Prediction of Thermal Transport Properties of Pristine and BN-Substituted Holey Graphynes

Abstract

The merging of pore designs is a potential strategy for achieving ultra-low lattice thermal conductivity (κ), for which phonon anharmonicity and size effect are indispensable for discovering novel functional materials in thermal applications. In this study, monolayer holey graphyne (HGY) and boron nitride holey graphyne (BN-HGY) were examined for their phonon thermal transport properties through first-principles calculation and phonon Boltzmann function. HGY exhibits an intrinsic lattice thermal conductivity (κ) of 38.01 W/mK at room temperature, which exceeds BN-HGY’s 24.30 W/mK but is much lower than 3550 W/mK for BTE graphene. The phonon–phonon scattering behavior of BN-HGY is obviously increased compared to HGY due to the enhancement of anharmonicity, which leads to a shorter phonon lifetime and lower κ. Additionally, at room temperature, the representative mean free path (rMFP) of BN-HGY is substantially higher than that of HGY, and the κ of BN-HGY decreases faster at a larger rMFP (within a unit nm). This work will be constructive to further the application of HGY and BN-HGY as thermal management materials.

1. Introduction

Carbon possesses a distinctive structural framework that can be hybridized into sp³, sp², and sp configurations, which enables the formation of numerous stable allotropes. Graphite and diamond are two common early natural carbon allotropes that have sp³ and sp² hybrid carbon atoms, respectively [1,2,3,4,5]. In past decades, many carbon allotropes have been synthesized and characterized, like graphene [6], fullerene [7], carbon nanotube [8], graphyne [9,10,11,12,13], and graphdiyne [10,13,14]. These allotropes have excellent properties, including high thermal conductivity [15,16,17,18], superconductivity [19,20,21,22], quantum conductance [23,24], field emission [25,26], half-metallicity [27,28], and good transport properties [29,30]. Experimental observations and theoretical calculations have revealed that graphene exhibits a number of unusual properties [31,32,33], suggesting it to be a promising material for use in electronic circuits [34]. However, the zero-bandgap electronic structure of graphene imposes constraints on its potential applications as a semiconductor material [5]. It has been demonstrated by related studies that the band gap of graphene can be regulated by doping, defects, and external electric fields [35,36,37,38]. Boron (B) and nitrogen (N) can combine with carbon to form various new 2D materials due to the fact that B/N has electronic structures adjacent to that of carbon (C). In previous studies, investigations were conducted on amorphous boron carbide and carbon nitride thin films, which exhibited varied carbon-to-boron (C/B) and carbon-to-nitrogen (C/N) ratios. These composite semiconductor nanostructures were utilized for a diverse array of thermal property applications [39,40,41,42].

In recent years, a novel two-dimensional porous carbon structure with a periodic pattern of six- and eight-dimensional rings has been successfully synthesized. This structure is referred to as holey graphyne (HGY) [5]. The material under consideration has been demonstrated to exhibit uniform periodic holes. These holes have been shown to be useful in the context of energy storage applications. Gao et al. [43] explored hydrogen storage in Li- and Sc-modified HGY structures. Mahamiya et al. [44] also predicted a stable porous boron nitride porous graphene similar to HGY (namely BN-HGY). Desyatkin et al. [45] reported a scalable synthesis route for multilayer γ-graphyne using a crystallization-assisted cross-coupling polymerization strategy, representing a breakthrough in graphyne synthesis. Most recently, Aliev et al. [46] demonstrated the formation of a novel zero-bandgap, nongraphitic sp² carbon phase derived from γ-graphyne through a selective low-temperature intrasheet thermal transformation, confirmed by both theoretical and experimental analysis. The results indicate that the BN counterpart of the HGY structure is stable in terms of energy and dynamics with a tunable band gap, and the optical absorption of HGY and BN-HGY exhibits a broad spectrum across the visible and ultraviolet regions.

Porous structures have relatively low thermal conductivity and can be used as a good thermal insulator or thermoelectric material [47,48]. HGY and BN-HGY have the potential to have relatively low thermal conductivity, thus contributing to the design of thermoelectric nanodevices [49,50]. Their thermal transport properties, as a part of the important intrinsic properties of HGY and BN-HGY, are crucial for their applications in optoelectronics and nanoelectronics. Recently, Deb et al. [51] investigated the thermoelectric properties of pristine and BN-doped graphyne materials, revealing electronic structure modulation and enhanced thermoelectric efficiency via doping strategies. While their work provides valuable insights into electronic and thermoelectric performance, it primarily focuses on electronic transport characteristics and figure-of-merit (ZT) evaluations in doped systems. In contrast, our study emphasizes phonon-mediated lattice thermal conductivity in idealized monolayer graphyne systems, aiming to uncover intrinsic thermal transport behavior rooted in structural and bonding characteristics. Such an approach is critical for understanding fundamental phonon scattering mechanisms in 2D porous carbon materials and complements prior studies by shifting the focus from electronic to phononic contributions. Surprisingly, compared to the extensive theoretical research on the aforementioned properties of other two-dimensional materials, limited investigation has been conducted on the thermal conductivity of BN-HGY and HGY. In this study, the phonon thermal transport properties of monolayer HGY and BN-HGY were investigated using the density-functional theory (DFT) associated with the Boltzmann transport equation (BTE). Potential mechanisms controlling thermal transport, including group velocity, phonon relaxation time, phonon branching contribution, phonon discordance, and size effects, are discussed in depth.

2. Results and Discussions

2.1. Structural Stability and Phonon Dispersion

Firstly, the structural stability of HGY and BN-HGY has been studied by using DFT computations. The final optimized structures of HGY and BN-HGY with a 2 × 2 × 1 supercell are shown in Figure 1a,b. HGY has a hexagonal lattice structure and P6/mmm symmetry (space group ID 191). The lattice constants of the HGY crystal cell are a = b = 10.86 Å. The hexagonal BN-HGY structure has P6/m symmetry (space group ID 175), which is different from that of HGY. The lattice constants of BN-HGY are a = b = 10.96 Å. BN-HGY has four different BN bond length values corresponding to boron and nitrogen atoms hybridized with sp²-sp², sp²-sp, and sp-sp. There are two sp²-sp² hybrid bonds with bond lengths of 1.49 and 1.45 Å, and one sp²-sp or sp-sp hybrid bond has a length of 1.39 or 1.28 Å. For BN-HGY, boron and nitrogen atoms are located in positions similar to C atoms in HGY due to CC and BN being isoelectronic, and the changes in bond length and angle in BN-HGY are close to HGY.

The phonon frequencies of HGY and BN-HGY in BZ are validated through phonon dispersion calculations, as depicted in Figure 1c,d. Each primitive cell of HGY (or BN-HGY) has 3 acoustic and 69 optical phonon (OP) branches, originating from that which consists of 24 carbon atoms (or 12 nitrogen and 12 boron atoms). Off-plane (ZA), transverse (TA), and longitudinal (LA) modes are the three acoustic modes that originate from the Γ point. Among them, ZA mode is the softest vibrational phonon in any system. It can be seen that HGY and BN-HGY obviously have dynamic thermal stability, which holds no imaginary frequency. In particular, the acoustic branches of HGY are relatively steep, suggesting higher phonon group velocities, which are closely related to its high thermal conductivity. Moreover, the BN-HGY system shows more dispersed modes in the low-frequency optical region, implying stronger phonon scattering behavior that further affects its thermal transport performance. Additionally, most of the phonon dispersion branches are almost linear due to structural symmetry around a high-symmetry point. Phonon energy spectra of HGY and BN-HGY are very similar, with the only exception being the highest phonon frequencies: 2189 cm⁻¹ for HGY and 1993 cm⁻¹ for BN-HGY. This difference suggests that the bonding in HGY is stronger than in BN-HGY.

2.2. Phonon Transport Properties

2.2.1. Lattice Thermal Conductivity

Figure 2 shows the lattice thermal conductivities of monolayer HGY and BN-HGY. The temperature range is from 300 K to 800 K, with the temperature interval set to be 100 K during calculations. Thermal conductivity (κ) of HGY or BN-HGY is 38.01 W/mK and 24.30 W/mK at room temperature. The κ values of HGY and BN-HGY are considerably low during temperatures within the region [400 K–800 K] owing to phonon activity surging greatly at high temperatures. The Umklapp scattering process plays a more significant role in the phonon–phonon scattering process, resulting in a noticeable decrease in κ. The thermal conductivities of HGY and BN-HGY decrease with temperature, roughly following a T⁻¹ relation, indicating that anharmonic phonon–phonon interactions are the main contributors to thermal conductivity. The κ values of HGY and BN-HGY compared to other typical 2D materials are summarized in

Table 1. Summary of conductivity (κ) of HGY, BN-HGY, graphene [41], h-BN [52], and MoS₂ [53] at room temperature.

Materials	HGY	BN-HGY	Graphene [41]	h-BN [52]	MoS2 [53]
Conductivity κ(W/mK)	38.01	24.30	3550	250	100

, from which it can be found that HGY and BN-HGY have relatively low thermal conductivity.

2.2.2. Mode Level Analysis

In order to further understand the thermal transport behavior of HGY, the contributions of ${\kappa }_{ZA}$ (black curve), ${\kappa }_{TA}$ (red curve), ${\kappa }_{LA}$ (blue curve), and total optical modes ${\kappa }_{optic}$ (magenta curve) to the total thermal conductivity ($\mathsf{\kappa }$) have been investigated, as shown in

Table 2. Phonon branch contributions to thermal conductivity of HGY, BN-HGY, and graphene (300 K).

	ZA	TA	LA	Optical
HGY	31.89%	11.73%	5.23%	51.14%
BN-HGY	78.77%	8.92%	0.51%	11.80%
graphene	82.92%	10.1%	6.7%	1.0%

. It is evident that ZA and OP modes play a pivotal role in the thermal transport of HGY. Specifically, optical mode accounts for 51.2% of the total thermal conductivity, and TA and LA modes contribute comparatively less (16.9%). Notably, the temperature variation shows a minimal influence on the relative contributions of these phonon branches, as detailed in Table S1 of the Supporting Information. For BN-HGY, ZA mode plays a dominant role for κ_tol compared to LA, TA, and optical mode. For BN-HGY, ${\kappa }_{ZA}$ contributes almost 78.7% of κ_tol, while ${\kappa }_{TA}$ and ${\kappa }_{LA}$ give 0.1% and 0.5% at 300 K. In the context of graphene, acoustic phonon modes exhibit a predominant influence on thermal conductivity, accounting for 80.1% of the total conductivity. In contrast, the contribution from optical phonon modes is negligible, amounting to 1.1% [41] only. However, for HGY and BN-HGY, the contribution of optical phonon modes is particularly noteworthy, indicating their significant role in the thermal transport properties of these materials.

According to the definition of thermal conductivity, $\mathsf{\kappa }={\displaystyle \sum }_{\mathrm{i}}{{\mathrm{cv}}_{\mathrm{i}}}^2{\mathsf{\tau }}_{\mathrm{i}}$, group velocity (v) and phonon relaxation time (τ) as a function of frequency are plotted for each phonon branch of HGY and BN-HGY, as shown in Figure 3. It can be observed that the v values of HGY are comparable to those of BN-HGY. For long wavelengths, the v of TA and LA modes in HGY are approximately 5.46 km/s and 13.72 km/s. In contrast, the v values for BN-HGY are around 7.99 km/s for TA and 10.36 km/s for LA. Notably, the group velocity of the ZA mode in BN-HGY is significantly higher than that in HGY. In contrast, the frequency ranges (about 65–2186 cm⁻¹) of the optical modes in HGY are much broader than those of BN-HGY (as shown in Figure 1).

As shown in Figure 3c,d, the τ for each phonon branch of HGY and BN-HGY has been calculated. The results indicate that, for both materials, the ZA mode exhibits a longer relaxation time compared to other phonon modes. This phenomenon can be attributed to the material’s optimal planar configuration and its inherent geometric symmetry.

HGY is composed solely of carbon atoms, which have a lower atomic mass and a stronger C–C bonding compared to the B–N and B–C bonds in BN-HGY. This results in stronger phonon coupling and less phonon scattering in HGY, facilitating more efficient phonon transport. In addition, HGY exhibits higher lattice symmetry and structural regularity than BN-HGY. The introduction of B and N atoms into the HGY framework leads to mass disorder and bond heterogeneity, which significantly increases phonon scattering and impedes thermal transport. As shown in Figure 3, HGY presents more dispersive acoustic phonon branches, indicating higher group velocities. These high group velocities contribute to the larger thermal conductivity observed in HGY.

Compared with BN-HGY, HGY exhibits superior thermal conductivity, mainly for two reasons. First, the phonon group velocities of HGY—particularly for the acoustic modes (ZA, TA, LA)—are higher, which enables more efficient heat transfer. Second, the phonon lifetimes of HGY are longer over a wide range of frequencies, indicating less phonon–phonon scattering. In contrast, the interfacial mismatch and mass variation in BN-HGY lead to reduced velocities and enhanced scattering, thus lowering the thermal transport efficiency. Therefore, the higher thermal conductivity of HGY arises from its intrinsically favorable phonon transport properties.

The ZA mode of BN-HGY and graphene dominates the contribution to their total thermal conductivity. Additionally, the phonon relaxation time of three-phonon modes is greatly reduced, mainly due to strong hybridization between each optical phonon branch in HGY and BN-HGY. As a result, compared with graphene, the thermal conductivity (κ) of HGY and BN-HGY is significantly reduced, which can be attributed to their lower group velocity (v) and enhanced phonon–phonon scattering. Specifically, the optical phonon contribution in graphene is only 1.1%, whereas in HGY and BN-HGY, it rises to 51.2% and 20.7%, respectively. Since optical phonons are prone to strong scattering, this increased contribution leads to a noticeable decline in κ. This trend is consistent with our previous findings on C₃N [41], which also exhibits lower thermal conductivity than graphene due to enhanced optical phonon scattering.

To understand the origin of the reduced thermal conductivity observed in doped graphyne systems, we analyzed the hybridization characteristics at the atomic scale. In particular, the coexistence of the sp and sp² bonding configurations introduces localized vibrational modes and strong phonon scattering centers. The sp-bonded carbon atoms, which exhibit linear geometry, disrupt the continuity of the phonon transport pathways established by the sp² network, thus impeding coherent phonon propagation. This mixed bonding nature is evident in our calculated phonon dispersion relations, which show softened acoustic modes and increased low-frequency optical branches, indicative of enhanced anharmonicity and scattering.

2.2.3. Phonon Scattering Process

Total three-phonon scattering processes (P3) involve two distinct channels: absorption (P3⁽⁺⁾) and emission (P3⁽⁻⁾) processes. A more confined phase space results in a higher scattering rate, which ultimately reduces thermal conductivity. As illustrated in Figure 4, BN-HGY has a slightly larger phase space compared to HGY, particularly in the optical phonon region. This difference is consistent with the lower thermal conductivity observed in BN-HGY when compared to HGY. In Figure 4a,b, it is evident that low-frequency optical phonon branches in both HGY and BN-HGY exhibit a substantial scattering phase space. During the absorption process in the three-phonon scattering phase space (P3⁽⁺⁾), optical phonon branches are the primary absorption pathways for both materials, as shown in Figure 4c. These branches predominantly contribute to three-phonon interactions, such as the interaction of TA/LA/ZA modes with optical phonons (O), resulting in the transition: TA/LA/ZA + O → O.

As shown in Figure 4e,f, the three-phonon emission channel (P3⁽⁻⁾) of HGY and BN-HGY is mainly controlled by the optical phonon branch. During the emission process, the optical phonon branches of HGY and BN-HGY exhibit more emission channels than the acoustic phonon branch. As a result, the emission process O → LA/TA/LA+O will be instrumental in elucidating the phonon transport behavior of HGY and BN-HGY.

The Grüneisen parameter (γ) [54,55] is a quantitative tool that can be used to approximate the anharmonic nature of a structure. The Grüneisen parameter (γ) was calculated for both materials to evaluate the phonon anharmonicity of monolayer HGY and BN-HGY, as shown in Figure 5. In HGY and BN-HGY, γ is completely negative for the ZA branch, while γ is negative and partially positive for the TA, LA, and OP branches. The large negative γ of the ZA branch is typical for 2D materials and arises from membrane effects [56]. It is important to stress that the scattering behavior of the ZA branch is very restricted due to the symmetry-based selection rule [57]. In the context of one-atom-thick materials, reflection symmetry has been shown to result in the cancellation of third-order force constants involving an even number of out-of-plane directions. Consequently, scattering processes such as ZA + ZA ↔ ZA and ZA + LA/TA ↔ LA/TA cannot occur. Compared to HGY, the magnitude of γ for BN-HGY is slightly larger, especially in three-phonon branches, mainly due to the stronger non-harmonic phonon properties of BN-HGY. This results in stronger phonon–phonon scattering behavior originating from the enhancement of anharmonicity, leading to smaller phonon lifetimes and lower κ for BN-HGY compared to HGY.

2.2.4. Size Effects

The influence of the size effect on the phonon transport properties of HGY and BN-HGY can be discussed by calculating the accumulated κ of HGY and BN-HGY relative to the mean free path (MFP), as shown in Figure 6. BN-HGY exhibits a wider phonon MFP spectrum than HGY, spanning from a few nanometers to nearly 10⁵ nm. Additionally, phonons with short MFPs play a significant role in the total thermal transport of BN-HGY. This phenomenon is completely different from HGY, in which phonons with long MFP contributions play a dominant role in the thermal conductivity of HGY. By detecting the 50% accumulation κ, the representative mean free path (rMFP) can be obtained. The rMFP of HGY calculated at room temperature is 43 nm, while the rMFP of BN-HGY is 2600 nm, which is much higher than that of HGY. It is obvious that the κ of BN-HGY will decrease faster with a larger rMFP. In addition, nanostructures such as the formation of nanoribbons or the merging of pores can be used to effectively reduce κ and expand its application in thermoelectric fields.

Moreover, the size effect in reducing the lattice thermal conductivity can be further enhanced by fabricating nanoribbon structures of HGY and BN-HGY. Recent advances in focused ion beam (FIB) techniques have enabled the direct patterning of periodic nanoribbons in 2D TMDs with high spatial resolution [58]. Given the planar nature and robust framework of HGY-based materials, such techniques could potentially be extended to them. In addition, chemical etching and nanotube unzipping methods, previously demonstrated in graphene nanoribbons [59], may also serve as viable routes to obtain well-defined HGY nanoribbons for thermoelectric applications.

3. Calculation Method

The Vienna Ab Simulation Package (VASP) was used for first-principles calculations [60,61]. The plane wave expansion of the electron wave function was cut off at 500 eV energy. The exchange-correlation functional applied is the Generalized Gradient Approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [62]. The goal was to minimize the Hellmann–Feynman force acting on each atom. A rigorous process was initiated to optimize the internal coordinates and lattice parameters. This process was pursued until the force acting on each atom was reduced to 10⁻² eV/Å. A convergence criterion of 10⁻⁵ eV between consecutive steps was applied. The Brillouin zone was sampled using a 5 × 5 × 1 Monkhorst–Pack k-point grid. To effectively avoid interactions along non-periodic directions, a vacuum space of 10 Å was included.

The ShengBTE [63] program was used to calculate lattice thermal conductivity (κ) by solving the BTE. Second-order (harmonic) and third-order (anharmonic) atomic force constants (IFCs) were utilized through an iterative self-consistent approach. This approach has been demonstrated to be an effective means of predicting the lattice thermal conductivity of a variety of materials [41,42,64,65]. Phonon frequencies and harmonic interaction force constants (IFCs) were obtained by using density functional perturbation theory (DFPT) within the PHONOPY program [66]. In our calculations, a Γ-centered regular grid of 50 × 50 × 1 q-points was employed to discretize the Brillouin zone (BZ) for both HGY and BN-HGY. The nominal layer thickness was set to be 3.4 Å for both materials.

4. Conclusions

The phonon transport properties and thermal conductivities (κ) of HGY and BN-HGY were analyzed by integrating DFT calculations with BTE theory. The lattice thermal conductivity of HGY and BN-HGY is 38.01 W/mK and 24.30 W/mK, which is considerably lower than that of graphene. The distribution of acoustic and optical phonon modes, such as ZA/TA/LA+O → O processes, plays a pivotal role in phonon transport for both HGY and BN-HGY. It was found that the phonon non-harmonicity of BN-HGY is slightly stronger than that of HGY. The size effect of HGY was compared with that of BN-HGY, and it was found that BN-HGY has a wider phonon MFP spectrum due to the enhancement of anharmonicity. In BN-HGY, short MFP phonons are an important part of the heat transfer behavior. This phenomenon is completely different from HGY, in which phonons with long MFP play a dominant role in thermal conductivity. HGY and BN-HGY have the potential to exhibit relatively low thermal conductivity, which could assist in the design of thermoelectric nanodevices.