Functional renormalization group study of superconductivity in doped Sr 2 IrO 4
Yang Yang, Wan-Sheng Wang, Jin-Guo Liu, Huo Chen, Jian-Hui Dai, Qiang-Hua Wang
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Functional renormalization group study of superconductivity in doped Sr IrO Yang Yang, Wan-Sheng Wang, Jin-Guo Liu, Hua Chen, Jian-Hui Dai, and Qiang-Hua Wang National Laboratory of Solid State Microstructures & School of Physics, Nanjing University, Nanjing, 210093, China Zhejiang Institute of Modern Physics & Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Physics, Hangzhou Normal University, Hangzhou 310036, China
Using functional renormalization group we investigated possible superconductivity in dopedSr IrO . In the electron doped case, a d ∗ x − y -wave superconducting phase is found in a narrowdoping region. The pairing is driven by spin fluctuations within the single conduction band. Incontrast, for hole doping an s ∗± -wave phase is established, triggered by spin fluctuations within andacross the two conduction bands. In all cases there are comparable singlet and triplet componentsin 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 transitiontemperature. Experimental perspectives are discussed. PACS numbers: 71.10.Fd, 74.20.-z, 74.20.Rp, 71.27.+a
I. INTRODUCTION
Recently, the iridium oxide Sr IrO has been subjectto extensive investigations. In the parent compoundthe Ir atom is in the 5 d configuration. The spin-orbitalcoupling (SOC) splits the t g -manifold into filled J = 3 / J = 1 / J = 1 / and a cantedantiferromagnetic (AFM) order was found in X-ray scat-tering and neutron diffraction measurements. In anal-ogy to cuprates, an intriguing issue is whether supercon-ductivity (SC) could be realized by doping the parentinsulator.
Theoretically, a variational Monte Carlo (VMC) studyof Sr IrO suggests d -wave SC may appear but onlywithin a narrow region of electron doping. The absenceof SC in the hole doped side is not straightforward to un-derstand. In fact, by sufficient hole doping, both of thetwo higher bands are cut by the Fermi level (see Fig.1),forming Fermi pockets around the Γ and M points in theBrillouine zone. (In this case the band structure ques-tions the notion of doped Mott insulator for Sr IrO .) In-stead, the Fermi surface topology is closely similar to thatin iron pnictides, where inter-pocket scattering proves tobe very efficient to drive s ± -wave superconductivity. However, this does not seem to be the case in the VMCresults. Given the unavoidable bias in VMC, we thinkit beneficial to perform a complementary, yet unbiasedsearch for SC in doped Sr IrO .In this paper we resort to functional renormalizationgroup (FRG). This is because FRG treats all electronicinstabilities on equal footing without a priori assumptionof the candidate order parameters. It proves success-ful in doped cuprates and iron pnictides.
We limitourselves to sufficient electron/hole doping so that FRGhas a better chance to be reliable, as in the practice for −1.5−0.50.5−1.0 E ( e V ) (a) (b)DOS (a.u.)X ΓΓ MA BDC
FIG. 1: (Color online) (a) The Electronic structure of Sr IrO described by H = H Kin + H SOC , with H Kin from Ref. .Each band remains to be doubly degenerated. The horizontallines indicate Fermi levels addressed in the text. The Fermienergy of the undoped compound is set to zero (line-B). (b)Normal state density of states. doped cuprates. Since the three bands overlap withinan energy window of order 1eV, as seen in Fig.1, we in-clude all of the t g orbitals, and apply the recently de-veloped singular-mode functional renormalization group(SMFRG). Compared to the other FRG schemes, ithas the additional advantage to deal with orbital andspin degrees of freedom and the SOC among them in amore straightforward manner.Our main findings are as follows: In the electrondoped case, a d ∗ x − y -wave superconducting phase isfound in a narrow doping region close to the van Hovesingularity, in agreement to VMC. The pairing is drivenby spin-like fluctuations within the single conductionband. In contrast, for hole doping an s ∗± -wave phase isestablished, triggered by spin fluctuations within andacross the two conduction bands. In all cases thereare comparable singlet and triplet components in thepairing 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 proposethat hole doping is more promising to achieve a highertransition temperature. Experimental perspectives arediscussed. II. MODEL AND METHOD
We begin with specification of the model hamilto-nian H . The free part H of H contains the spin-invariant kinetic part, H Kin , and an atomic SOC part, H SOC = − λ P j ψ † j L · σψ j , where ψ j is the annihilationfield operator at site j , and L and σ/ L = ( L x , L y , L z ) in the orbitalbasis ( d xz , d yz , d xy ) are, L x = − L x = L y = − L y = L z = − L z = i. (1)We take H Kin suggested in Refs. , where the effect oflattice distortions has been taken into account. ForSOC we set λ = 0 . H = H Kin + H SOC areshown in Fig.1(a) and (b), respectively. (Notice that eachband remains two-fold degenerate.) The horizontal line-B corresponds to the undoped Fermi level, and the otherlines to the doped cases to be addressed specifically later.The interacting part H I of H contains intra-orbitalrepulsion U , inter-orbital repulsion U ′ , Hund’s rule spinexchange J and pair hopping J ′ . The explicit form of H I is standard and can be found elsewhere. We apply theKanamori relations U = U ′ + 2 J and J = J ′ to reducethe number of independent parameters. According to anestimate by constrained random phase approximation, we limit ourselves in the parameter ranges U = 2 ∼ J/U = 0 . ∼ . V α,β ; γ,δ k , k ′ , q → X m S m ( q ) φ α,βm ( k , q )[ φ γ,δm ( k ′ , q )] ∗ , (2)either in the particle-particle (p-p) or particle-hole (p-h)channel. Here, ( α, β, γ, δ ) are dummy labels for orbitaland spin indices, q is the collective momentum, and k (or k ′ ) is an internal momentum of the Fermionbilinears ψ † k + q ,α ψ †− k ,β and ψ † k + q ,α ψ k ,β in the p-p andp-h channels, respectively. The fastest growing eigen-value S ( Q ) implies an emerging order associated witha collective wave vector Q and eigenfunction (or formfactor) φ ( k , Q ). In the p-p channel Q = 0 is alwaysrealized at low energy scale due to the Cooper mecha-nism. More technical details can be found elsewhere. FIG. 2: (Color online) Results for n = 5 .
20. (a) Fermi surfaceand gap function ∆( k ) (color scale). (b) FRG flow of 1 /S ph,pp ,the inverse of the leading attractive interactions, versus therunning energy scale Λ. Notice that 1 /S pp,ph → − if S pp,ph diverges. The arrows indicate snapshots of the leading mo-mentum Q (divided by π ) in the p-h channel. The inset showsln | S ph ( q ) | in the Brillouine zone at the final energy scale. III. ELECTRON DOPING
We first discuss the electron doped case with the bandfilling n = 5 .
20, corresponding to line-A in Fig.1. TheFermi surface is contributed by the upper band alone, asshown in Fig.2(a), but we should emphasize that our SM-FRG includes virtual excitations from all bands. Fig.2(b)shows the FRG flow of the leading eigenvalues S pp,ph ver-sus the running energy scale Λ (the infrared cutoff of theMatsubara frequency) for U = 2 . J/U = 0 . S ph is close to Q = ( π, π ). Theinset shows S ph ( q ) versus q at the final energy scale.There is a broad peak around Q . We checked that theassociated form factors describes site-local spins alignedin the plane. Thus AFM spin fluctuations with easy-plane anisotropy exist. The enhancement of such spinfluctuations can be ascribed to the quasi-nesting of theFermi surface shown in Fig.2(a) and the proximity tothe van Hove singularity near X (see Fig.1). The easy-plane anisotropy is from SOC, and appears to be con-sistent with the easy-plane AFM order in the parentcompound, although FRG can not access the Mottlimit.From Fig.2(b), As S ph is enhanced below Λ = 0 . S pp to increase and eventually diverge. Thereforethe driving force of pairing here is the AFM spin fluctua-tion discussed above. We write the (matrix) pairing formfactor as φ pp ( k ) = ( g k + γ k ) iσ , (3)with singlet and triplet parts g k and γ k , respectively. Todescribe the momentum dependence, we introduce thelattice harmonics c x = cos k x , c y = cos k y . (4)The non-vanishing elements of g k and γ k in the orbitalbasis are, g / k ∼ ( ∓ . ± . c y/x ∓ . c x/y ) σ ,g k ∼ . c y − c x ) σ , (5) γ k ∼ (0 . c x − . c y ) L x σ + (0 . c x − . c y ) L y σ +0 . c x − c y ) L z σ . (6)Combining the transformation property of the d -orbitals, we see g k transforms as d x − y . The symme-try is consistent with the fact that spin fluctuations atthe wave vector Q = ( π, π ) overlap with the d x − y -wavesinglet pairing interaction in square lattices. The tripletparts mainly arise from nearest-neighbor bonds, and areorbital-singlets (i.e., odd in orbital space). We noticethat γ k is comparable to g k , and is a result of signifi-cant SOC. Under point group operations of spin, orbitaland momentum, γ k also transforms as d x − y . Accord-ing to Ref. we dub the symmetry of the total pairingfunction as d ∗ x − y . The pairing function respects time-reversal symmetry, which would have been anticipatedsince the d -wave representations on square lattices arenon-degenerate. We project the pairing function in theband basis as∆ k = h k | φ pp ( k )( | − k i ) ∗ = h k | g k + γ k | k i , (7)where | k i is a Bloch state and | − k i = iσ K | k i is thetime-reversal of | k i . The gap function ∆ k is shown inFig.2(a) (color scale) on one of the doubly degeneratedFermi surfaces, revealing the d -wave sign structure con-sistent with the above symmetry analysis in the spin-orbital basis. We notice that the gap function doesn’tchange between the degenerate Fermi surfaces. This isbecause any band dependence is determined by h k | γ k | k i ,but γ k is of the same form of SOC, which nonethelessdoes not break the degeneracy. We notice in passingthat the pairing function in the orbital basis in this pa-per would be useful in further VMC studies. IV. HOLE DOPING
We now discuss the hole doped cases. First considera band filling n = 4 .
83 associated with line-C in Fig.1.The Fermi surface topology changes drastically. A largeΓ-pocket from the upper band and a small M -pocketfrom the middle band appear, as shown in Fig.3(a).For reasons to be clearer later, we set U = 2 . J/U = 0 . Q vec-tor for the leading S ph evolves from ( π, π ) at high en-ergy scales to small momenta at moderate and low en-ergy scales. The inset shows S ph ( q ) versus q at the finalenergy scale. Incommensurate peaks around the zonecenter are obvious. The fact that they are stronger atlow energy scales suggests that they arise from intra-pocket scattering around M . We checked that suchfluctuations are also spin-like, but now the fluctuating FIG. 3: (Color online) The same plot as Fig.2, except that n = 4 . spins are aligned along the out-of-plane directions. Thushole doping leads to ferromagnetic-like spin fluctuationswith easy-axis anisotropy. The difference to the electrondoped case can be easily checked, e.g., by neutron scat-tering. On the other hand, there are secondary peaks at Q ′ ∼ ( π, π/
4) and its symmetry images in S ph ( q ). Theyare 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 en-ergy window, attractive pairing interaction S pp is inducedrapidly, and eventually diverge. At this stage, we find thefollowing non-vanishing elements for φ pp ( k ), g / k ∼ (0 . c x/y + 0 . c y/x ) σ ,g k ∼ [0 . − . c x + c y )] σ , (8) γ k ∼ . L x σ + L y σ )+[0 . − . c x + c y )] L z σ . (9)Symmetry analysis similar to the previous case showsthat the gap function transforms as s -wave. The singletand triplet parts are comparable in magnitude, and bothare time-reversal invariant. The projection of φ pp ( k ), or∆( k ), is shown in Fig.3(a) (color scale). We see that ∆( k )is roughly isotropic on each pocket, but changes sign fromΓ to M pocket. Combined with the admixture of singletsand triplets in the orbital basis, we dub the global pair-ing symmetry as s ∗± -wave. For the singlet part the signchange across the pockets enjoys the scattering providedby the secondary spin fluctuations near Q ′ mentionedabove. We conclude that pairing is driven by spin fluctu-ations within the hole-like band, and further enhanced bythe inter-pocket scattering in the two conduction bands.The reason that the inter-pocket scattering is not lead-ing is because the electron and hole pockets are poorlynested.We find the above picture also applies for higher lev-els of hole doping, except that the wavevector Q of theleading spin fluctuations becomes larger (since the holepocket is enlarged), and Q ′ for the sub-leading ones be-comes closer to ( π, π ) (since the quasi-nesting betweenthe pockets is improved). Instead of repeating the dis-cussions, we provide the pairing function for n = 4 .
25 (in
U=2.2 eVU=2.4 eVU=2.8 eV Λ c ( − e V ) J/U
U=2.2 eVU=2.4 eVU=2.8 eV (b)(a)
FIG. 4: (Color online) The superconducting critical scale Λ c versus J/U for various U . (a) The d ∗ x − y -wave pairing atelectron doping n = 5 .
20. (b) The s ∗± -wave pairing at holedoping n = 4 . view of potential application in VMC), associated withline-D in Fig.1, g / k ∼ [ − . − . c x + c y )] σ ,g k ∼ [ − . − . c x + c y )] σ , (10) γ k ∼ (0 . − . c x − . c y ) L x σ +(0 . − . c x − . c y ) L y σ +[0 .
19 + 0 . c x + c y )] L z σ . (11)obtained under the same parameters U and J as above.The pairing symmetry remains to be s ∗± -wave. We no-tice that at this level of hole doping, the hole pocket isquasi-nested, and this leads to stronger intra-pocket spinfluctuations and hence stronger SC (see below). V. SYSTEMATICS
We have performed systematic calculations by varyingthe bare interaction parameters. Fig.4 shows the crit-ical scale Λ c , the energy scale at which the supercon-ducting instability occurs, versus J/U for various valuesof U . For a fixed J/U , Λ c increases with U . The ef-fect of J for a fixed U is highly nontrivial, however. Inthe electron doped case, Fig.4(a) shows that the Hund’scoupling J suppresses Λ c for d ∗ x − y -wave pairing in theelectron doped case. In the contrary, in the hole dopedcase s ∗± -wave pairing is enhanced by J , as shown inFig.4(b). The systematics is consistent with the factthat the Hund’s rule coupling favors spin fluctuationsat smaller wavevectors. Judging from Fig.4 we concludethat hole doping is more promising to achieve a highertransition temperature for a reasonable Hund’s rule cou-pling (e.g., J/U ≥ . n . Fig.5 showsthe n dependence of Λ c . The grayed region is not consid-ered since it is too close to the Mott insulating limit forFRG to be reliable. We are interested in sufficient elec-tron/hole doing away from this region. We set U = 2 . J −2 −1 n Λ c ( m e V ) s * ± d *x −y FIG. 5: (Color online) The superconducting critical scale Λ c versus doping. Here U = 2 . J/U = 0 .
055 (0 . to have a fair comparison between electron and hole dop-ing. However, since J is badly unfavorable in the electrondoped case ( n > J/U = 0 .
055 just in orderto have a sizable Λ c . Even in this case, SC exists onlywithin a narrow doping region around n = 5 . n < J/U = 0 .
175 for definiteness. We see theSC phase extends for all n ≤ .
83, and Λ c is enhancedup to Λ c ∼ n = 4 .
25. This pairing scale is ofthe same order of that in iron pnictides, and we concludethat the deeply hole-doped Sr IrO could be a high- T c superconductor. VI. EXPERIMENTAL PERSPECTIVES
We discuss some experimental consequences regard-ing the pairing functions obtained so far. Since the d ∗ x − y -wave pairing has a nodal gap on the Fermi sur-face, while the s ∗± -wave pairing is fully gapped, they canbe easily differentiated by low temperature thermody-namic measurements (such as the specific heat and super-fluid density) and by spectroscopic measurements (suchas angle-resolved photoemission and scanning tunnelingmicroscopy). The change of spin anisotropy can be eas-ily probed by neutron diffraction. However, since bothtypes of pairing involve comparable mixing of singletsand triplets, the difference in the spin susceptibility is notas straightforward. We performed mean field calculationsin both cases, with the pairing interaction derived fromSMFRG (slightly before the divergence scale), and calcu-lated the direction-resolved spin susceptibilities χ xx,yy,zz versus temperature T . The results are shown in Fig.6(a)for d ∗ x − y - and (b) for s ∗± -wave pairing for n = 5 .
20 and n = 4 .
83, respectively. In both cases the susceptibili-ties are above 40% of the normal state value as T → χ xx , χ yy ) versus χ zz .Such behaviors, combined with the spectroscopic mea-surements, would provide an unambiguous probe of the c χ ( a r b . un it . ) c χ xx,yy χ zz χ xx,yy χ zz (a) (b) FIG. 6: (Color online) Spin susceptibilities χ xx,yy,zz as a func-tion of temperature for (a) d ∗ x − y -wave pairing in electrondoped case n = 5 .
20, and (b) s ∗± -wave pairing in the holedoped case n = 4 .
83. Here T c is the mean field transitiontemperature. novel pairing functions predicted here. VII. CONCLUSIONS AND REMARKS
To conclude, in electron (or hole) doped Sr IrO , a d ∗ x − y -wave (or s ∗± -wave) superconducting phase is pos-sible. They are triggered by in-plane AFM spin fluc- tuations for electron doping, and by out-of-plane spinfluctuations within the hole pocket as well as from inter-pocket scattering for hole doping. In all cases there arecomparable singlet and triplet components. The effect ofHund’s rule coupling J suppresses (enhances) SC in theelectron (hole) doped region significantly. A reasonablevalue of J/U ≥ . Whilefurther efforts are needed, our results for hole dopingstimulate a new direction.
Experimentally, hole dopingcan be achieved by substituting K or Na for Sr in Sr IrO .Presently isovalent substitution of Ca or Ba for Sr, andpartial substitution of Ru for Ir are reported. Acknowledgments
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