## 3.1 Cubic β-W1N1

We first examine the effect of the atomic vacancy on the electrical and thermal conductivity of cubic β-W1N1. For convenience, the electrical conductivity and thermal conductivity of the perfect system are taken as references. As displayed in Fig. 3(a), for the β-W1N1 system containing nitrogen (N) vacancies, the value of electrical conductivity decreases slightly with the increase of the vacancy concentration. Meantime, as shown in Fig. 3(b), their thermal conductivity also falls off slowly, which means that N vacancies have a slight inhibitory effect on the thermal transport property of the β-W1N1 compound. In contrast, for the system containing W vacancies, both electrical and thermal conductivity curves drop markedly; the thermal conductivity of the W-vacancy compound declines about 40% and 52% when nv(W) = 3.70% and 7.41%, respectively. Therefore, W vacancies can significantly depress the thermal transport properties of the system. For the system containing both W and N vacancies, the evolution of either electrical conductivity or thermal conductivity with the vacancy concentration is similar to those in the W-vacancy system. Therefore, the above three kinds of vacancy defects we considered all reduce the heat transport capacity of β-W1N1.

Next, the mechanism of the effect of W vacancies on electronic transport properties is discussed by calculating the EREC and ΔAPEC for the perfect system and the W-vacancy structure. As shown in Fig. 4(a), for the perfect system, the EREC value of tungsten (W-EREC) is 89.1%, while that of nitrogen (N-EREC) is only 10.9%. Apparently, the tungsten element predominantly contributes to the electrical conductivity of the perfect β-W1N1. In the structure containing three W atomic vacancies, the W-EREC significantly decreases to 53.1%, while the N-EREP remains unchanged almost. These results suggest that the influence of the W vacancies on the electronic transport of the N element in β-W1N1 is not obvious, but W vacancies are very detrimental to the electronic transport of the W element. Furthermore, the degradation of electron transport of the W element is apparently an important reason for the reduction of the thermal transport performance of the W-vacancy system.

Figure 4(b) shows the variation of ΔAPEC values for W and N atoms varying with distance from the atom to the nearest W vacancy. First, we found that the ΔAPEC values are negative for all tungsten atoms, which means that the electrical transport behaviors of all W atoms become worse after the appearance of the W vacancies. And this result is consistent with the previously mentioned decrease in W-EREC caused by W vacancies. Furthermore, it is found that the effective range of the inhibitory effect is long range: the ΔAPEC values are still significant even for W atoms at a distance of 6–7 Å away from W vacancies. In the case of N atoms, the ΔAPEC values are positive for several N atoms closer to W vacancies and negative for the N atoms far away from W vacancy sites. Thus, the effect of a W vacancy on N atoms exhibits distance-dependent feature: W vacancies can promote the electrical conductivity of N atoms in their vicinity, while they have a weak inhibitory effect on the electrical transport ability of distant N atoms.

To go further, we calculated the DOS of the β-W1N1 supercell before and after the introduction of W vacancies. First, we focused on the atom displaying the most significant changes in electronic transport properties, specifically those with notable values of ΔAPEC. As illustrated in Fig. 4(b), these atoms are marked as 1#W atom and 2#N. Clearly, the DOS of 1#W atom in the W-vacancy supercell is significantly smaller than that in the perfect system at the Fermi level. Since the electronic transport behavior in metallic materials is dominated by the electronic states near the Fermi level, the reduction of DOS at the Fermi level implies that fewer electronic states are involved in electronic transport, which is detrimental to both the electronic transport and the thermal conduction. Therefore, the electronic transport properties of the 1#W atom become deteriorated after the presence of W vacancies. For the 2#N atom, its DOS near the Fermi level in the W-vacancy supercell is significantly higher than that in the perfect system, so more electronic states are involved in electronic transport, which is favorable for the electrical conduction and thermal conduction. This is also consistent with the positive ΔAPEC value of 2#N atom.

In Fig. 4(d), the DOS for the W and N elements is presented for both the perfect system and the W-vacancy structure. After introducing W vacancies, the DOS of the W element decreases at the Fermi level. As a result, there are fewer electronic states associated with electronic transport, which is unfavorable for the electrical conductivity of the system. And this result is also consistent with the decrease in W-EREC. In contrast, the DOS of the N element in the W-vacancy supercell is almost the same as that in the perfect system, which means that the number of electronic states of the N element involved in conduction hardly changes after the introduction of W vacancies.

Similarly, we also investigated the mechanism of the effect of N vacancies on the β-W1N1 system by calculating EREC, ΔAPEC, and DOS.

The impact of N vacancies on the electronic transport properties of the W element is reflected in Fig. 5(a), where the value of W-EREC in the N-vacancy structure is reduced by about 5% compared to the perfect system, implying N vacancies have a certain suppressive effect on the electronic transport of the W element in β-W1N1. In Fig. 5(b), the ΔAPEC values are positive for the W atoms in the first proximal region near the N vacancies, in particular, the W atom (labeled as 3#W) closest to the N-vacancy has the largest ΔAPEC value; While the ΔAPEC values of the W atoms far away from vacancies are negative. These aspects suggest that the impact of N vacancies on the electronic transport of W atoms is distance-dependent: N-vacancies improve the electrical transport properties of nearby W atoms, but debase the electronic transport of distant W atoms. On the other hand, the local DOS of the 3#W atom in the N-vacancy structure is significantly larger than that in the perfect system at Fermi level, as shown in Fig. 5(c). This indicates that the presence of the N-vacancies enhances the electronic transport ability of the W atoms nearby. For comparison, the DOS of the W element, W-DOS, in the entire supercell containing N vacancies and in the whole perfect system are shown in Fig. 5(d). Strikingly, the W-DOS at Fermi level is almost the same in the two systems. Therefore, the number of states provided by the W element associated with the electronic conduction is almost unchanged after the introduction of N vacancies in the compound.

For the case of the N element in β-W1N1, the difference in N-EREC between the two structures is very small (seen in Fig. 5(a)), and the ΔAPEC values of N atoms displayed in Fig. 5(b) fluctuate around zero. In addition, the DOS of all N atoms in the perfect system and in the N-vacancy supercells are almost the same at Fermi level (Fig. 5(d)). These observations strongly suggest that the influence of N vacancies on the electron transport behavior of the N element in β-W1N1 is weak.

## 3.2 Hexagonal h-W2N1

We now focus on the h-W2N1 compound. As shown in Fig. 6, both the electrical conductivity and the thermal conductivity remain almost constant when the concentration of N vacancies increases from 0 to 3.7%, followed by a slight decrease when the concentration of N vacancies increases to 7.4%. Therefore, the high concentration of N vacancies is detrimental to the thermal transport of this compound.

For the case of the compound containing W vacancies, the electrical conductivity in the x&y and z directions gradually decreases with the increase of the vacancy concentration (Fig. 6(a, c)), and the conductivity in the z-direction drops more rapidly. On the other hand, the thermal conductivity of the W-vacancy system in the x&y directions remained almost constant as the vacancy concentration from 0 to 3.7%, and then decreased when nV(W) ranging from 3.7–8% (Fig. 6(b)). However, the thermal conductivity in the z-direction is monotonically and gradually decreasing (Fig. 6(d)). These results show that W vacancies can inhibit the thermal transport of the material to some extent.

When both W-vacancies and N-vacancies coexist in this compound, the electrical conductivity and the thermal conductivity of the compound drop more rapidly than those of the compound containing W vacancies. In this case, the coexisted W and N vacancies synergistically worsens the thermal transport of the electrons in the compound.

We furthermore studied the mechanism underlying the effect of W vacancies on h-W2N1 system by calculating the EREC, ΔAPEC and DOS. As shown in Fig. 7(a), the calculated values of ERPC clearly indicate that tungsten plays a dominant role (> 90%) in the electrical conductivity of the perfect h-W2N1 system. For the W-vacancy supercell, the W-EREC decrease to 87.4% in the x&y directions and to 78.8% in the z direction; Meanwhile, the N-EREC of nitrogen remains at around 7%. This means that W vacancies depress the electron transport of the W element, but have little influence on N element.

Figures 7(b, c) display the ΔAPEC values of the atoms after the introduction of W vacancies in the supercell. In the case of x&y directions (Fig. 7(b)), the ΔAPEC values of the W and N atoms exhibit fluctuations around zero. This means that the electronic transport properties contributed by some atoms improve, while others deteriorate. In the z direction (Fig. 7(c)), the ΔAPEC value of N atoms is distributed around zero, and the ΔAPEC values of most W atoms are negative. Thus, the decrease in the electrical conductivity in the z direction of the W vacancy system comes from the deterioration of the electronic transport in the z direction of these W atoms.

Figure 7(d) depicts the DOS of the perfect h-W2N1 system and the structure containing W vacancies. It is evident that the DOS of N element nearly unchanged in both structures. However, the DOS of the W element in the W-vacancy supercell is slightly lower at the Fermi level compared to the perfect system, indicating a reduction of electron transport states. As mentioned earlier, these reductions in electronic states are unfavorable for thermal transport.

Similar analysis was also performed for the system containing N vacancies. As shown in Fig. 8(a), the values of W-EREC and N-EREC in the N-vacancy supercell are almost unchanged compared to the perfect system. In addition, Figs. 7(b) and 7(c) show that the ΔAPEC values of both W and N atoms in the N-vacancy supercell exhibit a distribution around zero in both the x&y and z directions. Furthermore, as shown in Fig. 7(d), the values of N-DOS and W-DOS at the Fermi level in the N-vacancy supercell are nearly identical to those in the perfect system. These findings suggest that N vacancies do not significantly affect the electronic transport behavior of tungsten and nitrogen in the h-W2N1 system.

## 3.3 Hexagonal h-W2N3

We also examined the predicted most energetically stable tungsten nitride, h-W2N331. In the x&y directions (Fig. 9(a, b)), both the electrical conductivity and thermal conductivity grew rapidly as the concentration of three types of vacancies gradually increased from 0–8.33%, reaching several times that of the perfect system. Besides, the system containing both W and N vacancies showed the fastest increase in thermal conductivity, followed by the W-vacancy supercells and N-vacancy supercells. Similarly, in the z direction (Fig. 9(c, d)), the electrical and thermal conductivities also increase rapidly with increasing the concentration of the three types of vacancies. Specifically, the presence of N vacancies with a high concentration (8.33%) significantly enhanced the thermal transport performance of the material in the z-direction. In summary, all three types of vacancies considered in our study greatly improve the thermal transport properties of h-W2N3.

In the case of the perfect h-W2N3 system, as shown in Fig. 10(a), the primary control over electrical conductivity lies with tungsten, similar to the W1N1 and W2N1 compounds discussed previously. However, the introduction of W-vacancies into the supercell leads to a significant increase in both the W-EREC and N-EREC values in both x&y direction and z direction. This observation indicates that W vacancies enhance the electronic transport properties of both the W and N elements in the h-W2N3 system. Figures 10(b, c) provide the components of ΔAPEC in x&y and z directions, respectively. In the x&y directions, most W and N atoms exhibit positive values of ΔAPEC, implying that W vacancies enhance the electrical transport behavior associated with these atoms. Similar trends are observed in the z direction. Moreover, Fig. 10(d) presents the DOS for the h-W2N3 system before and after the introduction of W vacancies. It is evident that the DOS of both W and N elements within the W-vacancy supercell is considerably higher than that of the perfect system at the Fermi level. These findings suggest an increased number of states involved in electronic transport behavior within the W-vacancy supercell, which is beneficial for enhancing thermal conductivity.

Moving on to the h-W2N3 system containing N-vacancies, both the W-EREC and N-EREC values exhibit a substantial increase, as shown in Fig. 10(e). This indicates an enhanced electronic transport behavior of both W element and N element. In Fig. 10(f, g), the values of ΔAPEC are predominantly positive for most W atoms and N atoms, suggesting improved electrical transport behaviors contributed by these atoms in the N vacancy system. Figure 10(h) displays the DOS curves for both W and N elements. It is observed that the DOS of both W and N elements increase significantly at the Fermi level in N-vacancy supercell. As mentioned above, these newly emerged electronic states contribute to the improvement of thermal conductivity.

Before closing this paper, it is crucial to underline the behavior of N atoms in WNx compounds within the scope of nuclear fusion reactors [16, 35, 37, 53]. In a nuclear fusion reactor, the removal of N atoms is promoted through irradiation sputtering, thereby resulting in an increased presence of N vacancies in WNx materials. In the present study, our calculations show that the impact of N vacancies on the thermal conductivity of β-W1N1 and h-W2N1 systems is comparatively marginal. Moreover, we have found that N vacancies can potentially enhance the thermal conductivity of h-W2N3. Therefore, the presence of N vacancies in WNx compounds does not lead to a significant reduction in their thermal transport properties. This characteristic is notably advantageous for sustaining heat transfer stability on the surface of the deflector within nuclear fusion reactors and possesses significant implications for their design and operation.