arX

iv:1

711.

0540

5v1

[co

nd-m

at.m

es-h

all]

15

Nov

201

7

Crossed Andreev effects in two-dimensional quantum Hall systems

Zhe Hou,1 Yanxia Xing,2 Ai-Min Guo,3 and Qing-Feng Sun1, 4, ∗

1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China2Department of Physics, Beijing Institute of Technology, Beijing 100081, China3Department of Physics, Harbin Institute of Technology, Harbin 150001, China4Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

We study the crossed Andreev effects in two-dimensional conductor/superconductor hybrid sys-tems under a perpendicular magnetic field. Both graphene/superconductor hybrid system andelectron gas/superconductor one are considered. It is shown that an exclusive crossed Andreevreflection, with other Andreev reflections being completely suppressed, is obtained in a high mag-netic field because of the chiral edge states in the quantum Hall regime. Importantly, the exclusivecrossed Andreev reflection not only holds for a wide range of system parameters, e.g., the size ofsystem, the width of central superconductor, and the quality of coupling between the graphene andthe superconductor, but also is very robust against disorder. When the applied bias is within thesuperconductor gap, a perfect Cooper-pair splitting process with high-efficiency can be realized inthis system.

PACS numbers: 73.23.-b, 73.43.-f, 74.45.+c

I. INTRODUCTION

The Cooper pairs in a superconductor can be usedas a natural source of nonlocal Einstein-Podolsky-Rosen(EPR) electron pairs.1,2 By splitting the electrons in aCooper pair, one obtains two spatially separated elec-trons which still keep their spin and momentum entan-gled. The Cooper pair splitting is the inverse processof crossed Andreev reflection (CAR) which has been ex-tensively studied for many years.3–20 The CAR processoccurs at the interface between a normal conductor andthe superconductor, where an electron is injected fromone terminal of the normal conductor and is then re-flected out as a hole at the other terminal, and a Cooperpair forms in the superconductor. Except for the CAR,there also exists local Andreev reflection (LAR), wherethe hole is reflected into the same terminal. The CARprocess usually competes with the LAR one. To split aCooper pair efficiently, the LAR process has to be sup-pressed.

Up to now, many proposals have been put for-ward to realize the splitting of Cooper pairs3–18 forits promising applications in quantum communica-tion and quantum computing.9,21,22 For example, someCooper-pair splitters have been theoretically proposedin the system by coupling a superconductor with quan-tum dots,9,11,12 carbon nanotubes,13,14 Luttinger liq-uid wires,15 and graphene.16–18 On the experimentalsides,23–25 the Cooper-pair splitters have been realizedin the system by coupling a superconductor with twoquantum dots, where a central superconducting finger isconnected with two quantum dots and each quantum dotis coupled with a metallic lead. In the Coulomb block-ade regime, the electrons in a Cooper pair can tunnelinto different leads coherently from the superconductor,and the LAR process can be considerably suppressed bytuning the energy levels of the quantum dots. However,in this Cooper-pair splitter, the LAR process cannot be

completely suppressed, and thus the entangled electronsare not spatially separated completely.7 Furthermore, toimprove the efficiency of the CAR process, the parame-ters, such as the bias and the gate voltage of the quantumdots, have to be accurate. And it may be difficult for ex-perimental implementation of the Cooper-pair splitter.

Recently, due to the emergence of topological insula-tors and Majorana Fermion, some Cooper-pair splittersare proposed based on the hybrid system of the super-conductor and the topological insulators or the Majo-rana Fermion.7,26–30 For example, in a two-dimensional(2D) topological insulator-superconductor-2D topologi-cal insulator junction, an all-electric Cooper-pair splitterwas proposed by Reinthaler et al..30 James et al.

7 pro-posed an exclusive CAR device by inducing supercon-ductivity on a AIII class topological insulator wire whichsupports two topological phases with one or two Majo-rana fermion end states. In the phase with two Majo-rana fermions, the LAR is completely suppressed at thenormal lead/topological superconductor interface at zerobias, resulting in correlated and spin-polarized currentsin the leads.

However, all of the previous Cooper-pair splitters haveseveral disadvantages. Since the incoming electron andthe outgoing hole in the CAR process locate in spa-tially separated terminals, the width L of the centralsuperconductor is required to be less than the super-conducting coherent length ξ, and the CAR coefficientwould decay quickly to zero when L > ξ. In usual, theLAR also occurs inevitably, in which the outgoing holecomes back to the same terminal as the incoming elec-tron. In many Cooper-pair splitters, the LAR is usu-ally much larger than the CAR and the efficiency of theCooper-pair splitting is quite low. Furthermore, manyCooper-pair splitters are too elaborate to be realized ex-perimentally. And they can work only under certain spe-cial parameters, and are usually not robust against dis-orders and impurities, which exist inevitably in the ex-

2

periments. As a result, the CAR process is strongly sup-pressed and the Cooper-pair splitting efficiency is verylow. Here, we also notice that very recently, Zhang et

al. proposed a Cooper-pair splitter based on a quantumanomalous Hall insulator (QAHI).31 Due to the unidi-rectionality of the chiral edge states in the QAHI, theLAR can be suppressed completely and only the CAR oc-curs in the QAHI-superconductor-QAHI junction. Con-sequently, this QAHI-based Cooper-pair splitter can bevery efficient and be robust against the disorders, andcan work even if the size of the superconductor electrodeis much larger than the superconducting coherent length.However, it is very difficult to fabricate the QAHI in theexperiment, although the QAHI has been successfully re-alized in the magnetic topological insulator with the tem-perature being at the order of mK.32,33 Therefore, it isan urgent task to propose a perfect Cooper-pair split-ter which is of high-efficiency and is robust against thedisorder, and works in large-superconductor size.

Graphene is a 2D material with a unique bandstructure.34,35 Electrons in graphene exhibit relativistic-like behavior near the Dirac point. One of the peculiarproperties of graphene is the half integer quantum Halleffect with the chiral edge states. In this paper, we in-vestigate the electron transport through a three-terminalgraphene/superconductor system, and propose a perfectCooper-pair splitter with the aid of the chiral edge statesin the quantum Hall system. In general, the quantumHall effect and the superconductivity are mutually ex-clusive, because the former phenomenon exists in thepresence of strong magnetic field, whereas the latter onewill be destroyed by the strong magnetic field. However,with modern progress in materials science, the quantumHall effect can be observed at much smaller magneticfield, which ensures the possibility of coexistence of thequantum Hall effect and the superconductivity.36–40 Forexample, both the quantum Hall effect and the super-conductivity have been successfully realized in the junc-tion which consists of the 2D electron gas and the Nbcompounds, where the 2D electron gas with high mobil-ity possesses the quantum Hall regime under sufficientlysmall magnetic field and the Nb compounds have a highcritical magnetic field.38 The coexistence of the quantumHall effect and the superconductivity has also been ob-served in the graphene/superconductor hybrid system.39

By using the tight-binding model and the non-equilibrium Green’s function method, we obtain expres-sions of Andreev reflection coefficients and normal trans-mission coefficients under different magnetic fields. Instrong magnetic field, the chiral edge states form in thegraphene, and the electrons (holes) can be reflected uni-directionally as the holes (electrons) at the interface be-tween the graphene and the superconductor. Because ofthe unidirectionality of the chiral edge states, the outgo-ing hole will be transmitted to the other graphene ter-minal and only the CAR occurs, as shown in Fig.1(a).Notice that the LAR happens only if the outgoing holeis scattered from one edge of the graphene to the other

FIG. 1: (Color online) Schematic diagram for three-terminalhybrid system of (a) graphene/superconductor and (b) two-dimensional electron gas/superconductor. The red lines de-note the edge states due to the strong magnetic field B. Withthe aid of the unidirectional chiral edge states, the Cooperpairs in the superconductor lead can be split into two sepa-rated terminals.

edge. In the quantum Hall regime, the scattering betweenthe two edges is almost impossible, and hence the LARis completely inhibited and an exclusive CAR emerges.Fig.2 shows the main results in this paper, where onlythe CAR coefficient T13A has large value and the otherAndreev reflection coefficients are almost zero. As a re-sult, this device can serve as a perfect Cooper-pair split-ter with high-efficiency. This Cooper-pair splitter can bevery robust against the disorder. As long as the chiraledge states are present, the Cooper-pair splitter can workwell. Furthermore, it works well even when the width Lof the superconductor is larger than the superconductingcoherent length ξ and the width L breaks through the sizelimitation of previous Cooper-pair splitters. In addition,the exclusive CAR process, with the LAR being com-pletely suppressed, can hold for a wide range of systemparameters, such as the width of the graphene nanorib-bon, the coupling between the superconductor and thegraphene. Finally, a 2D electron gas is considered in-stead of the graphene and similar results are obtaineddue to the emergence of the chiral edge states, which areinduced by external magnetic field (see Fig.1(b)).The rest of the paper is constructed as follows. In

Sec.II, the theoretical model is presented and the expres-sions of the normal transmission coefficients and the An-dreev reflection coefficients are derived. In Sec.III, wenumerically investigate the transmission coefficient, theCAR coefficient, and the LAR coefficient, and discuss thecharacteristics of the proposed Cooper-pair splitter in thegraphene/superconductor hybrid system. In Sec.IV, wechange the Dirac point of the graphene to demonstratethe regime of the CAR and the LAR. Finally, the re-sults are summarized in Sec.V. The Cooper-pair splitterin the 2D electron gas/superconductor hybrid system ispresented in Appendix.

II. MODEL AND FORMULISM

We consider a three-terminal system by coupling azigzag edged graphene nanoribbon with a superconduc-tor lead (as shown in Fig.1(a)). The central region withwidth N and length L, as labeled by the rectangular area

3

-2 -1 0 1 20.00.10.20.30.40.50.6

T 13A,T

11A,T

31A&T 3

3A

T11A

T31A

T33A

T13A

FIG. 2: (Color online) Andreev reflection coefficients TA

versus incident energy ω for the graphene/superconductor hy-brid system at φ = 0.003. Due to the chiral edge states, theexclusive CAR coefficient T13A is very large, with the otherAndreev reflection coefficients being prohibited. The param-eters are listed as follows: the Dirac point energy E0 = −5∆,the width of the graphene nanoribbon N = 50, the lengthof the central region L = 30, the size of the covered area ofthe superconductor M = 15, the coupling strength betweenthe graphene and the superconductor tc = t, and the disorderstrength W=0.

in Fig.1(a), is partly covered by the superconductor lead.The covered region is described by width M and lengthL with 2M(2L− 1) carbon atoms. In Fig.1(a), it showsa system with N = 5, L = 7, and M = 2.The Hamiltonian of the system is

H = HG +HS +HC , (1)

where HG, HS, and HC are the Hamiltonians of thegraphene nanoribbon, the superconductor lead, and thecoupling between them, respectively. In the tight-bindingrepresentation, HG is:41,42

HG =∑

i,σ

Eia†iσaiσ −

∑

〈ij〉,σ

teiφija†iσajσ , (2)

where a†iσ and aiσ are the creation and annihilation op-erators at the discrete site i, and Ei = E0 + ωi de-notes the on-site energy. E0 is the Dirac point energywhich can be controlled experimentally by gate voltageand ωi is the on-site disordered energy. In the experi-ment, disorder and impurity exist inevitably. The dis-order in graphene p-n junction can result in several ex-tra conductance plateaus.42–44 Furthermore, the chargepuddle disorder has been confirmed in the graphene byrecent experiments.45,46 Here, we consider that the dis-order only exists in the central scattering region. ωi = 0at the left and right graphene terminals; while for thecentral region, ωi is uniformly distributed in the range[−W/2,W/2] with W being the disorder strength. Thesecond term in HG describes the nearest-neighbor hop-ping. Because of a uniform perpendicular magnetic fieldon the graphene nanoribbon, a phase φij is added in the

hopping element and φij =∫ j

i~A · d~l/φ0 with the vec-

tor potential ~A = (−By, 0, 0) and φ0 = h/e. Note thatin the superconductor lead and the covered graphene re-gion, no magnetic field exists due to the Meissner effectand φij = 0 in these regions.As for the superconductor lead, we consider the BCS

Hamiltonian and HS is:

HS =∑

k,σ

ǫkb†kσbkσ +

∑

k

(

∆b†k↑b

†−k↓ +∆b−k↓bk↑

)

, (3)

where ∆ is the superconductor gap and b†kσ(bkσ) is the

creation (annihilation) operator in the superconductorlead with momentum k = (kx, ky). The coupling betweenthe superconductor and the graphene is described by theHamiltonian:

HC =∑

i,σ

tca†iσbiσ + h·c·, (4)

where biσ is the annihilation operator at site i andbiσ =

∑

keik·ribkσ. Here, ri is the position of the ith

carbon atom in real space and tc is the coupling param-eter between the superconductor and the graphene.By using the non-equilibrium Green’s function

method, we can obtain the normal transmission coeffi-cient Tnm(m 6=n)N , the CAR coefficient Tnm(m 6=n)A, andthe LAR coefficient Tnm(m=n)A between the graphene

terminals m = 1, 3 and n = 1, 3:47–49

T σnm(m 6=n)N(ω) = Tr [GrΓn,σG

aΓm,σ] , (5)

T σnmA(ω) = Tr [GrΓn,σG

aΓm,σ] , (6)

where σ represents spin-up electron (↑) and spin-downhole (↓) in the 2 × 2 Nambu space, and σ =↑ (↓)for σ =↓ (↑). Since the Pauli matrices σx,y,z arecommutative with the Hamiltonian H , the transmis-sion coefficient and the Andreev reflection one satisfy

T ↑nmN = T ↓

nmN and T ↑nmA = T ↓

nmA, and we ignore thesuperscript “σ =↑, ↓” in the following for representa-tion simplicity. T13N (T31N ) represents normal tunnel-ing possibility from terminal-3 (terminal-1) to terminal-1 (terminal-3). The LAR coefficient TmmA (CAR co-efficient Tnm(m 6=n)A) is the probability of an electroncoming from the graphene terminal-m and getting An-dreev reflected as a hole into the same terminal-m (dif-ferent terminal-n). The linewidth function Γn(ω) ≡i[Σr

n − (Σrn)

†] and Σrn(ω) is the retarded self-energy

due to the coupling between the central region and theterminal-n. Gr(a)(ω) is the retarded (advanced) Green’sfunction of the central region in the Nambu space, andGr(ω) = [Ga(ω)]† = (ωI − HC − ∑

n=1,2,3Σrn(ω))

−1

with HC being the Hamiltonian matrix for the centralregion. For the self-energies Σr

1(3)(ω) of the left and

right leads, we have Σr1(3),ij(ω) = tgr

1(3),ij(ω)t, where

gr1(3),ij(ω) is the surface Green’s function of terminal-1

(terminal-3) which can be numerically calculated.50 For

4

the self-energy of the superconductor terminal, we haveΣr

2,ij(ω) = tcgr2,ij(ω)tc and gr

2,ij(ω) is:47

gr2,ij(ω) = −iπρβ(ω)J0(kF |ri − rj |)

⊗

(

1 ∆/ω∆/ω 1

)

, (7)

where ρ represents normal density of states for thesuperconductor and J0(kF |ri − rj |) is the 0th orderBessel function with kF being the Fermi wave vec-tor. β(ω) = −iω/

√∆2 − ω2 for |ω| < ∆ and β(ω) =

|ω|/√ω2 −∆2 for |ω| > ∆.47–49 For simplicity, we as-

sume that J0(kF |ri − rj |) = 1 for i = j and otherwiseJ0(kF |ri − rj |) = 0 for i 6= j. This assumption is reason-

able because kF is usually in the order of A−1

. After thisassumption, the superconductor lead seems to be madeup of one dimensional wires and each carbon atom in thecentral region connects independently with a supercon-ductor, that has been assumed in Ref.[51]. Then, Eq.(7)can be reduced as:

gr2,ij(ω) = −iπρβ(ω)δij

⊗

(

1 ∆/ω∆/ω 1

)

. (8)

By employing these transmission coefficients, the currentflowing from terminal 3 into the central region can beobtained straightforwardly:47,52

I3 =2e

h

∫

dω{ T23N (f3+ − f2) + T13N(f3+ − f1+)

+T13A(f3+ − f1−) + T33A(f3+ − f3−)} (9)

Here, fα±(ω) = 1/{ exp[(ω∓eVα)/kBT ]+1} is the Fermidistribution function for the terminal α, with the temper-ature T and the bias Vα. The bias of the superconductorlead V2 is set to zero.In the following numerical calculations, we set the hop-

ping energy t = 2.75eV, the nearest-neighbor carbon-carbon distance a = 0.142nm, and the superconductorgap ∆=1meV. The magnetic field B is expressed in termsof φ ≡ (3

√3/4)a2B/φ0 and (3

√3/2)a2B is the magnetic

flux in the honeycomb lattice. In the presence of disorder,the curves are averaged over 500 random configurations.

III. THREE-TERMINALGRAPHENE/SUPERCONDUCTOR HYBRID

SYSTEM

We first study the electron transport properties of thegraphene/superconductor hybrid system under differentmagnetic fields. Fig.3 shows the Andreev reflection coef-ficients TA and the normal transmission coefficients TN

versus the energy ω of the incident electron. In the ab-sence of magnetic field (φ = 0) or for weak magneticfield (φ = 0.0005 and 0.001), both the LAR process andthe CAR one can occur for the incident electron fromterminal-1 or from terminal-3 (see Figs.3(a)-3(d)). TheLAR coefficients T11A and T33A can be quite large, and

0.00.20.40.60.81.0

-2 -1 0 1 20.00.20.40.60.81.00.00.20.40.60.81.0

-2 -1 0 10.00.20.40.60.81.0

0.00

0.01

0.02

0.030.00.20.40.60.81.0

T 11A

(a)

(f)

T 13N

(d)

T 33A

(e)

T 31N

(c)

T 31A

(b)

T 13A

FIG. 3: (Color online) Andreev reflection coefficients TA andnormal transmission coefficients TN versus incident energy ω

for the graphene/superconductor hybrid system under differ-ent magnetic fields φ . The other parameters are the same asthose of Fig.2.

the CAR coefficients T31A and T13A are relatively smallwhen the energy ω is within the superconductor gap, i.e.,|ω| < ∆. Both the LAR and CAR coefficients decayquickly when ω is beyond the superconductor gap, whichis similar to usual normal-superconductor junction.53

The normal transmission coefficients T13N and T31N arelarge when |ω| < ∆ and are decreased when |ω| > ∆,because the tunneling from the graphene to the super-conductor can happen when |ω| > ∆. When the mag-netic field is gradually increased to φ = 0.003, one no-tices the following features. (i) For the incident electronfrom terminal-1, the Andreev reflection coefficients T11A

and T31A are gradually declined to zero, and the normaltransmission coefficient T31N is gradually increased toone (see Figs.3(a), 3(c), and 3(e)). This indicates that theelectrons tunnel directly from terminal-1 into terminal-3without being Andreev reflected by the superconductor.(ii) For the incident electron from terminal-3, the LARcoefficient T33A is shrunk to zero and the CAR coefficientT13A is increased to a remarkable value (see Figs.3(b) and3(d)), and T13A + T13N = 1 when |ω| < ∆ (see Figs.3(b)and 3(f)). This implies that when the electrons encounterthe superconductor, they get Andreev reflected and nobackscattering occurs. It should be mentioned that thecorresponding magnetic field B is about 75 Tesla whenφ = 0.003. Since the width of the graphene nanorib-bon in Fig.3 is very narrow (when N = 50, the width isabout (3N − 1)a ≈ 21nm), the magnetic field should besufficiently strong so that the quantum Hall effect couldappear. While for a wide graphene nanoribbon, the mag-netic field can be much weaker.

Now we explain the numerical results in Fig.3. In the

5

-2 -1 0 10.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

T 13N

T 13A,T

11A,T

31A&T 3

3A(a)

(b)

FIG. 4: (Color online) (a) Andreev reflection coefficients TA

and (b) transmission coefficient T13N versus incident energyω in the regime of strong magnetic fields φ. In panel (a), onlyT13A has large value; T11A, T31A and T33A are almost zero,and their curves overlap together and are shown by the blackbold lines. The other parameters are the same as those ofFig.2.

weak or zero magnetic field, the wave function of the in-cident electron can extend over the whole bulk of thegraphene nanoribbon. When the electron encounters thesuperconductor, the Andreev reflections occur at the in-terface of the graphene and the superconductor and giverise to either the LAR or the CAR. Since the Fermi en-ergy of the superconducting lead is EF = 0, which is faraway from the Dirac point E0 = −5∆ of the graphene inFig.3, the incident electron and the reflected hole locatein the same band (see Fig.8(c)). As a result, the Andreevretro-reflection dominates in this situation and the LARis more pronounced than the CAR.52 This accounts forthe phenomenon that the LAR coefficient is much largerthan the CAR one in the weak magnetic field. While inthe relatively strong magnetic field (e.g., φ = 0.003), boththe Landau levels and the edge states form, as depictedby the red lines in Fig.1. In this case, the incident elec-tron from terminal-1 transports along the bottom red lineof Fig.1(a) and tunnels directly into terminal-3 withoutencountering the superconductor. Thus, no backscatter-ing and Andreev reflections occur, and T11A = T31A = 0and T31N = 1 which equals to the filling factor ν of theLandau levels. On the other hand, the incident elec-tron from terminal-3 transports along the top red line ofFig.1(a) and will be Andreev reflected at the interfaceof the graphene and the superconductor. Since the re-flected hole lies in the same band as the incident electronmentioned above and their edge states have the same chi-rality, the reflected hole moves along the same directionof the incident electron. Then, only the CAR process oc-curs and the LAR process is completely suppressed. Ad-ditionally, since the normal tunneling from the grapheneinto the superconductor is prohibited when the energy ωis within the superconductor gap, and the backscatter-ing at the interface is also prohibited due to the chiraledge states, we have T13A + T13N = 1 because of currentconservation.

Next we study the transport properties of thegraphene/superconductor hybrid system in the regime

of high magnetic fields. In Fig.4, we plot the Andreevreflection coefficients TnmA and the normal transmissioncoefficients T13N versus the incident energy ω for differ-ent strong magnetic fields. The T11A, T33A, and T31A arecompletely suppressed and are almost zero, and only T13A

and T13N are remarkable. In addition, T13A + T13N = 1when ω is within the superconductor gap ∆. These aresimilar to the results of φ = 0.003 in Fig.3. When thebias of the terminal-1 is equal to that of the terminal-3, i.e., V1 = V3, the normal tunneling term T13N doesnot contribute to the current (see Eq.(9)). In the case ofsmall bias (|eV | < ∆), the tunneling from the grapheneto the superconductor is also prohibited (T23N = 0) dueto the existence of the superconductor gap. Then, thecurrent in Eq.(9) can be reduced to

I3 = (2e/h)

∫

dω{ T13A(f3+− f1−)+T33A(f3+− f3−)} ,

(10)in which only the Andreev reflection contributes to thecurrent. Because T11A = T33A = 0 at strong mag-netic field, the differential conductance is G3 = dI3

dV=

2e2

h[T13A(eV ) + T13A(−eV )] at zero temperature. Thus,

in the regime of strong magnetic field, an exclusive CART13A is obtained, with both the LAR and the normal tun-nelling being completely prohibited. Notice that since

the relation T ↑13A = T ↓

13A always holds, and the spin-upand spin-down electrons from a Cooper pair transportto the terminal-1 and the terminal-3 randomly, the twospatially separated electrons can keep their spin and mo-mentum entangled. By setting the bias of the supercon-ductor slightly higher than one of the graphene terminals,the current can be driven from the superconductor to thegraphene, and the Cooper pair can be split into two sep-arated electrons which will flow into different leads (seeFig.1(a)). This generates two spatially separated electronflows with entangled spin and momentum.Here, we emphasize that the efficiency of this Cooper-

pair splitter is very high, although the value of the CARshows oscillation behavior. From Eq.(10), we can definesplitting efficiency as:

η =

∫

dω{T13A(f3+ − f1−)}∫

dω{T13A(f3+ − f1−) + T33A(f3+ − f3−)}. (11)

Here, η describes the probability to obtain two spatiallyseparated electrons when a Cooper pair is split. Be-cause the LAR is completely suppressed with T33A = 0in the quantum Hall regime, the splitting efficiency η ofa Cooper pair is always 100%.Now let us study how the exclusive CAR coefficient

T13A in the graphene/superconductor system is affectedby the system parameters. Fig.5(a) shows T13A versus ωfor different length L of the central region. We find thatT13A can always reach large value by varying L from 30to 1000. In addition, T13A oscillates with the energy ω,and the oscillation frequency is increased by increasingthe length L,54,55 because the Fabry-Perot-like interfer-ence occurs in this three-terminal system. In particu-

6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-1 0 1

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1

0.0

0.2

0.4

0.6

0.8

1.0

L=30 L=100 L=1000

T 13A

(a) M=1 M=5 M=15 M=30

(b)

N=30 N=50 N=100

T 13A

(c)

tc=0.05t tc=0.5t tc=1t tc=2t

(d)

FIG. 5: (Color online) Energy-dependent CAR coefficientT13A by considering different L (a), M (b), N (c), and tc(d) in the graphene/superconductor hybrid system. If theparameters are not shown in the legend, they are as follows:L = 30, N = 50, M = 15, tc = t, E0 = −5∆, φ = 0.01, andW = 0. Here, T11A, T31A, and T33A are not shown in thefigure because they are all zero.

lar, for L = 1000, the length of the central region is√3aL ≈ 246nm which is much greater than the super-

conducting coherent length. However, the CAR coeffi-cient T13A is still very large in this case. As comparedwith previously proposed Cooper-pair splitters that thelength of the central region should be less than the su-perconductor coherent length ξ, we show in the presentstudy that the CAR process is not confined by the lengthL, because of the unidirectional chiral edge states. Theexclusive CAR coefficient T13A can also be quite largeby changing the width N of the graphene nanoribbon(as shown in Fig.5(c)). In fact, as long as the graphenenanoribbon is wide enough so that the top and bottomchiral edge states don’t mix together, an exclusive CARprocess can always be obtained. Figs.5(b) and 5(d) showhow T13A be affected by the covered area and the couplingstrength between the superconductor and the graphene.By varying the width M of the covered area of the super-conductor on the graphene and the coupling strength tc,T13A can still have large value. These results show thatthe exclusive CAR process in our system can hold verywell in a wide range of the system parameters, which ishelpful for experimental implementation of the Cooper-pair splitter.Let us study the effect of the disorder on the exclusive

CAR. Here, we consider the on-site Anderson disorderwhich only exists in the central region. Fig.6 shows theAndreev reflection coefficients TA and the normal trans-mission coefficient TN versus the incident energy ω fordifferent disorder strengths W . By increasing the disor-der strength W from 1meV to 100meV, the above coeffi-cients are almost unaffected, indicating that our Cooper-pair splitter is very robust. By further increasing W to

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 10.00

0.05

0.10

0.15

0.20

0.25

-2 -1 0 1 20.00

0.05

0.10

0.15

0.20

0.25 W= W=10 W=100 W=

T13A

(a)

T 13N

(b)

T11A

(c)

T33A

(d)

FIG. 6: (Color online) TA and TN versus ω for differentdisorder strengths W . The parameters are E0 = −5∆, N =50, L = 30,M = 15, tc = t, and φ = 0.01.

1000meV, the LAR coefficients T11A and T33A becomenonzero, and the oscillation behavior of T13A and T13N

disappears. This is due to the fact that the system goesinto the diffusive regime at large disorder and the edgestates are destroyed by the disorder. Therefore, as longas the edge states survive, the exclusive CAR T13A canpersist. This means that the exclusive CAR, i.e., theCooper-pair splitter, is robust owing to the quantum Halleffect.In Appendix, we show the electron transport proper-

ties of a three-terminal 2D electron gas/superconductorhybrid system and obtain similar results as thegraphene/superconductor hybrid system. Although thetwo systems have different band structures and electronicbehaviors, both of them can work as a perfect Cooper-pair splitter, owing to the chiral edge states which are in-duced by strong magnetic field. However, in general, thegraphene/superconductor system has more advantages ascompared with the 2D electron gas/superconductor one.This is attributed to the fact that the graphene has aunique band structure with a linear dispersion relationnear the Dirac point, and the zeroth Landau level at theDirac point is well separated from the first Landau level.

IV. GRAPHENE/SUPERCONDUCTORSYSTEM WITH DIRAC POINT NEAR FERMI

ENERGY

Let us change the Dirac point energy E0 from −5∆ to−0.5∆ to see the interesting behavior of the LAR and theCAR in the graphene/superconductor system. In the ex-periment, the Dirac point energy can be easily tuned bythe gate voltage. At zero magnetic field, the Andreev re-flection in the graphene/superconductor interface can bedivided into the retro-reflection and the specular reflec-tion, according to the situation that the incoming elec-tron and the reflected hole locate in the same bands or

7

0.00.20.40.60.81.0

0.00.20.40.60.81.0

0.00.10.20.30.40.5

0.00.20.40.60.81.0

-2 -1 0 10.00.20.40.60.81.0

-2 -1 0 1 20.00.20.40.60.81.0

T 11A

(a)

T 13A

(b)

T 31A

(c)

T 33A

(d)

T 31N

(e)

T 13N

(f)

FIG. 7: (Color online) Andreev reflection coefficients TA

and normal transmission coefficients TN versus energy ω forthe graphene/superconductor hybrid system under differentmagnetic fields, with the Dirac point energy E0 = −0.5∆.The other parameters are N = 50, L = 30, M = 15, tc = t,and W = 0.

different ones (see Figs.8(b) and 8(c)).6,52,56 When theDirac point is near the Fermi energy, the specular reflec-tion is dominant. While in the presence of the magneticfield, the movement of the reflected hole will be changedand the results are totally different. Fig.7 shows the An-dreev reflection coefficients TA and the normal transmis-sion coefficients TN versus the incident electron energyω under different magnetic fields. Similar to the caseof E0 = −5∆, the edge states gradually form in thethree-terminal system by increasing the magnetic fieldand some unique properties appear. Here, we emphasizethe following two facts: (i) When |ω| > |E0| = 0.5∆, theincident electron and the reflected hole locate in differ-ent bands, i.e., the valence band (E < E0) with negativechirality and the conduction band (E > E0) with posi-tive chirality (see Fig.8(b)). Thus, under strong magneticfield, the direction of the unidirectional chiral edge statesis opposite for the incident electron and the reflected hole.(ii) When |ω| < |E0|, the incident electron and the re-flected hole locate in the same band (see Fig.8(c)), andthe direction of the chiral edge states is the same.

Now we focus on the case of high magnetic field (e.g.,φ = 0.007) in Fig.7. For the incident electron from theterminal-1, the electron transports along the top edgestate from the left to the right when ω < E0 = −0.5∆(see Fig.8(a)), and then it meets the superconductor andwill be Andreev reflected. The reflected hole transportsalong the opposite direction as compared with the in-cident electron. As a result, T11A is considerably largeand T31A = 0 (see Figs.7(a) and 7(c)). In this case, thetwo electrons in the Cooper pair get into the same termi-

FIG. 8: (Color online) (a) Schematic diagram of positivechiral edge states (red lines) for the conduction band andnegative chiral edge states (blue lines) for the valence band inthe three-terminal graphene/superconductor system. (b) and(c) Schematic view of the specular reflection and the retro-reflection in the energy space. While in the presence of themagnetic field and the chiral edge states, the specular reflec-tion and the retro-reflection are changed into the LAR andthe CAR, respectively.

nal (see Fig.8(a)). When the incident energy ω is aboveE0, the electron goes along the bottom edge state fromthe left to the right and does not encounter the super-conductor. Then, T11A = T31A = 0 and T31N = 1, asshown in Figs.7(a), 7(c), and 7(e). Similar results canbe obtained for the incident electron from the terminal-3. When ω < E0, the electron goes along the bottomedge state from the right to the left without meeting thesuperconductor, and T13A = T33A = 0 and T13N = 1(see Figs.7(b), 7(d), and 7(f)); When E0 < ω < −E0

(see Fig.8(c)), both the incident electron and the re-flected hole are in the conduction band, and go alongthe top edge states from the right to the left. This corre-sponds to the exclusive CAR process and T13A has largevalue with T33A = 0 (see Figs.7(b) and 7(d)). When−E0 < ω, the incident electron and the reflected holeare in different bands, and only the LAR happens inthe graphene/superconductor interface. Note that inFigs.7(a)-7(d), the LAR and the CAR can be separatedcompletely by tuning the bias. Therefore, this providesus a good way to control and investigate these processes.

V. CONCLUSION

In summary, we investigate the electron transport in athree-terminal graphene/superconductor hybrid system.In high magnetic field, an exclusive crossed Andreev re-flection is obtained with the aid of the edge states in thequantum Hall regime, with the other Andreev reflectionsbeing prohibited. This exclusive crossed Andreev reflec-tion can hold by varying the size of the system and thecoupling strength between the graphene and the super-conductor. In particular, it can also work well for largewidth of the central superconductor and is very robustagainst the disorder. As a result, a perfect Cooper-pairsplitter with high efficiency is proposed in this study.Finally, a two-dimensional electron gas/superconductorquantum Hall system is also considered, where similarresults are obtained due to the chiral edge states in thesystem.

8

ACKNOWLEDGMENTS

We gratefully acknowledge the financial support fromNBRP of China (2012CB921303 and 2015CB921102),NSF-China under Grants No. 11274364, 11504066,and 11574007, and FRFCU under Grant No.AUGA5710013615. Y. Xing is supported by thegrant of Trans-Century Training Programme Founda-tion for the Talents by the State Education Commission(No. NCET-13-0048).

APPENDIX

In this appendix, we provide the electron transportproperties of a three-terminal system, which is composedof a 2D electron gas nanoribbon and a superconductorlead (see Fig.1(b)). In the tight-binding representation,the Hamiltonian of the 2D electron gas nanoribbon is:

HEG =∑

i,σ

Eia†iσaiσ −

∑

〈ij〉,σ

teiφija†iσajσ. (12)

Here, t = h2

2ma2 is the kinetic energy, Ei = Eb + 4t + wi

with Eb the bottom of the conduction band, and themagnetic field is described by the magnetic flux φ in asquare lattice. The Hamiltonians of the superconduc-tor and the coupling between the superconductor andthe electron gas are the same as Eqs.(3) and (4), respec-tively. Then, the Andreev reflection coefficients TA andthe normal transmission coefficients TN can be calculatedfrom Eqs.(5) and (6). In the numerical calculation, weset the superconductor gap ∆ = t/200, the conductionband bottom Eb = −0.02t, and the Fermi energy of thesuperconductor EF = 0.Fig.9 shows the transport properties under different

magnetic fields. In zero or weak magnetic field, both theLAR process and the CAR one occur (see Figs.9(a)-9(d)).By increasing the magnetic field to φ = 0.003, the edgestates gradually form just as the graphene based hybridsystem. Since φ = a2B/φ0, the corresponding magneticfield B is about 0.078 Tesla for φ = 0.003 and a = 5nm.Due to the edge states, an exclusive CAR T13A can beobtained, with the other Andreev reflection coefficientsT11A, T31A, and T33A being zero [see Figs.9(a)-9(d)]. Inaddition, some Hall plateaus emerge in the curve of T31N

[see Fig.9(e)], where the plateau values are determined bythe filling factor ν of the Landau levels. This indicatesthat the incident electron from the terminal-1 tunnelsdirectly into the terminal-3 without the interface scat-tering. With the aid of the edge states, these results canalso be well understood. Now both the incident elec-tron and the reflected hole are in the conduction band,and move anticlockwise along the edge of the electrongas under high magnetic field, as shown in Fig.1(b). Forthe incident electron from the terminal-3, both the CART13A and the direct tunneling T13N occur. For the in-cident electron from the terminal-1, only the direct tun-

0.00.20.40.60.81.01.2

0.00.20.40.60.81.0

0.00.10.20.30.40.5

0.0

0.3

0.6

0.9

1.2

-2 -1 0 1

2

4

6

8

10

-2 -1 0 1 2

2

4

6

8

10

T 11A

(a)

T 13A

(b)

T 31A

(c)

T 33A

(d)

T 31N

(e)

T 13N

(f)

FIG. 9: (Color online) Andreev reflection coefficients TA andnormal transmission coefficients TN versus incident energy ω

for a 2D electron gas/superconductor system under differentmagnetic fields. The parameters are Eb = −0.02t, N = 100,L = 50, M = 20, tc = 0.03t, and W = 0.

0.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 10.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

T 13A

(a)

tc=0.03t tc=0.05t tc=0.1t tc=0.3t

T 13A

(c)

L=50 L=100 L=500

(b)

(d)

W=0t W=0.1t W=0.2t

FIG. 10: (Color online) T13A versus ω for different φ (a),L (b), tc (c), and W (d) in the electron gas/superconductorhybrid system with Eb = −0.02t, N = 100, and M = 20. Theother parameters are: (a) L = 50, tc = 0.03t, and W = 0; (b)tc = 0.03t, φ = 0.005, and W = 0; (c) L = 50, φ = 0.005,and W = 0; and (d) L = 50, tc = 0.03t, and φ = 0.01. T11A,T31A, and T33A are almost zero so they are not shown in thefigure.

neling T31N occurs. The LAR is completely prohibited,regardless of the terminal where the electron is injected.Therefore, the Cooper-pair splitter is also very efficient inthe 2D electron gas/superconductor system. In fact, aslong as the edge states form, one can always demonstratea perfect Cooper-pair splitter in such quantum Hall sys-tems.Finally, we study the exclusive CAR coefficient T13A

9

by considering the influence of the magnetic field φ, thelength L of the central region, and the coupling strengthtc between the superconductor and the electron gas.Figs.10(a)-10(c) show T13A under large magnetic field,where the Hall edge states emerge in the system. It canbe seen that T13A is quite large for a wide range of thesystem parameters, and the other Andreev reflection co-efficients T11A, T31A, and T33A are almost zero. Thus,

the proposed Cooper-pair splitter based on the quantumHall chiral edge states can work well for the large widthof the central superconductor and for a wide range of thesystem parameters. We also consider the situation whenthe on-site energy disorder is introduced in the centralregion, as illustrated in Fig.10(d). It is evident that T13A

is very robust against the disorder. These results are sim-ilar to the case of the graphene/superconductor system.

∗ sunqf@pku.edu.cn1 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47,777 (1935).

2 M.D. Reid, P.D. Drummond, W.P. Bowen, E.G. Caval-canti, P.K. Lam, H.A. Bachor, U.L. Andersen, and G.Leuchs, Rev. Mod. Phys. 81, 1727 (2009).

3 J. Liu, J. Wang, and F.-C. Zhang, Phys. Rev. B 90, 035307(2014).

4 J.M. Byers and M.E. Flatte, Phys. Rev. Lett. 74, 306(1995).

5 D. Beckmann, H.B. Weber, and H.v. Lohneysen, Phys.Rev. Lett. 93, 197003 (2004).

6 C.W.J. Beenakker, Phys. Rev. Lett. 97, 067007 (2006).7 J.J. He, J. Wu, T.-P. Choy, X.-J. Liu, Y.Tanaka and K.T.Law, Nat. Commun. 5, 3232 (2014).

8 D. Chevallier, J. Rech, T. Jonckheere, and T. Martin,Phys. Rev. B 83, 125421 (2011).

9 P. Recher, E.V. Sukhorukov, and D. Loss, Phys. Rev. B63, 165314 (2001).

10 M. Veldhorst and A. Brinkman, Phys. Rev. Lett. 105,107002 (2010).

11 P. Recher and D. Loss, Phys. Rev. Lett. 91, 267003 (2003).12 P. Trocha and I. Weymann, Phys. Rev. B 91, 235424

(2015).13 C. Bena, S. Vishveshwara, L. Balents, and M.P.A. Fisher,

Phys. Rev. Lett. 89, 037901 (2002).14 L.G. Herrmann, F. Portier, P. Roche, A. Levy Yeyati,

T. Kontos, and C. Strunk, Phys. Rev. Lett. 104, 026801(2010).

15 P. Recher and D. Loss, Phys. Rev. B 65, 165327 (2002).16 J. Wang and S. Liu, Phys. Rev. B 85, 035402 (2012).17 A. Schroer, P.G. Silvestrov, and P. Recher, Phys. Rev. B

92, 241404(R) (2015).18 Y.S. Ang, L.K. Ang, C. Zhang, and Z. Ma, Phys. Rev. B

93, 041422(R) (2016).19 Dmitri S. Golubev, Mikhail S. Kalenkov, and Andrei D.

Zaikin, Phys. Rev. Lett 103, 067006 (2009).20 Minchul Lee, Heunghwan Khim, and Mahn-Soo Choi,

Phys. Rev. B 89, 035309 (2014).21 G.B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B

24, 287-290 (2001).22 N.M. Chtchelkatchev, G. Blatter, G.B. Lesovik, and T.

Martin, Phys. Rev. B 66, 161320 (2002).23 S.Russo, M. Kroug, T.M. Klapwijk, and A.F.Morpurgo,

Phys. Rev. Lett. 95, 027002 (2005).24 L. Hofstetter, S. Csonka, J. Nygard, and C. Schonenberger,

Nature 461, 960 (2009).25 L. Hofstetter, S. Csonka, A. Baumgartner, G.Fulop, S.

d’Hollosy, J. Nygard, and C. Schonenberger, Phys. Rev.Lett. 107, 136801 (2011).

26 J. Wang, L. Hao, and K.S. Chan, Phys. Rev. B 91, 085415

(2015).27 J. Nilsson, A.R. Akhmerov, and C.W.J. Beenakker, Phys.

Rev. Lett. 101, 120403 (2008).28 W. Chen, R. Shen, L. Sheng, B.G. Wang, and D.Y. Xing,

Phys. Rev. B 84, 115420 (2011).29 W. Chen, R. Shen, L. Sheng, B.G. Wang, and D.Y. Xing,

Phys. Rev. Lett. 109, 036802 (2012).30 R.W. Reinthaler, P. Recher, and E.M. Hankiewicz, Phys.

Rev. Lett. 110, 226802 (2013).31 Y.-T. Zhang, X. Deng, Q.-F. Sun, and Z. Qiao, Sci. Rep.

5, 14892 (2015).32 C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M.

Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng,S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He,Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340,167 (2013).

33 J.G. Checkelsky, R. Yoshimi, A. Tsukazaki, K.S. Taka-hashi, Y. Kozuka, J. Falson, M. Kawasaki, and Y. TokuraNat. Phys. 10, 731 (2014).

34 C.W.J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008).35 A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S.

Novoselov, and A.K. Geim Rev. Mod. Phys. 81, 109(2009).

36 H. Hoppe, U. Zolicke, and Gerd Schon, Phys. Rev. Lett.84, 1804 (2000).

37 N.M. Chtchelkatchev, Pis’ma v ZhETF 73, 100 (2001)[JETP Lett. 73, 94 (2001)].

38 J. Eroms and D. Weiss, Phys. Rev. Lett. 95, 107001 (2005).39 Peter Rickhaus, Markus Weiss, Laurent Marot, and Chris-

tian Schonenberger, Nano. Lett. 12, 1942 (2012).40 Pierre Carmier, Phys. Rev. B 88, 165415 (2013).41 D.N. Sheng, L. Sheng, and Z.Y. Weng, Phys. Rev. B 73,

233406 (2006).42 W. Long, Q.-F. Sun, and J. Wang, Phys. Rev. Lett. 101,

166806 (2008).43 J.R. Williams, L. DiCarlo, and C.M. Marcus, Science 317,

638 (2007).44 J.-C. Chen, T.C.A. Yeung, and Q.-F. Sun, Phys. Rev. B

81, 245417 (2010).45 J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J.H.

Smet, K. von Klitzing, and A. Yacoby, Nat. Phys. 4, 144(2007).

46 Y. Zhang, V.W. Brar, C. Girit, A. Zettl, and M.F. Crom-mie, Nat. Phys. 5, 722 (2009).

47 Q.-F. Sun and X.C. Xie, J. Phys. Condens. Matter 21,344204 (2009).

48 Q.-F. Sun, J. Wang, and T.-H. Lin, Phys. Rev. B 59, 3831(1999).

49 Q.-F. Sun, J. Wang, and T.-H. Lin, Phys. Rev. B 59, 13126(1999).

50 D.H. Lee and J.D. Joannopoulos, Phys. Rev. B 23, 4997

10

(1981); M.P. Lopez Sancho, J.M. Lopez Sancho, and J.Rubio, J. Phys. F 14, 1205 (1984);15, 851 (1985).

51 R.Melin, F.S. Bergeret, and A. Levy Yeyati, Phys. Rev. B79, 104518 (2009)

52 S.-G. Cheng, Y. Xing, J. Wang, and Q.-F. Sun, Phys. Rev.Lett. 101, 167003 (2009).

53 G. E. Blonder, M. Tinkham, and T.M. Klapwijk, Phys.Rev. B 25, 4515 (1982); G. Deutscher, Rev. Mod. Phys.

77, 109 (2005).54 J. Cayssol, Phys. Rev. Lett. 100, 147001 (2008).55 J. Linder, M. Zareyan, and A. Sudbø, Phys. Rev. B 80,

014513 (2009).56 Y. Xing, J. Wang, and Q.-F. Sun, Phys. Rev. B 83, 205418

(2011).