Microscopic mechanism of nucleation and growth of helium bubbles in monovacancy in tungsten: helium regulates the charged states of tungsten atoms

In fusion reactor, tungsten (W) has been selected as a candidate for plasma-facing materials due to its excellent properties. However, W-PFMs suffer from helium (He) bubbles where He atoms are produced during deuterium tritium fusion in fusion reactors. To date, there have been few contributions to uncovering the formation of He bubbles from the perspective of the microscopic electronic structure of He-mediated tungsten. In this work, we develop a tight-binding potential model for the W–He interaction to study He atom aggregation and nucleation in the electronic ground state as well as in different electronic excited states. The most important finding of this paper is that caused by the He atoms in the vacancy, some d-orbital electrons of the W atoms at the inner wall of the vacancy are transferred to the W atoms farther away from the vacancy, leading to the feature of positively charged W ions at the inner wall of the vacancy. As the number of He atoms in the vacancy increases, these W ions become more cationic. Under the repulsion between these adjacent cationic ions, the volume of vacancies increases, and more He atoms tend to gather and nucleate there. At the same time, the enhancement of the electronic excitation can also promote the abovementioned electron transfer between W atoms and further increase the vacancy volume, which increases the self-aggregation of the He atoms in the vacancy. Our results shed new light on understanding He self-aggregation in many different metal materials.

In fusion reactor, tungsten (W) has been selected as a candidate for plasma-facing materials due to its excellent properties. However, W-PFMs suffer from helium (He) bubbles where He atoms are produced during deuterium tritium fusion in fusion reactors. To date, there have been few contributions to uncovering the formation of He bubbles from the perspective of the microscopic electronic structure of He-mediated tungsten. In this work, we develop a tight-binding potential model for the W-He interaction to study He atom aggregation and nucleation in the electronic ground state as well as in different electronic excited states. The most important finding of this paper is that caused by the He atoms in the vacancy, some d-orbital electrons of the W atoms at the inner wall of the vacancy are transferred to the W atoms farther away from the vacancy, leading to the feature of positively charged W ions at the inner wall of the vacancy. As the number of He atoms in the vacancy increases, these W ions become more cationic. Under the repulsion between these adjacent cationic ions, the volume of vacancies increases, and more He atoms tend to gather and nucleate there. At the same time, the enhancement of the electronic excitation can also promote the abovementioned electron transfer between W atoms and further increase the vacancy volume, which increases the self-aggregation of the He atoms in the vacancy. Our results shed new light on understanding He self-aggregation in many different metal materials. * Author to whom any correspondence should be addressed.
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Experiments have demonstrated that after high-energy He irradiates W-PFMs, He particles gradually gather and form He bubbles therein [20,21]. In addition, a series of voids were generated in the system during the growth and evolution of He bubbles. The fuzzy nanostructure of W was observed on its surface. It could reduce the thermal conductivity of the W-PFM surfaces by several orders of magnitude, worsen the mechanical properties of W-PFM and cause embrittlement of W-PFM [1,[22][23][24][25][26][27][28][29]. This case significantly reduces the resistance of W-PFMs to irradiation damage and seriously affects their service life.
Although a series of physical behaviors of He bubbles on the W surfaces can be observed in experiments, it is difficult to obtain the real-time evolution process of the microstructure of the system during the nucleation and growth of He bubbles under the irradiation of energetic particles. Instead, it only speculates the physical mechanism of He bubble growth according to the experiments. Therefore, theoretical calculations are required further to study the physical mechanism of He bubble formation. Previous kinetic Monte Carlo simulations have shown that local He nucleation growth is critical for forming fuzzy nanostructures on W surfaces [30][31][32]. To deeply explore the nature of the aggregation, nucleation, and growth of He bubbles in W material, various mechanisms have been proposed in terms of empirical molecular dynamics (MD) calculations [33][34][35][36][37][38][39][40][41]. For example, the 'dislocation loop punching mechanism' has been suggested to understand the nucleation growth of He bubbles [33,37,[39][40][41]. The effect of binding energy on the evolution of the W self-interstitial dislocation loop around He clusters was presented from the energy perspective [33]. In some previous simulations, the growth rate of He bubbles was considered, corresponding to the gradual change in the He bubble growth mechanism from isotropic growth induced by generating W Frenkel pairs to the 'dislocation loop punching mechanism' [41]. Some researchers have proposed that the initial radius of the He bubble is also a factor associated with the growth mechanism of the He bubble [39]. On the one hand, the accuracy of empirical MD calculations is much lower than that of density functional theory (DFT). On the other hand, critical electronic structures in microphysical systems are not included in the empirical MD calculations. Therefore, those simulations only focus on the structural evolution during the growth of He bubbles. It is worth noting that these proposed mechanisms are based on a basic assumption that the atoms act as hard spheres without consideration of the electronic structure of the system during the growth of He bubbles.
To accurately gain deep insight into the formation mechanism of He bubbles in the concerned W-PFMs, it is necessary to employ the state-of-the-art method to study the microphysical mechanism of He bubble growth from the perspective of microscopic electronic structure. Typically, calculations based on DFT not only produce high-precision results but also provide electronic structures of a concerned system. Previous DFT calculations have predicted that the most stable interstitials for He atoms in the W lattice are the tetrahedral interstitial sites (TIS), and the migration energy barrier of a He atom between neighboring TIS is approximately 0.06 eV [42][43][44][45][46]. At the same time, some works also showed that He atoms could be easily trapped in a W vacancy [44,45]. More importantly, DFT calculations predicted that the sites occupied by He atoms are in the regions with lower electron density. Thus, the He atoms exhibit electrophobicity [43,46]. Although the DFT method can provide electronic structures of systems, it is difficult to treat large systems. For example, due to the limitation of computing resources, Song et al treated only the cases of He n with n = 1−13 in a vacancy of W crystal at the DFT level. Thus, the case of the system containing more He atoms is still a puzzle [46].
To study complex physical systems on a large scale under the framework of quantum mechanics, the tight-binding (TB) potential model is just a compromise between DFT calculations and empirical potential calculations. In the TB method, the multicenter bonding integrals in the system are simplified, thereby realizing the large-scale simulation of a system containing complex defects [47]. In this paper, based on the previous TB potential model of W, we develop a TB potential model for the interaction between W and He. The microstructural evolution of He atoms aggregated in a single vacancy is intensively studied using this potential model. Our calculations show that the He bubble makes the surrounding W atoms more cationic. The cation of these W ions increases with increasing He atoms in the vacancy, and the electrostatic repulsion between the W ions on the inner wall of the vacancy further increases the vacancy volume, causing the He bubbles to develop into giant ones. At the same time, during the operation period of the fusion reactor, the continuous irradiation of the high-energy photons excites some electrons of the W-PFMs, making the systems in excited electronic states. The nucleation behavior of the He bubbles in W-PFMs is also revealed in excited electronic states.

W-He TB potential model
In quantum theory, the energy eigen equation of a solid system is: Using the TB approximation, the eigenwave function ψ ⃗ k ( ⇀ r ) in the above equation is expanded in the representation based on the Bloch sum function composed of the localized atomic orbital wavefunction [47,48]: where N is the number of primitive unit cells, ϕ α ( is the αth orbital of atom i, C i,α is the expansion coefficient of the corresponding atomic orbital, and ⃗ R i is the position vector of atom i. The sum of i covers all N primitive unit cells. The Hamiltonian matrix elements and the overlap matrix elements are respectively expressed as (4) Let, In the TB potential model, the orthonormal basis approximation can be used, where the overlapping matrix becomes the identity matrix. Equation (5) includes the Coulomb integral term h iα,iα within the same atom and the hopping integral term between different atoms. The Coulomb integral approximately equals the value of the corresponding atomic orbital energy: For the hopping integral term h jβ,iα , multicenter integrals are included, which make it difficult to handle and thus need to be simplified. In our TB potential model, the double-center integral approximation proposed by Slater and Koster is employed [47]. In this scheme, hopping integral terms are expressed by defining different types of bond integrals, where bond integrals are expressed as empirical formulas with parameters. For W, we adopt the atomic orbital 5d6s6p as the basis. The expression and corresponding parameters of its bond integrals and the Coulomb terms have been reported in [49][50][51]. For the He system, we choose the 1s orbital as the atomic orbital basis set. In our TB potential, the 1s orbital energy of He is −17.573 675 eV. Although there is no chemical bond between He atoms, we still define the V ssσ bond integral between He atoms The bond integrals between W and He are: In formula (9), l ′ is the s state of He; l is the 5d, 6s or 6p of W. Therefore, V WHe ll ′ σ contains V WHe ssσ , V WHe psσ and V WHe dsσ .The parameters α WHe 1 , α WHe 2 , α WHe 3 , and α WHe 4 need to be determined through fitting. The term 1 represents the truncation function of the bond integral function, which rapidly decays to zero when the distance exceeds r WHe cut . Here, r WHe cut is the truncation distance that considers the interaction between atoms within this range. The sensitivity factor ω WHe controls the decay rate of the truncation function 1 . The corresponding parameters are listed in table 1. By solving the energy Eigen equations, the energy eigenvalues and the corresponding eigenfunctions can be obtained. Then, according to the band structure energy of the system is achieved. Here, ε m represents the mth eigenvalue, and f m is the Fermi-Dirac distribution function.
In the total energy calculations of the TB potential, we need to calculate the repulsion energy between atoms in the system in addition to the band structure energy. The repulsion energy is expressed as the sum of the pair potentials between atoms [51], namely: where f is a function expressed as a fourth-order polynomial with argument x = ∑ j ϕ (r ij ). The coefficients in formula (12) have been reported in [51]. ϕ WW represents the pair potential between W-W, the expression and related parameters have been also reported in [51]. We take the Lennard-Jones potential formula to express the pair potential between He-He: The pair potential formula between He-W is: All of the potential parameters between He-He and W-He are listed in table 2. In addition, the dimensionless parameters e WHe = 2.799 849 and e HeW = 0.006 116.
Finally, the total energy of the system can be written as [50,51]:

Theoretical treatment of nonthermal effects in TB calculations
In the real environment of a fusion reactor where W-PFMs serve, there are a large number of high-energy photons and high-energy ions irradiating the W-PFMs, causing strong electronic excitation in the W-PFMs. The excited electrons reach an equilibrium state through the interaction between electrons and the coupling between the electrons and phonons. The relaxation time of the interaction between the excited electrons is on the order of femtoseconds (fs), which is much shorter than that of the electron-phonon interaction (on the order of ps) [52][53][54][55]. Therefore, before the entire system, including the excited electrons and the lattice, reaches thermal equilibrium, the excited electrons reach their equilibrium state in advance, which is represented by the Fermi-Dirac distribution law with the electron temperature (T e ): In this situation, the excited electrons are at a high electron temperature, and the lattice is cooling. Therefore, the whole system, including the electrons and the lattice, is in a nonequilibrium state. In this state, the strongly excited electrons will change the interaction between ions in the lattice through Hellmann-Feynman forces, resulting in peculiar physical phenomena, such as coherent phonons [56], ultrafast solid-solid phase transitions [57,58], and nonthermal melting [59,60].
Generally, the time scale required for electrons to transition from the ground state to the excited state is on the order of fs. However, the energy transfer process from the excited state electrons to the ions through nonthermal effects (without electron-phonon interaction) is almost instantaneous. Within the framework of quantum theory, the electron excitation and de-excitation under ultrashort time intervals can be regarded as a kind of sudden perturbation [55], by which the free energy of the system is composed of the total energy (E tot ) of the system and the contribution of the electron entropy S e resulting from the excitation of some electrons [52]: with The evolution of atoms in the system in real space is driven by force acting on the atoms. The force on each atom can be expressed as the partial derivative of the free energy of the system to the coordinates of the atomic position: (20) where ⃗ R i represents the position vector of atom i, − ∂Erep ∂ ⃗ Ri is the repulsive force, and the Hellmann-Feynman force is expressed as: |ψ k ⟩ represents the electron energy eigenvector of the occupied k state, and H (k) is the corresponding Hamiltonian.
By this method, we can more accurately simulate the structural evolution and physical properties of the system caused by nonthermal effects in excited electronic state generated by ultrafast irradiation.

Results and discussions
We first examine the validity of our W-He TB potential model. To this end, we construct a 9 × 9 × 9 bic supercell of a perfect W system containing 1458 W atoms. After full relaxation in the electronic ground state, the lattice constant of the supercell is 28.49 • A. In the electronic ground state, a He atom is respectively added at one TIS and one octahedral interstitial site (OIS) in this supercell to assess the relative stability of the He atom at these two interstitial sites. As shown in table 3, our calculations show that the formation energies of a He atom occupying TIS and OIS are 6.10 eV and 6.37 eV respectively, which are close to the related values obtained from the DFT calculations. Meanwhile, the energy of the He atom occupying TIS is lower by 0.27 eV than that of OIS, which is consistent with the DFT values of 0.21-0.25 eV reported by Kong et al [44][45][46][61][62][63].
Since the formation of He bubbles in W-PFM should accompany the migration of He atoms, we utilized the climbing image nudged elastic band (CI-NEB) method to calculate the migration energy barrier of a He atom between the neighboring two TIS [64]. As seen in figure 1, the calculated migration energy barrier of 0.052 eV is very close to the energy barrier value (0.06 eV) predicted at the DFT level [42]. These features indicate that our developed W-He TB potential model is reliable for handling the W-He interaction.
3.1. Self-aggregation behavior of He in the W crystal in the electronic ground state 3.1.1. Self-aggregation behavior of He in tetrahedral sites. Figure 1 shows that the migration energy barrier of He in the W crystal is quite small, which indicates that a single He atom can easily migrate between different tetrahedral sites in the electronic ground state. This phenomenon indicates that the probability of He atoms meeting in the two adjacent TISs in an interstitial is very high. Therefore, we calculated the successive trapping energies of several He atoms at adjacent  [46]. b Huang et al [61]. c Becquart and Domain [62]. d Nguyen-Manh and Dudarev [63]. tetrahedral sites in interstitials in a W crystal. Here, the trapping energy can be defined as follows [65]: where E perfectTISn presents the total energy of the system when n He atoms are located in n neighboring tetrahedral sites, and E perfect the total energy of the perfect W system. E trap TISn is the energy pay when the nth He atom is attracted by the He cluster consisting of (n − 1) He atoms (He n − 1 ). If the trapping energy is negative, the nth He atom favorably merges into the He n − 1 cluster rather than occupying the TIS away from the He n − 1 cluster. Figure 2 shows the trapping energy as a function of the number of He atoms in an interstitial.
From figure 2(a), we find that as the number of He atoms increases from 1 to 6 in an interstitial, the trapping energy is negative and gradually decreases. This phenomenon indicates that as the number of He atoms increases, it becomes easier for an interstitial to trap He. Furthermore, we analyze the microstructure around the He cluster during the He atom aggregation process. As shown in figure 2(b), when the number of He atoms is less than 3, these He atoms almost locate the tetrahedral sites and hardly cause a structural change of the surrounding W atoms; when the number of He atoms reaches 3, the aggregation of these He atoms makes a neighboring W atom undergo a large displacement, becoming an interstitial W atom and leaving a vacancy behind. Therefore, from the perspective of atomic structure, the aggregation of He atoms in the adjacent tetrahedral sites can be converted into aggregation in the vacancy as the number of He atoms in the tetrahedral sites increases. Similar phenomena have also been observed from DFT calculations [66]. Undoubtedly, by further adding He atoms close to the He cluster, some of the newly added He atoms can gather at the new vacancy created above. Therefore, the subsequent He aggregation becomes the issue of He aggregation at W vacancy, which is discussed in the following subsection.

Self-aggregation of He atoms in a W single vacancy.
We sequentially introduce He atoms to a single vacancy until the number of He atoms reaches 36. The trapping energy of each He atom at the electronic ground state is shown in figure 3. Here, the trapping energy is defined as follows [65]: where E VacHen presents the total energy of the system when n He atoms are located in a single vacancy. Apparently, with the increasing number of He atoms in the single vacancy, each trapping energy is always negative, and the absolute values of these trapping energies are quite large. This result strongly implies that the energetic stability of He atoms in a vacancy is better than that of He atoms in a tetrahedral site away from the vacancy. Moreover, we calculated the energy barrier of one He atom migrating from a tetrahedral site near the vacancy to the vacancy, and no energy barrier was found. So, the single vacancy can spontaneously trap nearby He atoms. Figure 4 shows the stable structures of He n (n = 1-36) clusters in a single vacancy (notated as VacHe n , n = 1-36). One He atom is positioned exactly in the center of the single vacancy. When there are two He atoms there, they form a dumbbell with a length of 1.63 Å at the vacancy center, and this dumbbell is oriented along one of the <100> directions. Three He atoms in the vacancy form an equilateral triangle with sides of 1.70 Å on a {100} plane. Four He atoms form a regular tetrahedron with side lengths of 1.77 Å. The configuration of the VacHe 5 cluster is a right square pyramid, and the quadruple axis is along one of the <100> directions. Six He atoms form a regular octahedron around the vacancy center with a distance of 1.80 Å between the nearest neighboring He atoms, and the quadruple axis is along one of the <100> directions. After seven He atoms, each He cluster in the W vacancy exhibits a close-packed structure and does not possess any apparent geometric symmetry.
Generally, the self-aggregation of He atoms in a single W vacancy should change the microstructure of the system  to some extent, which is mainly reflected in the structural changes of the eight atoms at the inner wall of the vacancy. For convenience, we notated these eight W atoms as first-nearest-neighbor W (1NNW) of the He bubble. We found that the He atoms in the vacancies realize the growth of He bubbles by continuously pushing the W atoms around the vacancy outward. Figure 5(a) displays the displacements of these eight W atoms during the self-aggregation process of He atoms in the vacancy. The growth process of He bubbles can be divided into three stages: (  figure 4). Therefore, the He atoms are still trapped by the single vacancy. (3) The third stage is when 16 or more He atoms are introduced in the single W vacancy. At this stage, the disparity of the displacements of the eight W atoms around the vacancy is more obvious, among which the displacements of some W atoms become very large, and these W atoms have moved to the interstitial region (figure 4), resulting in double vacancies or even multiple vacancies to capture more He atoms.
According to the analysis above, we realize that an ideal single vacancy can trap up to 15 He atoms.
We now turn to understand why the growth of the He bubble causes the W atoms at the inner wall of the vacancy to emit outward. For this purpose, we calculated the formation energies for each W atom at the inner wall of a single vacancy to become a vacancy based on the definition as follows [45]: Here, E tot (Vac m He n ) presents the energy of a system containing n He atoms in the void consisting of m vacancies, E W−interstitial the energy of the W crystal in which there is an interstitial W atom, and E tot (Vac m+1 He n ) the energy of the system with n He atoms in the void consisting of (m + 1) vacancies. If the vacancy formation energy, E f (Vac) , is negative, a W atom at the inner wall prefers to stay in its original site. Otherwise, this W atom can easily move to the interstitial region around the vacancy. Figure 5(b) shows the relationship between the vacancy formation energy of the W atoms concerned above and the number of He atoms in the original vacancy. We found that as the number of He atoms in the original vacancy increases, the vacancy formation energies of W atoms at the inner wall of the vacancy vary from −4 eV to 8 eV. This result shows that the stability of these W atoms worsens during He bubble growth. In detail, when the number of filled He atoms increases from 1 to 4 (the first stage), the vacancy formation energies of the eight W atoms at the inner wall are all negative and are almost the same. This situation means that these W atoms cannot easily leave their sites. When 5-15 He atoms are contained in the original vacancy (the second stage), the vacancy formation energies of these W atoms become positive, and they differ slightly. This phenomenon means that the W atoms at the inner wall of the vacancy can easily leave their original sites. When more than 16 He atoms are contained in the original vacancy (the third stage), the W atoms at the inner wall are more unstable due to the high vacancy formation energies. They thus can leave their initial sites more easily and become interstitial W atoms. At the same time, the differences among the vacancy formation energies of these eight W atoms are more significant, and they are less equivalent compared to the case of the second stage. In total, as the number of He atoms in the single W vacancy increases, the stability of the W atoms at the inner wall gradually deteriorates.
As mentioned above, He atoms can self-aggregate in the neighboring tetrahedral sites of an ideal lattice or a single-atom vacancy. By comparing the trapping energy of the same number of He atoms in figures 2(a) and 3, we found that the trapping energy of He atoms placed in the TIS is systematically higher than that of He atoms placed in the vacancy. This result indicates that compared with TIS, He atoms are more willing to self-aggregate in the single vacancy of W.

Self-aggregation behavior of He at W single vacancy in excited electronic states
During the tokamak operation, many high-energy photons and high-energy ions generated in the plasma core region irradiate the W-PFM. The W-PFM is not in the electronic ground state but in excited electronic states. Therefore, considering the physical behavior of He bubble growth in an excited electronic state is closer to the real environment of the tokamak operation. We first revealed the pure effect of excited electronic states on the migration of He atoms between tetrahedral sites in the W crystal. In our treatment, we applied electronic temperatures of 1000 K, 5000 K, and 10 000 K to the system without considering lattice temperature. The energy barrier of a He atom migrating between neighboring tetrahedral sites was calculated using the CI-NEB method at each considered electronic temperature [64]. Figure 6(a) displays the energy profiles associated with the migration of a He atom in the concerned path at the three electronic temperatures, where the case of T e = 0 K is also shown for comparison.
As seen in figure 6(a), the energy barrier of the He atom migrating between tetrahedral sites decreases with enhancing electronic excitation. Thus, the migration of He atoms in the W lattice becomes easier at higher electronic excitations. Next, we evaluated the migration energy barrier for a He atom located in a tetrahedral site near a vacancy to migrate toward this vacancy in different excited electronic states. Our calculations indicate that no migration energy barrier is required for such a concerned migration, similar to the electronic ground state. Therefore, in excited electronic states, a He atom at a neighboring tetrahedral site of the vacancy still spontaneously enters the vacancy. From the energy landscape, the initial TIS, at which a He atom is located near the vacancy, corresponds to an unstable point or a saddle point on the potential energy surface. When the system is in different excited states, the slope of the potential surface over which a He atom slides may be different, which can be reflected in the energy difference of the system before and after the slide of He. Figure 6(b) shows that such an energy difference decreases as the electronic temperature increases, suggesting that the concerned self-aggregation behavior is weakened with increasing electronic excitation.  Now, we turn to the effect of electronic temperature on the ability of vacancies to trap He atoms. To this end, we perform structural relaxation for the cases where 1-36 He atoms are sequentially placed in the W vacancy at electronic temperatures of 1000 K, 5000 K, and 10 000 K, respectively. We found that, like the case of the electronic ground state, there was no significant change in the growth performance of the helium bubble in different excited electronic states from the perspective of structure. The growth of the He bubble in the vacancy is still achieved by constantly pushing the W atoms around the vacancy outward. The calculated trapping energies of He atoms are shown in figure 7(a). We found that the calculated trapping energies are all negative in the range of electronic temperatures we considered with large absolute values. Therefore, the He atoms still stably stay in the vacancy when the system is in these excited states. At the same time, the trapping energies of the He atoms in the highly excited electronic states increase with increasing electronic temperature, indicating that the probability of He atoms being trapped by vacancies decreases with increasing electronic temperature.
To gain deep insight into the electronic temperature effect shown above, we decompose the trapping energy into the band structure energy, the repulsion energy, and the energy contributed by the electron entropy, which correlates respectively with the electronic structure, atomic structure and degree of disorder distribution of electrons in the excited system. We take the case where the W vacancy contains one He atom as an example for the following discussion. As shown in figure 7(b), the contribution of electron entropy increases rapidly with increasing electron temperature. In contrast, the sum of the band structure energy and repulsion energy decreases gently with increasing electronic temperature. Since the electron entropy represents the degree of disorder distribution of electrons, our results shown in figure 7(b) mean that the increase in electron disorder arising from electronic excitation is the main reason for the increment of vacancy trapping energy.

Electronic structures in the electronic ground state
Helium is an inert element whose valence orbital is fully occupied by electrons. Due to this character, the interaction between He atoms and between the W atom and the He atom characterizes the feature of van der Waals interaction with no chemical bonds between them. From the perspective of electronic structure, the d-electron interaction between W atoms is one of the key factors that affect the change in vacancy structure. If He atoms are present in the vacancy, what effect do these He atoms have on the electronic structure of W atoms? For this concern, we have selected some typical W atoms for our analysis. These typical W atoms include those in 1NNW, 2NNW and 3NNW around the vacancy. The positions of representative atoms in 1NNW, 2NNW and 3NNW are shown in figure 8(a).
For comparison, we removed the He atoms in VacHe 1 -VacHe 20 with keeping the atomic structures of the vacancy. In these He-free systems, the average electron charges in the d-orbitals of the eight W atoms in 1NNW, six in 2NNW and twelve in 3NNW were computed, and by taking those as a reference, we calculated the changes in the average electron charges in the d-orbitals of the abovementioned W atoms as the number of He atoms in the vacancy increased. Figure 8(c) displays the calculated changes in these concerned electron charges, where the positive value corresponds to the gain of electron charges and the negative value to the depletion of electron charges. As displayed in figure 8(c), with the increasing number of He atoms, the W atoms in 1NNWs and 2NNWs around the VacHe n cluster lose more electron charges, but those in 3NNWs gain electron charges. When n reaches 20, the average d-electron charges on the 1NNWs and 2NNWs decrease by 0.47 e and 0.28 e, respectively, and that on the 3NNWs increases by 0.04 e.
In the three cases, we considered, the distance between 1NNW and the center of the vacancy is the smallest and that between 3NNW and the center of the vacancy is the largest. By combining these different distances with the associated changes in electron charges above, we immediately found that the distribution of electron charges in W atoms around the Hefilled vacancy depends on the distance between the W atom and the center of the vacancy. On the other hand, the structure of the W crystal is not isotropic, and the spatial distribution of the electron charges may also show the characteristics of anisotropy. To examine this feature, some W atoms that locate in the and [1 11 ] directions around the vacancy (see figure 8(b)) are selected, where the distances between these atoms and the center of the vacancy are different (see table 4). Clearly, as seen in figure 8(d), the changes of the electron charges in the concerned W atoms along different directions are different. This result becomes striking when more He atoms locate in the vacancy.
As pointed out before, He is an inert gas atom, which almost does not transfer electron charges between the He atom and the connected W atoms. Instead, when a He atom connects with one or more W atoms, some of the valence electrons of W electrostatically polarize the electronic cloud of the He atom, causing a dipole moment in it. As such, there are dipole moment interactions between the He atom and its neighboring W atoms, redistributing the electron charges of the W atoms. Therefore, the He atoms in the vacancy drive the transfer of electron charges between W atoms in the system. The schematic diagram of this mechanism is concisely represented in figure 8(e). Such van der Waals forces, which regulate electronic structures, have also been exhibited in twisted graphene [67].
Importantly, the He-induced changes in the electron charges of W atoms around the He-filled vacancy can be used to interpret the three stages of structural evolution of the He bubble described above. When the number of He atoms in the vacancy varies from 1 to 4 (corresponding to the first stage of the growth of the He bubble), as shown in figure 8(c), the average d-orbital charge in 1NNWs around VacHe n changes slightly (as shown with the black squares). Hence, the 1NNWs characterize a weakly ionic feature attributed to the weak interaction between the He atoms in the vacancy and the 1NNWs of VacHe n . In the second stage (corresponding to the case of 5-15 He atoms), the changes in the average d-orbital charges of 1NNW atoms around VacHe n become negative dramatically (as shown with the blue squares). Accordingly, the 1NNWs around VacHe n rapidly become more cationic. At this moment, the electrostatic repulsion between these cationic 1NNWs is greatly enhanced, leading to large positional deviations in these W atoms. By further introducing He atoms into the original vacancy, the d-orbital charges in 1NNWs around VacHe n are further reduced (as shown with the green squares), and the electrostatic repulsion between 1NNWs is correspondingly further strengthened, increasing the volume of the vacancy and promoting the growth of He bubbles at the vacancy.

Electronic structures in excited electronic states
As mentioned in the Introduction, during fusion reactor operation, the W-PFM is not in the electronic ground state but in excited electronic states. As electronic excitation is enhanced, the electronic structure of the system changes. Therefore, in addition to the He bubbles, the electronic excitation also leads to the redistribution of electron charges in the W lattice. At an electronic temperature corresponding to an excited state, the electrons of W occupy the energy states below the chemical potential. First, for the case of a single vacancy without He atom (VacHe 0 ), with respect to the electron charges of 1NNWs at the ground state, the changes in the average electron charges of 1NNWs in different excited electronic states can be estimated by using where µ the chemical potential, f excited (E) and f ground (E) are the Fermi-Dirac distribution functions of the VacHe 0 system in different excited electronic states and the electronic ground state respectively, and d excited (E) and d ground (E) are the local electronic density of states of 1NNWs around VacHe 0 in different excited electronic states and the electronic ground state. As shown in figure 9(a), with increasing electronic temperature, the changes in the average electron charges of 1NNWs around the VacHe 0 cluster are negative, and the curve shows a downward trend. This phenomenon indicates that the average electron charges of 1NNWs around VacHe 0 decrease in excited electronic states compared with the electronic ground state. Therefore, the enhancement of electronic excitation can promote the transfer of electron charges mentioned above.
With the enhancement of electronic excitation, the cationic nature of 1NNWs increases slightly, and the repulsion between these W ions strengthens weakly, leading to an increase in vacancy volume. From the perspective of the volume of the vacancy, electronic excitation seems to promote the ability of vacancies to accommodate He atoms weakly. Furthermore, we explore the effect of increasing the electronic temperature on the transfer of electron charges during the growth of He bubbles. Taking the average charges of 1NNWs around VacHe 0 as a reference, we calculated the changes in the average charges of 1NNWs around VacHe n (n = 1-20) in the electronic ground state and different excited electronic states. Two striking aspects can be observed in figure 9(b). First, with increasing the number of He atoms in the vacancy, the changes in the average charges of 1NNWs around VacHe n (n = 0-20) are all negative, and the curves show sharp downward trends. This result indicates that the growth of He bubbles in the vacancy can strikingly promote the transfer of electron charges from the W atoms near the vacancy to the W atoms far away from the vacancy, whether the system is in the electronic ground state or different electronic excited states. Second, as the electronic temperature increases, the absolute value of the average charge change on 1NNWs around VacHe n becomes slightly large for each given number of He atoms in the vacancy. Based on the above two aspects, we realize that the He atom filled in the vacancy and the electronic excitation of the system synergistically enhance the ionic nature of the W atoms on 1NNW around the vacancy, increasing in the volume of the vacancy. We should note that although the electronic excitation can expand the volume of the vacancy to benefit the accommodation of He atoms in the vacancy, the electron entropy arising from the electronic excitation contributes to a significant increase in energy for the accommodation of He atoms in the vacancy. As a result, the electronic excitation weakens the self-aggregation behavior of He in the system.

Conclusion
In this paper, the self-aggregation behavior of He atoms at TIS and vacancies in W crystals in both the electron ground state and the different electronic excited states were intensively investigated at the level of TB theory. In the ground state, He atoms can self-aggregate at TISs and vacancies. We propose that the initial nucleation and growth of a He bubble can be divided into three stages. Our calculated energetic stability of W atoms at the inner wall of the He-filled vacancy confirmed the reasonability of these three stages. More importantly, the calculated electronic structures of the systems revealed that during the growth of the He bubble, the filled He atoms in the vacancy can cause the transfer of electron charges from the W atoms at the inner wall of the vacancy to the W atoms far away from the vacancy, and the W atoms surrounding the vacancy show cationic nature. The repulsive interaction between the cations around the He-filled vacancy expands the volume of the vacancy, thus promoting the growth of He bubbles in the vacancy. This feature is further slightly enhanced when the system is electronically excited. However, the effect of electron entropy arising from the electronic excitation suppresses the self-aggregation of He atoms in the vacancy to some extent. Our results shed new light on understanding He self-aggregation in many different metal materials.