Thermal Transport Properties of Na₂X (X = O and S) Monolayers

Abstract

Motivated by the excellent functional thin film devices made from two-dimensional materials, we investigated the thermal transport properties of Na₂X (X = O and S) monolayers using first-principle calculations. The thermal conductivity at room temperature was 1.055 W/mK and 1.822 W/mK for the Na₂O monolayers and Na₂S monolayers, respectively. The high thermal conductivity for the Na₂S monolayers is mainly contributed to by in-plane transverse acoustic (TA) phonons. The group velocity for the Na₂S monolayers exhibits lower group velocity and a larger phonon relaxation time than the Na₂O monolayers. Our results are helpful for functional thin film devices made using Na₂X (X = O and S) monolayers.

1. Introduction

Two-dimensional (2D) materials usually consist of multiple thin layers that are connected by weak interlayer van der Waals forces. Owing to their ultra-thin structures, 2D materials have attracted increasing amounts of attention in a number of research areas, including in thermoelectric [1], photodetector [2], and spintronic devices [3]. In practice, various synthesis techniques are applied to build 2D material-based functional nanoscale devices [4,5,6]. As a typical representative of 2D materials, graphene shows great potential in photonic and electronic devices [7]. For example, Wang et al. [8] utilized graphene as a high-impedance surface to build reconfigurable terahertz leaky-wave antennae. The tunable conductivity of graphene can effectively modulate the reflection phase, resonant frequency, and radiation pattern. The unique physical and chemical properties of 2D materials make them well-suited for use in functional nanoscale devices.

Investigating new 2D functional materials has become one of the more popular topics in materials science [9]. Some 2D materials that have already been applied to fabricate thin film membranes include graphene [10], MoS₂ [11], g-C₃N₄ [12], h-BN [13], and MXene [14]. Due to its zero band gap [15] and high thermal conductivity [16], graphene has been found to be inappropriate for photovoltaic and thermoelectric applications. Moreover, these 2D materials are easily oxidized in the air. Therefore, plenty of efforts have made to investigate and fabricate stable 2D materials [17,18].

Among these new 2D materials, dialkali-metal monochalcogenide monolayers provide new opportunities and challenges for thermoelectric and photovoltaic devices. Hua et al. [19] reported that dialkali-metal monochalcogenide monolayers exhibit a 2D gate-tunable magnetism. They also found that dialkali-metal monoxide monolayers host multiple symmetry-protected topological phases that can be tuned by strain engineering [20]. Recently, Rawat et al. [21] investigated the photovoltaic properties of M₂X (M = Na, K, and Cs; X = O, S, Se, and Te) monolayers using density functional theory. They found that the van der Waals heterostructures of K₂O/Cs₂O, K₂S/Cs₂S, and K₂Se/Cs₂S exhibit high power conversion efficiency in tandem solar cells. These theoretical works prove that dialkali-metal monochalcogenide monolayers exhibit good semiconductor properties and show great potential in energy harvesting.

To achieve thermoelectric and photovoltaic applications, thermal transport properties are essential. Since there are many types of dialkali-metal monochalcogenide monolayers, we chose Na₂O and Na₂S monolayers as the research objects after a structure stability test. In this work, we present a theoretical prediction of the thermal transport properties in Na₂X (X = O and S) monolayers. Our results show that Na₂O and Na₂S monolayers exhibit low thermal conductivity.

2. Materials and Methods

All first-principle calculations were completed using the Quantum ESPRESSO package [22,23]. The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) [24] function was utilized, and the corresponding pseudopotential files were obtained from the standard solid-state pseudopotentials library [25]. To obtain the fully relaxed structures, the energy and force convergence threshold were set as 10⁻⁶ Ry and 10⁻⁴ Ry, respectively. The kinetic energy cutoff was set as 80 Ry, and the k-mesh was set as 13 × 13 × 1. We set the vacuum spacing to be 15 Å to avoid interlayer interactions. Spin–orbit coupling (SOC) has negligible effects and was not included in this work. The Heyd–Scuseria–Ernzerhof (HSE06) hybrid function [26] was added while calculating band structure.

The phononic thermal conductivity κ_ph was calculated using the phonon Boltzmann transport theory [27]. In the phonon Boltzmann transport theory, κ_ph is obtained by

$${\kappa }_{ph}=\frac{1}{NV}{\displaystyle \sum }_{\lambda }{\kappa }_{\lambda }=\frac{1}{NV}{\displaystyle \sum }_{\lambda }{C}_{\lambda }{v}_{\lambda }⨂v_{\lambda }{\tau }_{\lambda }$$

(1)

where N is the total number of phonon modes, $V$ is the volume, $C_{\lambda }$ is the heat capacity, $v_{\lambda }$ is the phonon group velocity, and ${\tau }_{\lambda }$ is the phonon relaxation time. For small displacements, the total potential energy can be expanded into

$$E=E_0+\frac{1}{2!}{\displaystyle \sum }_{ij,\alpha \beta }{\mathsf{\Phi }}_{ij}^{\alpha \beta }{u}_i^{\alpha }{u}_j^{\beta }+\frac{1}{3!}{\displaystyle \sum }_{ijk,\alpha \beta \gamma }{\mathsf{\Phi }}_{ijk}^{\alpha \beta \gamma }{u}_i^{\alpha }{u}_j^{\beta }{u}_k^{\gamma }+\dots$$

(2)

where $u_i^{\alpha }$ is the displacement of atom $i$ along the α(x, y, z) direction. The coefficients $\mathsf{\Phi }$ in the n-th order term are the corresponding n-th order interatomic force constants (IFCs). Calculating higher-order interatomic force constants requires a lot of time [28]. Generally, third-order IFCs are enough to obtain suitable results. A supercell-based finite-difference method was used to calculate the third-order IFCs. Under a small displacement $h$, ${\mathsf{\Phi }}_{ijk}^{\alpha \beta \gamma }$ is expressed as

$${\mathsf{\Phi }}_{ijk}^{\alpha \beta \gamma }=\frac{{\partial }^{3}E}{\partial u_i^{\alpha }\partial u_j^{\beta }\partial u_k^{\gamma }}\cong \frac{1}{2h}\left[\frac{{\partial }^{2}E}{\partial u_j^{\beta }\partial u_k^{\gamma }}\left(u_i^{\alpha }=h\right)-\frac{{\partial }^{2}E}{\partial u_j^{\beta }\partial u_k^{\gamma }}\left(u_i^{\alpha }=-h\right)\right]$$

(3)

The Grüneisen parameter $\gamma$ can also be calculated from third-order IFCs.

$$\gamma =-\frac{1}{6\omega }{\displaystyle \sum }_{ijk,\alpha \beta \gamma }{\mathsf{\Phi }}_{ijk}^{\alpha \beta \gamma }\frac{e_i^{\alpha *}{e}_j^{\beta }}{\sqrt{m_i{m}_j}}exp\left(i\mathit{q}·{\mathit{R}}_j\right)u_k^{\gamma }$$

(4)

where $m_i$ is the mass of atom $i$, $e$ is the phonon eigenvector, and $\mathit{R}$ is the unit cell vector. Details about phonon Boltzmann transport theory can be found in Refs. [27,28,29].

Phononic thermal conductivity was calculated using the ShengBTE code [27]. The phonon dispersions and harmonic interatomic force constants (second-order IFCs) were calculated using the PHONOPY package [30] with a 7 × 7 × 1 supercell (147 atoms in total). The anharmonic interatomic force constants (third-order IFCs) were calculated using the maximum atomic interaction distance of the ninth neighbor with a 7 × 7 × 1 supercell. The q-point grid was set to 60 × 60 × 1, and the smearing parameter was set as 1.0.

3. Results

In Figure 1a, we present different views of Na₂X (X = O and S) monolayers. The structure of the Na₂X (X = O and S) monolayers is the same as that of the MoS₂ monolayers. The space group is P-6m2 (No. 187), and the point group symmetry is D_3h. Along the z-axis, the atomic layers follow the sequence Na-X-Na (X = O and S). As depicted in Figure 1a, we defined the thickness d for the distance between two Na atoms along the z-axis. The relaxed lattice parameter a = 3.451 Å and 4.249 Å and the relaxed thickness d = 2.306 Å and 2.581 Å for the Na₂O and Na₂S monolayers, respectively. Since the transport properties along the x-axis and y-axis are the same, we only present the transport properties along the x-axis in the following discussion.

We present the band structures in Figure 1b,c. It can be noted that the band structures for the Na₂O and Na₂S monolayers are nearly the same, except for the band gaps. The band gaps are 0.251 eV and 1.467 eV for the Na₂O and Na₂S monolayers, respectively. The Na₂O monolayers exhibit stronger metallicity than the Na₂S monolayers. The band gap can reflect the energy needed to break the bond. This confirms the normalized electron localization function shown in Figure 1d,e. The values of 0 and 1 represent accumulated and vanishing electron densities [31]. It can be seen that the electron densities are highly localized around the X (X = O and S) atoms and that the distributions between the Na atoms and X (X = O and S) atoms are nearly negligible. This means that the Na-X (X = O and S) bonds are ionic. Apparently, the strength of a Na-O bond is weaker than that of a Na-S bond. The weaker bond strength corresponds to a lower κ_ph [32].

To evaluate the thermal transport properties, we first calculated the phonon dispersions of the Na₂X (X = O and S) monolayers in Figure 2a,b. Obviously, there is no imaginary mode that implies the dynamic stability of the Na₂O and Na₂S monolayers. It can be noted that the phonon dispersion in the Na₂O monolayers is similar to that of Na₂S monolayers. There is a small gap in the Na₂O monolayers, and this small gap disappears in the Na₂S monolayers. The cutoff frequencies are 400 cm⁻¹ and 275 cm⁻¹ for the Na₂O and Na₂S monolayers, respectively. We also calculated the corresponding projected density of states (DOS) in Figure 2c,d. In the low frequency range, the DOS is contributed to by Na and X (X = O and S) atoms equally. For the middle frequency range, the DOS is mainly contributed to by Na atoms. At the high frequency range, the DOS for the Na₂O monolayers is occupied by O atoms, and the DOS for the Na₂S monolayers is occupied by Na and S atoms equally.

After obtaining the second- and third-order IFCs, we can calculate the thermal conductivity κ_ph as a function of temperature, as shown in Figure 3a. Generally, the κ_ph vlaies of the Na₂S monolayers are higher than those of the Na₂O monolayers. For example, the κ_ph values for the Na₂O and Na₂S monolayers are 1.055 W/mK and 1.822 W/mK at room temperature, respectively. These κ_ph values are much lower than those of MoS₂ monolayers [33], which implies the potential thermoelectric properties of Na₂X (X = O and S) monolayers. The κ_ph decreases as the temperature increases and follows a $1/T$ dependence. This is caused by the enhanced phonon–phonon scattering with temperature. When the temperature increases to 700K, the κ_ph is reduced to 0.452 W/mK and 0.783 W/mK for the Na₂O and Na₂S monolayers, respectively. We also discussed the effect of system size on the phononic thermal transport properties. When the size of the system reduces to the mean free path level, boundary scattering is enhanced and leads to a lower κ_ph. Figure 3b displays the cumulative κ_ph as a function of the mean free path. The κ_ph for the Na₂O monolayers increases faster than that of the Na₂S monolayers as the mean free path increases. This means that reducing the κ_ph of Na₂O monolayer requires a smaller particle size than Na₂S monolayers do. To provide further discussion, we display the κ_ph contributed to by the out-of-plane flexural acoustic (ZA), in-plane transverse acoustic (TA), in-plane longitudinal (LA) acoustic, and optical modes in Figure 3c. The TA mode occupies the main part of the κ_ph for the Na₂X (X = O and S) monolayers. The κ_ph values occupied by optical modes for the Na₂X (X = O and S) monolayers are similar.

To analyze the differences in the thermal transport properties, we present the phononic-related parameters for the Na₂X (X = O and S) monolayers in Figure 4. Because the shapes of the phonon dispersions for the Na₂O and Na₂S monolayers are similar, the group velocity also shows similar shapes, as shown in Figure 4a. Na₂O monolayers possess higher group velocity than the Na₂S monolayers. According to the Equation (1), a higher group velocity means higher thermal conductivity. However, this is not consistent with the above κ_ph results (the κ_ph of Na₂S is higher than that of Na₂O). The phonon relaxation time is the main reason for this. In most frequency ranges, the phonon relaxation time for the Na₂S monolayers is higher than that of the Na₂O monolayers, as seen in Figure 4b.

The scattering rate $1/\tau$ can be reflected by the P₃ parameter and Grüneisen parameter $\gamma$ [34]. In three-phonon scattering processes, energy $\left({\omega }_j\left(\mathit{q}\right)\pm {\omega }_{j^{\prime }}\left({\mathit{q}}^{\prime }\right)={\omega }_{j^{″}}\left({\mathit{q}}^{″}\right)\right)$ and momentum $\left(\mathit{q}\pm {\mathit{q}}^{\prime }={\mathit{q}}^{″}+\mathit{G}\right)$ conservation are required [35]. This limits the phase space available for the three-phonon scattering processes. The P₃ parameter represents the available phase space for anharmonic three-phonon scattering. The square of the $\gamma$ parameter represents the strength of anharmonic scattering [27]. We present the P₃ and $\gamma$ parameters contributed by different modes in Figure 5. The overall P₃ parameter of the Na₂S monolayers is much higher than that of the Na₂O monolayers, especially for the acoustic part. The phonon modes for the Na₂S monolayers provide more channels for anharmonic scattering than the Na₂O monolayers. As for scattering strength, it can be noticed that the $\gamma$ from the ZA modes of the Na₂S monolayers is negative, which is common in 2D materials [36]. However, the $\gamma$ from the ZA modes of the Na₂S monolayers is much higher than in the other cases, as mentioned above. We noticed that the phonon dispersion of the ZA modes from Γ to K in the Na₂S monolayers is softer compared to in the Na₂O monolayers. Thus, the high κ_ph value from TA modes in the Na₂S monolayers arises from the increased available phase space for anharmonic scattering by S atoms.

4. Conclusions

In summary, we calculated the thermal conductivity of Na₂X (X = O and S) monolayers using first-principle calculations. The shapes of the band structures and the phonon dispersions for the two structures were similar but resulted in different thermal transport properties. The thermal conductivity of the Na₂S monolayers is higher than that of the Na₂O monolayers and mainly contributes to the TA mode. The room temperature κ_ph values are 1.055 W/mK and 1.822 W/mK for the Na₂O and Na₂S monolayers, respectively. This work provides a theoretical guide for thermal management in electronic or thermoelectric devices based on Na₂O and Na₂S monolayers.