Electrostatic gate-controlled quantum interference in a high-mobility two-dimensional electron gas at the (La$_{0.3}$Sr$_{0.7}$)(Al$_{0.65}$Ta$_{0.35}$)O$_3$/SrTiO$_3$ interface

This study reports the observation of electrostatic gate-controlled Altshuler-Aronov-Spivak quantum interference oscillations in a high-mobility two-dimensional electron gas at the (La$_{0.3}$Sr$_{0.7}$)(Al$_{0.65}$Ta$_{0.35}$)O$_3$/SrTiO$_3$ interface, attributed to closed-loop paths along SrTiO$_3$ domain walls and demonstrating a long phase coherence length that highlights the potential of complex oxide interfaces for quantum technologies.

The Gist

Imagine a bustling city where millions of tiny cars (electrons) are driving on a special, invisible highway. Usually, when you turn on a magnetic field (like a giant magnet), these cars get confused, bump into each other, and their movement becomes chaotic. This is what happens in most materials.

But in this research, scientists discovered something magical happening at the border between two very specific types of ceramic materials: (La0.3Sr0.7)(Al0.65Ta0.35)O3 and SrTiO3. Let's call this the "Super-Highway Interface."

Here is the story of what they found, explained simply:

1. The "Ghostly" Traffic Jams (Quantum Interference)

In the quantum world, electrons don't just act like cars; they act like waves, similar to ripples in a pond. When these waves travel, they can overlap. Sometimes, the waves line up perfectly and boost each other (constructive interference), making the traffic flow smoothly. Other times, they crash into each other and cancel out (destructive interference), causing a traffic jam.

Usually, this "wave behavior" is very fragile. If the temperature gets a little warm or the road gets a little bumpy, the waves lose their rhythm, and the effect disappears.

The Discovery:
The scientists found that at this specific ceramic interface, these electron waves stay in sync even when the road is bumpy and the temperature rises to 10 Kelvin (which is still very cold, but "warm" for quantum physics). They saw a rhythmic pattern in the electricity flowing through the material that repeated every time the magnetic field changed by a specific amount.

2. The "Magic Loop" vs. The "Straight Road"

To understand why this is special, imagine two scenarios:

• Scenario A (The Ring): You build a perfect circular racetrack. If you send a car around it, it can go clockwise or counter-clockwise. If the track is perfect, the waves interfere beautifully. This is the famous Aharonov-Bohm effect.
• Scenario B (The Messy City): You have a giant, messy city with no specific tracks. You wouldn't expect waves to interfere here because the paths are random.

The Twist:
The scientists didn't build a perfect ring. They built a straight "Hall bar" (a rectangular strip). Yet, they still saw the interference patterns!

• The Analogy: Imagine a river flowing through a forest. Even though the river is just a straight line, the water flows around thousands of tree trunks, creating tiny, hidden loops and eddies. The electrons in this material are doing the same thing. They are getting trapped in tiny, naturally formed loops created by defects in the material (specifically, "domain walls" or cracks in the atomic structure of the crystal).
• These loops act like thousands of tiny, invisible racetracks. The electrons travel around them, and because the loops are so small and the electrons are so fast, the waves interfere, creating the rhythmic pattern the scientists saw. This is called the Altshuler–Aronov–Spivak (AAS) effect.

3. The "Volume Knob" (Electrostatic Gating)

The researchers had a remote control for this system: a gate voltage. Think of this like a volume knob for the number of electrons on the highway.

• Low Volume (Low Voltage): When they turned the knob down (low electron density), the "ghostly" interference patterns were loud and clear. The electrons were sparse enough to find these tiny loops and dance in sync.
• High Volume (High Voltage): As they turned the knob up (adding more electrons), the patterns started to fade. Eventually, when the highway was too crowded, the patterns disappeared completely.
• Why? It's like trying to hear a whisper in a quiet room versus a stadium. When the room is quiet (few electrons), you can hear the subtle interference. When the stadium is packed (many electrons), the noise drowns out the delicate wave patterns.

4. Why This Matters

This discovery is a big deal for two reasons:

1. It's Robust: Usually, these quantum effects are so fragile they vanish if you look at them wrong. Here, they survived up to 10 Kelvin and worked in a messy, disordered material. This suggests that nature has found a way to protect these quantum states.
2. The "Coherence" Length: The scientists calculated that the electrons could stay "in sync" (coherent) for a distance of about 1.8 micrometers. To put that in perspective, that's like a runner staying perfectly in step with a drummer for a marathon, even though the track is full of potholes. This is incredibly long for this type of material.

The Big Picture

This paper tells us that complex ceramic materials (oxides) are not just boring insulators or conductors. They are like quantum playgrounds where nature builds its own tiny, invisible racetracks.

By using a simple "volume knob" (voltage), we can turn these quantum effects on and off. This opens the door to building new types of quantum sensors (super-sensitive detectors) and quantum computers that rely on these stable, wave-like behaviors, potentially using materials that are easier to make than the perfect rings we usually need.

In short: They found that even in a messy, straight road of atoms, electrons can find hidden loops to dance in, and we can control this dance with a simple electrical switch.

Technical Summary

1. Problem Statement

Quantum interference phenomena, such as the Aharonov-Bohm (AB) and Altshuler–Aronov–Spivak (AAS) effects, are crucial for understanding electron phase coherence and developing quantum technologies (e.g., sensors and interferometers). While these effects have been well-studied in fabricated ring structures using III-V semiconductors and graphene, their observation in complex oxide interfaces remains challenging.

• The Gap: Most oxide interface studies focus on Shubnikov–de Haas (SdH) oscillations (periodic in $1/B$). Observing B-periodic oscillations (indicative of quantum interference) in oxide interfaces without artificially fabricated mesoscopic rings is rare.
• The Challenge: Understanding the origin of these oscillations in disordered, high-mobility 2D electron gases (2DEGs) and determining if they can be tuned via electrostatic gating to control quantum coherence.

2. Methodology

The researchers investigated the quantum transport properties of a high-mobility 2DEG formed at the interface between (La$_{0.3}$Sr$_{0.7}$)(Al$_{0.65}$Ta$_{0.35}$)O$_3$ (LSAT) and SrTiO$_3$ (STO).

• Sample Fabrication:
  • Grown using Pulsed Laser Deposition (PLD) on TiO$_2$-terminated STO (001) substrates.
  • The LSAT/STO system was chosen over the conventional LaAlO$_3$/STO system due to its smaller lattice mismatch (~1% vs. ~3%), enabling ultra-high mobility (up to 50,000 cm$^2$V$^{-1}$s$^{-1}$).
  • Devices were patterned into Hall bars (160 $\mu$m $\times$ 50 $\mu$m) using photolithography.
• Experimental Setup:
  • Cryogenics: Measurements performed in a dilution refrigerator down to 100 mK and in a He-4 cryostat up to 9 K.
  • Magnetic Fields: Static fields up to 16 T and pulsed fields.
  • Electrostatic Gating: A back-gate configuration was used (silver paint on the rear of the STO substrate) to tune carrier density ($n$) by varying gate voltage ($V_g$) from 0 to 50 V.
  • Measurements: Simultaneous longitudinal resistance ($R_{xx}$) and Hall resistance ($R_{yx}$) measurements.
• Analysis Techniques:
  • Fast Fourier Transform (FFT) to identify oscillation frequencies.
  • Lifshitz–Kosevich (L-K) formalism to extract cyclotron mass and quantum mobility.
  • Temperature and angular dependence analysis to distinguish between AB and AAS effects.

3. Key Contributions & Results

A. Observation of B-Periodic Oscillations

• The study observed pronounced quantum oscillations in magnetoresistance that are periodic in the magnetic field ($B$), distinct from the $1/B$ periodicity of SdH oscillations.
• Characteristics:
  • Field Range: Dominant in low fields (0–7 T).
  • Temperature Stability: Persist up to 9 K (significantly higher than typical SdH oscillations in this system which vanish above 1.2 K).
  • Periodicity: The primary period is $\Delta B \approx 1.7$ T (frequency $F \approx 0.58$ T$^{-1}$), with a second harmonic at 1.16 T$^{-1}$.
  • Amplitude: The oscillation amplitude is $\sim$40% of the quantum conductance ($e^2/h$) and decays exponentially with both temperature and magnetic field.

B. Identification of the AAS Effect

The authors attribute these oscillations to the Altshuler–Aronov–Spivak (AAS) effect rather than the AB effect, based on:

1. Geometry: The device is a Hall bar, not a ring. AAS oscillations survive in diffusive transport and ensemble averaging, whereas AB oscillations typically require ballistic transport in ring geometries.
2. Scattering Regime: Calculations show the mean free path ($l \approx 65$ nm) is much smaller than the effective loop circumference ($L \approx 123$ nm), confirming a diffusive transport regime.
3. Mechanism: The oscillations arise from coherent backscattering along time-reversed paths in naturally formed closed-loop networks.

C. Origin: Interconnected Domain Walls

The paper proposes that the closed-loop paths are formed by interconnected quasi-one-dimensional conduction channels along SrTiO$_3$ domain walls.

• Evidence:
  • The oscillation period varies between devices (Device 1: ~1.7 T; Device 2: ~3.0 T), suggesting a random, non-uniform origin (consistent with domain walls) rather than a fixed lattice property (like a moiré pattern).
  • Resistance anomalies at ~100 K (cubic-to-tetragonal transition) and ~40 K (domain wall formation) correlate with the emergence of these effects.
  • The oscillations vanish when carrier density is high enough to homogenize the charge distribution, destroying the domain wall network.

D. Electrostatic Gating and Tunability

• Carrier Density Control: Increasing $V_g$ increases carrier density ($n$) from $2.3 \times 10^{12}$ cm$^{-2}$ to a saturation of $\sim 7.5 \times 10^{12}$ cm$^{-2}$.
• Suppression of Oscillations:
  • As $V_g$ increases, the amplitude of B-periodic oscillations decreases systematically.
  • The frequency (period) shifts to lower $B$ values.
  • Complete Suppression: Oscillations vanish entirely for $V_g > 20$ V (where $n > 5 \times 10^{12}$ cm$^{-2}$).
• Physical Interpretation: Higher carrier density reduces the Fermi wavelength ($\lambda_F$) and homogenizes the charge distribution, reducing the size and connectivity of the naturally formed closed loops required for AAS interference.

E. Phase Coherence Length ($L_\phi$)

• By analyzing the temperature dependence of the oscillation amplitude, the authors estimated the phase coherence length.
• Result: $L_\phi \approx$ 1.84 $\mu$m at 0.1 K.
• Significance: This is exceptionally long for an oxide interface (roughly 100 times longer than in comparable LAO/STO devices with shorter channel lengths), indicating high-quality phase coherence. The temperature dependence ($L_\phi \propto T^{-1.14}$) suggests decoherence is driven primarily by electron-electron interactions.

4. Significance

• Platform for Quantum Interference: The LSAT/STO interface is demonstrated as a robust platform for studying quantum interference in complex oxides, offering higher mobility and longer coherence lengths than traditional LAO/STO systems.
• Gate-Tunable Quantum Phenomena: The ability to electrostatically tune and completely switch off quantum interference effects via carrier density provides a new degree of control for device engineering.
• Natural Nanostructures: The findings highlight that naturally occurring domain walls in oxides can act as effective mesoscopic interferometers without the need for complex nanofabrication.
• Future Applications: The long phase coherence length and gate tunability make this system a promising candidate for mesoscopic interferometers, quantum sensors, and quantum computing components based on oxide electronics.

Conclusion

The paper successfully identifies and characterizes B-periodic quantum oscillations in a high-mobility LSAT/STO 2DEG. By attributing these to the AAS effect arising from interconnected SrTiO$_3$ domain walls, the authors demonstrate a gate-tunable quantum interference system with a remarkably long phase coherence length, paving the way for advanced quantum devices in complex oxide materials.