Josephson effect in a multi-orbital model for Sr$_{2}$RuO$_{4}$

TL;DR

This paper investigates Josephson effects in Sr$_{2}$RuO$_{4}$, analyzing how different pairing symmetries influence current-phase relations and magnetic flux dependence, providing potential experimental methods to identify the superconductor's pairing symmetry.

Contribution

It introduces detailed calculations of Josephson currents considering multi-orbital band structures and spin-orbit interactions, distinguishing chiral and helical p-wave states via their magnetic flux responses.

Findings

Chiral p-wave pairing shows a $ ext{cos}()$ term due to TRS breaking.

Helical p-wave states lack the $ ext{cos}()$ term, preserving TRS.

Corner junctions reveal distinct $I_c()$ asymmetries for chiral and helical states.

Abstract

We study Josephson current between s-wave/spin-triplet superconductor junctions by taking into account details of band structures in Sr$_{2}$RuO$_{4}$ such as three conduction bands, spin-orbit interaction in the bulk and that at the interface. We assume five superconducting order parameters in Sr$_{2}$RuO$_{4}$: a chiral p-wave symmetry and four helical p-wave symmetries. We calculate current-phase relationship $I(\varphi)$ in these junctions, where $\varphi$ is the macroscopic phase difference between two superconductors. The results for a chiral p-wave pairing symmetry show that $\cos(\varphi)$ term appears in the current-phase relation due to time-reversal symmetry (TRS) breaking. On the other hand, $\cos(\varphi)$ term is absent in the helical pairing states which preserve the TRS. We also study the dependence of maximum Josephson current $I_c$ on an external magnetic flux $\Phi$ in a corner junction. The calculated results of $I_c(\Phi)$ show a relation $I_{c}(\Phi) \neq I_{c}(-\Phi)$ in a chiral state and $I_{c}(\Phi)=I_{c}(-\Phi)$ in a helical state. We calculate $I_c(\Phi)$ in a corner and a symmetric SQUIDs geometry. In a symmetric SQUID geometry, the relation $I_{c}(\Phi)=I_{c}(-\Phi)$ is satisfied for all the pairing states and it is impossible to distinguish chiral state from helical one. On the other hand, results for a corner SQUID always show $I_{c}(\Phi) \neq I_{c}(-\Phi)$ and $I_{c}(\Phi)=I_{c}(-\Phi)$ for a chiral and a helical states, respectively. Experimental tests of these relations in a corner junctions and SQUIDs may serve as a tool for unambiguous determination of the pairing symmetry in Sr$_{2}$RuO$_{4}$.

Tables

Table 1. Table 1: Fourier series of current-phase relation in the absence of interface Rashba spin-orbit interaction. √ square-root \surd ( − - ) denotes coefficients with a nonzero (zero) value.

	$I_1^s$	$I_1^c$	$I_2^s$	$I_2^c$
(a)Chiral(single-band)	$-$	$-$	$\surd$	$-$
(b)Chiral(multi-band)	$\surd$	$\surd$	$\surd$	$\surd$
(c)Helical(single-band)	$-$	$-$	$\surd$	$-$
(d)Helical(multi-band)	$\surd$	$-$	$\surd$	$-$

Table 2. Table 2: Fourier series of the current-phase relation in the presence of interface Rashba spin-orbit interaction

	$I_1^s$	$I_1^c$	$I_2^s$	$I_2^c$
(a) Chiral(single-band)	$-$	$\surd$	$\surd$	$-$
(b) Chiral(multi-band)	$\surd$	$\surd$	$\surd$	$\surd$
(c) Helical(single-band)	$-$	$-$	$\surd$	$-$
(d) Helical(multi-band)	$\surd$	$-$	$\surd$	$-$

Table 3. Table 3: Relations between I a ​ ( φ a ) subscript 𝐼 𝑎 subscript 𝜑 𝑎 I_{a}(\varphi_{a}) and I b ¯ ​ ( φ b ¯ ) subscript 𝐼 ¯ 𝑏 subscript 𝜑 ¯ 𝑏 I_{\overline{b}}(\varphi_{\overline{b}}) shown in Fig. 1 for chiral( E u subscript 𝐸 𝑢 E_{u} ), helical ( A 1 ​ u subscript 𝐴 1 𝑢 A_{1u} , B 2 ​ u subscript 𝐵 2 𝑢 B_{2u} ), and helical ( A 2 ​ u subscript 𝐴 2 𝑢 A_{2u} , B 1 ​ u subscript 𝐵 1 𝑢 B_{1u} ) pairings.

Type of pairing	Relation between $I_a\left({\phi }_a\right)$ and $I_{\overline{b}}\left({\phi }_{\overline{b}}\right)$
Chiral($E_u$)	$I_a\left({\phi }_a\right)=-I_{\overline{b}}\left(-{\phi }_{\overline{b}}+\pi /2\right)$
Helical($A_{1u}$, $B_{2u}$)	$I_a\left({\phi }_a\right)=I_{\overline{b}}\left({\phi }_{\overline{b}}\right)$
Helical($A_{2u}$, $B_{1u}$)	$I_a\left({\phi }_a\right)=I_{\overline{b}}\left({\phi }_{\overline{b}}+\pi \right)$

Table 4. Table 4: Φ Φ \Phi dependence and zero points of I ​ ( Φ ) 𝐼 Φ I(\Phi) in an SRO/NM/ s 𝑠 s -wave corner junction for (b) chiral( E u subscript 𝐸 𝑢 E_{u} ), (c) helical( A 1 ​ u subscript 𝐴 1 𝑢 A_{1u} , B 2 ​ u subscript 𝐵 2 𝑢 B_{2u} ), and (d) helical( A 2 ​ u subscript 𝐴 2 𝑢 A_{2u} , B 1 ​ u subscript 𝐵 1 𝑢 B_{1u} ) pairings in SRO. Schematic illustration of corner junction (a).

Type of pairing	$\Phi$ dependence	zero points of $I\left(\Phi \right)$
(a)Chiral($E_u$)	asymmetric	$\pm 2{\Phi }_0,\pm 4{\Phi }_0,\cdots$
(b)Helical($A_{1u}$, $B_{2u}$)	symmetric	$\pm {\Phi }_0,\pm 2{\Phi }_0,\cdots$
(c)Helical($A_{2u}$, $B_{1u}$)	symmetric	$\pm 2{\Phi }_0,\pm 4{\Phi }_0,\cdots$

Table 5. Table 5: Φ Φ \Phi dependence and period of the maximum Josephson current I c subscript 𝐼 𝑐 I_{c} in a corner SQUID

Type of pairing	$\Phi$ dependence	Period
(a) Chiral	asymmetric	${\Phi }_0$
(b, c) Helical	symmetric	${\Phi }_0$

Table 6. Table 6: Φ Φ \Phi dependence and period of the maximum Josephson current I c subscript 𝐼 𝑐 I_{c} in a symmetric SQUID.

Type of pairing	$\Phi$ dependence	Period
(a) Chiral	symmetric	${\Phi }_0$
(b) Helical	symmetric	${\Phi }_0$

Equations

$${\mathcal{H}}_{\mathrm{kin}}=\sum _{\mathbf{k},\sigma }{\hat{c}}_{\mathbf{k}\sigma }^{†}\left(\begin{array}{ccc}{\epsilon }_{yz}(\mathbf{k})& g(\mathbf{k})& 0\\ g(\mathbf{k})& {\epsilon }_{zx}(\mathbf{k})& 0\\ 0& 0& {\epsilon }_{xy}(\mathbf{k})\end{array}\right)c_{\mathbf{k}\sigma },$$

$${\epsilon }_{xy}(\mathbf{k})$$

$${\epsilon }_{yz}(\mathbf{k})$$

$${\epsilon }_{zx}(\mathbf{k})$$

$$g(\mathbf{k})$$

$${\mathcal{H}}_{\mathrm{soi}}$$

$$\displaystyle\mathcal{H}_{\rm pair}=\sum_{\ell}{\hat{c}^{\ell\dagger}\left(\begin{array}[]{cc}\hat{0}&\hat{\Delta}^{\ell}(\bm{k})\\ -\hat{\Delta}^{\ell\ast}(-\bm{k})&\hat{0}\\ \end{array}\right)\hat{c}^{\ell}},$$

$$\{\begin{array}{l}{\mathbf{d}}_{Eu}^{yz}=\hat{z}{\Delta }_1(\delta \sin k_x+i\sin k_y),\\ {\mathbf{d}}_{Eu}^{zx}=\hat{z}{\Delta }_1(\sin k_x+i\delta \sin k_y),\\ {\mathbf{d}}_{Eu}^{xy}=\hat{z}{\Delta }_2(\sin k_x+i\sin k_y),\end{array}$$

$$\{\begin{array}{l}{\mathbf{d}}_{A1u}^{yz}=\hat{x}\delta {\Delta }_1\sin k_x+\hat{y}{\Delta }_1\sin k_y,\\ {\mathbf{d}}_{A1u}^{zx}=\hat{x}{\Delta }_1\sin k_x+\hat{y}\delta {\Delta }_1\sin k_y,\\ {\mathbf{d}}_{A1u}^{xy}=\hat{x}{\Delta }_2\sin k_x+\hat{y}{\Delta }_2\sin k_y,\end{array}$$

$$\{\begin{array}{l}{\mathbf{d}}_{A2u}^{yz}=\hat{x}{\Delta }_1\sin k_y-\hat{y}\delta {\Delta }_1\sin k_x,\\ {\mathbf{d}}_{A2u}^{zx}=\hat{x}\delta {\Delta }_1\sin k_y-\hat{y}{\Delta }_1\sin k_x,\\ {\mathbf{d}}_{A2u}^{xy}=\hat{x}{\Delta }_2\sin k_y-\hat{y}{\Delta }_2\sin k_x,\end{array}$$

$$\{\begin{array}{l}{\mathbf{d}}_{B1u}^{yz}=\hat{x}{\Delta }_1\sin k_x-\hat{y}\delta {\Delta }_1\sin k_y,\\ {\mathbf{d}}_{B1u}^{zx}=\hat{x}\delta {\Delta }_1\sin k_x-\hat{y}{\Delta }_1\sin k_y,\\ {\mathbf{d}}_{B1u}^{xy}=\hat{x}{\Delta }_2\sin k_x-\hat{y}{\Delta }_2\sin k_y,\end{array}$$

$$\{\begin{array}{l}{\mathbf{d}}_{B2u}^{yz}=\hat{x}\delta {\Delta }_1\sin k_y+\hat{y}{\Delta }_1\sin k_x,\\ {\mathbf{d}}_{B2u}^{zx}=\hat{x}{\Delta }_1\sin k_y+\hat{y}\delta {\Delta }_1\sin k_x,\\ {\mathbf{d}}_{B2u}^{xy}=\hat{x}{\Delta }_2\sin k_y+\hat{y}{\Delta }_2\sin k_x.\end{array}$$

$${\mathcal{H}}_{\mathrm{NM}}=\sum _{\mathbf{k}\sigma }({\epsilon }_{\mathbf{k}}-\mu )c_{\mathbf{k}\sigma }^{†}{c}_{\mathbf{k}\sigma },$$

$${\epsilon }_k=-2t_1(\cos (k_x)+\cos (k_y))-4t_2\cos (k_x)\cos (k_y)-{\mu }_n$$

$${\mathcal{H}}_{\mathrm{RSOI}}={\lambda }_R\sin k_y{\hat{\sigma }}_z.$$

$${\mathcal{H}}_{\mathrm{s}-\mathrm{wave}}=\sum _{\mathbf{k}}\Delta e^{i\phi }{c}_{\mathbf{k}\uparrow }^{†}c{†}_{-\mathbf{k}\downarrow }+c.c.,$$

$${\hat{G}}_n^L(k_y,i{\omega }_l)=(i{\omega }_l-{\hat{\epsilon }}_n(k_y)-{\hat{t}}_{n,n-1}{\hat{G}}_{n-1}^L(k_y,i{\omega }_l){\hat{t}}_{n-1,n}{)}^{-1},$$

$${\hat{G}}_n^R(k_y,i{\omega }_l)=(i{\omega }_l-{\hat{\epsilon }}_n(k_y)-{\hat{t}}_{n,n+1}{\hat{G}}_{n+1}^L(k_y,i{\omega }_l){\hat{t}}_{n+1,n}{)}^{-1},$$

$${\hat{G}}_{00}(k_y,i{\omega }_l)=(({\hat{G}}_0^L(k_y,i{\omega }_l){)}^{-1}-{\hat{t}}_{01}{\hat{G}}_1^R(k_y,i{\omega }_l){\hat{t}}_{10}{)}^{-1},$$

$${\hat{G}}_{11}(k_y,i{\omega }_l)=(({\hat{G}}_1^R(k_y,i{\omega }_l){)}^{-1}-{\hat{t}}_{10}{\hat{G}}_0^L(k_y,i{\omega }_l){\hat{t}}_{01}{)}^{-1}.$$

$${\hat{G}}_{01}(k_y,i{\omega }_l)$$

$${\hat{G}}_{10}(k_y,i{\omega }_l)$$

$${\hat{G}}_{01}(k_y,\tau )$$

$${\hat{G}}_{10}(k_y,\tau )$$

$${\hat{G}}_{01}(k_y,\tau )$$

$${\hat{G}}_{10}(k_y,\tau )$$

$${\hat{C}}_0^{†}$$

$${\hat{C}}_1^{†}$$

$$\displaystyle I(\varphi)=\frac{iet}{\hbar}\int_{-\pi}^{\pi}\mathrm{Tr^{\prime}}\Bigl{[}\hat{G}_{01}(k_{y},\tau=-0,\varphi)$$

$$-{\hat{G}}_{10}(k_y,\tau =-0,\phi )$$

$$\displaystyle=\frac{iet}{\hbar}\int_{-\pi}^{\pi}\mathrm{Tr^{\prime}}\frac{1}{\beta}\sum_{l}\Bigl{[}\hat{G}_{01}(k_{y},i\omega_{l},\varphi)$$

$$-{\hat{G}}_{10}(k_y,i{\omega }_l,\phi )$$

$$I(\phi )=\sum _{n=1}^{\infty }{I}_n^s\sin (n\phi )+I_n^c\cos (n\phi ).$$

$$\{\begin{array}{l}{\mathbf{d}}_{Eu}^{yz}=\hat{z}{\Delta }_1(\delta \sin k_x-i\sin k_y),\\ {\mathbf{d}}_{Eu}^{zx}=\hat{z}{\Delta }_1(\sin k_x-i\delta \sin k_y),\\ {\mathbf{d}}_{Eu}^{xy}=\hat{z}{\Delta }_2(\sin k_x-i\sin k_y),\end{array}$$

$$\mathbf{A}=A_y(x)\mathbf{y}.$$

$$\nabla \gamma =\frac{m^{\ast }{\mathbf{v}}_s}{\hslash }+\frac{2\pi }{{\Phi }_0}\mathbf{A}.$$

$$\gamma (y)=\gamma (0)+\frac{2\pi }{{\Phi }_0}A(\infty )y$$

$$\phi (y)=\phi (0)+\frac{2\pi }{{\Phi }_0}\left[A_2(\infty )-A_1(\infty )\right]y.$$

Full text

I Introduction

Strontium ruthenate (Sr2RuO4, or SRO) has attracted much interest for its unconventional superconductivity below the critical temperature $T_{c}\sim 1.5$ K Maeno et al. (1994). The constancy of the Knight shift across $T_{c}$ is strongly indicative of spin-triplet pairing order Ishida et al. (1998); Tou et al. (1998); Murakawa et al. (2004); Mackenzie and Maeno (2003); Maeno et al. (2012). Many theoretical studies have examined the microscopic mechanism of spin-triplet pairings in this material Rice and Sigrist (1995); Miyake and Narikiyo (1999); Kuwabara and Ogata (2000); Nomura and Yamada (2000, 2002a, 2002b); Arita et al. (2004); Nomura and Yamada (2005); Nomura et al. (2008); Yanase and Ogata (2003); Raghu et al. (2010); Sato and Kohmoto ; Kuroki et al. (2001); Takimoto (2000); Tsuchiizu et al. (2015). Exotic phenomena specific to spin-triplet superconductors Tanaka and Kashiwaya (2004); Tanaka et al. (2005a); Asano et al. (2006); Tanaka et al. (2005b); Tanaka and Golubov (2007) are therefore naturally expected in SRO. Although several studies have focused on the superconducting order parameter, the symmetry of a Cooper pair is not yet fully understood. Five spin-triplet pairing states are compatible with the tetragonal crystal structure of SRO Mackenzie and Maeno (2003). One of these is a spin-triplet chiral $p$-wave state (denoted the $E_{u}$ state in the Mulliken notation) where the $d$-vector is parallel to $c$-axis of the crystal. The other four candidates are called spin-triplet helical states (denoted $A_{1u}$, $A_{2u}$, $B_{1u}$, and $B_{2u}$ in the Mulliken notation), where the $d$-vectors lie in the $ab$-plane of the crystal.

According to the recently proposed topological classificationSchnyder et al. (2008); Qi and Zhang (2011); Tanaka et al. (2012); Alicea (2012), all of the proposed superconducting states are topologically nontrivial. Consequently, topologically protected Andreev bound states are expected at an SRO surface Kashiwaya et al. (2011). Some experimental results are consistent with the proposed pair potential. It has been suggested that the maximum Josephson current in Au0.5In0.5-SRO superconducting quantum interference devices (SQUID) displays an odd-parity pairing state Nelson et al. (2004).

Tunneling spectroscopy experiments also suggest the formation of a dispersive surface Andreev bound state (SABS) at the in-plane edges of SRO Laube et al. (2000); Kashiwaya et al. (2011); Wang et al. (2015). The dispersive SABSs Furusaki et al. (2001); Sengupta et al. (2002) are distinguishable from the dispersionless SABS in a $d$-wave superconductor. The former generates a broad zero-bias conductance peak (ZBCP) Yamashiro et al. (1997, 1998); Honerkamp and Sigrist (1998), whereas the latter forms a sharp ZBCP Tanaka and Kashiwaya (1995); Kashiwaya and Tanaka (2000); Hu (1994). Because SRO is a multi-band superconductor, the numerically determined energy dispersion of an SABS in a multi-band model is more complicated than that in a single-orbital model Imai et al. (2012, 2013). Yada, et al. successfully explained the variety of conductance spectra observed in experiments Kashiwaya et al. (2011) in terms of the three-band degrees of freedom Yada et al. (2014). Several Josephson-junction experiments suggested the presence of domain structures, detected from an anomalous current-switching behavior Kidwingira et al. (2006); Kambara et al. (2008, 2010); Anwar et al. (2013); Saitoh et al. (2015). These experimental findings are consistent with the existence of both chiral and helical $p$-wave pairing symmetries in SRO.

A chiral state is qualitatively different from the four helical states because it breaks the time-reversal symmetry (TRS), whereas the helical states preserve TRS Qi et al. (2009). Although the presence or absence of TRS in SRO is an important issue, experimental results remain controversial. TRS breaking can be verified by observing a spontaneous magnetic field or a spontaneous edge current. Theoretical studies have shown that the amplitude of the spontaneous magnetization is detectable experimentally Matsumoto and Sigrist (1999) and that the edge current is robust with respect to surface roughness Suzuki and Asano (2016). Measurements of muon spin resonance and of the Kerr effect have detected the presence of an internal magnetic field Luke et al. ; Xia et al. (2006), which in turn suggests a chiral $p$-wave symmetry. On the other hand, scanning SQUID experiments have not shown any signs of a spontaneous magnetic field Kirtley et al. (2007); Hicks et al. (2010), which suggests a helical $p$-wave symmetry. Several theoretical proposals have been put forward to explain the absence of the edge current in SRO Huang et al. (2014); Lederer et al. (2014); Ashby and Kallin (2009); Raghu et al. (2010); Tada et al. (2009). A resolution of this paradox requires an experimental test able to distinguish unambiguously between a chiral and a helical pairing symmetry.

In this paper, we present a theory of the Josephson effect between a spin-singlet $s$-wave superconductor and a spin-triplet $p$-wave superconductor by taking into account the three bands of the SRO in addition to the spin-orbit interaction in the bulk and at the interface. The importance of multi-orbital effects are apparent in various physical quantities Nomura and Yamada (2002c); Nomura et al. (2008). Since spin-orbit coupling influences the current-phase relation fundamentally, it is necessary that our theory consider a three-band model. We calculated the current-phase relation $I(\varphi)$ in Josephson junctions, where $\varphi$ is the macroscopic phase difference between the two superconductors. We found that $\cos(\varphi)$ appears in $I(\varphi)$ for chiral $p$-wave pairing, owing to TRS breaking, to ensure consistency with previous results Asano et al. (2003). However, $\cos(\varphi)$ is absent for helical pairing, thus reflecting time-reversal invariance. In the case of helical pairing, $\sin(\varphi)$ appears only in a three-band model. We also studied the dependence of the maximum Josephson current $I_{c}$ on an external magnetic flux $\Phi$ in two types of SQUID geometries: a corner SQUID and a symmetric SQUID. In a corner Josephson junction and a corner SQUID, we found $I_{c}(\Phi)\neq I_{c}(-\Phi)$ for a chiral state, whereas $I_{c}(\Phi)=I_{c}(-\Phi)$ holds true for a helical state. We show that the three-band character affects the oscillation period of $I_{c}(\Phi)$. It is possible to determine the pairing symmetry unambiguously by testing these relations in SRO-based corner junctions and SQUIDs. In a symmetric SQUID, the relation $I_{c}(\Phi)=I_{c}(-\Phi)$ is satisfied in both chiral and helical cases.

II Model and Formulations

This section introduces a model Hamiltonian for an SRO/normal metal (NM)/$s$-wave superconductor junction system. First, we explain the Hamiltonian for bulk SRO, which consists of three terms $\mathcal{H}_{\rm kin}$, $\mathcal{H}_{\rm soi}$ and $\mathcal{H}_{\rm pair}$. The first term $\mathcal{H}_{\rm kin}$ expresses the kinetic energy. ARPES measurements and first-principles calculations have shown that SRO has three two-dimensional Fermi surfaces Oguchi (1995); Singh (1995); Noce and Cuoco (1999); Haverkort et al. (2008). These Fermi surfaces were reproduced by considering three orbitals, $i.e.$, the $d_{xy}$, $d_{yz}$, and $d_{zx}$ orbitals, in SRO. We can therefore consider a three-band two-dimensional Hamiltonian constructed using the tight-binding model:

[TABLE]

where ${\bm{k}}$ is a wavenumber, $\sigma$ is the spin, and $\hat{c}_{{\bm{k}}\sigma}=(c_{{\bm{k}},\sigma}^{yz},c_{{\bm{k}},\sigma}^{zx},c_{{\bm{k}},-\sigma}^{xy})^{T}$ is the annihilation operator. The matrix components of Eq. (1) are given by

[TABLE]

where $t_{1}$, $t_{2}$, $t_{3}$, $t_{4}$, and $t_{5}$ are the hopping integrals up to next nearest-neighbor sites. The second term $\mathcal{H}_{\rm soi}$ denotes the spin-orbit interaction in bulk SRO,

[TABLE]

where $s_{\sigma}=1$ ($s_{\sigma}=-1$) for $\sigma=\uparrow$ ($\sigma=\downarrow$). This term mixes the spin and orbital degrees of freedom. The third term $\mathcal{H}_{\rm pair}$ expresses the pair potential in SRO. We chose spin-triplet chiral and helical $p$-wave pairings in the following analysis. In the chiral $p$-wave case, we considered a pair potential which belongs to the $E_{u}$ irreducible representation. In the helical $p$-wave case, we considered two kinds of pair potentials belonging to the $A_{u}$ and $B_{u}$ irreducible representations. Using the orbital-dependent $d$ vector $d^{\ell}(\bm{k})$, the pair potential can be expressed as

[TABLE]

with $\hat{c}^{\ell}=(c_{{\bm{k}},\uparrow}^{\ell},c_{{\bm{k}},\downarrow}^{\ell},c_{{-\bm{k}},\uparrow}^{\ell\dagger},c_{{-\bm{k}},\downarrow}^{\ell\dagger})^{T}$ , and $\hat{\Delta}^{\ell}(\bm{k})=i\bm{d}^{\ell}(\bm{k})\cdot\bm{\sigma}\sigma_{y}$, where $\ell$ denotes the orbital index. The five kinds of $d$ vectors are given by

[TABLE]

In these pair potentials, we only considered the intra-orbital pairing cases. Furthermore, we introduced anisotropy in the pair potential in quasi-one-dimensional $d_{yz}$ and $d_{zx}$ orbitals by setting $\delta<1$. In addition, the crystalline symmetry of SRO allows different magnitudes of the pair potential for the two-dimensional $d_{yz}$ orbital ($\Delta_{1}$) and the quasi-one-dimensional $d_{yz}$ and $d_{zx}$ orbitals ($\Delta_{2}$).

In the NM region between an SRO and an $s$-wave superconductor, we considered a single-orbital model given by

[TABLE]

where $c_{{\bm{k}}\sigma}$ is the annihilation operator for an electron in the NM. The energy dispersion of the NM is given by

[TABLE]

where $t$ is the hopping integral between nearest-neighbor sites. We took into account the interface Rashba spin-orbit coupling in the NM layer next to the SRO, which is given by

[TABLE]

In the spin-singlet $s$-wave superconductor region, we considered the on-site pair potential as well as the kinetic-energy term in Eq. (15):

[TABLE]

where $\varphi$ is the macroscopic phase of the pair potential relative to the interface normal of the $p$-wave superconductor. These three parts are coupled via hopping at the interface. The magnitude of the hopping at the interface between the NM and the $s$-wave superconductor was chosen to be the same as in the NM. The SRO-NM interface displays three kinds of hopping: $t_{xy}$, $t_{yz}$, and $t_{zx}$. The first, $t_{xy}$, corresponds to the hopping between the NM and the $d_{xy}$ orbital of SRO. Likewise, $t_{yz}$ ($t_{zx}$) also denotes the interface hopping between NM and $d_{yz}$ ($d_{zx}$) orbital of SRO.

We calculated the current-phase relation of the Josephson current in the single junction (see FIG. 1. (a)) based on a lattice Green’s function method that takes into account the Andreev reflection and Andreev bound states at the interface Furusaki and Tsukada (1991); Tanaka and Kashiwaya (1997). For that purpose, we calculated the Green’s function in the superconducting SRO/NM/$s$-wave superconductor junction. These three regions are aligned in the (100) direction, with the boundaries for the $s$-wave superconductor and SRO located at $x\leq x_{-2}$ and $x\geq x_{3}$, respectively. In the numerical calculations, four NM layers are inserted between these two superconductors at $x_{-1}\leq x\leq x_{2}$. Since we are considering flat interfaces in the ballistic limit, $k_{y}$ is a conserved quantity. In order to obtain the Green’s function in this junction, we first calculated the surface Green’s functions of the semi-infinite SRO and spin-singlet $s$-wave superconductor, where the surfaces are not coupled to the NM layer. These calculations were based on the recursive Green’s function method, using M$\ddot{\rm o}$bius transformation Umerski (1997). Next, we added the two NM layers on these surfaces with the following recursive equation:

[TABLE]

where $G_{n}^{LI}(k_{y},i\omega_{l})$ stands for the surface Green’s function for the system on the left (right) side of the interface, with $x\leq x_{n}$ ($x\geq x_{n}$). The operators $\hat{\varepsilon}_{n}(k_{y})$ and $\hat{t}_{n,n-1}$ represent the local and non-local parts of the Hamiltonian. Then, we obtained two surface Green’s function, defined for $x\leq x_{0}$ and $x\geq x_{1}$. These two systems are combined in the equation

[TABLE]

Then, we obtained the non-local Green’s functions in the $s$-wave/NM/SRO junction as follows:

[TABLE]

The Fourier-transforms of $\hat{G}_{01}(k_{y},i\omega_{l})$ and $\hat{G}_{10}(k_{y},i\omega_{l})$ are given by

[TABLE]

with $\beta=1/(k_{B}T)$ and where $T$ is the temperature. The above formulas for $\hat{G}_{01}(k_{y},\tau)$ and $\hat{G}_{10}(k_{y},\tau)$ can be expressed as

[TABLE]

with

[TABLE]

Thus, we obtained the current-phase relation $I(\varphi)$ by using these $\hat{G}_{01}(k_{y},\tau)$ and $\hat{G}_{10}(k_{y},\tau)$:

[TABLE]

where $\mathrm{Tr^{\prime}}$ is a partial sum of the diagonal elements of the Hamiltonian, including only those matrix elements that refer to the electron space.

Below, we define the model parameters that were used in the calculations. For the hopping parameters in SRO, we assumed $t_{2}/t_{1}=0.395$, $t_{3}/t_{1}=1.25$, $t_{4}/t_{1}=0.125$, and $t_{5}/t_{1}=0.15$, based on first-principles calculations. Here, $t_{1}$ is the nearest-neighbor hopping parameter in the $d_{xy}$ orbital in SRO, which first-principles calculations estimate as being approximately 230 meV Mackenzie and Maeno (2003). Furthermore, the chemical potentials in each orbital in SRO, $\mu_{yz}$, $\mu_{zx}$, and $\mu_{xy}$, were chosen to yield the following numbers of electron: $n_{yz}=n_{zx}=n_{xy}=2/3$. The chemical potential in the normal metal, $\mu_{n}$, was chosen so that the number of electron is $2/3$. The magnitude of the spin-orbit interaction in the bulk SRO, expressed as $\lambda$, changes these values. We set $\lambda=0.3$ for consistency with quasiparticle spectra obtained by angle-resolved photoemission spectroscopy Mackenzie and Maeno (2003). We chose the magnitudes of the pair potential for the $d_{yz}$ and $d_{zx}$ orbitals in SRO to exceed that of the $d_{xy}$ orbital, as determined previously by tunneling spectroscopy Yada et al. (2014); Kashiwaya et al. (2011). The magnitude of the pair potential in the $d_{yz}$ and $d_{zx}$ orbitals was set to $\Delta_{1}=0.001t_{1}$. We set the magnitude of the pair potential for the $d_{xy}$-orbital to $\Delta_{2}=0.4\Delta_{1}$. For the quasi-one-dimensional nature of the pair potential for $d_{xy}$-orbital, we set $\delta=0.1$, based on the ratio of $t_{3}$ to $t_{4}$.

We assumed that an $s$-wave superconductor and an NM are described by the same single-orbital model as that of the $d_{xy}$ orbital in SRO. We set their chemical potentials $\mu_{n}$ to the same level as the $d_{xy}$ orbital in SRO, in the absence of spin-orbit interaction in the bulk SRO. The magnitude of the pair potential of the $s$-wave superconductor was set to $\Delta_{s}=10\Delta_{1}$. The magnitude of the Rashba spin-orbit interaction at the interface between NM and SRO, $\lambda_{R}$, depends on the microscopic electronic properties of the junction and was set to $0.3$ in this study.

III Results

III.1 current phase relation

Figure 2 shows the current-phase relation in the absence of interface Rashba spin-orbit interaction. Here, the Josephson current $I(\varphi)$ is decomposed into the Fourier series

[TABLE]

It is then normalized by $I_{0}$, the maximum value of the Fourier coefficients. Table 1 shows which of the Fourier coefficients have nonzero values.

As shown in Figs. 2(a) and (c), the Josephson current is almost proportional to $\sin(2\varphi)$ in the case where the first-order Josephson coupling is absent. In fact, Table 1 shows that only the sinusoidal terms with an even-number order are nonzero. On the other hand, odd-order terms are nonzero in the case of the multi-band model, as shown in Fig. 2 and Table 1 (b) and (d). We confirmed that these odd-order terms are zero in the absence of spin-orbit interaction (LS coupling) in bulk SRO. We note that the cosine terms appear in the chiral $p$-wave case but are absent in the helical $p$-wave case. The cosine terms in the chiral $p$-wave case are nonzero even in the absence of Rashba spin-orbit coupling $\lambda_{R}$. This is because the hopping integral $t_{5}$ ($i.e.$, corresponding to inter-orbital hopping between the $d_{y}z$ and $d_{x}z$ orbitals) is nonzero and spin-orbit coupling in bulk SRO $\lambda$ enhances the magnitude of the cosine terms. When the opposite chirality of the pair potential is chosen with

[TABLE]

the signs of $I_{1}^{c}$ and $I_{2}^{c}$ are reversed.

We plot the current-phase relations in the presence of interface Rashba spin-orbit coupling ($\lambda_{R}>0$) in Fig. 3. Figure 3 (c) shows no qualitative difference between the Josephson currents in the presence or absence of interface Rashba spin-orbit interaction, in the single-band model and in the case of helical pairing. On the other hand, cosine terms appear as a result of the interface Rashba spin-orbit coupling in the case of the chiral $p$-wave shown in Fig. 3 (a) Asano et al. (2003). By contrast, there is no qualitative difference between the current-phase relations in the presence or absence of interface Rashba spin-orbit interaction in the multi-band model, as shown in Figs. 3 (b,d) and Table 2 (b,d). In the most general case, where both the interface Rashba spin-orbit interaction and bulk LS coupling in the multi-band model exist, we observe a qualitative difference between the chiral and helical $p$-wave cases. The cosine terms $I_{1}^{c}$ and $I_{2}^{c}$ appear only in the case of chiral $p$-wave pairing. This difference is due to the broken TRS that occurs in chiral $p$-wave pairing. In the following calculations for various junctions, we considered the interface Rashba spin-orbit interactions and used the multi-band model.

In order to take into account the corner structure of the junction, we show the relation between the current phase relations in different orientations in Table III. The orientation dependence affects the maximum Josephson current in a corner junction or SQUID when it is written as a function of the external magnetic flux $\Phi$. Although the calculation of the $\Phi$ dependence will be shown in next subsection, we first show the relation between $I_{a}(\varphi_{a})$ and $I_{\overline{b}}(\varphi_{\overline{b}})$ indicated in Fig. 4. This relation depends on the pairing symmetries specified in TABLE 3. This relation in chiral $p$-wave pairing is different from that in helical $p$-wave pairing. Furthermore, in the helical $p$-wave cases, the relation between $I_{a}(\varphi_{a})$ and $I_{\overline{b}}(\varphi_{\overline{b}})$ depends on the irreducible representations of the pair potentials. This affects the properties of the corner junction or corner SQUID, as shown in the next subsection. Next, we show the relation between the $I_{a}(\varphi_{a})$ and $I_{\overline{a}}(\varphi_{\overline{a}})$ indicated in Fig. 1. The equation $I_{a}(\varphi)=I_{\overline{a}}(\varphi+\pi)$ is valid for all pairings. This fact influences the properties of a symmetric SQUID.

III.2 Magnetic-field dependence of the maximum Josephson current

in various junctions

In this subsection, we calculate the magnetic-field dependence of the maximum Josephson current in corner junctions, corner SQUIDs, and symmetric SQUIDs.

We calculated the relation between the external magnetic flux $\Phi$ and the maximum Josephson current $I_{c}$ by a standard method. In the Josephson junctions shown in Figs. 1 and 4, we assumed that the external magnetic field was applied parallel to the $z$-axis. The vector potential is then given by

[TABLE]

On the other hand, the phase $\gamma$ of the pair potential obeys

[TABLE]

Since the magnetic field is screened inside the superconductor because of the Meissner effect, $A_{y}(x)$ takes the constant value $A_{y}(\infty)$ found at locations far from the interface. Using these properties, we integrated both sides of the $y$ component of Eq. (36) with respect to $y$.

[TABLE]

The phase difference between the $s$-wave superconductor and the SRO is therefore given by

[TABLE]

Here, $A_{1}$ and $A_{2}$ represent the vector potentials far from the interface in the SRO and $s$-wave superconductor, respectively. The Fourier components of the Josephson current, $I_{n}^{s}$ and $I_{n}^{c}$, defined in eq. (33), were obtained in the previous subsection in the absence of a magnetic field. In the presence of a magnetic field, the Josephson current becomes a function of $y$. We integrated this function with respect to $y$:

[TABLE]

where, $Y$ and $Z$ are the sizes of the junction. It is evident that Eq.(III.2) displays a periodicity of 2$\pi$ with respect to $\varphi(0)$. Therefore, by changing $\varphi(0)$ over the range $-\pi\leqq\varphi(0)\leqq\pi$, the maximum Josephson current $I_{c}$ can be obtained as a function of the external magnetic flux $\Phi$.

Next, we calculated the maximum Josephson current $I_{c}$ in the corner junction shown in Fig. 5 as a function of $\Phi$, using a similar approach to that described in [Van Harlingen, 1995]. We obtained the current-phase relations $I_{a}(\varphi_{a})$ and $I_{\overline{b}}(\varphi_{\overline{b}})$ indicated in Fig. 5. By calculating the following equation instead of Eq. (III.2), we obtained the maximum Josephson current $I_{c}$ as a function of the external magnetic flux $\Phi$ based on $I(\Phi,\varphi(0))$, given by

[TABLE]

Finally, we calculated the maximum Josephson current $I_{c}$ as a function of the external magnetic flux $\Phi$ in the two types of SQUID shown in Fig. 6. The macroscopic phase differences of the two superconductors $\varphi_{a}$ and $\varphi_{\overline{b}}$ obey the following relation:

[TABLE]

The total current in these parallel circuits is therefore given by

[TABLE]

By evaluating the maximum value of Eq. (41) for a given external magnetic flux $\Phi$, we obtained the maximum Josephson current as a function of $\Phi$.

The $I_{c}$ functions for the corner junction of SRO are plotted in Fig. 7. In the cases of the helical $p$-wave, the positions of the minima depend on the $d$-vector as shown in Figs. 5(b) and (c). This is because the relation between the Josephson currents $I_{a}(\varphi_{a})$ and $I_{\overline{b}}(\varphi_{\overline{b}})$ in Fig. 7 is different for each pairing symmetry. In particular, $I_{a}(\varphi)=I_{\overline{b}}(\varphi)$ for the $A_{1u}$ and $B_{2u}$ pairings, while $I_{a}(\varphi)=I_{\overline{b}}(\varphi+\pi)$ for the $A_{2u}$ and $B_{1u}$ pairings. For all the helical $p$-wave cases, the Fraunhofer patterns are symmetric functions of $\Phi$. On the other hand, $I(\Phi)$ is not a symmetric function of $\Phi$ for chiral $p$-wave pairing. This difference results from the existence of the cosine terms in the current-phase relation. In other words, the broken TRS causes the asymmetry of $I_{c}=I_{c}(\Phi)$, $i.e.$, $I_{c}(\Phi)\neq I_{c}(-\Phi)$. These results are summarized in Table 4. As seen from this table, there are qualitative differences between the helical and chiral $p$-wave pairings. The asymmetry of the Josephson current is due to the existence of cosine terms in the current-phase relation for the chiral $p$-wave pairings. These cosine terms can be nonzero unless both $\lambda$ and $\lambda_{R}$ are nonzero owing to the presence of inter-orbital hopping in the multi-band model. The magnitudes of these cosine terms and the resulting asymmetry of $I(\Phi)$ are enhanced by the spin-orbit interactions, expressed through $\lambda$ and $\lambda_{R}$.

Next, we discuss $I_{c}$ in the corner SQUID shown in Fig. 8. This $I_{c}$ is symmetric or asymmetric with respect to $\Phi$ for the helical and chiral cases, respectively. As in the case of the SRO/NM/$s$-wave corner junction, the existence of the cosine terms in the current-phase relation in chiral pairing causes the asymmetry of $I_{c}(\Phi)$. The chiral pairing is consistent with a previous study based on a single-band model Asano et al. (2005). In the cases of helical pairing, the position of the maximum or minimum in $I_{c}(\Phi)$ depends on the pairing symmetry (irreducible representation), $i.e.$, the $d$-vector as shown in Figs. 8(b) and (c). We note that the $\Phi_{0}$ periodicity in the helical pairing case appears only for a three-band model.

Finally, we consider the case of the so-called symmetric SQUID Geshkenbein et al. (1987). Figure 9 shows the $\Phi$ dependence of $I_{c}$ in the symmetric SQUID shown in Fig. 6(b). In this junction, there is no qualitative difference between the cases of chiral and helical pairing since $I_{a}(\varphi)=I_{\overline{b}}(\varphi+\pi)$ is satisfied. The resulting Josephson current $I_{c}$ is symmetric for both chiral and helical pairings, including in the presence of the cosine terms. Thus, we do not find any qualitative difference in $I_{c}$ for the chiral and helical pairings in this symmetric SQUID.

IV Discussion and Summary

Here, we discuss the multi-band effect on the Josephson current in the present calculations, starting with the chiral $p$-wave case. As shown in TABLES 1 and 2, the spin-orbit interaction in the bulk SRO ($\lambda$) and the interface Rashba spin-orbit interaction ($\lambda_{R}$) generate $I_{1}^{s}$ for chiral $p$-wave pairing. We found that the coefficient of the $\sin(\phi)$ term $I_{1}^{s}$ has the form

[TABLE]

which is confirmed by Fig. 10. This form suggests that $\lambda$ directly induces $I_{1}^{s}$, whereas the existence of inter-orbital hopping $t_{5}$ is needed to produce $I_{1}^{s}$ from $\lambda_{R}$. In the single-band model, $I_{1}^{s}$ is absent while $I_{1}^{c}$ is induced by $\lambda_{R}$ in chiral $p$-wave pairing. In the multi-orbital model, $t_{5}$ induces the effective phase shift of the pair potential. A part of $I_{1}^{c}$ is then converted to $I_{1}^{s}$ by $t_{5}$. Thus, we can conclude that the existence of $I_{1}^{s}$ results from the multi-band model in SRO. This term becomes dominant in the limit of low transmissivities, where the higher-order Josephson couplings are strongly suppressed.

Next, we discuss the helical $p$-wave case, where $I_{1}^{s}$ is given by

[TABLE]

This is because $I_{1}^{c}$ does not exist in the single-band model owing to the TRS of the helical $p$-wave pairing. Since $t_{5}$ only gives the effective phase shift of the pair potential, $I_{1}^{s}$ cannot be produced by $\lambda_{R}$. On the other hand, $\lambda$ directly induces $I_{1}^{s}$ in a similar manner as in the case of the chiral $p$-wave pairing.

In summary, we have studied Josephson currents in SRO/NM/$s$-wave junctions. We found that the first-order Josephson coupling is induced by the spin-orbit interaction for the cases of both chiral and helical $p$-wave pairings. Note that the $\sin(\varphi)$ term, which is absent in the single-band model, appears as a result of the spin-orbit interaction and inter-band hopping. In the case of helical pairing, the first-order Josephson term appears only in the three-band model. Owing to the existence of the first-order Josephson coupling, the period of the Josephson current, as the magnetic flux $\Phi$ is varied, is expected to become the period of the conventional junctions. For the case of chiral $p$-wave pairing, the Josephson current shows asymmetric behavior in the corner junction and the corner SQUID, owing to broken TRS. This asymmetry is enhanced by the spin-orbit interaction in the bulk SRO or at the interface in the junction. Since the magnitude of the spin-orbit interaction in SRO is not very small, it is possible to detect the asymmetry experimentally if the TRS breaking by chiral pairing is realized.

In this paper, we assumed ballistic junctions with flat interfaces. Surface roughness and impurity scattering are known to influence charge transport in spin-triplet $p$-wave superconductor junctions Bakurskiy et al. (2014); Lu et al. (2016). In particular, the odd-frequency spin-triplet $s$-wave component generated near the interface induces an anomalous proximity effect Tanaka and Kashiwaya (2004); Tanaka et al. (2005a), and the resulting Josephson current displays a low-temperature anomaly Tanaka and Kashiwaya (2004); Tanaka et al. (2005a); Asano et al. (2006). Taking into account the impurity-scattering effect in the multi-band model is an interesting prospect for future work.

Acknowledgements.

This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas, Topological Material Science (Grants No. JP15H05851, No. JP15H05852, and No. JP15H05853), a Grant-in-Aid for Scientific Research B (Grant No. JP15H03686), a Grant-in-Aid for Challenging Exploratory Research (Grant No. JP15K13498) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan (MEXT); Japan-RFBR JSPS Bilateral Joint Research Projects/Seminars (Grants No. 15-52-50054 and No. 15668956); Dutch FOM; the Ministry of Education and Science of the Russian Federation, Grant No. 14.Y26.31.0007; and by the Russian Science Foundation, Grant No. 15- 12-30030.