Possible superconductivity in Sr2IrO4 probed by quasiparticle interference

Based on the possible superconducting (SC) pairing symmetries recently proposed, the quasiparticle interference (QPI) patterns in electron- and hole-doped Sr2IrO4 are theoretically investigated. In the electron-doped case, the QPI spectra can be explained based on a model similar to the octet model of the cuprates while in the hole-doped case, both the Fermi surface topology and the sign of the SC order parameter resemble those of the iron pnictides and there exists a QPI vector resulting from the interpocket scattering between the electron and hole pockets. In both cases, the evolution of the QPI vectors with energy and their behaviors in the nonmagnetic and magnetic impurity scattering cases can well be explained based on the evolution of the constant-energy contours and the sign structure of the SC order parameter. The QPI spectra presented in this paper can be compared with future scanning tunneling microscopy experiments to test whether there are SC phases in electron- and hole-doped Sr2IrO4 and what the pairing symmetry is.

Based on the possible superconducting (SC) pairing symmetries recently proposed, the quasiparticle interference (QPI) patterns in electron-and hole-doped Sr 2 IrO 4 are theoretically investigated. In the electron-doped case, the QPI spectra can be explained based on a model similar to the octet model of the cuprates while in the hole-doped case, both the Fermi surface topology and the sign of the SC order parameter resemble those of the iron pnictides and there exists a QPI vector resulting from the interpocket scattering between the electron and hole pockets. In both cases, the evolution of the QPI vectors with energy and their behaviors in the nonmagnetic and magnetic impurity scattering cases can well be explained based on the evolution of the constant-energy contours and the sign structure of the SC order parameter. The QPI spectra presented in this paper can be compared with future scanning tunneling microscopy experiments to test whether there are SC phases in electron-and hole-doped Sr 2 IrO 4 and what the pairing symmetry is. R ecently, a very interesting material, the 5d transition metal oxide Sr 2 IrO 4 has attracted much attention 1-14 . In this material, the energy bands close to the Fermi level are mainly contributed by the t 2g orbitals of Ir and it is in the (t 2g ) 5 configuration. On the one hand, due to the extended nature of 5d orbitals, Coulomb interaction U for 5d electrons (1-3 eV) is expected to be smaller than that for 3d electrons (5-7 eV) 7 . On the other hand, the spin-orbit coupling (SOC) is considerably larger by a factor of 10 in 5d than in 3d 7 . In this case, the strong SOC splits the t 2g orbitals into an upper J 5 1/2 band and lower J 5 3/2 bands. In the parent compound, the J 5 3/2 bands are fully occupied while the J 5 1/2 band is half-filled. Meanwhile, the bandwidth of this J 5 1/2 band is much smaller than the original one in the absence of the SOC. Therefore, even a small U can lead the system into a Mott insulator with pseudospin 1/2 antiferromagnetic (AFM) order, making Sr 2 IrO 4 an analog to the parent compound of the cuprates. This J 5 1/2 AFM Mott insulating state is supported by several experiments [4][5][6]8,[10][11][12][13] . Therefore, Sr 2 IrO 4 is an ideal candidate to perform comparative studies with the cuprates. The question is, whether doping Sr 2 IrO 4 can induce superconductivity in analogy to the cuprates?
To resolve this issue, Refs. 15 and 16 theoretically investigated the superconducting (SC) properties in both electron-and hole-doped Sr 2 IrO 4 . They found that, in the electron-doped case, a SC phase indeed exists and the pairing contains both intraorbital and interorbital components as well as both singlet and triplet components of t 2g electrons, while the pairing symmetry on the Fermi surface is d x 2 {y 2 -wave (or d Ã x 2 {y 2 -wave as denoted by Ref. 16) and the pairing function respects time-reversal symmetry (TRS), similar to the cuprates. On the other hand, in the hole-doped case, the Fermi surface topology changes and resembles that of the iron pnictides, with an electron pocket around the C point and a hole pocket around the M point. In this case, Ref. 15 found that there is no SC phase while Ref. 16 concluded that a SC phase can also exist while the pairing function still respects TRS and the pairing symmetry is s Ã + -wave, similar to that of the iron pnictides 17 . In this paper, in order to search for an experimental test of the above two theories, we propose to measure the quasiparticle interference (QPI) patterns in both electron-and hole-doped Sr 2 IrO 4 by scanning tunneling microscopy (STM). As we know, the QPI patterns are strongly influenced by the shape and evolution of the constant-energy contour (CEC), as well as the relative sign of the SC order parameter of the states connected by the QPI wave vectors [18][19][20][21][22] . Therefore, by measuring the QPI patterns, we can not only determine whether the SC phase exists in the electron-and hole-doped cases, but also the SC pairing symmetry. 0 and E 1k~{ 2t 4 cos k x {2t 5 cos k y {m, Here c { k1: , c { k2: and c { k3: create a spin-up electron with momentum k in the d xz , d yz and d xy orbitals, respectively. A k stands for the tight-binding part of the Hamiltonian in the presence of the SOC, with l being the SOC strength. (t 1 , t 2 , t 3 , t 4 , t 5 , m xy , l) 5 (0.36, 0.18, 0.09, 0.37, 0.06, 20.36, 0.5) and m is the chemical potential which is adjusted according to the electron filling n. D k describes the pairing term of the Hamiltonian whose explicit expression is given later and we set D 0 5 0.05 (unless otherwise specified).
When a single impurity is located at the origin, the impurity Hamiltonian can be written as with N being the system size (396 3 396 throughout the paper) and s s 5 1(21) for s 5" (#). We consider both nonmagnetic and magnetic impurity scattering, diagonal in the orbital basis and with a scattering strength V s and V m for the nonmagnetic and magnetic cases, respectively. For definiteness, V s and V m are both taken to be 0.04. Following the standard T-matrix procedure 23 , the Green's function matrix is defined as and Here g 0 (k, v) is the Green's function in the absence of the impurity and can be written as where I is a 12 3 The experimentally measured local density of states (LDOS) is expressed as and its Fourier transform is defined as r q,v ð Þ~X r r r,v ð Þe iq:r . Since the system is even under k R 2k (D k is also an even function of k as can be seen later), it can be written as and the contribution from the spin up and spin down electrons can be expressed as Here we need to clarify what to measure in the STM experiment. If the impurity scattering is weak, then T(v) / V. In this case, if V s ? 0 and V m 5 0, we have r " (q, h) 5 r # (q, v) since the system respects TRS. On the other hand, if V s 5 0 and V m ? 0, TRS is broken and now for q ? 0, we have r " (q, v) 5 2r # (q, v), leading to r(q, v) 5 0. Therefore, in the STM experiment, people should measure the spin-resolved LDOS, either r " (r, v) or r # (r, v), to get a nontrivial QPI spectrum.

Results and discussion
At n 5 5.2, the electron-doped case, the pairing functions g k and c k in equation (3) can be expressed as 16 g 1 k~{ 0:35z0:2 cos k y {0:08 cos k x , g 2 k~0 :35{0:2 cos k x z0:08 cos k y , The LDOS in the absence of the impurity is homogeneous in real space and is shown in Fig. 1(a). Two SC coherence peaks are located at 6D where D < 0.4D 0 and the spectrum is V-shaped in the vicinity of v 5 0, indicating the nodal gap structure, consistent with the d Ã x 2 {y 2 -wave pairing symmetry.
In the presence of the impurity, we plot jr " (q, v)j in Fig. 2 and several QPI wave vectors can be identified. For nonmagnetic impurity scattering [from Figs. 2(a) to 2(f)], three QPI wave vectors q 1 , q 2 and q 6 can be clearly seen evolving with energy. q 1 is located along the (61, 61) directions and moves away from the origin as jvj increases. q 2 and q 6 are not located along the high-symmetry directions and they overlap after a 90 degree rotation. Furthermore, they are not so obvious at v/D 5 0.75 since they are masked by the highintensity spots around them. In contrast, for magnetic impurity scattering [from Figs. 2(g) to 2(l)], q 1 , q 2 and q 6 become less clear and instead, another two vectors q 3 and q 7 can be identified evolving with energy. They are both located along the (0, 61) and (61, 0) directions and move towards the origin as jvj increases.
The appearance and evolution of the above five QPI wave vectors can be understood from the evolution of the CEC. As we can see from Fig. 1(b), the CEC of the electron-doped Sr 2 IrO 4 is similar to the octet model of the cuprates [18][19][20][21]23 and the expected QPI vectors should be those connecting the tips of the CEC, i.e., q 1 , q 2 , …, q 7 in this case. For example, at jvj/D 5 0.5, q 1 , q 2 , …, q 7 shown in Fig. 1   www.nature.com/scientificreports and for all the energies we investigated, q 4 and q 5 cannot be clearly seen, thus we neglect these two vectors in the following.
Next we discuss the implication of the QPI vectors on the sign of the SC order parameter. As we know, due to the effect of the SC coherence factors, those scattering between the states with the opposite (same) sign of the SC order parameters will be enhanced (suppressed) by nonmagnetic impurity. For magnetic impurity scattering, the situation is reversed. In electron-doped Sr 2 IrO 4 , since the pairing symmetry is assumed to be d Ã x 2 {y 2 -wave and the sign of the SC order parameter on the CEC is shown in Fig. 1(b) as 1 and 2.
As we can see, q 1 , q 2 and q 6 are sign-reversing scattering processes while q 3 and q 7 are sign-preserving ones. Therefore, q 1 , q 2 and q 6 should be more discernable in the nonmagnetic impurity scattering case while q 3 and q 7 should be more distinct in the magnetic impurity scattering case. This is exactly what we obtain here as can be seen from Fig. 2. Therefore, the evolution of the QPI vectors with energy together with their different behaviors in the nonmagnetic and magnetic impurity scattering cases can help to determine whether the pairing symmetry is d Ã cos k x {cos k y À Á and a { k1: creates a pseudospin up electron with momentum k in the J 5 1/2 band. If we set D 0 5 0.02 here, then the LDOS in the absence of the impurity is qualitatively the same as that shown in Fig. 1(a) and now we have D 5 D 0 . In addition, the evolution of the CEC and the QPI spectra obtained are also similar to those in Figs. 1(b) and 2, respectively, indicating that the pairing functions adopted in Refs. 15 and 16 share the same characteristics. As we can see in the limit of large SOC (l R '), . Although the pairing is a pseudospin singlet, it contains both intraorbital and interorbital components as well as both singlet and triplet components of t 2g electrons and it respects the same symmetry as that shown in equation (12). Therefore, for electron-doped The LDOS in the absence of the impurity is shown in Fig. 3(a) and two pairs of SC coherence peaks are located at 60.4D 0 and 60.52D 0 , as denoted by the black and red arrows, respectively, with a U-shaped profile close to v 5 0, indicating the full gap opening at this doping level. The pairing function D k projected onto the Fermi surface is shown in Fig. 3(b). As we can see, the pairing order parameter on the electron pocket around C is negative and almost isotropic while on the hole pocket around M, it is positive and anisotropic. Therefore, the Fermi surface topology and the sign change of the pairing order parameter between the electron and hole pockets are very similar to the iron pnictides 17,22 and this pairing symmetry is dubbed as s Ã + -wave. The evolution of the CEC with energy at this doping level is shown in Figs. 3(c) and 3(d). As can be seen, at low energies (jvj 5 0.2D 0 and 0.3D 0 ), the CEC exists around the M point and the characteristic QPI vectors should be q 1 , q 2 , …, q 7 as shown in Fig. 3(c). q 1 and q 5 are located along the (61, 61) directions. q 1 moves towards the origin as jvj increases while q 5 hardly evolves with energy. q 2 and q 6 are not located along the high-symmetry directions while q 3 and q 7 are both located along the (0, 61) and (61, 0) directions. In addition, q 3 should move towards the origin with increasing jvj while the situation for q 7 is reversed. As jvj increases to 0.4D 0 , the tips of the two adjacent CECs touch each other. So in this case, q 1 should disappear while q 2 , q 3 , q 6 and q 7 become equivalent. When jvj $ 0.5D 0 , the CEC around M evolves into closed contours where no tips exist, thus the above mentioned QPI vectors disappear. Here q 1 , q 2 , …, q 7 are all sign-preserving scattering processes, therefore they should be more discernable in the magnetic impurity scattering case. As jvj increases to 0.6D 0 , another CEC shows up around the C point. In this case, a large portion of the CECs around the M and C points are quasinested with each other by a nesting vector (61.15p, 61.15p), as can be seen from Fig. 3(d). In this case, there should exist a QPI vector located at around (60.85p, 60.85p) in the first Brillouin zone and since it is a sign-reversing scattering process, it should be more distinct in the nonmagnetic impurity scattering case 22 .
To verify the above expectations, the QPI spectra are calculated and are plotted in Fig. 4. For magnetic impurity scattering [see Figs. 4(a) to 4(j)], indeed we can identify the QPI vectors q 1 , q 2 , …, q 7 , except that q 4 cannot be clearly seen. The evolution of these vectors is also consistent with that derived from Fig. 3(c), i.e., q 1 locates along the (61, 61) directions and moves towards the origin with increasing jvj. q 3 and q 7 both locate along the (0, 61) and (61, 0) directions while they become equivalent with q 2 and q 6 at jvj/D 0 5 0.4. Meanwhile, q 5 barely evolves with energy and at jvj/D 0 $ 0.5, the above mentioned QPI vectors disappear. On the other hand, for nonmagnetic impurity scattering, as we can see from Figs. 4(k) to 4(t), q 1 , q 2 , …, q 7 become less clear and instead, at v/D 0 5 0.5 and 0.6 [see Figs. 4(s) and 4(t)], another QPI vector q 8 shows up at around (60.85p, 60.85p), which is resulted from the interpocket scattering between the electron and hole pockets as we mentioned above. Therefore, the locations of these QPI vectors and their behaviors in the nonmagnetic and magnetic impurity scattering cases are consist-  Fig. 1(a). (b) The pairing function D k projected onto the Fermi surface. (c) The CECs at | v | 5 0.2D 0 (black) and 0.3D 0 (red). (d) The CEC at | v | 5 0.6D 0 . The red curves in (d) are displaced by (1.15p, 1.15p)  ent with what we expected from the evolution of the CEC and the sign structure of the SC order parameter.
Here we need to point out that, in real STM experiments, both nonmagnetic and magnetic scatterers inevitably coexist in the same sample and are difficult to control. In this case, a magnetic field is usually applied to introduce additional scatters into the system. When an external magnetic field is applied, the main effects are the formation of vortices and the Zeeman splitting. In the following, we discuss these two effects separately. As pointed out in Refs. 21 and 22, the introduction of vortices causes the phase of the SC gap to precess by 2p around each vortex, whereas the amplitude of the gap vanishes at its core. Both the phase gradient and the inhomogeneity in the SC gap amplitude can scatter quasiparticles. The inhomogeneous superflow about the vortex (resulting from the phase gradient) produces Doppler-shift scattering that is odd under time reversal like magnetic impurities, while the spatial inhomogeneity in the SC gap amplitude causes inhomogeneous Andreev scattering. Although the vortex core is not a simple magnetic impurity as shown in equation (4), all of these scatterings selectively activate the sign-preserving q points. Especially, from Table S1 in the supporting online material for Ref. 21 we can see clearly that both the phase gradient and the gap amplitude scatterings enhance the same q i (i 5 1, 4, 5) as the magnetic impurity does. This is further confirmed by Maltseva and Coleman 24 who found that both the Andreev scattering and the  resonant scattering (whose coherence factors are the same as those of the phase-gradient scattering) are equally effective in qualitatively modeling the observations, i.e., they both enhance the same q i (i 5 1, 4, 5) as the magnetic impurity does. On the other hand, strictly speaking, in order to study the effect of vortices on the QPI, we should solve the Bogoliubov-de Gennes (BdG) equations in real space to get the phase and amplitude variation of the SC gap selfconsistently by introducing a Peierls phase factor in the hopping integral t. However in our work, the pairing function is given in momentum space and it is very difficult to use a real-space attraction to simulate this momentum-space pairing function. Therefore it is impossible for us to solve the real-space BdG equations to exactly investigate the effects of vortices on the QPI. Thus in our work, the scattering off vortices can be approximated as the scattering off the magnetic impurity. In Fig. 5 we show the difference of the QPI spectra between the magnetic and nonmagnetic impurity scattering cases, defined as Djr " (q, v)j 5 jr " (q, v)j mag 2 jr " (q, v)j nonmag , which can be viewed as the magnetic-field-induced weight transfer as shown in Fig. 3(A) of Ref. 21 and Fig. 4 of Ref. 22. As we can see, the behaviors of the QPI vectors indeed meet our expectations, that is, the sign-preserving and sign-reversing q i are enhanced and suppressed by the introduction of vortices, respectively.
As to the Zeeman splitting, it is expected to be gm B B , 0.9 meV at B 5 8T (g 5 2). Supposing the SC gap in Sr 2 IrO 4 to be D , 9 meV, then the Zeeman energy is estimated to be 0.1D. In this case, the chemical potentials in equation (3) for the spin up and down electrons differ by this Zeeman energy 25 and we again calculate the QPI spectra (not shown here). We found that, at this value of the Zeeman energy, the main effects are a tiny splitting of the CECs and a slight displacement of the QPI vectors. In the meantime, the intensity of the QPI vectors stays almost unchanged. Therefore, experimentally the effect of the Zeeman splitting can be neglected.
Then we explain the reason why we adopt a scattering matrix V that is diagonal in the orbital basis. As we know, in single-band superconductors like the cuprates, the intensity of the QPI vectors is solely determined by the coherence factor C(k i , k f ), which is a combination of the BCS coefficients u k and v k . On the contrary, in multi-orbital superconductors, the intensity of the QPI vectors is determined not only by the coherence factor, but also by the matrix elements of the unitary transformation between the orbital and band bases. This can be seen from equation (7), where the above mentioned two factors are both incorporated in the expression of g 0 (k, v). Therefore, first of all, although we considered an impurity scattering matrix which is diagonal in the orbital basis [that is, we ignore the orbital effects in T(v)], the orbital effects still affect the QPI spectra through g 0 (k, v). Secondly, the orbital effects sometimes may blur the QPI vectors. For example, at n 5 5.2, from the CEC shown in Fig. 1(b), if we neglect all the orbital effects, then the most pronounced QPI vectors should be q 1 ,q 2 , Á Á Á ,q 7 since the tip to tip scatterings generally have the largest joint density of states. Of course, the intensity of these QPI vectors are influenced by the coherence factor C(k i , k f ). However, if the orbital effects are taken into account, then the joint density of states associated with the tip to tip scatterings is affected by the orbital to band transformation (i.e., the vari-ation of the orbital content along the CECs) and the intensity of the QPI vectors is influenced not only by C(k i , k f ), but also by this transformation and this may be the reason why q 4 and q 5 cannot be clearly seen. In this case, if we further consider the orbital effect of the impurity scattering [that is, we add some off-diagonal elements in equation (4)], then the QPI vectors may be further blurred. Thirdly, to the best of our knowledge, in multi-orbital systems, the exact orbital effects of the impurity scattering have not been determined, either experimentally or from first principles calculations. Therefore, in our work, we made the simplest approximation and predicted some QPI vectors which can be observed in experiments in the ideal case.
At last, we would like to mention that we have also calculated the spectra for Z " (q, v), which is the Fourier transform of Z " (r, v) 5 r " (r, v)/r " (r, 2v). Experimentally this procedure can eliminate extrinsic effects associated with the scanning feedback loop 21,22 and suppress the checkerboard signal 21 . However in our theoretical investigation, the above two factors do not exist and the spectra for Z " (q, v) are qualitatively the same as those for r " (q, v). Therefore the spectra for Z " (q, v) are not shown here.
In summary, we have studied the QPI spectra in both electron-and hole-doped Sr 2 IrO 4 , by assuming the pairing symmetries proposed by Refs. 15 and 16. In the electron-doped case, we found that the pairing functions in Refs. 15 and 16 are qualitatively the same and the QPI spectra can be explained based on a model similar to the octet model of the cuprates. On the other hand, for hole-doped Sr 2 IrO 4 , the QPI spectra in the SC phase resemble those of the iron pnictides where the interpocket scattering between the electron and hole pockets leads to a QPI vector locating at the nesting vector of these two pockets. In both cases, the evolution of the QPI vectors and their different behaviors in the nonmagnetic and magnetic impurity scattering cases can well be explained based on the evolution of the CEC and the sign structure of the SC order parameter. The QPI spectra presented in this paper can thus be compared with future STM experiments to test whether there are SC phases in electron-and hole-doped Sr 2 IrO 4 and what the SC pairing symmetry is.