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Fano-Kondo Spin Filter A. C. Seridonio1,2 , F. M. Souza∗1,3 , J. Del Nero4,5 and I. A. Shelykh1,6 1 International Center for Condensed Matter Physics, Universidade de Bras´ ılia, 70904-970, Bras´ ılia, DF, Brazil 2 Instituto de F´ısica, Universidade Federal Fluminense, 24310-346, Niter´i, RJ, Brazil o 3 Instituto de F´ ısica, Universidade Federal de Uberlˆndia, 38400-902, Uberlˆndia, MG, Brazil a a 4 Departamento de F´ ısica, Universidade Federal do Par´, 66075-110, Bel´m, PA, Brazil a e 5 Instituto de F´ ısica, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil 6 Science Department, University of Iceland, Dunhaga 3, IS-107, Reykjavik, IcelandAbstractThe spin-dependent scanning tunneling spectrum generated by a ferromagnetic tip on a metallic host witha Kondo adatom is discussed. The combined eﬀects of the Kondo screening, quantum interference, and tipmagnetism are studied. For short tip-adatom separations, the askew lineshape characteristic of quantuminterference in Kondo systems is split in two spin components. For the majority-spin component, the Fanoantiresonance is strongly enhanced. With the tip atop the adatom, a window of bias voltages is foundin which the tunneling current is nearly 100% spin polarized. In the vicinity of a Kondo adatom, theferromagnetic scanning tunneling microscope constitutes a powerful spin polarizer.1. Introduction electrode magnetizations, and on the external mag- netic ﬁeld [14, 15, 16, 17, 18]. Current technology The Kondo eﬀect is one of the most fascinat- allows the construction of single QD devices anding manifestations of strong correlations in elec- routine observation of all such phenomena [19, 20].tronic systems [1]. First observed as an enhance- In an alternative nanoscale setup, a magneticment of the resistance of dilute magnetic alloys atom is adsorbed on a metallic host and probed bybelow a characteristic temperature TK , the eﬀect a scanning tunneling microscope (STM) tip, eitherfound a more controlled environment in the early normal or ferromagnetic. The quantum interference90s, following the development of nanoscale minia- between currents ﬂowing from the tip to the hostturization techniques. Kondo eﬀects in quantum directly or by way of the adatom gives rise to a richdots (QDs) and QD arrays coupled to metallic or variety of transport regimes. In particular, at tem-semiconductor electrodes were reported [2]. A va- peratures below TK , in analogy with the distortionriety of mesoscopic phenomena associated with the of a Lorentzian into a antiresonance in conventionalnarrow resonance in the density of states of a QD Fano systems [21], the interference distorts the con-coupled to conduction-electron reservoirs were re- tribution of the Kondo resonance to the adatomported [3, 4, 5]. In particular, the Kondo resonance spectral function. Antiresonances with such charac-was shown to an enhance the conductance at low teristics have been widely reported in experimentstemperatures (T < TK ). with nonmagnetic tips [22, 23, 24, 25, 26] and dis- The mismatch between the electrode Fermi lev- cussed theoretically [27, 28, 29, 30].els resulting from the application of an externalbias splits the Kondo peak into two resonances The development of FM tips [31] makes adatom[6, 7, 8, 9, 10, 11, 12, 13]. If the electrodes are systems potentially important actors in the rapidlyferromagnetic, the large local exchange ﬁeld splits developing ﬁeld of spintronics [32]. Recently, fol-further the resonances. The separation between the lowing a report by Patton et al of a theoreticalpeaks now depends on the relative directions of the study of the conductance for a ferromagnetic tip in the Kondo regime [33], the present authors inves- tigated the dependence of the conductance on the ∗ Corresponding Author: fmsouza@inﬁs.ufu.br tip-adatom separation in a full range of the Fano pa-Preprint submitted to Physica E: Low Dimensional Systems and Nanostructures September 14, 2009
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rameters, deﬁned as the ratio between tip-adatomand tip-host couplings [34]. Here, we analyze a spin-resolved conductance forFM tip-adatom-host metal system in the weak-coupling regime, in which tunneling amplitudescan be treated as perturbations. To account forthe Kondo eﬀect, we refer to the Doniach-Sunjic-like expression for the adatom spectral function[35, 36, 37, 38]. We ﬁnd that very weakly mag-netized tips can generate strongly spin-polarizedcurrents [39, 40]. For certain conﬁgurations, in anappropriate bias interval, the current can be fullyspin-polarized, an eﬀect due to the suppression ofone spin component by the spin splitting of the Figure 1: Scanning Tunneling Microscope (STM) deviceFano-Kondo resonance. The inﬂuence of the tip- with a ferromagnetic (FM) tip with an adsorbed magneticadatom separation on this spin-polarizing device is atom (Kondo adatom) on the surface of a normal metal (Host surface). In the Host surface, the adatom site (c0 ) is hy-also investigated. bridized with the metal conduction band (f0 ) through the hopping v. The interference between the hopping elements t0R and t leads to the Fano-Kondo proﬁle in the conduc-2. Theoretical Model tance. We consider the following model Hamiltonian We focus the Kondo regime, which favors the H = HSIAM + Htip + HT , (1) formation of a localized magnetic moment at the adatom, i. e., the conditions ε0 < εF , ε0 + U > εFwhere (Htip ) represents the ferromagnetic tip, the and Γ |ε0 |, ε0 + U . At temperatures T TK ,HT , the host-tip coupling, and (HSIAM ) is the spin the antiferromagnetic correlations introduced bydegenerate Anderson Hamiltonian representing the the coupling to the host electrons screen the impu-host and the magnetic impurity coupled to it [41]: rity moment and add a narrow resonance, of half- width kB TK , centered at the Fermi level, to the HSIAM = εp a† apσ + ε0 c† c0σ adatom spectral density. The Kondo temperature pσ 0σ pσ σ is approximately given by the expression [42] + U n0↑ n0↓ ΓU † kB TK = exp[πε0 (ε0 + U )/2ΓU ]. (4) + v f0σ c0σ + H.C. , (2) 2 σ We neglect the electron- electron interactions inwhere apσ is a fermionic operator for a conduc- the FM tip and writetion electron with momentum p and spin σ; c0σ Htip = (εk + eV )b† bkσ , (5)is a fermionic operator for an electron localized on kσadatom state; and ε0 and U are the energy of the kσadatom level and Coulomb repulsion between two where the bkσ ’s are he tip conduction states and Velectrons with opposite spins at the adatom site, is the tip bias voltage. For a ferromagnetic tip, therespectively. spin-dependent density of states is For simplicity, we consider a structureless con-duction band with density of states ρ0 . The half ρσ = ρ0 (1 + σp) , tip (6)bandwidth is denoted D. To describe the hybridiza- where p is the magnetization. The tip-host-adatomtion between the band and the impurity, we intro- tunneling Hamiltonian readsduced the shorthand 1 f0σ = √ apσ , (3) HT = b† t ϕp (R)apσ + t0R c0σ N p kσ kσ pwhere N is the number of conduction states. + H.C., (7) 2
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where t and t0R = t0 e−kF R are the tip-host and results from a competition between the two con-tip-adatom hopping amplitudes, respectively. Fol- duction paths. As explained in detail in Ref. [34],lowing [29], we let t0R decay exponentially with the Eq.(10) can be rewritten astip-adatom lateral distance R. The amplitude t de- 2pends on the conduction wave function ϕp (R) at Fk F R 2 ρσ (eV ) = ρ0 0R 1− − (˜R ) qthe tip position R. ρ0 × πΓρσ (eV ) ad3. The Conductance Formula FkF R + 2Γ qR ˜ GRet (eV ) adσ , ρ0 To derive an expression for the conductance, we (14)treat the tunneling Hamiltonian (7) to linear or-der in perturbation theory [27, 28, 29, 30] under wheretypical experimental conditions eV D. The low-temperature conductance is then FkF R qR = qR + ˜ q (15) ρ0 ρσ tip ρσ 0R G(eV, T TK , R) = G0 , (8) is an eﬀective Fano parameter, σ ρ0 ρ0 1where k eV −εk 1 eV + D q= = ln (16) πρ0 π eV − D 2 e2 G0 = (2πtρ0 ) (9) is the Fano parameter for the Anderson Hamilto- h nian (2) decoupled from the tip [27, 28, 29, 30] andis the background conductance and 1 Fk F R 2 ρσ (eV ) ˜ ρσ (eV ) ad = − GRet (eV ) adσ ρσ (eV ) = ρ0 1− + 0R (10) π 0R 1 ρ0 ρ0 1 iΓK 2 =is the density of states of the host-adatom-tip sys- πΓ (eV + σδ) + iΓKtem. The function 1 = sin2 δeV σ (17) πΓ Fk F R = ϕk (R)δ(εk −εF ) = ρ0 J0 (kF R)(11) is the adatom spectral density, here expressed in k three alternative ways: as the imaginary part of thewhere J0 is the Bessel function of order zero, ac- interacting adatom Green’s function GRet (eV ),; by adσcounts for the dependence of the conductance on an analytical Doniach-Sunjic-like expression [35, 36,the tip-adatom separation R. The last term within σ 37, 38]; or by the conduction-state phase shift δeV ,curly brackets on the right-hand side of Eq. (10) is respectively. The ferromagnetic exchange interaction between FkF R ρσ (eV ) = ˜0R | m| f0σ the tip and the magnetic adatom [14, 17] lifts the m ρ0 spin degeneracy of the adatom level ε0 and hence + πρ0 vqR c0σ |Ω |2 splits the Kondo resonance into two peaks sepa- rated by an energy δ. A “poor man’s” scaling anal- × δ(Em − EΩ − eV ), (12) ysis [14, 33] yields an estimate of δ:where |Ω (|m ) is the ground-state (an eigenstate) δ = ε0↓ − ε0↑of the Anderson Hamiltonian (2). To highlight the ↑ ↓interference between the tunneling paths t0R and t γtip + γtipbuilt in Eq. (10), we have deﬁned the Fano param- = p ln(D/U ) 2πeter [29] γtip = ln(D/U ) p exp (−2kF R) , (18) π −1 t0R qR = (πρ0 v) = qR=0 e−kF R . (13) where t σ 2With qR 1 (qR 1) the tip-adatom (tip-host) γtip = πρσ |t0R | tipcurrent is dominant. With qR ≈ 1, the current = γtip (1 + σp) exp (−2kF R) . (19) 3
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Since eV D, the right-hand side of Eq.(16) 1.0 p=1is very small, which shows that it is safe to neglect 0.8 p=0.8the last term on the right-hand side of Eq.(15). The 0.6 p=0.6 Spin polarizationspin-resolved conductance is therefore 0.4 p=0.4 0.2 σ ρσ tip ρσ (eV ) 0.0 p=0.2 G /Gmax = , (20) p=0 ρ0 ρ0 -0.2 -0.4where -0.6 2 Gmax = 1 + qR G0 (21) -0.8 -1.0and -40 -30 -20 -10 0 10 20 30 40 Bias voltage ρσ (eV ) = ρσ (eV )/ 1 + qR 0R 2 (22) Figure 2: Spin polarization (℘) at R = 0 as a function of To simplify Eq.(20), we now let the equivalence the bias voltage eV in units of the Kondo half-width ΓKtan δqR ≡ qR deﬁne a Fano-parameter phase shift for qR=0 = 1 and tip magnetizations ranging from p = 0δqR and note that to p = 1. For nonzero p’s, ℘ reaches a unitary plateau in a magnetic polarization dependent range centered at eV = σ GRet (eV ) adσ 0. At negative voltages, the spin polarization dips to ℘ = tan δeV = − . (23) −1. Following Refs. [28] and [33], we have set ε0 = −0.9eV, GRet (eV ) adσ γtip = Γ = 0.2eV, U = 2.9eV, D = 5.5eV, TK = 50K, and kF = 0.189˚−1 . AFor R = 0 we then ﬁnd that Gσ /Gmax = (1 + σp) cos2 (δeV − δqR=0 ) , (24) σ The tip voltage V controls the spin polarizationwhich gives access to the spin polarization of the of the current, which can be tuned from −1 totunneling current: +1. The literature contains a number of proposed spin ﬁlters [43, 44, 45, 46] and spin diodes [47], a G↑ − G↓ few of which have been constructed in the labora- ℘= . (25) G↑ + G↓ tory [48]; dynamic control of the current polariza- tion by time-dependent bias voltages has also been4. Results proposed [49, 50]. To the best of our knowledge, however, ours is the ﬁrst demonstration that the To illustrate our discussion we have chosen the interplay between the Kondo eﬀect and quantumfollowing set of model parameters: qR=0 = 1, ε0 = interference can control and amplify the spin polar-−0.9eV, γtip = Γ = 0.2eV, U = 2.9eV, D = 5.5eV, ization of the current.TK = 50K and kF = 0.189˚−1 [28, 33]. The Kondo A Figure (3) shows the spin-resolved conductancestemperature was estimated using Eq.(4). G↑ and G↓ . For p = 0 we ﬁnd G↑ = G↓ . The Figure (2) shows the spin polarization ℘ as a interference between the two tunneling paths ac-function of the bias voltage for diﬀering tip mag- counts for the asymmetric Kondo resonance, a linenetization parameter p. As expected, ℘ vanishes shape frequently found with nonmagnetic metallicfor p = 0. For p = 0, two features of ℘ stand out: tips [22, 23, 24, 25, 26]. As p grows, the conduc-(i) a plateau ℘ ≈ +1, with p-dependent width, cen- tances become increasingly distinct: G↑ (G↓ ) shiftstered at eV = 0; and (ii) a narrow dip, which sinks to the left (right) and opens a window of completelyto -1, centered at a negative, p-dependent bias. As spin polarized conductance with G↓ ≈ 0, which cor-p is increased, the plateau broadens, while the dip responds to the ℘ = 1 plateau in Fig. (2).is shifted toward negative biases. For p = 1 (fully The narrow ℘ = −1 dip in Fig. (2) stems frompolarized tip), again as expected, the spin polar- the vanishing of G↑ at a negative voltage in eachization curve saturates at ℘ = 1. Remarkably, the panel of Fig. (3). As the ferromagnetic polarization100% polarization plateau persists even at weak tip p increases, while the amplitude of G↑ increases, G↓magnetizations, e. g., p = 0.2. For suﬃciently high is suppressed. This contrast is due to the densityvoltages, the spin polarization becomes constant, of states at the tip: for σ = −1 the prefactor on℘ = p. the right-hand side of Eq.(24) vanishes for p → 1, 4
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G /G max 1,0 2 (b) up (a) down 0,8 R=0 1 R=0 R=1A 0,6 R=2A Spin Polarization 0,4 0 -40 -20 0 20 40 2 (c) 0,2 R=1A 0,0 1 V V V -0,2 1A 2A 0 -40 -20 0 20 40 2 -0,4 (d) -0,6 R=2A 1 -0,8 -1,0 0 -40 -20 0 20 40 -40 -20 0 20 40 Bias VoltageFigure 3: Spin-resolved conductances (24) for tip atop the Bias voltageadatom (R = 0), as functions of the bias voltage eV in unitsof the Kondo half-width ΓK for qR=0 = 1. For a nonmag- Figure 4: (a) Spin polarization (℘) as a function of biasnetic tip (p = 0) the identical spin resolved conductances G↑ voltage eV in units of the Kondo half-width ΓK for ﬁxed tipand G↓ display the Fano-Kondo line shape. A ferromagnetic magnetization p = 0.4, qR=0 = 1 and diﬀering tip-adatomtip (p = 0) displaces the spin-resolved antiresonances to dif- lateral displacements R. Large separations R wash out theferent energies. A window of energies in which G↓ ≈ 0 while marked structure of the spin polarization visible at R = 0,G↑ is maximum results. In this region, the spin polarization and force the polarization into coincidence with the ferro-℘ is unitary. Model parameters as in Fig. (2). magnetic polarization p. Panels (b)-(d) show the evolution of the spin resolved conductances, G↑ and G↓ , with the dis- tance. As R increases, the energy splitting between the G↑ and G↓ antiresonances and the prominence of each antireso- nance is reduced. A sketch of the experimental setup appears in the inset. Model parameters as in Fig. (2).which forces G↓ → 0. 5. Conclusions Finally, Fig. (4) shows the spin polarization ℘as a function of the tip-adatom separation R. For We studied theoretically the spin-resolved con-large R , the minimum and the maximum ℘ are ductance for a ferromagnetic STM tip coupled to anreduced, the current polarization ℘ is governed by adatom on a nonmagnetic metallic host. For eachthe tip polarization p, and the polarization curve spin component, the spin-resolved conductance dis-becomes ﬂat. To explain the behavior at the po- plays a Fano antiresonance. The ferromagnetic po-larization at smaller separations, the (b)-(d) on the larization nonetheless tends to shift the two antires-right-side of the ﬁgure show the spin-resolved con- onances to diﬀerent energies. A window of bias volt-ductances as functions of the bias voltage at three ages arises in which G↓ ≈ 0, while G↑ is maximum.diﬀerent separations R behavior. The two strongly In that window, the spin polarization ℘ remainsasymmetric Fano-Kondo resonances in panel (b) close to unity. The application of an appropriately(R = 0) become less pronounced as R increases; tuned bias voltage can therefore produce fully spinmoreover, the splitting between G↑ and G↓ is re- polarized currents. A ferromagnetic STM tip there-duced. As R increases, therefore, the resulting po- fore constitutes a tunable spin polarizer.larization loses its structure and approaches a con-stant. We note that Ref. [22], which studied an 6. Acknowledgmentsunpolarized tip, reported a similar R dependenceof the Fano-Kondo line shape: a Fano antireso- We thank L. N. Oliveira for revising the text ofnance for R ≈ 0, a less pronounced antiresonance the ﬁnal version and M. Yoshida for fruitful discus-for intermediate R and an approximately ﬂat pro- sions. This work was supported by the Brazilianﬁle (background) for large R, a trend analogous to agencies IBEM, CAPES, FAPESPA and FAPERJ.the evolution of the solid curves in panels (b)-(d)and also to the evolution of the dashed curves. Dueto the ferromagnetism of the tip, however, G↑ is Referencesdominant over G↓ ; the resulting polarization ℘ thus [1] A. C. Hewson, The Kondo Problem to Heavy Fermions,approaches the background polarization p. Cambridge University Press, Cambridge (1993). 5
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