Coherent control of thermal transport with pillar-based phononic crystals

This paper demonstrates that pillar-based phononic crystals on suspended silicon nitride membranes can coherently control thermal transport at sub-Kelvin temperatures, achieving up to an order-of-magnitude reduction in thermal conductance, although the coherent theory fails to predict results for larger lattice constants due to diffusive scattering caused by surface roughness.

The Gist

The Big Idea: Stopping Heat with a "Traffic Jam"

Imagine heat as a crowd of tiny, invisible runners (called phonons) trying to sprint across a flat, suspended trampoline (a silicon nitride membrane). Usually, these runners move freely and quickly, carrying heat energy from one side to the other.

The scientists in this paper wanted to stop these runners without building a wall. Instead, they built a giant obstacle course right on the trampoline. They placed thousands of tiny aluminum pillars (like little metal trees) in a perfect grid pattern.

Their goal was to see if this obstacle course could slow down the heat so much that it becomes "coherent"—meaning the runners start moving in a synchronized, wave-like pattern that accidentally cancels itself out, effectively stopping the heat flow.

The Experiment: The "Forest" of Pillars

The researchers created four different versions of this obstacle course, changing the distance between the "trees" (the pillars):

1. The Dense Forest: Pillars very close together (0.3 micrometers apart).
2. The Medium Forest: Pillars a bit further apart (1 micrometer).
3. The Sparse Forest: Pillars quite far apart (3 and 5 micrometers).

They cooled everything down to near absolute zero (colder than outer space) and measured how much heat could get through.

The Results: What Happened?

1. The Small Forests Worked Like Magic (Coherent Control)

For the two smallest forests (0.3 µm and 1 µm), the experiment was a huge success.

• The Analogy: Imagine the runners trying to run through a forest where the trees are spaced just right. As they run, their footsteps create waves. Because the trees are arranged perfectly, these waves crash into each other and cancel out, like noise-canceling headphones for heat.
• The Outcome: The heat flow dropped by up to 10 times (an order of magnitude) compared to a plain trampoline with no trees. The "traffic jam" was so effective that the runners barely moved.

2. The Big Forests Failed (The Breakdown)

For the larger forests (3 µm and 5 µm), the magic stopped working.

• The Analogy: Imagine the trees are now so big and their bark is so rough that instead of running in a synchronized wave, the runners start tripping over the bark, slipping on the rough ground, and scattering in random directions. They stop behaving like waves and start behaving like a chaotic crowd bumping into things.
• The Outcome: The heat flow went back up. The "roughness" of the aluminum pillars acted like a bumpy road, destroying the delicate wave pattern needed to stop the heat.

Why This Matters

1. A New Way to Build "Heat Insulators"
Usually, to stop heat, you use thick, heavy materials like wool or foam. This paper shows you can stop heat using very thin, lightweight materials if you arrange them in a specific pattern. This is like building a super-insulating coat that is as thin as a piece of paper.

2. The "Roughness" Problem
The study taught us a valuable lesson: Perfection matters. To make this "wave-cancelling" trick work, the surfaces of the pillars need to be incredibly smooth. If they are even slightly rough, the magic disappears. This is a challenge for engineers: they need to build smoother pillars to make this work on larger scales.

3. Future Applications
This technology could be a game-changer for:

• Super-sensitive detectors: Devices that need to stay super cold to detect faint signals (like from space) without heat leaking in.
• Quantum computers: These computers need to be kept at near-zero temperatures to function. Better heat control means more stable quantum computers.

Summary

The scientists built a microscopic forest of pillars to trap heat. When the trees were small and the spacing was perfect, they successfully created a "heat traffic jam," reducing heat flow by 90%. However, when the trees got too big and their surfaces too rough, the runners got distracted by the bumps, and the heat escaped. It's a proof that with the right design, we can control heat like we control sound or light.

Technical Summary

1. Problem Statement

Phononic crystals (PnCs) are artificial periodic structures capable of manipulating elastic waves (phonons) to control heat flow. While previous research successfully demonstrated coherent thermal conduction control (where wave interference, not just scattering, dictates transport) using hole-based PnCs (membranes perforated with periodic holes), these structures suffer from mechanical fragility and fabrication challenges, especially for complex designs.

Conversely, pillar-based PnCs (periodic arrays of pillars on a solid membrane) are mechanically robust but have historically shown only modest thermal conductivity reductions (a few percent to ~40%) at higher temperatures. These reductions were typically attributed to incoherent scattering (diffusive processes) rather than coherent wave effects. Furthermore, previous sub-Kelvin experiments with pillar structures failed to conclusively demonstrate coherent control, often attributing results to electron-phonon interactions in metal pillars or incoherent scattering.

The Core Challenge: Can pillar-based PnCs achieve significant, coherent reduction of thermal conductance at sub-Kelvin temperatures, comparable to hole-based designs, and what limits this coherence?

2. Methodology

Device Design and Fabrication:

• Substrate: 320 nm thick amorphous silicon nitride (SiNx) membranes suspended over a silicon wafer.
• Pillars: Periodic arrays of cylindrical polycrystalline Aluminum (Al) pillars (300 nm tall).
• Geometry: Four samples were fabricated with square lattice constants ($a$) of 0.3, 1, 3, and 5 µm. All maintained a constant area filling factor of 0.65.
• Material Choice: Aluminum was chosen because it becomes superconducting below $T_c \approx 1.2$ K. This suppresses electronic excitations and inelastic electron-phonon scattering, ensuring that any thermal transport reduction is due to phonon-structure interactions rather than electronic effects.
• Measurement Setup: A custom heater-thermometer geometry was employed. A superconducting Niobium (Nb) heater surrounds a Superconductor-Insulator-Normal metal (SINIS) thermometer. This "encircling" geometry ensures the thermometer equilibrates with the heater temperature with minimal geometric correction factors, improving measurement accuracy.

Experimental Techniques:

• Thermal Conductance: Measured in a dilution refrigerator (base temperature ~50 mK) by applying power to the heater and measuring the temperature rise of the SINIS thermometer.
• Brillouin Light Scattering (BLS): Used to experimentally verify the phonon dispersion relations (band structure) of the smallest period (0.3 µm) sample.
• Material Characterization: Rutherford Backscattering Spectrometry (RBS) and BLS were used to determine precise material densities and elastic constants (Young's modulus, Poisson ratio) for the specific SiNx and Al films used, rather than relying on bulk literature values.

Simulations:

• Coherent Model: Finite Element Method (FEM) simulations solved the 3D elasticity equations to calculate phonon band structures, group velocities, and density of states (DOS). A modified theoretical model for emitted phonon power from a heater wire was derived, assuming isotropic emission from the wire element.
• Incoherent Model: Monte Carlo ray-tracing simulations were performed to model the incoherent (diffusive) limit, accounting for surface roughness and specularity parameters.

3. Key Contributions

1. Demonstration of Coherent Control in Pillar PnCs: The study provides the first conclusive experimental evidence that pillar-based PnCs can achieve coherent thermal conductance reduction at sub-Kelvin temperatures, rivaling the performance of fragile hole-based PnCs.
2. Mechanism Identification: The reduction is driven by band flattening due to local pillar resonances hybridizing with membrane modes. This drastically reduces phonon group velocities, which outweighs the concurrent increase in the Density of States (DOS).
3. Coherence Breakdown Threshold: The paper identifies a critical lattice constant threshold ($\approx 1$ µm) where the transport regime transitions from coherent to incoherent.
4. Surface Roughness Impact: It is demonstrated that surface roughness on the Al pillars acts as a diffusive scattering mechanism that destroys coherence for larger pillar sizes, explaining why previous studies (often at higher temperatures or with rougher surfaces) failed to see coherent effects.

4. Key Results

Thermal Conductance Reduction:

• 0.3 µm and 1.0 µm Periods: The experimental thermal conductance showed a strong reduction compared to an unpatterned membrane.
  • The 1.0 µm sample achieved a reduction of up to one order of magnitude (factor of ~10) compared to the unaltered membrane.
  • The temperature dependence ($P \propto T^{3.4}$) for the 1.0 µm sample qualitatively matched the coherent theory, though absolute values were slightly lower.
• 3.0 µm and 5.0 µm Periods: The thermal conductance increased with larger lattice constants, deviating from coherent theory predictions. The data followed a $P \propto T^4$ dependence, characteristic of incoherent 3D bulk phonon transport or diffusive surface scattering (Casimir limit).

Band Structure and Physics:

• Band Flattening: FEM simulations and BLS data confirmed that the pillars introduce flat bands (localized resonant modes) in the phonon dispersion.
• Group Velocity vs. DOS: While the flat bands increased the Density of States (which would theoretically increase conductance), they suppressed the group velocity ($\partial\omega/\partial k$) to near zero. The suppression of group velocity was the dominant factor, leading to a net decrease in thermal power emission.
• Agreement: For the 0.3 µm and 1.0 µm samples, the experimental data agreed well with coherent FEM simulations. For the 3.0 µm and 5.0 µm samples, the data aligned with incoherent Monte Carlo simulations, confirming the breakdown of coherence.

Role of Roughness:

• The Al pillars exhibited significant surface roughness (RMS ~8 nm top, ~70 nm sidewalls).
• For smaller periods (0.3 µm, 1 µm), the dominant phonon wavelengths were larger than the roughness features, allowing for specular (coherent) scattering.
• For larger periods, the effective scattering became diffusive, destroying the phase coherence required for bandgap formation and group velocity suppression.

5. Significance

• Robustness for Applications: Pillar-based PnCs offer a mechanically robust alternative to hole-based PnCs. This is crucial for applications requiring durable nanostructures, such as ultrasensitive radiation detectors (bolometers) and quantum technology devices operating at millikelvin temperatures.
• Design Guidelines: The study establishes that to achieve coherent thermal control with pillars, the lattice constant must be small enough (sub-micron to ~1 µm) such that the dominant thermal phonon wavelengths exceed the surface roughness scale.
• Fundamental Physics: It resolves the long-standing question of whether pillar resonances can effectively control heat flow via coherent wave interference, confirming that with proper fabrication (minimizing roughness) and geometry, they can outperform incoherent scattering mechanisms.
• Future Directions: The authors suggest that further reducing membrane thickness and improving pillar sidewall smoothness could lead to even greater reductions in thermal conductance, potentially approaching the limits predicted by atomistic simulations.