Emergence of strong-coupling superconductivity and quantum criticality correlated with Lifshitz transitions in the alternate stacking compound $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$

$4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$ is a naturally formed quasi-two-dimensional heterojunction material composed of alternating monolayers of insulating ($1T$-) and superconducting ($1H$-) $\mathrm{Ta}{\mathrm{S}}_2$. We report on a comprehensive high-pressure study on the interplay between charge-density wave (CDW) and superconductivity (SC) in $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$. The results uncover a dome-shaped strong-coupling superconductor on the border of a 3 × 3 commensurate CDW at $P_{\mathrm{c}1}\approx 2.0$ GPa: (1) nearly one order enhancement of upper critical field $B_{c2}$(0); (2) the derived $2{\mathrm{\Delta }}_0/k_{\mathrm{B}}{T}_c$ beyond the BCS theory, decaying to a conventional one above 4.5 GPa; (3) the exponent of normal-state resistivity $n\approx 1.50$ and triply enhanced electronic effective mass. Under pressure, a third CDW emerging from 2.0 GPa is related to the original two CDWs, then disappears above 11.5 GPa. The temperature dependence of $B_{c2}\left(T\right)$ collapses into a universal curve and the comparison of $B_{c2}\left(T\right)$ to a polar-state function in $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$. Theoretical calculations proposed the stronger interlayer coupling and band hybridization responsible for strong-coupling SC; the suppression of new CDWs is related to band inversions between the $1T$-Ta-$d_{xy}$ (GM1+) and S-$p_z$ bands (GM2-) at the $\mathrm{\Gamma }A$ point; above 5.0 GPa, the coexisting weak-coupling SC and linear magnetoresistance can be attributed to the formation of electronic bands along the $M-K$ lines and the $\mathrm{\Gamma }A$ point. Our discovery provides an excellent example to demonstrate the interplay of the strong-coupling superconducting state and topological electronic state in van der Waals heterojunctions.

Figure 1

(a)–(c) Crystal structures and lattice parameters ($a, c$, and $c/a$) of $1T$-, $2H$-, and $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$. Temperature dependence of (d) electrical resistivity ρ($T$) and (e) magnetic susceptibility are compared. The arrows indicate the CDW transitions.

Figure 2

(a)–(c) ρ($T$) in a PCC and a palm-type CAC for $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$ (1#,2#). The data in (a) and (b) are shifted a constant along the longitudinal coordinate for comparison. (d) The derivation dρ($T$)/$\mathit{dT}$ to map the evolution of the CDW transition; it changes from a sharp peak to a broad trough at 4.50 GPa. All three CDW transitions and the superconducting transition are marked by $T_{\mathrm{CDW}1}, T_{\mathrm{CDW}2}, T_{\mathrm{CDW}3}$, and $T_{\mathrm{c}}$, respectively.

Figure 3

(a) ρ($T$) below 8 K for $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$ (1#); the arrows indicate $T_{\mathrm{c}}^{\mathrm{onset}}$ and $T_{\mathrm{c}}^{\mathrm{zero}}$, respectively. (b) $T$ dependence of ac susceptibility in PCC. (c) $H$ dependence of resistivity at 2 K in PCC and CAC. The across points are defined as superconducting transition temperature. The $H$ dependence of ρ($T$) below 6 K under various pressures and magnetic fields: (d) 2.0 GPa and (e) 3.0 GPa. The $T$ dependence of the upper critical field of $B_{c2}\left(T\right)$. The $B_{c2}\left(0\right)$ was extracted by the WHH model, viz., $B_{c2}\left(T\right)=B_{c2}\left(0\right)[(1-t)-C_1{(1-t)}^2-C_2{(1-t)}^4]/0.73$ with $C_1=0.135$ and $C_2=0.134$, respectively. (d) The normalized upper critical field $h^{*}=B_{c2}\left(0\right)/T_{\mathrm{c}} \left[{(-d{B}_{\mathrm{c}2}/dT)}_{Tc}\right]$ versus the reduced temperature $t=T/T_{\mathrm{c}}$, and the fittings by an orbital limited $s$ wave and a polar-state function, respectively.

Figure 4

Normal-state resistivity just above the $T_{\mathrm{c}}$ and its polynomial fitting by $\rho \left(T\right)={\rho }_0+A{T}^n$ in a range of $T_{\mathrm{c}}^{\mathrm{onset}}\le T\le 20\mathrm{K}$ for $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$: (a) PCC run1, 1# and (b) CAC, 1#. (c) The $P$ dependence of CDW transitions; the solid lines indicate the trends. The pressure dependence of $T_{\mathrm{CDW}2}$ and $T_{\mathrm{CDW}2}$ was the fittings by $T_{\mathrm{CDW}}\left(P\right)=T_0{(1-P/P_c)}^{\beta }$, where the $T_0, P_c$, and β represent $T_{\mathrm{CDW}}$ at AP, the critical pressure for the collapse of CDW, and the exponent characterizing the suppression of CDW, respectively. The $P$ dependence of characteristic parameters: (d) the residual resistivity ${\rho }_0$, (e) the temperature coefficient $A$, and (f) the exponent $n$. The $A$ is linearly fitted with $\rho \left(T\right)={\rho }_0+A{T}^2$. The black solid lines show the trends.

Figure 5

Field dependence of MR under various pressures at 10 K in the range of (a) 0.06–3.0 GPa and (b) 3.0–11.5 GPa, respectively. (c) Temperature dependence of electrical resistivity was investigated under 0 and 7.0 T at a fixed pressure of 7.0 GPa. (d) The estimated MR based on the data in Fig. S5 and (c) under various pressures. The arrows indicate the CDW transition with an enhanced positive MR and 60 K where the MR≈0.

Figure 6

(a) Hall resistivity ${\rho }_{xy}$($H$) of $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$ at 10 K under pressure; (b) Hall conductivity ${\sigma }_{xy}$($H$) at 10 K and the mathematical fittings by the two-band model. Pressure dependence of characteristic parameters at 10 K: (d) carrier concentration, the electron-type $n_e$, and the hole-type $n_h$; (d) the mobility, the ${\mu }_e$, and ${\mu }_h$, respectively. Solid lines across the experimental data indicate the changing trends.

Figure 7

The characteristic parameters of normal/superconducting state: (a) $T_{\mathrm{c}}^{\mathrm{onset}}$; (b) $B_{\mathrm{C}2}$(0); (c) the normalized 2Δ(0)/$k_{\mathrm{B}}{T}_c$ and effective electronic mass $m^{*}/m^{*}$(AP); (d) the electron- and hole-type carriers concentration; (e) the MR (10 K) at 5.0 T under pressure. The solid lines across the data indicate the changing trends.

Figure 8

The electronic band structures from first-principles calculations for (a) AP, (b) 5 GPa, (c) 10 GPa, and (d) 15 GPa. The dashed lines in (a) are the band structures of $4H_b\text{−}\mathrm{Ta}{\mathrm{S}}_2$. The Brillouin zone and Fermi surfaces for (e) 5.0 and (f) 15 GPa, respectively. We note that the new hole pocket emerges at 15 GPa in the $\mathrm{\Gamma }A$ points.