Microscopic theory of Cooper pair beam splitters based on carbon nanotubes

P. Burset1, W. J. Herrera2 and A. Levy Yeyati11Departamento de Fısica Teorica de la Materia Condensada C-V,

Universidad Autonoma de Madrid, E-28049 Madrid, Spain2Departamento de Fısica, Universidad Nacional de Colombia, Bogota, Colombia

(Dated: September 18, 2021)

We analyze microscopically a Cooper pair splitting device in which a central superconductinglead is connected to two weakly coupled normal leads through a carbon nanotube. We determinethe splitting efficiency at resonance in terms of geometrical and material parameters, including theeffect of spin-orbit scattering. While the efficiency in the linear regime is limited to 50% and decayexponentially as a function of the width of the superconducting region we show that it can rise upto ∼ 100% in the non-linear regime for certain regions of the stability diagram.

PACS numbers: 73.63.-b, 74.45.+c, 73.63.Fg

Introduction: Producing entangled electron pairs in asolid state device from the splitting of Cooper pairs [1] isa challenging possibility which is starting to generate anintense experimental effort [2–4]. A basic splitting deviceis a three terminal system with a central superconduct-ing lead (S) in between two normal (N) ones as depictedin the upper panel of Fig. 1. When a Cooper pair isinjected from the S lead it can either be transmitted as awhole to one of the N leads by means of a local Andreevprocess (Fig. 1a) or split so that each of the electrons inthe pair is transmitted to a different lead, which corre-sponds to a crossed Andreev process (CAR) (Fig. 1b) [5].While initial experimental devices were based on nano-litographically defined diffusive samples [2] more recentexperiments are oriented towards tunable double quan-tum dots systems based either on carbon nanotubes [3]or InAs nanowires [4]. In spite of the difference in mate-rials the systems realized in both experiments were con-ceptually equivalent. They did correspond, however, todifferent physical regimes: while in Ref [3] the hybridiza-tion by direct tunneling between the dots was dominant,in Ref. [4] the direct tunneling appeared to be negligi-ble. In both works the Cooper pair splitting action wasdemonstrated indirectly by analyzing the changes in thebehavior of the conductance when going from the normalto the superconducting state. Both works pointed out anunexpectedly high efficiency for CAR, much higher thanwhat would be predicted by theories which do not takeinto account the direct inter-dot tunneling [1]. Still fur-ther experimental and theoretical efforts are needed inorder to demonstrate the splitting unambiguously and toreach the nearly 100% efficiency which could be necessaryfor entanglement detection [6].

In the present work we analyze microscopically the caseof double quantum dots (DQD) defined on single-walledcarbon nanotubes (SWCNTs) and show that the tworegimes of Refs. [3, 4] can be reached in metallic or semi-conducting tubes. We consider the situation illustratedin Fig. 1c and 1d where the central electrode modifies theelectrostatic potential and induces a pairing amplitude on

FIG. 1: (Color online) Schematic representation of local (a)and non-local (b) Andreev processes in a generic Cooper pairsplitter. (c) Specific geometry considered in this work: a finiteSWCNT coupled to normal leads at its ends and a centralsuperconducting lead. The lower image (d) illustrates thepotential profile along the tube.

the portion of the tube underneath without breaking itscontinuity. In agreement with the experimental obser-vations we show that in this case the splitting efficiencydecays rather weakly with the width of the central elec-trode [4]. Our results also suggest how to increase thesplitting efficiency up to a level close to 100% by operat-ing the devices in the non-linear regime. We begin witha SWCNT in the normal state without e-e interactions inorder to analyze the inter-dot coupling. Subsequently, weswitch on superconductivity in the central electrode andstudy the probability of the CAR processes and the split-ting efficiency. Finally, we use a minimal model to an-alyze the effect of the electron-electron interactions andthe non-linear regime, where the efficiency can be highlyenhanced.

Basic modeling: We focus on zig-zag SWCNTs which

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allow to consider both metallic and semiconducting cases[8]. If the coupling to the central lead is sufficientlysmooth on the atomic scale we may assume that inter-valley scattering is weak and the K-K’ degeneracy is pre-served. For this case and when the radius is of the orderof 1 nm or smaller it is important to consider curvatureeffects which produce a finite band gap in metallic tubesand enhances the effect of spin-orbit (SO) interactions.We use two complementary approaches for describing theelectronic states in the zig-zag SWCNT: a tight-binding(TB) model and a continuous description based on theBogoliubov-de Genes-Dirac equations. While the latterallows analytical insight, the numerical TB calculationsallow to identify effects due to deviations from linear dis-persion, disorder or arbitrary spatial variation of the elec-trostatic potential along the tube.

Within the continuous description the system is char-acterized by the equations(

Heτ,s − EF ∆(x)∆(x) EF −He

τ,s

)(uτ,svτ,s

)=Eτ,s

(uτ,svτ,s

)(1)

where Heτ,s = −i~vF∂x ·σx+τ~vF qnσy+τδ0s−τδ1sσy+

V (x) is the normal state effective Hamiltonian for then mode (corresponding to a quantized momenta qnaround the tube), ∆(x) is the induced pairing ampli-tude and V (x) the electrostatic potential profile alongthe tube. In these equations σµ are Pauli matrices insublattice space, and τ, s = ± correspond to the val-ley and spin indexes respectively. Finally, the terms inδ0 and δ1 take into account the SO interaction as inRefs. [10, 11]. The quantized momenta take the valuesqn = 2π

3Na0

(n± p

3

)− qcurv, with p = Nmod3 = 0,±1

and N being the number of atoms in the cross sec-tion. Depending on whether p = 0 or p = ±1 thetube is metallic or semiconductor, respectively. The cur-vature effect is included in qcurv = Ecurv/~vF , whereEcurv ' π2|Vppπ|/4N2 with |Vppπ| ' 2.7eV . In the nor-mal homogeneous case the corresponding energy levelsfor longitudinal wavevector k are thus given by Enτ,s(k) =

~vF√

(qn + τsδ1)2 + k2+δ0τs. The transport propertiescan be expressed in terms of Green functions which sat-isfy (E −Hτ,s(x))Gτ,s(x, x

′) = δ(x − x′), where Hτ,s(x)denotes the full Hamiltonian on the left hand side of Eq.(1). We obtain these quantities by solving first for theuniform finite regions and then matching the result usingthe method of Ref. [7].

Normal state: We start by analyzing the linear con-ductance along the tube, GLR, when the central lead isin the normal state. In Fig. 2 we show a map of GLR inthe VgL−VgR plane, obtained using the TB model in theusual nearest neighbors approximation with a hoppingparameter t ≡ Vppπ. As a first approximation the poten-tial profile along the tube is assumed to change discontin-uously as represented by the dashed lines in Fig. 1d. Thelateral leads are modeled by ideal 1D channels weakly

FIG. 2: (Color online) Left panel: Conductance map for ametallic tube (N = 12) in the normal state for the p-n-p region with (lower) and without (upper) SO interactions.Right panel: Same for the semiconducting tube (N = 11)but in a logarithmic scale. The geometrical parameters areW = WL,R = 170nm.

coupled to the ends of the tube, as represented schemati-cally in Fig. 1c. We fix the tunneling rates to these leadsto a value ΓL,R ∼ 0.01t which is consistent with the con-ductance values observed in Ref. [9] for the lowest energystates of a SWCNT quantum dot. To model the effect ofthe central lead we rely on the results of ab-initio calcu-lations for the case of Al electrodes [12, 13]. Accordingto Ref. [12] these produce a n-doping effect, leading toa shift of the tube bands EFs ∼ −0.5eV for an ideal in-terface. On the other hand in the normal state it wouldinduce a broadening of the tube levels of the order of a fewmeV [13] which suggests a typical value ΓS ∼ 1meV forthe corresponding tunneling rate. As in the experimentsof Ref. [3] the length of the central region is set initiallyto ∼ 200nm. We consider tubes with N = 11, 12 whichcorresponds to radii R ∼ 0.43, 0.47nm for the semicon-ducting and metallic cases respectively. In the metalliccase curvature effects lead to the opening of a narrowgap, which can be estimated as Eg ' Ecurv ' 45meV .The curvature gap is apparent in the upper left panel ofFig. 2. On the other hand, in the semiconducting casethe gap is Eg ' 412meV (top right panel of Fig. 2). Itshould be noticed that for these diameters and for gatepotentials of the order of 0.5eV only the lowest energymode, corresponding to n = 0, gives a significant contri-bution to the transport properties of the tubes that wediscuss below.

For positive VgL, VgR, i.e. in the p-n-p regime the con-ductance displays a DQD behavior as shown in Fig. 2.The metallic case (left panels) exhibits an anticrossingpattern similar to the one found in the experiments ofRef. [3]. As the gate potentials VgL, VgR become increas-ingly positive the conductance map exhibits resonances

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along lines VgL + VgR ∼ const indicating the delocaliza-tion of the electronic states due to Klein tunneling. Theconfinement of the dot states is much more pronouncedin the semiconducting case where the Klein tunneling isless significant. We have used a logarithmic scale in thiscase in order to enhance the visibility of the conductancepeaks.

When SO scattering is introduced (lower panels in Fig.2) there is a general splitting of the conductance peaksof the order of ∼ 2meV due to the breaking of the spin-valley degeneracy. Close to the gap edges this splittingis of the same order as the mean level separation.

Superconducting state: When superconductivity in thecentral lead is switched-on pairing correlations withinthe tube are induced by proximity effect. The size ofthe induced gap ∆i is set by ΓS (i.e. of the order of1meV ). We shall assume that temperature is zero andthat the energy E of the injected electrons from the nor-mal leads is smaller than ∆i. Then RAL(E) and RAR(E)denote the local Andreev reflection probabilities at theL,R leads while TCAR(E) correspond to the CAR pro-cesses. When operated as a beam splitter a finite voltagedifference V is applied between the S and the N leadsand the non-linear conductance is given by GL(R)(V ) =G0(TCAR(V ) + TCAR(−V ) + 2RAL(R)(V )), with G0 =2e2/h (notice that at finite energy in general TCAR(E) 6=TCAR(−E) due to the breaking of the electron-hole sym-metry). Thus we can define the splitting efficiency asη = G0(TCAR(V ) + TCAR(−V ))/(GL(V ) +GR(V )).

Within both the Dirac and TB models it is found thatthe CAR coefficients decay exponentially on the scaleξ(q) = ~vF /∆i

√1− (~vfq/EFs)2, where q = q0±δ1/~vF

exhibiting oscillations on the scale λF = ~vF /|EFs|, asillustrated for the linear regime, V = 0, in Fig. 3. Inthese plots we have fixed the gate voltages at the valuesindicated by the circles in Fig. 2. The CAR probabilitydecays more slowly in the metallic case (N = 12) due tothe longer effective coherence length ξ(q) of this case. Inboth cases, however, the decay is remarkably slower thanin the case of a 3D bulk BCS superconductor where theprediction is ∼ exp (−2W/ξ)/(kFW )2 [14], indicated bythe dashed lines in Fig. 3. This is a consequence of thesingle channel character of the connection between thedots in the present system and explain the rather largeefficiency values estimated in recent experiments [4]. Ascan be observed the efficiency η decreases from 0.4 atW ∼ 200nm to < 0.05 at W ∼ 700nm in the semicon-ducting case, while it varies between 0.5 and 0.2 for themetallic tube within the same W range. For sufficientlylarge W the overall evolution of η is well described bythe expression η ∼ 1/(1+exp 2W/ξ(q)). This qualitativebehavior is also found for a smoother potential profile[15].

Coulomb interactions and non-linear regime: For an-alyzing the effect of electron-electron interactions wefirst map the system into a minimal model in which we

FIG. 3: (Color online) Evolution of the CAR probability(blue) and the splitting efficiency η (red) in the linear regimeas a function of the length W of the central electrode for ametallic SWCNT with N = 12 (upper panel) and a semicon-ducting one with N = 11 (lower panel). The gate potentialsVgL,gR are fixed at the points indicated by the circles in Fig.2. The dashed lines represent the decay of the CAR prob-ability for a 3D bulk superconductor multiplied by a factor1000.

keep just one two-fold degenerate electron level EL,R ineach dot [16]. In the combined dot-Nambu space themodel properties can be expressed in terms of bispinorfields Ψµ = (dµ↑, d

†µ↓) where µ = L,R and d†µσ creates

dot electrons. In the absence of interactions this re-duced model is characterized by a retarded Green func-

tion matrix of the form G(0) =[E−h0+iΓ−Σ(E)

]−1,

where (h0)µν,αβ = Eµδµνδαβ(−1)α+1, with µ, ν ≡ L,R,

α, β ≡ 1, 2 are the Nambu indexes, (Γ)µν,αβ =Γµδµνδαβ ,

with Γµ=Γµa0/Wµ, correspond to the effective tunneling

rates to the normal leads and Σ is a matrix self-energydescribing the coupling with the central superconductingregion. The whole analytical expression for Σ is given inthe supplementary information.

Interactions are introduced by assuming a constantcharging energy UL,R � ∆i acting on each dot level.We take them into account within the equation of mo-tion (EOM) technique with a Hartree-Fock decouplingat the level of the two-body Green functions [15]. Thisapproximation is valid when Kondo and exchange corre-lations between the dots can be neglected. Further sim-plification is achieved in the limit Uµ → ∞ where we

find G=[g−1+iΓ−Σ

]−1, with g= (E− h0)−1[1− A∞]

and (A∞)µν,αβ = nµδµνδαβ . The evaluation of themean values nµ =< d†µσdµσ > must be performed self-

4

consistently. The relevant transport coefficients are fi-nally computed as TCAR(E) = 4ΓLΓR|GLR,12(E)|2 and

RAL(R) =4Γ2L(R)|GLL(RR),12|2.

FIG. 4: (Color online) Color map of the splitting efficiencywithin the minimal model with parameters corresponding toa semiconducting tube with W ∼ 700nm (indicated by thearrow in Fig. 3) in the linear (a) and non-linear (b) regimes.In the latter case the dot levels are varied along the lineEL ∼ −ER indicated by the dashed red line of (a). Thewhite dashed lines indicate the maxima in the spectral den-sity which is shown in (c) for the two dots along this line.Local and non-local Andreev processes at finite V are indi-cated by the black arrows.

The main effect of interactions within this approxi-mation is to shift the resonances and to reduce theirwidth, roughly as (1 − nµ)Γµ. Then, the CAR andthe local Andreev probabilities are reduced by a factor(1−n1)(1−n2) and (1−nµ)2 respectively, which thereforedoes not modify significantly the efficiency at resonance.The color map in Fig. 4a shows the efficiency in thelinear regime corresponding to the semiconducting casewith W ∼ 700nm (arrow in Fig. 3) and for the region ofgate voltages indicated by the circle in the right panel ofFig. 2. As can be observed, η exhibits maximum valuesat the crossing point between the resonances of the orderof 0.1 which are slightly higher than the values found inthe non-interacting case. The efficiency reaches a maxi-mum of the same magnitude along the line EL ∼ −ER(red dashed line). What is much more remarkable is thatthe efficiency along this line can rise up to 100% in thenon-linear regime V 6= 0. This is illustrated in Fig. 4b.The high efficiency regions lie within the dot resonances(indicated by the dashed white lines) which are shiftedfrom zero energy due to the presence of an induced mini-gap by the proximity with the superconducting lead.

The origin of these high efficiency regions can be un-derstood qualitatively from the spectral density on eachdot, which is shown in Fig. 4c. At any given point along

the line EL ∼ −ER electron and hole states are splitdue to hybridization between the dots. Furthermore, theelectron-hole symmetry in the local spectral density islost along this line. Crossed Andreev processes like theone sketched as the CAR arrow in Fig. 4c combines elec-tron and hole states on each dot with high spectral den-sity. These inter-dot transitions are then more favorablethan the intra-dot electron-hole conversions (arrows ARand AL), in which either the electron or the hole statehas low spectral density. As a consequence, local An-dreev processes become suppressed while non-local CARprocesses are enhanced, thus explaining the efficiency in-crease.

Conclusions: We have analyzed the splitting efficiencyof SWCNT double quantum dot devices in terms of ma-terial and geometrical parameters. The single channelcharacter of the connection between the dots in this con-figuration explains the weak decay of CAR with distancewhich is consistent with the experimental observations.Furthermore we have shown how the splitting efficiencycan rise up to ∼ 100% by working in the non-linearregime. We expect that our analysis can guide future ex-periments for the production of entangled electron pairsusing these devices .

The authors would like to thank T. Kontos, A. Cottetand R. Egger for fruitful discussions. This work was sup-ported by MICINN-Spain via grant FIS2008-04209 (PBand ALY) and DIB from Universidad Nacional de Colom-bia, project 12170 (WJH).

[1] P. Recher, E.V. Sukhorukov and D. Loss, Phys. Rev.B 63, 165314 (2001); N.M. Chtchelkatchev, G. Blatter,G.B. Lesovik and T. Martin, Phys. Rev. B 66, 161320(2002); C. Bena, S. Vishveshwara, L. Balents and M.P.A.Fisher, Phys. Rev. Lett. 89, 037901 (2002); P. Samuels-son, E.V. Sukhorukov and M. Buttiker, Phys. Rev. Lett.91, 157002 (2003).

[2] D. Beckmann, H.B. Weber and H.v. Lohneysen, Phys.Rev. Lett. 93, 197003 (2004); S. Russo, M. Kroug,T.M. Klapwijk and A.F. Morpurgo, Phys. Rev. Lett.95, 027002 (2005); P. Cadden-Zimansky and V. Chan-drasekhar, Phys. Rev. Lett. 97, 237003 (2006).

[3] L. Herrmann et al., Phys. Rev. Lett. 104, 026801 (2010).[4] L. Hofstetter et al., Nature 461, 960 (2009).[5] G. Deutscher and D. Feinberg, Appl. Phys. Lett. 76, 487

(2000).[6] V. Bouchiat et al., Nanotechnology 14, 77 (2003).[7] W.J. Herrera et al., J. Phys: Condens. Matter 22, 275304

(2010).[8] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 226801

(2005).[9] F. Kuemmeth et al., Nature 452, 448 (2008).

[10] J.-S. Jeong and H-W. Lee, Phys Rev. B 80, 075409(2009).

[11] S. Weiss et al., Phys. Rev. B 82, 165427 (2010).[12] G. Giovannetti et al., Phys. Rev. Lett. 101, 026803

5

(2008).[13] S. Barrasa-Lopez et al., Phys. Rev. Lett. 104, 076807

(2010).[14] G. Falci, D. Feinberg, and F. W. J. Hekking, Europhys.

Lett. 54, 255 (2001).

[15] See supplementary information.[16] The effect of SO interactions is to break the four-fold

degeneracy and is thus implicitly taken into account inthe minimal model.