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
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.
C
opyright © 2013 SciRes. Graphene

P. BURSET ET AL. 36
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

FF
vq EE and

FF
v
q
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

S
F
E

F
E~ on the normal side (N). For the analysis
it is instructive to include an artificial intermediate nor-
mal region with 0

 and 
I
S
F
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
EE

. The
DBdG equation reads
F
F
HE E
EH








(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


F
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

2
22
ΔFF EEvkq 2 .
we define the component of the momentum perpen-
dicular to the interface as


22
,
SS
Feh FF
vk Evq,
with 22
ED
2

 the conserved component of the
momentum parallel to the interface. The basis of scatter-
ing states in the normal region with
0xΔ0

is


1, e, 0,0ee
T
iik
iqy
ee
e


x
, (2)


0, 0,1,eee
T
iikx
qy
hh
h



 , (3)
for states moving towards the interface and


1,e, 0, 0ee
Tik x
iiqy
ee
e


 
 , (4)

 ,0, 0,1,eee
T
iikx
iqy
hh
h



, (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
x
d
with 0

 and S
F
E


reads


,e ,,eee
SSS
eee
T
iiikx
Si
euu vv


qy
(6)

P. BURSET ET AL. 37

,e,,e ee
SSS
hhh
T
iiikx
S
hvv uu


i
qy

(7)
where
 
,
,
e
S
eh
iS
Feh F
iqvk E

 and with


22 2uv E


the BCS coherence factors, normalized so that
. Finally, for the intermediate region
22
1uv
0
x
d we use the normal state basis changing the
doping level to
I
F
E.
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
F
EE

or
F
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
,,
,
,
,,
ee
e
ee
I
ieh ieh
I
ieh
eh I
ii
eh eh
r








, (8)
where
 

,arcsin
II
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 (
,,
I
F
EEvq
0
I
eh
 
FF
vq EE),
re,h be- come a pure phase factor , with

,eh
i 

exp
 
,,
 2sgnarctan exp
ehF eh
qEE 


 


and



,sgn argcosh ehF F
qvqE  E

.
For the Andreev reflection coefficient between regions I
and S one has A

exp
A
ri

, where


arccos
AE
 ,
as it corresponds to the Andreev reflection at an ideal N-S
interface with [24]. In the limit the
total phase accumulated is thush

S
F
E0d
2
Ae

 , from
which one obtains the following dispersion relation




22
22
gn e
s cos


eh
F
eh
EE


es
2co



eh
E

. (9)
This dispersion simplifies to

22

 
FF
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
F
E
q0
0

x in the top panel of Figure 1) is
set by


,
exp eh x 
, 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 v


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.

F
E

F
E
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,
††
g
ij iii
ij i
H
tcc Vcc 

, (10)
where 0
232.6


gF
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


i
VV V
i

is the potential applied to the lat-
tice, where 0
V is a uniform on-site doping and i V

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
0
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. 38
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
M matrix with elements ,, 
R
R
ijij v



 ,
where ,1,,4


,1,ij
,
label the atomic sites within the
unit cell, label the unit cells in the super-
lattice and
,M
,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
RR
i
 

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,
32
12
32
LL
BCS
LL LL
ee eehhhh
LLLL
ehehheheBCS
g
f
 



,
(11)
2
BCS
Ef EE
2

the BCS amplitudes and the
superconducting gap.
The spectral properties of the system are calculated
using the local retarded Green function , where

ˆ,
r
ii
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
 

2
π/ˆ
Tr Im
π/
2π
r SC SC q
ii
SC
dd
Ed
d




i
G



, (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


F
SC
vd is given by

2
2π
SC F
dv.
The results thus obtained do not depend on the ratio

g
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
3

y
dma, with n ≤ N and m ≤ M.
At each supercell, we define an electrostatic potential
i
V

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
V

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
0
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
xy
dd
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
dg
Vt

 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
10


xy
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. 40
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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