In this section, we present and discuss the results of our ab initio modelling of TaSSe, providing an extensive comparison with its parent compounds, TaS2 and TaSe2. The properties of a normal (undistorted) phase and of a PLD (distorted) phase exhibiting CDW are considered separately.
3.1. Normal Phase
Let us start the discussion of the system in question from the undistorted monolayer structure of a 1T polymorph, modelled using a 1 × 1 cell in the calculations, which we call a normal phase.
The structure of the studied TaSSe as well as of its parent compounds, TaS
2 and TaSe
2, is shown in
Figure 1, presenting side and top views of all the structures, with an in-plane unit cell marked with a solid rhombus (with the side length equal to the lattice constant
a). The values of structural parameters predicted by the present DFT calculations are summarized in
Table 1, containing the lattice constants, the monolayer thickness
d (defined as the distance between the upper and the lower plane containing chalcogen atoms), the relevant bond lengths between Ta atom and chalcogen atoms
and
, as well as the bond angles. The meaning of all the parameters is explained in a detailed way in
Figure 2 (see [
93]).
In general, a good consistency between the calculated and the measured lattice parameter
a values can be found, with the trend that the lattice constant for TaSe
2 is larger than for TaS
2 (as the atomic radius is larger for Se than for S); for the Janus structure, the predicted
a value is actually exactly an average of the lattice constants of the parent compounds. Also, the layer thickness takes a larger value for Se-containing TMD than for S-based TMD. In
Table 1, we offer some comparison with the experimental results. It must be emphasized that the experimental values after Ref. [
57] concern XRD measurements performed on the bulk samples. Moreover, the TaSSe data in Ref. [
57] do not correspond to the Janus structure, but to TaS
2−xSe
x random alloy with
. Therefore, such quantities as the bond lengths and the bond angles would be regarded as the averages of relevant parameters for Ta-S and Ta-Se bonds in the TaSSe Janus structure. On the other hand, the values reported in Ref. [
94] relate to the monolayers (but they result from the STM topography measurements, thus having significantly larger uncertainties than the XRD-based ones).
An interesting quantity for two-dimensional systems is the planar-averaged electrostatic potential as a function of the co-ordinate normal to the surface. It gives information about the electric field distribution inside the structure. The planar-averaged electrostatic potential
is plotted as a function of the co-ordinate
z perpendicular to the heterostructure plane in
Figure 3 for the studied Janus TaSSe and for the reference structures, TaS
2 and TaSe
2. The value of
corresponds to the position of the plane of Ta atoms. The asymmetric shape of the function
with two deep minima of unequal depth, corresponding to the planes of S and Se atoms, marks the presence of an in-built electric field in the Janus heterostructure (
Figure 3b), whereas for TaS
2 (
Figure 3a) and TaSe
2 (
Figure 3c), the analogous functions are symmetric (even) and no such in-built field is present. The difference in electrostatic potential at the position of S and Se plane can be estimated as 3.77 eV, being considerably larger than those predicted for PtSSe or other Pt-based Janus monolayers [
95], WSSe [
96], or MnSSe [
97]. The presence of an intrinsic electric field is a distinct feature of the Janus structures, leading to the interesting physical properties. As an example, the mirror symmetry breaking and the presence of an intrinsic electric field lead to inducing of the Rashba spin–orbit coupling [
8,
98].
The value of the potential
at a large distance from the surface relative to the Fermi level is equal to the work function of the material
. The values of predicted work functions are collected in
Table 2. It is notable that for Janus TaSSe, the values of work function at both sides of the layer differ by 0.368 eV, being a manifestation of an in-built electric field. This value is slightly smaller than the one predicted for MoSSe and WSSe [
18,
99] or PtSSe [
95]. On the other hand, when the values of
for TaSSe at each side are compared with
for TaS
2 and TaSe
2, it follows that the value at the S side is slightly lower with respect to the TaS
2 case, whereas the value at the Se side indicates an opposite tendency when compared with the result for TaSe
2. The work function values for TaS
2 are in accordance with the experimental results for bulk 1T-TaS
2 [
100]. In addition, it should be noted that a recent calculation of work function for monolayer 1T-TaSe
2 gave values consistent with our results [
101].
The in-built electric field in the Janus structure stemming from the difference in electronegativity between the chalcogen atoms at both sides is connected with the emergence of an electric dipole moment
, which can be calculated within the DFT formalism. In TaSSe, the dipole points from the S layer to the Se layer and its value is predicted to be equal to 0.0205 eV·Å (or 0.0983 Debye), whereas no dipole moment emerges for the parent compounds, TaS
2 and TaSe
2. The value obtained for TaSSe is somehow smaller than the one calculated by the DFT, for example, for MoSSe [
18,
99,
102], WSSe [
18,
99], PtSSe [
95,
103], or RhSSe [
104].
It is particularly interesting to investigate the effect of an external, perpendicular electric field on the selected quantities characterizing a Janus TaSSe monolayer. Similar computational studies in the literature involved such Janus monolayers as MnSSe [
97], SnSSe [
105], MoSSe [
102], PtSSe [
106], or HfSSe [
107].
The dependence of the electric dipole moment on the external electric field
F is shown in
Figure 4a. Note that the positive field in our convention points from the S plane to the Se plane. The observed dependence is piecewise linear. The inset shows the magnified region of the linear dependence of
on
F in the range of fields between −0.40 and 0.675 V/Å. In this particularly interesting range, a linear function,
can be fitted to the DFT data, resulting in estimation of the polarizability
equal to 0.290 e·
/V (or 4.17
if expressed as the polarizability volume). The electric dipole moment vanishes when the external field
of −0.0705 V/Å is applied (i.e., the in-build electric field is compensated by the external field equal to
).
It is instructive to additionally analyze the total energy of the structure obtained from the DFT calculations as a function of the external electric field, as shown in
Figure 4b. The energy scale is adjusted such that the maximum energy equals 0. The position of the maximum is exactly consistent with the vanishing of the dipole moment. A similar effect can be noticed, for example, for MoSSe [
102]. The energy of the induced electric dipole in an external electric field can be written as
, finally yielding
if the induced dipole moment
is expressed by Equation (
1). This parabolic approximation is valid in the range of electric fields between −0.4 and 0.675 V/Å, and the relevant parabolic function with
and
determined from fitting Equation (
1) to the DFT data for a dipole moment is shown in the inset in
Figure 4b, showing excellent agreement with the DFT data.
The charge redistribution in the structure can be analyzed based on the Bader charge [
108], like in other Janus monolayers [
97,
109,
110]. The values of Bader charge transfer on each constituent atom for TaSSe and its parent compounds are collected in
Table 3 (the values of charge transfer are relative to the thirteen valence electrons of Ta and the six valence electrons for S and Se). The results indicate that the electrons are transferred from the Ta atom to chalcogen atoms. As the electronegativity of S is slightly larger than that of Se, the Bader charge of the S atom exceeds somewhat the one attributed to the Se atom. Moreover, the electron gain of a given chalcogen atom (S or Se) is similar in a Janus structure and in a non-Janus typical dichalcogenide system (however, S in TaSSe gains slightly more electrons than in TaS
2, while the opposite is true for Se). The unequal value of the charge transfer for S and Se atoms results in the emergence of the in-built electric field in a Janus structure.
It is also interesting to analyze the Bader charge transfer dependence on the external electric field (see [
97]), which is plotted in
Figure 5. The results show an approximately linearly decreasing Bader charge as a function of the field for the range of fields below 0.675 V/Å for Ta atoms (
Figure 5a), whereas the trend is reversed for the stronger fields. For the chalcogen atoms, S tends to lower its Bader charge if the field is increased (with the exception of strong negative fields; see
Figure 5b), while an exactly opposite tendency is visible for Se atoms (
Figure 5c). Therefore, the field pointing from the S plane to the Se plane tends to push the electrons from Se to S atoms. It should be observed that the magnitude of the field-induced variation in Bader charge for the Ta atom is significantly lower than for the chalcogen atoms, so that the field mainly causes the charge transfer between both chalcogen layers.
The electronic structure of TaSSe and its parent compounds in the absence of an external electric field can be viewed in
Figure 6 (where the Fermi level is set to zero and marked with a dashed line), showing the calculation along the
–K–M–
path in the first Brillouin zone (see
Figure A1c). In all the cases, the monolayer has metallic properties, with a dispersive band crossing the Fermi level in between the K and the M high-symmetry point of the first Brillouin zone. This contrasts with such monolayers as MoSSe [
18,
99,
102], WSSe [
18,
99], or PtSSe [
95,
103], which exhibit significant energy gaps in the band structures. A notable feature of TaSSe is lifting the band degeneracy resulting in band splitting due to the mirror symmetry breaking, which results in the presence of spin–orbit coupling. A result of the corresponding calculation without taking into account the spin–orbit coupling can be viewed in
Figure A1a, where no band splitting occurs. The TaSSe band structure generally interpolates between TaS
2 and TaSe
2 cases. It is particularly visible for the
point and the position of valence-like bands below the Fermi level; the bands are separated by a gap from the conduction-like band touching the Fermi level for TaS
2, whereas they almost merge for TaSe
2; for the Janus structure, we deal with an intermediate gap value.
The characterization of the physical properties of monolayer TaSSe in normal (undistorted) phase can be supplemented with the calculation of phonon energies, the result of which is shown in
Figure 7 (for the case of TaSSe as well as its parent compounds). It should be emphasized that in the data presentation, the negative values correspond to the imaginary frequencies of phonon modes, giving rise to a structural instability. A striking feature of all the results is the presence of evident imaginary frequency modes along the
–K and
–M path. It should be emphasized that the most pronounced imaginary frequency mode appears in our calculations for a similar wavevector in all three cases—TaS
2, TaSSe, and TaSe
2. Such a situation is responsible for the instability of the systems towards a structural distortion causing PLD and entailing the emergence of CDW. Moreover, the close correspondence between the wavevectors at which the most pronounced imaginary modes occur and the characteristic wavevectors of CDW can be found, as emphasized, for example, by [
111] for the bulk case or by [
112] for the monolayer case of TaS
2. The characteristic wavevector of CDW phase for the
reconstruction is marked in
Figure A1c, showing the relevant first Brillouin zones. The vector is rotated anticlockwise by
from the
–M direction and its magnitude is equal approximately to 0.555 of the distance from
to M point. Let us mention here the calculations of phonon dispersion relations for monolayer TaS
2 [
30,
33,
113] and TaSe
2 [
33,
114] existing in the literature.
In order to simulate the influence of the increasing temperature on the phonon energies, we have performed the relevant DFT calculations using Fermi–Dirac smearing (see, for example, Ref. [
114] or [
115]).
Figure 8 shows the phononic dispersion relation obtained for TaSSe for the smearing temperature of 6000 K (thick lines) and for 100 K (thin lines). Let us emphasize that the electronic temperature used as a parameter for smearing relates only to the electronic subsystem, not to the ionic one, and is not equivalent to the physical equilibrium temperature of the studied system. Therefore, it is used to qualitatively illustrate the mechanism of thermally induced changes in the phononic frequencies. It is visible that the imaginary modes related to the acoustic phonons at low temperatures vanish when the temperature is elevated, so that the undistorted structure becomes stabilized by the increasing temperature. This mechanism is similar to the one observed in TiSe
2 in Ref. [
114].
3.2. Distorted Phase
After a thorough analysis of the normal phase properties, we present a discussion focused on the results obtained for a
supercell. The supercell geometry involves its rotation with respect to the 1 × 1 cell by an angle of
. For such a system, the atomic position relaxation leads to a distorted phase with PLD, giving rise to a commensurate CDW for all the three studied compounds. A top and side view of structures obtained for the supercell calculations is shown in
Figure 9. In the top view, four supercells are shown (each marked with a solid rhombus). A distinct feature of PLD in the discussed class of TMD is the star-of-David shape of the cluster formed by Ta atoms (which experience the most pronounced in-plane shifts). A supercell contains 13 Ta atoms; the central Ta atom of the star (situated at the edge of the supercell in our calculations) remains unshifted. An inner ring and an outer ring of Ta atoms (containing six atoms each and marked with solid circles in
Figure 9) can be singled out. The radii of the rings are reduced in PLD phase with respect to the values in normal phase, and this behavior is common to TaSSe and its parent compounds. The effect can be quantitatively tracked using the data collected in
Table 4, where the radii of the inner ring (
) and outer ring (
) are given for both phases, together with radii differences
and
between the phases. It is visible that the magnitudes of radius reduction in TaSSe are close to the values predicted for TaS
2, whereas the values found for TaSe
2 are lower (all the relative magnitudes being at the level of a few percent).
The values of the difference in total energy between the normal phase (without PLD) and the CDW phase (with PLD), calculated in a
supercell and normalized per formula unit, are collected in
Table 5. The results are expressed both in energy units and in temperature units (the latter being a sort of rough estimate of the transition temperature). It is notable that in all the cases, the CDW phase has significantly lower energy than the normal phase; the energetic stability of the CDW phase in TaSSe might be comparable to the case of TaS
2. It is worth mentioning here that the transition temperature for monolayer TaSe
2 was experimentally determined as 530 K [
46] (while it amounts to 180 K for bulk TaS
2 according to the same source).
In order to illustrate the suggested phase transition between a CDW and a normal phase when the temperature is increased, we additionally studied the dependence of the total energy per formula unit on the smearing temperature used in the DFT calculations (using Fermi–Dirac smearing for the scalar relativistic calculations). A similar approach was adopted in Ref. [
115] for the case of TiSe
2. The results for TaSSe and its parent compounds are shown in
Figure 10. In all the cases, for low smearing temperatures, the phase with CDW has significantly lower energy than the undistorted phase. When the smearing temperature is increased, the energy difference between the phases diminishes and, above some critical smearing temperature, the lowest energy phase is the undistorted phase. This sort of behavior indicates that the low-temperature stable phase should be a CDW phase.
In the side view of the structures in
Figure 9, some buckling of the Ta plane can also be observed, followed by a more distinct buckling of the upper and lower planes of chalcogen atoms.
The manifestation of buckling of the chalcogen planes is a non-uniform pattern of the local density of states (LDOS) in CDW phase. A calculated LDOS map for TaSSe is shown in
Figure 11 for the energy window centered at the Fermi level and having a width of 0.1 eV; the map covers a square 30 × 30
.
Figure 11a presents a LDOS map for the plane located 3 Å below the outermost atom from the S layer, whereas
Figure 11b presents a map for the plane located 3.5 Å over the outermost atom from Se layer. In both plots, the characteristic triangular patterns contributed by the chalcogen atom states are visible, as in the case of the parent compounds TaS
2 and TaSe
2 and other
supercell-forming TMDs [
34,
36,
46,
48,
94,
116,
117,
118,
119]. Such a modelling can be qualitatively compared to the STM image and verified experimentally [
34]. It is visible that our results indicate the presence the of pattern characteristic of a CDW phase with star-of-David clusters.
The emergence of PLD is connected with the formation of CDW. The charge redistribution can be tracked first on the basis of the Bader charge transfer for all three studied compounds.
Table 6 summarizes the Bader charge transfer values (charge relative to the number of valence electrons of a given atom) for the central Ta atom of a star-of-David deformation (
), for Ta atoms from the inner ring (
) and from the outer ring (
). Moreover, the average values for Ta, S, and Se atoms are given (
,
, and
, respectively). The average values are considerably close to the values for the normal phase, collected in
Table 3. It can be concluded that the charge redistribution during the formation of a CDW phase takes place mainly among the Ta atoms, whereas the contribution of the chalcogen atoms is less important. Namely, the central Ta atom gains the electrons and so do the Ta atoms from the inner ring, whereas the Ta atoms from the outer ring become depleted of electrons. The process is visualized in
Figure 12, which shows the differences in Bader charge values between the CDW phase and the normal phase. In the figure, the plots are centered at the unshifted Ta atom of a star-of-David distortion (located at the edge of the supercell), and the radii of discs are proportional to the Bader charge difference (with different scale for each compound shown in panels
Figure 12a–c). In the case of TaSSe (
Figure 12b) and TaS
2 (
Figure 12a), the central Ta atom gains most of the electrons (also, the inner ring of Ta atoms is enriched with electrons). For TaSe
2 (
Figure 12c), the charge gain by the central Ta atom is even more pronounced when compared to the inner ring of Ta atoms, whereas the outer ring loses a more significant number of the electrons (which are also redistributed to Se atoms in this case). The calculations support the picture of the central, unshifted Ta atom gaining the charge [
24,
37,
120,
121] under the formation of a CDW phase. The picture of charge gain by the central Ta atom is additionally supported by the calculation of LDOS at the Fermi level for the plane containing Ta atoms, as shown in
Figure 11c. There, the global LDOS maxima correspond to the positions of central Ta atoms, proving the tendency towards charge localization at these positions.
The band structure calculated for a
supercell for TaSSe and its parent compounds is shown in
Figure 13, along the
–K–M–
path in the supercell’s first Brillouin zone (note the difference between the supercell and the 1 × 1 cell case, as illustrated in
Figure A1c). A notable feature of all three plots is the presence of a weakly dispersive, almost flat band located at the Fermi level, which was not present for the undistorted structures (
Figure 6) and has a significant meaning for the CDW physics [
28,
122,
123]. This band is contributed mainly by
d orbitals of Ta atoms, mostly by the central, unshifted Ta atom of a star-of-David cluster [
28,
123]. The flat band is separated from the conduction-like bands with a significant gap. For TaS
2 and Janus TaSSe, the valence-like bands also lie considerably lower in energy than the flat band, whereas for TaSe
2, the mentioned bands almost merge at the
point. Like in the undistorted phase case, the band for TaSSe is additionally split due to the mirror symmetry breaking causing a spin–orbit coupling. As a reference, the band structure calculated without accounting for spin–orbit coupling is shown in
Figure A1b; no band splitting is visible in these results.
The band structure close to the Fermi level is plotted separately in
Figure 14. For TaS
2 and TaSe
2, there is a band minimum at the
point, whereas for Janus structure TaSSe, this minimum is shifted away from the
point, supporting the picture of the importance of Rashba spin–orbit coupling for the band structure.
Focusing on the weakly dispersive bands at the Fermi level, we have performed the calculations of the dispersion relations close to the
point in the directions
–K and
–M (as shown in
Figure 15a). A clear shift of the band minima from the
point (
) to the non-zero wavevector
can be observed in the data (with an anisotropic behavior in the wavevector space), suggesting the presence of Rashba spin–orbit coupling. The minima correspond to the energy
. This sort of behavior is expected in Janus TMD monolayers due to an in-built electric field perpendicular to the layer, owing to the different electronegativity of chalcogen atoms in the top and bottom layers and to the mirror symmetry breaking. Moreover, a non-parabolic dispersion is clearly visible for both bands, suggesting the presence of the cubic Rashba spin–orbit coupling contribution in addition to the usual linear term. In order to quantify the Rashba spin–orbit coupling, we have fitted our DFT data for the difference between upper and lower band energies (see
Figure 15a) with the following equation [
124,
125,
126,
127,
128,
129]:
with Rashba coefficients
and
. The results are collected in
Table 7 for the directions
–K and
–M. The fit quality can be assessed on the basis of
Figure 15b,c, where the solid lines show the fitted model with
and
, the dashed lines correspond to the model with
, and the points mark our DFT data. The data for
could be compared with the calculations presented in Ref. [
130] for the other Janus monolayer TMDs. It can be observed that
values are close to the ones predicted for MoSSe and WSSe, with somehow more pronounced anisotropy in the wavevector space for our case of TaSSe.
In order to illustrate the influence of the electric field on the CDW state of TaSSe monolayer, in
Figure 16, we show its band structure calculated for the field values of ±0.4 V/Å. The general band structure, visualized in
Figure 16a, shows that the influence of the electric field on the bands below the Fermi level is not significant in this energy scale. On the contrary, above the Fermi level, some additional bands appear for the negative field; moreover, the energy differences between both cases are more pronounced. In order to focus on the weakly dispersive band close to the Fermi level, we present
Figure 16b, which confirms that for both studied electric field values, the flat band behaves in a similar manner as in the absence of the field (see
Figure 14b).
It can be mentioned here that our calculations involve only a monolayer Janus system. The physics of monolayer and bulk TMDs (such as TaS
2) differs, for example, at the level of the band structure (see, for example, Ref. [
31]). For instance, in the CDW phase of bulk TaS
2, a band exhibiting the dispersion along an out-of-plane direction is present [
31,
118], contrary to the monolayer case. Moreover, in multilayer (or bulk) TMDs such as TaS
2, various stackings of the individual layers are possible. This can take place at two levels. The first one is connected with the mutual position of TMD monolayers in (undistorted) multilayer (see, for example, the various geometries investigated in Ref. [
131] or Ref. [
132] for W- and Mo-based Janus TMDs). The second one is related to the relative phase of CDW in monolayers composing a multilayer system [
23,
133,
134,
135,
136,
137,
138,
139]. These possibilities give rise to yet another factor shaping the already complex physical picture in the systems and enriching the phase diagram due to interlayer interactions. The stacking-related degree of freedom is even proved to provide the possibility to engineer the CDWs in TaS
2 by stackingtronics [
113]. As a consequence, a similar situation could also be envisaged in Janus TaSSe systems composed of more than one layer. Furthermore, the physics of Janus multilayers would be enriched in comparison to the parent compounds due to the presence of an in-built electric dipole and its thickness dependence. Nevertheless, it must be emphasized that for TaS
2 and TaSe
2, both bulk and monolayer systems exhibit the experimentally confirmed presence of CDWs with the same
reconstruction. Some examples of the effect of transition from monolayer to multilayer structure for Janus TMDs are presented in Refs. [
131,
132], where such parameters as the Rashba spin–orbit coupling coefficient or the in-built dipole are influenced.