Formation of Interface Bound States on a Graphene-Superconductor Junction in the Presence of Charge Inhomogeneities

doi:10.4236/graphene.2013.21005

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Graphene, 2013, 2, 35-41

http://dx.doi.org/10.4236/graphene.2013.21005 Published Online January 2013 (http://www.scirp.org/journal/graphene)

Formation of Interface Bound States on a

Graphene-Superconductor Junction in the Presence of

Charge Inhomogeneities

Pablo Burset1*, William Javier Herrera2, Alfredo Levy Yeyati1

1Departamento de Física Teórica de la Materia Condensada and Instituto Nicolás Cabrera,

Universidad Autónoma de Madrid, Madrid, Spain

2Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia

Email: *[email protected]

Received November 12, 2012; revised December 17, 2012; accepted January 14, 2013

ABSTRACT

Interface bound states have been theoretically predicted to appear at isolated graphene-superconductor junctions. These

states are formed at the interface due to the interplay between virtual Andreev and normal reflections and provide

long-range superconducting correlations on the graphene layer. We describe in detail the formation of these states from

combining the Dirac equation with the Bogoliubov-de Gennes equations of superconductivity. On the other hand, fluc-

tuations of the low-energy charge density in graphene have been confirmed as the dominating type of disorder. For

analyzing the effect of disorder on these states we use a microscopic tight-binding model. We show how the formation

of these states is robust against the presence of disorder in the form of electron charge inhomogeneities in the graphene

layer. We numerically compute the effect of disorder on the interface bound states and on the local density of states of

graphene.

Keywords: Graphene; Superconducting Hybrid Structures; Andreev Reflection

1. Introduction

The peculiar electronic band structure of graphene has

been the focus of an intense research activity [1]. In gra-

phene, the electronic low-energy properties are governed

by a massless relativistic Dirac Hamiltonian which makes

the carriers moving in it develop very interesting proper-

ties like an electronic spectrum linear with the wave

vector and electronic states which are chiral with respect

to the pseudospin defined by the two atoms of the crystal

unit cell. As a result, graphene exhibits exotic effects like

the Klein paradox—perfect transmission through poten-

tial barriers [2].

Of particular interest is the case of a graphene layer in

contact with a superconducting electrode, where the in-

terplay between superconductivity and the relativistic

behavior of charge carriers in graphene can be tested [3].

Graphene is not intrinsically a superconductor but it can

easily inherit the bulk properties of other materials when

in contact with them. Good contacts can be achieved

between lithographically defined superconducting elec-

trodes and graphene layers [4-7]. In such hybrid devices,

a superconducting gap is induced by proximity effect on

the graphene region underneath the metallic electrodes;

in these conditions, Andreev processes featuring conver-

sion of electrons into holes in the normal region and the

creation of a Cooper pair in the superconductor take

place [8]. In a recent experiment using planar Pb probes a

high-quality tunneling spectroscopy has been performed

when the Pb was in the superconducting regime [9].

Several theoretical studies of the electronic and trans-

port properties of graphene-superconductor hybrid struc-

tures have been reported in the recent years. The differ-

ential conductance and the spectral properties of gra-

phene-superconductor junctions and graphene-based Jo-

sephson junctions have been studied within a tight-

binding framework and solving the Dirac-Bogoliubov-de

Gennes (DBdG) equations under the approximation of

neglecting inter-valley scattering [10-19]. The influence

of electron-hole inhomogeneities on the Andreev reflec-

tion on graphene-superconductor hybrid structures was

studied in [20]. The properties of bound states arising

from multiple Andreev reflections in a graphene Joseph-

son junction have been analyzed in [21,22]. However, the

special electronic properties of graphene are such that

bound states can be formed even at isolated single junc-

tions. The existence of interface bound states (IBS) was

demonstrated in [23] for different edge orientations and

doping conditions of the graphene layer. In this work we

*Corresponding author.

P. BURSET ET AL.

study how robust these bound states are to the effect of

disorder on the graphene sample.

This article is organized as follows: First, we discuss

in Section 2 the formation of interface bound states at a

graphene-superconductor interface by matching of the

solutions of the DBdG equations. In Section 3, we intro-

duce the microscopic tight-binding model which includes

a superlattice potential that is used to simulate the effects

of disorder. Finally, in Section 4 we present our main

results and end with some conclusions.

2. Interface Bound States

The special electronic properties of graphene are such

that bound states can be formed at isolated graphene-

superconductor junctions [23]. The mechanism for the

emergence of these states can be understood from the

scheme depicted in the left panel of Figure 1. As is usu-

ally assumed the junction can be modeled as an abrupt

Figure 1. (a) Simple model for the emergence of IBSs illus-

trating the scattering processes taking place at a gra-

phene-superconductor interface with an intermediate heav-

ily doped normal graphene region of width d. Cases (i) and

(ii) correspond to the situation with



vq EE and



EE, respectively, where D> E > EF. (b) Map

of the spectral density of states where the light regions in-

dicate a higher spectral density showing the emergence of

the IBS from the Dirac cone. The spectral density has been

calculated using the microscopic tight-binding model for an

undoped (EF = 0) semi-infinite region of graphene without

disorder coupled to a superconductor. The distance from

the interface is equivalent to the superconducting coherence

length x. The white dashed line corresponds to the disper-

sion relation of (9).

discontinuity between two regions described by the

DBdG equation, taking a finite superconducting order

parameter



and large doping on the super-

conducting side (S) and zero order parameter and small

doping





E~ on the normal side (N). For the analysis

it is instructive to include an artificial intermediate nor-

mal region with 0



 and 

F, whose width, d,

can be taken to zero at the end of the calculation. This

intermediate region allows to spatially separate normal

reflection due to the Fermi energy mismatch from the

Andreev reflection associated to the jump in



. The

DBdG equation reads

HE E

















(1)

with the excitation energy E positive unless otherwise

specified. The pair potential couples electron and hole-

like excitations from different valleys (opposite momen-

tum), described by the same Hamiltonian





xx yy

Hiv





.

We assume in (1) a s-wave pairing which leads to a con-

stant gap



which is diagonal in sublattice space.

Whenever the pair potential is assumed constant, the low

energy spectrum is given by



ΔFF

EEvkq 2

we define the component of the momentum perpen-

dicular to the interface as



Feh FF

vk Evq,

with 22





the conserved component of the

momentum parallel to the interface. The basis of scatter-

ing states in the normal region with

0xΔ0







1, e, 0,0ee

iik

iqy





x

, (2)





0, 0,1,eee

iikx







 , (3)

for states moving towards the interface and





1,e, 0, 0ee

Tik x

iiqy







 , (4)





,0, 0,1,eee

iikx

iqy







, (5)

for states moving away from the interface. We define



eieh

Feh F

kiqvE



E. Analogously, the basis

of scattering states in the superconducting region

d

with 0



 and S





reads





,e ,,eee

SSS

eee

iiikx

euu vv





qy

(6)

P. BURSET ET AL. 37



,e,,e ee

SSS

hhh

iiikx

hvv uu





i



(7)

where





Feh F

iqvk E



 and with



22 2uv E





the BCS coherence factors, normalized so that

. Finally, for the intermediate region

1uv

d we use the normal state basis changing the

doping level to

As shown in Figure 1 (case i), an incident electron

from the normal side with energy E and parallel momen-

tum such that

qFF

vq EE is partially trans-

mitted into the intermediate region and after a sequence

of normal and Andreevreflections would be reflected as a

hole. This process can either correspond to retro or spe-

cular Andreev reflection depending on whether



EE [8].

For FF

vq EE neither electron or holes can

propagatewithin the graphene normal region. However,

virtual processes like theone depicted in Figure 1 (case ii)

would be present. These correspond to sequences of An-

dreev and normal reflections within the intermediate re-

gion. A bound state emergeswhen the total phase



accumulated in such processes reach the resonancecondi-

tion 2πn



.

It is quite straightforward to determine the dispersion

relation for the IBS from the model represented in the top

panel of Figure 1. The phase accumulated by a sequence

of normal and Andreev reflections in the intermediate

region can be obtained from the corresponding coeffi-

cients re, rh and rA. From the matching of the solutions of

the DBdG equations at the interface between the gra-

phene and the intermediate regions one obtains

ieh ieh

ieh

eh I

eh eh















, (8)

where

 



,arcsin

ehF F

vqE E











.

The condition F

allows to take

,. On the other hand, in the region of evanescent

electron and hole states for graphene (

EEvq





vq EE),

re,h be- come a pure phase factor , with



,eh





exp





2sgnarctan exp

ehF eh

qEE





 



and







,sgn argcosh

ehF F

qvqE





For the Andreev reflection coefficient between regions I

and S one has A



exp



, where





arccos



,

as it corresponds to the Andreev reflection at an ideal N-S

interface with [24]. In the limit the

total phase accumulated is thush



E0d

2





, from

which one obtains the following dispersion relation







gn e

s cos







2co









. (9)

This dispersion simplifies to





 

Evqvq

at the charge neutrality point (i.e. for ). In this

case the IBS approaches zero energy for and

tend asymptotically to the superconducting gap for large

q. Notice also that the decay of the states into the gra-

phene bulk region (

0

q0



x in the top panel of Figure 1) is

set by





exp eh



, where





sinh

eh FFeh

vEE





for the electron and hole components respectively, which

can be clearly much larger than the superconducting co-

herence length 0F





when . It is also in-

teresting to notice that the IBSs survive when ,

i.e. in the regime corresponding to the usuSSal Andreev

retroreflection, but with a much smaller spatial extension.





3. Microscopic Model

In order to describe the interface more microscopically,

we analyze the electronic states of a graphene layer using

the tight-binding approximation,

††

ij iii

ij i

tcc Vcc 



, (10)

where 0

232.6





tva eV denotes the hopping ele-

ment between nearest carbon atoms on the hexagonal

lattice and 00.142



a nm is the smallest carbon-carbon

distance. 0





VV V



is the potential applied to the lat-

tice, where 0

V is a uniform on-site doping and i



represents small fluctuations over the doping level (i.e.

charge inhomogeneities as introduced below). The spin

degree of freedom has been omitted due to degeneracy.

We assume that the graphene region is a strip with

armchair edges along the y-direction as sketched in Fig-

ure 1. We model the strip by repeating a unit cell com-

posed of four atoms N times along the x direction and M

times along the y direction (see Figures 2(a) and (b)). As

a consequence, the length of the graphene layer is

3LNa. For describing the limit we im-

pose periodic boundary conditions in the y direction and

define

WL

π,π









qdd

SCSC as the corresponding wave

vector, with dSC the vertical length of the supercell.

We connect the leftmost graphene armchair edge to a

superconducting electrode and the rightmost to a normal

lead. We maintain the graphene sublattice structure at the

edges, thus representing the experimental situation where

P. BURSET ET AL.

the electrodes are deposited on top of a graphene layer

[16,25-28]. The presence of the superconducting corre-

lations requires introducing the Nambu space, describing

electron and hole propagation within the graphene layer.

The self-energy on the graphene sites at the layer edge

coupled to the normal lead is approximated by a

88

M matrix with elements ,,



ijij v









where ,1,,4





,1,ij

,

label the atomic sites within the

unit cell, label the unit cells in the super-

lattice and

,eh





label the matrix elements in

Nambu space. Following the geometry depicted in Fig-

ure 2, the elements of the self-energy matrix are explic-

itly defined as

11, 44,32

 



and 14, 41,14,41,12

RR RR

ee eehhhh

 

(see more

details in [16,28]).

Analogously, the self-energy describing the coupling

to the superconducting electrode on the left armchair

edge is described by a matrix with elements with gBCS =

Figure 2. (a) Unit cell of the superlattice with four carbon

atoms; (b) Graphene region showing the axis selection for N

= 17 horizontal cells and M = 9 vertical cells. The normal

and superconducting electrodes would be coupled to the left

and right armchair edges, respectively; (c) Example of a

two dimensional superlattice potential with horizontal pe-

riod dx and vertical period dy extended over a graphene

region with N = 72 and M = 48 cells. A disordered superlat-

tice potential takes random values in the range [−Vd, Vd] at

every square region given by dx × dy.

22, 33,

23, 33,23,32,

22,33, 22,33,

BCS

LL LL

ee eehhhh

LLLL

ehehheheBCS

 







,

(11)

BCS

Ef EE



the BCS amplitudes and the

superconducting gap.

The spectral properties of the system are calculated

using the local retarded Green function , where



ˆ,

GqE

1, ,



iN labels the horizontal sites of the layer. We

can thus define the local density of states (LDOS) at the

site i as

 



π/ˆ

Tr Im

π/

2π

SC SC q















, (12)

which is normalized to one electron per site and spin. To

remove the dependence on the width of the superlattice

dSC, it is convenient to normalize the LDOS with the

density of a bulk graphene layer with zero doping at



E, 0



, which for





vd is given by



2π

SC F

dv.

The results thus obtained do not depend on the ratio



t used in our tight-binding calculations. In Figure

1(b) we show a map of the spectral density of states (the

integrand of (11)) for an undoped (EF = 0) semi-infinite

region of graphene coupled to a superconductor. The

energy-momentum map has been calculated at a distance

to the interface comparable to the superconducting co-

herence length



F

v. The IBSs are located outside

of the bulk band of graphene. The white dashed lines

indicate the solution of (9).

Model for Disorder Due to Charge Puddles

The electron-hole inhomogeneity in graphene is modeled

using a two-dimensional superlattice potential. As it is

sketched in Figure 2(c), we divide the graphene layer

into square supercells of horizontal length 0

d3

xna

and vertical length 0



dma, with n ≤ N and m ≤ M.

At each supercell, we define an electrostatic potential



which takes random values in the range [−Vd, Vd],

with Vd the disorder strength. This random superlattice

potential is added to the uniform electrostatic potential V0.

Consequently, V0 stands for the doping level of the gra-

phene layer and i



represents the charge inhomoge-

neities characteristic of graphene (i.e., charge puddles

[29,30]). Following [30], the best estimation for the

maximum strength of the charge puddles is of 30 meV,

over a region of typical size no greater than 30 nm.

4. Effect of Disorder

When studying the low-energy physics of graphene close

to the Dirac point, the electron-hole inhomogeneity in

P. BURSET ET AL. 39

graphene (i.e., charge puddles [29,30]) is a type of charge

disorder that has to be taken into account.

The results presented in Figure 1(b) correspond to a

semi-infinite layer of pristine graphene connected to a

superconductor. A more realistic model should include

size effects such as a graphene layer with a finite length,

and the possibility of electron-density inhomogeneities.

Although the effect of having a finite length in the nor-

mal region can be treated within both the continuous and

the TB models, the latter is more suitable to explore the

effect of disorder.

The results for a graphene layer of horizontal length

3800La with armchair edges coupled to a su-

perconductor are presented in Figure 3. Figure 3(a) has

been calculated with the strength of the disorder potential

set to zero. The results for the LDOS (right panel) are

similar to the undoped case presented in [16]: a clear

peak at , which rapidly decays inside the normal

region after a few times the superconducting coherence

length ξ. The finite length of the graphene layer is mani-

fested by the appearance of energy bands close to the gap

edge. The spectral density of states is plotted in the left

E

Figure 3. LDOS for a graphene layer of length L = 197 nm

(800 armchair cells) coupled to a superconductor on an

armchair edge calculated within the tight-binding model.

For the simulation, the superconducting gap has been cho-

sen to be D = 0.005 tg. (a) The results for the case without

disorder. On the left, the spectral density of states, calcu-

lated at a distance from the interface comparable to the

superconducting coherence length x, in a color map where

dark blue is the absence of states. On the right, the LDOS

as a function of the energy and the distance to the inter-

face; (b) The same as before with the introduction of a ran-

dom superlattice potential of strength Vd = 0.01 tg, with spa-

tial periods nm. 10

panel, calculated at a distance ξ



from the interface. The

IBS is clearly distinguishable outside of the Dirac cone

(dark red in the color plot, the dark blue corresponds to

the absence of spectral density). The finite armchair layer

considered now presents a gap in the energy of the nor-

mal state. The Dirac cone is not uniform but is formed by

discrete energy levels (light blue areas in the color plot)

due to the finite size of the layer.

When the disorder is taken into account (Figure 3(b)),

the peak at the edge of the gap remains unaltered and the

decay is comparable to the non-disordered case (right

panel). The weight of the states below the gap becomes

more important in the spectral density, as it is shown in

the left panel. The energy levels are deformed and ac-

quire a bigger weight in the spectral density (light blue in

the color plot).

For this strength of the disorder, the envelope of the

LDOS is still comparable to the non-disordered case, but

small fluctuations appear. The main effect of disorder

can be seen as an effective doping, which does not alter

deeply the LDOS. In Figure 4 we show the LDOS cal-

culated at a distance from the interface 6



, where the

IBS has completely decayed. The solid red line corre-

sponds to the case where the disorder strength is set to

zero. The LDOS presents a soft modulation and is sym-

metric with respect to the energy—as corresponds to an

undoped case. The introduction of disorder breaks this

electron-hole symmetry and clearly affects the modu-

lation of the LDOS (blue dashed line).

The strength of the random superlattice potential used

in these simulations is 0.01 2.7



 meV, which is

comparable to the greatest estimation for the measured

strength of the charge inhomogeneities in graphene [30].

The periodicity of the superlattice potential is



dd nm, which is slightly smaller than the typi-

Figure 4. LDOS with the same parameters as in Figure 3

calculated at a distance 6× from the interface for the case

without disorder (red solid line) and with disorder strength

Vd = 0.01 tg (blue dashed line).

P. BURSET ET AL.

cal length of a charge puddle in graphene, with an aver-

age size of 30 nm. In spite of this, the chosen values are

close enough to assume that a bigger period for the su-

perlattice potential would not affect considerably the

LDOS profiles.

In conclusion, we have shown that interface bound

states appear at isolated graphene-superconductor junc-

tions. The presence of charge inhomogeneities in the

normal region induces strong fluctuations in the LDOS

profile and breaks the electron-hole symmetry of the

LDOS. However, the IBS modifies more intensely the

LDOS and thus this electron-hole symmetry cannot be

appreciated at a distance from the interface comparable

to 2 - 3 ξ. For a longer distance, the IBS have decayed

and the effect of the disorder is clearly shown in the

LDOS. The formation of IBSs and their effect on the

profile of the LDOS is robust against a disorder strength

comparable to the measured strength of the charge pud-

dles in graphene.

5. Acknowledgements

This work was supported by MICINN-Spain via grant

FIS2008-04209 and EU project SE2ND (PB and ALY)

and COLCIENCIAS, project 110152128235 (WJH).

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