1. No category

CNR IMM-Bologna
Università dell!Aquila
How to make graphene superconducting
Gianni Profeta
Physics Department, University of L`Aquila, Italy
&
Max-Planck Institute of Microstructure Physics, Halle, Germany
Matteo Calandra and Francesco Mauri
IMPMC, University of Paris 6, CNRS, Jussieu , Paris France
Graphene: physical realization of fundamental phenomena in solid-state-physics
Graphene: physical realization of fundamental phenomena in solid-state-physics
• Ultra-relativistic particles near EF
E = ±vF p, vF = c/300
A. H. Castro Neto et al. RMP (2009)
Novoselov et al. Nature (2005)
Lee et al., Science (2008)
Novoselov et al, Science (2007)
Graphene: physical realization of fundamental phenomena in solid-state-physics
• Ultra-relativistic particles near EF
E = ±vF p, vF = c/300
• Effective theory around EF
−ivF σ · ∇ψ(r) = Eψ(r)
A. H. Castro Neto et al. RMP (2009)
Novoselov et al. Nature (2005)
Lee et al., Science (2008)
Novoselov et al, Science (2007)
plateaux in r xy and therefore occur at half-integer n; see Figs 2 and 4),
in agreement with theory14–17. Note, however, that from another
perspective the phase shift can be viewed as the direct manifestation
of Berry’s phase acquired by Dirac fermions moving in magnetic
field20,21.
Finally, we return to zero-field behaviour and discuss another
feature related to graphene’s relativistic-like spectrum. The spectrum
E = vanishing
±vF p,concentrations
vF = c/300
implies
of both carriers near the Dirac
point E ¼ 0 (Fig. 3e), which suggests that low-T resistivity of the
emerge only at expon
absence of localization,
explained qualitatively
free path l of charge carr
wavelength lF. Then, j
cannot be smaller than
is known to have failed
which disorder leads t
behaviour22,23. For 2D D
and, accordingly, Mott’
broad theoretical conse
should exhibit a minim
tion was not expected
value of ,e 2/ph, in dis
Thus, graphene exhi
for a 2D gas of particles
the Schro¨dinger equatio
Graphene: physical realization of fundamental phenomena in solid-state-physics
• Ultra-relativistic particles near EF
• Effective theory around EF
−ivF σ · ∇ψ(r) = Eψ(r)
• Cyclotron mass with carriers
√
mc ∝ n
Figure 3 | Dirac fermions of graphene. a, Dependence of B F on carrier
concentration n (positive n corresponds to electrons; negative to holes).
N isH.
theCastro
number
b, Examples of fan diagrams used in our analysis7 to find B F.A.
Neto et al. RMP (2009)
associated with different minima of oscillations. The lower and upper curves
are for graphene (sample of Fig. 2a) and a 5-nm-thick film ofNovoselov
graphite with
et al.a Nature (2005)
similar value of B F, respectively. Note that the curves extrapolate to different
origins, namely to N ¼ 1/2 and N ¼ 0. In graphene, curvesLee
foretallal.,
n Science (2008)
extrapolate to N ¼ 1/2 (compare ref. 7). This indicates a phase shift of p with
al, Science (2007)
respect to the conventional Landau quantization in metals.Novoselov
The shift isetdue
14,20
to Berry’s phase . c, Examples of the behaviour of SdHO amplitude Dj
plateaux in r xy and therefore occur at half-integer n; see Figs 2 and 4),
in agreement with theory14–17. Note, however, that from another
perspective the phase shift can be viewed as the direct manifestation
of Berry’s phase acquired by Dirac fermions moving in magnetic
field20,21.
Finally, we return to zero-field behaviour and discuss another
feature related to graphene’s relativistic-like spectrum. The spectrum
E = vanishing
±vF p,concentrations
vF = c/300
implies
of both carriers near the Dirac
point E ¼ 0 (Fig. 3e), which suggests that low-T resistivity of the
emerge only at expon
absence of localization,
explained qualitatively
free path l of charge carr
wavelength lF. Then, j
cannot be smaller than
is known to have failed
which disorder leads t
behaviour22,23. For 2D D
and, accordingly, Mott’
broad theoretical conse
should exhibit a minim
tion was not expected
value of ,e 2/ph, in dis
Thus, graphene exhi
for a 2D gas of particles
the Schro¨dinger equatio
Graphene: physical realization of fundamental phenomena in solid-state-physics
• Ultra-relativistic particles near EF
• Effective theory around EF
−ivF σ · ∇ψ(r) = Eψ(r)
• Cyclotron mass with carriers
√
Castro Neto
mc ∝
net al.: The electronic properties of graphene
116
Energy
!I!r" =
D
• Klein paradox
E
x
D
I
φ
III
II
θ
1
2 se
i#
ei!kxx+kyy" +
# $
r
2 se
1
i!$−#"
%
ei!−kxx+kyy" ,
!25"
V0
y
# $ %
1
with # = arctan!ky / kx", kx = kF cos #, ky = kF sin #, and kF
the Fermi momentum. In region II, we have
!II!r" =
φ
%
# $
a
1
2 s !e
i"
ei!qxx+kyy" +
# $
b
2 s !e
1
i!$−""
%
ei!−qxx+kyy" ,
!26"
x
with " = arctan!ky / qx" and
FIG. 6. !Color online" Klein tunneling in graphene. Top: sche!27"
qx = #!V0 − E"2/!v2F" − k2y ,
matic of the scattering of Dirac electrons by a square potential.
Bottom: definition of the angles # and " used in the scattering
and finally in region III we have a transmitted wave only,
formalism in regions I, II, andFigure
III. 3 | Dirac fermions of graphene. a, Dependence of B F on carrier
$ %
concentration n (positive n corresponds to electrons;
to holes).
1
t negative
i!kxx+kyy"
7
!
!r"
=
e
,
!28"
III
to
N isH.
theCastro
number
b, Examples of fan diagrams used in our analysis
Neto et al. RMP (2009)
#2find
1
1
seiB# F.A.
with different minima
!K!k" =
!24" of oscillations. The lower and upper curves
#2 ±ei"k . associated
are for graphene (sample of Fig. 2a) and
filmsof
with
a Nature
=graphite
sgn!E − V
The
coefficients
witha s5-nm-thick
= sgn!E" and
!Novoselov
et0".al.
(2005)r, a,
respectively.
curves
extrapolate
to different
similar
value of
B F, not
andthe
t are
determined
from
the continuity of the wave
We further assume that the
scattering
does
mix the Noteb,that
namely
to 6,
N we
¼ 1/2
and N ¼function,
0. In graphene,
forthat
nthe
Lee
etallal.,
Science
(2008) has to
which curves
implies
wave function
In Fig.
depict
momenta around K and Korigins,
! points.
p with
to Nbarrier
¼ 1/2 (compare
7). This
indicates
a phase!Ishift
!x = 0of, y"
= !II!x = 0 , y" and !II!x
the
conditions
the scattering process due extrapolate
to the square
of width ref.obey
Novoselov
et
al, Science (2007)
respect
to
the
conventional
Landau
quantization
in
metals.
The
shift
is
Schrödinger
equation,
= D , y" = !III!x = D , y". Unlike the due
D.
14,20
. c, Examples
the behaviour
amplitude
Djfunction but not its
Berry’s phase
only needoftoSdHO
match
the wave
The wave function in thetodifferent
regions can
be writ- of we
$ %
plateaux in r xy and therefore occur at half-integer n; see Figs 2 and 4),
in agreement with theory14–17. Note, however, that from another
perspective the phase shift can be viewed as the direct manifestation
of Berry’s phase acquired by Dirac fermions moving in magnetic
field20,21.
Finally, we return to zero-field behaviour and discuss another
feature related to graphene’s relativistic-like spectrum. The spectrum
E = vanishing
±vF p,concentrations
vF = c/300
implies
of both carriers near the Dirac
point E ¼ 0 (Fig. 3e), which suggests that low-T resistivity of the
emerge only at expon
absence of localization,
explained qualitatively
free path l of charge carr
wavelength lF. Then, j
cannot be smaller than
is known to have failed
which disorder leads t
behaviour22,23. For 2D D
and, accordingly, Mott’
broad theoretical conse
should exhibit a minim
tion was not expected
value of ,e 2/ph, in dis
Thus, graphene exhi
for a 2D gas of particles
the Schro¨dinger equatio
Graphene: physical realization of fundamental phenomena in solid-state-physics
• Ultra-relativistic particles near EF
• Effective theory around EF
−ivF σ · ∇ψ(r) = Eψ(r)
• Cyclotron mass with carriers
√
Castro Neto
mc ∝
net al.: The electronic properties of graphene
116
Energy
!I!r" =
D
• Klein paradox
E
x
D
I
• High stiffness and high mobilityφ
III
II
θ
1
2 se
i#
ei!kxx+kyy" +
# $
r
2 se
1
i!$−#"
%
ei!−kxx+kyy" ,
!25"
V0
y
# $ %
1
with # = arctan!ky / kx", kx = kF cos #, ky = kF sin #, and kF
the Fermi momentum. In region II, we have
!II!r" =
φ
%
# $
a
1
2 s !e
i"
ei!qxx+kyy" +
# $
b
2 s !e
1
i!$−""
%
ei!−qxx+kyy" ,
!26"
x
with " = arctan!ky / qx" and
FIG. 6. !Color online" Klein tunneling in graphene. Top: sche!27"
qx = #!V0 − E"2/!v2F" − k2y ,
matic of the scattering of Dirac electrons by a square potential.
Bottom: definition of the angles # and " used in the scattering
and finally in region III we have a transmitted wave only,
formalism in regions I, II, andFigure
III. 3 | Dirac fermions of graphene. a, Dependence of B F on carrier
$ %
concentration n (positive n corresponds to electrons;
to holes).
1
t negative
i!kxx+kyy"
7
!
!r"
=
e
,
!28"
III
to
N isH.
theCastro
number
b, Examples of fan diagrams used in our analysis
Neto et al. RMP (2009)
#2find
1
1
seiB# F.A.
with different minima
!K!k" =
!24" of oscillations. The lower and upper curves
#2 ±ei"k . associated
are for graphene (sample of Fig. 2a) and
filmsof
with
a Nature
=graphite
sgn!E − V
The
coefficients
witha s5-nm-thick
= sgn!E" and
!Novoselov
et0".al.
(2005)r, a,
respectively.
curves
extrapolate
to different
similar
value of
B F, not
andthe
t are
determined
from
the continuity of the wave
We further assume that the
scattering
does
mix the Noteb,that
namely
to 6,
N we
¼ 1/2
and N ¼function,
0. In graphene,
forthat
nthe
Lee
etallal.,
Science
(2008) has to
which curves
implies
wave function
In Fig.
depict
momenta around K and Korigins,
! points.
p with
to Nbarrier
¼ 1/2 (compare
7). This
indicates
a phase!Ishift
!x = 0of, y"
= !II!x = 0 , y" and !II!x
the
conditions
the scattering process due extrapolate
to the square
of width ref.obey
Novoselov
et
al, Science (2007)
respect
to
the
conventional
Landau
quantization
in
metals.
The
shift
is
Schrödinger
equation,
= D , y" = !III!x = D , y". Unlike the due
D.
14,20
. c, Examples
the behaviour
amplitude
Djfunction but not its
Berry’s phase
only needoftoSdHO
match
the wave
The wave function in thetodifferent
regions can
be writ- of we
$ %
plateaux in r xy and therefore occur at half-integer n; see Figs 2 and 4),
in agreement with theory14–17. Note, however, that from another
perspective the phase shift can be viewed as the direct manifestation
of Berry’s phase acquired by Dirac fermions moving in magnetic
field20,21.
Finally, we return to zero-field behaviour and discuss another
feature related to graphene’s relativistic-like spectrum. The spectrum
E = vanishing
±vF p,concentrations
vF = c/300
implies
of both carriers near the Dirac
point E ¼ 0 (Fig. 3e), which suggests that low-T resistivity of the
emerge only at expon
absence of localization,
explained qualitatively
free path l of charge carr
wavelength lF. Then, j
cannot be smaller than
is known to have failed
which disorder leads t
behaviour22,23. For 2D D
and, accordingly, Mott’
broad theoretical conse
should exhibit a minim
tion was not expected
value of ,e 2/ph, in dis
Thus, graphene exhi
for a 2D gas of particles
the Schro¨dinger equatio
Graphene: physical realization of fundamental phenomena in solid-state-physics
!
• Ultra-relativistic particles near EF
• Effective theory around EF
−ivF σ · ∇ψ(r) = Eψ(r)
• Cyclotron mass with carriers
√
Castro Neto
mc ∝
net al.: The electronic properties of graphene
LETTERS
116
Energy
NATURE|Vol 438|10 November 2005
# $ %
1
• Klein paradox
# $
1
1
%
r
i!kxx+kyy"
i!−kxx+kyy"
The unusual response of massless fermions to a magnetic field is! !r" =
e
+
,
i#
i!$−#" e
D
highlighted
further by their behaviour in the high-field limit, at I
se
se
2
2
which SdHOs evolve into the quantum Hall effect (QHE). Figure 4
shows the Hall conductivity j xy of graphene plotted as a function of
!25"
electron and V
hole
0 concentrations in a constant B. Pronounced QHE
plateaux are visible, but
E they do not occur in the expected sequence
with # = arctan!ky / kx", kx = kF cos #, ky = kF sin #, and kF
On the contrary, the plateaux
j xy ¼ (4e 2/h)N, where N is integer.
x
correspond to half-integer n so that the first plateau occurs at
2e 2/hFermi momentum. In region II, we have
the
2
and the
D sequence is (4e /h)(N þ 1/2). The transition from the lowest
hole (n ¼ 21/2) to the lowest electron (n ¼ þ1/2) Landau level (LL)
1
1
a
b
in graphene
requires
III 22 the same number of carriers (Dn ¼ 4B/
II
i!qxx+kyy"
i!−qxx+kyy"
12
!
!r"
=
e
+
,
f 0 < 1.2 £ 10 cm ) as the transition between other nearest levels II
i"
i!$−"" e
s !e
s !e
2
2
(compare the distances between minima in r xx). This results in a
φ
ladderθof equidistant
steps in j xy that are not interrupted when
!26"
passing through zero. To emphasize this highly unusual behaviour,
Fig. 4 also shows j xy for a graphite film consisting of only two
x
graphene layers, in which the sequence of plateaux returns to with
normal " = arctan!k / q " and
y
x
and the first plateau is at 4e 2/h, as in the conventional QHE. We
FIG. 6. !Color online" Klein
tunneling
in graphene.
Top:graphene
sche- and its twoattribute
this qualitative
transition between
2
2
layer
counterpart
to
the
fact
that
fermions
in
the
latter exhibit a finiteqx = !V0 − E"2/!v " − k ,
!27"
matic of the scattering of Dirac electrons by a square potential.
F
y
mass near n < 0 and can no longer be described as massless Dirac
Bottom: definition of theparticles.
angles # and " used in the scattering
and finally in region III we have a transmitted wave only,
half-integer
QHE
in graphene
has recently
been suggested bya, Dependence of B on carrier
formalism in regions I, II, The
andFigure
III. 315,16
|
Dirac
fermions
of
graphene.
F
two theory groups , stimulated by our work on thin graphite films7
concentration
n (positive
n corresponds
to holes).
1
but unaware
of the present experiment.
The effect
is single-particleto electrons;
t negative
i!kxx+kyy"
7
!
!r"
=
e
,
!28"
and is b,
intimately
related
to
subtle
properties
of
massless
Dirac
III
to find
of fan diagrams used in our analysis
1 fermions, Examples
1
seiB# F. N is the number
in particular to the existence of both electron-like and
2
14–17
different
minima
of oscillations.
The lower and upper curves
!K!k" =
. associated
!24"
. The latter
can be
Landau stateswith
at exactly
zero energy
i"k hole-like
±e
2
viewedare
as a direct
consequence
of
the
Atiyah–Singer
index
theorem
for graphene (sample of Fig. 2a) and
filmsof
with
a
sgn!E − V
coefficients r, a,
witha s5-nm-thick
= sgn!E" and
! =graphite
0". The
that is important in quantum field theory and the theory of super18,19
respectively.
curves
extrapolate
to different
similar
B F,fermions,
.scattering
For 2Dvalue
masslessof
Dirac
the theorem
guarantees
strings
b,that
andthe
t are
determined
from
the continuity of the wave
We further assume that
the
does
not
mix
the Note
the existence
of
Landau
states
at
E
¼
0
by
relating
the
difference
in
namely
to 6,
N we
¼ 1/2
and N ¼function,
0. In graphene,
all nthe wave function has to
which curves
impliesforthat
In with
Fig.
depict
momenta around K and
Korigins,
! points.
the number
of such states
opposite
chiralities
to the total flux
p with
to Nbarrier
¼ 1/2
(compare
7). This
indicates
a phase!Ishift
system
(magnetic
field
can be
!x = 0of, y"
= !II!x = 0 , y" and !II!x
the
conditions
the scattering process through
due extrapolate
tothethe
square
ofinhomogeneous).
width ref.obey
where S(E) ¼ pk 2 is the area in k-space of the orbits at the Fermi
energy E(k) (ref. 13). If these expressions are combined with the
experimentally found dependences m c / n 1/2 and B F ¼ (h/4e)n it is
straightforward to show that S must be proportional to E 2, which
yields E / k. The data in Fig. 3 therefore unambiguously prove the
linear dispersion E ¼ h! kc * for both electrons and holes with a
common origin at E ¼ 0 (refs 11, 12). Furthermore, the above
equations also imply m c ¼ E/c 2* ¼ (h 2n/4pc 2* )1/2 and the best fit
to our data yields c * < 106 m s21, in agreement with band structure
y
calculations11,12. The semiclassical model employed is fully justified by a recent theory for graphene14, which shows that SdHO’s
I
amplitude can indeed be described by the above expression
T/sinh(2p2k BTm c/!heB) with m c ¼ E/c 2* . Therefore, even though
the linear spectrum of fermions in graphene (Fig. 3e) implies zero
φ
rest mass, their cyclotron mass is not zero.
• High stiffness and high mobility
#
$ %
#
$ %A. H. Castro Neto et al. RMP (2009)
#
$
%
#
• Ambipolar
#
$ %
Novoselov et al. Nature (2005)
Lee et al., Science (2008)
al, Science (2007)
respect to the conventional Landau quantization
in=metals.
The shiftthe
isetdue
D , y". Novoselov
Unlike
Schrödinger
equation,
= D , y" = !III!x
D.
14,20
. c, Examples
the behaviour
amplitude
Djfunction but not its
Berry’s phase
only needoftoSdHO
match
the wave
The wave function in thetodifferent
regions can
be writ- of we
plateaux in r xy and therefore occur at half-integer n; see Figs 2 and 4),
in agreement with theory14–17. Note, however, that from another
perspective the phase shift can be viewed as the direct manifestation
of Berry’s phase acquired by Dirac fermions moving in magnetic
field20,21.
Finally, we return to zero-field behaviour and discuss another
feature related to graphene’s relativistic-like spectrum. The spectrum
E = vanishing
±vF p,concentrations
vF = c/300
implies
of both carriers near the Dirac
point E ¼ 0 (Fig. 3e), which suggests that low-T resistivity of the
emerge only at expon
absence of localization,
explained qualitatively
free path l of charge carr
wavelength lF. Then, j
cannot be smaller than
is known to have failed
which disorder leads t
behaviour22,23. For 2D D
and, accordingly, Mott’
broad theoretical conse
should exhibit a minim
tion was not expected
value of ,e 2/ph, in dis
Thus, graphene exhi
for a 2D gas of particles
the Schro¨dinger equatio
Graphene: physical realization of fundamental phenomena in solid-state-physics
!
• Ultra-relativistic particles near EF
Room-Temperature Quantum Hall Effect in Graphene
−iv
σ , ·G.S.∇ψ(r)
=& Eψ(r)
theory
around
EF, H.L. Stormer , U. Zeitler
•K.S.Effective
Novoselov , Z. Jiang
, Y. Zhang
, S.V. Morozov
, J.C.F
Maan
Boebinger , P. Kim
A.K. Geim
1
2, 3
2
1
2
4
4
3
2*
1*
1Department
of Physics, University of Manchester, M13 9PL, Manchester, UK
of Physics, Columbia University, New York, New York 10027, USA
3National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
4High Filed Magnet Laboratory, Radboud University Nijmegen, 6525 ED Nijmegen, Netherlands
116
Castro Neto et al.: The electronic properties of graphene
2Department
√
monc a ∝
n scale, has been attracting intense
Cyclotron
mass
with
The •
quantum
Hall effect (QHE),
one example
of a carriers
quantum phenomenon that occurs
truly macroscopic
LETTERS
NATURE|Vol 438|10 November 2005
$ %
$
%
interest since its discovery in 1980 (1). The QHE is exclusive to two-dimensional (2D) metals and has elucidated many important aspects of
quantum physics and deepened
our understanding of interactingEnergy
systems. It has also led to the establishment of a new metrological standard,1
1
1
r
i!kxx+kyy"
where S(E) ¼ pk 2 is2 the area in k-space of the orbits at the Fermi
The unusual response of massless fermions to a magnetic field is! !r" =
e
+
ei!−kxx+kyy" ,
that
contains
only
fundamental
constants
of
the
electron
charge
e
and
the
Planck
constant
h
(2).
As
many
other
the resistance
quantum
h/e
i
#
i!
$
−
#
"
D
energy E(k) (ref. 13). If these expressions are combined with the
highlighted
further by their behaviour in the high-field limit, at I
#
#2 se
1/2 usually requires low temperatures T, typically below the boiling point of liquid helium (1).2 se
quantum phenomena,
observation
ofmthe
QHE
and B F ¼ (h/4e)n it is
experimentallythe
found
dependences
which SdHOs evolve into the quantum Hall effect (QHE). Figure 4
c / n
, whichsemiconductors
as a function
straightforward
show that S must
be by,
proportional
to E 2using
shows the Hallwith
conductivity
j xy of graphene
Efforts to extend
the QHEto temperature
range
for example,
small effective
massesplotted
of charge
carriersofhave so far
!25"
yields
E
/
k.
The
data
in
Fig.
3
therefore
unambiguously
prove
the
electron
and
hole
concentrations
in
a
constant
B.
Pronounced
QHE
V
failed to reach T above 30K (3,4). These efforts are driven by both innate desire to0 observe apparently fragile quantum phenomena
under
linear dispersion E ¼ h! kc * for both electrons and holes with a
plateaux are visible, but
not occur in the expected sequence
E they dotemperatures.
ambient conditions
to perform
metrology
at roomjor,¼at(4e
least,
liquid-nitrogen
More robust
quantum
2
# =states,
arctan!ky / kx", kx = kF cos #, ky = kF sin #, and kF
with
/h)N,
where N is integer.
On the contrary,
the plateaux
common and
originthe
at pragmatic
E ¼ 0 (refsneed
11, 12).
Furthermore,
the above
x
xy
2
2
2
1/2
2
implied by equations
their persistence
higher
would
added
to investigate
finer features
and,
possibly,
allow higher
(h n/4also
pc * ) provide
and the
best freedom
fit
/h
correspond
to half-integer
n so that of
thethe
firstQHE
plateau
occurs
at
2e
also implyto m
c ¼ E/cT,
* ¼
the
Fermi momentum. In region II, we have
6
21
m sshow
, in that
agreement
with band
structure
/h)(N þ
1/2). The
transition
from the
lowestlattice – the
to our
data yields
cHere,
andof
the
sequence
is (4e 2tightly
quantization
accuracy
(2).
in
graphene
–
a
single
layer
carbon
atoms
packed
in
a
honeycomb
crystal
* < 10we
y
D
11,12
The at
semiclassical
model employed
fully tojusticalculations
hole (n
¼ 21/2)nature
to the lowest
electron
(n ¼ þ1/2)
Landau level
(LL) behave as
QHE can be
observed .even
room temperature.
This isis due
the highly
unusual
of charge
carriers
in graphene,
which
1
1
a
b
fied by a recent theory for graphene14, which shows that SdHO’s
in graphene
requires
the same number of carriers (Dn ¼ 4B/
III
I
II
i!qxx+kyy"
12
22
massless relativistic
particles
(Dirac
fermions)
and
move
with
little
scattering
under
ambient
conditions
(5).
!
!r"
=
e
+
ei!−qxx+kyy" ,
amplitude can indeed be described by the above expression
f 0 < 1.2 £ 10 cm ) as the transition between other nearest levels II
i
"
i!
$
−
"
"
2
2
#2 s ! e
#2 s ! e
p kroom-T
heB)
withinmgraphene.
though V(compare
theclear
distances between minima in r xx). This results in a
T/sinh(2the
Figure 1 shows
QHE
The Hall even
conductivity
xy reveals
BTm c/!
c ¼ E/c * . Therefore,
φ
θ
that
are
not
interrupted
when
the
linear
spectrum
of
fermions
in
graphene
(Fig.
3e)
implies
zero
ladder
of
equidistant
steps
in
j
xy
conductivity Uxx
plateaux at 2e2/h for both electrons and holes, while the longitudinal
φ passing
!26"
through zero. To emphasize this highly unusual behaviour,
rest mass, their cyclotron mass is not zero.
approaches zero (<10:) exhibiting an activation energy 'E |600K Fig.
(Fig.4 also
1B).shows
The j xy for a graphite film consisting of only two
x
graphene
layers,
in which the sequence of plateaux returns to with
normal " = arctan!k / q " and
Fig. 1C).
The
quantization in Vxy is exact within an experimental accuracy of |0.2% (see
2
y
x
/h,
as
in
the
conventional
QHE.
We
and
the
first
plateau
is
at
4e
survival of the QHE to such high temperatures can be attributed to the large cyclotron
FIG. 6. !Color online" Klein
tunneling
in graphene.
Top:graphene
sche- and its twoattribute
this qualitative
transition between
quantization
into the fact that fermions in the latter exhibit a finiteq = !V − E"2/!v2 " − k2 ,
gaps !Z characteristic to Dirac fermions in graphene. Their energy layer
counterpart
!27"
x
0
matic of the scattering of Dirac electrons by a square potential.
F
y
mass
near
n
<
0
and
can
no
longer
be
described
as
massless
Dirac
6
magnetic field B is described by E N v FBottom:
is the #
Fermi
| 2e!BNdefinition
| where vFof|10
them/s
angles
and " used in the scattering
• Klein paradox
$ %
• High stiffness and high mobility
$
%
#
and finally in region III we have a transmitted wave only,
velocity and N an integer Landau level (LL)formalism
number (5). in
Theregions
expression
yields
an
energy3QHE
The
half-integer
in graphene
has recently
been suggested bya, Dependence of B on carrier
I, II,
and
III.
Figure
|
Dirac
fermions
of
graphene.
F
15,16
, stimulated by our work on thin graphite films7
twoLandau
theory groups
gap 'E |2800K at B =45T if the Fermi energy EF lies between the lowest
level
concentration
n (positive
n corresponds
to holes).
1
but unaware
of the present experiment.
The effect
is single-particleto electrons;
t negative
i!kxx+kyy"
N=0 and the first excited one N=±1 (inset, Fig. 1B). This implies that, in our
7
!
!r"
=
e
,
!28"
and experiments
is b,
intimately
related
to
subtle
properties
of
massless
Dirac
III
to
N isH.
theCastro
number
Examples
of fan diagrams used in our analysis
Neto et al. RMP (2009)
#2find
1 offermions,
1 a factor
seiB# F.A.
10. Importantly,
at room temperature, !Z exceeded the thermal energy kBT by
in particular to the existence of both electron-like and
14–17
different
minima
of oscillations.
The lower and upper curves
!k" = of other factors
. thatassociated
!24"
. The latter
can be
hole-like
Landau
at exactly
zero energy
in addition to the large cyclotron gap, there are!aKnumber
help
thestateswith
#2 ±ei"k viewed
as
a
direct
consequence
of
the
Atiyah–Singer
index
theorem
are
forforgraphene
(sample of Fig. 2a) and
filmsof
with
a Nature
=graphite
sgn!E − V
The
coefficients
witha s5-nm-thick
= sgn!E" and
QHE in graphene to survive to so high temperatures. First, graphene devices
allow
!Novoselov
et0".al.
(2005)r, a,
that is important
in quantum field theory and the theory of super13
-2
18,19
respectively.
curves
extrapolate
to different
similar
B F,fermions,
cm )further
with only aassume
single 2D that
subband
occupied,
very high carrier concentrations (up to 10 We
.scattering
For 2Dvalue
masslessof
Dirac
the theorem
guarantees
strings
b,that
andthe
t are
determined
from
the continuity of the wave
the
does
not
mix
the Note
existence
of
Landau
states
at
E
¼
0
by
relating
the
difference
in
which is essential to fully populate the lowest LL even in ultra-high B. Thisthe
is in
contrast
to
namely
to 6,
N we
¼ 1/2
and N ¼function,
0. In graphene,
forthat
nthe
Lee
etallal.,
Science
(2008) has to
which curves
implies
wave function
In with
Fig.
depict
momenta around K and
Korigins,
! points.
the number
of such states
opposite
chiralities
to the total flux
traditional 2D systems (for example, GaAs heterostructures) which are either
depopulated
p with
to N field
¼ 1/2
(compare
ref.obey
7). This
indicates
a phase!Ishift
throughextrapolate
system
(magnetic
can be
inhomogeneous).
!x = 0of, y"
= !II!x = 0 , y" and !II!x
the
conditions
the scattering
tothethe
already in moderate B or exhibit multiple subband
occupation process
leading to due
the reduction
ofsquare barrier of width
Novoselov
et
al, Science (2007)
respect
to
the
conventional
Landau
quantization
in
metals.
The
shift
is
Schrödinger
equation,
= D , y" = !III!x = D , y". Unlike the due
the effective energy gap to values well D.
below !Z . Second, the mobility P of Dirac
14,20
. c, Examples
the behaviour
amplitude
Djfunction but not its
Berry’s phase
only needoftoSdHO
match
the wave
wave function
in thetodifferent
regions can
be writ- of we
fermions in our samples does not changeThe
appreciably
from liquid-helium
to room
particles.
• Ambipolar
• Room Temperature QHE
$ %
$ %
...but in the long list of graphene remarkable properties, a fundamental block is missing:
SUPERCONDUCTIVITY
Vol 446 | 1 March 2007 | doi:10.1038/nature05555
...but in the long list of graphene remarkable properties, a fundamental block is missing:
LETTERS
NATURE | Vol 446 | 1 March 2007
mesoscopic samples, the QHE can be used to identify single layer
devices.
75
30
Cooling down the devices below the critical temperature of the
electrodes (Tc < 1.3 K) leads to proximity-induced superconductivity
in the graphene layer. A direct proof of induced superconductivity is
20
0
I
the observation of a Josephson supercurrent20. Figure 2a shows the
current–voltage (I–V) characteristics of a single layer device. The curVG = –50 V
10
rent flows without resistance (no voltage drop at finite current) below
VG = –7.5 V
–7 5
the critical current, Ic (what we actually measure is the switching
VG = +1 V
current; the intrinsic Ic may be higher20,21). In our devices, Ic ranged
0
– 250
0
250
–10
0
10
from ,10 nA to more than 800 nA (at high VG). Remarkably, we have
B (mT)
I (n A )
measured proximity-induced supercurrents in all the devices that we
c 1.2
d 60
100
tested (17 flakes in total, with several devices on some flakes), including four flakes that were unambiguously identified as single layer
2 ∆/3
30
graphene via QHE (the rest being probably two to four layers thick).
1
1
0
1.0
This clear observation demonstrates the robustness of the Josephson
0
effect in graphene junctions. (The data shown are representative of the
–100
general behaviour observed; the measurements shown in Figs 1, 2a, c,
=
4
.
5
G
H
z
f
0.8
– 30 P1/2 (a.u.)
d and 3 have been taken on the same device, whereas those in Figs 2b
and 4 correspond to a different single layer device, shown in Fig. 1a.)
∆ 2∆
– 60
To investigate further the superconducting properties of our
– 150
– 75
0
75
– 700
– 350
0
350
700
V (µV)
I (n A)
devices, we measured the dependence of Ic on magnetic field
(Fig. 2b). The critical current exhibits an oscillatory Fraunhofer-like
density only in the bottom one or two layers. For single Figure
layers,
the
2 | Josephson effect in graphene. a, Voltage, V, versus current bias,
pattern, with at least six visible side lobes, which is indicative of a
I, characteristics
at various VG, showing a modulation of the critical current.
position of the resistance maximum corresponds to the gate
voltage
Vol 446 | 1 March 2007 | doi:10.1038/nature05555
uniform supercurrent density distribution22. The periodicity of the
Inset, current bias sweeps from negative to positive (red) and vice versa
wethat the asymmetry in the main panel is due to hystereticPUBLISHED
at which the Fermi energy is located at the Dirac point, V(blue),
oscillations
is in the ideal case expected to be equal to a flux quantum
showing
D, and
ONLINE: 6 FEBRUARY 2011 | DOI: 10.1038/NPHYS1911
W
divided
by the junction area. The area that corresponds to the
behaviour
(the
retrapping
current
is
smaller
than
the
switching
current,
as
o
the
typically find that jVDj , 20 V. We unambiguously determine
is typical for an underdamped junction). b, Colour-scale representation of
2.5 6 0.5 mT period is 0.8 6 0.2 mm2, which is in good agreement
dV/dI(I,B)
at T 5 30 mK (yellow-orange is zero, that is, the supercurrent
single layer character of a device by quantum Hall effect
(QHE)
with the measured device area (0.7 6 0.2 mm2) determined from
region, and red corresponds to finite dV/dI). The critical current exhibits a
the atomic force microscope image. Applying a radio-frequency field
measurements. Because the superconducting proximity
series ofeffect
oscillations described by a Fraunhofer-like pattern. c, Differential
to the sample results in the observation of quantized voltage steps,
resistance,
dV/dI, versus V, showing multiple Andreev reflection dips below
requires two closely spaced electrodes, we can only perform
magneknown as Shapiro steps, in the I–V characteristics20. The voltage
the superconducting energy gap. The dips in dV/dI occur at values of
steps, of amplitude "v/2e (v is the microwave frequency), are a
toconductance measurements in a two terminal configuration.
In n is an integer number. d, a.c. Josephson effect. The
V 5 2D/en, where
manifestation of the a.c. Josephson effect, and are evident in
Shapiro steps in the I–V characteristics appear when the sample is irradiated
general, the conductance, G, measured in this way is a mixture
of In the example, we applied 4.5 GHz microwaves, resulting
Fig. 2d.Taylor
The induced
superconductivity
itselfUchoa,
also at finite
with microwaves.
Travis
Dirks,
L. Hughes,
Siddharthamanifests
Lal † , Bruno
Yung-Fu Chen, Cesar Chialvo,
1
1
1
1
Hubert
B.
Heersche
*,
Pablo
Jarillo-Herrero
*,
Jeroen
B.
Oostinga
,
Lieven
M.
K.
Vandersypen
in
9.3
mV
voltage
steps.
Inset,
colour-scale
plot
showing
the
characteristic
bias
in
the
form
of
subgap
structure
in
the
differential
resistance
due
ongitudinal and Hall 1 signals, but at high fields G < jGHallj (this
†
Paul M.toGoldbart
and Nadya
Mason
23 *
microwave amplitude (P1/2; a.u., arbitrary units) dependence of the a.c.
& Alberto F. Morpurgo
multiple Andreev
reflections
, as shown in Fig. 2c. This subgap
19
measureapproximation is exact at the Hall plateaus ). Indeed, theJosephson
effect (orange and red correspond respectively to zero and finite
structure consists of a series of minima at source–drain voltages
When a low-energy
is incident
on anenables
interfaceus
between
Fig. 2a the
(discussed
dV/dI).
V 5 2D/enelectron
(n 5 1,2…),
which
to determine
super-below). The devices consist of normal-metal end
ment of G versus VG at B 5 10 T shows clearly identifiable Hall plaa metal and superconductor, it causes the injection of a contacts (2 nm/50 nm Cr/Au bilayer) and SC tunnel probes18,19
conducting
D. We findand
D 5the
125generation
meV, as expected
ournm
Ti/Al
in the 2007
bottom one or two layers. For single layers, the
recently discovered form of graphite
only one
Cooper pair
into 18the gap,
superconductor
of (200for
nm/30
Pb/In bilayer) (see Methods for details). The
| Voldensity
446 | 1only
March
LETTERS
(Fig.
1d),
characteristic
of the
teaus atGraphene—a
half-integer
multiples of 4e2/hNATURE
bilayers
. For
. 2D/e,
the differential
resistancetunnelling
returns resistances
to the through the SC probes were typically 10–100
position of the resistance maximum corresponds to the gate voltage
atomic layer thick1—constitutes a new model system in condensed
a hole that
reflects
backVinto
the metal—a
process known
a
b
V (µV)
I (nA)
V
SUPERCONDUCTIVITY
Some experimental discoveries
V (µV)
I (nA)
ostinga , Lieven M. K. Vandersypen
dV/dI (kΩ)
phene
LETTERS
LETTERS
Transport through Andreev bound states in a
graphene quantum dot
Bipolar supercurrent in graphene
LETTERS
2,3
at which thethat,
Fermi energy
matter
physics, layer
because itgraphene
is the first material
in which
charge
. This
demonstrates
evenis located
in at the Dirac point, V
QHE in
single
dI/dV (µS)
IcRn (µV)
I (nA)
I (nA)
µV)
V (µV)
(kΩ)
T = 30 mK
G (mS)
2.0
as Andreev reflection. In confined geometries, this process times larger than the end-to-end resistances.
In all samples, the charge neutral point, as seen in the end-to-end
can give rise to discrete Andreev bound states (ABS),
which can b
enable transport of supercurrents through non- conductance versus back-gate measurements, shows a large offset
2
mesoscopic
the
QHE
canbeen
beproposed
useda to toidentify
single
the positive-gate
sidelayer
(see Fig. 1c,d for samples A and B, respecsuperconducting samples,
materials and
have
recently
120realizing solid-state qubits1–3 . Here, we report
tively). For example, sample A shows an asymmetric cone around
as a2means of
devices.
4
transport measurements of sharp, gate-tunable ABS formed in the Dirac point at Vg ∼ +17.5 V. The asymmetry and direction of
are consistent with
work-function mismatch (�W ) at the
a superconductor–quantum
(QD)–normal
system
Cooling down thedotdevices
below
therealized
criticaloffset
temperature
of athe
on an exfoliated graphene sheet. The QD is formed in graphene Cr/Au–graphene interface, which results in a transfer of charge that
electrodes
(Tc < 1.3 K)contact
leadsastoaproximity-induced
3 superconductivity
the surface potentials4 ; similar charge transfer effects have
beneath a superconducting
result of a work- equalizes
80 4,5 . Individual ABS form when the discrete
previously
been predicted4 and
function
mismatch
in
the
graphene
layer.
A
direct
proof
of
induced
superconductivity
isobserved5 . Because of its low density
1
QD levels are proximity-coupled to the superconducting20 of states, graphene is efficiently doped by this charge transfer. The
the
observation
of density
a Josephson
Figure
2a shows
2
interfacethe
of the end contacts is dominated by the
contact.
Owing to the low
of states ofsupercurrent
graphene and the . Cr/Au–graphene
work function
of the
Au. cur(This is expected because the Cr layer is so
sensitivity of the QD levels
to an
applied gate voltage,
the
ABS layer
current–voltage
(I–V)
characteristics
of
a
single
device.
The
VG = –50 V
spectra are narrow
be continuously tuned down to zero thin as to be discontinuous. It is also similar to the Ti/Au–graphene
10
0
40 and can
a
b
rent
flows
resistance
(no voltage drop atcontacts
finite
current)
VG = –7.5
1 µmV
measured in below
ref. 5.) Therefore, �W ranges from 0.14 to
energy
by thewithout
gate voltage.
1
0critical
1.04
eV
(see
Methods
for calculation). The positive �W indicates
Although
signatures
of Andreev
reflection
and
bound
states
in
the
current,
I
(what
we
actually
measure
is
the
switching
c
VG = +1 V
conductance have been widely reported6 , it has been difficult
to hole (that is, p-type) doping of the graphene under the end leads.
VD
20,21
current;
the
intrinsic
be higher
). In our0The
devices,
Ic ranged
c may
0
superconducting
tunnel probe also leads to a local doping of
directly probe
individual
ABS. IIn
superconductor
(SC)–graphene
0
250
–10
0
10
– 1 00
graphene.
However,
for
the
Pb–graphene
interface,
�W
structures,
most
previous
work
has
focused
on
the
nature
of
0
from
,10
nA
to
more
than
800
nA
(at
high
V
).
Remarkably,
we
have
0
2
4 is −1.15
6 to
¬6
¬4
¬2
G
20the supercurrent in well-coupled
– 40
– 2Josephson
0
0junctions7–10
20 . SC–QD
– 4 0 B (mT
0
– 2) 0
I (n A )
−0.25 eV and is negative, implying local electron (n-type) doping.
V
(mV)
V
(
V
)
V G (V )
measured
proximity-induced
supercurrents
in all
the devices
g
As illustrated
in Figs that
1b andwe
2a,tunnel
the �W doping generates a potential
hybrids in graphene
have not been studied,
although recent work
d 60
11–14
15
well
underneath
the
SC tunnel probe, which acts as a confining
has predicted
and demonstrated
that(V
ABS) is
can
bedevices
isolated
by
100
tested
(17
flakes
in
total,
several
on
some
flakes),
includb
Figure 3 | Bipolar supercurrent transistor
behaviour and finite
finite.
Note
that
the
critical
current
is
supercurrent
at the
Diracwith
point
D
0.8
for a QD—that is, a pn junction. Quantum dots formed by
coupling them to discrete QD energy levels. However,
the ABS peaks potential
asymmetry isas
notsingle layer
notflakes
symmetric
withwere
respect unambiguously
to VD. The origin of this
supercurrent at the Dirac point. a, Colour-scale plot of dV/dI(VG,I). Yellow
ing
four
that
have been observed in carbon nanotubes20 , by means
in previous
SC–QD experiments
were strongly
broadened, eitheridentified
by pn junctions
B=1
0
T
means
zero,
that
is,
the
supercurrent
region,
and
finite
dV/dI
increases
via
known,
but
a
similar
asymmetry
is
seen
in
the
normal
state
conductance
15
16
2 ∆/3
3
0
the
large
lead
density
of
states
of
work-function
doping,
and in graphene21 , where Klein tunnelling
or
by
the
lack
of
a
tunnel
barrier
.
In
d
graphene
via
QHE
(the rest
being
probably
two
to four
layers
thick).
c
b, Letter,
Product
of the
critical
current times
the normal
state
orange
swept
and
(blue
curve).
0 negative to positive values,
= 10 Tfrom
B =to
35dark
mT red. The current is B
2.0
the
work
described
in
this
sharp
subgap
conductance
peaks
through
smooth
barriers
can
lead to the confinement of carriers22,23 .
0.6
.5 the hysteresis associated with an underdamped
normal state resistance
is measured
at
resistance
versus into
VG. The
is asymmetric owing2to
This
clear
observation
the
robustness
ofzero
are obtained
by
tunnelling
ademonstrates
proximity-coupled
QD formed
The QD
isthe
alsoJosephson
proximity coupled to the superconducting lead.
T = 30 mK
D, and we
typically find that jVDj , 20 V. We unambiguously determine the
12
–2
a Hall effect (QHE)n (10 cm )
single layer character of a device by quantum
–2
0
measurements. Because the superconducting proximity
effect
200
b we can only
requires two closely spaced electrodes,
perform magnetoconductance measurements in a two terminal configuration. In
30this way is a mixture of
general, the conductance, G, measured in
longitudinal and Hall signals, but at high fields G < jGHallj (this
approximation is exact at the Hall plateaus19). Indeed,
100 the measurement of G versus VG at B 5 10 T shows clearly
20 identifiable Hall pla2
teaus
I at half-integer multiples of 4e /h (Fig. 1d), characteristic of the
QHE in single layer graphene2,3. This demonstrates that, even in
I (nA)
c
V
a
carriers behave as massless chiral relativistic particles. The anomalous quantization of the Hall conductance2,3, which is now understood theoretically4,5, is one of the experimental signatures of the
peculiar transport properties of relativistic electrons
b ain graphene.
Other unusual phenomena, like 1
theµm
finite conductivity 7
of5order
4e2/h (where e is the electron charge and h is Planck’s constant) at
the charge neutrality (or Dirac) point2, have come as a surprise and
remain to be explained5–13. Here we experimentally study the
Josephson effect14 in mesoscopic junctions consisting of a gra0
phene layer contacted by two closely spaced superconducting
electrodes15. The charge density in the graphene layer can be controlled by means of a gate electrode. We observe a supercurrent
that, depending on the gate voltage, is carried by either electrons in
the conduction band or by holes in the valence band. More impor–7 5 of
tantly, we find that not only the normal state conductance
graphene is finite, but also a finite supercurrent can flow at zero
charge density. Our observations shed light on the special– role
2 5 0of
time reversal symmetry in graphene, and demonstrate phase
coherent electronic transport at the Dirac point.
Owing to the Josephson effect14,16, a supercurrent can flow through
c 1.2
a normal conductor placed between two closely spaced superconducting electrodes. For this to happen, transport indthe normal conductor must be phase coherent and time reversal symmetry (TRS)
B = 35 mT
must be present. In graphene, the Josephson effect can be investigated
2 .5
in the ‘relativistic’ regime15, where the supercurrent is carried
1.0 by
Superconductivity: Is it expected?
Superconductivity: Is it expected?
The total electron-phonon coupling is :
λ=
N (0)D 2
M ω2
Superconductivity: Is it expected?
The total electron-phonon coupling is :
N (0) is vanishingly small, however it can be varied by rigid doping
but it increases very slowly
λ=
N (0)D 2
M ω2
Superconductivity: Is it expected?
The total electron-phonon coupling is :
λ=
N (0) is vanishingly small, however it can be varied by rigid doping
but it increases very slowly
The deformation potential, D, for in-plane modes is high, but 1/ω2 is low
N (0)D 2
M ω2
Superconductivity: Is it expected?
The total electron-phonon coupling is :
λ=
N (0) is vanishingly small, however it can be varied by rigid doping
but it increases very slowly
The deformation potential, D, for in-plane modes is high, but 1/ω2 is low
By symmetry out-of-plane modes do not couple with πelectrons
N (0)D 2
M ω2
Superconductivity in Graphite
Superconductivity in Graphite
Superconductivity in Graphite
The interlayer state
Superconductivity in Graphite
The interlayer state
The electron-phonon coupling:
Cz coupling
M. Calandra and F. Mauri, Phys. Rev. Lett (2005)
Superconductivity in Graphite
The interlayer state
The Cambridge criterion
M. S. Dresselhaus and G. Dresselhaus, Adv. in Phys. (2002)
G. Csanyi et al, Nat. Phys. (2005)
I. MazinPhilosophical
and A. V. Balatsky
Mag. Lett. (2010)
MagazinePhil.
Letters
735
The electron-phonon coupling:
Cz coupling
M. Calandra and F. Mauri, Phys. Rev. Lett (2005)
4. Correlation between the position of nearly free band, brought down to chemical
The interlayer state
It is a strongly 3D dispersive state living between carbon layers*
* M. Posternak et al. Phys. Rev. Lett. (1983)
L. Boeri et al. Phys. Rev. B (2007)
10
5
Energy (eV)
0
-5
-10
-15
-20
K
M
K
K
The interlayer state
It is a strongly 3D dispersive state living between carbon layers*
* M. Posternak et al. Phys. Rev. Lett. (1983)
L. Boeri et al. Phys. Rev. B (2007)
But it exists even in multiwall nanotubes, fullerenes, BN, MgB2 and.........
10
10
5
5
0
0
Energy (eV)
Energy (eV)
.......graphene
-5
-5
-10
-10
-15
-15
-20
Γ≡Γ’
Γ≡K’
-20
K
M
K
K
K
M
K
K
Use the interlayer state: put it at EF
Interlayer state is a truly 2D free-electron state in xy direction, ψ(x, y) = ei(kx x+ky y)
However, in the z direction is not exactly “free”.
a
Kronig-Penney model of bulk GIC
a
En =
2 �2 π 2
n 2m a2
The interlayer state energy decreases removing the quantum confinement along z
a→∞
that is Metal adatoms on Graphene
% dpotential
!k +(r,
q,k"
lascreened
namely,
termined
intothe
so-calle
0) in the
golden
rule,
ij!k,k + q"dV
jiSCF
tions
is
identical
the
numbe
G.
Phonon
frequencies
interpolation
over
'q(Fermi
full width
half
own equation for the phonon linewidth
Specifically,
if
the
band-structure
is
!T
"
N
this
work we
choose
to
with
well
controlled
error.
k ph
group
of
bands,
then
the
mat
phonon
momentum:
Practical
imple
inN2kEq.
29,
we
obtain
Allen
ng
ωqνa!FWHM",
ximum
that
is,
f
!T
"
−
f
!T
"
posite group of bands and the number
(T )
ki ph
k+qj phdoped Graphene
Metal
!
Wannier
functions
(ML
conditions
illustrated
in Refs. 29,30, the
− certain'
4πωqν
trix.
On
the
contrary
if
the
d
y,= Under
ν
2
The
practical
implementation
of
the
abo
tions
is
identical
to
the
number
of
band
Nk!Tph" |gkijnm (k, k#ki+−q)|
#k+qjδ(εkn )δ(εk+qm
)
nierization
schemes
are
phonon
linewidth
can
be
reduced
to:
Nk!T"
in
a
larger
manifold,
a
prelim
N
(T
)
mulation
for
the
phonon
frequencies
proce
k
group
of
bands,
then
the
matrix
U
(k
We performed
first-principles
Density Functional Theory
calculation
of structural,
electronic,
mn
k,n,m
s 4#
r
Nk (T ) Fermi
(
matrices
Ulinear-response
(k) are m
ob
2
nm
!k
+
q,k"
!23"
cedure
from
the
manifold
! 'q(% d=ij!k,k + q"d
ji
!1"
Perform
a
standard
(g
!k,k
+
q"(
%
nmNk (T )
trix.
On the contrary
if the
desired
bag
(30)
ν
2
!
N
!T"
of
Refs.
14,28,
which
dynamical
andk
properties
of)metal adatoms
on Graphene
|gnm (k,
+ k,m,n
q)|
4πωδ(ε
k superconducting
the
square
matrix
Umna(k)
is p
given
phonon
momentum
q using
Nk!T
k+qm
qν kn )δ(ε
ν
2
2,29
heAllen-formula
following γ
definition
of
the
deformation
potential
in
a
larger
manifold,
a
preliminary
dise
˜qν = is widely used,
|gnmand
(k, krepresents
+ q)| ×and
a smearing calization
of
the
final
W
for
the
electronic
inte
T
k,n,m
The
deformation
potential
ph
N
(T
)
k
element: of"the
!f knphonon-linewidth
− f k+qm"k,m,n
%!&k+qm − &due
!q(electron", cedure
!28"
from
the
manifold will
must
be
pe
estimation
kn − to
quirement
be
essen
!q
,
0
,
T
".
C
The electron-phonon matrix element
(30)
sr
ph
q,
k) (seeU Eq.(k)24)
are calcu2
the
square
matrix
is
obtained
n
effects,
the
absence
of
anharmonic
effects
and
mn
2,29 inδ(%
!2"
In
order
to
perform
the secon
of
the
electron-phonon
)δ(%
−
%
−
ω
)
(29)
&
v
kn
k+qm
kn
qν
SCF
widely
used,
a
mula
sand represents
( ds is
s
scheme.
In
this
calculation
q,k"
qm&
, kprocesses.
+ q"
=!
%=s(k
eq(+In
·d
!k + q&kn)
, k"the
/ -2M
ere gnm!k
The
potential
matrix-e
q Specifically,
mn!k
scattering
addition,
Allen-formula
s!q!24"
(.
, ! , T0" #Eq.
!23"$,
calculateif
deformat
"sr!qdeformation
mn
the ba
&
u
of
the
phonon-linewidth
due
to
electronfor
the
deformation
potenti
qs
absence
of the electron-electron
aobtain
Eq. interaction
29, by
weinelastic
Allen
neglecting
ωqν asin measured
element
of24)
Eq.are
!24"
on of
thebands
electron
q,similar
k)trix(see
Eq.
calculated
usin
snInFinally,
the
phonon-linewidth,
posite
group
a
s
the
absence
of
anharmonic
effects
and
k+q
d
(k,
k),
as
explained
in
se
29
here
is the
periodic
the
Bloch
wave
func-external
pression
involving
only part
the of
derivative
of
the bare
k points
using
a the
smen
composed
ofthis
Ntions
formula
namely,
scheme.
Inmn
particula
k!Tcalculation
ph" is
on
or&kn)
X-ray
measurements,
to
the
electron-phonon
identical
to
ik·r
processes.
In&kn)
addition,
the
Allen-formula
*Nk andusing
It is wave
crucial
to note
that
t
Phonon
linewidth
'
)
=
e
/
u
is
the
Fourier
trans.e.,
&
!3"
Generate
functions
&kn)
and
en
ential
is
easily
obtained
Fermi
golden
rule.
In
the
kn
qs
k
for
the
deformation
potential
at
zon
ng as:
group
of bands,
then
th
s
N
(T
)
on-linewidth,
as
measured
by
inelastic
|k
+
qm#
entering
in
d
(k
+
k
.
The
integration
in
Eq.
of
the
phonon
displacement
u
!T
"
electron-momentum
k-points
N
s denser
Ls
sence of electron-electron
interaction
our
formalism
jusmn
k
0
!
dmn (k, k), as explained
in
sec.
III B. if
4πωqν
trix.
On
the
contrary
ν
2
γ
is
understood
to
be
on
the
unit
cell.
The
quantity
vbare
y the
measurements,
to the
the|g
electron-phonon
!4")Perform
a second nonself-consistent
same wavefunctions
used for
qν (k, k +
=
q)|
δ(ε
)δ(ε
γqν
SCF
replacement
of
derivative
of
the
external
kn
k+qm
nm
Itdeformation-potential
is crucial to
thatmanifold,
the
period
λthe
,
(31)
in note
a larger
ap
qν
Nofk (T
) =
eriodic
part
static
self-consistent
potential,
2
matrix
element
on
dure.
If
this
is
not
the
case,
s
2πω
N
k,n,m
ential by the derivative
of
the
static
screened
potential
s
s
qν
entering
in
d
(k
+
q,
k)
hav
cedure
from
the
manifo
mn
y, VSCF!r" = vSCF!r"eiq·r. This static self-consistent po-|k +Nqm#
!T
"
k-points
grid
using
smearing
T
k 0(30)
0 and
appear
in
the
|kn#’s
(i.e.
a
!r
,
0"
in
the
Fermi
golden
rule
with
a
well-controlled
CF
γqν standard
same
wavefunctions
used
for
the
Wan
the
square
matrix
U
(
is
calculated
using
linear
response
at
tem!T
"
grid
to
obtain
the
first
the
N
mn
2,29
k
ph
N
is
the
electronic
density
of
states
per
spin
at
λ
,
(31)
=
routine or other c
is widely used, and representsnalization
a
sThe Allen-formula
or.
2
.
re Tphqν
deformation
dure.
the case,
spuriouspot
(u
!qthis
, ! , Tis
"srIf
2πω
N
0". notThe
s
qν
rmi
energy.
good certain
estimation
of the illustrated
phonon-linewidth
to electronUnder
conditions
in Refs. due
31 and
32, the localization properties o
q, k) (see
24) are
appear in the |kn#’s
(i.e. Eq.
a phase
add
hphonon
this
definition,
the
isotropic
Eliashberg-function
phononlinewidth
effects, in
absence
of anharmonic effects and
real space are completely lo
canthe
reduced
electronic
density
ofbestates
pertospin at 165111-5
scheme.
In computati
this calcu
nalization
routine
or
other
other scattering processes. In addition, the Allen-formula
condition
is
to
use
a
uniform
y. The Eliashberg functionN !T"
for
the
deformation
po
the
localization
properties
of
the
Wan
!
!
relates the phonon-linewidth,
as measured by inelastic
k
k
+
q
=
!
s k + G with k sti
4
#!
1
d
(k, k),
as
explained
nition,
q( Eliashberg-function
real
space
are
completely
lost.
A wa
2 the isotropic
(
ω
2
1.04(1+λ)
mn
log
Neutron
X-ray
measurements,
to
the
electron-phonon
˜'or
α F (ω)
=
λ
ω
δ(ω
−
ω
)
(32)
=
(g
!k,k
+
q"(
grid.
In
this
way
|k+qn#
can
qνnmqν
qν
%
T
=
exp{−
}
q(
∗
c
It
is
crucial
to
note
1.20
λ−µ
(1+0.62λ)
2N
condition is to use a uniform grid cent
q qν
k!T"
k,m,n
coupling as: N
ply
by a phase
sd
! multiplying
!entering
|k
+
qm#
in
d
k + q = k + G with k still belongim
1 !
translation and no additional
Metal doped Graphene
• Ca is the first choice (CaC6 has Tc=11.5 K)
• Ca adsorbs in the hollow site at 2.24 Å
•
IL at EF ?
Metal doped Graphene
• Ca is the first choice (CaC6 has Tc=11.5 K)
• Ca adsorbs in the hollow site at 2.24 Å
•
IL at EF ?
Monolayer
2
2
1
1
Energy (eV)
Energy (eV)
Bulk
0
-1
-2!K
0
-1
M
K
!K
-2!K
M
K
!K
Ca on graphene:
The interlayer state
CaC6 mono
CaC6 bulk
|sK!IL|
2
0.6
0.4
0.2
00
1
2
3
4
5
6
7
8
z distance (Å)
|sK!IL|
2
0.6
0.4
LiC6 mono
LiC6 bulk
0.2
The IL moves away from graphene, reducing the electron-phonon coupling
00
7
1
2
3
4
8
5
6
with Cz modes
z distance (Å)
Ca on graphene: electron-phonon coupling
1600
1400
_ F(t"#$h%t"
1000
800
!
-1
Frequency (cm )
1200
600
1.5
1
0.5
0
400
CaC6
0
1000
500
-1
Frequency (cm )
0!K
M
K
!K
_ F(t"#$h%t"
200
1.5
1
LiC6
1500
!
_ F Bulk
!
_ F Monolayer
h%t" Bulk
h%t" Monolayer
!
The coupling with Cz modes, present in the bulk compound, is small in the monolayer case
0.5
The total electron-phonon coupling is λ= 0.40 with an extimated Tc= 1.4 K ( λ= 0.70 and Tc= 11.5 K)
0
0
1000
500
1500
-1
Frequency (cm )
Removal of quantum confinement reduces λ: Bulk better than Film
Lithium doped Graphene
• There is at least one case among GICs that can be further explored, stage-1 Lithium GIC, LiC6.
• IL state is completely empty:
the strong quantum confinement prevents its occupation
• It is not superconducting
Removal of confinement along c direction should bring the IL at the Fermi level
Energy (eV)
2
1
0
-1
-2!K
M
K
!K
Lithium doped Graphene
• There is at least one case among GICs that can be further explored, stage-1 Lithium GIC, LiC6.
• IL state is completely empty:
the strong quantum confinement prevents its occupation
• It is not superconducting
2
2
1
1
Energy (eV)
Energy (eV)
Removal of confinement along c direction should bring the IL at the Fermi level
0
0
-1
-1
-2!K
-2!K
M
K
!K
M
K
!K
Lithium doped Graphene
400
200
0
!K
M
K
!K
Phonon DOS
2
Energy (eV)
-1
Frequency (cm )
600
1
• IL state at the Fermi level
0.8
0.6
0
M
K
• Softening of Li and Cz modes
0.4
0.2
0
-1
• Increase of DOS at EF
-0.2
-0.4
-2!K
-0.6
-0.8
M
K
!K
!
_ F(t"#$h%t"
0
Lithium doped Graphene
1000
500
Electron-phonon coupling
-1
Frequency (cm )
0
1.5
LiC6
1500
!
_ F Bulk
!
_ F Monolayer
h%t" Bulk
h%t" Monolayer
1
0.5
0
0
1000
500
-1
Frequency (cm )
1500
• Bulk LiC6 is not superconducting (no interlayer state)
•The interlayer state allows the coupling with “dormant” Cz modes
• Increases the DOS at the Fermi level
• Allows an additional intra-band and 2 inter-band scattering channels
• Increases the total electron-phonon coupling to λ= 0.61 and
Tc= 8.1 K
Lithium on both sides of Graphene
2
1600
Energy (eV)
1
-1
Frequency (cm )
1200
800
0
-1
-2
-3
-4
!K
M
K
DOS (states/eV)
400
λ= 1.0 and Tc= 18 K
0
!K
M
K
!K 0.5 1 1.5
"
!_ F(t#$!h%t#
!K 2 4
Conclusions
• Rigid doping can not induce electron-phonon driven superconductivity
• Metal adatoms on graphene promotes the interlayer state at the Fermi level
• The coupling depends on the particular adatom, very different from the bulk counterpart
• Lithium on graphene has a sizable electron-phonon coupling
• and can induce superconductivity.
“How to make graphene superconducting”
G. Profeta, M. Calandra and F. Mauri
cond-mat arXiv: xxx.xxxx
Thank you
for your attention