Zero Energy Peak and Triplet Correlations in Nanoscale SFF Spin-Valves
ZZero Energy Peak and Triplet Correlations in Nanoscale
SFF
Spin-Valves
Mohammad Alidoust, ∗ Klaus Halterman, † and Oriol T. Valls ‡ Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Michelson Lab, Physics Division, Naval Air Warfare Center, China Lake, California 93555, USA School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA (Dated: September 24, 2018)Using a self-consistent Bogoliubov-de Gennes approach, we theoretically study the proximity-induced den-sity of states (DOS) in clean
SF F spin-valves with noncollinear exchange fields. Our results clearly demon-strate a direct correlation between the presence of a zero energy peak (ZEP) in the DOS spectrum and thepersistence of spin-1 triplet pair correlations. By systematically varying the geometrical and material param-eters governing the spin-valve, we point out to experimentally optimal system configurations where the ZEPsare most pronounced, and which can be effectively probed via scanning tunneling microscopy. We complementthese findings in the ballistic regime by employing the Usadel formalism in the full proximity limit to investigatetheir diffusive
SF F counterparts. We determine the optimal normalized ferromagnetic layer thicknesses whichresult in the largest ZEPs. Our results can serve as guidelines in designing samples for future experiments.
PACS numbers: 74.50.+r, 74.25.Ha, 74.78.Na, 74.50.+r, 74.45.+c, 74.78.FK, 72.80.Vp, 68.65.Pq, 81.05.ue
I. INTRODUCTION
The interplay of ferromagnetism and superconductivity inhybrid superconductor ( S ) ferromagnet ( F ) structures ( S/F structures) constitutes a controllable system in which to studyfundamental physics, including prominently that of compet-ing multiple broken symmetries.
The proximity of a con-ventional s -wave superconductor with non-aligned ferromag-netic layers, or a textured ferromagnet, induces both spin-singlet and odd frequency (or equivalently odd-time ) spin-triplet correlations with and ± spin projections along aspin quantization axis. These triplet pairs stem from bro-ken time reversal and translations symmetries. This kindof spin-triplet pairings originally suggested as a possible pair-ing mechanism in He, has reportedly been observed in in-termetallic compounds such as Sr RuO . SF heterostruc-tures are particularly simple and feasible experimental sys-tems which allow for direct studies of the intrinsic behavior ofdiffering superconducting pairings. Unlike the opposite-spincorrelations, spin-1 pairing correlations are rather insensitiveto the pair-breaking effects of ferromagnetic exchange split-ting, and hence to the thickness of the magnetic layers, tem-perature, and magnetic scattering impurities. The amplitudesof the opposite-spin correlations pervading the adjacent fer-romagnet, undergo damped oscillations as a function of po-sition which reveals itself in - π transitions of the supercur-rent. Since about a decade ago, several proposals havebeen put forth to achieve attainable and practical platformsthat isolate and utilize the proximity-induced superconduct-ing triplet correlations in SF hybrids. The signatures of the proximity-induced electronic den-sity of states (DOS) in the F layers of these hybrid struc-tures can reveal the existence and type of superconductingcorrelations in the region. One promising prospect for un-ambiguously detecting triplet correlations experimentally in-volves tunneling spectroscopy experiments which can probethe local single particle spectra encompassing the proximity-induced DOS.
Nonetheless, competing effects canmake analysis of the results of such a ‘direct’ probe of spin- triplet superconducting correlations problematic. The DOS in
SN S junctions and
SF S heterostructures where the magne-tization pattern of the F layer can be either uniform or tex-tured (including domain wall and nonuniform textures, suchas the spiral magnetic structure of Holmium) has been ex-tensively studied. It was found that the DOS ina normal metal sandwiched between two s -wave supercon-ducting banks shows a minigap which closes by simply tun-ing the superconducting phase differences up to the value of π . In contrast, the DOS can exhibit anomalous behav-ior in inhomogeneous magnetic layers. Namely, upon modu-lating the superconducting phase difference a peak arisesat zero energy, at the center of what was a minigap. It wasalso shown that the zero energy peak (ZEP) in the DOS fora simple textured
SF S junction can be maximized at a π bias. The minigap-to-peak behavior of the DOS at zeroenergy is an important signature of the emergence of tripletcorrelations.
Recently it was theoretically proposed thatthe minigap-to-peak phenomenon be leveraged for functional-ity in device platforms such as SQUIDs, to enhance their per-formance and as ultrasensitive switching devices, including a singlet-triplet superconducting quantum magnetometer. An important spectroscopic tool for investigating proxim-ity effects on an atomic scale with sub-meV energy resolu-tion is the scanning tunneling microscope (STM). As shownin Fig. 1, an
SF F spin valve structure can be probed exper-imentally by positioning a nonmagnetic STM tip at the edgeof the sample to measure the tunneling current ( I ) and volt-age ( V ) characteristics. This technique yields a direct probeof the available electronic states with energy eV near the tip.Therefore, the differential conductance dI ( V ) /dV over theenergy range of interest is proportional to the local DOS. Nu-merous experiments have reported signatures of the energyspectra in this manner. When ferromagneticelements are present, the superconducting proximity-inducedDOS reveals a number of peculiarities due to the additionalspin degree of freedom that arises from the magnetic layers.However, the experimental signatures of the odd-frequencyspin-triplet correlations can be washed out by more dominate a r X i v : . [ c ond - m a t . s t r- e l ] J un singlet correlations. When the exchange splitting h of themagnetic layers is large ( ∼ ε F , i.e. close to the half metal-lic limit), the characteristic length scale ξ F that describes thepropagation length of opposite-spin pairs in the ferromagnetsis extremely small. These types of proximity-induced correla-tions can thus only be experimentally observed in weak mag-netic alloys h (cid:28) ε F (such as Cu x Ni y ) or thin F layers so that d F / ξ F is sufficiently large to allow the opposite-spin super-conducting correlations to propagate in the ferromagnet with-out being completely suppressed. Since spin-1 tripletpairs are not destroyed by the ferromagnetic exchange fieldin strong magnets, there should exist certain system parame-ters, e.g., ferromagnet widths and exchange fields, that resultin regions whereby equal-spin pairs are the only pair correla-tions present. This scenario was explored in a S /Ho bilayer ,where phase-periodic conductance oscillations were observedin Ho wires connected to an ordinary s-wave superconductor.This behavior was qualitatively explained in terms of the long-range penetration of proximity-induced spin-1 triplet pairingsdue to the helical structure of the magnetization. In prac-tice however, simpler structures involving SF hybrids withuniform exchange fields are often preferable from both an ex-perimental and theoretical perspective. Therefore,the primary aim of this work is the determination of experi-mentally optimal parameters for probing odd-frequency spin-1 triplet correlations with DOS signatures in nanoscale
SF F spin valves.Nearly all of the past theoretical works on
SF F struc-tures have considered the diffusive case, where impuritiesstrongly scatter the quasiparticles. The clean regime has beenstudied, using a self-consistent solution of the Bogoliubov-de Gennes (BdG) equations, in Ref. 37. That work, how-ever, focused largely on the transition temperature oscilla-tions. The results for these oscillations were found to agreewith experiment and to be consistent with other experimen-tally established results. In the present work, we usethe same general methods used there to study a simple SF F structure with noncollinear exchange fields in the ballisticregime, but we focus on a very different quantity which isreadily accessible experimentally, namely the local DOS andits detailed low-energy structure. We strongly emphasize therelation between the ZEP and the triplet pairing amplitudes. Inparticular, given the assertion that variations in the transitiontemperature in these valve structures are quantitatively relatedto the average triplet pair amplitudes in the outer F layer, wewill search for, (and, as will be seen, find) correlations be-tween the ZEP and these averages. This BdG study is com-plemented with a briefer investigation of the correspondingdiffusive case. By considering both regimes, we will be ableto provide some general guidelines for future experiments.The structure we study is schematically depicted in Fig. 1,where the STM tip is positioned at the outermost F layer,near the vacuum boundary. In the ballistic regime, we employthe full microscopic BdG equations within a self-consistentframework. From the solutions, we calculate the local DOSover a broad range of experimentally relevant parameters, andstudy its behavior at low energies. For the diffusive regime,we make use of the quasiclassical Usadel approach to study the diffusive SF F counterparts in the full proximity limit.Our systematic investigations thus provide a comprehensiveguide into such spin valves. Utilizing experimentally realis-tic parameters, we determine favorable thicknesses for the F layers to induce maximal ZEPs, which occurs when the pop-ulation of triplet correlations in the outer layer, dominates thesinglets.The paper is organized as follows. In Sec. II we outlinethe theoretical approaches used. In Sec. III, we present ourresults in two subsections, pertaining to the ballistic and dif-fusive regimes. In the ballistic case, we study the local DOSfor differing exchange field misalignments, exchange field in-tensities, and interface scattering strengths. We also investi-gate the singlet and triplet pairing correlations for similar pa-rameters to determine how ZEPs in the DOS correlate withthe triplet correlations. In the diffusive case, we present two-dimensional maps of the ZEP at different exchange field mis-alignments, and SF interface opacity. Finally, we summarizewith concluding remarks in Sec. IV. II. METHODS AND THEORETICAL TECHNIQUES
In this section, we first discuss the theoretical frameworkused to study clean samples. We then outline the Usadel tech-nique in the full proximity regime, which properly describesdirty samples.
FIG. 1. (Color online) Schematic of the
SF F spin-valve structure.The ferromagnetic layers have uniform exchange fields located inthe yz plane. The exchange field of each layer is defined by (cid:126)h , = h (0 , sin β , , cos β , ) in which β , are the angle of the exchangefields with respect to the z direction. The ferromagnets ( F , F )and superconductor ( S ) are stacked in the x direction with thickness d F , d F , and d S , respectively. The STM tip is located at edge ofthe SF F spin-valve.
A. Microscopic approach: Bogoliubov-de Gennes equation
For the ballistic regime, we use the microscopic BdG equa-tions to study
SF F spin valve nanostructures. We solve theseequations in a fully self-consistent manner. A schematicof the spin valve configuration is depicted in Fig. 1. The gen-eral spin-dependent BdG equations for the quasiparticle ener-gies, (cid:15) n , and quasiparticle amplitudes, u nσ , v nσ is written: H − h z − h x + ih y − h x − ih y H + h z ∆ 00 ∆ ∗ − ( H − h z ) − h x − ih y ∆ ∗ − h x + ih y − ( H + h z ) u n ↑ u n ↓ v n ↑ v n ↓ = (cid:15) n u n ↑ u n ↓ v n ↑ v n ↓ , (1)where the pair potential, ∆( x ) , is calculated self-consistentlyas explained below. This quasi one-dimensional system is de-scribed by the single particle Hamiltonian H ( x ) as, H ( x ) ≡ m (cid:0) − ∂ x + k y + k z (cid:1) − E F + U ( x ) , (2)where E F is the Fermi energy, and U ( x ) is the spin-independent interface scattering potential which we take tobe of the form U ( x ) = H [ δ ( x − d F ) + δ ( x − d F − d F )] .The in-plane wavevector components, k y and k z , arise fromthe translational invariance in the y and z directions. The sys-tem is finite in the x direction, with widths of each F and S layer shown in the schematic. Our method permits arbitraryorientations and magnitudes of the magnetic exchange fields, (cid:126)h i ( i = 1 , ), in each of the ferromagnet regions. Specifically,we fix the exchange field in F to be aligned in the z direction,while in F , its orientation is described by the angle β : (cid:126)h = (cid:40) (cid:126)h = h (0 , sin β , cos β ) , in F (cid:126)h = h ˆ z , in F , (3)where we consider the experimentally appropriate situation ofan in-plane Stoner-type exchange field interaction.The spin-splitting effects of the exchange field coupled withthe pairing interaction in the S regions, results in a nontrivialspatial dependence of the pair potential ∆( x ) . In general, itis necessary to calculate the pair potential in a self consistentmanner by an appropriate sum over states: ∆( x ) = g ( x )2 (cid:88) n [ u n ↑ ( x ) v ∗ n ↓ ( x ) + u n ↓ ( x ) v ∗ n ↑ ( x )] tanh (cid:16) (cid:15) n T (cid:17) , (4)where g ( x ) is the attractive interaction that exists solely insidethe superconducting region and the sum is restricted to thosequantum states with positive energies below an energy cutoff, ω D .We now discuss the appropriate quantities that characterizethe induced triplet correlations. We define the following triplet pair amplitude functions in terms of the field operatorsin the Heisenberg picture, f ( x, t ) = 12 [ (cid:104) ψ ↑ ( x, t ) ψ ↓ ( x, (cid:105) + (cid:104) ψ ↓ ( x, t ) ψ ↑ ( x, (cid:105) ] , (5a) f ( x, t ) = 12 [ (cid:104) ψ ↑ ( x, t ) ψ ↑ ( x, (cid:105) − (cid:104) ψ ↓ ( x, t ) ψ ↓ ( x, (cid:105) ] , (5b)where t is the relative time. With the quantization axis alignedalong the z direction, the time-dependent triplet amplitudes, f ( x, t ) and f ( x, t ) , can be written in terms of the quasipar-ticle amplitudes: f ( x, t ) = 12 (cid:88) n (cid:0) f ↑↓ n ( x ) − f ↓↑ n ( x ) (cid:1) ζ n ( t ) , (6) f ( x, t ) = 12 (cid:88) n (cid:0) f ↑↑ n ( x ) + f ↓↓ n ( x ) (cid:1) ζ n ( t ) , (7)where we define f σσ (cid:48) n ( x ) = u nσ ( x ) v ∗ nσ (cid:48) ( x ) , and the time fac-tor ζ n ( t ) is written, ζ n ( t ) = cos( (cid:15) n t ) − i sin( (cid:15) n t ) tanh (cid:16) (cid:15) n T (cid:17) . (8)Experimentally accessible information regarding the quasi-particle spectra is contained in the local density of one particleexcitations in the system. This includes the zero-energy sig-natures in the density of states (DOS), which present a pos-sible experimental avenue in which to detect the emergenceof equal-spin triplet correlations within the outer ferromag-net. The total DOS, N ( x, (cid:15) ) , is the sum N ↑ ( x, (cid:15) ) + N ↓ ( x, (cid:15) ) ,involving the spin-resolved local density of states (DOS), N σ ,which are written, N σ ( x, (cid:15) ) = − (cid:88) n (cid:110) [ u σn ( x )] f (cid:48) ( (cid:15) − (cid:15) n )+[ v σn ( x )] f (cid:48) ( (cid:15) + (cid:15) n ) (cid:111) , (9)where σ denotes the spin ( = ↑ , ↓ ), and f (cid:48) ( (cid:15) ) = ∂f /∂(cid:15) is thederivative of the Fermi function. B. Quasiclassical approach: Usadel equation
When the system contains a strong impurity concentration,then for sufficiently small energy scales, the superconductingcorrelations are governed by the Usadel equation. FollowingRef. 12, the Usadel equation compactly reads: D (cid:104) ∂, G ( r , (cid:15) ) (cid:2) ∂, G ( r , (cid:15) ) (cid:3)(cid:105) + i (cid:104) (cid:15)ρ + diag (cid:2) h ( r ) · σ , (cid:0) h ( r ) · σ (cid:1) T (cid:3) , G ( r , (cid:15) ) (cid:105) = 0 , (10)in which ρ and σ = (cid:0) σ x , σ y , σ z (cid:1) are × and × Pauli ma-trices, respectively, and D represents the diffusive constant ofthe magnetic region. The quasiclassical approach employedin this section supports ferromagnets with arbitrary exchangefield directions; h ( r ) = (cid:0) h x ( r ) , h y ( r ) , h z ( r ) (cid:1) . In Eq. (10), G represents the total Green’s function which is made of Ad-vanced ( A ), Retarded ( R ), and Keldysh ( K ) blocks. There-fore, the total Green’s function can be expressed by: G ( r , (cid:15) ) = (cid:18) G R G K G A (cid:19) , G R ( r , (cid:15) ) = (cid:18) G F−F ∗ −G ∗ (cid:19) . (11)In the presence of ferromagnetism, the components of ad-vanced block, G A ( r ) , of total Green’s function G can be writ-ten as: F ( r , (cid:15) ) = (cid:18) f (cid:20) f ↑↓ f ↓↑ f (cid:21) (cid:19) , G ( r , (cid:15) ) = (cid:18) g (cid:20) g ↑↓ g ↓↑ g (cid:21) (cid:19) . (12)In this paper, however, we assume stationary conditions forour systems under consideration, and hence the three blockscomprising the total Green’s function are related to each otherin the following way: G A ( r , (cid:15) ) = − (cid:2) ρ G R ( r , (cid:15) ) ρ (cid:3) † , and G K ( r , (cid:15) ) = tanh( β(cid:15) ) (cid:2) G R ( r , (cid:15) ) − G A ( r , (cid:15) ) (cid:3) , where β ≡ k B T / .The SF interface controls the proximity effect. There-fore, appropriate boundary conditions should be consideredto properly model the system. In our work, we considerthe Kupriyanov-Lukichev boundary conditions at the SF interface which controls the induced proximity correlationsusing a parameter ζ as the barrier resistance: ζG ( r , (cid:15) ) ∂G ( r , (cid:15) ) = [ G BCS ( θ, (cid:15) ) , G ( r , (cid:15) )] . (13)The solution for a bulk even-frequency s -wave superconduc-tor G R BCS reads, ˆ G R BCS ( θ, (cid:15) ) = (cid:18) cosh ϑ ( (cid:15) ) iσ y sinh ϑ ( (cid:15) ) iσ y sinh ϑ ( (cid:15) ) − cosh ϑ ( (cid:15) ) (cid:19) , (14)where ϑ ( (cid:15) ) = arctanh ( | ∆ | /(cid:15) ) .The system local density of states, N ( r , (cid:15) ) , can be ex-pressed by the following equation: N ( r , (cid:15) ) = N Re (cid:104) Tr (cid:8) G ( r , (cid:15) ) (cid:9)(cid:105) , (15)in which N is the density of state normal state. III. RESULTS AND DISCUSSION
In this section, we describe our results. We start with thosefor a ballistic
SF F structure and then present the predictionsof Usadel formalism for diffusive samples.
A. Ballistic Regime
In this subsection we present the self-consistent results forthe ballistic regime. The numerical method used here to it-eratively solve in a self consistent way Eqs. (1) and (4) hasbeen extensively described elsewhere, and details neednot be repeated here. In the calculations, the temperature T is held constant at T = 0 . T c , where T c is the transi-tion temperature of a pure bulk S sample. All length scalesare normalized by the Fermi wavevector, so that the coordi-nate x is written X = k F x , and the F and F widths arewritten D F i ≡ k F d F i , for i = 1 , . The ferromagnet F and superconductor are set to fixed values, corresponding to D F = 400 , and D S = 600 , respectively. We also assume acoherence length corresponding to k F ξ = 100 . One of ourmain objectives in this paper is to study the triplet correla-tions, which are odd in time. To accomplish this, we employthe expressions in Eqs. (6) and (7), which describe the spa-tial and temporal behavior of the triplet amplitudes. At t = 0 the triplet correlations vanish because of the Pauli exclusionprinciple. At finite t , the triplet correlations generated nearthe S/F interface tend to increase in amplitude and spreadthroughout the structure. We normalize the time t accordingto τ = ω D t , and we set it to a representative value of τ = 4 .We can then study the behavior of the triplet amplitudes f and f throughout the junction. To explore the proximity inducedsignatures in the single-particle states, which is the main pur-pose of this work, we then present a systematic investigationof the experimentally relevant local DOS. All DOS resultspresented are local values taken at a fixed position near theedge of the sample in the F region. We characterize inter-face scattering, when present, by delta functions of strength H , which we write in terms of the dimensionless parameter H B ≡ H/v F . Finally, we use natural units, e.g., (cid:126) = k B = 1 throughout. Triplet and singlet pair correlations
Here we present results for both the triplet and singlet cor-relations, calculated using Eqs. (6)-(7). For the cases shownbelow, the absolute value of the singlet and triplet complexquantities are averaged over the region of interest, which inthis case is the experimentally probed F region. An impor-tant reason for focusing on those spatially averaged (over theouter magnet) quantities, rather than the spatial profiles dis-cussed in Ref. 37, is that it was experimentally shown thatthese triplet averages perfectly anticorrelate with the transi-tion temperatures, i.e. the spin valve effect. We also normal-ize all pair correlations to the value of the bulk singlet pairamplitude. We begin by showing, in Fig. 2, the spatially aver-aged absolute value of the complex triplet amplitudes | f , avg | (with spin projection m = 0 ), and | f , avg | (with spin pro-jection m = ± ) along with the singlet | f , avg | , (note that f ( x ) ≡ ∆( x ) /g ( x ) ) as functions of D F . Each row of pan-els corresponds to a different exchange field value: from top tobottom rows, we have h/ε F = 0 . , . , and . . Examiningthe opposite spin correlations, f and f , damped oscillatorybehavior with D F is evident: this is related to the spatial os-cillation of the Cooper pairs due to their acquiring a center ofmass momentum when entering the magnet. Therefore, thewavelength of these oscillations varies inversely with the ex-change field in F (this is why the D F range for the weakerexchange fields is extended). Quantum interference effectsgenerate peaks in f and f that occur approximately when D F f avg D F f avg -2 x10 -2 D F f avg -2 (a) (b) (c) D F f avg D F f avg -2 x10 -2 D F f avg -2 (d) (f)(e) D F f avg D F f avg -2 x10 -2 D F f avg -2 (i)(h)(g) FIG. 2. (Color online) The absolute value of the normalized tripletand singlet pair correlations, averaged over the F region, as a func-tion of D F . The exchange field strengths are (from top to bottom): h/ε F = 0 . , . , . . The relative exchange field orientations areorthogonal, with β = π/ , and β = 0 . d F /ξ F = nπ , (i.e. D F = nπ ( h/ε F ) − ). In the ballisticregime, the length scale that characterizes the damped oscil-lations is ξ F = v F / (2 h ) , where v F is the Fermi velocity. Theequal-spin amplitudes f , are seen to behave oppositely, witha phase offset of approximately π/ . Their magnitude de-clines more rapidly with D F , compared to the behaviors of f and f . This is consistent with f triplet generation beingoptimal for highly asymmetric ferromagnetic layer widths. It is notable that the periodic occurrence of peaks in f whenvarying D F , evolves into a single maximum as h is reducedfurther.One of the strengths of the microscopic BdG formalismis having the ability to properly include the full microscopicrange of length and energy scales inherent to the problem.This includes the exchange energy h , which in our BdGframework can span the limits from a nonmagnetic normal h F f avg h F f avg -2 x10 -2 h F f avg -2 (c)(b)(a) h F f avg h F f avg -2 x10 -2 h F f avg -2 (f)(e)(d) h F f avg h F f avg -2 x10 -2 h F f avg -2 (i)(h)(g) FIG. 3. (Color online) The spatially averaged (in F ) normalizedtriplet and singlet pair correlations as a function of h/ε F . As inFig. 2, the magnitude of each quantity is taken and averaged overthe F region. Each row of panels corresponds to a different F width, with D F = 15 (top row), D F = 10 (middle row), and D F = 5 (bottom row). The relative exchange field orientations areorthogonal, with β = π/ , and β = 0 . metal ( h/ε F = 0 ) to a half-metallic ferromagnet ( h/ε F = 1 ).It is particularly useful to consider the behavior of the sin-glet and triplet correlations over this broad range of strengthsof h/ε F . Thus, in Fig. 3, we show the same quantities asFig. 2, plotted now as a function h/ε F . Again, we have or-thogonal relative exchange field orientations, with β = π/ ,and β = 0 . Each three-panel row corresponds to a differ-ent F width: D F = 15 , , and (from top to bottom).The central column reveals that the averaged equal spin am-plitudes | f , avg | displays regularly occurring prominent peaks,the number of which varies with the length of the F region.For the exchange fields and F widths considered in Fig. 2,the triplet f was generally weaker than either the singlet f or triplet f . For the system parameters used in Fig. 3 how- f avg f avg -2 x10 -2 f avg -2 h / F = 0.1 h / F = 0.5 h / F = 0.35 h / F = 1 (c)(a) (b) FIG. 4. (Color online) Plots of the averaged singlet and triplet com-ponents as a function of magnetic orientation β . Here D F = 10 ,and results for several magnetic strengths are shown, ranging fromweak to half-metallic. ever, we find that for narrow widths D F and sufficiently largeexchange fields, the equal-spin triplet component f can dom-inate the other pair correlations. In particular, for strong ferro-magnets with h/ε F ≈ . , and thin F layers with D F = 5 ,panels (g) and (i) illustrate that the f and f amplitudes con-sisting of opposite spin pairs, are negligible due to the pairbreaking effects of the strong magnet. On the other hand, theequal-spin pairs shown in panel (h) are seen to survive in thislimit. This has important consequences for isolating and mea-suring this triplet component in experiments.Having seen how the magnitude of the exchange field h affects the singlet and triplet correlations, we next investi-gate the effects of changing its direction. Therefore, we ex-amine in Fig. 4, the behavior of the averaged singlet andtriplet amplitudes when changing the magnetic orientation an-gle, β . We again consider a broad range of exchange fieldstrengths, as shown in the legend. One of the more obviousfeatures is that the maximum of | f , avg | typically does notoccur for orthogonal relative exchange fields, for smaller β (cid:46) ◦ , especially for stronger magnets. This is in agree-ment with previous experimental and theoreticalresults. Due to the non-monotonicity of | f , avg | with h [seeFig. 3(e)], the h/ε F = 0 . case seen in Fig. 4(b) is larger forall β than for the weaker h/ε F = 0 . case. The singlet f and triplet f amplitudes are largest for antiparallel configu-rations ( β = 180 ◦ ), where the opposite exchange fields areeffectively weakened, with reduced spin-splitting effects onthe opposite-spin Cooper pairs. This is a well-known result.The results also show that the relative magnetic orientationangles leading to the minima of these two quantities are an-ticorrelated with the angles at which the f correlations aremaximal. As seen in Figs. 4(a) and (c), | f , avg | and | f , avg | decay much more abruptly as the value of h in the magnets ap-proaches the half-metallic limit: this is consistent with the dis-cussions above. Therefore, SF F structures involving strongferromagnets ( h ∼ ε F ) with β at or near orthogonal ori-entations, can host larger generated triplet pair correlationswhereby | f | (cid:29) {| f | , | f |} , thus allowing for direct prob-ing of the spin triplet superconducting correlations in experi-ments. X R e f ] -0.04-0.020.000.020.040.06 35710 D F X R e f ] -0.03-0.02-0.010.000.010.02 (a)(b) FIG. 5. (Color online) Local spatial profiles of the real parts of thetriplet components f and f in the F region for a few different F widths, D F . The exchange field in the ferromagnets corresponds to h/ε F = 0 . , and the relative exchange field orientations are orthog-onal, with β = π/ , and β = 0 . More detailed information regarding the triplet amplitudes,can be obtained from the spatial profiles of the local tripletcorrelations within the F region. In Fig. 5, we present thereal parts of the normalized f ( x ) and f ( x ) triplet compo-nents in terms of the dimensionless coordinate X . Results areplotted at four different values of D F as indicated in the leg-end. The exchange fields in the ferromagnets has magnitudecorresponding to h/ε F = 0 . , and the directions are mutuallyorthogonal, with β = 90 ◦ , and β = 0 . For the time scaleconsidered here, the imaginary part of f is typically muchsmaller than its real part. As to f , its imaginary part is usuallynot negligible, but it exhibits trends that are similar to thosefor the real part. Examining the top panel, it is evident that f exhibits the trademark damped oscillatory spatial depen-dence arising from the difference in the spin-up and spin-downwavevectors of the Cooper pairs. The oscillatory wavelengthis thus governed by the quantity πk F ξ F = 2 π ( h/ε F ) − ,which for our parameters corresponds to π . The modulat-ing f has the same wavelength for each D F , although eachcurve can differ in phase. The averaged f amplitudes are con-sistent with this local behavior: Fig. 2(a) demonstrated thatwhen D F ≈ and D F ≈ , there is an enhancementof the f component, while for D F ≈ , it is substantiallyreduced. The equal-spin f amplitudes, are shown in the bot-tom panel of Fig. 5. Near the interface at X = 0 , the f correlations are created, and then they subsequently increasein magnitude until deeper within the ferromagnet, where theyclearly exhibit a gradual long-ranged decay. The trends ob-served here are opposite to those in the top panel, where forinstance the D F = 3 case leads to maximal f triplet gener-ation, in agreement with Fig. 2(b). Local density of states
After the discussion of the salient features of the singletand triplet pair correlations in the outer F layer, we now turnto the main topic of the paper: the local density of statesmeasured in F . This is the experimentally relevant quan-tity that can reveal the signatures of these correlations. Thedamped oscillatory behavior of the pair correlations can leadto spectroscopic signatures in the form of DOS inversions, and multiple oscillations. In the quasiclassical approxima-tion, a ZEP can emerge from the long-range tripletcorrelations in SF F systems. However, this approxima-tion is not appropriate for experimental conditions involvingstrong magnets and clean interfaces. It would be beneficialexperimentally to characterize the ZEP relation to the singletand triplet correlations and see how the ZEP may be a use-ful fingerprint in identifying the existence of the long-rangetriplet component. To properly do this over the broad range ofparameters found in experimental conditions, a microscopicself-consistent theory that can accommodate the wide rang-ing length and energy scales is needed. In this subsection, wetherefore present an extensive microscopic study of the ZEPas a function of parameters such as F layer thicknesses, ex-change energy, or interface transparency. These results arethen correlated with the self-consistent singlet and triplet paircorrelations in the previous subsection. In what follows, theDOS is normalized by the DOS at the Fermi level N F , andplotted vs the normalized energy ε/ ∆ , where ∆ is the bulkvalue of the pure S material gap at zero temperature. Our em-phasis will be on energies within the subgap region ε ≤ ∆ ,where the ZEP phenomenon arises. Since the DOS is a localquantity that depends on position [see Eq. (9)], in our calcula-tions we assume the location to be near the edge of the samplejust below the STM tip as shown in Fig. 1.To correlate the triplet correlations in Fig. 3 with the ZEP,we begin by studying in Fig. 6 the sensitivity of the DOS toa broad range of exchange field strengths h . Each panel cor-responds to a different F width, D F . The range of h con-sidered in each panel varies since the largest ZEP depends onthe relative values of h and D F . The top panel ( D F = 5 )clearly shows the progression of the ZEP with h : Beginningwith the smallest exchange field, h/ε F = 0 . , a moderatepeak is observed that increases to its maximum height and nar-rower width when h/ε F = 0 . . Further increases in h contin-uously diminish the ZEP, broadening its width, until eventu- -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S D F = 5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S D F = 10 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S D F = 15 h F (a)(b)(c) FIG. 6. (Color online) Normalized (see text) local DOS. The multi-ple curves in each panel are for different values of h/ε F . Each panelcorresponds to a different value of D F (as labeled). The ferromag-nets have exchange fields with orthogonal relative directions. ally it is effectively washed away. This non-monotonic behav-ior is consistent with the ZEP being related to the presence ofthe f triplet amplitude near the edge of the ferromagnet. Thiscan be seen by reexamining the triplet amplitudes in Fig. 3(h),where the exchange field leading to the highest ZEP occurswhen | f , avg | is largest, at h/ε F ≈ . . The same consistencyis found between Figs. 2(e) and (b) and the middle and lowerpanels of Fig. 3, respectively. For both the D F = 10 , cases, the average value of | f | is largest near h/ε F = 0 . .However, the secondary peak structure in Fig. 2 is not clearlyrefected in the DOS.Next we study the DOS counterpart to Fig. 2. The normal-ized DOS, and the corresponding ZEP, are shown in Fig. 7 for -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S h / F = 0.05 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S h / F = 0.1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S h / F = 0.5 D F D F D F (a)(b)(c) FIG. 7. (Color online) Normalized local DOS as a function of thenormalized energy. The curves in each panel are for different val-ues of the width D F . Each panel corresponds to a different h/ε F : h/ε F = 0 . , . , . , and we consider orthogonal relative ex-change fields. a broad range of widths D F . The parameter values here aresimilar to those used in Fig. 2, where each panel correspondsto a different exchange field. In panel (a) with h/ε F = 0 . ,the most prominent ZEP occurs for D F = 25 , coincidingwith the F width that yields a local maximum for the m = 0 triplet amplitude f (see Fig. 2(g)). By comparison, the f component observed in Fig. 2(h), is smaller and lacks the mul-tiple peak structure found for f , at this weaker exchange field.Therefore, the largest ZEP in the case of weak exchange fields,does not necessarily occur when the triplet f is maximal; asFig. 2(h) demonstrated, | f , avg | peaks at D F = 10 beforerapidly declining. For these weaker fields, it follows fromFig. 2 that the magnitude of f exceeds that of f . It wouldappear then that it is the larger triplet component which de- -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S FIG. 8. (Color online) Variation of the normalized local DOS withthe in-plane exchange field angle β . The exchange field is fixedalong z in F . Also, D F = 10 , and h/ε F = 0 . . termines the ZEP structure. This is consistent with the knownresult that the total value of the triplet component is corre-lated with T c . The next case in panel (b) corresponds also toa moderately weak magnet with h/ε F = 0 . , or double theexchange field considered in panel (a). Since the frequencyof the oscillations involving the opposite-spin f amplitudes[see Fig. 2(d)] also doubles, the maximum ZEP at D F = 12 ,occurs at about half the F width found for the maximumZEP in (a). The equal-spin triplet correlations f were seenin Fig. 2(e) to exhibit a single peak structure, but their magni-tude is larger than at weaker fields. This is because typicallystronger magnets in this situation lead to an enhancement ofthe f amplitudes. Lastly, we consider (bottom panel) a rel-atively strong ferromagnet with h/ε F = 0 . . For this case,there are additional subgap peaks flanking the main ZEP. Thelarger ZEP arises at smaller widths ( D F = 7 . , . ) than forweaker exchange fields, due to an increase in the frequency ofthe oscillations as a function of D F for the f and f com-ponents as seen in Fig. 2(a) and (b). Thus, the ZEP tends toexhibit a structure that dampens and widens for strong mag-nets, while the opposite is true for weaker ones and is cor-related with the stronger of the m = 0 and m = ± tripletcomponents present.Having established the behavior of the ZEP for differing h/ε F , we now fix the magnitude of the exchange fields ineach magnet and investigate the effects of varying their rela-tive orientation. Figure 8 illustrates the normalized DOS forthe specific case D F = 10 , and h/ε F = 0 . . Accordingto Fig. 4(b), the equal-spin triplet component f is greatestwhen β ≈ ◦ . Thus we would expect the ZEP to alsobe maximal at this angle. Figure 8 shows that this is indeedthe case. There the normalized DOS is shown for a range of ◦ ≤ β ≤ ◦ in increments of ◦ . Clearly the orthog-onal relative exchange field ( β = 90 ◦ ) configuration resultsin the most prominent ZEP. When β deviates from this angle / -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 D O S H B FIG. 9. (Color online) Evolution of the ZEP with scattering strength H B : The normalized local DOS is shown as a function of the di-mensionless energy. Each curve depicts results for a different scat-tering strength H B (see text). The system parameters correspond to D F = 10 , and h/ε F = 0 . . The exchange fields in the ferromag-nets are mutually orthogonal with β = 90 ◦ , and β = 0 ◦ . towards the P ( β = 0 ◦ ) or AP ( β = 180 ◦ ) alignments, boththe triplet amplitude f and the ZEP decline until β = 0 ◦ or ◦ , whereby f = 0 , and the ZEP has vanished.Finally, in Fig. 9 we examine the effects of interface scat-tering on the self-consistent energy spectra. We assume thateach interface has the same delta function potential barrierwith dimensionless scattering strength H B . We consider abroad range of H B , from transparent interfaces with H B = 0 ,to very high interfacial scattering, with H B = 1 . . By allow-ing H B to vary, we effectively control the proximity effects:a small H B results in stronger proximity coupling betweenthe F and S regions, while a large H B results in isolation ofeach segment, and weak proximity effects. This is evident inthe DOS, as seen in Fig. 9, which has its largest ZEP when H B = 0 . The width and height of the ZEP is strongly in-fluenced by the presence of interface scattering. Increasing H B results in the ZEP widening while gradually diminishingin height. Eventually, when the scattering strength reaches H B ≈ . , the peak begins to split. Further increments in H B causes the peaks to separate and eventually proximity ef-fects are so weakened that the DOS becomes that of an iso-lated bulk ferromagnet. The two secondary subgap peaks thatlie symmetrically about the ZEP are seen to also decline in amonotonic fashion as H B becomes larger. B. Diffusive Regime
In this section, we consider a diffusive
SF F junction inthe full proximity limit. We employ the Usadel approach de-scribed in Sec. II to investigate the local DOS. As remarked earlier, the quasiclassical method is limited to energies closeto the Fermi level. Hence, our discussion here will be limitedto relatively weak ferromagnets. As in the ballistic regime,we consider heterostructures where the magnetic layers aremade of identical materials so that the ferromagnetic coher-ence lengths are the same, ξ F = ξ F ≡ ξ F , and we considerthe low temperature regime where T = 0 . T c . Prior to cal-culating the DOS, we normalize the Usadel equation by ξ F ,which in the diffusive regime is written, ξ F = (cid:112) D/h . Us-ing this normalization scheme, the explicit dependency on theexchange field is removed and the Usadel equation now in-volves terms containing the ratio d F /ξ F . This approach canlead to easier pinpointing of regions in parameter space wherethe ZEP is most prominent, and it also permits a broad rangeof this ratio to be studied. We assume that the magnetic orien-tation angle is fixed at β = 0 , or equivalently h = (0 , , h z ) .We numerically solve the Usadel equation, Eq. (10), to-gether with the mentioned boundary conditions. To find thetotal Green’s function, we substitute the solution into Eq. (10)and obtain the DOS. To determine the optimal geometry inwhich the ZEPs are most pronounced, we present in Fig. 10the ZEP at the topmost edge of the SF F structure, corre-sponding to the location x = d F + d F . The two-dimensionalcolor mapping depicts the strength of the DOS at zero en-ergy (the ZEP) as a function of the normalized F thicknesses, d F /ξ F and d F /ξ F . In the top row panels, the internal fieldof the F layers have a misalignment angle of β = π/ ,while for the bottom row β = π/ . The left, middle, andright columns are for different opacities at the SF interface: ζ = 1 . , . , and . , respectively. By increasing ζ , the over-all strength of the proximity effects is effectively weakened:it is evident that transparent SF contacts yield stronger ZEPs,that persist in thicker F layers. It is also apparent that theorthogonal case β = π/ has more extensive regions in theparameter space spanned by the F thicknesses with enhancedZEPs, as compared to the β = π/ case. An important as-pect of the ZEP that all cases investigated in Fig. 10 share, isthat it is strongest when d F (cid:28) d F . This finding is fully con-sistent with low proximity bilayer SFF hybrids. Therefore,for the parameters considered here, the ZEPs are strongest for ζ = 1 , . ξ F (cid:46) d F (cid:46) ξ F , and . ξ F (cid:46) d F (cid:46) . ξ F .The ratio of the F thickness to the length scale ξ F is an im-portant dimensionless quantity that appears in the normalizedUsadel equations, and consequently thinner d F and d F , al-low for stronger ferromagnets when studying the DOS in thediffusive limit.Finally, we study the sensitivity of the ZEPs to both the ori-entation angle β and interface transparency parameter ζ . Wethus show in Fig. 11 the ZEP as a function of β over a widerange of ζ , as shown in the legend. The geometric parameterscorrespond to d F = 0 . ξ F and d F = 3 . ξ F , which resideswithin the range of system widths studied in Fig. 10 resultingin the largest ZEPs. In calculating the ZEP, we again considerthe DOS at the edge of the sample (see also Fig. 1). It is seenthat the interface transparency can significantly alter the be-havior of the ZEP as the relative exchange field angle sweepsfrom the P ( β = 0 ◦ ) to AP ( β = 180 ◦ ) orientations. Forexample, when ζ = 1 . , the maximal ZEP is offset from the0 d F ξ F d F / ξ F d F / ξ F d F ξ F d F / ξ F ( d )( a ) ( b ) ( c )( f )( e ) FIG. 10. (Color online) Zero energy peakin the DOS spectrum of diffusive
SF F spinvalves as a function of the normalized F layerthicknesses d F /ξ F and d F /ξ F . Each col-umn corresponds to a different SF trans-parency, with ζ = 1 , . , (from left toright). The top row of panels shows the evo-lution of the ZEP for the misalignment angle β = π/ , while the bottom row of panelsare for β = π/ . For both cases, the in-ternal field of the F layer is along the z di-rection, β = 0 . The ZEPs are computed at x = d F + d F (at the top most F /vacuuminterface). β / π Z e r o E n e r gy P ea k ( ZE P ) ζ = FIG. 11. (Color online) Zero energy peak of the DOS spectrum indiffusive
SF F spin valves as a function of exchange field orientation β for several values of ζ , which controls the opacity of the SF interface. We set β = 0 and rotate the exchange field direction of F from the parallel ( β = 0 ) to antiparallel ( β = π ) orientations.We have chosen representative values of d F = 0 . ξ F and d F =3 . ξ F , in accordance with the system parameters used in Fig. 10. orthogonal configuration, occurring at β ≈ . π . By increas-ing the barrier strength, this peak shifts towards larger β , un-til the relative exchange fields are nearly antiparallel. Thereis also a simultaneous reduction in amplitude, due to the F and S regions becoming decoupled as the proximity effectsdiminish. Interestingly, as ζ increases, there is a splitting ofthe main peak: weaker secondary peaks emerge. Eventuallyhowever, for sufficiently large ζ , the opacity of the interfacecauses the low energy DOS to be insensitive to β , and theZEP flattens out. The ZEPs are also observed to disappearwhen the relative exchange fields are collinear, correspondingto the situation when the triplet amplitudes vanish in both thediffusive and ballistic regimes (see also Fig. 4). IV. SUMMARY AND CONCLUSIONS
In summary, we have employed a microscopic self-consistent wavefunction approach to study the low energyproximity induced local DOS in
SF F spin valves with non-collinear exchange fields in the clean limit. Our emphasis hasbeen on the results of STM methods that probe the outer F layer. To identify the physical source of the correspondingZEPs that occurs for such data in these systems, we also cal-culated the absolute value of the triplet pair correlations, av-eraged over the outer F layer. We have done so for a broadrange of experimentally relevant parameters, including the ex-change field strength and orientation, as well as thicknessesof the ferromagnets. Our results demonstrate a direct linkbetween the spin-1 triplet correlations and the appearance ofZEPs in the local DOS spectra, and point to system parame-ters and configurations which would support larger equal-spintriplet superconducting correlations. These correlations couldthen be probed indirectly via single-particle signatures that aremeasurable using local spectroscopy techniques. Our resultsare consistent with findings relating the average strength oftriplet correlations to the angular dependence of the transi-tion temperature. Our findings suggest that the ZEPs arisingfrom the spin-1 triplet amplitudes can be effectively isolatedin SF F systems with strong ferromagntets, with the outer onebeing very thin. This asymmetric geometry not only producesgreater equal-spin triplet generation, but it can also filter outthe rapidly decaying opposite-spin pairs deep within the sam-ple. We also considered the same valve structure in the diffu-sive regime utilizing a Green function method within the fullproximity limit. Our investigations yielded a broad range of F layer thicknesses and relative exchange fields orientationsthat lead to observable signatures in the low energy DOS, thusalso giving useful guidelines for future experiments. ACKNOWLEDGMENTS
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