Intrinsic finite-energy Cooper pairing in $j=3/2$ superconductors

We show that Cooper pairing can occur intrinsically away from the Fermi surface in $j=3/2$ superconductors with strong spin-orbit coupling and equally curved bands in the normal state. In contrast to conventional pairing between spin-$1/2$ electrons, we derive that pairing can happen between interband electrons having different magnetic quantum numbers, for instance, $m_j=1/2$ and $m_j=3/2$. Such superconducting correlations manifest themselves by a pair of indirect gaplike structures at finite excitation energies. An observable signature of this exotic pairing is the emergence of a pair of symmetric superconducting coherence peaks in the density of states at finite energies. Moreover, the angular-momentum-resolved density of states in the presence of a perturbative Zeeman field reflects the $m_j$ composition of the Cooper pairs. We argue that such finite-energy pairing is a generic feature of $j=3/2$ superconductors, both in the presence and absence of inversion symmetry.

Figure 1

BdG spectra along the [0,0,1] direction in absence of pairing for (a) $\beta =0$ and (b) $\beta =0.2\left|\alpha \right|$, respectively. The spectra are independent of $\gamma$. BdG spectrum in the (c) [0,0,1] and (d) [1,1,0] (i.e., $k_x=k_y=k_1$) directions for $\gamma =\beta$ in the presence of septet pairing with amplitude $\mathrm{\Delta }/E_{0}a=4.15$. The color denotes the probability of electronic states $|{\mathrm{\Psi }}_e{|}^2$ in both (c) and (d). Other parameters are $\mu /E_0=-5, k_0={10}^{-2}{a}^{-1}, E_0={10}^{-3}\left|\alpha \right|a^{-2}$, and $\alpha =-20$. $a$ is the lattice constant in a tight-binding version of the continuum model.

Figure 2

BdG spectra in (a) [0,0,1] and (b), (c) [1,0,1] (with $k_x=k_z=k_2$) directions for (a) $(\mathrm{\Delta }/E_{0}a,\delta /E_{0}a)=(3.5,10)$, (b) (1.5,2.5), and (c) (0.15,2.5), respectively. (a) and (b) correspond to septet pairing, and (c) corresponds to $A_{1g}$ pairing. The corresponding density of states $N$ normalized with respect to its maximum value are presented in the right panel of each spectrum. $\gamma =-0.05\left|\alpha \right|$ and other parameters are the same as those in Fig. 1.