Can a NodeLess Wave Function Have Higher Energy than NodeFull Ones?
© The Physical Society of Japan
This article is on
Discovery of Peculiar Electronic Structures of Decavacancy V10 in Silicon Crystal
(JPSJ Editors' Choice)
J. Phys. Soc. Jpn. 91, 064709 (2022).
“The energy level of an electron state increases as the number of nodes in its wave function increases.” The preceding statement, often found in textbooks, was challenged by our largescale DFT (densityfunctional theory) calculations performed for the decavacancy in Si crystal.
In quantum mechanics textbooks, a nodeless wave function is described as having lower energy than that with nodal planes, because the kinetic energy increases with an increasing number of nodes. This statement has been thought to be universal.
However, we have recently found an exceptional case in the decavacancy of Si crystal, where nodeless mixing of four danglingbond orbitals (φ_{A}, φ_{B}, φ_{C}, and φ_{D}) in the singlet state leads to a higher energy than nodefull mixing in the triplet states. That is,
where
(+φ_{A} φ_{B }φ_{C }+φ_{D})/2,
(+φ_{A}φ_{B }+φ_{C }φ_{D})/2.
This unexpected energy ordering is the enigma discussed and solved in our study.
Decavacancy V_{10}. is one of the magic number vacancies, obtained by removing a Si_{10 }cluster from an otherwise perfect Si crystal. Although the vacancy is accompanied by 16 dangling bonds, 12 of them are rebonded with adjacent ones. Thus, we have only four dangling bonds remaining in V_{10}. The four danglingbond orbitals (φ_{A}, φ_{B}, φ_{C}, and φ_{D}) are arranged under the T_{d} symmetry, which mix to generate the singlet and triplet electron states in the band gap, as described by Eq. (2). Our finding in Eq. (1) is surprising, because it seems to contradict a universal rule stated in the textbooks.
To clarify the underlying physics, we constructed a model to reproduce the energy ordering in Eq. (1), and successfully showed that such a nonintuitive electronic structure originates from the slight hybridization of the four danglingbond states in the band gap with the 12 rebond states outside the band gap. Here, the “rebond states” refers to the six bonding states in the valence band and the six antibonding states in the conduction band, generated when 12 dangling bonds are paired with adjacent ones. Although such pairing makes the rebond states inactive, we found that the passivation was not perfect. They are slightly hybridized with the four danglingbond states in the band gap, which make the energy of the nodeless singlet higher than that of the nodefull triplet.
We argue that this peculiar electronic structure is reflected in the JahnTeller instability of the system. We also note that largescale DFT calculations are paramount for this work, because the supercell size must be large enough for the results of calculations to converge into Eq. (1).
(Written by K. Uchida on behalf of all authors.)
Discovery of Peculiar Electronic Structures of Decavacancy V10 in Silicon Crystal
(JPSJ Editors' Choice)
J. Phys. Soc. Jpn. 91, 064709 (2022).
Share this topic
Fields
Related Articles

What Determines NonNewtonian Flow Behavior in GlassForming Liquids?
Crossdisciplinary physics and related areas of science and technology
Structure and mechanical and thermal properties in condensed matter
2023320
Even minute structural changes can lead to significant reductions in the flow resistances of glassforming liquids. Here, possible scenarios and predictions for two different classes of glassforming liquids are provided.

Clockwise or Anticlockwise, That is the Question: Phonons with Angular Momentum in Chiral Crystals
Structure and mechanical and thermal properties in condensed matter
202337
Chiral crystals have lattice structures with no mirror or inversion symmetries.
A few basic questions about their unique phonon excitations with intrinsic angular momentum are answered. 
AngleResolved Photoelectron Spectroscopy Microscopy: A Tool to Accelerate Nanomaterials Research
Electronic structure and electrical properties of surfaces and nanostructures
Structure and mechanical and thermal properties in condensed matter
Crossdisciplinary physics and related areas of science and technology
2023210
Researchers have published a practical guide on new uses of photoelectron microscopy combined with valence band dispersion analysis. They visualized severalmicrometerswide graphite facets and precisely characterized the band structure.

Powered by Machine Learning: Obtaining Spectral Conductivity and Chemical Potential of Thermoelectric Materials from Experimental Data
Structure and mechanical and thermal properties in condensed matter
2023131
We propose a machinelearning method to obtain the fundamental physical quantity, namely, the spectral conductivity, from experimental data of thermoelectric coefficients. Our study introduces a new datadriven approach that reveals the underlying lowenergy electronic states of highperformance thermoelectric materials.

Phonon Simulations and Its Applications
Structure and mechanical and thermal properties in condensed matter
20221226
Computation of phonons using first principles has many applications for understanding crystal properties. This review provides an overview of the present capabilities of such simulations using finite displacement supercell approach.