Berry Phase Effects on Electronic Properties (0907.2021v1)

Abstract: Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

• The paper presents a semiclassical formulation of electron dynamics, demonstrating how Berry curvature induces anomalous Hall currents.
• It elucidates the quantum Hall effect by linking non-zero Chern numbers in band structures to quantized conductivities.
• The study extends its framework to orbital magnetization and thermoelectric effects, paving the way for innovative topological material applications.

Summary of "Berry Phase Effects on Electronic Properties"

The paper "Berry Phase Effects on Electronic Properties" by Di Xiao, Ming-Che Chang, and Qian Niu presents a comprehensive overview of the influence of Berry phase on various electronic properties of materials. The focus is predominantly on the field of solid-state physics, where the Berry phase has been identified as a pivotal component in understanding phenomena such as ferroelectricity, orbital magnetism, and different Hall effects.

Key Contributions and Discussions

The paper is structured around several primary themes that illustrate how the Berry phase enriches our understanding of electronic dynamics:

1. Semiclassical Formulation of Electron Dynamics: A central theme of the paper is the semiclassical approach, which remains a robust method for examining electron dynamics under electromagnetic fields and other perturbations. The authors emphasize the significant impact of the Berry curvature on electron velocity, which results in a transverse anomalous velocity influencing Hall currents.
2. Berry Phase in Solid State Physics: The narrative on Berry phase in solid state physics underscores its emergence in diverse material properties. The paper describes how Berry curvature transforms the conventional perception of electron dynamics, accentuating its role in phenomena like the anomalous and quantum Hall effects.
3. Quantum Hall Effect (QHE): The paper revisits the QHE, illustrating that a non-zero Chern number in band structures accounts for the quantized Hall conductivity, a key feature of two-dimensional insulators. This topological interpretation aligns with the geometric nature of the Berry phase.
4. Anomalous Hall Effect (AHE): The authors explore the intrinsic and extrinsic contributions to AHE, pointing to the Berry curvature-induced anomalous velocity as a key intrinsic mechanism, distinct from skew scattering and side jump effects.
5. Orbital Magnetization and Thermoelectric Effects: A critical insight provided is the derivation of orbital magnetization using Berry curvature and Berry-phase-induced modifications to density of states. The extension to thermoelectric effects, such as anomalous Nernst, demonstrates the versatility of the theoretical framework in describing cross-phenomena between spin, thermal, and charge channels.
6. General Perturbations and Deformed Crystals: The discussion on electron dynamics under more general perturbations, including lattice deformations, and the role of Berry curvature in such contexts, marks the paper as a valuable resource for exploring strained systems and spin textures.
7. Re-quantization of Semiclassical Theory: The paper doesn't shy away from connecting the semiclassical theory to a fully quantized treatment. This is particularly highlighted in contexts such as Bloch oscillations and magnetic Bloch bands, showing how Berry phase affects both classical trajectories and quantum state densities.

Implications and Future Directions

The work is a testament to the ubiquity and profound influence of Berry phase in theoretical and computational material science. It suggests several implications and future paths, such as:

• Topological Materials and Devices: The understanding of Berry phase effects informs the development of topological insulators and spintronic devices, providing pathways for engineering new materials with unique electronic properties.
• First-principles Calculations: The methodology outlined could enrich first-principles studies by incorporating Berry phase and curvature effects, potentially leading to more accurate predictions of material properties.
• Broadening the Impact Across Fields: The approaches discussed can extend beyond solid-state physics to broader material systems, such as photonic crystals or ultracold atomic systems, modeling novel quantum phenomena with topological characteristics.

In conclusion, the paper offers a comprehensive exploration of Berry phase, urging further inquiry into this fundamental phenomenon that bridges quantum mechanics with tangible material properties. Its pedagogical approach provides both breadth and depth, serving as an essential reference in the field of condensed matter physics.