Anomalous phase shift and superconducting diode effect in Josephson junctions via thin films of rare-earth intermetallic magnets

This paper theoretically demonstrates that Josephson junctions incorporating ultra-thin films of the rare-earth intermetallic magnet $\mathrm{GdIr_2Si_2}$ exhibit a significant anomalous phase shift and a tunable superconducting diode effect, offering promising prospects for superconducting memory and logic applications.

The Gist

Imagine a superconductor as a super-highway where electricity flows without any friction or traffic jams. Usually, if you connect two of these super-highways with a short bridge (a "Josephson junction"), the traffic flows equally well in both directions. It's like a two-way street where the speed limit is the same going north or south.

However, this paper explores a special kind of bridge that breaks the rules. It creates a "one-way street" for electricity, even without any external magnets or batteries pushing it. This is called the Josephson diode effect.

Here is how the researchers built this special bridge and what they found, explained simply:

1. The Special Bridge Material

To build this one-way bridge, the researchers didn't use a standard metal. They used a very thin, microscopic film of a material called GdIr₂Si₂ (Gadolinium-Iridium-Silicon).

• Think of it like this: Imagine a sandwich where the bread is the superconductor (the highway) and the filling is this special magnetic film.
• This filling is made of rare-earth metals. It has two special "superpowers" that make it unique:
  1. Strong Magnetism: It acts like a tiny, internal magnet.
  2. Spin-Orbit Coupling: This is a fancy way of saying the electrons inside are "twisting" as they move, like a corkscrew.

2. The "Anomalous Phase Shift" (The Tilted Starting Line)

In a normal bridge, the "ground state" (the resting position) is perfectly straight. But in this special bridge, the resting position is slightly tilted.

• The Analogy: Imagine a pendulum. In a normal clock, it hangs straight down. In this special junction, the pendulum naturally wants to hang slightly to the left or right, even when no one is pushing it.
• The researchers found that this "tilt" (called the phase shift, $\phi_0$) isn't fixed. It changes depending on which way the internal magnet inside the bridge is pointing. If you rotate the magnet slightly, the tilt changes.

3. The One-Way Street (The Diode Effect)

Because of that tilt and the twisting electrons, the bridge behaves like a diode (a one-way valve).

• The Analogy: Imagine a turnstile at a subway station. It's easy to push through in one direction, but hard to push through in the other.
• In this junction, the maximum amount of electricity that can flow without resistance is different depending on the direction.
  • Flowing "North": You can push a lot of current.
  • Flowing "South": You can only push a little bit before it gets stuck.
• The researchers calculated that this difference is significant (about 30% efficiency), meaning it's a very effective one-way street for super-currents.

4. The "Knob" for Control

The most exciting part is that you can control this one-way street just by turning a "knob."

• The Analogy: Imagine a dimmer switch for a light, but instead of making the light brighter or dimmer, you are changing which direction the traffic flows.
• By slightly rotating the direction of the magnet inside the GdIr₂Si₂ film, the researchers can:
  • Change how strong the one-way effect is.
  • Even flip the direction (make the "easy" direction become the "hard" direction).
• This happens because the electrons in this material are very sensitive to the angle of the magnet. It's like a lock and key where the key (the current) only fits if the lock (the magnet) is turned to the exact right angle.

5. Why This Matters (According to the Paper)

The paper suggests this discovery is a big deal for future technology because:

• Memory and Logic: You could use this to build super-fast, super-efficient computer memory. Since the "one-way" direction depends on the magnet's state, you could store a "0" or a "1" by setting the magnet's direction.
• No External Magnets Needed: Unlike other systems that need a giant magnet outside to work, this one has its own internal magnet, making it self-contained.
• Tunability: Because the effect changes so dramatically with small rotations of the magnet, it offers a very precise way to control electrical flow.

Summary

The researchers used a computer to simulate a microscopic bridge made of a rare-earth magnetic film sandwiched between superconductors. They found that this bridge naturally tilts its resting state and acts as a one-way street for electricity. By simply rotating the internal magnet of the bridge, they can control how strong this one-way effect is and which way it points. This creates a new type of switch that could be used for advanced, ultra-fast computing and memory devices.

Technical Summary

Technical Summary: Anomalous Phase Shift and Superconducting Diode Effect in Josephson Junctions via Thin Films of Rare-Earth Intermetallic Magnets

Problem Statement
The implementation of the zero-field Josephson diode effect and anomalous ground state phase shifts ($\phi_0 \neq 0, \pi$) in superconductor/ferromagnet/superconductor (S/F/S) Josephson junctions (JJs) is a critical goal for superconducting memory and logic applications. While previous strategies utilizing proximity-induced magnetism in spin-orbit coupled (SOC) layers (e.g., Pt, Ta, W) have demonstrated field-free diode effects, they lack the capability for genuine Josephson-current-driven magnetization dynamics. This limitation arises because the induced magnetic order is weak compared to the magnetic anisotropy of the adjacent insulating ferromagnet. Conversely, theoretical models for intrinsic ferromagnetic interlayers often rely on simplified quasiclassical approximations that neglect realistic electronic structures, such as strong exchange splitting, SOC comparable to the band width, and the presence of multiple electronic bands at the Fermi level. There is a need for a targeted search for materials that intrinsically possess strong SOC and exchange splitting to enable tunable, non-volatile $\phi_0$-JJs.

Methodology
This study employs a multi-scale theoretical approach combining Density Functional Theory (DFT) with Bogoliubov-de Gennes (BdG) calculations to investigate a specific candidate material: an ultra-thin film of the rare-earth intermetallic magnet GdIr$_2$Si$_2$.

1. Electronic Structure and Magnetic Properties (DFT):

   • Calculations were performed using the VASP package with the projector augmented-wave (PAW) method and Generalized Gradient Approximation (GGA).
   • To account for strong correlations in Gd-4f and Ir-5d electrons, the GGA+U method was applied ($U^* = 6$ eV for Gd, $3.5$ eV for Ir).
   • Van der Waals corrections (DFT-D3) were included for structural optimization.
   • The study analyzed symmetric slabs of GdIr$_2$Si$_2$ with different crystal phases (I-phase and P-phase) and surface terminations (Si or Ir).
   • Magnetic ordering (ferromagnetic vs. antiferromagnetic, in-plane vs. out-of-plane) was evaluated by calculating total energies per magnetic atom, including stray field energies.

2. Effective Tight-Binding Hamiltonian (TBH):

   • An effective TBH was constructed to describe the electronic subsystem of the GdIr$_2$Si$_2$ slab.
   • The model focuses on two conduction bands crossing the Fermi level, each exhibiting spin-orbit splitting (linear Rashba-type for the lower band, cubic for the upper).
   • The Hamiltonian parameters (hopping amplitudes, exchange fields, and SOC constants) were fitted to minimize the mean square error against the DFT-calculated band dispersions along high-symmetry directions.
   • The model includes 12 nearest-neighbor hopping elements and accounts for both intra-band and inter-band interactions, as well as a fixed $z$-axis exchange field component to correctly describe spin splitting independent of magnetization rotation.

3. Josephson Current Calculation (BdG):

   • The S/GdIr$_2$Si$_2$/S junction was modeled on a square lattice using the constructed TBH for the weak link and a standard tight-binding model for superconducting leads.
   • The BdG equations were diagonalized to calculate the current-phase relationship (CPR).
   • Calculations were performed at $T = 0.1\Delta_0$ with specific parameters for the superconducting gap ($\Delta_0 = 10$ meV) and interface hopping ($t_{SF} = 50$ meV), chosen to ensure robust qualitative features despite computational constraints.

Key Results

• Material Properties: The DFT analysis identifies the I-phase GdIr$_2$Si$_2$ film with Si-termination as the most energetically favorable configuration. It exhibits easy-plane magnetism with ferromagnetic alignment within layers and ferromagnetic interlayer ordering. The magnetic moment is constrained to the in-plane direction due to a total anisotropy energy favoring the in-plane orientation by $\approx 1$ meV/magnetic atom.
• Anomalous Phase Shift ($\phi_0$): The calculated CPRs demonstrate a pronounced anomalous ground state phase shift $\phi_0$ of the order of unity. Crucially, $\phi_0$ exhibits a highly non-monotonic dependence on the in-plane magnetization component $m_y$. This contrasts with simple one-band models where $\phi_0 \propto m_y$. The non-monotonicity arises from the interference of multiple electronic bands with distinct spin-momentum locking characteristics.
• Josephson Diode Effect: The system exhibits a significant superconducting diode effect with an efficiency $\eta \lesssim 0.3$. The diode efficiency is also non-monotonic with respect to $m_y$ and can undergo sign reversal.
• Anisotropy and Tunability: Both the critical currents and the diode efficiency show strong anisotropy relative to the orientation of the in-plane magnetization. Because the film is an easy-plane magnet, small rotations of the magnetization vector allow for the tuning of the anomalous phase shift and diode efficiency.
• Robustness: Sensitivity analysis in the Appendix indicates that while quantitative values of the diode efficiency depend on superconducting parameters ($\Delta_0$, $t_{SF}$) and temperature, the key qualitative features—specifically the non-monotonic dependence of $\phi_0$ and $I_c$ on $m_y$—remain robust. These features are determined solely by the intrinsic electronic band structure of the interlayer.

Significance and Claims
The paper claims that GdIr$_2$Si$_2$ thin films represent a viable candidate for realizing $\phi_0$-S/F/S Josephson junctions with intrinsic, non-volatile diode effects. The primary significance lies in the demonstration that realistic material properties (strong exchange splitting and SOC in a multi-band system) lead to complex, non-monotonic dependencies of the Josephson characteristics on magnetization direction. This behavior suggests that the coupling between the superconducting phase and magnetization is richer than predicted by simplified models, potentially enabling more complex control over magnetization dynamics via Josephson currents.

The authors posit that the LnT$_2$X$_2$ family (where Ln is a lanthanide, T is a transition metal, and X is a p-element) offers a "playground" for tuning these properties by varying the transition metal to adjust SOC strength and the lanthanide to control magnetic moment orientation. The work provides a computational blueprint for identifying materials where the Josephson current can effectively control magnetization and where the diode effect polarity can be regulated by the magnetization direction, which is essential for superconducting memory and logic circuit applications. The study does not propose specific experimental fabrication protocols but highlights the feasibility of growing such films via molecular beam epitaxy (MBE) based on recent progress in similar ThCr$_2$Si$_2$-type structures.