Physics in 2D Materials

Taro WAKAMURA (Université Paris-Saclay)

Lecture 1

Overview of the lectures

5 lectures (2hrs for each)

1st lecture: Graphene 1

2nd lecture: Graphene 2

3rd lecture: Transition Metal DiChalcogenides (TMDCs) 1

4th lecture: TMDCs 2/hexagonal Boron-Nitride (h-BN)/Black Phosphorus

5th lecture: Xene/2D heterostructures

Slides of the lectures will be uploaded on the web

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Today’s Topics

Lecture 1: Graphene 1

1.1 Overview of 2D materials

1.2 Fundamental physical properties of mono and bilayer graphene

1.3 How to fabricate 2D materials -Examples for graphene-

1.4 Graphene in magnetic fields

What is “2D materials”?Different dimensionality in diverse systems

3D materials (bulk) 2D materials (layer, film) 1D materials (wire)

What is graphene?

Birth of atomically thin 2D materials: Graphene

Graphite

Layered structures!

When microscopically observed...

Conventional 2D system: Thin films, quantum wells...

Is it possible to make one-atom-thick real 2D systems?

What is graphene?

Intralayer: Covalent bonding (strong)

Interlayer: Van-der-Waals coupling（weak）

Graphite

Then, is it possible to take out one layer of graphite?

Graphite is composed of 2D sheets weakly coupled three-dimensionally

Something interesting in monolayer limit!

Experimentally demonstrated by

Prof. Geim and Novoselov in 2004

+You can get monolayer graphene by cleaving with Scotch tape!

What is graphene?

Nobel Prize in Physics in 2010

“Monolayer graphite (=graphene)”

How to make graphene?

Mechanical Exfoliation

Graphite

1 cm

10 mm

Graphite flakes

+ Scotch tape

After

fold&unfold

Stamp on

a chip

Why is graphene so interesting?

Electronic structure of monolayer graphene

Monolayer graphene

Honeycomb lattice structure

There are two inequivalent points A and B

in the unit cell (sublattice degree of freedom).

In the Brillouin zone: There are two inequivalent points

K and K’ (valley degree of freedom).

Hamiltonian up to the next nearest neighbors

A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

Electronic structure of monolayer graphene

(via the Fourier transformation) The energy dispersion is expressed as

(q is the wave vector from K or K’)

Linear Dispersion!

Electrons in solids follow the Schroedinger equation

In free electron case (V=0)

Parabolic Dispersion

Electronic structure of monolayer graphene

(via the Fourier transformation) The energy dispersion is expressed as

(q is the wave vector from K or K’)

Linear Dispersion!

The linear dispersion between the energy and the wave vector reminds us...

The Dirac equation (Free electron case)

a, b : Dirac matrices

For relativistic electrons

Electrons in solids follow the relativistic wave equation!A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

Electronic structure of monolayer graphene

The Dirac equation

The kinetic energy reads

If m = 0

The speed of light

Linear Dispersion!

In the case of graphene

The Fermi velocityA. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

Massless Dirac fermion

Electronic structure of monolayer graphene

Indeed, electrons follow the Dirac equation with m=0

Massless Dirac fermions

s: Sublattice spin

Because electrons in graphene follow the Dirac equation,

they exhibit many distinct properties.

Examples

Klein tunneling Absence of localization Specular Andreev reflection

=

Because it is two dimensional system, the Hamiltonian is written as

x=1 for the valley K, -1 for the valley K’

A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

Example: Klein paradox

For classical particles

When the energy of a particle e < V0

It is bounced back by the potential barrier.

The wave function of the electron

exponentially decays inside the potential barrier.

Electronic structure of monolayer graphene

For electrons

(nonrelativistic quantum mechanics)

Reflected

Distinct properties of Dirac fermions

For a potential barrier V0 > mc2

Regardless of the thickness of the barrier, electrons

exhibit perfect transmission (T=1).

Klein paradox

In reality, it is not easy to make a potential barrier

with the height of mc2 in the Compton wave length（It needs E > 1016 Vcm-1)

For Dirac fermions in graphene, becausem=0 we can neglect the term mc2

Electronic structure of monolayer graphene

Example: Klein paradox

For electrons

(relativistic quantum mechanics=Dirac equation)

Why is the Klein tunneling possible?

Hamiltonian for graphene

Sublattice spin

Momentum of an electron strongly couples

sublattice spin

Unless sublattice spin is flipped, the electron

cannot be back-scattered (chirality effect)

= Unless atomically sharp defects exist at the interface with the barrier,

electrons are not backscattered

Electronic structure of bilayer graphene

Natural bilayer graphene takes “Bernal stacking” form

Interlayer hopping between two A sites (=g1)

Four atoms in the unit cell Four bands

forms a “dimer”, and pushes out two states to higher energy

Full Hamiltonian (for 4 sites)

Band structure of bilayer graphene

E. McCann and V. I. Fal’ko, Rep. Prog. Phys. 76, 056503 (2013).

Electronic structure of bilayer graphene

The solution

a=1, 2

a=2 Higher energy states

From g1

Hamiltonian for the low-energy states A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

Asymmetry between two layers

Electronic structure of bilayer graphene

From g1

In the intermediate energy range

, where

Parabolic band

At lowest energy ( )

term becomes important

The isoenergetic line breaks into four pockets,

three ellipses and one central circle

E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805 (2006).

“Trigonal warping”

A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).1

Electronic structure of graphene

Monolayer graphene vs Bilayer graphene

Hamiltonian for monolayer Hamiltonian for bilayer

More generically, Hamiltonian for “J-layer graphite” is expressed as

or since ,

for bilayer , where

Pseudospin prefers being parallel to the momentum vector

= Chiral nature of Dirac fermions

In the case of bilayer graphene, s is “which-layer” sublattice spin

Electronic structure of graphene

where ,

When a quasiparticle adiabatically propagate along the closed loop,

it acquires the Berry phase Jp

Monolayer graphene Berry phase p

Bilayer graphene Berry phase 2p

Berry phase plays important roles in many intriguing phenomena

(e. g. Quantum Hall effect)

Electronic structure of graphene

What is the Berry phase?

1) Assume a sphere and a vector tangent to the sphere

2) Assume parallel transport of the vector as 1 → 2 → 3 → 1

3) When there is a curvature on the sphere, the direction of

the vector changes at 1 before/after the parallel transport

1

2 3

This idea is related to differential geometry,

but also applicable to the Hilbert space

Berry phase

Differential geometry

Direction of a vector

Quantum mechanics

Phase of a wave function

Electronic structure of graphene

nth eigenstate of the Hamiltonian with a parameter

at time t is expressed as

When adiabatically changes from t = 0, time evolution of the eigenstate follows

the relation

where

Berry phase (geometrical phase)

Electronic structure of graphene

Example: spin ½ coupled to a magnetic field

Hamiltonian is written as

R: Magnetic field s : spin

RWhen R circles once along the loop C, an electron obtains

C

The Berry phase is expressed by using the solid angle

swept by a spin

The sign of the electron’s wave function changes after a circulation!

Electronic structure of graphene

Hamiltonian for monolayer graphene

Analogy to the spin ½ system

R

C

k is now on a plane (2D) Berry phase:

or since ,

where

General Hamiltonian

e.g. Bilayer graphene (J=2): When changes from 0 to 2p, nJ circles twice

・ O

(When C circles around k = (0,0))

Electronic structure of graphene

C・ O

General form of the Berry phase (when circulates back to at )

R

:Berry connection

:Berry curvature

Vector potential

Magnetic field

Brief summary

Graphene is monolayer (or a few-layer) graphite, and experimentally

obtained by mechanical exfoliation (scotch tape method).

Electrons in monolayer graphene follow the massless Dirac equation,

and the Fermi velocity is one order of magnitude larger than that of

conventional metals and semiconductors.

Electrons in bilayer graphene exhibit different properties than those of

monolayer graphene and have parabolic band dispersion.

Graphene is also important for applications

Large scale fabrications, high mobility, large gap...

Epitaxial growth

Heating SiC under vacuum or Ar atmosphere (T > 1000 deg)

Only Si atoms leave due to the difference in the vapor pressure between Si

and C.

Remaining C forms epitaxial graphene spontaneously on the surface.

W. Norimatsu and M. Kusunoki, Phys. Chem. Chem. Phys. 16, 3501 (2014).

Various fabrications methods of graphene

STEP 1: Annealing polycrystalline Ni film in Ar/H2 atmosphere at 900-1000 deg

STEP 2: Exposed to H2/CH4 gas mixture. CH4 decomposes and carbon atoms

dissolve into the Ni film.

STEP 3: Samples are cooled down in argon gas

Temperature dependent solubility of carbon for Ni causes precipitation of graphene

films on the surface of the Ni.

Cu is also good (better than Ni) catalyst.

Y. Zhang et al., Acc. Chem. Rch. 46, 2329 (2013).

Chemical Vapor Deposition (CVD)

Various fabrications methods of graphene

For transport measurements for example, Ni (or Cu) layer has to be removed.

PMMA resist and metal etchant are used to transfer graphene from the

metallic film to the top of a substrate (see below).

Y. Zhang et al., Acc. Chem. Rch. 46, 2329 (2013).

Various fabrications methods of graphene

Enhancing mobility of graphene

Zero effective mass near the Dirac point High mobility of carriers in graphene

“Mobility”: How mobile carriers are.

High mobility Important for fast information processing

Mobility of graphene (200000 cm-1V-1s-1):more than 100 times larger than that of Si

-50 0 500

1

2

Vg [V]

R [k

]

T = 200 mK

A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

s = nem

n = cgVg𝜇 =

1

𝑒𝑐𝑔

𝜎

𝑉

-50 0 500

1

2

Vg [V]

R [k

]

T = 200 mK

𝜇 =1

𝑒𝑐𝑔

𝑑𝜎

𝑑𝑉

Cg: capacitance of the gate insulator, n: carrier density, m: mobility

In practice,

Steepest point

Doped silicon

Vg

Enhancing mobility of grapheneHow can we calculate mobility?

Slope is important

Why not resistance diverges?

Band structure Density of states

Why is the resistance finite at the Dirac point?

S. Das Sarma et al., Rev. Mod. Phys. 83, 407 (2011).A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

Why is the peak (Dirac peak) so broad?

Why not resistance diverges?

In fabrication process

Graphene is exfoliated on SiO2 substrate

Charge traps (due to oxygen vacancies etc.)

may exist

To form electrical contacts, some resist (polymer)

is coated

Residues of resist may behave as disorders

Reduced mobility?

Why not resistance diverges?

Spatial map of charge density on graphene at the Dirac point

Electron doped regionHole doped region

Histogram of charge density

Trapped charges on a SiO2 substrate, residue of resist etc.

J. Martin et al., Nat. Phys. 4, 144 (2008)

How to enhance mobilities?

Hexagonal boron nitride (h-BN) as a substrate for graphene

Mobility of graphene on top of SiO2 substrates

are limited by charged impurities scatterings and

surface roughness etc.

Can we find alternative better substrates?

Hexagonal boron nitride

Two dimensional van-der Waals insulator

Atomically flat, less charge traps, small lattice mismatch with graphene

Good candidate as a substrate for graphene!

How to enhance mobilities?

Hexagonal boron nitride as a substrate for graphene

Mechanically-exfoliated graphene is transferred

onto mechanically-exfoliated h-BN

High mobilities (~ 60000 cm2V-1s-1) are observed!

C. R. Dean et al., Nat. Nanotech. 5, 722 (2010).

How to enhance mobility?

Disorders that reduce mobility also come

from resist residues

Graphene protected from external

environment should have better mobility

Graphene encapsulated by h-BNs

L. Wang, Science 342, 614 (2013).

Hexagonal boron-nitride (h-BN) is an ideal

material as a substrate for graphene: flat,

flee from charge inhomogeneity

Graphene encapsulated from two h-

BNs should be flee from resist residues,

charged impurities.

L. Wang, Science 342, 614 (2013).

How to enhance mobilities?

Graphene is fully encapsulated between h-BNs

How can we make contacts to graphene?

One-dimensional contact technique

Graphene as well as h-BNs are etched by dry etching,

and deposit metals to make side contacts.

Advantages

Bulk graphene is protected from resists and other

contaminations.

At the sides graphene has dangling bonds, thus orbital

coupling between carbon atoms and metals becomes

stronger.

Brief summary

Graphene is monolayer (or a few-layer) graphite, and experimentally

obtained by mechanical exfoliation (scotch tape method).

Electrons in monolayer graphene follow the massless Dirac equation,

and the Fermi velocity is one order of magnitude larger than that of

conventional metals and semiconductors.

Graphene is also important for applications

Large scale fabrications, high mobility, large gap...

How can we gap out graphene?

Electronic structure of bilayer graphene

Electrical energy gap control in bilayer graphene

Sublattice spin degree of freedom of bilayer graphene

“Which layer” degree of freedom

By applying a perpendicular electric field, the sublattice

symmetry is broken

Introducing a “mass term” that opens a gap between

the conduction and valence band

Top gate + back gate device structure

Independent control of the carrier density and

perpendicular electric field

Y. Zhang et al., Nature 459, 820 (2009).

Electronic structure of bilayer graphene

Electric field from the bottom gate: Db

Electric field from the top gate: Dt

Carrier doping Db-Dt

Breaking layer symmetry (Db+Dt)/2

Y. Zhang et al., Nature 459, 820 (2009).

Electronic structure of bilayer graphene

Optical detection of gap-opening by the electric field

Transition I: Corresponds to the gap between the valence band and conduction band

edge

As a function of the net electric field , the absorption peak shifts

to the higher energy Increase of the gap

Y. Zhang et al., Nature 459, 820 (2009).

Magnetic field effects on graphene

Graphene in weak magnetic fields

G. Bergmann, Phys. Rep. 107, 1 (1984).

Important effects: weak (anti)localization

Time-reversed pair of the closed path of diffusive electrons

Constructive interference (Weak localization (WL))

With SOI

Additional phase (Berry phase)

Distractive interference (Weak antilocalization)

Without SOI

Magnetic field breaks constructive interference

Decrease of resistance (DR < 0)

Magnetic field breaks distractive interference

Increase of resistance (DR > 0) 46

DR

[]

WL

WAL

(WAL)

Graphene in weak magnetic fields

Hamiltonian for graphene

Sublattice spin

Remember the spin-orbit Hamiltonian...

Real spin

Graphene acquire p Berry phase due to sublattice spin

Weak antilocalization

p Berry phase is acquired due to spin rotation Weak antilocalization

Graphene in weak magnetic fields

Chirality conservation Weak antilocalization

Weak localization can be observed if the chirality is broken.

Atomically sharp defects

Break the sublattice symmetry

& induce intervally scattering

Edges of graphene play an important role

Shape dependence of Ds (B) at low T

B: Long narrow sample

D: Square sample

F1, F2: Rectangle sample

F. V. Tikhonenko et al., Phys. Rev. Lett. 100, 056802 (2008).

Graphene in weak magnetic fields

Sample D

Dirac point

Electron doped

Electron doped

Sample B, Electron doped

Ds (B) is expressed as

: Intervalley scattering time

, : Digamma function

Upturn contribution

(weak localization)

: Downturn contribution (weak antilocalization)

: Intravalley scattering time

F. V. Tikhonenko et al., Phys. Rev. Lett. 100, 056802 (2008).

Graphene in weak magnetic fields

Sample D

Dirac point

Electron doped

Electron doped

Sample B, Electron doped

Ds (B) is expressed as

Sample D, F1, F2: Downturn is observed at

higher B.

Effect of weak antilocalization

Sample B: No downturn is observed

Due to strong intervalley scatterings at the edges,

weak antilocalization is suppressedF. V. Tikhonenko et al., Phys. Rev. Lett. 100, 056802 (2008).

Graphene in weak magnetic fields

Crossover between WL and WAL by temperature

Intervalley scattering: atomically sharp impurities

Phase coherence time: Temperature dependent

: Intervalley scatterings are effective

: Intervalley scatterings are negligible in

the interference process

Temperature independent

By modulating a temperature, crossover

between weak localization (WL) and

weak antilocalization (WAL) may occur

F. V. Tikhonenko, Phys. Rev. Lett. 103, 226801 (2009).

Graphene in weak magnetic fields

Crossover between WL and WAL by temperature

F. V. Tikhonenko et al., Phys. Rev. Lett. 103, 226801 (2009).

Depending on the temperature and

gate voltage, WL to WAL crossover

is clearly seen!

Graphene in high magnetic fields

53

Be-

e-

e-e-e- e-

Non magnetic materialCurrent: le

Lorenz force:

F = evF B V

Conventional Hall effect

Electrons are deflected by the Lorenz force.

Radius of cyclotron orbits depends on the magnetic field: 𝑟𝑐 =𝑚𝑣

𝑒𝐵

As a function of magnetic field, cyclotron orbits become smaller.

Graphene in high magnetic fieldsQuantum Hall effect

Semiclassical picture: In high fields, all electrons

form cyclotron orbits, and those at the edges move

via a skipping motion.

Chiral edge states (move in one direction)

Hamiltonian with a magnetic field

where commutation relation follows

Here a dynamical momentum is defined

Introduction of quantum Hall effect

Then the Hamiltonian becomes

Now the Hamiltonian can be rewritten by using raising and lowering operators:

,

The same Hamiltonian as that for a harmonic oscillator!

Energy levels are quantized Landau quantization

,

Then the Hamiltonian reads

Cyclotron frequency

Introduction of quantum Hall effect

For realistic samples: The system contains impurities

Landau level broadening

When a certain Landau level (LL) is considered,

a path with the energy above/below the LL circulates

around the potential.

Localized states

A path at a LL is extended.

It can carry a current

Due to the energy gap between LLs,

the current is allowed to flow only perpendicular

to the applied current.

The Hall voltage thus the Hall conductivity

Introduction of quantum Hall effect

The edges of the sample is modeled as a steeply rising

potential.

More electrons are injected at the right edge than

In this case, the longitudinal current can be written by Dm as

Hall conductivity quantization

Chemical potential difference Dm

Dm

Introduction of quantum Hall effect

At each LL, one delocalized state exists

Landau levels below EF can contribute to

transport.

With fixed EF, the number of LLs changes as a

function of B.

Each state provides to the Hall

conductivity

takes integer multiples of as a function of B

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 k

von Klitzing constant