Microscopic theory of tunneling spectroscopy in Sr 2 RuO 4
Keiji Yada, Alexander A. Golubov, Yukio Tanaka, Satoshi Kashiwaya
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Journal of the Physical Society of Japan
FULL PAPERS
Microscopic theory of tunneling spectroscopy in Sr RuO Keiji Yada , Alexander A. Golubov , , Yukio Tanaka and Satoshi Kashiwaya Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Faculty of Science and Technology and MESA + Institute of Nanotechnology, University of Twente, 7500 AE,Enschede, The Netherlands Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow Region, Russia National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba 305-8568, Japan
We study the surface Andreev bound state (ABS) of superconducting Sr RuO , which is a candidate material for therealization of the chiral p -wave superconducting state. In order to clarify the role of chiral edge modes as ABSs, thesurface density of states and the tunneling conductance is calculated in the normal metal / Sr RuO junction within theframework of recursive Green’s function method, while taking into account the orbital degrees of freedom (includingSpin-Orbit interactions) with realistic material parameters. In Sr RuO , there are two bands α and β originating fromquasi-one-dimensional orbitals d yz and d zx and a two-dimensional band γ originating from d xy orbital. We discuss aboutthe contributions of various electronic bands to LDOS and the influence of atomic spin-orbit interaction (SOI). In thelight of our calculations, quasi-one-dimensional model with dominant pair potentials in α and β bands is consistent withconductance measurements in Au / Sr RuO junctions.
1. Introduction Sr RuO has attracted much interest for its unconventionalsuperconductivity appearing at T c ∼ . There havebeen several experimental reports consistent with spin-tripletpairing with broken time reversal symmetry. The mostpromising candidate for the pairing symmetry of the pair po-tential is the so-called spin-triplet chiral p -wave state, whosepair potential can be represented as ∆ ˆ z ( k x ± ik y ) in the freeelectron model. This state is the two-dimensional analog ofthe A -phase of superfluid He, and can be categorized as topo-logical superconductors.
8, 9
In topological superconductors,gapless Andreev bound states (ABSs) appear at their edgesdue to bulk-edge correspondence which are symmetricallyprotected by the bulk energy gap.
For the chiral p -wavecase, a gapless ABS with linear dispersion is generated.
9, 15
Although there have been several attempts to observe the chi-ral edge modes directly, their existence was not confirmed ex-perimentally yet.
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One of the few e ffi cient ways to detect the chiral edgemodes indirectly is tunneling conductance between normalmetal / superconducting junctions.
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For example, a sharpzero-bias conductance peak (ZBCP) has been predicted inspin-singlet d -wave superconductors by the ABS with flat dis-persion.
18, 20
Due to this flat-band dispersion, ZBCP ubiqui-tously emerges in actual experiments in high T C cuprates. On the other hand, performing tunneling spectroscopy ex-periments in Sr RuO junctions is not an easy task. Al-though Sr RuO has extremely fragile surface, tunnelingspectroscopy experiments have been performed on i) c-axissurface, ii) in-plane,
34, 35 and iii) 3K-phase.
36, 37
As for c-axis tunneling, several reliable data are obtained using scan-ning tunneling microscopy / spectroscopy (STM / S) on in-situ cleaved surfaces. The most important feature observed incommon is that conductance spectra exhibit gap feature ex-cept for zero-bias peak obtained at the vortex cores. Thegap spectra show residual conductance at the zero-bias about85% and 50%, while a fully opened gap has been de-tected in the Al tip case. The variety of gap amplitudes forthree di ff erent bands obtained in the specific heat measure-ments are not discernible for these experiments. In the caseof ab-plane tunneling spectroscopy, due to the extreme dif-ficulty in surface treatment, the number of experiments re-ported are quite limited.
34, 35
The data of point contact spec-troscopy and thin film junction formed on cleaved surfacein vacuum, shows the presence of the zero-bias conductancepeaks in common, which suggest the formation of gaplessABS at the in-plane edges. Comparing with the zero-biasconductance peaks obtained in high- T C cuprates, the dome-like peak spreading in the whole gap amplitude observed onSr RuO indicates the formation of dispersive edge states ex-pected for chiral p -wave superconductors. The shapes ofthe peak show variation depending on the junction (see Fig.1).Compared to the fragile 1.5K phase surfaces, the 3K phases ofSr RuO tends to have stable surfaces due to the inclusion ofinert surface of Ru. Conductance spectra obtained on these 3Kphase commonly exhibit a sharp narrow peak near zero-bias inaddition to the broad peak spreading in the whole gap ampli-tude
36, 37 (see curve (c) in Fig. 1). However, the origin of thistwo-step peak has not been clarified. One of the proposals forthe superconducting state in 3K-phase is the inhomogeneousgap structure near the Ru-inclusions. However, it is still anexperimentally unresolved problem. Since two-step peaks in1.5K-phase were recently observed by one of the author, webelieve that the two-step peaks do not originate from the in-
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FULL PAPERS homogeneous gap structure but the ABSs between Sr RuO and a Ru-inclusion on the surface. Fig. 1. (Color online) Conductance spectra detected using tunneling junc-tions formed between in-situ cleaved surface of Sr RuO (SRO) and Auat T ∼ . Beside the above experimental studies, a number oftheories of tunneling spectroscopy in normal metal / spin-triplet chiral p -wave superconductor junctions were formu-lated. It was shown that the line shape of a tunnel-ing conductance in chiral p -wave superconductor junctionshas a broad ZBCP (see Fig. 2) due to the ABS with lineardispersion. These theories could explain the experimental re-sults like curve (b) in Fig. 1. However, this simple picturecan not give reasonable explanation for zero bias dip likecurve (a) and two-step peak like curve (c) in Fig. 1. Whilethe origin of zero bias dip can be explained by mismatchof the Fermi surface or the anisotropy of the pair potentialin momentum space, there are no theories which elucidatethe origin of two-step peak in conductance. If the two-steppeak in conductance comes from intrinsic nature of the su-perconducting Sr RuO , then one possibility for this originis the multi-band e ff ect. It is well known that this materialhas a two-dimensional electronic structure where cylindricalFermi surfaces are formed by three bands, α , β , and γ sheets,which mainly originate from the Ru 4 d orbitals. The γ -bandis mainly composed of two-dimensional d xy -orbital while α and β -bands are mainly composed of quasi-one-dimensional d yz and d zx -orbitals. Due to this di ff erence of the orbital na-ture, it is natural to consider that the energy gap in each or- −1 0 1012 eV/ ∆∆∆∆ σσσσ Fig. 2. (Color online) Normalized conductance based on e ff ective massapproximation for high transmissivity (dashed line) and low transmissivity(solid line). bital is di ff erent. This di ff erence of the energy scale might pro-duce the two-step peak. Actually, several microscopic mecha-nisms of superconductivity in Sr RuO are proposed based onmulti-orbital model as well as single band model. Someof these theories stress the superconductivity which stemsfrom the two-dimensional γ -band, while there are theo-ries where the quasi-one-dimensional α and β -bands were ad-dressed. Still, a theory of conductance considering multi-orbital e ff ect is lacking.In taking into account the multi-orbital e ff ect on the con-ductance quantitatively, one has to work with realistic bandstructures based on microscopic considerations. A numberof theoretical studies addressed superconductivity in the d -wave or p -wave pairing in the framework of tight bindingmodel. However, this approach su ff ers from the problemof the existence of two di ff erent energy scales: transfer inte-gral and the pair potential. Usually, the magnitude of transferintegral is two to three orders of magnitude larger than that ofthe pair potential. Owing to the limitations of computationalresources, one can not obtain reliable data of conductance in afinite size system if we choose the realistic values of ∆ , since,the coherence length is large and finite-size e ff ect becomesdistinct. Since many exotic properties have been predicted inspin-triplet p -wave superconductor junctions, it is quiteurgent task to calculate SDOS and σ s in spin-triplet chiral p -wave superconductors based on a microscopic model whiletaking into account electronic structures of Sr RuO and real-istic magnitude of pair potentials.In this paper, we calculate the SDOS and σ s of Sr RuO based on the three-band model using the Green’s functionmethod for semi-infinite system. This approach is free fromthe problem of finite-size e ff ect, and therefore one can choosethe realistic magnitude of superconducting pair potential. For γ band, we assume two-dimensional chiral p -wave pair poten-tial for all the cases we have studied. For α and β bands, westudy two kinds of pair potentials: two-dimensional pair po-
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FULL PAPERS tentials and quasi-one-dimensional ones. In two-dimensionalmodel, the calculated SDOS ( σ s ) shows a zero energy (zerobias) dip. In the presence of a spin-orbit interaction (SOI),from atomic origins, a small zero energy (zero bias) peakinside dip-like structure in SDOS ( σ s ) appears for the two-dimensional model. For the quasi-one-dimensional model,where the pair potential from γ -band is dominant, the ob-tained SDOS ( σ s ) shows a zero energy (zero bias) peak inthe absence of SOI. On the other hand, this zero energy (zerobias) peak of the SDOS( σ s ) is suppressed by the SOI. Inthe case of quasi-one-dimensional model where the pair po-tentials from α and β -bands are dominant, the resulting σ s shows a two-step zero bias peak. The experimentally obtainedtwo-step structure with sharp ZBCP can be explained by thisquasi-one-dimensional model.The organization of this paper is as follows. In Sec. 2 wediscuss the general formulation of the recursive Green’s func-tion. In Sec. 3, we show the calculated results of SDOS and σ s by the recursive Green’s function. We also show Andreevbound states obtained in finite system to understand SDOSand σ s in detail. Sec. 4 provides the summary of our paper.
2. Model and Formulations
In this section, we introduce a three-band model forSr RuO and briefly review the recursive Green’s functionmethod combined with M¨obius transformation proposed byUmerski. RuO In Sr RuO , three cylindrical Fermi surface called α -, β -and γ -bands are obtained by the first principle calculationsor ARPES measurements. A tight-binding model consideringthe d xy -, d yz - and d zx -orbitals can describe these band struc-tures, H = H kin + H soi + H pair . (1)The first term expresses the kinetic energy, H kin = X k ,σ ˆ c † k σ ε yz ( k ) g ( k ) 0 g ( k ) ε zx ( k ) 00 0 ε xy ( k ) c k σ , (2)where, ˆ c k σ = ( c yz k ,σ , c zx k ,σ , c xy k , − σ ) T is the annihilation operatorswith momentum k and spin σ (- σ ) for yz - and zx -orbitals ( xy -orbital). Considering the hopping integral up to next nearestneighbor sites, the components of Eq. (2) are given by, ε xy ( k ) = − t (cos k x + cos k y ) − t cos k x cos k y − µ xy , (3) ε yz ( k ) = − t cos k x − t cos k y − µ yz , (4) ε zx ( k ) = − t cos k x − t cos k y − µ zx , (5) g ( k ) = − t sin k x sin k y . (6) The second term describes an atomic SOI which causes a mix-ture of spin and orbital, H soi = λ X k ,σ ˆ c † k σ is σ − s σ − is σ i − s σ − i ˆ c k σ , (7)where, s σ = s σ = −
1) for σ = ↑ ( σ = ↓ ). The lastterm denotes the condensation energy due to the formationof Cooper pairs. In the superconducting state of Sr RuO , themost promising candidate of the pair potential belongs to the E u irreducible representation.
7, 16
In this irreducible represen-tation, we consider the intraorbital pairing, H pair = X k ,ℓ ∆ ∗ ℓ ( k ) c ℓ k , ↑ c ℓ − k , ↓ + h . c ., (8)where, ℓ denotes the orbital index. Considering the crystalsymmetry of Sr RuO and the orbital nature, we study twokinds of pair potentials. The first case is the two-dimensionalpair potentials, ( ∆ yz ( k ) = ∆ zx ( k ) = ∆ (sin k x + i sin k y ) , ∆ xy ( k ) = ∆ (sin k x + i sin k y ) . (9)Since the d yz - and d zx -orbitals have quasi-one-dimensional na-ture, hence we choose another possibility which we call thequasi-one-dimensional pair potentials, ∆ yz ( k ) = i ∆ sin k y , ∆ zx ( k ) = ∆ sin k x , ∆ xy ( k ) = ∆ (sin k x + i sin k y ) . (10) Starting from the Hamiltonian introduced in the previoussection, we calculate local Green’s function at the surface ofsemi-infinite Sr RuO layer. For this purpose, we use recur-sive Green’s function method using M¨obius transformationproposed by Umerski. For semi-infinite layer, we considerthe clean and homogeneous system with a flat (100) surfaceat x = x . Then, we can assume that the momentum parallelto the surface, k y , is a good quantum number and the systemis one-dimensional chain for each k y . To calculate the surfaceGreen’s function at k y and complex frequency z , we first de-fine the following matrix, X = ˆ0 ˆ t − i + , i ( k y ) − ˆ t i , i + ( k y ) ( z ˆ I − ˆ h loc ( k y ))ˆ t − i + , i ( k y ) ! , (11)where, ˆ I is a unit matrix. ˆ h loc ( k y ) and ˆ t i , j ( k y ) are matrices of lo-cal term and non-local term in the Hamiltonian, respectively.The size of the Hamiltonian is 12 ×
12 including three, twoand two degrees of freedom for orbital, spin and electron-holespaces, respectively, i.e., the size of the matrix X is 24 × X by the eigenmatrix O , O − XO = λ λ . . . λ , (12)
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FULL PAPERS with | λ | < | λ | < · · · < | λ | , we obtain the surface Green’sfunction ˆ G s ( k y , z ) in the following form,ˆ G s ( k y , z ) = O O − , (13)where O and O are the submatrices of OO = O O O O ! . (14)From the poles of this surface Green’s function inside the en-ergy gap, we can evaluate the dispersion of the ABSs. We canalso evaluate the SDOS by the imaginary part of the retardedGreen’s function, ρ ( ω ) = − π Z π − π Im { Tr ′ [ ˆ G s ( k y , ω + i η )] } dk y , (15)where η is an infinitesimal imaginary part. For the trace inEq. (15), we only sum up the electronic part and drop thecontribution due to the holes. Next, we calculate the con-ductance in normal metal / superconducting Sr RuO junction.In the normal metal, we consider a two-dimensional single-band model with the energy dispersion given by ε ( k ) = − t n (cos k x + cos k y ) − t ′ n cos k x cos k y − µ n , where t n ( t ′ n ) isthe transfer integral between (next) nearest neighbor sites. Tocalculate the conductance, we first obtain the surface Green’sfunctions ˆ G s ( k y , z ) for normal metal at x = x and ˆ G s ( k y , z )for Sr RuO at x = x in the absence of the interface hop-pings, and we obtain the local and the non-local Green’s func-tions in normal metal / superconducting Sr RuO junction,ˆ G ( k y , z ) = { ˆ G s ( k y , z ) − − ˆ t ˆ G s ( k y , z )ˆ t } − , (16)ˆ G ( k y , z ) = { ˆ G s ( k y , z ) − − ˆ t ˆ G s ( k y , z )ˆ t } − , (17)ˆ G ( k y , z ) = ˆ G s ( k y , z )ˆ t ˆ G ( k y , z ) , (18)ˆ G ( k y , z ) = ˆ G s ( k y , z )ˆ t ˆ G ( k y , z ) , (19)where, ˆ t and ˆ t are the hopping matrices at the interface.From these Green’s functions, we obtain the conductance σ S by using the Kubo formula.
77, 78
Since we start from the BdGHamiltonian, which is not the eigenstate of the particle num-ber, for Sr RuO , the electric current is not conserved in-side the superconductor unless either a source term is addedor self-consistency due to proximity e ff ect is included. Toavoid this problem, in the actual calculation, we calculate theexpectation values of the current operator inside the normalmetal, where the source term is absent. Note that known sin-gle band results in the absence of SOI are reproduced in thepresent three band model if we just introduce the interfacehopping between normal metal and d xy -orbital in Sr RuO .
3. Results
In this section, we show the calculated results of the dis-persion of ABSs, SDOS and conductance. Prior to that, weexplain the choice of parameters. We have set t = RuO , wechoose t = . t = . t = .
125 and t = .
15 asadopted in the previous studies.
56, 80
The obtained Fermi sur-faces using these parameters are shown in Fig. 3, in which the Fermi momenta in k y -direction k F α , k F β and k F γ are de-fined for convenience. For the SOI λ , we adopt the value es-timated from the quasiparticle spectra of angle resolved pho-toemission spectroscopy et al . The obtained values of λ inRef. 81 is λ/ t ≃ .
40 and λ/ t ≃ .
22, which correspondto λ ≃ .
40 and 0 .
28, respectively, by using the values of t and t in the above. Though λ ≃ . λ =
0, 0 . . ff ectof the SOI. The chemical potentials are determined to set thenumber of electron to be n xy = n yz = n zx = / ∆ = . RuO .For the hopping parameters and the chemical potential in thenormal metal, we choose t n = t ′ n = .
395 and µ n = . xy -orbital without the SOI ( γ -band). In the later subsections, we present the results for fol- −1 0 1−101 k x / π k y / π αγ β (0, k F β )(0, k F γ ) ( π , k F α ) Fig. 3. (Color online) Fermi surfaces for t = t = . t = . t = . t = .
15 and λ =
0. These for λ = . lowing three cases: The first one is the two-dimensional pairpotential as shown in Eq.(9) with ∆ = . ∆ and ∆ = ∆ .This is a simple expansion of a single band model consider-ing γ -band.
48, 49, 82
The second is the quasi-one-dimensionalpair potential as shown in Eq.(10) with ∆ = . ∆ and ∆ = ∆ . This pair potential includes quasi-one-dimensionalnature of α and β -bands, while the dominant pairing is com-posed of γ -band.
53, 54, 56
Finally, the quasi-one-dimensionalpair potential is same as the second one but with ∆ = ∆ and ∆ = . ∆ . In this case, the dominant pairing comes from α and β -bands.
61, 68, 69
In this subsection, we consider the case of two-dimensionalpair potential, where the pair potential of each orbital is shownin Eq. (9). For the magnitude of the pair potential, we choose ∆ = . ∆ and ∆ = ∆ in accordance with theoretical resultsby Nomura and Yamada.
56, 80
The interface hoppings in nor-mal metal / Sr RuO junction are chosen as t xy = t yz = t zx = t xy = t yz = t zx = .
2) for high (low) transmissivity.
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FULL PAPERS −101−1 0 1−101 0 1 k y / π E / ∆ k y / π E / ∆ (a) (b)(d)(c) Fig. 4. (Color online) Energy spectrum for two-dimensional pair potentialwith ∆ = . ∆ and ∆ = ∆ for (a) λ =
0, (b) λ = .
1, (c) λ = . λ = .
3. Thin broken lines show the bulk energy gap on the Fermi surfaces.Thick solid lines show the dispersion of ABS inside the bulk energy gap.Thick dotted lines show the ABS in the continuum levels ( λ = −2 0 20123 E / ∆ ρ s ( E ) / ρ n ( ) λ =0 λ =0.3 Fig. 5. (Color online) Normalized SDOS for two-dimensional pair poten-tial with ∆ = . ∆ and ∆ = ∆ for λ = λ = . The calculated dispersion of ABSs is shown in Fig. 4.Since Sr RuO has a mirror symmetry, the Hamiltonian ofSr RuO is divided into two mirror sectors. In Fig. 4, weshow the ABSs for one of the mirror sectors. The ABSs forthe other sector is obtained by the particle-hole transforma-tion, i.e. E ( k y ) → − E ( − k y ). In the absence of SOI, there arethree ABS originating from α , β , and γ -bands as shown in
012 −1 0 1012 −1 0 1 eV/ ∆ σ s / σ n eV/ ∆ (b)(a) t xy =t yz =t zx =1 t xy =t yz =t zx =0.2 σ s / σ n (d)(c) Fig. 6. (Color online) Normalized conductance for two-dimensional pairpotential with ∆ = . ∆ and ∆ = ∆ for (a) λ =
0, (b) λ = .
1, (c) λ = . λ = .
3. Interface hoppings are chosen as t xy = t yz = t zx = t xy = t yz = t zx = . Fig. 4 (a). Dispersions of these three ABS is represented by ∆ sin k y ( ∆ sin k y ) for α and β -bands ( γ -band). Note that theABSs that originate from α and β -bands are degenerate at k F α < | k y | < k F β ; where, k F α and k F β are the Fermi momentashown in Fig. 3. This degeneracy is lifted by SOI and one ofthe ABS disappears at λ = . λ ≥ .
2. Next, we show calculated results for the SDOS andconductance in Figs. 5 and 6, respectively. Without the SOI,SDOS shows a zero energy dip surrounded by four peaks ataround E = ± ∆ and ± ∆ . The position of the peaks corre-sponds to the energy levels of the ABSs at around k y = ± . π where their slope become to be zero. These features are es-sentially the same as the single band results considering thepair potential ∆ ( k ) = ∆ (sin k x + i sin k y ). Further, σ s in lowtransmissivity has a zero bias dip and four peaks at eV = ± ∆ , ± ∆ similar to the SDOS as seen from solid line in Fig. 6(a).Though these four peaks are smeared in high transmissivity, σ s still has a dip-like structure at eV ∼
0. On the other hand,SDOS in the presence of SOI has zero bias peak since thediepersion of ABS at k F α < | k y | < k F β is close to the Fermilevels. It also have many small spikes reflecting the complexstructure of the energy dispersion of the ABS. σ s in the pres-ence of SOI has a zero bias peak for both high and low trans-missivity as seen from Fig. 6(b)-(d). In the case of low trans-
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FULL PAPERS missivity, as compared to σ s without SOI, the positions ofpeaks in σ s with SOI other than zero bias peak shift to lowerenergy and the height of the peaks becomes lower due to thesuppression of the bulk energy gap. In the case of high trans-missivity, three peaks merge and σ s shows a single ZBCP. Here, we consider the case of quasi-one-dimensional pairpotential where the pair potential of each orbital is given byEq. (10). We consider two choices for the magnitude of thepair potentials. One is the case with ∆ = . ∆ , ∆ = ∆ .This pair potential well reproduces Nomura and Yamada’sresults
56, 80 not only for the magnitude of pair potential butfor the momentum dependence of the pair potential on theFermi surface. The other case is ∆ = ∆ , ∆ = . ∆ to em-phasize the importance of the quasi-one-dimensional band assuggested by some microscopic models.
61, 68, 69 −101−1 0 1−101 0 1 k y / π E / ∆ k y / π E / ∆ (a) (b)(d)(c) Fig. 7. (Color online) Energy spectrum for quasi-one-dimensional pair po-tential with ∆ = . ∆ and ∆ = ∆ for (a) λ =
0, (b) λ = .
1, (c) λ = . λ = .
3. Thin broken lines show the bulk energy gap on the Fermisurfaces. Thick solid lines show the dispersion of ABS inside the bulk energygap. Thick dotted lines show the ABS in the continuum levels ( λ = First, we consider the case for ∆ = . ∆ and ∆ = ∆ without the SOI. Without SOI, bulk energy gap and disper-sion of ABS (7(a)) originating from γ -band are the same asin the case of two-dimensional pair potential (Fig. 4(a)) be-cause the pair potential for xy -orbital are the same. On theother hand, the dispersion of chiral edge modes originatingfrom α and β -bands are almost flat at around k y = ± π .This can be understood as follows: On the Fermi surface of α - −2 0 20123 E / ∆ ρ s ( E ) / ρ n ( ) λ =0 λ =0.3 Fig. 8. (Color online) Normalized SDOS for quasi-one-dimensional pairpotential with ∆ = . ∆ and ∆ = ∆ for λ = λ = .
012 −1 0 1012 −1 0 1 eV/ ∆ σ s / σ n eV/ ∆ (b)(a) t xy =t yz =t zx =1 t xy =t yz =t zx =0.2 σ s / σ n (d)(c) Fig. 9. (Color online) Normalized conductance for quasi-one-dimensionalpair potential with ∆ = . ∆ and ∆ = ∆ for (a) λ =
0, (b) λ = . λ = . λ = .
3. Interface hoppings are chosen as t xy = t yz = t zx = t xy = t yz = t zx = . and β -bands, where azimuthal angles measured from the cen-ter of the Fermi surfaces are from − π/ π/
4, i.e. at around k y = β -band) and ± π ( α -band), their dominant componentsof orbitals are zx -orbital, whose pair potential is proportionalto sin k x like p x -wave. It is known that the system has zeroenergy ABS on (100) surface for p x -wave pair potential dueto the π -phase shift of the pair potential between incident and
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FULL PAPERS reflected quasi-particles.
64, 84–86
This one-dimensional natureremains even in the presence of the mixture of yz and zx -orbitals. For this reason, the energy levels of the ABS near k y = ± π is nearly zero. With increasing the magnitude ofthe SOI, quasi-one-dimensional nature of the ABS disappearsdue to the coupling with xy -orbital as seen from Figs. 7(b)-(d).Due to the level repulsion between the ABSs originating from γ -band and β -bands, the group velocity of the ABS of β -bandsat k y ∼ σ s in low bias voltage (e.g. | E / ∆ | < . | eV / ∆ | < .
5, respectively) drastically change as comparedto those with two-dimensional pair potential. SDOS in the ab-sence of the SOI shows a sharp zero energy peak due to thequasi-one-dimensional nature of the ABS as seen from dashedline in Fig. 8. A similar line shape is also found in σ s in thelow transmissivity for λ up to 0.2(Fig. 9(a)-(c)). In contrast,this sharp zero energy (bias) peak in SDOS ( σ s in low trans-missivity) suppressed for λ = .
3, and it shows a small zerobias dip as shown in Fig. 8 (Fig. 9(d)). In high transmissivity, σ s shows a single zero bias peak regardless of the magnitudeof the SOI. −101−1 0 1−101 0 1 k y / π E / ∆ k y / π E / ∆ (a) (b)(d)(c) Fig. 10. (Color online) Energy spectrum for quasi-one-dimensional pairpotential with ∆ = ∆ and ∆ = . ∆ for (a) λ =
0, (b) λ = .
1, (c) λ = . λ = .
3. Thin broken lines show the bulk energy gap on theFermi surfaces. Thick solid lines show the dispersion of ABS inside the bulkenergy gap. Thick dotted lines show the ABS in the continuum levels ( λ = Next, we consider another choice of the magnitude of thepair potential; ∆ = ∆ and ∆ = . ∆ . The dispersions ofthe ABSs without the SOI shown in Fig. 10(a) are identical −2 0 2024 E / ∆ ρ s ( E ) / ρ n ( ) λ =0 λ =0.3 Fig. 11. (Color online) Normalized SDOS for quasi-one-dimensional pairpotential with ∆ = ∆ and ∆ = . ∆ for λ = λ = . eV/ ∆ σ s / σ n eV/ ∆ (b)(a) t xy =t yz =t zx =1 t xy =t yz =t zx =0.2 σ s / σ n (d)(c) Fig. 12. (Color online) Normalized conductance for quasi-one-dimensional pair potential with ∆ = ∆ and ∆ = . ∆ for (a) λ = λ = .
1, (c) λ = . λ = .
3. Interface hoppings are chosen as t xy = t yz = t zx = t xy = t yz = t zx = . with those for the previous case (Fig. 7(a)) except for the en-ergy scales of ∆ and ∆ . Thus, the SDOS and σ s show asharp zero energy peak and a ZBCP similar to the case with ∆ > ∆ as shown in Fig. 11 and Fig. 12(a), respectively. Thedispersions of the ABSs in the presence of the SOI are shownin Figs. 10(b)-(d). In comparison with the case of ∆ < ∆
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FULL PAPERS in Fig. 11, the e ff ect of the SOI on the group velocity of theABSs at around k y = ± π is small. This is because theinduced two-dimensionality for the ABS originating from α - β -bands is small since the magnitude of the two-dimensionalpair potential in xy -orbital is small. As a result, the SDOS and σ s still have zero energy peak and ZBCP up to λ = . σ s originating from γ -band under cover of strong ZBCP due to α - β -bands. Thus, σ s shows the two-step peaks regardless of the magnitude ofthe SOI as shown in solid lines in Fig. 12.
4. Discussion and Summary pair potential dominant pair line shape of conductance2D( α, β, γ ) γ Three peaksQ1D( α, β ) + γ ) γ Tiny zero bias dipQ1D( α, β ) + γ ) α, β Two-step peak k x + ik y - single ZBCPsin k x + i sin k y - Zero bias dip Table I.
Line shapes of the conductance for low transmissivity in thepresent multi-band model for λ = . In table I, we show our results of the line shapes of conduc-tance for λ = .
39, 40, 46
Here, wecompare the calculated results with experimental data shownin Fig. 1. Two-step ZBCP structure like curve (c) in Fig. 1only appears for quasi-one-dimensional pair potential with | ∆ | > | ∆ | (see Fig. 12). We shall discuss the strength of theSOI here. In Ref. 87, the band dispersion of Sr RuO has beenstudied by the first principle calculation including the SOI.The obtained value of the energy-level splitting at Γ -point be-tween α and β -bands is about 90meV. In the present model,by choosing the unit of the energy t to be 230meV, we canreproduce the energy level of the α and β -bands at Γ -point.The obtained values of the energy-level splitting are 48, 98and 152 meV for λ = .
1, 0.2 and 0.3, respectively. If wesimply see this result, λ ∼ . | ∆ | < | ∆ | as shown inFig. 9(c). However, the electron correlation which is not fullytaken into account in the local density approximation mightinduce the renormalization of the quasiparticle energy bandsand the e ff ective values of λ/ t becomes about 0 . λ/ t = . .
3, two-stepZBCP never appears for the two-dimensional pair potentialbut for the quasi-one-dimensional pair potential.As for a broad ZBCP like curve (b) in Fig. 1, a junctionwith high transmissivity shows this for all pairings consideredin the present paper since the Andreev reflection process gov-erns the conductance in the energy gap. A single ZBCP alsoappears when the size of the Fermi surface is small and / or the insulating barrier potential is high. In these cases, the contri-bution of the perpendicular injection ( k y =
0) is enhanced,where the energy levels of the ABSs are close to zero for allthe pairing considered in the present study. However, the for-mer is not likely in the present case since it is known that thesize of the Fermi surface of Gold is large.Besides, we have also confirmed that dip-like structure asshown in curve (a) in Fig. 1 is reproduced if we ignore theinterface hoppings to d yz - and d zx -orbitals, though the mecha-nism to realize this situation is unclear. Nevertheless, if onlythe γ -band contributes to the conductance, dip-like structureappears as explained in Ref. 46. Another possibility of theemergence of a zero bias dip is the c -axis tunneling, wherethere is no ABS and a conventional gap structure appears.However, the junctions which are used in the measurementsof the conductance, have been made at the position withoutthe c -axis tunneling by measuring the interface by the scan-ning ion microscopy. Therefore, the contribution of the c -axistunneling is excluded below the resolution.In this paper, we have studied ABSs, SDOS and σ s ofSr RuO based on the three band model using recursiveGreen’s function. For d xy -orbital, we assume two-dimensionalchiral p -wave pair potential ∆ (sin k x + i sin k y ) for all caseswe have studied. For d yz and d zx -orbital we have studied twokinds of pair potentials: two-dimensional pair potentials andquasi-one-dimensional ones. In the former case, we have cho-sen ∆ yz ( k ) = ∆ zx ( k ) = ∆ (sin k x + i sin k y ), and for the lat-ter, ∆ yz ( k ) = i ∆ sin k y and ∆ zx ( k ) = ∆ sin k x . For a two-dimensional model, the calculated SDOS ( σ s ) shows a zeroenergy (bias) dip without the SOI. In the presence of theSOI, small zero energy (bias) peak inside dip-like structurein SDOS( σ s ) appears. While, in the quasi-one-dimensionalmodel with | ∆ | < | ∆ | , the obtained SDOS ( σ s ) shows a zeroenergy (bias) peak. This zero energy (bias) peak in SDOS ( σ s )is suppressed by the SOI. In the case of | ∆ | > | ∆ | , the result-ing σ s shows a two-step zero bias peak with sharp ZBCP andbroad one. The last one can reasonably explain the experimen-tal data. We do not consider the roughness of the surface anddisorder. Since the pairing symmetry is influenced by thesee ff ects, it is necessary to construct more realistic theory tak-ing account of these e ff ects in the future.We gratefully acknowledge M. Sato, A. Yamakage, I. A.Devyatov, A. V. Burmistrova and S. Onari for valuable dis-cussions, and we thank A. Dutt for critical reading of themanuscript. This work was supported in part by a Grant-inAid for Scientific Research from MEXT of Japan, ”Topo-logical Quantum Phenomena” Grants No. 22103005 and No.20654030 (Y.T.), Dutch Foundation for Fundamental Re-search on Matter (FOM) and by EU-Japan program ”IRONSEA”, and Russian Ministry of Education and Science.
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