Physics in 2D Materials
Taro WAKAMURA (Université Paris-Saclay)
Lecture 2
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Today’s Topics
Lecture 2: Graphene 2
2.1 Graphene superlattices
2.2 Superconductivity and graphene
2.3 Graphene spintronics (Maybe in the next lecture)
1.4 (Integer) quantum Hall effect in graphene (from Lecture 1)
Introduction of quantum Hall effect
Relation between the resistivity tensor and the conductivity tensor
When the Fermi level is inside the gap (between LLs), no carriers are excited.
Summary
When the Fermi level is in between LLs, is quantized as a integer multiple of
At the same time, the longitudinal conductivity and resistivity ( , ) are both 0.
Introduction of quantum Hall effect
Quantum Hall effect in conventional 2D electron gas
Resis
tance
Magnetic field
Hall resistivity plateau
Zero
Hall resistivity is quantized as
= 25.8 kW
von Klitzing constant
Quantum Hall effect in grapheneConventional 2DEG vs Graphene
Nth Landau level
Hall conductivity
C 2DEG Mono graphene Bi graphene
Number of degeneracy (spin 2 x valley 2)Spin degeneracy
Quantum Hall effect in grapheneConventional 2DEG vs Graphene
Nth Landau level
Hall conductivity
C 2DEG Mono graphene Bi graphene
Landau level
Y. Barlas et al., Nanotechnology 23, 052001 (2012).
Quantum Hall effect in graphene
Conventional 2DEG Mono-graphene Bi-graphene
Conventional 2DEG Landau level does not exist at zero energy
Monolayer graphene Four-fold Landau level exists at zero energy
Bilayer graphene Eight-fold Landau level exists at zero energy
Landau level
Y. Barlas et al., Nanotechnology 23, 052001 (2012).
Quantum Hall effect in grapheneQuantum Hall effect for monolayer graphene
Clear plateau and zero longitudinal resistance are
observed at B = 14 T, T = 4 K (right figure)
Energy for Nth Landau level:
Energy gap DE between N=0 and N=1: 600 K at 29 T
Quantum Hall effect at room temperature!
K. S. Novoselov et al., Nature 438, 197 (2005).
Quantum Hall effect in grapheneQuantum Hall effect for bilayer graphene
is quantized as a integer multiple of
Due to the additional orbital degeneracy, there is a
step around the zero filling factor
Zero longitudinal resistivity is also observed as a function
of a carrier density (n) with a fixed magnetic field.
Double with of the central peak is a signature of the eight-
fold degeneracy of the Landau level
K. S. Novoselov et al., Nat. Phys. 2, 177 (2006).
Fractional Quantum Hall effect in graphene
n=1/3
K. I. Bolotin et al., Nature 462, 196 (2009).C. R. Dean et al., Nat. Phys. 7, 693 (2011).
Graphene superlattices
How to enhance mobility?
Hexagonal boron nitride as a substrate for graphene
Mechanically-exfoliated graphene is transferred
onto mechanically-exfoliated h-BN
High mobility (~ 60000 cm2V-1s-1) are observed!
C. R. Dean et al., Nat. Nanotech. 5, 722 (2010).
Graphene/h-BN superlattices
hexagonal boron-nitride (h-BN):
Ideal substrate for graphene (flat & small charge
inhomogeneity)
Small lattice mismatch with graphene: 1.8 %
When two layers interact with each other,
what happens?
The answer is...
Moiré pattern
Modulation of the band structure
Modulation of the density of states (DOS)...
Graphene/h-BN superlattices
Graphene in a 1D potential
Dirac electrons in 1D potential
The group velocity vk is strongly modulated
when an electron propagates along the
potential boundary (qk:the angle between vk and x)
The group velocity is not affected when
an electron moves normal to the bound-
ary (x-direction)
Owing to the chirality
vy can become zero depending on the
strength of the potential
C. -H. Park et al., Nat. Phys. 4, 213 (2008).
Graphene/h-BN superlattices
Graphene in a triangular potential
At M points in the supercell Brillouin zone,
the energy separation between the 1st and 2nd
bands (DE) closes
Secondary Dirac cones at M points
(new massless fermions are generated)
C. -H. Park et al., Phys. Rev. Lett. 101, 126804 (2008).
Graphene/h-BN superlattices
Graphene in a triangular potential
At M points in the supercell Brillouin zone,
the energy separation between the 1st and 2nd
bands (DE) closes
Secondary Dirac cones at M points
(new massless fermions are generated)
DOS vanishes at this energy (new electron and
hole states are generated around the new Dirac
points)
The group velocity of new massless fermions is
anisotropic
C. -H. Park et al., Phys. Rev. Lett. 101, 126804 (2008).
Graphene/h-BN superlattices
Experimental observation of moiré pattern by STM
Moiré wave length l a: lattice constant of graphene
d: lattice mismatch between
graphene and h-BN
f: relative angle between h-BN
and graphene
Tunneling conductance (=DOS) clearly shows the periodic
modulation (moiré pattern)M. Yankowitz et al., Nat. Phys. 8, 382 (2012).
Graphene/h-BN superlattices
Transport measurements of graphene on h-BN
Additional peaks are observed in Rxx
Signature of the secondary Dirac points away
from the original Dirac point
Sign changes of Rxy around the secondary
Dirac points
Switch between electron & hole nature of mass-
less fermions around the secondary Dirac points
Moiré pattern is clearly observed by AFM images
M. Yankowitz et al., Nat. Phys. 8, 382 (2012).
Graphene/graphene superlattices
Graphene + h-BN
h-BN: Band insulator with a large band gap (~6.0 eV)
No carriers in h-BN around the Fermi energy of graphene
What will happen if we produce a superlattice with two
graphenes?
Intrinsic superconductivity in graphene superlattices
Electron correlation in Hamiltonian
Kinetic term Potential term
When the kinetic term is much smaller than the potential term,
electron-electron (e-e) interaction becomes important.
Electron-electron interaction Potential term
Kinetic term is related to the Fermi velocity of electrons
When the Fermi velocity is small, effects from e-e interaction becomes large
Moderate band dispersions = flat bands are required
Intrinsic superconductivity in graphene superlattices
Theoretical prediction on flat bands for twisted bilayer graphene
Tight-binding calculations’ results
Red dotted bands: band dispersions for monolayer graphene (for reference)
At large angles Linear dispersion close to the Dirac point
Green dashed bands: band dispersions for bilayer graphene (for reference)
Flat band appears at around 1.5○!
Gap closes at 1.5○ Different electronic regimes below/above this angle
vF=0 around 1.4○ Electronic correlation effects may be dominant
Intrinsic superconductivity in graphene superlattices
Band-engineering in twisted bilayer graphene
Two graphene lattice with relative angle q
Due to the interlayer interaction, the energy
bands are modulated as a function of q
“Flat bands” emerge when q = 1.05○
“Magic angle”
R. Bistrizer and A. H. MacDonald, PNAS 108, 12233 (2011).
Intrinsic superconductivity in graphene superlattices
Intrinsic superconductivity is observed around the magic angle
in twisted bilayer graphene!Y. Cao et al., Nature 556, 43 (2018).
Intrinsic superconductivity in graphene superlatticesExperimental results
For two-layer graphene devices with twisted
angle q superconducting transition observed
at 1.7 K at maximum
Y. Cao et al., Nature 556, 43 (2018).
Intrinsic superconductivity in graphene superlatticesExperimental results
For two-layer graphene devices with twisted
angle q superconducting transition observed
at 1.7 K at maximum
Superconductivity is observed when EF is slightly
above or below half-filling.
* Energy at half-filling (hole) is slightly below the flat band
around E = 0
*Four electrons per the Brillouin zone of the supercell(considering 2x spin + 2x valley degeneracy)
Y. Cao et al., Nature 556, 43 (2018).
Intrinsic superconductivity in graphene superlattices
n = -2
Conductance map as a function of carrier density
Superconductivity
Full-filling (insulating)
Y. Cao et al., Nature 556, 43 (2018).
Intrinsic superconductivity in graphene superlattices
Mott insulator state around the half-filling
At half-filling the system makes a transition into
Mott-insulator state
Mott insulator
Band insulator: (Large) band gap suppresses electron
excitation.
Full filling
Y. Cao et al., Nature 556, 43 (2018).
Intrinsic superconductivity in graphene superlattices
Mott insulator state around the half-filling
Mott insulator:
Hubbard model
Hopping On-site
interaction
When U >> t, strong electron-electron interaction
prevents electrons from hopping between sites.
Half-filling antiferromagnetic insulating ground state
Intrinsic superconductivity in graphene superlattices
Narrow band (e.g. d-band) is half-filled
Mott insulator state around the half-filling
Mott insulator:
Band descriptionNot filled
Because of large U, the d-band splits into
two with a charge gap
Example of Mott insulator: Cuprate high-Tc
superconductors
Upper Hubbard band
Lower Hubbard band
Intrinsic superconductivity in graphene superlattices
Phase diagram: High Tc SC vs twisted graphene
Superconducting domes appear around the (antiferromagnetic) Mott insulator
phase in both system
Exotic superconducting state (d-wave paring)???
Y. Cao et al., Nature 556, 43 (2018).
Intrinsic superconductivity in graphene superlattices
“Strange Metal” phase around half-filling
Y. Cao et al., arxiv:1901.03710.
Intrinsic superconductivity in graphene superlattices
(Slightly) Twisted bilayer graphene
Y. Cao et al., Nature 556, 80 (2018).
STM measurements
High density of states at AA site
Three-fold rotational symmetry (outside of the flat band)
Y. Jiang et al., arxiv:1904.10153.
Highly-doped region
Intrinsic superconductivity in graphene superlattices
Charge stripe phase at n=0
Y. Jiang et al., arxiv:1904.10153.
Around the charge neutral point, broken C3 symmetry is observed
d-wave like charge-ordered state as a ground state at Mott gap
Ferromagnetism in twisted bilayer graphene
n~0.75ns
Rxx
Rxy
Anomalous Hall effect is observed!A. L. Sharpe et al., arxiv:1901.03520
q=1.17○
Ferromagnetism in twisted bilayer graphene
Tc~5 K Electron-hole transition
A. L. Sharpe et al., arxiv:1901.03520
Ferromagnetism in twisted bilayer graphene
DC current bias strongly modulates the anomalous Hall signal
Domain wall motion driven by spin-transfer torque?
A. L. Sharpe et al., arxiv:1901.03520
Ferromagnetism in twisted bilayer graphene
Ferromagnetism Graphene aligned with h-BN
(relative angle 0.83○)
Aligned sample Misaligned sample
Ferromagnetic behavior is not observed for the misaligned sample
Interaction with h-BN may play an important role
A. L. Sharpe et al., arxiv:1901.03520
Correlated electrons in trilayer graphene
Trilayer graphene Normally ABA stacking
You can sometimes find ABC stacking!
Electronic properties are different
(e.g. band structures)
G. Chen et al., arxiv:1901.04621.
Correlated electrons in trilayer graphene
Signature of a flat band
by applying an electric field
G. Chen et al., arxiv:1901.04621.
Correlated electrons in trilayer graphene
Tunable Mott-state by a transverse electric field
Depending on the transverse electric field, insulating gap (dis)appearsG. Chen et al., arxiv:1901.04621.
Superconductivity in trilayer grapheneD = -0.54 V/nm D = -0.17 V/nm
Superconductivity is observed close to ¼ and ½ fillings,
at appropriate electric fields
G. Chen et al., arxiv:1901.04621.
Superconductivity in trilayer graphene
Signatures of superconductivity
Tc ~ 0.65 K
Clear temperature drop is observed, but superconductivity seems
rather weak...
G. Chen et al., arxiv:1901.04621.
Ferromagnetism in trilayer graphene
G. Chen et al., arxiv:1905.06535.
Ferromagnetism at ¼ filling
Ferromagnetism in trilayer graphene
G. Chen et al., arxiv:1905.06535.
Ferromagnetism at ¼ filling
Chern insulator (C=2)?
Superconductivity in twisted bilayer-bilayer graphene
Twisted bilayer-bilayer graphene
Electron correlation can be modulated
by an transverse electric field
Large resistance peak is observed at ½ ns
C. Shen et al., arxiv:1903.06952. X. Liu et al., arxiv:1903.08130.
Superconductivity in twisted bilayer-bilayer graphene
Strong suppression of conductivity at full
and half-filling
X. Liu et al., arxiv:1903.08130.
Effective mass for the valence and
conduction band is one order of magnitude
larger than that of Bernal-stacked bilayer
graphene
Signature of electron correlations
Temperature dependence of resistance
Energy gap of the half-filled state ~ 3 meV
Superconductivity in twisted bilayer-bilayer graphene
½ ns peak appears/disappears depending on the net transverse electric field
(top gate + bottom gate)
Electron-hole asymmetric
Change of the carrier type
C. Shen et al., arxiv:1903.06952.
Superconductivity in twisted bilayer-bilayer graphene
Superconductivity onsets around 12 K
close to n = ½ ns
The resistance peak at n = ½ ns emerges by applying
an inplane magnetic field
Signature of a spin-polarized insulating state
C. Shen et al., arxiv:1903.06952.
Superconductivity in twisted bilayer-bilayer graphene
X. Liu et al., arxiv:1903.08130.
Half-filled state enlarges
with increasing parallel fields
Quarter-filled states appear
with increasing parallel fields
Spin polarized filled states
at half-filling
&
Spin and valley polarized
states at ¼ filling
Superconductivity in twisted bilayer-bilayer graphene
X. Liu et al., arxiv:1903.08130.
Critical temperature enhances
by an inplane magnetic field
Superconductivity in twisted bilayer-bilayer graphene
TBBG at a modest angle (q =0.84○)
Flat bands
Increasing D
1
23
4
5
6
Electrons in the range from -3ns to +3 ns may be strongly correlated
Y. Cao et al., arxiv:1903.08596.
Superconductivity in twisted bilayer-bilayer graphene
Resistance vs perpendicular magnetic field
Half- & quarter-filling states emerge as B increases
Signature of spin-polarized ground states at these fillings
Y. Cao et al., arxiv:1903.08596.
)sin( 21 ff cII
What is Josephson junction?
f1f2 : phase difference of two superconductors.
Superconducting order parameter Y:fieYY
where f is macroscopic phase of Y.
When two superconductor with different phase couple through weak link,
supercurrent flows according to relation:
What happens when two superconductors are weakly coupled
through a junction?
Superconductivity and graphene
Andreev reflection
In superconductors: electrons form “Cooper pairs”
An electron in a normal metal needs a partner
to transmit into a neighboring superconductor
A hole is reflected back to the normal metal
Superconductor/Normal metal/Superconductor
(SNS) Josephson junction
Andreev reflection occurs at both interfaces
and form “Andreev bound states”
Superconductivity and graphene
Superconductivity and graphene
Graphene Not an intrinsic superconductor
Superconductivity can be induced via the proximity effect
(i.e. in contact with a superconductor)
First report on graphene Josephson junction
Ti/Al superconducting contacts
H. B. Heesche et al., Nature 446, 56 (2007).
Superconductivity and graphene
H. B. Heesche et al., Nature 446, 56 (2007).
Zero supercurrent and Fraunhofer pattern (diffraction pattern) are observed
Signature of a Josephson junction
Superconductivity and graphene
Enhancing the quality of graphene Josephson junctions
Interesting phenomena such as specular Andreev
reflection or phase-coherent supercurrents
Clean graphene Josephson junctions are essential
hBN/Graphene/hBN + 1D superconducting contacts
Flee from charged impurities and resist residues
MoRe superconductor
High Tc (~ 8 K) + High Hc2 (~ 8 T) + Good electrical
contacts to graphene
Ideal superconducting contacts for graphene
V. E. Calado et al., Nat. Nanotech. 10, 761 (2015).
Superconductivity and graphene
Enhancing the quality of graphene Josephson junctions
Asymmetry in R vs Vgate curve
Graphene is n-doped close to the MoRe contacts
V. E. Calado et al., Nat. Nanotech. 10, 761 (2015).
Superconductivity and graphene
Enhancing the quality of graphene Josephson junctions
Fabry-Pérot oscillations of the critical current
and normal state conductance
Signatures of phase-coherent transport
In the normal state, Fabry-Pérot oscillations are observed
only when graphene is p-doped
Fabry-Pérot oscillations are due to scatterings at
the p-n junctions, not at the interfaces between graphene
and MoRe
In the superconducting state, Fabry-Pérot oscillations
are observed only when graphene is p-doped
V. E. Calado et al., Nat. Nanotech. 10, 761 (2015).
Superconductivity and graphene
Andreev retroreflection and Specular Andreev reflection
“Specular reflection” (in the normal state)
x- and y- components of momenta px, py
py: conserved at reflection
px: changes sign at reflection
“Andreev retroreflection” (conventional)
py: conserved at the reflection
px: changes sign at the reflection
However, a conduction-band hole moves opposite to its
momenta.
The reflected hole traces the same path as
that of the incident electron.
C. W. J. Beenakker, Phys. Rev. Lett. 97, 067007 (2006).
Superconductivity and graphene
“Specular Andreev reflection” (for Dirac fermions)
py: conserved at the reflection
px: changes sign at the reflection
The reflected hole traces the specular path
as that of the incident electron.
A valence-band hole moves the same direction to its
momenta.
Namely, the specular Andreev reflection is the interband process.
C. W. J. Beenakker, Phys. Rev. Lett. 97, 067007 (2006).
Superconductivity and graphene
Normal metal/Superconductor junctions
Both an incident electron and a reflected
hole belong to the conduction band.
Andreev retroreflection regime
Graphene/Superconductor junctions
Because of the zero-gap semiconductor
nature, if the Fermi level is close to the
Dirac point a reflected hole can belong to
the valence band.
Specular Andreev reflection
is possible!
D. K. Efetov et al., Nat. Phys. 12, 328 (2016).
Superconductivity and graphene
Right figure:
Solid lines: Conduction band
Dashed lines: Valence band
For e (=eV) < EF
An incident electron in the conduction band
is reflected as a hole in the conduction band.
Andreev retroreflection
For e (=eV) > EF
An incident electron in the conduction band
is reflected as a hole in the valence band.
Specular Andreev reflection
Enhanced conductance at high bias
Enhanced conductance at low bias
EF << D0
EF >> D0
C. W. J. Beenakker, Phys. Rev. Lett. 97, 067007 (2006).
Superconductivity and graphene
Experimental observation of specular Andreev reflection
Josephson junctions with superconducting metals
suffer from p-n junctions close to the superconducting
contact
NbSe2/bilayer graphene heterojunction
NbSe2: Intrinsic 2D superconductor (Tc ~ 7 K, D ~ 1.2 meV)
D. K. Efetov et al., Nat. Phys. 12, 328 (2016).
The condition: D > e (=eV) > EF is necessary
Bilayer graphene Smaller charge inhomogeneity
around the Dirac point due to large
DOS (stronger screening)
Superconductivity and graphene
The junction conductance (G) as a function of eF
and the bias voltage (Vns)
For |eVns| < EF
D. K. Efetov et al., Nat. Phys. 12, 328 (2016).
&
Enhanced conductance (blue region)
Suppressed conductance (red region)
For |eVns| > EF
For |eVns| ~ EF
Superconductivity and graphene
The junction conductance (G) as a function of eF
and the bias voltage (Vns)
For |eVns| < EF
D. K. Efetov et al., Nat. Phys. 12, 328 (2016).
: Intraband transition
Suppressed conductance (red region)
For |eVns| > EF
For |eVns| ~ EF
Far from the Dirac point, large DOS
Enhanced conductance (blue region)
: Interband transition
Around the Dirac point, small DOS
Superconductivity and graphene
D. K. Efetov et al., Nat. Phys. 12, 328 (2016).
For |eVns| < EF : Intraband transition
py: conserved at the reflection
Because the effective mass of a hole in the conduction
band is negative, vy should be negative
Andreev retroreflection
For |eVns| > EF : Interband transition
py: conserved at the reflection
Because the effective mass of a hole in the conduction
band is negative, vy should be positive
Specular Andreev retroreflection
Summary for today
Specular Andreev reflection is observed in graphene, specific to Dirac fermions.
Graphene/h-BN superlattices exhibit satellite Dirac cones, which emerge as
gate-dependent resistance peaks, and also moiré pattern.
Twisted bilayer graphene, trilayer graphene, twisted double bilayer graphene
are model systems to investigate electron correlations of Dirac fermions.
These systems have similar properties to those of high-Tc cuprates and they
exhibit superconducting transition.
Dirac electrons do not localize due to chirality. QHE in graphene is different from
that of conventional 2D electron gas.