Pair-mixing induced Time-reversal-breaking superconductivity

Abstract

Experimental evidences of spontaneous time-reversal (TR) symmetry breaking have been reported for the superconducting ground state in the transition metal dichalcogenide (TMD) superconductor 4H$_b$-TaS$_2$ or chiral molecule intercalated TaS$_2$ hybrid superlattices, and is regarded as evidence of emergent chiral superconductivity. However, the $T_c$ of these TMD superconductors is of the same order as pristine 1H or 2H-TaS$_2$, which do not show any signature of TR breaking and are believed to be conventional Bardeen-Cooper-Schrieffer superconductors. To resolve this puzzle, we propose a new type of pair-mixing state that mixes the dominant conventional s-wave pairing channel with the subdominant chiral p-wave pairing channel via a finite Cooper-pair momentum, based on symmetry analysis within the Ginzburg-Landau theory. Our analysis shows that the fourth-order terms in the chiral p-wave channel can lead to a variety of pair-mixing states with spontaneous TR breaking. These TR-breaking superconducting states also reveal a zero-field, junction-free superconducting diode effect that is observed in chiral molecule intercalated TaS$_2$ superlattices.

Figures (7)

• Figure 1: (a) Schematics of 1H-TaS$_2$ layer sandwiched between chiral layers formed by either 1T-TaS$_2$ layer or chiral molecule layers. The s-wave singlet pairing in 1H-TaS$_2$ layer are mixed with p-wave triplet pairing in chiral layers at a finite momentum $\mathbf{k}_0$. (b) $T_c^{(0)}(k)-1$ as a function of $k$ with $T_{c,A}=2.7K, T_{c,E}=2K, \gamma_{A}=0.076, \gamma_{E}=0.01$ at different values of $\zeta_1$. (c) Color denotes $T_c(\mathbf{k})-1$ as a function of $\mathbf{k}$ with $\zeta_2=\zeta_4=0.13$, $\zeta_1=0.3643$, and all other parameters set to be same as (b). $\mathbf{Q}_n$ with $n\in \mathbb{Z}_6$ labels six momenta for the maximal $T_c(\mathbf{k})$. (d) The blue and red curves depict the $k_0$ and the ratio $\left|\frac{\eta_E}{\eta_A}\right|$ as a function of $\zeta_{1}$, respectively, with the parameters to be the same as (b). All quantities above have been regularized to be dimensionless.
• Figure 2: (a) Phase diagram showing the stripe and vortex-antivortex phases for $c_1=c_2<0,c_0=1,c_4=0$. (b) Minimal free energy $\mathcal{F}_{min}$ for phase 2c (blue curve for $\mathcal{F}_{min,2c}$) and phase 3d (yellow curve for $\mathcal{F}_{min,3d}$) across the dotted path in (a). Here $\mathcal{F}_{min}$ is in units of $T_{c,A}$. The red curve represents the SDE coefficient $\eta$ across the dotted path in (a). (c) and (d) represent the real space distribution $|\Psi(\mathbf{r})|$ of pair function for the Phase 2c and 3d, respectively. In (d), the black hexagon denotes the unit cell of vortex-antivortex lattice for Phase 3d, the red arrows depict the direction and magnitude of local supercurents, and the cyan and orange hexagons show the regions for vortcies and antivortices, respectively. (e) Distribution of $L_z(\mathbf{r})$ for Phase 3d. (f) Distribution of normalized critical current $\mathcal{J}(\theta)$ as a function of angle $\theta$. Here we choose $\zeta_1=0.3647$ and all other parameters are same as Fig.\ref{['fig:pairmixing']}(b). All in-plane coordinates $\{x,y\}$ are in units of $\pi/k_0$.
• Figure 3: The transformation $\mathcal{KT}C_{3z}$ that leads to cyclic permutations $\mathcal{C}_1$, the emergent symmetry of the free energy.
• Figure 4: Spatial distribution of different order parameter $\Psi(\mathbf{r})$ in striped phase 2c. All axes represent the $x,y$ coordinates in scales of $\frac{\pi}{k_0}$. (a) Real-space distribution of the A-component $|\eta_{A}(\mathbf{r})|$(b)Real-space distribution of the E components $|\eta_{E}(\mathbf{r})|$, defined in Eq.\ref{['eq:Eampli']}(c)The total amplitude of the order parameter, $|\Psi(\mathbf{r})|$, defined in Eq.\ref{['eq:totampli']}. All numerical parameters are same as Fig 1. in main-text.
• Figure 5: The spatial distribution and angular momentum properties of the order parameter $\Psi(\mathbf{r})$ in vortex-antivortex phase 3d. All axes represent the $x,y$ coordinates in scales of $\frac{\pi}{k_0}$. $\phi_1=\phi_2=0$ for all settings, since these Goldstone modes only act in translation. All black hexagons represent the unit cells appearing in the corresponding quantity distributions. (a) Spatial distribution of $|\eta_A(\mathbf{r})|$ (b) Spatial distribution of $|\eta_E(\mathbf{r})$ in Eq.\ref{['eq:Eampli']} (c)Spatial distribution of overall amplitude $|\Psi(\mathbf{r})|$. Arrows represent local current vector $\frac{1}{2e\gamma}\mathbf{j}_{S}(\mathbf{r})$ in Eq.\ref{['eq:localcurrent']}, it forms a hexagonal vortex-antivortex lattice. The bigger hexagon represents the unit cell and the yellow and cyan colored hexagonal plaquettes represent the domain of the vortices and anti-vortices, respectively. (d)Local distribution of intrinsic angular momenta $\mathbf{L}(\mathbf{r})$. Arrows represent the $L_x(\mathbf{r}),L_y(\mathbf{r})$ components and the color-bar represents $L_z(\mathbf{r})$ (e)Local distribution of normalized intrinsic angular momenta $\mathbf{n}(\mathbf{r})$. Arrows represent the $n_x(\mathbf{r}),n_y(\mathbf{r})$ components and the color-bar represents $n_z(\mathbf{r})$ (f) Distribution of the local winding number $\mathbf{n}(\mathbf{r})\cdot (\partial_x\mathbf{n}(\mathbf{r})\times\partial_\mathbf{n}(\mathbf{r}))$. For all plots, the same parameters as Fig. 1(c) in maintext is used.