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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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LPS Physique Mesoscopique Enseignement
Registration on ADUM: Maybe possible...
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