Josephson scanning tunneling spectroscopy in superconducting phases coexisting with pair-, charge- and spin-density-waves

Abstract

We demonstrate that the recent observations of spatial oscillations in the energy position of the superconducting coherence peaks in the cuprate, transition metal dichalcogenide, iron-based and heavy-fermion superconductors are consistent with the possible presence of a pair-, charge- or spin-density-wave phase. We show that for all three cases, the spatial oscillations of the superconducting order parameter, $Δ({\bf r})$ can be imaged via the critical Josephson current, $I_c({\bf r})$ measured in Josephson scanning tunneling spectroscopy experiments. Finally, we show that the spatial oscillations of the density waves and of $Δ({\bf r})$ exhibit relative phase shifts of $Δφ= 0$ or $π$ for a charge-density wave, and $Δφ=π/2$ for the spin-density wave.

Figures (5)

• Figure 1: Spatial and energy dependence of the LDOS for a mixed PDW phase with four different wavelengths: (a) $\lambda = 200 a_0 = 8\xi$, (b) $\lambda = 100 a_0 = 4\xi$, (c) $\lambda = 50 a_0 = 2\xi$, (d) $\lambda = 25 a_0 = \xi$ and (e) $\lambda = 4 a_0 = 0.16\xi$. (f)-(j) show the spatial dependence of the normalized Josephson critical current $I_{c}(\mathbf{r})/I_{c}^{\text{avg}}$ with $\Delta_{\text{tip}} = \Delta_0$ (dashed red) and of $\Delta(\mathbf{r})/\Delta_{0}$ (solid black) for the corresponding cases in (a)-(e). Parameters are $(\Delta_0, \Delta_\mathrm{PDW}, \mu_0, \Delta \mu, JS) = (0.05, 0.015, -3.6, 0,0)t$. For (a)-(e), we set $\Gamma = 0.0025t$, while for (f)-(j) $\Gamma = 0.1t$.
• Figure 2: Spatial and energy dependence of the LDOS for a mixed CDW phase with four different wavelengths:(a) $\lambda = 200 a_0 = 8\xi$, (b) $\lambda = 100 a_0 = 4\xi$, (c) $\lambda = 50 a_0 = 2\xi$, (d) $\lambda = 25 a_0 = \xi$, and (e) $\lambda = 4 a_0 = 0.16\xi$. (f)-(j) show the spatial dependence of the normalized Josephson critical current $I_{c}(\mathbf{r})/I_{c}^{\text{avg}}$ with $\Delta_{\text{tip}} = \Delta_0$ (dashed red) and of $\Delta(\mathbf{r})/\Delta^{\text{avg}}$ (solid black) and charge density $\rho(\mathbf{r})$ (dashed blue) for the corresponding cases in (a)-(e). Parameters are $(\Delta_0, \Delta_\mathrm{PDW}, \mu_0, \Delta \mu, JS) = (0.05, 0, -3.6, 0.45, 0)t$. For (a)-(e), we set $\Gamma = 0.0025t$, while for (f)-(j) $\Gamma = 0.1t$.
• Figure 3: (a) Spatial and energy dependence of the LDOS for a CSC phase with $\lambda = 200 a_0 = 8\xi$, for a more than half-filled electronic band. (b) Spatial dependence of the normalized Josephson critical current $I_{c}(\mathbf{r})/I_{c}^{\text{avg}}$ with $\Delta_{\text{tip}} = \Delta_0$ (dashed red) and of $\Delta(\mathbf{r})/\Delta^{\text{avg}}$ (solid black) and $\rho(\mathbf{r})$ (dashed blue), corresponding to (a). Parameters are $(\Delta_0, \Delta_\mathrm{PDW}, \mu_0, \Delta \mu, JS) = (0.05, 0, 3.6, 0.45, 0)t$. For (a), we set $\Gamma = 0.0025t$, while for (b) $\Gamma = 0.1t$.
• Figure 4: Spatial and energy dependence of the LDOS for a mixed SDW phase with four different wavelengths: (a) $\lambda = 200 a_0 = 8\xi$, (b) $\lambda = 100 a_0 = 4\xi$, (c) $\lambda = 50 a_0 = 2\xi$, (d) $\lambda = 25 a_0 = \xi$, and (e) $\lambda = 8 a_0 = 0.32\xi$. (f)-(j) show the spatial dependence of the normalized Josephson critical current $I_{c}(\mathbf{r})/I_{c}^{\text{max}}$ with $\Delta_{\text{tip}} = \Delta_0$ (dashed red) and of $\Delta(\mathbf{r})/\Delta^{\text{max}}$ (solid black) for the corresponding cases in (a)-(e). The spatial dependence of the spin polarization $S(\mathbf{r})$ (dashed blue) is additionally shown in (f) and (g). Parameters are $(\Delta_0, \Delta_\mathrm{PDW}, \mu_0, \Delta \mu, JS) = (0.05, 0.0, -3.6, 0.0, 0.2)t$. For (a)-(e), we set $\Gamma = 0.0025t$, while for (f)-(j) $\Gamma = 0.1t$.
• Figure 5: (a) Mixed PDW for wavelength $\lambda = 50 a_0 = 2\xi$ with same parameters as Fig. \ref{['fig:Fig2']}. (b) Mixed CDW for wavelength $\lambda = 50 a_0 = 2\xi$ with same parameters as Fig. \ref{['fig:Fig3']}. (c) Mixed SDW for wavelength $\lambda = 100 a_0 = 4\xi$ with same parameters as Fig. \ref{['fig:Fig4']}. (d) Figure 2(g) of Ref.Kong2025.