Electron Crystals’ Melting Points Accurately Predicted Using New Theoretical Model

Scientists have long sought to understand the melting behaviour of two-dimensional quantum crystals, a process governed by the creation and movement of defects within the material. Now, H. Xia, Qianhui Xu, and Jiasen Niu from the Leiden Institute of Physics, Leiden University, alongside Jian Sun and Yang Liu working with colleagues at the International Center for Quantum Materials, Peking University, and Pengjie Wang, Bo Yang, and Xi Lin from the Department of Physics, University of Illinois Urbana-Champaign, present a significant advance in predicting the melting temperature of these systems. Their research, combining transport experiments on ultraclean GaAs/AlGaAs wells with advanced theoretical modelling, quantitatively predicts the melting point of quantum Hall bubble crystals across multiple Landau levels. This agreement validates the bubble-crystal interpretation of these states and establishes a robust framework for understanding defect-mediated melting in strongly interacting electronic solids, offering a readily adaptable approach for investigating other electronic crystals such as those found in moiré Chern bands.

This work focuses on quantum Hall electron bubble phases, exotic states of matter formed in ultra-clean semiconductor structures, and establishes a predictive framework for understanding the behaviour of strongly interacting electronic solids.

Researchers combined precise transport experiments with advanced theoretical modelling to quantitatively capture the solid-liquid phase transition boundaries across a range of conditions. The study validates the interpretation of these phases as “bubble crystals” and demonstrates that their melting is driven by the proliferation and unbinding of topological defects, imperfections in the crystal structure.
Predicting the melting point of any solid is difficult, but becomes particularly complex at extremely low temperatures where quantum fluctuations dominate. Two-dimensional materials present an additional hurdle, as true long-range crystalline order is fundamentally impossible at any finite temperature according to the Mermin-Wagner theorem.

The Kosterlitz, Thouless, Halperin, Nelson, Young (KTHNY) theory describes how these 2D materials melt through the creation of defects, but accurately predicting the melting temperature has remained elusive, especially in strongly correlated systems. This research overcomes these limitations by employing a sophisticated approach that incorporates Landau-level quantization, the energy levels electrons occupy in a magnetic field, and many-body interactions.

The investigation utilised high-mobility gallium arsenide/aluminium gallium arsenide quantum wells to create the electron bubble phases, probing them using Corbino-geometry transport experiments. These experiments, combined with Hartree, Fock elasticity and the full KTHNY melting criterion, allowed for a detailed analysis of the melting process.

The theoretical calculations, incorporating a finite-temperature renormalization-group calculation, yielded melting temperatures that precisely matched the experimental measurements for Landau levels 2 through 5. This agreement confirms that the observed melting is indeed driven by defects and provides a robust method for studying the energetics of these imperfections.

This work not only advances our understanding of quantum Hall physics but also opens avenues for exploring other electronic crystals, including Wigner crystals found in twisted van der Waals materials. The ability to accurately predict and control the melting behaviour of these materials could pave the way for novel electronic devices and a deeper understanding of strongly correlated electron systems. The approach developed in this study provides a quantitatively reliable method for probing screening and defect physics in quantum Hall solids, offering a powerful tool for future research in this field.

Reentrant integer quantum Hall states reveal electron bubble phase formation in a GaAs/AlGaAs heterostructure

Corbino-geometry transport measurements underpin this work, employing an ultraclean GaAs/AlGaAs heterostructure to investigate electron bubble phases within Landau levels 2 through 5. This technique, utilising a circular Hall bar geometry, allows for precise determination of the material’s response to magnetic fields and isolates the effects of electron-electron interactions.

The 2DEG, confined within a 30nm-wide AlGaAs/GaAs/AlGaAs quantum well doped with a delta layer to achieve a carrier density of 2.6x 10 11cm -2 and mobility of 2.8x 10 7cm 2 /V s, was chosen for its exceptional material quality and demonstrated reentrant integer quantum Hall states. These RIQH states signify the formation of the electron bubble phases central to the study.

To complement the experimental approach, Hartree-Fock elasticity calculations were performed, a method that approximates the many-body interactions between electrons by replacing them with an average field. This simplification allows for a tractable theoretical description of the solid phase, capturing the essential energetic contributions to lattice stability.

Crucially, these calculations were projected onto the Landau levels present in the 2DEG, accounting for the quantization of electron motion in a strong magnetic field and the resulting reshaping of electron wavefunctions. This projection accurately models the spatial separation of electrons and the weakening of short-range Coulomb repulsion observed in high Landau levels.

A key methodological innovation lies in the application of the full Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) melting criterion, a theoretical framework describing continuous phase transitions driven by topological defects. This criterion was extended with a finite-temperature renormalization-group calculation, enabling a robust prediction of the melting temperature by accounting for thermal fluctuations. By combining these theoretical tools with the precise experimental data, the research establishes a predictive framework for understanding the behaviour of strongly interacting electronic solids.

Melting temperatures and Landau level transitions in two-dimensional electron bubble phases

Measurements reveal solid-liquid transition temperatures reaching 150 millikelvin for electron bubble phases confined within a two-dimensional electron gas. These temperatures were determined by tracking conductivity minima as a function of temperature, identifying peaks in the non-monotonic temperature dependence as the melting point.

The study focused on Landau levels 2 through 5, observing that the maximum thermal stability, and thus the highest melting temperature, occurred at an optimal filling factor before decreasing towards adjacent values. Analysis of the data demonstrates a discrete jump in melting temperature when transitioning between Landau levels, with ratios of 1.35 observed for the 2 to 3 transition and 1.17 for the 3 to 4 transition.

These values align reasonably with theoretical predictions of 1.49 and 1.07, respectively, derived from a classical Kosterlitz, Thouless, Halperin, Nelson, Young (KTHNY) melting criterion assuming an effective charge related to the number of electrons per bubble. While this classical model captures the qualitative inter-level jumps, it underestimates absolute melting temperatures by an order of magnitude and fails to accurately describe the functional dependence within a single Landau level.

Specifically, experimental data shows a nearly linear relationship between melting temperature and magnetic field, contrasting with the square-root scaling predicted by the classical theory. The research establishes that the observed melting temperatures are significantly higher than those predicted by simple classical models, indicating the importance of considering more complex interactions within the electron bubble phases. This work provides a quantitative validation of defect-mediated melting as a predictive framework for strongly correlated quantum Hall electron solids.

The Bigger Picture

Scientists have long struggled to predict when two-dimensional materials will transition from a solid, ordered state to a disordered, liquid-like one. This isn’t merely an academic puzzle; understanding this ‘melting’ process is crucial for designing new electronic devices that rely on the precise control of electrons confined to ultra-thin layers.

Conventional theories often fall short because they struggle to account for the complex interactions between electrons in these materials, particularly the role of defects, imperfections in the crystal structure. This research offers a significant step forward by successfully predicting the melting temperature of an ‘electron bubble crystal’ formed in a semiconductor heterostructure.

The breakthrough lies in a sophisticated combination of experiment and theory, linking direct measurements of electrical conductivity with a detailed model of how defects drive the melting process. By accurately calculating the energy associated with these defects, researchers have validated a long-standing theoretical framework, the Kosterlitz, Thouless, Halperin, Nelson, Young (KTHNY) theory, in a real material system.

This agreement isn’t just about confirming existing ideas; it demonstrates the power of using bulk transport measurements to probe the subtle energetics of topological defects. However, the model, while successful for this specific system, still relies on approximations of electron interactions. The extent to which these findings generalise to other, more complex electronic crystals, like those found in moiré materials, remains an open question.

Future work will likely focus on extending this approach to these novel materials and refining the theoretical models to capture even more nuanced interactions. Ultimately, a deeper understanding of defect-mediated melting could unlock new avenues for manipulating and controlling electronic behaviour at the nanoscale, paving the way for advanced electronic technologies.