

• Fundamental Physics: The work provides a robust theoretical basis for understanding spin dynamics in unitary triplet superconductors, resolving the paradox of how to generate spin currents in systems with zero equilibrium spin polarization. It establishes a direct analogy between charge dynamics in $s$-wave superconductors and spin dynamics in $p$-wave systems.
• Technological Impact:
  • New Device Paradigm: It proposes a highly efficient method to generate and manipulate dissipationless spin currents without requiring non-unitary (spin-polarized) superconductors, which are harder to realize and control.
  • Energy Efficiency: Since the current is supercurrent (dissipationless), it offers a pathway to ultra-low-power spintronic devices.
  • Experimental Feasibility: The proposed mechanism relies on standard magnetic resonance techniques (FMR) and known material interfaces (LaAlO$_3$/KTaO$_3$), making it a viable target for experimental verification.
• Broader Applicability: The general principle derived can be applied to other unconventional superconductors, potentially opening new avenues for quantum computing and spin-based information processing.

In summary, Li and Yu demonstrate that magnetization dynamics can pump a dissipationless spin supercurrent in unitary triplet superconductors by exploiting the triplet order parameter as a spin sink, effectively converting quasiparticle spin into Cooper pair spin, thereby extending the concept of spin pumping well beyond conventional limits.