Non-equilibrium spin-current detection with a single Kondo impurity
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Non-equilibrium spin-current detection with a single Kondo impurity
Jong Soo Lim
Institut de F´ısica Interdisciplin`aria i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain andSchool of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Rosa L´opez
Institut de F´ısica Interdisciplin`aria i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain andDepartament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
Laurent Limot
Institut de Physique et Chimie des Mat´eriaux de Strasbourg, Universit´e de Strasbourg, CNRS, 67034 Strasbourg, France
Pascal Simon
Laboratoire de Physique des Solides, CNRS UMR-8502, Univ. Paris Sud, 91405 Orsay Cedex, France (Dated: March 6, 2018)We present a theoretical study based on the Anderson model of the transport properties of a Kondo impurity(atom or quantum dot) connected to ferromagnetic leads, which can sustain a non-equilibrium spin current. Weanalyze the case where the spin current is injected by an external source and when it is generated by the voltagebias. Due to the presence of ferromagnetic contacts, a static exchange field is produced that eventually destroysthe Kondo correlations. We find that such a field can be compensated by an appropriated combination of thespin-dependent chemical potentials leading to the restoration of the Kondo resonance. In this respect, a Kondoimpurity may be regarded as a very sensitive sensor for non-equilibrium spin phenomena.
PACS numbers: 72.10.Fk, 72.15.Qm,73.63.-b,68.37.Ef
I. INTRODUCTION
In the last decades, there has been a revived interest inKondo physics. This many-body effect is produced by high-order correlated tunneling events consisting of electronicspins hopping in and out a localized impurity, which ulti-mately lead to an efficient screening of the impurity spin. TheKondo effect has been extensively investigated for the anoma-lous behavior it produces on the resistivity versus temperaturein bulk metals possessing magnetic impurities. Experimen-tal advances allow nowadays probing the Kondo effect in sin-gle objects through the detection of a zero-bias peak knownas Kondo resonance. It is now possible to tackle non-trivialmany-body effects in a controlled environment. The Kondoresonance has been investigated through scanning tunnelingmicroscopy (STM) in single atoms either isolated or cou-pled to other atoms, in single-atom contacts, and in sin-gle molecules.
It has also been successfully evidenced innanoscale devices, in particular quantum dots, carbon nanotubes, and nanowires. Of particular interest—especially in the context of spintron-ics, is the issue of screening in the presence of a magneticenvironment such as spin-polarized electrodes, and spin-polarized edge states. A spin-dependent hybridization forthe spin-up and spin-down energy levels of the impurity isthen predicted, resulting in an effective static magnetic fieldat the impurity site (this field can eventually be compensatedby an external magnetic field).
In the presence of fer-romagnetism, the Kondo resonance therefore splits apart asconfirmed experimentally.
While such a splitting is well understood, the impact of anon-equilibrium spin current on the Kondo resonance has so
Figure 1. Color on line.(a) Sketch of a quantum dot connected to anormal lead (right) and to a spin-accumulated lead (left). The spincurrent is injected by external means. The symbols Γ L/R representthe hybridization between the dot level and the left/right leads. I L, ↑ and I L, ↓ denote the spin currents for electrons with spin ↑ and ↓ .(b)Schematic diagram of a magnetic atom adsorbed on a surface in con-tact with a spin-accumulated tip. (c) Spin accumulation gives rise tospin-dependent chemical potentials and eventually to a spin polariza-tion. Both phenomena may lead to a splitting of the impurity levels.The color code is as follows: blue (resp. purple) denote electronswith spin ↑ (resp. ↓ ). far been little addressed in correlated nanostructures. This re-mains an open question since a decade ago it was shown that aspin current flowing from a Co wire through a Cu(Fe) wire isable to strongly suppress the resistivity of the Cu(Fe) Kondoalloy near the interface. As demonstrated by Johnson (seealso Refs. 40–42), a spin current induces spin accumulationyielding spin-dependent chemical potentials µ ↑ = µ ↓ , whichis equivalent to a spin bias. Using the equation of motion ap-proach, Qi et al. analyzed the fate of a Kondo resonance inthe presence of spin accumulation. They showed in particu-lar that the Kondo resonance is split into two peaks attachedto the two spin-dependent chemical potentials. Kobayashi et al. recently validated this prediction by studying exper-imentally a Kondo quantum dot in contact with a spin accu-mulated electrode and two normal electrodes. A simplifiedgeometry related to this experiment is sketched in Fig. 1a. Be-sides demonstrating that the Kondo splitting can be controlledthrough spin accumulation, they also showed that the Kondoresonance may be restored through an external magnetic field.Single-atom contacts with STM are another appealing wayfor investigating the interplay of a spin current with a Kondoimpurity. In STM tunneling spectra, the Kondo resonance isdetected as a Fano line shape due to the interference betweenelectrons tunneling into the conduction band of the substrateand those involved into the Kondo state. When the tipis brought into contact with the atom such a picture remainsvalid although the Kondo resonance becomes more symmet-ric and of order the conductance unit e /h . As shown re-cently, it it possible to introduce a spin current in the single-atom contact by using a ferromagnetic tip coated with a thicknormal copper spacer (see Fig. 1b for a sketch of the setup).As in macroscopic spintronic devices, the copper spaceraims at minimizing the direct or indirect magnetic exchangeinteractions between the cobalt atom and the tip. The Kondosplitting observed can then be assigned to spin accumulationin the copper spacer. In this respect, the Kondo resonance actsas a very sensitive local sensor for spin current. We want toemphasize that in the STM setup the tip is simultaneously thesource of the spin accumulation and also the transport probe.Therefore the spin current becomes voltage-dependent con-trary to Ref. 44 where the spin current was supplied by an ex-ternal spin-accumulated electrode while the differential con-ductance was probed using two different leads.The purpose of this paper is to provide a microscopic de-scription of a magnetic Kondo impurity embedded between aspin-polarized electrode able to carry a spin current and a nor-mal metallic electrode. The impurity can be either artificial,such as a quantum dot (see Fig. 1a) or a genuine magneticatom adsorbed on a surface (see Fig. 1b). We consider bothcases in which the spin current is either driven by an externalsource (and therefore constant) or driven by the same elec-trode (and therefore voltage dependent). By using the equa-tion of motion techniques and comparing various trun-cation methods to obtain a consistent picture, we show thatthe Kondo ground state depends sensitively on the spin po-larization of the electrode and the spin accumulation that itgenerates. We investigate both the spin-resolved spectral den-sity of the localized spin and the nonlinear conductance. Inthe case of a constant spin current, we demonstrate that spinaccumulation leads to a splitting of the impurity Kondo reso-nance as shown previously and schematically summarizedin Fig. 1c. Taking into account both the static spin polariza-tion and the spin accumulation, we show additionally that botheffects can actually compensate each other and therefore theKondo resonance can be restored. The case of a voltage de- pendent spin current turns out to be more subtle. Through aphenomenological approach we show that a non-linear depen-dence of the spin current with voltage bias is required to splitthe Kondo resonance. As we show in this work, our finite Q approximation (which amounts to voltage-independent spin-dependent chemical potentials at large bias) turns out to berather accurate when the impurity is easy to spin polarize.The plan of the paper is as follows: In Sec. II, we intro-duce our model consisting of an impurity in contact with aspin-polarized electrode and a nomagnetic electrode. We alsodiscuss the method and approximation we use to tackle sucha non-equilibrium interacting problem. In Sec. III, we studyboth analytically and numerically the case where one elec-trode has a finite spin polarization and sustains a constant spinaccumulation. In Sec. IV, we investigate the more subtle casewhere the spin accumulation becomes bias dependent. Fi-nally, in Sec. V we provide a summary of our main results anddiscuss some perspectives. Details on the truncated equationof motion approach used are presented in Appendix A. II. MODEL HAMILTONIAN AND METHOD
We consider an impurity—a quantum dot or atom, coupledto left and right electrodes as depicted in Fig. 1. We have inmind situations in which the quantum dot is used as a detectorof a non-equilibrium spin accumulation, therefore we focus onthe asymmetric situation in which one electrode (the left onein Fig 1a or the STM tip in Fig 1b) may be partially polarizedand able to sustain a spin current.We assume that the impurity (the quantum dot or the mag-netic adatom) correspond to a spin S = 1 / impurity. In orderto model the magnetic impurity, we consider an Anderson-type Hamiltonian H = X α,k,σ ( ε αkσ − µ ασ ) c † αkσ c αkσ + X σ ε σ d † σ d σ + U n ↑ n ↓ + X α,k,σ ( V αkσ c † αkσ d σ + h.c ) . (1)Here, c † αkσ ( c αkσ ) denotes the creation (annihilation) operatorin contact α and d † σ ( d σ ) is the corresponding operator in thedot. V αkσ describes a tunneling matrix element between con-tacts and localized levels and can eventually be spin depen-dent. U and ε σ parametrize the on-site Coulomb interactionand the spin-dependent localized energy level, respectively.Notice that an initial energy difference between localized lev-els, ∆ Z = ε ↑ − ε ↓ , may model an external magnetic field.As we emphasized in the introduction, a non-equilibrium spinaccumulation entails spin-dependent chemical potentials µ ασ and polarizations. The spin polarization in the contacts islumped into spin dependent hybridization functions Γ ασ . Fol-lowing Refs. 29–31 we write a spin-polarization parameter as P α = Γ α ↑ − Γ α ↓ Γ α ↑ + Γ α ↓ , (2)where Γ ασ = Γ α (1 + σP α ) with Γ α = π P k | V αkσ | ρ σ . ρ σ is the spin-dependent lead DOS at the Fermi energy, whichis assumed flat for both electrodes. Similarly, we introduce aspin bias parameter which is defined as Q α = µ α ↑ − µ α ↓ µ α ↑ + µ α ↓ . (3)We consider the commonly used wideband limit for the tun-neling rates where the hybridization Γ ασ are constant. Fol-lowing Ref. [52], the spin-dependent current I σ reads I σ = − eh Γ Lσ Γ Rσ Γ Lσ + Γ Rσ Z dω [ f Lσ ( ω ) − f Rσ ( ω )] ℑ (cid:2) G rσ,σ ( ω ) (cid:3) . (4)Here, e is the elementary (positive) unit charge. and ℑ denotesthe imaginary part. Although the spin current expression maylook simple, it is worth underlining that the retarded Greenfunction G r is the exact bias-dependent Green function. Com-puting such quantity remains a tremendous task. Since we aredealing with a non-equilibrium interacting problem, we mustrely on some approximate approach able to capture qualita-tively the physics. We have employed the equation of mo-tion technique to calculate the retarded Green function. Thistheoretical approach uses a truncated system of equations ofmotion for the retarded Green’s function. There are severalschemes for the truncation in order to obtain a close set ofequations. In our case, we follow Refs. 53 and 54 in orderto compute the Keldysh Green functions. This procedure hasbeen demonstrated to be suitable to treat systems with spin-polarized contacts. Details of the truncation scheme we haveused can be found in the appendix A. III. KONDO RESONANCE IN THE PRESENCE OF ACONSTANT SPIN CURRENT AND POLARIZATION
In this section, we focus on a quantum dot connectedto spin-polarized electrodes that are able to sustain a non-equilibrium spin current. In order to provide a qualitative un-derstanding of the physics, it turns out to be useful to firstperform a second order perturbation theory in the tunnel-ing matrix elements, which generates an effective but non-equilibrium local Zeeman term in the dot Hamiltonian.
A. Effective magnetic fields
Before presenting the results of our numerical calculations,we would like to present an extension of the heuristic argu-ment developed in [55] aiming at interpreting the effect of afinite polarization and/or finite spin current in a lead as an ef-fective local exchange magnetic field viewed/felt by the spinimpurity. In order to calculate this effective magnetic field,we proceed as in [55] to investigate the functional dependence of the effective magnetic field on temperature and gate volt-age. To do so, we derive an effective Hamiltonian H eff usingsecond-order perturbation theory. Physically, the split Kondopeak can be understood in terms of the dot valence instability(virtual charge fluctuation) and spin-dependent tunneling am-plitudes. To deal with this instability, we perform a Schrieffer-Wolff-type transformation of the Hamiltonian given by Eq. (1)and obtain H spin = X α,k X β,q (cid:20) V αk ↑ V βq ↑ ε d ↑ − ε βq ↑ c αk ↑ c † βq ↑ − V αk ↓ V βq ↓ ε d ↓ − ε βq ↓ c αk ↓ c † βq ↓ + V αk ↑ V βq ↑ U + ε d ↑ − ε βq ↑ c † αk ↑ c βq ↑ − V αk ↓ V βq ↓ U + ε d ↓ − ε βq ↓ c † αk ↓ c βq ↓ (cid:21) S z + [ · · · ] , (5)where [ · · · ] includes the usual terms corresponding to thespin-flip terms and potential scatterings that show up inthe Kondo Hamiltonian. At this point, unlike in the usualSchrieffer-Wolff transformation, we employ a mean-field ap-proximation for the lead electrons: h c αkσ c † βqσ i = [1 − f ( ε αkσ )] δ α,β δ k,q , and h c † βqσ c αkσ i = f ( ε αkσ ) δ α,β δ k,q . Theresulting effective Hamiltonian can be written as H eff = − B eff S z . The effective magnetic field generated by havingspin-accumulation, i.e., spin-dependent chemical potentialsand spin-polarized contacts then reads B eff ∝ X α Z dω (cid:20) Γ α ↑ [1 − f α ↑ ( ω )] ω − ε d ↑ − Γ α ↓ [1 − f α ↓ ( ω )] ω − ε d ↓ + Γ α ↑ f α ↑ ( ω ) ω − ε d ↑ − U − Γ α ↓ f α ↓ ( ω ) ω − ε d ↓ − U (cid:21) , (6)As emphasized in the introduction, we will mainly focus onthe asymmetrical situation and assume that the spin accumu-lation and polarization occurs only in the left contact. Thegeneral case can be trivially extended. We also assume spin-degenerate localized levels ε d ↑ = ε d ↓ = ε d . Therefore thespin-dependent chemical potentials are parametrized as fol-lows: µ L ↑ = µ L (1 + Q ) , and µ L ↓ = µ L (1 − Q ) ,µ R ↑ = µ R ↓ = 0 , (7)and the lead polarization P is defined by: Γ L ↑ = Γ L (1 + P ) , and Γ L ↓ = Γ L (1 − P ) , Γ Rσ = Γ R . (8)With this parametrization, the effective field can be then writ-ten as B eff ∝ Γ L Z dω (cid:20) (1 + P )[1 − f L ↑ ( ω )] ω − ε d − (1 − P )[1 − f L ↓ ( ω )] ω − ε d + (1 + P ) f L ↑ ( ω ) ω − ε d − U − (1 − P ) f L ↓ ( ω ) ω − ε d − U (cid:21) . (9)Up to leading order in P and Q , the previous expression simplifies to B eff ∝ − P Γ L ℜ (cid:26) Ψ (cid:18) − iβ ( ε d − µ L )2 π (cid:19) − Ψ (cid:18) − iβ ( ε d + U − µ L )2 π (cid:19)(cid:27) + 2 Qµ L Γ L Z dω f ′ ( ω − µ L ) (cid:20) ω − ε d − ω − ε d − U (cid:21) , (10)with β = 1 /k B T and Ψ denotes the digamma function defined as the logarithmic derivative of the gamma function. Taking thelimit T → , the effective magnetic field takes the compact form B eff ∝ − P Γ L ln (cid:12)(cid:12)(cid:12)(cid:12) ε d − µ L ε d + U − µ L (cid:12)(cid:12)(cid:12)(cid:12) + 2 Qµ L Γ L U ( ε d − µ L )( ε d + U − µ L ) . (11)Note that for the particle-hole symmetry point ( | ε d − µ L | = U/ ) the effective field due to the finite spin polarization van-ishes, while there still remains the effective field due to thespin-dependent chemical potentials. Such effective magneticfield can be cancelled by applying an external static magneticfield B ext such that B ext + B eff = 0 as this has been shownexperimentally by Kobayashi et al. . However, we want to stress that this is not the only wayto cancel this effective magnetic field. The two contributionsrelated to the spin accumulation and static polarization mayindeed have different sign. The sign of the former term isdetermined by the difference of the spin-dependent chemicalpotentials while the sign of the latter is fixed by the differenceof the density of states at the Fermi energy between up anddown electrons. Therefore this heuristic argument suggeststhat we can control the spin current independently of the po-larization or vice-versa and thus restore the Kondo resonance.This could for example be achieved by controlling the polar-ization of the left lead. We check that this is indeed the casein the next subsection.
B. Spectral weights and differential conductance
We now present our results for the total and spin-resolvedspectral weights which have been obtained with the trun-cated equation of motion approach.
We work in units of Γ L + Γ R = Γ = 1 . We adopt the following set of param-eters ǫ d = − . and D = 50 . For simplicity, the U → ∞ limit is considered, but our results can be generalized and re-main qualitatively correct in the finite U limit. Now, we vary P and Q and show the total and spin-resolved spectral den-sity evolutions in Fig. 2. First, when both contacts are normal( P = Q = 0 ), the low energy spectral density shows a singlepeak at the Fermi energy corresponding to the Kondo singu-larity [see Fig. 2(a)]. When there is a finite polarization butno spin-dependent chemical potentials ( P = 0 , Q = 0 ), thespin- ↑ spectral density moves towards negative frequencies,whereas spin- ↓ does the opposite, resulting in a split Kondoresonance. This behavior is shown in Fig. 2(b). Next, we con-sider the situation with some degree of spin-dependent chem-ical potentials, but no spin polarization ( P = 0 , Q = 0 ).Such a situation applies when a spin current is injected from A σ (a) P=0Q=0 µ L =0 A ↑ A ↓ A ↑ +A ↓ (b) P=0.2Q=0 µ L =0 A σ (c) P=0Q=0.5 µ L =0.2 (d) P=0.2Q=0.5 µ L =0.2 A σ ω / Γ (e) P=0.2Q=0.5 µ L =0.5 -1.5 -0.75 0 0.75 1.5 ω / Γ (f) P=-0.18Q=0.5 µ L =0.2 Figure 2. Spectral weights vs P and Q . Parameters: Γ L = Γ R =0 . , ε d = − . , D = 50 , µ Rσ = 0 , T = T K , and U → ∞ . an external terminal (see Ref. [44]). We observe that thepeaks in the spin-resolved spectral densities are located at ω ↑ ( ↓ ) ≈ µ L ↓ ( ↑ ) , µ R ; this is illustrated in Fig. 2(c) for whichwe use Q = 0 . , and µ L = 0 . leading to µ L ↑ = 0 . , and µ L ↓ = 0 . while µ R = µ R ↑ = µ R ↓ = 0 . The position ofthe peaks is determined by the poles of the impurity retardedgreen function (see Appendix A particularly Eq. (A21)). Wefind that the real part of the denominator of G rσ,σ ( ω ) has zeroesat µ α ¯ σ when the Kondo correlation develops. With a finitepolarization, the renormalized levels become spin-dependentso that the peak positions are no longer at µ ασ but dependon the degree of polarization P . In general, when P = 0 and Q = 0 , the total spectral density shows a four peakstructure. However, if two of the four peaks encountered for A ↑ and A ↓ coincide, the total spectral function displays onlythree peaks. This situation is depicted in Fig. 2(e) for which P = 0 . , and Q = 0 . with µ L = 0 . . The three peaks canbe also designed by considering P = − . and Q = 0 . d I / d V ( e / h ) (a) Q=0 P=0.00=0.25=0.50=0.75 (b)
P=0Q=0.00=0.25=0.50=0.75 d I / d V ( e / h ) V/ Γ (c) P=0.2Q=0.00=0.25=0.50=0.75 -0.5 -0.25 0 0.25 0.5 V/ Γ (d) P=-0.18Q=0.00=0.25=0.50=0.75
Figure 3. Differential conductance vs P and Q . Parameters: Γ L ↑ =0 . P ) , Γ L ↓ = 0 . − P ) , Γ Rσ = 0 . , µ L = 0 . , ε d = − . , D = 50 , T = 1 . T K , and U → ∞ . and µ L = 0 . as shown in Fig. 2(f). Whereas the peak split-ting in the spectral weights due to the static polarization canoccur both under equilibrium and non equilibrium conditions,we want to stress again that the split spectral weights due tothe spin-dependent chemical potentials can only occur undernon-equilibrium conditions.In addition, we investigate the nonlinear conductance whichis an experimentally accessible quantity. For practical pur-poses, it turns out to be more convenient to configure the bi-ases in such a way that the spin-dependent chemical potentialsof the left contact are fixed, while the spin-independent chem-ical potential of the right contact, µ Rσ = µ L + eV , is varied.The differential conductance reads: dI σ dV = 4 e h Γ Lσ Γ Rσ Γ Lσ + Γ Rσ Z dω (cid:26) df R ( ω ) d ( eV ) ℑ (cid:2) G rσ,σ ( ω ) (cid:3) − [ f Lσ ( ω ) − f R ( ω )] ℑ (cid:20) dG rσ,σ ( ω ) d ( eV ) (cid:21)(cid:27) . (12)Figure 3(a) illustrates the nonlinear conductance in the ab-sence of spin-dependent chemical potentials ( Q = 0 ). Theobserved splitting is attributed to the effective field gener-ated by the presence of the spin-polarized contacts. Noticethat the two peaks in the nonlinear conductance are almostsymmetrically located and the splitting (therefore the peakpositions) grows with P . In the absence of spin polariza-tion ( P = 0 ) but with spin-dependent chemical potentials,the nonlinear conductance shows similarly two peaks. Con-trary to the P = 0 case, we easily identify that the dI/dV conductance shows two peaks at eV = µ L ↑ ( / ↓ ) − µ L [Fig.3(b)]. This corresponds to adjusting the spin-independentchemical potential of the right lead with the spin-dependentchemical potentials for the left lead. This can be under-stood as follows. The leading part of the differential con-ductance is given by the first term in Eq. (12). At zero tem-perature, this term is proportional to the spin-dependent localDOS of the impurity ( A σ ( ω ) = − π ℑ (cid:2) G rσ,σ ( ω ) (cid:3) ). When weneglect the second term in Eq. (12), we therefore find that dI σ /dV ∝ A dσ ( µ L + eV ) . The total differential conductance dI/dV = P σ dI σ /dV will be maximum when the condition eV = ± ( µ L ↑ − µ L ↓ )2 = ± µ L Q , (13)is satisfied. In the presence of both spin bias and polarization ( Q = 0 ,and P = 0 ), the nonlinear conductance for positive P also ex-hibits two peaks and the splitting between two peaks widenscompared with the P = 0 cases. On the contrary, for negative P , the splitting observed in the dI/dV is reduced and even-tually vanishes at some particular value of Q . This is shownin Fig. 3(d) where the nonlinear conductance shows a singleresonance for P = − . and Q = 0 . . In this case, theKondo effect is restored. From the heuristic argument givenin Sec. III A, we can interprete this restoration of the reso-nance as the compensation of the effective fields generated bythe spin dependent polarization and the spin bias. Our numer-ical calculation therefore confirm nicely the qualitative resultswe discuss in Sec. III A that the accumulation spin currentand the static polarization may have antagonist effect on theKondo resonance which results in its restoration. IV. KONDO RESONANCE IN THE PRESENCE OF ABIAS-DEPENDENT SPIN ACCUMULATION
In the previous section, we studied the case where the spincurrent is injected by an external terminal as in Ref. 44. Thiswas inherently a non-equilibrium situation even at V = 0 . Letus now consider the two-terminal situation with the left leadspin-polarized. We assume that at equilibrium (for V = 0 )no spin current is generated but only a static magnetic spin-polarization P . This implies µ L ↑ = µ L ↓ at V = 0 . A finite V generates both charge and spin currents. The purpose of thissection is to analyze what will be the effect of such a bias-dependent spin current on the Kondo resonance. As we haveseen, a finite spin current generates spin-dependent chemicalpotentials that are now encoded through the function Q ( V ) verifying Q (0) = 0 .Let us first discuss the case where a static induced Zee-man field is not present, i.e., B eff = 0 . Note that this can beachieved by fine tuning the gate voltage of a quantum dot tothe particle-hole symmetric point according to Eq. (11) where B eff = 0 . In the STM setup, this corresponds to the situa-tion where P = 0 . This was implemented experimentally inRef. 47 by coating a magnetic tip with several layers of cop-per, in other words by introducing a normal metallic spacerbetween the spin-polarized tip and the atom.One may first try to expand the function Q ( V ) in powers of V which should correctly capture the behavior at low voltage.Keeping the first linear order in V such that Q ( V ) ≈ aV ,we found numerically by solving the equation of motion foreach value of the bias that the Kondo resonance does not split(we take P = 0 ). Indeed, this is consistent with the conditiongiven in Eq. (13) which implies a = 0 . Moreover, at smallbias, we expect a small non-equilibrium effective magneticfield according to Eq. (11) which does not split the Kondoresonance for B eff . T K . From Eq. (4), we see that the spincurrent I spin = I L ↑ − I L ↓ is obviously a function of the bias V but also of Q ( V ) . However, in the left lead, we expect ∆ µ L = µ L ↑ − µ L ↓ = 2 µ L Q to be proportional to I spin . Therefore, the function Q ( V ) is highly non-linear and needs apriori to be determined self-consistently. However, this turnsout to be a very difficult task (this is a non-linear and non-equilibrium interacting problem). We have chosen a differentand simpler strategy by assuming various phenomenologicalforms for Q ( V ) taking into account the constraints imposedon the function Q ( V ) at small and large bias.Since all the current must proceed through the single im-purity states, the spectral weight of the impurity necessarilylimits the total amount of current. In other words, a finitespin current entails a finite spin accumulation which cannotgrow infinitely in a nanoscale structure. Therefore, the func-tion Q ( V ) is upper bounded and must converge asymptoti-cally to a constant Q at large V .In order to reconcile both the small and large V limits wediscussed, we have tested the following phenomenologicalforms for Q ( V ) given by Q ( V ) = Q tanh ( | V | /V c ) , (14a) Q ( V ) = Q [1 − exp ( −| V | /V c )] . (14b)At large | V | , Q i ( V ) → Q , with i = 1 , . We have used V c as a phenomenological energy scale which is related to theimpurity orbital non-equilibrium polarization. The larger V c ,the less susceptible to be spin polarized the impurity is. When V c is small compared to the voltage bias range explored, onecan neglect the exponential and Q / ( V ) ≈ Q . We thereforerecover the constant Q case studied in Sec. III, which as wehave seen leads to a splitting of the Kondo resonance. Sincethe main energy scale entering into our problem is the bareKondo temperature T K , one has to compare V c to T K . Notehere, that the bare Kondo temperature is the Kondo temper-ature the impurity would acquire in absence of polarizationor spin accumulation. We have first computed the differen-tial conductance for Q = 0 . without any static polarization( P = 0 ) for different values of V c using the function Q ( V ) .The results are summarized in Fig. 4. We found that for V c . T K , the constant- Q approximation provides resultsqualitatively similar to the constant- Q approximation with apeak splitting. Only for large value of V c ≫ T K , do werecover a Kondo resonance. Indeed for large V c ≫ T K , wecan expand the functions Q / ( V ) in V /V c since we are inter-ested in bias of order of a few T K . One can check that keepingthe lowest terms of the expansion does not lead to a splittingof the peak using Eq. (13). We have also performed the samecalculations with the function Q ( V ) defined in Eq. (14b).The results are almost similar to the ones in Fig. 4.We have also considered the case of a finite polarization P .When P and Q have the same sign, the Kondo resonance isalways split as explained in Sec. III. We have computed thedifferential conductance for P = − . and Q = 0 . fordifferent values of V c using Q ( V ) . Our results are shown inFig. 5b. We found that for V c . T K , as in the constantQ approximation, the static polarization and the spin bias-dependent chemical potential provide opposite effects which d I / d V ( e / h ) V / Γ P=0Q =0.5 µ L =0.2V c /T K =0.1=1 =10 =100 Figure 4. Differential conductance for P = 0 and in presence ofbias dependent spin chemical potentials determined by Q ( V ) = Q tanh ( −| V | /V c ) for different values of V c . The other parametersare the same as before. d I / d V ( e / h ) V/ Γ P=-0.18Q =0.5 µ L =0.2 V c /T K =0.1=1=10=100 Figure 5. Differential conductance for P = − . and Q ( V ) = Q [1 − exp ( | V | /V c )] for different values of V c . The other parame-ters are the same as before. result in a restoration of the Kondo peak. Only for large valueof V c ≫ T K , are we dominated by the static polarization,which entails a splitting of the Kondo peak. V. CONCLUSION
In this paper we have studied a ferromagnetic-impurity-normal geometry paying attention to the fact that the ferro-magnetic electrode is able to also sustain a spin current. Sucha generic geometry can describe an artificial impurity (a quan-tum dot, a molecule, a carbon nanotube, etc.) or a genuineimpurity contacted between a spin-polarized electrode and anormal one. For STM experiments, this corresponds to amagnetic adatom adsorbed on a metallic surface and in con-tact with a spin-polarized tip. We have first considered thecase in which the spin current is injected in one electrodeand therefore maintained constant as realized experimentallyin Ref. 44. By computing the spectral functions and the dif-ferential conductance, we found that this leads to a splittingof the Kondo resonance. In that respect, the Kondo effectturns out to be a very sensitive phenomenon to detect a micro-scopic non-equilibrium spin current. We also studied the casewhere the spin current is injected into a spin-polarized elec-trode. Since the non-equilibrium spin accumulation and theequilibrium polarization have different origin, they can haveantagonist effects. We have shown both analytically and nu-merically that the Kondo resonance can be restored when thisis the case. Since the effect of the static polarization on an ar-tificial impurity can be controlled by a gate voltage, this offersa knob to observe such a restoration of the Kondo resonance.We have also considered the case where a spin current is gen-erated when a voltage is applied between the two electrodes.Under this hypothesis, the spin-dependent chemical potentialsbecome voltage dependent. Assuming simple phenomenolog-ical functions for this dependence that match the low-voltageand large-voltage case, we were able to show that the split-ting of the Kondo resonance depends on a energy scale V c which can be related to the polarisability of the impurity. For V c /T K ≫ , no splitting is found which corresponds to thecase where the generated spin current at a fixed voltage V istoo small to split the resonance of width of order T K . In theother limit, we recover results similar to the constant spin ac-cumulation. We think that this semi-phenomenological treat-ment captures the essential physics. Future work will be nec-essary to describe this phenomenon microscopically withoutfurther assumptions.We have analyzed in the paper an anisotropic situation inwhich only one electrode is spin polarized. A natural exten-sion of the present analysis is the more isotropic case in whichthe magnetic impurity is coupled to two spin-polarized elec-trodes that are able to sustain a spin current. Such a situationmay apply to atomic contacts made from ferromagnetic mate-rials where the observation of Kondo-Fano line shapes in theconductance have been reported. ACKNOWLEDGMENTS
We would like to thank M.V. Rastei for fruitful discus-sions. J.-S.L. and R.L. were supported by MICINN Grant No.FIS2011-2352. P.S. has benefited from financial support fromthe ANR under Contract No. DYMESYS (ANR 2011-IS04-001-01).
Appendix A: Explicit expressions for the retarded Green’sfunction G rσ,σ ( ω ) In this appendix we derive the retarded Green’s function tobe used for the calculation of the differential conductance. For that purpose we employ the equation of motion technique and in particular the truncation schemes proposed in Refs. 53and 54. These truncation procedures have been demonstratedto describe properly systems attached to spin-polarized con-tacts. The retarded impurity Green function is defined as G rσ,σ ( ω ) ≡ hh d σ , d † σ ii rω = Z dt e iωt hh d σ , d † σ ii rt , (A1)where hh d σ , d † σ ii rt = − i Θ( t ) (cid:10) [ d σ ( t ) , d † σ (0)] (cid:11) . (A2)For a generic two particle operator hh A, B ii rω we have for itsequation of motion ω hh A, B ii rω + hh [ H , A ] , B ii rω = h [ A, B ] + i . (A3)When A = d σ , B = d † σ the previous expression gives risethe equation of motion for the impurity Green function in thefrequency domain. The imaginary part i + going alongside ω is implicitly assumed. To simplify the notations, hereafter wewrite the retarded Green’s functions hh A, d † σ ii rω as hh A ii . Thefirst equations of motion in the hierarchy are ( ω − ε σ ) hh d σ ii = 1+ U hh d σ n ¯ σ ii + X α,k V αkσ hh c αkσ ii , ( ω − ε αkσ ) hh c αkσ ii = V αkσ hh d σ ii , (A4)where we take V αkσ as real. Then, we have ( ω − ε σ − Σ σ ( ω )) hh d σ ii = 1 + U hh d σ n ¯ σ ii , (A5)where we define Σ σ ( ω ) as the hopping selfenergy Σ σ ( ω ) = X α,k | V αkσ | ω − ε αkσ . (A6)To go to the next order we need to calculate hh d σ n ¯ σ ii as well: ( ω − ε σ − U ) hh d σ n ¯ σ ii = h n ¯ σ i + X α,k V αkσ hh c αkσ n ¯ σ ii− X α,k V αk ¯ σ hh c † αk ¯ σ d ¯ σ d σ ii + X α,k V αk ¯ σ hh d † ¯ σ c αk ¯ σ d σ ii . (A7)The next step is to consider the equation of motion foreach of the three higher order Green’s functions hh c αkσ n ¯ σ ii , hh c † αk ¯ σ d ¯ σ d σ ii , and hh d † ¯ σ c αk ¯ σ d σ ii that appear on the r.h.s. ofEq. (A7). Below we write these three equation of motionsapproximated as ( ω − ε αkσ ) hh c αkσ n ¯ σ ii = V αkσ hh d σ n ¯ σ ii − X β,q V βq ¯ σ hh c αkσ c † βq ¯ σ d ¯ σ ii + X β,q V βq ¯ σ hh c αkσ d † ¯ σ c βq ¯ σ ii , (A8) ( ω − ε αk ¯ σ + ε ¯ σ − ε σ ) hh d † ¯ σ c αk ¯ σ d σ ii = D d † ¯ σ c αk ¯ σ E + V αk ¯ σ hh d σ n ¯ σ ii− X β,q V βq ¯ σ hh c † βq ¯ σ c αk ¯ σ d σ ii + X β,q V βqσ hh d † ¯ σ c αk ¯ σ c βqσ ii , (A9) ( ω + ε αk ¯ σ − ε σ − ε ¯ σ − U ) hh c † αk ¯ σ d ¯ σ d σ ii = D c † αk ¯ σ d ¯ σ E − V αk ¯ σ hh d σ n ¯ σ ii + X β,q V βqσ hh c † αk ¯ σ d ¯ σ c βqσ ii + X β,q V βq ¯ σ hh c † αk ¯ σ c βq ¯ σ d σ ii , (A10)In the next step we truncate the system of equations keeping the Kondo correlations. To abbreviate the notation, we introduce ashorthand F σa ; b ≡ (cid:10) c † aσ c bσ (cid:11) . We consider the onset of Kondo correlations by approximating Eqs. (A8), (A10), and (A9) in thefollowing way: X α,k V αkσ hh c αkσ n ¯ σ ii ≈ X α,k | V αkσ | ω − ε αkσ hh d σ n ¯ σ ii = Σ σ ( ω ) hh d σ n ¯ σ ii , (A11) X α,k V αk ¯ σ hh d † ¯ σ c αk ¯ σ d σ ii ≈ Σ σ ( ω, ¯ σ ; σ, αk ¯ σ ) hh d σ n ¯ σ ii + X α,k V αk ¯ σ D d † ¯ σ c αk ¯ σ E ω − ε αk ¯ σ + ε ¯ σ − ε σ (1 + Σ σ ( ω ) hh d σ ii ) − X α,k X β,q V βq ¯ σ V αk ¯ σ F ¯ σβq ; αk ω − ε αk ¯ σ + ε ¯ σ − ε σ hh d σ ii , (A12) X α,k V αk ¯ σ hh c † αk ¯ σ d ¯ σ d σ ii ≈ − Σ σ ( ω, αk ¯ σ ; σ, ¯ σ, U ) hh d σ n ¯ σ ii + X α,k V αk ¯ σ D c † αk ¯ σ d ¯ σ E ω + ε αk ¯ σ − ε σ − ε ¯ σ − U (1 + Σ σ ( ω ) hh d σ ii ) + X α,k X β,q V αk ¯ σ V βq ¯ σ F ¯ σαk ; βq ω + ε αk ¯ σ − ε σ − ε ¯ σ − U hh d σ ii , (A13)where the self-energies appearing in Eqs. (A11), (A12), and(A13) are defined accordingly Σ σ ( ω, ¯ σ ; σ, αk ¯ σ ) = X α,k | V αk ¯ σ | ω − ε αk ¯ σ + ε ¯ σ − ε σ , (A14a) Σ σ ( ω, αk ¯ σ ; σ, ¯ σ, U ) = X α,k | V αk ¯ σ | ω + ε αk ¯ σ − ε σ − ε ¯ σ − U . (A14b) Using the previous truncated high order propagators we canshow that Eq. (A7) takes the expression hh d σ n ¯ σ ii = h n ¯ σ i − Σ σ ( ω ) hh d σ ii ω − ε σ − U − Σ σ ( ω ) − Σ σ ( ω ) , (A15)where we have defined h n ¯ σ i = h n ¯ σ i + X α,k V αk ¯ σ D d † ¯ σ c αk ¯ σ E ω − ε αk ¯ σ + ε ¯ σ − ε σ − X α,k V αk ¯ σ D c † αk ¯ σ d ¯ σ E ω + ε αk ¯ σ − ε σ − ε ¯ σ − U , (A16)together with the following self-energies Σ σ ( ω ) = Σ σ ( ω, ¯ σ ; σ, αk ¯ σ ) + Σ σ ( ω, αk ¯ σ ; σ, ¯ σ, U ) , (A17) Σ σ ( ω ) = X α,k X β,q V βq ¯ σ V αk ¯ σ F ¯ σβq ; αk ω − ε αk ¯ σ + ε ¯ σ − ε σ + X α,k X β,q V αk ¯ σ V βq ¯ σ F ¯ σαk ; βq ω + ε k ¯ σ − ε σ − ε ¯ σ − U − X α,k V αk ¯ σ D d † ¯ σ c αk ¯ σ E ω − ε αk ¯ σ + ε ¯ σ − ε σ − X α,k V αk ¯ σ D c † αk ¯ σ d ¯ σ E ω + ε k ¯ σ − ε σ − ε ¯ σ − U Σ σ ( ω ) , (A18)Taking back Eq. (A5) into Eq. (A7) the dot retarded Green’s function is finally obtained hh d σ ii = 1 − h n ¯ σ i ω − ε σ − Σ σ ( ω ) + U Σ σ ( ω ) ω − ε σ − U − Σ σ ( ω ) − Σ σ ( ω ) + h n ¯ σ i ω − ε σ − U − Σ σ ( ω ) + U [Σ σ ( ω ) − Σ σ ( ω )] ω − ε σ − Σ σ ( ω ) − Σ σ ( ω ) . (A19)It is worth to consider the limit U → ∞ where the impurity Green function expression (A19) is greatly simplified as hh d σ ii = 1 − h n ¯ σ i − P α,k V αk ¯ σ h d † ¯ σ c αk ¯ σ i ω − ε αk ¯ σ + ε ¯ σ − ε σ ω − ε σ − Σ σ ( ω ) − P α,k P β,q V βq ¯ σ V αk ¯ σ F ¯ σβq ; αk ω − ε αk ¯ σ + ε ¯ σ − ε σ + P α,k V αk ¯ σ h d † ¯ σ c αk ¯ σ i ω − ε αk ¯ σ + ε ¯ σ − ε σ Σ σ ( ω ) . (A20)Now we follow Ref. [50] to obtain a much simple truncatedimpurity Green function by setting (cid:10) d † σ c kσ (cid:11) = 0 and F σqk = δ k,q f ( ε kσ ) . 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