Superconductivity and strain-enhanced phase stability of Janus tungsten chalcogenide hydride monolayers

In this section, we investigate the structures of the 2H and 1T phases of WSeH and WSH, and compared them with 2H-WSH from a previous work [36]. In addition, we investigated the effects of the applied in-plane strain on the crystal structures. In this work, we concentrated on only one layer of WSeH or WSH, called a monolayer. However, in a WSeH or WSH monolayer, it can be divided into three sub-layers, which are the tungsten, hydrogen, and chalcogenide sub-layers. We chose the tungsten sublayer as the plane of reference. We chose the hydrogen sublayer as the top sublayer and the chalcogenide sublayer as the bottom sublayer. The crystal structures of WSeH are shown in Fig. 1 as an example. With periodic boundary conditions, the structure belongs to the 3D trigonal space group P3m1 (No.156). For the 2H phase, the Wyckoff positions of tungsten are at (0,0) in the in-plane 2D coordinates, and hydrogen and selenium are at the same Wyckoff positions at (1/3,2/3) in the 2D coordinates. This is called ABA stacking. For the 1T phase, the Wyckoff position of hydrogen changes to (2/3,1/3), and the stacking order becomes ABC.

For the non-strained study, the optimization of Janus 2H-WSeH gives the lattice constant of $a$= 3.070 Å, the distance between the top sublayer and the tungsten sublayer is $h_{tf}$= 1.008 Å, and the bottom sublayer and the tungsten sublayer is $h_{bf}$= 1.764 Å. When spin-orbit coupling (SOC) is activated, we obtain slightly different values of the cell parameters, which are $a$= 3.070, $h_{tf}$= 1.012 Å, and $h_{bf}$= 1.769 Å. For the biaxial strain study, we analyzed various amounts of strain for WSeH and WSH. We found that $h_{tf}$ and $h_{bf}$ decrease as the tensile (positive) biaxial strain is applied, but $h_{tf}$ and $h_{bf}$ increase as the compressive (negative) biaxial strain is applied. All crystal parameters of the non-strain and biaxial strain of the 1T and 2H phases are also summarized in Table 1.

In this section, we report the results of the electronic structures of WSeH and WSH. There are a few combinations of different lattice structures, such as 2H and 1T structures, and the different electronic structures, such as spin orbit coupling (SOC) and non-SOC electronic structures. The electronic dispersions and the corresponding total electronic density of states (TDOS) from 2H-WSeH with non-SOC (black lines) and SOC (red lines) calculations are shown in Fig.2 (a). It is readily seen that the 2H-WSeH monolayer exhibits metallic behavior as a result. The bands crossing the Fermi level are the W-$d$ orbitals: they have only a small projection onto the H-$s$ and Se-$p$ orbitals, The d-bands can be grouped into $A^{\prime}(d_{z^{2}})$, $E^{\prime}(d_{xy},d_{x^{2}-y^{2}})$ and $E^{\prime\prime}(d_{yz},d_{xz})$ at the $\Gamma$ point. The corresponding Fermi surfaces are shown in Fig.2 (b) and (c) for the case of non-SOC and SOC calculations, respectively. In general, the electronic structures near the Fermi level are very sensitive to the spin-orbit coupling (SOC) effect. In the non-SOC calculation (black lines in Fig.2 (a)), the band dispersions become flat near the $\Gamma$ point, resulting in a sharp peak of TDOS close to the Fermi level, which could be identified as the van Hove singularity (vHs). When we activate the SOC effect, the band degeneracies are removed, especially near the $\Gamma$ point and the Fermi level. Consequently, the vHs peak splits into two smaller vHs peaks near the Fermi level, as shown by the red lines in Fig.2 (a). The SOC effect significantly changes the topology of the Fermi surfaces, as shown in Fig.2 (c) compared to the non-SOC Fermi surfaces in (b). The topology of the Fermi surface is one of the key ingredients in superconductivity. For a conventional superconductor with strong electron-phonon interaction, a high electronic density of states at the Fermi level and strong Fermi surface nesting are the most expected features from the electronic part. The Fermi surface nesting describes the connection between any two different points on the Fermi surfaces. In these compounds, the Fermi surfaces are in the form of several enclosed loops. These loops have some certain spacing between them in the k-space. These features will also manifest themselves in the features of the phonon linewidths. We will discuss these features again when we consider the phonon linewidths in the next sections.

As the SOC effect is the most accurate representation of the electronic structures, we performed all of our calculations with SOC from now on, unless otherwise stated. Fig.3 (a)-(d) shows the comparison of the electronic structures between the 1T and 2H phases of WSeH and WSH. In general, the electronic structures show metallic behavior, and the bands are dominated by the W-$d$ orbitals that cross the Fermi level. The electronic structures between the 2H and 1T phases are very similar. The dispersions of 1T-WSeH (Fig.3 (c)) and 1T-WSH (Fig.3 (d)) below the Fermi level show greater dips to lower energy along $\Gamma$ to $K$, and $\Gamma$ to $M$, compared to 2H-WSeH (Fig.3 (a)) and 2H-WSH (Fig.3 (b)). These dips indicate stronger bonding between the H atom at the 1T site and the W atom. These findings should explain why the 1T structures have slightly lower energy than the 2H structures; see Fig. 4. However, when we performed the phonon calculations, we found that the 2H phonons are all stable, but some of the 1T phonons are unstable. Therefore, we applied the biaxial compressive strain to the 1T structure. We found that a small amount of strain can stabilize the 1T phase in both WSeH and WSH, shifting the bands at $\Gamma$ away from the Fermi surface as shown in Fig.3. The dynamical stability will be discussed in more detail in the next section. An interesting feature is observed along the $K-M$ directions at +1eV: the bands cross in the Se compounds, but not in the S compounds. These bands have $d$-orbital character, so in WSeH they hybridize between W and Se, breaking symmetry and opening a pseudogap. By contrast, in sulfur has no $d$-electrons, so the symmetry is unbroken and the bands cross.

At this stage, we discuss the static phase stability of WSeH and WSH. In Fig.4, we found that the 1T-WSeH and 1T-WSH monolayers are more energetically favorable than their 2H phases. For the WSeH monolayer, the energy of the 1T phase is lower than that of the 2H phase by 0.1283 eV. For the WSH monolayer, the energy of the 1T phase is lower than that of the 2H phases by 0.1369 eV. However, we also found that the 1T-WSeH and 1T-WSH structures are not dynamically stable unless compressive biaxial strain is applied. The strain-stabilized 1T structure will be referred to as $1T_{s}$. We found that $1T_{s}$-WSeH with compressive biaxial strain of 4% has a 0.0109 eV lower energy than that of 2H-WSeH, and $1T_{s}$-WSH with compressive biaxial strain of 2% is 0.1048 eV has a lower energy than that of 2H-WSH. The energy barriers for WSeH and WSH for the phase transition from 2H to 1T and a strain-stabilized 1T are shown in Fig. 4. It is clearly seen that relatively small amounts of energy can drive the phase transformation from the 2H to the 1T phase. It is also worth noting that both the 2H and 1T structures may not be true ground-state structures. However, the search for the ground-state structure is beyond the scope of this work. Instead, we tried to show that both phases could possibly be synthesized as in the case of MoSH [31]. Hence, we determined the formation energy of the crystal structure given by

where $E_{\text{W}}$, $E_{\text{Se/S}}$ and $E_{\text{H}}$ are the energy of the isolated tungsten, chalcogenide (Se/S) and hydrogen atoms, respectively. The formation energies per formula unit of 2H-WSeH and 2H-WSH are -15.86 and -17.24eV, while the formation energies of 1T-WSeH with strain -4% and 1T-WSH with strain -2% are -15.87 and -17.35eV, respectively.

Firstly, we discussed the phonons of the 2H WSeH. The calculations were performed using both non-SOC and SOC schemes. Fig.5(a) shows that all phonon modes are stable for this compound with the P3m1 trigonal space group. As the primitive cell contains three atoms, the phonon bands consist of three acoustic branches ($\nu=1-3$) and six optical branches ($\nu=4-9$). The three acoustic branches can be divided into an in-plane longitudinal mode (LA), an in-plane transverse mode (TA), and an out-of-plane flexural mode (ZA). These acoustic phonon bands will be referred to as band $\nu=1-3$. These bands occupy the frequency range of 0-15meV. The dispersion of the LA and TA modes must have linear behavior at the long-wavelength limit, while the dispersion of the ZA must exhibit a quadratic relation, as expected in all 2D materials [76, 77, 78, 79, 73]. For the six optical modes, they can be categorized into two sets of one $A_{1}$ and two $E$ subgroups. The first set of the optical bands will be referred to as band $\nu=4-6$. These bands occupy the frequency range of 20-35meV. The second set of the optical bands will be referred to as band $\nu=7-9$. The band 7 and 8 occupy the frequency range of 85-100meV, while the band 9 occupies the frequency range of around 125 meV. All bands and their associated frequency ranges are shown in Fig.5 (b). The absence of the 3D inversion symmetry causes all modes to be Raman and IR active. All important information is summarized in Table 2 for the SOC scheme. The most significant effects of the SOC scheme on the phonons are that the frequencies of the ZA modes around the $M$ point are a little higher than those of the non-SOC scheme, and the frequencies of the $E$ optical modes (the whole band 7 and 8) are a little lower than those of the non-SOC scheme. These behaviors are similar to those of 2H-WSH [36]. The noticeable differences between the phonons of WSeH and WSH are the frequencies of the band $\nu=4-6$, where the eigenvectors are dominated by the vibrations of the S/Se atoms. As the S atom is 2.46 times lighter than the Se atom, the frequencies of these modes (band $\nu=4-6$) in WSH are $\sqrt{2.46}=1.56$ times higher than those of WSeH; see Table 2 for comparison. We also studied the effects of strain on the 2H phase. We found that the 2H phase can withstand small amounts of both tensile and compressive strain, maintaining positive phonon frequencies. The corresponding superconducting quantities, computed under these strains, are shown in Table 3.

Next, we investigate the 1T structures. From Fig.6 (b) and (d), it is obvious that the unstrained 1T structures (green lines) are dynamically unstable in both WSeH and WSH. Thus, this leads us to consider the strained 1T structures. In this work, we consider only the biaxial strain. For the 1T structures, we found that -4% biaxial strain (blue lines) is needed to stabilize the 1T-WSeH structure, and -2% biaxial strain (blue lines) is needed to stabilize the 1T-WSH structure, as shown in Fig.6 (b) and (d). If we try to increase compression, the frequencies of the phonons tend to increase slightly, as shown by the red lines in Fig.6 (b) and (d). The main differences between the 1T and 2H structures are the frequencies of the phonon band 7 and 8. From the eigenvectors, the vibrations of these bands are dominated by the movement of the H atom, and somehow the 1T structures give higher frequencies of these modes than their 2H structures; see Fig.6 (a) and (b), and Fig.6 (c) and (d), and also Table 2 for comparison. These higher frequencies should come from the stronger bonding between the H atom at the 1T site and the W atom, as discussed in the section on electronic structures. The stable phonons of the 1T phase of -4% strained WSeH (blue lines) and -2% strained WSH (red lines) are compared again in Fig. 7 for clarification. The electron-phonon coupling ($\lambda_{\mathbb{q}\nu}$) is also projected onto the phonon dispersion in the 2H phase of WSeH and WSH, as shown in Fig.6 (a) and (c), which will be discussed in the next section along with Fig. 10. It shows that one of the most strongly coupled phonons arises from the $M$ point, which will play a key role in superconductivity as the electron-phonon coupling ($\lambda_{\mathbb{q}\nu}$) is averaged to $\lambda$.

As the unstrained 1T has unstable phonons, we investigated anharmonicity using ab initio molecular dynamics (AIMD) simulations. This is to determine whether thermal effects can stabilize the high-symmetry phase. We use the Born-Oppenheimer approximation which means that the zero-point motion is ignored. Zero-point motion can stabilize phonons which are unstable (”imaginary”) in lattice dynamics [80]. It also means that tunnelling through barriers is not possible. Here, we will demonstrate that the imaginary phonons are dynamically stabilised by temperature: this gives an upper bound since zero point would only make them more stable. BOMD may overestimate the metastability of the 2H phase, but the barrier calculation (Fig. 4) suggest that the tunnelling barrier is both high and wide.

The results of the AIMD simulations in the NVT ensemble are shown in Fig. 8. The 2H phase of both WSH and WSeH remain stable at 200K for 10ps, while for the 1T phase, both WSH and WSeH, with applied strains of -2% and -4%, respectively, shows thermal stability at room temperature of 300K where the result of WSH is shown in Fig. 8 (c) and (d).

At 300K, for both WSH and WSeH, we observed that the hydrogen atoms begin to migrate from the 2H sites to the 1T sites. This indicates that the 2H phase starts to undergo a phase transformation to the 1T phase, see Fig. 8 (a) and (b). For the 1T phase, we found that with a suitable lattice constant, the 1T phase of WSeH and WSH is stable up to room temperature. This finding is consistent with the calculations of the static structures, which has 1T as the lower energy phase and a barrier of around 150meV as shown in Fig. 4.

In this section, we evaluated the phonon linewidth, $\gamma_{\mathbb{q}\nu}$, and the electron-phonon coupling. $\lambda_{\mathbb{q}\nu}$, as a function of phonon wave vector $\mathbb{q}$ of band $\nu$, where $\nu=1-9$. From these two equations, it can be derived that $\lambda_{\mathbb{q}\nu}\propto\frac{\gamma_{\mathbb{q}\nu}}{\omega^{2}_{\mathbb{q}\nu}}$. Due to the electron-phonon interaction, the phonons are treated as a quasiparticle with a finite lifetime. This lifetime is directly related to $\gamma_{\mathbb{q}\nu}$. From Eq.(2), $\gamma_{\mathbb{q}\nu}$ also gives the description of the weighted connection between any two points on the Fermi surfaces by the phonon wave vector $\mathbb{q}$ of the band $\nu$. The Fermi surface connection can be written in analytic form as

and weighted by the transition propability $|g_{\bm{k}+\bm{q},\bm{k}}^{\bm{q}\nu,mn}|^{2}$, where $m$ and $n$ are the electronic band index. The Fermi surface connection can be divided into two groups; an intraband and an interband connections. For the intraband, $m=n$, the connecting vector $\mathbb{q}$ can have a magnitude roughly around $0\lesssim q<2k_{F}$. It appears that there are more connections of low q than those of large q, see the peaks near $\Gamma$ point in Fig.9. All these peaks are from the optical modes, $\nu=4-9$. The total magnitude of these peaks near $\Gamma$ point is truncated in the color mapping, otherwise it will overwhelm the features of the other peaks in the brillouin zone. However, for the acoustic mode, $\nu=1-3$, there is a special feature in which $\gamma_{\mathbb{q}\nu}\rightarrow 0$ as $\mathbb{q}\rightarrow 0$. For the interband, $m\neq n$, the connecting vector $\mathbb{q}$ can only have some discrete values, depended on the inter spacing between the different Fermi surfaces. This is because the topology of the Fermi surfaces in these compounds is in the form of several enclosed loops, where there are specific values of spacing between the different surfaces. The connection between these surfaces requires a specific value of $\mathbb{q}$. Thus, these features will manifest themselves as peaks in $\gamma_{\mathbb{q}\nu}$ at specific values of $\mathbb{q}$ only; see examples of the peaks away from the $\Gamma$ point in Fig.9, or the red spots in the color mappings.

For $\lambda_{\mathbb{q}\nu}\propto\frac{\gamma_{\mathbb{q}\nu}}{\omega^{2}_{\mathbb{q}\nu}}$, we can discuss general behaviors in the acoustic and optical modes. As $\lambda=\sum_{\mathbb{q}\nu}\lambda_{\mathbb{q}\nu}$, we would expect to find a high value of $\lambda_{\mathbb{q}\nu}$ everywhere. However, due to specific electronic and phonon structures, these lead to some noticeable patterns in $\lambda_{\mathbb{q}\nu}$. For the acoustic mode, $\nu=1-3$, $\gamma_{\mathbb{q}\nu}\rightarrow 0$ and $\omega_{\mathbb{q}\nu}\rightarrow 0$ as $\mathbb{q}\rightarrow 0$, and it has a small variation elsewhere. If $\gamma_{\mathbb{q}\nu}\rightarrow 0$ fast enough, $\lambda_{\mathbb{q}\nu}$ will go to zero as $\mathbb{q}\rightarrow 0$ as well. Thus, $\lambda_{\mathbb{q}\nu}$ can be large only if $\omega_{\mathbb{q}\nu}$ have comparatively low frequencies, as occurred around those softening points in WSeH, see peaks around the M point in Fig. 10, both in the dispersions and in the color mappings. For the optical mode, $\nu=4-9$, $\omega_{\mathbb{q}\nu}$ usually have high frequencies. Thus, $\lambda_{\mathbb{q}\nu}$ can be large near the $\Gamma$ point, where $\gamma_{\mathbb{q}\nu}$ is large, as seen in Fig. 10. These mappings of $\lambda_{\mathbb{q}\nu}$ give the description of how strong the electron-phonon coupling is in the phonon $\mathbb{q}$-space.

From the Eliashberg spectral function, $\alpha^{2}F(\omega)$, as shown in Fig. 5 (c) and 7 (c), we can compute the electron-phonon coupling constant $\lambda$ by using Eq.(6) and the logarithmic average of the phonon energy $\omega_{\text{ln}}$ by using Eq.(7). Then, the superconducting critical temperature can be evaluated by using Eq.(5). We assumed $\mu^{*}=0.1$. All superconducting data are shown in Table 3. There are several notable points, as follows;

1. The data show that $\lambda$ in the non-SOC scheme is slightly larger than that of the SOC scheme of the same phase of the same compound.

2. $\omega_{\text{ln}}$ in the non-SOC scheme is slightly lower than that of the SOC scheme of the same phase of the same compound.

3. Strain is important in stabilizing the phonons, and $T_{c}$ is slightly higher in the phase under tensile strain compared to compressive strain. But the effect on T_c is small (Table III).

4. $T_{c}$ appears to be rather unaffected by spin-orbit coupling. For 2H-WSeH, $T_{c}$ is 11.58K and 11.60K in the non-SOC and SOC schemes, respectively. In the case of 2H-WSH the values are 12.2K (SOC) and 11.6K (non-SOC)[36].