Computational
Condensed Matter Theory
Read about my work under Dr. Jianxin Zhu at the Physics of Condensed Matter and Complex Systems Theoretical Division (T-4) of Los Alamos National Laboratory studying the Magnetic Properties of 2D van der Waals Heavy Fermion Materials
Magnetic Properties of 2D van der Waals Heavy Fermion Materials
The advent of integrated circuits in the 1950s revolutionized electronics by enabling the creation of smaller, faster, and more powerful devices. Integrated solid-state electronics have become one of the most successful technologies in human history, but their growing energy consumption and environmental impact underscore the need for more energy- efficient alternatives. Two-dimensional (2D) quantum materials, with thicknesses on the atomic scale, oHer a promising solution. The reduction in dimensionality enhances confinement and interaction strength, making quantum eHects more pronounced compared to their bulk counterparts.
Spintronics devices, which use electron spin as an information carrier, are predicted to be a core technology in the next information industry revolution. To support this innovation, investigating intrinsic low-dimensional magnetism in 2D van der Waals (vdW) materials is crucial. The discovery of the first 2D van der Waals material, graphene, in 2004 opened new doors to studying the control of emergent quantum phenomena; while heavy fermion behavior is not intrinsic to the system, stacking twisted sheets of graphene can induce unconventional superconductivity. This breakthrough prompted the question of whether an intrinsic strongly correlated 2D van der Waals material can be realized. Electronic structure calculations have identified our material of interest as an ideal 2D vdW heavy-fermion candidate. Additionally, experimental evidence confirms that our material of interest exhibits intrinsic 2D quantum properties, providing a new platform to explore quantum critical points, unconventional superconductivity, and other emergent phenomena.
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In this project, we apply the WIEN2k code and density functional theory (DFT) to theoretically probe the magnetic ground state of a strongly correlated 2D van der Waals system. To account for the strong localization of d and f electrons in this material, we employ DFT+U, which adds a Hubbard-like term to correctly treat the strong on-site Coulomb interactions of these localized electrons, while applying standard DFT approximations to the rest. Ultimately, this project can be extended to analyze the electronic structure of this material in response to defect/vacancy effects, with the goal of uncovering potential quantum phase transitions and advancing the development of novel, energy-efficient electronic devices.
Theoretical Foundation: Density Functional Theory
When describing a many-electron system, several factors contribute to the effective potential experienced by each electron. These include a known external potential v(r) and the Hartree potential derived from the electrostatic (Coulomb) force between electrons, represented by the integral in Equation 2.6 (Fig. 1). Given the electron density of the system, these terms can be calculated exactly. However, knowledge of these potentials alone is generally not enough to fully characterize such a physical situation, as they only account for its non-interacting aspects. In DFT, the exchange and correlation energy, (E_XC), represents the part of the total energy that is unaccounted for by the Hartree approximation described above. The exchange energy is a correction that accounts for the Pauli exclusion principle, which excludes certain electron configurations that would otherwise introduce a large repulsive energy. The correlation energy accounts for how the motion of each electron depends on surrounding electrons, rather than solely considering the electrostatic case. Since there is no general expression for these energies, they are combined into EXC; the goal of DFT is to approximate this term. Although its general form is unknown, there is one instance in which the exchange and correlation energy can be exactly derived. In a homogeneous electron gas, i.e. a gas of free electrons with uniform density at all points in space, this term can be calculated from the density using the formula in Equation 2.3 (Fig. 1). In this equation, the term εXC represents the exchange energy per electron in such a system. Suppose the density of a system is slowly varying (i.e. rs/r0 << 1). At a given point in time, each electron can be treated as a particle in a homogeneous electron gas, with density given by that of its immediate surroundings.
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Then, the exchange and correlation energy for such a gas can be precisely calculated; the resulting value is a suitable approximation for the actual EXC, provided that the slowly varying assumption is valid. This is the above mentioned local density approximation, and is foundational to early DFT calculations for materials such as monovalent metals and alloys, which can be reasonably approximated with the homogeneous electron gas model except at their boundaries. With knowledge of the electron density of a system, we can use this assumption to describe the exchange and correlation energy of an inhomogeneous, interacting system by treating it as a homogenous gas at each point in time and space.
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A pivotal feature of the Kohn-Sham equations is their inherent self-consistency; the electron density used to calculate the effective potential is the same as the density derived by solving the KS equations using that potential.
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The computational procedure to solve the many-electron problem using Kohn-Sham DFT is an iterative loop. First, an initial guess for the electron density, n(r), is established. This guess is inserted into Equations 2.6 and 2.7 in order to calculate the effective potential experienced by an electron at each spatial point. The terms of Equation 2.6 represent the known external field and the Hartree approximation for electrostatic interactions. Equation 2.7 describes the exchange-correlation potential for a homogeneous electron gas with density given by n(r) at a given point, which approximates the true exchange and correlation effects at that point if the LDA holds.
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The sum of these two equations gives the "effective potential", which is subsequently substituted into the Schrödinger equation (Equation 2.8). This yields the system's wave function for a given position in space and serves as the basis for computing the electron density distribution of the system at any location r, as governed by Equation 2.9. This generated density is then compared to the initial guess. If the two densities agree, the process is complete; if they differ, the calculated density is used as a new initial guess and the iterative process repeats until the effective potential and electron density have stabilized, therefore obtaining a self-consistent solution for the electronic structure of the system. The iterative self-consistent nature of KS DFT provides a computationally efficient means to describe the quantum mechanical behavior of electrons in various materials.
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(Above is an excerpt from PHYS 4E (Quantum Mechanics) Final paper, written by Alison Rhoads, Mel Conti-Chen and Noah Lopez)
