Introduction

The study of the Hubbard model has greatly contributed to our understanding of various physical phenomena, from ferromagnetism to superconductivity. This model has been implemented in quantum simulations using quantum dots in semiconductors, providing a platform for experimental validation of theoretical results. One intriguing result is the prediction of Nagaoka ferromagnetism in single-band systems that are nearly half-filled with strongly interacting electrons. This phenomenon has been experimentally demonstrated in square GaAs quantum dot arrays, affirming the validity of theoretical predictions. However, the realization of similar systems in other materials, such as black phosphorene, still presents challenges.

Recent developments in black phosphorene, a monolayer form of black phosphorus, have attracted significant attention due to its strongly correlated phenomena and tunable optical properties. The anisotropy of black phosphorene’s carrier effective masses makes it an ideal candidate for studying electron hopping within quantum dot arrays. Furthermore, the high effective mass of electrons in phosphorene enhances the impact of electron-electron interactions, making it an attractive platform for investigating Nagaoka ferromagnetism driven by interactions.

**FAQ:**

**Q: What is Nagaoka ferromagnetism?**

A: Nagaoka ferromagnetism is the complete spin polarization of a system with strongly interacting electrons that are nearly half-filled. This state is energetically favorable as it reduces the kinetic energy of the system.

**Q: What is black phosphorene?**

A: Black phosphorene is a monolayer form of black phosphorus, a two-dimensional material with unique electronic and optical properties.

**Q: What are quantum dot arrays?**

A: Quantum dot arrays are arrangements of semiconductor quantum dots, which are nanoscale structures capable of confining a small number of electrons.

**Theory**

In this study, we focus on exploring Nagaoka ferromagnetism in phosphorene quantum dot arrays. Given the anisotropic nature of the effective mass in phosphorene, the hopping integrals between quantum dots cannot be isotropic like in GaAs. We develop an anisotropic effective mass model and optimize the geometry and parameters of the array to maximize the Nagaoka gap, which represents the energy required to disrupt the ferromagnetic state.

**FAQ:**

**Q: What is the Nagaoka gap?**

A: The Nagaoka gap is the energy difference between the ferromagnetic ground state and other possible states in a system exhibiting Nagaoka ferromagnetism. A larger Nagaoka gap indicates greater stability of the ferromagnetic state.

**Conclusion**

In conclusion, this study highlights the potential for exploring Nagaoka ferromagnetism in phosphorene quantum dot arrays. By considering the anisotropic nature of the effective mass in phosphorene, we demonstrate the strong stability of Nagaoka ferromagnetism and its large Nagaoka gap. The results obtained pave the way for experimental verification of Nagaoka ferromagnetism in phosphorene quantum dot arrays and offer insights into the underlying physics of this phenomenon.

Sources:

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