Light-induced Andreev phase coherence and tunneling Hall effect in semi-Dirac systems

This paper theoretically demonstrates that off-resonant circularly polarized light applied to a semi-Dirac normal metal/superconductor junction induces a phase-coherent, transversely asymmetric Andreev reflection, thereby generating a switchable tunneling Hall effect that reverses direction with the light's handedness.

The Gist

Imagine a busy highway where cars (electrons) are trying to drive from one city to another. Usually, this highway is perfectly symmetrical: if you drive straight, you go straight. If you turn slightly left, you turn slightly left. But in this paper, the researchers are studying a very special, weird kind of highway made of a material called a Semi-Dirac material.

Think of this material as a road that behaves differently depending on which way you look at it. If you drive North-South, the road is smooth and straight (like a normal highway). But if you drive East-West, the road is bumpy and curved (like a rollercoaster).

Here is the simple story of what happens in this paper:

1. The Setup: A Traffic Jam at a Border

The researchers set up a traffic system with three zones:

• Zone A (Left): A normal city where cars drive freely.
• Zone B (Middle): A construction zone where the road is weird (the Semi-Dirac material).
• Zone C (Right): A special "Super-City" (a Superconductor) where cars can pair up and dance together without any friction.

Normally, when a car from Zone A hits the border of Zone C, it doesn't just bounce back. It gets "converted" into a mirror-image car (a hole) and bounces back, while a pair of cars enters the Super-City. This is called Andreev Reflection. In a normal, boring world, this bounce-back is perfectly symmetrical. If you send a car in from the left, it bounces back to the left. No traffic ever drifts sideways.

2. The Twist: The "Magic Flashlight"

Now, the researchers shine a special circularly polarized light (like a spinning flashlight beam) onto the middle construction zone (Zone B).

Think of this light not as a lamp, but as a spinning wind that pushes the cars.

• If the wind spins clockwise, it pushes cars moving slightly to the left harder than cars moving to the right.
• If the wind spins counter-clockwise, it does the opposite.

This light doesn't just push the cars; it gives them a secret "phase" or a mental tag. It's like giving every car a tiny sticker that says, "I was pushed by a clockwise wind."

3. The Magic: The "Bouncing Ball" Effect

Here is the clever part. When a car hits the Super-City border and bounces back, it has to travel through the middle construction zone again. It might bounce off the left wall, then the right wall, then the Super-City, and bounce back again.

Because of the "secret sticker" (the light-induced phase) the car picked up, every time it bounces, it remembers the spin of the light.

• If the car was moving slightly left, the light makes its path change in a specific way.
• If the car was moving slightly right, the light changes its path differently.

This creates a skewed reflection. Instead of bouncing straight back, the cars start drifting sideways, like a billiard ball that hits a cushion and suddenly curves off to the side.

4. The Result: The "Tunneling Hall Effect"

Because all the cars are drifting sideways in the same direction, a sideways current is created!

• The Longitudinal Current (Forward): The amount of traffic moving forward doesn't change much, no matter which way the light spins. It's just a little faster or slower.
• The Transverse Current (Sideways): This is the big discovery. If you spin the light clockwise, the traffic drifts Left. If you flip the light to counter-clockwise, the traffic instantly drifts Right.

It's as if you have a tunnel, and by simply changing the spin of a light in the middle, you can force all the cars to exit the tunnel on the left side or the right side, without changing the tunnel's shape.

Why Does This Matter?

In the real world, creating a sideways electrical current usually requires strong magnets or complex materials with special "spin" properties. This paper shows that you can do it just by shining the right kind of light on a specific material.

The Big Picture:
The researchers found a way to use light to act as a traffic director for electrons in a superconductor. By controlling the "handedness" (spin) of the light, they can control the direction of the electrical current. This could lead to new types of super-fast, light-controlled electronic switches for future computers, where you don't need magnets to steer electricity—you just need a flashlight.

In a nutshell:
They used a spinning light to make electrons in a weird material "dance" sideways, creating a new way to control electricity using light instead of magnets.

Technical Summary

1. Problem Statement

The paper addresses the lack of understanding regarding superconducting transport properties in Semi-Dirac Materials (SDMs) under light irradiation.

• Context: SDMs exhibit anisotropic dispersion (linear in one direction, quadratic in the orthogonal direction). While circularly polarized light (CPL) can open a Haldane-like mass gap in conventional Dirac materials (like graphene), it cannot open a gap in SDMs.
• Gap: Previous studies focused on the normal-state transport of SDMs under light, but their superconducting transport (specifically Andreev reflection and tunneling) remained largely unexplored.
• Goal: The authors aim to investigate whether off-resonant CPL applied to a normal metal/superconductor junction based on SDMs can induce new transport phenomena, specifically a transverse tunneling current (Tunneling Hall Effect) without requiring spin-orbit coupling or valley degrees of freedom.

2. Methodology

The study employs a theoretical framework combining tight-binding models, Floquet theory, and the Non-Equilibrium Green's Function (NEGF) formalism.

• Model System: A Normal metal/Normal metal/Superconductor (NNS) double-junction structure based on a 2D SDM lattice (honeycomb lattice with specific hopping parameters $t$ and $t'$).
  • Region 1 ($x < 0$): Normal lead.
  • Region 2 ($0 < x < L$): Central normal region irradiated by off-resonant circularly polarized light (CPL).
  • Region 3 ($x > L$): Superconducting electrode with $s$-wave pairing.
• Hamiltonian Construction:
  • The low-energy effective Hamiltonian of the SDM is derived near the semi-Dirac point ($M$ point in the Brillouin zone), yielding a form with quadratic dispersion in $k_x$ and linear in $k_y$.
  • Light-Matter Interaction: The effect of CPL is incorporated via the Peierls substitution ($k \to k + eA(t)$). Using Floquet theory in the high-frequency limit ($\hbar\omega \gg ev_A$), an effective time-independent Hamiltonian is derived.
  • Key Result of Light: The irradiation induces an effective mass term proportional to $k_x \tau_z$ (where $\tau$ is the pseudospin matrix), which acts as a momentum-dependent phase shift.
• Transport Calculation:
  • The Bogoliubov-de Gennes (BdG) Hamiltonian is constructed to describe electron-hole mixing.
  • The system is mapped onto a 1D chain along the $x$-direction with periodic boundary conditions in $y$.
  • NEGF Formalism: The retarded and lesser Green's functions are calculated using the recursive Green's function algorithm to determine:
    • Andreev reflection coefficients ($|a_A|^2$).
    • Longitudinal conductance ($G$).
    • Transverse current ($I_y$) and transverse conductance ($G_T$).

3. Key Contributions and Mechanisms

The paper establishes a novel phase-coherence mechanism for generating transverse currents:

• Light-Induced Phase Coherence: The CPL irradiation in the central region induces an additional phase shift in the back-reflected states. Crucially, this phase depends on the transverse momentum ($k_y$) and the handedness of the light ($\eta = \pm 1$).
• Asymmetric Andreev Reflection: Due to multiple reflections within the central spacer, the light-induced phase leads to constructive or destructive interference that is asymmetric with respect to $k_y$.
  • The total reflection phase $\theta$ acquires a term proportional to $\lambda k_y$ (where $\lambda$ is the illumination parameter).
  • This breaks the symmetry of Andreev reflection: $|a_A(k_y, \lambda)|^2 \neq |a_A(-k_y, \lambda)|^2$.
• Generation of Tunneling Hall Effect: The transverse asymmetry in Andreev reflection results in a net charge imbalance flowing perpendicular to the bias voltage, generating a finite transverse tunneling current (Tunneling Hall Effect).
• Distinct Mechanism: Unlike previous tunneling Hall effects driven by spin-orbit coupling, valley polarization, or chirality, this effect arises purely from light-induced phase coherence in the absence of additional internal degrees of freedom.

4. Key Results

• Transverse Conductance ($G_T$):
  • $G_T$ is zero at zero bias due to particle-hole symmetry.
  • Under finite bias, $G_T$ exhibits a non-zero value in the subgap regime.
  • Handedness Reversal: Switching the circular polarization from right-handed ($\lambda > 0$) to left-handed ($\lambda < 0$) reverses the sign of $G_T$, indicating a reversal of the transverse current direction.
  • $G_T$ shows a peak at incident energy $eV \approx 0.85\Delta$ and decreases thereafter.
• Longitudinal Conductance ($G$):
  • $G$ is insensitive to the handedness of the light (symmetric for $\pm \lambda$) because it involves summing over all $k_y$ modes, which cancels out the odd-parity transverse effects.
  • However, $G$ exhibits a small phase shift (Fabry-Pérot oscillation shift) as the light intensity increases, attributed to the light-induced phase accumulation.
• Hall Angle ($\Theta = G_T/G$):
  • The Hall angle scales approximately linearly with the illumination parameter $\lambda$.
  • It increases with bias voltage but vanishes at zero bias.
• Zero-Bias Anomaly: The transverse current vanishes at zero bias ($eV=0$) because the kinetic phase accumulated in the spacer vanishes, leaving only the light-induced phase which, in the limit of zero energy, results in symmetric reflection.

5. Significance and Applications

• Fundamental Physics: The work demonstrates that light can induce phase-coherent transport phenomena in superconducting junctions without opening a topological gap or requiring spin-orbit coupling. It highlights the unique role of the semi-Dirac dispersion in mediating light-matter interactions.
• Superconducting Electronics: The ability to control the direction and magnitude of a transverse supercurrent simply by switching the handedness of light offers a new mechanism for superconducting diodes and Hall sensors.
• Device Potential: The proposed mechanism suggests a route to realize tunable tunneling Hall devices in superconducting circuits, potentially leading to low-dissipation electronic components where transport direction is optically controlled.

In summary, the paper theoretically predicts a light-controlled Tunneling Hall Effect in semi-Dirac superconducting junctions, driven by the momentum-dependent phase coherence of Andreev reflections induced by circularly polarized light.