# Phonon Transport and Thermoelectric Properties of Imidazole-Graphyne

^{1}

^{2}

^{*}

## Abstract

**:**

^{12}cm

^{−2}hole concentration, which is much higher than the value for many other carbon-based materials. This work demonstrates that changing structural units from hexagonal to pentagonal can significantly reduce the lattice thermal conductivity and enhance the thermoelectric performance of carbon-based materials.

## 1. Introduction

^{2}σT/(k

_{e}+ k

_{l}), which depends on the synergetic effect of the Seebeck coefficient (S), electrical conductivity (σ), absolute temperature (T), electronic thermal conductivity (k

_{e}), and lattice thermal conductivity (k

_{l}). However, most commercial thermoelectric materials are based on elements that are relatively scarce and/or toxic, such as Bi

_{2}Te

_{3}[1], PbTe [2], and Sb

_{2}Te

_{3}[3]. Therefore, there is a need to find other earth-abundant and environmentally friendly materials with a good thermoelectric performance.

^{2}covalent bonds, hindering its application in the thermoelectric field. As an allotrope of graphene, the recently synthesized γ-graphyne [12] has provided a new possibility for the application of carbon-based materials in the thermoelectric field due to its high Seebeck coefficient of 690 μV/K [13] and low k

_{l}of 106.24 W/mK [14], which are superior to the corresponding values of graphene.

## 2. Computational Methods

^{−1}, is used to sample the Brillouin zone for integration in the reciprocal space. All atomic positions are fully optimized with convergence thresholds of 10

^{−8}eV and 10

^{−6}eV/Å for the total energy and force component, respectively. During the calculations of geometry optimization and band structure, 2D periodic boundary conditions along the x and y directions are applied to ID-GY, while a vacuum region of 16.59 Å is set along the z direction to exclude the mirror interactions between adjacent images.

## 3. Results and Discussion

#### 3.1. Phonon Spectrum and Band Structure

_{1}, C

_{2}

_{,}and N atoms are in sp

^{2}hybridization while C

_{3}, C

_{4}, and C

_{5}atoms are in sp hybridization. Unlike γ-graphyne, composed of hexagonal units, ID-GY is composed of pentagonal units connected by acetylenic linkers. Compared with the highly symmetric γ-graphyne, the complex geometric structure and hybridized bonding in ID-GY make it a promising material with a low lattice thermal conductivity, just like graphyne and graphdiyne [29,30].

_{3}and C

_{4}atoms. The bond length of C

_{3}-C

_{4}is 1.23 Å, showing the characteristics of alkyne bonds. Because the large portion of heat is carried by low-frequency phonons, especially the acoustic phonons, the low-frequency region of the phonon spectrum is magnified, and the acoustic phonon branches are highlighted in red. The longitudinal acoustic (LA) and transverse acoustic (TA) branches of ID-GY are linear when the wave vector q is close to the Γ point, while the out-of-plane acoustic (ZA) branch exhibits parabolic dispersion, which is a characteristic of monolayer 2D materials [31]. The highest frequency of the acoustic phonon is relatively low (<5 THz), lower than that of γ-graphyne (about 8 THz) [14]. The low frequency of the acoustic phonons is associated with a low acoustic Debye temperature, as discussed in the following paragraph. Moreover, there is a strong overlap between the acoustic and low-frequency optical branches. These characteristics indicate that the lattice thermal conductivity of ID-GY might be low.

_{11}> 0, C

_{66}> 0 and C

_{11}> C

_{12}. The Young’s modulus Y, Poisson’s ratio ν, bulk modulus B, and shear modulus G were also calculated and are presented in Table 1. It was found that the stiffness (122.20 N/m) of ID-GY is only half that of graphene (342 N/m) [33], owing to weak in-plane bonds. Moreover, the sound velocity, which is usually used to measure the speed of phonons propagating through the lattice, can be determined from bulk modulus B and shear modulus G by the following formulas [34]: longitudinal sound velocity v

_{l}= $\sqrt{\frac{B+G}{\rho}}$, transverse sound velocity v

_{t}= $\sqrt{\frac{G}{\rho}}$, and average sound velocity vs. = 1/$\sqrt[3]{\frac{1}{3}(\frac{1}{{v}_{l}^{2}}+\frac{2}{{v}_{t}^{2}})}$, where ρ is the mass density. Based on the sound velocity, we obtained the Debye temperature using θ

_{D}= $\frac{\hslash {v}_{s}}{{k}_{B}}{\left(\frac{4\pi N}{S}\right)}^{1/2}$, where N is the number of atoms in the cell and S is the area of the unit cell. Debye temperature measures the temperature above which all modes begin to be excited; therefore, a high θ

_{D}indicates weak three-phonon scattering and hence a high k

_{l}. The calculated Debye temperature of ID-GY is 647 K, which is much lower than the corresponding value of 805 K of γ-graphyne. Consequently, it is natural to expect that ID-GY possesses a lower lattice thermal conductivity than γ-graphyne.

_{2}[36].

#### 3.2. Thermal Transport Properties

_{l}) of ID-GY was calculated for different temperatures. As shown in Figure 2a, the lattice thermal conductivity of ID-GY is 10.76 W/mK at 300 K, which is two orders of magnitude lower than that of graphene (3151.53 W/mK) [10] and much lower than that of many other 2D carbon hexagonal structures, including α-graphyne (21.11 W/mK) [14], β-graphyne (22.3 W/mK) [14], γ-graphyne (106.24 W/mK) [14], and graphdiyne (22.3 W/mK) [29], at the same temperature. This shows the importance of structural units in affecting the thermal conductivity of a material. We fitted the relationship of k

_{l}with temperature and found that k

_{l}is proportional to 1/T

^{1.05}, indicating that the three-phonon scattering is dominant in ID-GY, as is the case with graphene [38] and penta-graphene [15]. This was further confirmed by comparing the scattering rates of three-phonon scattering with those of the isotopic scattering process. The calculated results for these two scattering processes are plotted in Figure S1 in the Supplementary Materials. We found that the three-phonon scattering rates are nearly 100 times larger than the isotopic scattering ones. The calculated cumulative k

_{l}as a function of frequency is plotted in Figure S2, which shows that the phonons with frequencies lower than 20 THz contribute about 85% to the lattice thermal conductivity. Therefore, we focus on the low-frequency phonon branches (<20 THz) in the following discussion.

_{l}of ID-GY. k

_{l}can be expressed in the following form through the summation of the contribution of all of the phonon modes λ(q, j) with the wave vector q and branch index j:

^{3}K at 300 K. The variation in the phonon volumetric-specific heat with temperature is plotted in Figure 2b. It is worth mentioning that the phonon-specific heat value usually is not different from one material to another [39]. For instance, the phonon-specific heat value of γ-GY is 1.68 J/cm

^{3}K [29], which is almost same as that of ID-GY. The change in group velocity of ID-GY with frequency is given in Figure 2c, which shows that at the long-wavelength limit, the group velocity reaches the highest value of 17.5 km/s, close to that of γ-GY (~17.9 km/s) [14]. In the low-frequency region (below 20 THz), the overall group velocity is only slightly lower than that of γ-GY. The variation in phonon lifetime with frequency at room temperature is plotted in Figure 2d. The lifetime for most low-frequency phonon modes (0~20 THz) is about 2 ps, while that for γ-GY is larger than 10 ps [14]. Therefore, the short phonon lifetime in ID-GY is the main reason for the low lattice thermal conductivity.

_{3}and C

_{4}atoms vibrate strongly around their equilibrium positions. The atomic trajectory shows that -C

_{3}$\equiv $C

_{4}- pairs in acetylenic linkers are weakly bonded to the pentagonal rings. These conclusions are further confirmed by the atomic displacement parameter (ADP) and the bond energy curve, which can provide a visualization of the anharmonicity. As shown in Figure 3c, the ADPs of the C

_{3}and C

_{4}atoms are much larger. The bond energy curve shows the vibration of a relative energy change ΔE (in eV/per atom) when the bond length changes and reveals the phonon anharmonicity. By fitting the bond energy curve (Figure 3d), we obtain the anharmonic parameters (a

_{3}) for different type of bonds in ID-GY. The data in Table 2 show that the strong anharmonicity in ID-GY mainly originates from the -C

_{3}$\equiv $C

_{4}- pairs, which are weakly bonded to the pentagonal rings. The single N-C

_{4}and C

_{1}-C

_{3}bonds are too weak to yield an inefficient thermal transport by lattice vibration. The inhomogeneous bond environment and large lattice vibrational mismatch between the pentagonal rings and the acetylenic linkers hinder the transport of heat.

#### 3.3. Electrical Transport Properties

_{1}is the deformation-potential constant. The effective mass of carrier m* was calculated from the curvature of the conduction band minimum or valence band maximum by the parabolic fitting of the band edge using the formula m* = $\hslash {\left[{\partial}^{2}E/\partial {k}^{2}\right]}^{-1}$. These calculated results are summarized in Table 3.

_{e}, indicating that ID-GY could have a considerably high carrier mobility. The obtained carrier mobilities of ID-GY are 7933 and 1826 cm

^{2}/Vs for electrons and holes, respectively; accordingly, the electron relaxation time (τ

_{e}) is much longer than the hole relaxation time (τ

_{h}). It is worth noting that although the deformation potential approximation [41] has been widely used for predicting the carrier mobility of new thermoelectric materials [42,43], the carrier mobility is usually overestimated as compared to the experimental result due to the neglect of scattering between the carrier and either the defect or the substrate [44]. Conversely, these scattering processes can also reduce the thermal conductivity. Thus, the overestimated electrical conductivity and overestimated thermal conductivity may cancel each other out to some extent, resulting in a more reliable prediction.

^{2}σ), an optimum chemical potential is needed. The maximum PF value for p-type ID-GY is 22.14 mW/mK

^{2}at 300 K, while that for p-type ID-GY is 20.18 mW/mK

^{2}.

_{e}= LσT, where the Lorenz number L is equal to 2.44 × 10

^{−8}WΩ/K

^{2}[45]. As shown in Figure 4d, electronic thermal conductivity has a similar tendency to electrical conductivity.

^{12}and 1.75 × 10

^{12}cm

^{−2}, respectively, which are higher than the ZT values of many other 2D carbon materials [30], including graphene (0.01), α-graphyne (0.03), β-graphyne (0.12), γ-graphyne (0.17), and 6,6,12-graphyne (0.05). While at 800 K, the ZT values are 2.20 and 2.21 for p-type and n-type ID-GY.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Geometric structure, (

**b**) electronic band structure, (

**c**) phonon spectrum, and (

**d**) partial phonon density of states (PhDOS) of ID-GY.

**Figure 2.**(

**a**) Lattice thermal conductivity (k

_{l}), (

**b**) phonon volumetric-specific heat (C

_{v}), (

**c**) group velocity (v

_{g}), and (

**d**) phonon lifetime (τ) of ID-GY.

**Figure 3.**(

**a**) Weighted phase space (WP3), (

**b**) trajectory of the atoms in the xy-plane from ab initio molecular dynamics simulations at 800 K, (

**c**) atomic displacement parameter (ADP), and (

**d**) bond energy curves of ID-GY.

**Figure 4.**(

**a**) Seebeck coefficient (S), (

**b**) electrical conductivity (σ), (

**c**) power factor (PF), and (

**d**) electronic thermal conductivity (k

_{e}) of ID-GY as a function of chemical potential μ, respectively.

**Figure 5.**Variation in the thermoelectric figure of merit (ZT) of ID-GY with the chemical potential (μ).

**Table 1.**Calculated elastic coefficients C

_{ij}(in N/m), Young’s modulus Y (in N/m), Poisson’s ratio ν, bulk modulus B (in N/m), shear modulus G (in N/m), longitudinal sound velocity v

_{l}(in km/s), transverse sound velocity v

_{t}(in km/s), average sound velocity vs. (in km/s), and Debye temperature θ

_{D}(in K) for ID-GY. For comparison, the corresponding values for γ-graphyne (γ-GY) are also listed here.

C_{11} | C_{12} | C_{66} | Y | ν | B | G | v_{l} | v_{t} | v_{s} | θ_{D} | |
---|---|---|---|---|---|---|---|---|---|---|---|

ID-GY | 164.26 | 83.12 | 12.79 | 122.20 | 0.51 | 124.69 | 40.46 | 17.17 | 8.5 | 4.59 | 647 |

γ-GY | - | - | - | - | 0.41 * | 122.73 * | 77.04 * | 18.53 | 11.51 | 5.50 | 805 |

C_{3}-C_{4} | N-C_{4} | N-C_{1} | C_{1}-C_{3} | C_{5}-C_{5} | N-C_{2} | C_{2}-C_{5} | |
---|---|---|---|---|---|---|---|

a_{3} | 15.41 | 7.13 | 5.90 | 4.28 | 3.28 | 0.68 | 0.60 |

**Table 3.**In-plane elastic constant C (in N/m), deformation-potential constant E

_{1}(in eV), effective mass of carrier m* (in m

_{e}), carrier mobility μ (in cm

^{2}/Vs), and carrier relaxation time τ (in 10

^{−14}s) of ID-GY at 300 K.

Carrier Type | C | E_{1} | m* | μ | τ |
---|---|---|---|---|---|

electron | 164.26 | 4.94 | 0.11 | 7932.54 | 49.56 |

hole | 164.26 | 4.53 | 0.25 | 1826.31 | 25.93 |

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**MDPI and ACS Style**

Chen, Y.; Sun, J.; Kang, W.; Wang, Q.
Phonon Transport and Thermoelectric Properties of Imidazole-Graphyne. *Materials* **2021**, *14*, 5604.
https://doi.org/10.3390/ma14195604

**AMA Style**

Chen Y, Sun J, Kang W, Wang Q.
Phonon Transport and Thermoelectric Properties of Imidazole-Graphyne. *Materials*. 2021; 14(19):5604.
https://doi.org/10.3390/ma14195604

**Chicago/Turabian Style**

Chen, Yanyan, Jie Sun, Wei Kang, and Qian Wang.
2021. "Phonon Transport and Thermoelectric Properties of Imidazole-Graphyne" *Materials* 14, no. 19: 5604.
https://doi.org/10.3390/ma14195604