Influence of the Fermi surface geometry on a Josephson effect between an iron-pnictide and conventional superconductors

We study Josephson junctions between a multi-band iron-pnictide Ba1-xNaxFe2As2 and conventional s-wave superconductors Nb and Cu/Nb bilayer. We observe that junctions with a Cu interlayer exhibit much larger IcRn, despite a weaker proximity-induced superconductivity. This counterintuitive result is attributed to the difference in Fermi surface geometries of Nb and Cu, which leads to a selective one-band tunneling from Cu and a non-selective multi-band tunnelng from Nb. The latter leads to a mutual cancellation of supercurrents due to the sign-reversal s+- symmetry of the order parameter in the pnictide. Our results indicate that Fermi surface geometries play a crucial role for pnictide-based junctions. This provides a new tool for phase sensitive studies and paves a way to a conscious engineering of such junctions.

Electronic structure of superconductors is usually quite complicated, even for low-T c materials, such as the transition metal Nb. Nevertheless, a simple description of Josephson effects, which does not take into account complex Fermi surface (FS) geometry, works remarkably well for conventional superconductors [1,2]. This happens because probabilities of electron and Cooper-pair tunneling are similar [3]. Together with a momentum-independent s-wave energy gap, ∆, it leads to the inverse relationship between the normal resistance, R n , and the critical current, I c . Thus, the I c R n product becomes a universal function of ∆, independent of FS geometry [4].
In this work we fabricate and study high-quality Josephson junctions (JJ's) between single crystals of an iron-pnictide Ba 1−x Na x Fe 2 As 2 (BNFA) and conventional low-T c superconductors made of either Nb film or Cu/Nb bilayer. Both types of JJ's exhibit clean and clear Josephson phenomena. However, JJ's with a Cu interlayer exhibit almost two order of magnitude larger I c R n , despite a weaker proximity-induced superconductivity in Cu. This counterintuitive result is attributed to the difference in FS geometries of Nb [multiple FS's at various parts of the Brillouin zone (BZ)] and Cu [a single quasispherical FS]. Therefore, tunneling from Nb takes place into all bands of BNFA. Due to the sign-reversal s ± order parameter in BNFA, this leads to a mutual cancellation of supercurrents and a very small I c R n ∼ 3µV. To the contrary, tunneling from Cu occurs predominantly into one sub-band avoiding such cancellation and leading to a significantly larger I c R n 200 µV. Our results indi-cate that FS geometries play a crucial role for JJ's with multi-band, sign-reversal superconductors. This provides a new tool for fundamental studies of unconventional superconductivity and opens a possibility for optimization and adjustment of junction characteristics. Figure 1 (a) represents a scanning electron microscope (SEM) image of the BNFA-Cu/Nb sample. Our samples contain six junctions made on a freshly cleaved BNFA single crystal. Fig. 1 (b) shows a closeup on the junction. Here the vertical strip represents the window in SiO 2 isolation layer and the horizontal strip -the top contact electrode. Micrometer-size JJ's are formed at the overlap between the two strips. As the top electrode we use either pure Nb film (∼ 200 nm thick) or Cu(15 nm)/Nb(180 nm) bilayer deposited by magnetron sputtering in a single cycle without breaking vacuum. Details of sample fabrication, experimental setup and a list of JJ parameters can be found in the Supplementary [15].
Multiterminal configuration of our samples allows simultaneous measurements of junction and crystal characteristics [23,24]. The blue line in Fig. 1 (c) shows the in-plane resistive transition of BNFA. At T ∼ 150 K there is a kink in R(T ), corresponding to a structural transition and spin-density-wave (SDW) ordering [12,14]. The superconducting transition occurs at T c (BN F A) 30 K. Observation of the SDW state and the slightly suboptimal T c indicate that the BNFA crystal is moderately underdoped. The red line in Fig. 1 (c) shows a simultaneously measured resistive transition of a junction. It has two steps, first at T c (BN F A) and the second at B = 15 mT. It is seen that I c is completely suppressed by a small parallel field, much smaller than the upper critical fields of BNFA [23,25] and Nb [26]. Therefore, in such a field we can carefully measure temperature dependence R n (T ), as shown in Fig. 1 (f). The modest upturn of R n with decreasing T is quite common for c-axis characteristics of high-T c superconductors, commonly associated with a pseudogap [27].
Suppression of I c by small parallel field is caused by flux quantization in the junction. Figure 2  Figure 2 (c) shows I-V curves of a BNFA-Cu/Nb JJ without (black) and with (red) applied high-frequency electromagnetic radiation at f 74 GHz at H = 0 and T 0.4 K. A clear Shapiro step is seen at V 1 = hf /2e. Fig. 2 (d) shows the normalized differential conductance for this I-V as a function of V 1 /V . It reveals nu-merous subharmonic Shapiro steps. This indicates the strongly non-sinusoidal current-phase relation in the JJ [28], which is indeed anticipated for s-s ± JJ's [7,8]. On the other hand, the non-sinusoidality may also be caused by the proximity effect in the Cu/Nb bilayer [29].
Thus, our JJ's exhibit clean and clear dc-and ac-Josephson effects. The high quality of the JJ's together with a good reproducibility of junction parameters (see the Supplementary [15]) allows us to investigate genuine characteristics of composing them superconductors (as opposed to interface defects). Figs. 2 (e) and (f) show temperature dependencies of (e) the critical current density J c and (f) the I c R n product for both types of junctions. Despite similarities in behavior, the same BNFA crystal [15] and fabrication procedure, the two types of JJ's exhibit largely (almost by two orders of magnitude) different I c R n values. BNFA-Nb JJ's have a very small I c R n 3 µV [24], much smaller than ∆/e > 1 mV in both suuperconductors, while for BNFA-Cu/Nb JJ's I c R n 200 µV. The difference can be clearly seen in the I-V curves from Figs. 1 (d) and (e). The reported remarkable influence of the thin Cu interlayer is the key observation of this work.  [4]. The dashed black line shows a simulated dependence for an s-s± junctions from Ref. [8].
The increase of I c R n in BNFA-Cu/Nb JJ's is associated with the increase of R n . The latter indicates that the interface transparency, β, between BNFA and Cu is reduced compared to BNFA-Nb. Yet, as mentioned above, this does not explain the increase of I c R n because usually I c ∝ 1/R n and I c R n is independent of β.
The proximity induced superconducting order parameter in Cu at the junction interface is Φ N βΨ S exp(d N /ξ N ), where Ψ S is the order parameter in Nb, d N = 15 nm is the Cu layer thickness and ξ N is the coherence length in Cu. According to Ref. [30], for thin sputtered Cu films ξ N 18 T c (N b)/T (nm), where T c (N b) 9 K. It gives ξ N (0.3K) 100 nm, ξ N (3K) 30 nm and ξ N (T c (N b)) 18 nm. Thus, our Cu interlayer is always thinner than ξ N . Although Cu and Nb films are deposited without breaking vacuum, the Cu/Nb interface transparency is modest, β 0.4 [30], predominantly due to the FS mismatch between Nb and Cu. Thus the proximity induced order parameter in Cu is smaller than in Nb. For SINS JJ's (I -insulator, N-normal metal) made of s-wave superconductors, the proximity effect leads to the reduction of I c R n [31]. This is opposite to our observation. This discrepancy points out that the unconventional (non-s-wave) symmetry of the order parameter in BNFA plays an essential role. In particular, the extremely small I c R n of BNFA-Nb junctions provides evidence for the sign-reversal s ± symmetry in BNFA, due to which supercurrents from bands with opposite signs of ∆ cancel each other [24].
For a more quantitative understanding we consider FS geometries of involved metals. Figures 3 (a-c) show DFT calculated three-dimensional images of Fermi surfaces for Cu (a), Nb (b) [32,33] and BNFA (c) [34]. FS of Cu is simple quasi-spherical. The transition metal Nb has a very complex FS with many small pockets spread over the BZ. BNFA has two bunches of the FS sheets, the three large quasi-cylinders in the center and the propeller-like FS's at the corners of the BZ [35,36]. Those bunches are believed to have opposite signs of ∆ [12,13].
Electron tunneling between two metals usually con-  serves the in-plain momentum k = (k x , k y ). Therefore, the total single-electron current can be written as where T 12 is the tunneling matrix element between initial and final states, (k z1 , k x , k y ) and (k z2 , k x , k y ), in the two electrodes, A 1,2 are the spectral functions (momentumdependent density of states) and f 1,2 are the corresponding distribution functions. The key band-structure-dependent factor is the density of states projection on the junction plane, which can be integrated independently

Figs. 3 (d) and (e) show such projections for Cu and Nb.
The corresponding projection for BNFA is pretty similar to the pattern, shown in Fig. 3 (i).
For comparison with experiment we must take into account the polycrystalline structure of Cu and Nb elec-trodes and make an average with respect to random crystalline orientation. This is similar to averaging with respect to rotation of k x,y axes. Figs 3 (f) and (g) show thus averaged projections, < N i (k x , k y ) >, for polycrystalline Cu and Nb. The key difference is that due to the quasi-spherical FS of Cu, the polycrystalline density of states projection keeps the circular shape with the radius given by the Fermi momentum. To the contrary, averaging for multi-band polycrystalline Nb leads to a more uniform distribution of the density of states.
Figs. 3 (h) and (i) show a product of the density of state projections (h) < N Cu (k x , k y ) > N BN F A (k x , k y ) for BNFA-Cu/Nb and (i) < N N b (k x , k y ) > N BN F A (k x , k y ) for BNFA-Nb junctions. It gives a hint about contribution of the two BNFA bands in electrical current through the junction. For BNFA-Nb JJ's both BNFA bands participate approximately equally due to fairly uniform distribution of < N N b (k x , k y ) > in the BZ projection, Fig.  3 (g). To the contrary, the highly non-uniform, circularshape < N Cu (k x , k y ) >, Fig. 3 (f), blocks tunneling into the central band of BNFA.
For calculation of supercurrent, A i should be replaced by A i Ψ i , where Ψ i is the superconducting order parameter in the corresponding metal. For Nb and Cu/Nb with s-wave order parameter Ψ is just a number. However, for the unconventional two-band superconductor BNFA, which likely has the s ± symmetry, Ψ changes sign between central and corner bands. For BNFA-Nb JJ's with similar transport contribution of the two bands this leads to an almost complete cancellation of the total supercurrent [24]. However, for BNFA-Cu/Nb JJ's the cancellation is much smaller because tunneling from central bands is suppressed. Therefore, such analysis qualitatively explains larger values of both R n and I c R n in BNFA-Cu/Nb junctions.
To conclude, we fabricated and studied high-quality Josephson junctions between an iron-pnictide superconductor Ba 1−x Na x Fe 2 As 2 and either a conventional low-T c superconductors Nb or a Cu/Nb bilayer. Remarkably, we observed that addition of a very thin (15 nm) Cu interlayer changes drastically junction properties and, in particular, increases the I c R n product by almost two orders of magnitude. The latter is opposite to expectations for proximity-coupled junctions made of conventional swave superconductors [31]. This counterintuitive result adds to evidence for the unconventional s ± symmetry of the order parameter in the pnictide. The phenomenon is explained qualitatively taking into account particular Fermi surface geometries of involved metals. It is shown that the multi-band structure of Nb leads to similar contributions of both pnictide bands into electron transport, which due to the sign-reversal s ± superconducting order parameter in the two electronic bands of the pnictide, leads to the cancellation of the total supercurrent and results in a very small I c R n 3 µV [24]. To the contrary, the simple quasi-spherical Fermi sur-face of Cu supports tunneling predominantly from only one band, avoiding the supercurrent cancellation and resulting in much larger I c R n 200 µV. Our results indicate that unlike for junctions made of conventional swave superconductors, for junctions with unconventional sign-reversal superconductors the Fermi surface geometry plays a crucial role. This provides a new tool for phase sensitive studies of such materials and could probably explain some of reported variations of I c R n values in pnictide JJ's [8,24,37]. The reported material-dependence of tunneling into pnictide superconductors can be used for optimization and conscious engineering of pnictide-based Josephson junctions.