Enhanced Superconductivity in Bilayer PtTe₂ by Alkali-Metal Intercalations

Transition metal dichalcogenides (TMDCs) exhibit intriguing physical properties including superconductivity, charge-density wave, Dirac semimetals, among which the two-dimensional (2D) superconductivity has received growing attention, where the reduced dimensionality leads to unique behaviors compared to their bulk counterparts. One notable example is monolayer NbSe₂: (1) the enhanced charge-density wave and reduced superconductivity in going from bulk to the monolayer [1]; (2) the transition from a two-gap superconductor (bulk) to a single-gap one (monolayer) [2, 3], owning to the competition between charge-density wave and superconductivity [4]; (3) suppressed magnetic instability by charge-density wave [5]; (4) Ising superconductivity, where the inversion-symmetry-broken crystal leads to large Zeeman type spin-orbit coupling (SOC), resulting in the extremely large in-plane upper critical field [6, 7]. Besides the inversion-symmetry-broken Ising superconductivity, the large critical fields were also observed in 2D centrosymmetric systems known as type-II Ising superconductivity, arising from multiple degenerate orbitals with spin-orbital locking [8, 9, 10]. Clearly, realizing 2D superconductivity in layered TMDCs is desirable to offer potential platforms for studying the novel quantum physics and the interplay among different orders in 2D limit.

The group VIII TMDCs, MTe₂, where M=Ni, Pd, Pt, recently attach increasing attentions for the experimentally verified type-II Dirac semimetal [11, 12, 13], pressure induced superconductivity in their bulk phases [14, 15], and novel physics in their ultrathin films [9, 16, 17, 18, 19, 20, 21]. In particular, monolayer NiTe₂ was predicted to be an intrinsic superconductor, and lithium intercalation can boost the superconducting transition temperature (henceforth $T_{\text{c}}$) of bilayer NiTe₂ up to 11.3 K [16]. Few-layer PdTe₂ were experimentally verified to be type-II Ising superconductors though with $T_{\text{c}}$$<$ 1 K [21]. Very recently, their homologues PtTe₂ crystals were reported to be synthesized with controllable thickness down to monolayer limit [18]. Different from NiTe₂, monolayer PtTe₂ is an intrinsic semiconductor with band gap about 0.8 eV. Interlayer coupling in PtTe₂ is expected to be stronger, due to the smaller interlayer spacing in bulk PtTe₂. Furthermore, SOC is supposed to be stronger in PtTe₂, as Pt element is much heavier than Pd and Ni. The above differences, combined with the longing for the 2D superconductor motivate us to study the possibility of the emergence of superconductivity in 2D PtTe₂ crystals.

This Letter reports an $ab~{}initio$ study on electron-phonon coupling (EPC) and superconducting properties of PtTe₂. We show that the weak superconductivity in bilayer PtTe₂(Pt₂Te₄) can be significantly boosted by alkali-metal intercalations. In particular, rubidium (Rb) intercalation leads to the formation of a thermodynamically stable crystal with the stoichiometry of RbPt₂Te₄, where the Rb occupy all the octahedral sites. Based on anisotropic Midgal-Eliashberg formalism, the $T_{\text{c}}$ of the RbPt₂Te₄ is computed to be 8 K with SOC, which is very high among TMDCs. The homologues KPt₂Te₄ is shown to have comparable $T_{\text{c}}$ to RbPt₂Te₄. The mechanism of the remarkable boosted superconductivity and the effect of SOC are systematically analyzed.

Density-functional theory and density-functional perturbation theory calculations were performed with the exchange-correlation functional of PBE [22] to study the crystal structures, electronic structures, EPC of bulk, and few-layer  PtTe₂ before and after alkali-metal intercalations [23, 24]. The norm-conserving pseudopotentials of FHI98 [25] and ONCV  [26] were used to describe the interaction between valance and core electrons. The Kohn-shame valance states were expanded as plane waves below 80 Rydberg. A 18$\times$18 (18$\times$18$\times$12) $\mathbf{k}$-mesh and a 6$\times$6 (6$\times$6$\times$4) $\mathbf{q}$-mesh were adopted to calculate the ground states of charge density and phonons for few-layer (bulk) systems, whereupon the electron-phonon coupling (EPC) matrix element $g_{mn,\nu}(\mathbf{k},\mathbf{q})$ are calculated, which quantifies the scattering amplitude between the electronic states with wavevector $\mathbf{k}$, band index $m$ ($\mathbf{k}$, $m$), and ($\mathbf{k}$+$\mathbf{q}$, $n$) via a phonon with branch $\nu$ and wavevector $\mathbf{q}$. Then above quantities are interpolated to the ${\mathbf{k}}$-grid of 120$\times$120 (60$\times$60$\times$36) and ${\mathbf{q}}$-grid of $60\times 60$ (30$\times$30$\times$20) [28, 27], based on which the mass-enhancement parameter [$\alpha^{2}F(\omega)/\omega$] are computed, where $\alpha^{2}F(\omega)$ is the Eliashberg spectrum, defined as:

where $\lambda_{\mathbf{q}\nu}$ are phonon-momentum-resolved EPC constant [27], $\Omega_{\mathrm{BZ}}$ is the volume of the 1st Brillouin zone, and $\delta\left(\omega-\omega_{\mathbf{q}\nu}\right)$ is replaced with a gaussian function with a broadening of 0.5 meV. Using the same ${\mathbf{k}}$- and ${\mathbf{q}}$-grids, the temperature-dependent superconducting gaps [$\Delta(\mathbf{k},T)$] are obtained by solving the anisotropic Midgal-Eliashberg equations with the Matsubara frequencies below 0.23 eV on imaginary axis [27, 29], followed by performing analytic continuation to the real axis with Padé functions.

Monolayer PtTe₂ is composed of Te-Pt-Te triatomic layers, where each Pt is octahedral-coordinated by six Te atoms. Its bulk counterpart, 1$T$-PtTe₂  is formed by the AA stacking of such monolayers along $z$ direction, separated by an interlayer spacing $d={\color[rgb]{0,0,0}2.57}$ Å [30], with the hexagonal lattice constants $a$ = 4.01 and $c={\color[rgb]{0,0,0}5.24}$ Å [18]. The optimized lattice constants are $a$ =4.08, and $c$ = 5.27 Å using the FHI98 pseudopotential [25], in nice agreement with the experiment. To study the SOC effect on EPC and superconducting properties, we also adopt the ONCV pseudopotential [26], which yields the optimized $a=4.10$, and $c={\color[rgb]{0,0,0}5.38}$ Å, slightly larger the experimental one. Nonetheless, we found that the above two pseudopotentials yield consistent computational results (see Sec. S1 [31] for details). In addition to the different pseudopotentials, the van der Waals (vdW) correlations [32, 33, 34] are also found to have little influence on the electronic structures (see Sec. S1[31]). Therefore, we mainly report in the main text, the results of the ONCV without vdW correction. It should also be mentioned that we involve the projector augmented wave type of pseudopotentials for more efficient calculations of $ab~{}initio$ molecular dynamic (AIMD), and the determination of convex hull for the Rb intercalated bilayer PtTe₂, which will be discussed soon. More details of the motivation of using the above methods can be found [35]. The interlayer distance $d={\color[rgb]{0,0,0}2.57}$ Å of the bulk PtTe₂ is much smaller than that of other layered TMDCs (e.g., 2.90 Å for 2$H$-NbSe₂ [36], 2.89 Å for $T_{d}$-WTe₂ [37]) and even smaller than its homologues NiTe₂ (2.63 Å) [38], indicating the potential strong interlayer coupling. Indeed, it was reported that the dispersive bandstructure along $c^{*}$ suppresses its superconductivity in bulk PtTe₂ [39]. Besides, our computational results show that the electron-doped monolayer PtTe₂ can exhibit overall large EPC strength ($\lambda$) and isotropic superconducting $T_{\text{c}}$ as shown in Sec. S2 [31]. Therefore, it is expected that the superconductivity can be emerged in PtTe₂ by electron doping and weakening the interlayer coupling. Thus, to enhance the superconductivity in layered PtTe₂, the intercalation of alkali-metal atoms should be a reasonable way, as the intercalants can act as electron donors and relieve the interlayer coupling by expanding the interlayer spacing. To realize 2D superconductivity in PtTe₂, we then study the alkali-metal intercalated bilayer PtTe₂.

We begin by studying the energy favorable geometry structures of Rb-intercalated bilayer PtTe₂, whose stoichiometry can be generated as Rb_m(Pt₂Te₄)_n, where $m$ and $n$ are integers, counting the number of Rb and Pt₂Te₄ units, respectively. We use a 3$\times$3 supercell to calculate the formation energies $\Delta E$ with respect to the Rb concentration fraction $x=m/(m+n)$, defined as $\Delta E(x)$= $E$[Rb_x(Pt₂Te₄)_1-x] - $x$ E [Rb] - (1-$x$) $E$[Pt₂Te₄], where $E$[Rb] and $E$[Pt₂Te₄] are the energies of a body-center cubic Rb crystal per atom and a bilayer PtTe₂ per Pt₂Te₄ unit, respectively. Fig. 1(b) displays the corresponding results, where $\Delta E(x)$ decrease as the increase of $x$ and reach the minimum at $x=0.5$, corresponding to the crystal with the stoichiometry RbPt₂Te₄. Further increasing the number of Rb, the $\Delta E$ begins to increase. The resultant convex hull shown in Fig. 1(b) suggests that the RbPt₂Te₄ crystal is thermodynamically stable with respect to any other stoichiometry. Fig. 1(a) displays the crystal structure of the RbPt₂Te₄, where the Rb atoms occupy all the octahedral sites at the midpoints of the nearest pairs of Pt atoms in the adjacent monolayers, which is similar to the case of lithium intercalated bilayer NiTe₂ [16]. AIMD simulations [24, 40] further suggest that the RbPt₂Te₄ crystal is stable at room temperature without structure distortion in 10 picoseconds (see Sec. S3 [31]).

Figs. 2(a) and S4(a) display the electronic bandstructures of bilayer (henceforth Pt₂Te₄) and monolayer PtTe₂, respectively. The monolayer is computed to have a band gap of 0.8 eV, which agrees well with the experiment [18]. On going from the monolayer to the bilayer, a semiconductor-to-metal transition is seen with the strong modification of the electronic structure around Fermi energy ($\epsilon_{\text{F}}$). The mostly remarkable change is a band derived mainly from the $p_{z}$-like orbitals of Te drops down and crosses $\epsilon_{\text{F}}$ (see Sec. S4 for the projected bandstructures [31]). This can be assigned to the overlap between the $p_{z}$-like orbitals of the Te atoms in the bilayer, suggesting the strong interlayer coupling. The calculated bandstructure of the Pt₂Te₄ is again consistent with the ARPES measurements [18]. After Rb intercalation, the interlayer spacing expands and the interlayer coupling weakens, accompanied by the electrons transfer from Rb to its adjacent Te layers. The synergy effects of Rb-mediated interlayer coupling and electron doping make the bandstructure of the RbPt₂Te₄ resemble that of the electron doped monolayer PtTe₂, but with slight band splitting [compare Figs. 2(b) and S4(a)]. One can see from Fig. 2(b) that without SOC, there are two conduction bands crossing the $\epsilon_{\text{F}}$, leading to the formation of several electron pockets depicted by the solid black circles in Figs. 2(c)– 2(d). The two bands intersect at the Brillouin zone corners, K and K’, at the energy $\sim$12 meV above $\epsilon_{\text{F}}$, which is responsible for the emergent tiny cirles centered at K and K’. When the SOC is included, though the Kramers degeneracy is preserved due to the inversion symmetry at the site of Rb, the double degenerated bands at K and K’ are split, leading to the avoided crossing near $\epsilon_{\text{F}}$  and a $\sim$57 meV gap opening at K and K’, giving rise to a slight reduced density of states at $\epsilon_{\text{F}}$ [$N(0)$]. In addition to the small Fermi circles, the clover-shaped pockets centered at K and K’ are also noted, each of them consists of three separate petals. Centered at M and $\Gamma$ points, there are also two electron pockets. The calculated projected electronic density of states (DOS) [right panel of Fig. 2(b)] and the momentum-resolved DOS for the states near $\epsilon_{\text{F}}$ [Figs. 2(c)–2(d)] suggest that these states near K, K’ are significantly contributed by the $p$-like orbitals, and the remaining states are assigned to the hybridization of the $d$- and $p$-like orbitals.

The computed phonon dispersions ($\omega_{\boldsymbol{q}\nu}$) of the RbPt₂Te₄ [Figs. 3(a) and  S5(b)] and Pt₂Te₄ [Fig. S5(a)] show that all the $\omega_{\mathbf{q}\nu}\geq 0$, suggesting the dynamically stabilities of the crystals. The SOC  is found to have little influence on $\omega_{\boldsymbol{q}\nu}$ of RbPt₂Te₄ [Fig. S5(b)], and slightly decrease the intensity of the mass-enhancement parameter [$\alpha^{2}F(\omega)/\omega$; see Fig. 3(b)], but the main features of the $\alpha^{2}F(\omega)/\omega$ is similar to the case without SOC. For easier analysis, we first focus on the case without SOC. Fig. 3(b) displays the comparison of the $\alpha^{2}F(\omega)/\omega$ and the corresponding accumulated EPC strength $\lambda(\omega)=2\int_{0}^{\omega}\alpha^{2}F(\omega)/\omega d\omega$ (dashed lines) between RbPt₂Te₄ and Pt₂Te₄. It is found that after the intercalation of Rb, the $\lambda(\omega)$ are remarkably enhanced in the energy region between 5 – 13 meV. Particularly, three intensive peaks are found located at 6.6, 9.6 and 12.3 meV, respectively. To further study the mechanism of the enhanced EPC, we decomposed the $\alpha^{2}F(\omega)$  into the contributions from in-plane ($xy$) and out-of-plane ($z$) vibrations of each atom by computing

where $e^{j}_{\mathbf{q},\nu}$ is the component of atom $j$ (Pt, Rb, Te) in the eigenvector of the dynamic matrix with phonon momentum ${\mathbf{q}}$ and modes $\nu$, and $\hat{n}$ is the unit projection direction vector, which is chosen along in-plane ($xy$) and out-of-plane ($z$) directions. The computed $\alpha^{2}F(\omega,j,\hat{n})/\omega$ and the corresponding accumulated EPC constants $\lambda(j,\hat{n})=2\int_{0}^{\infty}d\omega\alpha^{2}F(\omega,j,\hat{n})/\omega$ are shown in Figs. 3(c) and  3(e), respectively. The most remarkable change is the doubled $\lambda(\text{Te}_{xy})$ to 0.56 after the Rb intercalation [Fig. 3(e)]. This is consistent with the computed $\alpha^{2}F(\omega,\text{Te}_{xy})/\omega$ [Fig. 3(c)], where $\alpha^{2}F(\omega,\text{Te}_{xy})/\omega$ dominates the total spectrum of $\alpha^{2}F(\omega)/\omega$ in 5 – 13 meV, suggesting the strongly coupling of the in-plane vibration of Te atoms with the electrons. In particular, the two pairs of double degenerated modes E_g and E_u can be seen at $\Gamma$ with energy of 12.08 and 12.11 meV, respectively [Fig. 3(a)], which are derived from Te_xy vibrations [see Fig. S6(a) [31]]. Among them, the two E_g modes exhibit large $\lambda_{\boldsymbol{q}\nu}$. The vibrational pattern for one of the E_g is shown in the inset of Fig. 3(d). By comparing the bandstructures with and without such phonon displacements imposed on the equilibrant RbPt₂Te₄ crystal, one can see that the Te_xy displacements have strong influences on the bands near $\epsilon_{\text{F}}$, especially around K, where a Lifshitz transition is noted [Fig. 3(d)]. The secondary contributions to $\lambda$ arise from the Te_z, Pt_xy and Rb_xy vibrations as shown in Fig. 3(e). By further examining the corresponding spectra in Fig. 3(c), one can see that the contribution of Te_z vibration is mainly associated with the peaks at 6.6 and 12.3 meV, while Pt_xy exhibits a relatively uniform contribution in the whole energy region. Interestingly, the Rb also exhibits moderate contribution to $\lambda$, as the computed $\lambda(\text{Rb})=0.27$ is comparable to $\lambda(\text{Pt})$ = 0.31 [see Fig. 3(e)]. The Rb vibrations promote the $\alpha^{2}F(\omega)/\omega$ peak at 9.6 meV [Fig. 3(c)], being on par with the contribution of Te_xy at the same energy.

We turn to study the mechanism of the enhanced EPC in RbPt₂Te₄ by analyzing the electronic states involved in EPC. We computed the $\mathbf{k}$-resolved EPC constant $\lambda_{\mathbf{k}}$, defined as follow in this work:

where $\Omega_{\text{BZ}}$ is the volume of the first Brillouin zone, $g_{mn,\nu}(\mathbf{k},\mathbf{q})$ is the EPC matrix element, and the $\delta$ functions related to electron and phonon energies are replaced by gaussian functions with broadening of 10 and 0.5 meV, respectively. The $\lambda_{\mathbf{k}}$ is related to total EPC constant by $\lambda=\int\frac{d\mathbf{k}}{\Omega_{\text{BZ}}}\lambda_{\mathbf{k}}$. By comparing $\lambda_{\mathbf{k}}$ of Pt₂Te₄ and RbPt₂Te₄ shown in Figs. 4(a) and  4(b), one can see the clover-shaped circles at K and K’ are expanded, and the additional Fermi circles emerge after Rb intercalation, promoting the $N(0)$ from 1.54 to 4.09 states/eV/Pt. Consequently, the $\lambda$ in RbPt₂Te₄  is remarkably boosted to 1.4, as more pairs of electronic states can be involved in the EPC. Furthermore, the contribution of the $\lambda$ of RbPt₂Te₄ are almost arising from the $\mathbf{k}$ near K and K’ as shown in Fig. 4(b), which are mainly derived from the electronic states of Te atoms [Fig. 2(d)]. By similar analysis, the states in the remaining sections of the Fermi surface, derived from the $p$-$d$ hybridization of Te and Pt atoms, also contribute to $\lambda$. The SOC is found to reduce the $\lambda_{\mathbf{k}}$ near K, K’ [Fig. 4(c)], attributable to the reduced $N(0)$ arising from band splitting at K, K’[Fig. 2(b)]. Consequently, $\lambda$ decreases to 1.3 with SOC. Except that, the main feature of the computed $\lambda_{\mathbf{k}}$ is unchanged. Therefore, it is evident that the large $\lambda_{\mathbf{k}}$ in RbPt₂Te₄ is mainly contributed by the pairing of electronic states of Te atoms near K, K’ Fermi pockets.

The significantly enhanced $\lambda$ indicates the boost of the superconducting $T_{\text{c}}$. Indeed, using McMillan-Allen-Dynes approach [41, 42, 43] based on the calculated $\lambda$ = 1.30, the logarithmic average of the phonon frequencies $\omega_{\text{log}}=104.5~{}$K, and $\mu^{*}=0.205$ evaluated by $\mu^{*}\approx[0.26N(0)/[1+N(0)]$ [44], the isotropic superconducting $T_{\text{c}}$ of RbPt₂Te₄ is calculated to be 6.6 K with SOC. In contrast, the computed $T_{\text{c}}$ for Pt₂Te₄ are only 1.1 K. The reliability of the computed isotropic $T_{\text{c}}$ is also examined by computing bulk PtTe₂ as discussed in Sec. S7, where one can see the computed $\lambda$ for bulk PtTe₂ is 0.35 and the corresponding $T_{\text{c}}$ is 0 K, in nice agreement with previous study that the bulk PtTe₂ is non-superconducting with $\lambda=0.33$ [39]. To obtain more reliable superconducting $T_{\text{c}}$ for the systems with reduced dimensionality and anisotropic Fermi surface [29, 45, 46], we further evaluate the $T_{\text{c}}$ of RbPt₂Te₄ by solving the full Midgal-Eliashberg gap equation [29, 45]. Compared to the slight decrease of the aforementioned isotropic $\lambda$, the SOC has relatively noticeable influences on the anisotropic superconducting properties. Fig. 4(d) shows the evolution of the energy distribution of superconducting gaps ($\Delta_{\mathbf{k}}$) with respect to temperatures. At $T$ = 2 K without SOC, the $\Delta_{\mathbf{k}}$ are distributed in the energy range between 0.9 – 2.4 meV with an average value of 1.6 meV, showing strong anisotropy. When SOC is involved, the overall decreased $\Delta_{\mathbf{k}}$  and the suppressed anisotropic distribution of $\Delta_{\mathbf{k}}$  are seen, as the average value of $\Delta_{\mathbf{k}}$ decreases to $\sim$1.2 meV, and the energy distribution range is reduced to 0.6 – 1.6 meV. The corresponding distribution of $\Delta_{\mathbf{k}}$ in an extended Brillouin zone is shown in Fig. 4(e), where one can see that the  $\Delta_{\mathbf{k}}$  exhibit anisotropic distribution to some extents, in which the relatively large  $\Delta_{\mathbf{k}}$  comes from the Fermi pockets around K and K’, dominated by the electronic states of Te atoms as analyzed before. As the temperature increases, the values of  $\Delta_{\mathbf{k}}$ gradually reduce and finally vanish at $T=8~{}K$, suggesting the $T_{\text{c}}$ of the RbPt₂Te₄ is $\sim$8 K, which is higher than the isotropic one (6.6 K) due to the anisotropic $\Delta_{\mathbf{k}}$. We also study the intercalation of other alkali metal elements including lithium (Li), sodium(Na), potassium(K) and caesium(Cs), in which the KPt₂Te₄ is expected to have comparable $T_{\text{c}}$ to the RbPt₂Te₄, followed by the $T_{\text{c}}$ of CsPt₂Te₄, whereas the LiPt₂Te₄ and NaPt₂Te₄ are with relatively low $T_{\text{c}}$ [see Sec. S8 for details].

In summary, we have predicted the enhanced EPC and superconducting $T_{\text{c}}$  in RbPt₂Te₄ and understood the corresponding mechanisms from $ab~{}initio$ calculations. Firstly, according to the computed convex hull, phonon dispersions and $ab~{}initio$ molecular dynamic simulations, the intercalated Rb are energetically favorable to occupy all the octahedral sites in the interlayer gallery, forming the thermodynamically stable RbPt₂Te₄ crystal. The $T_{\text{c}}$ of RbPt₂Te₄  is computed to be 8 K with the anisotropic superconducting gaps based on the anisotropic Midgal-Eliashberg formalism in the presence of SOC, though the pristine Pt₂Te₄  have a very low $T_{\text{c}}$. Such a remarkable enhancement is assigned to the effects of the Rb intercalations from two sides. On one hand, the synergy effect of Rb-mediated interlayer coupling and electron doping lead to the significant promotion of $N(0)$, accompanied by the markedly enhanced EPC of Te phonons. On the other hand, the Rb directly contributes to the EPC by significantly increasing the intensity of $\alpha^{2}F(\omega)/\omega$ peak at 9.6 meV. The SOC reduces the $T_{\text{c}}$ and the anisotropic superconducting gaps by splitting the band degeneracy near $\epsilon_{\text{F}}$. The KPt₂Te₄ is proposed to have comparable $T_{\text{c}}$ to the RbPt₂Te₄. The reliability of our predictions is supported by the consistent computational results of electronic structures of few-layer PtTe₂, and the EPC and superconductivity of bulk PtTe₂ with previous studies.

Considering bilayer PtTe₂ has been experimentally synthesized  [18], and the intercalation of alkali-metal atoms therein can be experimentally realized [47, 48, 49, 50], our predictions can be straightforwardly probed. Therefore, the experimental accessibility, combined with the relatively high superconducting $T_{\text{c}}$ with SOC will make these superconductors promising platforms to investigate intriguing novel quantum physics associated with 2D superconductivity. For example, as we have shown before, the RbPt₂Te₄ crystal has three-fold rotational symmetries, which are preserved at high symmetry momenta of K and K’, at which the double degenerated electronic bands very close to $\epsilon_{\text{F}}$ are noted without SOC, and they will be further split by SOC [Fig. 2(b)]. The KPt₂Te₄ also exhibits similar results (Fig. S10). The above results suggest that the alkali-metal intercalated bilayer PtTe₂ are potential candidates for realizing the type-II Ising superconductivity with relatively high superconducting $T_{\text{c}}$ [8], which calls for further studies.