Tunneling conductance of $d+ip$ -wave superconductor

Unconventional superconductors (SCs) can host surface bound states, the Andreev bound states (ABSs), forming a zero-energy flat band Buchholtz and Zwicknagl (1981); Hara and Nagai (1986); Hu (1994); Matsumoto and Shiba (1995a); Kashiwaya and Tanaka (2000); Löfwander et al. (2001). The zero-energy ABSs can be observed as a zero-bias conductance peak (ZBCP) in the quasiparticle tunneling spectra of the junctions of a normal metal and a high-$T_{c}$ cuprate (i.e., a spin-singlet $d$-wave SC) Tanaka and Kashiwaya (1995); Kashiwaya et al. (1995a, 1998); Covington et al. (1997); Alff et al. (1997); Wei et al. (1998); Biswas et al. (2002); Chesca et al. (2008); Bouscher et al. (2020); Saadaoui et al. (2013). In addition, the ABSs induce a Josephson current with a low-temperature anomaly Tanaka and Kashiwaya (1996, 1997); Barash et al. (1996) and a paramagnetic Meissner current Higashitani (1997); Barash et al. (2000); Tanaka et al. (2005); Asano et al. (2011); Yokoyama et al. (2011); Suzuki and Asano (2014); Higashitani (2014); Suzuki and Asano (2015); Espedal et al. (2016); Lee et al. (2017). The origin of the ABSs has been clarified from the view point of the topological invariant defined using the bulk Hamiltonian Sato et al. (2011).

The zero-energy ABSs may be fragile against perturbations because of the high degeneracy of the flat band. The surface $s$-wave subdominant order originating this instability was proposed Matsumoto and Shiba (1995b); Kuboki and Sigrist (1996) in 1995. Theoretically, the subdominant $s$-wave component splits the zero-energy peak in the local density of states (LDOS) Kashiwaya et al. (1995b); Tanuma et al. (1999, 2001a) and gives rise to a spontaneous surface current by breaking the time reversal symmetry (TRS) Matsumoto and Shiba (1995c). In experiments, however, neither such peak splitting nor TRS breaking has been observed Kashiwaya et al. (1995a, 1998); Alff et al. (1997); Wei et al. (1998); Biswas et al. (2002); Chesca et al. (2008); Bouscher et al. (2020); Saadaoui et al. (2013), except for a few cases Covington et al. (1997); Fogelström et al. (1997); Krupke and Deutscher (1999). More seriously, the induced $s$-wave pair potential requires an on-site attractive interaction, in contradiction to the strong repulsive interaction in the cuprate. The instability of the ABSs is not still conclusive, even though other possibilities have been pointed out, such as surface ferromagnetism Potter and Lee (2014), spin density wave Honerkamp and Sigrist (1998), staggered flux phase Kuboki (2014, 2015), and translational symmetry breaking Vorontsov (2009); Higashitani and Miyawaki (2015); Miyawaki and Higashitani (2015, 2017); Håkansson et al. (2015); Holmvall et al. (2018); Miyawaki and Higashitani (2018); Holmvall et al. (2019).

The spin-triplet $p$-wave subdominant order has recently been proposed using the finite-size two-dimensional Hubbard model with the random-phase approximation Matsubara and Kontani (2020a). The ferromagnetic fluctuation caused by the ABSs can stabilize such a $p$-wave subdominant order, which breaks the TRS Matsubara and Kontani (2020b). The obtained $d+ip$-wave pairing shows different properties compared with those of the $d+is$-wave: there is no clear zero-energy splitting in the LDOS and no spontaneous surface current. Although a number of papers have studied the $d+is$-wave state Matsumoto and Shiba (1995b, c); Tanuma et al. (1999, 2001a), the unique properties of the $d+ip$-wave state have not yet been clarified. In particular, the mixture of the spin-singlet and spin-triplet pairs would cause non-trivial phenomena.

We here study the conductance spectra of the normal-metal/$d_{xy}$-wave SC junctions with a subdominant $p_{y}$-wave pair potential at the interface and compare the results with those for the well-known $d+is$-wave superconducting junction. We consider a ballistic planar junction, as shown in Fig. 1, where the barrier potential is present at the interface. Utilizing the quasiclassical Eilenberger formalism, we obtain the differential conductance using the pair potential obtained by solving the self-consistency equation. The calculated results show that the ZBCP of the $d_{xy}$-wave SC can survive under the spin-triplet $p_{y}$-wave subdominant pair potential, even though it is fragile against the spin-singlet $s$-wave subdominant pair potential. From the spin-resolved conductance spectra, we show that the transport properties of the $d+ip$-wave junction strongly depend on the spin of an injected electron because of the coexistence of the spin-triplet and singlet pairs near the interface.

We also investigate whether the ZBCP can survive, even if the $p$-wave surface attractive potential is short-range and strong, as pointed out in Ref. Matsubara and Kontani, 2020a. The ZBCP is demonstrated to be robust even against such a short-range attractive potential. Comparing our results with experimental data, we conclude that our results, which take the subdominant $p$-wave order into account phenomenologically, support recent theoretical predictions based on microscopic calculations.

We consider the ballistic normal-metal/$d_{xy}$-wave SC junction shown in Fig. 1, where the SC and the normal-metal (N) occupy $x\geq 0$ and $x<0$ respectively. In a ballistic SC, Green’s function obeys the Eilenberger equation Eilenberger (1968):

where $\hat{g}$ and $\hat{f}$ are the normal and anomalous Green’s functions, $v_{F_{x}}$ is the $x$ component of the Fermi velocity $v_{F}$, $\omega_{n}$ is the Matsubara frequency with the integer $n$, the direction of the momentum is characterized by the angle $\phi$ ($k_{x}=\alpha\cos\phi$, and $k_{y}=\sin\phi$ with $\alpha=\pm 1$ and $-\pi/2\leq\phi\leq\pi/2$), $\check{\Delta}_{\alpha}=\check{\Delta}_{\alpha}(\phi,x)$, and $\check{\tau}_{j}$ ($j=1$, $2$, or $3$) are the Pauli matrices in the particle-hole space. In this study, the symbols $\check{\cdot}$ and $\hat{\cdot}$ denotes the matrices in the particle–hole and spin space, respectively. The pair-potential matrix is defined as

where we have omitted the index $\alpha$. The spin structure of the pair potential is parametrized as

where $\Delta_{\mu}$ ($\mu=s$, $p$, and $d$) is the amplitude of the $\mu$-wave pair potential and $\hat{\sigma}_{j}$ ($j=1$, $2$, or $3$) are the Pauli matrices in spin space. The momentum dependence of the pair potentials are given by

The pair potentials are determined by the self-consistency equation:

where the angular brackets means the angle average on the Fermi surface: $\langle\cdots\rangle=\sum_{\alpha}\int_{-\pi/2}^{\pi/2}\cdots({d\phi}/{2\pi})$ and $n_{c}$ is the cutoff integer which is decided by the relation $2n_{c}+1<\omega_{c}/\pi T\leq 2n_{c}+3$ with $\omega_{c}$ being the cutoff energy. The attractive potential depends on the pairing symmetry:

The coupling constant $\Lambda_{\mu}$ is

where $T_{\mu}$ is the effective critical temperature. Namely, the ratio $T_{p}/T_{d}$ and $T_{s}/T_{d}$ characterize the amplitude of the subdominant pair potential.

The microscopic theoriesMatsubara and Kontani (2020a, b) suggest that the attractive potential for the $p$-wave channel may be short-range. To model such a short-range potential, we introduce the spatial-dependent attractive potential $\Lambda_{\mu}^{\prime}(x)$ as

where $\kappa$ is the decay parameter which characterizes the length scale of the attractive potential. Replacing $\Lambda_{\mu}$ in Eq. (16) by $\Lambda_{\mu}^{\prime}(x)$, we can self-consistently calculate the subdominant pair potential under the short-range attractive potential.

When the pair-potential matrix does not have the diagonal component as in Eq. (12), the $4\times 4$ Eilenberger equation can be decomposed into two $2\times 2$ ones:

The spin-reduced Green’s function $\tilde{g}^{X}$ and the pair potential are defined as

where we omit the direction index $\alpha$ and the index $X=O$ and $I$ means the outer and inner components in the Nambu space respectively. Hereafter we make the index $X$ explicit only when necessary. Using the Riccati parametrization Schopohl and Maki (1995); Eschrig (2000, 2009), the quasiclassical Green’s functions $\hat{g}^{X}_{\alpha\alpha}$ can be expressed as

where $D_{\alpha}=D_{\alpha}^{X}$ and $F_{\alpha}=F_{\alpha}^{X}$ are the so-called Riccati amplitudes. The Riccati amplitudes obey the following Riccati-type differential equations:

The Eilenberger equation is supplemented by the boundary conditions Ashida et al. (1989); Nagato et al. (1993); Eschrig (2000); Tanuma et al. (2001b); Hirai et al. (2003); Eschrig (2009). The boundary conditions at the N/SC interface are given by:

where $R$ is the reflection probability amplitude and $s_{\omega}=\mathrm{sgn}[\omega_{n}]$.

We have confirmed that our conclusions on the ZECP do not depend on the details of the potential at the interface. We have replaced the $\delta$-function insulating barrier with the rectangular one. The results with the rectangular potential (not shown) are qualitatively the same as those with the $\delta$-function one [e.g., Fig. 7(a)]; the ZBCP of the $d+ip$-wave junction does not split. Therefore, our conclusions are valid even when the interface potential has a finite width (i.e., more realistic model than the $\delta$-function barrier model).

The conductance spectra of cuprate superconductors observed in experiments to date have a peak at the zero energy Kashiwaya et al. (1995a, 1998); Alff et al. (1997); Wei et al. (1998); Biswas et al. (2002); Chesca et al. (2008); Bouscher et al. (2020), even though several experiments have reported splitting of the ZBCP Covington et al. (1997); Fogelström et al. (1997); Krupke and Deutscher (1999). Our theoretical study, where the $p$-wave surface attractive interaction is phenomenologically taken into account, demonstrates that the $p$-wave subdominant pair potential does not split the zero-bias peak. In particular, our results show that the microscopic theory Matsubara and Kontani (2020a, b) can be consistent with experimental results obtained to date.

The conductance spectrum of a $d+ip$-wave junction depends on the spin of the injected particle because of the mixture of the $p$-wave spin-triplet and $d$-wave spin-singlet pairs. Replacing the normal-metal electrode with a ferromagnetic metal will provide useful information for detecting the surface subdominant pair potential. Investigating how the $p$- and $s$-wave subdominant pair potentials modify the conductance spectra would be interesting.

In this paper, we have studied the transport property of the $d+ip$-wave junction in the ballistic limit by calculating the tunneling conductance. The induced $p$-wave pair, however, is more fragile against impurity scatterings than are $s$-wave pairs Suzuki and Asano (2015); Yamada et al. (1996); Poenicke et al. (1999); Zare et al. (2008); Suzuki and Asano (2016). Thus, in the presence of disorder, the differential conductance of the $d+ip$- and $d+is$-wave junctions may be different. Clarifying the effect of disorder would be important for applying our theory to experimental results.

We have theoretically studied the conductance spectroscopy of normal-metal/$d_{xy}$-wave superconductor junctions with the spin-triplet $p_{y}$-wave subdominant order at the interface utilizing the quasiclassical Eilenberger formalism. We have considered the ballistic junction, where a $\delta$-function-type insulating barrier exists at the interface. The conductance spectra are calculated using the self-consistent pair potentials.

The calculated conductance spectra show that the ZBCP originating from the $d_{xy}$-wave pair potential is not split by the $p_{y}$-wave subdominant pair potential at the interface, in contrast to the $s$-wave subdominant component, which is known to split the ZBCP. The $p_{y}$-wave pair potential has nodes at $k_{y}=0$, which does not affect the zero-energy states around $k_{y}=0$. The contributions from these channels form the ZBCP in the conductance spectra, even in the presence of the $p_{y}$-wave subdominant pair potential.

In addition, we have studied the effect of the $p_{y}$-wave subdominant pair potential on the dispersion of the surface states of the $d_{xy}$-wave SC. The $p_{y}$-wave subdominant component changes the zero-energy flat band formed with the Andreev bound states to a V-shape or inverted V-shape, depending on the spin subspace. The spin-subspace dependence stems from the coexistence of the spin-singlet and spin-triplet pair potentials near the interface.