# Faculty of Engineering

Research Highlights

## Theory of Relativity Meets Mechanical and Electrical Systems

A material is the aggregate of its atoms, which accordingly consist of many electrons. The behavior of these electrons determines the various properties or characteristics of a material.

Quantum mechanics, which was initiated in the early 20th century, has been a strong tool for clarifying the behavior of electrons. In particular, the Schrödinger equation has been core concept as the fundamental equation in quantum mechanics theory, because the behavior of electrons can be described by finding an approximate solution to the Schrödinger equation, and various material characteristics including mechanical, electrical, magnetic, optical, thermal, and chemical properties can be derived from the techniques employed in quantum physics and quantum chemistry.

On the other hand, the Schrödinger equation is just a non-relativistic limit in quantum mechanics; namely, no relativistic effects are included in the information obtained from the Schrödinger equation. Some physical and chemical characteristics cannot be practically described via non-relativistic treatment. For examples, the equilibrium distance of metal dimer, the chemical shift of nuclear magnetic resonance, and that the photoelectron spectrum cannot be evaluated quantitatively for some systems without considering the relativistic effect are all well-known.

In Dr. Nakamura’s Quantum Materials Physics and Chemistry (QMPC) Laboratory at KUAS, researchers are studying not only analyzing the electronic processes in chemical reactions but multiplex topics related to their material properties. For example, thermoelectric and electromechanical properties are important research targets from the viewpoint of quantum mechanics. Piezoresistivity is a typical electromechanical property in materials representing a change in the electrical resistivity when mechanical strain or stress is applied, and it is a widely utilized principle for mechanical sensors such as accelerometers and gyroscopes. The sensing element of piezoresistors is generally silicon, the most fundamental semiconductor material, but the behavior of holes (that is, the lack of electrons) in positively doped silicon cannot be exactly expressed via the non-relativistic Schrödinger equation because of the spin-orbit coupling at the valence-band top introduced naturally from relativistic quantum mechanics. Thus, for some material systems, the exact mechanism of piezoresistive sensors cannot be clarified without considering Einstein’s theory of relativity.

In these cases, the Dirac equation is available for use as the fundamental equation in quantum mechanics expanded to the theory of special relativity. Relativistic quantum mechanics based on the Dirac equation in material systems can derive material properties that non-relativistic techniques cannot explain. More qualitative and quantitative analysis of sensor mechanisms and of material properties can be achieve through novel techniques with relativistic quantum mechanics introduced by the QMPC Lab. For example, the Dirac equation is generally described with a 4-component spinor that consists of two of the large components representing the electronic state principally and two of the small components representing the positronic state principally. The 4-component spinor requires too much of a higher computational cost compared to a general wave function under the Schrödinger equation for application to large atomistic models corresponding to material systems. Thus, Dr. Nakamura has introduced the relativistic 2-component Dirac-Kohn-Sham theory for material systems via the combination of the density functional theory and the elimination of the small component method. For the application of this relativistic quantum-mechanical technique, transition-metal dichalcogenides become the most important research targets for the QMPC Lab members as post-graphene 2-dimensional materials with effective spin-orbit couplings, leading to giant piezoresistivity and other important material properties.

Quantum mechanics, which was initiated in the early 20th century, has been a strong tool for clarifying the behavior of electrons. In particular, the Schrödinger equation has been core concept as the fundamental equation in quantum mechanics theory, because the behavior of electrons can be described by finding an approximate solution to the Schrödinger equation, and various material characteristics including mechanical, electrical, magnetic, optical, thermal, and chemical properties can be derived from the techniques employed in quantum physics and quantum chemistry.

On the other hand, the Schrödinger equation is just a non-relativistic limit in quantum mechanics; namely, no relativistic effects are included in the information obtained from the Schrödinger equation. Some physical and chemical characteristics cannot be practically described via non-relativistic treatment. For examples, the equilibrium distance of metal dimer, the chemical shift of nuclear magnetic resonance, and that the photoelectron spectrum cannot be evaluated quantitatively for some systems without considering the relativistic effect are all well-known.

In Dr. Nakamura’s Quantum Materials Physics and Chemistry (QMPC) Laboratory at KUAS, researchers are studying not only analyzing the electronic processes in chemical reactions but multiplex topics related to their material properties. For example, thermoelectric and electromechanical properties are important research targets from the viewpoint of quantum mechanics. Piezoresistivity is a typical electromechanical property in materials representing a change in the electrical resistivity when mechanical strain or stress is applied, and it is a widely utilized principle for mechanical sensors such as accelerometers and gyroscopes. The sensing element of piezoresistors is generally silicon, the most fundamental semiconductor material, but the behavior of holes (that is, the lack of electrons) in positively doped silicon cannot be exactly expressed via the non-relativistic Schrödinger equation because of the spin-orbit coupling at the valence-band top introduced naturally from relativistic quantum mechanics. Thus, for some material systems, the exact mechanism of piezoresistive sensors cannot be clarified without considering Einstein’s theory of relativity.

In these cases, the Dirac equation is available for use as the fundamental equation in quantum mechanics expanded to the theory of special relativity. Relativistic quantum mechanics based on the Dirac equation in material systems can derive material properties that non-relativistic techniques cannot explain. More qualitative and quantitative analysis of sensor mechanisms and of material properties can be achieve through novel techniques with relativistic quantum mechanics introduced by the QMPC Lab. For example, the Dirac equation is generally described with a 4-component spinor that consists of two of the large components representing the electronic state principally and two of the small components representing the positronic state principally. The 4-component spinor requires too much of a higher computational cost compared to a general wave function under the Schrödinger equation for application to large atomistic models corresponding to material systems. Thus, Dr. Nakamura has introduced the relativistic 2-component Dirac-Kohn-Sham theory for material systems via the combination of the density functional theory and the elimination of the small component method. For the application of this relativistic quantum-mechanical technique, transition-metal dichalcogenides become the most important research targets for the QMPC Lab members as post-graphene 2-dimensional materials with effective spin-orbit couplings, leading to giant piezoresistivity and other important material properties.