Crossed Andreev reflection in graphene normal–superconductor–normal structures with pseudo-diffusive interfaces

Physics Letters A 375 (2011) 1339–1343

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Crossed Andreev reflection in graphene normal–superconductor–normal structures with pseudo-diffusive interfaces Hakimeh Mohammadpour a,∗ , Asghar Asgari b,c,1 a b c

Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, 45195 Zanjan, Iran Research Institute for Applied Physics, University of Tabriz, 51665-163, Tabriz, Iran School of Electrical, Electronics and Computer Engineering, the University of Western Australia, Crawley, WA 6009, Australia

a r t i c l e

i n f o

Article history: Received 8 September 2010 Received in revised form 11 January 2011 Accepted 12 January 2011 Available online 15 January 2011 Communicated by R. Wu Keywords: Graphene Crossed Andreev reflection Superconductor Pseudodiffusive conduction

a b s t r a c t In this Letter graphene normal–superconductor–normal heterostructures are modeled for studying the crossed Andreev reflection. A thin layer of undoped graphene with Fermi energy at the Dirac point at is assumed the interface between superconductor layer and each normal lead. The resulting contribution of the crossed Andreev reflection to the nonlocal conductance equals that of the electron elastic cotunneling. We explain this as another figure of merit for pseudodiffusive conduction at the Dirac point of the undoped layers. Also structures with only one undoped layer at the interface between the superconductor and one of the normal leads, as well as structures in which one of the leads is ferromagnetic, show pseudodiffusive conduction at the Dirac points. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Mesoscopic contacts between superconductor (S) and normal (N) or ferromagnetic (F) metals have been subject of vast studies [1–5]. They reveal quantum correlations between the coherence effects in the normal metal and the intrinsic coherence of the superconductivity provided by Andreev reflection (AR) in which an electron-like quasiparticle in the normal metal is reflected at the NS interface as a hole-like quasiparticle and a Cooper pair is carried into the superconductor [1]. However, there is a novel quantum phenomenon, called Crossed Andreev Reflection (CAR) in the presence of two normal metals separated by a superconductor of width of the order of the superconducting coherence length. An electron-like quasiparticle of energy less than the superconducting gap in one of the normal metals that hits the NS interface forms Cooper pair in S by capturing another electron from the other normal metal [6–10]. In this process a hole is scattered to the second normal metal the wavefunction-amplitude of which determines the CAR probability. This nonlocal coherent quantum effect is one of the main sources of the solid state entanglement that is an essential aspect of the quantum physics and proposed for exciting applications in the quantum computing [11,12].

*

Corresponding author. Tel.: +98 914 4076401, fax: +98 411 3347050. E-mail addresses: [email protected] (H. Mohammadpour), [email protected] (A. Asgari). 1 Tel.: +98 411 3393005; fax: +98 411 3347050. 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.01.026

A simple model for studying CAR is composed of a threeterminal NSN structure in which by applying voltage between S and one of the normal metals, a voltage is induced in the other normal metal. When an electron from one of the normal leads (say N 1 ) hits the S interface, four types of scattering may be emerged; it may be reflected as either an electron (normal reflection) or a hole (AR) at the same normal metal (N 1 ), as well as it may undergo scattering as an electron (elastic cotunneling, CT) or a hole (CAR) at the second normal lead (N 2 ). In ballistic NSN structures the sub-gap transport is governed mainly by the processes of AR and CT, while CAR has negligible contribution to the conductance [13]. The CAR amplitude is larger at FSF structures with ferromagnetic leads of anti-parallel magnetization directions [7,8]. Intrinsic spin-singlet Cooper pairs in S hinders AR and CT, because AR demands both of the spin-up and spin-down states at the first contact and CT occurs between sub-bands of the same spin-species of the two lateral contacts. It has also been shown that in diffusive NSN, the processes of CT and CAR contribute equally to the nonlocal conductance [6,9,10]. Graphene composed of carbon atoms arranged in the twodimensional hexagonal lattice, on account of its atomic-size thickness, can be superconducting or ferromagnetic by proximity to a superconductor or ferromagnetic electrode on top of it [14–18]. This has triggered vast studies in the electronic transport in the NS and FS hetero-structures of graphene [19–25]. The conduction and valence bands in graphene touch each other, making it a gapless semiconductor. So, the carrier type (electron, n, or hole, p) and its density can be easily tuned in graphene in a controllable

1340

H. Mohammadpour, A. Asgari / Physics Letters A 375 (2011) 1339–1343

where H 0 = −i h¯ v F (σ x ∂x + σ y ∂ y ) − E FI is the Dirac Hamiltonian and u and v are the BCS coherence factors belonging to different valleys of the k-space. σ x and σ y are the Pauli matrices in the pseudospin space of the two sublattices and I is the two-dimensional unit matrix. v F is the Fermi velocity and S, N , N E F ≡ E F 1 2 is the Fermi energy in S, N 1 and N 2 , respectively. It N

N

takes zero value at the undoped D regions and relations E F 1 = E F 2 , N EF 1

E SF = + U hold. The large U in the S lets us assume a step-like variation of the superconducting pair potential that takes nonzero value  only inside S [19]. ε > 0 is the excitation energy measured from the Fermi level that in terms of the two-dimensional wave-vector in the normal regions k ≡ (k, q) is:

Fig. 1. Modeled NDSDN structure on graphene in the xy plane.

manner by local electrostatic gates or chemical doping. By employing this property in a ballistic graphene n–S–p structure, channels for the CT and AR processes are suppressed resulting in enhanced CAR [24]. Previous studies have shown that in a ballistic graphene NSN structure, as in the normal metals, CT is the dominant process in the sub-gap nonlocal transport while CAR has less contribution to it [20,21]. The conductance and shot noise measurements for normal graphene, as well as the Josephson current and magneto-transport measurements manifest pseudodiffusive electronic transport in graphene at the Dirac point of zero density of states [26–31]. While the ballistic graphene NSN structure has been studied [20,21] but the pseudodiffusive CAR has not been considered so far. Motivated by this, in this Letter we consider a graphene NSN structure having a thin un-doped strip, D, in the interface between S and each N contact (NDSDN). We have addressed the question of whether pseudodiffusive transport manifests itself in the nonlocal conductance of graphene NDSDN structures as in diffusive normal NSN structures. Our calculations show that as in the diffusive metallic NSN structures, in the graphene NDSDN, pseudodiffusive transport results in the equal CT and CAR probabilities in the nonlocal conductance. Examining the case in which only one undoped interface layer is assumed between S and one of the normal contacts (NDSN and NSDN) shows a pseudodiffusive electronic transport, as well. Studying the ferromagnetic counterpart by switching on the ferromagnetic correlations in one of the normal leads (FDSN and FSDN) reveals a pseudodiffusive electronic transport. 2. Model and basic equations

H0

I

    I u u =ε v v −H 0



(2)

where ± refers to the valence and the conduction bands, respectively, and n = N 1 , N 2 , D 1 , D 2 . Dirac equation is solved for an incident electron to the N 1 D boundary in N 1 with sub-gap energy ε <  and incidence angle ϕeN 1 = arcsin( h¯ v FNq1 ), where q is the ε+ E F

y-component of the wavevector. From the conservation of q under scattering, we obtain the following relations for the scattering angles: N (N2 , D 1 , D 2 )

sin ϕh 1

=

N1

sin ϕe

ε + E NF 1 ε − E NF 1 (N 2 , D 1 , D 2 ) ε + E NF 1

N2 (D 1 ,D 2 )

sin ϕe

=

N1

sin ϕe

(1)

(3)

(D1,D2) ε + E N2 F

However, the magnitude of the momentum is changed by different scattering processes (see Eq. (2)). By substituting Eq. (2) into Eq. (3) for an electron with energy ε above the Fermi level hitting N the junction in an angle ϕe 1 the following relations are obtained for the x components of the momentum:

kh 1 = (h¯ v F )−1 N

N ke(2h)



−1

= (h¯ v F )



ε − E NF 1

 

2

2    − ε + E NF 1 sin2 ϕeN 1

ε + (−) E NF 2

2

2    − ε + E NF 1 sin2 ϕeN 1

(4)

These relations define critical angles for each scattering process above which the corresponding wavefunction is evanescent. These critical angles read:





We consider a NDSDN structure on graphene in the xy plane shown in Fig. 1 in which a wide superconducting strip (S) of length L is the interconnect between two normal leads (N 1 and N 2 ) with thin undoped graphene layers (D 1 and D 2 ) at the interfaces. The Fermi energy at each region can be modulated by doping or using electrostatic gates. Highly doped superconductor and the resulting large Fermi energy mismatch, U , between the superconductor strip and other parts of the structure is assumed. Transport properties of the quasiparticles in this system are studied in the scattering formalism by solving the Dirac–Bogoliubov–de Gennes (DBdG) equation which describes superconducting correlations between massless Dirac fermions of different valley indices in graphene [19]. On account of the valley degeneracy in wide ballistic graphene, we work only on one set of the four-dimensional (electron and hole two-component pseudospin) equations which describes coupling of an electron from one valley to a hole from the other valley. It takes the following form





ε =  E nF ± h¯ v F |k|



ϕhN,1critic = arcsin ε − E NF 1  ε + E NF 1  





ϕeN(2h),critic = arcsin ε + (−) E NF 2  ε + E NF 1 

(5) D ,D

respectively. Inside undoped D-regions with E F 1 2 = 0, the critical angle for ε = 0 is zero. Above the critical angle, one should replace ϕhN 1 and ϕen(h) (n = N 2 , D 1 , D 2 ) with

 

 sin(ϕeN 1 )   sgn ϕ sgn ε −  N 2 sin(ϕh,1critic ) 

  sin(ϕeN 1 )   N    π  sgn ϕe 1 × sgn ε + (−) E nF − i cosh−1  2 sin(ϕen(h),critic )  

N1  × e



π



N  E F 1 − i cosh−1 

(6) Inside the normal regions, the wavefunctions are linear combinations of the four eigenfunctions of the Dirac Hamiltonian:

    n ψen± ∝ exp iqy ± ikne x ⊗ e ∓i ϕe , ±1, 0, 0     n ψhn± ∝ exp iqy ± iknh x ⊗ 0, 0, e ∓i ϕh , ∓1 .

(7)

H. Mohammadpour, A. Asgari / Physics Letters A 375 (2011) 1339–1343

1341

Propagation directions along and opposite to the x axis are denoted by ±, respectively. (For electrons in the valance band and holes in the conduction band ϕen(h) and kne(h) are changed to −ϕen(h) and

−kne(h) .) The wavefunctions in the two lateral leads read:

Ψ N 1 = ψeN+1 + r ψeN−1 + r A ψhN−1 Ψ N 2 = t e ψeN+2 + th ψhN+2

(8)

r and r A are the amplitudes of the electron normal and Andreev reflections in N 1 . t e and th are the amplitudes of scattering as an electron and hole into N 2 that manifest CT and CAR processes, respectively. The wavefunction in S is a linear combination of the eigenfunctions of Eq. (1) inside S. These eigenfunctions are:

    ψhS± ∝ exp iqy + ikhS ± x ⊗ exp(−i β), ∓ exp(−i β), 1, ∓1     ψeS± ∝ exp iqy + ikeS± x ⊗ exp(i β), ± exp(i β), 1, ±1

(9)

where β = arccos(ε /), keS± = ±(k0 + ik), khS ± = ∓(k0 − ik) and

k = sin β/ξ, k0 = (¯h v F )−1 × ( E SF + ε ). For ε   it describes the wavefunctions with damping amplitudes in the propagation direction. The scattering probabilities R = |r |2 , R A = |r A |2 , T e = |t e |2 , T h = |th |2 are obtained by imposing continuity condition for the pseudospins at the four interfaces. Relation R + R A + T e + T h = 1 ensures current conservation. We note that contrary to the normal regions, the wavefunctions in S are propagating only for ε   (imaginary k); for ε <  (real k) the wavefunctions are exponentially evanescent along their propagation direction within a scale of the order of ξ = h¯ v F / inside S. Thus in the limit L /ξ  1, the sub-gap transmission contains only r A while t e = th = 0. By decreasing the superconducting length to the values around L /ξ = 1, the electron elastic cotunneling through S comes into play and meanwhile the local proximity is reduced in favor of the nonlocal proximity to the second normal lead via CAR. Nonequilibrium behavior of the system is studied under the condition that there is a voltage difference V between S and normal lead N 1 . The contributions of the CT and CAR to the nonlocal differential conductance of the system, G CT and G CAR , are calculated by the generalized Blonder–Tinkham–Klapwijk [32] formula as: N ϕh,2critic



N

N

dϕe 1 cos ϕe 1 |th |2

G CAR = g 0 0 N2 ϕe,critic



N

N

dϕe 1 cos ϕe 1 |t e |2

G CT = g 0

(10)

0

g 0 = (2e /π h¯ ) N 0 (eV ). The density of states is determined by N N 0 (ε ) = |ε + E F 1 | W /(π h¯ v F ) where W is the width of the junction. 2

3. Results Comparing Eqs. (10) for the contributions from CT and CAR to the nonlocal conductance reveals the ballistic or pseudodiffusive properties of the quasiparticle transport. We first analyze the transport behavior of the structure when the lateral leads are in normal state, then the ferromagnetic correlation is taken into account. In a ballistic NSN structure on graphene, the CT, which is the pseudospin-conserving scattering process, is dominant. However, in pseudodiffusive NDSDN, CT equals CAR. This behavior is best explained in terms of the pseudospin conservation. The inci-

Fig. 2. G CAR and G CT versus L /ξ with U / = 15, E F / = 5, ε / = 0, d/λ = 0.2, 0.5.

dent propagating wavefunction at N 1 has a definite pseudospin deN termined by ϕe 1 . Inside undoped regions D, the wavefunction has imaginary k and

D

ϕe(h1()2) . Therefore except for the lowest mode of

ϕeN 1 = 0, the initial pseudospin is destroyed inside D 1 and D 2 . The phenomena of evanescent transmission and the pseudospin destruction are more pronounced in higher modes. So due to the lack of definite chirality for electron-like and hole-like quasiparticles inside undoped D-regions, CT and CAR act equally in matching the wave functions in the interfaces. This gives rise to the equal amplitudes for the CT and CAR in the nonlocal conductance. The rest of the Letter is organized as follows: in Section 3.1, NDSDN structure is studied and in Section 3.2, we study NDSN and NSDN structures in which only one of the NS interfaces is an undoped layer. Finally in Section 3.3 the case with one ferromagnetic lead is studied in N N the FDSN and FSDN structures. We assume E F 1 = E F 2 = E F as well as

ϕeN(1h) = ϕeN(2h) = ϕe(h) .

3.1. NDSDN structure Let’s analyze the behavior of the different scattering probabilities to the differential conductance of a symmetric NDSDN structure. Fig. 2 shows G CAR and G CT versus the thickness of the superconducting strip, L /ξ for U / = 15, E F / = 5, ε / = 0, d/λ = 0.2, 0.5. d is the thickness of the undoped layers and λ is the wavelength of the electron at the doped lateral normal leads. This is the characteristic length-scale for decaying of the wavefunction in D. Quantum oscillations with varying L /ξ are present at zero bias, ε / = 0. The period of these oscillations scales with the wavelength of the wavefunctions inside S and is proportional to (U + E F ). At small thicknesses that AR and CAR cannot be developed, CT is the dominant process. For thicknesses of L /ξ ≈ 1, G CAR and G CT are equal and by more increasing the thickness, L /ξ , both of these nonlocal processes are suppressed totally. Comparison of G CT and G CAR between structures of different d/λ indicates that they not only decrease for larger d/λ, but also saturate in large amounts of L /ξ . The first issue is due to the large backscattering at the first undoped layer. The second property is best understood recalling that for the larger lengths of the undoped regions, the wavefunctions are more confined in S and therefore the reflection probability as an electron (r) or a hole (r A ) does not increase by increasing the superconductor length. The large oscillation amplitudes of G CAR in NDSDN is due to the formation of the bound states in S, because the two undoped regions give rise to equal amplitudes to the interfered wavefunctions inside S.

1342

H. Mohammadpour, A. Asgari / Physics Letters A 375 (2011) 1339–1343

Fig. 3. Distributions of CT and CAR amplitudes versus incidence angle lengths of the undoped layers d/λ = 0.2, 0.5.

ϕeN 1 for two

Fig. 5. G CAR and G CT versus L /ξ in FDSN and FSDN for the same values of the parameters as in Fig. 4.

3.3. FSDN and FDSN structures

Fig. 4. G CAR and G CT versus L /ξ in NDSN and NSDN for the same values of the parameters as in Fig. 2.

Fig. 3 exhibits the distribution of the scattering probabilities N T e and T h , versus the incidence angle ϕ ≡ ϕe 1 for two different lengths of the undoped layers, d/λ = 0.2, 0.5. In ballistic NSN, the reflectionless tunneling of the normally incident electrons through the potential well favors CT in small angles and renders CAR to the larger angles. By increasing d/λ, the pseudodiffusive property plays a crucial role, so that CAR is developed in small angles around the normal to the interface. As a matter of the fact that the chirality of the normal incidence is not affected by any scattering process, T e is independent of the value of d/λ at ϕ = 0. 3.2. NSDN and NDSN structures In this section we study structures with only one undoped layer in the NS interface, i.e. NDSN and NSDN. Fig. 4 shows the normalized G CAR and G CT versus L /ξ for the same values of the parameters of Fig. 2. For d/λ = 0.5 we have G CAR = G CT at L /ξ ≈ 1. By increasing L /ξ , these two components of the nonlocal conductance tend to zero in an oscillatory manner. Comparison between Figs. 2 and 4 manifests the reduced amplitudes of the oscillations in NDSN and NSDN in which one of the normal leads has ballistic junction to the S strip. This is because of the less confinement of the wavefunctions to the S strip. For the case with d/λ = 0.2 the effect of the pseudodiffusive transport at the undoped layer is weak and system behaves in a manner similar to the NSN, resulting in large G CT and small G CAR .

In Fig. 5, G CAR and G CT are plotted versus L /ξ for systems with only one undoped interface in which the first electrode is a half metallic ferromagnetic (FDSN and FSDN). In these systems because the spin sub-band for AR is blocked we have r A = 0. Therefore, the existence of the D layer is not of importance and the FDSN behaves like a ballistic FSN with reduced G CAR in favor of G CT . There is also a π phase shift between the contributions of the electrons and the holes to the nonlocal conductance. This is related to the Berry phase between the two channels at this structure which is a characteristic of the ballistic transport regime. Investigation of the inset of Fig. 5, however, reveals that the FSDN structure benefits from the pseudodiffusive conduction through the D layer so that we observe enhanced G CAR as well as oscillation amplitudes larger than that observed for NSDN. As in the NDSDN, these large oscillation amplitudes indicate confinement of the quasiparticle wavefunctions to the superconducting strip via D layer. 4. Conclusions In this Letter we studied nonlocal conductance in graphene NSN structures with thin undoped layers at the interfaces. As in the normal diffusive metals, in our system, G CAR = G CT which is another manifestation for the pseudodiffusive electronic transport in a monolayer graphene with Fermi energy at the Dirac point. We also addressed the case with only one undoped layer in one of the NS interfaces and observed a similar property. Replacing one of the contacts with ferromagnetic graphene shows pseudodiffusive behavior, too. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

A.F. Andreev, Sov. Phys. JETP 19 (1964) 1228. F.W.J. Hekking, Yu.V. Nazarov, Phys. Rev. Lett. 71 (1993) 1625. A.I. Buzdin, Rev. Mod. Phys. 77 (2005) 935. F.S. Bergeret, A.F. Volkov, K.B. Efetov, Rev. Mod. Phys. 77 (2005) 1321. M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. Lett. 74 (1995) 1657. G. Falci, D. Feinberg, F.W.J. Hekking, Europhys. Lett. 54 (2001) 255. G. Deutscher, D. Feinberg, Appl. Phys. Lett. 76 (2000) 487. J.P. Morten, D. Huertas-Hernando, W. Belzig, A. Brataas, Phys. Rev. B 78 (2008) 224515. A. Brinkman, A.A. Golubov, Phys. Rev. B 74 (2006) 214512. J.M. Byers, M.E. Flatté, Phys. Rev. Lett. 74 (1995) 306. N.M. Chtchelkatchev, G. Blatter, G.B. Lesovik, T. Martin, Phys. Rev. B 66 (2002) 161320(R). L.B. Ioffe, V.B. Geshkenbein, M.V. Feigelman, A.L. Fauchere, G. Blatter, Nature (London) 398 (1999) 679.

H. Mohammadpour, A. Asgari / Physics Letters A 375 (2011) 1339–1343

[13] M. Bozovic, Z. Radovic, arXiv:cond-mat/0207375v1. [14] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [15] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [16] H.B. Heersche, P. Jarillo-Herrero, J.B. Oostinga, L.M.K. Vandersypen, A.F. Morpurgo, Nature 446 (2007) 56. [17] G. Fagas, G. Tkachov, A. Pfund, K. Richter, Phys. Rev. B 71 (2005) 224510. [18] N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Nature 448 (2007) 571. [19] C.W.J. Beenakker, Phys. Rev. Lett. 97 (2006) 067007. [20] C. Benjamin, J.K. Pachos, Phys. Rev. B 78 (2008) 235403. [21] H. Mohammadpour, M. Zareyan, Iranian J. Phys. Res. 8 (2) (2008) 81.

1343

[22] M. Zareyan, H. Mohammadpour, A.G. Moghaddam, Phys. Rev. B 78 (2008) 193406. [23] Y. Asano, T. Yoshida, Y. Tanaka, A.A. Golubov, Phys. Rev. B 78 (2008) 014514. [24] Q. Zhang, D. Fu, B. Wang, R. Zhang, D.Y. Xing, Phys. Rev. 101 (2008) 047005. [25] J. Cayssol, Phys. Rev. Lett. 100 (2008) 147001. [26] M.I. Katsnelson, Eur. Phys. J. B 51 (2006) 157. [27] J. Tworzydlo, B. Trauzette, M. Titov, A. Rycerz, C.W.J. Beenakker, Phys. Rev. Lett. 96 (2006) 246802. [28] A.R. Akhmerov, C.W. Beenakker, Phys. Rev. B 75 (2007) 045426. [29] M. Titov, C.W.J. Beenakker, Phys. Rev. B 74 (2006) 041401(R). [30] J.C. Cuevas, A. Levy Yeyati, Phys. Rev. B 74 (2006) 180501(R). [31] E. Prada, P. San-Jose, B. Wunsch, F. Guinea, Phys. Rev. B 75 (2007) 113407. [32] G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515.