Quantum transport in graphene normal-metal superconductor- normal-metal structures
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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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