How Surfaces Grow: Research Team Demonstrates Universal 2D Growth

Overview

Crystals, bacterial colonies, flame fronts: the growth of surfaces was first described in the 1980s by the Kardar–Parisi–Zhang equation. Since then, it has been regarded as a fundamental model in physics, with implications for mathematics, biology, and computer science. Now—forty years later—a Würzburg-based research team from the Cluster of Excellence ctd.qmat has achieved the first experimental demonstration of KPZ behavior on 2D surfaces in space and time. This was made possible by sophisticated materials engineering and a bold experimental approach: researchers injected polaritons—hybrid particles composed of light and matter—into the material. The results have been published in Science.

 

Forty Years of Universality in Growth 

The question of how surfaces grow is one of the most fundamental problems in physics. In 1986, three physicists laid the foundation for a universal theory of growth with the Kardar–Parisi–Zhang (KPZ) equation—a framework with wide-ranging applications across physics, mathematics, biology, and computer science. From the dynamics of crystal formation and mathematical system analysis to the growth of cells, populations, and flame fronts—and even the development of machine-learning algorithms—the KPZ universality class applies wherever growth processes are modeled.

 

After the model was first experimentally confirmed for one-dimensional systems based on polaritons in 2022, a Würzburg research team has now tested this powerful framework again in the lab, delivering the world’s first experimental proof for two-dimensional systems and interfaces.

 

Würzburg Research Team Achieves Breakthrough in 2D Quantum System 

“When surfaces grow—whether crystals, bacteria, or flame fronts—the process is always nonlinear and random. In physics, we describe such systems as being out of equilibrium,” explains Siddhartha Dam, a postdoctoral researcher in the Würzburg–Dresden Cluster of Excellence ctd.qmat at the University of Würzburg’s Chair of Technical Physics. “Engineering a system capable of simultaneously measuring how a non-equilibrium process evolves in space and time is extremely challenging—especially because these processes unfold on ultrashort timescales. That’s why verifying the KPZ model in two dimensions has taken so long. We have now succeeded in controlling a non-equilibrium quantum system in the laboratory—something that has only recently become technically feasible.” 

 

To achieve this, the researchers cooled a semiconductor sample based on gallium arsenide (GaAs) to −269.15°C and continuously excited it with a laser. Through precise materials engineering, polaritons—hybrid particles made of photons (light) and excitons (matter)—formed within a specific layer of the structure. Polaritons exist only under non-equilibrium conditions: they are generated by laser excitation and decay within just a few picoseconds before leaving the system.

 

“We can precisely track where the polaritons are in the material. When we pump the system with light, polaritons are created—they grow. Using advanced experimental techniques, we were able to quantify both the spatial and temporal evolution of this growing quantum system and found that it follows the KPZ model,” Dam explains.

 

The key idea—testing a universal theory of growth in a quantum system using polaritons, which themselves exist only within a highly dynamic growth process—was developed by Sebastian Diehl, a professor at the Institute for Theoretical Physics at the University of Cologne and a member of the research team. The theoretical groundwork dates back to 2015. In 2022, a research group in Paris provided the first experimental evidence of KPZ behavior—but only in a one-dimensional system. “The experimental demonstration of KPZ universality in two-dimensional material systems highlights just how fundamental this equation is for real non-equilibrium systems,” says Diehl, commenting on the Würzburg team’s achievement.

 

Targeted Materials Design Enables Injection of Polaritons 

To inject polaritons into the material, the researchers engineered a highly complex sample structure. Mirror layers confine photons within a central “quantum film” layer where they can couple with excitons in the gallium arsenide to form polaritons, grow, and be measured.

 

“By precisely controlling the thickness of individual material layers using molecular beam epitaxy, we were able to tune their optical properties and hence fabricate the necessary highly reflective mirrors under ultra-high vacuum conditions,” explains Simon Widmann, a doctoral researcher at the Chair of Engineering Physics, who conducted the experiments together with Siddhartha Dam. “We control how the material grows atom by atom and can fine-tune all experimental parameters—for example, the laser, which must excite the sample with micrometer precision. This level of control was essential for successfully demonstrating KPZ universality.”