Phase-controlled spin and charge currents in superconductor-ferromagnet hybrids
Ali Rezaei, Robert Hussein, Akashdeep Kamra, Wolfgang Belzig
PPhase-controlled spin and charge currents in superconductor-ferromagnet hybrids
Ali Rezaei, Robert Hussein, Akashdeep Kamra, and Wolfgang Belzig ∗ Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: August 27, 2019)We investigate spin-dependent quasiparticle and Cooper-pair transport through a central nodeinterfaced with two superconductors and two ferromagnets. We demonstrate that voltage biasingof the ferromagnetic contacts induces superconducting triplet correlations on the node and reversesthe supercurrent flowing between the two superconducting contacts. We further find that suchtriplet correlations can mediate a tunable spin current flow into the ferromagnetic contacts. Ourkey finding is that unequal spin-mixing conductances for the two interfaces with the ferromagnetsresult in equal-spin triplet correlations on the node, detectable via a net charge current betweenthe two magnets. Our proposed device thus enables the generation, control, and detection of thetypically elusive equal-spin triplet Cooper pairs.
Cooper-pairs from a superconductor (S) placed in thevicinity of a ferromagnet (F) may diffuse into the latter,thus, modifying their electronic properties [1–5]. Theengineering of this proximity effect in hybrid structuresgenerated over the last few years considerable interest inthermoelectricity [6–16], spin calorics [17, 18], and topo-logical superconductivity [19–24]. Somehow unexpectedis that not only ferromagnets [25–28] and antiferromag-netic insulators [29] can cause spin imbalances into super-conductors, but also normal metals [30] exploiting that asuperconductor itself may serve as a spin filter.By sandwiching a normal metal or ferromagnet be-tween two superconductors, one can realize a Joseph-son junction featuring a current of Cooper-pairs be-tween them, which is characterized by the junctions’free energy [31–33]. Its global minimum determines theground state of the system. While a ground state atzero phase difference indicates a Josephson current me-diated by singlet Cooper-pairs, a shifted groundstateabout π —occurring in magnetic Josepshon junctions [34–41]—signals triplet superconductivity [42–44] manifest-ing in a reversed current phase relation (CPR) [45–47]. Such magnetic Josepshon junctions are interestingfor quantum computation [48, 49] and cryogenic memo-ries [50, 51].Recently, the generation of equal-spin triplet pairs hasbeen demonstrated in S/F structures utilizing inhomo-geneous magnetic fields [52], as originally predicted byKadigrobov [53] and Bergeret [54, 55] et al. Mixed-spin triplet pairs with a zero spin projection on the z -axis, however, already arise for homogeneous magnetiza-tion. It is the immunity of equal-spin triplet correlationsagainst internal magnetic fields which causes long pen-etration length into ferromagnets compared to the onesof singlet and mixed-triplet correlations [56–58]. Thisproperty of equal-spin triplet pairs makes them particu-lar attractive for low-power spintronics [59–61].In this Letter, we study spin and charge transport in afour-terminal S/F system consisting of two s-wave super-
0, allows the generation and detection of equal-spintriplet correlations (encircled arrows of the same color) bymeasuring a net charge current between the ferromagneticleads. a r X i v : . [ c ond - m a t . s up r- c on ] A ug (a) (b)0 10 00 0 . . . . − − − eV = 0 eV = 0 . eV & ∆ const ∝ ǫ Th ∝ Vθ/πI S [50 G N ǫ Th /e ] I zS [ ~ G N ǫ Th /e ] I zF [ ~ G N ∆ /e ] | I S | [ G N ∆ / e ] ǫ Th [∆] V [∆ /e ] FIG. 2. (a) Modulus of the net supercurrent I S as a function of the Thouless energy (cid:15) Th for different voltages V . Thedivergencies occurring for large voltages, V (cid:38) ∆ /e , indicate 0– π -transitions. Parameters are θ = 0, ϕ = π/
2, and G φ = G φ = 0.The thin purple line indicates the corresponding critical current I C = max ϕ I S for V (cid:38) ∆ /e . (b) Net supercurrent I S , and netspin currents I zS and I zF between the superconducors/ferromagnets for θ = ϕ = π/ (cid:15) Th (cid:28) ∆. The inset shows the netspin currents as a function of θ for V = 1 . /e , as indicated by the horizontal dotted line in panel (b). conducting ( n = S , S ) and two ferromagnetic ( n = F , F )terminals with a common contact region, Fig. 1(a). Ourkey finding is that a voltage applied to the two ferromag-netic leads in combination with noncollinear magnetiza-tions can induce triplet superconducting correlations inthe system and, therewith, transitions in the CPR. Weshow that the relative magnetization angle θ qualifies forvoltages above the superconducting gap as a control knobfor spin currents in the ferromagnetic and superconduct-ing contacts [Fig. 1(b)]. Finally, we highlight that asym-metric spin-mixing conductances (e.g. G φ = 0, G φ > < θ < π ,but otherwise symmetric configuration, indeed, a finitenet charge current between the ferromagnetic contacts[Fig. 1(c)]. This effect is attributed to the generation ofequal-spin triplet correlations in the central node, andallows for their experimental detection and exploitationin a convenient manner. Method.—
We study diffusive transport, within a semi-classical [62–66] circuit theory [67, 68]. In this frame-work, hybrid structures are discretized as a network ofnodes, terminals, and connectors. Here, we map our sys-tem to a layout consisting of a central node which is inter-faced with two superconducting and two ferromagneticterminals via corresponding connectors, see Fig. 1(a). Anadditional leakage terminal can account for losses of su-perconducting correlations. The Green function ˇ G c ofthe node, which is an 8 × = (cid:80) n ˇ I n , together withthe normalization condition ˇ G c = determines ˇ G c and,therewith, the individual matrix currents ˇ I n ≡ [ ˇ M n , ˇ G c ] between terminal n ∈ { S , S , F , F , Leak } and thecentral node [69].The superconducting contacts are characterized by theBCS bulk Green functions (2 /G S ) ˇ M Sα = ˇ G Sα with G S being the conductance of the corresponding con-nector to the central node, and α = 1 ,
2. Its re-tarded/advanced component reads in the spinor basis { Ψ †↑ , Ψ †↓ , Ψ ↓ , − Ψ ↑ } [6, 7]ˆ G R,ASα = ± sgn( (cid:15) ) (cid:112) ( (cid:15) ± i Γ) − | ∆ α | (cid:18) ± i Γ + (cid:15) ∆ α − ∆ ∗ α ∓ i Γ − (cid:15) (cid:19) ⊗ , (1)whereby ∆ , ≡ ∆ exp[ ± iϕ/
2] denotes the superconduct-ing gap with phase difference ϕ across the junction. Asmall imaginary component in the denominator of Eq. (1)(here Γ = 10 − ∆) accounts for a finite lifetime (cid:126) / Γ of thequasiparticles, with energy (cid:15) , smearing out the supercon-ducting gap [70]. The ferromagnetic contacts are gov-erned by ˇ M F α = ( G N / + P ˇ κ α ) ˇ G F − iG φα ˇ κ α ] with G N ( G φα ) being the normal (spin-mixing) conductancesof the corresponding connectors. P denotes the contactspin polarization, and ˇ κ α = ⊗ σ z ⊗ ( m α · σ ) is te spinmatrix which is diagonal in Keldysh space. In the last ex-pression, σ = { σ x , σ y , σ z } labels the vector of Pauli ma-trices and m α is the magnetization vector correspondingto the ferromagnet F α . Here, we fix m = (0 , ,
1) in z -direction and consider m = (sin θ, , cos θ ) tilted by anarbitrary angle θ . We further consider fully polarized fer-romagnetic contacts, i.e. P = 1. The retarded/advancedcomponent of the ferromagnetic Green function is givenby ˆ G R,AF = ± σ z ⊗ . Finally, the leakage terminalis described by ˇ M Leak = − iG S ( (cid:15)/(cid:15) Th ) ⊗ σ z ⊗ with (cid:15) Th being the Thouless energy. The Keldysh com-ponent of the S and F Green functions follows fromˆ G Kn = ˆ G Rn ˆ h n − ˆ h n ˆ G An with the distribution function ˆ h n =diag ( tanh[( (cid:15) − eV n ) / k B T ] , tanh[( (cid:15) + eV n ) / k B T ] ) ⊗ .Hereafter, we assume all contacts at zero temperature, k B T = 0, the superconductors at zero (reference) voltage, V S = V S = 0, equally biased ferromagnetic contacts, V ≡ V F = V F , and equal conductances G S = G N .The Keldysh component of the matrix currents ˇ I n leads to the charge currents I n = 18 e (cid:90) ∞−∞ d(cid:15) tr[( σ z ⊗ ) ˆ I Kn ( (cid:15) )] , (2)and the z -polarized spin currents I zn = (cid:126) e (cid:90) ∞−∞ d(cid:15) tr[( ⊗ σ z ) ˆ I Kn ( (cid:15) )] (3)[68, 71–73]. In particular, we will analyze the charge, I X = I X − I X , and spin net current, I zX = I zX − I zX between the superconductors/ferromagnets ( X = S, F ).The matrix elements f ss (cid:48) = (cid:104) Ψ s | ˆ G Kc | Ψ s (cid:48) (cid:105) with s, s (cid:48) = ↑ , ↓ contain the spectral information about spin-pair correla-tions in nonequilibrium. Here, we consider the integratedquantities over positive energies (cid:15) > (cid:15) ), to quantify singlet, F S = (cid:82) d(cid:15) ( f ↑↓ − f ↓↑ ) / √
2, mixed-spin triplet, F T = (cid:82) d(cid:15) ( f ↑↓ + f ↓↑ ) / √ F T s = (cid:82) d(cid:15) f ss . Scaling.—
Before analyzing the interplay betweenCooper-pair and quasiparticle transport, let us recall thatin equilibrium only a Josephson current I eq S = I C sin ϕ may flow between both s-wave superconductors, whichhas a purely sinusoidal CPR—all other currents requirequasiparticle excitations. Depending on the effective size L of the central node, the Josephson current scales inthe diffusive regime for (cid:15) Th (cid:28) ∆ (large-island) with theThouless energy (cid:15) Th ≡ (cid:126) D/L , where D is the diffusionconstant [see black solid line in Fig. 2(a)]. For (cid:15) Th (cid:29) ∆(small-island), however, it is characterized by the super-conducting gap ∆ [33]. While the superconducting con-densate is in equilibrium entirely formed by spin-singletCooper-pairs, finite voltages may cause triplet correla-tions in the system. They can lead for voltages below thegap [dashed line in Fig. 2(a)] to a reduction of the Joseph-son current—here, we consider parallel collinear magne-tization. For voltages above the gap and intermediatevalues of the Thouless energy (cid:15) Th , such triplet correla-tions can even induce current reversals in the Josephsoncurrent I S . In Fig. 2(a), showing the modulus of I S on adouble logarithmic scale, these zero-crossings (at whichthe logarithm diverges) result in the two sharp dips ofthe dash-dotted curve. Also the corresponding criticalcurrent I C = max ϕ I S [purple line in Fig. 2(a)] indicateswith the kinks the presence of triplet correlations.For Thouless energies much smaller (larger) than thesuperconducting gap, the corresponding net supercurrentstays always positive and is dominated by spin-singlet (a)(b)01 − . . . θ/πI S [ G N ∆ /e ] I F [ G N ∆ /e ] I zS [ ~ G N ∆ /e ] I zF [4 ~ G N ∆ /e ] | F S / ∆ || F T / ∆ | | F T ↑ / ∆ || F T ↓ / ∆ | FIG. 3. (a) Net charge I S , I F , and net spin currents I zS , I zF ,as a function of θ for (cid:15) Th (cid:29) ∆, ϕ = π/ V = 2∆ /e , G φ = 0,and G φ = 2 G N . (b) Modulus of the spin-singlet, F S , and thespin-triplet pairing functions F T , F T ↑ , F T ↓ . Notice, the fi-nite spin-mixing conductance G φ induces a net charge current I F (top panel) following the curve progression of the equal-spin triplet correlations, thus, making them experimentallyattainable by standard current measurements [see Fig. 1(c)for θ ≈ . π , where | F T ↓ | < | F T ↑ | ]. correlations. We show in the following that appliedvoltages in combination with non-parallel magnetization, θ (cid:54) = 0, can induce triplet correlations and transitions inthe CPR also in the regimes (cid:15) Th (cid:28) ∆ and (cid:15) Th (cid:29) ∆. Phase transitions and spin current control.—
First, letus consider the large-island regime, (cid:15) Th (cid:28) ∆, where theCooper-pair transport is characterized by the Thoulessenergy, i.e. I S , I zS ∝ (cid:15) Th . While the net current betweenboth superconductors follows in equilibrium the usual si-nusoidal CPR, I S ∝ sin ϕ , nonequilibrium in combina-tion with non-collinear magnetization, 0 < θ < π , caninduce 0– π -transitions for voltages V > I S saturates. In this regime, a finite net spincurrent I zF ∝ V (red dot-dashed line) emerges betweenthe ferromagnetic contacts for a nonzero magnetizationangle, irrespective of ϕ . For the chosen symmetric con-figuration, however, no corresponding net charge current I F flows.A special feature of our setup is the occurrence of afinite net spin current I zS between both superconductors(blue dashed line) for voltages V (cid:38) ∆ /e . This effectis maximal for parallel magnetization, θ = 0, as can beseen in the inset, for which only mixed-triplet and singletcorrelations are present. It vanishes for antiparallel ori-entation, θ = π , where no triplet correlations arise, andwhen the Josephson phase ϕ is a multiple of π . Noticethat I zS is antisymmetric in ϕ , and I zS as well as I zF aresymmetric in θ . While voltages V above the gap can trig-ger net spin currents I zF and I zS , the magnetization angle θ can control their ratio (see inset), making the proposedsetup, thus, attractive for future applications. Spin-mixing induced charge current.—
Let us now turnto the small-island regime, (cid:15) Th (cid:29) ∆, where losses ofsuperconducting coherences become irrelevant, and theJosephson transport is characterized by the gap energy,i.e. I S , I zS ∝ ∆. Here, we find that non-antiparallel mag-netization θ (cid:54) = π can, indeed, induce triplet correlationsfor sufficiently large voltages, which results in an asym-metric sinusoidal CPR. However, it cannot induce 0– π -transitions. To cure this circumstance, we consider in thefollowing finite spin-mixing for voltages above the gap.Indeed, figure 3(a) indicates that the system is for paral-lel magnetization, θ = 0, but finite spin-mixing G φ > π -phase featuring a negative net supercurrent I S (blacksolid line). Roughly at θ = π/
4, the CPR undergoes a π –0-transition. Similar to the large-island regime [insetof Fig. 2(b)], a net spin current I zF may flow between theferromagnets [red dot-dashed line in Fig. 3(a)] which isessentially unaffected by spin-mixing. The net spin cur-rent I zS between the superconductors (dashed blue line),on the contrary, is apart from θ = 0 modified by spin-mixing, and features a current reversal about θ = π/ | F T ↑ | and | F T ↓ | , see Fig. 3(b). A distinctive feature, however,is that these equal-spin triplet correlations can induce afinite net charge current I F into the ferromagnets [dot-ted green line in Fig. 3(a)] for asymmetric spin-mixing, G φ (cid:54) = G φ . Where, this feature is attributed to the cre-ation of an imbalance in the ferromagnetic spin chan-nels, see Fig. 1(c). This effect also persists for vanishingJosephson phase ϕ . Under a mutual exchange of thespin-mixing conductances ( G φ ↔ G φ ), the net chargecurrent I F just inverts. An experimentally measurablecharge current I F serves also in the large-island regimeas a signature of equal-spin triplet correlations. It fea-tures in this regime a similar curve progression, but scalesinstead with (cid:15) Th . Conclusions.—
Spin-dependent quasiparticle and Cooper-pair transport have been analyzed in aproximity-coupled multi-terminal S/F-heterostructurein nonequilibrium. We have shown that 0– π -transitionscan be induced in the CPR by biasing the ferromagneticcontacts and bearing non-collinear magnetic moments,as long as the loss of superconducting coherences islarge, (cid:15) Th (cid:28) ∆. In this limit, voltages exceeding thesuperconducting gap, V (cid:38) ∆ /e , trigger net spin currentsinto the ferromagnets/superconductors, which can becontrolled by the relative magnetization angle θ . Thesmall-island regime, however, requires additionally finitespin-mixing to induce CPR. The considered heterostruc-ture qualifies as an ideal platform for the generationof triplet correlations of different spin projection, andas a voltage- and phase-controlled switch for spin andelectron currents. 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