Surface-angle dependence of the tunneling spectroscopy in iron-based superconductors: sign-reversing s-wave scenarios
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Surface-angle dependence of the tunneling spectroscopy in iron-basedsuperconductors: sign-reversing s -wave scenarios Yuki Nagai a,b , Nobuhiko Hayashi c,d , Masahiko Machida b,d,e a Department of Physics, University of Tokyo, Tokyo 113-0033, Japan b JST, TRIP, Chiyoda, Tokyo, 102-0075, Japan c Nanoscience and Nanotechnology Research Center (N2RC), Osaka Prefecture University, 1-2 Gakuen-cho, Sakai 599-8570, Japan d CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan e CCSE, Japan, Atomic Energy Agency, 6-9-3 Higashi-Ueno, Tokyo 110-0015, Japan
Abstract
We discuss the surface Andreev bound states in Fe-based superconductors with the use of an e ff ective five-bandmodel and investigate the surface-angle dependence of the tunneling spectroscopy by a quasiclassical approach foran isotropic and an anisotropic ± s -wave gap superconductivity. We show that information on the normal state isimportant for the Andreev bound state and its peak positions do not depend on the gap amplitude anisotropy. Key words:
Iron-pnictides, Andreev bound states, tunneling spectroscopy, theory
PACS: ± s -wave pair-ing symmetry has been theoretically proposed as one ofthe candidates for the pairing symmetry in Fe-based su-perconductors. The ± s -wave symmetry means that thesymmetry of pair potentials on each Fermi surface is s -wave and the relative phase between them is π [2, 3, 4].The Fe-based superconductors are known to be multi-band systems and have multiple Fermi surfaces.It is important for the identification of the ± s -wavesymmetry to detect the sign change in the pair po-tentials between Fermi surfaces. As demonstrated instudies of high- T c cuprates, Andreev bound states areformed at a surface or a junction when the quasiparti-cles feel di ff erent signs of the pair potential before andafter scattering [5]. Motivated by the expectation thatone can extract the information on the relative phasethrough such Andreev bound states, several theoreticalstudies on junctions and surfaces have been reported re-cently [6]. Andreev bound states at zero energy havebeen experimentally observed as a zero-bias conduc-tance peak (ZBCP) in tunneling spectroscopy for Fe-based superconductors [7].In this paper, to investigate Andreev bound states wecalculate the local density of states (LDOS) at a spec- ular surface with the use of the extended Matsumoto-Shiba method for n -band superconductors [8, 9]. Wediscuss the surface-angle dependence of the LDOS withthe e ff ective five-band model by Kuroki et al. [3] and ± s -wave pairing symmetry.We consider the surface situated at x = U ( r ) written as ˇ U ( r ) = U δ ( x )ˇ τ . Here, ˇ τ i ( i = , ,
3) denote Pauli ma-trices in Nambu space, r is the position in the realspace and we take the x ( y )-axis perpendicular (paral-lel) to the surface. The surface is actually representedin the limit U → ∞ . We use units in which ~ = G R ( x , x ′ , k y ) in the presentsystem is obtained as ˇ G R ( x , x ′ , k y ) = ˇ G R ( x , x ′ , k y ) − ˇ G R ( x , , k y )[ ˇ G R (0 , , k y )] − ˇ G R (0 , x ′ , k y ). Assuming thatintra-band pairings are dominant, ˇ G R can be divided intoa sum of the Green functions defined on each band [9]:ˇ G R ( x , x ′ , k y ) = X i Z dk x π e ik x ( x − x ′ ) ˇ G i ( k x , k y ) , (1)where i is the band index andˇ G i ≡ ( E + λ i ) ˆ M i ∆ i ˆ M i ∆ ∗ i ˆ M i ( E − λ i ) ˆ M i ! −| ∆ i | + E − λ i , (2)with [ ˆ M i ] jk = [ ˆ P ] ji [ ˆ P ] ∗ ki . Here, ˆ P is the unitary ma- Preprint submitted to Physica C October 30, 2018 rix consisting of the eigenvectors that diagonalize thenormal state Hamiltonian [9] represented with orbitalbasis, and λ i ( i = , , · · · , n ) denote the eigenvalues. ∆ i are the superconducting pair potentials. Then, the k x integration can be performed on each band inde-pendently. The surface LDOS at x = N ( E ) = − Im h Tr R dk y π ˇ G R ( x = , x ′ = , k y ) i /π .First, let us consider the ZBCP in the surface LDOS.With the use of a quasiclassical approximation proce-dure described in Refs. [8, 9], one can obtain the appear-ance condition of the ZBCP from the above formulationas det − ˆ I ˆ L ˆ L ˆ I ! = , (3)where ˆ L ≡ − i P i ∈ Q , l ˆ M i ( k i , lFx )sgn { ∆ i ( k i , lFx ) } / | v i , lFx | and ˆ I ≡ P i < Q , l π R dk x λ i ( k x ) ˆ M i ( k x ), as defined in Ref. [9]. Eq. (3)shows that the appearance condition does not dependon the anisotropy of the pair potentials ∆ i and it dependsonly on the signs of them. This result signifies that in-formation on the normal state (i.e., the matrices ˆ M i andthe Fermi velocity v i , lFx ) is important for the ZBCP toappear.Next, on the basis of the above general formula-tion [9], we perform numerical calculations of theLDOS on specific Fermi surfaces obtained from thefive-band model [3] of Kuroki et al. In Figs. 1 ( E F = . E F = . nm
0] denotes the surface normal vector). Compar-ing the left and right panels in Fig. 1, it appears that thepeak positions of Andreev bound states do not dependon whether the pair potential amplitude is anisotropic ornot. Comparison of the results for the [210] surface inFigs. 1 (left panel) and 2 indicates that the appearancecondition of the ZBCP indeed depends on informationon the normal state, namely it depends on the Fermi en-ergy E F here. Because the peaks due to Andreev boundstates do not appear for an s -wave pairing without signchange, our results also suggest that the mid-gap peaksin point-contact spectroscopy experiments may be theevidence of the ± s -wave superconductivity.In conclusion, we calculated the surface LDOS for ± s -wave pair potentials with the e ff ective five-bandmodel. We showed that the peak positions do not de-pend on the anisotropy of the pair potential ampllitudes,but depend on the normal-state properties. Acknowledgments
We thank Y. Kato, N. Nakai, H. Nakamura, M. Oku-mura, Y. Ohta, C. Iniotakis, M. Sigrist, Y. Tanaka and S. N s / N E/ ∆ [100][110][210][310][410] 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2E/ ∆ [100][110][210][310][410] Figure 1: Angular dependence of the surface density of states for thefive-band model with the isotropic (Left) and the anisotropic (Right) ± s -wave pair potential defined in Ref. [4]. The smearing factor is η = . ∆ and the Fermi energy is E F = . ∆ is the maximumgap amplitude.) N s / N E/ ∆ [100][110][210][310][410] Figure 2: Angular dependence of the surface density of states for thefive-band model with the isotropic ± s -wave pair potential. η = . ∆ and E F = . Onari for helpful discussions and comments. Y.N. ac-knowledges support by Grand-in-Aid for JSPS Fellows(204840).
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