Spin-polarized Andreev molecules and anomalous nonlocal Josephson effects in altermagnetic junctions

This paper demonstrates that weakly coupled altermagnetic Josephson junctions host spin-polarized Andreev molecules, which induce anomalous nonlocal Josephson effects including tunable phase transitions and a nonreciprocal diode effect, thereby establishing altermagnetism as a promising platform for nonlocal spin Josephson phenomena.

The Gist

Imagine you are trying to build a super-fast, super-smart electronic circuit. To do this, you need to understand how electricity flows without resistance (superconductivity) and how magnetic materials behave.

For a long time, scientists had two main players in this game: Superconductors (materials that conduct electricity perfectly) and Magnets (materials that create magnetic fields). Usually, these two don't get along well; magnets tend to kill superconductivity.

But recently, scientists discovered a new, weird type of magnet called an Altermagnet. Think of an Altermagnet as a "ghost magnet." It has magnetic properties that can split electrons based on their spin (like a tiny internal compass), but unlike a normal magnet, it has zero net magnetism. It's like a crowd of people where half are facing North and half are facing South, so the crowd as a whole isn't pulling in any direction, but the individuals are still very organized.

This paper explores what happens when you put these "ghost magnets" inside a special circuit called a Josephson Junction.

The Setup: The "Sandwich"

Imagine a sandwich:

1. Bread 1 (Left): A superconductor.
2. Filling (Middle): An Altermagnet.
3. Bread 2 (Right): Another superconductor.

In this paper, the scientists didn't just make one sandwich. They made two sandwiches and stuck them together side-by-side, sharing a middle piece of bread.

• Left Sandwich: Superconductor — Altermagnet — Superconductor.
• Right Sandwich: Superconductor — Altermagnet — Superconductor.
• The Middle: They share a central superconductor layer.

The Magic: "Andreev Molecules"

Inside these sandwiches, electrons do a weird dance called "Andreev reflection." When an electron hits the boundary between a normal metal and a superconductor, it bounces back as a "hole" (a missing electron), and a partner electron is left behind. These pairs form energy states called Andreev Bound States (ABS).

Usually, if you have two separate sandwiches, these dances happen independently. But in this paper, the scientists made the middle layer very thin. This allowed the "dances" in the left sandwich and the right sandwich to talk to each other.

When they talk, they merge into a single, giant dance called an Andreev Molecule.

• Analogy: Imagine two separate groups of dancers on opposite sides of a room. If the room is huge, they dance to their own music. If you shrink the room so they can hear each other, they start dancing in sync, forming a new, combined routine.
• The Twist: Because of the "ghost magnet" (Altermagnet), this new dance is spin-polarized. This means the dancers are sorted by their internal compass (spin). The left side might be dancing with "North" spins, while the right side dances with "South" spins, even though they are part of the same molecule.

The Result: The "Remote Control" Effect

Here is the most exciting part. Because these two sandwiches are now dancing as one molecule, changing the "music" (phase) in the Right Sandwich instantly changes the flow of electricity in the Left Sandwich.

This is called the Nonlocal Josephson Effect.

• Analogy: Imagine you have two light switches in different rooms. Usually, flipping the switch in Room A only turns on the light in Room A. But in this new system, flipping the switch in Room A magically changes the brightness of the light in Room B, even though no wire connects them directly.

The Superpower: The "Diode" Effect

In normal electronics, a diode is a one-way street for electricity. It lets current flow forward but blocks it backward.

The scientists found that their "Andreev Molecule" system acts like a nonlocal diode.

• How it works: By tweaking the "ghost magnet's" strength or the phase in the right sandwich, they can make the current flow easily in one direction but get stuck in the other.
• The Cool Part: You can flip the direction of this one-way street just by changing the settings on the other side of the device. It's like having a traffic light that you can control from a different city.

Why Does This Matter?

This discovery is a big deal for the future of Quantum Computing.

1. New Materials: It shows that "ghost magnets" (Altermagnets) are a powerful tool for building new types of electronic devices.
2. Spintronics: It allows us to control the "spin" of electrons (which is like a tiny magnetic switch) over long distances without wires.
3. Efficiency: It creates a way to build circuits that can act as switches or diodes without needing external magnetic fields, making them smaller and more energy-efficient.

In a nutshell: The paper shows that by using a special "ghost magnet" to connect two superconducting circuits, we can create a "molecular" link where controlling one side instantly controls the other. This link acts like a smart, remote-controlled one-way valve for electricity, opening up new possibilities for super-fast, low-energy quantum computers.

Technical Summary

Here is a detailed technical summary of the paper "Spin-polarized Andreev molecules and anomalous nonlocal Josephson effects in altermagnetic junctions" by Sayan Mondal and Jorge Cayao.

1. Problem Statement

The paper addresses the intersection of two emerging fields in condensed matter physics: altermagnetism and coherently coupled Josephson junctions (JJs).

• Altermagnetism (AM): A recently discovered magnetic phase characterized by zero net magnetization, collinear-compensated ordering, and anisotropic spin-polarized Fermi surfaces. While AMs have been shown to induce spin-polarized Andreev bound states (ABSs) and anomalous Josephson effects in single junctions, their impact on coupled systems remains unexplored.
• Coherent Coupling: When two JJs are coupled via a shared superconductor, their ABSs hybridize to form "Andreev molecules." This enables nonlocal Josephson effects (NJE), where the supercurrent in one junction is controlled by the phase of the other.
• The Gap: Previous research on Andreev molecules and NJE has focused primarily on superconductor-semiconductor systems. The role of altermagnetism in creating spin-polarized Andreev molecules and enabling nonlocal spin functionalities in Josephson circuits has not been investigated.

2. Methodology

The authors employ a tight-binding lattice model to simulate a system of three coupled superconductors:

• System Geometry: Two outer superconductors (Left, $L$; Right, $R$) and a middle superconductor ($M$), all coupled via a $d$-wave altermagnet. The system is modeled as finite along the $x$-axis and periodic along the $y$-axis.
• Hamiltonian: The total Hamiltonian includes:
  • Kinetic terms with spin-orbit coupling induced by the altermagnet ($J_1$ for $d_{xy}$ and $J_2$ for $d_{x^2-y^2}$ symmetries).
  • $s$-wave spin-singlet superconducting pairing potentials ($\Delta_\alpha e^{i\phi_\alpha}$) in the three superconductors.
  • Coupling terms between adjacent regions.
• Parameters: The study varies the length of the middle superconductor ($L_M$) relative to the coherence length ($\xi$) to tune the coupling strength between the junctions (weak coupling for $L_M \gg \xi$; strong coupling for $L_M \lesssim \xi$).
• Analysis: The authors calculate the low-energy spectrum (Andreev bound states), spin polarization ($\langle S_z \rangle$), and the Josephson current ($I_L$) as a function of the superconducting phase differences ($\phi_L, \phi_R$) and altermagnetic strength.

3. Key Contributions

1. Prediction of Spin-Polarized Andreev Molecules: The paper demonstrates that in the strong coupling regime ($L_M \lesssim \xi$), the hybridization of spin-polarized ABSs from the two junctions forms spin-polarized Andreev molecules. These are distinct from standard Andreev molecules due to the spin-splitting induced by the altermagnet.
2. Anomalous Nonlocal Josephson Effect: The authors show that these spin-polarized molecules enable a nonlocal Josephson effect where the current in the left junction ($I_L$) is strongly modulated by the phase of the right junction ($\phi_R$).
3. Tunable Junction Transitions: The system exhibits tunable transitions between 0-junction, $\pi$-junction, and $\phi_0$-junction behaviors in the nonlocal current, driven by the altermagnetic symmetry and strength.
4. Nonlocal Josephson Diode Effect: The study identifies a nonreciprocal critical current ($I_c^+ \neq I_c^-$) in the nonlocal channel, establishing a nonlocal Josephson diode effect. Crucially, the polarity of this diode can be tuned nonlocally by $\phi_R$ or locally by the altermagnetic field strength.

4. Detailed Results

A. Formation of Spin-Polarized Andreev Molecules

• Weak Coupling ($L_M \gg \xi$): The left and right junctions behave independently. The altermagnet splits the ABSs into spin-polarized branches. The left junction's spectrum depends on $\phi_L$, while the right junction's spectrum is dispersionless with respect to $\phi_L$ (and vice versa).
• Strong Coupling ($L_M \lesssim \xi$): The ABSs from both junctions hybridize.
  • $d_{x^2-y^2}$ symmetry: The hybridized states form spin-polarized Andreev molecules that are asymmetric with respect to $\phi_L = \pi$. Depending on the field strength, these states can cross zero energy, leading to robust zero-energy crossings.
  • $d_{xy}$ symmetry: The hybridization creates molecules with distinct spin polarizations at the same energy, exhibiting asymmetric behavior around $\phi_L = \pi$.
  • Real-space behavior: Wavefunction analysis confirms that for short $L_M$, the wavefunctions of the left and right junctions overlap significantly, forming a single molecular state spanning the entire structure.

B. Anomalous Nonlocal Josephson Current

• Phase Control: In the strong coupling regime, the current $I_L(\phi_L, \phi_R)$ becomes highly sensitive to $\phi_R$.
• Symmetry Breaking: Unlike standard junctions where $I_L$ is an odd function of $\phi_L$ alone, here $I_L(\phi_L, \phi_R) = -I_L(-\phi_L, -\phi_R)$.
• Junction Types:
  • $\phi_0$-junction: A finite current flows even when $\phi_L = 0$ (controlled by $\phi_R$).
  • $\pi$-junction: The current-phase relation shifts such that the ground state corresponds to a phase difference of $\pi$.
  • The specific behavior (0, $\pi$, or $\phi_0$) is tunable by the altermagnetic strength ($J_{1,2}$) and the phase $\phi_R$.

C. Nonlocal Josephson Diode Effect

• Nonreciprocity: The critical currents for positive and negative bias directions are unequal ($I_c^+ \neq I_c^-$), satisfying the condition for a diode effect.
• Quality Factor ($\eta$): The diode efficiency is quantified by $\eta = (I_c^+ - I_c^-)/(I_c^+ + I_c^-)$.
  • $\eta$ is non-zero for most $\phi_R$ values (vanishing only at $\phi_R = n\pi$).
  • Polarity Control: The sign of $\eta$ (polarity) can be reversed by:
    1. Changing the superconducting phase $\phi_R$ (nonlocal control).
    2. Varying the altermagnetic field strength $J$ (local control).
• Symmetry Dependence: While both $d_{xy}$ and $d_{x^2-y^2}$ symmetries support the effect, $d_{x^2-y^2}$ generally supports higher diode efficiencies at larger field strengths, whereas $d_{xy}$ requires lower fields to switch polarity.

5. Significance and Implications

• New Platform for Quantum Devices: This work establishes altermagnetism as a key ingredient for designing nonlocal spin-Josephson phenomena. It offers a pathway to create Josephson diodes without external magnetic fields, utilizing the intrinsic properties of the material.
• Tunability: The ability to control the diode polarity and junction type (0/$\pi$/$\phi_0$) via superconducting phases and material parameters provides a high degree of flexibility for circuit design.
• Quantum Computing Applications: Since Josephson junctions are the building blocks of superconducting qubits, these findings suggest new ways to engineer qubit couplings, protect quantum states, and implement non-reciprocal elements (diodes) in quantum circuits.
• Fundamental Physics: The study bridges the gap between altermagnetism and many-body quantum phenomena, demonstrating how time-reversal symmetry breaking (without net magnetization) can drive complex nonlocal quantum transport.

In summary, the paper theoretically predicts that coupling Josephson junctions through altermagnets creates spin-polarized Andreev molecules, which in turn enable a highly tunable, nonlocal Josephson diode effect with switchable polarity, opening new avenues for spintronic and quantum computing applications.