Nuclear magnetic relaxation rates of unconventional superconductivity in doped topological insulators

We study the temperature dependence of nuclear magnetic relaxation (NMR) rates to detect unconventional superconductivity in doped topological insulators, such as $M(=\mathrm{Cu},\text{Nb},{\mathrm{Sr})}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x\mathrm{Te}$. The Hebel-Slichter coherence effect below a critical temperature $T_{\mathrm{c}}$ depends on the superconducting states predicted by a minimal model of doped topological insulators. In a nodal anisotropic topological state similar to the ABM phase in ${}^3\mathrm{He}$, the NMR rate has a conventional $s\text{-wave-like}$ coherence peak below $T_{\mathrm{c}}$. In contrast, in a fully-gapped isotropic topological superconducting state, this rate below $T_{\mathrm{c}}$ exhibits an antipeak profile. Moreover, in a twofold in-plane anisotropic topological superconducting state, there is no coherence effect, which is similar to that in a chiral $p\text{-wave}$ state. We also claim in a model of ${\text{Cu}}_x{\text{Bi}}_2{\text{Se}}_3$ that a signal of the fully-gapped odd-parity state is attainable from the change of the antipeak behavior depending on doping level. Thus, we reveal that the NMR rates shed light on unconventional superconductivity in doped topological insulators.

Figure 1

The Fermi surface and the spectral gap ${\mathrm{\Delta }}^{\mathrm{spec},1}\left(\mathit{k}\right)$ defined in Eq. (24) with $k_z=0$ (a) without and (b) with the hexagonal warping term $(\lambda =1)$ to treat the sixfold crystal structure.

Figure 2

Temperature dependence of nuclear magnetic relaxation rates in (a) an even-parity gap ${\mathrm{\Delta }}^4$ $\left(A_{1g}\right)$ $\propto \sum \langle c_{-\mathit{k}\downarrow }^1{c}_{\mathit{k}\uparrow }^1+c_{-\mathit{k}\downarrow }^2{c}_{\mathit{k}\uparrow }^2\rangle$, a fully-gapped isotropic odd-parity gap ${\mathrm{\Delta }}^5$ $\left(A_{1u}\right)$ $\propto \sum \langle c_{-\mathit{k}\downarrow }^2{c}_{\mathit{k}\uparrow }^1+c_{-\mathit{k}\uparrow }^2{c}_{\mathit{k}\downarrow }^1\rangle$, (c) a nodal odd-parity gap ${\mathrm{\Delta }}^3$ $\left(A_{2u}\right)$ $\propto \sum \langle c_{-\mathit{k}\downarrow }^1{c}_{\mathit{k}\uparrow }^1-c_{-\mathit{k}\downarrow }^2{c}_{\mathit{k}\uparrow }^2\rangle$, and (d) a nodal odd-parity gap ${\mathrm{\Delta }}^1$ $\left(E_u\right)$ $\propto \sum \langle c_{-\mathit{k}\uparrow }^1{c}_{\mathit{k}\uparrow }^2-c_{-\mathit{k}\downarrow }^1{c}_{\mathit{k}\downarrow }^2\rangle$ . We set the chemical potential $\mu =0.8$, the Dirac mass $m=0.4$, and the gap amplitude ${\mathrm{\Delta }}_0=0.01$. We ignore the warping term. Equation (23) is used as the phenomenological temperature dependence of the gap amplitude.

Figure 3

Temperature dependence of nuclear magnetic relaxation rates with the hexagonal warping term in (a) a fully-gapped isotropic odd-parity gap ${\mathrm{\Delta }}^5$ $\left(A_{1u}\right)$ and (b) an anisotropic odd-parity gap ${\mathrm{\Delta }}^1$ $\left(E_u\right)$. The parameters are the same as in Fig. 2.

Figure 4

Doping dependence of the Fermi surface in the model for ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. The unit of the energy is eV.

Figure 5

Temperature dependence of nuclear magnetic relaxation rates in the model for ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ with (a) an even-parity gap ${\mathrm{\Delta }}^4$ $\left(A_{1g}\right)\propto \sum \langle c_{-\mathit{k}\downarrow }^1{c}_{\mathit{k}\uparrow }^1+c_{-\mathit{k}\downarrow }^2{c}_{\mathit{k}\uparrow }^2\rangle$, a fully-gapped isotropic odd-parity gap ${\mathrm{\Delta }}^5$ $\left(A_{1u}\right)$ $\propto \sum \langle c_{-\mathit{k}\downarrow }^2{c}_{\mathit{k}\uparrow }^1+c_{-\mathit{k}\uparrow }^2{c}_{\mathit{k}\downarrow }^1\rangle$, (c) a nodal odd-parity gap ${\mathrm{\Delta }}^3$ $\left(A_{2u}\right)\propto \sum \langle c_{-\mathit{k}\downarrow }^1{c}_{\mathit{k}\uparrow }^1-c_{-\mathit{k}\downarrow }^2{c}_{\mathit{k}\uparrow }^2\rangle$, and (d) a nodal odd-parity gap ${\mathrm{\Delta }}^1$ $\left(E_u\right)$ $\propto \sum \langle c_{-\mathit{k}\uparrow }^1{c}_{\mathit{k}\uparrow }^2-c_{-\mathit{k}\downarrow }^1{c}_{\mathit{k}\downarrow }^2\rangle$ . The model parameters are shown in Eq. (25). The doping dependence is considered as the chemical potential dependence $\mu$. The minimum and maximum values of the chemical potentials are $\mu =0.33$ [eV] and $\mu =1.03$ [eV], respectively. At a higher doping level, larger ${\left(T_{1}T\right)}^{-1}$ at $T>T_{\mathrm{c}}$. We set the gap amplitude ${\mathrm{\Delta }}_0=0.05$ [eV]. Equation (23) is used as the phenomenological temperature dependence of the gap amplitude.