Spin-polarized multiple Andreev reflections in spin-split superconductors
SSpin-polarized multiple Andreev reflections in spin-split superconductors
Bo Lu, Pablo Burset, and Yukio Tanaka Center for Joint Quantum Studies and Department of Physics, Tianjin University, Tianjin 300072, China Department of Applied Physics, Aalto University, 00076 Aalto, Finland Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan (Dated: July 18, 2019)We study the transport properties of a voltage biased contact between two spin-split supercon-ductors separated by an insulating barrier of arbitrary transparency. At low transparency, thecontribution of multiple Andreev reflections leads to a subharmonic gap structure that cruciallydepends on the amplitude and relative angle of the spin-splitting fields of each superconductor. Fornon-collinear fields, we find an interesting even-odd effect on the bound states within the gap, wherethe odd order multiple Andreev reflections split, but the even order ones remain at their expectedpositions. By computing the current-voltage characteristics, we determine the transparency requiredfor the emergence of a subharmonic gap structure and show that the splitting of the odd boundstates is associated with different threshold energies of spin-polarized Andreev processes. Our find-ings provide a tool to experimentally determine the amplitude and orientation of Zeeman fields inspin-split superconductors.
Introduction.—
Hybrid structures between supercon-ductors and magnetic materials reveal many interest-ing phenomena originated from the coexistence and in-terplay between ferromagnetism and superconductivity.As a result, the new field of superconducting spintron-ics has emerged [1, 2], aiming at incorporating super-conducting order into modern spintronic devices. Thecreation of spin-triplet Cooper pairs and spin-polarizedquasiparticles with long spin-coherence lengths [3, 4] sug-gest future applications based on a reliable and effi-cient manipulation of spin-polarized currents [5]. In thiscontext, superconductors with spin-split energy bands,commonly known as spin-split (or Zeeman-split) su-perconductors (SSc), are attracting considerable inter-est [6]. SSc can be realized either in a thin ferromagnet–superconductor (FM-S) junction via proximity effect [7–10], cf. Fig. 1(a), or in a thin-film superconductor subjectto a parallel (in-plane) magnetic field [11–14]. Highlyspin-polarized currents can be generated in SSc hybridjunctions [4, 10, 15] and large thermoelectric effects havebeen predicted [6, 16–20].The transport properties of hybrid junctions involv-ing SSc has thus become a topic of fundamental interestin superconducting spintronics. For example, the phasedifference in Josephson junctions between SSc has beenused as a source to manipulate the spin-polarized super-currents [8, 21–24]. Recently, the tunneling quasiparticlecurrent between two SSc linked by a spin-polarized bar-rier has been analyzed using quasi-classical Green func-tion techniques, showing good agreement between theoryand experiments [25]. Here, we go one step further andanalyze voltage biased SSc Josephson junctions of arbi-trary transparency, where multiple Andreev reflections(MAR) play an important role, and find that the sub-harmonic gap structure (SGS) is very sensitive to thespin-splitting fields of the superconductors.When two superconductors are in electric contact, MAR [26–28] take place at voltages eV < eV as they travel across the interface, until es-caping to the reservoirs for energies above the supercon-ducting gap. This phenomenon results in the so-calledSGS, a series of resonant conductance peaks at volt-ages V n = 2∆ / ( en ) in the current-voltage characteristics,where n is an integer [26–31]. The peaks reveal the sin-gular density of states at the superconducting energy gapedges, see Fig. 1(b-e). The study of the SGS has provento be useful for identifying properties of high-Tc super-conductors [32, 33], topological superconductors [34, 35],and other mesoscopic hybrid junctions [36–42].In this paper, we study the transport properties in theMAR regime of a junction between two SSc, as depictedin Fig. 1(a). For simplicity, we only consider junctionswhere both superconductors have the same pair ampli-tude, ∆, and strength of the spin-splitting field, h . How-ever, we allow the junction to be asymmetric by changingthe relative orientation, α , of the in-plane spin-splittingfields. Without including phonon-induced spin-flip [43],we find that the SGS due to MAR is highly tunableand presents a spin-dependent shift proportional to theZeeman field h . Additionally, depending on the relativeorientation of the spin-splitting fields, we find an even-odd effect on the conductance peaks forming the SGS.Fig. 1 illustrates how MAR are modified by the spin-splitting fields. For n -order MAR, a quasiparticle under-goes n − odd order ( n an odd integer) involvequasiparticles transferred between superconductors, for even order processes ( n an even integer), quasiparticles a r X i v : . [ c ond - m a t . s up r- c on ] J u l FIG. 1. (a) Schematic diagram of the multilayer FM-S-I-S-FM device used to develop a SSc-I-SSc Josephson junction,where I is an insulating barrier. The moments M L,R of theFMs are confined to the x − z plane, with a relative angle α ,while transport takes place along the y -direction. (b)(c)(d)Schematic representation of the 3rd-order MAR. The super-conducting gap is ∆ and the energy bands are split by theZeeman fields. Solid (dashed) line represent electron (hole)trajectories. The relative angle α affects the positions of thethreshold voltages for odd order MAR. (e) 2nd-order MAR,with threshold voltage at eV = ∆, is independent of α . instead return to the same superconductor. Therefore,only odd order MAR are sensitive to the relative orien-tation of the splitting fields. The threshold energies forquasiparticles traveling from one side to the other are 2∆for parallel magnetization ( α = 0) and 2 (∆ ± h ) for an- tiparallel magnetization ( α = π ). Arbitrary values of α lead to spin-mixing and result in three possible channels2∆ and 2 (∆ ± h ). Thus, the odd subharmonics become2∆ /n for α = 0 [see Fig. 1(b)], 2 (∆ ± h ) /n for α = π [see Fig. 1(d)], and 2∆ /n , 2 (∆ ± h ) /n for α (cid:54) = 0 , π [seeFig. 1(c)]. However, for even order processes, quasiparti-cles travel back to the same side, with spin conserved, andthus the threshold energy gains or loses between the twogap edges are independent of the splitting field, as shownin Fig. 1(e). Therefore, the position of the conductancepeaks for even subharmonics, 2∆ /n , is not altered by thespin-splitting fields. The SGS thus provides useful infor-mation about the strength and relative orientation of thespin-splitting fields at Josephson junctions between SSc. Model.—
The system we study consists of two semi-infinite superconductors in a point contact geometry.Spin-splitting fields are induced on both sides via proxim-ity effect to a FM region. The phenomenological Hamil-tonian of the system is written as ˆ H = ˆ H L + ˆ H R + ˆ H T ( t ).ˆ H L and ˆ H R describe the bulk SSc on the left and rightside: ˆ H j = L,R = 12 (cid:88) k ˆΨ † jk (cid:34) H jk ˆ∆ˆ∆ † − H Tjk (cid:35) ˆΨ jk , (1)where ˆΨ jk = [ ψ jk, ↑ , ψ jk, ↓ , ψ † j ( − k ) , ↑ , ψ † j ( − k ) , ↓ ] T is the spinorin Nambu-spin space. The noninteracting Hamiltonian is H jk = (cid:2) k / (2 m ) − µ (cid:3) ˆ σ − gµ B ˆ σ · M j . (2)We assume that the orientation of the Zeeman fieldslie in the x − z plane and parametrize the moments as M L = M e z and M R = M [sin α e x + cos α e z ], with g , µ B and M L,R the effective Land´e g -factor, Bohr magneton,and induced magnetic fields, respectively. Other spin ori-entations can be accounted for by an appropriate rotationwithout affecting our results. The gap matrix in Eq. (1)is ˆ∆ = i ˆ σ y ∆ for spin-singlet s -wave pairing, where thePauli matrices ˆ σ ,x,y,z operate in spin space. The tun-neling term ˆ H T ( t ) is given byˆ H T ( t ) = 12 (cid:88) k,k (cid:48) (cid:104) ˆΨ † Lk ˆ T kk (cid:48) ( t ) ˆΨ Rk (cid:48) + h . c . (cid:105) , (3)with ˆ T kk (cid:48) ( t ) = t kk (cid:48) ˆ τ z e iχ ( t )ˆ τ z / and ˆ τ x,y,z the Pauli matri-ces in particle-hole space. In the presence of a voltagebias V , the superconducting phase difference χ ( t ) = 2 ω J t is time-dependent, with ω J = 2 eV the Josephson fre-quency. The resulting time-dependent current followsthe Josephson relation ∆ → ∆ e i eV t [44], with I ( t ) = (cid:80) n I n e inω J t . For simplicity, we set t kk (cid:48) to be t kk (cid:48) = t δ k,k (cid:48) and denote ˆ T ( t ) = t ˆ τ z e iχ ( t )ˆ τ z / . Following the quasi-classical approximation [45, 46], we average the summa-tion over channels k without affecting the characteris-tics of the SGS, but simplifying calculations. By the so-called ξ -integration, the retarded/advanced Green func- FIG. 2. Electric current and differential conductance as a function of the voltage, for relative angles α = 0 , π/ , π , andtransmissions T = 0 . , . , .
3. In all cases, h/ ∆ = 0 .
15. The dashed vertical lines show the n -th order SGS at n = 1 (black) n = 2 (green) and n = 3 (red). tion ˆ G r/aj ( ω ) in the bulk state adopts the form [47]ˆ G r/aL = g r/a ↑ f r/a ↑ g r/a ↓ f r/a ↓ f r/a ↓ g r/a ↓ f r/a ↑ g r/a ↑ (cid:44) (cid:34) ˚Υ r/a ˚Υ r/a ˚Υ r/a ˚Υ r/a (cid:35) , (4)where ˆ G r/aR = ˆ U ˆ G r/aL ˆ U † (cid:44) (cid:34) ˚Γ r/a ˚Γ r/a ˚Γ r/a ˚Γ r/a (cid:35) , (5) g r/aσ ( ω ) = − ( ω + ζ σ h ± i + ) W (cid:112) ∆ − ( ω + ζ σ h ± i + ) , (6) f r/aσ ( ω ) = ∆ W (cid:112) ∆ − ( ω + ζ σ h ± i + ) , (7)with ζ σ = ↑ , ↓ = ± U = e − i ˆ σ y α/ ˆ τ . The mag-nitude of the proximity-induced spin-splitting fields is h = gµ B M and W is a band parameter [36]. We choosethe magnetic field below the so-called Chandrasekhar-Clogston limit [48, 49], h max = ∆ / √
2. The KeldyshGreen function ˆ G Kj = L,R ( ω ) is given by ˆ G Kj ( ω ) = [ ˆ G rj ( ω ) − ˆ G aj ( ω )] tanh[ ω/ (2 k B T )] [50].Following the transfer-matrix approach and doubleFourier transformation [31, 36, 51], we express the chargecurrent in the following form: I ( t ) = eν (0) v F (cid:88) m (cid:90) dω π T r [ˆ τ z I m ] e imω J t/ , (8)with ν (0) the density of states at the Fermi level in the normal state. The components I m are defined as I m = (cid:88) n ˆ G rR, [ T an ] † ˆ G KL,n T anm + T r n ˆ G KR,n [ T rmn ] † ˆ G aL,m − [ T an ] † ˆ G KL,n T anm ˆ G aR,m − ˆ G rL, T r n ˆ G KR,n [ T rmn ] † , (9)with ˆ G r,a,Kj,n = ˆ G r,a,Kj ( ω + nω J / m = 0 harmonic inEq. (8). Experimentally, it relates to the average elec-tric current in the long time limit. The transfer-matrixsatisfies T r/anm ( ω ) = T r/an − m, ( ω + mω J /
2) and can be de-termined by the recursive relationˆ T r/anm = t (cid:20) ˆ σ
00 0 (cid:21) δ n, − + t (cid:20) − ˆ σ (cid:21) δ n, (10)+ ˆ (cid:15) r/an ˆ T r/anm + ˆ V r/an,n +2 ˆ T r/an +2 ,m + ˆΛ r/an,n − ˆ T r/an − ,m , with ˆ (cid:15) r/an = t (cid:34) ˚Γ r/a ,n +1 ˚Υ r/a ,n ˚Γ r/a ,n +1 ˚Υ r/a ,n ˚Γ r/a ,n − ˚Υ r/a ,n ˚Γ r/a ,n − ˚Υ r/a ,n (cid:35) , ˆ V r/an,n +2 = − t (cid:20) ˚Γ r/a ,n +1 ˚Υ r/a ,n +2 ˚Γ r/a ,n +1 ˚Υ r/a ,n +2 (cid:21) , ˆΛ r/an,n − = − t (cid:20) r/a ,n − ˚Υ r/a ,n − ˚Γ r/a ,n − ˚Υ r/a ,n − (cid:21) . Eq. (10) is numerically solved by introducing ladderoperators T r/an,m = z r/an, ± T r/an ∓ ,m and using cut-off values z r/an, ± = 0 for a sufficient large | n | = n N , where T r/a ± n N , areassumed to vanish. We normalize the current in unitsof σ N ∆ /e , where σ N is the conductance when both elec-trodes are in non-superconducting states without mag-netic elements. FIG. 3. Normalized differential conductance as a function of the voltage. (a) Plot of the conductance for several relative angles,with fixed h/ ∆ = 0 .
15. (b,c) Plot of the conductance for various spin-splitting fields, with fixed α = π/ α = π for(c). In all cases, the curves have been vertically displaced for clarity and the dashed vertical lines show the n -th order SGS at n = 1 (black) n = 2 (green) and n = 3 (red). Josephson dc current —
We now compute the current–voltage characteristics of a dc-biased contact between twoSSc. As explained before, we set the gap of both super-conductors to be the same, ∆, and the spin-splitting fieldalso equal, h . We show in Fig. 2 the current and differen-tial conductance, σ S , for different values of the transmis-sion, T , and relative orientation angle α . In the case ofa high-transmission barrier, i.e., T (cid:46)
1, the current looksalmost featureless, but its derivative displays small peaksdue to MAR, which are more prominent at low voltages(blue lines in Fig. 2). This is a result of the enhancedprobability for Andreev reflections at high transmissions.For lower transmissions, T (cid:46) .
5, the subgap Andreev re-flections are suppressed and thus the SGS becomes morevisible as kinks in the current and peaks in the differen-tial conductance (red and black lines in Fig. 2). Thesefeatures are now more visible at higher voltages, due tothe quasiparticle transfer at the gap edges in the tun-neling regime. These are common features of MAR insuperconducting junctions; the novel effect comes fromthe angle α determining the relative orientations of thespin-splitting fields. Fig. 2 shows the three representa-tive cases with α = 0 , π (parallel and anti-parallel) and α (cid:54) = 0 , π (non-collinear). As observed in Fig. 2(a,d), wherethe fields are parallel, the SGS shows the usual distribu-tion with peaks at eV = 2∆ /n . This result is similarto previous works [31] without induced spin-split field insuperconductors. By contrast, when the fields are anti-parallel, the odd order MAR at eV = 2∆ /n disappearwhile new peaks at eV = 2(∆ ± h ) /n emerge ( n is an odd integer), see Fig. 2(c,f). Finally, when the magnetic fieldis non-collinear, the odd order MAR feature both typesof peaks, at eV = 2∆ /n and at eV = 2(∆ ± h ) /n , as shownin Fig. 2(b,e). Importantly, the splitting of the odd orderpeaks for α (cid:54) = 0 is directly proportional to the amplitudeof the Zeeman field h .From this result, we conclude that the odd SGS arevery sensitive to the configuration of the two spin-splitfields. Thus, the I − V characteristics could provide a wayto measure the amplitude and relative orientation of thespin-splitting fields. To clearly show these dependenceon the spin-split fields, we plot the normalized differen-tial conductance for a junction with average transmission T = 0 . h andchange the relative angle in Fig. 3(a). The odd orderMAR split into three peaks when α (cid:54) = 0. The peaks at eV = 2(∆ ± h ) /n reach their maximum as α approaches π . By contrast, the peaks at eV = 2∆ /n are significantlyreduced until they disappear for α = π . Importantly, theeven order MAR remain unchanged as α varies. Next,we show in Fig. 3(b)(c) the SGS for several values of h in the representative cases of non-collinear ( α = π/
2) andanti-parallel ( α = π ) fields. The splitting of the odd orderMAR becomes wider as h increases. This result indicatesthe great tunability of the SGS by both the relative angleand the magnitude of spin-split field. And the tunabilityis attributed to the spin-polarization of odd order MAR. Conclusions —
We have theoretically analyzed theSGS in a Josephson junction between two spin-split su-perconductors with arbitrary amplitude and orientationof their Zeeman fields. Our results show an interestingeven-odd effect of the bound states within the gap. TheSGS induced by odd MAR are split by the Zeeman fieldsand they are strongly influenced by their relative angle.The SGS from even MAR, instead, are independent ofthe spin-splitting fields. The analysis of the SGS is thusa useful tool to determine with great precision the mag-nitude and relative orientation of the Zeeman fields in ex-periments. We remark that, although we consider a sim-plified single-channel superconducting weak link model,the splitting of odd-order MAR resonances is also presentin common setups between spin-split superconductors,such as in a two-dimensional planar junction.
Acknowledgments —
We thank J. Cayao, S. Suzuki,S. Tamura and A. Yamakage for insightful discussions.We acknowledge support from the Horizon 2020 researchand innovation programme under the Marie Sk(cid:32)lodowska-Curie Grant No. 743884 and the Academy of Fin-land (project 312299); Topological Material Science(Grants No. JP15H05851, No. JP15H05853, and No.JP15K21717); Grant-in-Aid for Scientific Research B(Grant No. JP18H01176) from the Ministry of Educa-tion, Culture, Sports, Science, and Technology, Japan(MEXT). [1] Jacob Linder and Jason W.A. Robinson, “Superconduct-ing spintronics,” Nat. Phys. , 307–315 (2015).[2] Matthias Eschrig, “Spin-polarized supercurrents for spin-tronics: a review of current progress,” Reports onProgress in Physics , 104501 (2015).[3] Hyunsoo Yang, See-Hun Yang, Saburo Takahashi,Sadamichi Maekawa, and Stuart S. P. Parkin, “Ex-tremely long quasiparticle spin lifetimes in superconduct-ing aluminium using MgO tunnel spin injectors,” NatureMaterials , 586 (2010).[4] F. 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