Spin-resolved Andreev transport through double-quantum-dot Cooper pair splitters
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Spin-resolved Andreev transport through double-quantum-dot Cooper pair splitters
Piotr Trocha ∗ and Ireneusz Weymann † Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna´n, Poland (Dated: November 14, 2018)We investigate the Andreev transport through double quantum dot Cooper pair splitters withferromagnetic leads. The analysis is performed with the aid of the real-time diagrammatic techniquein the sequential tunneling regime. We study the dependence of the Andreev current, the differentialconductance and the tunnel magnetoresistance on various parameters of the model in both the linearand nonlinear response regimes. In particular, we analyze the spin-resolved transport in the crossedAndreev reflection regime, where a blockade of the current occurs due to enhanced occupation of thetriplet state. We show that in the triplet blockade finite intradot correlations can lead to considerableleakage current due to direct Andreev reflection processes. Furthermore, we find additional regimesof current suppression resulting from enhanced occupation of singlet states, which decreases therate of crossed Andreev reflection. We also study how the splitting of Andreev bound states,triggered by either dot level detuning, finite hopping between the dots or magnetic field, affectsthe Andreev current. While in the first two cases the number of Andreev bound states is doubled,whereas transport properties are qualitatively similar, in the case of finite magnetic field furtherlevel splitting occurs, leading to a nontrivial behavior of spin-resolved transport characteristics, andespecially that of tunneling magnetoresistance. Finally, we discuss the entanglement fidelity betweensplit Cooper pair electrons and show that by tuning the device parameters fidelity can reach unity.
PACS numbers: 73.23.-b,73.21.La,74.45.+c,72.25.-b
I. INTRODUCTION
Quantum dots coupled to superconducting and normalleads provide very promising systems to study the inter-play between the superconducting correlations and themesoscopic electronic transport.
Such hybrid nanos-tructures have recently attracted a lot of attention due tothe possibility to control and split the Cooper pairs.
When the quantum dot is attached to normal and su-perconducting lead and for voltages smaller than the su-perconducting energy gap ∆, the current flows throughthe system due to Andreev reflection. More specifically,transport occurs then through sub-gap, Andreev boundstates (ABS), which were recently probed experimentallyusing bias spectroscopy.
For three-terminal systems,e.g. with one superconducting and two normal leads, theCooper pair, when leaving the superconductor, can en-ter either to the same normal lead or can be split whenthe two electrons forming Cooper pair end in differentleads. The former process is known as direct Andreevreflection (DAR), whereas the latter one is referred toas crossed Andreev reflection (CAR). Usually both pro-cesses contribute to the Andreev current, however, undercertain conditions, by properly changing device param-eters, one can tune the contributions due to CAR andDAR processes or even suppress one of them. This canbe obtained in the case when the normal leads are fer-romagnetic. Then, in the antiparallel magnetic configu-ration of the device, for leads with large degree of spinpolarization, only CAR processes contribute, since eachlead supports electrons of opposite spin.
Another interesting system, which allows for con-trollable manipulation of Cooper pairs, can be madeof double quantum dots (DQDs). Recent experiments have shown that DQDs can work as Cooper pair beamsplitters, whose operation can be controlled by gatevoltages.
In contrast to single quantum dot hybridsystems, the DQD setup allows to study pure CAR trans-port regime by considering suitable system’s parameters.Specifically, in real double quantum dots the intradotCoulomb repulsion can be much larger than other energyscales, which for a wide range of applied bias volt-ages prevents double occupancy of each dot. As a con-sequence, the direct Andreev reflection processes, whichrequire simultaneous transfer of two electrons with oppo-site spins by the same dot, become suppressed. However,Andreev reflection processes can still occur through CARprocesses, in which the electrons forming the Cooper pairare transferred simultaneously through the two dots. An-other advantage of DQDs is the possibility of independentlevel tuning of each individual dot. Due to this ability, ithas been shown experimentally that Cooper pair splittingcan be dominant on resonance, whereas out of resonanceelastic cotunneling processes dominate.
Transport properties of DQD Cooper pair beamsplitters have already been addressed in severalpublications, which, among others, addressedthe problem of coherence and entanglement of splitCooper pairs and their probing, as well as thenoise correlations and Cooper pair microwavespectroscopy.
These investigations were performedfor DQD Cooper pair beam splitters with nonmagneticleads. However, because using ferromagnetic leads canbe important to estimate entanglement between splitelectrons, providing comprehensive study of transportproperties of DQD Cooper pair splitters with ferromag-netic contacts seems desirable. The analysis of Andreevtransport through such systems is thus the goal of thepresent paper. Furthermore, DQDs coupled to a su- (cid:0) ! ✁ ✂ ! ! " ! ✄ !" ☎ ! ✆ ! ! " ! ✝ ✞ ✟✠ !" ✡ ! ! " ! ! ! " FIG. 1. (Color online) Schematic of a double quantumdot Cooper pair splitter with ferromagnetic leads. The left(right) dot is coupled to the left (right) lead with the cou-pling strength Γ σL (Γ σR ) and each dot is coupled to a commonsuperconducting lead with coupling Γ SL and Γ SR for the left andright dot, respectively. The magnetizations of the ferromag-netic leads can form either parallel or antiparallel magneticconfiguration. The level energy and Coulomb repulsion indot i are denoted by ε i and U i , while t and U LR describe thehopping and the Coulomb correlations between the two dots. perconductor and to two ferromagnetic contacts, canexhibit a considerable tunnel magnetoresistance (TMR)and generate large spin current. Therefore, such nanos-tructures are also interesting for spin nanoelectronicsand understanding their magnetoresistive properties isof great importance. Transport properties of quan-tum dots attached to ferromagnetic leads have alreadybeen broadly investigated both experimentally andtheoretically. However, spin-resolved transport prop-erties of hybrid dots, consisting of quantum dots cou-pled to ferromagnetic and superconducting leads, havebeen mainly studied in the case of single quantumdots, while the case of double quantum dots islargely unexplored.
In this paper, we therefore investigate spin-dependentAndreev transport through two single-level quantumdots, coupled to one superconducting and two ferromag-netic leads. Our analysis is performed with the aid of thereal-time diagrammatic technique in the lowest-order ex-pansion with respect to the coupling to ferromagneticleads, while the coupling to superconductor can be ar-bitrarily strong. First, by assuming infinite correlationsin the dots, we analyze the pure CAR regime where thetriplet blockade of the current occurs. We then showthat even very large but finite intradot correlations canlead to considerable leakage current in the triplet block-ade. We thoroughly study the behavior of the Andreevcurrent, differential conductance and TMR, deriving ap-proximate zero-temperature formulas for TMR in appro-priate transport regimes. We also analyze the effect of fi-nite splitting of Andreev bound states on transport prop-erties, discussing the splitting caused by either level de-tuning, finite hopping between the dots or finite magneticfield. For finite correlations in the dots, we study trans-port properties in the full parameter space and identifyadditional transport regimes where the current suppres-sion occurs. At the end, we also consider the entangle- ment fidelity between split electrons forming Cooper pairand show that, depending on parameters, fidelity canreach unity.The paper is organized in the following way: Sec.II contains the description of the DQD model and themethod used in calculations. The numerical results andtheir discussion in the crossed Andreev reflection regimeare presented in Sec. III. In Sec. IV we analyze how thesplitting of Andreev bound states affects the transportproperties. The next section is devoted to the analysis oftransport in the full parameter space where both CARand DAR processes are present. The behavior of entan-glement fidelity on bias and gate voltages is studied inSec. VI and, finally, the paper is concluded in Sec. VII.
II. THEORETICAL FRAMEWORKA. Model Hamiltonian
We consider double quantum dot Cooper pair split-ter, which is schematically displayed in Fig. 1. It con-sists of two single-level quantum dots, each attached toits own ferromagnetic lead, and both coupled to a com-mon s -wave superconductor. The magnetizations of fer-romagnetic leads are assumed to form either parallel (P)or antiparallel (AP) configuration. Switching betweenthese two configurations can be obtained upon applyinga small external magnetic field B s . We assume that thisfield is so small that it does not lead to the splitting of thedot’s level, neither affects it the superconducting phase.The total system is modeled by the following effectiveHamiltonian: H = X β = L,R H β + H S + H DQD + H T , (1)where the first term, H β , describes the left ( β = L ) and right ( β = R ) ferromagnetic electrodes inthe noninteracting quasiparticle approximation, H β = P k σ ε k βσ c † k βσ c k βσ . Here, c † k βσ ( c k βσ ) is the creation (an-nihilation) operator of an electron with the wave vector k and spin σ in the lead β , whereas ε k βσ denotes thecorresponding single-particle energy. The second term inEq. (1) describes the s -wave BCS superconducting (S)lead in the mean field approximation H S = X k σ ε k Sσ c † k Sσ c k Sσ +∆ X k ( c k S ↓ c − k S ↑ + h . c . ) (2)with ε k Sσ denoting the relevant single-particle energyand ∆ standing for the order parameter of the super-conductor, which is assumed to be real and positive.The third term of the Hamiltonian (1) describes thetwo single-level quantum dots and acquires the follow-ing form: H DQD = X i = L,R X σ ε i d † iσ d iσ + B z S iz + U i n i ↑ n i ↓ ! + X σ,σ ′ U LR n Lσ n Rσ ′ + t X σ ( d † Lσ d Rσ + d † Rσ d Lσ ) , (3)where d † iσ creates a spin- σ electron in dot i of energy ε i , n iσ = d † iσ d iσ , U L ( U R ) is the Coulomb correlation energyof the left (right) dot, and B z denotes external magneticfield in units of gµ B ≡ S iz = ( n i ↑ − n i ↓ ) / U LR and t stand for the interdot Coulomb repulsion and thehopping between the dots, respectively.The last term of the Hamiltonian describes tunnelingof electrons between the leads ( L, R, S ) and the two dots H T = X k σ X i = L,R ( V i k σ c † k iσ d iσ + h . c . ) (4)+ X k σ X i = L,R ( V Si k σ c † k Sσ d iσ + h . c . ) , with V i k σ ( V Si k σ ), for i = L, R , denoting the relevant tun-neling matrix elements between the two dots and ferro-magnetic leads (the superconducting lead). In the fol-lowing, we assume that these matrix elements are k and σ independent, V i k σ ≡ V i and V Si k σ ≡ V Si . The cou-pling of the dots to respective ferromagnetic leads can beparametrized by, Γ σi = 2 π | V i | ρ σi , where ρ σi is the spin-dependent density of states of lead i . Within the wideband approximation these couplings become energy inde-pendent and constant. Introducing the spin polarizationof the i -th lead, p i = ( ρ + i − ρ − i ) / ( ρ + i + ρ − i ), where ρ + i ( ρ − i )is the spin majority (minority) density of states, the cou-plings can be written in the form, Γ σi = (1 + σp i )Γ i , withΓ i = (Γ ↑ i + Γ ↓ i ) /
2. Generally, each dot can be coupledto its lead with different strength and the two leads canhave different spin polarizations, here, however, we re-strict our analysis to symmetric systems and note thatthe presented results are also qualitatively valid for sys-tems with weak asymmetry in the couplings. We thusassume p L = p R ≡ p and Γ L = Γ R ≡ Γ /
2. More-over, we also assume that the dots’ levels are degenerate ε L = ε R ≡ ε and the dots’ Coulomb energies are equal, U L = U R ≡ U , unless stated otherwise.Since in this paper we are only interested in Andreevtransport, we can take the limit of an infinite supercon-ducting gap, ∆ → ∞ . Then, the quantum dot systemcoupled to the superconducting lead can be described bythe effective Hamiltonian H eff DQD = H DQD − X i = L,R Γ Si (cid:16) d † i ↑ d † i ↓ + h . c . (cid:17) (5)+ Γ SLR (cid:16) d † R ↑ d † L ↓ + d † L ↑ d † R ↓ + h . c . (cid:17) , where Γ SLR = q Γ SL Γ SR . The superconducting proxim-ity effects are included in the last two terms of Eq. (5). The first term describes local proximity effects on eachdot and arises due to direct Andreev reflection, whereasthe second term describes creation of nonlocal entangledstates between the two dots. These nonlocal correlationsare responsible for crossed Andreev reflection. The effec-tive pair potential Γ Si ( i = L, R ) is the coupling strengthbetween the i -th dot and superconducting electrode andacquires the form, Γ Si = 2 π | V Si | ρ S , where ρ S denotesthe density of states of the superconductor in the normalstate. We assume that the couplings between the dotsand superconductor are equal, Γ SL = Γ SR ≡ Γ S .The device is biased in the following way: The elec-trochemical potential of the superconducting lead is as-sumed to be grounded, µ S = 0, see Fig. 1, while thepotentials of the left and right leads are kept the same, µ L = µ R ≡ µ = eV . In this way the net current betweenthe left and right ferromagnetic lead is zero. In the follow-ing, we use the convention that for positive bias, eV > eV <
0, the Cooper pairs are extractedfrom the superconducting electrode.We would like to note that while the assumption ofinfinite superconducting gap allows us to exclude normaltunneling processes and study only the Andreev trans-port, it needs to be taken with some care. This is be-cause in real systems the gap can be large, but is clearlyfinite.
However, for relatively low bias voltages, asconsidered in this paper, one can expect normal tunnel-ing processes to be negligible and, thus, the assumptionof large superconducting energy gap is reasonable.
B. Method
In order to calculate the transport characteristics ofthe considered system, we employ the real-time diagram-matic technique (RTDT), which is based on system-atic perturbation expansion of the reduced density ma-trix and operators of interest with respect to the couplingstrength Γ. Within the RTDT, in the stationary state thereduced density matrix ˆ ρ can be found from X χ ′ W χ,χ ′ P χ ′ = 0 , (6)where the elements W χ,χ ′ of the self-energy matrix W describe transitions between the states | χ i and | χ ′ i onthe Keldysh contour, with | χ i denoting the many-bodyeigenstate of H eff DQD , H eff DQD | χ i = ε χ | χ i . P is the vectorof diagonal density matrix elements P χ = h χ | ˆ ρ | χ i , whichcan be found from Eq. (6) together with the normaliza-tion condition. The current flowing from the ferromag-netic lead i can be found from I i = e ~ Tr (cid:8) W I i P (cid:9) , (7)where W I i denotes the modified self-energy matrix W ,which takes into account the number of electrons trans-ferred through the junction i .To find the occupation probabilities and the cur-rent, we perform the perturbation expansion of the self-energies, occupation probabilities and the current withrespect to the coupling strength to ferromagnetic leads Γ.In our studies, we consider the weak coupling regime andtake into account only the first-order tunneling processes,which correspond to sequential tunneling. We first deter-mine the self-energies using the respective diagrammaticrules and then calculate the occupation probabilitiesand the current by using Eqs. (6) and (7).We also note that when Γ ≪ Γ S , as considered in thepresent paper, and taking into account only the lowest-order tunneling processes in the coupling to ferromag-netic leads, the reduced density matrix becomes diagonalin the eigenbasis of the effective Hamiltonian. This iswhy Eq. (6) includes only the diagonal elements of the re-duced density matrix. Moreover, we would like to empha-size that while the perturbation expansion with respectto the coupling strength Γ is performed, no assumptionon the strength of the Coulomb correlation parameters isimposed, and they are treated in an exact way. Further-more, the assumption of the weak coupling regime impliesthat the Kondo temperature of the system is exponen-tially small. Thus, at temperatures considered in calcu-lations, the correlations leading to the Kondo effect areirrelevant and do not need to be taken into account.
C. Quantities of interest
The main quantity of interest is the current flowingthrough the system due to Andreev reflection processes.By calculating the currents I L and I R flowing through theleft and right junctions, the total current flowing into thesuperconductor can be simply obtained from the Kirch-hoff’s law I S = I L + I R , (8)together with the corresponding differential conductance, G S = dI S /dV .Since the normal leads are ferromagnetic, the Andreevcurrent depends on the magnetic configuration of the de-vice, which is assumed to be either parallel or antiparal-lel. We thus also calculate the tunnel magnetoresistanceassociated with the change of magnetic configuration ofthe system, which is defined as TMR = I APS − I PS I PS , (9)where I PS and I APS denote the Andreev current flowinginto the superconductor in the parallel and antiparal-lel magnetic configurations, respectively. Note that thisdefinition is opposite to that in the case of the Jullieremodel. This is because for hybrid quantum dots withsuperconducting and ferromagnetic leads, the Andreevcurrent in the antiparallel configuration is usually largerthan that in the parallel configuration. In the following we present and discuss the results onthe Andreev transport through DQD Cooper pair split-ters with ferromagnetic leads obtained within the sequen-tial tunneling approximation. We systematically studythe behavior of the Andreev current, the associated dif-ferential conductance and the TMR in both the linearand nonlinear response regimes, exploring basically thewhole parameter space of the considered model. In par-ticular, to analyze the transport regime where CAR pro-cesses are dominant, we first assume infinite Coulombcorrelations on the dots, so that DAR processes are to-tally suppressed. We then relax this condition and allowfor finite correlations in the dots to study the transportproperties basically in the whole parameter space. Wealso analyze the effects of finite hopping between the dots,nonzero detuning of DQD levels and finite external mag-netic field. In addition, we calculate the entanglementfidelity between the split electrons forming Cooper pair.
III. RESULTS IN THE CROSSED ANDREEVREFLECTION REGIME
When the charging energy of each dot is much largerthan Coulomb correlations between the dots and the ap-plied bias voltage, the rate of direct Andreev reflectionis suppressed and only CAR processes are possible. Thiscondition is in fact one of the main requirements for thesystem to work as Cooper pair beam splitter. Thereforeto analyze the Andreev transport in the case when DARprocesses are absent we now assume U → ∞ . For thesake of simplicity of the following discussion, let us alsoat this point neglect the hopping term assuming t = 0(the role of finite hopping will be considered further).In the limit of infinite intradot correlations, double oc-cupation of each dot is forbidden and only nine states(out of 16) of the effective Hamiltonian (5) are relevantfor transport. These are the following states: emptystate | , i , four singly occupied states | σ, i , | , σ i , andfour doubly occupied states | σ, σ ′ i for σ, σ ′ = ↑ , ↓ , where | α, β i ≡ | α i L | β i R , with | i i and | σ i i denoting emptyand singly occupied states of dot i . Due to the hop-ping and additional terms in Eq. (5) resulting from theproximity effect, the Hamiltonian is not diagonal in theabove basis. Diagonalizing the effective Hamiltonian, onefinds a new basis consisting of the following states: foursingly occupied states | σ, i , | , σ i , three triplet states | T i = ( | ↓ , ↑i + | ↑ , ↓i ) / √ | T σ i = | σ, σ i , and two states |±i = 1 √ r ∓ δ ε A | , i ∓ r ± δ ε A | S i ! , (10)being linear combinations of empty state and singletstate, | S i = ( | ↓ , ↑i − | ↑ , ↓i ) / √
2. We note that thetriplet states become decoupled from the superconduc-tor since Cooper pairs consist of two electrons with com-pensated spin, i.e., S pair = 0, and there is no couplingbetween the singlet and triplet states. The correspond-ing eigenenergies of the eigenstates given by Eq. (10) |I PS |/I (cid:1) /U LR e V / U L R e V / U L R (a)(b) |I SAP |/I -2 -1 0 1 2 (cid:0) /U LR e V / U L R e V / U L R FIG. 2. (Color online) The absolute value of the Andreevcurrent calculated for (a) the parallel ( I PS ) and (b) antiparallel( I APS ) magnetic configurations as a function of detuning δ =2 ε + U LR and the applied bias eV . The parameters are: Γ S =0 . T = 0 . . t = 0, B z = 0, with U LR ≡ p = 0 .
5. The current is plotted in units I = e Γ / ~ . are E ± = δ/ ± ε A , where δ = 2 ε + U LR denotes de-tuning between the singlet states and the empty state,while 2 ε A = p δ + 2Γ S measures the energy differencebetween the states | + i and |−i .The Andreev bound state (ABS) energies are definedas E ABS αβ = α U LR β q δ + 2Γ S , (11)where α, β = ± . These energies are the excitation ener-gies between doublet and singlet states of the double dotdecoupled from the ferromagnetic leads.Due to the assumption, U → ∞ , double occupancyof each dot is forbidden and direct Andreev tunnelingbecomes totally suppressed. The only way to transfercharge between the double dot and the superconductor isby crossed Andreev reflection, the process which involvestwo electrons with opposite spins coming from different ferromagnetic leads. For eV >
0, the two electrons tun-nel to the superconductor, while for eV <
0, the Cooperpairs are extracted from the superconducting electrodeand entangled pairs of electrons are transmitted to ferro-magnetic leads (each electron ends in different lead).
A. Andreev current and differential conductance
In Fig. 2 we show the dependence of the absolute valueof the Andreev current on the applied bias eV and thedetuning parameter δ = 2 ε + U LR for the parallel andantiparallel magnetic configurations of the system. Atlow bias, the Andreev current is generally suppresseddue to the Coulomb blockade, except for two values of δ , | δ | = p U LR − S , where E ABS − + = E ABS+ − = 0 and thecorresponding Andreev bound states are at resonance.With increasing the bias, the current starts flowing once | eV | > | E ABS − + | . As the Andreev reflection becomesoptimized for parameters corresponding to particle-holesymmetry point, the Andreev current reaches maximumvalue for small detuning, δ ≈
0. These two features aresimilar to those observed in a tree-terminal system in-cluding a single quantum dot. However, in the presentcase the Andreev current exhibits a striking difference,as it does not reveal the symmetry with respect to thebias reversal, which has been present for the single dotsystem. In the present case the absolute value of theAndreev current reveals a very strong asymmetry withrespect to the sign change of the bias. As can be seen inFig. 2, for positive bias, for which the double dot becomesoccupied by two electrons, the Andreev current ceases toflow. In fact, for eV > | ( δ + U LR ) / | , the double dot isin the triplet state, which explains the vanishing of theAndreev current, as the symmetry of the triplet statedoes not match the symmetry of the s-wave supercon-ductor. The blockade region is therefore independent ofthe magnetic configuration of the system – the Andreevcurrent stops flowing due to the triplet blockade in boththe parallel and antiparallel alignments, see Fig. 2.Figure 3 shows the dependence of the Andreev differ-ential conductance G S = dI S /dV on the bias voltage eV and detuning parameter δ . The sudden drop of the An-dreev current around the bias voltage eV ≈ ( δ + U LR ) / δ/U LR > − δ , see e.g. δ/U LR = 2 in Figs. 2 and 3. This asymmetry especiallyreveals in the differential conductance when comparingthe intensity of the low-bias peaks, i.e. the peak associ- δ /U LR e V / U L R e V / U L R (a)(b) |I SAP |/I -2 -1 0 1 2 δ /U LR e V / U L R e V / U L R G PS (e /h)G SAP (e /h) FIG. 3. (Color online) The differential conductance G S = dI S /dV of the Andreev current in the parallel ( G PS ) and an-tiparallel ( G APS ) magnetic configurations as a function of de-tuning δ and applied bias voltage eV . Parameters are thesame as in Fig. 2. ated with the Andreev level E ABS − + for eV > δ > E ABS+ − for eV < δ > δ/U LR = 2.The Andreev processes can occur in the system if theoccupation of states | + i and/or |−i is finite. For largedetuning the state | + i is high in energy and does not playany role in the considered bias voltage regime. The statewhich is relevant for Andreev transport is the state |−i .One should also note that for large value of δ the state |−i contains only relatively small admixture of the singletstate, cf. Eq. (10), while the empty state is mostly occu-pied. This generally leads to small values of the currentfor large δ .For bias voltages such that E ABS+ − < eV < E ABS − + , thereare no Andreev levels in the transport window and thecurrent flowing into/out of the superconductor is sup-pressed. When eV crosses one of those levels, either forpositive or negative bias, the current starts to flow and a (cid:1)(cid:2) (cid:1)(cid:3) (cid:4) (cid:3) (cid:2)(cid:4)(cid:5)(cid:4)(cid:4)(cid:4)(cid:5)(cid:4)(cid:6)(cid:4)(cid:5)(cid:4)(cid:7) (cid:8)(cid:9)(cid:10) (cid:11) (cid:11) (cid:12) (cid:13) (cid:11) (cid:8) (cid:14) (cid:2) (cid:15) (cid:16) (cid:10) (cid:14)(cid:17)(cid:15)(cid:18) (cid:19)(cid:20) (cid:1)(cid:4)(cid:5)(cid:3)(cid:2)(cid:1)(cid:4)(cid:5)(cid:4)(cid:7)(cid:1)(cid:4)(cid:5)(cid:4)(cid:6)(cid:4)(cid:5)(cid:4)(cid:4)(cid:4)(cid:5)(cid:4)(cid:6) (cid:11) (cid:11) (cid:11) (cid:21) (cid:13) (cid:15)(cid:21) (cid:4) (cid:11)(cid:22)(cid:11)(cid:23)(cid:22) (cid:8)(cid:24)(cid:10) (cid:1) (cid:15)(cid:18) (cid:19)(cid:20) (cid:25)(cid:2) FIG. 4. (Color online) The Andreev current (a) in the parallel(solid line) and antiparallel (dashed line) magnetic configura-tion and the corresponding differential conductance (b) as afunction of the bias voltage calculated for detuning parameter δ/U LR = 2. The other parameters are the same as in Fig. 2. peak appears in the differential conductance, see Figs. 2and 3. It can be seen that the system becomes moretransparent for eV <
0, when the Cooper pairs are ex-tracted from the superconductor, than for eV >
0, whenone injects electron pairs into superconducting lead. Thisis because for positive bias voltage singly occupied statesbecome populated and, since the singlet state is requiredfor the current to flow into the superconductor, the An-dreev current is decreased. More specifically, when pass-ing the energy ε (note that ε = U LR / δ = 2 U LR ) theprobability of finding the double dot in state |−i becomesstrongly suppressed at the cost of enhanced occupation ofone-electron states, decreasing the Andreev current. Onthe other hand, for negative bias voltage the singly occu-pied states play little role in transport and the current isthen larger compared to the case of eV > δ . Figure 4 displays the bias dependence ofthe current and differential conductance in both mag-netic configurations calculated for δ/U LR = 2. Thecurrent as a function of eV exhibits well-defined stepscorresponding to consecutive Andreev bound states be-ing active in transport, while the differential conduc-tance shows the respective peaks. One can also see thatthe differential conductance exhibits negative value dueto the triplet blockade, which occurs for bias voltage eV ≈ ( δ + U LR ) / δ/U LR > −
1. Interestingly, thedifferential conductance in the parallel magnetic align-ment reveals additional negative differential conductance,which is not related to the triplet blockade, see Figs. 3(a)and 4(b). This negative differential conductance developsfor δ > p U LR − S and for bias voltages eV ≈ E ABS − + .The effect of negative differential conductance in theparallel configuration can be explained bearing in mindthat formation of Cooper pairs involves two electronswith opposite spins. In the parallel alignment there aremore electrons with one spin orientation than with theother one, thus, the rate of electron pairs is determinedby the density of states of minority carriers. When thedouble dot starts to be occupied by odd number of elec-trons (here, singly occupied), the occupation probabilityof electrons with spin-up orientation increases whereasthat of electrons with spin-down decreases. With fur-ther increase of the bias voltage the occupation of thespin-up level becomes greatly enhanced, whereas that ofspin-down level becomes strongly suppressed, giving riseto nonequilibrium spin accumulation. As a consequenceof spin accumulation, the Andreev current becomes alsosuppressed, which reveals as negative values in the differ-ential conductance. For reversed bias voltages the spinaccumulation becomes irrelevant since the double dot isin the state |−i , being rather empty with small admix-ture of singlet state. Thus, Cooper pairs can be moreeasily extracted from the superconductor compared tothe opposite bias polarization.The Andreev current and differential conductance asa function of bias voltage in the absence of detuning areshown in Fig. 5. Now, one can clearly see that while fornegative bias the current displays typical steps accompa-nied with peaks in dI S /dV , for positive bias voltage thecurrent first increases but then drops and becomes fullysuppressed due to the triplet blockade, see Fig. 5(a). Theassociated negative differential conductance is clearly vis-ible in both magnetic configurations, see Fig. 5(b). B. Tunnel magnetoresistance
To observe more subtle differences between the paralleland antiparallel magnetic configurations one needs to usequantity which is more sensitive to a change of magneticalignment of ferromagnetic leads. In Fig. 6 we presentthe dependence of TMR on detuning δ and the biasvoltage eV . In the region determined by the equation eV & | ( δ + U LR ) / | , where the current ceases to flow dueto the triplet blockade, the TMR becomes indeterminate.This region is marked by white area in Fig. 6. Moreover,the TMR is strongly suppressed for δ > p U LR − S and for the bias voltage E ABS+ − < eV < E ABS − + . For thistransport regime however the first-order processes aresuppressed and to obtain correct value of TMR higher-order tunneling events should be considered. As has been (cid:1)(cid:2) (cid:1)(cid:3) (cid:4) (cid:3) (cid:2)(cid:1)(cid:4)(cid:5)(cid:6)(cid:1)(cid:4)(cid:5)(cid:2)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:2)(cid:4)(cid:5)(cid:6)(cid:4)(cid:5)(cid:7) (cid:8)(cid:9)(cid:10) (cid:11) (cid:11) (cid:12) (cid:13) (cid:11) (cid:8) (cid:14) (cid:2) (cid:15) (cid:16) (cid:10) (cid:14)(cid:17)(cid:15)(cid:18) (cid:19)(cid:20) (cid:1)(cid:3)(cid:5)(cid:4)(cid:1)(cid:4)(cid:5)(cid:21)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:21) (cid:1) (cid:15)(cid:18) (cid:19)(cid:20) (cid:22)(cid:4) (cid:11) (cid:11) (cid:11) (cid:23) (cid:13) (cid:15)(cid:23) (cid:4) (cid:11)(cid:24)(cid:11)(cid:25)(cid:24) (cid:8)(cid:26)(cid:10) FIG. 5. (Color online) The same as in Fig. 4 calculated fordetuning parameter δ = 0. (cid:4)(cid:5)(cid:6) (cid:7)(cid:8) (cid:9)(cid:10)(cid:11) -(cid:12)(cid:13)(cid:14) -1 (cid:15)(cid:16)(cid:17)(cid:18) -2 -1 0 1 2 (cid:19) /U L(cid:20) e V / U (cid:21)(cid:22) e V / U (cid:23)(cid:24) T(cid:25)(cid:26) FIG. 6. (Color online) The tunnel magnetoresistance TMR asa function of detuning δ and the bias voltage eV . The whiteregion indicates the range of parameters where the TMR isundetermined since the current vanishes in both configura-tions due to the triplet blockade. The parameters are thesame as in Fig. 2. shown recently, the cotunneling processes can lead toenhancement of TMR in this transport regime.Since the TMR takes well-defined values for parame-ters corresponding to plateaus in the current, it is pos-sible to find some approximate analytical formulas forthe TMR. This can be done assuming very low tempera-tures when the Fermi functions can be replaced by stepfunctions. The formula for TMR in the transport regimecorresponding to δ/U LR = 2 and describing the TMR atthe plateau U LR / . eV . U LR / ε A )(7 + 9 ε A ) 2 p − p , (12)while the TMR for bias voltages eV . − U LR / ε A − ε A −
1) 2 p − p . (13)For δ/U LR = 2, the first coefficient, 8(1 + ε A ) / (7 + 9 ε A ),is very close to unity, while the second coefficient, 2( ε A − / (3 ε A − δ = 0 and for negative bias voltage, − U LR / − Γ S / √ 2, the TMR has a plateau of width √ S and is given byTMR = 23 2 p − p , (14)while for eV < − U LR / − Γ S / √ 2, the TMR readsTMR = 12 2 p − p . (15)For positive bias voltage and for δ = 0, the TMR exhibitsa plateau for U LR / − Γ S / √ < eV < U LR / S / √ 2, at which it is given byTMR = 89 2 p − p . (16)Note that the TMR is always positive and smaller than2 p / (1 − p ), see Fig. 6. C. The influence of intradot correlations Results presented in previous sections were obtainedin the limit of infinite Coulomb correlations in the dots,so that the Andreev current was mediated only by CARprocesses. We now relax this condition and allow forfinite intradot Coulomb correlations, and study their in-fluence on the Andreev current and the TMR, focusingon the triplet blockade regime. Finite Coulomb correla-tions allow for nonzero current due to DAR processes,which can lead to a nonzero leakage current in the tripletblockade. Thus, once the current is finite, one can ana-lyze the behavior of the TMR, which is now well definedin the whole range of considered bias voltage.Before proceeding with the discussion of the TMR, inFig. 7 we first study the bias dependence of the Andreevcurrent for zero detuning δ = 0. To elucidate the roleof finite intradot correlations the current is plotted inthe logarithmic scale. This figure clearly shows how fi-nite Coulomb correlations affect the current in the triplet (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3) (cid:5)(cid:2)(cid:1)(cid:4)(cid:1) (cid:6)(cid:7) (cid:4)(cid:1) (cid:6)(cid:8) (cid:4)(cid:1) (cid:6)(cid:9) (cid:4)(cid:1) (cid:6)(cid:10) (cid:4)(cid:1) (cid:6)(cid:3) (cid:4)(cid:1) (cid:6)(cid:11) (cid:4)(cid:1) (cid:6)(cid:12) (cid:4)(cid:1) (cid:6)(cid:5) (cid:4)(cid:1) (cid:6)(cid:4) (cid:13)(cid:14)(cid:15)(cid:5)(cid:13)(cid:14)(cid:15)(cid:3)(cid:13)(cid:14)(cid:15)(cid:4)(cid:1)(cid:13)(cid:14)(cid:15)(cid:4)(cid:1) (cid:5) (cid:13)(cid:14)(cid:15)(cid:4)(cid:1) (cid:12) (cid:13)(cid:14)(cid:15)(cid:4)(cid:1) (cid:11) (cid:13) (cid:13) (cid:16) (cid:17)(cid:18) (cid:19) (cid:20) (cid:21) (cid:22)(cid:20) (cid:1)(cid:23) (cid:24)(cid:25)(cid:22)(cid:14) (cid:26)(cid:27) FIG. 7. (Color online) The logarithm of the Andreev currentin the parallel configuration for δ = 0 as a function of the biasvoltage for different Coulomb correlations U , as indicated.The other parameters are the same as in Fig. 2. (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:6)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:7)(cid:6)(cid:2)(cid:1) (cid:8) (cid:8) (cid:9)(cid:10)(cid:2)(cid:1) (cid:9)(cid:6)(cid:2)(cid:7) (cid:9)(cid:6)(cid:2)(cid:1) (cid:9)(cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:7) (cid:6)(cid:2)(cid:1) (cid:6)(cid:2)(cid:7) (cid:10)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:10)(cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:6)(cid:2)(cid:1) (cid:8)(cid:11)(cid:12)(cid:10)(cid:8)(cid:11)(cid:12)(cid:7)(cid:8)(cid:11)(cid:12)(cid:6)(cid:1)(cid:8)(cid:11)(cid:12)(cid:6)(cid:1) (cid:10) (cid:8)(cid:11)(cid:12)(cid:6)(cid:1) (cid:13) (cid:8)(cid:11)(cid:12)(cid:6)(cid:1) (cid:3) (cid:8) (cid:8) (cid:14) (cid:15) (cid:16) (cid:17)(cid:18)(cid:19)(cid:11) (cid:20)(cid:16) FIG. 8. (Color online) The tunnel magnetoresistance as afunction of the bias voltage calculated for δ = 0 and for dif-ferent values of Coulomb correlations U , as indicated. Theother parameters are the same as in Fig. 2. blockade regime. Since the dependence of the current isqualitatively similar in both magnetic configurations, weconsider only the case of parallel alignment. One can seethat in the (unphysical) case of infinite correlations, thecurrent for eV > U LR / I S ∝ exp( − eV /T ). However, even relativelylarge values of U lead to finite current in the tripletblockade. For semiconductor double quantum dots theinterdot correlations are typically an order of magni-tude smaller than the intradot correlations. Althoughfor recently-implemented Cooper pair splitters based onnanowire DQDs the capacitive coupling between the dotsis even smaller, it can still play a role. As can be seenin Fig. 7, the influence of direct Andreev reflection on thetriplet blockade is clearly nontrivial, for experimentallyrelevant parameters it leads to relatively large leakagecurrent in the triplet blockade, see e.g. the curves for U/U LR . δ = 0 and for different values of in-tradot Coulomb correlations. First of all, one can see thatthe behavior of TMR for eV < U LR / U . Since the values of the TMR in this trans-port regime were discussed in the previous section, let usonly focus on the range of bias voltages, eV > U LR / p / (1 − p ). Such large values of TMR indi-cate that for finite intradot correlations not only the rateof DAR processes becomes considarable, but also that ofCAR processes increases. This can be simply understoodby realizing that with lowering U the occupation of thetriplet state decreases at the cost of other states of theDQD, so that finite current due to both types of An-dreev reflection processes can flow. Since DAR processesare not sensitive to a change of magnetic configuration ofthe device (the two electrons tunnel always to the same,either left or right, lead), the TMR provides an indirectinformation about CAR processes in the system, the rateof which is clearly dependent on magnetic configuration. IV. SPLITTING ANDREEV BOUND STATES We now study how the transport properties of theDQD Cooper pair splitters change when one allows forfinite splitting of Andreev bound states. Such splittingcan be induced in various ways, e.g. by detuning theDQD levels, allowing for hopping between the dots orapplying finite magnetic field. We again focus on thesame parameter space as in previous sections, i.e. onthe transport regime where mainly CAR processes arepresent. To gain a deeper insight of how the ABS be-come affected, let us consider only 9 states of the DQD(the limit of U → ∞ ). The analytical formula for An-dreev bound states’ energies can be then written as E ABS αβγδ = E ABS αβ + γ p t + (∆ ε ) + δ B z , (17)where ∆ ε = ε L − ε R denotes the detuning of DQD lev-els, γ, δ = ± , and E ABS αβ is given by Eq. (11). Notethat the splitting of ABS results only from the split-ting of single-electron states, while the states | + i and |−i are not affected. Moreover, one can also notice thatfinite level detuning ∆ ε can have a similar effect as fi-nite hopping between the dots when ∆ ε = 2 t . In thepresence of either level detuning or hopping each of thefour ABS states splits into two, which results in eightAndreev bound states. When external magnetic field isadditionally present, these states split again and thereare sixteen ABS states. One can thus expect that theconsequences of ABS splitting will reveal as nontrivialfeatures in transport characteristics. In the following westudy the spin-resolved transport properties for finite de-tuning, hopping and, finally, for finite magnetic field.Although we focus on transport regime where doublyoccupied states play negligible role, in calculations weassume large but finite intradot correlations to be able ((cid:27)(cid:28) |(cid:29) PS (cid:30)(cid:31)! " %&’ )*+ ,./1 -1 -2 -1 0 1 2 (b) :;<=>?@A BCDEFGHIJK -0.20.1-0.1 G PS (e /h) MNO PQRS /U UV e V / U WX YZ[ \]^ _‘ab -1 cdef e V / U gh -2 -1 0 1 2 i /U jklmn opq rstu -1 vwxy e V / U z{ -2 -1 0 1 2 } /U ~(cid:127) FIG. 9. (Color online) The bias voltage and detuning depen-dence of (a) absolute value of the Andreev current and (b)the respective differential conductance in the parallel mag-netic configuration as well as (c) the TMR calculated for∆ ε/U LR = 0 . 4. The parameters are the same as in Fig. 2with U/U LR = 10. to determine the detuning and bias voltage dependenceof the TMR in the considered parameter space. We thusassume U/U LR = 10, if not stated otherwise.0 (cid:128)(cid:129)(cid:130) (cid:131)(cid:132) PS (cid:133)(cid:134)(cid:135) (cid:136)(cid:137)(cid:138) (cid:139)(cid:140)(cid:141) (cid:142)(cid:143)(cid:144) (cid:145)(cid:146)(cid:147)(cid:148) -1 (cid:149)(cid:150)(cid:151)(cid:152) -2 -1 0 1 2 (b) (cid:153)(cid:154)(cid:155) (cid:156)(cid:157)(cid:158) (cid:159)(cid:160)¡¢ -1 £⁄¥ƒ -2 -1 0 1 2 0.3 §¤'“ «‹›fifl(cid:176)–† ‡·(cid:181)¶•‚„”»… -0.20.1-0.1 G PS (e /h) ‰(cid:190)¿(cid:192)`´ ˆ˜¯ ˘˙¨(cid:201) -1 ˚¸(cid:204)˝ -2 -1 0 1 2 1.110.200.30.4 ˛ˇ— (cid:209)(cid:210)(cid:211)(cid:212)(cid:213)(cid:214)(cid:215) /U (cid:216)(cid:217)(cid:218) /U (cid:219)(cid:220)(cid:221) /U (cid:222)(cid:223) e V / U (cid:224)Æ e V / U (cid:226)ª e V / U (cid:228)(cid:229) FIG. 10. (Color online) The bias voltage and detuning depen-dence of (a) absolute value of the Andreev current and (b)the respective differential conductance in the parallel mag-netic configuration as well as (c) the TMR calculated in thepresence of finite hopping between the dots t/U LR = 0 . 2. Theother parameters are the same as in Fig. 2 with U/U LR = 10. A. Finite level detuning or hopping From Eq. (17), one could simply expect finite t and ∆ ε to have the same effect on transport properties. This ishowever not entirely true, as we show in the following.The main difference results form the bonding and anti-bonding states that form in the case of finite t and are absent if the splitting of ABS is caused only by detuningof the DQD levels. The Andreev current, related differ-ential conductance in the parallel configuration and theTMR are shown in Fig. 9 in the case of finite ∆ ε and inFig. 10 in the case of t = ∆ ε/ 2. The fact that t = ∆ ε/ δ exhibitsmore Coulomb steps compared to the case in the absenceof ABS splitting. The region of the triplet blockade canbe clearly visible for both finite ∆ ε and t , see Figs. 9(a)and 10(a). The main difference with the case shown inFig. 3 (absence of splitting) is the shift of the tripletline in the ( eV, δ )-plane by a factor of the induced ABSsplitting towards larger bias voltages. Moreover, the neg-ative differential conductance for δ/U LR & G PS ,see Figs. 9(b) and 10(c). Note that for δ & 1, in thecase of finite ∆ /U LR ε , there are four regions of negativedifferential conductance, while for finite hopping t , thereare only three. Similar asymmetry can be observed fornegative detuning δ/U LR . − 1, where for ∆ ε = 0, onefinds two regions of current suppression, which occur for eV < 0, while for t = 0 there is only one, cf. Figs. 9and 10. However, the effect of negative differential con-ductance, which is associated with spin accumulation indoublet states, is not that spectacular as in the case of thetriplet blockade, where the current suppression is muchmore pronounced (note the nonlinear color scale used inFigs. 9 and 10).Although there are small differences in the behavior ofthe current and differential conductance in the case ofnonzero ∆ ε and t , the behavior of the TMR is essentialthe same in both cases, see Figs. 9(c) and 10(c). One canobserve a large TMR in the triplet blockade, TMR > p / (1 − p ),cf. Fig. 6 and Eq. (12). Now, however, this region iswider when changing the bias voltage and the splittingof ABS reveals as stripes in TMR as a function of the biasvoltage and detuning, which are most visible for δ & B. Finite magnetic field Here, we release the assumption about the smallnessof external magnetic field B s needed to switch magneticconfiguration of the ferromagnetic leads. Now, we as-sume that this field is strong enough to induce splittingof the ABS states, which can be achieved using ferromag-nets with sufficiently large coercive field. The splittingcaused by external magnetic field has a larger influenceon spin-resolved transport compared to the splitting due1 (cid:230)(cid:231)Ł ØŒ PS º(cid:236)(cid:237) (cid:238)(cid:239)(cid:240) æ(cid:242)(cid:243) (cid:244)ı(cid:246) (cid:247)łøœ -1 ß(cid:252)(cid:253)(cid:254) -2 -1 0 1 2 (b) (cid:255) (cid:0) (cid:1)(cid:2)(cid:3) -(cid:4)(cid:5)(cid:6) -1 (cid:7)(cid:8)(cid:9)(cid:10) -2 -1 0 1 2 0.3 (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) -0.20.1-0.1 G PS (e /h) (!" &’) *+,. -1 /123 -2 -1 0 1 2 2 :;<=T>?@ /U LAB /U CDE /U FG e V / U HI e V / U JK e V / U MN FIG. 11. (Color online) The bias voltage and detuning depen-dence of (a) absolute value of the Andreev current and (b)the respective differential conductance in the parallel mag-netic configuration as well as (c) the TMR calculated in thepresence of external magnetic field B z /U LR = 0 . 2. The otherparameters are the same as in Fig. 2 with U/U LR = 10. to either finite ∆ ε or t . The Andreev current, differ-ential conductance in the parallel configuration and theTMR for finite B z are shown in Fig. 11. One can seethat the triplet blockade region is rather not affected bymagnetic field, see Fig. 11(a). This is due to the factthat while finite B z splits the components of the triplet,the total occupancy of all the triplet components does not change. However, the finite magnetic field changesthe range of bias voltage for which the triplet blockadeoccurs. The negative differential conductance associatedwith the triplet blockade is thus also clearly visible, seeFig. 11(b). Interestingly, the negative differential con-ductance due to the spin accumulation in the doubletstates in now present only for negative values of detun-ing δ/U LR . − 1, cf. Figs. 3 and 11. This can be un-derstood by realizing that while for δ/U LR . − eV < 0, splitting caused by finite magnetic field addi-tionally enhances the spin accumulation, in the case of δ/U LR & eV > 0, magnetic field diminishes thespin accumulation. Consequently, the current suppres-sion is reduced in the latter case, while in the former oneit is enhanced.The bias voltage and detuning dependence of the TMRis shown in Fig. 11(c). First of all, one can noticea strong asymmetry with respect to the bias reversal,which is most visible for small eV and δ/U LR . 1. For δ/U LR & eV = 0. On the other hand, when | δ/U LR | . 1, the DQD is singly occupied and the TMRreveals then a strong asymmetry with respect to the biasreversal. For positive bias voltage, there is a large pos-itive TMR, while for negative bias, the TMR becomesnegative. Such asymmetry is associated with the split-ting of the doublet ground state of the DQD. It stronglyaffects CAR processes in the antiparallel configuration,since the Cooper pair electrons tunnel then to either ma-jority or minority spin bands of the ferromagnets. Be-cause for one bias polarization the electron occupyingthe DQD is the majority-band electron, while for the op-posite bias polarization, this electron belongs to the spinminority channel, it effectively leads to large asymme-try of the flowing current with respect to the bias re-versal, which is most visible in the TMR. The effect ofsign change of the TMR can be even more pronouncedin the case of δ/U LR . − 1, where the DQD is occupiedby two electrons. For eV > 0, one then finds TMR ≈ eV < 0, TMR ≈ − / 2. The mechanism lead-ing to this asymmetry is similar to that described above.Note, however, that in the blockade regions the current,and thus the TMR, can be still modified by cotunnelingprocesses. Out of the Coulomb blockade region, the changes inTMR are not that spectacular, however, there are stillconsiderable differences compared to the case of B z = 0.First of all, negative values of the TMR can be observedfor eV . − U LR and δ/U LR . − 1. Moreover, althoughin the triplet blockade regime, one observes a large pos-itive TMR, similarly as in the absence of magnetic field,cf. Figs. 6 and Fig. 11(c), for finite B z there is an addi-tional region of an enhanced TMR. It occurs for voltagesaround eV ≈ ( δ + U LR ) / > OPQ |R PS SUV 3 2 1 0 -1 -2 -3-8 -5 -2 1 2-7 -6 -4 -3 -1 0 WXY TMR Z /U LR e V / U L R G PS (e /h) (b) -8 -5 -2 1 2-7 -6 -4 -3 -1 0 [ /U LR 3 2 1 0 -1 -2 -3 e V / U L R 3 2 1 0 -1 -2 -3 e V / U L R -8 -5 -2 1 2-7 -6 -4 -3 -1 0 \ /U LR FIG. 12. (Color online) The absolute value of the Andreevcurrent (a) calculated for the parallel ( I PS ) magnetic config-uration, the corresponding differential conductance (b) andthe tunnel magnetoresistance (c) as a function of detuning δ = 2 ε + U LR and the applied bias eV . The parameters arethe same as in Fig. 2 with U/U LR = 2. V. RESULTS IN THE FULL PARAMETERSPACE To complete the analysis of Andreev transport throughDQD Cooper pair splitters, in this section we extend thediscussion to the full parameter space. Figure 12 presentsthe bias voltage and detuning dependence of the abso- lute value of the Andreev current and the correspondingdifferential conductance in the parallel magnetic configu-ration as well as the TMR. Transport characteristics arenow symmetric with respect to the particle-hole symme-try point of the DQD Hamiltonian, ε ph = − U LR − U/ δ ph = − U LR − U ), with immediate sign change of thebias voltage, I S ( eV, δ > δ ph ) = − I S ( − eV, δ < δ ph ), G S ( eV, δ > δ ph ) = G S ( − eV, δ < δ ph ), and TMR( eV, δ >δ ph ) = TMR( − eV, δ < δ ph ), see Fig. 12. Note that to en-able direct comparison with previous results we still plottransport characteristics as a function of δ = 2 ε + U LR .Moreover, we assumed relatively large capacitive cou-pling between the two dots, U = 2 U LR , to be able to showtransport properties in the whole range of detuning δ ina single panel and not to obscure the features discussedpreviously, which occur around δ ≈ 0. However, resultsare qualitatively the same for larger intradot correlations,as checked numerically (not shown), the main differenceis in the distance between the resonances occurring nowfor δ/U LR ≈ − δ/U LR ≈ − 5, see Fig. 12, whichincreases with increasing U .Due to finite Coulomb correlations in the dots, thereare more Andreev states available for transport, whichgenerally reveals as steps in the bias dependence of thecurrent and corresponding peaks in the differential con-ductance. One can clearly see the regime of the tripletblockade, which occurs for ( δ + U LR ) / . eV . ( δ + U LR +2 U ) and δ & δ . − U − U LR . Note that thereis a relatively large leakage current in the triplet block-ade due to finite intradot correlations, which allow DARprocesses to participate in transport. Interestingly, thereare also another regions where the current becomes sup-pressed and negative differential conductance occurs, seeFigs. 12(a) and (b). To present and discuss these effectsin more detail, let us show the relevant cross-sections ofFig. 12. Since all transport features display an appropri-ate symmetry with respect to δ = δ ph , in the followingwe will only analyze the results for δ ≥ δ ph .The current and differential conductance in both mag-netic configurations as well as the resulting TMR in thecase of δ/U LR = − δ = δ ph . Moreover,one can see that the current does not increase in a mono-tonic way, see Fig. 13(a). For | eV | /U LR & 1, the Andreevcurrent becomes suddenly suppressed and the system ex-hibits a pronounced negative differential conductance,see Fig. 13(b), which is present in both magnetic con-figurations. The decrease of the current is related withan enhanced occupation of doublet states. More pre-cisely, for positive bias voltage eV /U LR & | ↑ , d i−| d, ↑i ) / √ | ↓ , d i−| d, ↓i ) / √ (cid:1)(cid:2) (cid:1)(cid:3) (cid:4) (cid:3) (cid:2)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:6)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:7)(cid:4)(cid:5)(cid:2)(cid:4)(cid:5)(cid:8)(cid:4)(cid:5)(cid:9)(cid:4)(cid:5)(cid:10) (cid:11)(cid:12)(cid:13) (cid:14) (cid:14) (cid:15) (cid:16) (cid:17) (cid:18)(cid:19)(cid:20)(cid:21) (cid:22)(cid:17) (cid:1)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:2)(cid:4)(cid:5)(cid:9) (cid:11)(cid:23)(cid:13) (cid:14)(cid:24)(cid:14)(cid:25)(cid:24) (cid:14) (cid:14) (cid:26) (cid:27) (cid:14) (cid:11) (cid:18) (cid:3) (cid:20) (cid:28) (cid:13) (cid:1)(cid:6)(cid:5)(cid:4)(cid:1)(cid:4)(cid:5)(cid:8)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:8)(cid:6)(cid:5)(cid:4) (cid:14) (cid:14) (cid:29) (cid:27) (cid:20)(cid:29) (cid:4) (cid:14)(cid:24)(cid:14)(cid:25)(cid:24)(cid:11)(cid:30)(cid:13) (cid:1) (cid:20)(cid:21) (cid:22)(cid:17) (cid:31)(cid:1)(cid:7) FIG. 13. (Color online) The Andreev current (a) and thecorresponding differential conductance (b) in both magneticconfigurations together with the tunnel magnetoresistance (c)as a function of applied bias eV for δ/U LR = − δ = δ ph ).The other parameters are the same as in Fig. 12. occupied. Consequently, the Andreev current becomessuppressed and the system exhibit negative differentialconductance. However, with further increase of the biasvoltage, | eV | /U LR & 2, the occupation of the above dou-blet states decreases and the current raises again, chang-ing then monotonically with the bias voltage.The above-described behavior is present in both mag-netic configurations, however, in the parallel configura-tion there is a strong spin accumulation in the doubletstates and the current is more suppressed compared tothe antiparallel configuration. This is reflected in the be-havior of the TMR on the bias voltage, which is shown inFig. 13(c). At low voltage the TMR is negligible, whichis related to the fact that the system is occupied by thesinglet state α ( | , i − | d, d i ) + β ( | ↑ , ↓i − | ↓ , ↑i ), and ther-mally activated transport through this state is insensitive (cid:1)(cid:2) (cid:1)(cid:3) (cid:4) (cid:3) (cid:2)(cid:1)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:2)(cid:4)(cid:5)(cid:6)(cid:4)(cid:5)(cid:7) (cid:8)(cid:9)(cid:10) (cid:11) (cid:11) (cid:12) (cid:13) (cid:14) (cid:15)(cid:16)(cid:17)(cid:18) (cid:19)(cid:14) (cid:1)(cid:4)(cid:5)(cid:3)(cid:1)(cid:4)(cid:5)(cid:20)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:20)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:21) (cid:8)(cid:22)(cid:10) (cid:11)(cid:23)(cid:11)(cid:24)(cid:23) (cid:11) (cid:11) (cid:25) (cid:26) (cid:11) (cid:8) (cid:15) (cid:3) (cid:17) (cid:27) (cid:10) (cid:1)(cid:4)(cid:5)(cid:2)(cid:1)(cid:4)(cid:5)(cid:3)(cid:4)(cid:5)(cid:4)(cid:4)(cid:5)(cid:3) (cid:11) (cid:11) (cid:28) (cid:26) (cid:17)(cid:28) (cid:4) (cid:11)(cid:23)(cid:11)(cid:24)(cid:23)(cid:8)(cid:29)(cid:10) FIG. 14. (Color online) The same as in Fig. 13 calculated for δ/U LR = − to the change in magnetic configuration. With increasing eV , the TMR increases at the first step in the current. In-terestingly, for voltages where the current is suppressed,the TMR takes large values due to spin accumulation inthe doublet states, but then drops again with next stepin the current when the voltage is increased further on.When moving away from the particle-hole symmetrypoint, δ = δ ph , there is a large change in the transportcharacteristics, see Fig. 12. First of all, a pronouncedasymmetry with respect to the sign change of the biasvoltage occurs. Moreover, transport characteristics be-come more complex, since the number of negative differ-ential conductance regions increases and one can also findtransport regimes where negative TMR occurs. Here,let us discuss in somewhat greater detail the case of δ/U LR = − 2, which is presented in Fig. 14.The Andreev current as a function of bias voltage isshown in Fig. 14(a). Its bias dependence is not mono-tonic irrespective of bias polarization and magnetic con-4figuration of the device. For both positive and nega-tive bias voltage, there are two regions of current sup-pression accompanied with respective negative differen-tial conductance, see Fig. 14(b). For positive bias, thecurrent first decreases once eV /U LR ≈ / 2, which is as-sociated with enhanced occupation of the doublet state[ | ↑ , d i − | d, ↑i ] / √ 2. In this transport regime DAR pro-cesses are suppressed and transport is mainly due to CARprocesses. With increasing the bias voltage, I S increasesfor eV /U LR ≈ / eV /U LR ≈ 4, wherethe DQD becomes mainly occupied by the state | d, d i .Then, the rate of both Andreev reflection processes be-comes decreased. Similar features can be observed fornegative bias voltage and the mechanism leading to cur-rent suppression and negative differential conductance isbasically the same. The first negative differential conduc-tance occurs due to enhanced occupation of the doubletstate [ | ↓ , i − | , ↓i ] / √ − / & eV /U LR & − / | , i (for eV /U LR . − / . eV /U LR . / 2, that is at the first plateau for positivebias voltage, see Fig. 14(c). This is related with nonequi-librium spin accumulation in the parallel configuration,due to which the occupation of the triplet component | ↑ , ↑i becomes enhanced and the Andreev current drops.This triplet blockade is absent in the antiparallel config-uration, which leads to large difference in the currents inboth configurations and thus to large TMR effect.Another interesting feature is the negative TMR,which occurs at very low positive bias voltage, seeFig. 14(c). In this transport regime the Andreev pro-cesses are suppressed due to the fact that the DQD isoccupied by the doublet state [ | ↓ , i − | , ↓i ] / √ α [ | ↑ , i + | , ↑i ] + β [ | ↑ , d i + | d, ↑i ]. It is thisstate that allows for finite Andreev current in the parallelconfiguration, giving rise to negative TMR effect. VI. ENTANGLEMENT FIDELITY In this section we study the entanglement fidelity be-tween split electrons forming a Cooper pair. Since the ]^_ ‘ (b) a /U bcd /U ef e V / U gh e V / U ij k FIG. 15. (Color online) Fidelity F in the parallel magneticconfiguration as a function of detuning δ and the bias volt-age eV calculated for (a) ∆ ε = 0 and (b) ∆ ε/U LR = 0 . F = 1. The otherparameters are the same as in Fig. 2. Cooper pair is split and each electron tunnels to differ-ent arm of the device in a CAR process, we again focus onthe transport regime where DAR processes are excludedby assuming infinite intradot Coulomb correlations.To analyze the fidelity let us consider the Wernerstate, which has the following form, W ( F ) = F | S ih S | + (1 − F ) − | S ih S | , (18)where F denotes Werner fidelity. For F ≤ / 2, theWerner state is unentangled, whereas for 1 / < F ≤ 1, there exist purification protocols, which can extractstates with arbitrary large entanglement. Werner fidelityfor the considered system is given by the formula, F = P S P S + P T , (19)where P T = P T + P T ↑ + P T ↓ and P S are occupation proba-bilities for the triplet and singlet states, respectively. The5latter can be expressed by P ± = h±| ˆ ρ |±i as P S = (cid:18) αβ + βα (cid:19) − (cid:20) P + β + P − α (cid:21) , (20)with α = (1 / √ p − δ/ (2 ε A ) and β =(1 / √ p δ/ (2 ε A ). Roughly speaking, when P S ≫ P T , fidelity reaches its maximal value F ≈ δ and eV is shown in Fig. 15 (a) in the absenceof detuning, ∆ ε = 0, and (b) for ∆ ε/U LR = 0 . 4. One canclearly observe transport regimes where F is either equalto one or zero. More specifically, for eV < | ( δ + U LR ) / | ,one has P S ≫ P T , and fidelity reaches its maximal valuewith F ≈ 1. Thus, the transmitted pairs of electrons canbe considered as entangled. On the other hand, for biasvoltages, eV > | ( δ + U LR ) / | , the situation is just oppo-site and one obtains F ≈ 0. Consequently, the consideredDQD setup guarantees that, by properly tuning the de-vice parameters, fully entangled pairs of electrons can beextracted from the superconductor and transmitted intonormal leads. This effect is insensitive to the value of theleads’ spin polarization p and the magnetic configurationof the system (results not shown). It is also interesting tonotice that fidelity provides information about the flow-ing current. The current does not flow due to the tripletblockade ( P S = 0 and P T = 1), i.e. when F = 0, cf.Figs. 2 and 15.The situation becomes more complex when finite de-tuning of DQD levels is present. In this case the map of fi-delity possesses richer structure, see Fig. 15(b). The maindifference is in the splitting of the line, eV ≈ ( δ + U LR ) / F = 1 for smaller bias voltages forgiven detuning δ & − U LR / 2, compared to the case of∆ ε = 0. In fact, the voltage is smaller by a factor oflevel splitting ∆ ε . Moreover, with increasing the biasvoltage and for δ & − U LR / F does not drop to zeroimmediately, but becomes suppressed in a nonmonotonicway in the transition region, eV ≈ ( δ + U LR ) / ± ∆ ε , ofwidth 2∆ ε , see Fig. 15(b). Finally, we note that similarsplitting of the transition line separating the regions with F = 0 and F = 1 also occurs in the case of finite hoppingbetween the dots or finite magnetic field. VII. CONCLUSIONS In this paper we have studied the spin-resolved An-dreev transport through double quantum dot-basedCooper pair splitters with ferromagnetic leads. The con-sidered device consisted of two single-level quantum dots coupled to a common s -wave superconductor and eachdot coupled to its own ferromagnetic lead. The calcula-tions were performed with the aid of the real-time dia-grammatic technique, assuming weak coupling betweenDQD and ferromagnets and taking into account the se-quential tunneling processes. We have analyzed the biasvoltage and DQD level dependence of the Andreev cur-rent and the differential conductance in the parallel andantiparallel configurations, as well as the resulting tunnelmagnetoresistance.In the case of infinite correlations in the dots, we havediscussed the behavior of spin-dependent characteristicsin the transport regime where only crossed Andreev re-flection processes are possible. For certain DQD levels’configuration and applied bias voltage, the current is thensuppressed due to the triplet blockade. We showed thateven relatively large intradot correlations can lead to fi-nite leakage current in the triplet blockade. We found anenhanced TMR in the triplet blockade, which indicatesthe role of CAR processes in transport. We have alsoanalyzed the effect of splitting the Andreev bound statesby either finite DQD level detuning, finite hopping be-tween the dots or finite magnetic field. While in the firsttwo cases each Andreev bound state becomes split intotwo, finite magnetic field further splits the ABS, resultingin more complex transport characteristics, with negativedifferential conductance and negative TMR occurring incertain transport regimes.Moreover, assuming finite correlations in the dots, wehave studied transport properties in the full parameterspace, where both DAR and CAR processes are relevant.We found transport regimes where additional currentsuppression accompanied with negative differential con-ductance occurs. These suppression regimes are due toenhanced occupation of certain many-body DQD states,which diminishes the rate of either CAR or DAR pro-cesses, depending on transport region.Finally, in the CAR transport regime we have also ana-lyzed the entanglement fidelity between electrons formingCooper pairs. We showed that the fidelity of split Cooperpair electrons can be tuned by bias and gate voltages andfor certain parameters F can reach unity. 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