Interfering trajectories in a ballistic Andreev cavity

This paper presents the first study of ballistic Andreev transport in a two-dimensional HgTe quantum well cavity, demonstrating that two distinct finite-bias conductance peaks arise from different classes of interfering trajectories—one dominated by normal reflection and the other by retro-reflected Andreev processes—which exhibit contrasting responses to magnetic fields due to Aharonov-Bohm and Doppler shift effects.

The Gist

Imagine a tiny, high-tech billiard table. This isn't a normal table, though. One side of the table is lined with a special "magic wall" (a superconductor), and the other side is a regular wall (a normal metal). The table itself is made of a super-smooth material (a high-quality quantum well) where tiny particles called electrons can zoom around without bumping into anything, like a ghost gliding through a hallway.

This is the story of a new experiment that looked at what happens when these electrons try to cross from the regular side to the magic side.

The Old Way of Thinking

For decades, scientists thought of this setup like a pinhole camera. They imagined the electrons squeezing through a single, tiny dot to get from one side to the other. In this old view, the size and shape of the room didn't matter; only the tiny hole did. It was a simple, one-dimensional story: In, bounce, out.

The New Discovery: A Ballroom Dance

This paper says, "Wait a minute! That's too simple."

In this experiment, the room is actually a large, two-dimensional ballroom. The electrons aren't squeezing through a pinhole; they are entering through a wide doorway. Because the floor is so smooth, the electrons don't just bounce straight back and forth. They zoom across the room, hit the walls, and bounce around in complex patterns before deciding to leave.

The researchers found that when these electrons interact with the "magic wall" (the superconductor), they do something called Andreev reflection.

• The Analogy: Imagine an electron (a single dancer) running into the magic wall. Instead of bouncing back as a dancer, it gets "promoted" into a pair (a Cooper pair) that stays in the magic wall. To conserve the rules of physics, it leaves behind a "hole" (an empty spot that acts like a ghost dancer) which runs back out the door.
• Sometimes, this ghost dancer runs back, hits the other wall, and comes back to the magic wall to get promoted again.

The Two Types of Paths

The researchers discovered that the electrons take two very different types of paths, which create two distinct "peaks" in the electrical signal:

1. The Open Path (The Sprinter):
   Some electrons zoom in, bounce off the magic wall once, and run straight back out. They don't really loop around.

   • The Metaphor: Think of a sprinter running to a wall and turning around immediately.
   • The Result: This path is stubborn. Even if you put a magnet near the table, this sprinter doesn't care. Their path doesn't change, so their signal stays strong.

2. The Closed Loop (The Figure-Eight Skater):
   Other electrons get trapped in a loop. They bounce off the magic wall, turn into a ghost, bounce off the other wall, turn back into a real electron, and return to the exact spot they started. They create a perfect circle or figure-eight.

   • The Metaphor: Think of a figure skater spinning in a perfect circle on the ice.
   • The Result: This path is very sensitive. If you bring a magnet near the table, it's like a strong wind blowing across the ice. The skater gets pushed off their perfect circle. The "interference" that makes this path special gets messed up, and the signal disappears.

The "Magic" of the Magnet

The coolest part of the experiment was using a magnet.

• When they turned on a weak magnetic field, the Figure-Eight Skater (the inner peak) vanished. The magnet disrupted their perfect loop.
• The Sprinter (the outer peak) kept running strong. The magnet didn't bother them because they weren't doing a loop.

This proved that the electrons were indeed taking these complex, looping paths inside the room, not just squeezing through a tiny hole.

Why Does This Matter?

For a long time, scientists used the "pinhole" model to understand how electricity moves between normal metals and superconductors. It worked fine for messy, slow-moving electrons (like traffic in a crowded city).

But in the future, we want to build computers using "topological" materials where electrons move like ghosts (ballistic transport). In these high-speed, clean environments, the shape of the room matters just as much as the door.

The Takeaway:
This paper is like realizing that to understand a dance, you can't just look at the doorway; you have to watch the whole dance floor. By understanding how the shape of the room and the magnetic field change the dancers' paths, we can build better, more powerful quantum devices in the future. It's a shift from thinking of electrons as particles in a tunnel to thinking of them as dancers in a ballroom.

Technical Summary

Here is a detailed technical summary of the paper "Interfering trajectories in a ballistic Andreev cavity."

1. Problem Statement

The conventional description of charge transport at a normal-metal/superconductor (N/S) interface relies on the Blonder-Tinkham-Klapwijk (BTK) model. This model treats the system as a zero-dimensional, Sharvin-type point contact, effectively reducing the problem to one dimension and ignoring device geometry.

• Limitation: While successful for diffusive transport regimes, the BTK model fails in ballistic transport regimes typical of high-mobility semiconductor-superconductor hybrid structures (e.g., HgTe quantum wells).
• The Gap: In ballistic mesoscopic devices, transport occurs through a two-dimensional (2D) cavity with extended one-dimensional interfaces. The conventional model neglects:
  1. The 2D nature of the cavity.
  2. The interplay between dynamical phases and Andreev phases.
  3. The specific role of the second interface (Normal contact) in a cavity bounded by both Normal and Superconducting leads.
  4. Magnetic field effects on closed-loop trajectories (Aharonov-Bohm and Doppler shifts).

2. Methodology

Experimental Setup:

• Material: High-mobility HgTe quantum wells (embedded in (Cd,Hg)Te layers) with a carrier density of $\sim 7.8 \times 10^{11} \text{ cm}^{-2}$.
• Device Geometry: Rectangular mesas ($L \approx 1 \mu\text{m}$, $W = 5.5 \mu\text{m}$) patterned via wet-etching. The length is shorter than the electron mean free path ($l \approx 3 \mu\text{m}$), ensuring ballistic transport.
• Contacts: Side-contacted by a normal gold (Au) electrode and a superconducting Molybdenum-Rhenium (MoRe) electrode ($T_c \approx 9.6 \text{ K}$, gap $\Delta \approx 1.44 \text{ meV}$).
• Measurement: Differential conductance ($dI/dV$) spectroscopy was performed at base temperature ($\sim 25 \text{ mK}$) and above $T_c$ ($13 \text{ K}$). Measurements included varying perpendicular magnetic fields ($H_z$) from 0 to 2 mT.

Theoretical Modeling:

• Approach: A semi-classical scattering matrix formalism within a Landauer-Buttiker framework.
• Key Innovation: The model treats the device as a 2D ballistic cavity with extended 1D interfaces, rather than a point contact.
• Trajectory Classification: The model categorizes electron/hole trajectories into two classes:
  1. Open Trajectories: Electrons reflect normally or undergo single Andreev reflection but do not return to their origin.
  2. Closed Trajectories: Electrons undergo retro-reflection (Andreev reflection) at the S-interface, travel back, reflect at the N-interface, and undergo a second Andreev reflection, returning to the origin.
• Parameters: The model incorporates a distribution of injection angles ($\theta$) and an inhomogeneous interface transparency parameter ($\gamma$), representing patches of high and low transparency.

3. Key Results

Experimental Observations:

• Double Peak Structure: In the sub-gap regime ($V_b < \Delta$), the differential conductance exhibits two distinct finite-bias peaks at approximately 180 $\mu$V and 600 $\mu$V, alongside a zero-bias dip.
• Magnetic Field Response:
  • The outer peak (~600 $\mu$V) is robust against perpendicular magnetic fields up to 2 mT.
  • The inner peak (~180 $\mu$V) is strongly suppressed and washed out by fields as low as 500 $\mu$T.
• Device Length Dependence: Measurements on devices with shorter lengths ($L=650$ nm and $860$nm) confirmed the phenomenology. The outer peak shifted to higher bias (consistent with$E \propto 1/L$), while the inner peak position remained relatively stable (dominated by the Andreev phase).

Theoretical Interpretation:

• Origin of Peaks:
  • The Outer Peak arises from open trajectories (Process 1 & 2 in the paper). These involve normal reflection or single Andreev reflection. Their resonance condition depends on the dynamical phase ($2EL/\hbar v_F$) and is independent of magnetic flux, explaining their robustness.
  • The Inner Peak arises from closed trajectories (Process 3). These involve two Andreev reflections, creating a loop where the electron returns to its origin.
• Suppression Mechanism: The closed trajectories are sensitive to magnetic fields due to two effects:
  1. Aharonov-Bohm (AB) Effect: The loop encloses a magnetic flux, introducing a phase shift $\phi_{AB} \propto H_z$.
  2. Doppler Shift: Interaction with Meissner screening currents at the S-interface imparts a momentum shift, altering the kinetic energy of the charge-conjugated particle.
  • These effects shift the resonance condition and, for higher-order reflections, cause the energy gain to exceed the superconducting gap, effectively cutting off the contribution of these trajectories.

4. Key Contributions

1. Beyond BTK: The paper provides the first comprehensive study of Andreev transport in a 2D ballistic cavity with extended interfaces, moving beyond the oversimplified Sharvin point-contact model.
2. Trajectory Classification: It successfully attributes distinct experimental features (the two bias peaks) to specific classes of ballistic trajectories (open vs. closed loops).
3. Magnetic Field Sensitivity: It elucidates how closed-loop Andreev trajectories are uniquely suppressed by magnetic fields via AB and Doppler mechanisms, a phenomenon not captured in 1D models.
4. Inhomogeneous Interface Model: The authors demonstrate that a realistic description requires modeling the N/S interface as inhomogeneous (patches of varying transparency) to reproduce the experimental double-peak structure.

5. Significance

• Fundamental Physics: The work clarifies the role of geometry and interference in mesoscopic superconducting hybrids, showing that the "second interface" (the normal contact) is crucial for defining the transport regime in ballistic cavities.
• Topological Materials: Since many topological materials (like HgTe) exhibit long mean free paths and are often combined with superconductors to search for Majorana states, understanding the interplay of ballistic transport, geometry, and Andreev reflection is critical.
• Experimental Design: The findings provide a framework for interpreting transport data in hybrid structures. Researchers must account for cavity geometry and trajectory interference rather than assuming simple point-contact behavior, which is essential for correctly identifying exotic states (like Majorana zero modes) versus geometric interference effects.