🔬 mesoscale physics

Localized and quasi-localized energy levels in the electron spectrum of graphene with isolated boron and nitrogen substitutions

This paper analyzes the local Green functions of low-concentration boron and nitrogen impurities in graphene to determine the conditions and characteristics for the formation of discrete energy levels outside the band and quasi-localized states near the Fermi level, providing analytical expressions for boron-induced levels and investigating nitrogen-induced states.

Original authors: S. B. Feodosyev, V. A. Sirenko, E. S. Syrkin, E. V. Manzhelii, I. S. Bondar, K. A. Minakova

Published 2026-11-03

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Original authors: S. B. Feodosyev, V. A. Sirenko, E. S. Syrkin, E. V. Manzhelii, I. S. Bondar, K. A. Minakova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a sheet of carbon atoms, arranged in a perfect honeycomb pattern, so thin that it is essentially two-dimensional. This material, known as graphene, behaves like a semiconductor with no gap between its energy levels, allowing electrons to zip through it with almost no resistance. In a perfect sheet, these electrons move freely, but the real world is rarely perfect. When scientists introduce foreign atoms into this lattice—replacing a carbon atom with something else—they create a disturbance. This disturbance can trap electrons in specific spots, creating what physicists call localized states. These trapped electrons are significant because they alter how the material conducts electricity and could potentially be used to build new types of electronic devices or even induce superconductivity, a state where electricity flows with zero resistance. Understanding exactly how these impurities change the behavior of graphene is crucial for the future of nanotechnology.

A team of researchers from Ukraine set out to map out exactly how these trapped electron states form when specific impurities are added to graphene. They focused on two common scenarios: replacing a carbon atom with boron, and replacing it with nitrogen. Using advanced mathematical models to simulate the behavior of electrons in a grid of atoms, they calculated how the energy levels of these electrons shift when an impurity is present. Their work distinguishes between two types of trapped states. The first type involves electrons that are completely stuck in a specific energy range outside the normal flow of the material, creating discrete, isolated energy levels. The second type involves electrons that are only partially trapped, hovering near the material's natural energy limit, known as the Fermi level, where they can still interact with the rest of the electron sea.

For the boron impurity, the researchers found that the disturbance is strong enough to push electrons into these isolated, discrete energy levels that sit entirely outside the normal range of motion for electrons in graphene. They developed a set of straightforward rules to predict exactly where these energy levels would appear and how strong the signal of these trapped electrons would be. Their calculations showed that these levels are not just theoretical; they are robust features that depend on how much the boron atom differs from the carbon atom it replaces. Because these levels are so distinct and separated from the rest of the electron spectrum, the researchers suggest that boron-doped graphene could act as a "two-level" system. This is a specific configuration that is highly desirable for quantum computing, where information is stored in discrete states rather than continuous flows.

In contrast, when they looked at nitrogen impurities, the story changed. Nitrogen does not push electrons out of the normal range. Instead, it creates states that are "quasi-localized," meaning the electrons are loosely held near the Fermi level, the specific energy point where the material's electronic activity is most intense. The researchers found that these nitrogen-induced states pile up right around this critical energy level. This accumulation is significant because a higher density of electrons at this specific energy could, in theory, make the graphene more likely to transition into a superconducting state. While the paper does not claim to have created a superconductor, it provides a clear mechanism showing how introducing nitrogen could increase the population of electrons in the exact region where superconductivity might emerge.

The researchers validated their mathematical predictions by comparing them with more complex, high-resolution simulations. The results matched closely, giving them confidence in their simplified formulas. They demonstrated that by measuring the energy of these trapped states in a real experiment, scientists could work backward to determine the exact nature of the impurity and how it interacts with the surrounding carbon atoms. This ability to decode the defect structure from the energy signature is a powerful tool for designing nanomaterials with specific properties. Ultimately, the study clarifies that not all impurities behave the same way: some, like boron, create sharp, isolated islands of energy useful for quantum logic, while others, like nitrogen, create a crowded zone of activity that might unlock the potential for superconductivity in graphene nanostructures.

Technical Summary: Localized and Quasi-Localized Energy Levels in the Electron Spectrum of Graphene with Isolated Substitutional Impurities

Problem Statement
Graphene is a two-dimensional semiconductor with a zero band gap, characterized by a linear (relativistic) electron dispersion near the Dirac points ( $K$ ) and a V-shaped density of states (DOS) singularity at the Fermi level. While ideal graphene possesses translational symmetry, real-world applications involve graphene on substrates where controlled introduction of defects, such as isolated substitutional impurities, is possible. The presence of these impurities can alter the electronic structure by creating localized states. Specifically, the paper addresses two distinct phenomena:

The formation of Local Discrete Levels (LDL) outside the quasi-continuous spectrum (QCS) band, which act as isolated bound states.
The formation of Quasi-Localized States (QLS) with energies near the Fermi level, which may increase the population of states at the Fermi energy and potentially influence superconducting transitions.

While LDLs have been studied for decades in various quasiparticle spectra, a rigorous analytical description for realistic graphene models (specifically for substitutional impurities like Boron and Nitrogen) has been lacking.

Methodology
The authors employ the tight-binding approximation, considering interactions only between nearest neighbors. The system is modeled using a Hamiltonian where impurity atoms differ from carbon atoms in two parameters: the on-site energy ( $\epsilon_0$ ) and the overlap integral ( $J_0 = J(1+\eta)$ ).

To analyze the electron spectrum, the authors utilize the method of Jacobian matrices (J-matrix method). This approach allows for the direct calculation of local partial Green's functions without explicitly determining dispersion laws, making it highly effective for systems with violated translational symmetry.

Green's Function Analysis: The local Green's function (LGF) is calculated using a generating vector corresponding to the impurity excitation. The Hamiltonian is represented as a tridiagonal J-matrix.
Analytical Approximation: The authors derive an analytical approximation for the LGF using the "two-moment approximation" (rank $n=1$ of the J-matrix). This approximation is shown to be highly accurate for energies outside the QCS band, despite converging slowly near van Hove singularities within the band.
Lifshitz Equation: The conditions for the existence of impurity states are determined by solving the Lifshitz equation, $Re G(\epsilon) = S(\epsilon, \Lambda)$ , where $S$ is determined by the perturbation operator.
Specific Impurities: Calculations are performed for Boron (forming LDLs outside the band) and Nitrogen (forming QLS near the Fermi level).

Key Contributions and Results

Analytical Expressions for Local Discrete Levels (Boron):
- The paper derives simple analytical expressions for the energy, intensity, and damping parameters of LDLs formed by Boron impurities.
- Energy: The energies of the LDLs are determined as the poles of the perturbed Green's function. The authors provide explicit formulas (Eq. 13) relating these energies to the impurity parameters ( $\epsilon_0, \eta$ ) and the lattice bandwidth.
- Existence Conditions: The analysis reveals that LDLs exist in a wide range of $\epsilon$ and $\eta$ values. A threshold exists for the overlap integral parameter $\eta$ ; if $\eta > \sqrt{3/2} - 1$ , at least one local discrete level is guaranteed to form.
- Intensity and Decay: The paper calculates the "intensity" (residue) of the LDLs on the impurity atom and derives the decay law of the state intensity with distance from the impurity. The intensity decays as a geometric progression.
- Validation: The analytical results derived from the two-moment approximation show excellent agreement with numerical calculations using high-rank J-matrices ( $n \geq 100$ ).
Quasi-Localized States (Nitrogen):
- For Nitrogen impurities, the real part of the Green's function intersects the perturbation function within the QCS band near the Fermi level.
- This leads to the formation of quasi-localized states (QLS) rather than discrete levels outside the band.
- The local density of states (LDOS) for Nitrogen shows quasi-local maxima near the Fermi energy, distinct from the Boron case where the LDOS area is reduced by the formation of discrete levels.
Reconstruction of Defect Parameters:
- The derived analytical formulas allow for the inverse problem to be solved: experimental measurements of LDL energies (e.g., via scanning tunneling microscopy) can be used to reconstruct the specific defect parameters ( $\epsilon_0$ and $\eta$ ) of the impurity.

Significance and Claims
The paper claims that the primary significance of this work lies in providing simple analytical expressions that relate the main characteristics of localized states (energy, intensity, damping) directly to the physical parameters of the defect and the lattice. This bridges the gap between general theoretical descriptions and specific, realistic models of graphene.

For Boron: The formation of two local discrete levels separated by more than the width of the quasi-continuous spectrum suggests that Boron-doped graphene can be considered a "two-level" system. The authors note this is of interest for nanoelectronics and quantum computing.
For Nitrogen: The analysis of QLS near the Fermi level supports the hypothesis that defect-induced states can increase the population of the Fermi level. The authors state this retains "hope" for the discovery of a superconducting transition in graphene nanostructures, though they do not claim to have proven superconductivity.
Methodological: The work demonstrates the efficacy of the J-matrix method combined with the two-moment approximation for describing defect states in 2D systems, offering a computationally efficient alternative to full numerical diagonalization for determining impurity characteristics.

The authors conclude that these findings provide a theoretical basis for extracting defect structure information from experimental data and for the potential engineering of nanomaterials with predetermined spectral characteristics.

Technical Summary: Localized and Quasi-Localized Energy Levels in the Electron Spectrum of Graphene with Isolated Substitutional Impurities

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