Orbital-resolved tuning of electronic thermal conductivity in monolayer h-B2O via doping in the diffusive regime

TL;DR

This paper studies how doping affects the thermal conductivity of a 2D material called h-B2O, showing that it can be tuned in a direction-dependent way.

Contribution

The first calculation of electronic thermal conductivity in h-B2O using the Kubo-Greenwood formalism, revealing orbital-resolved and anisotropic behavior.

Findings

h-B2O shows strong anisotropy in electronic thermal conductivity with higher values along the zigzag direction.

n-type doping enhances ETC via the Pz orbital, while p-type doping causes only minor changes.

Orbital symmetry and spatial orientation significantly influence thermal transport in h-B2O.

Abstract

The highly stable two-dimensional monolayer honeycomb borophene oxide (h-B2O) has attracted considerable interest due to its unique topological features and potential superconducting behavior. In this study, a tight-binding Hamiltonian is constructed by incorporating the Py and Pz orbitals of boron, effectively capturing the essential physics governing the material’s low-energy electronic behavior. Additionally, for the first time, the electronic thermal conductivity (ETC) of monolayer h-B2O is calculated using the Kubo-Greenwood formalism within the diffusive transport regime. The results reveal strong anisotropy (\documentclass[12pt]{minimal}
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\begin{document}$$\:{}^{-1}$$\end{document}along the armchair \documentclass[12pt]{minimal}
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\begin{document}$$\:({\kappa\:}_{xy}$$\end{document}) directions, respectively. Furthermore, we systematically investigate the impact of impurity-induced disorder on ETC in h-B2O under both n-type and p-type doping, employing the T-matrix approximation. In the n-type regime, increasing impurity concentration \documentclass[12pt]{minimal}
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\begin{document}$$\:{n}_{i}$$\end{document} = 2%, 4%, 6% leads to a significant enhancement of the ETC associated with the out-of-plane Pz orbital, attributed to its favorable spatial orientation and higher carrier occupancy. Conversely, the in-plane Py orbital exhibits a reduction in ETC due to increased localization and enhanced electron-electron scattering. Despite this orbital contrast, the total ETC rises along all crystallographic directions, governed by the dominant contribution of the Pz orbital, thereby revealing strong orbital-resolved behavior and pronounced directional anisotropy. In contrast, p-type doping induces only modest changes: the ETC contribution from the Py orbital slightly increases, while that of the Pz orbital is marginally reduced, resulting in an overall weak response of the total ETC. These findings highlight the crucial role of orbital symmetry, spatial orientation, and dopant type in shaping the anisotropic and tunable thermal transport properties of h-B2O. The thermal resilience under p-type doping, alongside the direction-dependent enhancement under n-type doping, positions h-B2O as a promising candidate for nanoscale thermoelectric and thermal management technologies.

Figures

Crystal structure of h-B $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document}$ O without oxygen atoms, showing motif atoms A and B in the unit cell and the basis vectors $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{

Calculated BS at high-symmetry points (left panel) and total DOS (right panel) for pristine monolayer h-B $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document}$ O: (a) Over a wide energy range, (b) In a narrow energy range around the Fermi level, (c) Projected DOS for P $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \use

Schematic of impurity doping on the h-B_2_O surface, showing the random distribution of dopants.

Electronic thermal conductivity of h-B_2_O as a function of temperature with impurity concentration $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathrm{n}}_{\mathrm{i}}$$\end{document}$ = 2, 4, and 6%, and $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin

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Full text

Introduction

In recent decades, two-dimensional (2D) materials have gained a special place in the science community. These materials comprise atomically thin sheets with in-plane covalent and weak interlayer bonding. Additionally, due to the reduction of dimensions, they exhibit unique physical, chemical, and electronic properties. Graphene is known as the first 2D material that was synthesized in 2004 by Geim and his colleagues. It boasts a planar honeycomb structure of carbon atoms^1,2^. After this groundbreaking discovery, efforts continued, revealing that various elements and mixtures can form two-dimensional materials, each exhibiting unique characteristics. Among the materials actively explored in this field are hexagonal boron nitride (h-BN)^3,4^, phosphorene^5,6^, transition metal dichalcogenides (TMDs)^7,8^, stanene^9,10^, germanene^11–13^, arsenene^14^, and silicene^15,16^. Nevertheless, many researchers remain focused on discovering new materials that match or even surpass the exceptional properties of graphene. As the left-hand neighbor of carbon in the periodic table, boron exhibits non-local covalent bonds due to its electron deficiency, leading to interesting properties^17^. Additionally, its capacity to accept sp \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{2}$$\end{document} hybridization and short covalent radius create hope for the development of graphene analogs and potentially superior materials^18,19^.

Borophene is a single layer of boron atoms, and many studies have been conducted on it. The pioneering work involved the prediction of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\alpha\:$$\end{document} -sheets and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\beta\:$$\end{document} -sheets of monolayer B based on DFT calculations^20^. For the first time, Piazza et al. succeeded in synthesizing a highly stable quasi-planar boron cluster using photoelectron spectroscopy, leading to the coining of the name ’borophene’^21^. Many other attempts were made later, and various other phases were successfully synthesized by different research groups, such as the Pmmn phase^22^, and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\chi\:}_{3}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\beta\:}_{12}$$\end{document} phases of borophene^23^. It is worth mentioning that all of these phases were synthesized using the molecular beam epitaxy (MBE) method on an Ag(111) substrate. Researchers remain interested in identifying graphene-like (honeycomb) phases due to their unique properties, including Dirac fermions^24,25^. However, there is a fundamental challenge. Boron, with only three valence electrons, cannot form a stable honeycomb structure through sp \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{2}$$\end{document} bonding. This limitation leads to the incomplete filling of bond states in the two-dimensional structure, typically making it unstable. Despite this inherent instability, researchers have made efforts to overcome this limitation to exploit boron’s potential in various nanotechnology applications. As presented by Zhang et al.^26^ and Shirodkar et al.^27^, electron deficiency in the hexagonal lattice can be compensated through the electron doping process, allowing these additional electrons to partially or fully occupy the empty \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{p}_{z}$$\end{document} orbital in boron. However, in 2018, Li et al.^28^ successfully synthesized pristine, flat, non-buckled, graphene-like honeycomb borophene, as demonstrated by scanning tunneling microscopy (STM) images. A noteworthy aspect of this research was the use of an unconventional Al(111) substrate. According to^27^, approximately 0.7 electrons are transferred from the Al(111) substrate to each boron atom in borophene. Borophene demonstrates improved stability when exposed to oxygen adsorption due to strong B-O interactions^29^. This stability mechanism indicates the potential for synthesizing 2D boron oxides. Motivated by this potential, Zhang et al. conducted a comprehensive investigation into the electronic properties of various two-dimensional boron oxide crystals, including B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{4}$$\end{document} O, B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{5}$$\end{document} O, B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{6}$$\end{document} O, B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{7}$$\end{document} O, and B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{8}$$\end{document} O, using first-principles calculations^30^.

A turning point in this field occurred in 2019 with Zhong et al.’s prediction of a new family of boron oxides on honeycomb borophene^31^. They established that by adding oxygen atoms to the structure, the oxygen atoms tend to occupy bridge sites, similar to those in graphene or silicon oxides^32–35^. However, in this case, the oxygen atoms form a completely planar structure known as honeycomb borophene oxide (h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O). Considering that honeycomb borophene hydride (h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} ) is synthesized with two hydrogen atoms located on both sides of the honeycomb borophene bridge^36^. h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} has a formation energy of -0.462, and given that the formation energy for h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O is more than three times higher, it can be concluded that h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O has strong potential for synthesis^31^. Furthermore, the oxidation process can be facilitated using a laser, similar to the method employed for black phosphorene^37^. In recent years, different research groups have reported unique properties of this new and exciting two-dimensional material. For example, the h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O exhibits high mechanical and thermal stability. It also has a nodal loop protected by mirror symmetry near the Fermi surface and undergoes a topological phase transition when strain is applied^31^. Yan et al. have predicted the existence of intrinsic superconductivity with a transition temperature of 10.3 K in a single layer of graphene using DFT. This transition temperature can be increased to 14.7 K by applying a 1% tensile strain^38^. Using DFT, the hydrogen storage performance of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O has been studied by decorating it with different materials. For example, the storage capacity of NLi \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{4}$$\end{document} -decorated h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O (NLi \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{4}$$\end{document} @B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O) can reach 9.1 wt %, making it a highly effective medium for hydrogen storage^39^. Another study shows that several alkali metal-decorated h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O structures exhibit increased H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} binding energies compared to the pristine h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O structure. Notably, the Li-decorated variant exhibits excellent results and achieves a gravimetric hydrogen density of 8.3 wt %^40^. As a very good achievement, the theoretical study of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O’s Hydrogen Evolution Reaction (HER) catalytic activity has revealed Gibbs free energy values comparable to those of platinum. Furthermore, these values have been adjusted using strain^41^.

Thermal conductivity in materials generally comprises two fundamental parts: the electronic component^42,43^ and the phononic component^44,45^. These aspects are essential for characterizing heat transport behavior and have received significant attention in recent studies, particularly involving two-dimensional (2D) materials. Graphene, for example, has been the subject of numerous investigations concerning both its electronic thermal conductivity (ETC)^46^ and phonon thermal conductivity (PTC)^47^. Comparable analyses have also been conducted on various borophene allotropes^48^ as well as on topological crystalline insulators, such as monolayer SnTe (001), where both ETC^49,50^ and PTC^51^ have been reported. Notably, in SnTe (001), the electronic and phononic contributions to thermal conductivity are found to be of similar magnitude. Similar studies have also evaluated the electronic and phononic thermal conductivities of monolayer and bilayer hexagonal boron phosphide^52,53^. This study is motivated by the fact that, although phonon-based thermal conductivity in h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O has already been calculated^54^, a complete picture of its thermal transport performance also requires insight into its electronic contribution.

In this study, we employed the TB approximation for the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals of boron atoms, along with the Green’s function method, to investigate the electronic properties of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, including its BS and DOS. Additionally, for the first time, we applied the fully quantum mechanical Kubo-Greenwood (KG) transport formalism to calculate the ETC in the diffusive transport regime, providing a more comprehensive understanding of the material’s thermal behavior. In the next phase of our investigation, we consider both n-type and p-type doping on the surface, and calculate the perturbed Green’s function using the T-matrix approximation and self-energy corrections. We then evaluate the effect of these impurities, at different concentrations, on the ETC along all crystallographic directions for both the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O. Impurity doping is one of the most common approaches for tuning various material properties^55–59^.

This paper is organized as follows: Sect. 2.1 introduces the TB Hamiltonian for the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals of boron atoms in monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O. Section 2.2 presents an analysis of the key electronic properties, including the BS, obtained by diagonalizing the TB Hamiltonian, and the DOS, which is computed using the Green’s function approach. Section 3 discusses the ETC within the framework of Onsager transport coefficients, calculated through the Kubo-Greenwood (KG) formalism, to investigate the ETC of all crystallographic components. Finally, Sect. 4 explores both n-type and p-type doping scenarios, analyzing their respective effects on the ETC contributions from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals in monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O.

Theory and method

TB Hamiltonian

The tight-binding (TB) approximation has gained increasing attention as a computationally efficient and conceptually robust framework for investigating the electronic and transport properties of two-dimensional materials^60,61^. This methodology constructs the electronic structure from Hamiltonian matrices parameterized by a limited set of physical quantities, such as on-site energies and hopping integrals. In comparison to more resource-intensive approaches like density functional theory (DFT), TB models offer significant reductions in computational cost. Moreover, when combined with ab initio calculations, the TB approach enables direct extraction of parameters from first-principles data, thereby improving both the accuracy and physical relevance of the model^62^.

To analyze the electronic behavior of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, we construct a tailored TB Hamiltonian grounded in the material’s geometric and symmetry properties. The monolayer exhibits a planar honeycomb structure aligned along the x-axis, situated within an orthorhombic crystal lattice. This configuration belongs to the Cmmm space group (No. 65) and displays C \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2v}$$\end{document} symmetry. The primitive unit cell comprises two boron atoms and one oxygen atom. Structural optimization yields lattice constants of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\alpha\:$$\end{document} = 7.127 Å and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\beta\:$$\end{document} = 2.806 Å. The B-B and B-O bond lengths are 1.706 Å and 1.341 Å, respectively. The bond angles, measured at 107.8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{\circ\:}$$\end{document} and 126.1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{\circ\:}$$\end{document} ^38^. This distortion not only influences real-space configuration but also alters the shape of the first Brillouin zone (FBZ), which adopts a rotated and non-regular hexagonal form (see Fig. 1a and b). The relevant atomic orbitals near the Fermi level are identified among the s, P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{x}$$\end{document} , P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} , and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} states. We focus primarily on the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals, as they dominate low-energy excitations. While oxygen atoms are structurally present, they are excluded from the TB basis set to simplify the model without compromising the system’s symmetry or electronic accuracy^31,63^ (see Fig. 2).

Fig. 1(a) Top view of the crystal structure of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document} O, where boron atoms are shown in blue and oxygen atoms in red. (b) First Brillouin zone (FBZ) of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document} O in reciprocal space, showing high-symmetry points and principal reciprocal lattice vectors.

Fig. 2. Crystal structure of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document} O without oxygen atoms, showing motif atoms A and B in the unit cell and the basis vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathrm{a}}_{1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathrm{a}}_{2}$$\end{document} , with additional structural features highlighted.

Our TB Hamiltonian is constructed using the second quantization formalism. Due to the orthogonality of the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals, they are treated independently. We begin with the out-of-plane P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital, where the Hamiltonian is separated into on-site and hopping contributions (see Fig. 3b):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{H}}_{{P}_{z}}^{\overrightarrow{k}}={\mathcal{H}}_{0}+{\mathcal{H}}_{1}$$\end{document}

Fig. 3(a) P_y_ orbital of boron atoms, illustrating interactions with nearest and next-nearest neighbor atoms. (b) P_z_ orbital of boron atoms, highlighting interactions with nearest neighbors.

here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{H}}_{0}$$\end{document} includes on-site potentials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\epsilon\:}_{i}$$\end{document} for boron atoms, while \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{H}}_{1}$$\end{document} represents the nearest-neighbor hopping terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{ij}$$\end{document} . The full expression is:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{{{P_z}}}^{{\vec {k}}}=\mathop \sum \limits_{i} {\varepsilon _i}h_{i}^{\dag }{h_i}+\mathop \sum \limits_{{\left\langle {ij} \right\rangle }} {t_{ij}}h_{i}^{\dag }{h_j}+H.c.$$\end{document}

where H.c. stands for the Hermitian conjugate. In matrix representation, the Hamiltonian takes the form:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{H}}_{{P}_{z}}^{\overrightarrow{k}}=\left[\begin{array}{cc}{\epsilon\:}_{A}&\:{f}_{\overrightarrow{k}}\\\:{f}_{\overrightarrow{k}}^{\mathrm{*}}&\:{\epsilon\:}_{B}\\\:&\:\end{array}\right]$$\end{document}

assuming equivalent on-site energies ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\epsilon\:}_{A}={\epsilon\:}_{B}$$\end{document} ), the off-diagonal coupling term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{f}_{\overrightarrow{k}}$$\end{document} becomes:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{f}_{\overrightarrow{k}}={t}_{1}{e}^{i{k}_{x}{L}_{2}}+{t}_{2}{e}^{-i{k}_{x}x}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{k}_{y}{L}_{3}}{2}\right)$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{L}_{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{L}_{3}$$\end{document} denote the real-space vectors between interacting sites. Operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{i}^{\dag }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{h}_{i}$$\end{document} are the creation and annihilation operators at boron sites \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:A$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:B$$\end{document} . Notably, neighbor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\delta\:}_{1}$$\end{document} corresponds to the hopping parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{1}$$\end{document} , whereas neighbors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\delta\:}_{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\delta\:}_{3}$$\end{document} correspond to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{2}$$\end{document} .

We next construct the Hamiltonian for the in-plane P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:y$$\end{document} orbital. Owing to its in-plane geometry, this orbital exhibits not only nearest-neighbor interactions ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{2}$$\end{document} ) but also substantial next-nearest-neighbor couplings, represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{3}$$\end{document} (associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\Delta\:}}_{3}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\Delta\:}}_{4}$$\end{document} ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{4}$$\end{document} (associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\Delta\:}}_{1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\Delta\:}}_{2}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\Delta\:}}_{5}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\Delta\:}}_{6}$$\end{document} ) (Fig. 3a). The Hamil We begin with an analysis of the tonian matrix is given by:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{H}}_{{P}_{y}}^{\overrightarrow{k}}=\left[\begin{array}{cc}{\epsilon\:}_{A}&\:{g}_{\overrightarrow{k}}\\\:{g}_{\overrightarrow{k}}^{\mathrm{*}}&\:{\epsilon\:}_{B}\\\:&\:\end{array}\right]$$\end{document}

the off-diagonal term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{g}_{\overrightarrow{k}}$$\end{document} is defined as:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{g}_{\overrightarrow{k}}={t}_{1}{e}^{i{k}_{x}{L}_{2}}+{t}_{2}{e}^{-i{k}_{x}x}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{k}_{y}{L}_{3}}{2}\right)+2{t}_{3}\mathrm{c}\mathrm{o}\mathrm{s}\left({k}_{y}{L}_{3}\right)+4{t}_{4}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{k}_{y}{L}_{3}}{2}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha\:{k}_{x}\right)$$\end{document}

a complete list of hopping integrals and on-site energy values used in these Hamiltonians is provided in Table 1. These parameters were obtained through optimization against first-principles calculations to ensure fidelity to the underlying electronic structure.

Table 1. Optimized parameters, such as on-site energies and hopping integrals, for the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{\mathrm{y}}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{\mathrm{z}}$$\end{document} orbitals of the boron atom, with all values expressed in eV.Orbital \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\epsilon\:}_{\alpha\:}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{1}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{2}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{3}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{t}_{4}$$\end{document} P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} −4.12−0.6722.7180.134−0.098P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} 3.111.333−1.835––

Electronic properties

This section is devoted to exploring the electronic characteristics of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O via the TB Hamiltonian framework. The analysis commences with the calculation of the BS, followed by an assessment of the DOS. The TB-derived electronic bands are found to be in strong agreement with DFT results reported earlier^63^, confirming the reliability and effectiveness of the TB model. This consistency provides a solid foundation for extending the TB formalism to investigate transport properties, particularly the ETC, which constitutes the core subject of this study.

We begin with an analysis of the BS. As established in prior works^63^, the states near the Fermi level are predominantly contributed by the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals of boron atoms. Consequently, the TB Hamiltonian is formulated by considering only these orbitals. The resulting BS, over a wide energy range and in a narrow energy range around the Fermi level, shown in the left panels of Fig. 4a and b, are obtained by diagonalizing the Hamiltonian matrix, which leads to the dispersion relation:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\left|{\mathcal{H}}^{\overrightarrow{k}}-{\mathcal{E}}_{\left(\lambda\:\right)}^{\overrightarrow{k}}\widehat{\mathrm{I}}\right|=0\:\:;\:\:\lambda\:\in\:\{1,2\}$$\end{document}

Fig. 4. Calculated BS at high-symmetry points (left panel) and total DOS (right panel) for pristine monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document} O: (a) Over a wide energy range, (b) In a narrow energy range around the Fermi level, (c) Projected DOS for P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{\mathrm{y}}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{\mathrm{z}}$$\end{document} orbitals separately, (d) Atom-resolved DOS of B and O atoms and their total in monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document} O, highlighting the dominant contribution of B atoms in the vicinity of the Fermi energy.

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\widehat{\mathrm{I}}$$\end{document} denotes the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:2\times\:2$$\end{document} identity matrix, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\lambda\:$$\end{document} labels the resulting bands. A key feature observed is the presence of band crossings around the Fermi energy ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{E}}_{F}=0$$\end{document} ) along the high-symmetry paths \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\Gamma\:}$$\end{document} –Y and Y–M. These crossings, originating from interactions between P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals, reflect the metallic behavior of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, in line with other borophene phases. Notably, these crossings do not form isolated Dirac points; instead, they constitute a type-I nodal loop centered around the Y point in the Brillouin zone^64^. This loop is stabilized by mirror symmetry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{M}_{z}$$\end{document} , under which the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital is odd and the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital is even^31^, thus preserving the band crossings due to symmetry constraints (see Fig. 4b). Additionally, Dirac-like features are associated with the on-site energy levels of P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:-4.12$$\end{document} eV) and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:3.11$$\end{document} eV), visible in the valence and conduction bands, respectively Fig. 4a. For a more comprehensive representation of the electronic structure, Fig. 5a displays the three-dimensional (3D) BS of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O. The plot clearly reveals a Dirac-cone-like dispersion, indicating linear band crossings and suggesting the presence of massless Dirac fermions. This feature is particularly significant, as it implies the potential for high carrier mobility and intriguing topological properties. To complement this, Fig. 5b presents the corresponding contour plot, which provides a top-down view of the 3D band structure and shows the locations of high-symmetry points within the first Brillouin zone (FBZ). The visualization of the FBZ clarifies the symmetry environment in reciprocal space and highlights the specific momentum points where key electronic features, such as band crossings, occur.

Fig. 5(a) The three-dimensional BS of h-B_2_O reveals a Dirac-cone-like dispersion. (b) The contour plot of h-B_2_O shows the location of high-symmetry points within the FBZ.

We next apply the Green’s function approach to evaluate both the total and projected density of states (PDOS). While the total DOS includes contributions from all energy states across all orbitals, the PDOS isolates the contributions from specific orbitals, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\xi\:\in\:\{{P}_{y},{P}_{z}\}$$\end{document} . The PDOS is calculated by projecting the eigenfunctions onto localized atomic orbitals and summing over all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\overrightarrow{k}$$\end{document} -points in the FBZ:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{D}}_{\xi\:}\left(\mathcal{E}\right)=-\frac{1}{\pi\:{N}_{a}}\sum\:_{\overrightarrow{k}\in\:\mathrm{FBZ}}\mathrm{Im}\left[\mathrm{Tr}\hspace{0.17em}G(\overrightarrow{k},\mathcal{E})\right]$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{N}_{a}$$\end{document} represents the number of atoms in the unit cell, and the trace is taken over the orbital subspace. The retarded Green’s function is given by:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:G\left(\mathcal{E}\right)=\frac{1}{\mathcal{E}-{\mathcal{E}}_{n}+i\eta\:}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{E}}_{n}$$\end{document} denoting the eigenenergies, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\eta\:=5$$\end{document} meV being the broadening parameter. The total DOS profile highlights not only the on-site energy contributions from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals but also several Van Hove singularities (VHSs), which correspond to extremum/saddle points in the band structure (see the right panels of Fig. 4a and b). These singularities play a pivotal role in understanding both electronic transitions and potential instabilities. Specifically, when the Fermi level crosses a Van Hove singularity in the DOS, the resulting high density of electronic states can drive the system toward instability, potentially giving rise to emergent phases such as superconductivity^65–67^. The PDOS for the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals is shown separately in Fig. 4c, clearly demonstrating a greater contribution from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital near the Fermi level compared to the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital. This is indicated in the diagram by a double-headed arrow at the Fermi energy, where DOS \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{{P}_{z}}$$\end{document} > DOS \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{{P}_{y}}$$\end{document} .

It is worth noting that we have included the atom-resolved PDOS for both B and O atoms, obtained from DFT calculations (see Fig. 4d). This plot clearly shows that the low-energy states in the vicinity of the Fermi energy are overwhelmingly dominated by B-derived orbitals, whereas O-related states contribute only a very small spectral weight within this energy window. This finding supports the validity of our effective TB description, in which only B orbitals are retained. It also indicates that the influence of oxygen is implicitly encoded in the renormalized TB parameters-particularly in the hopping amplitudes (with |t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} |> |t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{1}$$\end{document} |), obtained from first principles calculations^63^.

Electronic thermal conductivity

In this section, we examine the fundamental mechanisms governing thermal conductivity. Thermal energy transport in materials is primarily driven by two mechanisms: electronic conduction and phonon-mediated heat transfer, expressed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{total}={\kappa\:}_{el}+{\kappa\:}_{ph}$$\end{document} )^68^. The electronic contribution arises from the movement of free electrons, while the phonon contribution originates from lattice vibrations. In this work, we present the first calculation of the electronic thermal conductivity (ETC) of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O along three directions: xx (armchair), yy (zigzag), and xy, the latter corresponding to the anomalous Righi-Leduc effect. The Righi-Leduc effect is the thermal counterpart of the Hall effect, where the heat current takes the place of the electric current, and the temperature gradient replaces the electric field^69–71^. This electron-driven thermal transport complements the lattice-driven heat conduction, providing a more comprehensive understanding of the system’s overall thermal response. To describe electronic transport properties, we employ linear response theory and Onsager transport coefficients, which serve as the theoretical foundation for quantifying charge and heat transport under external perturbations. A rigorous analysis of these coefficients yields valuable insights into the material’s electronic and thermal conductivity, as well as its thermoelectric properties^72–75^. Within this formalism, the charge flux \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathbf{J}}_{1}^{\boldsymbol{\upmu\:}}$$\end{document} and heat flux \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathbf{J}}_{2}^{\boldsymbol{\upmu\:}}$$\end{document} in direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\boldsymbol{\upmu\:}$$\end{document} can be derived by considering the influence of the electric field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathrm{E}}^{\boldsymbol{\upupsilon\:}}$$\end{document} and temperature gradient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\nabla\:}^{\boldsymbol{\upupsilon\:}}\mathrm{T}$$\end{document} in direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\boldsymbol{\upupsilon\:}$$\end{document} . The matrix form can be represented as follows^76–78^:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\left[\begin{array}{c}{\mathbf{J}}_{1}^{\boldsymbol{\upmu\:}}\\\:{\mathbf{J}}_{2}^{\boldsymbol{\upmu\:}}\end{array}\right]=\left[\begin{array}{cc}{\mathcal{L}}_{11}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}&\:{\mathcal{L}}_{12}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}\\\:{\mathcal{L}}_{21}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}&\:{\mathcal{L}}_{22}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}\end{array}\right]\left[\begin{array}{c}{\mathrm{E}}^{\boldsymbol{\upupsilon\:}}\\\:{\nabla\:}^{\boldsymbol{\upupsilon\:}}\mathrm{T}\end{array}\right]$$\end{document}

Consequently

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{J}}_{1}^{{\mathbf{\mu }}}=\mathcal{L}_{{11}}^{{{\mathbf{\mu \upsilon }}}}{{\mathrm{E}}^{\mathbf{\upsilon }}}+\mathcal{L}_{{12}}^{{{\mathbf{\mu \upsilon }}}}{\nabla ^{\mathbf{\upsilon }}}{\mathrm{T}}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{J}}_{2}^{{\mathbf{\mu }}}=\mathcal{L}_{{21}}^{{\mu \upsilon }}{{\mathrm{E}}^{\mathbf{\upsilon }}}+\mathcal{L}_{{22}}^{{{\mathbf{\mu \upsilon }}}}{\nabla ^{\mathbf{\upsilon }}}{\mathrm{T}}$$\end{document}

in these equations, the terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{L}}_{mn}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}[m,n\in\:(1,2\left)\right]$$\end{document} represent the Onsager coefficients. The connection between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{L}}_{mn}^{\mu\:\upsilon\:}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{L}}_{nm}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}$$\end{document} is defined by Onsager’s reciprocal relation:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{L}}_{mn}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}={\mathcal{L}}_{nm}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}$$\end{document}

this association illustrates the symmetrical nature of transport coefficients within the framework of linear response theory. It is now opportune to construct the ETC using the Onsager transport coefficients:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}=\frac{1}{T}\left({\mathcal{L}}_{22}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}-\frac{({\mathcal{L}}_{12}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}{)}^{2}}{{\mathcal{L}}_{22}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}}\right)$$\end{document}

Now, our focus shifts to acquiring the coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{L}}_{mn}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}$$\end{document} in the Kubo-Greenwood (KG) approach^79,80^. This process entails the examination of time-correlation functions and depends on the utilization of Kubo’s quantum-statistical linear response theory^81,82^. Resulting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}_{{mn}}^{{{\mathbf{\mu \upsilon }}}}\left( {i\omega } \right)=\frac{{{k_B}T}}{{i\omega \Omega }}\mathop \smallint \limits_{0}^{{1/{k_B}T}} d\tau {e^{i\omega \tau }}\left\langle {{T_\tau }{\mathbf{J}}_{m}^{{\mathbf{\mu }}}\left( \tau \right){\mathbf{J}}_{n}^{{\mathbf{\nu }}}\left( 0 \right)} \right\rangle$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\varOmega\:$$\end{document} is volume. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{k}_{B}$$\end{document} represents the Boltzmann constant, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\tau\:$$\end{document} corresponds to imaginary time, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\omega\:$$\end{document} signifies the Matsubara frequency. Here, we concentrate on the static limit ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\omega\:\to\:0$$\end{document} ), thereby \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{L}}_{mn}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}$$\end{document} = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\underset{\omega\:\to\:0}{\mathrm{l}\mathrm{i}\mathrm{m}}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:Im{\mathcal{L}}_{mn}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}(i\omega\:\to\:\omega\:+i\eta\:)$$\end{document} . In the static limit, we incorporate a broadening factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\eta\:=5$$\end{document} meV in our numerical computations. Through this approach and by assessing the time-auto-correlation function in the Heisenberg picture, we arrive at the conclusion that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{c}} {\mathcal{L}_{{mn}}^{{{\mathbf{\mu \upsilon }}}}\left( T \right)=\frac{{h{{( - e)}^{4 - m - n}}{k_B}T}}{{d\Omega }}\mathop \smallint \limits_{{ - \infty }}^{\infty } dE{\mathcal{E}^{m+n - 2}}\left( {\frac{{\partial {\mathcal{N}_{FD}}\left( {\mathcal{E},T} \right)}}{{\partial \mathcal{E}}}} \right)} \\ {\mathop \sum \limits_{{\vec {k} \in FBZ}} \mathop \sum \limits_{\lambda } V_{\lambda }^{{\mathbf{\mu }}}\left( {\vec {k}} \right)V_{\lambda }^{{\mathbf{\upsilon }}}\left( {\vec {k}} \right)Tr{{\left( {{\mathbf{Im}}G\left( {\vec {k},\mathcal{E}} \right)} \right)}^2}} \end{array}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathcal{N}}_{FD}(\mathcal{E},T)=1/[1+\mathrm{e}\mathrm{x}\mathrm{p}(\mathcal{E}/{k}_{B}T\left)\right]$$\end{document} represents the Fermi-Dirac distribution function, whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{{\mathbf{\mu }}}^{\lambda }\left( {\vec {k}} \right)={h^{ - 1}}\frac{{\partial {\mathcal{E}^\lambda }\left( {\vec {k}} \right)}}{{\partial {k_{\mathbf{\mu }}}}}$$\end{document} denotes the group velocity of fermions ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\lambda\:$$\end{document} is number of bands). The next section will focus on the results of the ETC.

To investigate the temperature-dependent behavior of ETC in pristine h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, we analyze its variation as a function of temperature, expressed in watts per meter per Kelvin (W m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} ). Given their significant influence on BS and DOS, the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals play a crucial role in shaping ETC. This study examines their contributions along the primary transport directions: diagonal (xx, yy) and off-diagonal (xy) components. To systematically interpret ETC trends, we classify the results into three temperature intervals, as variations in directional and orbital contributions follow a consistent qualitative pattern. As depicted in Fig. 6, the electronic thermal conductivity (ETC) initially increases gradually from 0 K to approximately 40 K. In this low-temperature regime, thermal energy is insufficient to activate a significant number of charge carriers. Consequently, both carrier mobility and group velocity ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{V}_{\boldsymbol{\upmu\:}}^{\lambda\:}\left(\overrightarrow{k}\right)$$\end{document} ) remain low, leading to limited charge excitation and diminished transport contributions. The electronic states near the Fermi level also remain largely unoccupied due to thermal suppression, resulting in minimal thermal energy transport. Between 40 K and 200 K, the ETC exhibits a sharp increase with rising temperature, reaching a pronounced peak around 200 K. This temperature range marks a transition from a transport-limited regime to one dominated by thermally activated conduction. As thermal energy becomes sufficient to excite electrons into higher energy states, the population of mobile carriers increases markedly, resulting in more efficient transport. The steep rise in ETC within this intermediate temperature range highlights the system’s sensitivity to thermal excitation and the critical role of temperature in modulating electronic transport behavior. At higher temperatures (200–1300 K), the ETC begins to decline gradually. In this regime, electron-electron scattering processes become increasingly dominant. These scattering mechanisms, arising from the reduced mean free path of electrons, introduce resistance to heat transport, thereby lowering overall conductivity. Moreover, as energy states become increasingly populated, the available phase space for further excitation narrows due to the Pauli exclusion principle. This saturation effect, combined with enhanced scattering mechanisms at elevated temperatures, contributes to the observed reduction in ETC. The combined influence of scattering and energy-state occupancy thus defines a high-temperature regime characterized by suppressed electronic thermal transport despite continued thermal input. This trend is evident in the calculated total \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{\mathrm{total}}^{\boldsymbol{\upmu\:}\boldsymbol{\upupsilon\:}}$$\end{document} values for the xx, yy, and xy components, which reach \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:6.8\times\:{10}^{-2}$$\end{document} mW m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} , 1.2 mW m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} , and 0.19 mW m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} , respectively, at the critical temperature, as indicated by a vertical line in the diagram. Such behavior is expected upon entering the diffusive transport regime^83^ and has also been observed in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\beta\:}_{12}$$\end{document} -borophene phase^84^ and graphene^46^. This trend resembles PTC, where Umklapp scattering induces phonon-phonon interactions, a phenomenon previously studied in graphene^85^. To ensure numerical stability and physical relevance, we limit the energy range to the order of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{k}_{B}T$$\end{document} , as the Fermi-Dirac derivative decays rapidly beyond this range. Additionally, high-energy electrons remain tightly bound to the nucleus, requiring substantial energy for excitation. Consequently, their contribution to thermal transport is negligible within the accessible thermal window. By restricting our analysis to this limited energy range, we capture the most influential electronic states governing transport properties^86^.

Fig. 6(a) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{x}\mathrm{x}}$$\end{document} , (b) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{y}\mathrm{y}}$$\end{document} , and (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{x}\mathrm{x}}$$\end{document} of pristine h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{2}$$\end{document} O as a function of temperature for P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{\mathrm{y}}$$\end{document} , P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}_{\mathrm{z}}$$\end{document} , and their sum.

After conducting a qualitative analysis of the ETC plots as a function of temperature, we now delve deeper into the physical implications. The total ETC values (P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} + P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} ) are found to be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:5.9\times\:{10}^{-2}$$\end{document} mW m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} , 1 mW m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} , and 0.17 mW m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}^{-1}$$\end{document} along the xx, xy, and yy directions at room temperature (T = 300 K), respectively. From the results, significant physical insights emerge. The pronounced anisotropy in the h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O monolayer is confirmed by the markedly higher ETC along the zigzag direction compared to the armchair direction ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{total}^{\mathbf{y}\mathbf{y}}>{\kappa\:}_{total}^{\mathbf{x}\mathbf{x}}$$\end{document} ). This behavior reflects the intrinsic geometric structure of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, which adopts a deformed hexagonal configuration rather than an ideal isotropic honeycomb lattice. As a result, the B-B bonds and the local B-B coordination differ along the armchair and zigzag directions, leading to anisotropic overlap of the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbitals of the boron atoms that dominate the low-energy electronic states and, in our TB model, to direction-dependent hopping amplitudes (with |t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} |>|t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{1}$$\end{document} |). According to Eq. [eq15], this directly translates into larger group velocities and stronger charge-carrier propagators (Green’s functions) along the zigzag direction, and therefore into a larger \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{yy}$$\end{document} than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{yy}$$\end{document} . Notably, this conclusion is consistent with previous first-principles calculations, which reported more efficient phonon heat transport along the zigzag direction than along the armchair direction in h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, underscoring the same structural origin of the anisotropy^54^. Additionally, the ETC value for the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital is notably higher than that of the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital. The P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital, which extends out of the plane and provides greater freedom for charge carriers, exhibits a higher ETC than the in-plane P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital. To better understand the underlying mechanism, we refer to the DOS diagram. While the BS indicates the importance of both orbitals in the electronic properties of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, it does not quantify their relative contributions. The PDOS analysis shows that the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital has a higher density of available carriers (i.e., more occupied states) and dominates across nearly all energy ranges, especially near the Fermi level, which is critical for transport calculations (see Fig. 4c).

The ETC of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, obtained from the KG formalism, is several orders of magnitude smaller than that of conventional metallic 2D systems. This ultra low value can be traced back to both the peculiar electronic structure and the transport regime considered here. First, h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O behaves as a low carrier density, Dirac like metal rather than a high carrier density, parabolic band metal: in our tight binding band structure, the Fermi level lies close to Dirac cone like crossings, and the Fermi surface is composed of relatively small pockets in momentum space. Consequently, the phase space of thermally activated carriers around the Fermi level is severely restricted, so that the energy integral in the KG expression, which is weighted by the velocity matrix elements and spectral functions, yields a strongly suppressed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{e}$$\end{document} despite the large band velocities near the crossing points (see Eq. [eq15]). Second, our calculations explicitly address the diffusive regime, in which scattering is encoded in the Green’s functions, further reducing both charge and heat transport compared with 2D metals in the ballistic transport regime (such as pristine graphene or some borophene phases). Taken together, the combination of a low effective carrier density near Dirac like nodes and carrier scattering in the diffusive transport regime provides a natural explanation for why \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{e}$$\end{document} in h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O is orders of magnitude smaller than in typical high carrier density metallic 2D materials (see Table 2). It will also be important to examine ultra low thermal conductivities and their microscopic origins in other 2D systems. For instance, Cui et al.^87^ reported that the ultra low lattice thermal conductivity of the natural van der Waals compound KP15 stems from inherently weak couplings between its quasi one dimensional tubes, which in turn is beneficial for its thermoelectric performance. Hao et al.^88^ showed that the conventional deformation potential framework markedly inflates the calculated electron lifetime and, as a result, exaggerates the thermoelectric performance of Janus penta PdXY (XY = SeTe, SSe, STe) monolayers. When scattering is treated within an electron phonon averaged approximation, strong electron phonon interactions emerge, leading to a pronounced suppression of the electronic lifetime. In the following section, to further align the study with practical applications, we will explore the impact of surface modifications on the monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O through n-type and p-type doping, and investigate how these modifications influence the ETC.

Table 2. Electronic thermal conductivity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upkappa\:}$$\end{document} at room temperature (300 K) for selected 2D materials. All values are given in W m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}^{-1}$$\end{document} K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{}}^{-1}$$\end{document} .Material \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{xx}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{yy}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{xy}$$\end{document} Ref.Graphene \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:48\times\:{10}^{-3}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:48\times\:{10}^{-3}$$\end{document} – ^48^

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\delta\:}_{5}$$\end{document} 1.521.52– ^48^

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\alpha\:$$\end{document} 0.320.32– ^48^

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\chi\:}_{3}$$\end{document} 1.71.28– ^48^

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\delta\:}_{6}$$\end{document} 6.528.61– ^48^ h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:5.9\times\:{10}^{-5}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:1.0\times\:{10}^{-3}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:0.17\times\:{10}^{-3}$$\end{document} This work

Impurity doping

It is now time to investigate the effects of impurities on the ETC of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O. The details of this investigation can provide valuable insights into future experimental studies of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, as most impurities are introduced into the material during the synthesis process, typically originating from the substrate or the surrounding environment. Additionally, there is another type of impurity that we will focus on in this study: intentionally introduced dilute charged impurities added to the sample surface. In this study, we focus on two classes of dopants: n-type impurities (which introduce extra electrons) and p-type impurities (which generate holes). These low-concentration impurities are modeled as short-range charged potentials, represented by the Dirac delta function in reciprocal space. We keep the number of impurities constant while their positions are randomly distributed throughout the system (see Fig. 7). Additionally, the final results are obtained by averaging over all possible impurity configurations. We will also employ the Born approximation and the T-matrix method to study these impurities. The Born approximation is effective when the impurity density is low and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:$$\end{document} is small. However, when localized states form near the impurities, the Born approximation becomes inadequate, and the T-matrix approach must be used, as the T-matrix provides an exact solution for any value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:$$\end{document} ^89^. Therefore, the impurity potential, represented in reciprocal space, is present at both sites within a unit cell. In other words, the impurities considered here have the following form and same amplitude, given by:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:={\upsilon\:}_{i}\left[\begin{array}{cc}1&\:0\\\:0&\:1\\\:&\:\end{array}\right]$$\end{document}

Fig. 7. Schematic of impurity doping on the h-B_2_O surface, showing the random distribution of dopants.

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upsilon\:}_{i}$$\end{document} is a constant, its nonzero components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:$$\end{document} signify the presence of an impurity near or directly on the atomic sites. This method is particularly effective for modeling dopants such as silicon, oxygen, carbon, and sulfur, which introduce scattering effects^90^. Given that Green’s function encapsulates the system’s correlations, incorporating impurities requires computing a perturbed Green’s function to describe the system. This results in^89^:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:G(\overrightarrow{k},{\overrightarrow{k}}^{{\prime\:}},\mathcal{E})={G}_{0}(\overrightarrow{k},\mathcal{E})+{G}_{0}(\overrightarrow{k},\mathcal{E}){\tau\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:}}}{G}_{0}({\overrightarrow{k}}^{{\prime\:}},\mathcal{E})$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\tau\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:}}}$$\end{document} is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\tau\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:}}}=\hspace{0.25em}{\upsilon\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:}}}+\sum\:_{{\overrightarrow{k}}^{{\prime\:\prime\:}}}{\upsilon\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:\prime\:}}}{G}_{0}({\overrightarrow{k}}^{{\prime\:\prime\:}},\mathcal{E}){\upsilon\:}_{{\overrightarrow{k}}^{{\prime\:\prime\:}}{\overrightarrow{k}}^{{\prime\:}}}+...=\hspace{0.25em}{\upsilon\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:}}}+\sum\:_{{\overrightarrow{k}}^{{\prime\:\prime\:}}}{\upsilon\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:\prime\:}}}{G}_{0}({\overrightarrow{k}}^{{\prime\:\prime\:}},\mathcal{E}){\tau\:}_{{\overrightarrow{k}}^{{\prime\:\prime\:}}{\overrightarrow{k}}^{{\prime\:}}}$$\end{document}

here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upsilon\:}_{\overrightarrow{k}{\overrightarrow{k}}^{{\prime\:}}}$$\end{document} describes the interaction between two momenta: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\overrightarrow{k}$$\end{document} , representing the initial momentum of surface electrons, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\overrightarrow{k}}^{{\prime\:}}$$\end{document} , which corresponds to the momentum change induced by impurity scattering along a given energy contour \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\mathcal{E}$$\end{document} . For local impurities and isotropic scattering, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\tau\:$$\end{document} depends solely on energy, So

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\tau\:\left(\mathcal{E}\right)=\upsilon\:+\upsilon\:{\widehat{G}}_{0}\left(\mathcal{E}\right)\tau\:\left(\mathcal{E}\right)=\frac{\upsilon\:}{1-\upsilon\:{\widehat{G}}_{0}\left(\mathcal{E}\right)}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\tau\:$$\end{document} obeys the self-consistent equation and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\widehat{G}}_{0}\left(\mathcal{E}\right)=\frac{1}{{N}_{c}}\sum\:_{\overrightarrow{k}\in\:\mathrm{FBZ}}{G}_{0}(\overrightarrow{k},\mathcal{E})$$\end{document} (N \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{c}$$\end{document} represents the number of unit cells). After averaging over the random impurity distribution, the perturbed Green’s function is described by a single momentum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\overrightarrow{k}$$\end{document} , while the overall effect of the impurities is encapsulated in the self-energy^91,92^.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\widehat{{\Sigma\:}}\left(\mathcal{E}\right)={n}_{i}\tau\:\left(\mathcal{E}\right)=\frac{{n}_{i}\upsilon\:}{I-\frac{\upsilon\:}{{N}_{c}}\sum\:_{\overrightarrow{k}\in\:\mathrm{FBZ}}{G}_{0}(\overrightarrow{k},\mathcal{E})}$$\end{document}

the impurity concentration is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{n}_{i}=N/{\varOmega\:}_{c}$$\end{document} ​, where N is the total number of impurities and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\varOmega\:}_{c}$$\end{document} represents the unit cell volume. This formulation yields the perturbed Green’s function via Dyson’s equation^76^

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:G(\overrightarrow{k},\mathcal{E})={G}_{0}(\overrightarrow{k},\mathcal{E})+{G}_{0}(\overrightarrow{k},\mathcal{E})\widehat{{\Sigma\:}}\left(\mathcal{E}\right)G(\overrightarrow{k},\mathcal{E})$$\end{document}

having obtained the perturbed Green’s function, we note that it depends on two key parameters characterizing the impurities introduced to the surface: the impurity concentration (n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{i}$$\end{document} ) and the scattering potential ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upsilon\:}_{i}$$\end{document} ). The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upsilon\:}_{i}$$\end{document} plays a crucial role in distinguishing between n-type and p-type doping. A positive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upsilon\:}_{i}$$\end{document} indicates the addition of an electron to the surface, leading to electron-electron interactions (n-doped), whereas a negative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upsilon\:}_{i}$$\end{document} signifies the introduction of a hole, resulting in electron-hole interactions (p-doped)^91^. In this study, we consider dilute impurity concentrations of 2%, 4%, and 6%, setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\nu\:}_{i}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:-1.5$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:1.5$$\end{document} .

Indeed, in this work we adopt an effective impurity model formulated at the TB level. Short-range dopants and defects are represented by local on-site potentials vi acting near or on the boron sites, which mimic the impurity-induced shift of the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital energies caused by substitutional atoms, vacancies, or strongly bound adsorbates. This strategy is analogous in spirit to the defect-engineered design of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\gamma\:$$\end{document} -graphyne nanoribbons, where different defect configurations are encoded directly in an effective p-orbital TB Hamiltonian and their impact on thermoelectric transport is explored using nonequilibrium Green’s function^93^. The chosen magnitude | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:$$\end{document} |=1.5 eV is of the same order as the on-site energies and hopping amplitudes in our model, corresponding to an intermediate scattering strength that produces pronounced but not destructive modifications of the Green’s function and the resulting electronic thermal conductivity.

We systematically examine the impact of impurity-induced disorder on the ETC of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, with particular emphasis on orbital-resolved contributions and directional anisotropy. To this end, we explore both n-type and p-type doping scenarios and analyze their respective effects on the ETC contributions from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbitals along the crystallographic xx-, yy-, and xy-directions. The ETC is computed as a function of temperature, with orbital-specific scattering treated independently for each impurity concentration. These combined results provide a comprehensive picture of the underlying transport mechanisms.

In the n-type doping scenario, modeled via a positive scattering potential ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:$$\end{document} = 1.5 eV), we observe a strongly orbital-selective response, highlighting the anisotropic and orbital-dependent character of thermal transport in h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O. Figure 8 presents nine subpanels: panels (a-c) show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{xx}$$\end{document} , panels (d-f) show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{yy}$$\end{document} , and panels (g-i) show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{xy}$$\end{document} , each plotted for P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} , P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} , and their combined contribution, respectively. Specifically, with increasing impurity concentration ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{n}_{i}$$\end{document} = 2%, 4%, and 6%), the ETC associated with the out-of-plane P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital increases consistently across all transport directions. Conversely, the contribution from the in-plane P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital exhibits only mild suppression. This dichotomous behavior, underscores the fundamentally distinct transport roles and scattering sensitivities of the two orbitals under electron doping. This contrasting response arises from differences in orbital geometry and their interactions with charge carriers and impurity scattering centers. The P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital, oriented perpendicular to the plane, provides enhanced conduction channels that directly benefit from the increased carrier density introduced by n-type doping. This leads to elevated occupation of conduction band (CB) states and improved carrier mobility. In contrast, the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital, which lies primarily within the plane, has a reduced capacity to accommodate CB carriers and is more prone to localization effects and enhanced electron-electron (e-e) scattering. These factors promote the formation of localized states, thereby diminishing its contribution to the ETC. When both orbital contributions are combined, the total ETC exhibits a net increase in all directions under n-type doping, following the trend set by the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital. However, the magnitude of this enhancement is direction-dependent: the yy-direction shows the smallest gain, while the xx-direction experiences the most significant improvement, reflecting the anisotropic interplay between orbital character and crystallographic orientation. Moreover, this enhancement remains stable across a wide temperature range, indicating that ETC under n-type doping is largely insensitive to thermal fluctuations. This intrinsic stability underscores the effectiveness of n-type doping as a reliable strategy for optimizing the thermal transport properties of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O and enhancing ETC along all directions, albeit at varying rates. The progressive increase in ETC with doping concentration further highlights the interplay between carrier density and thermal transport, emphasizing the potential of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O to be tailored for applications requiring precise thermal management. For example, thermoelectric energy-harvesting systems could leverage its high thermal conductivity to achieve more efficient heat-to-energy conversion^94^.

Fig. 8. Electronic thermal conductivity of h-B_2_O as a function of temperature with impurity concentration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathrm{n}}_{\mathrm{i}}$$\end{document} = 2, 4, and 6%, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upupsilon\:}$$\end{document} = 1.5 eV. Panels show: (a–c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{x}\mathrm{x}}$$\end{document} , (d–f) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{y}\mathrm{y}}$$\end{document} , and (g–i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{x}\mathrm{y}}$$\end{document} for P_y_, P_z_, and their sum.

In contrast, p-type doping, implemented by inverting the scattering potential to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\upsilon\:$$\end{document} = -1.5 eV, induces more modest modifications to the ETC. As shown in Fig. 9, the figure presents nine subpanels: panels (a-c) show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{xx}$$\end{document} , panels (d-f) show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{yy}$$\end{document} , and panels (g-i) show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{xy}$$\end{document} , each plotted for P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:y$$\end{document} , P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:z$$\end{document} , and their combined contribution across doping concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{n}_{i}=2\mathrm{\%}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:4\mathrm{\%}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:6\mathrm{\%}$$\end{document} . As the doping concentration increases, the ETC contribution from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:y$$\end{document} orbital increases slightly, whereas that from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:z$$\end{document} orbital decreases. However, the overall changes remain relatively small, resulting in minimal variation in the total ETC across doping levels. Despite the subdued overall response, directional anisotropy persists: I. Along the xx-direction, the total ETC decreases marginally, primarily due to residual contributions from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital. II. Along the yy-direction, the total ETC increases, indicating the dominant role of the enhanced P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital in this direction. III. Along the off-diagonal xy-direction, the total ETC increases with doping, but the trend is non-monotonic: the value at 6% exceeds that at 2%, yet remains lower than at 4%. These results underscore the remarkable robustness of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O to p-type doping and stand in sharp contrast to the more pronounced, directionally dependent modifications observed under n-type doping. Overall, our findings highlight the crucial role of orbital symmetry, spatial orientation, and dopant character in shaping the thermal transport behavior of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O materials. The isotropic thermal response of p-doped h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O ensures stable heat transport properties, making it a suitable candidate for two-dimensional electronic devices that require uniform heat dissipation to prevent thermal hotspots. This characteristic, combined with the tunability of thermal conductivity via doping concentration, enables precise control over heat flow and paves the way for advanced applications in emerging technologies such as thermal cloaking and quantum devices^95^.

Fig. 9. Electronic thermal conductivity of h-B_2_O as a function of temperature with impurity concentration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\mathrm{n}}_{\mathrm{i}}$$\end{document} = 2, 4, and 6%, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\upupsilon\:}$$\end{document} = -1.5 eV. Panels show: (a–c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{x}\mathrm{x}}$$\end{document} , (d–f) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{y}\mathrm{y}}$$\end{document} , and (g–i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{{\upkappa\:}}_{\mathrm{x}\mathrm{y}}$$\end{document} for P_y_, P_z_, and their sum.

Conclusions

In conclusion, we have conducted a theoretical investigation into the electronic and thermal transport properties of highly stable h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O, an emerging two-dimensional material known for its intriguing topological features and potential multifunctionality. Utilizing a tight-binding approach that incorporates the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals of boron, along with Green’s function techniques, we accurately captured the key characteristics of the BS and DOS. Our findings indicate that h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O exhibits metallic behavior and hosts four Dirac points: two arise independently from the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} and P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbitals due to differing on-site energies, while the other two originate from hybridization between these orbitals along the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\Gamma\:}\to\:$$\end{document} Y and Y \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:\to\:$$\end{document} M paths. Furthermore, we have calculated the ETC of monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:2$$\end{document} O for the first time, employing the Kubo–Greenwood formalism within the diffusive regime. The results reveal pronounced anisotropy in thermal transport, with significantly higher ETC observed along the zigzag direction compared to the armchair direction ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\kappa\:}_{yy}\gg\:{\kappa\:}_{xx}$$\end{document} ). This direction-dependent behavior is attributed to lattice distortions in the hexagonal framework, which modulate charge carrier mobility and thermal conduction pathways. Charge transport is found to be predominantly governed by the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital of boron, due to its higher carrier occupancy relative to the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{y}$$\end{document} orbital. Additionally, through a detailed analysis of impurity-induced disorder using the T-matrix approximation and self-energy calculations, we identified that n-type doping leads to a significant enhancement in ETC across all directions, driven by the favorable transport characteristics of the P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{z}$$\end{document} orbital. In contrast, p-type doping results in only minor changes, indicating thermal robustness under hole doping. These findings underscore the critical role of orbital symmetry, spatial orientation, and doping character in determining the anisotropic and tunable ETC of h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O. Overall, our results position monolayer h-B \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{}_{2}$$\end{document} O as a promising platform for future applications in thermoelectric energy conversion, thermal management, and quantum device technologies.