Physics in 2D Materials - equipes.lps.u-psud.fr · Introduction of quantum Hall effect Relation...

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 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 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 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).


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).


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


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


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 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).


n = -2

Conductance map as a function of carrier density

Superconductivity

Full-filling (insulating)

Y. Cao et al., Nature 556, 43 (2018).


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).



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


Narrow band (e.g. d-band) is half-filled


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


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).


“Strange Metal” phase around half-filling

Y. Cao et al., arxiv:1901.03710.


(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


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○


Tc~5 K Electron-hole transition

A. L. Sharpe et al., arxiv:1901.03520


DC current bias strongly modulates the anomalous Hall signal

Domain wall motion driven by spin-transfer torque?



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


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.


Signature of a flat band

by applying an electric field



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


Superconductivity in trilayer graphene

Signatures of superconductivity

Tc ~ 0.65 K

Clear temperature drop is observed, but superconductivity seems

rather weak...


Ferromagnetism in trilayer graphene


Ferromagnetism at ¼ filling

Ferromagnetism in trilayer graphene


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.


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


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 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.



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



Critical temperature enhances

by an inplane magnetic field


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



Resistance vs perpendicular magnetic field

Half- & quarter-filling states emerge as B increases

Signature of spin-polarized ground states at these fillings


)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”



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).


H. B. Heesche et al., Nature 446, 56 (2007).

Zero supercurrent and Fraunhofer pattern (diffraction pattern) are observed

Signature of a Josephson junction


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).



Asymmetry in R vs Vgate curve

Graphene is n-doped close to the MoRe contacts




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



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).


“Specular Andreev reflection” (for Dirac fermions)


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.



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).


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



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)


The condition: D > e (=eV) > EF is necessary

Bilayer graphene Smaller charge inhomogeneity

around the Dirac point due to large

DOS (stronger screening)


The junction conductance (G) as a function of eF

and the bias voltage (Vns)

For |eVns| < EF


: 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



For |eVns| < EF : Intraband transition


Because the effective mass of a hole in the conduction

band is negative, vy should be negative

Andreev retroreflection

For |eVns| > EF : Interband transition


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.

Physics in 2D Materials - equipes.lps.u-psud.fr · Introduction of quantum Hall effect Relation...

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