Page 1

Iranian Journal of Physics Research, Vol. 8, No. 2, 2008

Quantum transport in graphene

normal-metal superconductor- normal-metal structures

H Mohammadpour and M Zareyan

Institute for Advanced Studies in Basic Sciences (IASBS), P. O. Box 45195-1159, Zanjan 45195, Iran

(Received 30 January 2008)

Abstract

We study the transport of electrons in a graphene NSN structure in which two normal regions are connected by a superconducting strip

of thickness d. Within Dirac-Bogoliubov-de Gennes equations we describe the transmission through the contact in terms of different

scattering processes consisting of quasiparticle cotunneling, local and crossed Andreev reflections. Compared to a fully normal structure

we show that the angular dependence of the transmission probability is significantly modified by the presence of superconducting

correlations. This modifation can be explained in terms of the interplay between Andreev reflection and Klein tunneling of chiral

quasiparticles. We further analyze the energy dependence of the resulting differential conductance of the structure. The subgap

differential conductance shows features of Andreev reflection and cotunelling processes, which tends to the values of an NS structure for

large ds. Above the gap, the differential conductance shows an oscillatory behavior with energy even at very large ds.

Keywords: graphene, quantum transport, superconducting proximity, Klein tunneling, Andreev reflection

1. Introduction

Recently graphene a new material composed of carbon

atoms arranged in two-dimensional honeycomb lattice

which is a single atomic layer pulled out of bulk

graphite, was synthesized by Geim's group in

Manchester University [1]. Compared to the metallic and

semiconducting materials, graphene has shown many

intriguing electronic properties which have led to an

explosion of studies in recent years [2, 3, 4, 5].

Due to its peculiar electronic band structure electrons

in graphene behave like two dimensional mass less Dirac

fermions [6]. In the hexagonal reciprocal space of

graphene there are two non equivalent points

the so called Dirac points, at the corners of the first

Brillouin zone [7]. Around each of Dirac points, low

energy electrons and holes have linear dispersion

( )

F

kvk

ε= ±

?

, versus two-dimensional wave vector

?

with Fermi velocity v

conduction (electrons) and valence (holes) conical bands

touch at the points

,KK

nonequivalent valleys. This makes graphene a gapless

semiconductor with relativistic-like dispersion of the

excitations [8].

An electron in graphene structure is described by 4-

,KK

+−,

??

k

6

10/

F

m s

≈

[7]. Thus the

+−( ( )

ε

0k

=

?

) producing two

component spinor (

,,,

AB

B

A

ψψψψ

++−−) in which,

( )

A B

ψ+

is

referred to the amplitude of the electron wave function on

sublattice A(B) of the honeycomb structure with wave

vector centered around the valley K+;

corresponding wave functions for the valley K−[6,8].

This spinor satisfies Dirac equation of the form [6,8]:

0

,

0

H

±

=∂ ±∂

where

,

xy

σσ

are Pauli matrices describing pseudo spin

space of two sublattices A and B.

Already anomalies of several quantum transport

effects have been found in graphene. The integer QHE in

graphene occurs with Hall conductivity plateaus

appearing at half integer multiples of four (two spin and

two valley degeneracy) times the quantum conductance

2

eh [2,3]. This quantization rule is caused by the

quantum anomaly of the lowest Landau level in

graphene which has a twice smaller degeneracy than the

higher levels and its energy does not depend on the

magnetic field [9].

Most of the studies are focused on the anomalies at

( )A B

ψ−

are the

,

F

x xy y

H

i v

?

H

ψ

εψ

σσ

+

−

−=

(1)

Page 2

82 H Mohammadpour and M Zareyan IJPR Vol. 8, No.2

the limit of zero doping (zero charge carrier density)

where the Fermi level is located close to the Dirac

points. A finite conductance of order

in graphene samples at Dirac point [10, 11, 12, 13]. This

is surprising since one expects zero conductivity at this

point where the electronic density of states vanishes.

The relativistic-like dynamics of electrons in

graphene also affects the current fluctuations. Very

recently the prediction of a finite shot noise in an

undoped ballistic graphene was confirmed in the

experiment [14]. For a wide ballistic graphene strip the

Fano factor (the ratio of the noise power and the mean

current) has the value

1/ 3F =

shot noise power in a diffusive metallic contact [15, 16].

While graphene itself is not superconducting, but due

to its atomic size thickness it can be superconducting by

depositing a superconducting electrode on top of it [17].

Proximity induced Josephson effect between two

superconducting graphene regions was predicted

theoretically [18, 19, 20, 21] and observed in the

experiment [22, 23]. Superconductivity in graphene can

be described by Dirac-Bogoliubov-de Gennes (DBdG)

equation [19] which takes into account electron-hole

correlations, induced by the superconducting pair

potential∆ . Within this equation it was found that a

nonzero supercurrent can flow through a mesoscopic

graphene sample even at the Dirac point with zero

carrier concentration. Interestingly the Josephson current

possesses a bipolar characteristic close to Dirac point

where depending on the gate voltage the supercurrent

carriers could be either the conduction band electrons or

the valence band holes [18].

In Ref. [19] the superconducting proximity effect

was studied in graphene NS contacts. As it is well

known a special process called Andreev reflection (AR)

[24] is responsible for the proximity effect in NS

interfaces. AR is the conversion of an electron excitation

into its time reversed hole when it hits the NS interface

from N side with an energy lower than the

superconducting energy gap∆ .

conductors is a retro reflection in which the electron and

the reflected hole have opposite velocities [24]. In

contrast it was found that in undoped graphene AR

occurs in a specular manner where only the component

perpendicular to the interface changes sign. This has a

pronounced effect in the current-voltage characteristics

of a graphene NS contact as explained in [19].

In addition to several quantum transport phenomena

studied so far, there has been important studies of

relativistic quantum electrodynamics phenomena in the

context of graphene. The most famous effect is the

reflection less tunneling of an electron from a potential

barrier [25]. This is a condensed matter analog of the so

called Klein paradox of relativistic quasi particles, that

arises from the spinor nature of the wave functions in

graphene and the relativistic linear spectrum [4]. This

unique property of absence of back scattering could be

regarded as responsible for extremely high mobility in

2

eh is measured

which corresponds to the

AR in ordinary

graphene layers even at room temperature [2, 3]. The

angular-dependence of the transmission probability

through a potential barrier with perfect transmission at

normal incidence has been explained theoretically [4].

In this paper we study a graphene NSN structure to

see the interplay between AR taking place at the NS

interfaces by the superconducting pair potential ∆ and

the Klein tunneling from the potential difference

between S and N regions. Within scattering formalism

and using DBdG equation

superconducting regions we calculate the transmission

amplitudes of electrons through the structure. The

transmission processes contain different mechanisms

which are electron cotunneling (CT), AR and crossed

AR (CAR) [26, 27]. While AR occurs at an individual

NS-interface, in CAR, incident electron and reflected

hole emerge from different NS-interfaces.

Employing these transmission amplitudes in the

Blonder-Tinkham-Klapwijk (BTK) formula [28], we

have calculated the differential conductance of the

structure. We analyze the energy-dependence of the

differential conductance

corresponding to different values of the chemical

potential in N and S regions compared to ∆ and thickness

of the S region d. We consider the limit of large

mismatch between chemical potentials of S and N

regions. We explain difference of angular-dependence of

the transmission probabilities for low and high chemical

potentials of the N regions where, respectively, specular

and retro AR dominates the electron-hole conversion.

Compared to the fully normal structure [4] we show that

the angular-dependence of transmission is modified by

the presence of the superconductivity.

The resulting differential conductance shows an

oscillatory behavior at energies above ∆ which persists

even at thicknesses larger than the superconducting

coherence length. At subgap energies, the behavior of the

differential conductance is dominated by CAR and CT for

thinner S region and by AR for thicker S region. We give

a full analysis of differential conductance for different

thicknesses of the S region and chemical potentials of the

N regions.

2. Model and basic equations

We consider a graphene NSN structure occupying xy

plane, as shown in Fig. 1. A wide superconducting strip

(S) of thickness d connects two normal leads (N1 and

N2). The normal leads are held at a voltage difference V

and the voltage at S is zero. Using several electrostatic

local gates, the chemical potential in different regions

can be modulated. We consider the case where there is a

chemical potential difference U between S and N

regions. The potential profile then reads

<<

=

in normal and

in different regimes

,0

otherwise

( )

0,

Uxd

U r

?

(2)

In the presence of superconducting correlations we use

the DBdG equation which takes the form of two

Page 3

IJPR Vol. 8, No.2 Quantum transport in grapheme normal-metal-superconductor …

83

Figure 1. NSN structure on graphene

decoupled sets of equations as;

+−

∆

with u and v being, respectively, two-component wave

functions of electron-like and hole-like quasiparticles

from two different valleys. Note that superconducting

correlations couple an electron in one valley to its time-

reversed hole in the other valley.

in normal leads and

0

ε >

measured from the chemical potential.

In N1 and N2 the pair potential ∆ = 0 and neglecting

the suppression of superconductivity in S close to NS

interfaces we take ∆ to be real and constant inside S.

This assumption is most valid when the Fermi wave

length in region S is much smaller than the Fermi wave

length in region N, namely when

the valley degeneracy, we will consider only one set of

the four-dimensional equations (3) which describes

coupling of electrons from valleyK+

valleyK−.

Due to the one dimensional nature of the applied

potential, the wave vector in the direction parallel to the

boundaries (q ) is constant in the three regions.

Assuming an incident electron with probability

amplitude 1 from N1, the solution of Eq. (3) inside N1, S

and N2 are, respectively, written as the followings

rr

ψψψψ=++

,

*

( )

( )

F

F

u

v

u

v

H U rE

EU rH

ε

±

±

∆

=

−−

?

?

(3)

F

E

is the Fermi energy

is the excitation energy

F

UE

>>

. Because of

to holes from

1NNeNeANh

+−−

SSeSeShSh

abcd

ψψψψψ

+−+−

=+++

,

2Ne

ψ

Neh

ψ

Nh

tt

ψ

where

electron-like (e) and hole-like (h) quasiparicles which

propagate in ± directions of x axis. Correspondingly in

N1 and N2,

,Ne Nh

are the electron and hole bases

propagating in x

±

directions. The bases are given by

++

=+

,(4)

,Se Sh

ψ±

are bases of DBdG in S which describe

ψ±

0

0

exp(

exp(

)

)

exp(

),

1

exp( )

exp(

exp(

)

)

exp(

),

1

exp()

Se

Sh

i

γ

ii

iqy

ik x kx

i

i

ii

iqy

ik xkx

i

β

β

ψ

γ

−

β

γβ

ψ

γ

±

±

±±+

=

±

±

−

=

±

−

∓

∓∓

∓

∓

∓

(5)

exp(

exp(

±

2)

2)

exp()

,

0

0

cos

0

0

i

±

exp()

,

exp(

exp(

∓

2)

2)

cos

e

Ne

h

Nh

i

i

iqyik x

α

iqyik x

α

′

i

α

α

ψ

ψ

α

α

±

±

±

±

=

±

=

′

′

∓

∓

(6)

where

222

0

2

0

arccos( ),

,

arccos ( ),

arcsin(/),

() / (),

()

sin

()

FF

F

−

F

F

v

F

if

ihif

v qEU

kEU

∆

vq

E

?

U

k

k

ε

∆

ε

β

ε

∆

ε

γπ

β

< ∆

=

−

+

> ∆

=−

=−−

=

?

?

and

k

1

1

()()cos ,

()()cos

eFF

hFF

vE

kvE

εα

εα

−

−

=+

′

=−

?

?

(7)

Here

α

′ =

that an electron and hole wave vectors make with x

axes. We note that beyond a critical angle of incidence

defined by

arcsin( (

cFF

EE

αεε=−+

α′ becomes imaginary and so there is no AR and CAR.

The coefficients , , a b c and d are relative amplitudes of

different wave functions in S. r and rAare, respectively,

amplitudes of electron normal and Andreev reflections in

N1. For an incident electron in N1 , et

amplitudes of transmission of an electron and hole into

N2 which describe CT and CAR processes respectively.

Note that while in N1 and N2 the wave functions are

propagating for allε , in S they are propagating only for

arcsin(

v q

?

/ ())[-2,2]

FF

v q

?

/ ( -

ε

E

αε

))

ππ=+

are, respectively, the angles

∈

and

arcsin(

FF

E

)) ,(8)

and h tare the

Page 4

84 H Mohammadpour and M Zareyan IJPR Vol. 8, No.2

(a) (b)

Figure 2.

(b) specular and retro reflection regimes by

( )T α

in NSN structures with

0.122,0d ξε=∆ =

and (a)

243,0

F

U

.

E

∆ =∆=

(dashed curve),

0.01

F

E

∆=

(solid curve) ,

9

20, 0.01, 10

F

EU

∆=∆ =

ε > ∆ . For ε < ∆ the wave functions are evanescent

inside S where they have an exponential decay along x

direction within a scale of order

In contrast to the parabolic energy spectrum in

ordinary two dimensional electron gas (2DEG) which

requires the continuity of the derivatives of the wave

functions as well as their amplitudes, for chiral electrons

in graphene the velocity is constant irrespective of the

energy; so, one has to impose only the continuity of the

wave functions for each sublattices which also conserves

the chirality. Imposing the continuity condition at N1S

and SN2 interfaces we have the relations

rra

ψψψψ++=

at

0

x =

and

abcd

ψψψψ+++

at xd

=

. These relations constitute a system of 8 linear

equations whose solutions give us r,

F

v

ξ =∆

?

.

NeANhSeSh

Ne

Se

Sh

bc

ψ

d

ψψ

+−−+−+−

+++

(9)

SeShe

ψ

h

Ne

Se

Sh Nh

tt

ψ

+−+−++

=+

(10)

A r ,

et and h t . The

relation

current conservation.

The differential conductance of the system, I

for ε =e V

×

is calculated by the BTK formula as:

2222

1

Aeh

rrtt

+++=

holds for ensuring

V

∂ ∂

22

0

0

( )(1(, )

α

(, ) )cos

α

c

A

I

V

g Vr eVreVd

α

∫

α α

∂

∂

=−+

2

0( )

( )N

ε

4

E

(),

v()

F

F

g Veh

ε+

N eV

W

π

=×

=

?

(11)

where

a ballistic graphene strip of width W . In our model we

consider a wide geometry of W

0

gis the conductance of N transverse modes in

d

>>

for that 1N >> .

3. Results and discussion

First we analyze the angular dependence of the

transmission probability

( )

T

α

We note that for

F

E

ε <<

, in contrast to the amplitude of

AR, amplitude of CAR is small since it requires a

change of pseudospin.

In Fig. 2a we have plotted

0

ε =

when 243

U ∆ =

(0.01

F

E

∆=

) and normal strip (

normal case there is always a perfect transmission at

normal incident

0

α =

due to Klein tunneling [4].

Additional perfect transmissions can occur at larger

angles due to the resonances which for given U and

d are defined by the solutions of the following equation

22

4sin sinsin2 sin2

2sincos sin2 kd

αγ−−

2

1- ( )

r

α=

for electrons.

( )

T α

at the Fermi level

for both superconducting

0

∆ = ) cases. In the

2

4sin sin sin

α

0

kdkd

kd

γγ

γ

+

=

(12)

Introducing superconductivity leaves the transmission

probability intact at almost normal incidence, but

suppresses the transmission at larger angles. In the case of

Fig. 2a the resonance at

superconductivity. We can understand this behavior by

noting that a retro AR conserves the pseudospin and

thereby the reflection-less tunneling through S is

preserved at small angles. However the resonant

transmission at larger angles is affected by evanescent

nature of the subgap quasiparticles in S strip leading to a

suppression of transmission probability.

Fig. 2b represents an example of comparison of

( )T α

in two limits of retro (

2

απ≈

disappears by

0.01

F

E

∆=

) and

specular (

ε ∆ =

difference in the angular dependence in two cases which

20

F

E

∆

d ξ =

=

0.122

) AR regimes for

. This shows a remarkable

9

10 , U ∆ =

0,

Page 5

IJPR Vol. 8, No.2 Quantum transport in grapheme normal-metal-superconductor …

85

(a) (b) (c)

Figure 3.

oscillation. The values of

1

0 gIV

−∂ ∂

in different regimes for

are written beside each plot.

9

10U ∆ =

, ford ξ = (a) 0.1, (b)1, (c) 10; scales are different to reveal the d -dependent period of the

F

E

∆

(a) (b)

I

(c)

Figure 4. Separate contribution of each propagation probability in

1

0 gV

−∂ ∂

, with

0.1

F

E

∆=

for d ξ =(a) 0.1, (b) 1, (c) 10 when

9

10U ∆ =

.

depends on d and U .

Now let us analyze the behavior of the differential

conductance, IV

∂ ∂

which is given by Eq. (11). Fig. 3

shows

thicknesses,

cover whole range from pure specular to pure retro AR.

For a thin strip with

d ξ =

is dominated by CT which results in a normalized

differential conductance

subgap differential conductance shows small deviations

due to weak AR amplitude. Above the gap, by increasing

the voltage,

0

gIV

increases to unity in an

oscillatory manner. The period of oscillations is

inversely proportional to the thickness.

Increasing the thickness of S strip leads to an increase

in the amplitude of AR and thus enhancing deviations

from normal differential conductance at subgap voltages.

This is seen in Fig. 3b where

d ξ = . Above the gap

interference oscillations with smaller period compared to

0.1

d ξ =

. Note that the period of oscillations does not

depend on the doping of N regions given by

1

0 gIV

−∂ ∂

versus eV ∆ when

0.1,1,10 and for different

9

10

E

U ∆ =

for

to

d ξ =

F

∆

0.1(Fig. 3a) the transport

1

0 g IV

−∂ ∂

close to unity. The

1

−∂ ∂

1

0

I

gIV

−∂ ∂

is plotted for

1

1

0 g V

−∂ ∂

shows quantum

F

E

∆

−∂ ∂

. In

contrast to this, below the gap, behavior of

strongly depends on

∆

noting that contributions of two different types of retro

and specular ARs in electron transmission depend on

1

0

gIV

F

E

. This can be understood by

F

E

∆

retro reflection at

at

F

E

∆

By further increasing d ξ , the contribution of CT

becomes even smaller at subgap voltages. The subgap

differential conductance is determined only by AR. As a

result,

0

gIV

takes zero value at

for that the AR is forbidden since the critical angle

0

c

α =

(Eq.(8)). Therefore the subgap conductance of

very large d ξ is very similar to that of NS system [19].

−∂ ∂

is oscillating with energy as

two previous cases. The oscillations persist even at very

large thicknesses due to the ballistic feature of the

contact.

To give more insight into different scattering

processes, in Fig. 4 we have plotted contributions of CT,

AR, CAR and R to differential conductance for different

thicknesse and for

0.1

F

E

∆=

above given explanations. Note that the contribution of

CAR is always small because in contrast to AR, CAR

requires a change of pseudospin which is prevented in

the absence of psudospin flip scattering.

4. Conclusions

In this paper we have studied quantum transport of

electrons through a graphene superconducting strip which

connects two normal leads. Using DBdG equations we

have calculated amplitudes of different scattering

. The Andreev transmission ranges from purely

1

F

E

∆<<

to purely specular reflection

1

>>

[19].

1

−∂ ∂

F

eVE

=

(Fig. 3c)

Above the gap,

1

0

gIV

. These plots confirm the

Page 6

86 H Mohammadpour and M Zareyan IJPR Vol. 8, No.2

processes which are normal reflection, direct and crossed

Andreev reflections and cotunneling. First we have

observed superconducting induced changes in the angular

dependence of the electron transmission which can be

explained in terms of interplay between Klein tunneling,

Andreev reflection and resonance transmission. Within

BTK formalism we have analyzed the resulting

differential conductance for different thicknesses of the S

strip and doping degrees of the N electrodes. The subgap

1eV ∆ <

differential conductance is determined by the

competition of Andreev reflection and cotunneling. While

in the limits of

is respectively, dominated by purely cotunneling and

Andreev reflection processes, at intermediate thicknesses

1d ξ ≈ , it is determined by superposition of both

processes. Above the gap

conductance shows oscillatory behavior with eV ∆ . We

have given a full analysis of the period and the amplitude

of this quantum interference oscillations in terms of d ξ

and doping of N regions.

1d ξ <<

and 1d ξ >>

the transmission

1eV ∆ > , the differential

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