Functional renormalization group study of superconductivity in doped Sr{}_{2}IrO{}_{4}

Functional renormalization group study of superconductivity in doped SrIrO

Yang Yang National Laboratory of Solid State Microstructures School of Physics, Nanjing University, Nanjing, 210093, China    Wan-Sheng Wang National Laboratory of Solid State Microstructures School of Physics, Nanjing University, Nanjing, 210093, China    Jin-Guo Liu National Laboratory of Solid State Microstructures School of Physics, Nanjing University, Nanjing, 210093, China    Hua Chen Zhejiang Institute of Modern Physics Department of Physics, Zhejiang University, Hangzhou 310027, China    Jian-Hui Dai Department of Physics, Hangzhou Normal University, Hangzhou 310036, China    Qiang-Hua Wang National Laboratory of Solid State Microstructures School of Physics, Nanjing University, Nanjing, 210093, China
Abstract

Using functional renormalization group we investigated possible superconductivity in doped SrIrO. In the electron doped case, a -wave superconducting phase is found in a narrow doping region. The pairing is driven by spin fluctuations within the single conduction band. In contrast, for hole doping an -wave phase is established, triggered by spin fluctuations within and across the two conduction bands. In all cases there are comparable singlet and triplet components in the pairing function. The Hund’s rule coupling reduces (enhances) superconductivity for electron (hole) doping. Our results imply that hole doping is more promising to achieve a higher transition temperature. Experimental perspectives are discussed.

pacs:
71.10.Fd, 74.20.-z, 74.20.Rp, 71.27.+a

I Introduction

Recently, the iridium oxide SrIrO has been subject to extensive investigations. Randall ; Crawford ; Cao ; BJKim-1 ; BJKim-2 ; SJMoon ; xray1 ; xray2 ; xray3 ; Cetin ; neutron In the parent compound the Ir atom is in the configuration. The spin-orbital coupling (SOC) splits the -manifold into filled multiplets and half-filled doublets, leading to a band structure as shown in Fig.1. Since the top band is half-filled and the width is narrowed down to the scale of local interactions, the parent compound was argued to be a Mott insulator. Indeed, transport measurements revealed insulating behavior,Cao and a canted antiferromagnetic (AFM) order was found in X-ray scattering and neutron diffraction measurements.BJKim-2 ; xray1 ; xray2 ; xray3 ; neutron In analogy to cuprates, an intriguing issue is whether superconductivity (SC) could be realized by doping the parent insulator.xray2 ; wangfa

Theoretically, a variational Monte Carlo (VMC) study of SrIrOvmc-2 suggests -wave SC may appear but only within a narrow region of electron doping. The absence of SC in the hole doped side is not straightforward to understand. In fact, by sufficient hole doping, both of the two higher bands are cut by the Fermi level (see Fig.1), forming Fermi pockets around the and points in the Brillouine zone. (In this case the band structure questions the notion of doped Mott insulator for SrIrO.) Instead, the Fermi surface topology is closely similar to that in iron pnictides, where inter-pocket scattering proves to be very efficient to drive -wave superconductivity.iron-SC ; s+- However, this does not seem to be the case in the VMC results. Given the unavoidable bias in VMC, we think it beneficial to perform a complementary, yet unbiased search for SC in doped SrIrO.

Figure 1: (Color online) (a) The Electronic structure of SrIrO described by , with from Ref.vmc-1 ; vmc-2 . Each band remains to be doubly degenerated. The horizontal lines indicate Fermi levels addressed in the text. The Fermi energy of the undoped compound is set to zero (line-B). (b) Normal state density of states.

In this paper we resort to functional renormalization group (FRG).wetterich This is because FRG treats all electronic instabilities on equal footing without a priori assumption of the candidate order parameters. It proves successful in doped cuprates and iron pnictides.Honerkamp ; frg We limit ourselves to sufficient electron/hole doping so that FRG has a better chance to be reliable, as in the practice for doped cuprates.CuO Since the three bands overlap within an energy window of order eV, as seen in Fig.1, we include all of the orbitals, and apply the recently developed singular-mode functional renormalization group (SMFRG). wws1 ; xyy1 ; xyy2 ; wws2 ; xyy3 ; yy ; sro ; wws3 Compared to the other FRG schemes, it has the additional advantage to deal with orbital and spin degrees of freedom and the SOC among them in a more straightforward manner.

Our main findings are as follows: In the electron doped case, a -wave superconducting phase is found in a narrow doping region close to the van Hove singularity, in agreement to VMC. The pairing is driven by spin-like fluctuations within the single conduction band. In contrast, for hole doping an -wave phase is established, triggered by spin fluctuations within and across the two conduction bands. In all cases there are comparable singlet and triplet components in the pairing function. The Hund’s rule coupling reduces (enhances) superconductivity in the electron (hole) doped case. In view of reasonable Hund’s rule coupling, the doping range and the pairing scale, we propose that hole doping is more promising to achieve a higher transition temperature. Experimental perspectives are discussed.

Ii Model and method

We begin with specification of the model hamiltonian . The free part of contains the spin-invariant kinetic part, , and an atomic SOC part, , where is the annihilation field operator at site , and and are the operators for the orbital and spin angular momenta. To be specific, the nonzero elements of in the orbital basis are,

(1)

We take suggested in Refs.vmc-2 ; vmc-1 , where the effect of lattice distortions Crawford ; wangfa has been taken into account. For SOC we set eV. The corresponding band structure and density of states (DOS) for are shown in Fig.1(a) and (b), respectively. (Notice that each band remains two-fold degenerate.) The horizontal line-B corresponds to the undoped Fermi level, and the other lines to the doped cases to be addressed specifically later.

The interacting part of contains intra-orbital repulsion , inter-orbital repulsion , Hund’s rule spin exchange and pair hopping . The explicit form of is standard and can be found elsewhere.yy We apply the Kanamori relations and to reduce the number of independent parameters. According to an estimate by constrained random phase approximation,crpa we limit ourselves in the parameter ranges eV and .

The bare interactions, upon full anti-symmetrization, provide the initial values of the running interaction vertices (versus a decreasing energy scale) in SMFRG. A general interaction vertex function can be decomposed as

(2)

either in the particle-particle (p-p) or particle-hole (p-h) channel. Here, are dummy labels for orbital and spin indices, is the collective momentum, and (or ) is an internal momentum of the Fermion bilinears and in the p-p and p-h channels, respectively. The fastest growing eigenvalue implies an emerging order associated with a collective wave vector and eigenfunction (or form factor) .note In the p-p channel is always realized at low energy scale due to the Cooper mechanism. More technical details can be found elsewhere.xyy1 ; wws1

Figure 2: (Color online) Results for . (a) Fermi surface and gap function (color scale). (b) FRG flow of , the inverse of the leading attractive interactions, versus the running energy scale . Notice that if diverges. The arrows indicate snapshots of the leading momentum (divided by ) in the p-h channel. The inset shows in the Brillouine zone at the final energy scale.

Iii Electron doping

We first discuss the electron doped case with the band filling , corresponding to line-A in Fig.1. The Fermi surface is contributed by the upper band alone, as shown in Fig.2(a), but we should emphasize that our SMFRG includes virtual excitations from all bands. Fig.2(b) shows the FRG flow of the leading eigenvalues versus the running energy scale (the infrared cutoff of the Matsubara frequency) for eV and . Apart from some intermediate deviations the momentum associated with is close to . The inset shows versus at the final energy scale. There is a broad peak around . We checked that the associated form factors describes site-local spins aligned in the plane. Thus AFM spin fluctuations with easy-plane anisotropy exist. The enhancement of such spin fluctuations can be ascribed to the quasi-nesting of the Fermi surface shown in Fig.2(a) and the proximity to the van Hove singularity near (see Fig.1). The easy-plane anisotropy is from SOC, and appears to be consistent with the easy-plane AFM order in the parent compound,BJKim-2 ; xray1 ; xray2 ; xray3 ; neutron although FRG can not access the Mott limit.

From Fig.2(b), As is enhanced below eV, it triggers to increase and eventually diverge. Therefore the driving force of pairing here is the AFM spin fluctuation discussed above. We write the (matrix) pairing form factor as

(3)

with singlet and triplet parts and , respectively. To describe the momentum dependence, we introduce the lattice harmonics

(4)

The non-vanishing elements of and in the orbital basis are,

(5)
(6)

Combining the transformation property of the -orbitals,wy we see transforms as . The symmetry is consistent with the fact that spin fluctuations at the wave vector overlap with the -wave singlet pairing interaction in square lattices. The triplet parts mainly arise from nearest-neighbor bonds, and are orbital-singlets (i.e., odd in orbital space). We notice that is comparable to , and is a result of significant SOC. Under point group operations of spin, orbital and momentum, also transforms as . According to Ref.yy we dub the symmetry of the total pairing function as . The pairing function respects time-reversal symmetry, which would have been anticipated since the -wave representations on square lattices are non-degenerate. We project the pairing function in the band basis as

(7)

where is a Bloch state and is the time-reversal of . The gap function is shown in Fig.2(a) (color scale) on one of the doubly degenerated Fermi surfaces, revealing the -wave sign structure consistent with the above symmetry analysis in the spin-orbital basis. We notice that the gap function doesn’t change between the degenerate Fermi surfaces. This is because any band dependence is determined by , but is of the same form of SOC, which nonetheless does not break the degeneracy. We notice in passing that the pairing function in the orbital basis in this paper would be useful in further VMC studies.

Iv Hole doping

We now discuss the hole doped cases. First consider a band filling associated with line-C in Fig.1. The Fermi surface topology changes drastically. A large -pocket from the upper band and a small -pocket from the middle band appear, as shown in Fig.3(a). For reasons to be clearer later, we set eV and , with a larger Hund’s rule coupling. The FRG flow is shown in Fig.3(b). In this case, the vector for the leading evolves from at high energy scales to small momenta at moderate and low energy scales. The inset shows versus at the final energy scale. Incommensurate peaks around the zone center are obvious. The fact that they are stronger at low energy scales suggests that they arise from intra-pocket scattering around . We checked that such fluctuations are also spin-like, but now the fluctuating spins are aligned along the out-of-plane directions. Thus hole doping leads to ferromagnetic-like spin fluctuations with easy-axis anisotropy. The difference to the electron doped case can be easily checked, e.g., by neutron scattering. On the other hand, there are secondary peaks at and its symmetry images in . They are also spin-like by checking the associated form factors. These spin fluctuations can only come from inter-pocket (thus inter-band in our case) scattering. From Fig.3(b), as spin fluctuations are enhanced in the intermediate energy window, attractive pairing interaction is induced rapidly, and eventually diverge. At this stage, we find the following non-vanishing elements for ,

(8)
(9)

Symmetry analysis similar to the previous case shows that the gap function transforms as -wave.wy The singlet and triplet parts are comparable in magnitude, and both are time-reversal invariant. The projection of , or , is shown in Fig.3(a) (color scale). We see that is roughly isotropic on each pocket, but changes sign from to pocket. Combined with the admixture of singlets and triplets in the orbital basis, we dub the global pairing symmetry as -wave.spm For the singlet part the sign change across the pockets enjoys the scattering provided by the secondary spin fluctuations near mentioned above. We conclude that pairing is driven by spin fluctuations within the hole-like band, and further enhanced by the inter-pocket scattering in the two conduction bands. The reason that the inter-pocket scattering is not leading is because the electron and hole pockets are poorly nested.

Figure 3: (Color online) The same plot as Fig.2, except that .

We find the above picture also applies for higher levels of hole doping, except that the wavevector of the leading spin fluctuations becomes larger (since the hole pocket is enlarged), and for the sub-leading ones becomes closer to (since the quasi-nesting between the pockets is improved). Instead of repeating the discussions, we provide the pairing function for (in view of potential application in VMC), associated with line-D in Fig.1,

(10)
(11)

obtained under the same parameters and as above. The pairing symmetry remains to be -wave. We notice that at this level of hole doping, the hole pocket is quasi-nested, and this leads to stronger intra-pocket spin fluctuations and hence stronger SC (see below).

Figure 4: (Color online) The superconducting critical scale versus for various . (a) The -wave pairing at electron doping . (b) The -wave pairing at hole doping .

V Systematics

We have performed systematic calculations by varying the bare interaction parameters. Fig.4 shows the critical scale , the energy scale at which the superconducting instability occurs, versus for various values of . For a fixed , increases with . The effect of for a fixed is highly nontrivial, however. In the electron doped case, Fig.4(a) shows that the Hund’s coupling suppresses for -wave pairing in the electron doped case. In the contrary, in the hole doped case -wave pairing is enhanced by , as shown in Fig.4(b). The systematics is consistent with the fact that the Hund’s rule coupling favors spin fluctuations at smaller wavevectors. Judging from Fig.4 we conclude that hole doping is more promising to achieve a higher transition temperature for a reasonable Hund’s rule coupling (e.g., .)

On the other hand, we have also performed systematic calculations by varying the filling level . Fig.5 shows the dependence of . The grayed region is not considered since it is too close to the Mott insulating limit for FRG to be reliable. We are interested in sufficient electron/hole doing away from this region. We set eV here for illustration. In principle we also need to fix to have a fair comparison between electron and hole doping. However, since is badly unfavorable in the electron doped case (), we set just in order to have a sizable . Even in this case, SC exists only within a narrow doping region around (close to the van Hove filling), in agreement to the VMC result. Instead, in the hole doped case (), we set a reasonable value for definiteness. We see the SC phase extends for all , and is enhanced up to meV for . This pairing scale is of the same order of that in iron pnictides, and we conclude that the deeply hole-doped SrIrO could be a high- superconductor.

Figure 5: (Color online) The superconducting critical scale versus doping. Here eV, and () for electron (hole) doping.

Vi Experimental perspectives

We discuss some experimental consequences regarding the pairing functions obtained so far. Since the -wave pairing has a nodal gap on the Fermi surface, while the -wave pairing is fully gapped, they can be easily differentiated by low temperature thermodynamic measurements (such as the specific heat and superfluid density) and by spectroscopic measurements (such as angle-resolved photoemission and scanning tunneling microscopy). The change of spin anisotropy can be easily probed by neutron diffraction. However, since both types of pairing involve comparable mixing of singlets and triplets, the difference in the spin susceptibility is not as straightforward. We performed mean field calculations in both cases, with the pairing interaction derived from SMFRG (slightly before the divergence scale), and calculated the direction-resolved spin susceptibilities versus temperature . The results are shown in Fig.6(a) for - and (b) for -wave pairing for and , respectively. In both cases the susceptibilities are above of the normal state value as , and there are anisotropy between versus . Such behaviors, combined with the spectroscopic measurements, would provide an unambiguous probe of the novel pairing functions predicted here.

Figure 6: (Color online) Spin susceptibilities as a function of temperature for (a) -wave pairing in electron doped case , and (b) -wave pairing in the hole doped case . Here is the mean field transition temperature.

Vii Conclusions and remarks

To conclude, in electron (or hole) doped SrIrO, a -wave (or -wave) superconducting phase is possible. They are triggered by in-plane AFM spin fluctuations for electron doping, and by out-of-plane spin fluctuations within the hole pocket as well as from inter-pocket scattering for hole doping.s2d In all cases there are comparable singlet and triplet components. The effect of Hund’s rule coupling suppresses (enhances) SC in the electron (hole) doped region significantly. A reasonable value of makes hole doping more promising to achieve a higher transition temperature. Experimental perspectives are discussed.

We notice that superconductivity has not been observed yet experimentally by electron doping.Korneta While further efforts are needed, our results for hole doping stimulate a new direction. Experimentally, hole doping can be achieved by substituting K or Na for Sr in SrIrO. Presently isovalent substitution of Ca or Ba for Sr,Shimura and partial substitution of Ru for Ir are reported.Ru4Ir

Acknowledgements.
The project was supported by NSFC (under grant No.11023002 and No.11274084) and the Ministry of Science and Technology of China (under grant No.2011CBA00108 and 2011CB922101).

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